Hendrickson Walking Beam Suspension Diagram

for -

SCIENTISTS & ENGINEERS with Modern Physics

DOUGLAS C. GIANCOLI

•

Upper Saddle River, New Jersey 07458

Library of Congress Cataloging-in-Publication Data

Giancoli, Douglas C. Physics for scientists and engineers with modern physics / Douglas C. Giancoli.-4th ed. p.cm. Includes bibliographical references and index. ISBN 0-13-149508-9 - 1. Physics-Textbooks. I. Title. — QC21.3.GS39 2008 — 530---dc22 2006039431

President, Science: Paul Corey Sponsoring Editor: Christian Botting Executive Development Editor: Karen Karlin Production Editor: Clare Romeo Senior Managing Editor: Scott Disanno Art Director and Interior & Cover Designer: John Christiana Manager, Art Production: Sean Hogan Copy Editor:Jocelyn Phillips Proofreaders: Karen Bosch, Gina Cheselka, Traci Douglas, Nancy Stevenson, and Susan Fisher Senior Operations Specialist: Alan Fischer Art Production Editor: Connie Long Illustrators: Audrey Simonetti and Mark Landis Photo Researchers: Mary Teresa Giancoli and Truitt & Marshall Senior Administrative Coordinator:Trisha Tarricone Composition: Emilcomp/Prepare Inc.; Pearson Education/Lissette Quinones, Clara Bartunek Photo credits appear on page A-72 which constitutes a continuation of the copyright page.

•

A II rights reserved. No part of this book may be reproduced, in any form or by any means, wi1hout permission in writing from the publisher.

Pearson Prentice Hall™ is a trademark of Pearson Education, Inc. Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

ISBN-13: 978 - 0-13-149508 -1 ISBN-10: D-13-149508 - 9 Pearson Education LID., London Pearson Education Australia PTY, T,imitcd, Sydney Pearson Education Singapore, Pte. Ltd. Pearson Education North Asia Ltd., Hong &mg Pearson Education Canada, Ltd., Toronto Pearson Educaci6n de Mexico, S.A. de C.V. Pearson Education-Japan, Tokyo — Pearson Education Malayida, Ptc. I ,td.

Contents

3 3-1 3-2 3-3 -

3--4 3-5 3-6 3-7 3-8 3-9 -

KINEMATICS IN Two OR THREE DIMENSIONS; VECTORS Vectors and Scalars Methods Addition of Vectors-Graphical Subtraction of Vectors, and Multiplication of a Vector by a Scalar Adding Vectors by Components Unit Vectors Vector Kinematics Projectile Motion Solving Problems: Projectile Motion Relative Velocity SUMMARY PROBLEMS

xiv xviii xix

PRl:EA.CI::

To SrnoE~rs USE.OFCoLOR

Volume 1

1 1-1 1-2 1-3 1-4 1-5 1-6 *l-7

The Nature of Science Models, Theories, and Laws Measurement and Uncertainty; Significant Figures Units, Standards, and the SI System Converting Units Order of Magnitude: Rapid Estimating Dimensioni; and Dimensional Analyi;is SUMMARY 14 QUESTIONS 14 PROBLEMS 14 GENERAL PROBLEMS 16

DESCRIBING MOTION: KINEMATICS IN ONE DIMENSION

2-1 -

Reference Frames and Displacement Average Velocity Instantaneous Velocity Acceleration Motion at Constant Acceleration Solving Problems Freely Falling Objects Variable Acceleration; Integral Calcului; Graphical Analysis and Numerical Integration

2-2 2-3 2-4 2-5 2-6 2-7 *2-8

*2-9

SUMMARY

QUESTIONS

PROBLEMS

GENERAL PROBLEMS

59 59 62 64 71

4-1 -

Force 84 Newton's First Law of Motion 84 Mass 86 86 Newton's Second Law of Motion 89 Newton's '!bird Law of Motion Weight-the — JbrceofGravity;the Nonna! Force 92 Solving Problems with Ne\Vton's Laws: 95 Free-Body Diagrams Problem Solving-A General Approach l02 SUMMARY 102 OUESTrOKS 103 PROBLEMS 104 GEJ\ERAL PROBLEMS 109

5-1

QUESTlOKS 75 GEJ\"ERAL PROBLEMS

54 55

4-8 -

INTRODUCTION, MEASUREMENT, ESTIMATING

52 52

DYNAMICS: NEWTON'S LAws OF MOTION

4-2 4-3 4--4 4-5 4-6 4-7 -

xii

APPLICATIONS IJST

5-2

6 8

5--4 -

«5-5

*5-6

5-3

USING NEWroN's l.Aws: FruCTioN, CIRCUIAR MOTION, DRAG FORCES 112 Applications of Newton's Laws Involving Friction 113 Uniform Circular Motion-Kinematici; — 119 Dynamics of Uniform Circular Motion 122 Highway Curves: Banked and Unbanked 126 Nonuniform Circular Motion 128 Velocity-Dependent Forces: Drag and Terminal Velocity 129 SUMMARY 130 QUESTlOKS 131 PROBLEMS 132 GEKilRAL PROBLEMS 136

18 19 20 22 24 28 30 34 39 40

43 48

Ill

GRAVITATION AND NEWTON'S ' SYNTHESIS

6-1 6-2

Newton's Law of Univeri;al Gravitation 140 Vector Form of Newton's Law of Universal Gravitation 143 Gravity Near the Earth's Surface; Geophysical Applications 143 Satellites and "Weightlessness" 146 Kepler's Laws and Newton's Synthesis 149 Gravitational Field 154 Types of Forces in Nature 155 Principle of Equivalence; Curvature of Space; Black Holes 155 SCMMARY 157 QUESTIONS 157 PROBLEMS 158 OE:>IERAL PROBLEMS HiO

6-3 6-4 6-5 "'6-6

6-7 -

*6-8 -

139

~i!~

LINEAR MOMENTUM

Momentum and Its Relation to Force Conservation of Momentum 9-3 Collisions and Impulse 9-4 Conservation of Energy and Momentum in Collisions 9-5 Elastic Collisions in One Dimension 9-6 Jnelastic Collisioni; 9-7 Collisions in Two or Three Dimensions 9-8 Center of Mass (C:\-1) 9-9 Center of Mass and Translational Motion *9-10 Systems of Variable~; Rocket Propulsion SUMMARY 239 Qt;RSTIONS 239 PROBLEMS

240

GENERAL PROBLEMS

10 ROTATIONAL MOTION

214

9-1 9-2 -

215 217 220 222 222 225 227 230 234 236

245

248

10-1 10-2 10-3 10-4 10-5

7 7-1 7-2 7-3 7--4 -

8 8-1 8-2 8-3 8-4 8-5 8-6 8-7 8-8 -

*8-9

WORK AND ENERGY

163

Work Done by a Constant Force 164 Scalar ProdU(,1 of Two Vectors 167 Work Done by a Varying Force 168 Kinetic Energy and the Work-Energy Principle 172 SL"l,,.IMARY 176 QUESTIONS 177 PROBLEMS 177 GE:>IERAL PROBLEMS 180

CONSERVATION OF ENERGY

183

Conservative and Nonconservative Forces Potential Energy Mechanical Energy and Its Conservation Problem Solving Using Conservation of Mechanical Energy 111e Law of Conservation of Energy Energy Conservation with Dissipative Forces: Solving Problems Gravitational Potential Energy and Escape Velocity Power Potential Energy Diagrams; Stable and Unstable Equilibrium SD1MARY PROillEMS

iv CONTENTS

205 207

QUESTIONS 205 GE~ERAL PROBLEMS

211

184 186 189

190 196 197 199 201 204

Angular Quantities 249 254 Vector Nature of Angular Quantities Constant Angular Acceleration 255 Torque 256 Rotational Dynamics; Torque and Rotational Inertia 258 1()-6 Solving Problems in Rotational Dynamics 26() 10-7 Determining Moments of Inertia 2(i3 10-8 Rotational Kinetic Energy 265 10-9 Rotational Plus Translational Motion; Rolling 267 *10-10 Why Doel$ a Rolling Sphere Slow Down? 273 SUMMARY PROBLEMS

274 276

Ql."ESTIONS 275 GilNERAL PROBLnMS

ANGUlAR MOMENrnM; GENERAL ROTATION

281

284

11-1 Angular Momentum-Objects — Rotating About a Fixed Axis 11-2 Vector Cross Product;Torque as a Vector 11-3 Angular Momentum of a Particle 11-4 Angular Momentum and Torque for a System of Particles; General Motion 11-5 Angular Momentum and Torque for a Rigid Object 11-6 Conservation of Angular Momentum *11-7 The Spinning Top and Gyroscope *11-8 Rotating Frames of Reference; Inertial Forces *11-9 The Coriolis Effect SUMMARY 302 OUl:iSTIO:,jS 303 PRORI.RMS 303 GENERAL PROHLl:i.MS

308

285 289 291 292

294 297

299 300 301

14 0SCIU.ATIONS

369

14-1 Oscillations of a Spring 14-2 Simple Harmonic Motion 14-3 Energy in the Simple Harmonic Oscillator 14-4 Simple Harmonic Motion Related to Uniform Circular Motion 14-5 The Simple Pendulum *14-6 The Physical Pendulum and the Torsion Pendulum 14-7 Damped Harmonic Motion 14-8 Forced Oscillations; Resonance SUMMARY PROBLEMS

STATIC EQUILIBRIUM;

12 EIASTICfIY AND FRACTIJRE 12-1 12-2 12-3 12-4 12-5 *12-6 *12-7

The Conditions for Equilibrium Solving Statics Problems Stability and Balance Elasticity; Stress and Strain Fracture Trusses and Bridges Arches and Domes SUMMARY

329

QUESTIONS

PROBI.RM$

330

GENERAL PRORI.F.MS

13 FLums 13-1 13-2 13-3 13-4 -

311 312 313 317 318 322 324 327

329

334

339

Phases of Matter 340 Density and Specific Gravity 340 341 Pressure in Fluids Atmospheric Pressure and 345 Gauge Pressure 13-5 Pascal's Principle 346 l 3-6 - Measurement of Pressure; Gauges and the Barometer 346 348 13-7 Buoyancy and Archimedes' Principle 13-8 - Fluids in Motion; Flow Rate 352 and the Equation of Continuity 13-9 354 - Bernoulli's Equation 13-10 Applications of Bernoulli's Principle: 356 Torricelli, Airplanes, Baseballs, TIA *13-11 Viscosity 358 *13-12 Flow in Tubes: Poiseuille's Equation, Blood Flow 358 *13-13 Surface Tension and Capillarity 359 *13-14 Pumps, and the Heart 361 SUMMARY 361 QUESTIONS 362 PROBIBMS 363 GENERAL PROBLEMS 367

387 388

QUEST1O1'S 388 GEJ\"ERAL PROBLEMS

370 372 377 379 379 381

382 385 392

395

WAVE MOTION

15-1 ChaTacteristic:, of Wave Motion 15-2 Types of Waves: Ttansven;e and Longitudinal 15-3 Energy Transported by Waves 15-4 Mathematical Representation of a Traveling Wave * 15-5 The Wave Equation 15-6 The Principle of Superposition 15-7 Reflection and Transmission 15-8 Interference 15-9 Standing Waves; Resonance ~15-10 Refraction * 15-11 Diffraction SUMMARY PROBLEMS

417

418

QUESTrOl\'S 417 GEI\ERJ\L PROBLEMS

396 398

402 404

406 408 409 410 412

415 416 422

424

SOUND

16-1 Characteristics of Sound 16-2 Mathematical Representation of Longitudinal Waves 16-3 Intensity of Sound: Decibels 16-4 Sources of Sound: Vibrating Strings and Air Columns *16-5 Quality of Sound, and Noise; Superposition 16-6 Interference of Sound Waves; Beats 16-7 Doppler Effect "16-8 Shock Waves and the Sonic Boom * 16-9 Applicati.oni;: Sonar, Ultrasound, and Medical Imaging SUMMARY PROBLEMS

446 448

QUESTlOKS 447 GEJ\"ERAL PROBLEMS

425 426 427 431 436 437 439 443

444 451

CONTENTS

eat a n d t h e

T herm od

F ir s t L a w n a m ic s

496

19-1 19-2 19-3 19-4 19-5 19-6 19-7

Heat as Energy Transfer 497 Internal Energy 498 Specific Heat 499 alorimetry — Solving Problems 500 Latent Heat 502 The First Law of Thermodynamics 505 The First Law of Thermodynamics Applied; alculating the Work 507 19-8 Molar Specific Heats for Gases, and the Equipartition of Energy 511 19-9 Adiabatic Expansion of a Gas 514 19-10 Heat Transfer: onduction, onvection, Radiation 515 SUMMARY 520 QUESTIONS 521 PROBLEMS 522 GENERAL PROBLEMS 526

20 't f j

T e m p er tu r e, T h e r m l E x p n s io n ,

1 /

a n d t h e I d e a l G as Law

454

17-1 Atomic Theory of Matter 455 17-2 Temperature and Thermometers 456 17-3 Thermal Equilibrium and the Zeroth Law of Thermodynamics 459 17-4 Thermal Expansion 459 *17-5 Thermal Stresses 463 17-6 The Gas Laws and Absolute Temperature 463 17-7 The Ideal Gas Law 465 17-8 Problem Solving with the Ideal Gas Law 466 17-9 Ideal Gas Law in Terms of Molecules: Avogadro's Number 468 *17-10 Ideal Gas Temperature Scale— a Standard 469 SUMMARY 470 QUESTIONS 471 PROBLEMS 471 GENERAL PROBLEMS 474

K in e t ic T h e o r

G a se s

476

18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature 476 18-2 Distribution of Molecular Speeds 480 482 18-3 Real Gases and hanges of Phase 18-4 Vapor Pressure and Humidity 484 486 *18-5 Van der Waals Equation of State *18-6 Mean Free Path 487 *18-7 Diffusion 489 SUMMARY 490 QUESTIONS 491 PROBLEMS 492 GENERAL PROBLEMS 494

C NTENTS

S e c o n d La w o f T h e r m o d n a m ic s

528

20-1 The Second Law of Thermodynamics— Introduction 529 20 -2 Heat Engines 530 20-3 Reversible and Irreversible Processes; the arnot Engine 533 20 -4 Refrigerators, Air onditioners, and Heat Pumps 536 20-5 Entropy 539 20 -6 Entropy and the Second Law of 541 Thermodynamics 544 20-7 Order to Disorder 20-8 Unavailability of Energy; Heat Death 545 *20-9 Statistical Interpretation of Entropy and the Second Law 546 *20-10 Thermodynamic Temperature; Third Law of Thermodynamics 548 *20-11 Thermal Pollution, Global Warming, and Energy Resources 549 SUMMARY 551 QUESTIONS 552 PROBLEMS 552 GENERAL PROBLEMS 556

Volume 2 /■%« - E lectr Z 1 E lectr

c c

C harge F eld

607

E lec tr ic P o t e n t ia l

23 -1 Electric Potential Energy and

and

559

21-1 Static Electricity; Electric harge and Its onservation 560 21-2 Electric harge in the Atom 561 21-3 Insulators and onductors 561 21-4 Induced harge; the Electroscope 562 21-5 oulomb's Law 563 21-6 The Electric Field 568 21-7 Electric Field alculations for ontinuous harge Distributions 572 21-8 Field Lines 575 21-9 Electric Fields and onductors 577 21-10 Motion of a harged Particle in an Electric Field 578 21-11 Electric Dipoles 579 *21-12 Electric Forces in Molecular Biology; DNA 581 *21-13 Photocopy Machines and omputer Printers Use Electrostatics 582 SUMMARY 584 QUESTIONS 584 PROBLEMS 585 GENERAL PROBLEMS 589

Potential Difference

607

23 - 2 Relation between Electric Potential

and Electric Field Electric Potential Due to Point harges Potential Due to Any harge Distribution Equipotential Surfaces Electric Dipole Potential E Determined from V Electrostatic Potential Energy; the Electron Volt *23 -9 athode Ray Tube: TV and omputer Monitors, Oscilloscope 23 - 3 23 - 4 23 - 5 23 - 6 23 - 7 23 - 8

SUMMARY PROBLEMS

622 623

QUESTIONS 622 GENERAL PROBLEMS

C a p a c it a n c e , D ie le c t r ic s , E lec tr ic E n e r g S t o r a g e

24 -1 24 - 2 24 - 3 24 - 4 24 - 5 *24 -6

apacitors Determination of apacitance apacitors in Series and Parallel Electric Energy Storage Dielectrics Molecular Description of Dielectrics SUMMARY PROBLEMS

643 644

QUESTIONS 643 GENERAL PROBLEMS

628 630 633 636 638 640 648

651

The Electric Battery Electric urrent Ohm's Law: Resistance and Resistors Resistivity Electric Power Power in Household ircuits Alternating urrent Microscopic View of Electric urrent: urrent Density and Drift Velocity *25 -9 Superconductivity *25 -10 Electrical onduction in the Nervous System 25 -1 25 - 2 25 - 3 25 - 4 25 - 5 25 - 6 25 - 7 25 - 8

26 G auss ' s Law 22-1 22-2 22-3 *22-4

___

591

Electric Flux 592 Gauss's Law 593 Applications of Gauss's Law 595 Experimental Basis of Gauss's and oulomb's Laws 600 SUMMARY 601 QUESTIONS 601 PROBLEMS 601 GENERAL PROBLEMS 605

QUESTIONS 671 GENERAL PROBLEMS

699

666 668 669

677

D C C ir c u it s

EMF and Terminal Voltage Resistors in Series and in Parallel Kirchhoffs Rules Series and Parallel EMFs; Battery harging ircuits ontaining Resistor and apacitor (RC ircuits) 26 - 6 Electric Hazards *26-7 Ammeters and Voltmeters 698

652 654 655 658 660 662 664

675

26 -1 26 - 2 26 - 3 26-4 26 - 5

SUMMARY PROBLEMS

620

628

E lec tr ic C u r r e n t s R e sist a n c e

671 672

619

626

and

SUMMARY PROBLEMS

610 612 614 616 617 617

QUESTIONS 698 GENERAL PROBLEMS

678 679 683 686 687 692 695

704

C NTENTS

vii

E lectromagnet c I n d u c t o n and Faraday' s Law

758

29-1 29-2 29-3 29-4 *29-5

Induced EMF 759 Faraday's Law of Induction; Lenz's Law 760 EMF Induced in a Moving onductor 765 Electric Generators 766 Back EMF and ounter Torque; Eddy urrents 768 29-6 Transformers and Transmission of Power 770 29 -7 A hanging Magnetic Flux Produces an Electric Field 773 *29-8 Applications of Induction: Sound Systems, omputer Memory, Seismograph, GF I 775 SUMMARY 111 QUESTIONS 111 PROBLEMS 778 GENERAL PROBLEMS 782

707

M a g n e t is m

27-1 Magnets and Magnetic Fields 707 27-2 Electric urrents Produce Magnetic Fields 710 27-3 Force on an Electric urrent in a Magnetic Field; Definition of B 710 27-4 Force on an Electric harge Moving in a Magnetic Field 714 27-5 Torque on a urrent Loop; Magnetic Dipole Moment 718 *27-6 Applications: Motors, Loudspeakers, Galvanometers 720 27-7 Discovery and Properties of the Electron 721 27-8 The Hall Effect 723 724 *27-9 Mass Spectrometer SUMMARY 725 QUESTIONS 726 PROBLEMS 727 GENERAL PROBLEMS 730

S ources o f M agnet

eld

733

28-1 Magnetic Field Due to a Straight Wire 734 28-2 Force between Two Parallel Wires 735 28-3 Definitions of the Ampere and the oulomb 736 28-4 Ampere's Law 737 28-5 Magnetic Field of a Solenoid and 741 a Toroid 28-6 Biot-Savart Law 743 28-7 Magnetic Materials— Ferromagnetism 746 *28-8 Electromagnets and Solenoids— Applications 747 *28-9 Magnetic Fields in Magnetic Materials; Hysteresis 748 *28-10 Paramagnetism and Diamagnetism 749 SUMMARY 750 QUESTIONS 751 PROBLEMS 751 GENERAL PROBLEMS 755

viii

C NTENTS

Inductance , E lectromagnet c O sc llat ons, and AC C rcu ts 7 8 5 30-1 30-2 30-3 30-4 30-5 30-6 30-7 30-8 30-9 *30-10 *30-11

Mutual Inductance 786 Self-Inductance 788 Energy Stored in a Magnetic Field 790 LR ircuits 790 LC ircuits and Electromagnetic Oscillations 793 LC Oscillations with Resistance (.LRC ircuit) 795 A ircuits with A Source 796 LRC Series A ircuit 799 Resonance in A ircuits 802 Impedance Matching 802 Three-Phase A 803 SUMMARY 804 QUESTIONS 804 PROBLEMS 805 GENERAL PROBLEMS 809

M axwell' s E quat ons and E lectromagnet c W aves 31-1 31-2 31-3 31-4 31-5 31-6 31-7 31-8 31-9 31-10

812

hanging Electric Fields Produce Magnetic Fields; Ampere's Law and Displacement urrent 813 Gauss's Law for Magnetism 816 Maxwell's Equations 817 Production of Electromagnetic Waves 817 Electromagnetic Waves, and Their Speed, from Maxwell's Equations 819 Light as an Electromagnetic Wave and the Electromagnetic Spectrum 823 Measuring the Speed of Light 825 Energy in EM Waves; the Poynting Vector 826 Radiation Pressure 828 Radio and Television; Wireless ommunication 829 SUMMARY 832 QUESTIONS 832 PROBLEMS 833 GENERAL PROBLEMS 835

L g h t : R eflect on and R efract on

837

32-1 The Ray Model of Light 838 32-2 Reflection; Image Formation by a Plane Mirror 838 32-3 Formation of Images by Spherical 842 Mirrors 32-4 Index of Refraction 850 32-5 Refraction: Snell's Law 850 852 32-6 Visible Spectrum and Dispersion 32-7 Total Internal Reflection; Fiber Optics 854 *32-8 Refraction at a Spherical Surface 856 SUMMARY 858 QUESTIONS 859 PROBLEMS 860 GENERAL PROBLEMS 864

34 rm

y j i i ' w n in H r .

th e n h ( « l K- p l4 A 'J .11 lltL> N irti I t u p r o d lM U a v ic tu a l im u jw . h W h

> ^ C L S m n i r i t o n n f rw irt j f n ( . ^ m p j f j s o n u l p jT L { J J

ift

IMIw c j* « ictmed.

K K tly U I V f ^ a l p u ir L

/viewed t ihe ne r point wirfiSL**vcn ' >™ 'w the in m gniricttliim or mttgnifying power, rr' ngle subtended by n object when u s i n X ^ S j f c ^ c r,JhlJ ,lf lHc un ided eye, with the object t the ne ^P^^u^itn^duMnicih.L: norm l eye): »" ,h4: lN fDr *

M w here 0 nd flr rs shown in Fig. 3 3 - 3 3 . length by noting th t 0 = h / N (Fig, 3.1-

* « .he heigh, of Ihe object and we ^buir suits and lungems, ff the eye is j^t infinity and the ot>ji.>el will h> ^

(»-£i vritcjWin[LinnL»1lilt t«*l ' " /tl" (Hj(. 3? .l.lij,wliLii; ' lc*#tcwialtk>rtandflrcijiml J -3J

T he W ave N ature I nterference

ght ;

900

34-1 Waves Versus Particles; Huygens' 901 Principle and Diffraction 34-2 Huygens' Principle and the Law of 902 Refraction 34-3 Interference — Young's Double-Slit 903 Experiment *34-4 Intensity in the Double-Slit 906 Interference Pattern 909 34-5 Interference in Thin Films 914 *34-6 Michelson Interferometer 915 *34-7 Luminous Intensity SUMMARY 915 QUESTIONS 916 PROBLEMS 916 GENERAL PROBLEMS 918

nd fl' = h / f .

35 33

Lenses a nd O pt I nstruments

33-1 33-2 33-3 *33-4 33-5 33-6 33-7 33-8 *33-9 *33-10

Thin Lenses; Ray Tracing 867 The Thin Lens Equation; Magnification 870 ombinations of Lenses 874 Lensmaker's Equation 876 ameras: Film and Digital 878 The Human Eye; orrective Lenses 882 Magnifying Glass 885 Telescopes 887 ompound Microscope 890 Aberrations of Lenses and Mirrors 891 SUMMARY 892 QUESTIONS 893 PROBLEMS 894 GENERAL PROBLEMS 897

cal

866

D ffract on and P olar zat on

921

35-1 Diffraction by a Single Slit or Disk 922 *35-2 Intensity in Single-Slit Diffraction Pattern 924 *35-3 Diffraction in the Double-Slit Experiment 927 35-4 Limits of Resolution; ircular Apertures 929 35-5 Resolution of Telescopes and Microscopes; the ALimit 931 *35-6 Resolution of the Human Eye and Useful Magnification 932 35-7 Diffraction Grating 933 35-8 The Spectrometer and Spectroscopy 935 *35-9 Peak Widths and Resolving Power for a Diffraction Grating 937 35-10 X-Rays and X -Ray Diffraction 938 35-11 Polarization 940 *35-12 Liquid rystal Displays (L D) 943 *35-13 Scattering of Light by the Atmosphere 945 su m m a r 945 q u e s t io n s 946 PROBLEMS 946 GENERAL PROBLEMS 949 C NTENTS

Volume 3

S pec

T heory

R elat v ty 951

952 36-1 Galilean - Newtonian Relativity *36-2 The Michelson -Morley Experiment 954 36-3 Postulates of the Special Theory of Relativity 957 958 36-4 Simultaneity 960 36-5 Time Dilation and the Twin Paradox 964 36-6 Length ontraction 967 36-7 Four-Dimensional Space -Time 968 36-8 Galilean and Lorentz Transformations 971 36-9 Relativistic Momentum and Mass 974 36-10 The Ultimate Speed 974 36-11 E = me2; Mass and Energy 978 *36-12 Doppler Shift for Light 980 36-13 The Impact of Special Relativity SUMMARY 981 QUESTIONS 981 PROBLEMS 982 GENERAL PROBLEMS 985

E arly Q uantum T heory M odels o f the A tom

and

987

37-1 Blackbody Radiation; Planck's Quantum Hypothesis 987 37-2 Photon Theory; Photoelectric Effect 989 37-3 Photon Energy, Mass, and Momentum 993 37-4 ompton Effect 994 37-5 Photon Interactions; Pair Production 996 37-6 Wave-Particle Duality; the Principle of omplementarity 997 37-7 Wave Nature of Matter 997 *37-8 Electron Microscopes 1000 37-9 Early Models of the Atom 1000 37-10 Atomic Spectra: Key to Atomic Structure 1001 37-11 The Bohr Model 1003 37-12 deBroglie's Hypothesis Applied to Atoms 1009 SUMMARY 1010 QUESTIONS 1011 PROBLEMS 1012 GENERAL PROBLEMS 1014

Q uantum M echan

1017

38-1 Quantum Mechanics— A New Theory 1018 38-2 The Wave Function and Its Interpretation; the Double-Slit Experiment 1018 38-3 The Heisenberg Uncertainty Principle 1020 38-4 Philosophic Implications; Probability Versus Determinism 1024 38-5 The Schrodinger Equation in One Dimension — Time-Independent Form 1025 *38-6 Time -Dependent Schrodinger Equation 1027 38-7 Free Particles; PlaneWaves and Wave Packets 1028 38-8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) 1030 38-9 Finite Potential Well 1035 38-10 Tunneling through a Barrier 1036 SUMMARY 1039 QUESTIONS 1039 PROBLEMS 1040 GENERAL PROBLEMS 1042 X

C NTENTS

Q uantum M echan A toms

cs o f

1044

39-1 Quantum -Mechanical View of Atoms 1045 39-2 Hydrogen Atom: Schrodinger Equation and Quantum Numbers 1045 39-3 Hydrogen Atom Wave Functions 1049 39-4 omplex Atoms; the Exclusion Principle 1052 39-5 Periodic Table of Elements 1053 39-6 X -Ray Spectra and Atomic Number 1054 *39-7 Magnetic Dipole Moment; Total Angular Momentum 1057 39-8 Fluorescence and Phosphorescence 1060 39-9 Lasers 1061 *39-10 Holography 1064 SUMMARY 1066 QUESTIONS 1066 PROBLEMS 1067 GENERAL PROBLEMS 1069

M olecules

and

S ol

1071

40-1 Bonding in Molecules 1071 40-2 Potential-Energy Diagrams for Molecules 1074 40-3 Weak (van der Waals) Bonds 1077 40-4 Molecular Spectra 1080 40-5 Bonding in Solids 1085 40-6 Free -Electron Theory of Metals; Fermi Energy 1086 40-7 Band Theory of Solids 1090 40-8 Semiconductors and Doping 1093 40-9 Semiconductor Diodes 1094 40-10 Transistors and Integrated ircuits ( hips) 1097 SUMMARY 1098 QUESTIONS 1099 PROBLEMS 1099 GENERAL PROBLEMS 1102

41 41-1 41-2 41-3 41-4 41-5 41-6 41-7 41-8 41-9 41-10 41-11

N uclear P hys R ad oact v ty

cs and

1104

Structure and Properties of the Nucleus 1105 Binding Energy and Nuclear Forces 1108 Radioactivity 1110 Alpha Decay 1111 1114 Beta Decay Gamma Decay 1116 onservation of Nucleon Number and Other onservation Laws 1117 Half-Life and Rate of Decay 1117 Decay Series 1121 1122 Radioactive Dating Detection of Radiation 1124 SUMMARY 1126 QUESTIONS 1126 PROBLEMS 1127 GENERAL PROBLEMS 1129

N uclear E nergy; E ffects and U ses o f R ad at on

E lementary Part

cles

1164

43-1 High-Energy Particles and Accelerators 1165 43-2 Beginnings of Elementary Particle Physics—Particle Exchange 1171 43-3 Particles and Antiparticles 1174 43-4 Particle Interactions and onservation Laws 1175 43-5 Neutrinos—Recent Results 1177 43-6 Particle lassification 1178 43-7 Particle Stability and Resonances 1180 43-8 Strangeness? harm? Towards a New Model 1181 1182 43-9 Quarks 43-10 The Standard Model: Q D and 1184 Electroweak Theory 43-11 Grand Unified Theories 1187 43-12 Strings and Supersymmetry 1189 SUMMARY 1189 QUESTIONS 1190 PROBLEMS 1190 GENERAL PROBLEMS 1191

1131

42-1 Nuclear Reactions and the Transmutation of Elements 1132 42-2 ross Section 1135 42-3 Nuclear Fission; Nuclear Reactors 1136 42-4 Nuclear Fusion 1141 42-5 Passage of Radiation Through Matter; Radiation Damage 1146 42-6 Measurement of Radiation — Dosimetry 1147 *42-7 Radiation Therapy 1150 *42-8 Tracers in Research and Medicine 1151 *42-9 Imaging by Tomography: AT Scans and Emission Tomography 1153 *42-10 Nuclear Magnetic Resonance (NMR); Magnetic Resonance Imaging (MRI) 1156 SUMMARY 1159 QUESTIONS 1159 PROBLEMS 1160 GENERAL PROBLEMS 1162

A strophys

cs and

C osmology 1193

44-1 Stars and Galaxies 1194 44-2 Stellar Evolution: Nucleosynthesis, and the Birth and Death of Stars 1197 44-3 Distance Measurements 1203 44-4 General Relativity: Gravity and the urvature of Space 1205 44-5 The Expanding Universe: Redshift and Hubble's Law 1209 44-6 The Big Bang and the osmic Microwave Background 1213 44-7 The Standard osmological Model: Early History of the Universe 1216 44-8 Inflation 1219 44-9 Dark Matter and Dark Energy 1221 44-10 Large-Scale Structure of the Universe 1224 44-11 Finally ... 1224 SUMMARY 1225 QUESTIONS 1226 PROBLEMS 1226 GENERAL PROBLEMS 1227

A ppend

ces

A B C D

M athem atical F o r m u l a s A -1 D erivatives a n d I n te g r a ls A -6 A -8 M o r e o n D im e n s io n a l A nal sis G ravitational F o r c e d u e t o a S p h e r ic a l M a ss D ist r ib u t io n A -9 E D ifferential Fo r m o f M axwell's E q uations A - 1 2 F S elected I s o t o p e s A -14 A n sw e r s t o O d d - N u m b e r e d P ro bl em s A - 18 A -4 7 In d e x A -7 2 P h o t o C r e d it s

C NTENTS

A pplic

tions

(s e le c te d )

Ch pter 1 The 8000-m peaks 8 Estimating volume of a lake 10 Height by triangulation 11 Radius of the Earth 11 Heartbeats in a lifetime 12 Particulate pollution (Pr30) 15 Global positioning satellites (Pr39) 16 Lung capacity (Pr65) 17 Ch pter 2 Airport runway design 29 Automobile air bags 31 Braking distances 32 CD error correction (PrlO) 44 CD playing time (Prl3) 45 Golfing uphill or down (Pr79) 48 Rapid transit (Pr83) 49 Ch pter 3 Kicked football 66,69 Ball sports (Problems) 77,81,82 Extreme sports (Pr41) 77 Ch pter 4 Rocket acceleration 90 What force accelerates a car? 90 How we walk 90 Elevator and counterweight 99 Mechanical advantage of pulley 100 Bear sling (Q24) 104 High-speed elevators (Prl9) 105 Mountain climbing (Pr31,82,83) 106,110 ity planning, cars on hills (Pr71) 109 Bicyclists (Pr72,73) 109 "Doomsday" asteroid (Pr84) 110 Ch pter 5 Push or pull a sled? 116 entrifugation 122 Not skidding on a curve 126-7 Banked highways 127 Simulated gravity (Q18, Pr48) 131,134 "Rotor-ride" (Pr82) 136 Ch pter 6 Oil/mineral exploration 144,420 Artificial Earth satellites 146 Geosynchronous satellites 147 Weightlessness 148 Free fall in athletics 149 Planet discovery, extrasolar planets 152 Black holes 156 Asteroids (Pr44,78) 159,162 Navstar GPS (Pr58) 160 Black hole, galaxy center (Pr61,64) 160,161 Tides (Pr75) 162 Ch pter 7 ar stopping distance of v2 174 Lever(Pr6) 177 Spiderman (Pr54) 179 Bicycling on hills, gears (Pr85) 181 hild safety in car (Pr87) 181 Rock climber's rope (Pr90) 182 Ch pter 8 Downhill ski runs 183 Rollercoaster 191,198 Pole vault 192-3 Toy dart gun 193 x ii

APPLICATI NS

Escape velocity from Earth or Moon 201 Stair climbing power 202 Power needs of car 202-3 ardiac treadmill (Prl04) 213 Ch pter 9 Tennis serve 216 Rocket propulsion 219,236-8 Rifle recoil 220 Karate blow 221 Billiards/bowling 223,228 Nuclear collisions 225,228 Ballistic pendulum 226 onveyor belt 237 Gravitational slingshot (Prl05) 246 rashworthiness (Prl09) 247 Asteroids, planets (PrllO, 112,113) 247 Ch pter 10 Hard drive and bit speed 253 Wrench/tire iron 256 Flywheel energy 266,281 Yo-yo 271 ar braking forces 272-3 Bicycle odometer calibration (Ql) 275 Tightrope walker (Q ll) 275 Triceps muscle and throwing (Pr38,39) 278 D speed (Pr84) 281 Bicycle gears (Pr89) 281 Ch pter 11 Rotating skaters, divers 284,286,309 Neutron star collapse 287 Auto wheel balancing 296 Top and gyroscope 299-300 oriolis effect 301-2 Hurricanes 302 SUV possible rollover (Pr67) 308 Triple axel jump (Pr79) 309 Bat's "sweet spot" (Pr82) 310 Ch pter 12 Tragic collapse 311,323 Lever's mechanical advantage 313 antilever 315 Biceps muscle force 315 Human balance with loads 318 Trusses and bridges 324-6,335 Architecture: arches and domes 327-8 Forces on vertebrae (Pr87) 337 Ch pter 13 Lifting water 345,348 Hydraulic lift, brakes 346 Pressure gauges 346-7 Hydrometer 351 Helium balloon lift 352,368 Blood flow 353,357,361 Airplane wings, lift 356 Sailing against the wind 357 Baseball curve 357 Blood to the brain, TIA 357 Blood flow and heart disease 359 Surface tension, capillarity 359-60 Walking on water 360 Pumps and the heart 361 Reynolds number (Pr69) 366 Ch pter 14 ar shock absorbers 383 Resonance damage 386

Ch pter 15 Echolocation by animals Earthquake waves

400 401,403,416

Ch pter 16 Distance from lightning 425 Autofocus camera 426 Wide range of human hearing 427-8,431 Loudspeaker response 428 Stringed instruments 432-3 433-6 Wind instruments Tuning with beats 439 Doppler blood flow meter 442,453 AAA Sonar: sonic boom 445-6 Ultrasound medical imaging 448 Motion sensor (Pr5) Ch pter 17 Hot air balloon 454 Expansion joints, highways 456,460,463 Gas tank overflow 462 Life under ice 462 old and hot tire pressure 468 Molecules in a breath 469 Thermostat (Q10) 471 Scuba/snorkeling (Pr38,47,82,85) 473,475 Ch pter 18 hemical reactions, temperature dependence 481 Superfluidity 483 Evaporation cools 484,505 Humidity, weather 485-6 hromatography 490 Pressure cooker (Pr35) 493 Ch pter 19 Working off the calories 498 old floors 516 Heat loss through windows 516 How clothes insulate 516-7 i?-values for thermal insulation 517 onvective house heating 517 Human radiative heat loss 518 Room comfort and metabolism 519 Radiation from Sun 519 Medical thermography 519 Astronomy—size of a star 520 Thermos bottle (Q30) 521 Weather, air parcel, adiabatic lapse rate (Pr56) 525 Ch pter 20 Steam engine 530 Internal combustion engine 531,535-6 ar efficiency 532 Refrigerators, air conditioners 537-8 Heat pump 538 Biological evolution, development 545 Thermal pollution, global warming 549-51 Energy resources 550 Diesel engine (Pr7) 553 Ch pter 21 Static electricity 560,589 (Pr78) Photocopiers 569,582-3 Electric shielding, safety 577 DNA structure and replication 581-2 Biological cells: electric forces and kinetic theory 581-2,617 Laser & inkjet printers 583

Ch pter 23 Breakdown voltage 612 Lightning rods, corona 612 RT, oscilloscopes, TV monitors 620-1,723 Photocell (Pr75) 626 Geiger counter (Pr83) 627 Van de Graaff (Pr84) 627,607 Ch pter 24 apacitor uses 628,631 Very high capacitance 631 omputer key 631 amera flash 636 Heart defibrillator 638 DRAM (PrlO, 57) 644,647 Electrostatic air cleaner (Pr20) 645 MOS circuits (Pr53) 647 Ch pter 25 Light bulb 651,653,660 Battery construction 653 Loudspeaker wires 659 Resistance thermometer 660 Heating elements, bulb filament 660 Why bulbs burn out at turn on 661 Lightning bolt 662 Household circuits, shorts 662-3 Fuses, circuit breakers 662-3,747,776 Extension cord danger 663 Nervous system, conduction 669-70 Strain gauge (Pr 24) 673 Ch pter 26 ar battery charging, jump start 686,687 RC applications: flashers, wipers 691 Heart pacemaker 692,787 Electric hazards 692-4 Proper grounding 693-4 Heart fibrillation 692 Meters, analog and digital 695-7 Potentiometers and bridges (Pr) 704,705 Ch pter 27 ompass and declination 709 Aurora borealis 717 Motors, loudspeakers, galvonometers 720-1 Mass spectrometer 724-5 Electromagnetic pumping (Q14) 726 yclotron (Pr66) 731 Beam steering (Pr67) 731 Ch pter 28 oaxial cable 740,789 Solenoid switches: car starters, doorbell 747 ircuit breakers, magnetic 747,776 Relay (Q16) 751 Atom trap (Pr73) 757 Ch pter 29 Induction stove 762 EM blood-flow meter 765 Power plant generators 766-7 ar alternators 768 Motor overload 769 Airport metal detector 770 Eddy current damping 770 Transformers and uses, power 770-3 ar ignition, bulb ballast 772,773 Microphone 775 Read/write on disks and tape 775 Digital coding 775 redit card swipe 776

Ground fault circuit interrupter (GF I) 776 Betatron (Pr55) 782 Search coil (Pr68) 783 Inductive battery charger (Pr81) 784 Ch pter 30 Spark plug 785 Pacemaker 787 Surge protector 792 794,802 LC oscillators, resonance apacitors as filters 799 Loudspeaker cross-over 799 Impedance matching 802-3 Three-phase A 803 0 -value (Pr86,87) 810 Ch pter 31 Antennas 824,831 Phone call lag time 825 Solar sail 829 Optical tweezers 829 Wireless: AM/FM, TV, tuning, cell phones, remotes 829-32 Ch pter 32 How tall a mirror do you need 840-1 lose up and wide-view mirrors 842,849,859 Where you can see yourself in a concave mirror 848 Optical illusions 851,903 Apparent depth in water 852 Rainbows 853 olors underwater 854 Prism binoculars 855 Fiber optics in telecommunications 855-6,865 Medical endoscopes 856 Highway reflectors (Pr86) 865 Ch pter 33 Where you can see a lens image 869 ameras, digital and film 878 amera adjustments 879-80 Pixels and resolution 881 Human eye 882-5,892 orrective lenses 883-5 ontact lenses 885 Seeing under water 885 Magnifying glass 885-7 Telescopes 887-9,931-2 Microscopes 890-1,931,933 Ch pter 34 Bubbles, reflected color 900,912-3 Mirages 903 olors in thin soap film, details 912-3 913-4 Lens coatings Multiple coating (Pr52) 919 Ch pter 35 Lens and mirror resolution 929-30 Hubble Space Telescope 930 Eye resolution, useful magnification 930,932-3 Radiotelescopes 931 Telescope resolution, Arule 931 Spectroscopy 935-6 X-ray diffraction in biology 939 Polarized sunglasses 942 L Ds—liquid crystal displays 943-4 Sky color 945

Ch pter 36 Space travel 963 Global positioning system (GPS) 964 Ch pter 37 Photocells 992 Photodiodes 992 Photosynthesis 993 Measuring bone density 995 Electron microscopes 1000 Ch pter 38 Tunneling through a QM barrier 1038 Scanning tunneling electron microscope 1038-9 Ch pter 39 Fluorescence analysis 1060 Fluorescent bulbs 1060 Phosphorescence, watch dials 1061 Lasers 1061-5 DVD and D players 1063 Barcodes 1063 Laser surgery 1064 Holography 1064-5 Ch pter 40 ell energy—activation energy,ATP 1075-7 Weak bonds in cells, DNA 1077-8 Protein synthesis 1079-80 Transparency 1092 Semiconductor diodes, transistors 1094-8 Rectifier circuits 1096 LED displays; photodiodes 1096 Integrated circuits ( hips) 1098 Ch pter 41 Smoke detectors 1114 arbon-14 dating 1122-3 Archeological, geological dating 1123-4 Oldest Earth rocks and earliest life 1124 Ch pter 42 Nuclear reactors and power plants 1138^40 Manhattan Project 1141 Stellar fusion 1142-3 Fusion energy reactors 1131,1144-6 Biological radiation damage 1146-7 Radiation dosimetry 1147-9 Radon 1148,1150 Human radiation exposure 1148-9 Radiation sickness 1149 Radiation therapy 1150-1 Proton therapy 1151 Tracers in medicine and biology 1151-2 X-ray imaging 1153 AT scans 1153-5 Emission tomography: PET and SPET 1156 NMR and MRI 1156-9 Ch pter 43 Antimatter 1174-5,1188 Ch pter 44 Stars and galaxies 1194-9 Star evolution 1200-2 Supernovae 1201,1202,1203 Star distances 1194,1203^1 Black holes 1202,1208-9 urved space 1207-8 Big Bang 1212,1213-6 Evolution of universe 1216-9 Dark matter and dark energy 1221-3 APPLICATI NS x iii

Preface I was motivated from the beginning to write a textbook different from others that present physics as a sequence of facts, like a Sears catalog: "here are the facts and you better learn them." Instead of that approach in which topics are begun formally and dogmatically, I have sought to begin each topic with concrete observations and experiences students can relate to: start with specifics and only then go to the great generalizations and the more formal aspects of a topic, showing w y we believe what we believe. This approach reflects how science is actually practiced.

Why

Fourth Edition?

Two recent trends in physics texbooks are disturbing: (1) their revision cycles have become short—they are being revised every 3 or 4 years; (2) the books are getting larger, some over 1500 pages. I don't see how either trend can be of benefit to students. My response: (1) It has been 8 years since the previous edition of this book. (2) This book makes use of physics education research, although it avoids the detail a Professor may need to say in class but in a book shuts down the reader. And this book still remains among the shortest. This new edition introduces some important new pedagogic tools. It contains new physics (such as in cosmology) and many new appealing applications (list on previous page). Pages and page breaks have been carefully formatted to make the physics easier to follow: no turning a page in the middle of a derivation or Example. Great efforts were made to make the book attractive so students will want to read it. Some of the new features are listed below.

Wh t's New

F, y, B

xiv

PREFACE

C apter-Opening Questions: Each hapter begins with a multiple-choice question, whose responses include common misconceptions. Students are asked to answer before starting the hapter, to get them involved in the material and to get any preconceived notions out on the table. The issues reappear later in the hapter, usually as Exercises, after the material has been covered. The hapter-Opening Questions also show students the power and usefulness of Physics. APPROACH paragrap in worked-out numerical Examples .A short introductory paragraph before the Solution, outlining an approach and the steps we can take to get started. Brief NOTES after the Solution may remark on the Solution, may give an alternate approach, or mention an application. Step -by -Step Examples: After many Problem Solving Strategies (more than 20 in the book), the next Example is done step-by-step following precisely the steps just seen. Exercises within the text, after an Example or derivation, give students a chance to see if they have understood enough to answer a simple question or do a simple calculation. Many are multiple choice. Greater clarity : No topic, no paragraph in this book was overlooked in the search to improve the clarity and conciseness of the presentation. Phrases and sentences that may slow down the principal argument have been eliminated: keep to the essentials at first, give the elaborations later. Vector notation, arrows: The symbols for vector quantities in the text and Figures now have a tiny arrow over them, so they are similar to what we write by hand. Cosmological Revolution: With generous help from top experts in the field, readers have the latest results.

Page layout: more than in the previous edition, serious attention has been paid to how each page is formatted. Examples and all important derivations and arguments are on facing pages. Students then don't have to turn back and forth. Throughout, readers see, on two facing pages, an important slice of physics. New Applications'. L Ds, digital cameras and electronic sensors ( D, MOS), electric hazards, GF Is, photocopiers, inkjet and laser printers, metal detectors, underwater vision, curve balls, airplane wings, DNA, how we actually see images. (Turn back a page to see a longer list.) Examples modified: more math steps are spelled out, and many new Examples added. About 10% of all Examples are Estimation Examples. T is Book is S orter than other complete full-service books at this level. Shorter explanations are easier to understand and more likely to be read.

Content nd Org niz tion l Ch nges • •

• • • • •

•

• •

Rot tion l Motion: hapters 10 and 11 have been reorganized. All of angular momentum is now in hapter 11. First l w of thermodyn mics, in hapter 19, has been rewritten and extended. The full form is given: AK + AU + AEint = Q — W, where internal energy is Ete, and U is potential energy; the form Q — W is kept so that dW = P dV. Kinematics and Dynamics of ircular Motion are now treated together in hapter 5. Work and Energy, hapters 7 and 8, have been carefully revised. Work done by friction is discussed now with energy conservation (energy terms due to friction). hapters on Inductance and A ircuits have been combined into one: hapter 30. Graphical Analysis and Numerical Integration is a new optional Section 2 -9. Problems requiring a computer or graphing calculator are found at the end of most hapters. Length of an object is a script £ rather than normal /, which looks like 1 or I (moment of inertia, current), as in F = IIB. apital L is for angular momentum, latent heat, inductance, dimensions of length [L\. Newton's law of gravitation remains in hapter 6. Why? Because the 1/r2 law is too important to relegate to a late chapter that might not be covered at all late in the semester; furthermore, it is one of the basic forces in nature. In hapter 8 we can treat real gravitational potential energy and have a fine instance of using U = - JF •di. New Appendices include the differential form of Maxwell's equations and more on dimensional analysis. Problem Solving Strategies are found on pages 30, 58, 64, 96,102,125,166, 198,229,261,314,504,551,571, 685,716,740,763,849, 871, and 913.

Org niz tion Some instructors may find that this book contains more material than can be covered in their courses. The text offers great flexibility. Sections marked with a star * are considered optional. These contain slightly more advanced physics material, or material not usually covered in typical courses and/or interesting applications; they contain no material needed in later hapters (except perhaps in later optional Sections). For a brief course, all optional material could be dropped as well as major parts of hapters 1, 13, 16, 26, 30, and 35, and selected parts of hapters 9,12,19,20, 33, and the modern physics hapters. Topics not covered in class can be a valuable resource for later study by students. Indeed, this text can serve as a useful reference for years because of its wide range of coverage.

Versions of this Book Complete version: 44 hapters including 9 hapters of modern physics. Cl ssic version: 37 hapters including one each on relativity and quantum theory. 3 Volume version: Available separately or packaged together (Vols. 1 & 2 or all 3 Volumes): Volume 1: hapters 1 -20 on mechanics, including fluids, oscillations, waves, plus heat and thermodynamics. Volume 2: hapters 21-35 on electricity and magnetism, plus light and optics. Volume 3: hapters 36-44 on modern physics: relativity, quantum theory, atomic physics, condensed matter, nuclear physics, elementary particles, cosmology and astrophysics. PREFACE

Th nks Many physics professors provided input or direct feedback on every aspect of this textbook. They are listed below, and I owe each a debt of gratitude. Mario Affatigato, oe ollege Lorraine Allen, United States oast Guard Academy Zaven Altounian, McGill University Bruce Barnett, Johns Hopkins University Michael Barnett, Lawrence Berkeley Lab Anand Batra, Howard University ornelius Bennhold, George Washington University Bruce Birkett, University of alifornia Berkeley Dr. Robert Boivin, Auburn University Subir Bose, University of entral Florida David Branning, Trinity ollege Meade Brooks, ollin ounty ommunity ollege Bruce Bunker, University of Notre Dame Grant Bunker, Illinois Institute of Technology Wayne arr, Stevens Institute of Technology harles hiu, University of Texas Austin Robert oakley, University of Southern Maine David urott, University of North Alabama Biman Das, SUNY Potsdam Bob Davis, Taylor University Kaushik De, University of Texas Arlington Michael Dennin, University of alifornia Irvine Kathy Dimiduk, University of New Mexico John DiNardo, Drexel University Scott Dudley, United States Air Force Academy John Essick, Reed ollege assandra Fesen, Dartmouth ollege Alex Filippenko, University of alifornia Berkeley Richard Firestone, Lawrence Berkeley Lab Mike Fortner, Northern Illinois University Tom Furtak, olorado School of Mines Edward Gibson, alifornia State University Sacramento John Hardy, Texas A&M J. Erik Hendrickson, University of Wisconsin Eau laire Laurent Hodges, Iowa State University David Hogg, New York University Mark Hollabaugh, Normandale ommunity ollege Andy Hollerman, University of Louisiana at Lafayette William Holzapfel, University of alifornia Berkeley Bob Jacobsen, University of alifornia Berkeley TerukiKamon, Texas A&M Daryao Khatri, University of the District of olumbia Jay Kunze, Idaho State University

Jim LaBelle, Dartmouth ollege M.A.K. Lodhi, Texas Tech Bruce Mason, University of Oklahoma Dan Mazilu, Virginia Tech Linda McDonald, North Park ollege Bill McNairy, Duke University Raj Mohanty, Boston University Giuseppe Molesini, Istituto Nazionale di Ottica Florence Lisa K. Morris, Washington State University Blaine Norum, University of Virginia Alexandria Oakes, Eastern Michigan University Michael Ottinger, Missouri Western State University Lyman Page, Princeton and WMAP Bruce Partridge, Haverford ollege R. Daryl Pedigo, University of Washington Robert Pelcovitz, Brown University Vahe Peroomian, U LA James Rabchuk, Western Illinois University Michele Rallis, Ohio State University Paul Richards, University of alifornia Berkeley Peter Riley, University of Texas Austin Larry Rowan, University of North arolina hapel Hill indy Schwarz, Vassar ollege Peter Sheldon, Randolph-Macon Woman's ollege Natalia A. Sidorovskaia, University of Louisiana at Lafayette James Siegrist, U Berkeley, Director Physics Division LBNL George Smoot, University of alifornia Berkeley Mark Sprague, East arolina University Michael Strauss, University of Oklahoma Laszlo Takac, University of Maryland Baltimore o. Franklin D.Trumpy, Des Moines Area ommunity ollege Ray Turner, lemson University Som Tyagi, Drexel University John Vasut, Baylor University Robert Webb, Texas A&M Robert Weidman, Michigan Technological University Edward A. Whittaker, Stevens Institute of Technology John Wolbeck, Orange ounty ommunity ollege Stanley George Wojcicki, Stanford University Edward Wright, U LA Todd Young, Wayne State ollege William Younger, ollege of the Albemarle Hsiao-Ling Zhou, Georgia State University

I owe special thanks to Prof. Bob Davis for much valuable input, and especially for working out all the Problems and producing the Solutions Manual for all Problems, as well as for providing the answers to odd -numbered Problems at the end of this book. Many thanks also to J. Erik Hendrickson who collaborated with Bob Davis on the solutions, and to the team they managed (Profs. Anand Batra, Meade Brooks, David urrott, Blaine Norum, Michael Ottinger, Larry Rowan, Ray Turner, John Vasut, William Younger). I am grateful to Profs. John Essick, Bruce Barnett, Robert oakley, Biman Das, Michael Dennin, Kathy Dimiduk, John DiNardo, Scott Dudley, David Hogg, indy Schwarz, Ray Turner, and Som Tyagi, who inspired many of the Examples, Questions, Problems, and significant clarifications. rucial for rooting out errors, as well as providing excellent suggestions, were Profs. Kathy Dimiduk, Ray Turner, and Lorraine Allen. A huge thank you to them and to Prof. Giuseppe Molesini for his suggestions and his exceptional photographs for optics.

xvi

PREFACE

For hapters 43 and 44 on Particle Physics and osmology and Astrophysics, I was fortunate to receive generous input from some of the top experts in the field, to whom I owe a debt of gratitude: George Smoot, Paul Richards, Alex Filippenko, James Siegrist, and William Holzapfel (U Berkeley), Lyman Page (Princeton and WMAP), Edward Wright (U LA and WMAP), and Michael Strauss (University of Oklahoma). I especially wish to thank Profs. Howard Shugart, hair Frances Heilman, and many others at the University of alifornia, Berkeley, Physics Department for helpful discussions, and for hospitality. Thanks also to Prof. Tito Arecchi and others at the Istituto Nazionale di Ottica, Florence, Italy. Finally, I am grateful to the many people at Prentice Hall with whom I worked on this project, especially Paul orey, Karen Karlin, hristian Botting, John hristiana, and Sean Hogan. The final responsibility for all errors lies with me. I welcome comments, corrections, and suggestions as soon as possible to benefit students for the next reprint. D. .G. email: Paul. [email protected] Post: Paul orey One Lake Street Upper Saddle River, NJ 07458

About the Author Douglas . Giancoli obtained his BA in physics (summa cum laude) from the University of alifornia, Berkeley, his MS in physics at the Massachusetts Institute of Technology, and his PhD in elementary particle physics at the University of ali fornia, Berkeley. He spent 2 years as a post-doctoral fellow at U Berkeley's Virus lab developing skills in molecular biology and biophysics. His mentors include Nobel winners Emilio Segre and Donald Glaser. He has taught a wide range of undergraduate courses, traditional as well as innovative ones, and continues to update his texbooks meticulously, seeking ways to better provide an understanding of physics for students. Doug's favorite spare-time activity is the outdoors, especially climbing peaks (here on a dolomite summit, Italy). He says climbing peaks is like learning physics: it takes effort and the rewards are great.

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M steringPhysics™ (www.m steringphysics.com) is sophistic ted online tutoring nd homework system devel oped speci lly for courses using c lculus - b sed physics. Origin lly developed by D vid Pritch rd nd coll bor tors t MIT, M steringPhysics provides students with individu lized online tutoring by responding to their wrong nswers nd providing hints for solving multi-step problems when they get stuck. It gives them immedi te nd up -to -d te ssessment of their progress, nd shows where they need to pr ctice more. M steringPhysics provides instructors with f st nd effective w y to ssign tried nd-tested online homework ssignments th t comprise r nge of problem types. The powerful post - ssignment di gnostics llow instructors to ssess the progress of their cl ss s whole s well s individu l students, nd quickly identify re s of difficulty.

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PREFACE

xvii

To Students HOW TO STUDY 1. Read the hapter. Learn new vocabulary and notation. Try to respond to questions and exercises as they occur. 2. Attend all class meetings. Listen. Take notes, especially about aspects you do not remember seeing in the book. Ask questions (everyone else wants to, but maybe you will have the courage). You will get more out of class if you read the hapter first. 3. Read the hapter again, paying attention to details. Follow derivations and worked-out Examples. Absorb their logic. Answer Exercises and as many of the end of hapter Questions as you can. 4. Solve 10 to 20 end of hapter Problems (or more), especially those assigned. In doing Problems you find out what you learned and what you didn't. Discuss them with other students. Problem solving is one of the great learning tools. Don't just look for a formula — it won't cut it. NOTES ON THE FORMAT AND PROBLEM SOLVING 1. Sections marked with a star (*) are considered option l. They can be omitted without interrupting the main flow of topics. No later material depends on them except possibly later starred Sections. They may be fun to read, though. 2. The customary conventions are used: symbols for quantities (such as m for mass) are italicized, whereas units (such as m for meter) are not italicized. Symbols for vectors are shown in boldface with a small arrow above: F. 3. Few equations are valid in all situations. Where practical, the limit tions of important equations are stated in square brackets next to the equation. The equations that represent the great laws of physics are displayed with a tan background, as are a few other indispensable equations. 4. At the end of each hapter is a set of Problems which are ranked as Level I, II, or III, according to estimated difficulty. Level I Problems are easiest, Level II are standard Problems, and Level III are "challenge problems." These ranked Problems are arranged by Section, but Problems for a given Section may depend on earlier material too. There follows a group of General Problems, which are not arranged by Section nor ranked as to difficulty. Problems that relate to optional Sections are starred (*). Most hapters have 1 or 2 omputer/Numerical Problems at the end, requiring a computer or graphing calculator. Answers to odd-numbered Problems are given at the end of the book. 5. Being able to solve Problems is a crucial part of learning physics, and provides a powerful means for understanding the concepts and principles. This book contains many aids to problem solving: (a) worked-out Ex mples and their solutions in the text, which should be studied as an integral part of the text; (b)some of the worked-out Examples are Estim tion Ex mples, which show how rough or approximate results can be obtained even if the given data are sparse (see Section 1 -6); (c) special Problem Solving Str tegies placed throughout the text to suggest a step-by-step approach to problem solving for a particular topic— but remember that the basics remain the same; most of these "Strategies" are followed by an Example that is solved by explicitly following the suggested steps; (d) special problem-solving Sections; (e) "Problem Solving" marginal notes which refer to hints within the text for solving Problems; (f) Exercises within the text that you should work out imme diately, and then check your response against the answer given at the bottom of the last page of that hapter; (g) the Problems themselves at the end of each hapter (point 4 above). 6. Conceptu l Ex mples pose a question which hopefully starts you to think and come up with a response. Give yourself a little time to come up with your own response before reading the Response given. 7. M th review, plus some additional topics, are found in Appendices. Useful data, conversion factors, and math formulas are found inside the front and back covers.

xviii

PREFACE

USE OF

OLOR

Vectors A gener l vector result nt vector (sum) is slightly thicker components of ny vector re d shed Displ cement (D, ?) Velocity (v) Acceler tion ( ) Force (F) Force on second or third object in s me figure Momentum (p ormv) Angul r momentum (L) Angul r velocity (to) Torque ( f ) Electric field (E) M gnetic field (B)

Electricity nd m gnetism

Electric circuit symbols Wire, with switch S

Equipotenti l lines

Resistor

M gnetic field lines

C p citor + ) or • +

Electric ch rge (+)

Electric ch rge ( - )

or • -

Inductor B ttery

S -v w v -

1 l T T

Electric field lines

Ground

Object Re l im ge (d shed)

Other

Optics Light r ys

- /n n n p -

— *— 1

Energy level ( tom, etc.) Me surement lines

4 ■ ■ ■

P th of moving object

Virtu l im ge 4 (d shed nd p ler) ■ ■ ■

Direction of motion or current

h—1.0 m —H ------------------ ►

PREFACE

Image of the Earth from a NASA satellite. The sky appears blackfrom out in space because there are so few molecules to reflect light (Whythe sky appears blue to us on Earth has to do with scattering of light by molecules of the atmosphere, as

• * 4

Chapter 35.) Note the storm off the coast of Mexico.

Introduction, Measurement, Estimating CHAPTER-OPENING QUESTION — Guess now! Suppose you wanted to actually measure the radius of the Earth, at least roughly, rather than taking other people's word for what it is. Which response below describes the best approach? ( ) Give up; it is impossible using ordinary means. (b) Use an extremely long measuring tape. (c) It is only possible by flying high enough to see the actual curvature of the Earth. (d) Use a standard measuring tape, a step ladder, and a large smooth lake. (e) Use a laser and a mirror on the Moon or on a satellite. \We start eac C apter wit a Question, like t e one above. Try to answer it rig t away. Don 't worry about getting t e rig t answer now — t e idea is to get your preconceived notions out on t e table. If t ey are misconceptions, we expect t em to be cleared up as you read t e C apter. You will usually get anot er c ance at t e Question later in t e C apter w en t e appropriate material as been covered. T ese C apter-Opening Questions will also elp you to see t e power and usefulness of p ysics. ]

CONTENTS 1 - 1 The N ture of Science 1 - 2 Models, Theories, nd L ws 1 - 3 Me surement nd Uncert inty; Signific nt Figures 1 - 4 Units, St nd rds, nd the SI System 1 - 5 Converting Units 1 - 6 Order of M gnitude: R pid Estim ting :1 - 7 Dimensions nd Dimension l An lysis

hysics is the most basic of the sciences. It deals with the behavior and structure of matter. The field of physics is usually divided into classicalp ysics which includes motion, fluids, heat, sound, light, electricity and magnetism; and modem p ysics which includes the topics of relativity, atomic structure, condensed matter, nuclear physics, elementary particles, and cosmology and astrophysics. We will cover all these topics in this book, beginning with motion (or mechanics, as it is often called) and ending with the most recent results in our study of the cosmos. An understanding of physics is crucial for anyone making a career in science or technology. Engineers, for example, must know how to calculate the forces within a structure to design it so that it remains standing (Fig. 1 - la). Indeed, in hapter 12 we will see a worked-out Example of how a simple physics calculation— or even intuition based on understanding the physics of forces—would have saved hundreds of lives (Fig. 1 - lb). We will see many examples in this book of how physics is useful in many fields, and in everyday life.

1—1 The N ture of Science

(b) FIGURE 1 - 1 ( ) This Rom n queduct w s built 2000 ye rs go nd still st nds, (b) The H rtford Civic Center coll psed in 1978, just two ye rs fter it w s built.

The principal aim of all sciences, including physics, is generally considered to be the search for order in our observations of the world around us. Many people think that science is a mechanical process of collecting facts and devising theories. But it is not so simple. Science is a creative activity that in many respects resem bles other creative activities of the human mind. One important aspect of science is observ tion of events, which includes the design and carrying out of experiments. But observation and experiment require imagination, for scientists can never include everything in a description of what they observe. Hence, scientists must make judgments about what is relevant in their observations and experiments. onsider, for example, how two great minds, Aristotle (384-322 b .c .) and Galileo (1564-1642), interpreted motion along a horizontal surface. Aristotle noted that objects given an initial push along the ground (or on a tabletop) always slow down and stop. onsequently, Aristotle argued that the natural state of an object is to be at rest. Galileo, in his reexamination of horizontal motion in the 1600s, imagined that if friction could be eliminated, an object given an initial push along a horizontal surface would continue to move indefinitely without stopping. He concluded that for an object to be in motion was just as natural as for it to be at rest. By inventing a new approach, Galileo founded our modern view of motion ( hapters 2,3, and 4), and he did so with a leap of the imagination. Galileo made this leap conceptually, without actually eliminating friction. Observation, with careful experimentation and measurement, is one side of the scientific process. The other side is the invention or creation of theories to explain and order the observations. Theories are never derived directly from observations. Observations may help inspire a theory, and theories are accepted or rejected based on the results of observation and experiment. The great theories of science may be compared, as creative achievements, with great works of art or literature. But how does science differ from these other creative activities? One important difference is that science requires testing of its ideas or theories to see if their predictions are borne out by experiment. Although the testing of theories distinguishes science from other creative fields, it should not be assumed that a theory is "proved" by testing. First of all, no measuring instrument is perfect, so exact confirmation is not possible. Further more, it is not possible to test a theory in every single possible circumstance. Hence a theory cannot be absolutely verified. Indeed, the history of science tells us that long-held theories can be replaced by new ones.

1 -2 Models, Theories, nd L ws 2

CHAPTER 1

When scientists are trying to understand a particular set of phenomena, they often make use of a model. A model, in the scientist's sense, is a kind of analogy or mental image of the phenomena in terms of something we are familiar with. One

example is the wave model of light. We cannot see waves of light as we can water waves. But it is valuable to think of light as made up of waves because experiments indicate that light behaves in many respects as water waves do. The purpose of a model is to give us an approximate mental or visual picture— something to hold on to —when we cannot see what actually is happening. Models often give us a deeper understanding: the analogy to a known system (for instance, water waves in the above example) can suggest new experiments to perform and can provide ideas about what other related phenomena might occur. You may wonder what the difference is between a theory and a model. Usually a model is relatively simple and provides a structural similarity to the phenomena being studied. A theory is broader, more detailed, and can give quantitatively testable predictions, often with great precision. It is important, however, not to confuse a model or a theory with the real system or the phenomena themselves. Scientists give the title l w to certain concise but general statements about how nature behaves (that energy is conserved, for example). Sometimes the state ment takes the form of a relationship or equation between quantities (such as Newton's second law, F = ma). To be called a law, a statement must be found experimentally valid over a wide range of observed phenomena. For less general statements, the term principle is often used (such as Archimedes' principle). Scientific laws are different from political laws in that the latter are prescriptive: they tell us how we ought to behave. Scientific laws are descriptive: they do not say how nature s ould behave, but rather are meant to describe how nature does behave. As with theories, laws cannot be tested in the infinite variety of cases possible. So we cannot be sure that any law is absolutely true. We use the term "law" when its validity has been tested over a wide range of cases, and when any limitations and the range of validity are clearly understood. Scientists normally do their research as if the accepted laws and theories were true. But they are obliged to keep an open mind in case new information should alter the validity of any given law or theory.

1 -3 Me surement nd Uncert inty; Signific nt Figures In the quest to understand the world around us, scientists seek to find relationships among physical quantities that can be measured.

Uncert inty Reliable measurements are an important part of physics. But no measurement is absolutely precise. There is an uncertainty associated with every measurement. Among the most important sources of uncertainty, other than blunders, are the limited accuracy of every measuring instrument and the inability to read an instrument beyond some fraction of the smallest division shown. For example, if you were to use a centimeter ruler to measure the width of a board (Fig. 1 -2), the result could be claimed to be precise to about 0.1 cm (1 mm), the smallest division on the ruler, although half of this value might be a valid claim as well. The reason is that it is difficult for the observer to estimate (or interpolate) between the smallest divisions. Furthermore, the ruler itself may not have been manufactured to an accuracy very much better than this. When giving the result of a measurement, it is important to state the estim ted uncert inty in the measurement. For example, the width of a board might be written as 8.8 ± 0.1 cm. The ± 0.1 cm ("plus or minus 0.1 cm") represents the estimated uncertainty in the measurement, so that the actual width most likely lies between 8.7 and 8.9 cm. The percent uncert inty is the ratio of the uncertainty to the measured value, multiplied by 100. For example, if the measurement is 8.8 and the uncertainty about 0.1 cm, the percent uncertainty is

FIGURE 1-2 Measuring the width of a board with a centimeter ruler. The uncertainty is about ± 1 mm.

1%, where ~ means "is approximately equal to.'

SECTI N 1- 3

Often the uncertainty in a measured value is not specified explicitly. In such cases, the uncertainty is generally assumed to be one or a few units in the last digit specified. For example, if a length is given as 8.8 cm, the uncertainty is assumed to be about 0.1 cm or 0.2 cm. It is important in this case that you do not write 8.80 cm, for this implies an uncertainty on the order of 0.01 cm; it assumes that the length is probably between 8.79 cm and 8.81 cm, when actually you believe it is between 8.7 and 8.9 cm.

Signific nt Figures

(b)

(a)

FIGURE 1 - 3 These two calculators show the wrong number of significant figures. In (a), 2.0 was divided by 3.0. The correct final result would be 0.67. In (b), 2.5 was multiplied by 3.2. The correct result is 8.0.

p PR

BLEM

LVING

Significant figure rule: N um ber o f significant figures in final result s ould be sam e as t e least significant input value

The number of reliably known digits in a number is called the number of signific nt figures. Thus there are four significant figures in the number 23.21 cm and two in the number 0.062 cm (the zeros in the latter are merely place holders that show where the decimal point goes). The number of significant figures may not always be clear. Take, for example, the number 80. Are there one or two signif icant figures? We need words here: If we say it is roug ly 80 km between two cities, there is only one significant figure (the 8) since the zero is merely a place holder. If there is no suggestion that the 80 is a rough approximation, then we can often assume (as we will in this book) that it is 80 km within an accuracy of about 1 or 2 km, and then the 80 has two significant figures. If it is precisely 80 km, to within + 0.1 km, then we write 80.0 km (three significant figures). When making measurements, or when doing calculations, you should avoid the temptation to keep more digits in the final answer than is justified. For example, to calculate the area of a rectangle 11.3 cm by 6.8 cm, the result of multiplication would be 76.84 cm2. But this answer is clearly not accurate to 0.01 cm2, since (using the outer limits of the assumed uncertainty for each measurement) the result could be between 11.2 cm X 6.7 cm = 75.04 cm2 and 11.4 cm X 6.9 cm = 78.66 cm2. At best, we can quote the answer as 77 cm2, which implies an uncertainty of about 1 or 2 cm2. The other two digits (in the number 76.84 cm2) must be dropped because they are not significant. As a rough general rule (i.e., in the absence of a detailed consideration of uncertainties), we can say that t e final result o f a multiplication or division s ould ave only as many digits as t e number wit t e least number o f significant figures used in t e calculation. In our example, 6.8 cm has the least number of significant figures, namely two. Thus the result 76.84 cm2 needs to be rounded off to 77 cm2. EXERCISE A The area of a rectangle 4.5 cm by 3.25 cm is correctly given by (a) (b) 14.63 cm2; (c) 14.6 cm2; (d) 15 cm2.

AUT I ON

Calculators err wit significant figures

I PR

BLEI VI

LVING

R eport only t e p ro p er num ber o f significant figures in t e final result. Keep extra digits during t e calculation

FIGURE 1 - 4 Example 1-1. A protractor used to measure an angle.

CHAPTER 1

14.625 cm2;

When adding or subtracting numbers, the final result is no more precise than the least precise number used. For example, the result of subtracting 0.57 from 3.6 is 3.0 (and not 3.03). Keep in mind when you use a calculator that all the digits it produces may not be significant. When you divide 2.0 by 3.0, the proper answer is 0.67, and not some such thing as 0.666666666. Digits should not be quoted in a result, unless they are truly significant figures. However, to obtain the most accurate result, you should normally keep one or more extra significant figures t roug out a calculation, and round o ff only in t e final result. (With a calculator, you can keep all its digits in intermediate results.) Note also that calculators sometimes give too few significant figures. For example, when you multiply 2.5 X 3.2, a calculator may give the answer as simply 8. But the answer is accurate to two significant figures, so the proper answer is 8.0. See Fig. 1 -3. C NCEPTUAL EXAMPLE 1 - 1 | Significant figures. Using a protractor (Fig. 1-4), you measure an angle to be 30°. (a) How many significant figures should you quote in this measurement? (b) Use a calculator to find the cosine of the angle you measured. RESP NSE (a) If you look at a protractor, you will see that the precision with which you can measure an angle is about one degree (certainly not 0.1°). So you can quote two significant figures, namely, 30° (not 30.0°). (b) If you enter cos 30° in your calculator, you will get a number like 0.866025403. However, the angle you entered is known only to two significant figures, so its cosine is correctly given by 0.87; you must round your answer to two significant figures. N TE osine and other trigonometric functions are reviewed in Appendix A.

| EXERCISE B

Do 0.00324 and 0.00056 have the same number of significant figures?

Be careful not to confuse significant figures with the number of decimal places. EXERCISE C For each of the following numbers, state the number of significant figures and the number of decimal places: {a) 1.23; (b) 0.123; (c) 0.0123.

Scientific_Not tion We commonly write numbers in "powers of ten," or "scientific" notation — for instance 36,900 as 3.69 X 104, or 0.0021 as 2.1 X 10-3. One advantage of scientific notation is that it allows the number of significant figures to be clearly expressed. For example, it is not clear whether 36,900 has three, four, or five significant figures. With powers of ten notation the ambiguity can be avoided: if the number is known to three significant figures, we write 3.69 X 104, but if it is known to four, we write 3.690 X 104. Write each of the following in scientific notation and state the number of | significant figures for each: (a) 0.0258, (b) 42,300, (c) 344.50.

I EXERCISE D

Percent Uncert inty versus Signific nt Figures The significant figures rule is only approximate, and in some cases may underestimate the accuracy (or uncertainty) of the answer. Suppose for example we divide 97 by 92: 97 — = 1.05 « 1.1. 92 Both 97 and 92 have two significant figures, so the rule says to give the answer as 1.1. Yet the numbers 97 and 92 both imply an uncertainty of + 1 if no other uncertainty is stated. Now 92 + 1 and 97 + 1 both imply an uncertainty of about 1% (1/92 « 0.01 = 1%). But the final result to two significant figures is 1.1, with an implied uncertainty of + 0.1, which is an uncertainty of 0.1/1.1 « 0.1 ~ 10%. In this case it is better to give the answer as 1.05 (which is three significant figures). Why? Because 1.05 implies an uncertainty of + 0.01 which is 0.01/1.05 « 0.01 ~ 1%, just like the uncertainty in the original numbers 92 and 97. SUGGESTION: Use the significant figures rule, but consider the % uncer tainty too, and add an extra digit if it gives a more realistic estimate of uncertainty.

Approxim tions Much of physics involves approximations, often because we do not have the means to solve a problem precisely. For example, we may choose to ignore air resistance or friction in doing a Problem even though they are present in the real world, and then our calculation is only an approximation. In doing Problems, we should be aware of what approximations we are making, and be aware that the precision of our answer may not be nearly as good as the number of significant figures given in the result.

Accur cy versus Precision There is a technical difference between "precision" and "accuracy." Predsion in a strict sense refers to the repeatability of the measurement using a given instrument. For example, if you measure the width of a board many times, getting results like 8.81 cm, 8.85 cm, 8.78 cm, 8.82 cm (interpolating between the 0.1 cm marks as best as possible each time), you could say the measurements give a precision a bit better than 0.1 cm. Accur cy refers to how close a measurement is to the true value. For example, if the ruler shown in Fig. 1 - 2 was manufactured with a 2% error, the accuracy of its measurement of the board's width (about 8.8 cm) would be about 2% of 8.8 cm or about + 0.2 cm. Estimated uncertainty is meant to take both accuracy and precision into account. SECTI N 1- 3

Measurement, Uncertainty; Significant Figures

1—4 Units, St nd rds, nd the SI System TABLE 1-1 Some Typical Lengths or Distances (order of magnitude) Length (or D ist nce)

Meters ( pproxim te)

Neutron or proton (di m eter)

10- 15

A tom (di m eter) -7

Virus [see Fig. l - 5 ] Sheet o f p per (thickness) Finger width

10 - 4 10"2

Footb ll field length

102

H eight o f Mt. Everest [see Fig. l - 5 b ] E rth di meter

104 107

E rth to Sun

1011

E rth to ne rest st r E rth to ne rest g l xy

1016 1022

E rth to f rthest g l xy visible

1026

FIGURE 1 - 5 Som e lengths: ( ) viruses ( bout 10 - 7 m long) tt cking cell; (b) Mt. E verest 's height is on the order of 104 m (8850 m, to be precise).

The measurement of any quantity is made relative to a particular standard or unit, and this unit must be specified along with the numerical value of the quantity. For example, we can measure length in British units such as inches, feet, or miles, or in the metric system in centimeters, meters, or kilometers. To specify that the length of a particular object is 18.6 is meaningless. The unit must be given; for clearly, 18.6 meters is very different from 18.6 inches or 18.6 millimeters. For any unit we use, such as the meter for distance or the second for time, we need to define a st nd rd which defines exactly how long one meter or one second is. It is important that standards be chosen that are readily reproducible so that anyone needing to make a very accurate measurement can refer to the standard in the laboratory.

Length The first truly international standard was the meter (abbreviated m) established as the standard of length by the French Academy of Sciences in the 1790s. The stan dard meter was originally chosen to be one ten-millionth of the distance from the Earth's equator to either pole,f and a platinum rod to represent this length was made. (One meter is, very roughly, the distance from the tip of your nose to the tip of your finger, with arm and hand stretched out to the side.) In 1889, the meter was defined more precisely as the distance between two finely engraved marks on a particular bar of platinum -iridium alloy. In 1960, to provide greater precision and reproducibility, the meter was redefined as 1,650,763.73 wavelengths of a particular orange light emitted by the gas krypton-86. In 1983 the meter was again redefined, this time in terms of the speed of light (whose best measured value in terms of the older definition of the meter was 299,792,458 m/s, with an uncertainty of lm /s). The new definition reads: "The meter is the length of path traveled by light in vacuum during a time interval of 1/299,792,458 of a second."* British units of length (inch, foot, mile) are now defined in terms of the meter. The inch (in.) is defined as precisely 2.54 centimeters (cm; 1 cm = 0.01 m). Other conversion factors are given in the Table on the inside of the front cover of this book. Table 1 - 1 presents some typical lengths, from very small to very large, rounded off to the nearest power of ten. See also Fig. 1 - 5. [Note that the abbreviation for inches (in.) is the only one with a period, to distinguish it from the word "in".]

Time

(a)

The standard unit of time is the second (s). For many years, the second was defined as 1/86,400 of a mean solar day (24h/day X 60min/h X 60s/min = 86,400 s/day). The standard second is now defined more precisely in terms of the frequency of radi ation emitted by cesium atoms when they pass between two particular states. [Specifically, one second is defined as the time required for 9,192,631,770 periods of this radiation.] There are, by definition, 60 s in one minute (min) and 60 minutes in one hour (h). Table 1 - 2 presents a range of measured time intervals, rounded off to the nearest power of ten.

M ss The standard unit of m ss is the kilogr m (kg). The standard mass is a particular platinum -iridium cylinder, kept at the International Bureau of Weights and Measures near Paris, France, whose mass is defined as exactly 1 kg. A range of masses is presented in Table 1 -3. [For practical purposes, 1 kg weighs about 2.2 pounds on Earth.]

(b)

CHAPTER 1

tModern me surements of the E rth's circumference reve l th t the intended length is off by bout one-fiftieth of 1%. Not b d! *The new definition of the meter h s the effect of giving the speed of light the ex ct v lue of 299,792,458 m/s.

Introduction, Measurement, Estimating

TABLE 1 - 2 Some Typical Time Intervals Time Interv l

TABLE 1 -3 Some Masses Seconds ( pproxim te)

Object

Kilogr ms ( pproxim te)

Lifetim e o f very unst ble sub tomic p rticle

1 0 - 23 s

Electron

1(T30 kg

Lifetim e o f r dio ctive elem ents

10~22 s to 1028 s

Proton, neutron

10 - 27 kg 1(T17 kg

Lifetim e o f muon

1(T6 s

D N A m olecule

Time betw een hum n he rtbe ts

10°

s ( = 1 s)

B cterium

1(T15 kg

O ne d y

105

M osquito

1(T5 kg

O ne ye r

3 X 107

Plum

10" 1 kg

Hum n life sp n

2 X 109

Hum n

102

108

Length o f recorded history

1011

Ship

Hum ns on E rth

1014 1017 1018

E rth

6 X 1024 kg

Sun

2 X 1030 kg

Life on E rth A g e of U niverse

G l xy

When dealing with atoms and molecules, we usually use the unified tomic m ss unit (u). In terms of the kilogram, l u = 1.6605 X 10-27kg. The definitions of other standard units for other quantities will be given as we encounter them in later hapters. (Precise values of this and other numbers are given inside the front cover.)

Unit Prefixes In the metric system, the larger and smaller units are defined in multiples of 10 from the standard unit, and this makes calculation particularly easy. Thus 1 kilometer (km) is 1000 m, 1 centimeter is ifem, 1 millimeter (mm) is or ^cm , and so on. The prefixes "centi-," "kilo-," and others are listed in Table 1 - 4 and can be applied not only to units of length but to units of volume, mass, or any other metric unit. For example, a centiliter (cL) is ^ liter (L)> and a kilogram (kg) is 1000 grams (g).

Systems of Units When dealing with the laws and equations of physics it is very important to use a consistent set of units. Several systems of units have been in use over the years. Today the most important is the Systeme Intern tion l (French for International System), which is abbreviated SI. In SI units, the standard of length is the meter, the standard for time is the second, and the standard for mass is the kilogram. This system used to be called the MKS (meter-kilogram-second) system. A second metric system is the cgs system, in which the centimeter, gram, and second are the standard units of length, mass, and time, as abbreviated in the title. The British engineering system has as its standards the foot for length, the pound for force, and the second for time. We use SI units almost exclusively in this book.

B se versus Derived Qu ntities Physical quantities can be divided into two categories: base quantities and derived quantities. The corresponding units for these quantities are called base units and derived units. A b se qu ntity must be defined in terms of a standard. Scientists, in the interest of simplicity, want the smallest number of base quantities possible consistent with a full description of the physical world. This number turns out to be seven, and those used in the SI are given in Table 1-5. All other quantities can be defined in terms of these seven base quantities/ and hence are referred to as derived qu ntities. An example of a derived quantity is speed, which is defined as distance divided by the time it takes to travel that distance. A Table inside the front cover lists many derived quantities and their units in terms of base units. To define any quantity, whether base or derived, we can specify a rule or procedure, and this is called an oper tion l definition. trThe only exceptions re for ngle (r di ns—see Ch pter 8) nd solid ngle (ster di n). No gener l greement h s been re ched s to whether these re b se or derived qu ntities.

SECTI N 1- 4

1041 kg

TABLE 1 -4 Metric (SI) Prefixes Abbrevi tion

V lue

yott

Prefix

1024

zett

1021

1018

pet

1015

ter

1012

gig m eg

109

106

kilo

103

hecto

102

dek

101

deci

KT1

centi

1(T2

milli

1(T3

microf

V n

1(T6

P f

1(T12

n no pico fem to

K T9 1(T15

KT18

tto zepto

1(T21

yocto

KT24

fju, is the Greek letter "mu."

TABLE 1 -5 SI Base Quantities and Units Unit Abbrevi tion

Qu ntity

Unit

Length

m eter

Time

second

M ss

kilogr m

Electric current Temper ture

mpere kelvin

A K

A m ount of subst nce m ole

m ol

Luminous intensity

c ndel

Units, Standards, and the SI System

1—5 Converting Units Any quantity we measure, such as a length, a speed, or an electric current, consists of a number and a unit. Often we are given a quantity in one set of units, but we want it expressed in another set of units. For example, suppose we measure that a table is 21.5 inches wide, and we want to express this in centimeters. We must use a conversion f ctor, which in this case is (by definition) exactly 1 in. = 2.54 cm or, written another way, 1 = 2.54 cm/in. Since multiplying by one does not change anything, the width of our table, in cm, is 21.5 inches = (21.5

^ 2 .5 4 ^ ^

= 54.6 cm.

Note how the units (inches in this case) cancelled out. A Table containing many unit conversions is found inside the front cover of this book. Let's consider some Examples. 0

PHYSICS

APPLIED

T e world's tallest peaks

EXAMPLE 1 - 2 The 8000 - m peaks. The fourteen tallest peaks in the world (Fig. 1 - 6 and Table 1 - 6) are referred to as "eight-thousanders," meaning their summits are over 8000 m above sea level. What is the elevation, in feet, of an elevation of 8000 m? APPR ACH We need simply to convert meters to feet, and we can start with the

conversion factor 1 in. = 2.54 cm, which is exact. That is, 1 in. = 2.5400 cm to any number of significant figures, because it is defined to be. S LUTI N One foot is 12 in., so we can write

cm 1 ft = (1 2 is.)(2 .5 4 — J = 30.48 cm = 0.3048 m, FIGURE 1 - 6 The w orld 's second highest pe k, K2, w h ose sum m it is considered the m ost difficult o f the "8000 - ers. " K2 is seen h ere from the north (C hin ).

which is exact. Note how the units cancel (colored slashes). We can rewrite this equation to find the number of feet in 1 meter:

lm TABLE 1 - 6 The 8000 - m Peaks Pe k

= a U s

= 3'28084ft

We multiply this equation by 8000.0 (to have five significant figures): H eigh t (m )

8000.0m = (8000.0 l .) ^ 3 .2 8 0 8 4 ; ^

= 26,247ft.

Mt. E verest

8850

8611

K ngchenjung

8586

An elevation of 8000 m is 26,247 ft above sea level.

L hotse

8516

N TE We could have done the conversion all in one line:

M k lu

8462

C ho O yu

8201

D h ul giri

8167

M n slu

8156

N ng P rb t

8125

A nn purn

8091

G sherbrum I

8068

B ro d Pe k

8047

G sherbrum II

8035

Shish P ngm

8013

CHAPTER 1

annum -

- 26O T «.

The key is to multiply conversion factors, each equal to one ( = 1.0000), and to make sure the units cancel. EXERCISE E There re only 14 eight - thous nd - m eter pe ks in the world (see E x m ple 1 - 2 ), nd their n m es nd elev tion s re given in T ble 1 - 6 . They re ll in the H im l y m ou n t in r nge in Indi , P kist n, Tibet, nd Chin . D eterm in e the elev tion o f the w orld 's three highest pe ks in feet.

Introduction, Measurement, Estimating

EXAMPLE 1 - 3 Apartment area. You have seen a nice apartment whose floor area is 880 square feet (ft2). What is its area in square meters? APPR ACH We use the same conversion factor, 1 in. = 2.54 cm, but this time

we have to use it twice. S LUTI N Because lin. = 2.54cm = 0.0254m, then lft2 = (12 in.)2(0.0254 m/in.)2 = 0.0929 m2. So 880 ft2 = (880ft2)(0.0929 m2/ft2) « 82 m2. N TE As a rule of thumb, an area given in ft2 is roughly 10 times the number of square meters (more precisely, about 10.8 X) . EXAMPLE 1 - 4 Speeds. Where the posted speed limit is 55 miles per hour (mi/h or mph), what is this speed (a) in meters per second (m/s) and (b) in kilometers per hour (km/h)? APPR ACH We again use the conversion factor 1 in. = 2.54 cm, and we recall

that there are 5280 ft in a mile and 12 inches in a foot; also, one hour contains (60min/h) X (60s/min) = 3600 s/h. S LUTI N (a) We can write 1 mile as 2.54

1 mi = (5280ir)(

jGirr 1m 'TRv. / \ 100 jGfTf

= 1609 m.

We also know that 1 hour contains 3600 s, so 55 — = h

'mi. ir

1609

1 JT

~mL J V3600 s

= 25" , s

where we rounded off to two significant figures. (b) Now we use 1 mi = 1609 m = 1.609 km; then 55 — = h

' m i.

1.609

km 'm i

_km

88- .

N TE Each conversion factor is equal to one. You can look up most conversion factors in the Table inside the front cover. EXERCISE F Would limit?

driver tr veling t 15 m /s in

BLEM

LVING

Conversion factors = 1

35 m i/h zone be exceeding the speed

When changing units, you can avoid making an error in the use of conversion factors by checking that units cancel out properly. For example, in our conversion of 1 mi to 1609 m in Example 1 - 4(a), if we had incorrectly used the factor ( n ^ ) instead of (ujoSn), the centimeter units would not have cancelled out; we would not have ended up with meters.

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Unit conversion is w ron g if units do n ot cancel

1—6 Order of M gnitude: R pid Estim ting We are sometimes interested only in an approximate value for a quantity. This might be because an accurate calculation would take more time than it is worth or would require additional data that are not available. In other cases, we may want to make a rough estimate in order to check an accurate calculation made on a calculator, to make sure that no blunders were made when the numbers were entered. A rough estimate is made by rounding off all numbers to one significant figure and its power of 10, and after the calculation is made, again only one significant figure is kept. Such an estimate is called an order-of-m gnitude estim te and can be accurate within a factor of 10, and often better. In fact, the phrase "order of magnitude" is sometimes used to refer simply to the power of 10. SECTI N 1- 6

^ \ \ PR

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H o w to m ake a roug estimate

rder of Magnitude: Rapid Estimating

r = 500 in 10 m

(b) FIGURE 1 - 7 E x m ple 1 - 5 . ( ) H ow m uch w ter is in this l ke? (P h oto is o f on e o f the R e L kes in the Sierr N ev d o f C liforni .) (b) M od el o f the l ke s cylinder. [We could go on e step further nd estim te the m ss or w eight o f this l ke. W e w ill see l ter th t w ter h s density o f 1000 k g /m 3, so this l ke h s m ss o f bout (I0 3 k g /m 3) ( l0 7 m 3) « 1010 kg, which is bout 10 b illion kg or 10 m illion m etric tons. (A m etric ton is 1000 kg, bout 2200 lbs, slightly l rger th n British ton, 2000 lbs.)]

APPLIED

E stim atin g t e vo lu m e (or m ass) o f a lake; see also Fig. 1 - 7

EXAMPLE 1 - 5 ESTIMATE I Volume of a lake. Estimate how much water there is in a particular lake, Fig. l - 7 a , which is roughly circular, about 1 km across, and you guess it has an average depth of about 10 m. APPR ACH No lake is a perfect circle, nor can lakes be expected to have a

perfectly flat bottom. We are only estimating here. To estimate the volume, we can use a simple model of the lake as a cylinder: we multiply the average depth of the lake times its roughly circular surface area, as if the lake were a cylinder (Fig. l - 7 b ). S LUTI N The volume V of a cylinder is the product of its height times the area of its base: V = irr2, where r is the radius of the circular base.f The radius r is \ km = 500 m, so the volume is approximately V =

irr2 M (10 m) X (3) X (5 X 102m)2 m 8 X 106m3 « 107m3,

where tt was rounded off to 3. So the volume is on the order of 107m3, ten million cubic meters. Because of all the estimates that went into this calculation, the order-of-magnitude estimate (l07m3) is probably better to quote than the 8 X 106m3 figure. N TE To express our result in U.S. gallons, we see in the Table on the inside front cover that 1 liter = 10-3 m3 « \ gallon. Hence, the lake contains (8 X 106m3)(l gallon/4 X 10_3m3) « 2 X 109 gallons of water. EXAMPLE 1 - 6

ESTIMATE I Thickness of a page. Estimate the thickness

of a page of this book. APPR ACH At first you might think that a special measuring device, a micrometer PR

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Use sy m m e try w en p o ssib le

(Fig. 1 - 8), is needed to measure the thickness of one page since an ordinary ruler clearly won't do. But we can use a trick or, to put it in physics terms, make use of a symmetry, we can make the reasonable assumption that all the pages of this book are equal in thickness. S LUTI N We can use a ruler to measure hundreds of pages at once. If you measure the thickness of the first 500 pages of this book (page 1 to page 500), you might get something like 1.5 cm. Note that 500 numbered pages, fFormul s like this for volume, re , etc., re found inside the b ck cover of this book.

CHAPTER 1

Introduction, Measurement, Estimating

counted front and back, is 250 separate sheets of paper. So one page must have a thickness of about 1.5 cm 250 pages

10-3 cm = 6 X 10"2mm,

6 X

or less than a tenth of a millimeter (0.1 mm). EXAMPLE 1 - 7 ESTIMATE- ! Height by triangulation. Estimate the height of the building shown in Fig. 1 - 9, by "triangulation," with the help of a bus-stop pole and a friend. APPR ACH By standing your friend next to the pole, you estimate the height of

the pole to be 3 m. You next step away from the pole until the top of the pole is in line with the top of the building, Fig. l - 9a. You are 5 ft 6 in. tall, so your eyes are about 1.5 m above the ground. Your friend is taller, and when she stretches out her arms, one hand touches you, and the other touches the pole, so you estimate that distance as 2 m (Fig. l - 9a). You then pace off the distance from the pole to the base of the building with big, 1-m-long steps, and you get a total of 16 steps or 16 m. S LUTI N Now you draw, to scale, the diagram shown in Fig. l - 9 b using these measurements. You can measure, right on the diagram, the last side of the triangle to be about x = 13 m. Alternatively, you can use similar triangles to obtain the height x : 1.5 m 2m

18 m

FIGURE 1 - 8 Ex mple 1 - 6. Micrometer used for me suring sm ll thicknesses. FIGURE 1 - 9 Ex m ple 1 - 7 . Di gr ms re re lly useful!

13 4m.

Finally you add in your eye height of 1.5 m above the ground to get your final result: the building is about 15 m tall. EXAMPLE 1 - 8 ESTIMATE I Estimating the radius of Earth. Believe it or not, you can estimate the radius of the Earth without having to go into space (see the photograph on page 1). If you have ever been on the shore of a large lake, you may have noticed that you cannot see the beaches, piers, or rocks at water level across the lake on the opposite shore. The lake seems to bulge out between you and the opposite shore— a good clue that the Earth is round. Suppose you climb a stepladder and discover that when your eyes are 10 ft (3.0 m) above the water, you can just see the rocks at water level on the opposite shore. From a map, you estimate the distance to the opposite shore as d ~ 6.1 km. Use Fig. 1 -10 with = 3.0 m to estimate the radius R of the Earth.

18m

APPR ACH We use simple geometry, including the theorem of Pythagoras,

c2 = a2 + b2, where c is the length of the hypotenuse of any right triangle, and a and b are the lengths of the other two sides. S LUTI N For the right triangle of Fig. 1 -10, the two sides are the radius of the Earth R and the distance d = 6.1 km = 6100 m. The hypotenuse is approxi = 3.0 m. By the Pythagorean theorem, mately the length R + , where R2 + d2 « (R +

FIGURE 1 - 1 0 Ex m ple 1 - 8 , but not to sc le. You c n see sm ll rocks t w ter level on the opposite shore o f l ke 6.1 km wide if you st nd on stepl dder.

« R2 + 2 R + 2. We solve algebraically for R , after cancelling R 2 on both sides: d2 2

(6100 m)2 - (3.0 m)2 = 6.2 X 106m = 6200 km. 6.0 m

N TE Precise measurements give 6380 km. But look at your achievement! With a few simple rough measurements and simple geometry, you made a good estimate of the Earth's radius. You did not need to go out in space, nor did you need a very long measuring tape. Now you know the answer to the hapter-Opening Question on p. 1. SECTI N 1- 6

rder of Magnitude: Rapid Estimating

EXAMPLE 1 - 9

ESTIMATE- ! Total

number

heartbeats. Estimate the

total number of beats a typical human heart makes in a lifetime. APPR ACH A typical resting heart rate is 70beats/min. But during exercise it

can be a lot higher. A reasonable average might be 80 beats/min. S LUTI N One year in terms of seconds is (24h)(3600s/h)(365 d) « 3 X 107s. If an average person lives 70 years = (70yr)(3 X 107s/yr) « 2 X 109s, then the total number of heartbeats would be about min / \ 60 s

:) ( 2 x 109 s) « 3 x 109,

or 3 trillion.

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Estim ating o w m any pian o tuners t ere are in a city

Another technique for estimating, this one made famous by Enrico Fermi to his physics students, is to estimate the number of piano tuners in a city, say, hicago or San Francisco. To get a rough order-of-magnitude estimate of the number of piano tuners today in San Francisco, a city of about 700,000 inhabitants, we can proceed by estimating the number of functioning pianos, how often each piano is tuned, and how many pianos each tuner can tune. To estimate the number of pianos in San Francisco, we note that certainly not everyone has a piano. a guess of 1 family in 3 having a piano would correspond to 1 piano per 12 persons, assuming an average family of 4 persons. As an order of magnitude, let's say 1 piano per 10 people. This is certainly more reasonable than 1 per 100 people, or 1 per every person, so let's proceed with the estimate that 1 person in 10 has a piano, or about 70,000 pianos in San Francisco. Now a piano tuner needs an hour or two to tune a piano. So let's estimate that a tuner can tune 4 or 5 pianos a day. A piano ought to be tuned every 6 months or a year— let's say once each year. A piano tuner tuning 4 pianos a day, 5 days a week, 50 weeks a year can tune about 1000 pianos a year. So San Francisco, with its (very) roughly 70,000 pianos, needs about 70 piano tuners. This is, of course, only a rough estimated It tells us that there must be many more than 10 piano tuners, and surely not as many as 1000.

1—7 Dimensions nd Dimension l An lysis When we speak of the dimensions of a quantity, we are referring to the type of base units or base quantities that make it up. The dimensions of area, for example, are always length squared, abbreviated [L2], using square brackets; the units can be square meters, square feet, cm2, and so on. Velocity, on the other hand, can be measured in units of km/h, m/s, or mi/h, but the dimensions are always a length [L] divided by a time [T\: that is, [L/T]. The formula for a quantity may be different in different cases, but the dimen sions remain the same. For example, the area of a triangle of base b and height is A = \b , whereas the area of a circle of radius r is A = irr2. The formulas are different in the two cases, but the dimensions of area are always [L2]. Dimensions can be used as a help in working out relationships, a procedure referred to as dimension l n lysis. One useful technique is the use of dimensions to check if a relationship is incorrect. Note that we add or subtract quantities only if they have the same dimensions (we don't add centimeters and hours); and the quantities on each side of an equals sign must have the same dimensions. (In numerical calculations, the units must also be the same on both sides of an equation.) For example, suppose you derived the equation v = v0 + I at2, where v is the speed of an object after a time t, v0 is the object's initial speed, and the object undergoes an acceleration a. Let's do a dimensional check to see if this equation tA check of the S n Fr ncisco Yellow P ges (done fter this c lcul tion) reve ls bout 50 listings. E ch of these listings m y employ more th n one tuner, but on the other h nd, e ch m y lso do rep irs s well s tuning. In ny c se, our estim te is re son ble. *Some Sections of this book, such s this one, m y be considered optional t the discretion of the instructor, nd they re m rked with n sterisk (*). See the Pref ce for more det ils.

CHAPTER 1

Introduction, Measurement, Estimating

could be correct or is surely incorrect. Note that numerical factors, like the \ here, do not affect dimensional checks. We write a dimensional equation as follows, remembering that the dimensions of speed are [L/T\ and (as we shall see in hapter 2) the dimensions of acceleration are [L /T 2]:

.?]1M+

The dimensions are incorrect: on the right side, we have the sum of quantities whose dimensions are not the same. Thus we conclude that an error was made in the derivation of the original equation. A dimensional check can only tell you when a relationship is wrong. It can't tell you if it is completely right. For example, a dimensionless numerical factor (such as \ or 2 t t ) could be missing. Dimensional analysis can also be used as a quick check on an equation you are not sure about. For example, suppose that you can't remember whether the equa tion for the period of a simple pendulum T (the time to make one back-and-forth swing) of length i is T = 2 V tj g or T = 2 V g /l, where g is the acceleration due to gravity and, like all accelerations, has dimensions [L /T 2]. (Do not worry about these formulas—the correct one will be derived in hapter 14; what we are concerned about here is a person's recalling whether it contains £/g or g/L) A dimensional check shows that the former (i/g ) is correct:

[r|

J\ S k ]

whereas the latter (g/l) is not:

'[l / 2] = m

n r

= 1

[l ] \I[ r - } [ ] Note that the constant 2 has no dimensions and so can't be checked using dimensions. Further uses of dimensional analysis are found in Appendix . Planck length. The smallest meaningful measure of length is called the "Planck length," and is defined in terms of three fundamental constants in nature, the speed of light c = 3.00 X 108m/s, the gravitational constant G = 6.67 X 10-11 m3/kg • s2, and Planck's constant = 6.63 X 10_34kg*m2/s. The Planck length AP (A is the Greek letter "lambda") is given by the following combination of these three constants:

[2 5 J 2 I2 H H H

AP — Show that the dimensions of APare length [L], and find the order of magnitude of AP. APPR ACH We rewrite the above equation in terms of dimensions. The dimen sions of c are [L/T], of G are [L3/M T 2], and of are [ML2/T]. S LUTI N The dimensions of AP are | L, / "

r l - v P I - w

which is a length. The value of the Planck length is /Gfc

(6.67 X 10-11m3A g - s2)(6.63 X 10-34kg - m2/s) ~ (3.0 X 10sm/s)3

Ap = A/ —r = a / ------------------- ;----------- :----- ^ V

4 X 10 "m ,

which is on the order of 10-34 or 10-35m. N TE Some recent theories ( hapters 43 and 44) suggest that the smallest particles (quarks, leptons) have sizes on the order of the Planck length, 10_35m. These theories also suggest that the "Big Bang," with which the Universe is believed to have begun, started from an initial size on the order of the Planck length. *SECTI N 1 - 7

Dimensions and Dimensional Analysis

Summ ry [The Summary that appears at the end of each hapter in this book gives a brief overview of the main ideas of the hapter. The Summary cannot serve to give an understanding of the material, which can be accomplished only by a detailed reading of the hapter.] Physics, like other sciences, is a creative endeavor. It is not simply a collection of facts. Important theories are created with the idea of explaining observ tions. To be accepted, theories are tested by comparing their predictions with the results of actual experiments. Note that, in general, a theory cannot be "proved" in an absolute sense. Scientists often devise models of physical phenomena. A model is a kind of picture or analogy that helps to describe the phenomena in terms of something we already know. A theory, often developed from a model, is usually deeper and more complex than a simple model. A scientific l w is a concise statement, often expressed in the form of an equation, which quantitatively describes a wide range of phenomena.

Me surements play a crucial role in physics, but can never be perfectly precise. It is important to specify the uncert inty of a measurement either by stating it directly using the ± notation, and/or by keeping only the correct number of signific nt figures. Physical quantities are always specified relative to a partic ular standard or unit, and the unit used should always be stated. The commonly accepted set of units today is the Systeme Intern tion l (SI), in which the standard units of length, mass, and time are the meter, kilogr m, and second. When converting units, check all conversion f ctors for correct cancellation of units. Making rough, order-of-m gnitude estim tes is a very useful technique in science as well as in everyday life. [*The dimensions of a quantity refer to the combination of base quantities that comprise it. Velocity, for example, has dimensions of [length/time] or [L/T]. Dimension l n lysis can be used to check a relationship for correct form.]

Questions 1. What are the merits and drawbacks of using a person's foot as a standard? onsider both (a) a particular person's foot, and (ib) any person's foot. Keep in mind that it is advantagous that fundamental standards be accessible (easy to compare to), invariable (do not change), indestructible, and reproducible. 2. Why is it incorrect to think that the more digits you represent in your answer, the more accurate it is? 3. When traveling a highway in the mountains, you may see elevation signs that read "914 m (3000 ft)." ritics of the metric system claim that such numbers show the metric system is more complicated. How would you alter such signs to be more consistent with a switch to the metric system? 4. What is wrong with this road sign: Memphis 7 mi (11.263 km)? 5. For an answer to be complete, the units need to be speci fied. Why?

6. Discuss how the notion of symmetry could be used to estimate the number of marbles in a 1-liter jar. 7. You measure the radius of a wheel to be 4.16 cm. If you multiply by 2 to get the diameter, should you write the result as 8 cm or as 8.32 cm? Justify your answer. 8. Express the sine of 30.0° with the correct number of significant figures. 9. A recipe for a souffle specifies that the measured ingredients must be exact, or the souffle will not rise. The recipe calls for 6 large eggs. The size of "large" eggs can vary by 10%, according to the USDA specifications. What does this tell you about how exactly you need to measure the other ingredients? 10. List assumptions useful to estimate the number of car mechanics in (a) San Francisco, (b) your hometown, and then make the estimates. 11. Suggest a way to measure the distance from Earth to the Sun. *12. an you set up a complete set of base quantities, as in Table 1 - 5, that does not include length as one of them?

| Problems [The Problems at the end of each hapter are ranked I, II, or III according to estimated difficulty, with (I) Problems being easiest. Level (III) Problems are meant mainly as a challenge for the best students, for "extra credit." The Problems are arranged by Sections, meaning that the reader should have read up to and including that Section, but not only that Section—Problems often depend on earlier material. Each hapter also has a group of General Problems that are not arranged by Section and not ranked.]

1-3 Measurement, Uncertainty, Significant Figures {Note: In Problems, assume a number like 6.4 is accurate to +0.1; and 950 is + 10 unless 950 is said to be "precisely" or "very nearly" 950, in which case assume 950 + 1.) 1. (I) The age of the universe is thought to be about 14 billion years. Assuming two significant figures, write this in powers of ten in (a) years, (b) seconds.

HAPTER 1

Introduction, Measurement, Estimating

2. (I) How many significant figures do each of the following numbers have: (a) 214, (b) 81.60, (c) 7.03, (d) 0.03, (e) 0.0086, ( /) 3236, and (g) 8700? 3. (I) Write the following numbers in powers of ten notation: (a) 1.156, (b) 21.8, (c) 0.0068, (d) 328.65, (e) 0.219, and (/) 444. 4. (I) Write out the following numbers in full with the correct number of zeros: (a) 8.69 X 104, (b) 9.1 X 103, (c) 8.8 X 10_1, (d) 4.76 X 102, and (e) 3.62 X 10"5. 5. (II) What is the percent uncertainty in the measurement 5.48 ± 0.25 m? 6. (II) Time intervals measured with a stopwatch typically have an uncertainty of about 0.2 s, due to human reaction time at the start and stop moments. What is the percent uncertainty of a hand-timed measurement of (a) 5 s, (b) 50 s, (c) 5 min? 7. (II) Add (9.2 X 103s) + (8.3 X 104s) + (0.008 X 106s).

8. (II) Multiply 2.079 X 102m by 0.082 X 10-1, taking into account significant figures. 9. (Ill) For small angles 6, the numerical value of sin 0 is approximately the same as the numerical value of tan0. Find the largest angle for which sine and tangent agree to within two significant figures. 10. (Ill) What, roughly, is the percent uncertainty in the volume of a spherical beach ball whose radius is r = 0.84 ± 0.04 m?

28. (II) Estimate how long it would take one person to mow a football field using an ordinary home lawn mower (Fig. 1-11). Assume the mower moves with a 1 - km/h speed, and has a 0.5-m width.

1-4 and 1-5 Units, Standards, SI, Converting Units 11. (I) Write the following as full (decimal) numbers with stan dard units: (a) 286.6 mm, (b) 85 /jlV, ( c ) 760 mg, (d) 60.0 ps, (e) 22.5 fm, ( /) 2.50 gigavolts. 12. (I) Express the following using the prefixes of Table 1 -4: (a) 1 X 106volts, (b) 2 X 10-6 meters, (c) 6 X 103days, (id) 18 X 102bucks, and (e) 8 X 10-8 seconds. 13. (I) Determine your own height in meters, and your mass in kg. 14. (I) The Sun, on average, is 93 million miles from Earth. How many meters is this? Express (a) using powers of ten, and (b) using a metric prefix. 15. (II) What is the conversion factor between (a) ft2 and yd2, (b) m2 and ft2? 16. (II) An airplane travels at 950 km /h. How long does it take to travel 1.00 km? 17. (II) A typical atom has a diameter of about 1.0 X 10-10m. (a) What is this in inches? (b) Approximately how many atoms are there along a 1.0-cm line? 18. (II) Express the following sum with the correct number of significant figures: 1.80 m + 142.5 cm + 5.34 X 105/xm. 19. (II) Determine the conversion factor between (a) km /h and m i/h, (b) m /s and ft/s, and (c) km /h and m/s. 20. (II) How much longer (percentage) is a one-mile race than a 1500-m race ("the metric mile")? 21. (II) A lig t-year is the distance light travels in one year (at speed = 2.998 X 108m /s). (a) How many meters are there in 1.00 light-year? (b) An astronomical unit (AU) is the average distance from the Sun to Earth, 1.50 X 108km. How many AU are there in 1.00 light-year? (c) What is the speed of light in A U /h? 22. (II) If you used only a keyboard to enter data, how many years would it take to fill up the hard drive in your computer that can store 82 gigabytes (82 X 109bytes) of data? Assume "normal" eight-hour working days, and that one byte is required to store one keyboard character, and that you can type 180 characters per minute. 23. (Ill) The diameter of the Moon is 3480 km. (a) What is the surface area of the Moon? (b) How many times larger is the surface area of the Earth?

1-6

rder-of-Magnitude Estimating

{Note: Remember that for rough estimates, only round numbers are needed both as input to calculations and as final results.) 24. (I) Estimate the order of magnitude (power of ten) of: (a) 2800, (b) 86.30 X 102, (c) 0.0076, and (d) 15.0 X 108. 25. (II) Estimate how many books can be shelved in a college library with 3500 m2 of floor space. Assume 8 shelves high, having books on both sides, with corridors 1.5 m wide. Assume books are about the size of this one, on average. 26. (II) Estimate how many hours it would take a runner to run (at 10 km/h) across the United States from New York to alifornia. 27. (II) Estimate the number of liters of water a human drinks in a lifetime.

FIGURE 1-11

Problem 28. 29. (II) Estimate the number of dentists (a) in San Francisco and (b) in your town or city. 30. (Ill) The rubber worn from tires mostly enters the atmos phere as particulate pollution. Estimate how much rubber (in kg) is put into the air in the United States every year. To get started, a good estimate for a tire tread's depth is 1 cm when new, and rubber has a mass of about 1200 kg per m3 of volume. 31. (Ill) You are in a hot air balloon, 200 m above the flat Texas plains. You look out toward the horizon. How far out can you see — that is, how far is your horizon? The Earth's radius is about 6400 km. 32. (Ill) I agree to hire you for 30 days and you can decide between two possible methods of payment: either (1) $1000 a day, or (2) one penny on the first day, two pennies on the second day and continue to double your daily pay each day up to day 30. Use quick estimation to make your decision, and justify it. 33. (Ill) Many sailboats are moored at a marina 4.4 km away on the opposite side of a lake. You stare at one of the sailboats because, when you are lying flat at the water's edge, you can just see its deck but none of the side of the sailboat. You then go to that sailboat on the other side of the lake and measure that the deck is 1.5 m above the level of the water. Using Fig. 1-12, where = 1.5 m, esti mate the radius R of the Earth. h-------- d

FIGURE 1 - 12 Problem 33. You see a sailboat across a lake (not to scale). R is the radius of the Earth. You are a distance d = 4.4 km from the sailboat when you can see only its deck and not its side. Because of the curvature of the Earth, the water "bulges out" between you and the boat.

34. (Ill) Another experiment you can do also uses the radius of the Earth. The Sun sets, fully disappearing over the horizon as you lie on the beach, your eyes 20 cm above the sand. You immediately jump up, your eyes now 150 cm above the sand, and you can again see the top of the Sun. If you count the number of seconds ( = t) until the Sun fully disappears again, you can estimate the radius of the Earth. But for this Problem, use the known radius of the Earth and calculate the time t. Problems

1-7 Dimensions *35. (I) What are the dimensions of density, which is mass per volume? *36. (II) The speed v of an object is given by the equation v = A t3 — Bt, where t refers to time, (a) What are the dimensions of A and 5 ? (b) What are the SI units for the constants A and 5 ? *37. (II) Three students derive the following equations in which x refers to distance traveled, v the speed, a the acceleration (m /s2), t the time, and the subscript zero (0) means a quantity at time t = 0: (a) x = vt2 + 2at, (b) x = v0t + \a t2, and (c) x = v0t + 2at2. Which of these could possibly be correct according to a dimensional check?

1538. (II) Show that the following combination of the three funda mental constants of nature that we used in Example 1 - 10 (that is G, c, and ) forms a quantity with the dimensions of time: =

This quantity, tP, is called the Planck time and is thought to be the earliest time, after the creation of the Universe, at which the currently known laws of physics can be applied.

| Gener l Problems___________ 39. Global positioning satellites (GPS) can be used to deter mine positions with great accuracy. If one of the satellites is at a distance of 20,000 km from you, what percent uncertainty in the distance does a 2-m uncertainty represent? How many significant figures are needed in the distance? 40. Computer c ips (Fig. 1 - 13) etched on circular silicon wafers of thickness 0.300 mm are sliced from a solid cylindrical silicon crystal of length 25 cm. If each wafer can hold 100 chips, what is the maximum number of chips that can be produced from one entire cylinder?

48. Estimate the number of gumballs in the machine of Fig. 1-14.

FIGURE 1 - 14 Problem 48. Estimate the number of gumballs in the machine.

Problem 40. The wafer held by the hand (above) is shown below, enlarged and illuminated by colored light. Visible are rows of integrated circuits (chips). F IG U R E ! - 13

41. {a) How many seconds are there in 1.00 year? (b) How many nanoseconds are there in 1.00 year? (c) How many years are there in 1.00 second? 42. American football uses a field that is 100 yd long, whereas a regulation soccer field is 100 m long. Which field is longer, and by how much (give yards, meters, and percent)? 43. A typical adult human lung contains about 300 million tiny cavities called alveoli. Estimate the average diameter of a single alveolus. 44. One hectare is defined as 1.000 X 104m2. One acre is 4.356 X 104ft2. How many acres are in one hectare? 45. Estimate the number of gallons of gasoline consumed by the total of all automobile drivers in the United States, per year. 46. Use Table 1 - 3 to estimate the total number of protons or neutrons in (a) a bacterium, (b) a DNA molecule, (c) the human body, (d) our Galaxy. 47. An average family of four uses roughly 1200 L (about 300 gallons) of water per day (l L = 1000 cm3). How much depth would a lake lose per year if it uniformly covered an area of 50 km2 and supplied a local town with a population of 40,000 people? onsider only population uses, and neglect evaporation and so on.

HAPTER 1

Introduction, Measurement, Estimating

49. Estimate how many kilograms of laundry soap are used in the U.S. in one year (and therefore pumped out of washing machines with the dirty water). Assume each load of laundry takes 0.1 kg of soap. 50. How big is a ton? That is, what is the volume of something that weighs a ton? To be specific, estimate the diameter of a 1 -ton rock, but first make a wild guess: will it be 1 ft across, 3 ft, or the size of a car? [Hint: Rock has mass per volume about 3 times that of water, which is 1 kg per liter (lO3 cm3) or 62 lb per cubic foot.] 51. A certain audio compact disc ( D) contains 783.216 megabytes of digital information. Each byte consists of exactly 8 bits. When played, a D player reads the D's digital information at a constant rate of 1.4 megabits per second. How many minutes does it take the player to read the entire D? 52. Hold a pencil in front of your eye at a position where its blunt end just blocks out the Moon (Fig. 1 - 15). Make appro priate measurements to estimate the diameter of the Moon, given that the Earth - M oon distance is 3.8 X 105 km. FIGURE 1 - 15 Problem 52. How big is the Moon?

53. A heavy rainstorm dumps 1.0 cm of rain on a city 5 km wide and 8 km long in a 2-h period. How many metric tons (l metric ton = 103kg) of water fell on the city? (1 cm3 of water has a mass of 1 g = 10-3 kg.) How many gallons of water was this?

54. Noah's ark was ordered to be 300 cubits long, 50 cubits wide, and 30 cubits high. The cubit was a unit of measure equal to the length of a human forearm, elbow to the tip of the longest finger. Express the dimensions of Noah's ark in meters, and estimate its volume (m3). 55. Estimate how many days it would take to walk around the world, assuming 10 h walking per day at 4 km/h. 56. One liter (1000 cm3) of oil is spilled onto a smooth lake. If the oil spreads out uniformly until it makes an oil slick just one molecule thick, with adjacent molecules just touching, estimate the diameter of the oil slick. Assume the oil mole cules have a diameter of 2 X 10-10 m. 57. Jean camps beside a wide river and wonders how wide it is. She spots a large rock on the bank directly across from her. She then walks upstream until she judges that the angle between her and the rock, which she can still see clearly, is now at an angle of 30° downstream (Fig. 1 - 16). Jean measures her stride to be about 1 yard long. The distance back to her camp is 120 strides. About how far across, both in yards and in meters, is the river? \m v

60. The diameter of the Moon is 3480 km. What is the volume of the Moon? How many Moons would be needed to create a volume equal to that of Earth? 61. Determine the percent uncertainty in 6, and in sin 6, when (a) 6 = 15.0° ± 0.5°, (b) 6 = 75.0° ± 0.5°. 62. If you began walking along one of E arth's lines of longi tude and walked north until you had changed latitude by 1 minute of arc (there are 60 minutes per degree), how far would you have walked (in miles)? This distance is called a "nautical mile." 63. Make a rough estimate of the volume of your body (in m3). 64. Estimate the number of bus drivers (a) in Washington, D. ., and (b) in your town. 65. The American Lung Association gives the following formula for an average person's expected lung capacity V (in liters, where 1 L = 103 cm3): V = 4.1 H - 0.018A - 2.69,

66.

67.

FIGURE 1 - 16

Problem 57.

120 Strides 68.

58. A watch manufacturer claims that its watches gain or lose no more than 8 seconds in a year. How accurate is this watch, expressed as a percentage? 59. An angstrom (symbol A ) is a unit of length, defined as 10-10 m, which is on the order of the diameter of an atom, (a) How many nanometers are in 1.0 angstrom? (b) How many femtometers or fermis (the common unit of length in nuclear physics) are in 1.0 angstrom? (c) How many angstroms are in 1.0 m? (d) How many angstroms are in 1.0 light-year (see Problem 21)?

69.

where H and A are the person's height (in meters), and age (in years), respectively. In this formula, what are the units of the numbers 4.1, 0.018, and 2.69? The density of an object is defined as its mass divided by its volume. Suppose the mass and volume of a rock are measured to be 8 g and 2.8325 cm3. To the correct number of significant figures, determine the rock's density. To the correct number of significant figures, use the infor mation inside the front cover of this book to determine the ratio of (a) the surface area of Earth compared to the surface area of the Moon; (b) the volume of Earth compared to the volume of the Moon. One mole of atoms consists of 6.02 X 1023 individual atoms. If a mole of atoms were spread uniformly over the surface of the Earth, how many atoms would there be per square meter? Recent findings in astrophysics suggest that the observable Universe can be modeled as a sphere of radius R = 13.7 X 109 light-years with an average mass density of about 1 X 10_26kg/m 3, where only about 4% of the Universe's total mass is due to "ordinary" matter (such as protons, neutrons, and electrons). Use this information to estimate the total mass of ordinary matter in the observable Universe. (1 light-year = 9.46 X 1015 m.)

Answers to Exercises A: (d). B: No: they have 3 and 2, respectively. : All three have three significant figures, although the number of decimal places is (a) 2, (b) 3, (c) 4.

D: (a) 2.58 X 10"2, 3; (b) 4.23 X 104, 3 (probably); (c) 3.4450 X 102, 5. Mt. Everest, 29,035 ft; K 2,28,251 ft; Kangchenjunga, 28,169 ft. F: No: 15 m /s ~ 34 mi/h.

General Problems

A high - speed c r h s rele sed p r chute to reduce its sp eed quickly. The directions o f the c r 's velocity nd cceler tion re show n by the green (v) nd gold ( ) rrows. M otion is described using the concepts o f velocity nd cceler tion. In the c se show n here, the cceler tion is in the op p osite direction from the velocity v, w hich m e ns the object is slow ing dow n. W e ex m ine in det il m otion w ith const nt cceler tion, including the vertic l m otion o f objects f lling under gr vity.

T £

Describing Motion: Kinematics in One Dimension CHAPTER -OPENING QUESTION — Guess now! ONTENTS 2 - 1 R eference Fr mes nd Displ cement 2 - 2 Aver ge Velocity 2 - 3 Inst nt neous Velocity 2 - 4 Acceler tion 2 - 5 M otion t Const nt Acceler tion 2 - 6 Solving Problems

[D o n 't w o r ry a b o u t getting t e rig t an sw er n o w — y o u w ill g et anot er c ance later in t e C apter. See also p. 1 o f C apter 1 f o r m o re explanation .]

Two small heavy balls have the same diameter but one weighs twice as much as the other. The balls are dropped from a second -story balcony at the exact same time. The time to reach the ground below will be: ( ) twice as long for the lighter ball as for the heavier one. (b) longer for the lighter ball, but not twice as long. (c) twice as long for the heavier ball as for the lighter one. (d) longer for the heavier ball, but not twice as long. (e) nearly the same for both balls.

2 - 7 Freely F lling Objects * 2 - 8 V ri ble Acceler tion; Integr l C lculus * 2 - 9 Gr phic l An lysis nd Numeric l Integr tion

he motion of objects — baseballs, automobiles, joggers, and even the Sun and M oon — is an obvious part of everyday life. It was not until the sixteenth and seventeenth centuries that our modern understanding of motion was established. Many individuals contributed to this understanding, particularly Galileo Galilei (1564-1642) and Isaac Newton (1642-1727).

The study of the motion of objects, and the related concepts of force and energy, form the field called mech nics. Mechanics is customarily divided into two parts: kinem tics, which is the description of how objects move, and dyn mics, which deals with force and why objects move as they do. This hapter and the next deal with kinematics. For now we only discuss objects that move without rotating (Fig. 2 - la). Such motion is called tr nsl tion l motion. In this hapter we will be concerned with describing an object that moves along a straight-line path, which is one-dimensional translational motion. In hapter 3 we will describe translational motion in two (or three) dimensions along paths that are not straight. We will often use the concept, or model, of an idealized p rticle which is considered to be a mathematical point with no spatial extent (no size). A point particle can undergo only translational motion. The particle model is useful in many real situations where we are interested only in translational motion and the object's size is not significant. For example, we might consider a billiard ball, or even a spacecraft traveling toward the Moon, as a particle for many purposes.

2 —1 R f r nc Fram s and Displac m nt Any measurement of position, distance, or speed must be made with respect to a reference fr me, or fr me of reference. For example, while you are on a train trav eling at 80 km/h, suppose a person walks past you toward the front of the train at a speed of, say, 5 km /h (Fig. 2 - 2). This 5 km/h is the person's speed with respect to the train as frame of reference. With respect to the ground, that person is moving at a speed of 80 km/h + 5 km/h = 85 km/h. It is always important to specify the frame of reference when stating a speed. In everyday life, we usually mean "with respect to the Earth" without even thinking about it, but the reference frame must be specified whenever there might be confusion.

% & «

% 0 At FIGURE 2 - 8 C r speedom eter showing m i/h in white, nd k m /h in or nge.

The notation limA^ 0 means the ratio A x/A t is to be evaluated in the limit of At approaching zero. But we do not simply set At = 0 in this definition, for then Ax would also be zero, and we would have an undefined number. Rather, we are considering the ratio A x/A t, as a whole. As we let At approach zero, Ax approaches zero as well. But the ratio A x/A t approaches some definite value, which is the instantaneous velocity at a given instant. In Eq. 2 - 3, the limit as At —> 0 is written in calculus notation as dx/dt and is called the derivative of x with respect to t: = lim ^ Af->0 At

FIGURE 2 - 9 Velocity o f c r s function o f time: ( ) t const nt velocity; (b) with v rying velocity. 60

&40 •3 20-> 0 0 ( )

H----- 1----- 1----- h 0.1

0.2 0.3 0.4 Time (h)

0.5

60Aver ge velocity

| 20J \ ^ n I---- 1----- u 0 .1 0.2 (b)

0.3 0.4 Time (h)

CHAPTER 2

0.5

(2-3)

dx dt

(2-4)

This equation is the definition of instantaneous velocity for one-dimensional motion. For instantaneous velocity we use the symbol v, whereas for average velocity we use v, with a bar above. In the rest of this book, when we use the term "velocity" it will refer to instantaneous velocity. When we want to speak of the average velocity, we will make this clear by including the word "average." Note that the instantaneous speed always equals the magnitude of the instan taneous velocity. Why? Because distance traveled and the magnitude of the displacement become the same when they become infinitesimally small. If an object moves at a uniform (that is, constant) velocity during a particular time interval, then its instantaneous velocity at any instant is the same as its average velocity (see Fig. 2 - 9a). But in many situations this is not the case. For example, a car may start from rest, speed up to 50 km/h, remain at that velocity for a time, then slow down to 20 km/h in a traffic jam, and finally stop at its destina tion after traveling a total of 15 km in 30 min. This trip is plotted on the graph of Fig. 2 - 9b. Also shown on the graph is the average velocity (dashed line), which is v = A x/A t = 15km/0.50h = 30 km/h.

Describing Motion: Kinematics in

ne Dimension

To better understand instantaneous velocity, let us consider a graph of the position of a particular particle versus time (x vs. t), as shown in Fig. 2 -10. (Note that this is different from showing the "path" of a particle on an x vs. y plot.) The particle is at position x 1 at a time tx, and at position x2 at time t2. Pi and P2 repre sent these two points on the graph. A straight line drawn from point Pi (x 1, to point P2(x2, t2) forms the hypotenuse of a right triangle whose sides are Ax and M. The ratio Ax/At is the slope of the straight line PiP2. But Ax/At is also the average velocity of the particle during the time interval At = t2 - tx. Therefore, we conclude that the average velocity of a particle during any time interval At = t2 — is equal to the slope of the straight line (or c ord) connecting the two points (xx, and (x2, t2) on an x vs. t graph. onsider now a time tx, intermediate between tx and t2, at which time the particle is at x{ (Fig. 2 -11). The slope of the straight line P ^ is less than the slope of PxP2in this case. Thus the average velocity during the time interval /• - tx is less than during the time interval t2 — tx. Now let us imagine that we take the point Pj in Fig. 2 -11 to be closer and closer to point Pj. That is, we let the interval tx — tx, which we now call At, to become smaller and smaller. The slope of the line connecting the two points becomes closer and closer to the slope of a line tangent to the curve at point . The average velocity (equal to the slope of the chord) thus approaches the slope of the tangent at point Px. The definition of the instantaneous velocity (Eq. 2 - 3) is the limiting value of the average velocity as At approaches zero. Thus the instantaneous velocity equals t e slope o f t e tangent to t e curve at that point (which we can simply call "the slope of the curve" at that point). Because the velocity at any instant equals the slope of the tangent to the x vs. t graph at that instant, we can obtain the velocity at any instant from such a graph. For example, in Fig. 2 - 12 (which shows the same curve as in Figs. 2 - 10 and 2 -11), as our object moves from x 1to x2, the slope continually increases, so the velocity is increasing. For times after t2, however, the slope begins to decrease and in fact reaches zero (so v = 0) where x has its maximum value, at point P3 in Fig. 2 -12. Beyond this point, the slope is negative, as for point P4. The velocity is therefore negative, which makes sense since x is now decreasing—the particle is moving toward decreasing values of x, to the left on a standard xy plot. If an object moves with constant velocity over a particular time interval, its instantaneous velocity is equal to its average velocity. The graph of x vs. t in this case will be a straight line whose slope equals the velocity. The curve of Fig. 2 -10 has no straight sections, so there are no time intervals when the velocity is constant. X

I I ------ 1-------------- 1---------------------- 1 tx t2

FIGURE 2 - 1 0 Gr ph of p rticle 's position x vs. time t. The slope of the str ight line Pi P2 represents the ver ge velocity of the p rticle during the time interv l At = t2 — t\ .

FIGURE 2 - 1 1 S me position vs. time curve s in Fig. 2 - 1 0 , but note th t the ver ge velocity over the time interv l t[ — tx (which is the slope of Pi Pi) is less th n the ver ge velocity over the time interv l t2 — 11. The slope o f the thin line t ngent to the curve t point Pj equ ls the inst nt neous velocity t time t\ . x

____ ii_____ ii ii 0]

l ~t 2

FIGURE 2 - 1 2 S me x vs. t curve s in Figs. 2 - 1 0 nd 2 - 1 1 , but here showing the slope t four different points: A t P3 , the slope is zero, so v = 0. A t P4 the slope is neg tive, so v < 0.

EXERCISE C W h t is your speed t the inst nt you turn round to m ove in the opposite direction? {a) D epends on how quickly you turn round; (b ) lw ys zero; (c) lw ys neg tive; (d) none o f the bove.

The derivatives of various functions are studied in calculus courses, and this book gives a summary in Appendix B. The derivatives of polynomial functions (which we use a lot) are:

where

4 - (C tn) = nCt" - 1 and ^ = 0, dt dt is any constant. SECTI N 2 - 3

Instantaneous Velocity

X (ml

4()

(a) T ngent t P2 whose slope is v2 = 21.0 m/s

EXAMPLE 2 - 3 Given x as a function of t . A jet engine moves along an experimental track (which we call the x axis) as shown in Fig. 2-13a. We will treat the engine as if it were a particle. Its position as a function of time is given by the equation x = A t2 + B, where A = 2.10 m/s2 and B = 2.80 m, and this equa tion is plotted in Fig. 2 -13b. (a) Determine the displacement of the engine during the time interval from to t2 = 5.00 s. (b) Determine the average velocity during this time interval, (c) Determine the magnitude of the instanta neous velocity at t = 5.00 s. APPR ACH We substitute values for tx and t2in the given equation for x to obtain

x 1 and x2. The average velocity can be found from Eq. 2 - 2. We take the deriva tive of the given x equation with respect to t to find the instantaneous velocity, using the formulas just given. S LUTI N (a) At tx = 3.00 s, the position (point Px in Fig. 2 - 13b) is X! = A t\ + B = (2.10m/s2)(3.00s)2 + 2.80 m = 21.7 m. At t2 = 5.00 s, the position (P2 in Fig. 2 -13b) is x2 = (2.10m/s2)(5.00s)2 + 2.80 m = 55.3 m. The displacement is thus FIGURE 2 - 1 3 Ex m ple 2 - 3 . ( ) Engine tr veling on str ight tr ck. (b) Gr ph of x vs. t: x = A t 2 + B.

x2 - x 1 = 55.3 m - 21.7 m = 33.6 m. (b) The magnitude of the average velocity can then be calculated as v =

33.6 m = 16.8 m/s. 2.00 s

tl-

This equals the slope of the straight line joining points and P2 shown in Fig. 2 -13b. (c) The instantaneous velocity at t = t2 = 5.00 s equals the slope of the tangent to the curve at point P2 shown in Fig. 2-13b. We could measure this slope off the graph to obtain v2. But we can calculate v more precisely for any time t, using the given formula x = A t2 + B, which is the engine's position x as a function of time t. We take the derivative of x with respect to time (see formulas at bottom of previous page): „ = ^

= l - [ A t2 + B)

2At.

We are given A = 2.10 m /s2, so for t = t2 = 5.00 s, v2 = 2A t = 2(2.10 m /s2)(5.00s) = 21.0 m/s.

2 —4 Acc l ration An object whose velocity is changing is said to be accelerating. For instance, a car whose velocity increases in magnitude from zero to 80 km/h is accelerating. Acceleration specifies how rapidly the velocity of an object is changing.

Aver ge Acceler tion Aver ge cceler tion is defined as the change in velocity divided by the time taken to make this change: change of velocity average acceleration = ---- ;------------- ----time elapsed In symbols, the average acceleration over a time interval A£ = t2 — tx during

CHAPTER 2

Describing Motion: Kinematics in

ne Dimension

which the velocity changes by Av = v2 — v1, is defined as v2 - V! Au a = ---------- = • t2 - tx At

^ (2 - 5)

Because velocity is a vector, acceleration is a vector too. But for one-dimensional motion, we need only use a plus or minus sign to indicate acceleration direction relative to a chosen coordinate axis. EXAMPLE 2 - 4 Average acceleration. A car accelerates along a straight road from rest to 90 km/h in 5.0 s, Fig. 2-14. What is the magnitude of its average acceleration? APPR ACH Average acceleration is the change in velocity divided by the elapsed

time, 5.0 s. The car starts from rest, so vx = 0. v2 = 90 km/h = 90 X 103m/3600 s = 25 m/s. S LUTI N From Eq. 2 - 5, the average acceleration is

The final velocity is

v2 ~ vx 25 m/s - Om/s ^ ^ m/s a = ---------- = --------—--------- = 5.0 -----t2 - tx 5.0 s s This is read as "five meters per second per second" and means that, on average, the velocity changed by 5.0 m /s during each second. That is, assuming the acceleration was constant, during the first second the car's velocity increased from zero to 5.0 m/s. During the next second its velocity increased by another 5.0 m/s, reaching a velocity of 10.0 m /s at t = 2.0 s, and so on. See Fig. 2 -14.

Acceleration \n - 5.0 m/s~l

v, = 0

;ii t * 1.0 s r = 5.0 m/s

iil I - 2.0s if = 10.0 m/s

at ( - t2 = 5,0 s v = t = 25 ni/s

FIGURE 2 - 1 4 Ex m ple 2 - 4 . The c r is shown t the st rt with vi = 0 t t\ = 0. The c r is shown three m ore times, t t = 1.0 s, t = 2.0 s, nd t the end of our time interv l, t2 = 5.0 s. We ssume the cceler tion is const nt nd equ ls 5.0 m /s2. The green rrows represent the velocity vectors; the length of e ch rrow represents the m gnitude of the velocity t th t m oment. The cceler tion vector is the or nge rrow. Dist nces re not to sc le.

We almost always write the units for acceleration as m /s2 (meters per second squared) instead of m/s/s. This is possible because: m/s _ m _ m s s •S s2 According to the calculation in Example 2 - 4, the velocity changed on average by 5.0 m/s during each second, for a total change of 25 m/s over the 5.0 s; the average acceleration was 5.0 m /s2. Note that acceleration tells us ow quickly t e velocity c anges, whereas velocity tells us ow quickly t e position c anges. SECTI N 2 - 4

Acceleration

C NCEPTUAL EXAMPLE 2 - 5 I Velocity and acceleration, (a) If the velocity of an object is zero, does it mean that the acceleration is zero? (b) If the acceleration is zero, does it mean that the velocity is zero? Think of some examples. RESP NSE A zero velocity does not necessarily mean that the acceleration is zero, nor does a zero acceleration mean that the velocity is zero, (a) For example, when you put your foot on the gas pedal of your car which is at rest, the velocity starts from zero but the acceleration is not zero since the velocity of the car changes. (How else could your car start forward if its velocity weren't changing— that is, accelerating?) (b) As you cruise along a straight highway at a constant velocity of 100 km/h, your acceleration is zero: a = 0, v # 0. EXERCISE D A powerful c r is dvertised to go from zero to 60 m i/h in 6.0 s. Wh t does this s y bout the c r: (a) it is f st (high speed); or (b ) it cceler tes well?

EXAMPLE 2 - 6 Car slowing down. An automobile is moving to the right along a straight highway, which we choose to be the positive x axis (Fig. 2 -15). Then the driver puts on the brakes. If the initial velocity (when the driver hits the brakes) is vx = 15.0 m/s, and it takes 5.0 s to slow down to v2 = 5.0 m/s, what was the car's average acceleration?

Acceleration t f, = 0

tt = -2,0 m/s2

■ S ffiS E at

= 5.0 s 5.0 m/s

APPR ACH We put the given initial and final velocities, and the elapsed time,

into Eq. 2 - 5 for a. S LUTI N In Eq. 2 - 5, we call the initial time tx = 0, and set t2 = 5.0 s. (Note that our choice of tx = 0 doesn't affect the calculation of a because only At = t2 — ti appears in Eq. 2 - 5.) Then

FIGURE 2 - 1 5 Ex m ple 2 - 6 , showing the position o f the c r t times t\ nd t2, s w ell s the c r 's velocity represented by the green rrows. The cceler tion vector (or nge) points to the left s the c r slows down while moving to the right.

5.0 m/s - 15.0 m/s = - 2.0 m/s' 5.0 s The negative sign appears because the final velocity is less than the initial velocity. In this case the direction of the acceleration is to the left (in the negative x direction)—even though the velocity is always pointing to the right. We say that the acceleration is 2.0 m /s2to the left, and it is shown in Fig. 2 -15 as an orange arrow.

Deceler tion

AUT I ON

Deceleration m eans t e m agnitude o f t e velocity is decreasing; a is not necessarily negative

When an object is slowing down, we can say it is deceler ting. But be careful: deceler ation does not mean that the acceleration is necessarily negative. The velocity of an object moving to the right along the positive x axis is positive; if the object is slowing down (as in Fig. 2-15), the acceleration is negative. But the same car moving to the left (decreasing x), and slowing down, has positive acceleration that points to the right, as shown in Fig. 2-16. We have a deceleration whenever the magnitude of the velocity is decreasing, and then the velocity and acceleration point in opposite directions.

FIGURE 2 - 1 6 The c r of Ex m ple 2 - 6 , now m oving to the left nd deceler ting. The cceler tion is v2 -

a = ------------

( - 5 .0 m / s ) -

( - 1 5 .0 m / s )

5.0 s —5.0 m /s + 15.0 m /s

5X)s

a =

+ 2.0 m /s.

EXERCISE E A c r m oves long the x xis. Wh t is the sign o f the c r 's cceler tion if it is m oving in the positive x direction with ( ) incre sing speed or (b) decre sing speed? Wh t is the sign of the cceler tion if the c r m oves in the neg tive direction with (c) incre sing speed or (d ) decre sing speed?

CHAPTER 2

Describing Motion: Kinematics in

ne Dimension

Inst nt neous Acceler tion The inst nt neous cceler tion, a, is defined as the limiting value o f t e average acceleration as we let At approac zero: Av dv = lim —— = — ■ (2- 6) ► o At dt This limit, dv/dt, is the derivative of v with respect to t. We will use the term "acceleration" to refer to the instantaneous value. If we want to discuss the average acceleration, we will always include the word "average." If we draw a graph of the velocity, v, vs. time, t, as shown in Fig. 2 -17, then the average acceleration over a time interval At = t2 - tx is represented by the slope of the straight line connecting the two points P1 and P2 as shown. [ ompare this to the position vs. time graph of Fig. 2 - 10 for which the slope of the straight line represents the average velocity.] The instantaneous acceleration at any time, say t\ , is the slope of the tangent to the v vs. t curve at that time, which is also shown in Fig. 2 -17. Let us use this fact for the situation graphed in Fig. 2 -17; as we go from time ti to time t2the velocity continually increases, but the acceleration (the rate at which the velocity changes) is decreasing since the slope of the curve is decreasing. Acceleration given x(t). A particle is moving in a straight line so that its position is given by the relation x = (2.10 m/s2)?2 + (2.80 m), as in Example 2 -3. alculate (a) its average acceleration during the time interval from ti = 3.00 s to t2 = 5.00 s, and (b) its instantaneous acceleration as a function of time. APPR ACH To determine acceleration, we first must find the velocity at tx and t2

by differentiating x: v = dx/dt. Then we use Eq. 2 - 5 to find the average acceleration, and Eq. 2 - 6 to find the instantaneous acceleration. S LUTI N (a) The velocity at any time t is 2.80 m ] = (4.20 m/s2)t,

Slope is ver ge cceler tion during At = t2 - ?i Slope is inst nt neous cceler tion

FIGURE 2 - 1 7 A gr ph of velocity v vs. time t. The ver ge cceler tion over time interv l A t = t2 - ti is the slope of the str ight line Pi P2 : a = A v / At. The inst nt neous cceler tion t time t\ is the slope of the v vs. t curve t th t inst nt. FIGURE 2 - 1 8 Ex m ple 2 - 7 . Gr phs of ( ) x vs. t, (b) v vs. t, nd (c) a vs. t for the m otion x = A t 2 + B. N ote th t v incre ses line rly with t nd th t the cceler tion a is const nt. A lso, v is the slope of the x vs. t curve, where s a is the slope o f the v vs. t curve.

as we saw in Example 2-3c. Therefore, at tx = 3.00 s, v1 = (4.20m/s2)(3.00s) = 12.6 m/s and at t2 = 5.00 s, v2 = 21.0 m/s. Therefore, _ Av _ 21.0 m/s - 12.6 m/s = 4.20 m/s2. At 5.00 s - 3.00 s (b) With v = (4.20 m /s2)?, the instantaneous acceleration at any time is U = ~^t =

= 4.20 m /s2.

The acceleration in this case is constant; it does not depend on time. Figure 2 -18 shows graphs of (a) x vs. t (the same as Fig. 2 - 13b), (b) v vs. t, which is linearly increasing as calculated above, and (c) a vs. t, which is a horizontal straight line because a = constant. Like velocity, acceleration is a rate. The velocity of an object is the rate at which its displacement changes with time; its acceleration, on the other hand, is the rate at which its velocity changes with time. In a sense, acceleration is a "rate of a rate." This can be expressed in equation form as follows: since a = dv/dt and v = dx/dt, then dv _ d f d x \ _ d2x dt d t\d t) dt2 Here d2x /d t2 is the second derivative of x with respect to time: we first take the derivative of x with respect to time (dx/dt), and then we again take the derivative with respect to time, (d/dt) (dx/dt), to get the acceleration. EXERCISE F The position of

■=-+ —1— 1— 1— 1— b - t ( s) 1 2 3 4 5 6 ( )

r '

i1 11 11 11 11 11 l 2 3 4 5 6

(b)

a = 4.20 m/s2

p rticle is given by the follow ing equ tion:

x = (2.00 m /s3)?3 + (2.50 m /s )t. Wh t is the cceler tion of the p rticle (c) 24.0 m /s2; (d) 2.00 m /s 2.

t = 2.00 s?

(a) 13.0 m /s2; (b ) 22.5 m /s2;

— 1— 1— 1— 1— 1— b - t { s) 1 2 3 4 5 6 (C)

SECTI N 2 - 4

Acceleration

v (km/li J

C NCEPTUAL EXAMPLE 2^8~1 Analyzing with graphs. Figure 2 - 19 shows the velocity as a function of time for two cars accelerating from 0 to 100 km/h in a time of 10.0 s. ompare (a) the average acceleration; (b) instantaneous acceleration; and (c) total distance traveled for the two cars. RESP NSE («) Average acceleration is A v/A t. Both cars have the same Av

( - M s)

(100 km/h) and the same At (10.0 s), so the average acceleration is the same for both cars. (b) Instantaneous acceleration is the slope of the tangent to the v vs. t curve. For about the first 4 s, the top curve is steeper than the bottom curve, so car A has a greater acceleration during this interval. The bottom curve is steeper during the last 6 s, so car B has the larger acceleration for this period, (c) Except at t = 0 and t = 10.0 s, car A is always going faster than car B. Since it is going faster, it will go farther in the same time.

2 - 5 Motion at Constant Acc l ration We now examine the situation when the magnitude of the acceleration is constant and the motion is in a straight line. In this case, the instantaneous and average accelerations are equal. We use the definitions of average velocity and acceleration to derive a set of valuable equations that relate x, v, a, and t when a is constant, allowing us to determine any one of these variables if we know the others. To simplify our notation, let us take the initial time in any discussion to be zero, and we call it t0: tx = t0 = 0. (This is effectively starting a stopwatch at t0.) We can then let t2 = t be the elapsed time. The initial position and the initial velocity (i^) of an object will now be represented by x0 and v0, since they represent x and v at t = 0. At time t the position and velocity will be called x and v (rather than x2 and v2). The average velocity during the time interval t — t0will be (Eq. 2 - 2) _ _ Ax _ x - x0 _ x - x0 V At t - t0 t since we chose t0 = 0. The acceleration, assumed constant in time, is (Eq. 2 - 5) v - v0

AUT I ON

A verage velocity, but only if a = constant

A common problem is to determine the velocity of an object after any elapsed time t, when we are given the object's constant acceleration. We can solve such problems by solving for v in the last equation to obtain: v = v0 + at. [constant acceleration] (2 -7) If an object starts from rest (v0 = 0) and accelerates at 4.0 m /s2, after an elapsed time t = 6.0 s its velocity will be v = at = (4.0 m/s2)(6.0 s) = 24 m/s. Next, let us see how to calculate the position x of an object after a time t when it undergoes constant acceleration. The definition of average velocity (Eq. 2 - 2) is v = (x — x0)/t, which we can rewrite as x = x0 + vt. (2 - 8) Because the velocity increases at a uniform rate, the average velocity, v, will be midway between the initial and final velocities: _ Vq + v v = — ----[constant acceleration] (2 -9) ( areful: Equation 2 - 9 is not necessarily valid if the acceleration is not constant.) We combine the last two Equations with Eq. 2 - 7 and find x = x0 + vt x0 . .

^0 + ^0 + at or v01 + \a t2.

CHAPTER 2

[constant acceleration] (2-10)

Equations 2 -7, 2 - 9, and 2 -10 are three of the four most useful equations for

motion at constant acceleration. We now derive the fourth equation, which is useful in situations where the time t is not known. We substitute Eq. 2 - 9 into Eq. 2 -8: v + v0 X = x0 + v t = x0 + Next we solve Eq. 2 - 7 for t , obtaining V - Vo and substituting this into the previous equation we have V2 - Vq v + vo) f v ~ vo x0 2 j\ a 2a We solve this for v2 and obtain [constant acceleration] (2-11) v2 = vl + 2a(x — x0), which is the useful equation we sought. We now have four equations relating position, velocity, acceleration, and time, when the acceleration a is constant. We collect these kinematic equations here in one place for future reference (the tan background screen emphasizes their usefulness):

v = v0 + at

[a = constant] (2 - 12 )

x = xQ+ v0t + \a t2

[a = constant] (2 -12b)

v2 = Vq + 2a(x - x0) v Vo

[a = constant] (2 - 12c)

Kinem atic equations fo r constant acceleration (w e 'll use t em a lot)

[a = constant] (2 -12d)

These useful equations are not valid unless a is a constant. In many cases we can set x0 = 0, and this simplifies the above equations a bit. Note that x represents posi tion, not distance, that x — x0is the displacement, and that t is the elapsed time. Runway design. You are designing an airport for small planes. One kind of airplane that might use this airfield must reach a speed before takeoff of at least 27.8 m/s (100 km/h), and can accelerate at 2.00 m /s2. (a) If the runway is 150 m long, can this airplane reach the required speed for takeoff? (b) If not, what minimum length must the runway have? APPR ACH The plane's acceleration is constant, so we can use the kinematic equations for constant acceleration. In (a), we want to find v, and we are given: Known *o = 0

(^a j

y s i c s

a p p l i e d

A irp o rt design

W nted V

x = 150 m a = 2.00 m /s2

S LUTI N (a) Of the above four equations, Eq. 2 - 12c will give us v when we know vQ, a, x, and x0: v2 = vl + 2a(x — x0) = 0 + 2(2.00 m /s2)(150m) = 600m2/s2

\PR

BLEM

LVING

E quations 2 - 1 2 are valid only w en t e acceleration is constant, w ic w e assume in t is Exam ple

v = \ / 600 m2/s 2 = 24.5 m/s. This runway length is not sufficient. (b) Now we want to find the minimum length of runway, x — x0, given v = 27.8 m/s and a = 2.00 m /s2. So we again use Eq. 2-12c, but rewritten as (27.8 m /s)2 - 0 = 193 m. (x - *o) = 2a 2(2.00 m /s2) A 200-m runway is more appropriate for this plane. N TE We did this Example as if the plane were a particle, so we round off our answer to 200 m. EXERCISE G A c r st rts from rest nd cceler tes t const nt 10 m /s2 during \ m ile (402 m) r ce. H ow f st is the c r going t the finish line? ( ) 8090 m /s; (b) 9 0m /s; (c) 81 m /s; (d) 809 m /s.

SECTI N 2 - 5

2 - 6 Solving Probl ms Before doing more worked-out Examples, let us look at how to approach problem solving. First, it is important to note that physics is not a collection of equations to be memorized. Simply searching for an equation that might work can lead you to a wrong result and will surely not help you understand physics. A better approach is to use the following (rough) procedure, which we put in a special "Problem Solving Strategy." (Other such Problem Solving Strategies, as an aid, will be found throughout the book.)

S.O

B i

c, O L V /

1. Read and rere d the whole problem carefully before trying to solve it. 2. Decide what object (or objects) you are going to study, and for what time interv l. You can often choose the initial time to be t = 0. 3. Dr w a di gr m or picture of the situation, with coordinate axes wherever applicable. [You can place the origin of coordinates and the axes wherever you like to make your calculations easier.] 4. Write down what quantities are "known" or "given," and then what you want to know. onsider quanti ties both at the beginning and at the end of the chosen time interval. 5. Think about which principles of physics apply in this problem. Use common sense and your own experi ences. Then plan an approach. 6. onsider which equ tions (and/or definitions) relate the quantities involved. Before using them, be sure their r nge of v lidity includes your problem (for example, Eqs. 2 - 12 are valid only when the accelera tion is constant). If you find an applicable equation that involves only known quantities and one desired unknown, solve the equation algebraically for the

unknown. Sometimes several sequential calculations, or a combination of equations, may be needed. It is often preferable to solve algebraically for the desired unknown before putting in numerical values. 7. arry out the c lcul tion if it is a numerical problem. Keep one or two extra digits during the calculations, but round off the final answer(s) to the correct number of significant figures (Section 1 -3). 8. Think carefully about the result you obtain: Is it re son ble? Does it make sense according to your own intuition and experience? A good check is to do a rough estim te using only powers of ten, as discussed in Section 1 -6. Often it is preferable to do a rough estimate at the start of a numerical problem because it can help you focus your attention on finding a path toward a solution. 9. A very important aspect of doing problems is keeping track of units. An equals sign implies the units on each side must be the same, just as the numbers must. If the units do not balance, a mistake has no doubt been made. This can serve as a check on your solution (but it only tells you if you're wrong, not if you're right). Always use a consistent set of units.

EXAMPLE 2 - 1 0 Acceleration of a car. How long does it take a car to cross a 30.0-m-wide intersection after the light turns green, if the car accelerates from rest at a constant 2.00 m/s2? APPR ACH We follow the Problem Solving Strategy above, step by step. S LUTI N FIGURE 2 - 20

Ex m ple 2 - 1 0 .

a - 2,00 m/s2

V =0

CHAPTER 2

a = 2.00 m/a2

30.0 m

1. Rere d the problem. Be sure you understand what it asks for (here, a time interval). 2. The object under study is the car. We choose the time interv l: t = 0, the initial time, is the moment the car starts to accelerate from rest (v0 = 0); the time t is the instant the car has traveled the full 30.0-m width of the intersection. 3. Dr w a di gr m: the situation is shown in Fig. 2 -20, where the car is shown moving along the positive x axis. We choose x0 = 0 at the front bumper of the car before it starts to move.

Describing Motion: Kinematics in

ne Dimension

Known

W nted

*o = 0 x = 30.0 m a = 2.00 m /s 2

o II 5*

4. The "knowns" and the "wanted" are shown in the Table in the margin, and we choose x0 = 0. Note that "starting from rest" means v = 0 at t = 0; that is, v0 = 0. 5. The physics: the motion takes place at constant acceleration, so we can use the kinematic equations, Eqs. 2 -12. 6. Equ tions: we want to find the time, given the distance and acceleration; Eq. 2 - 12b is perfect since the only unknown quantity is t. Setting v0 = 0 and jc0 = 0 in Eq. 2 -12b (x = xQ+ v0t + \a t2), we can solve for t: 2'

2x tl = — a so t = A— 7. The c lcul tion: 2(30.0 m)

= 5.48 s. 2.00 m /s2 This is our answer. Note that the units come out correctly. 8. We can check the re son bleness of the answer by calculating the final velocity v = at = (2.00m/s2)(5.48 s) = 10.96 m/s, and then finding x = x0 + vt = 0 + \ (10.96 m /s + 0)(5.48s) = 30.0 m, which is our given distance. 9. We checked the units, and they came out perfectly (seconds). N TE In steps 6 and 7, when we took the square root, we should have written t = ± \^ 2 x /a = ± 5.48 s. Mathematically there are two solutions. But the second solution, t = - 5.48 s, is a time before our chosen time interval and makes no sense physically. We say it is "unphysical" and ignore it. t = A— =

We explicitly followed the steps of the Problem Solving Strategy for Example 2-10. In upcoming Examples, we will use our usual "Approach" and "Solution" to avoid being wordy. EXAMPLE 2 - 11 ESTIMATE"! Air bags. Suppose you want to design an airbag system that can protect the driver at a speed of 100 km /h (60 mph) if the car hits a brick wall. Estimate how fast the air bag must inflate (Fig. 2 - 21) to effec tively protect the driver. How does the use of a seat belt help the driver? APPR ACH We assume the acceleration is roughly constant, so we can use

PHYSICS

APPLIED

Car safety — air bags

FIGURE 2-21 Ex m ple 2 - 1 1 . A n ir b g deploying on imp ct.

Eqs. 2 -12. Both Eqs. 2 - 12a and 2 -12b contain t, our desired unknown. They both contain a, so we must first find a, which we can do using Eq. 2 -12c if we know the distance x over which the car crumples. A rough estimate might be about 1 meter. We choose the time interval to start at the instant of impact with the car moving at v0 = 100 km/h, and to end when the car comes to rest (v = 0) after traveling 1 m. S LUTI N We convert the given initial speed to SI units: 100 km /h = 100 X 103m/3600 s = 28 m/s. We then find the acceleration from Eq. 2-12c: (28 m /s)2 a = - — = ----- — ------ = - 390 m /s. 2x 2.0 m This enormous acceleration takes place in a time given by (Eq. 2-12a): v 3 0 - 28 m /s 0.07 s. t = - 390 m /s2 a To be effective, the air bag would need to inflate faster than this. What does the air bag do? It spreads the force over a large area of the chest (to avoid puncture of the chest by the steering wheel). The seat belt keeps the person in a stable position against the expanding air bag. SECTI N 2 - 6

Solving Problems

FIGURE 2 -22 E x m p le 2 - 1 2 : stopping dist nce for br king c r.

T m vet during —

re ction litne v = const nt = 14 m/s t = 0.50 s a = 0

PHYSICS

APPLIED B ra k in g distances

P art 1: R eaction tim e

Known t Vq v a *o

= = = = =

0.50 s 14 m /s 14 m /s 0 0

W nted X

P art 2: B ra k in g

Known = Vq = v = a =

7.0 m 14 m /s 0 - 6 . 0 m /s 2

FIGURE 2 -23

W nted X

E x m p le 2 - 1 2 . G r phs o f (a) v vs. t nd (b ) x vs. t.

T nivel during

FOTmJ

v decre ses from 14 m/s to zero a = - 6.0 m /s2

EXAMPLE 2 - 1 2 ESTIMATE"! Braking distances. Estimate the minimum stopping distance for a car, which is important for traffic safety and traffic design. The problem is best dealt with in two parts, two separate time intervals. (1) The first time interval begins when the driver decides to hit the brakes, and ends when the foot touches the brake pedal. This is the "reaction time" during which the speed is constant, so a = 0. (2) The second time interval is the actual braking period when the vehicle slows down (a ^ 0) and comes to a stop. The stopping distance depends on the reaction time of the driver, the initial speed of the car (the final speed is zero), and the acceleration of the car. For a dry road and good tires, good brakes can decelerate a car at a rate of about 5 m /s2 to 8 m /s2. alculate the total stopping distance for an initial velocity of 50 km /h ( = 14 m /s « 31m i/h) and assume the acceleration of the car is - 6 .0 m /s2 (the minus sign appears because the velocity is taken to be in the positive x direction and its magnitude is decreasing). Reaction time for normal drivers varies from perhaps 0.3 s to about 1.0 s; take it to be 0.50 s. APPR ACH During the "reaction time," part (1), the car moves at constant speed of 14 m/s, so a = 0. Once the brakes are applied, part (2), the acceleration is a = - 6 .0 m /s2 and is constant over this time interval. For both parts a is constant, so we can use Eqs. 2 - 12. S LUTI N Part (1). We take x0 = 0 for the first time interval, when the driver is reacting (0.50 s): the car travels at a constant speed of 14 m /s so a = 0. See Fig. 2 - 22 and the Table in the margin. To find x, the position of the car at t = 0.50 s (when the brakes are applied), we cannot use Eq. 2 - 12c because x is multiplied by a, which is zero. But Eq. 2 - 12b works:

x = v0t + 0 = (14m /s)(0.50s) = 7.0 m. Thus the car travels 7.0 m during the driver's reaction time, until the instant the brakes are applied. We will use this result as input to part (2). Part (2). During the second time interval, the brakes are applied and the car is brought to rest. The initial position is x 0 = 7.0 m (result of part (1)), and other variables are shown in the second Table in the margin. Equation 2 - 12a doesn't contain x; Eq. 2 - 12b contains x but also the unknown t. Equation 2 -12c, v2 — vl = 2a(x — jc0), is what we want; after setting x0 = 7.0 m, we solve for x, the final position of the car (when it stops): x0

2a - 196 m /s 0 - (14 m /s)2 = 7.0 m H----- ----------- ——- = 7.0 m H------ —— —r~ - 1 2 m /s2 2( - 6.0 m /s2) = 7.0 m + 16 m = 23 m. The car traveled 7.0 m while the driver was reacting and another 16 m during the braking period before coming to a stop, for a total distance traveled of 23 m. Figure 2 - 23 shows graphs of (a) v vs. t and (b) x vs. t.

CHAPTER 2

N TE From the equation above for x, we see that the stopping distance after the driver hit the brakes (= x - x 0) increases with the square of the initial speed, not just linearly with speed. If you are traveling twice as fast, it takes four times the distance to stop.

EXAMPLE 2 - 13

ESTIMATE"! Two Moving

bjects: Police and Speeder.

A car speeding at 150 km/h passes a still police car which immediately takes off in hot pursuit. Using simple assumptions, such as that the speeder continues at constant speed, estimate how long it takes the police car to overtake the speeder. Then estimate the police car's speed at that moment and decide if the assump tions were reasonable. APPR ACH When the police car takes off, it accelerates, and the simplest

assumption is that its acceleration is constant. This may not be reasonable, but let's see what happens. We can estimate the acceleration if we have noticed automobile ads, which claim cars can accelerate from rest to 100 km/h in 5.0 s. So the average acceleration of the police car could be approximately 100 km/h km/h 11000 m lh cip — = 20 = 5.6 m /s2. 5.0 s 1 km j \ 3600 s S LUTI N We need to set up the kinematic equations to determine the unknown

quantities, and since there are two moving objects, we need two separate sets of equations. We denote the speeding car's position by xs and the police car's position by xP. Because we are interested in solving for the time when the two vehicles arrive at the same position on the road, we use Eq. 2 -12b for each car: xs = vost + ^ast2 = (150 km/h)? = (42 m/s )? xP = v0Pt + \a Yt2 = ^(5.6 m /s2)?2, where we have set t>0P = 0 and as = 0 (speeder assumed to move at constant speed). We want the time when the cars meet, so we set xs = xF and solve for ?: (42 m/s)? = (2.8 m /s2)?2. The solutions are 0 and

? =

42 m/s = 15 s. 2.8 m /s2

The first solution corresponds to the instant the speeder passed the police car. The second solution tells us when the police car catches up to the speeder, 15 s later. This is our answer, but is it reasonable? The police car's speed at ? = 15 s is Vp = -%> + aFt = 0 + (5.6m/s2)(15 s) = 84 m/s or 300 km/h (« 190 mi/h). Not reasonable, and highly dangerous. N TE More reasonable is to give up the assumption of constant acceleration. The police car surely cannot maintain constant acceleration at those speeds. Also, the speeder, if a reasonable person, would slow down upon hearing the police siren. Figure 2 -24 shows (a) x vs. ? and (b) v vs. ? graphs, based on the original assumption = constant, whereas (c) shows v vs. ? for more reasonable assumptions. of

AUT I ON

Initial assum ptions need to be c ecked ou t fo r reasonableness

FIGURE 2 - 24

(a)

(b)

Ex m ple 2 - 1 3 .

Solving Problems

2 - 7 Fr ly Falling Obj cts

FIGURE 2 - 25

G lileo G lilei

(1564 - 1642).

/j\ CAUTION_______ A freely falling object increases in speed, but not in proportion to its m ass or weig t

FIGURE 2 - 26 Multifl sh photogr ph o f f lling pple, t equ l time interv ls. The pple f lls f rther during e ch successive interv l, which m e ns it is cceler ting.

One of the most common examples of uniformly accelerated motion is that of an object allowed to fall freely near the Earth's surface. That a falling object is accel erating may not be obvious at first. And beware of thinking, as was widely believed before the time of Galileo (Fig. 2 -25), that heavier objects fall faster than lighter objects and that the speed of fall is proportional to how heavy the object is. Galileo made use of his new technique of imagining what would happen in idealized (simplified) cases. For free fall, he postulated that all objects would fall with the same constant acceleration in the absence of air or other resistance. He showed that this postulate predicts that for an object falling from rest, the distance traveled will be proportional to the square of the time (Fig. 2-26); that is, d oc t2. We can see this from Eq. 2-12b; but Galileo was the first to derive this mathematical relation. To support his claim that falling objects increase in speed as they fall, Galileo made use of a clever argument: a heavy stone dropped from a height of 2 m will drive a stake into the ground much further than will the same stone dropped from a height of only 0.2 m. learly, the stone must be moving faster in the former case. Galileo claimed that all objects, light or heavy, fall with the same acceleration, at least in the absence of air. If you hold a piece of paper horizontally in one hand and a heavier object—say, a baseball—in the other, and release them at the same time as in Fig. 2-27a, the heavier object will reach the ground first. But if you repeat the experiment, this time crumpling the paper into a small wad (see Fig. 2-27b), you will find that the two objects reach the floor at nearly the same time. Galileo was sure that air acts as a resistance to very light objects that have a large surface area. But in many ordinary circumstances this air resistance is negli gible. In a chamber from which the air has been removed, even light objects like a feather or a horizontally held piece of paper will fall with the same acceleration as any other object (see Fig. 2 -28). Such a demonstration in vacuum was not possible in Galileo's time, which makes Galileo's achievement all the greater. Galileo is often called the "father of modern science," not only for the content of his science (astronomical discoveries, inertia, free fall) but also for his approach to science (idealization and simplification, mathematization of theory, theories that have testable consequences, experiments to test theoretical predictions). Galileo's specific contribution to our understanding of the motion of falling objects can be summarized as follows: t given loc tion on the E rth nd in the bsence of ir resist nce, ll objects f ll with the s me const nt cceler tion. We call this acceleration the cceler tion due to gr vity on the surface of the Earth, and we give it the symbol g. Its magnitude is approximately

Acceleration due to gravity

g = 9.80 m /s2.

[at surface of Earth]

In British units g is about 32 ft/s2. Actually, g varies slightly according to latitude and elevation, but these variations are so small that we will ignore them for most FIGURE 2 - 27 ( ) A b ll nd light piece of p per re dropped t the s me time, (b) R epe ted, with the p per w dded up.

<
mg.

= 2.0 N upward. See Fig. 4-16. We apply Newton's second law and see that the box moves upward with an acceleration 'Zfy _ 2.0 N m 10.0 kg = 0.20 m /s2.

EXAMPLE 4 - 8 Apparent weight loss. A 65-kg woman descends in an elevator that briefly accelerates at 0.20g downward. She stands on a scale that reads in kg. (a) During this acceleration, what is her weight and what does the scale read? (b) What does the scale read when the elevator descends at a constant speed of 2.0 m/s? FIGURE 4 - 1 7 Ex m ple 4 - 8 . The cceler tion vector is shown in gold to distinguish it from the red force vectors.

APPR ACH Figure 4 - 17 shows all the forces that act on the woman (and only

those that act on her). The direction of the acceleration is downward, so we choose the positive direction as down (this is the opposite choice from Examples 4 - 6 and 4 - 7). S LUTI N {a) From Newton's second law, 2 F = ma

mg — FN = m(0.20g). We solve for FN: Fn = mg - 0.20mg = 0.80mg, and it acts upward. The normal force FN is the force the scale exerts on the person, and is equal and opposite to the force she exerts on the scale: Fn = 0.80mg downward. Her weight (force of gravity on her) is still mg = (65 kg)(9.8m/s2) = 640 N. But the scale, needing to exert a force of only 0.80mg, will give a reading of 0.80m = 52 kg. (b) Now there is no acceleration, a = 0, so by Newton's second law, mg - FN = 0 and FN = mg. The scale reads her true mass of 65 kg. N TE The scale in (a) may give a reading of 52 kg (as an "apparent mass"), but her mass doesn't change as a result of the acceleration: it stays at 65 kg.

CHAPTER 4

Dynamics: Newton's Laws of Motion

4 —7 Solving Problems with Newton's L ws: Free-Body Di gr ms Newton's second law tells us that the acceleration of an object is proportional to the net force acting on the object. The net force, as mentioned earlier, is the vector sum of all forces acting on the object. Indeed, extensive experiments have shown that forces do add together as vectors precisely according to the rules we developed in hapter 3. For example, in Fig. 4 - 18, two forces of equal magni tude (100 N each) are shown acting on an object at right angles to each other. Intuitively, we can see that the object will start moving at a 45° angle and thus the net force acts at a 45° angle. This is just what the rules of vector addition give. From the theorem of Pythagoras, the magnitude of the resultant force is Fr = V ( 100N)2 + (100 N)2 = 141N. EXAMPLE 4 - 9 Adding force vectors. alculate the sum of the two forces exerted on the boat by workers A and B in Fig. 4-19a.

(a)

(b)

FIGURE 4 - 18

( ) Two forces, FA nd Fb , exerted by workers A nd B, ct on cr te, (b) The sum, or result nt, of FA nd FB is FR.

FIGURE 4 - 19 Ex m ple 4 - 9 : Two force vectors ct on bo t.

APPR ACH We add force vectors like any other vectors as described in

hapter 3. The first step is to choose an xy coordinate system (see Fig. 4-19a), and then resolve vectors into their components. S LUTI N The two force vectors are shown resolved into components in Fig. 4-19b. We add the forces using the method of components. The components of FAare Fax =

cos

Fa = 40.0

45.0° = (40.0 N) (0.707) = 28.3 N,

FAy = Fa sin 45.0° = (40.0 N)(0.707) = 28.3 N. The components of FBare FBx = +Fb cos 37.0° = +(30.0 N)(0.799) = +24.0 N, FBy = ~ F b sin 37.0° = -(30.0N)(0.602) = - 18.1 N. FBy is negative because it points along the negative y axis. The components of the resultant force are (see Fig. 4 - 19c) Frx = FAx + FBx = 28.3 N + 24.0 N = 52.3 N, FBy = 28.3 N - 18.1 N = 10.2 N. FAy

FRy

To find the magnitude of the resultant force, we use the Pythagorean theorem ?2 + F ly V(52.3)2 + (10.2)2N = 53.3 N. \ F* = V f Rx The only remaining question is the angle 0 that the net force FRmakes with the x axis We use: tan 0

r Ry

F rx

10.2 N = 0.195, 52.3 N

and tan 1(0.195) = 11.0°. The net force on the boat has magnitude 53.3 N and acts at an 11.0° angle to the x axis. When solving problems involving Newton's laws and force, it is very important to draw a diagram showing all the forces acting on each object involved. Such a diagram is called a free-body di gr m, or force di gr m: choose one object, and draw an arrow to represent each force acting on it. Include every force acting on that object. Do not show forces that the chosen object exerts on ot er objects. To help you identify each and every force that is exerted on your chosen object, ask yourself what other objects could exert a force on it. If your problem involves more than one object, a separate free-body diagram is needed for each object. For now, the likely forces that could be acting are gravity and contact forces (one object pushing or pulling another, normal force, friction). Later we will consider air resistance, drag, buoyancy, pressure, as well as electric and magnetic forces. SECTI N 4 - 7

PRQBLEM

LVING

Free-body diagram

Solving Problems with Newton's Laws: Free-Body Diagrams

Motion

FIGURE 4 - 2 0 Ex m ple 4 - 1 0 . W hich is the correct free - body di gr m for hockey puck sliding cross frictionless ice?

Molicm

Molion

J (h) *

C NCEPTUAL EXAMPLE 4 - 1 0 I The hockey puck. A hockey puck is sliding at constant velocity across a flat horizontal ice surface that is assumed to be frictionless. Which of the sketches in Fig. 4 - 20 is the correct free-body diagram for this puck? What would your answer be if the puck slowed down? RESP NSE Did you choose ( o At dt where Av is the change in velocity during the short time interval At. We will eventually consider the situation in which At approaches zero and thus obtain the instantaneous acceleration. But for purposes of making a clear drawing (Fig. 5-11), we consider a nonzero time interval. During the time interval At, the particle in Fig. 5-1 la moves from point A to point B, covering a distance A£ along t e arc which subtends an angle A0.The change in the velocity vector is v2 - vx = Av, and is shown in Fig. 5 - llb . Now we let At be very small, approaching zero. Then At and A0 are also very small, and v2 will be almost parallel to vx (Fig. 5 - 1 lc); Av will be essentially perpendicular to them. Thus Av points toward the center of the circle. Since , by definition, is in the same direction as Av, it too must point toward the center of the circle. Therefore, this acceleration is called centripet l cceler tion ("centerpointing" acceleration) or r di l cceler tion (since it is directed along the radius, toward the center of the circle), and we denote it by aR. We next determine the magnitude of the radial (centripetal) acceleration, aR. Because A in Fig. 5 - l l a is perpendicular to v1? and B is perpendicular to v2, it follows that the angle A0, defined as the angle between A and B, is also the angle between and v2. Hence the vectors vl 5v2, and Av in Fig. 5 - llb form a triangle that is geometrically similar* to triangle AB in Fig. 5 - lla . If we take A0 to be very small (letting At be very small) and setting v = v1 = v2 because the magnitude of the velocity is assumed not to change, we can write Av Ai or Av

-A t. r This is an exact equality when At approaches zero, for then the arc length A£ equals the chord length AB. We want to find the instantaneous acceleration, aR, so we use the expression above to write Av v At aR = lim = lim — ■ a?— »o r At Ar->o At Then, because FIGURE 5 - 1 1 Determining the ch nge in velocity, Av, for p rticle moving in circle. The length A t is the dist nce long the rc, from A to B.

/j\ CAUTION____________ In uniform circularmotion, t espeed is constant, but t e acceleration isnot zero FIGURE 5 - 1 2 For uniform circul r motion, a is lw ys perpendicul r to v.

A t —>0 At is just the linear speed, v, of the object, we have for the centripetal (radial) acceleration

[centripetal (radial) acceleration] (5 -1)

flu —

Equation 5 - 1 is valid even when v is not constant. To summarize, an object moving in a circle o f radius r at constant speed v as an acceleration w ose direction is toward t e center o f t e circle and w ose magni tude is aR = v2/r. It is not surprising that this acceleration depends on v and r. The greater the speed v, the faster the velocity changes direction; and the larger the radius, the less rapidly the velocity changes direction. The acceleration vector points toward the center of the circle. But the velocity vector always points in the direction of motion, which is tangential to the circle. Thus the velocity and acceleration vectors are perpendicular to each other at every point in the path for uniform circular motion (Fig. 5 -12). This is another example that illustrates the error in thinking that acceleration and velocity are always in the same direction. For an object falling vertically, and v are indeed parallel. But in circular motion, and v are perpendicular, not parallel (nor were they parallel in projectile motion, Section 3 -7). EXERCISE C C n Equ tions 2 - 1 2 , the kinem tic equ tions for const nt cceler tion, be used for uniform circul r m otion? For ex mple, could Eq. 2 - 1 2 b be used to c lcul te the time for the revolving b ll in Fig. 5 - 1 2 to m ke one revolution?

120

CHAPTERS

fAppendix A cont ins

review of geometry.

ircular motion is often described in terms of the frequency / , the number of revolutions per second. The period T of an object revolving in a circle is the time required for one complete revolution. Period and frequency are related by T " 7 (5" 2) For example, if an object revolves at a frequency of 3 rev/s, then each revolution takes \ s. For an object revolving in a circle (of circumference 2irr) at constant speed v, we can write 2irr v = -=- » T since in one revolution the object travels one circumference. ■ Acceleration of a revolving ball. A 150-g ball at the end of a string is revolving uniformly in a horizontal circle of radius 0.600 m, as in Fig. 5-10 or 5-12. The ball makes 2.00 revolutions in a second. What is its centripetal acceleration? APPR ACH The centripetal acceleration is aR = v2/r. We are given r, and we can find the speed of the ball, v, from the given radius and frequency. S LUTI N If the ball makes two complete revolutions per second, then the ball travels in a complete circle in a time interval equal to 0.500 s, which is its period T. The distance traveled in this time is the circumference of the circle, 2irr, where r is the radius of the circle. Therefore, the ball has speed 2irr 277(0.600 m) v = —— = — .— = 7.54 m/s. T (0.500 s) ' The centripetal acceleration is v2 (7.54 m /s)2 — — = 94.7 m /s2. aR = — = r (0.600 m) EXERCISE D If the r dius is doubled to 1.20 m but the period st ys the s me, by wh t f ctor will the centripet l cceler tion ch nge? ( ) 2, (b) 4, (c) \ , (d) \ , (e) none of these.

EXAMPLE 5 - 9 Moon's centripetal acceleration. The Moon's nearly circular orbit about the Earth has a radius of about 384,000 km and a period T of 27.3 days. Determine the acceleration of the Moon toward the Earth. APPR ACH Again we need to find the velocity v in order to find aR. We will need to convert to SI units to get v in m/s. S LUTI N In one orbit around the Earth, the Moon travels a distance 2irr, where r = 3.84 X 108m is the radius of its circular path. The time required for one complete orbit is the Moon's period of 27.3 d. The speed of the Moon in its orbit about the Earth is v = 2irr/T. The period T in seconds is T = (27.3 d) (24.0 h/d) (3600 s/h) = 2.36 X 10 s. Therefore,

(2irr f

47r2r

T2r

4it-2(3.84 X 108m) (2.36 X 10 s)2 = 0.00272 m/s2 = 2.72 X 10"3m /s2.

We can write this acceleration in terms of g = 9.80 m/s2 (the acceleration of gravity at the Earth's surface) as a = 2.72 X 10"3m/s2( ----- r) = 2.78X 10"V \ 9.80 m /s2/ N TE The centripetal acceleration of the Moon, a = 2.78 X 10-4 g, is not the acceleration of gravity for objects at the Moon's surface due to the Moon's gravity. Rather, it is the acceleration due to the Eart 's gravity for any object (such as the Moon) that is 384,000 km from the Earth. Notice how small this acceleration is compared to the acceleration of objects near the Earth's surface. differences in the fin l digit c n depend on whether you keep ll digits in your c lcul tor for v (which gives R = 94.7 m/s2), or if you use v = 7.54 m /s in which c se you get R = 94.8 m /s2. Both results re v lid since our ssumed ccur cy is bout + 0.1 m /s (see Section 1 - 3).

A CAUTION________ Distinguis t e M o o n 's gravity on objects at its surface, from t e E art 's gravity acting on t e M oon (t is Exam ple)

SECTI N 5 - 2

121

P H Y S 1C S

APPLIED Centrifuge Force exerted

*Centrifug tion entrifuges and very high speed ultracentrifuges, are used to sediment materials quickly or to separate materials. Test tubes held in the centrifuge rotor are accelerated to very high rotational speeds: see Fig. 5 -13, where one test tube is shown in two positions as the rotor turns. The small green dot represents a small particle, perhaps a macromolecule, in a fluid-filled test tube. At position A the particle has a tendency to move in a straight line, but the fluid resists the motion of the particles, exerting a centripetal force that keeps the particles moving nearly in a circle. The resistive force exerted by the fluid (liquid, gas, or gel, depending on the application) usually does not quite equal mv2/r , and the particles move slowly toward the bottom of the tube. A centrifuge provides an "effective gravity" much larger than normal gravity because of the high rotational speeds, thus causing more rapid sedimentation. Ultracentrifuge. The rotor of an ultracentrifuge rotates at 50,000 rpm (revolutions per minute). A particle at the top of a test tube (Fig. 5-13) is 6.00 cm from the rotation axis. alculate its centripetal acceleration, in "g's."

FIGURE 5 - 1 3 Two positions of rot ting test tube in centrifuge (top view). A t A , the green dot represents m cromolecule or other p rticle being sedimented. It would tend to follow the d shed line, he ding tow rd the bottom of the tube, but the fluid resists this m otion by exerting force on the p rticle s shown t point B.

APPR ACH We calculate the centripetal acceleration from aR = v2/r. S LUTI N The test tube makes 5.00 X 104revolutions each minute, or, dividing

by 60 s/min, 833 rev/s. The time to make one revolution, the period T, is T = /0„„ 1 ■, = 1.20 X 10"3s/rev. (833 rev/s) ' At the top of the tube, a particle revolves in a circle of circumference 277r = (2 )(0.0600 m) = 0.377 m per revolution. The speed of the particle is then 2irr _ ( 0.377 m/rev T \ 1.20 X 10-3 s/rev The centripetal acceleration is v2

(3.14 X 102m /s)2

0.0600m = 7 = t 4 X 10 m/s2 which, dividing by g = 9.80 m /s2, is 1.67 X 105 g's = 167,000 g's.

FIGURE 5 - 1 4 A force is required to keep n object moving in circle. If the speed is const nt, the force is directed tow rd the circle 's center.

5— 3 Dyn mics ofUniform Circul rMotion According to Newton's second law (SF = ma), an object that is accelerating must have a net force acting on it. An object moving in a circle, such as a ball on the end of a string, must therefore have a force applied to it to keep it moving in that circle. That is, a net force is necessary to give it centripetal acceleration. The magnitude of the required force can be calculated using Newton's second law for the radial component, 2 FR = maR, where aR is the centripetal acceleration, aR = v2/ r>and 2 Fr is the total (or net) force in the radial direction: v

EFr = maR = m — •

/j\ CAUTION_________ Centripetal force is n ot a n ew kin d o f force (E very force m ust be exerted by an object)

122

CHAPTER 5

[circular motion] (5 -3)

For uniform circular motion (v = constant), the acceleration is aR, which is directed toward the center of the circle at any moment. Thus the net force too must be directed toward t e center o f t e circle, Fig. 5-14. A net force is necessary because if no net force were exerted on the object, it would not move in a circle but in a straight line, as Newton's first law tells us. The direction of the net force is continually changing so that it is always directed toward the center of the circle. This force is sometimes called a centripetal ("pointing toward the center") force. But be aware that "centripetal force" does not indicate some new kind of force. The term merely describes the direction of the net force needed to provide a circular path: the net force is directed toward the circle's center. The force must be applied by ot er objects. For example, to swing a ball in a circle on the end of a string, you pull on the string and the string exerts the force on the ball. (Try it.)

Using Newton's Laws: Friction, Circular Motion, Drag Forces

There is a common misconception that an object moving in a circle has an outward force acting on it, a so-called centrifugal ("center-fleeing") force. This is incorrect: t ere is no outward force on the revolving object. onsider, for example, a person swinging a ball on the end of a string around her head (Fig. 5-15). If you have ever done this your self, you know that you feel a force pulling outward on your hand. The misconception arises when this pull is interpreted as an outward "centrifugal" force pulling on the ball that is transmitted along the string to your hand. This is not what is happening at all. To keep the ball moving in a circle, you pull inwardly on the string, and the string exerts this force on the ball. The ball exerts an equal and opposite force on the string (Newton's third law), and t is is the outward force your hand feels (see Fig. 5-15). The force on t e ball is the one exerted inwardly on it by you, via the string. To see even more convincing evidence that a "centrifugal force" does not act on the ball, consider what happens when you let go of the string. If a centrifugal force were acting, the ball would fly outward, as shown in Fig. 5 -16a. But it doesn't; the ball flies off tangentially (Fig. 5 -16b), in the direction of the velocity it had at the moment it was released, because the inward force no longer acts. Try it and see! EXERCISE E Return to the Ch pter-Opening Question, p ge 112, nd nswer it g in now. Try to expl in why you m y h ve nswered differently the first time.

EXAMPLE 5 - 11 ESTIMATE~| Force on revolving ball (horizontal). Estimate the force a person must exert on a string attached to a 0.150-kg ball to make the ball revolve in a horizontal circle of radius 0.600 m. The ball makes 2.00 revolutions per second (T = 0.500 s), as in Example 5 -8. Ignore the string's mass.

/ t\ CAUT I ON T ere is no real "centrifugalforce" Force on b ll N

FIGURE 5 -15 the end of

Swinging string.

b ll on

FIGURE 5 - 16 If centrifug l force existed, the revolving b ll would fly outw rd s in ( ) when rele sed. In f ct, it flies off t ngenti lly s in (b). For ex mple, in (c) sp rks fly in str ight lines t ngenti lly from the edge of rot ting grinding wheel.

APPR ACH First we need to draw the free-body diagram for the ball. The forces

acting on the ball are the force of gravity, mg downward, and the tension force Fx that the string exerts toward the hand at the center (which occurs because the person exerts that same force on the string). The free-body diagram for the ball is as shown in Fig. 5-17. The ball's weight complicates matters and makes it impos sible to revolve a ball with the cord perfectly horizontal. We assume the weight is small, and put « 0 in Fig. 5 -17. Thus Fx will act nearly horizontally and, in any case, provides the force necessary to give the ball its centripetal acceleration. S LUTI N We apply Newton's second law to the radial direction, which we assume is horizontal: (2 F )r = maR, where aR = v2/r and v = 2irr/T = 27r(0.600m)/(0.500s) = 7.54 m/s. Thus v2 (7.54 m /s): FT = m — = (0.150kg) : r v (0.600 m)

14 N.

N TE We keep only two significant figures in the answer because we ignored the ball's weight; it is mg = (0.150 kg)(9.80m/s2) = 1.5 N, about ^ of our result, which is small but not so small as to justify stating a more precise answer for FT. N TE To include the effect of r g, resolve Fx in Fig. 5 - 17 into components, and set the horizontal component of Fx equal to mv2/r and its vertical component equal to mg.

FIGURE 5 -17

Ex m ple 5 - 1 1 .

Dynamics of Uniform Circular Motion

123

EXAMPLE 5 - 1 2 Revolving ball (vertical circle). A 0.150-kg ball on the end of a 1.10-m-long cord (negligible mass) is swung in a vertical circle, (a) Determine the minimum speed the ball must have at the top of its arc so that the ball continues moving in a circle. (b) alculate the tension in the cord at the bottom of the arc, assuming the ball is moving at twice the speed of part (a). APPR ACH The ball moves in a vertical circle and is not undergoing uniform circular motion. The radius is assumed constant, but the speed v changes because of gravity. Nonetheless, Eq. 5 - 1 is valid at each point along the circle, and we use it at the top and bottom points. The free-body diagram is shown in Fig. 5 - 18 for both positions. S LUTI N (a) A t the top (point 1), two forces act on the ball: rag, the force of gravity, and Fti> the tension force the cord exerts at point 1. Both act down ward, and their vector sum acts to give the ball its centripetal acceleration aR. We apply Newton's second law, for the vertical direction, choosing downward as posi tive since the acceleration is downward (toward the center): ( I F ) R = maR

m gf

mgT

FIGURE 5 -18 Example 5-12. Freebody diagrams for positions 1 and 2. / t\ C A

TIO N

Circular motion only if cord is under tension

F i + mg

m— [at top] r From this equation we can see that the tension force Ft1 at point 1 will get larger if Vx (ball's speed at top of circle) is made larger, as expected. But we are asked for the minimum speed to keep the ball moving in a circle. The cord will remain taut as long as there is tension in it. But if the tension disappears (because is too small) the cord can go limp, and the ball will fall out of its circular path. Thus, the minimum speed will occur if Ft1 = 0, for which we have

mg = ra m —• [minimum speed at top] r We solve for v-l, keeping an extra digit for use in (b): Vi = V g? = V (9 - 80m /s2)(1.10m) = 3.283 m/s. This is the minimum speed at the top of the circle if the ball is to continue moving in a circular path. (b) When the ball is at the bottom of the circle (point 2 in Fig. 5 - 18), the cord exerts its tension force Ft 2 upward, whereas the force of gravity, rag, still acts downward. hoosing upward as positive, Newton's second law gives: (^ F )R -

maR

[at bottom] m— r The speed v2 is given as twice that in (a), namely 6.566 m/s. We solve for F\T 2 F 2 ~ mg

V22 = m ----- 1- mg r (6.566 m /s)' + (0.150 kg)(9.80 m /s2) = 7.35 N. = (0.150 kg) (1 .10 m)

EXERCISE F A rider on a Ferris wheel moves in a vertical circle of radius r at constant speed v (Fig. 5-19). Is the normal force that the seat exerts on the rider at the top of the wheel (a) less than, (b) more than, or (c) the same as, the force the seat exerts at the bottom of the wheel?

FIGURE 5-19

124

CHAPTERS

Exercise F.

EXAMPLE 5 - 13 Conical pendulum. A small ball of mass m, suspended by a cord of length £, revolves in a circle of radius r = I sin 0, where 0 is the angle the string makes with the vertical (Fig. 5 -20). (a) In what direction is the acceler ation of the ball, and what causes the acceleration? (b) alculate the speed and period (time required for one revolution) of the ball in terms of 1, 0, g, and m. APPR ACH We can answer (a) by looking at Fig. 5 -20, which shows the forces

on the revolving ball at one instant: the acceleration points horizontally toward the center of the ball's circular path (not along the cord). The force responsible for the acceleration is the net force which here is the vector sum of the forces acting on the mass m : its weight FG(of magnitude FG = mg) and the force exerted by the tension in the cord, FT. The latter has horizontal and vertical components of magnitude FTsin 0 and FTcos 0, respectively. S LUTI N (b) We apply Newton's second law to the horizontal and vertical directions. In the vertical direction, there is no motion, so the acceleration is zero and the net force in the vertical direction is zero:

FIGURE 5 - 20 Ex m ple 5 - 1 3 . Conic l pendulum.

FTcos 6 - mg = 0. In the horizontal direction there is only one force, of magnitude FTsin 0, that acts toward the center of the circle and gives rise to the acceleration v2/r. Newton's second law tells us: v2 F sin 0 = m — r We solve the second equation for v, and substitute for FT from the first equation (and use r = £ sin 0): rFTsin 0 m

v =

mg sin0 cos 0 £g sin20 cos 0

The period T is the time required to make one revolution, a distance of 2irr = 2irl sin 0. The speed v can thus be written v = lirl sin 6/T; then T =

2ir£ sin 6 v

2i £sin 6 Ig sin26 cos 6 = 2

N TE The speed and period do not depend on the mass m of the ball. They do

depend on i and 6. %O L v , *O B Z

Uniform Circular Motion 1. Dr w free-body di gr m, showing all the forces acting on each object under consideration. Be sure you can identify the source of each force (tension in a cord, Earth's gravity, friction, normal force, and so on). Don't put in something that doesn't belong (like a centrifugal force). 2. Determine which of the forces, or which of their compo nents, act to provide the centripetal acceleration—that

is, all the forces or components th t ct r di lly, toward or away from the center of the circular path. The sum of these forces (or components) provides the centripetal acceleration, aR = v2/r. 3. Choose convenient coordin te system, preferably with one axis along the acceleration direction. 4. Apply Newton's second l w to the radial component: v2 (2 F )r = maR = m — •

SECTI N 5 - 3

[radial direction]

Dynamics of Uniform Circular Motion

125

Force on car

FIGURE 5-21

The ro d exerts n inw rd force on c r (friction g inst the tires) to m ke it m ove in circle. The c r exerts n inw rd force on the p ssenger.

Isum of friel mil forces acting on each lire)

Forcc on

___ m aide ncy for passenger lo go straight

p ssenger

5 —4 Highw y Curv s: Bank dand Unbank d

FIGURE 5 - 22 R ce c r he ding into curve. From the tire m rks we see th t most c rs experienced sufficient friction force to give them the needed centripet l cceler tion for rounding the curve s fely. But, we lso see tire tr cks of c rs on which there w s not sufficient force — nd which unfortun tely followed more ne rly str ight-line p ths. FIGURE 5 - 23 Ex mple 5 - 14. Forces on c r rounding curve on fl t ro d, ( ) Front view, (b) top view.

Fn 1

An example of circular dynamics occurs when an automobile rounds a curve, say to the left. In such a situation, you may feel that you are thrust outward toward the right side door. But there is no mysterious centrifugal force pulling on you. What is happening is that you tend to move in a straight line, whereas the car has begun to follow a curved path. To make you go in the curved path, the seat (friction) or the door of the car (direct contact) exerts a force on you (Fig. 5-21). The car also must have a force exerted on it toward the center of the curve if it is to move in that curve. On a flat road, this force is supplied by friction between the tires and the pavement. If the wheels and tires of the car are rolling normally without slipping or sliding, the bottom of the tire is at rest against the road at each instant; so the fric tion force the road exerts on the tires is static friction. But if the static friction force is not great enough, as under icy conditions or high speed, sufficient friction force cannot be applied and the car will skid out of a circular path into a more nearly straight path. See Fig. 5-22. Once a car skids or slides, the friction force becomes kinetic friction, which is less than static friction. EXAMPLE 5 - 1 4 Skidding on a curve. A 1000-kg car rounds a curve on a flat road of radius 50 m at a speed of 15 m/s (54 km/h). Will the car follow the curve, or will it skid? Assume: (a) the pavement is dry and the coefficient of static fric = 0.25. tion is ijls = 0.60; (b) the pavement is icy and APPR ACH The forces on the car are gravity mg downward, the normal force FN exerted upward by the road, and a horizontal friction force due to the road. They are shown in Fig. 5 -23, which is the free-body diagram for the car. The car will follow the curve if the maximum static friction force is greater than the mass times the centripetal acceleration. S LUTI N In the vertical direction there is no acceleration. Newton's second law tells us that the normal force FNon the car is equal to the weight mg: Fn = mg = (1000 kg)(9.80 m/s2) = 9800 N.

In the horizontal direction the only force is friction, and we must compare it to the force needed to produce the centripetal acceleration to see if it is sufficient. The net horizontal force required to keep the car moving in a circle around the curve is < )

F = "'fi

v2 (15 m /s)2 (Z F )R = maR = m — = (1000 kg) = 4500 N. Now we compute the maximum total static friction force (the sum of the friction forces acting on each of the four tires) to see if it can be large enough to provide a safe centripetal acceleration. For (a), /jls = 0.60, and the maximum friction force attainable (recall from Section 5 - 1 that Ffr < /asFn) is (^fr)m x =

*5 - 6 Velocity-Dependent Forces *65. (I) Use dimensional analysis (Section 1 - 7) in Example 5 - 17 to determine if the time constant r is r = m /b or r = b/m . *

FIGURE 5 - 47

Problem 55.

. (II) The terminal velocity of a 3 X 10 5 kg raindrop is about 9 m/s. Assuming a drag force FD = —bv, determine (a) the value of the constant b and (b) the time required for such a drop, starting from rest, to reach 63% of terminal velocity.

*67. (II) An object moving vertically has v = v0 at t = 0. Determine a formula for its velocity as a function of time assuming a resistive force F = —bv as well as gravity for two cases: (a) v0 is downward and (b) v0 is upward.

Problems

135

8. (Ill) The drag force on large objects such as cars, planes, and sky divers moving through air is more nearly Fd = ~ bv2. (a) For this quadratic dependence on v, determine a formula for the terminal velocity V j, of a vertically falling object, (b) A 75-kg sky diver has a terminal velocity of about 60 m/s; determine the value of the constant b. (c) Sketch a curve like that of Fig. 5 - 27b for this case of Fd oc v2. For the same terminal velocity, would this curve lie above or below that in Fig. 5 - 27? Explain why. 69. (Ill) A bicyclist can coast down a 7.0° hill at a steady 9.5 km/h. If the drag force is proportional to the square of the speed v, so that FD = —cv2, calculate (a) the value of the constant c and (b) the average force that must be applied in order to descend the hill at 25 km/h. The mass of the cyclist plus bicycle is 80.0 kg. Ignore other types of friction. 70. (Ill) Two drag forces act on a bicycle and rider: FDi due to rolling resistance, which is essentially velocity independent; and F d2 due to air resistance, which is proportional to v2. For a specific bike plus rider of total mass 78 kg, Fd1 « 4.0 N; and for a speed of 2.2 m/s, FD2 ~ 1.0 N. (a) Show that the total drag force is Fd = 4.0 + 0.21v2, where v is in m/s, and FD is in N and opposes the motion. (b) Determine at what slope angle 0 the bike and rider can coast downhill at a constant speed of 8.0 m/s.

*71. (Ill) Determine a formula for the position and acceleration of a falling object as a function of time if the object starts from rest at t = 0 and undergoes a resistive force F = —bv, as in Example 5 - 17. *72. (Ill) A block of mass m slides along a horizontal surface lubricated with a thick oil which provides a drag force proportional to the square root of velocity: Fd = —bv2. If v = v0 at t = 0, determine v and x as functions of time. *73. (Ill) Show that the maximum distance the block in Problem 72 can travel is 2m Vq/2/3 b. *74. (Ill) You dive straight down into a pool of water. You hit the water with a speed of 5.0 m/s, and your mass is 75 kg. Assuming a drag force of the form FD = - (1.00 X 104kg/s) v, how long does it take you to reach 2% of your original speed? (Ignore any effects of buoyancy.) *75. (Ill) A motorboat traveling at a speed of 2.4 m /s shuts off its engines at t = 0. How far does it travel before coming to rest if it is noted that after 3.0 s its speed has dropped to half its original value? Assume that the drag force of the water is proportional to v.

| Gener l Problems 76. A coffee cup on the horizontal dashboard of a car slides forward when the driver decelerates from 45 km /h to rest in 3.5 s or less, but not if she decelerates in a longer time. What is the coefficient of static friction between the cup and the dash? Assume the road and the dashboard are level (horizontal). 77. A 2.0-kg silverware drawer does not slide readily. The owner gradually pulls with more and more force, and when the applied force reaches 9.0 N, the drawer suddenly opens, throwing all the utensils to the floor. What is the coefficient of static friction between the drawer and the cabinet? 78. A roller coaster reaches the top of the steepest hill with a speed of 6.0 km /h. It then descends the hill, which is at an average angle of 45° and is 45.0 m long. W hat will its speed be when it reaches the bottom? Assume = 0.12. 79. An 18.0-kg box is released on a 37.0° incline and accelerates down the incline at 0.220 m /s2. Find the friction force impeding its motion. How large is the coefficient of friction? 80. A flat puck (mass M ) is revolved in a circle on a frictionless air hockey table top, and is held in this orbit by a light cord which is connected to a dangling mass (mass m) through a central hole as shown in Fig. 5 - 48. Show that the speed of the puck is given by v = V m g R /M .

81. A motorcyclist is coasting with the engine off at a steady speed of 20.0 m /s but enters a sandy stretch where the coef ficient of kinetic friction is 0.70. Will the cyclist emerge from the sandy stretch without having to start the engine if the sand lasts for 15 m? If so, what will be the speed upon emerging? 82. In a "Rotor - ride" at a carnival, people rotate in a vertical cylindrically walled "room." (See Fig. 5 - 49). If the room radius was 5.5 m, and the rotation frequency 0.50 revo lutions per second when the floor drops out, what minimum coefficient of static friction keeps the people from slipping down? People on this ride said they were "pressed against the wall." Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides nausea)? [Hint: Draw a free -body diagram for a person.]

FIGURE 5 - 4 9

FIGURE 5 - 4 8

136

HAPTER 5

Problem 80.

Problem 82.

83. A device for training astronauts and jet fighter pilots is designed to rotate the trainee in a horizontal circle of radius 11.0 m. If the force felt by the trainee is 7.45 times her own weight, how fast is she rotating? Express your answer in both m /s and rev/s.

Using Newton's Laws: Friction, ircular Motion, Drag Forces

84. A 1250-kg car rounds a curve of radius 72 m banked at an angle of 14°. If the car is traveling at 85 km /h, will a friction force be required? If so, how much and in what direction? 85. Determine the tangential and centripetal components of the net force exerted on a car (by the ground) when its speed is 27 m/s, and it has accelerated to this speed from rest in 9.0 s on a curve of radius 450 m. The car's mass is 1150 kg. 8 . The 70.0-kg climber in Fig. 5 - 50 is supported in the "chimney" by the friction forces exerted on his shoes and back. The static coefficients of fric tion between his shoes and the wall, and between his back and the wall, are 0.80 and 0.60, respectively. What is the minimum normal force he must exert? Assume the walls are vertical and that the static friction forces are both at their maximum. Ignore his grip on the rope.

FIGURE 5 - 5 0

Problem 8 . 87. A small mass m is set on the surface of a sphere, Fig. 5 - 51. If the coefficient of static friction is fis = 0.70, at It 7 what angle would the mass start sliding?

An airplane traveling at 480 km /h needs to reverse its course. The pilot decides to accomplish this by banking the wings at an angle of 38°. (a) Find the time needed to reverse course. (b) Describe any additional force the passengers experience during the turn. [Hint: Assume an aerodynamic "lift" force that acts perpendicularly to the flat wings; see Fig. 5-53.] FIGURE 5 - 53

Problem 91. 92. A banked curve of radius R in a new highway is designed so that a car traveling at speed v0 can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly then it will slip toward the center of the circle. If it travels too fast, it will slip away from the center of the circle. If the coefficient of static friction increases, it becomes possible for a car to stay on the road while traveling at a speed within a range from vm\n to vmax. Derive formulas for vmjn and Umax as functions of fJLs,v 0, and R. 93. A small bead of mass m is constrained to slide without friction inside a circular vertical hoop of radius r which rotates about a vertical axis (Fig. 5 -54) at a frequency / . (a) Determine the angle 6 where the bead will be in equilibrium — that is, where it will have no tendency to move up or down along the hoop, (b) If / = 2.00 rev/s and r = 22.0 cm, what is 61 (c) an the bead ride as high as the center of the circle (0 = 90°)? Explain.

FIGURE 5 - 51

Problem 87.

FIGURE 5 - 54

Problem 93. 88. A 28.0-kg block is connected to an empty 2.00-kg bucket by a cord running over a frictionless pulley (Fig. 5 - 52). The coefficient of static friction between the table and the block is 0.45 and the coefficient of kinetic friction between the table and the block is 0.32. 28,0 kii Sand is gradually added to the bucket until the system just begins to move. (a) alculate the mass of sand added to the bucket. (b) alculate the accelera tion of the system.

FIGURE 5 - 52

Problem 88. 89. A car is heading down a slippery road at a speed of 95 km/h. The minimum distance within which it can stop without skidding is m. What is the sharpest curve the car can negotiate on the icy surface at the same speed without skidding? 90. What is the acceleration experienced by the tip of the 1.5-cm -long sweep second hand on your wrist watch?

94. Eart is not quite an inertial frame. We often make measure ments in a reference frame fixed on the Earth, assuming Earth is an inertial reference frame. But the Earth rotates, so this assumption is not quite valid. Show that this assumption is off by 3 parts in 1000 by calculating the acceleration of an object at Earth's equator due to Earth's daily rotation, and compare to g = 9.80 m /s2, the acceleration due to gravity. 95. While fishing, you get bored and start to swing a sinker weight around in a circle below you on a 0.45-m piece of fishing line. The weight makes a complete circle every 0.50 s. What is the angle that the fishing line makes with the vertical? [Hint: See Fig. 5 -20.] 96. onsider a train that rounds a curve with a radius of 570 m at a speed of 160 km /h (approximately 100 mi/h). (a) alcu late the friction force needed on a train passenger of mass 75 kg if the track is not banked and the train does not tilt. (b) alculate the friction force on the passenger if the train tilts at an angle of 8.0° toward the center of the curve. 97. A car starts rolling down a l-in - 4 hill (l- in - 4 means that for each 4 m traveled along the road, the elevation change is 1 m). How fast is it going when it reaches the bottom after traveling 55 m? (a) Ignore friction. (b) Assume an effective coefficient of friction equal to 0.10.

General Problems

1 37

98. The sides of a cone make an angle with the vertical. A small mass m is placed on the inside of the cone and the cone, with its point down, is revolved at a frequency / (revolutions per second) about its symmetry axis. If the coefficient of static friction is /xs, at what positions on the cone can the mass be placed without sliding on the cone? (Give the maximum and minimum distances, r, from the axis). 99. A 72-kg water skier is being accelerated by a ski boat on a flat ("glassy") lake. The coefficient of kinetic friction between the skier's skis and the water surface is = 0-25 (Fig. 5 -55). (a) What is the skier's acceleration if the rope pulling the skier behind the boat applies a horizontal tension force of magnitude FT = 240 N to the skier (0 = 0°)? (b) What is the skier's horizontal acceleration if the rope pulling the skier exerts a force of FT = 240 N on the skier at an upward angle 6 = 12°? (c) Explain why the skier's acceleration in part (b) is greater than that in part (a).

F = 240 N ___ 10= 0.25

FIGURE 5 - 5 5

Problem 99.

100. A ball of mass m = 1.0 kg at the end of a thin cord of length r = 0.80 m revolves in a vertical circle about point O, as shown in Fig. 5 -56. During the time we observe it, the only forces acting on the ball are gravity and the tension in the cord. The motion is circular but not uniform because of the force of gravity. The ball increases in speed as it descends and decelerates as it rises on the other side of the circle. A t the moment the cord makes an angle 0 = 30° below the horizontal, the ball's .___ speed is 6.0 m/s. At \ s \ / this point, determine \ the tangential accel eration, the radial acceleration, and the tension in the cord, FT . Take 0 increasing downward as shown.

FIGURE 5 - 5 6 Problem 100.

101. A car drives at a constant speed around a banked circular track with a diameter of 127 m. The motion of the car can be described in a coordinate system with its origin at the center of the circle. A t a particular instant the car's accel eration in the horizontal plane is given by a = ( —15.71 — 23.2j) m /s2. (a) What is the car's speed? (b) Where (* and y ) is the car at this instant?

18Num rical/Comput r *102. (Ill) The force of air resistance (drag force) on a rapidly falling body such as a skydiver has the form FD = —kv 2, so that Newton's second law applied to such an object is dv = mg — ki v2, m— dt where the downward direction is taken to be positive. (a) Use numerical integration [Section 2 - 9] to estimate (within 2%) the position, speed, and acceleraton, from t = 0 up to t = 15.0 s, for a 75-kg skydiver who starts from rest, assuming k = 0.22 kg/m. (ft) Show that the diver eventually reaches a steady speed, the terminal speed, and explain why this happens, (c) How long does it take for the skydiver to reach 99.5% of the terminal speed? "103. (Ill) The coefficient of kinetic friction between two surfaces is not strictly independent of the velocity of the object. A possible expression for for wood on wood is 0.20 (1 + 0.0020'y2) where v is in m/s. A wooden block of mass 8.0 kg is at rest on a wooden floor, and a constant horizontal force of 41 N acts on the block. Use numerical integration [Section 2 - 9] to determine and graph (a) the speed of the block, and (b) its position, as a function of time from 0 to 5.0 s. (c) Determine the percent difference for the speed and position at 5.0 s if /% is constant and equal to 0.20. '104. (Ill) Assume a net force F = - m g - k v 2 acts during the upward vertical motion of a 250-kg rocket, starting at the moment (t = 0) when the fuel has burned out and the rocket has an upward speed of 120 m/s. Let k = 0.65 kg/m. Estimate v and y at 1.0-s intervals for the upward motion only, and estimate the maximum height reached. ompare to free-flight conditions without air resistance (k = 0).

Answers to Exercises A: (c).

F: (a).

B: FPx is insufficient to keep the box moving for long.

G: (c).

: No — the acceleration is not constant (in direction). D: (a), it doubles.

H: Yes.

E: (d).

I: (a) No change; (b) 4 times larger.

138

HAPTER 5

Using Newton's Laws: Friction, ircular Motion, Drag Forces

The stron uts in the upper left of this photo re working on the Sp ce Shuttle. A s they orbit the E rth — t r ther high speed — they experience pp rent weightlessness. The M oon, in the b ckground, lso is orbiting the E rth t high speed. Wh t keeps the M oon nd the sp ce shuttle ( nd its stron uts) from m oving off in str ight line w y from E rth? It is the force of gr vity. N ew ton 's l w of univers l gr vit tion st tes th t ll objects ttr ct ll other objects with force proportion l to their m sses nd inversely proportion l to the squ re of the dist nce between them.

T £

Gravitation and Newton's Synthesis CHAPTER-OPENING QUESTIOr — Guess now! A space station revolves around the Earth as a satellite, 100 km above Earth's surface. What is the net force on an astronaut at rest inside the space station? (a) Equal to her weight on Earth. (b) A little less than her weight on Earth. (c) Less than half her weight on Earth. (d) Zero (she is weightless). (e) Somewhat larger than her weight on Earth. ir Isaac Newton not only put forth the three great laws of motion that serve as the foundation for the study of dynamics. He also conceived of another great law to describe one of the basic forces in nature, gravitation, and he applied it to understand the motion of the planets. This new law, published in 1687 in his book P ilosop iae Naturalis Principia Mat ematica (the Principia, for short), is called Newton's law of universal gravitation. It was the capstone of Newton's analysis of the physical world. Indeed, Newtonian mechanics, with its three laws of motion and the law of universal gravitation, was accepted for centuries as a mechanical basis for the way the universe works.

CONTENTS 6 - 1 Newton 's L w of Univers l Gr vit tion 6 - 2 Vector Form of Newton 's L w of Univers l Gr vit tion 6 - 3 Gr vity Ne r the E rth 's Surf ce; Geophysic l Applic tions 6 - 4 S tellites nd "Weightlessness" 6 - 5 Kepler 's L ws nd Newton 's Synthesis * 6 - 6 Gr vit tion l Field 6 - 7 Types of Forces in N ture * 6 - 8 Principle of Equiv lence; Curv ture of Sp ce; Bl ck Holes

139

6—1 Newton's L wof Univers l Gr vit tion

J V

FIGURE 6 - 1 Anyw here on E rth, whether in Al sk , Austr li , or Peru, the force o f gr vity cts downw rd tow rd the center o f the E rth.

Among his many great accomplishments, Sir Isaac Newton examined the motion of the heavenly bodies— the planets and the Moon. In particular, he wondered about the nature of the force that must act to keep the Moon in its nearly circular orbit around the Earth. Newton was also thinking about the problem of gravity. Since falling objects accelerate, Newton had concluded that they must have a force exerted on them, a force we call the force of gravity. Whenever an object has a force exerted on it, that force is exerted by some other object. But what exerts the force of gravity? Every object on the surface of the Earth feels the force of gravity, and no matter where the object is, the force is directed toward the center of the Earth (Fig. 6 -1). Newton concluded that it must be the Earth itself that exerts the gravitational force on objects at its surface. According to legend, Newton noticed an apple drop from a tree. He is said to have been struck with a sudden inspiration: If gravity acts at the tops of trees, and even at the tops of mountains, then perhaps it acts all the way to the Moon! With this idea that it is Earth's gravity that holds the Moon in its orbit, Newton devel oped his great theory of gravitation. But there was controversy at the time. Many thinkers had trouble accepting the idea of a force "acting at a distance." Typical forces act through contact—your hand pushes a cart and pulls a wagon, a bat hits a ball, and so on. But gravity acts without contact, said Newton: the Earth exerts a force on a falling apple and on the Moon, even though there is no contact, and the two objects may even be very far apart. Newton set about determining the magnitude of the gravitational force that the Earth exerts on the Moon as compared to the gravitational force on objects at the Earth's surface. At the surface of the Earth, the force of gravity accelerates objects at 9.80 m /s2. The centripetal acceleration of the Moon is calculated from aR = v2/r (see Example 5 - 9) and gives aR = 0.00272 m /s2. In terms of the acceleration of gravity at the Earth's surface, g, this is equivalent to 0.00272 m/s2 9.80 m/s2 °8

1 3600 8-

That is, the acceleration of the Moon toward the Earth is about ^ as great as the acceleration of objects at the Earth's surface. The Moon is 384,000 km from the Earth, which is about 60 times the Earth's radius of 6380 km. That is, the Moon is 60 times farther from the Earth's center than are objects at the Earth's surface. But 60 X 60 = 602 = 3600. Again that number 3600! Newton concluded that the gravitational force F exerted by the Earth on any object decreases with the square of its distance, r, from the Earth's center: FIGURE 6 - 2 The gr vit tion l force one object exerts on second object is directed tow rd the first object, nd is equ l nd opposite to the force exerted by the second object on the first.

Gr vit tion l force exerted on Moon by E rth

F oc

The Moon is 60 Earth radii away, so it feels a gravitational force only ^ = mo times as strong as it would if it were a point at the Earth's surface. Newton realized that the force of gravity on an object depends not only on distance but also on the object's mass. In fact, it is directly proportional to its mass, as we have seen. According to Newton's third law, when the Earth exerts its gravi tational force on any object, such as the Moon, that object exerts an equal and opposite force on the Earth (Fig. 6 - 2). Because of this symmetry, Newton reasoned, the magnitude of the force of gravity must be proportional to bot the masses. Thus F oc

E rth

Gr vit tion l force exerted on E rth by the Moon

CHAPTER 6

mEmB

where raE is the mass of the Earth, mB the mass of the other object, and r the distance from the Earth's center to the center of the other object.

Gravitation and Newton's Synthesis

Newton went a step further in his analysis of gravity. In his examination of the orbits of the planets, he concluded that the force required to hold the different planets in their orbits around the Sun seems to diminish as the inverse square of their distance from the Sun. This led him to believe that it is also the gravitational force that acts between the Sun and each of the planets to keep them in their orbits. And if gravity acts between these objects, why not between all objects? Thus he proposed his l w of univers l gr vit tion, which we can state as follows: Every p rticle in the universe ttr cts every other p rticle with force th t is proportion l to the product of their m sses nd inversely proportion l to the squ re of the dist nce between them. This force cts long the line joining the two p rticles. The magnitude of the gravitational force can be written as mi m2 F = G— rL

( 6- 1)

LAW OF UNIVERSAL GRAVITATION

where m 1 and m2 are the masses of the two particles, r is the distance between them, and G is a universal constant which must be measured experimentally and has the same numerical value for all objects. The value of G must be very small, since we are not aware of any force of attraction between ordinary-sized objects, such as between two baseballs. The force between two ordinary objects was first measured by Henry avendish in 1798, over 100 years after Newton published his law. To detect and measure the incred ibly small force between ordinary objects, he used an apparatus like that shown in Fig. 6 - 3. avendish confirmed Newton's hypothesis that two objects attract one another and that Eq. 6 - 1 accurately describes this force. In addition, because avendish could measure F, m 1,m 2, and r accurately, he was able to determine the value of the constant G as well. The accepted value today is G = 6.67 X 10-n N*m7kg2. (See Table inside front cover for values of all constants to highest known precision.) Strictly speaking, Eq. 6 -1 gives the magnitude of the gravitational force that one particle exerts on a second particle that is a distance r away. For an extended object (that is, not a point), we must consider how to measure the distance r. You might think that r would be the distance between the centers of the objects. This is true for two spheres, and is often a good approximation for other objects. A correct calculation treats each extended body as a collection of particles, and the total force is the sum of the forces due to all the particles. The sum over all these particles is often best done using integral calculus, which Newton himself invented. When extended bodies are small compared to the distance between them (as for the Earth - Sun system), little inaccuracy results from considering them as point particles. Newton was able to show (see derivation in Appendix D) that the gravitational force exerted on a particle outside a sp ere, wit a sp erically symmetric mass distribution, is t e same as if t e entire mass o f t e sp ere was concentrated at its center. Thus Eq. 6 -1 gives the correct force between two uniform spheres where r is the distance between their centers. ■

NEWTON'S

FIGURE 6 - 3 Schem tic di gr m of C vendish 's pp r tus. Two spheres re tt ched to light horizont l rod, which is suspended t its center by thin fiber. W hen third sphere (l beled A ) is brought close to one of the suspended spheres, the gr vit tion l force c uses the l tter to m ove, nd this twists the fiber slightly. The tiny m ovem ent is m gnified by the use of n rrow light be m directed t mirror m ounted on the fiber. The be m reflects onto sc le. Previous determin tion of how l rge force will twist the fiber given mount then llows the experim enter to determ ine the m gnitude of the gr vit tion l force betw een two objects.

Fiber

ESTIMATE I Can you attract another person gravitationally?

A 50-kg person and a 70-kg person are sitting on a bench close to each other. Estimate the magnitude of the gravitational force each exerts on the other. APPR ACH This is an estimate: we let the distance between the centers of the two people be ^m (about as close as you can get). S LUTI N We use Eq. 6 - 1, which gives (6.67 X 10-11 N - m2/kg2)(50 kg)(70 kg) (0.5 m): rounded off to an order of magnitude. Such a force is unnoticeably small unless extremely sensitive instruments are used. N TE As a fraction of their weight, this force is (10" N )/(70 kg)(9.8 m /s2) ~ 10 9. SECTI N 6 - 1

Newton's Law of Universal Gravitation

141

Motion

EXAMPLE 6 - 2 Spacecraft at 2rE. What is the force of gravity acting on a 2000-kg spacecraft when it orbits two Earth radii from the Earth's center (that is, a distance rE = 6380 km above the Earth's surface, Fig. 6 - 4)? The mass of the Earth is mE = 5.98 X 1024kg. APPR ACH We could plug all the numbers into Eq. 6 - 1, but there is a simpler

approach. The spacecraft is twice as far from the Earth's center as when it is at the surface of the Earth. Therefore, since the force of gravity decreases as the square of the distance (and ^ = J), the force of gravity on the satellite will be only one-fourth its weight at the Earth's surface. S LUTI N At the surface of the Earth, FG = mg. At a distance from the Earth's center of 2rE, FGis \ as great: FIGURE 6 - 4

Fg = I mg = \ (2000 kg)(9.80 m /s2) = 4900 N.

Ex m ple 6 - 2 .

FIGURE 6 - 5 Ex m ple 6 - 3 . O rient tion o f Sun (S), E rth (E ), nd M oon (M ) t right ngles to e ch other (not to sc le).

EXAMPLE __________________ 6 -3 Force on the Moon. Find the net force on the Moon

mM = 7.35 X 1022kg) due to the gravitational attraction of both the Earth mE = 5.98 X 1024kg) and the Sun (ras = 1.99 X 1030kg), assuming they are at right angles to each other as in Fig. 6 -5. APPR ACH The forces on our object, the Moon, are the gravitational force

Moon

exerted on the Moon by the Earth FME and the force exerted by the Sun Fms , as shown in the free-body diagram of Fig. 6 - 5. We use the law of universal gravi tation to find the magnitude of each force, and then add the two forces as vectors. S LUTI N The Earth is 3.84 X 105km = 3.84 X 108m from the Moon, so FME (the gravitational force on the Moon due to the Earth) is (6.67 X 10-11 N •m2/kg2)(7.35 X 1022kg)(5.98 X 1024kg) (3.84 X 108m)2

= 1.99 X 10 N.

The Sun is 1.50 X 108km from the Earth and the Moon, so FMS (the gravitational force on the Moon due to the Sun) is

Sun

(6.67 X 10-11 N •m2/kg2)(7.35 X 1022kg)(l.99 X 1030kg)

^MS —

(1.50 X 1011 m)

= 4.34 X 10 N.

The two forces act at right angles in the case we are considering (Fig. 6 - 5), so we can apply the Pythagorean theorem to find the magnitude of the total force: F = V(1.99 X 102ON)2 + (4.34 X 1020N)2 = 4.77 X lO^N. The force acts at an angle 0 (Fig. 6 - 5) given by 0 = tan _1(l.99/4.34) = 24.6°. N TE The two forces, FME and FMS, have the same order of magnitude (1020N). This may be surprising. Is it reasonable? The Sun is much farther from Earth than the Moon (a factor of 10n m/108m « 103), but the Sun is also much more massive (a factor of 1030kg/1023kg « 107). Mass divided by distance squared (107/ 106) comes out within an order of magnitude, and we have ignored factors of 3 or more. Yes, it is reasonable. Distinguis between N ew to n 's second law and t e law o f universal gravitation

Note carefully that the law of universal gravitation describes a particular force (gravity), whereas Newton's second law of motion (F = ma) tells how an object accelerates due to any type of force.

*Spheric l Shells Newton was able to show, using the calculus he invented for the purpose, that a thin uniform spherical shell exerts a force on a particle outside it as if all the shell's mass were at its center; and that such a thin uniform shell exerts zero force on a particle inside the shell. (The derivation is given in Appendix D.) The Earth can be modelled as a series of concentric shells starting at its center, each shell uniform but perhaps having a different density to take into account Earth's varying density in various layers. As a simple example, suppose the Earth were uniform throughout; what is 142

CHAPTER 6

Gravitation and Newton's Synthesis

the gravitational force on a particle placed exactly halfway from Earth's center to its surface? Only the mass inside this radius r = \ rE would exert a net force on this particle. The mass of a sphere is proportional to its volume V = f tt t 3, so the mass m inside r = \ rE is Q) = | the mass of the entire Earth. The gravitational force on the particle at r = \ rE, which is proportional to m /r (Eq. 6 - 1), is reduced to (!)/0 )2 = i the gravitational force it would experience at Earth's surface.

6—2 Vector Form of Newton's L w of Univers l Gr vit tion We can write Newton's law of universal gravitation in vector form as m2 (6 - 2) F 12 = - G r21> ri\ where Fl2 is the vector force on particle 1 (of mass exerted by particle 2 (of mass m2), which is a distance r21 away; r21 is a unit vector that points from particle 2 toward particle 1 along the line joining them so that r21 = ?2i/r21, where r21 is the displacement vector as shown in Fig. - . The minus sign in Eq. 6 - 2 is necessary because the force on particle 1 due to particle 2 points toward ra2, in the direction opposite to r21. The displacement vector r12 is a vector of the same magnitude as r21, but it points in the opposite direction so that r 12

— r 21

•

By Newton's third law, the force F21 acting on m2 exerted by m x must have the same magnitude as F12 but acts in the opposite direction (Fig. 6 - 7), so that mi r 2 „

y ^ r 21 Ai ^ m2m x . — {j 2 1*12• r\2 The force of gravity exerted on one particle by a second particle is always directed toward the second particle, as in Fig. - . When many particles interact, the total gravitational force on a given particle is the vector sum of the forces exerted by each of the others. For example, the total force on particle number 1 is F2i = —F12 = G

FIGURE 6 - 6 The displ cem ent vector r21 points from p rticle of m ss m 2 to p rticle of m ss m i . The unit vector shown, r21 is in the s me direction s r21, but is defined s h ving length one. FIGURE 6 - 7 B y N ew ton 's third l w, the gr vit tion l force on p rticle 1 exerted by p rticle 2, F12, is equ l nd opposite to th t on p rticle 2 exerted by p rticle 1, F2i ; th t is F21 = —F12.

mi '12

Fi = F12 + F13 + F14 + ••• + Fln = 2 ^ 1 ;

(6 - 3)

i=2

where Flf- means the force on particle 1 exerted by particle i, and n is the total number of particles. This vector notation can be very helpful, especially when sums over many particles are needed. However, in many cases we do not need to be so formal and we can deal with directions by making careful diagrams.

6—3 Gr vity Ne r the E rth's Surf ce; Geophysic l Applic tions When Eq. 6 - 1 is applied to the gravitational force between the Earth and an object at its surface, mx becomes the mass of the Earth raE, m2 becomes the mass of the object m, and r becomes the distance of the object from the Earth's center, which is the radius of the Earth rE. This force of gravity due to the Earth is the weight of the object, which we have been writing as mg. Thus, mmE mg = G — j— ri. We can solve this for g, the acceleration of gravity at the Earth's surface:

raF

g = G -^-

(6 - 4)

Thus, the acceleration of gravity at the surface of the Earth, g, is determined by mB and rE. (Don't confuse G with g; they are very different quantities, but are related by Eq. 6 - 4.) SECTI N 6 - 3

/ j \ AUT I ON Distinguis Gfrom g

Gravity Near the Earth's Surface; Geophysical Applications

143

Until G was measured, the mass of the Earth was not known. But once G was measured, Eq. 6 - 4 could be used to calculate the Earth's mass, and avendish was the first to do so. Since g = 9.80 m /s2 and the radius of the Earth is rB = 6.38 X 10 m, then, from Eq. 6 - 4, we obtain mE =

(9.80m/s2)(6.38 X 10 m)2

?r |

G~ ~~

6.67 X 10_u N •m2/kg2

= 5.98 X 1024kg for the mass of the Earth. Equation 6 - 4 can be applied to other planets, where g, m, and r would refer to that planet. EXAMPLE 6 - 4 ESTIMATE- ! Gravity on Everest. Estimate the effective value of g on the top of Mt. Everest, 8850 m (29,035 ft) above sea level (Fig. - 8). That is, what is the acceleration due to gravity of objects allowed to fall freely at this altitude? APPR ACH The force of gravity (and the acceleration due to gravity g) depends

on the distance from the center of the Earth, so there will be an effective value g' on top of Mt. Everest which will be smaller than g at sea level. We assume the Earth is a uniform sphere (a reasonable "estimate"). S LUTI N We use Eq. 6 - 4, with rE replaced by r = 6380 km + 8.9 km = 6389 km = 6.389 X 10 m: (6.67 X 10-11 N •m2/kg2)(5.98 X 1024kg) 8 = G— rL

(6.389 X 10 m = 9.77 m /s2

FIGURE 6 - 8 Ex m ple 6 - 4 . M ount Everest, 8850 m (29,035 ft) bove se level; in the foreground, the uthor with sherp s t 5500 m (18,000 ft).

which is a reduction of about 3 parts in a thousand (0.3%). N TE This is an estimate because, among other things, we ignored the mass accumulated under the mountaintop.

TABLE 6 -1 Acceleration Due to Gravity at Various Locations on Earth

Note that Eq. 6 - 4 does not give precise values for g at different locations because the Earth is not a perfect sphere. The Earth not only has mountains and valleys, and bulges at the equator, but also its mass is not distributed precisely uniformly (see Table 6 - 1). The Earth's rotation also affects the value of g (see Example 6 -5). However, for most practical purposes when an object is near the Earth's surface, we will simply use g = 9.80 m /s2 and write the weight of an object as mg.

Elev tion Loc tion

(m)

8,r

(m /s2)

N ew York

S n Fr ncisco

9.800

D enver

1650

9.796

Pikes Pe k

4300

9.789

Sydney, Austr li

9.798

Equ tor

9.780

North Pole (c lcul ted)

9.832

PHYSICS

9.803

APPLIED

G eology — mineral and oil exploration

144

CHAPTER 6

EXERCISE A Suppose you could double the m ss of pl net but kept its volum e the s me. H ow would the cceler tion o f gr vity, g, t the surf ce ch nge?

The value of g can vary locally on the Earth's surface because of the presence of irregularities and rocks of different densities. Such variations in g, known as "gravity anomalies," are very small— on the order of 1 part per 10 or 107 in the value of g. But they can be measured ("gravimeters" today can detect variations in g to 1 part in 109). Geophysicists use such measurements as part of their investigations into the structure of the Earth's crust, and in mineral and oil exploration. Mineral deposits, for example, often have a greater density than surrounding material; because of the greater mass in a given volume, g can have a slightly greater value on top of such a deposit than at its flanks. "Salt domes," under which petroleum is often found, have a lower than average density and searches for a slight reduction in the value of g in certain locales have led to the discovery of oil.

Gravitation and Newton's Synthesis

EXAMPLE 6 - 5 Effect of Earth's rotation on g. Assuming the Earth is a perfect sphere, determine how the Earth's rotation affects the value of g at the equator compared to its value at the poles. APPR ACH Figure 6 - 9 shows a person of mass ra standing on a doctor's scale

at two places on the Earth. At the North Pole there are two forces acting on the mass ra: the force of gravity, FG = rag, and the force with which the scale pushes up on the mass, w. We call this latter force w because it is what the scale reads as the weight of the object, and by Newton's third law it equals the force with which the mass pushes down on the scale. Since the mass is not accelerating, Newton's second law tells us mg — w = 0, so w = mg. Thus the weight w that the spring registers equals mg, which is no surprise. Next, at the equator, there is an acceleration because the Earth is rotating. The same magnitude of the force of gravity FG = mg acts downward (we are letting g represent the acceleration of gravity in the absence of rotation and we ignore the slight bulging of the equator). The scale pushes upward with a force w'; w' is also the force with which the person pushes down on the scale (Newton's third law) and hence is the weight registered on the scale. From Newton's second law we now have (see Fig. 6 - 9)

FIGURE 6 - 9

Ex m ple 6 - 5 .

mg — w' = m because the person of mass ra now has a centripetal acceleration due to Earth's rotation; rE = 6.38 X 10 m is the Earth's radius and v is the speed of ra due to the Earth's daily rotation. S LUTI N First we determine the speed v of an object at rest on the Earth's equator, rembering that Earth makes one rotation (distance = circumference of Earth = 2irrE) in 1 day = (24h)(60m in/h)(60s/m in) = 8.64 X 104s: 277TE

(6.283)(6.38 X 10 m) 1 day ~~ (8.64 x 104s) = 4.640 X 102m/s.

The effective weight is w' = mg' where g' is the effective value of g, and so g' = w '/m . Solving the equation above for w ', we have w

= m\ g

w' = m

so 8

V2 rE

Hence v2 Ag = g ~ g' = — rE

(4.640 X 102m/s) (6.38 X 10 m) = 0.0337 m /s2,

which is about Ag ~ 0.003g, a difference of 0.3%. N TE In Table 6 -1 we see that the difference in g at the pole and equator is actually greater than this: (9.832 - 9.780) m /s2 = 0.052 m /s2. This discrepancy is due mainly to the Earth being slightly fatter at the equator (by 21 km) than at the poles. N TE The calculation of the effective value of g at latitudes other than at the poles or equator is a two-dimensional problem because FG acts radially toward the Earth's center whereas the centripetal acceleration is directed perpendicular to the axis of rotation, parallel to the equator and that means that a plumb line (the effective direction of g) is not precisely vertical except at the equator and the poles. SECTI N 6 - 3

Gravity Near the Earth's Surface; Geophysical Applications

145

E rth s Inerti l Reference Fr me We often make the assumption that reference frames fixed on the Earth are inertial reference frames. Our calculation in Example 6 - 5 above shows that this assumption can result in errors no larger than about 0.3% in the use of Newton's second law, for example. We discuss the effects of Earth's rotation and reference frames in more detail in hapter 11, including the oriolis effect.

6 - 4 Sat llit s and "W ightl ssn ss" S tellite Motion PHYSICS

APPLIED

Artificial Eart satellites

Artificial satellites circling the Earth are now commonplace (Fig. 6 -10). A satellite is put into orbit by accelerating it to a sufficiently high tangential speed with the use of rockets, as shown in Fig. 6 - 11. If the speed is too high, the spacecraft will not be confined by the Earth's gravity and will escape, never to return. If the speed is too low, it will return to Earth. Satellites are usually put into circular (or nearly circular) orbits, because such orbits require the least takeoff speed.

27,00(1 km/h

circular

FIGURE 6 - 10 A s tellite, the Intern tion l Sp ce St tion, circling the E rth.

FIGURE 6 - 12 A m oving s tellite "f lls " out of str ight -line p th tow rd the E rth. Without gravity

With''- N gr vity s

FIGURE 6-11

.10,000 km/h elliptical

Artifici l s tellites l unched t

different speeds.

It is sometimes asked: "What keeps a satellite up?" The answer is: its high speed. If a satellite stopped moving, it would fall directly to Earth. But at the very high speed a satellite has, it would quickly fly out into space (Fig. 6 -12) if it weren't for the gravitational force of the Earth pulling it into orbit. In fact, a satellite is falling (accelerating toward Earth), but its high tangential speed keeps it from hitting Earth. For satellites that move in a circle (at least approximately), the needed acceleration is centripetal and equals v2/r. The force that gives a satellite this acceleration is the force of gravity exerted by the Earth, and since a satellite may be at a considerable distance from the Earth, we must use Newton's law of universal gravitation (Eq. 6 - 1) for the force acting on it. When we apply Newton's second law, 2 FR = maR in the radial direction, we find mmF v G — z— = m — 5 r r

(6 -5)

where m is the mass of the satellite. This equation relates the distance of the satellite from the Earth's center, r, to its speed, v, in a circular orbit. Note that only one force— gravity— is acting on the satellite, and that r is the sum of the Earth's radius rB plus the satellite's height above the Earth: r = rE + . 146

CHAPTER 6

Gravitation and Newton's Synthesis

EXAMPLE 6 - 6 Geosynchronous satellite. A geosync ronous satellite is one that stays above the same point on the Earth, which is possible only if it is above a point on the equator. Such satellites are used for TV and radio transmission, for weather forecasting, and as communication relays. Determine (a) the height above the Earth's surface such a satellite must orbit, and (b) such a satellite's speed, (c) ompare to the speed of a satellite orbiting 200 km above Earth's surface. APPR ACH To remain above the same point on Earth as the Earth rotates, the satellite must have a period of 24 hours. We can apply Newton's second law, F = ma, where a = v2/r if we assume the orbit is circular. S LUTI N (a) The only force on the satellite is the force of universal gravitation due to the Earth. (We can ignore the gravitational force exerted by the Sun. Why?) We apply Eq. 6 -5, assuming the satellite moves in a circle: mSatmE v2 G ----- ^— = mSat r This equation has two unknowns, r and v. But the satellite revolves around the Earth with the same period that the Earth rotates on its axis, namely once in 24 hours. Thus the speed of the satellite must be 2irr v = -=- > T where T = 1 day = (24h)(3600s/h) = 86,400 s. We substitute this into the "satellite equation" above and obtain (after canceling mSat on both sides) mE (2irr)2 t ^ t — r2 rT2 After cancelling an r, we can solve for r 3 .

PHYSICS

APPLIED

Geosync ronous satellites

GmET2 4

(6.67 X 10-11 N •m2/kg2)(5.98 X 1024kg)(86,400s): 47T = 7.54 X 1022m3. We take the cube root and find r = 4.23 X 107m, or 42,300 km from the Earth's center. We subtract the Earth's radius of 6380 km to find that a geosynchronous satellite must orbit about 36,000 km (about rE) above the Earth's surface. (b) We solve for v in the satellite equation, Eq. 6-5: GmE

(6.67 X 10-11 N •m2/kg2)(5.98 X 1024kg) --------- = 3070 m/s. (4.23 X 107m)

We get the same result if we use v = lirr/T. (c) The equation in part (b) for v shows v oc V l / r . So for r = rE + 6380 km + 200 km = 6580 km, we get = (3 0 7 0 m /s )^ g g ^

= 7780m /,

N TE The center of a satellite orbit is always at the center of the Earth; so it is not possible to have a satellite orbiting above a fixed point on the Earth at any latitude other than 0°. C NCEPTUAL EXAMPLE I P T l Catching a satellite. You are an astronaut in the space shuttle pursuing a satellite in need of repair. You find yourself in a circular orbit of the same radius as the satellite, but 30 km behind it. How will you catch up with it? RESP NSE We saw in Example - (or see Eq. 6 - 5) that the velocity is propor tional to 1 / V r . Thus you need to aim for a smaller orbit in order to increase your speed. Note that you cannot just increase your speed without changing your orbit. After passing the satellite, you will need to slow down and rise upward again. SECTI N 6 - 4

Satellites and "Weightlessness"

147

EXERCISE B Two s tellites orbit the E rth in circul r orbits of the s me r dius. O ne s tel lite is twice s m ssive s the other. Which of the following st tements is true bout the speeds of these s tellites? (a) The he vier s tellite m oves twice s f st s the lighter one. (b) The two s tellites h ve the s me speed, (c) The lighter s tellite m oves twice s f st s the he vier one. (d ) The he vier s tellite m oves four times s f st s the lighter one.

feJ

Weightlessness People and other objects in a satellite circling the Earth are said to experience apparent weightlessness. Let us first look at a simpler case, that of a falling elevator. In Fig. 6 - 13a, an elevator is at rest with a bag hanging from a spring scale. The scale reading indicates the downward force exerted on it by the bag. This force, exerted o n the scale, is equal and opposite to the force exerted b y the scale upward on the bag, and we call its magnitude w . Two forces act on the bag: the downward gravitational force and the upward force exerted by the scale equal to w . Because the bag is not accelerating (a = 0) when we apply 2 F = m a to the bag in Fig. 6 -13a we obtain

w -

where m g is the weight of the bag. Thus, w = m g , and since the scale indicates the force w exerted on it by the bag, it registers a force equal to the weight of the bag, as we expect. Now let the elevator have an acceleration, a. Applying Newton's second law, 2 F = m a , to the bag as seen from an inertial reference frame (the elevator itself is not an inertial frame) we have w — mg

Solving for

m a.

we have

[a is + upward]

m g + m a.

We have chosen the positive direction up. Thus, if the acceleration a is up, a is posi tive; and the scale, which measures w , will read more than m g . We call w the a p p a r e n t w e ig t of the bag, which in this case would be greater than its actual weight (m g ). If the elevator accelerates downward, a will be negative and w , the apparent weight, will be less than m g . The direction of the velocity v doesn't matter. Only the direction of the acceleration a (and its magnitude) influences the scale reading. Suppose, for example, the elevator's acceleration is \ g upward; then we find w (c)

| 5 - s (down)

FIGURE 6 - 1 3 ( ) A n object in n elev tor t rest exerts force on spring sc le equ l to its weight. (b) In n elev tor cceler ting upw rd t §g, the object 's pp rent weight is 1 \ times l rger th n its true weight. (c) In freely f lling elev tor, the object experiences "w eightlessness " : the sc le re ds zero.

148

CHAPTER 6

m g + m ( lg )

I m g.

That is, the scale reads i \ times the actual weight of the bag (Fig. 6-13b). The apparent weight of the bag is 1\ times its real weight. The same is true of the person: her apparent weight (equal to the normal force exerted on her by the elevator floor) is l \ times her real weight. We can say that she is experiencing l \ g's, just as astronauts experience so many g's at a rocket's launch. If, instead, the elevator's acceleration is a = - \ g (downward), then w = m g - \ m g = \ m g . That is, the scale reads half the actual weight. If the elevator is in f r e e f a l l (for example, if the cables break), then a = —g and w = m g - m g = 0. The scale reads zero. See Fig. 6 -13c. The bag appears weightless. If the person in the elevator accelerating at —g let go of a pencil, say, it would not fall to the floor. True, the pencil would be falling with acceleration g. But so would the floor of the elevator and the person. The pencil would hover right in front of the person. This phenomenon is called a p p a r e n t w e ig tle s s n e s s because in the reference frame of the person, objects don't fall or seem to have weight—yet gravity does not disappear. Gravity is still acting on each object, whose weight is still mg. The person and other objects seem weightless only because the elevator is accelerating in free fall, and there is no contact force on the person to make her feel the weight.

Gravitation and Newton's Synthesis

The "weightlessness" experienced by people in a satellite orbit close to the Earth (Fig. 6-14) is the same apparent weightlessness experienced in a freely falling elevator. It may seem strange, at first, to think of a satellite as freely falling. But a satellite is indeed falling toward the Earth, as was shown in Fig. 6-12. The force of gravity causes it to "fall" out of its natural straight-line path. The acceleration of the satellite must be the acceleration due to gravity at that point, since the only force acting on it is gravity. (We used this to obtain Eq. 6 -5.) Thus, although the force of gravity acts on objects within the satellite, the objects experience an apparent weightlessness because they, and the satellite, are accelerating together as in free fall. EXERCISE C Return to the Ch pter - Opening Q uestion, p ge 139, nd nswer it g in now. Try to expl in why you m y h ve nswered differently the first time.

Figure 6 -15 shows some examples of "free fall," or apparent weightlessness, experienced by people on Earth for brief moments. A completely different situation occurs if a spacecraft is out in space far from the Earth, the Moon, and other attracting bodies. The force of gravity due to the Earth and other heavenly bodies will then be quite small because of the distances involved, and persons in such a spacecraft would experience real weightlessness. EXERCISE D Could stron uts in bowling b ll (m = 7 kg)?

FIGURE 6 - 1 5

sp cecr ft f r out in sp ce e sily pl y c tch with

FIGURE 6 - 1 4 This stron ut is m oving outside the Intern tion l Sp ce St tion. H e must feel very free bec use he is experiencing pp rent weightlessness.

Experiencing "weightlessness " on E rth.

(a)

(c)

(b)

6 —5 Kepler's L ws nd Newton's

Synthesis More than a half century before Newton proposed his three laws of motion and his law of universal gravitation, the German astronomer Johannes Kepler (1571-1630) had worked out a detailed description of the motion of the planets about the Sun. Kepler's work resulted in part from the many years he spent exam ining data collected by Tycho Brahe (1546-1601) on the positions of the planets in their motion through the heavens. SECTI N 6 - 5

Kepler's Laws and Newton's Synthesis

149

FIGURE 6 -16 Kepler's first law. An ellipse is a closed curve such that the sum of the distances from any point P on the curve to two fixed points (called the foci, Fi and F2) remains constant. That is, the sum of the distances, FiP + F2P, is the same for all points on the curve. A circle is a special case of an ellipse in which the two foci coincide, at the center of the circle. The semimajor axis is s (that is, the long axis is 2s) and the semiminor axis is b, as shown. The eccentricity, e, is defined as the ratio of the distance from either focus to the center divided by the semimajor axis a. Thus es is the distance from the center to either focus, as shown. For a circle, e = 0. The Earth and most of the other planets have nearly circular orbits. For Earth e = 0.017.

Among Kepler's writings were three empirical findings that we now refer to as Kepler's l ws of pl net ry motion. These are summarized as follows, with addi tional explanation in Figs. 6 - 16 and 6 -17. FIGURE 6-17 Kepler's second law. The two shaded regions have equal areas. The planet moves from point 1 to point 2 in the same time as it takes to move from point 3 to point 4. Planets move fastest in that part of their orbit where they are closest to the Sun. Exaggerated scale.

Kepler's first law: The path of each planet about the Sun is an ellipse with the Sun at one focus (Fig. 6 - 16). Kepler's second law: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time (Fig. 6-17). Kepler's t ird law: The ratio of the squares of the periods of any two planets revolving about the Sun is equal to the ratio of the cubes of their semimajor axes. [The semimajor axis is half the long (major) axis of the orbit, as shown in Fig. 6 - 16, and represents the planet's mean distance from the Sun.1]That is, if 7\ and T2 represent the periods (the time needed for one revolution about the Sun) for any two planets, and sx and s2 represent their semimajor axes, then T\ T2 We can rewrite this as T\ meaning that s3/ T 2 should be the same for each planet. Present- day data are given in Table 6 - 2; see the last column.

TABLE 6 - 2 Planetary Data Applied to Kepler's Third Law

Mean Distance s*/T2 from Sun, s Period, J Planet (10 km) (Earth yr) ( » " $ Mercury 57.9 0.241 3.34 108.2 Venus 0.615 3.35 Earth 149.6 1.0 3.35 Mars 227.9 1.88 3.35 Jupiter 778.3 11.8 3.35 Saturn 1427 29.5 3.34 Uranus 2870 84.0 3.35 3.34 Neptune 4497 165 Pluto 5900 248 3.34

Kepler arrived at his laws through careful analysis of experimental data. Fifty years later, Newton was able to show that Kepler's laws could be derived m athe matically from the law of universal gravitation and the laws of motion. He also showed that for any reasonable form for the gravitational force law, only one that depends on the inverse square of the distance is fully consistent with Kepler's laws. He thus used Kepler's laws as evidence in favor of his law of universal gravitation, Eq. 6 - 1. We will derive Kepler's second law later, in hapter 11. Here we derive Kepler's third law, and we do it for the special case of a circular orbit, in which case the semimajor axis is the radius r of the circle. (Most planetary orbits are close to a circle.) First, we write Newton's second law of motion, 'ZF = ma. For F we use the law of universal gravitation (Eq. 6 - 1) for the force between the Sun and a planet of mass m 1, and for a the centripetal acceleration, v2/r. We assume the mass of the Sun Ms is much greater than the mass of its planets, so we ignore the effects of the planets on each other. Then 2F m 1Ms

ma m1

~W ~ tThe semim jor xis is equ l to the pl net's me n dist nce from the Sun in the sense th t it equ ls h lf the sum of the pl net's ne rest nd f rthest dist nces from the Sun (points Q nd R in Fig. 6 - 16). Most pl net ry orbits re close to circles, nd for circle the semim jor xis is the r dius of the circle.

CHAPTER 6

Gravitation and Newton's Synthesis

Here is the mass of a particular planet, rx its distance from the Sun, and vx its average speed in orbit; Ms is the mass of the Sun, since it is the gravitational attrac tion of the Sun that keeps each planet in its orbit. The period 7\ of the planet is the time required for one complete orbit, which is a distance equal to 27rrl5 the circumference of a circle. Thus 27771 Vl = " t T We substitute this formula for vx into the equation above: &

m 1Ms r2 rl

4t \ ml T2 11

We rearrange this to get T \ = _ 4^ _ A GMS

( 6 - 6)

We derived this for planet 1 (say, Mars). The same derivation would apply for a second planet (say, Saturn) orbiting the Sun, T\ = r\ GMs ' where T2 and r2 are the period and orbit radius, respectively, for the second planet. Since the right sides of the two previous equations are equal, we have T l/r\ = T \/r\ or, rearranging, T l\ 2

(6 - 7)

which is Kepler's third law. Equations - and 6 - 7 are valid also for elliptical orbits if we replace r with the semimajor axis 5. The derivations of Eqs. - and 6 - 7 (Kepler's third law) compared two planets revolving around the Sun. But they are general enough to be applied to other systems. For example, we could apply Eq. - to our Moon revolving around Earth (then Ms would be replaced by ME, the mass of the Earth). Or we could apply Eq. 6 - 7 to compare two moons revolving around Jupiter. But Kepler's third law, Eq. 6 - 7, applies only to objects orbiting the same attracting center. Do not use Eq. 6 - 7 to compare, say, the Moon's orbit around the Earth to the orbit of Mars around the Sun because they depend on different attracting centers. In the following Examples, we assume the orbits are circles, although it is not quite true in general.

/j\ CAUTI ON______ Com pare orbits o f objects only around t e sam e center

EXAMPLE 6 - 8 Where is Mars? Mars' period (its "year") was first noted by Kepler to be about 687 days (Earth -days), which is (687 d/365 d) = 1.88 yr (Earth years). Determine the mean distance of Mars from the Sun using the Earth as a reference. APPR ACH We are given the ratio of the periods of Mars and Earth. We can

find the distance from Mars to the Sun using Kepler's third law, given the Earth - Sun distance as 1.50 X 1011 m (Table 6 -2; also Table inside front cover). S LUTI N Let the distance of Mars from the Sun be rMS, and the Earth - Sun distance be rES = 1.50 X 10n m. From Kepler's third law (Eq. 6 -7):

s - d )! - ( ^ ) ! - So Mars is 1.52 times the Earth's distance from the Sun, or 2.28

1011m.

SECTI N 6 - 5

Kepler's Laws and Newton's Synthesis

151

PHYSICS

EXAMPLE 6 - 9 The Sun's mass determined. Determine the mass of the Sun given the Earth's distance from the Sun as rES = 1.5 X 1011 m. APPR ACH Equation - relates the mass of the Sun Ms to the period and distance of any planet. We use the Earth. S LUTI N The Earth's period is TE = ly r = (365^d)(24 h/d)(3600 s/h) = 3.16 X 107s. We solve Eq. - for M s : 4tt2(1.5 X 10n m)2 47 - 72rlES = = 2.0 X 1030kg. (6.67 X 10-11 N •m2/kg2)(3.16 X 107

APPLIED Determ ining t e Sun 's mass

EXERCISE E Suppose there w ere pl net in circul r orbit ex ctly h lfw y b etw een the orbits o f M rs nd Jupiter. W h t would its period be in E rth - ye rs? U se T ble 6 - 2 .

PHYSICS

APPLIED

Perturbations and discovery o f planets

PHYSICS

APPLIED Planets around ot er stars

Accurate measurements on the orbits of the planets indicated that they did not precisely follow Kepler's laws. For example, slight deviations from perfectly elliptical orbits were observed. Newton was aware that this was to be expected because any planet would be attracted gravitationally not only by the Sun but also (to a much lesser extent) by the other planets. Such deviations, or perturb tions, in the orbit of Saturn were a hint that helped Newton formulate the law of universal gravitation, that all objects attract gravitationally. Observation of other perturbations later led to the discovery of Neptune and Pluto. Deviations in the orbit of Uranus, for example, could not all be accounted for by perturbations due to the other known planets. areful calcu lation in the nineteenth century indicated that these deviations could be accounted for if another planet existed farther out in the solar system. The position of this planet was predicted from the deviations in the orbit of Uranus, and telescopes focused on that region of the sky quickly found it; the new planet was called Neptune. Similar but much smaller perturbations of Neptune's orbit led to the discovery of Pluto in 1930. Starting in the mid 1990s, planets revolving about distant stars (Fig. 6-18) were inferred from the regular "wobble" of each star due to the gravitational attraction of the revolving planet(s). Many such "extrasolar" planets are now known. The development by Newton of the law of universal gravitation and the three laws of motion was a major intellectual achievement: with these laws, he was able to describe the motion of objects on Earth and in the heavens. The motions of heavenly bodies and objects on Earth were seen to follow the same laws (not recognized previously). For this reason, and also because Newton integrated the results of earlier scientists into his system, we sometimes speak of Newton's synthesis. The laws formulated by Newton are referred to as c us l l ws. By c us lity we mean the idea that one occurrence can cause another. When a rock strikes a window, we infer that the rock causes the window to break. This idea of "cause and effect" relates to Newton's laws: the acceleration of an object was seen to be caused by the net force acting on it. As a result of Newton's theories the universe came to be viewed by many scientists and philosophers as a big machine whose parts move in a deterministic way. This deterministic view of the universe, however, had to be modified by scientists in the twentieth century ( hapter 38).

FIGURE 6 - 1 8 Our sol r system ( ) is comp red to recently discovered pl nets orbiting (b) the st r 47 Urs e M joris nd (c) the st r U psilon A ndrom ed e with t le st three pl nets, mj is the m ss of Jupiter. (Sizes not to sc le.)

Jupiter

(a)

Sun

Pl net

(b)

Urs e M joris

3mj

(c)

Upsilon Andromed e

> 0.7mj

152

CHAPTER 6

Gravitation and Newton's Synthesis

C 2mj

4mt

EXAMPLE 6 - 10 Lagrange Point. The mathematician Joseph-Louis Lagrange discovered five special points in the vicinity of the Earth's orbit about the Sun where a small satellite (mass ra) can orbit the Sun with the same period T as Earth's ( = 1 year). One of these "Lagrange Points," called LI, lies between the Earth (mass ME) and Sun (mass Ms), on the line connecting them (Fig. 6-19). That is, the Earth and the satellite are always separated by a distance d. If the Earth's orbital radius is RES, then the satellite's orbital radius is (RES — d). Determine d. APPR ACH We use Newton's law of universal gravitation and set it equal to the

mass times the centripetal acceleration. But how could an object with a smaller orbit than Earth's have the same period as Earth? Kepler's third law clearly tells us a smaller orbit around the Sun results in a smaller period. But that law depends on only the Sun's gravitational attraction. Our mass ra is pulled by both the Sun and the Earth. S LUTI N Because the satellite is assumed to have negligible mass in comparison to the masses of the Earth and Sun, to an excellent approximation the Earth's orbit will be determined solely by the Sun. Applying Newton's second law to the Earth gives M e (27 Res)2 gmem s w V2 n2 ---= M* j T le s ^ES LES or 477 GMs ES

Sun

\ \

' \

> -d -€) m»* ^

E rth

^ -*--71 lES/

FIGURE 6 - 19 Finding the position of the L gr nge Point LI for s tellite th t c n rem in long the revolving line betw een the Sun nd E rth, t dist nce d from the E rth. Thus m ss m t LI h s the s me period round the Sun s the E rth h s. (N ot to sc le.)

^ES Next we apply Newton's second law to the satellite ra (which has the same period T as Earth), including the pull of both Sun and Earth (see simplified form, Eq. (i)) GMS

GMe

4t72(R es - d)

(RES - d f

which we rewrite as GMs ( _ _ d _ Y 2 _ GMe _ 4tt2R ES R2es V * es/ d2 ~ T We now use the binomial expansion (1 + x )f x = d /R ES and assuming d « R ES, we have

^ES, 1 + nx, if x «

GMp d GMs lES 1+ 2 1 ^ES les ^ES Substituting GMS/R ES from Eq. (i) into Eq. (ii) we find GMS /

„ d \

(1+2^d

^ES Simplifying, we have

gme

GMS /

d \

Ris I

R es )

1. Setting

(ii)

FIGURE 6 - 20 A rtist 's rendition o f the Sol r nd H eliospheric Observ tory (SO H O ) s tellite in orbit.

GMs

^ES V ^ES We solve for d to find Me ES ,3 Ms Substituting in values we find d = 1.0 X IQ"2RES = 1.5 X 10 km. d =

N TE Since d /R ES = 10 2, we were justified in using the binomial expansion. N TE Placing a satellite at LI has two advantages: the satellite's view of the Sun

is never eclipsed by the Earth, and it is always close enough to Earth to transmit data easily. The LI point of the Earth - Sun system is currently home to the Solar and Heliospheric Observatory (SOHO) satellite, Fig. 6-20. SECTI N 6 - 5

Kepler's Laws and Newton's Synthesis

153

* 6-6 Gr vit tion l Field Most of the forces we meet in everyday life are contact forces: you push or pull on a lawn mower, a tennis racket exerts a force on a tennis ball when they make contact, or a ball exerts a force on a window when they make contact. But the gravitational force acts over a distance: there is a force even when the two objects are not in contact. The Earth, for example, exerts a force on a falling apple. It also exerts a force on the Moon, 384,000 km away. And the Sun exerts a gravitational force on the Earth. The idea of a force acting at a distance was a difficult one for early thinkers. Newton himself felt uneasy with this concept when he published his law of universal gravitation. Another point of view that helps with these conceptual difficulties is the concept of the field, developed in the nineteenth century by Michael Faraday (1791-1867) to aid understanding of electric and magnetic forces which also act over a distance. Only later was it applied to gravity. According to the field concept, a gr vit tion l field surrounds every object that has mass, and this field permeates all of space. A second object at a particular location near the first object experiences a force because of the gravitational field that exists there. Because the gravitational field at the location of the second mass is considered to act directly on this mass, we are a little closer to the idea of a contact force. To be quantitative, we can define the gr vit tion l field as the gravitational force per unit mass at any point in space. If we want to measure the gravitational field at any point, we place a small "test" mass m at that point and measure the force F exerted on it (making sure only gravitational forces are acting). Then the gravitational field, g, at that point is defined as F g = —

[gravitational field] (6 -8)

The units of g are N/kg. From Eq. - 8 we see that the gravitational field an object experiences has magnitude equal to the acceleration due to gravity at that point. (When we speak of acceleration, however, we use units m /s2, which is equivalent to N/kg, since 1 N = 1 kg - m /s2.) If the gravitational field is due to a single spherically symmetric (or small) object of mass M, such as when m is near the Earth's surface, then the gravitational field at a distance r from M has magnitude 1 mM M g — — G —T- — G —=-• m r2 r2 In vector notation we write . = _ GM ® r2 F'

T dUe to a [single mass M \

where r is a unit vector pointing radially outward from mass M, and the minus sign reminds us that the field points toward mass M (see Eqs. 6 - 1 ,6 - 2 , and 6 - 4). If several different bodies contribute significantly to the gravitational field, then we write the gravitational field g as the vector sum of all these contributions. In interplanetary space, for example, g at any point in space is the vector sum of terms due to the Earth, Sun, Moon, and other bodies that contribute. The gravitational field g at any point in space does not depend on the value of our test mass, m, placed at that point; g depends only on the masses (and locations) of the bodies that create the field there. 154

CHAPTER 6

Gravitation and Newton's Synthesis

6—7 Types o f Forces in N ture We have already discussed that Newton's law of universal gravitation, Eq. 6 - 1, describes how a particular type of force— gravity— depends on the distance between, and masses of, the objects involved. Newton's second law, 2F = m , on the other hand, tells how an object will accelerate due to any type of force. But what are the types of forces that occur in nature besides gravity? In the twentieth century, physicists came to recognize four different funda mental forces in nature: (1 ) the gravitational force; (2) the electromagnetic force (we shall see later that electric and magnetic forces are intimately related); (3) the strong nuclear force; and (4) the weak nuclear force. In this hapter, we discussed the gravitational force in detail. The nature of the electromagnetic force will be discussed in detail in hapters 21 to 31. The strong and weak nuclear forces operate at the level of the atomic nucleus; although they manifest themselves in such phenomena as radioactivity and nuclear energy ( hapters 41 to 43), they are much less obvious in our daily lives. Physicists have been working on theories that would unify these four forces—that is, to consider some or all of these forces as different manifestations of the same basic force. So far, the electromagnetic and weak nuclear forces have been theoretically united to form electroweak theory, in which the electromagnetic and weak forces are seen as two different manifestations of a single electroweak force. Attempts to further unify the forces, such as in grand unified t eories (GUT), are hot research topics today. But where do everyday forces fit into this scheme? Ordinary forces, other than gravity, such as pushes, pulls, and other contact forces like the normal force and friction, are today considered to be due to the electromagnetic force acting at the atomic level. For example, the force your fingers exert on a pencil is the result of electrical repulsion between the outer electrons of the atoms of your finger and those of the pencil.

*6-8 Principle of Equiv lence; Curv ture of Sp ce; Bl ck Holes We have dealt with two aspects of mass. In hapter 4, we defined mass as a measure of the inertia of a body. Newton's second law relates the force acting on a body to its acceleration and its inerti l m ss, as we call it. We might say that inertial mass represents a resistance to any force. In this hapter we have dealt with mass as a property related to the gravitational force— that is, mass as a quantity that determines the strength of the gravitational force between two bodies. This we call the gr vit tion l m ss. It is not obvious that the inertial mass of a body should be equal to its gravitational mass. The force of gravity might have depended on a different property of a body, just as the electrical force depends on a property called electric charge. Newton's and avendish's experiments indicated that the two types of mass are equal for a body, and modern experiments confirm it to a precision of about 1 part in 1012. Albert Einstein (1879-1955) called this equivalence between gravitational and inertial masses the principle of equiv lence, and he used it as a foundation for his general t eory o f relativity (c. 1916). The principle of equivalence can be stated in another way: there is no experiment observers can perform to distinguish if an acceleration arises because of a gravitational force or because their reference frame is accelerating. If you were far out in space and an apple fell to the floor of your spacecraft, you might assume a gravitational force was acting on the apple. But it would also be possible that the apple fell because your spacecraft accelerated upward (relative to an inertial system). The effects would be indistinguishable, according to the principle of equivalence, because the apple's inertial and gravitational masses—that determine how a body "reacts" to outside influences—are indistinguishable. *SECTI N 6 - 8

Principle of Equivalence; Curvature of Space; Black Holes

155

FIGURE 6-21 ( ) Light be str ight cross n elev tor th cceler ting, (b) The light be (ex gger ted) in n elev tor in n upw rd direction.

m goes t is not m bends cceler ting

(b)

FIGURE 6 - 22 ( ) Three st rs in the sky. (b) If the light from one of these st rs p sses very ne r the Sun, whose gr vity bends the light be m, the st r will ppe r higher th n it ctu lly is.

E rth

( )

App rent position o f st r

E rth

(b)

The principle of equivalence can be used to show that light ought to be deflected due to the gravitational force of a massive object. Let us consider a thought experiment in an elevator in free space where virtually no gravity acts. If a light beam enters a hole in the side of the elevator, the beam travels straight across the elevator and makes a spot on the opposite side if the elevator is at rest (Fig. 6-21a). If the elevator is accelerating upward as in Fig. 6 -21b, the light beam still travels straight as observed in the original reference frame at rest. In the upwardly accelerating elevator, however, the beam is observed to curve downward. Why? Because during the time the light travels from one side of the elevator to the other, the elevator is moving upward at ever-increasing speed. According to the equivalence principle, an upwardly accelerating reference frame is equivalent to a downward gravitational field. Hence, we can picture the curved light path in Fig. 6-21b as being the effect of a gravitational field. Thus we expect gravity to exert a force on a beam of light and to bend it out of a straight-line path! Einstein's general theory of relativity predicts that light should be affected by gravity. It was calculated that light from a distant star would be deflected by 1.75" of arc (tiny but detectable) as it passed near the Sun, as shown in Fig. 6-22. Such a deflection was measured and confirmed in 1919 during an eclipse of the Sun. (The eclipse reduced the brightness of the Sun so that the stars in line with its edge at that moment would be visible.) That a light beam can follow a curved path suggests that space itself is curved and that it is gravitational mass that causes the curvature. The curvature is greatest near very massive objects. To visualize this curvature of space, we might think of space as being like a thin rubber sheet; if a heavy weight is hung from it, it curves as shown in Fig. 6-23. The weight corresponds to a huge mass that causes space (space itself!) to curve. The extreme curvature of space-time shown in Fig. 6 -23 could be produced by a black hole, a star that becomes so dense and massive that gravity would be so strong that even light could not escape it. Light would be pulled back in by the force of gravity. Since no light could escape from such a massive star, we could not see it—it would be black. An object might pass by it and be deflected by its gravitational field, but if the object came too close it would be swallowed up, never to escape. Hence the name black holes. Experimentally there is good evidence for their existence. One likely possibility is a giant black hole at the center of our Galaxy and probably at the center of other galaxies.

FIGURE 6 -23

Rubber - sheet n logy for sp ce (technic lly sp ce - tim e) curved by m tter.

156

CHAPTER 6

Gravitation and Newton's Synthesis

Weight

Summ ry Newton's l w of univers l gr vit tion states that every particle in the universe attracts every other particle with a force propor tional to the product of their masses and inversely proportional to the square of the distance between them:

r2 The direction of this force is along the line joining the two parti cles, and is always attractive. It is this gravitational force that keeps the Moon revolving around the Earth and the planets revolving around the Sun. The total gravitational force on any object is the vector sum of the forces exerted by all other objects; frequently the effects of all but one or two objects can be ignored. Satellites revolving around the Earth are acted on by gravity, but "stay up" because of their high tangential speed.

Newton's three laws of motion, plus his law of universal gravitation, constituted a wide-ranging theory of the universe. With them, motion of objects on Earth and in the heavens could be accurately described. And they provided a theoretical base for Kepler's l ws of planetary motion. [*According to the field concept, a gr vit tion l field surrounds every object that has mass, and it permeates all of space. The gravitational field at any point in space is the vector sum of the fields due to all massive objects and can be defined as

where F is the force acting on a small "test" mass m placed at that point.] The four fundamental forces in nature are (1) the gravita tional force, (2) electromagnetic force, (3) strong nuclear force, and (4) weak nuclear force. The first two fundamental forces are responsible for nearly all "everyday" forces.

Questions 1. Does an apple exert a gravitational force on the Earth? If so, how large a force? onsider an apple (a) attached to a tree and (b) falling. 2. The Sun's gravitational pull on the Earth is much larger than the Moon's. Yet the M oon's is mainly responsible for the tides. Explain. [Hint: onsider the difference in gravita tional pull from one side of the Earth to the other.] 3. Will an object weigh more at the equator or at the poles? What two effects are at work? Do they oppose each other? 4. Why is more fuel required for a spacecraft to travel from the Earth to the Moon than it does to return from the Moon to the Earth? 5. The gravitational force on the Moon due to the Earth is only about half the force on the Moon due to the Sun (see Example 6-3). Why isn't the Moon pulled away from the Earth? . How did the scientists of Newton's era determine the distance from the Earth to the Moon, despite not knowing about spaceflight or the speed of light? [Hint: Think about why two eyes are useful for depth perception.] 7. If it were possible to drill a hole all the way through the Earth along a diameter, then it would be possible to drop a ball through the hole. When the ball was right at the center of the Earth, what would be the total gravitational force exerted on it by the Earth? 8. Why is it not possible to put a satellite in geosynchronous orbit above the North Pole? 9. Which pulls harder gravitationally, the Earth on the Moon, or the Moon on the Earth? Which accelerates more? 10. Would it require less speed to launch a satellite (a) toward the east or (b) toward the west? onsider the E arth's rota tion direction. 11. An antenna loosens and becomes detached from a satellite in a circular orbit around the Earth. Describe the antenna's motion subsequently. If it will land on the Earth, describe where; if not, describe how it could be made to land on the Earth. 12. Describe how careful measurements of the variation in g in the vicinity of an ore deposit might be used to estimate the amount of ore present.

13. The Sun is below us at midnight, nearly in line with the Earth's center. Are we then heavier at midnight, due to the Sun's gravitational force on us, than we are at noon? Explain. 14. When will your apparent weight be the greatest, as measured by a scale in a moving elevator: when the elevator (a) accelerates downward, (b) accelerates upward, (c) is in free fall, or (d) moves upward at constant speed? In which case would your apparent weight be the least? When would it be the same as when you are on the ground? 15. If the Earth's mass were double what it actually is, in what ways would the Moon's orbit be different? 16. The source of the Mississippi River is closer to the center of the Earth than is its outlet in Louisiana (since the Earth is fatter at the equator than at the poles). Explain how the Mississippi can flow "uphill." 17. People sometimes ask, "What keeps a satellite up in its orbit around the Earth?" How would you respond? 18. Explain how a runner experiences "free fall" or "apparent weightlessness" between steps. 19. If you were in a satellite orbiting the Earth, how might you cope with walking, drinking, or putting a pair of scissors on a table? 20. Is the centripetal acceleration of Mars in its orbit around the Sun larger or smaller than the centripetal acceleration of the Earth? 21. The mass of the planet Pluto was not known until it was discovered to have a moon. Explain how this enabled an estimate of Pluto's mass. 22. The Earth moves faster in its orbit around the Sun in January than in July. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a factor in producing the seasons — the main factor is the tilt of the E arth's axis relative to the plane of its orbit.] 23. Kepler's laws tell us that a planet moves faster when it is closer to the Sun than when it is farther from the Sun. What causes this change in speed of the planet? *24. Does your body directly sense a gravitational field? ( ompare to what you would feel in free fall.) *25. Discuss the conceptual differences between g as accelera tion due to gravity and g as gravitational field.

Questions

1 57

Probl ms 6 - 1 to 6 - 3 Law of Universal Gravitation 1. (I) alculate the force of E arth's gravity on a spacecraft 2.00 Earth radii above the Earth's surface if its mass is 1480 kg. 2. (I) alculate the acceleration due to gravity on the Moon. The Moon's radius is 1.74 X 106m and its mass is 7.35 X 1022kg. 3. (I) A hypothetical planet has a radius 2.3 times that of Earth, but has the same mass. What is the acceleration due to gravity near its surface? 4. (I) A hypothetical planet has a mass 1.80 times that of Earth, but the same radius. What is g near its surface? 5. (I) If you doubled the mass and tripled the radius of a planet, by what factor would g at its surface change? . (II) alculate the effective value of g, the acceleration of gravity, at (a) 6400 m, and (b) 6400 km, above the Earth's surface. 7. (II) You are explaining to friends why astronauts feel weightless orbiting in the space shuttle, and they respond that they thought gravity was just a lot weaker up there. onvince them and yourself that it isn't so by calculating how much weaker gravity is 300 km above the Earth's surface. 8. (II) Every few hundred years most of the planets line up on the same side of the Sun. alculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line, Fig. 6 -24. The masses are Mv = 0.815 ME, Mj = 318 M e , MSat = 95.1 ME, and the mean distances of the four planets from the Sun are 108, 150, 778, and 1430 million km. What fraction of the Sun's force on the Earth is this?

o FIGURE 6 - 2 4

13. (II) Suppose the mass of the Earth were doubled, but it kept the same density and spherical shape. How would the weight of objects at the E arth's surface change? 14. (II) Given that the acceleration of gravity at the surface of Mars is 0.38 of what it is on Earth, and that Mars' radius is 3400 km, determine the mass of Mars. 15. (II) A t what distance from the Earth will a spacecraft trav eling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull with equal and opposite forces? 16. (II) Determine the mass of the Sun using the known value for the period of the Earth and its distance from the Sun. [Hint: The force on the Earth due to the Sun is related to the centripetal acceleration of the Earth.] ompare your answer to that obtained using Kepler's laws, Example 6 - 9. 17. (II) Two identical point masses, each of mass M, always remain separated by a distance of 2R. A third mass m is then placed a distance x along the perpendicular bisector of the original two masses, as shown in Fig. 6 -26. Show that the gravitational force on the third mass is directed inward along jQ M the perpendicular bisector and R has a magnitude of m IG M m x f I-' o F = I (x2 + R 2)2 R

FIGURE 6 - 2 6 ± (j)M Problem 17. 18. (II) A mass M is ring shaped with radius r. A small mass m is placed at a distance x along the ring's axis as shown in Fig. 6-27. Show that the gravitational force on the mass m due to the ring is directed inward along the axis and has magnitude F =

Problem 8 (not to scale).

GMmx {x2 + r2)

9. (II) Four 8.5-kg spheres are located at the corners of a square of side 0.80 m. alculate the magnitude and direction of the gravitational force exerted on one sphere by the other three. 10. (II) Two objects attract each other gravitationally with a force of 2.5 X 10-10N when they are 0.25 m apart. Their total mass is 4.00 kg. Find their individual masses. 11. (II) Four masses are arranged as shown in Fig. 6 - 25. Determine the x and y components of the gravitational force on the mass at the origin (m). Write ^ the force in vector notation (i, j). 4m 3m x0

[Hint: Think of the ring as made up of many small point masses dM ; sum over the forces due to each dM, and use symmetry.] FIGURE 6 - 27 Problem 18. 19. (Ill) (a) Use the binomial expansion n (n r~-----X) x L2 +. (1 + x )n = 1 + nx + ----to show that the value of g is altered by approximately Ar Ag « - 2 g rE

^0 FIGURE 6 - 2 5 Problem 11.

---------------------------- 1k-------2m *0

12. (II) Estimate the acceleration due to gravity at the surface of Europa (one of the moons of Jupiter) given that its mass is 4.9 X 1022kg and making the assumption that its density is the same as E arth's.

158

HAPTER 6

Gravitation and Newton's Synthesis

at a height Ar above the E arth's surface, where rE is the radius of the Earth, as long as Ar « rE. (b) What is the meaning of the minus sign in this relation? (c) Use this result to compute the effective value of g at 125 km above the E arth's surface. ompare to a direct use of Eq. 6 - 1. 20. (Ill) The center of a 1.00 km diameter spherical pocket of oil is 1.00 km beneath the Earth's surface. Estimate by what percentage g directly above the pocket of oil would differ from the expected value of g for a uniform Earth? Assume the density of oil is 8.0 X 102kg/m 3.

21. (Ill) Determine the magnitude and direction of the effec tive value of g at a latitude of 45° on the Earth. Assume the Earth is a rotating sphere. *22. (Ill) It can be shown (Appendix D) that for a uniform sphere the force of gravity at a point inside the sphere depends only on the mass closer to the center than that point. The net force of gravity due to points outside the radius of the point cancels. How far would you have to drill into the Earth, to reach a point where your weight is reduced by 5.0%? Approximate the Earth as a uniform sphere.

6 - 4 Satellites and Weightlessness 23. (I) The space shuttle releases a satellite into a circular orbit 680 km above the Earth. How fast must the shuttle be moving (relative to E arth's center) when the release occurs? 24. (I) alculate the speed of a satellite moving in a stable circular orbit about the Earth at a height of 5800 km. 25. (II) You know your mass is 65 kg, but when you stand on a bathroom scale in an elevator, it says your mass is 76 kg. What is the acceleration of the elevator, and in which direction? 26. (II) A 13.0-kg monkey hangs from a cord suspended from the ceiling of an elevator. The cord can withstand a tension of 185 N and breaks as the elevator accelerates. What was the elevator's minimum acceleration (magnitude and direction)? 27. (II) alculate the period of a satellite orbiting the Moon, 120 km above the Moon's surface. Ignore effects of the Earth. The radius of the Moon is 1740 km. 28. (II) Two satellites orbit Earth at altitudes of 5000 km and 15,000 km. Which satellite is faster, and by what factor? 29. (II) What will a spring scale read for the weight of a 53-kg woman in an elevator that moves (a) upward with constant speed 5.0 m/s, (b) downward with constant speed 5.0 m/s, (c) upward with acceleration 0.33 g, (d) downward with acceleration 0.33 g, and (e) in free fall? 30. (II) Determine the time it takes for a satellite to orbit the Earth in a circular "near-Earth" orbit. A "near-Earth" orbit is at a height above the surface of the Earth that is very small compared to the radius of the Earth. [Hint. You may take the acceleration due to gravity as essentially the same as that on the surface.] Does your result depend on the mass of the satellite? 31. (II) What is the apparent weight of a 75-kg astronaut 2500 km from the center of the E arth's Moon in a space vehicle (a) moving at constant velocity and (b) accelerating toward the Moon at 2.3 m /s2? State "direction" in each case. 32. (II) A Ferris wheel 22.0 m in diameter rotates once every 12.5 s (see Fig. 5 -19). What is the ratio of a person's apparent weight to her real weight (a) at the top, and (b) at the bottom? 33. (II) Two equal-mass stars maintain a constant distance apart of 8.0 X 1011 m and rotate about a point midway between them at a rate of one revolution every 12.6 yr. (a) Why don't the two stars crash into one another due to the gravitational force between them? (b) What must be the mass of each star? 34. (Ill) (a) Show that if a satellite orbits very near the surface of a planet with period T, the density (= mass per unit volume) of the planet is p = m /V = 3 tt/G T 2. (b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of 85 min. Approximate the Earth as a uniform sphere. 35. (Ill) Three bodies of identical mass M form the vertices of an equilateral triangle of side I and rotate in circular orbits about the center of the triangle. They are held in place by their mutual gravitation. What is the speed of each?

36. (Ill) An inclined plane, fixed to the inside of an elevator, makes a 32° angle with the floor. A mass m slides on the plane without friction. What is its acceleration relative to the plane if the elevator (a) accelerates upward at 0.50 g, (b) accelerates downward at 0.50 g, (c) falls freely, and (0 is the integral of (F cos 0 d£) from point a to point b. The symbol for the integral, J, is an elongated S to indicate an infinite sum; and A£ has been replaced by d£, meaning an infinitesimal distance. [We also discussed this in the optional Section 2-9.] In this limit as A£ approaches zero, the total area of the rectangles (Fig. 7 - 9a) approaches the area between the (F cos 0) curve and the £ axis from a to b as shown shaded in Fig. 7 - 9b. That is, t e work done by a variable force in moving an object between two points is equal to t e area under t e (F cos 0) versus (£) curve between t ose two points. In the limit as A£ approaches zero, the infinitesimal distance d£ equals the magnitude of the infinitesimal displacement vector di. The direction of the vector di is along the tangent to the path at that point, so 0 is the angle between F and di at any point. Thus we can rewrite Eq. 7 - 6, using dot-product notation: ■d i

FIGURE 7 - 9 Work done by force F is ( ) pproxim tely equ l to the sum of the re s of the rect ngles, (b) ex ctly equ l to the re under the curve of F cos 0 vs. L

( )

A£j A£2 A£3 A£4 A£5 A£6 A£7b Dist nce, £

(7 - 7)

This is a general definition o f work. In this equation, a and b represent two points in space, (xa, ya, za) and (xb, yh, zb). The integral in Eq. 7 - 7 is called a line integral since it is the integral of F cos 0 along the line that represents the path of the object. (Equation 7 - 1 for a constant force is a special case of Eq. 7 - 7.) SECTI N 7 - 3

169

In rectangular coordinates, any force can be written F = Fx i + Fyj + Fzk and the displacement di is d t = dxi + dyj + dzk. Then the work done can be written r*b •*b

ryb m

rzb rzb

W = \ Fx dx + \ Fydy + \ Fz dz.

Jxa

Jv ly a

Jz«

To actually use Eq. 7 - 6 or 7 - 7 to calculate the work, there are several options: (1) If F cos 0 is known as a function of position, a graph like that of Fig. 7 - 9b can be made and the area determined graphically. (2) Another possibility is to use numerical integration (numerical summing), perhaps with the aid of a computer or calculator. (3) A third possibility is to use the analytical methods of integral calculus, when it is doable. To do so, we must be able to write F as a function of position, F(x, y, z), and we must know the path. Let's look at some specific examples.

Work Done by ( ) Unstretched

j-x -l

Spring Force

Let us determine the work needed to stretch or compress a coiled spring, such as that shown in Fig. 7 -10. For a person to hold a spring either stretched or compressed an amount x from its normal (relaxed) length requires a force FPthat is directly proportional to x. That is, FP = kx, where A: is a constant, called the spring constant (or spring stiffness constant), and is a measure of the stiffness of the particular spring. The spring itself exerts a force in the opposite direction (Fig. 7 - 10b or c): Fs = - k x .

(c) Compressed FIGURE 7 - 1 0 ( ) Spring in norm l (unstretched) position, (b) Spring is stretched by person exerting force FP to the right (positive direction). The spring pulls b ck with force Fs where Fs = —kx. (c) Person com presses the spring (x < 0), nd the spring pushes b ck with force Fs = —k x where Fs > 0 bec use x < 0.

FIGURE 7 - 1 1 Work done to stretch spring dist nce x equ ls the tri ngul r re under the curve F = kx . The re o f tri ngle is | X b se X ltitude, so W = \ { x ) { k x ) = \ k x 2. F

(7 -8)

This force is sometimes called a "restoring force" because the spring exerts its force in the direction opposite the displacement (hence the minus sign), and thus acts to return the spring to its normal length. Equation 7 - 8 is known as the spring equation or Hooke's law, and is accurate for springs as long as x is not too great (see Section 12 -4) and no permanent deformation occurs. Let us calculate the work a person does to stretch (or compress) a spring from its normal (unstretched) length, x a = 0, to an extra length, xb = x. We assume the stretching is done slowly, so that the acceleration is essentially zero. The force FP is exerted parallel to the axis of the spring, along the x axis, so FP and d l are parallel. Hence, since d l = dxi in this case, the work done by the person isf I.xb=* [x fx [FP(jt)i] • [dxi] = Fp(x)dx = kx dx = \k x 2 = \k x 2. J*a=o J° (As is frequently done, we have used x to represent both the variable of integra tion, and the particular value of x at the end of the interval xa = 0 to xb = x.) Thus we see that the work needed is proportional to the square of the distance stretched (or compressed), x. This same result can be obtained by computing the area under the graph of F vs. x (with cos 6 = 1 in this case) as shown in Fig. 7-11. Since the area is a triangle of altitude kx and base x, the work a person does to stretch or compress a spring an amount x is W = \(x ){k x ) = \ k x 2, which is the same result as before. Because W oc x2, it takes the same amount of work to stretch a spring or compress it the same amount x. tSee the T ble of Integr ls, Appendix B.

CHAPTER 7

Work and Energy

EXAMPLE 7 - 5 Work done on a spring, (a) A person pulls on the spring in Fig. 7 - 10, stretching it 3.0 cm, which requires a maximum force of 75 N. How much work does the person do? (b) If, instead, the person compresses the spring 3.0 cm, how much work does the person do? APPR ACH The force F = kx holds at each point, including xmax. Hence Fmax

occurs at x = %max. S LUTI N (a) First we need to calculate the spring constant k : * = ^

= -Z 5 IL

*m x

0 .0 3 0

= 2.5 X 103 N /m .

Then the work done by the person on the spring is W = \kx^ax = |(2.5 X 103N/m)(0.030m)2 = 1.1 J. (b) The force that the person exerts is still FP = kx, though now both x and FP are negative (x is positive to the right). The work done is r x = - 0 .0 3 0 m

rJC =- 0.030 m

—0.030 m

FP(x)dx = \ k x d x = \k x 0 Jx =0 Jo = \ (2.5 X 103N /m )( - 0.030 m)2 = 1.1 J,

which is the same as for stretching it. N TE We cannot use W = Fd (Eq. 7 - 1) for a spring because the force is not constant.

A More Complex Force L w —Robot Arm Force as function of x. A robot arm that controls the position of a video camera (Fig. 7 -12) in an automated surveillance system is manipulated by a motor that exerts a force on the arm. The force is given by

lx 2 F(x) = F0[ 1 + 7 3 6 xl where F0 = 2.0 N, x0 = 0.0070 m, and x is the position of the end of the arm. If the arm moves from x 1 = 0.010 m to x2 = 0.050 m, how much work did the motor do? APPR ACH The force applied by the motor is not a linear function of x. We can

determine the integral fF (x )d x , or the area under the F(x) curve (shown in Fig. 7-13). S LUTI N We integrate to find the work done by the motor: Wm = Fr

r Jxt

FIGURE 7 -12

R obot rm positions

video c mer .

FIGURE 7 -13

Ex m ple 7 - 6 .

20.0 T F (N)

dx = Fn [ dx + —^\ f : jXi 6x0JXl

We put in the values given and obtain 2.0 N (0.050 m - 0.010 m) +

(0.050 m)3 - (0.010 m)' (3) (6) (0.0070 m )A

= 0.36 J.

\x(m) 0.01 0.02 0.03 0.04 0.05

SECTI N 7 - 3

Work Done by a Varying Force

171

7—4 Kinetic Energy nd the Work-Energy Principle Energy is one of the most important concepts in science. Yet we cannot give a simple general definition of energy in only a few words. Nonetheless, each specific type of energy can be defined fairly simply. In this hapter we define translational kinetic energy; in the next hapter, we take up potential energy. In later hapters we will examine other types of energy, such as that related to heat ( hapters 19 and 20). The crucial aspect of all the types of energy is that the sum of all types, the total energy, is the same after any process as it was before: that is, energy is a conserved quantity. For the purposes of this hapter, we can define energy in the traditional way as "the ability to do work." This simple definition is not very precise, nor is it really valid for all types of energy. * It works, however, for mechanical energy which we discuss in this hapter and the next. We now define and discuss one of the basic types of energy, kinetic energy. A moving object can do work on another object it strikes. A flying cannonball does work on a brick wall it knocks down; a moving hammer does work on a nail it drives into wood. In either case, a moving object exerts a force on a second object which undergoes a displacement. An object in motion has the ability to do work and thus can be said to have energy. The energy of motion is called kinetic energy, from the Greek word kinetikos, meaning "motion." To obtain a quantitative definition for kinetic energy, let us consider a simple rigid object of mass ra (treated as a particle) that is moving in a straight line with an initial speed . To accelerate it uniformly to a speed v2, a constant net force -^net is exerted on it parallel to its motion over a displacement d, Fig. 7 -14.

FIGURE 7 - 1 4 A const nt net force Fnet cceler tes c r from speed Vi to speed v 2 over displ cem ent d. The net work done is Wnet = Fnet d.

net

Then the net work done on the object is Wnet = Fnetd- We apply Newton's second law, F,net = ma, and use Eq. 2 -12c (v2 = v\ + lad), which we rewrite as 2

V2 -

U = 2d where vx is the initial speed and v2 the final speed. Substituting this into f i i e t = ma, we determine the work done: (v j-v l Wnet = Fnetd = mad = m l — — — jd = m l ---- ----or Wnet

(7-9)

= \ m v \ - \m v\.

We define the quantity \m v 2 to be the tr nsl tion l kinetic energy, K, of the object: Kinetic energy (defined)

K = \m v .

(7-10)

(We call this "translational" kinetic energy to distinguish it from rotational kinetic energy, which we discuss in hapter 10.) Equation 7 - 9, derived here for one dimensional motion with a constant force, is valid in general for translational motion of an object in three dimensions and even if the force varies, as we will show at the end of this Section. tEnergy ssoci ted with he t is often not v il ble to do work, s we will discuss in Ch pter 20.

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We can rewrite Eq. 7 - 9 as: w„et = K2 - K, or W„et = A K = \m v \ - \m v \.

(7 -11)

W O R K - E N E R G Y P R IN C IP L E

Equation 7 -11 (or Eq. 7 - 9) is a useful result known as the work-energy principle. It can be stated in words: The net work done on n object is equ l to the ch nge in the object's kinetic energy. Notice that we made use of Newton's second law, -^net = ma, where -^net is the net force— the sum of all forces acting on the object. Thus, the work-energy principle is valid only if W is the net work done on the object— that is, the work done by all forces acting on the object. The work-energy principle is a very useful reformulation of Newton's laws. It tells us that if (positive) net work W is done on an object, the object's kinetic energy increases by an amount W. The principle also holds true for the reverse situation: if the net work W done on an object is negative, the object's kinetic energy decreases by an amount W. That is, a net force exerted on an object opposite to the object's direction of motion decreases its speed and its kinetic energy. An example is a moving hammer (Fig. 7 - 15) striking a nail. The net force on the hammer ( - F in Fig. 7 - 15, where F is assumed constant for simplicity) acts toward the left, whereas the displacement d of the hammer is toward the right. So the net work done on the hammer, Wh = (F)(d)(cos 180°) = - F d , is negative and the hammer's kinetic energy decreases (usually to zero). Figure 7 - 15 also illustrates how energy can be considered the ability to do work. The hammer, as it slows down, does positive work on the nail: Wn = (+ F )(+ d)(cos0°) = Fd and is positive. The decrease in kinetic energy of the hammer (= Fd by Eq. 7 -11) is equal to the work the hammer can do on another object, the nail in this case. The translational kinetic energy (= \m v 2) is directly proportional to the mass of the object, and it is also proportional to the square of the speed. Thus, if the mass is doubled, the kinetic energy is doubled. But if the speed is doubled, the object has four times as much kinetic energy and is therefore capable of doing four times as much work. Because of the direct connection between work and kinetic energy, energy is measured in the same units as work: joules in SI units. [The energy unit is ergs in the cgs, and foot-pounds in the British system.] Like work, kinetic energy is a scalar quantity. The kinetic energy of a group of objects is the sum of the kinetic energies of the individual objects. The work-energy principle can be applied to a particle, and also to an object that can be approximated as a particle, such as an object that is rigid or whose internal motions are insignificant. It is very useful in simple situations, as we will see in the Examples below. The work-energy principle is not as powerful and encompassing as the law of conservation of energy which we treat in the next hapter, and should not itself be considered a statement of energy conservation.

W O R K - E N E R G Y P R IN C IP L E

/?\ CAUTION____________ W ork - energy valid only f o r net w o rk

(on h mmer) (on n il)

FIGURE 7 - 1 5 A moving h mmer strikes n il nd com es to rest. The h mmer exerts force F on the n il; the n il exerts force —F on the h mmer (N ew ton 's third l w). The work done on the n il by the h mmer is positive (Wn = F d > 0). The work done on the h mmer by the n il is neg tive (W^ = —F d).

EXAMPLE 7 - 7 Kinetic energy and work done on a baseball. A 145-g baseball is thrown so that it acquires a speed of 25 m/s. (a) What is its kinetic energy? (b) What was the net work done on the ball to make it reach this speed, if it started from rest? APPR ACH We use K = \m v 2, and the work-energy principle, Eq. 7 -11. S LUTI N (a) The kinetic energy of the ball after the throw is K = \m v 2 = |(0.145 kg) (25 m /s)2 = 45 J. (b) Since the initial kinetic energy was zero, the net work done is just equal to the final kinetic energy, 45 J. SECTI N 7 - 4

Kinetic Energy and the Work-Energy Principle

173

EXAMPLE 7 - 8 V] ^ 20 m/s

ESTIMATE-! Work on a car, to increase its kinetic energy.

How much net work is required to accelerate a 1000-kg car from 20 m/s to 30 m/s (Fig. 7 -16)?

t?j = 30 m/s

APPR ACH A car is a complex system. The engine turns the wheels and tires FIGURE 7 - 16

which push against the ground, and the ground pushes back (see Example 4 - 4). We aren't interested right now in those complications. Instead, we can get a useful result using the work-energy principle, but only if we model the car as a particle or simple rigid object. S LUTI N The net work needed is equal to the increase in kinetic energy:

Ex m ple 7 - 8 .

W = K2 ~ K\ = \m v \ ~ \rnv\ = \ (1000 kg) (30 m /s)2 - \ (1000 kg) (20 m /s)2 = 2.5 X 105J. EXERCISE C {a) M ke guess: will the work needed to cceler te the c r in Ex m ple 7 - 8 from rest to 20 m /s be m ore th n, less th n, or equ l to the work lre dy c lcul ted to cceler te it from 20 m /s to 30 m /s? (b ) M ke the c lcul tion.

FIGURE 7 -17

Ex m ple 7 - 9 .

d {d = 20m)

(u)

1^ = 0

p, a 120 km/h

dW = ?)

(b)

C NCEPTUAL EXAMPLE 7 - 9 | Work to stop a car. A car traveling 60 km/h can brake to a stop within a distance d of 20 m (Fig. 7 -17a). If the car is going twice as fast, 120 km/h, what is its stopping distance (Fig. 7 -17b)? Assume the maximum braking force is approximately independent of speed. RESP NSE Again we model the car as if it were a particle. Because the net stopping force F is approximately constant, the work needed to stop the car, Fd, is proportional to the distance traveled. We apply the work-energy principle, noting that F and d are in opposite directions and that the final speed of the car is zero:

Wnet = Fd cos 180° = - F d . Then —Fd = AK = \m v \ — \m v\ =

PHYSICS

APPLIED

Car 's stoppin g distance oc initial speed squared

- \m v \.

Thus, since the force and mass are constant, we see that the stopping distance, d, increases with the square of the speed: d oc v2. If the car's initial speed is doubled, the stopping distance is (2)2 = 4 times as great, or 80 m. | EXERCISE D C n kinetic energy ever be neg tive? EXERCISE E (a) If the kinetic energy o f n rrow is doubled, by wh t f ctor h s its speed incre sed? (b) If its speed is doubled, by wh t f ctor does its kinetic energy incre se?

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Work and Energy

EXAMPLE 7 - 10 A compressed spring. A horizontal spring has spring constant k = 360 N/m. (a) How much work is required to compress it from its uncompressed length (x = 0) to x = 11.0 cm? (b) If a 1.85-kg block is placed against the spring and the spring is released, what will be the speed of the block when it separates from the spring at x = 0? Ignore friction, (c) Repeat part (b) but assume that the block is moving on a table as in Fig. 7 - 18 and that some kind of constant drag force FD = 7.0 N is acting to slow it down, such as friction (or perhaps your finger).

FIGURE 7-18

Example 7-10.

APPR ACH We use our result from Section 7 - 3 that the net work, W, needed to

stretch or compress a spring by a distance x is W = \ k x 2. In (b) and (c) we use the work-energy principle. S LUTI N (a) The work needed to compress the spring a distance x = 0.110 m is W = \ (360 N/m)(0.110 m)2 = 2.18 J, where we have converted all units to SI. (b) In returning to its uncompressed length, the spring does 2.18 J of work on the block (same calculation as in part (a), only in reverse). According to the work-energy principle, the block acquires kinetic energy of 2.18 J. Since K = \m v 2, the block's speed must be v = 2(2.18 J) ———— = 1.54 m/s. 1.85 kg ' (c) There are two forces on the block: that exerted by the spring and that exerted by the drag force, FD. Work done by a force such as friction is complicated. For one thing, heat (or, rather, "thermal energy") is produced —try rubbing your hands together. Nonetheless, the product FD • d for the drag force, even when it is friction, can be used in the work-energy principle to give correct results for a particle-like object. The spring does 2.18 J of work on the block. The work done by the friction or drag force on the block, in the negative x direction, is WD = - F d x = - (7.0 N) (0.110 m) = - 0.77 J. This work is negative because the drag force acts in the direction opposite to the displacement x. The net work done on the block is Wnet = 2.18 J - 0.77 J = 1.41 J. From the work-energy principle, Eq. 7-11 (with v2 = v and vx = 0), we have v =

m 2(1.41 J) T iJ k T

= 1'23m /s

for the block's speed at the moment it separates from the spring (x = 0). SECTI N 7 - 4

Kinetic Energy and the Work-Energy Principle

175

Gener l Deriv tion of the Work-Energy Principle We derived the work-energy principle, Eq. 7 -11, for motion in one dimension with a constant force. It is valid even if the force is variable and the motion is in two or three dimensions, as we now show. Suppose the net force Fnet on a particle varies in both magnitude and direction, and the path of the particle is a curve as in Fig. 7 -8. The net force may be considered to be a function of I, the distance along the curve. The net work done is (Eq. 7 -6): Wnet = | Fnet • d l = j"Fnet cos 6 di = where F\\represents the component of the net force parallel to the curve at any point. By Newton's second law,

A p rticle cted on by v ri ble force F, m oves long the p th shown from point to point b.

Fu =

dv m aw =

m ~^'

where fly, the component of a parallel to the curve at any point, is equal to the rate of change of speed, dvI dt. We can think of v as a function of i, and using the chain rule for derivatives, we have dv dt

dv di ~ didt

dv v, ~di

since di/dt is the speed v. Thus (letting 1 and 2 refer to the initial and final quantities, respectively): Wnet =

= [ mi

de =

2mv — d V dll AO

= I

mv dv,

which integrates to ^net = \m v 2 - \m v \ = AK. This is again the work-energy principle, which we have now derived for motion in three dimensions with a variable net force, using the definitions of work and kinetic energy plus Newton's second law. Notice in this derivation that only the component of ®net parallel to the motion, i*j|, contributes to the work. Indeed, a force (or component of a force) acting perpendicular to the velocity vector does no work. Such a force changes only the direction of the velocity. It does not affect the magnitude of the velocity. One example of this is uniform circular motion in which an object moving with constant speed in a circle has a ("centripetal") force acting on it toward the center of the circle. This force does no work on the object, because (as we saw in Example 7 -3) it is always perpendicular to the object's displacement di.

Summary Work is done on n object by force when the object m oves through dist nce, d. The work W done by const nt force F on n object whose position ch nges by displ cement d is given by W

cosd

F d ,

(7 -1 , 7 -3 )

where 6 is the ngle betw een F nd d. The l st expression is c lled the sc l r product of F nd d. In gener l, the sc l r product of ny two vectors A nd B is defined s A •B =

AB cosO

where e is the ngle betw een A n tes w e c n lso write A • B = A XBX The work W done by

176

CHAPTER 7

(7 - 2 ) nd B. In rect ngul r coordi

A y By

A ZB Z.

v ri ble force F on

Work and Energy

(7 - 4 )

n object th t

m oves from point W

to point b is =

F • dl =

jF c o stfd f,

(7 - 7 )

where d l represents n infinitesim l displ cem ent long the p th of the object nd e is the ngle betw een d l nd F t e ch point of the object 's p th. The tr nsl tion l kinetic energy, K, o f n object o f m ss m m oving with speed v is defined to be

K = \mv2.

(7 - 1 0 )

The work - energy principle st tes th t the net work done on n object by the net result nt force is equ l to the ch nge in kinetic energy o f the object: Wnet =

AK = \ m v \ - \m v \ .

(7 - 1 1 )

Questions 1. In what ways is the word "work" as used in everyday language the same as defined in physics? In what ways is it different? Give examples of both. 2. A woman swimming upstream is not moving with respect to the shore. Is she doing any work? If she stops swimming and merely floats, is work done on her? 3. an a centripetal force ever do work on an object? Explain. 4. Why is it tiring to push hard against a solid wall even though you are doing no work? 5. Does the scalar product of two vectors depend on the choice of coordinate system? 6. an a dot product ever be negative? If yes, under what conditions? 7. If A • C = B • C, is it necessarily true that A = B? 8. Does the dot product of two vectors have direction as well as magnitude? 9. an the normal force on an object ever do work? Explain. 10. You have two springs that are identical except that spring 1 is stiffer than spring 2 (k\ > /c2). On which spring is more work done: (a) if they are stretched using the same force; (ib) if they are stretched the same distance? 11. If the speed of a particle triples, by what factor does its kinetic energy increase?

12. In Example 7 - 10, it was stated that the block separates from the compressed spring when the spring reached its equilibrium length (x = 0). Explain why separation doesn't take place before (or after) this point. 13. Two bullets are fired at the same time with the same kinetic energy. If one bullet has twice the mass of the other, which has the greater speed and by what factor? Which can do the most work? 14. Does the net work done on a particle depend on the choice of reference frame? How does this affect the work -energy principle? 15. A hand exerts a constant horizontal force on a block that is free to slide on a frictionless surface (Fig. 7 -19). The block starts from rest at point A, and by the time it has traveled a distance d to point B it is traveling with speed vB. When the block has traveled another distance d to point , will its speed be greater than, less than, or equal to 2vB? Explain your reasoning.

FIGURE 7 - 19

Question 15.

| Problems 7 - 1 Work, Constant Force 1. (I) How much work is done by the gravitational force when a 280-kg pile driver falls 2.80 m? 2. (I) How high will a 1.85-kg rock go if thrown straight up by someone who does 80.0 J of work on it? Neglect air resistance. 3. (I) A 75.0-kg firefighter climbs a flight of stairs 20.0 m high. How much work is required? 4. (I) A hammerhead with a mass of 2.0 kg is allowed to fall onto a nail from a height of 0.50 m. What is the maximum amount of work it could do on the nail? Why do people not just "let it fall" but add their own force to the hammer as it falls? 5. (II) Estimate the work you do to mow a lawn 10 m by 20 m with a 50-cm wide mower. Assume you push with a force of about 15 N. 6. (II) A lever such as that shown in Fig. 7 - 20 can be used to lift objects we might not otherwise be able to lift. Show that the ratio of output force, Fo, to input force, Fi, is related to the lengths ^ and £0 from the pivot by F0 /Fi = l i / l o . Ignore friction and the mass of the lever, and assume the work output equals work input. FIGURE 7 - 20

A lever. Problem 6.

7. (II) What is the minimum work needed to push a 950-kg car 310 m up along a 9.0° incline? Ignore friction. 8. (II) Eight books, each 4.0 cm thick with mass 1.8 kg, lie flat on a table. How much work is required to stack them one on top of another? 9. (II) A box of mass 6.0 kg is accelerated from rest by a force across a floor at a rate of 2.0 m /s2 for 7.0 s. Find the net work done on the box. 10. (II) (a) What magnitude force is required to give a helicopter of mass M an acceleration of 0.10 g upward? (b) What work is done by this force as the helicopter moves a distance upward? 11. (II) A 380-kg piano slides 3.9 m down a 27° incline and is kept from accelerating by a man who is pushing back on it parallel to t e incline (Fig. 7 - 21). Determine: (a) the force exerted by the man, (b) the work done by the man on the piano, (c) the work done by the force of gravity, and (d) the net work done on the piano. Ignore friction.

FIGURE 7 - 21

Problem 11. 12. (II) A gondola can carry 20 skiers, with a total mass of up to 2250 kg. The gondola ascends at a constant speed from the base of a mountain, at 2150 m, to the summit at 3345 m. (a) How much work does the motor do in moving a full gondola up the mountain? (b) How much work does gravity do on the gondola? (c) If the motor is capable of generating 10% more work than found in (a), what is the acceleration of the gondola?

Problems

1 77

13. (II) A 17,000-kg jet takes off from an aircraft carrier via a cata pult (Fig. 7 - 22a). The gases thrust out from the jet's engines exert a constant force of 130 kN on the jet; the force exerted on the jet by the catapult is plotted in Fig. 7 - 22b. Determine: (a) the work done on the jet by the gases expelled by its engines during launch of the jet; and (b) the work done on the jet by the catapult during launch of the jet.

28. (II) Use the scalar product to prove the law o f cosines for a triangle: c2 = a2 + b2 - lab cos 0,

29.

F (kN) 1100 30.

31.

(a) FIGURE 7 - 2 2

Problem 13.

14. (II) A 2200-N crate rests on the floor. How much work is required to move it at constant speed (a) 4.0 m along the floor against a drag force of 230 N, and (b) 4.0 m vertically? 15. (II) A grocery cart with mass of 16 kg is being pushed at constant speed up a flat 12° ramp by a force FPwhich acts at an angle of 17° below the horizontal. Find the work done by each of the forces (mg, FN, FP) on the cart if the ramp is 15 m long.

7 - 2 Scalar Product 16. (I) What is the dot product of A = 2.0x2i — 4.0jcj + 5.0k and B = 11.Oi + 2.5xj? 17. (I) For any vector V = Vx i + Vy\ + Vz k show that Vx = i • V, 18. (I)

Vy = j • V,

y7 = k • v .

alculate the angle between the vectors:

A = 6.8i - 3.4j - 6.2k

and

B = 8.2i + 2.3j - 7.0k.

19. (I) Show that A • (-B ) = - A • B. 20. (I) Vector Vi points along the z axis and has magnitude V\ = 75. Vector V2 lies in the xz plane, has magnitude V2 = 58, and makes a —48° angle with the x axis (points below x axis). What is the scalar product Vi • V2? 21. (II) Given the vector A = 3.0i + 1.5j, find a vector B that is perpendicular to A. 22. (II) A constant force F = (2.0i + 4.0j) N acts on an object as it moves along a straight-line path. If the object's displace ment is d = (l.Oi + 5.0j) m, calculate the work done by F using these alternate ways of writing the dot product: (a) W = FdcosO; (b) W = Fx dx + Fy dy . 23. (II) If A = 9.0i - 8.5j, B = —8.01 + 7.1j + 4.2k, and = 6.8i - 9.2j, determine (a) A • (B + ); (b) (A + ) • B; (c) (B + A) • . 24. (II) Prove that A • B = A XBX + A y By + A ZBZ, starting from Eq. 7 - 2 and using the distributive property (p. 167, proved in Problem 33). 25. (II) Given vectors A = - 4 .8 i + 6.8j and B = 9.6i + 6.7j, determine the vector that lies in the xy plane perpendic ular to B and whose dot product with A is 20.0. 26. (II) Show that if two nonparallel vectors have the same magnitude, their sum must be perpendicular to their difference. 27. (II) Let V = 20.0i + 22.0j - 14.0k. What angles does this vector make with the x, y, and z axes?

178

HAPTER 7

Work and Energy

32. 33.

where a, b, and c are the lengths of the sides of a triangle and 0 is the angle opposite side c. (II) Vectors A and B are in the xy plane and their scalar product is 20.0 units. If A makes a 27.4° angle with the x axis and has magnitude A = 12.0 units, and B has magnitude B = 24.0 units, what can you say about the direction of B? (II) A and B are two vectors in the xy plane that make angles a and /3 with the x axis respectively. Evaluate the scalar product of A and B and deduce the following trigonometric identity: cos (a —/3) = cos a cos /3 + sin a sin /3. (II) Suppose A = l.Oi + l.Oj - 2.0k and B = - l.O i + l.Oj + 2.0k, (a) what is the angle between these two vectors? (b) Explain the significance of the sign in part (a). (II) Find a vector of unit length in the xy plane that is perpendicular to 3.0i + 4.0j. (Ill) Show that the scalar product of two vectors is distributive: A • (B + C) = A • B + A • C. [Hint: Use a diagram showing all three vectors in a plane and indicate dot products on the diagram.]

7 - 3 Work, Varying Force 34. (I) In pedaling a bicycle uphill, a cyclist exerts a downward force of 450 N during each stroke. If the diameter of the circle traced by each pedal is 36 cm, calculate how much work is done in each stroke. 35. (II) A spring has k = 65N /m . Draw a graph like that in Fig. 7 - 11 and use it to determine the work needed to stretch the spring from x = 3.0 cm to x = 6.5 cm, where x = 0 refers to the spring's unstretched length. 36. (II) If the hill in Example 7 - 2 (Fig. 7 - 4) was not an even slope but rather an irregular curve as in Fig. 7 - 23, show that the same result would be obtained as in Example 7 - 2: namely, that the work done by gravity depends only on the height of the hill and not on its shape or the path taken.

FIGURE 7 - 23

Problem 36. 37. (II) The net force exerted on a particle acts in the positive x direction. Its magnitude increases linearly from zero at x = 0, to 380 N at x = 3.0 m. It remains constant at 380 N from x = 3.0 m to x = 7.0 m, and then decreases linearly to zero at x = 12.0 m. Determine the work done to move the particle from x = 0 to x = 12.0 m graphically, by determining the area under the Fx versus x graph. 38. (II) If it requires 5.0 J of work to stretch a particular spring by 2.0 cm from its equilibrium length, how much more work will be required to stretch it an additional 4.0 cm? 39. (II) In Fig. 7 - 9 assume the distance axis is the x axis and that a = 10.0 m and b = 30.0 m. Estimate the work done by this force in moving a 3.50-kg object from a to b.

40. (II) The force on a particle, acting along the x axis, varies as shown in Fig. 7 - 24. Determine the work done by this force to move the particle along the x axis: (a) from x = 0.0 to x = 10.0 m; (b) from x = 0.0 to x = 15.0 m.

x (m) FIGURE 7 - 24

Problem 40. 41. (II) A child is pulling a wagon down the sidewalk. For 9.0 m the wagon stays on the sidewalk and the child pulls with a horizontal force of 22 N. Then one wheel of the wagon goes off on the grass so the child has to pull with a force of 38 N at an angle of 12° to the side for the next 5.0 m. Finally the wagon gets back on the sidewalk so the child makes the rest of the trip, 13.0 m, with a force of 22 N. How much total work did the child do on the wagon? 42. (II) The resistance of a packing material to a sharp object penetrating it is a force proportional to the fourth power of the penetration depth*; that is, F = k x 4i. alculate the work done to force a sharp object a distance d into the material. 43. (II) The force needed to hold a particular spring compressed an amount x from its normal length is given by F = k x + ax3 + bx4 How much work must be done to compress it by an amount X , starting from x = 0? 44. (II) A t the top of a pole vault, an athlete actually can do work pushing on the pole before releasing it. Suppose the pushing force that the pole exerts back on the athlete is given by F (x) = (1.5 X 102N /m )x - (1.9 X 102N /m 2)jt2 acting over a distance of 0.20 m. How much work is done on the athlete? 45. (II) onsider a force F\ = A / ^ / x which acts on an object during its journey along the x axis from x = 0.0 to x = 1.0 m, where A = 2.0 N - m 1/2. Show that during this journey, even though F1 is infinite at x = 0.0, the work done on the object by this force is finite. 46. (II) Assume that a force acting on an object is given by F = axi + by\, where the constants a = 3.0 N*m -1 and b = 4.0 N - m -1. Determ ine the work done on the object by this force as it moves in a straight line from the origin to r = (lO.Oi + 20.0j) m. 47. (II) An object, moving along the circumference of a circle with radius R, is acted upon by a force of constant magni tude F. The force is directed at all times at a 3o ;> ____ _ 30° angle with respect to the tangent to the circle as shown in Fig. 7 - 25. Determine the work done by this force when the object moves along the half circle from A to B. FIGURE 7 - 25

Problem 47. 48. (Ill) A 2800-kg space vehicle, initially at rest, falls vertically from a height of 3300 km above the E arth's surface. Determine how much work is done by the force of gravity in bringing the vehicle to the Earth's surface.

49. (Ill) A 3.0-m-long steel chain is stretched out along the top level of a horizontal scaffold at a construction site, in such a way that 2.0 m of the chain remains on the top level and 1.0 m hangs vertically, Fig. 7 - 26. A t this point, the force on the hanging segment is sufficient to pull the entire chain over the edge. Once the chain is moving, the kinetic friction is so small that it can be neglected. How much work is performed on the chain by the force of gravity as the chain falls from the point where 2.0 m remains on the scaffold to the point where the entire chain has left the scaffold? (Assume that the chain has a linear weight density of 18 N/m.) 1.0 m FIGURE 7 - 2 6

Problem 49.

7 - 4 Kinetic Energy; Work-Energy Principle 50. (I) A t room temperature, an oxygen molecule, with mass of 5.31 X 10_26kg, typically has a kinetic energy of about 6.21 X 10-21 J. How fast is it moving? 51. (I) (a) If the kinetic energy of a particle is tripled, by what factor has its speed increased? (b) If the speed of a particle is halved, by what factor does its kinetic energy change? 52. (I) How much work is required to stop an electron (m = 9.11 X 10-31 kg) which is moving with a speed of 1.40 X 106m/s? 53. (I) How much work must be done to stop a 1300-kg car traveling at 95 km /h? 54. (II) Spiderman uses his spider webs to save a runaway train, Fig. 7 - 27. His web stretches a few city blocks before the 104-kg train comes to a stop. Assuming the web acts like a spring, estimate the spring constant.

FIGURE 7 - 27

Problem 54. 55. (II) A baseball (m = 145 g) traveling 32 m /s moves a fielder's glove backward 25 cm when the ball is caught. What was the average force exerted by the ball on the glove? 56. (II) An 85-g arrow is fired from a bow whose string exerts an average force of 105 N on the arrow over a distance of 75 cm. What is the speed of the arrow as it leaves the bow? 57. (II) A mass m is attached to a spring which is held stretched a distance x by a force F (Fig. 7 -28), and then released. The spring compresses, pulling the mass. Assuming there is no friction, determine the speed of the mass m when the spring returns: (a) to its normal length (x = 0); (b) to half its orig in inal extension (x/2). |

x = 0

FIGURE 7 - 28

Problem 57.

Problems

1 79

58. (II) If the speed of a car is increased by 50%, by what factor will its minimum braking distance be increased, assuming all else is the same? Ignore the driver's reaction time. 59. (II) A 1200-kg car rolling on a horizontal surface has speed v = 66 km /h when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2 m. What is the spring constant of the spring? 60. (II) One car has twice the mass of a second car, but only half as much kinetic energy. When both cars increase their speed by 7.0 m/s, they then have the same kinetic energy. What were the original speeds of the two cars? 61. (II) A 4.5-kg object moving in two dimensions initially has a velocity vi = (lO.Oi + 20.0j) m/s. A net force F then acts on the object for 2.0 s, after which the object's velocity is v2 = (15.0i + 30.0j) m/s. Determine the work done by F on the object. 62. (II) A 265-kg load is lifted 23.0 m vertically with an acceler ation a = 0.150 g by a single cable. Determine (a) the tension in the cable; (b) the net work done on the load; (c) the work done by the cable on the load; (d) the work done by gravity on the load; (e) the final speed of the load assuming it started from rest. 63. (II) (a) How much work is done by the horizontal force Fp = 150 N on the 18-kg block of Fig. 7 - 29 when the force pushes the block 5.0 m up along the 32° frictionless incline? (b) How much work is done by the gravitational force on the block during this displacement? (c) How much work is done by the normal force? (d) What is the speed of the block (assume that it is zero initially) after this displacement? [Hint. Work -energy involves net work done.l

FIGURE 7 - 29

Problems 63 and 64. 64. (II) Repeat Problem 63 assuming a coefficient of friction = 0.10. 65. (II) A t an accident scene on a level road, investigators measure a car's skid mark to be 98 m long. It was a rainy day and the coefficient of friction was estimated to be 0.38. Use these data to determine the speed of the car when the driver slammed on (and locked) the brakes. (Why does the car's mass not matter?)

66. (II) A 46.0-kg crate, starting from rest, is pulled across a floor with a constant horizontal force of 225 N. For the first 11.0m the floor is frictionless, and for the next 10.0 m the coefficient of friction is 0.20. What is the final speed of the crate after being pulled these 21.0 m? 67. (II) A train is moving along a track with constant speed relative to the ground. A person on the train holds a ball of mass m and throws it toward the front of the train with a speed v2 relative to the train. alculate the change in kinetic energy of the ball (a) in the Earth frame of reference, and (b) in the train frame of reference, (c) Relative to each frame of reference, how much work was done on the ball? (d) Explain why the results in part (b) are not the same for the two frames — after all, it's the same ball. 68. (Ill) We usually neglect the mass of a spring if it is small compared to the mass attached to it. But in some applications, the mass of the spring must be taken into account. onsider a spring of unstretched length £ and mass Ms uniformly distributed along the length of the spring. A mass m is attached to the end of the spring. One end of the spring is fixed and the mass m is allowed to vibrate horizontally without friction (Fig. 7 - 30). Each point on the spring moves with a velocity proportional to the distance from that point to the fixed end. For example, if the mass on the end moves with speed vQ, the midpoint of the spring moves with speed V q/ 2 . Show that the kinetic energy of the mass plus spring when the mass is moving with velocity v is K = \M v 2 where M = m + |M S is the "effective mass" of the system. [Hint: Let D be the total length of the stretched spring. Then the velocity of a mass dm of a spring of length dx located at x is v(x) = v0(x/D ). Note also that dm = dx(Ms/D).]

- H11! dx

FIGURE 7 - 30

Problem 68.

69. (Ill) An elevator cable breaks when a 925-kg elevator is 22.5 m above the top of a huge spring (k = 8.00 X 104N /m ) at the bottom of the shaft. alculate (a) the work done by gravity on the elevator before it hits the spring; (b) the speed of the elevator just before striking the spring; (c) the amount the spring compresses (note that here work is done by both the spring and gravity).

| Gener l Problems__________ 70. (a) A 3.0-g locust reaches a speed of 3.0 m /s during its jump. What is its kinetic energy at this speed? (b) If the locust transforms energy with 35% efficiency, how much energy is required for the jump? 71. In a certain library the first shelf is 12.0 cm off the ground, and the remaining 4 shelves are each spaced 33.0 cm above the previous one. If the average book has a mass of 1.40 kg with a height of 22.0 cm, and an average shelf holds 28 books (standing vertically), how much work is required to fill all the shelves, assuming the books are all laying flat on the floor to start? 18

HAPTER 7

Work and Energy

72. A 75-kg meteorite buries itself 5.0 m into soft mud. The force between the meteorite and the mud is given by F{x) = (640N/m 3)x 3, where x is the depth in the mud. What was the speed of the meteorite when it initially impacted the mud? 73. A 6.10-kg block is pushed 9.25 m up a smooth 37.0° inclined plane by a horizontal force of 75.0 N. If the initial speed of the block is 3.25 m /s up the plane, calculate (a) the initial kinetic energy of the block; (b) the work done by the 75.0-N force; (c) the work done by gravity; (d) the work done by the normal force; (e) the final kinetic energy of the block.

74. The arrangement of atoms in zinc is an example of "hexagonal close -packed" structure. Three of the nearest neighbors are found at the following (x, y, z ) coordinates, given in nanometers (l0 - 9 m): atom 1 is at (0, 0, 0); atom 2 is at (0.230, 0.133, 0); atom 3 is at (0.077, 0.133, 0.247). Find the angle between two vectors: one that connects atom 1 with atom 2 and another that connects atom 1 with atom 3. 75. Two forces, Fj = (l.50i - 0.80j + 0.70k) N and F2 = ( —0.70i + 1.20j) N, are applied on a moving object of mass 0.20 kg. The displacement vector produced by the two forces is d = (8.0i + 6.0j + 5.0k) m. What is the work done by the two forces? 76. The barrels of the 16-in. guns (bore diameter = 16 in. = 41 cm) on the World War II battleship U.S.S. Massac usetts were each 15 m long. The shells each had a mass of 1250 kg and were fired with sufficient explosive force to provide them with a muzzle velocity of 750 m /s. Use the work-energy principle to determine the explosive force (assumed to be a constant) that was applied to the shell within the barrel of the gun. Express your answer in both newtons and in pounds. 77. A varying force is given by F = Ae ~ kx, where x is the position; A and k are constants that have units of N and m _1, respectively. What is the work done when x goes from 0.10 m to infinity? 78. The force required to compress an imperfect horizontal spring an amount x is given by F = 150x + 12x3, where x is in meters and F in newtons. If the spring is compressed 2.0 m, what speed will it give to a 3.0-kg ball held against it and then released? 79. A force F = (lO.Oi + 9.0j + 12.0k) kN acts on a small object of mass 95 g. If the displacement of the object is d = (5.0i + 4.0j) m, find the work done by the force. What is the angle between F and d? 80. In the game of paintball, players use guns powered by pressurized gas to propel 33-g gel capsules filled with paint at the opposing team. Game rules dictate that a paintball cannot leave the barrel of a gun with a speed greater than 85m /s. Model the shot by assuming the pressurized gas applies a constant force F to a 33-g capsule over the length of the 32-cm barrel. Determine F (a) using the work-energy principle, and (b) using the kinematic equations (Eqs. 2 - 12) and Newton's second law. 81. A softball having a mass of 0.25 kg is pitched horizontally at 110 km /h. By the time it reaches the plate, it may have slowed by 10%. Neglecting gravity, estimate the average force of air resistance during a pitch, if the distance between the plate and the pitcher is about 15 m. 82. An airplane pilot fell 370 m after jumping from an aircraft without his parachute opening. He landed in a snowbank, creating a crater 1.1 m deep, but survived with only minor injuries. Assuming the pilot's mass was 88 kg and his terminal velocity was 45 m /s, estimate: (a) the work done by the snow in bringing him to rest; (b) the average force exerted on him by the snow to stop him; and (c) the work done on him by air resistance as he fell. Model him as a particle. 83. Many cars have "5 m i/h (8 km /h) bumpers" that are designed to compress and rebound elastically without any physical damage at speeds below 8 km /h. If the material of the bumpers permanently deforms after a compression of 1.5 cm, but remains like an elastic spring up to that point, what must be the effective spring constant of the bumper material, assuming the car has a mass of 1050 kg and is tested by ramming into a solid wall?

84. What should be the spring constant A: of a spring designed to bring a 1300-kg car to rest from a speed of 90 km /h so that the occupants undergo a maximum acceleration of 5.0 g? 85. Assume a cyclist of weight mg can exert a force on the pedals equal to 0.90 mg on the average. If the pedals rotate in a circle of radius 18 cm, the wheels have a radius of 34 cm, and the front and back sprockets on which the chain runs have 42 and 19 teeth respectively (Fig. 7 - 31), determine the maximum steepness of hill the cyclist can climb at constant speed. Assume the mass of the bike is 12 kg and that of the rider is 65 kg. Ignore friction. Assume the cyclist's average force is always: (a) downward; (b) tangential to pedal motion.

FIGURE 7 - 31

Problem 85.

86. A simple pendulum consists of a small object of mass m (the "bob") suspended by a cord of length £ (Fig. 7 -32) of negli gible mass. A force F is applied in the horizontal direction (so F = Fi), moving the bob very slowly so the acceleration is essentially zero. (Note that the magnitude of F will need to vary with the angle 6 that the cord makes with the vertical at any moment.) (a) Determine the work done by this force, F, to move the pendulum from 0 = 0 to 6 = 60. (b) Deter mine the work done by the gravitational force on the bob, Fg = mg, and the work done by the force Ft that the cord exerts on the bob.

FIGURE 7 - 32

Problem 86. 87. A car passenger buckles himself in with a seat belt and holds his 18-kg toddler on his lap. Use the work -energy principle to answer the following questions, (a) While traveling 25 m /s, the driver has to make an emergency stop over a distance of 45 m. Assuming constant deceleration, how much force will the arms of the parent need to exert on the child during this deceleration period? Is this force achievable by an average parent? (b) Now assume that the car (v = 25 m /s) is in an accident and is brought to stop over a distance of 12 m. Assuming constant deceleration, how much force will the parent need to exert on the child? Is this force achievable by an average parent?

General Problems

181

88. As an object moves along the x axis from x = 0.0 m to x = 20.0 m it is acted upon by a force given by F = (100 - (x - 10)2)N. Determine the work done by the force on the object: (a) by first sketching the F vs. x graph and estimating the area under this curve; (b) by evaluating the integral dx. 89. A cyclist starts from rest and coasts down a 4.0° hill. The mass of the cyclist plus bicycle is 85 kg. After the cyclist has traveled 250 m, (a) what was the net work done by gravity on the cyclist? (b) How fast is the cyclist going? Ignore air resistance. 90. Stretchable ropes are used to safely arrest the fall of rock climbers. Suppose one end of a rope with unstretched length £ is anchored to a cliff and a climber of mass m is attached to the other end. When the climber is a height £ above the anchor point, he slips and falls under the influence of gravity for a distance 2£, after which the rope becomes taut and stretches a distance x as it stops the climber (see Fig. 7-33). Assume a stretchy rope behaves as a spring with spring constant k. (a) Applying the work-energy principle, show that x =

91. A small mass m hangs at rest from a vertical rope of length £ that is fixed to the ceiling. A force F then pushes on the mass, perpendicular to the taut rope at all times, until the rope is oriented at an angle 6 = 0O and the mass has been raised by a vertical distance (Fig. 7 - 34). Assume the force's magnitude F is adjusted so that the mass moves at constant speed along its curved trajectory. Show that the work done by F during this process equals mg , which is equivalent to the amount of work it takes to slowly lift a mass m straight up by a height . [Hint: When the angle is increased by dd (in radians), the mass moves along an arc length ds = £ d6.\

mg 4k£ 1 + J l + k mg

(b) Assuming m = 85 kg, determine x/£ (the fractional stretch of the rope) and k x /m g (the force that the rope exerts on the climber compared to his own weight) at the moment the climber's fall has been stopped.

£ = 8.0 m and k = 850 N/m,

18Num rical/Comput r 1592. (II) The net force along the linear path of a particle of mass 480 g has been measured at 10.0-cm intervals, starting at jc = 0.0, to be 26.0, 28.5, 28.8, 29.6, 32.8, 40.1, 46.6, 42.2, 48.8,52.6,55.8,60.2,60.6,58.2,53.7,50.3,45.6,45.2,43.2,38.9, 35.1, 30.8, 27.2, 21.0, 22.2, and 18.6, all in newtons. Determine the total work done on the particle over this entire range. H93. (II) When different masses are suspended from a spring, the spring stretches by different amounts as shown in the Table below. Masses are + 1.0 gram. Mass (g) 0 50 100 150 200 250 300 350 400 Stretch (cm) 0 5.0 9.8 14.8 19.4 24.5 29.6 34.1 39.2

FIGURE 7-33 Problem 90.

(a) Graph the applied force (in Newtons) versus the stretch (in meters) of the spring, and determine the best-fit straight line. (b) Determine the spring constant (N/m) of the spring from the slope of the best-fit line, (c) If the spring is stretched by 20.0 cm, estimate the force acting on the spring using the best-fit line.

Answers to Exercises D: No, because the speed v would be the square root of a

A: (c).

negative number, which is not real.

B: 0b). : (b) 2.0 X 105J (i.e., less).

182

HAPTER 7

Work and Energy

E: (a) \ f l , (b) 4.

A polevaulter running toward the high bar has kinetic energy. When he plants the pole and puts his weight on it, his kinetic energy gets transformed: first into elastic potential energy of the bent pole and then into gravitational potential energy as his body rises. As he crosses the bar, the pole is straight and has given up all its elastic potential energy to the athlete's gravitational potential energy. Nearly all his kinetic energy has disappeared, also becoming gravitational potential energy of his body at the great height of the bar (world record over 6 m), which is exactly what he wants. In these, and all other energy transformations that continually take place in the world, the total energy is always conserved. Indeed, the conservation of energy is one of the greatest laws of physics, and finds applications in a wide range of other fields.

onservation of Energy CHAPTER -OPENING QUESTION — Guess now!

ONTENTS

A skier starts at th e top of a hill. O n which run does her gravitational potential energy change the most: ( ), (b), (c), or (d); or are they (e) all the same? O n which run w ould h er speed at the b o tto m be th e fastest if the runs are icy and we assum e no friction? Recognizing th at th ere is always som e friction, answ er th e above tw o q u es tions again. L ist y o u r four answ ers now.

8 - 1 C on serv tive nd N o n co n se rv tiv e F orces 8 - 2 P oten ti l E n erg y 8 - 3 M ech n ic l E n erg y C on serv tion

nd Its

8 - 4 P rob lem S olvin g U sin g C on serv tion o f M ech n ic l E n ergy 8 - 5 T h e L w o f C o n serv tio n o f E n ergy 8 - 6 E n ergy C on serv tio n w ith D issip tiv e Forces: S olvin g P rob lem s

E sy Interm edi te D ifficu lt ♦ ♦ V ery d ifficu lt

8 - 7 G r vit tion l P oten ti l E n ergy nd E sc p e V elocity 8 - 8 P ow er * 8 - 9 P oten ti l E n erg y D i gr m s; S t b le nd U n st b le E q u ilib riu m

183

his chapter continues the discussion of the concepts of work and energy begun in hapter 7 and introduces additional types of energy, in particular potential energy. Now we will see why the concept of energy is so important. The reason, ultimately, is that energy is conserved—the total energy always remains constant in any process. That a quantity can be defined which remains constant, as far as our best experiments can tell, is a remarkable statement about nature. The law of conservation of energy is, in fact, one of the great unifying principles of science. The law of conservation of energy also gives us another tool, another approach, to solving problems. There are many situations for which an analysis based on Newton's laws would be difficult or impossible— the forces may not be known or accessible to measurement. But often these situations can be dealt with using the law of conservation of energy. In this hapter we will mainly treat objects as if they were particles or rigid objects that undergo only translational motion, with no internal or rotational motion.

8 - 1 Conserv tive nd Nonconserv tive Forces We will find it important to categorize forces into two types: conservative and nonconservative. By definition, we call any force a conserv tive force if the work done by the force on n object moving from one point to nother depends only on the initi l nd fin l positions of the object, nd is independent of the p rticul r p th t ken.

FIGURE 8 - 1 Object o f m ss m: ( ) f lls height vertic lly; (b) is r ised long n rbitr ry two - dim ension l p th.

A conservative force can be a function only o f position, and cannot depend on other variables like time or velocity. We can readily show that the force of gravity is a conservative force. The gravitational force on an object of mass m near the Earth's surface is F = mg, where g is a constant. The work done by this gravitational force on an object that falls a vertical distance is WG = Fd = mg (see Fig. 8 - la). Now suppose that instead of moving vertically downward or upward, an object follows some arbitrary path in the xy plane, as shown in Fig. 8 - lb . The object starts at a vertical height y1 and reaches a height y2, where y2 ~ y\ = . To calculate the work done by gravity, WG, we use Eq. 7 -7: Wn

■D. ( )

• di

mg cos 6 d t

We now let and dy = di cos (f), we have ry2 Wo = ~ I mg dy J yi

= ~ mg(y2 - yj).

(8 - 1)

Since (y2 — y j is the vertical height , we see that the work done depends only on the vertical height and does not depend on the particular path taken! Hence, by definition, gravity is a conservative force. Note that in the case shown in Fig. 8 - lb , y2 > y1and therefore the work done by gravity is negative. If on the other hand y2 < y1, so that the object is falling, then WGis positive. 184

CHAPTER 8

Conservation of Energy

We can give the definition of a conservative force in another, completely equivalent way: force is conserv tive if the net work done by the force on n object moving round ny closed p th is zero. To see why this is equivalent to our earlier definition, consider a small object that moves from point 1 to point 2 via either of two paths labeled A and B in Fig. 8-2a. If we assume a conservative force acts on the object, the work done by this force is the same whether the object takes path A or path B, by our first definition. This work to get from point 1 to point 2 we will call W. Now consider the round trip shown in Fig. 8-2b. The object moves from 1 to 2 via path A and our force does work W. Our object then returns to point 1 via path B. How much work is done during the return? In going from 1 to 2 via path B the work done is W, which by definition equals JxF • dl. In doing the reverse, going from 2 to 1, the force F at each point is the same, but dl is directed in precisely the opposite direction. onsequently F • dl has the opposite sign at each point so the total work done in making the return trip from 2 to 1 must be —W. Hence the total work done in going from 1 to 2 and back t o l i s W + ( ~ W ) = 0, which proves the equivalence of the two above definitions for a conservative force. The second definition of a conservative force illuminates an important aspect of such a force: the work done by a conservative force is recoverable in the sense that if positive work is done by an object (on something else) on one part of a closed path, an equivalent amount of negative work will be done by the object on its return. As we saw above, the force of gravity is conservative, and it is easy to show that the elastic force (F = —kx) is also conservative.

FIGURE 8 - 2 ( ) A tiny object m oves betw een points 1 nd 2 vi two different p ths, A nd B. (b) The object m kes round trip, vi p th A from point 1 to point 2 nd vi p th B b ck to point 1.

/ m .

m i

M l

FIGURE 8 - 3 A cr te is pushed t const nt speed cross rough floor from position 1 to position 2 vi two p ths, one str ight nd one curved. The pushing force FP is lw ys in the direction of motion. (The friction force opposes the m otion.) H ence for const nt m gnitude pushing force, the work it does is W = Fp d, so if d is gre ter ( s for the curved p th), then W is gre ter. The work done does not depend only on points 1 nd 2; it lso depends on the p th t ken.

Many forces, such as friction and a push or pull exerted by a person, are nonconserv tive forces since any work they do depends on the path. For example, if you push a crate across a floor from one point to another, the work you do depends on whether the path taken is straight, or is curved. As shown in Fig. 8 - 3, if a crate is pushed from point 1 to point 2 along the longer semicircular path, you do more work against friction than if you push it along the straight path. This is because the distance is greater and, unlike the gravitational force, the pushing force FPis always in the direction of motion. Thus the work done by the person in Fig. 8 - 3 does not depend only on points 1 and 2; it depends also on the path taken. The force of kinetic friction, also shown in Fig. 8-3, always opposes the motion; it too is a nonconservative force, and we discuss how to treat it later in this hapter (Section 8 - 6). Table 8 - 1 lists a few conservative and nonconservative forces. SECTI N 8 - 1

TABLE 8 -1 Conservative and Nonconservative Forces Conserv tive Forces

Nonconserv tive Forces

Gr vit tion l

Friction

El stic

Air resist nce

Electric

Tension in cord M otor or rocket propulsion Push or pull by person

Conservative and Nonconservative Forces

185

8 —2 Potenti l Energy In hapter 7 we discussed the energy associated with a moving object, which is its kinetic energy K = \m v 2. Now we introduce potenti l energy, which is the energy associated with forces that depend on the position or configuration of objects relative to the surroundings. Various types of potential energy can be defined, and each type is associated with a particular conservative force. The wound-up spring of a toy is an example of potential energy. The spring acquired its potential energy because work was done on it by the person winding the toy. As the spring unwinds, it exerts a force and does work to make the toy move.

Gr vit tion l Potenti l Energy ?2-

i_____ 3 d

(exerted ext ^ , jv by h nd)

rFG= mg y\

i_____

FIGURE 8 - 4 A person exerts n upw rd force Fext = m g to lift brick from y1 to y2 .

Perhaps the most common example of potential energy is gravitational potential energy. A heavy brick held above the ground has potential energy because of its position relative to the Earth. The raised brick has the ability to do work, for if it is released, it will fall to the ground due to the gravitational force, and can do work on a stake, driving it into the ground. Let us seek the form for the gravitational potential energy of an object near the surface of the Earth. For an object of mass m to be lifted vertically, an upward force at least equal to its weight, mg, must be exerted on it, say by a person's hand. To lift it without acceleration a vertical displacement of height , from position y1 to y2 in Fig. 8 - 4 (upward direction chosen positive), a person must do work equal to the product of the "external" force she exerts, Fext = mg upward, times the vertical displacement . That is, Wext = Fext • d = mg cos 0° = mg

= mg(y2 - y j

where both ^ext and d point upward. Gravity is also acting on the object as it moves from y1to y2, and does work on the object equal to WG = FG• d = mg cos 180° = - m g

= ~ mg(y2 — y j,

where 6 = 180° because FG and d point in opposite directions. Since FG is downward and d is upward, WGis negative. If the object follows an arbitrary path, as in Fig. 8 - lb , the work done by gravity still depends only on the change in vertical height (Eq. 8 -1): WG = ~ m g(y2 — y j = - m g . Next, if we allow the object to start from rest and fall freely under the action of gravity, it acquires a velocity given by v2 = 2g (Eq. 2 -12c) after falling a height . It then has kinetic energy \m v 2 = \m (2g ) = mg , and if it strikes a stake it can do work on the stake equal to mg . To summarize, raising an object of mass m to a height requires an amount of work equal to mg . And once at height , the object has the ability to do an amount of work equal to mg . Thus we can say that the work done in lifting the object has been stored as gravitational potential energy. Indeed, we can define the c ange in gravitational potential energy U, when an object moves from a height y1 to a height y2, as equal to the work done by a net external force to accomplish this without acceleration: AU = U2 - U, = Wext = mg(y2 - y,). Equivalently, we can define the change in gravitational potential energy as equal to the negative of the work done by gravity itself in the process: AU = U2 - U, = - W a = mg(y2 - y,). (8 - 2) Equation 8 - 2 defines the change in gravitational potential energy when an object of mass m moves between two points near the surface of the Earth.f The gravitational potential energy, U, at any point a vertical height y above some reference point (the origin of the coordinate system) can be defined as ^grav = mgy. [gravity only] (8 -3) Note that the potential energy is associated with the force of gravity between the Earth and the mass m. Hence t/grav represents the gravitational potential energy, not simply of the mass m alone, but of the mass - Earth system. 186

CHAPTER 8

fSection 8 - 7 de ls with the l / r 1 dependence of Newton's l w of univers l gr vit tion.

Gravitational potential energy depends on the vertical eig t of the object above some reference level, U = mgy. Sometimes you may wonder from what point to measure y. The gravitational potential energy of a book held high above a table, for example, depends on whether we measure y from the top of the table, from the floor, or from some other reference point. What is physically important in any situation is the c ange in potential energy, AU, because that is what is related to the work done, and it is AU that can be measured. We can thus choose to measure y from any reference point that is convenient, but we must choose the reference point at the start and be consistent throughout any given calculation. The c ange in potential energy between any two points does not depend on this choice. Potential energy belongs to a system, and not to a single object alone. Potential energy is associated with a force, and a force on one object is always exerted by some other object. Thus potential energy is a property of the system as a whole. For an object raised to a height y above the Earth's surface, the change in gravitational potential energy is mgy. The system here is the object plus the Earth, and properties of both are involved: object (ra) and Earth (g). In general, a system is one or more objects that we choose to study. The choice of what makes up a system is always ours, and we often try to choose a simple system. Below, when we deal with the potential energy of an object in contact with a spring, our system will be the object and the spring.

A AUTION C ange in poten tial energy is w at is p ysically m eaningful

/j\

AUT I ON

Potential energy belongs to a system, n ot to a single object

EXERCISE A Return to the Ch pter -Opening Q uestion, p ge 183, nd nswer it g in now. Try to expl in why you m y h ve nswered differently the first time.

EXAMPLE 8 - 1 Potential energy changes for a roller coaster. A 1000-kg roller-coaster car moves from point 1, Fig. 8 - 5, to point 2 and then to point 3. (a) What is the gravitational potential energy at points 2 and 3 relative to point 1? That is, take y = 0 at point 1. (b) What is the change in potential energy when the car goes from point 2 to point 3? (c) Repeat parts (a) and (b), but take the reference point (y = 0) to be at point 3. APPR ACH We are interested in the potential energy of the car - Earth system.

We take upward as the positive y direction, and use the definition of gravitational potential energy to calculate the potential energy. = 0), which means initially S LUTI N (a) We measure heights from point 1 that the gravitational potential energy is zero. At point 2, where y2 = 10 m, U

2 = mgy2 = (1000 kg)(9.8 m /s2)(10 m) = 9.8

FIGURE 8 - 5

Ex m ple 8 - 1 .

104J.

At point 3, y3 = - 1 5 m, since point 3 is below point 1. Therefore, U

3 = mgy3 = (1000 kg)(9.8 m /s2)( - 1 5 m) = - 1.5

(b) In going from point 2 to point 3, the potential energy change

(U fm3L\

105J. — /initiai) is

( - 1.5 X 105J) - (9.8 X 104J) = - 2.5 X 105J. The gravitational potential energy decreases by 2.5 X 105J. (c) Now we set y3 = 0. Then y1 = +15 m at point 1, so the potential energy initially (at point 1) is Ux = (1000 kg)(9.8 m/s2)(15 m) = 1.5 X 105J. U

At point 2, y2 = 25 m, so the potential energy is U

2 = 2.5

105J.

At point 3, y3 = 0, so the potential energy is zero. The change in potential energy going from point 2 to point 3 is U

2 = 0 - 2.5

105J = - 2.5 X 105J,

which is the same as in part (b). N TE Work done by gravity depends only on the vertical height, so changes in gravitational potential energy do not depend on the path taken. EXERCISE B B y how much does the potenti l energy ch nge when the top o f

1200 - kg c r climbs to 300 - m - t ll hill? (a) 3.6 X 105 J; (b) 3.5 X 106J; (c) 4 J; (d) 40 J; (e) 39.2 J.

SECTI N 8 - 2

187

Potenti l Energy in Gener l We have defined the change in gravitational potential energy (Eq. 8 - 2) to be equal to the negative of the work done by gravity when the object moves from height yi to y2, which we now write as A t/ = - W o = - j V

There are other types of potential energy besides gravitational. In general, we define the c ange in potential energy associated wit a particular conservative force F as t e negative o f t e work done by t at force:

Potential energy can be defined only fo r conservative forces

f • di (8 -4) AU = U2 - U x = -W . - - r However, we cannot use this definition to define a potential energy for all possible forces. It makes sense only for conservative forces such as gravity, for which the integral depends only on the end points and not on the path taken. It does not apply to nonconservative forces like friction, because the integral in Eq. 8 - 4 would not have a unique value depending on the end points 1 and 2. Thus the concept of potential energy cannot be defined and is meaningless for a nonconservative force.

El stic Potenti l Energy FIGURE 8 - 6 A spring ( ) c n store energy (el stic potenti l energy) w hen com pressed (b), which c n be used to do work when rele sed (c) nd (d).

x=0

(a)

We consider now potential energy associated with elastic materials, which includes a great variety of practical applications. onsider a simple coil spring as shown in Fig. 8 - 6, whose mass is so small that we can ignore it. When the spring is compressed and then released, it can do work on a ball (mass m). Thus the spring - ball system has potential energy when compressed (or stretched). Like other elastic materials, a spring is described by Hooke's law (see Section 7 - 3) as long as the displacement x is not too great. Let us take our coordinate system so the end of the uncompressed spring is at x = 0 (Fig. 8 - 6a) and x is positive to the right. To hold the spring compressed (or stretched) a distance x from its natural (unstretched) length requires the person's hand to exert a force Ff = kx on the spring (Fig. 8 - 6b), where k is the spring stiffness constant. The spring pushes back with a force (Newton's third law), Fs = - k x ,

(b)

Fig. 8 -6c. The negative sign appears because the force Fs is in the direction opposite to the displacement x. From Eq. 8 - 4, the change in potential energy when the spring is compressed or stretched from xx = 0 (its uncompressed position) to x2 = x (where x can be + or - ) is •2

AU = U(x) - 1/(0) = - j Fs - d l (C)

- j

(-k x )d x

Here, U(x) means the potential energy at x, and t/(0) means U at x = 0. It is usually convenient to choose the potential energy at x = 0 to be zero: t/(0) = 0, so the potential energy of a spring compressed or stretched an amount x from equilibrium is Uel(x) = \ k x 2.

(d)

[elastic spring] (8 -5)

Potenti l Energy Rel ted to Force ( 1-D ) In the one-dimensional case, where a conservative force can be written as a function of x, say, the potential energy can be written as an indefinite integral U(x) = where the constant 188

CHAPTER 8

Conservation of Energy

F(x) dx + C,

(8 - 6 )

represents the value of U at x = 0; we can sometimes

choose = 0. Equation 8 - 6 tells us how to obtain U(x) when given F(x). If, instead, we are given U(x), we can obtain F(x) by inverting the above equation: that is, we take the derivative of both sides, remembering that integration and differentiation are inverse operations:

Thus

-y- I F(x) dx = F(x). dx I dU(x) " M '

- s r 1 -

« - 7>

EXAMPLE 8 - 2 Determine F from U. Suppose U(x) = - a x /(b 2 + x2), where a and b are constants. What is F as a function of x l APPR ACH Since U(x) depends only on x, this is a one-dimensional problem. S LUTI N Equation 8 - 7 gives

ax b2 + x2

b2 + x 2

(b1 + x2)2

(b2 + x2)2

*Potenti l Energy in Three Dimensions In three dimensions, we can write the relation between F(jc, y, z) and U as: dU ----» dx

dU Fv — ------- » y dy

dU F7 — ------- ; dz

or N F(x, y ,z)

,d U dx

,d U dy

~ dU dz

- i - ---- j - ----- k—

Here, d/dx, d/dy and d/dz are called partial derivatives; d/dx, for example, means that although U may be a function of x, y, and z, written U(x, y, z), we take the derivative only with respect to x with the other variables held constant.

8—3 Mech nic l Energy nd Its Conserv tion Let us consider a conservative system (meaning only conservative forces do work) in which energy is transformed from kinetic to potential or vice versa. Again, we must consider a system because potential energy does not exist for an isolated object. Our system might be a mass m oscillating on the end of a spring or moving in the Earth's gravitational field. According to the work-energy principle (Eq. 7 -11), the net work Wnet done on an object is equal to its change in kinetic energy: W„et =

AX.

(If more than one object of our system has work done on it, then Wnet and AK can represent the sum for all of them.) Since we assume a conservative system, we can write the net work done on an object or objects in terms of the change in total potential energy (see Eq. 8 - 4) between points 1 and 2: At/U1.al = - J \ e t ■d t = ~ WM .

(8 - 8)

We combine the previous two equations, letting U be the total potential energy: AK + AU = 0

[conservative forces only] (8 -9a)

(K 2 — K-l) + (U2 — Ui) = 0.

[conservative forces only] (8 -9b)

SECTI N 8 - 3

Mechanical Energy and Its Conservation

189

We now define a quantity E, called the tot l mech nic l energy of our system, as the sum of the kinetic energy plus the potential energy of the system at any moment E = K + U. We can rewrite Eq. 8 - 9b as CONSERVATION OF MECHANICAL ENERGY

CONSERVATION OF MECHANICAL ENERGY

K2 + U2 = K x + Ui

[conservative forces only] (8-10 )

or E2 = E1 = constant. [conservative forces only] (8-10b) Equations 8 - 10 express a useful and profound principle regarding the total mechanical energy— it is a conserved qu ntity, as long as no nonconservative forces do work; that is, the quantity E = K + U at some initial time 1 is equal to K + U at any later time 2. To say it another way, consider Eq. 8 - 9a which tells us AU = - AK; that is, if the kinetic energy K increases, then the potential energy U must decrease by an equivalent amount to compensate. Thus the total, K + U, remains constant. This is called the principle of conserv tion of mech nic l energy for conservative forces: If only conserv tive forces re doing work, the tot l mech nic l energy of system neither incre ses nor decre ses in ny process. It st ys const nt—it is conserved. We now see the reason for the term "conservative force"—because for such forces, mechanical energy is conserved. If only one object of a system1has significant kinetic energy, then Eqs. 8-10 become E = \m v 2 + U = constant,

[conservative forces only] (8 - ll )

If we let Vx and Ux represent the velocity and potential energy at one instant, and v2 and U2 represent them at a second instant, then we can rewrite this as \m v 2 + Ux = \m v 2 + U2.

[conservative system] (8 - llb )

From this equation we can see again that it doesn't make any difference where we choose the potential energy to be zero: adding a constant to U merely adds a constant to both sides of Eq. 8 - llb , and these cancel. A constant also doesn't affect the force obtained using Eq. 8 - 7, F = —dU/dx, since the derivative of a constant is zero. Only changes in the potential energy matter.

8—4 Problem Solving Using Conserv tion of Mech nic l Energy FIGURE 8 - 7 The rock 's potenti l energy ch nges to kinetic energy s it f lls. N ote b r gr phs representing potenti l energy U nd kinetic energy K for the three different positions. U K

ll potenti l energy

L U K

h lf U, h lf K

U K

A simple example of the conservation of mechanical energy (neglecting air resistance) is a rock allowed to fall due to Earth's gravity from a height above the ground, as shown in Fig. 8-7. If the rock starts from rest, all of the initial energy is potential energy. As the rock falls, the potential energy mgy decreases (because y decreases), but the rock's kinetic energy increases to compensate, so that the sum of the two remains constant. At any point along the path, the total mechanical energy is given by E = K + U = \m v 2 + mgy where y is the rock's height above the ground at a given instant and v is its speed at that point. If we let the subscript 1 represent the rock at one point along its path (for example, the initial point), and the subscript 2 represent it at some other point, then we can write total mechanical energy at point 1 = total mechanical energy at point 2 or (see also Eq. 8 - llb ) \m v \ + mgyx = \m v \ + mgy2.

ll kinetic energy

CHAPTER 8

y2 = 0

[gravity only] (8-12)

Just before the rock hits the ground, where we chose y = 0, all of the initial potential energy will have been transformed into kinetic energy. fFor n object moving under the influence of E rth's gr vity, the kinetic energy of the E rth c n usu lly be ignored. For m ss oscill ting t the end of spring, the m ss of the spring, nd hence its kinetic energy, c n often be ignored.

EXAMPLE 8 - 3

Falling rock. If the original height of the rock in Fig. 8 - 7 is yx = = 3.0 m, calculate the rock's speed when it has fallen to 1.0 m above the ground. APPR ACH We apply the principle of conservation of mechanical energy, Eq. 8-12, with only gravity acting on the rock. We choose the ground as our reference level

(y =

°)-

S LUTI N At the moment of release (point 1) the rock's position is y1 = 3.0 m

and it is at rest: vx = 0. y2 = 1.0 m. Equation 8 -12 \m v \ + mgyi The ra's cancel out; setting

We want to find v2 when the rock is at position gives = \m v \ + mgy2. vx = 0 and solving for v2 we find

v2 = V ~ yi) = V 2(9 -8m/s2)[(3.0m) - (1.0m)] = 6.3m/s. The rock's speed 1.0 m above the ground is 6.3 m/s downward. N TE The velocity of the rock is independent of the rock's mass. I EXERCISE C In Ex m ple 8 - 3 , wh t is the rock 's speed just before it hits the ground? | ( ) 6.5 m /s;

(b) 7.0 m /s;

Equation 8 -12 can be applied to any object moving without friction under the action of gravity. For example, Fig. 8 - 8 shows a roller-coaster car starting from rest at the top of a hill, and coasting without friction to the bottom and up the hill on the other side. True, there is another force besides gravity acting on the car, the normal force exerted by the tracks. But this "constraint" force acts perpendicular to the direction of motion at each point and so does zero work. We ignore rotational motion of the car's wheels and treat the car as a particle undergoing simple translation. Initially, the car has only potential energy. As it coasts down the hill, it loses potential energy and gains in kinetic energy, but the sum of the two remains constant. At the bottom of the hill it has its maximum kinetic energy, and as it climbs up the other side the kinetic energy changes back to potential energy. When the car comes to rest again at the same height from which it started, all of its energy will be potential energy. Given that the gravitational potential energy is proportional to the vertical height, energy conservation tells us that (in the absence of friction) the car comes to rest at a height equal to its original height. If the two hills are the same height, the car will just barely reach the top of the second hill when it stops. If the second hill is lower than the first, not all of the car's kinetic energy will be transformed to potential energy and the car can continue over the top and down the other side. If the second hill is higher, the car will only reach a height on it equal to its original height on the first hill. This is true (in the absence of friction) no matter how steep the hill is, since potential energy depends only on the vertical height.

FIGURE 8 - 8 A roller - co ster c r m oving without friction illustr tes the conserv tion o f mech nic l energy.

I f r f M I J W n a Roller-coaster car speed using energy conservation. Assuming the height of the hill in Fig. 8 - 8 is 40 m, and the roller-coaster car starts from rest at the top, calculate {a) the speed of the roller-coaster car at the bottom of the hill, and (b) at what height it will have half this speed. Take y = 0 at the bottom of the hill. APPR ACH We choose point 1 to be where the car starts from rest (vx = 0) at the top of the hill ( ^ = 40 m). Point 2 is the bottom of the hill, which we choose as our reference level, so y2 = 0. We use conservation of mechanical energy. S LUTI N (a) We use Eq. 8 - 12 with vx = 0 and y2 = 0, which gives mgyx = \m v \ or v2 = V 2 gy1 = V /5 (9 ^ n 7 s2)(4^in) = 28 m/s. (b) We again use conservation of energy, \m v \ + mgyx = \m v \ + mgy2, but now v2 = \ (28 m/s) = 14 m/s, vx = 0, and y2is the unknown. Thus v\ yi = yx ~ Yg = 30m That is, the car has a speed of 14 m/s when it is 30 vertical meters above the lowest point, both when descending the left-hand hill and when ascending the right-hand hill. SECTI N 8 - 4

191

The mathematics of the roller-coaster Example 8 - 4 is almost the same as in Example 8 - 3. But there is an important difference between them. In Example 8 - 3 the motion is all vertical and could have been solved using force, acceleration, and the kinematic equations (Eqs. 2 -12). But for the roller coaster, where the motion is not vertical, we could not have used Eqs. 2 - 12 because a is not constant on the curved track; but energy conservation readily gives us the answer.

Pilll]

U K C NCEPTUAL EXAMPLE 8 - 5 1 Speeds on two water slides. Two water slides at a pool are shaped differently, but start at the same height (Fig. 8 -9). Two riders, Paul and Kathleen, start from rest at the same time on different slides, (a) Which rider, Paul or Kathleen, is traveling faster at the bottom? (b) Which rider makes it to the bottom first? Ignore friction and assume both slides have the same path length. RESP NSE (a) Each rider's initial potential energy mg gets transformed to kinetic energy, so the speed v at the bottom is obtained from \m v 2 = mg . The mass cancels and so the speed will be the same, regardless of the mass of the rider. Since they descend the same vertical height, they will finish with the same speed. (b) Note that Kathleen is consistently at a lower elevation than Paul at any instant, until the end. This means she has converted her potential energy to kinetic energy earlier. onsequently, she is traveling faster than Paul for the whole trip, and because the distance is the same, Kathleen gets to the bottom first. EXERCISE D Two b lls re rele sed from the s me height bove the floor. B ll A f lls freely through the ir, where s b ll B slides on curved frictionless tr ck to the floor. H ow do the speeds of the b lls comp re w hen they re ch the floor?

BLEM

LVING

Use energy, o r N ew ton ' s laws?

0 P H Y S I C S

APPLIED Sports

FIGURE 8 - 1 0 energy during

Tr nsform tion of pole v ult.

You may wonder sometimes whether to approach a problem using work and energy, or instead to use Newton's laws. As a rough guideline, if the force(s) involved are constant, either approach may succeed. If the forces are not constant, and/or the path is not simple, energy is probably the better approach. There are many interesting examples of the conservation of energy in sports, such as the pole vault illustrated in Fig. 8-10. We often have to make approximations, but the sequence of events in broad outline for the pole vault is as follows. The initial kinetic energy of the running athlete is transformed into elastic potential energy of the bending pole and, as the athlete leaves the ground, into gravitational potential energy. When the vaulter reaches the top and the pole has straightened out again, the energy has all been transformed into gravitational potential energy (if we ignore the vaulter's low horizontal speed over the bar). The pole does not supply any energy, but it acts as a device to store energy and thus aid in the transformation of kinetic energy into gravitational potential energy, which is the net result. The energy required to pass over the bar depends on how high the center of mass ( c m ) of the vaulter must be raised. By bending their bodies, pole vaulters keep their c m so low that it can actually pass slightly beneath the bar (Fig. 8 -11), thus enabling them to cross over a higher bar than would otherwise be possible. ( enter of mass is covered in hapter 9.)

FIGURE 8 - 1 1 By bending their bodies, pole v ulters c n keep their center of m ss so low th t it m y even p ss below the b r. By ch nging their kinetic energy (of running) into gr vit tion l potenti l energy ( = m g y ) in this w y, v ulters c n cross over higher b r th n if the ch nge in potenti l energy were ccomplished without c refully bending the body.

192

CHAPTER 8

Conservation of Energy

EXAMPLE 8 - 6 ESTIMATE I Pole vault. Estimate the kinetic energy and the speed required for a 70-kg pole vaulter to just pass over a bar 5.0 m high. Assume the vaulter's center of mass is initially 0.90 m off the ground and reaches its maximum height at the level of the bar itself. APPR ACH We equate the total energy just before the vaulter places the end of the pole onto the ground (and the pole begins to bend and store potential energy) with the vaulter's total energy when passing over the bar (we ignore the small amount of kinetic energy at this point). We choose the initial position of the vaulter's center of mass to be y1 = 0. The vaulter's body must then be raised to a height y2 = 5.0 m - 0.9 m = 4.1 m. S LUTI N We use Eq. 8 -12, \m v i + 0 = 0 + mgy2

K x = \m v \ = mgy2 = (70 kg)(9.8 m /s2)(4.1 m) = 2.8 X 103J. The speed is = 8.9 m/s ~ 9 m/s. 70 kg N TE This is an approximation because we have ignored such things as the vaulter's speed while crossing over the bar, mechanical energy transformed when the pole is planted in the ground, and work done by the vaulter on the pole. All would increase the needed initial kinetic energy. Vi =

As another example of the conservation of mechanical energy, let us consider an object of mass m connected to a horizontal spring (Fig. 8 - 6) whose own mass can be neglected and whose spring stiffness constant is k. The mass m has speed v at any moment. The potential energy of the system (object plus spring) is \ k x 2, where x is the displacement of the spring from its unstretched length. If neither friction nor any other force is acting, conservation of mechanical energy tells us that [elastic p e only] (8-13) \m v i + \k x \ = \m v \ + \ k x \ , where the subscripts 1 and 2 refer to the velocity and displacement at two different moments. Toy dart gun. A dart of mass 0.100 kg is pressed against the spring of a toy dart gun as shown in Fig. 8 - 12a. The spring (with spring stiffness constant k = 250 N/m and ignorable mass) is compressed 6.0 cm and released. If the dart detaches from the spring when the spring reaches its natural length (x = 0), what speed does the dart acquire? APPR ACH The dart is initially at rest (point 1), so K x = 0. We ignore friction and use conservation of mechanical energy; the only potential energy is elastic. S LUTI N We use Eq. 8 -13 with point 1 being at the maximum compression of the spring, so vx = 0 (dart not yet released) and x x = - 0.060 m. Point 2 we choose to be the instant the dart flies off the end of the spring (Fig. 8 - 12b), so x2 = 0 and we want to find v2. Thus Eq. 8 -13 can be written

FIGURE 8 - 1 2 Ex m ple 8 - 7 . ( ) A d rt is pushed g inst spring, compressing it 6.0 cm. The d rt is then rele sed, nd in (b) it le ves the spring t velocity v2.

«|=0

0 + \ k x \ = \m v 2 + 0. Then v\ =

U K

kx\ m

and

(250 N/m) ( - 0.060 m): = 3.0 m/s. (0.100 kg) N TE In the horizontal direction, the only force on the dart (neglecting friction) was the force exerted by the spring. Vertically, gravity was counterbalanced by the normal force exerted on the dart by the gun barrel. After it leaves the barrel, the dart will follow a projectile's path under gravity. v2

SECTI N 8 - 4

6. ) "cm*

I (h)£ = 4m r -

Problem Solving Using Conservation of Mechanical Energy

193

EXAMPLE 8 - 8 Two kinds of potential energy. A ball of mass m = 2.60 kg, starting from rest, falls a vertical distance = 55.0 cm before striking a vertical coiled spring, which it compresses an amount Y = 15.0 cm (Fig. 8-13). Deter mine the spring stiffness constant of the spring. Assume the spring has negligible mass, and ignore air resistance. Measure all distances from the point where the ball first touches the uncompressed spring (y = 0 at this point). APPR ACH The forces acting on the ball are the gravitational pull of the Earth and

the elastic force exerted by the spring. Both forces are conservative, so we can use conservation of mechanical energy, including both types of potential energy. We must be careful, however: gravity acts throughout the fall (Fig. 8-13), whereas the elastic force does not act until the ball touches the spring (Fig. 8-13b). We choose y positive upward, and y = 0 at the end of the spring in its natural (uncompressed) state. S LUTI N We divide this solution into two parts. (An alternate solution follows.) Part 1: Let us first consider the energy changes as the ball falls from a height yi = = 0.55 m, Fig. 8-13a, to y2 = 0, just as it touches the spring, Fig. 8-13b. Our system is the ball acted on by gravity plus the spring (which up to this point doesn't do anything). Thus \m v \ + mgy1 = \m v 2 + mgy2 = \m v 2 + 0. 0 + mg We solve for v2 = V2g = y / 2(9.80 m/s2)(0.550 m) = 3.283 m/s « 3.28 m/s. This is the speed of the ball just as it touches the top of the spring, Fig. 8-13b. Part 2: As the ball compresses the spring, Figs. 8-13b to c, there are two conservative forces on the ball—gravity and the spring force. So our conservation of energy equation is E2 (ball touches spring) = E3 (spring compressed) \m v \ + mgy2 + \k y \ = \m v \ + mgy3 + \k y \. Substituting y2 = 0, v2 = 3.283 m/s, v3 = 0 (the ball comes to rest for an instant), and y3 = —Y = - 0.150 m, we have \m v 2 + 0 + 0 = 0 - m gY + \ k ( - Y ) 2. We know m, v2, and Y, so we can solve for k\ k = - ^ H m v2 +

m gY]

= ^ [ v l + 2g Y ]

(2.60 kg) [(3.283 m /s)2 + 2(9.80 m/s2)(0.150m)] = 1590 N/m. (0.150 m)2 PR

BLEM

LVING

Alternate Solution

Alternate Solution Instead of dividing the solution into two parts, we can do it all at once. After all, we get to choose what two points are used on the left and right of the energy equation. Let us write the energy equation for points 1 and 3 in Fig. 8-13. Point 1 is the initial point just before the ball starts to fall (Fig. 8 -13a), so vx = 0, and y\ = = 0.550 m. Point 3 is when the spring is fully compressed (Fig. 8-13c), so v3 = 0, y3 = —Y = - 0.150 m. The forces on the ball in this process are gravity and (at least part of the time) the spring. So conservation of energy tells us \m v \ + mgyx + |A:(0)2 = \m v \ + mgy3 + \ k y 3 0

+ mg

+ 0

- m gY + \ k Y 2

where we have set y = 0 for the spring at point 1 because it is not acting and is not compressed or stretched. We solve for k : k =

2mg(

+ y)

2(2.60 kg)(9.80m/s2)(0.550m + 0.150 m)

just as in our first method of solution. 194

CHAPTER 8

Conservation of Energy

(0.150 m)2

1590 N/m

EXAMPLE 8 - 9 A swinging pendulum. The simple pendulum shown in Fig. 8 -14 consists of a small bob of mass m suspended by a massless cord of length I. The bob is released (without a push) at t = 0, where the cord makes an angle 0 = 0O to the vertical, (a) Describe the motion of the bob in terms of kinetic energy and potential energy. Then determine the speed of the bob (b) as a function of position 0 as it swings back and forth, and (c) at the lowest point of the swing. (d) Find the tension in the cord, FT. Ignore friction and air resistance. APPR ACH We use the law of conservation of mechanical energy (only the conser vative force of gravity does work), except in (d) where we use Newton's second law. S LUTI N (a) At the moment of release, the bob is at rest, so its kinetic energy K = 0. As the bob moves down, it loses potential energy and gains kinetic energy. At the lowest point its kinetic energy is a maximum and the potential energy is a minimum. The bob continues its swing until it reaches an equal height and angle (0O) on the opposite side, at which point the potential energy is a maximum and K = 0. It continues the swinging motion as U —» K —» U and so on, but it can never go higher than 0 = ± d 0 (conservation of mechanical energy). (b) The cord is assumed to be massless, so we need to consider only the bob's kinetic energy, and the gravitational potential energy. The bob has two forces acting on it at any moment: gravity, mg, and the force the cord exerts on it, Fx . The latter (a constraint force) always acts perpendicular to the motion, so it does no work. We need be concerned only with gravity, for which we can write the potential energy. The mechanical energy of the system is E = \m v 2 + m gy, where y is the vertical height of the bob at any moment. We take y = 0 at the lowest point of the bob's swing. Hence at t = 0, y = y0 = £ - £cos 0O = ^(l - cos0o) as can be seen from the diagram. At the moment of release E = mgy0, since v = v0 = 0. At any other point along the swing E = \m v 2 + mgy = mgy0. We solve this for v:

FIGURE 8 - 14 Ex m ple 8 - 9: simple pendulum; y is me sured positive upw rd.

v = V 2s ( » - y). In terms of the angle 0 of the cord, we can write (>b — y) = (i ~ Zcosdo) — (£ - tcosd) = £(cos 0 - cos0o) so v = '\/2g£(cos 0 - cos 0O) . (c) At the lowest point, y = 0, so » = V 2 gyi> _____________ v = \ / 2 g i ( l - cos 0O) . (d) The tension in the cord is the force FT that the cord exerts on the bob. As we've seen, there is no work done by this force, but we can calculate the force simply by using Newton's second law 2 F = ma and by noting that at any point the acceleration of the bob in the inward radial direction is v2/i, since the bob is constrained to move in an arc of a circle of radius t In the radial direction, Fx acts inward, and a component of gravity equal to mg cos 0 acts outward. Hence v2 m — = FT — mg cos 0. or

We solve for FT and use the result of part (b) for v2\ F

+ g c o s 0 j = 2mg(cos0 - cos0o) + mg cos 0

= (3 cos 0 — 2 cos 0o)mg. SECTI N 8 - 4

Problem Solving Using Conservation of Mechanical Energy

195

8 —5 The L w of Conserv tion of Energy

FIGURE 8 - 1 5 The burning of fuel ( chemic l re ction) rele ses energy to boil w ter in this ste m engine. The ste m produced exp nds g inst piston to do work in turning the wheels.

We now take into account nonconservative forces such as friction, since they are important in real situations. For example, consider again the roller-coaster car in Fig. 8 - 8, but this time let us include friction. The car will not in this case reach the same height on the second hill as it had on the first hill because of friction. In this, and in other natural processes, the mechanical energy (sum of the kinetic and potential energies) does not remain constant but decreases. Because frictional forces reduce the mechanical energy (but not the total energy), they are called dissip tive forces. Historically, the presence of dissipative forces hindered the formulation of a comprehensive conservation of energy law until well into the nineteenth century. It was not until then that heat, which is always produced when there is friction (try rubbing your hands together), was interpreted in terms of energy. Quantitative studies in the nineteenth-century ( hapter 19) demonstrated that if heat is considered as a transfer of energy (sometimes called therm l energy), then the total energy is conserved in any process. For example, if the roller-coaster car in Fig. 8 - 8 is subject to frictional forces, then the initial total energy of the car will be equal to the car's kinetic energy plus the potential energy at any subsequent point along its path plus the amount of thermal energy produced in the process. A block sliding freely across a table, for example, comes to rest because of friction. Its initial kinetic energy is all transformed into thermal energy. The block and table are a little warmer as a result of this process: both have absorbed some thermal energy. Another example of the transformation of kinetic energy into thermal energy can be observed by vigorously striking a nail several times with a hammer and then gently touching the nail with your finger. According to the atomic theory, thermal energy represents kinetic energy of rapidly moving molecules. We shall see in hapter 18 that a rise in temperature corresponds to an increase in the average kinetic energy of the molecules. Because thermal energy represents the energy of atoms and molecules that make up an object, it is often called intern l energy. Internal energy, from the atomic point of view, can include not only kinetic energy of molecules but also potential energy (usually electrical in nature) because of the relative positions of atoms within molecules. On a macroscopic level, thermal or internal energy corresponds to nonconservative forces such as friction. But at the atomic level, the energy is partly kinetic, partly potential corresponding to forces that are conservative. For example, the energy stored in food or in a fuel such as gasoline can be regarded as potential energy stored by virtue of the relative positions of the atoms within a molecule due to electric forces between atoms (referred to as chemical bonds). For this energy to be used to do work, it must be released, usually through chemical reactions (Fig. 8 -15). This is analogous to a compressed spring which, when released, can do work. To establish the more general law of conservation of energy, it required nineteenth -century physicists to recognize electrical, chemical, and other forms of energy in addition to heat and to explore if in fact they could fit into a conserva tion law. For each type of force, conservative or nonconservative, it has always been found possible to define a type of energy that corresponds to the work done by such a force. And it has been found experimentally that the total energy E always remains constant. That is, the change in the total energy, kinetic plus poten tial plus all other forms of energy, equals zero:

LAW OF CONSERVATION OF ENERGY

AK + AU + [change in all other forms of energy] = 0.

(8 -14)

This is one of the most important principles in physics. It is called the l w of conserv tion of energy and can be stated as follows: LAW OF CONSERVATION OF ENERGY 196

CHAPTER 8

The tot l energy is neither incre sed nor decre sed in ny process. Energy c n be tr nsformed from one form to nother, nd tr nsferred from one object to nother, but the tot l mount rem ins const nt.

Conservation of Energy

For conservative mechanical systems, this law can be derived from Newton's laws (Section 8 - 3) and thus is equivalent to them. But in its full generality, the validity of the law of conservation of energy rests on experimental observation. Even though Newton's laws have been found to fail in the submicroscopic world of the atom, the law of conservation of energy has been found to hold there and in every experimental situation so far tested.

8—6 Energy Conserv tion with Dissip tive Forces: Solving Problems In Section 8 - 4 we discussed several Examples of the law of conservation of energy for conservative systems. Now let us consider in detail some examples that involve nonconservative forces. Suppose, for example, that the roller-coaster car rolling on the hills of Fig. 8 - 8 is subject to frictional forces. In going from some point 1 to a second point 2, the energy dissipated by the friction force Ffr acting on the car (treating it as a particle) is Ji Ffr • dl. If Ffris constant in magnitude, the energy dissipated is simply Ffr£, where £is the actual distance along the path traveled by the object from point 1 to point 2. Thus we write our conservation of energy equation, Eq. 8-14, as AK + AU + Ffl£ = 0, or \m (v \ - v§ + mg(y2 - * ) + Fb l = 0. We can rewrite this, comparing the initial energy E1to the final energy E2: 1mv\ + mgy, = \m v \ + mgy2 + Fb l.

[fS S m g ]

(8" 15>

That is, = E2 initial energy = final energy (including thermal energy). On the left we have the mechanical energy of the system initially. It equals the mechanical energy at any subsequent point along the path plus the amount of thermal (or internal) energy produced in the process. Other nonconservative forces can be treated similarly. If you are not sure about the sign of the last term (J F • dt) on the right, use your intuition: is the mechanical energy increased or decreased in the process.

Work-En rgy v rsus En rgy Cons rvation The law of conservation of energy is more general and more powerful than the workenergy principle. Indeed, the work-energy principle should not be viewed as a statement of conservation of energy. It is nonetheless useful for some mechanical problems; and whether you use it, or use the more powerful conservation of energy, can depend on your c oice o f t e system under study. If you choose as your system a particle or rigid object on which external forces do work, then you can use the work-energy principle: the work done by the external forces on your object equals the change in its kinetic energy. On the other hand, if you choose a system on which no external forces do work, then you need to apply conservation of energy to that system directly. onsider, for example, a spring connected to a block on a frictionless table (Fig. 8-16). If you choose the block as your system, then the work done on the block by the spring equals the change in kinetic energy of the block: the work-energy principle. (Energy conservation does not apply to this system— the block's energy changes.) If instead you choose the block plus the spring as your system, no external forces do work (since the spring is part of the chosen system). To this system you need to apply conservation of energy: if you compress the spring and then release it, the spring still exerts a force on the block, but the subsequent motion can be discussed in terms of kinetic energy ( |mv2) plus potential energy ( | kx 2), whose total remains constant. SECTI N 8 - 6

FIGURE 8 - 1 6 A spring connected to block on frictionless t ble. If you choose your system to be the block plus spring, then

E = \mv2 + \k x 2 is conserved.

Energy Conservation with Dissipative Forces: Solving Problems

197

Problem solving is not a process that can be done by simply following a set of rules. The following Problem Solving Strategy, like all others, is thus not a prescription, but is a summary to help you get started solving problems involving energy.

Cons rvation of En rgy

o &

1. Dr w picture of the physical situation. 2. Determine the system for which you will apply energy conservation: the object or objects and the forces acting. 3. Ask yourself what quantity you are looking for, and choose initi l (point 1) nd fin l (point 2) positions. 4. If the object under investigation changes its height during the problem, then choose reference fr me with a convenient y = 0 level for gravitational potential energy; the lowest point in the problem is often a good choice. If springs are involved, choose the unstretched spring position to be x (or y) = 0 .

FIGURE 8 - 17 Ex m ple 8 - 1 0 . Bec use o f friction, roller - co ster c r does not re ch the origin l height on the second hill. (Not to sc le)

5. Is mech nic l energy conserved? If no friction or other nonconservative forces act, then conservation of mechanical energy holds: K x + Ux = K2 + U2. 6. Apply conserv tion of energy. If friction (or other nonconservative forces) are present, then an additional term of the form J F • dl will be needed. For a constant friction force acting over a distance I K\ + U\ = K2 + U2 + F L For other nonconservative forces use your intuition for the sign of J F • dl\ is the total mechanical energy increased or decreased in the process? 7. Use the equation(s) you develop to solve for the unknown quantity.

EXAMPLE 8 - 1 0 ESTIMATE- ! Friction on the roller-coaster car. The roller coaster car in Example 8 - 4 reaches a vertical height of only 25 m on the second hill before coming to a momentary stop (Fig. 8-17). It traveled a total distance of 400 m. Determine the thermal energy produced and estimate the average friction force (assume it is roughly constant) on the car, whose mass is 1000 kg. APPR ACH We explicitly follow the Problem Solving Strategy above. S LUTI N

1. Dr w picture. See Fig. 8-17. 2. The system. The system is the roller-coaster car and the Earth (which exerts the gravitational force). The forces acting on the car are gravity and friction. (The normal force also acts on the car, but does no work, so it does not affect the energy.) 3. Choose initi l nd fin l positions. We take point 1 to be the instant when the car started coasting (at the top of the first hill), and point 2 to be the instant it stopped 25 m up the second hill. 4. Choose reference fr me. We choose the lowest point in the motion to be y = 0 for the gravitational potential energy. 5. Is mech nic l energy conserved? No. Friction is present. 6. Apply conserv tion of energy. There is friction acting on the car, so we use conservation of energy in the form of Eq. 8 -15, with vx = 0, yx = 40 m, v2 = 0, y2 = 25 m, and I = 400 m. Thus 0 + (1000 kg) (9.8 m /s2)(40 m) = 0 + (1000kg)(9.8m/s2)(25m) + F i. 7. Solve. We solve the above equation for Ffr£, the energy dissipated to thermal energy: FfT£ = (1000 kg)(9.8 m /s2)(40 m - 25 m) = 147,000 J. The average force of friction was Ffr = (1.47 X 105j)/400m = 370 N. [This result is only a rough average: the friction force at various points depends on the normal force, which varies with slope.] 198

CHAPTER 8

Conservation of Energy

EXAMPLE 8 -11 Friction with a spring. A block of mass ra sliding along a rough horizontal surface is traveling at a speed v0 when it strikes a massless spring head-on (see Fig. 8 -18) and compresses the spring a maximum distance X. If the spring has stiffness constant k, determine the coefficient of kinetic friction between block and surface.

'I*

m m m m m (3 )

APPR ACH At the moment of collision, the block has K = \m v l and the

spring is presumably uncompressed, so U = 0. Initially the mechanical energy of the system is \m v §. By the time the spring reaches maximum compression, K = 0 and U = \ k X 2. In the meantime, the friction force (= (JiyiFN = /xkmg) has transformed energy FiTX = m g X into thermal energy. S LUTI N onservation of energy allows us to write

i—

'LIA

energy (initial) = energy (final) mB, then the c m is closer to the larger mass. If all the mass is concentrated at xB, so m A = 0, then *cm = (0*a + m Bx B)/{® + rnB) = x B, as we would expect. 23

CHAPTER 9

Linear Momentum

Now let us consider a system consisting of n particles, where n could be very large. This system could be an extended object which we consider as being made up of n tiny particles. If these n particles are all along a straight line (call it the x axis), we define the cm of the system to be located at rrixXi + m2x2 + + m nxn (9-10) — rrii + m2 + ■■■ + mn M where m 1,m 2, ...m n are the masses of each particle and x1,x 2,.. are their positions. The symbol 2"=i is the summation sign meaning to sum over all the particles, where i takes on integer values from 1 to n. (Often we simply write '2mixi , leaving out the i = 1 to n.) The total mass of the system is M = '2mi. X CM

CM three guys on a raft. Three people of roughly equal masses m o m . lightweight (air-filled) banana boat sit along the x axis at positions x A = 1.0 m, x B = 5.0 m, and x c = 6.0 m, measured from the left-hand end as shown in Fig. 9-24. Find the position of the cm . Ignore the boat's mass. APPR ACH We are given the mass and location of the three people, so we use three terms in Eq. 9-10. We approximate each person as a point particle. Equivalently, the location of each person is the position of that person's own cm . S LUTI N We use Eq. 9 - 10 with three terms: m xA + m xB + m xc

14) m

5,0 m 6.0 m

FIGURE 9 - 2 4

Ex m ple 9 - 1 4 .

FIGURE 9 - 2 5

Ex m ple 9 - 1 5 .

m {xA + x B + x c)

3m (1.0m + 5.0m .0m) 12.0 m = 4.0m. 3 3 The cm is 4.0 m from the left-hand end of the boat. This makes sense—it should be closer to the two people in front than the one at the rear. Note that the coordinates of the cm depend on the reference frame or coordinate system chosen. But the physical location of the cm is independent of that choice. EXERCISE F

the driver

alculate the cm of the three people in Example 9-14 taking the origin at 0) on the right. Is the physical location of the cm the same?

(x c =

If the particles are spread out in two or three dimensions, as for a typical extended object, then we define the coordinates of the cm as Sm/*,/n ^ _ 2 m ^_ Sm/Z/ * M— M y™ ~ M ZcM ~ M ' { ) where x yt , Zi are the coordinates of the particle of mass m t and again M = 'Zrrii is the total mass. Although from a practical point of view we usually calculate the components of the cm (Eq. 9-11), it is sometimes convenient (for example, for derivations) to write Eq. 9-11 in vector form. If ?*• = xt\ + + Z;k is the position vector of the zthparticle, and ? M = x Mi + y Mj + z Mk is the position vector of the center of mass, then M

(9-12)

EXAMPLE 9 - 15 Three particles in 2-D. Three particles, each of mass 2.50 kg, are located at the corners of a right triangle whose sides are 2.00 m and 1.50 m long, as shown in Fig. 9-25. Locate the center of mass. APPR ACH We choose our coordinate system as shown (to simplify calculations) with mA at the origin and mBon the x axis. Then raAhas coordinates xA = yA = 0; mB has coordinates xB = 2.0 m, yB = 0; and rac has coordinates xc = 2.0 m, yc = 1.5 m. S LUTI N From Eqs. 9 -11, (2.50 kg) (0) + (2.50 kg) (2.00 m) + (2.50 kg) (2.00 m) = 1.33 m Xriji 3(2.50 kg) (2.50 kg) (0) + (2.50 kg) (0) + (2.50 kg) (1.50 m) 0.50 m. 3^cm — 7.50 kg The cm and the position vector r Mare shown in Fig. 9 -25, inside the "triangle" as we should expect.

SECTI N 9 - 8

Center of Mass (c m )

231

EXERCISE G A diver does a high dive involving a flip and a half-pike (legs and arms straight, but body bent in half). What can you say about the diver's center of mass? (a) It accelerates with a magnitude of 9.8 m/s2 (ignoring air friction). (b) It moves in a circular path because of the rotation of the diver, (c) It must always be roughly located inside the diver's body, somewhere in the geometric center, (d) All of the above are true.

It is often convenient to think of an extended object as made up of a continuous distribution of matter. In other words, we consider the object to be made up of - yt , z ,, and we take n particles, each of mass Am, in a tiny volume around a point xz, the limit of n approaching infinity (Fig. 9 -26). Then Am, becomes the infinitesimal mass dm at points x, y, z. The summations in Eqs. 9 -11 and 9 - 12 become integrals: An extended object, here shown in only two dimensions, can be considered to be made up of many tiny particles (n), each having a mass Am,-. One such particle is shown located at a point fi = Xji + yi\ + n k. We take the limit of n —>oo so Am,- becomes the infinitesimal dm. FIGURE 9 - 26

FIGURE 9 -27

Example 9-16.

dm = Xdx

x -------- H dx

* CM = ~

\ x d m ' yctl = J t f y d m ' ZcM =

f z d m ' 0. To treat the general case of variable mass, let us consider the system shown in Fig. 9 - 33. At some time t, we have a system of mass M and momentum Mv. We also have a tiny (infinitesimal) mass dM traveling with velocity u which is about to enter our system. An infinitesimal time dt later, the mass dM combines with the system. For simplicity we will refer to this as a "collision." So our system has changed in mass from M to M + dM in the time dt. Note that dM can be less than zero, as for a rocket propelled by ejected gases whose mass M thus decreases. In order to apply Eq. 9 - 5, d P /d t = 2 Fext, we must consider a definite fixed system of particles. That is, in considering the change in momentum, dP we must consider the momentum of the same particles initially and finally. We will define our total system as including M plus dM. Then initially, at time t, the total momentum is Mv + u dM (Fig. 9 - 33). At time t + dt, after dM has combined with M, the velocity of the whole is now v + dy and the total momentum is (M + dM)(y + dy). So the change in momentum dP is dP = (M + dM)(y + dy) - (M y + u dM) = M dy + v dM + dM dy — u dM. The term dM dy is the product of two differentials and is zero even after we "divide by d t" which we do, and apply Eq. 9 - 5 to obtain dP

2F ext — —— — ext dt

M dy + y dM - u dM dt

Thus we get -* dy £ F ext = M j -

(a — v ) ^ -

(9 -19a)

Note that the quantity (u —v) is the relative velocity, vrei, of dM with respect to M. 236

CHAPTER 9

Linear Momentum

That is, Vrel =

U -

is the velocity of the entering mass dM as seen by an observer on M. We can rearrange Eq. 9-19a:

= 2Fext + vrelf l

(9-19b)

We can interpret this equation as follows. M dv/dt is the mass times the acceleration of M. The first term on the right, 2 F ext, refers to the external force on the mass M (for a rocket, it would include the force of gravity and air resistance). It does not include the force that dM exerts on M as a result of their collision. This is taken care of by the second term on the right, %ex(dM/dt), which represents the rate at which momentum is being transferred into (or out of) the mass M because of the mass that is added to (or leaves) it. It can thus be interpreted as the force exerted on the mass M due to the addition (or ejection) of mass. For a rocket this term is called the t rust, since it represents the force exerted on the rocket by the expelled gases. For a rocket ejecting burned fuel, dM /dt < 0, but so is vrei (gases are forced out the back), so the second term in Eq. 9 - 19b acts to increase v. EXAMPLE 9 - 1 9 Conveyor belt. You are designing a conveyor system for a gravel yard. A hopper drops gravel at a rate of 75.0 kg/s onto a conveyor belt that moves at a constant speed v = 2.20 m/s (Fig. 9-34). (a) Determine the additional force (over and above internal friction) needed to keep the conveyor belt moving as gravel falls on it. (b) What power output would be needed from the motor that drives the conveyor belt?

PHYSICS

APPLIED

M ovin g con veyor belt FIGURE 9 - 3 4 Ex m ple 9 - 1 9 . Gr vel dropped from hopper onto conveyor belt.

APPR ACH We assume that the hopper is at rest so u = 0, and that the hopper

has just begun dropping gravel so dM /dt = 75.0 kg/s. S LUTI N (a) The belt needs to move at a constant speed (dv/dt = 0), so Eq. 9 -19 as written for one dimension, gives:

ext " = 0 -(0 -v ) —

, , dv . . dM d f ~ { u ~ v ) ~dT n

= VW

\ d M

= (2-2° m/s) (75.0 kg/s) = 165 N.

(b) This force does work at the rate (Eq. 8 -21) dW * 9dM d T = F- ' v = v d T = 363 W, which is the power output required of the motor. N TE This work does not all go into kinetic energy of the gravel, since dK dt

d (1 w

= — - M r d t\ 2

1 dM , 2 dt

which is only half the work done by Fexf The other half of the external work done goes into thermal energy produced by friction between the gravel and the belt (the same friction force that accelerates the gravel). *SECTI N 9 - 1 0

Systems of Variable Mass; Rocket Propulsion

237

P HY S I CS APPLI ED R ocket propu lsion

^rocket

EXAMPLE 9 - 20 Rocket propulsion. A fully fueled rocket has a mass of 21,000 kg, of which 15,000 kg is fuel. The burned fuel is spewed out the rear at a rate of 190 kg/s with a speed of 2800 m/s relative to the rocket. If the rocket is fired vertically upward (Fig. 9 -35) calculate: (a) the thrust of the rocket; (b) the net force on the rocket at blastoff, and just before burnout (when all the fuel has been used up); (c) the rocket's velocity as a function of time, and (d) its final velocity at burnout. Ignore air resistance and assume the acceleration due to gravity is constant at g = 9.80 m /s2. APPR ACH To begin, the thrust is defined (see discussion after Eq. 9 - 19b) as

the last term in Eq. 9 - 19b, vTe\(dM/dt). The net force [for (b)] is the vector sum of the thrust and gravity. The velocity is found from Eq. 9-19b. S LUTI N (a) The thrust is: ^thrust =

= ( - 2800 m /s)( - 190 kg/s) = 5.3 X 105N,

where we have taken upward as positive so vrQ\ is negative because it is down ward, and dM /dt is negative because the rocket's mass is diminishing. (b) Fext = M g = (2.1 X 104kg)(9.80m/s2) = 2.1 X 105N initially, and at burnout Fext = (6.0 X 103kg)(9.80m/s2) = 5.9 X 104N. Hence, the net force on the rocket at blastoff is Fnet = 5.3 X 105N - 2.1 X 105N = 3.2 X 105N,

Vg ses

FIGURE 9 - 35

Ex mple 9 - 20;

vrel = vg ses — vrocket • M is the m ss of the rocket t ny inst nt nd is decre sing until burnout.

[blastoff]

and just before burnout it is Fnet = 5.3 X 105N - 5.9 X 104N = 4.7 X 105N.

[burnout]

After burnout, of course, the net force is that of gravity, - 5.9 X 104N. (c) From Eq. 9 -19b we have ^ M

M '

where ^ext = - M g , and M is the mass of the rocket and is a function of time. Since vTe\ is constant, we can integrate this easily: U JVn

, M dM - ( ' g d t + t)rei | — JO

or v(t) = v0 - gt + vxe\ In ——s where v(t) is the rocket's velocity and M its mass at any time t. Note that urel is negative ( - 2800 m /s in our case) because it is opposite to the motion, and that In (M /M q) is also negative because M0 > M. Hence, the last term — which represents the thrust—is positive and acts to increase the velocity. (d) The time required to reach burnout is the time needed to use up all the fuel (15,000 kg) at a rate of 190 kg/s; so at burnout, t

1.50 X 104kg 190 kg/s

---------------------- =

79 s

If we take v0 = 0, then using the result of part (c): v = - (9.80m /s2)(79 s) + ( - 2800 m/s) ^ln

238

CHAPTER 9

Linear Momentum

6000 kg 21,000 kg

= 2700 m/s.

Summ ry The line r momentum, p, of an object is defined as the product of its mass times its velocity, p = my.

(9 - 1)

In terms of momentum, Newton's second l w can be written as ^ dfi 2F = ~ (9 - 2) dt That is, the rate of change of momentum of an object equals the net force exerted on it. When the net external force on a system of objects is zero, the total momentum remains constant. This is the l w of conser v tion of momentum. Stated another way, the total momentum of an isolated system of objects remains constant. The law of conservation of momentum is very useful in dealing with the class of events known as collisions. In a colli sion, two (or more) objects interact with each other for a very short time, and the force each exerts on the other during this time interval is very large compared to any other forces acting. The impulse of such a force on an object is defined as J =

| F dt

and is equal to the change in momentum of the object as long as F is the net force on the object: Ap = pf - pi =

F dt = J.

(9 - 6)

SmjXj M

yCM —

or * M = T7 [ * dm, M

ycM = 7 7 [ y dm, M

z M = 77 \ z dm, M (9 - 13)

where M is the total mass of the system. The center of mass of a system is important because this point moves like a single particle of mass M acted on by the same net external force, 2F ext. In equation form, this is just Newton's second law for a system of particles (or extended objects): M arM = 2 Fext >

(9 - 17)

where M is the total mass of the system, a Mis the acceleration of the cm of the system, and 2 Fext is the total (net) external force acting on all parts of the system. For a system of particles of total linear momentum P = Sm/Vj = M vcm, Newton's second law is dP ~dt = 2Fext'

Total momentum is conserved in any collision: Pa + Pb = Pa + Pb The total energy is also conserved; but this may not be useful unless kinetic energy is conserved, in which case the collision is called an el stic collision: \ m A v \ + \ m BvI = \ m A v'l + %mBv g .

If two colliding objects stick together as the result of a collision, the collision is said to be com pletely inel stic. For a system of particles, or for an extended object that can be considered as having a continuous distribution of matter, the center o f m ss (cm) is defined as

(9 - 7)

If kinetic energy is not conserved, the collision is called inel stic.

(9_5)

[*If the mass M of an object is not constant, then = 2F ext +

(9 - 19b)

where v is the velocity of the object at any instant and vrei is the relative velocity at which mass enters (or leaves) the object.]

Questions 1. We claim that momentum is conserved. Yet most moving objects eventually slow down and stop. Explain. 2. Two blocks of mass m \ and m 2 rest on a frictionless table and are connected by a spring. The blocks are pulled apart, stretching the spring, and then released. Describe the subse quent motion of the two blocks. 3. A light object and a heavy object have the same kinetic energy. Which has the greater momentum? Explain. 4. When a person jumps from a tree to the ground, what happens to the momentum of the person upon striking the ground? 5. Explain, on the basis of conservation of momentum, how a fish propels itself forward by swishing its tail back and forth. . Two children float motionlessly in a space station. The 20-kg girl pushes on the 40-kg boy and he sails away at 1.0 m/s. The girl (a) remains motionless; (b) moves in the same direction at 1.0 m/s; (c) moves in the opposite direction at 1.0m/s; (d) moves in the opposite direction at 2.0 m/s; (1e) none of these.

7. A truck going 15 km /h has a head-on collision with a small car going 30 km /h. Which statement best describes the situation? (a) The truck has the greater change of momentum because it has the greater mass, (b) The car has the greater change of momentum because it has the greater speed, (c) Neither the car nor the truck changes its momentum in the collision because momentum is conserved, (d) They both have the same change in magnitude of momentum because momentum is conserved. (e) None of the above is necessarily true. 8. If a falling ball were to make a perfectly elastic collision with the floor, would it rebound to its original height? Explain. 9. A boy stands on the back of a rowboat and dives into the water. What happens to the rowboat as the boy leaves it? Explain. 10. It is said that in ancient times a rich man with a bag of gold coins was stranded on the surface of a frozen lake. Because the ice was frictionless, he could not push himself to shore and froze to death. What could he have done to save himself had he not been so miserly?

Questions

239

11. The speed of a tennis ball on the return of a serve can be just as fast as the serve, even though the racket isn't swung very fast. How can this be? 12. Is it possible for an object to receive a larger impulse from a small force than from a large force? Explain. 13. How could a force give zero impulse over a nonzero time interval even though the force is not zero for at least a part of that time interval? 14. In a collision between two cars, which would you expect to be more damaging to the occupants: if the cars collide and remain together, or if the two cars collide and rebound backward? Explain. 15. A superball is dropped from a height onto a hard steel plate (fixed to the Earth), from which it rebounds at very nearly its original speed, (a) Is the momentum of the ball conserved during any part of this process? (b) If we consider the ball and the Earth as our system, during what parts of the process is momentum conserved? (c) Answer part (b) for a piece of putty that falls and sticks to the steel plate. 16. ars used to be built as rigid as possible to withstand colli sions. Today, though, cars are designed to have "crumple zones" that collapse upon impact. What is the advantage of this new design? 17. A t a hydroelectric power plant, water is directed at high speed against turbine blades on an axle that turns an elec tric generator. For maximum power generation, should the turbine blades be designed so that the water is brought to a dead stop, or so that the water rebounds? 18. A squash ball hits a wall at a 45° angle as shown in Fig. 9 -36. What is the direction (a) of the change in momentum of the ball, (b) of the force on the wall?

FIGURE 9 - 3 6

Question 18. 19. Why can a batter hit a pitched baseball farther than a ball he himself has tossed up in the air? 20. Describe a collision in which all kinetic energy is lost.

21. Inelastic and elastic collisions are similar in that (a) momentum and kinetic energy are conserved in both; (b) momentum is conserved in both; (c) momentum and potential energy are conserved in both; (d) kinetic energy is conserved in both. 22. If a 20-passenger plane is not full, sometimes passengers are told they must sit in certain seats and may not move to empty seats. Why might this be? 23. Why do you tend to lean backward when carrying a heavy load in your arms? 24. Why is the cm of a 1-m length of pipe at its midpoint, whereas this is not true for your arm or leg? 25. Show on a diagram how your cm shifts when you move from a lying position to a sitting position. 26. Describe an analytic way of determining the cm of any thin, triangular-shaped, uniform plate. 27. Place yourself facing the edge of an open door. Position your feet astride the door with your nose and abdomen touching the door's edge. Try to rise on your tiptoes. Why can't this be done? 28. If only an external force can change the momentum of the center of mass of an object, how can the internal force of the engine accelerate a car? 29. A rocket following a parabolic path through the air suddenly explodes into many pieces. What can you say about the motion of this system of pieces? 30. How can a rocket change direction when it is far out in space and essentially in a vacuum? 31. In observations of nuclear /3-decay, the electron and recoil nucleus often do not separate along the same line. Use conservation of momentum in two dimensions to explain why this implies the emission of at least one other particle in the disintegration. 32. Bob and Jim decide to play tug-of-war on a frictionless (icy) surface. Jim is considerably stronger than Bob, but Bob weighs 160 lbs while Jim weighs 145 lbs. Who loses by crossing over the midline first? 33. A t a carnival game you try to knock over a heavy cylinder by throwing a small ball at it. You have a choice of throwing either a ball that will stick to the cylinder, or a second ball of equal mass and speed that will bounce backward off the cylinder. Which ball is more likely to make the cylinder move?

| Problems__________________ 9 -1

M o m e n tu m

1. (I) alculate the force exerted on a rocket when the propelling gases are being expelled at a rate of 1300 kg/s with a speed of 4.5 X 104m/s. 2. (I) A constant friction force of 25 N acts on a 65-kg skier for 15 s. What is the skier's change in velocity? 3. (II) The momentum of a particle, in SI units, is given by p = 4.812\ - 8.0j - 8.9 tk. What is the force as a function of time? 4. (II) The force on a particle of mass m is given by F = 26i - 1 2 12j where F is in N and t in s. What will be the change in the particle's momentum between t = 1.0 s and t = 2.0 s? 5. (II) A 145-g baseball, moving along the x axis with speed 30.0 m/s, strikes a fence at a 45° angle and rebounds along the y axis with unchanged speed. Give its change in momentum using unit vector notation. 24

HAPTER 9

Linear Momentum

. (II) A 0.145-kg baseball pitched horizontally at 32.0 m /s strikes a bat and is popped straight up to a height of 36.5 m. If the contact time between bat and ball is 2.5 ms, calculate the average force between the ball and bat during contact. 7. (II) A rocket of total mass 3180 kg is traveling in outer space with a velocity of 115 m /s. To alter its course by 35.0°, its rockets can be fired briefly in a direction perpendicular to its original motion. If the rocket gases are expelled at a speed of 1750 m/s, how much mass must be expelled? 8. (Ill) Air in a 120 -km/h wind strikes head -on the face of a building 45 m wide by 65 m high and is brought to rest. If air has a mass of 1.3 kg per cubic meter, determine the average force of the wind on the building.

9 - 2 Conservation of Momentum 9. (I) A 7700-kg boxcar traveling 18 m /s strikes a second car. The two stick together and move off with a speed of 5.0m/s. What is the mass of the second car?

10. (I) A 9150-kg railroad car travels alone on a level frictionless track with a constant speed of 15.0 m/s. A 4350-kg load, initially at rest, is dropped onto the car. What will be the car's new speed? 11. (I) An atomic nucleus at rest decays radioactively into an alpha particle and a smaller nucleus. What will be the speed of this recoiling nucleus if the speed of the alpha particle is 2.8 X 105m/s? Assume the recoiling nucleus has a mass 57 times greater than that of the alpha particle. 12. (I) A 130-kg tackier moving at 2.5 m/s meets head-on (and tackles) an 82-kg halfback moving at 5.0 m/s. What will be their mutual speed immediately after the collision? 13. (II) A child in a boat throws a 5.70-kg package out horizontally with a speed of 10.0 m/s, Fig. 9-37. alculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is 24.0 kg and that of the boat is 35.0 kg.

FIGURE 9 - 37

Problem 13. 14. (II) An atomic nucleus initially moving at 420 m/s emits an alpha particle in the direction of its velocity, and the remaining nucleus slows to 350 m/s. If the alpha particle has a mass of 4.0 u and the original nucleus has a mass of 222 u, what speed does the alpha particle have when it is emitted? 15. (II) An object at rest is suddenly broken apart into two frag ments by an explosion. One fragment acquires twice the kinetic energy of the other. What is the ratio of their masses? 16. (II) A 22-g bullet traveling 210 m/s penetrates a 2.0-kg block of wood and emerges going 150 m/s. If the block is stationary on a frictionless surface when hit, how fast does it move after the bullet emerges? 17. (II) A rocket of mass m traveling with speed v0 along the x axis suddenly shoots out fuel equal to one-third its mass, perpendicular to the x axis (along the y axis) with speed 2v0. Express the final velocity of the rocket in i, j, k notation. 18. (II) The decay of a neutron into a proton, an electron, and a neutrino is an example of a three-particle decay process. Use the vector nature of momentum to show that if the neutron is initially at rest, the velocity vectors of the three must be coplanar (that is, all in the same plane). The result is not true for numbers greater than three. 19. (II) A mass raA = 2.0 kg, moving with velocity \ A = (4.0i + 5.0j - 2.0k) m/s, collides with mass mB = 3.0 kg, which is initially at rest. Immediately after the collision, mass mA is observed traveling at velocity \ A = ( - 2 .0 i + 3.0k) m/s. Find the velocity of mass raB after the collision. Assume no outside force acts on the two masses during the collision. 20. (II) A 925-kg two-stage rocket is traveling at a speed of 6.60 X 103m/s away from Earth when a predesigned explo sion separates the rocket into two sections of equal mass that then move with a speed of 2.80 X 103m/s relative to each other along the original line of motion, (a) What is the speed and direction of each section (relative to Earth) after the explosion? (b) How much energy was supplied by the explosion? [Hint: What is the change in kinetic energy as a result of the explosion?]

21. (Ill) A 224-kg projectile, fired with a speed of 116 m/s at a 60.0° angle, breaks into three pieces of equal mass at the highest point of its arc (where its velocity is horizontal). Two of the fragments move with the same speed right after the explosion as the entire projectile had just before the explosion; one of these moves vertically downward and the other horizontally. Determine (a) the velocity of the third fragment immediately after the explosion and (b) the energy released in the explosion.

9 - 3 Collisions and Impulse 22. (I) A 0.145-kg baseball pitched at 35.0 m/s is hit on a hori zontal line drive straight back at the pitcher at 56.0 m/s. If the contact time between bat and ball is 5.00 X 10-3 s, calcu late the force (assumed to be constant) between the ball and bat. 23. (II) A golf ball of mass 0.045 kg is hit off the tee at a speed of 45 m/s. The golf club was in contact with the ball for 3.5 X 10 3s. Find (a) the impulse imparted to the golf ball, and (b) the average force exerted on the ball by the golf club. 24. (II) A 12-kg hammer strikes a nail at a velocity of 8.5 m/s and comes to rest in a time interval of 8.0 ms. (a) What is the impulse given to the nail? (b) What is the average force acting on the nail? 25. (II) A tennis ball of mass m = 0.060 kg and speed v = 25 m/s strikes a wall at a 45° angle and rebounds with the same speed at 45° (Fig. 9-38). What is the impulse (magnitude and direction) given to the ball? v V 45 \ 45° FIGURE 9 - 38

Problem 25. 26. (II) A 130-kg astronaut (including space suit) acquires a speed of 2.50 m/s by pushing off with his legs from a 1700-kg space capsule, (a) What is the change in speed of the space capsule? (b) If the push lasts 0.500 s, what is the average force exerted by each on the other? As the reference frame, use the position of the capsule before the push, (c) What is the kinetic energy of each after the push? 27. (II) Rain is falling at the rate of 5.0 cm/h and accumulates in a pan. If the raindrops hit at 8.0 m/s, estimate the force on the bottom of a 1.0 m2 pan due to the impacting rain which does not rebound. Water has a mass of 1.00 X 103kg per m3. 28. (II) Suppose the force acting on a tennis ball (mass 0.060 kg) points in the +x direction and is given by the graph of Fig. 9 -39 as a function of time. Use graphical methods to estimate (a) the total impulse given the ball, and (b) the velocity of the ball after being struck, assuming the ball is being served so it is nearly at rest initially.

FIGURE 9 - 39

Problem 28.

Problems

241

29. (II) With what impulse does a 0.50-kg newspaper have to be thrown to give it a velocity of 3.0 m/s? 30. (II) The force on a bullet is given by the formula F = [ 7 4 0 — (2 .3 X 1 0 5 s _ 1 ) ? ] N over the time interval t = 0 to t = 3 .0 X 1 0 - 3 s. (a) Plot a graph of F versus t for t = 0 to t = 3 .0 ms. (b) Use the graph to estimate the impulse given the bullet, (c) Determine the impulse by inte gration. (d) If the bullet achieves a speed of 2 6 0 m /s as a result of this impulse, given to it in the barrel of a gun, what must the bullet's mass be? (e) What is the recoil speed of the 4.5-kg gun? 31. (II) (a) A molecule of mass m and speed v strikes a wall at right angles and rebounds back with the same speed. If the collision time is At, what is the average force on the wall during the collision? (b) If molecules, all of this type, strike the wall at intervals a time t apart (on the average) what is the average force on the wall averaged over a long time? 32. (Ill) (a) alculate the impulse experienced when a 65-kg person lands on firm ground after jumping from a height of 3.0 m. (ib) Estimate the average force exerted on the person's feet by the ground if the landing is stiff-legged, and again (c) with bent legs. With stiff legs, assume the body moves 1.0 cm during impact, and when the legs are bent, about 50 cm. [Hint: The average net force on her which is related to impulse, is the vector sum of gravity and the force exerted by the ground.] 33. (Ill) A scale is adjusted so that when a large, shallow pan is placed on it, it reads zero. A water faucet at height = 2.5 m above is turned on and water falls into the pan at a rate R = 0.14 kg/s. Determine (a) a formula for the scale reading as a function of time t and (b) the reading for t = 9.0 s. (c) Repeat (a) and (b), but replace the shallow pan with a tall, narrow cylindrical container of area A = 20 cm2 (the level rises in this case).

9 - 4 and 9 - 5 Elastic Collisions 34. (II) A 0.060-kg tennis ball, moving with a speed of 4.50 m/s, has a head -on collision with a 0.090-kg ball initially moving in the same direction at a speed of 3.00 m /s. Assuming a perfectly elastic collision, determine the speed and direction of each ball after the collision. 35. (II) A 0.450-kg hockey puck, moving east with a speed of 4.80 m/s, has a head -on collision with a 0.900-kg puck initially at rest. Assuming a perfectly elastic collision, what will be the speed and direction of each object after the collision? 36. (II) A 0.280-kg croquet ball makes an elastic head -on collision with a second ball initially at rest. The second ball moves off with half the original speed of the first ball. (a) What is the mass of the second ball? (b) What fraction of the original kinetic energy (AK /K ) gets transferred to the second ball? 37. (II) A ball of mass 0.220 kg that is moving with a speed of 7.5 m /s collides head -on and elastically with another ball initially at rest. Immediately after the collision, the incoming ball bounces backward with a speed of 3.8 m/s. alculate (a) the velocity of the target ball after the collision, and (b) the mass of the target ball. 38. (II) A ball of mass m makes a head -on elastic collision with a second ball (at rest) and rebounds with a speed equal to 0.350 its original speed. What is the mass of the second ball? 242

HAPTER 9

Linear Momentum

39. (II) Determine the fraction of kinetic energy lost by a neutron (mi = 1.01 u) when it collides head -on and elastically with a target particle at rest which is (a) }H (m = 1.01 u); (b) 2H (heavy hydrogen, m = 2.01 u); (c) x\ (m = 12.00 u); (d) 2jgPb (lead, m = 208 u). 40. (II) Show that, in general, for any head -on one-dimensional elastic collision, the speeds after collision are 2m A fm B - mA vb = vA -----7 ----A + mB mA + raB + M \ m and 2m B ( m A — m B\ ( + % \ m A + m B/ where vA and vB are the initial speeds of the two objects of mass m A and m B. 41. (Ill) A 3.0-kg block slides along a frictionless tabletop at 8.0 m /s toward a second block (at rest) of mass 4.5 kg. A coil spring, which obeys Hooke's law and has spring constant k = 850 N /m , is attached to the second block in such a way that it will be compressed when struck by the moving block, Fig. 9 - 40. (a) What will be the maximum compression of the spring? (b) What will be the final velocities of the blocks after the collision? (c) Is the collision elastic? Ignore the mass of the spring. 3.0 kg

v = 8.0 m/s

FIGURE 9 - 4 0

MMM i f f w r 4.5 kg

Problem 41.

9 - 6 Inelastic Collisions 42. (I) In a ballistic pendulum experiment, projectile 1 results in a maximum height of the pendulum equal to 2.6 cm. A second projectile (of the same mass) causes the the pendulum to swing twice as high, 2 = 5.2 cm. The second projectile was how many times faster than the first? 43. (II) (a) Derive a formula for the fraction of kinetic energy lost, AK /K , in terms of m and M for the ballistic pendulum collision of Example 9 - 11. (b) Evaluate for m = 16.0 g and M = 380 g. 44. (II) A 28-g rifle bullet traveling 210 m /s buries itself in a 3.6-kg pendulum hanging on a 2.8-m-long string, which makes the pendulum swing upward in an arc. Determine the vertical and horizontal components of the pendulum's maximum displacement. 45. (II) An internal explosion breaks an object, initially at rest, into two pieces, one of which has 1.5 times the mass of the other. If 7500 J is released in the explosion, how much kinetic energy does each piece acquire? 46. (II) A 920-kg sports car collides into the rear end of a 2300-kg SUV stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 2.8 m before stopping. The police officer, estimating the coefficient of kinetic friction between tires and road to be 0.80, calculates the speed of the sports car at impact. What was that speed? 47. (II) You drop a 12-g ball from a height of 1.5 m and it only bounces back to a height of 0.75 m. What was the total impulse on the ball when it hit the floor? (Ignore air resistance). 48. (II) ar A hits car B (initially at rest and of equal mass) from behind while going 35 m /s. Immediately after the collision, car B moves forward at 25 m /s and car A is at rest. What fraction of the initial kinetic energy is lost in the collision?

49. (II) A measure of inelasticity in a head -on collision of two objects is the coefficient o f restitution, e, defined as Va ~ vb e = vb ~ v a where vA — vB is the relative velocity of the two objects after the collision and — vA is their relative velocity before it. (a) Show that e = 1 for a perfectly elastic collision, and e = 0 for a completely inelastic collision, (b) A simple method for measuring the coefficient of restitution for an object colliding with a very hard surface like steel is to drop the object onto a heavy steel plate, as shown in Fig. 9 -41. Determine a formula for e in terms of the original height and the maximum height ' I I I reached after collision. I I i ! 1i 1i n'

FIGURE 9 - 41 Problem 49. Measurement of coefficient of restitution.

vAf | v A

50. (II) A pendulum consists of a mass M hanging at the bottom end of a massless rod of length £, which has a frictionless pivot at its top end. A mass m, moving as shown N\ in Fig. 9 - 42 with velocity v, impacts \ \ M and becomes embedded. What is the smallest value of v sufficient \ to cause the pendulum (with \ _ embedded mass m ) to swing clear over the top of its arc? I / FIGURE 9 - 4 2

Problem 50.

51. (II) A bullet of mass m = 0.0010 kg embeds itself in a wooden block with mass M = 0.999 kg, which then compresses a spring (k = 120 N /m ) by a distance x = 0.050 m before coming to rest. The coefficient of kinetic friction between the block and table is fi = 0.50. (a) What is the initial speed of the bullet? (b) What fraction of the bullet's initial kinetic energy is dissipated (in damage to the wooden block, rising temperature, etc.) in the collision between the bullet and the block? 52. (II) A 144-g baseball moving 28.0 m /s strikes a stationary 5.25-kg brick resting on small rollers so it moves without significant friction. After hitting the brick, the baseball bounces straight back, and the brick moves forward at 1.10 m/s. (a) What is the baseball's speed after the collision? (b) Find the total kinetic energy before and after the collision. 53. (II) A 6.0-kg object moving in the +x direction at 5.5 m /s collides head - on with an 8.0-kg object moving in the —x direction at 4.0 m/s. Find the final velocity of each mass if: (a) the objects stick together; (b) the collision is elastic; (c) the .0-kg object is at rest after the collision; (d) the 8.0-kg object is at rest after the collision; (e) the .0-kg object has a velocity of 4.0 m /s in the —x direction after the collision. Are the results in (c), (d), and (e) "reasonable"? Explain.

9 - 7 Collisions in Two Dimensions 54. (II) Billiard ball A of mass mA = 0.120 kg moving with speed vA = 2.80 m /s strikes ball B, initially at rest, of mass me = 0.140 kg. As a result of the collision, ball A is deflected off at an angle of 30.0° with a speed vA = 2.10 m/s. (a) Taking the x axis to be the original direction of motion of ball A, write down the equations expressing the conservation of momentum for the components in the x and y directions separately, (b) Solve these equations for the speed, vB, and angle, 0'B , of ball B. Do not assume the collision is elastic. 55. (II) A radioactive nucleus at rest decays into a second nucleus, an electron, and a neutrino. The electron and neutrino are emitted at right angles and have momenta of 9.6 X 10_23kg - m /s and 6.2 X 10_23kg - m /s, respectively. Determine the magnitude and the direction of the momentum of the second (recoiling) nucleus. 56. (II) Two billiard balls of equal mass move at right angles and meet at the origin of an xy coordinate system. Initially ball A is moving upward along the y axis at 2.0 m /s, and ball B is moving to the right along the x axis with speed 3.7 m/s. After the collision (assumed elastic), the second ball is moving along the positive y axis (Fig. 9 -43). What is the final direction of ball A, and what are the speeds of the two balls? +x B 0 vB = 3.7 m/s va

= 2 .0

m/s

Problem 56. (Ball A after the collision is not shown.)

FIGURE 9 - 4 3

57. (II) An atomic nucleus of mass m traveling with speed v collides elastically with a target particle of mass 2m (initially at rest) and is scattered at 90°. (a) A t what angle does the target particle move after the collision? (b) What are the final speeds of the two particles? (c) What fraction of the initial kinetic energy is transferred to the target particle? 58. (II) A neutron collides elastically with a helium nucleus (at rest initially) whose mass is four times that of the neutron. The helium nucleus is observed to move off at an angle #He = 45°. Determine the angle of the neutron, 0'n , and the speeds of the two particles, v 'n and ^He >after the colli sion. The neutron's initial speed is 6.2 X 105 m/s. 59. (Ill) A neon atom (m = 20.0 u) makes a perfectly elastic collision with another atom at rest. After the impact, the neon atom travels away at a 55.6° angle from its original direction and the unknown atom travels away at a - 50.0° angle. What is the mass (in u) of the unknown atom? [Hint: You could use the law of sines.] 60. (Ill) For an elastic collision between a projectile particle of mass mA and a target particle (at rest) of mass mB, show that the scattering angle, 0A , of the projectile (a) can take any value, 0 to 180°, for mA < mB, but (b) has a maximum angle (f) given by cos2(f> = 1 — (mB/m A)2 for m A > mB. 61. (Ill) Prove that in the elastic collision of two objects of identical mass, with one being a target initially at rest, the angle between their final velocity vectors is always 90°.

Problems

243

9 - 8 Center o f Mass (cm )

9 - 9 CM and Translational Motion

62. (I) The cm of an empty 1250-kg car is 2.50 m behind the front of the car. How far from the front of the car will the cm be when two people sit in the front seat 2.80 m from the front of the car, and three people sit in the back seat 3.90 m from the front? Assume that each person has a mass of 70.0 kg. 63. (I) The distance between a carbon atom (m = 12 u) and an oxygen atom (m = 16 u) in the O molecule is 1.13 X 10- 10m. How far from the carbon atom is the center of mass of the molecule? 64. (II) Three cubes, of side £0, 2£0, and 3£0, are placed next to one another (in contact) with their centers along a straight line as shown in Fig. 9 - 44. What is the position, along this line, of the cm of this r _ q system? Assume the cubes are made of the same uniform material.

72. (II) Mass Ma = 35 kg and mass AfB = 25 kg. They have velocities (in m /s) vA = 12i - 16j and vB = - 2 0 i + 14j. Determine the velocity of the center of mass of the system. 73. (II) The masses of the Earth and Moon are 5.98 X 1024kg and 7.35 X 1022kg, respectively, and their centers are sepa rated by 3.84 X 108m. ( ) Where is the cm of this system located? (b) What can you say about the motion of the E arth - M oon system about the Sun, and of the Earth and Moon separately about the Sun? 74. (II) A mallet consists of a uniform cylindrical head of mass 2.80kg and a diameter 0.0800 m mounted on a uniform cylindrical handle of mass 0.500 kg and length 0.240 m, as shown in Fig. 9 - 47. If this mallet is tossed, spinning, into the air, how far above the bottom of the handle is the point that will follow a parabolic c trajectory? 24.0 cm

FIGURE 9 - 4 4 FIGURE 9 - 47

- V — 2f 0-

Problem 64.

65. (II) A square uniform raft, 18 m by 18 m, of mass 6200 kg, is used as a ferryboat. If three cars, each of mass 1350 kg, occupy the NE, SE, and SW corners, determine the cm of the loaded ferryboat relative to the center of the raft. . (II) A uniform circular plate of radius 2R has a circular hole of radius R cut out of it. The center ' of the smaller circle is a distance 0.80i? from the center of the larger circle, Fig. 9 - 45. What is the position of the center of mass of the plate? [Hint: Try subtraction.]

75.

76. 77.

78. FIGURE 9 - 4 5

Problem

67. (II) A uniform thin wire is bent into a semicircle of radius r. Determine the coordinates of its center of mass with respect to an origin of coordinates at the center of the "full" circle. 8. (II) Find the center of mass of the ammonia molecule. The chemical formula is N H 3. The hydrogens are at the corners of an equilateral triangle (with sides 0.16 nm) that forms the base of a pyramid, with nitrogen at the apex (0.037 nm verti cally above the plane of the triangle). 69. (Ill) Determine the cm of a machine part that is a uniform cone of height and radius R, Fig. 9 - 46. [Hint: Divide the cone into an infinite number ^ of disks of thickness dz, one of which is shown.l

/ FIGURE 9 - 4 6

Problem 69.

70. (Ill) Determine the cm of a uniform pyramid that has four triangular faces and a square base with equal sides all of length s. [Hint: See Problem 69.] 71. (Ill) Determine the cm of a thin, uniform, semicircular plate. 244

HAPTER 9

Linear Momentum

79.

Problem 74. *00 cm (II) A 55-kg woman and a 72-kg man stand 10.0 m apart on frictionless ice. (a) How far from the woman is their cm? (b) If each holds one end of a rope, and the man pulls on the rope so that he moves 2.5 m, how far from the woman will he be now? (c) How far will the man have moved when he collides with the woman? (II) Suppose that in Example 9 - 18 (Fig. 9 - 32), ran = 3m i. (a) Where then would ran land? (b) What if mi = 3mn ? (II) Two people, one of mass 85 kg and the other of mass 55 kg, sit in a rowboat of mass 78 kg. With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat, 3.0 m apart from each other, now exchange seats. How far and in what direction will the boat move? (Ill) A 280-kg flatcar 25 m long is moving with a speed of 6.0 m /s along horizontal frictionless rails. A 95-kg worker starts walking from one end of the car to the other in the direction of motion, with speed 2.0 m /s with respect to the car. In the time it takes for him to reach the other end, how far has the flatcar moved? (Ill) A huge balloon and its gondola, of mass M, are in the air and stationary with respect to the ground. A passenger, of mass m, then climbs out and slides down a rope with speed v, measured with respect to the balloon. With what speed and direction (relative to Earth) does the balloon then move? What happens if the passenger stops?

*9 - 10 Variable Mass *80. (II) A 3500-kg rocket is to be accelerated at 3.0 g at take -off from the Earth. If the gases can be ejected at a rate of 27 kg/s, what must be their exhaust speed? *81. (II) Suppose the conveyor belt of Example 9 - 19 is retarded by a friction force of 150 N. Determine the required output power (hp) of the motor as a function of time from the moment gravel first starts falling (t = 0) until 3.0 s after the gravel begins to be dumped off the end of the 22-m-long conveyor belt. *82. (II) The jet engine of an airplane takes in 120 kg of air per second, which is burned with 4.2 kg of fuel per second. The burned gases leave the plane at a speed of 550 m /s (relative to the plane). If the plane is traveling 270 m /s (600m i/h), determine: (a) the thrust due to ejected fuel; (b) the thrust due to accelerated air passing through the engine; and (c) the power (hp) delivered.

*83. (II) A rocket traveling 1850 m /s away from the Earth at an altitude of 6400 km fires its rockets, which eject gas at a speed of 1300 m /s (relative to the rocket). If the mass of the rocket at this moment is 25,000 kg and an acceleration of 1.5 m /s2 is desired, at what rate must the gases be ejected?

*84. (Ill) A sled filled with sand slides without friction down a 32° slope. Sand leaks out a hole in the sled at a rate of 2.0 kg/s. If the sled starts from rest with an initial total mass of 40.0 kg, how long does it take the sled to travel 120 m along the slope?

| Gener l Problems__________ 85. A novice pool player is faced with the corner pocket shot shown in Fig. 9-48. Relative dimensions are also shown. Should the player worry that this might be a "scratch shot," in which the cue ball will also fall into a pocket? Give details. Assume equal mass balls and an elastic collision. FIGURE 9 - 4 8

Problem 85. 8 . During a hicago storm, winds can whip horizontally at speeds of 120 km /h. If the air strikes a person at the rate of 45 kg/s per square meter and is brought to rest, calculate the force of the wind on a person. Assume the person is 1.60m high and 0.50 m wide. ompare to the typical maximum force of friction (/x ~ 1 .0) between the person and the ground, if the person has a mass of 75 kg. 87. A ball is dropped from a height of 1.50 m and rebounds to a height of 1.20 m. Approximately how many rebounds will the ball make before losing 90% of its energy? 88. In order to convert a tough split in bowling, it is necessary to strike the pin a glancing blow as shown in Fig. 9 - 49. Assume that the bowling ball, initially traveling at 13.0 m/s, has five times the mass of a pin and that the pin goes off at 75° from the original direction of the ball. alculate the speed (a) of the pin and (b) of the ball just after collision, and (c) calculate the angle through which the ball was deflected. Assume the colli sion is elastic and ignore any spin of the ball.

90. A hockey puck of mass 4 m has been rigged to explode, as part of a practical joke. Initially the puck is at rest on a frictionless ice rink. Then it bursts into three pieces. One chunk, of mass m, slides across the ice at velocity vi. Another chunk, of mass 2m, slides across the ice at velocity 2vj. Determine the velocity of the third chunk. 91. For the completely inelastic collision of two railroad cars that we considered in Example 9 - 3, calculate how much of the initial kinetic energy is transformed to thermal or other forms of energy. 92. A 4800-kg open railroad car coasts along with a constant speed of 8.60 m /s on a level track. Snow begins to fall verti cally and fills the car at a rate of 3.80 kg/min. Ignoring friction with the tracks, what is the speed of the car after 60.0 min? (See Section 9 - 2.) *93. onsider the railroad car of Problem 92, which is slowly filling with snow, (a) Determine the speed of the car as a function of time using Eqs. 9 - 19. ( b ) What is the speed of the car after 60.0 min? Does this agree with the simpler calculation (Problem 92)? 94. Two blocks of mass raA and mB, resting on a frictionless table, are connected by a stretched spring and then released (Fig. 9 - 51). (a) Is there a net external force on the system? ( b ) Determine the ratio of their speeds, v / v q . (c) What is the ratio of their kinetic energies? (d) Describe the motion of the cm of this system. (e) How would the pres ence of friction alter the above results?

M /V V W V ^ FIGURE 9 - 51

FIGURE 9 - 4 9

Problem 88. 89. A gun fires a bullet vertically into a 1.40-kg block of wood at rest on a thin horizontal sheet, 1.40 kg Fig. 9 - 50. If the bullet has a mass of 24.0 g and a speed of 310 m/s, how high will the block rise into the air after the bullet becomes (\v= 310 m/s embedded in it?

FIGURE 9 - 50

Problem 89.

Problem 94.

95. You have been hired as an expert witness in a court case involving an automobile accident. The accident involved car A of mass 1500 kg which crashed into stationary car B of mass 1100 kg. The driver of car A applied his brakes 15 m before he skidded and crashed into car B. After the collision, car A slid 18 m while car B slid 30 m. The coeffi cient of kinetic friction between the locked wheels and the road was measured to be 0.60. Show that the driver of car A was exceeding the 55 -mi/h (90 km /h) speed limit before applying the brakes. 96. A meteor whose mass was about 2.0 X 108kg struck the Earth (raE = 6.0 X 1024kg) with a speed of about 25km /s and came to rest in the Earth, (a) What was the Earth's recoil speed (relative to Earth at rest before the collision)? ( b ) What fraction of the meteor's kinetic energy was trans formed to kinetic energy of the Earth? (c) By how much did the Earth's kinetic energy change as a result of this collision?

General Problems

245

2KS Ax

97. Two astronauts, one of mass 65 kg and the other 85 kg, are initially at rest in outer space. They then push each other apart. How far apart are they when the lighter astronaut has moved 12 m? 98. A 22-g bullet strikes and becomes embedded in a 1.35-kg block of wood placed on a horizontal surface just in front of the gun. If the coefficient of kinetic friction between the block and the surface is 0.28, and the impact drives the block a distance of 8.5 m before it comes to rest, what was the muzzle speed of the bullet? 99. Two balls, of masses mA = 45 g and mB = 65 g, are suspended as shown in Fig. 9-52. The lighter ball is pulled away to a ° angle with the vertical and released. (a) What is the velocity of the lighter ball before impact? (b) What is the velocity of each ball after the elastic collision? (c) What will be the maximum G height of each ball after the elastic A\ 30 cm collision? /wa\

A :,I?J mA

FIGURE 9 - 52

Problem 99.

FIGURE 9 - 54

Problem 103.

103. A 0.25-kg skeet (clay target) is fired at an angle of 28° to the horizon with a speed of 25 m/s (Fig. 9-54). When it reaches the maximum height, , it is hit from below by a 15-g pellet traveling vertically upward at a speed of 230 m/s. The pellet is embedded in the skeet. (a) How much higher, ', did the skeet go up? (b) How much extra distance, Ax, does the skeet travel because of the collision? 104. A massless spring with spring constant k is placed between a block of mass ra and a block of mass 3 ra. Initially the blocks are at rest on a frictionless surface and they are held together so that the spring between them is compressed by an amount D from its equilibrium length. The blocks are then released and the spring pushes them off in opposite directions. Find the speeds of the two blocks when they detach from the spring. 105. T e gravitational slings ot effect. Figure 9 -55 shows the planet Saturn moving in the negative x direction at its orbital speed (with respect to the Sun) of 9.6 km/s. The mass of Saturn is 5.69 X 102 kg. A spacecraft with mass 825 kg approaches Saturn. When far from Saturn, it moves in the +x direction at 10.4 km/s. The gravitational attrac tion of Saturn (a conservative force) acting on the space craft causes it to swing around the planet (orbit shown as dashed line) and head off in the opposite direction. Esti mate the final speed of the spacecraft after it is far enough away to be considered free of Saturn's gravitational pull.

100. A block of mass ra = 2.20 kg slides down a 30.0° incline which is 3.60 m high. At the bottom, it strikes a block of mass M = 7.00 kg which is at rest on a horizontal surface, Fig. 9-53. (Assume a smooth transition at the bottom of the incline.) If the collision is elastic, and friction can be ignored, determine (a) the speeds of the two blocks after the collision, and (b) how far back up the incline the smaller mass will go. jti 3*60 Hi

FIGURE 9 - 53

Problems 100 and 101.

101. In Problem 100 (Fig. 9-53), what is the upper limit on mass ra if it is to rebound from M, slide up the incline, stop, slide down the incline, and collide with M again? 102. After a completely inelastic collision between two objects of equal mass, each having initial speed, v, the two move off together with speed v/3. What was the angle between their initial directions? 246

HAPTER 9

Linear Momentum

106. Two bumper cars in an amusement park ride collide elastically as one approaches the other directly from the rear (Fig. 9-56). ar A has a mass of 450 kg and car B 490 kg, owing to differ ences in passenger mass. If car A approaches at 4.50 m/s and car B is moving at 3.70 m/s, calculate (a) their velocities after the collision, m„ = mA = and (b) the change 490 kg in momentum of 450 kg each. FIGURE 9 - 56

Problem 106: (a) before collision, (b) after collision.

4.50 m/s

-V70 m/s

___ a*1"--*-__ (")

107. In a physics lab, a cube slides down shown in Fig. 9 -57 and elastically strikes another cube at the bottom that is only one-half its mass. If the incline is 35 cm high and the table is 95 cm off the floor, where does each cube land? [Hint: Both leave the incline moving horizontally.]

frictionless incline as

112. An extrasolar planet can be detected by observing the wobble it produces on the star around which it revolves. Suppose an extra solar planet of mass raBrevolves around its star of mass mA. If no external force acts on this simple two-object system, then its cm is stationary. Assume mA and raB are in circular orbits with radii rA and rB about the system's cm . (a) Show that

raB

95 cm

FIGURE 9 - 57

Problem 107. 108. The space shuttle launches an 850-kg satellite by ejecting it from the cargo bay. The ejection mechanism is activated and is in contact with the satellite for 4.0 s to give it a velocity of 0.30 m/s in the z-direction relative to the shuttle. The mass of the shuttle is 92,000 kg. (a) Determine the component of velocity Vf of the shuttle in the minus z-direction resulting from the ejection, (b) Find the average force that the shuttle exerts on the satellite during the ejection. 109. You are the design engineer in charge of the crashworthi ness of new automobile models. ars are tested by smashing them into fixed, massive barriers at 45 km/h. A new model of mass 1500 kg takes 0.15 s from the time of impact until it is brought to rest, (a) alculate the average force exerted on the car by the barrier, (b) alculate the average decel eration of the car. 110. Astronomers estimate that a 2.0-km-wide asteroid collides with the Earth once every million years. The collision could pose a threat to life on Earth, (a) Assume a spherical asteroid has a mass of 3200 kg for each cubic meter of volume and moves toward the Earth at 15km/s. How much destructive energy could be released when it embeds itself in the Earth? (b) For comparison, a nuclear bomb could release about 4.0 X 101 J. How many such bombs would have to explode simultaneously to release the destructive energy of the asteroid collision with the Earth? 111. An astronaut of mass 210 kg including his suit and jet pack wants to acquire a velocity of 2.0 m/s to move back toward his space shuttle. Assuming the jet pack can eject gas with a velocity of 35 m/s, what mass of gas will need to be ejected?

rB mA (b) Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, mB = 1.0 X 10_3mA and the planet has an orbital radius of 8.0 X 1011 m. Deter mine the radius rA of the star's orbit about the system's cm . (c) When viewed from Earth, the distant system appears to wobble over a distance of 2 rA . If astronomers are able to detect angular displacements 6 of about 1 milliarcsec (1 arcsec = ^ of a degree), from what distance d (in light-years) can the star's wobble be detected (l ly = 9.46 X 1015 m)? (d) The star nearest to our Sun is about 4 ly away. Assuming stars are uniformly distributed throughout our region of the Milky Way Galaxy, about how many stars can this technique be applied to in the search for extrasolar planetary systems? 113. Suppose two asteroids strike head on. Asteroid A (mA = 7.5 X 1012kg) has velocity 3.3km/s before the collision, and asteroid B (mB = 1.45 X 1013kg) has velocity 1.4 km/s before the collision in the opposite direction. If the asteroids stick together, what is the velocity (magni tude and direction) of the new asteroid after the collision?

*Numeric l/Computer *114. (Ill) A particle of mass raA traveling with speed v A collides elastically head-on with a stationary particle of smaller mass mB. (a) Show that the speed of mB after the collision is ,

2 va

Vb 1 + mB/m A (b) onsider now a third particle of mass mc at rest between mA and mB so that mA first collides head on with mc and then mc collides head on with mB. Both collisions are elastic. Show that in this case, _______ m c mA_______ , _ VB VA(mC + + m c) (c) From the result of part (b), show that for maximum v'B , mc = \J m A mB. (d) Assume mB = 2.0 kg, mA = 18.0 kg and v A = 2.0 m/s. Use a spreadsheet to calculate and graph the values of v'B from rac = 0.0 kg to mc = 50.0 kg in steps of 1.0 kg. For what value of mc is the value of v B maximum? Does your numerical result agree with your result in part (c)?

A n sw e rs to Ex ercise s

A: (c) because the momentum change is greater. B: Larger (Ap is greater). : 0.50 m/s. D: (a) 6.0 m/s; (b) almost zero; (c) almost 24.0 m/s.

E: (&);( [0 in radians] (10 - l ) R where R is the radius of the circle and £ is the arc length subtended by the angle 0, which is specified in radians. If £ = R, then 0 = 1 rad. The radian, being the ratio of two lengths, is dimensionless. We thus do not have to mention it in calculations, although it is usually best to include it to remind us the angle is in radians and not degrees. We can rewrite Eq. 10 - la in terms of arc length £: £ = Re. (10 - lb) Radians can be related to degrees in the following way. In a complete circle there are 360°, which must correspond to an arc length equal to the circumference of the circle, £ = 2 R. Thus 6 = £/R = 2i R /R = 2 rad in a complete circle, so 360° = 27rrad. One radian is therefore 360°/27r « 360°/6.28 « 57.3°. An object that makes one complete revolution (rev) has rotated through 360°, or 2 radians: 1 rev = 360° = 2ir rad.

FIGURE 1 0 - 1 Looking t wheel th t is rot ting counterclockwise bout n xis through the w heel 's center t O ( xis perpendicul r to the p ge). E ch point, such s point P, m oves in circul r p th; £ is the dist nce P tr vels s the wheel rot tes through the ngle 6.

FIGURE! 0 - 2 Showing the distinction betw een r (the position vector) nd R (the dist nce from the rot tion xis) for point P on the edge of cylinder rot ting bout the z xis.

/j\

AUT I ON

Use radians in calculating, n ot degrees

SECTI N 10-1

Angular Quantities

249

EXAMPLE 10-1 Birds of prey —in radians. A particular bird's eye can just distinguish objects that subtend an angle no smaller than about 3 X 10 4rad. (a) How many degrees is this? (b) How small an object can the bird just distinguish when flying at a height of 100 m (Fig. 10-3a)? APPR ACH For (a) we use the relation 360° = 2tt rad. For (b) we use Eq. 10 - lb,

£ = Rd, to find the arc length. S LUTI N (a) We convert 3 X 10"4rad to degrees: (3 X 10"4rad)

Chord

(b)

FIGURE 10-3 ( ) Ex m ple 10 - 1. (b) For sm ll ngles, rc length nd the chord length (str ight line) re ne rly equ l. For n ngle s l rge s 15°, the error in m king this estim te is only 1%. For l rger ngles the error incre ses r pidly.

FIGURE 10 -4 A w h e e l rot tes from ( ) initi l position to (b) fin l position 02 . The ngul r displ cem ent is A0 = 02 ~~ #i -

= 0.017°,

or about 0.02°. (b) We use Eq. 10 - lb, £ = Rd. For small angles, the arc length £ and the chord length are approximately the same (Fig. 10-3b). Since R = 100 m and 0 = 3 X 10"4 rad, we find

Arc length

(a)

360° 2 rad

£ = (100 m)(3 X 10"4rad) = 3 X 10"2 m = 3 cm. A bird can distinguish a small mouse (about 3 cm long) from a height of 100 m. That is good eyesight. N TE Had the angle been given in degrees, we would first have had to convert it to radians to make this calculation. Equation 10-1 is valid only if the angle is specified in radians. Degrees (or revolutions) won't work. To describe rotational motion, we make use of angular quantities, such as angular velocity and angular acceleration. These are defined in analogy to the corresponding quantities in linear motion, and are chosen to describe the rotating object as a whole, so they have the same value for each point in the rotating object. Each point in a rotating object also has translational velocity and acceleration, but they have different values for different points in the object. When an object, such as the bicycle wheel in Fig. 10 - 4, rotates from some initial position, specified by 0ly to some final position, 02, its ngul r displ cem ent is A0 = 0,

0i.

The angular velocity (denoted by a>, the Greek lowercase letter omega) is defined in analogy with linear (translational) velocity that was discussed in hapter 2. Instead of linear displacement, we use the angular displacement. Thus the ver ge ngul r velocity of an object rotating about a fixed axis is defined as the time rate of change of angular position: (0 =

A0 At

(10 - 2 )

where A0 is the angle through which the object has rotated in the time interval At. The inst nt neous ngul r velocity is the limit of this ratio as At approaches zero: A0 d6 = lim —— = —— A*-x> A* dt

(10 - 2b)

Angular velocity has units of radians per second (rad/s). Note that all points in a rigid object rotate wit t e same angular velocity, since every position in the object moves through the same angle in the same time interval. An object such as the wheel in Fig. 10 -4 can rotate about a fixed axis either clockwise or counterclockwise. The direction can be specified with a + or - sign, just as we did in hapter 2 for linear motion toward the +x or —x direction. The usual convention is to choose the angular displacement A0 and angular velocity w as positive when the wheel rotates counterclockwise. If the rotation is clockwise, then 0 would decrease, so A0 and (o would be negative. 250

CHAPTER 10

Rotational Motion

Angular acceleration (denoted by a, the Greek lowercase letter alpha), in analogy to linear acceleration, is defined as the change in angular velocity divided by the time required to make this change. The ver ge ngul r cceler tion is defined as co2 ~ (Oi Aa)

" =

= at

(10_3a)

where is the angular velocity initially, and co2 is the angular velocity after a time interval At. Inst nt neous ngul r cceler tion is defined as the limit of this ratio as At approaches zero: Aw do) a = lim —— = —— (10-3b) a ^o At dt Since to is the same for all points of a rotating object, Eq. 10-3 tells us that a also will be the same for all points. Thus, o) and a are properties of the rotating object as a whole. With co measured in radians per second and t in seconds, a has units of radians per second squared (rad/s2). Each point or particle of a rotating rigid object has, at any instant, a linear velocity v and a linear acceleration a. We can relate the linear quantities at each point, v and a, to the angular quantities, (o and a, of the rotating object. onsider a point P located a distance R from the axis of rotation, as in Fig. 10-5. If the object rotates with angular velocity w, any point will have a linear velocity whose direction is tangent to its circular path. The magnitude of that point's linear velocity is v = d£/dt. From Eq. 10 - lb, a change in rotation angle dO (in radians) is related to the linear distance traveled by d£ = R dd. Hence di v = — = dt

FIGURE 1 0 - 5 A point P on rot ting w heel h s line r velocity v t ny moment.

„d0 R— dt v = R(o,

(10 -4)

where R is a fixed distance from the rotation axis and a>is given in rad/s. Thus, although (o is the same for every point in the rotating object at any instant, the linear velocity v is greater for points farther from the axis (Fig. 10-6). Note that Eq. 10 -4 is valid both instantaneously and on the average. CONCEPTUAL EXAMPLE 10 - 2 I Is the lion faster than the horse? On a rotating carousel or merry-go-round, one child sits on a horse near the outer edge and another child sits on a lion halfway out from the center, (a) Which child has the greater linear velocity? (b) Which child has the greater angular velocity? RESPONSE (a) The linear velocity is the distance traveled divided by the time interval. In one rotation the child on the outer edge travels a longer distance than the child near the center, but the time interval is the same for both. Thus the child at the outer edge, on the horse, has the greater linear velocity. (b) The angular velocity is the angle of rotation divided by the time interval. In one rotation both children rotate through the same angle (360° = 2ir rad). The two children have the same angular velocity.

FIGURE 1 0 - 6 A w heel rot ting uniformly counterclockwise. Two points on the wheel, t dist nces R A nd R b from the center, h ve the s me ngul r velocity co bec use they tr vel through the s me ngle 0 in the s me time interv l. But the two points h ve different line r velocities bec use they tr vel different dist nces in the s m e time interv l. Since R B > R A , then Vb > vA (bec use v = Roy).

If the angular velocity of a rotating object changes, the object as a whole — and each point in it —has an angular acceleration. Each point also has a linear acceler ation whose direction is tangent to that point's circular path. We use Eq. 10 -4 (v = Rw) to show that the angular acceleration a is related to the tangential linear acceleration atan of a point in the rotating object by _ dv _ fltan ~ dF ~

d(o df

flt n = Ra(10 -5) In this equation, R is the radius of the circle in which the particle is moving, and the subscript "tan" in atan stands for "tangential."

SECTI N 10-1

Angular Quantities

251

FIGURE 1 0 - 7 On rot ting wheel whose ngul r speed is incre sing, point P h s both t ngenti l nd r di l (centripet l) components of line r cceler tion. (See lso Ch pter 5.)

TABLE 10-1 Linear and Rotational Quantities Line r

Rot Rel tion tion l (0 in r di ns)

Type

displ cement

velocity

cceler tion

x = R6 v = Ra)

^tan

FIGURE 1 0 - 8 Ex m ple 1 0 - 3. The tot l cceler tion vector = t n + R, t t = 8.0 s.

(a)

252

CHAPTER 10

The total linear acceleration of a point at any instant is the vector sum of two components: ** = ®tan + 3r , where the radial component, aR, is the radial or "centripetal" acceleration and its direction is toward the center of the point's circular path; see Fig. 10-7. We saw in hapter 5 (Eq. 5 - 1) that a particle moving in a circle of radius R with linear speed v has a radial acceleration aR = v2/R; we can rewrite this in terms of oj using Eq. 10-4: u2 (R (of = (o2R. (10- 6) aR ~ R R Equation 10 -6 applies to any particle of a rotating object. Thus the centripetal acceleration is greater the farther you are from the axis of rotation: the children farthest out on a carousel experience the greatest acceleration. Table 10-1 summarizes the relationships between the angular quantities describing the rotation of an object to the linear quantities for each point of the object. ■ Angular and linear velocities and accelerations. A carousel is initially at rest. At t = 0 it is given a constant angular acceleration a = 0.060 rad/s2, which increases its angular velocity for 8.0 s. At t = 8.0 s, determine the magnitude of the following quantities: (a) the angular velocity of the carousel; (b) the linear velocity of a child (Fig. 10-8a) located 2.5 m from the center, point P in Fig. 10-8b; (c) the tangential (linear) acceleration of that child; (d) the centripetal acceleration of the child; and (e) the total linear acceleration of the child. APPR ACH The angular acceleration a is constant, so we can use a = A (o/At to solve for (o after a time t = 8.0 s. With this a>and the given a, we determine the other quantities using the relations we just developed, Eqs. 10 - 4,10 - 5, and 10-6. S LUTI N (a) In Eq. 10-3a, a = (by

> =

2 tTf .

(10 -7)

The unit for frequency, revolutions per second (rev/s), is given the special name the hertz (Hz). That is 1 Hz = 1 rev/s. Note that "revolution" is not really a unit, so we can also write 1 Hz = 1 s-1. The time required for one complete revolution is the period T, and it is related to the frequency by T = ~

(10 -8)

If a particle rotates at a frequency of three revolutions per second, then the period of each revolution is ^s. EXERCISE A In Ex m ple 1 0 - 3 , w e found th t the c rousel, fter 8.0 s, rot tes t n ngul r velocity cd = 0.48 r d/s, nd continues to do so fter t = 8.0 s bec use the cceler tion ce sed. D eterm ine the frequency nd period of the c rousel fter it h s re ched const nt ngul r velocity.

EXAMPLE 10 - 4 Hard drive. The platter of the hard drive of a computer rotates at 7200 rpm (rpm = revolutions per minute = rev/min). (a) What is the angular velocity (rad/s) of the platter? (b) If the reading head of the drive is located 3.00 cm from the rotation axis, what is the linear speed of the point on the platter just below it? (c) If a single bit requires 0.50/xm of length along the direction of motion, how many bits per second can the writing head write when it is 3.00 cm from the axis?

( ^ PHYSICS

APPLIED

Hard drive and bit speed

APPR ACH We use the given frequency / to find the angular velocity a> of

the platter and then the linear speed of a point on the platter (v = Rco). The bit rate is found by dividing the linear speed by the length of one bit (v = distance/time). S LUTI N (a) First we find the frequency in rev/s, given / = 7200 rev/min: / = J

(7200 rev/min) = 120 rev/s = 120 Hz. , ■x (60 s/min) '

Then the angular velocity is o) = 2irf = 754 rad/s. (b) The linear speed of a point 3.00 cm out from the axis is given by Eq. 10-4: v = R(o = (3.00 X 10"2m)(754 rad/s) = 22.6 m/s. (c) Each bit requires 0.50 X 10 6m, so at a speed of 22.6 m/s, the number of bits passing the head per second is

22. m /s mA*. ------------- 7— -— = 45 X 10 bits per second, 0.50 X 10" m /bit F or 45 megabits/s (Mbps). SECTI N 10-1

Angular Quantities

253

EXAMPLE 10 - 5 Given o> as function of time. A disk of radius i? = 3.0m rotates at an angular velocity a> = (1.6 + 1.21) rad/s, where t is in seconds. At the instant t = 2.0 s, determine (a) the angular acceleration, and (b) the speed v and the components of the acceleration a of a point on the edge of the disk. APPR ACH We use a = da)/dt, v = Rcj, «tan = Ra, and aR = w2R, which

are Eqs. 10 -3b, 1 0 - 4,10 - 5 and 10 - 6. We can write a) explicitly showing units of the constants (in case we want to check later): o) = [1. s- 1 + (1.2 s~2)t] which will give us s- 1 (= rad/s) for each term. S LUTI N (a) The angular acceleration is a = - j — = ^ - ( 1. dt dt

+ 1.2 f)s -1 = 1.2 rad/s2.

(b) The speed v of a point 3.0 m from the center of the rotating disk at t = 2.0 s is, using Eq. 10 - 4, v = Rco = (3.0m)(1.6 + 1.2 f)s -1 = (3.0 m)(4.0 s-1) = 12.0 m/s. The components of the linear acceleration of this point at t = 2.0 s are atan = Ret = (3.0 m)(l.2 rad/s2) = 3.6 m /s2 = od2R = [(1.6 + 1.2£)s- 1]2(3.0m) = (4.0 s_1)2(3.0m) = 48 m /s2.

1 0 -2 Vector N ture of Angul r Qu ntities

(b)

FIGURE 10 - 9 (a) Rotating wheel, (b) Right-hand rule for obtaining direction of o).

FIGURE 10 - 10 (a) Velocity is a true vector. The reflection of v points in the same direction, (b) Angular velocity is a pseudovector since it does not follow this rule. As can be seen, the reflection of the wheel rotates in the opposite direction, so the direction of to is opposite for the reflection.

(a)

254

(b )

CHAPTER 10

Both to and a can be treated as vectors, and we define their directions in the following way. onsider the rotating wheel shown in Fig. 10 - 9a. The linear velocities of different particles of the wheel point in all different directions. The only unique direction in space associated with the rotation is along the axis of rotation, perpendicular to the actual motion. We therefore choose the axis of rotation to be the direction of the angular velocity vector, w. Actually, there is still an ambiguity since a> could point in either direction along the axis of rotation (up or down in Fig. 10 - 9a). The convention we use, called the right -h nd rule, is this: when the fingers of the right hand are curled around the rotation axis and point in the direction of the rotation, then the thumb points in the direction of is c lled pseudovector or axial vector. The ngul r cceler tion a is lso pseudovector, s re ll cross products of true vectors (Section 11 - 2). The difference between true vectors nd pseudovectors is import nt in element ry p rticle physics, but will not concern us in this book.

Rotational Motion

10—3 Const nt Angul r Acceler tion In hapter 2, we derived the useful kinematic equations (Eqs. 2 -12) that relate acceleration, velocity, distance, and time for the special case of uniform linear acceleration. Those equations were derived from the definitions of linear velocity and acceleration, assuming constant acceleration. The definitions of angular velocity and angular acceleration are the same as those for their linear counter parts, except that 6 has replaced the linear displacement x , to has replaced v , and a has replaced a. Therefore, the angular equations for const nt ngul r cceler tion will be analogous to Eqs. 2 - 12 with x replaced by 6, v by a>, and a by a , and they can be derived in exactly the same way. We summarize them here, opposite their linear equivalents (we have chosen x0 = 0 and 60 = 0 at the initial time t = 0): Angul r

Line r

(Oq + at

v = Vq + at

[constant a, a] (10-9 )

Kinematic equations

(O0t

x = Vqt + \a t2

[constant a, a] (10-9b)

for constant

a>o + 2ad 0) + (Oq

v2 = Vq + 2ax v + V

[constant a, a] (10-9c)

angular acceleration

[constant a, a] (10-9d)

(x0 = 0, 0O = 0)

Note that (o0 represents the angular velocity at t = 0, whereas 0 and a) represent the angular position and velocity, respectively, at time t. Since the angular acceleration is constant, a = a. Centrifuge acceleration. A centrifuge rotor is accelerated from rest to 20,000 rpm in 30 s. (a) What is its average angular acceleration? (b) Through how many revolutions has the centrifuge rotor turned during its acceleration period, assuming constant angular acceleration?

~ [2 5 E I2 H IiH S

PHYS1CS

APPLIED

Centrifuge

APPR ACH To determine a = Aco/At, we need the initial and final angular

velocities. For (b), we use Eqs. 10 -9 (recall that one revolution corresponds to 0 = 277 rad). S LUTI N (a) The initial angular velocity is co = 0. The final angular velocity is CO = 2 n f = (2ir rad/rev)

(20,000 rev/min) = 2100rad/ s' (60s/min)

Then, since a = Ac*/At and At = 30 s, we have 2100 rad/s - 0 - (Qq 70 rad/s . At 30 s That is, every second the rotor's angular velocity increases by 70 rad/s, or by (70/27t) = 11 revolutions per second. (b) To find 6 we could use either Eq. 10 -9b or 10-9c, or both to check our answer. The former gives a =

6 = 0 + |(7 0 rad /s2)(30s)2 = 3.15 X 104rad, where we have kept an extra digit because this is an intermediate result. To find the total number of revolutions, we divide by 2ir rad/rev and obtain ^ . 3.15 X 104rad — -------------- = 5.0 X 103 rev. 27t rad/rev N TE Let us calculate 6 using Eq. 10-9c:

£o2 — col (2100 rad/s)2 0 = 3.15 X 104 rad 2a 2(70 rad/s2) which checks our answer using Eq. 10 -9b perfectly. 6 =

SECTI N 10 - 3

Constant Angular Acceleration

255

1 0 -4 Torque

Fb |

FIGURE 1 0 - 1 1 Top view of door. Applying the s me force with different lever rms, R A nd R B . If R a = 3RB , then to cre te the s me effect ( ngul r cceler tion), FB needs to be three times FA , or FA = \ FB .

FIGURE 1 0 - 1 2 ( ) A tire iron too c n h ve long lever rm. (b) A plumber c n exert gre ter torque using wrench with long lever rm.

We have so far discussed rotational kinematics— the description of rotational motion in terms of angular position, angular velocity, and angular acceleration. Now we discuss the dynamics, or causes, of rotational motion. Just as we found analogies between linear and rotational motion for the description of motion, so rotational equivalents for dynamics exist as well. To make an object start rotating about an axis clearly requires a force. But the direction of this force, and where it is applied, are also important. Take, for example, an ordinary situation such as the overhead view of the door in Fig. 10-11. If you apply a force FA to the door as shown, you will find that the greater the magnitude, FA, the more quickly the door opens. But now if you apply the same magnitude force at a point closer to the hinge— say, FB in Fig. 10 -11 — the door will not open so quickly. The effect of the force is less: w ere the force acts, as well as its magnitude and direction, affects how quickly the door opens. Indeed, if only this one force acts, the angular acceleration of the door is proportional not only to the magnitude of the force, but is also directly proportional to t e perpendicular distance from t e axis o f rotation to t e line along w ic t e force acts. This distance is called the lever arm, or moment arm, of the force, and is labeled Ra and Rb for the two forces in Fig. 10-11. Thus, if Ra in Fig. 10-11 is three times larger than Rb , then the angular acceleration of the door will be three times as great, assuming that the magnitudes of the forces are the same. To say it another way, if Ra = 3Rb , then FB must be three times as large as FA to give the same angular acceleration. (Figure 10-12 shows two examples of tools whose long lever arms are very effective.) The angular acceleration, then, is proportional to the product of the force times t e lever arm. This product is called the moment o f t e force about the axis, or, more commonly, it is called the torque, and is represented by r (Greek lowercase letter tau). Thus, the angular acceleration a of an object is directly proportional to the net applied torque r: a oc t ,

Axis of rot tion

(a)

(b)

FIGURE 1 0 - 1 3 ( ) Forces cting t different ngles t the doorknob. (b) The lever rm is defined s the perpendicul r dist nce from the xis o f rot tion (the hinge) to the line of ction o f the force.

256

CHAPTER 10

and we see that it is torque that gives rise to angular acceleration. This is the rotational analog of Newton's second law for linear motion, a oc F. We defined the lever arm as the perpendicular distance from the axis of rotation to the line of action of the force—that is, the distance which is perpendicular both to the axis of rotation and to an imaginary line drawn along the direction of the force. We do this to take into account the effect of forces acting at an angle. It is clear that a force applied at an angle, such as Fc in Fig. 10-13, will be less effective than the same magnitude force applied perpendicular to the door, such as FA (Fig. 10-13a). And if you push on the end of the door so that the force is directed at the hinge (the axis of rotation), as indicated by FD, the door will not rotate at all. The lever arm for a force such as Fc is found by drawing a line along the direction of Fc (this is the "line of action" of Fc). Then we draw another line, perpendicular to this line of action, that goes to the axis of rotation and is perpendicular also to it. The length of this second line is the lever arm for Fc and is labeled Rc in Fig. 10-13b. The lever arm for FAis the full distance from the hinge to the door knob, Ra ; thus Rc is much smaller than Ra . The magnitude of the torque associated with Fc is then RCFC. This short lever arm R c and the corresponding smaller torque associated with Fc is consistent with the observation that Fc is less effective in accelerating the door than is FA. When the lever arm is defined in this way, experiment shows that the relation a oc t is valid in general. Notice in Fig. 10-13 that the line of action of the force FDpasses through the hinge, and hence its lever arm is zero. onsequently, zero torque is associated with FD and it gives rise to no angular acceleration, in accord with everyday experience.

Rotational Motion

In general, then, we can write the magnitude of the torque about a given axis as r = R± F, (lO -lOa) where R± is the lever arm, and the perpendicular symbol (_L) reminds us that we must use the distance from the axis of rotation that is perpendicular to the line of action of the force (Fig. 10-14a). An equivalent way of determining the torque associated with a force is to resolve the force into components parallel and perpendicular to the line that connects the axis to the point of application of the force, as shown in Fig. 10-14b. The component exerts no torque since it is directed at the rotation axis (its moment arm is zero). Hence the torque will be equal to F±times the distance R from the axis to the point of application of the force: t = RF±. (lO -lOb) This gives the same result as Eq. lO-lOa because F± = F sin 6 and R± = R sin 0. So r = RF sin 6 (lO-lOc) in either case. [Note that 0 is the angle between the directions of F and R (radial line from the axis to the point where F acts)]. We can use any of Eqs. 10-10 to calculate the torque, whichever is easiest. Because torque is a distance times a force, it is measured in units of m •N in SI units,f cm •dyne in the cgs system, and ft •lb in the English system. When more than one torque acts on an object, the angular acceleration a is found to be proportional to the net torque. If all the torques acting on an object tend to rotate it about a fixed axis of rotation in the same direction, the net torque is the sum of the torques. But if, say, one torque acts to rotate an object in one direction, and a second torque acts to rotate the object in the opposite direction (as in Fig. 10-15), the net torque is the difference of the two torques. We normally assign a positive sign to torques that act to rotate the object counterclockwise (just as 6 is usually positive counterclockwise), and a negative sign to torques that act to rotate the object clockwise, when the rotation axis is fixed.

Point of pplic tion

Axis o f v

rotation ' R I//

of force

(a)

(b) FIGURE 10-14 Torque = R ±F = RF±

EXAMPLE 10 - 7 Torque on a compound wheel. Two thin disk-shaped wheels, of radii RA = 30 cm and RB = 50 cm, are attached to each other on an axle that passes through the center of each, as shown in Fig. 10-15. alculate the net torque on this compound wheel due to the two forces shown, each of magnitude 50 N. APPR ACH The force FA acts to rotate the system counterclockwise, whereas FB

acts to rotate it clockwise. So the two forces act in opposition to each other. We must choose one direction of rotation to be positive— say, counterclockwise. Then FAexerts a positive torque, r A = RaFa , since the lever arm is RA. On the other hand, FBproduces a negative (clockwise) torque and does not act perpen dicular to RB, s o we must use its perpendicular component to calculate the torque it produces: r B = ~ R BFB± = - R BFB sin 0, where 0 = 60°. (Note that 0 must be the angle between FB and a radial line from the axis.) S LUTI N The net torque is - R b Fb sin60° = (0.30m)(50N) - (0.50m)(50N)(0.866) = - 6.7 m -N. This net torque acts to accelerate the rotation of the wheel in the clockwise direction.

FIGURE 10-15 Ex m ple 10 - 7 . The torque due to FA tends to cceler te the w heel counterclockwise, where s the torque due to FB tends to cceler te the w heel clockwise.

EXERCISE B Two forces (FB = 20 N nd FA = 30 N ) re pplied to m eter stick which c n rot te bout its left end, Fig. 10 - 16. Force FB is pplied perpendicul rly t the m idpoint. W hich force exerts the gre ter torque: FA , FB , or both the s me?

FIGURE 10-16

r a

Exercise B.

i i i i i i i i | i i i i i i i i i

^ ---------- 1------------ i fNote th t the units for torque re the s me s those for energy. We write the unit for torque here s m -N (in SI) to distinguish it from energy (N •m) bec use the two qu ntities re very different. An obvious differ ence is th t energy is sc l r, where s torque h s direction nd is vector ( s we will see in Ch pter 11). The speci l n me joule (1 J = 1 N •m) is used only for energy ( nd for work), never for torque.

Axis

T „

SECTI N 10 - 4

Torque

257

10—5 Rot tion l Dyn mics; Torque nd Rot tion l Inerti We discussed in Section 10 -4 that the angular acceleration a of a rotating object is proportional to the net torque t applied to it: a

FIGURE 10-17 A mass m rotating in a circle of radius R about a fixed point.

2 t,

where we write E r to remind us that it is the net torque (sum of all torques acting on the object) that is proportional to a. This corresponds to Newton's second law for translational motion, a oc 2 F, but here torque has taken the place of force, and, correspondingly, the angular acceleration a takes the place of the linear acceleration a. In the linear case, the acceleration is not only proportional to the net force, but it is also inversely proportional to the inertia of the object, which we call its mass, m. Thus we could write a = 'ZF/m. But what plays the role of mass for the rotational case? That is what we now set out to determine. At the same time, we will see that the relation a oc E r follows directly from Newton's second law, E F = ma. We first consider a very simple case: a particle of mass m rotating in a circle of radius R at the end of a string or rod whose mass we can ignore compared to m (Fig. 10-17), and we assume that a single force F acts on m tangent to the circle as shown. The torque that gives rise to the angular acceleration is r = RF. If we use Newton's second law for linear quantities, 2 F = ma, and Eq. 10-5 relating the angular acceleration to the tangential linear acceleration, atan = Ra, then we have F = ma = mRa, where a is given in rad/s2. When we multiply both sides of this equation by R, we find that the torque r = RF = R(mRa), or r = m R2a.

[single particle] (10-11)

Here at last we have a direct relation between the angular acceleration and the applied torque r. The quantity mR2 represents the rotational inertia of the particle and is called its moment o f inertia. Now let us consider a rotating rigid object, such as a wheel rotating about a fixed axis through its center, such as an axle. We can think of the wheel as consisting of many particles located at various distances from the axis of rotation. We can apply Eq. 10-11 to each particle of the object; that is, we write rt = m tR2a for the it particle of the object. Then we sum over all the particles. The sum of the various torques is just the total torque, 2 r, so we obtain: 2r,- = ('2miR2)a

[axis fixed] (10-12)

where we factored out the a since it is the same for all the particles of a rigid object. The resultant torque, 2 t, represents the sum of all internal torques that each particle exerts on another, plus all external torques applied from the outside: = 2 r ext + 2 tjnt. The sum of the internal torques is zero from Newton's third law. Hence E r represents the resultant external torque. The sum Drazi?2 in Eq. 10-12 represents the sum of the masses of each particle in the object multiplied by the square of the distance of that particle from the axis of rotation. If we give each particle a number (1,2,3,...), then 2 m tR 2 = m xR2 + m2R \ + m3R\ + •••. 258

CHAPTER 10

Rotational Motion

This summation is called the m om ent o f inerti (or rotational inertia) I of the object: I = 2 iriiR2 = m 1R l + m 2R \

(10 -13)

ombining Eqs. 10 -12 and 10 -13, we can write x'

_ T

I axis fixed in 1 [inertial reference frame ^

NEWTON'S SECOND LAW FOR ROTATION

This is the rotational equivalent of Newton's second law. It is valid for the rotation of a rigid object about a fixed axis.1" It can be shown (see hapter 11) that Eq. 10 -14 is valid even when the object is translating with acceleration, as long as I and a are calculated about the center of mass of the object, and the rotation axis through the c m doesn't change direction. (A ball rolling down a ramp is an example.) Then (2 r)
= 30.0 rad/s: A(o 30.0 rad/s - 0 = 10.0 rad/s2 " "" ~Kt 3.00 s We can now solve for I in Newton's second law (see step 7). 6. Other c lcul tions: None needed. 7. Solve for unknowns. We solve for I in Newton's second law for rotation = la , and insert our values for 2 t and a: 3.85 m - N = 0.385 kg •m . a 10.0 rad/s2 8. Do rough estim te. We can do a rough estimate of the moment of inertia by assuming the pulley is a uniform cylinder and using Fig. 10-20c: I « \M R l = \ (4.00 kg) (0.330 m)2 = 0.218 kg -m2. This is the same order of magnitude as our result, but numerically somewhat less. This makes sense, though, because a pulley is not usually a uniform cylinder but instead has more of its mass concentrated toward the outside edge. Such a pulley would be expected to have a greater moment of inertia than a solid cylinder of equal mass. A thin hoop, Fig. 10-20a, ought to have a greater I than our pulley, and indeed it does: I = M R q = 0.436 kg •m2.

FIGURE 10-21

Ex m ple 1 0 - 9 .

BLEM

LVING

Usefulness and p o w e r o f roug estimates

SECTI N 10 - 6

261

EXAMPLE 10 - 10 Pulley and bucket. onsider again the pulley in Fig. 10-21 and Example 10 -9 with the same friction. But this time, instead of a constant 15.0-N force being exerted on the cord, we now have a bucket of weight w = 15.0 N (mass m = w /g = 1.53 kg) hanging from the cord. See Fig. 10-22a. We assume the cord has negligible mass and does not stretch or slip on the pulley. (a) alculate the angular acceleration a of the pulley and the linear acceleration a of the bucket. (b) Determine the angular velocity a) of the pulley and the linear velocity v of the bucket at t = 3.00 s if the pulley (and bucket) start from rest at t = 0. APPR ACH This situation looks a lot like Example 10 -9, Fig. 10-21. But there is

a big difference: the tension in the cord is now an unknown, and it is no longer equal to the weight of the bucket if the bucket accelerates. Our system has two parts: the bucket, which can undergo translational motion (Fig. 10-22b is its free-body diagram); and the pulley. The pulley does not translate, but it can rotate. We apply the rotational version of Newton's second law to the pulley, = la , and the linear version to the bucket, 2 F = ma. S LUTI N (a) Let Ft be the tension in the cord. Then a force Fx acts at the edge of the pulley, and we apply Newton's second law, Eq. 10-14, for the rotation of the pulley:

( )

la =

= R0F — r fr,

[pulley]

Next we look at the (linear) motion of the bucket of mass m. Figure 10-22b, the free-body diagram for the bucket, shows that two forces act on the bucket: the force of gravity mg acts downward, and the tension of the cord FT pulls upward. Applying Newton's second law, E F = ma, for the bucket, we have (taking downward as positive):

njg (b) FIGURE 10-22 Ex m ple 10 - 10. ( ) Pulley nd f lling bucket of m ss r . (b) Free - body di gr m for the bucket.

mg — FT = ma.

[bucket]

Note that the tension FT, which is the force exerted on the edge of the pulley, is not equal to the weight of the bucket (= mg = 15.0 N). There must be a net force on the bucket if it is accelerating, so FT < mg. Indeed from the last equation above, FT = mg — ma. To obtain a, we note that the tangential acceleration of a point on the edge of the pulley is the same as the acceleration of the bucket if the cord doesn't stretch or slip. Hence we can use Eq. 10-5, atan = a = R0a. Substituting F = mg - ma = mg - mR0a into the first equation above (Newton's second law for rotation of the pulley), we obtain la = 2 t = R0FT — t fr = Ro(mg — mR0a) — Tfr = mgR0 — mR^a — Tfr . The variable a appears on the left and in the second term on the right, so we bring that term to the left side and solve for a: mgR0 ~ r fr I + mRl The numerator (mgR0 - Tfr) is the net torque, and the denominator (/ + mRl) is the total rotational inertia of the system. Then, since I = 0.385 kg •m2, m = 1.53 kg, and r fr = 1.10m - N (from Example 10-9), a

(15.0N)(0.330m) - 1.10m -N 0.385 kg - m2 + (1.53 kg) (0.330 m)2

The angular acceleration is somewhat less in this case than the 10.0 rad/s2 of Example 10-9. Why? Because FT (= mg — ma) is less than the 15.0-N weight of the bucket, mg. The linear acceleration of the bucket is a = R0a = (0.330 m) (6.98 rad/s2) = 2.30 m /s2. N TE The tension in the cord FT is less than mg because the bucket accelerates.

262

CHAPTER 10

Rotational Motion

(ib) Since the angular acceleration is constant, after 3.00 s o) = (o0 + at = 0 + (6.98rad/s2)(3.00s) = 20.9 rad/s. The velocity of the bucket is the same as that of a point on the wheel's edge: v = R0(o = (0.330 m) (20.9 rad/s) = 6.91 m/s. The same result can also be obtained by using the linear equation v = v0 + at = 0 + (2.30 m /s2)(3.00 s) = 6.90 m/s. (The difference is due to rounding off.) E E E E H M E H I I Rotating rod. A uniform rod of mass M and length £ can pivot freely (i.e., we ignore friction) about a hinge or pin attached to the case of a large machine, as in Fig. 10-23. The rod is held horizontally and then released. At the moment of release (when you are no longer exerting a force holding it up), determine (a) the angular acceleration of the rod and (b) the linear acceleration of the tip of the rod. Assume the force of gravity acts at the center of mass of the rod, as shown. APPR ACH (a) The only torque on the rod about the hinge is that due to

gravity, which acts with a force F = Mg downward with a lever arm 1/2 at the moment of release (the cm is at the center of a uniform rod). There is also a force on the rod at the hinge, but with the hinge as axis of rotation, the lever arm of this force is zero. The moment of inertia of a uniform rod pivoted about its end is (Fig. 10-20g) I = \ M f . In part (b) we use atan = Ra. S LUTI N We use Eq. 10-14, solving for a to obtain the initial angular acceler ation of the rod: a = j

m4 \M£2

____________ CM.

F' I Mg

FIGURE 10-23

Example 10-11.

3g 2£

As the rod descends, the force of gravity on it is constant but the torque due to this force is not constant since the lever arm changes. Hence the rod's angular acceleration is not constant. (b) The linear acceleration of the tip of the rod is found from the relation fltan = Rot (Eq. 10 -5) with R = £: «t n =

| g-

N TE The tip of the rod falls with an acceleration greater than g! A small object

balanced on the tip of the rod would be left behind when the rod is released. In contrast, the cm of the rod, at a distance £/2 from the pivot, has acceleration "t n = W 2 ) a = Jg -

10—7 Determining Moments of Inerti By Experiment The moment of inertia of any object about any axis can be determined experi mentally, such as by measuring the net torque D t required to give the object an angular acceleration a. Then, from Eq. 10-14, I = 'Er/a. See Example 10-9.

Using C lculus For simple systems of masses or particles, the moment of inertia can be calculated directly, as in Example 10-8. Many objects can be considered as a continuous distribution of mass. In this case, Eq. 10-13 defining moment of inertia becomes I = \ R2dm, =1

(10 - 16)

where dm represents the mass of any infinitesimal particle of the object and R is the perpendicular distance of this particle from the axis of rotation. The integral is taken over the whole object. This is easily done only for objects of simple geometric shape. SECTI N 10 - 7

Determining Moments of Inertia

263

EXAMPLE 10 - 12 Cylinder, solid or hollow, (a) Show that the moment of inertia of a uniform hollow cylinder of inner radius , outer radius R2, and mass M, is I = \M {R \ + R$), as stated in Fig. 10-20d, if the rotation axis is through the center along the axis of symmetry. (b) Obtain the moment of inertia for a solid cylinder. APPR ACH We know that the moment of inertia of a thin ring of radius R is

mR2. So we divide the cylinder into thin concentric cylindrical rings or hoops of thickness dR, one of which is indicated in Fig. 10-24. If the density (mass per unit volume) is p, then dm = p dV, FIGURE 1 0 - 2 4 D eterm ining the m om ent o f inerti o f hollow cylinder (Ex m ple 1 0 - 1 2 ).

where dV is the volume of the thin ring of radius R, thickness dR, and height . Since dV = (2iTR)(dR)( ), we have dm = lirp R dR. S LUTI N (a) The moment of inertia is obtained by integrating (summing) over

all these rings: f f R2 / = J R 2 dm = J lirp R3dR = 2irp

R 4 — 7?4 =

^ ( * 2

« l),

where we are given that the cylinder has uniform density, p = constant. (If this were not so, we would have to know p as a function of R before the integration could be carried out.) The volume V of this hollow cylinder is V = (j R\ - 7rR2) , so its mass M is M = pV = pi (R22 - R2) . Since {R\ - R § = (R\ - R $(R \ + RI), we have I = ^ ( « 2 - « l)(«2 + *l) = \M {R \ + Rl), as stated in Fig. 10-20d. (b) For a solid cylinder,

= 0 and if we set

> then

I = \M R l, which is that given in Fig. 10-20c for a solid cylinder of mass M and radius R0.

The P r llel-Axis Theorem There are two simple theorems that are helpful in obtaining moments of inertia. The first is called the p r llel- xis theorem. It relates the moment of inertia I of an object of total mass M about any axis, and its moment of inertia / Mabout an axis passing through the center of mass and parallel to the first axis. If the two axes are a distance apart, then I = / M + M 2. FIGURE 1 0 - 2 5

Ex m ple 10 - 13.

Axis _3 -A -

[parallel axis] (10-17)

Thus, for example, if the moment of inertia about an axis through the cm is known, the moment of inertia about any axis parallel to this axis is easily obtained. EXAMPLE 10 - 13 Parallel axis. Determine the moment of inertia of a solid cylinder of radius R0 and mass M about an axis tangent to its edge and parallel to its symmetry axis, Fig. 10-25. APPR ACH We use the parallel-axis theorem with / M = \M R \ (Fig. 10-20c).

= R0, Eq. 10-17 gives

S LUTI N Since

/ =

/c m

+ M 1 = IM Rl

EXERCISE C In Figs. 1 0 - 2 0 f nd g, the m om ents of inerti for thin rod bout two different xes re given. A re they rel ted by the p r llel- xis theorem ? Ple se show how.

264

CHAPTER 10

Rotational Motion

*Proof of the P r llel-Axis Theorem The proof of the parallel-axis theorem is as follows. We choose our coordinate system so the origin is at the cm , and / CM is the moment of inertia about the z axis. Figure 10-26 shows a cross section of an object of arbitrary shape in the xy plane. We let I represent the moment of inertia of the object about an axis parallel to the z axis that passes through the point A in Fig. 10-26 where the point A has coordinates xA and yA. Let xt , yt , and ra* represent the coordinates and mass of an arbitrary particle of the object. The square of the distance from this point to A is [ ( x j - xA)2 + (yt - }> )2]- S o the moment of inertia, I , about the axis through A is I = XmiKxi - xA)2 + (y, - yA)2] =

+ yf) - 2xA2 m,xi - 2yA2mjyf + ('Zml) (x 2A + y2A).

The first term on the right is just / M= Sm ,(x] + yf) since the cm is at the origin. The second and third terms are zero since, by definition of the c m , EmjXj = = 0 because x M = ycu = 0. The last term is M 2 since Em/ = M and (pc\ + y i) = 2 where is the distance of A from the c m . Thus we have proved / = / M+ M 2, which is Eq. 10-17.

FIGURE 1 0 - 2 6 Deriv tion of the p r llel- xis theorem.

*The Perpendicul r-Axis Theorem The parallel-axis theorem can be applied to any object. The second theorem, the perpendicul r - xis theorem, can be applied only to plane (flat) objects — that is, to two-dimensional objects, or objects of uniform thickness whose thickness can be neglected compared to the other dimensions. This theorem states that the sum of the moments of inertia of a plane object about any two perpendicular axes in the plane of the object, is equal to the moment of inertia about an axis through their point of intersection perpendicular to the plane of the object. That is, if the object is in the xy plane (Fig. 10-27), Iz = Ix + Iy .

FIGURE 1 0 - 2 7 xis theorem .

The perpendicul r -

[object in xy plane] (10-18)

Here Iz , Ix , Iy are moments of inertia about the z, x, and y axes. The proof is simple: since Ix = 'Zrriiy2, Iy = 'ZrriiX2, and Iz = 'Zm ^x2 + y2), Eq. 10-18 follows directly.

1 0-8 Rot tion l Kinetic Energy The quantity \m v 2 is the kinetic energy of an object undergoing translational motion. An object rotating about an axis is said to have rot tion l kinetic energy. By analogy with translational kinetic energy, we would expect this to be given by the expression \la>2 where I is the moment of inertia of the object and a) is its angular velocity. We can indeed show that this is true. onsider any rigid rotating object as made up of many tiny particles, each of mass mi. If we let Rt represent the distance of any one particle from the axis of rotation, then its linear velocity is vt = Rico. The total kinetic energy of the whole object will be the sum of the kinetic energies of all its particles: K = S ( |rriivj) = ^ n i i R j a ) 2) = \^{miR])(D2. We have factored out the \ and the (o2since they are the same for every particle of a rigid object. Since 2 rriiR2 = I, the moment of inertia, we see that the kinetic energy, K, of an object rotating about a fixed axis is, as expected, K = \lo)2.

[rotation about a fixed axis] (10-19)

If the axis is not fixed in space, the rotational kinetic energy can take on a more complicated form. SECTI N 10 - 8

Rotational Kinetic Energy

265

The work done on an object rotating about a fixed axis can be written in terms of angular quantities. Suppose a force F is exerted at a point whose distance from the axis of rotation is R, as in Fig. 10-28. The work done by this force is | f • d i = | f , R de. where dt is an infinitesimal distance perpendicular to R with magnitude d£ = R dd, and F± is the component of F perpendicular to R and parallel to di (Fig. 10-28). But F±R is the torque about the axis, so FIGURE 1 0 - 2 8 C lcul ting the work done by torque cting on rigid object rot ting bout fixed xis.

2 Je1

(10- 20)

is the work done by a torque t to rotate an object through the angle The rate of work done, or power P, at any instant is P =

dW dt

cW dt

02 —

#i-

(10- 21)

TO).

The work-energy principle holds for rotation of a rigid object about a fixed axis. From Eq. 10-14 we have t

Tda) dt

do) dO dd dt

da) dd

= l a = I —— = I —— — = l a ) ——'

where we used the chain rule and co = dd/dt. Then

dd = I codo) and

W = = fI trdd ed == fI Icodto = \lw \ - \lw \. J 0!

(10-22)

This is the work-energy principle for a rigid object rotating about a fixed axis. It states that the work done in rotating an object through an angle 02 — is equal to the change in rotational kinetic energy of the object. PHYSICS

APPLIED

Energy fro m a flyw eel

EXAMPLE 10 - 14 ESTIMATE"! Flywheel. Flywheels, which are simply large rotating disks, have been suggested as a means of storing energy for solarpowered generating systems. Estimate the kinetic energy that can be stored in an 80,000-kg (80-ton) flywheel with a diameter of 10 m (a three-story building). Assume it could hold together (without flying apart due to internal stresses) at 100 rpm. APPR ACH We use Eq. 10-19, K = \l(D2, but only after changing 100 rpm to cd

in rad/s. S LUTI N We are given

rev ,\ ( 1 min ,\ 0, = lOOrpm = (I*™ m —

ra d )\ = 10.5 rad/s.

The kinetic energy stored in the disk (for which I = \ MRf y is K = \l(o2 = \§ M R § w 2 = \ (8.0 X 104kg)(5m)2(10.5 rad/s)2 = 5.5 X 107J. N TE In terms of kilowatt-hours [l kWh = (1000 J/s)(3600s/h)(l h) = 3.6 X 10 j], this energy is only about 15 kWh, which is not a lot of energy (one 3-kW oven would use it all in 5 h). Thus flywheels seem unlikely for this application.

266

CHAPTER 10

Rotational Motion

EXAMPLE 10 -15 Rotating rod. A rod of mass M is pivoted on a frictionless hinge at one end, as shown in Fig. 10-29. The rod is held at rest horizontally and then released. Determine the angular velocity of the rod when it reaches the vertical position, and the speed of the rod's tip at this moment.

>CM

APPR ACH We can use the work-energy principle here. The work done is due to

gravity, and is equal to the change in gravitational potential energy of the rod. S LUTI N Since the cm of the rod drops a vertical distance £/2, the work done by gravity is

w = Mg The initial kinetic energy is zero. Hence, from the work-energy principle,

FIGURE 1 0 - 2 9

Ex m ple 10 - 15.

\lw 2 = M g ~ Since I = \ M f for a rod pivoted about its end (Fig. 10-20g), we can solve for a):

The tip of the rod will have a linear speed (see Eq. 10-4) v = £w = \/3g£. N TE By comparison, an object that falls vertically a height £ has a speed

v = V2g£. EXERCISE D Estim te the energy stored in the rot tion l m otion of hurric ne. M odel the hurric ne s uniform cylinder 300 km in di meter nd 5 km high, m de of ir whose m ss is 1.3 kg per m 3. Estim te the outer edge o f the hurric ne to m ove t speed o f 200 k m /h .

10-9 Rot tion l Plus Tr nsl tion l Motion; Rolling Rolling Without Slipping The rolling motion of a ball or wheel is familiar in everyday life: a ball rolling across the floor, or the wheels and tires of a car or bicycle rolling along the pavement. Rolling wit out slipping depends on static friction between the rolling object and the ground. The friction is static because the rolling object's point of contact with the ground is at rest at each moment. Rolling without slipping involves both rotation and translation. There is a simple relation between the linear speed v of the axle and the angular velocity o) of the rotating wheel or sphere: namely, v = Rco, where R is the radius, as we now show. Figure 10-30a shows a wheel rolling to the right without slipping. At the instant shown, point P on the wheel is in contact with the ground and is momentarily at rest. The velocity of the axle at the wheel's center is v. In Fig. 10-30b we have put ourselves in the reference frame of the wheel—that is, we are moving to the right with velocity v relative to the ground. In this reference frame the axle is at rest, whereas the ground and point P are moving to the left with velocity —v as shown. Here we are seeing pure rotation. We can then use Eq. 10 - 4 to obtain v = R(o, where R is the radius of the wheel. This is the same v as in Fig. 10-30a, so we see that the linear speed v of the axle relative to the ground is related to the angular velocity (o by v = Ro),

FIGURE 1 0 - 3 0 ( ) A w heel rolling to the right. Its center C m oves with velocity v. Point P is t rest t this inst nt, (b) The s me w heel s seen from reference fr me in which the xle of the w heel C is t rest — th t is, w e re m oving to the right with velocity v rel tive to the ground. Point P, which w s t rest in ( ), here in (b) is m oving to the left with velocity - v s shown. (See lso Section 3 - 9 on rel tive velocity.)

(a)

[rolling without slipping]

This is valid only if there is no slipping. SECTI N 10 - 9

Rotational Plus Translational Motion; Rolling

267

FIGURE 1 0 - 3 1 ( ) A rolling w heel rot tes bout the inst nt neous xis (perpendicul r to the p ge) p ssing through the point of cont ct with the ground, P. The rrows represent the inst nt neous velocity of e ch point. (b) Photogr ph o f rolling wheel. The spokes re m ore blurred where the speed is gre ter.

Inst nt neous Axis When a wheel rolls without slipping, the point of contact of the wheel with the ground is instantaneously at rest. It is sometimes useful to think of the motion of the wheel as pure rotation about this "instantaneous axis" passing through that point P (Fig. 10-31a). Points close to the ground have a small linear speed, as they are close to this instantaneous axis, whereas points farther away have a greater linear speed. This can be seen in a photograph of a real rolling wheel (Fig. 10—31b): spokes near the top of the wheel appear more blurry because they are moving faster than those near the bottom of the wheel. FIGURE 1 0 - 3 2 A wheel rolling without slipping can be considered as translation of the wheel as a whole with velocity v Mplus rotation about the cm.

Tr nsl tion

Rot tion

Tot l Kinetic Energy = K CM + K rot An object that rotates while its center of mass (cm) undergoes translational motion will have both translational and rotational kinetic energy. Equation 10-19, K = \l(o2, gives the rotational kinetic energy if the rotation axis is fixed. If the object is moving, such as a wheel rolling along the ground, Fig. 10-32, this equation is still valid as long as the rotation axis is fixed in direction. To obtain the total kinetic energy, we note that the rolling wheel undergoes pure rotation about its instantaneous point of contact P, Fig. 10-31. As we saw above relative to the discussion of Fig. 10-30, the speed v of the cm relative to the ground equals the speed of a point on the edge of the wheel relative to its center. Both of these speeds are related to the radius R by v = a)R. Thus the angular velocity co about point P is the same w for the wheel about its center, and the total kinetic energy is

tftot =

Rolling

\ 0)2,

where / P is the rolling object's moment of inertia about the instantaneous axis at P. We can write Ktot in terms of the center of mass using the parallel-axis theorem: = ^cm + M R2, where we have substituted = R in Eq. 10-17. Thus K«

= ilcu-cel11 B ox (slid in g )

Rotational Plus Translational Motion; Rolling

269

S p h e re , rutting U j ih c right — ►

FIGURE 10-35 A sphere rolling to the right on pl ne surf ce. The point in cont ct with the ground t ny m om ent, point P, is moment rily t rest. Point A to the left of P is moving ne rly vertic lly upw rd t the inst nt shown, nd point B to the right is moving ne rly vertic lly downw rd. A n inst nt l ter, point B will touch the pl ne nd be t rest moment rily. Thus no work is done by the force of st tic friction.

/j\

AUT I ON

W en is 2 t = l a valid?

FIGURE 10 -36

Ex m ple 10 - 18.

CHAPTER 10

If there had been little or no static friction between the rolling objects and the plane in these Examples, the round objects would have slid rather than rolled, or a combination of both. Static friction must be present to make a round object roll. We did not need to take friction into account in the energy equation for the rolling objects because it is static friction and does no work—the point of contact of a sphere at each instant does not slide, but moves perpendicular to the plane (first down and then up as shown in Fig. 10-35) as it rolls. Thus, no work is done by the static friction force because the force and the motion (displacement) are perpendicular. The reason the rolling objects in Examples 10-16 and 10-17 move down the slope more slowly than if they were sliding is not because friction slows them down. Rather, it is because some of the gravitional potential energy is converted to rotational kinetic energy, leaving less for the translational kinetic energy. EXERCISE E Return to the Ch pter -Opening Q uestion, p. 248, nd nswer it Try to expl in why you m y h ve nswered differently the first time.

g in now.

Using D tcm ~ I c m **cm We can examine objects rolling down a plane not only from the point of view of kinetic energy, as we did in Examples 10-16 and 10-17, but also in terms of forces and torques. If we calculate torques about an axis fixed in direction (even if the axis is accelerating) which passes through the center of mass of the rolling sphere, then 2 t cm = /cm^cm is valid, as we discussed in Section 10-5. See Eq. 10-15, whose validity we will show in hapter 11. Be careful, however: Do not assume 2 r = la is always valid. You cannot just calculate r, I, and a about any axis unless the axis is (1 ) fixed in an iner tial reference frame or (2 ) fixed in direction but passes through the cm of the object. EXAMPLE 10 - 18 Analysis of a sphere on an incline using forces. Analyze the rolling sphere of Example 10-16, Fig. 10-33, in terms of forces and torques. In particular, find the velocity v and the magnitude of the friction force, Ffr, Fig. 10-36. APPR ACH We analyze the motion as translation of the cm plus rotation about the cm . Ffr is due to static friction and we cannot assume Ffr = /jlsFn , only FfT — iisFtf. S LUTI N For translation in the x direction we have from ST7 = ma, Mg sin 0 — Ffr = Ma, and in the y direction Fn - Mg cos 0 = 0 since there is no acceleration perpendicular to the plane. This last equation merely tells us the magnitude of the normal force, Fn = Mg cos 0. For the rotational motion about the c m , we use Newton's second law for rotation 2 t cm = IcMacM (Eq. 10-15), calculating about an axis passing through the cm but fixed in direction: Fb r0 = (| Mrl)a. The other forces, FN and Mg, point through the axis of rotation ( cm ) , so have lever arms equal to zero and do not appear here. As we saw in Example 10-16 and Fig. 10-30, co = v/r0 where v is the speed of the cm . Taking derivatives of 00= v/r0 with respect to time we have a = a/r0\ substituting into the last equation we find F = \Ma. When we substitute this into the top equation, we get Mg sin 0 - \M a = Ma, or a = f g s in 0. We thus see that the acceleration of the cm of a rolling sphere is less than that for an object sliding without friction (a = gsin0). The sphere started from rest

Rotational Motion

at the top of the incline (height H). To find the speed v at the bottom we use Eq. 2 -12c where the total distance traveled along the plane is x = H / sin0 (see Fig. 10-36). Thus _________________ = \fla x

H sin0

= A/ygtf-

This is the same result obtained in Example 10-16 although less effort was needed there. To get the magnitude of the force of friction, we use the equations obtained above: Ffr = \M a = §M (fgsin0) = |M gsin0. N TE If the coefficient of static friction is sufficiently small, or 0 sufficiently large so that Ffr > /xsFN(that is, iff tan 0 > |/xs), the sphere will not simply roll but will slip as it moves down the plane.

*More Adv nced Ex mples Here we do three more Examples, all of them fun and interesting. When they use = la , we must remember that this equation is valid only if r, a, and I are calculated about an axis that either (1 ) is fixed in an inertial reference frame, or (2) passes through the cm of the object and remains fixed in direction. A falling yo-yo. String is wrapped around a uniform solid cylinder (something like a yo-yo) of mass M and radius R, and the cylinder starts falling from rest, Fig. 10-37a. As the cylinder falls, find (a) its acceleration and (b) the tension in the string. APPR ACH As always we begin with a free-body diagram, Fig. 10-37b, which

shows the weight of the cylinder acting at the cm and the tension of the string F x acting at the edge of the cylinder. We write Newton's second law for the linear motion (down is positive) Ma = 2 F = Mg - Ft . Since we do not know the tension in the string, we cannot immediately solve for a. So we try Newton's second law for the rotational motion, calculated about the center of mass: 2 t M = / M^ M FTR = \M R 2a. Because the cylinder "rolls without slipping" down the string, we have the additional relation that a = aR (Eq. 10-5). S LUTI N The torque equation becomes FrR

MRl \ —

FIGURE

so Substituting this into the force equation, we obtain Ma = Mg — Fx = Mg - \Ma. Solving for a, we find that a = \g . That is, the linear acceleration is less than what it would be if the cylinder were simply dropped. This makes sense since gravity is not the only vertical force acting; the tension in the string is acting as well. (b) Since a = §g, Fx = \M a = \Mg. EXERCISE F Find the cceler tion of yo - yo w hose spindle h s r dius \R . A ssum e the m om ent of inerti is still \M R 2 (ignore the m ss of the spindle). tFfr > ju,s^N is equiv lent to t n# > \ i i s bec use Ffr = fM gsin6

= /xsM gcos6.

SECTI N 10 - 9

Rotational Plus Translational Motion; Rolling

271

EXAMPLE 10 -20 What if a rolling ball slips? A bowling ball of mass M and radius r0 is thrown along a level surface so that initially (t = 0) it slides with a linear speed v0 but does not rotate. As it slides, it begins to spin, and eventually rolls without slipping. How long does it take to begin rolling without slipping?

CM "

FIGURE 1 0 - 3 8

_______

APPR ACH The free-body diagram is shown in Fig. 10-38, with the ball moving to

the right. The friction force does two things: it acts to slow down the translational motion of the cm; and it immediately acts to start the ball rotating clockwise. S LUTI N Newton's second law for translation gives

Ex m ple 10 - 20. M ax =

^Fx =

—Ffr =

~ t i k FN =

M g,

where is the coefficient of kinetic friction because the ball is sliding. Thus a x = —fJLkg- The velocity of the cm is ^ M = v0 + ax t = v0 - fLkgt. Next we apply Newton's second law for rotation about the cm, / Ma M = ^ t cm: | M r \ qicm =

Ffrr0

= VkMgr0. The angular acceleration is thus aCM = 5 fik g/2r0, which is constant. Then the angular velocity of the ball is (Eq. 10 -9a) - Mabove: Vo - fLtg

5 ^ k g ti 2r0 r0

so 2v0 7/^k g 0

PHYSICS

APPLIED

Braking distribution of a car FIGURE 1 0 - 3 9 Forces on br king c r (Ex m ple 1 0 - 2 1 ).

EXAMPLE 10 -21 ESTIMATE"! Braking a car. When the brakes of a car are applied, the front of the car dips down a bit; and the force on the front tires is greater than on the rear tires. To see why, estimate the magnitude of the normal forces, ^N1 and on the front and rear tires of the car shown in Fig. 10-39 when the car brakes and decelerates at a rate a = 0.50 g. The car has mass M = 1200 kg, the distance between the front and rear axles is 3.0 m, and its cm (where the force of gravity acts) is midway between the axles 75 cm above the ground. APPR ACH Figure 10-39 is the free-body diagram showing all the forces on the

car. Fi and F2 are the frictional forces that decelerate the car. We let Fx be the sum of the forces on both front tires, and F2 likewise for the two rear tires. -^Nl and ^N2 are the normal forces the road exerts on the tires and, for our estimate, we assume the static friction force acts the same for all the tires, so that Fx and F2 are proportional respectively to ^N1 and FmFi = pFm

272

CHAPTER 10

Rotational Motion

and

F2 = nFN2.

S LUTI N The friction forces F1 and F2 decelerate the car, so Newton's second

law gives Fi + F2 = Ma = (1200 kg)(0.50)(9.8 m /s2) = 5900 N.

(i)

While the car is braking its motion is only translational, so the net torque on the car is zero. If we calculate the torques about the cm as axis, the forces Fx, F2, and Fm all act to rotate the car clockwise, and only Fm acts to rotate it counterclock wise; so Ffil must balance the other three. Hence, FN1 must be significantly greater than ■^N2 - Mathematically, we have for the torques calculated about the cm: (1.5 m)FN1 - (1.5 m)FN2 - (0.75 m)F1 - (0.75 m)F2 = 0. Since Fx and F2 are proportional1 to FN1 and FN2 (Fx = /aFn1, F2 = fiFm ), we can write this as (1.5m)(FN1 - Fm ) - (0.75m)(M)(FN1 + FN2) = 0.

(ii)

Also, since the car does not accelerate vertically, we have F + F Mg = Fm + FN2 = - L- — -•

(iii)

omparing (iii) to (i), we see that fi = a/g = 0.50. Now we solve (ii) for ^N1 and use /jl = 0.50 to obtain

Fm =

=IFn2-

Thus Fm is 1f times greater than FN2• Actual magnitudes are determined from (iii) and (i): FN1 + FN2 = (5900 N)/(0.50) = 11,800 N which equals FN2(l + f ) ; so FN2 = 4400 N and FN1 = 7400 N. N TE Because the force on the front tires is generally greater than on the rear tires, cars are often designed with larger brake pads on the front wheels than on the rear. Or, to say it another way, if the brake pads are equal, the front ones wear out a lot faster.

10-10 Why Does Rolling Sphere Slow Down? A sphere of mass M and radius r0 rolling on a horizontal flat surface eventually comes to rest. What force causes it to come to rest? You might think it is friction, but when you examine the problem from a simple straightforward point of view, a paradox seems to arise. Suppose a sphere is rolling to the right as shown in Fig. 10-40, and is slowing down. By Newton's second law, 2F = M , there must be a force F (presumably frictional) acting to the left as shown, so that the acceleration will also point to the left and v will decrease. uriously enough, though, if we now look at the torque equation (calculated about the center of mass), S r M = ICMa, we see that the force F acts to increase the angular acceleration a, and thus to increase the velocity of the sphere. Thus the paradox. The force F acts to decelerate the sphere if we look at the translational motion, but speeds it up if we look at the rotational motion. fOur proportion lity const nt /jl is not equ l to the c r is just bout to skid.

/jls, the st

FIGURE 1 0 - 4 0 the right.

Sphere rolling to

tic coefficient of friction (Ffr < n sFN), unless

*SECTI N 10 - 10

Why Does a Rolling Sphere Slow Down?

273

The resolution of this apparent paradox is that some other force must be acting. The only other forces acting are gravity, Mg, and the normal force Fn (= —Mg). These act vertically and hence do not affect the horizontal transla tional motion. If we assume the sphere and plane are rigid, so the sphere is in contact at only one point, these forces give rise to no torques about the cm either, since they act through the cm. The only recourse we have to resolve the paradox is to give up our idealization that the objects are rigid. In fact, all objects are deformable to some extent. Our sphere flattens slightly and the level surface also acquires a slight depression where the two are in contact. There is an area of contact, not a point. Hence there can be a torque at this area of contact which acts in the opposite direction to the torque associated with F, and thus acts to slow down the rotation of the sphere. This torque is associated with the normal force FN that the table exerts on the sphere over the whole area of contact. The net effect is that we can consider FN acting vertically a distance £ in front of the cm as shown in Fig. 10-41 (where the deformation is greatly exaggerated). Is it reasonable that the normal force FNshould effectively act in front of the CM as shown in Fig. 10-41? Yes. The sphere is rolling, and the leading edge strikes the surface with a slight impulse. The table therefore pushes upward a bit more strongly on the front part of the sphere than it would if the sphere were at rest. At the back part of the area of contact, the sphere is starting to move upward and so the table pushes upward on it less strongly than when the sphere is at rest. The table pushing up more strongly on the front part of the area of contact gives rise to the necessary torque and justifies the effective acting point of FNbeing in front of the cm. When other forces are present, the tiny torque r N due to FN can usually be ignored. For example, when a sphere or cylinder rolls down an incline, the force of gravity has far more influence than r N, so the latter can be ignored. For many purposes (but not all), we can assume a hard sphere is in contact with a hard surface at essentially one point.

FIGURE 1 0 - 4 1 The norm l force, F n , exerts torque th t slows down the sphere. The deform tion of the sphere nd the surf ce it m oves on h s been ex gger ted for det il.

Summary W hen rigid object rot tes bout fixed xis, e ch point of the object m oves in circul r p th. Lines dr wn perpendicul rly from the rot tion xis to different points in the object ll sweep out the s m e ngle 0 in ny given time interv l. Angles re conveniently me sured in r di ns. One r di n is the ngle subtended by n rc whose length is equ l to the r dius, or 277 - r d = 360 °

lr d

57.3 °.

A ll p rts o f rigid object rot ting bout fixed xis h ve the s m e ngul r velocity to nd the s me ngul r cceler tion a t ny inst nt, where (10 - 2b )

dt do) dt

(10 - 3b )

The units o f o) nd a re r d /s nd r d /s2. The line r velocity nd cceler tion of ny point in n object rot ting bout fixed xis re rel ted to the ngul r qu ntities by v = R(o

«t n

R a

aR = (o2R

(1 0 - 4 ) (1 0 - 5 ) (1 0 - 6 )

where R is the perpendicul r dist nce o f the point from the rot tion xis, nd a tan nd aR re the t ngenti l nd r di l

CHAPTER 10

Rotational Motion

to = 2tr f

(1 0 - 7 )

T = 1/ / .

(1 0 - 8 )

Angul r velocity nd ngul r cceler tion re vectors. For rigid object rot ting bout fixed xis, both nd a point long the rot tion xis. The direction of to is given by the right-h nd rule. If rigid object undergoes uniformly cceler ted rot tion l m otion ( = const nt), equ tions n logous to those for line r m otion re v lid: to = o)q + at; a)2 = o)q + 2a0\

274

com ponents o f the line r cceler tion. The frequency / period T re rel ted to o) (r d /s) by

(Opt + \ a t 2;

0) + (Oq

(1 0 - 9 )

The torque due to force F exerted on rigid object is equ l to r = R ± F = RF± = R F sin 0, (10 - 1 0 ) where R ± , c lled the lever rm, is the perpendicul r dist nce from the xis of rot tion to the line long which the force cts, nd 6 is the ngle betw een F nd R. The rot tion l equiv lent of N ew ton 's second l w is S t = la ,

(10 - 1 4 )

where I = 'ZrriiR2 is the m om ent o f inerti o f the object bout the xis of rot tion. This rel tion is v lid for rigid object

r o t t in g w h en t, /,

bout nd

x is f i x e d in r e c lc u l te d

n in e r t i l r e f e r e n c e f r m e , o r b o u t th e c en te r o f m ss o f

The following Table summarizes angular (or rotational) quantities, comparing them to their translational analogs.

o b j e c t e v e n i f t h e c m is m o v in g .

The rot tion l kinetic energy of an object rotating about a fixed axis with angular velocity o) is K = \la>2.

(10 -19)

For an object undergoing both translational and rotational motion, the total kinetic energy is the sum of the translational kinetic energy of the object's c m plus the rotational kinetic energy of the object about its c m : K m = \M v 2ck + \ I VM0?

(10 -23)

as long as the rotation axis is fixed in direction.

Tr nsl tion X V a m F K = \m v 2 W = Fd 'IF = ma

Rot tion

Connection

x = R0 v = Rco a = Ra I = Zm R2 t = RF sin 0

(O a I I/O,2 W = t0 2 t = la

Questions 1. A bicycle odometer (which counts revolutions and is calibrated to report distance traveled) is attached near the wheel hub and is calibrated for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels? 2. Suppose a disk rotates at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk's angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change? 3. ould a nonrigid object be described by a single value of the angular velocity a>? Explain. 4. an a small force ever exert a greater torque than a larger force? Explain. 5. Why is it more difficult to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer this. . Mammals that depend on being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 10 -42). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

10. Two solid spheres simultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom? 11. Why do tightrope walkers (Fig. 10 -43) carry a long, narrow beam?

FIGURE 10 - 43

Question 11.

12. A sphere and a cylinder have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational kinetic energy? FIGURE 10 - 42

Question . A gazelle. 7. If the net force on a system is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zero? 8. Two inclines have the same height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain. 9. Two spheres look identical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which.

13. The moment of inertia of this textbook would be the least about which symmetry axis through its center? 14. The moment of inertia of a rotating solid disk about an axis through its c m is \M R 7 (Fig. 10 -20c). Suppose instead that a parallel axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller? 15. The angular velocity of a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Questions

275

Problems 10-1 Angular Quantities 1. (I) Express the following angles in radians: (a) 45.0°, (b) 60.0°, (c) 90.0°, (d) 360.0°, and (e) 445°. Give as numerical values and as fractions of . 2. (I) The Sun subtends an angle of about 0.5° to us on Earth, 150 million km away. Estimate the radius of the Sun. 3. (I) A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges at an angle 0 (Fig. 10 -44) of 1.4 X 10-5 rad. What diameter spot will it make on the Moon?

E rili

12. (II) A 64-cm -diameter wheel accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determine (a) its angular acceleration, and (b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. 13. (II) In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun's energy evenly. A t the start of their trip, they accel erated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine (a) the angular acceleration, and (b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 7.0 min after it started this acceleration. 14. (II) A turntable of radius R x is turned by a circular rubber roller of radius R2 in contact with it at their outer edges. What is the ratio of their angular velocities, «i/g>2?

10 - 2 Vector Nature of w and a FIGURE 10 - 44

Problem 3. 4. (I) The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 4.0 s. What is the angular acceleration as the blades slow down? 5. (II) (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. alculate its angular velocity in rad/s. (b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? . (II) A bicycle with tires 8 cm in diameter travels 7.2 km. How many revolutions do the wheels make? 7. (II) alculate the angular velocity of (a) the second hand, (ib) the minute hand, and (c) the hour hand, of a clock. State in rad/s. (d) What is the angular acceleration in each case? 8. (II) A rotating merry-go-round makes one complete revolu tion in 4.0 s (Fig. 10-45). (a) What is the linear speed of a child seated 1.2 m from the center? (b) What is her acceleration (give components)?

FIGURE 10 - 45

Problem 8. 9. (II) What is the linear speed of a point (a) on the equator, (b) on the Arctic ircle (latitude 66.5° N), and (c) at a latitude of 45.0° N, due to the E arth's rotation? 10. (II) alculate the angular velocity of the Earth (a) in its orbit around the Sun, and (b) about its axis. 11. (II) How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceler ation of 100,000 g's? 276

HAPTER 10

Rotational Motion

15. (II) The axle of a wheel is mounted on supports that rest on a rotating turntable as shown in Fig. 10 -46. The wheel has angular velocity a>i = 44.0 rad/s about its axle, and the turntable has angular velocity a)2 = 35.0 rad/s about a vertical axis. (Note arrows showing these motions in the figure.) (a) What are the directions of i and ia2 at the instant shown? (b) What is the resultant angular velocity of the wheel, as seen by an outside observer, at the instant shown? Give the magnitude and direction, (c) What is the magnitude and direction of the angular acceleration of the wheel at the instant shown? Take the z axis vertically upward and the direction of the axle at the moment shown to be the x axis pointing to the right.

FIGURE 10 - 46

Problem 15.

10 - 3 Constant Angular Acceleration 16. (I) An automobile engine slows down from 3500 rpm to 1200 rpm in 2.5 s. alculate (a) its angular acceleration, assumed constant, and (b) the total number of revolutions the engine makes in this time. 17. (I) A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time? 18. (I) Pilots can be tested for the stresses of flying high-speed jets in a whirling "human centrifuge," which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed, (a) What was its angular acceleration (assumed constant), and (b) what was its final angular speed in rpm? 19. (II) A cooling fan is turned off when it is running at 850 rev/min. It turns 1350 revolutions before it comes to a stop, (a) What was the fan's angular acceleration, assumed constant? (b) How long did it take the fan to come to a complete stop? 20. (II) Using calculus, derive the angular kinematic equations 10 - 9a and 10 - 9b for constant angular acceleration. Start with a = dco/dt.

21. (II) A small rubber wheel is used to drive a large pottery wheel. The two wheels are mounted so that their circular edges touch. The small wheel has a radius of 2.0 cm and accelerates at the rate of 7.2 rad/s2, and it is in contact with the pottery wheel (radius 21.0 cm) without slipping. alcu late (a) the angular acceleration of the pottery wheel, and (b) the time it takes the pottery wheel to reach its required speed of 65 rpm. 22. (II) The angle through which a rotating wheel has turned in time t is given by 6 = 8.51 - 15.012 + 1.6 tA, where 6 is in radians and t in seconds. Determine an expression (a) for the instantaneous angular velocity a) and (b) for the instan taneous angular acceleration a. (c) Evaluate oo and a at t = 3.0 s. (d) What is the average angular velocity, and (e) the average angular acceleration between t = 2.0 s and t = 3.0 s? 23. (II) The angular acceleration of a wheel, as a function of time, is a = 5.012 — 8.5 t, where a is in rad/s2 and t in seconds. If the wheel starts from rest (6 = 0, o) = 0, at t = 0), determine a formula for (a) the angular velocity co and (b) the angular position 6, both as a function of time. (c) Evaluate o>and 0 at t = 2.0 s.

29. (II) The bolts on the cylinder head of an engine require tightening to a torque of 75 m •N. If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end? If the six-sided bolt head is 15 mm across (Fig. 10 -49), estimate the force applied near each of the six points by a socket wrench. 2« cm

15 rM | WIYflL'h cmbolt FIGURE 10 - 49

Problem 29.

30. (II) Determine the net torque on the 2.0-m-long uniform beam shown in Fig. 10 -50. ,%N alculate about (a) point , , the cm, and (b) point P at \3(Yj one end.

65 N

10 - 4 Torque 24. (I) A 62-kg person riding a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. (a) What is the maximum torque she exerts? (b) How could she exert more torque? 25. (I) alculate the net torque about the axle of the wheel shown in Fig. 10 -47. Assume that a friction torque of 0.40 m •N opposes the motion. 35 N j

28 N FIGURE 10 - 50

52 N

Problem 30.

10 - 5 and 10 - 6 Rotational Dynamics FIGURE 10 - 47

Problem 25. 26. (II) A person exerts a horizontal force of 32 N on the end of a door 96 cm wide. What is the magnitude of the torque if the force is exerted (a) perpendicular to the door and (b) at a 60.0° angle to the face of the door? 27. (II) Two blocks, each of mass ra, are attached to the ends of a massless rod which pivots as shown in Fig. 10 -48. Initially the rod is held in the horizontal position and then released. alculate the magnitude and direction of the net torque on this system when it is first released. f, nt

FIGURE 10 - 48

Problem 27.

28. (II) A wheel of diameter 27.0 cm is constrained to rotate in the xy plane, about the z axis, which passes through its center. A force F = ( —31.0i + 43.4j) N acts at a point on the edge of the wheel that lies exactly on the x axis at a particular instant. What is the torque about the rotation axis at this instant?

31. (I) Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center. 32. (I) Estimate the moment of inertia of a bicycle wheel 67 cm in diameter. The rim and tire have a combined mass of 1.1 kg. The mass of the hub can be ignored (why?). 33. (II) A potter is shaping a bowl on a potter's wheel rotating at constant angular speed (Fig. 10-51). The friction force between her hands and the clay is 1.5 N total, (a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm? (b) How long would it take for the potter's wheel to stop if the only torque acting on it is due to the potter's hand? The initial angular velocity of the wheel is 1. rev/s, and the moment of inertia of the wheel and the bowl is 0.11 kg - m 2.

FIGURE 10 -51

Problem 33.

Problems

277

34. (II) An oxygen molecule consists of two oxygen atoms whose total mass is 5.3 X 10-26kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.9 X 10-4 kg - m 2. From these data, estimate the effective distance between the atoms. 35. (II) A softball player swings a bat, accelerating it from rest to 2.7 rev/s in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it. 36. (II) A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0.380 kg. alculate (a) its moment of inertia about its center, and (b) the applied torque needed to accelerate it from rest to 1750 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s. 37. (II) A small 650-g ball on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. alculate (a) the moment of inertia of the ball about the center of the circle, and (b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod's moment of inertia and air resistance. 38. (II) The forearm in Fig. 10 -52 accelerates a 3.6-kg ball at 7.0 m /s2 by means of the triceps muscle, as shown. alculate (a) the } torque needed, and (b) the force that must be exerted by the triceps muscle. Ignore the mass 3 1cm of the arm. 2,5 cm'

Axis of rotaiior (;it elbow)

FIGURE 10 - 52

Triceps

Problems 38 and 39.

TIiu sl ' I l-

39. (II) Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 10 -52. The ball is accelerated uniformly from rest to 8.5 m /s in 0.35 s, at which point it is released. alculate (a) the angular acceler ation of the arm, and (b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.7 kg and rotates like a uniform rod about an axis at its end. 40. (II) alculate the moment of inertia of the array of point objects shown in Fig. 10 -53 about (a) the vertical axis, and (b) the horizontal axis. Assume m = 2.2 kg, M = 3.1 kg, and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis, (c) About which axis would it be harder to accelerate this array?

.5 m

i.V I l

inA =■8,0 kg w p *

V w u = 10.0 kg

\32Q FIGURE 10 - 54

Problem 46.

47. (II) A helicopter rotor blade can be considered a long thin

rod, as shown in Fig. 10 -55. (a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 135 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation, (b) How much torque must the motor apply to bring the blades from rest up to a speed of 5.0 rev/s in 8.0 s?

1.5 1

41. (II) A merry -go -round accelerates from rest to 0.68 rad/s in 24 s. Assuming the merry -go -round is a uniform disk of radius 7.0 m and mass 31,000 kg, calculate the net torque required to accelerate it. 42. (II) A 0.72-m -diameter solid sphere can be rotated about an axis through its center by a torque of 10.8m - N which accelerates it uniformly from rest through a total of 180 revolutions in 15.0 s. What is the mass of the sphere? 43. (II) Suppose the force FT in the cord hanging from the pulley of Example 10 - 9, Fig. 10 -21, is given by the relation FT = 3.001 — 0.2012 (newtons) where t is in seconds. If the pulley starts from rest, what is the linear speed of a point on its rim 8.0 s later? Ignore friction. 44. (II) A dad pushes tangentially on a small hand -driven merry -go -round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry -go -round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. alculate the torque required to produce the acceleration, neglecting frictional torque. What force is required at the edge? 45. (II) Four equal masses M are spaced at equal intervals, £, along a horizontal straight rod whose mass can be ignored. The system is to be rotated about a vertical axis passing through the mass at the left end of the rod and perpendicular to it. (a) What is the moment of inertia of the system about this axis? (b) What minimum force, applied to the farthest mass, will impart an angular acceleration a l (c) What is the direction of this force? 46. (II) Two blocks are connected by a light string passing over a pulley of radius 0.15 m and moment of inertia I. The blocks move (towards the right) with an acceleration of 1.00m /s2 along their frictionless inclines (see Fig. 10 -54). (a) Draw free -body diagrams for each of the two blocks and the pulley, (b) Determine F ix and Ftb> the tensions in the two parts of the string, (c) Find the net torque acting on the pulley, and determine its moment of inertia, /.

M m = 135 kg FIGURE 10 - 55 FIGURE 10 - 53

278

HAPTER 10

Problem 40.

Rotational Motion

Problem 47.

48. (II) A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.20 m •N. If the mass of the rotor is 3.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest, and how long will it take? 49. (II) When discussing moments of inertia, especially for unusual or irregularly shaped objects, it is sometimes conve nient to work with the r dius o f gyr tion, k. This radius is defined so that if all the mass of the object were concentrated at this distance from the axis, the moment of inertia would be the same as that of the original object. Thus, the moment of inertia of any object can be written in terms of its mass M and the radius of gyration as I = M k 2. Determine the radius of gyration for each of the objects (hoop, cylinder, sphere, etc.) shown in Fig. 10 -20. 50. (II) To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 10-56. If the satellite has a mass of 3600 kg, a radius of 4.0 m, and the rockets each add a mass of 250 kg, what is the required steady force of each rocket if the ''R End view of satellite is to reach cylindrical /II s tellite 32 rpm in 5.0 min, starting from rest? FIGURE 10 - 56

Problem 50. 51. (Ill) An A tw ood's mac ine consists of two masses, m A and mB, which are connected by a massless inelastic cord that passes over a pulley, Fig. 10-57. If the pulley has radius R and moment of R O R inertia I about its axle, determine the acceleration of the masses m A and mB, and compare to the situation in which the moment of inertia of the pulley is v i ignored. [Hint: The tensions FTA and 1 1 jj?TA Ftb are not equal. We discussed the Atwood machine in Example 4 -13, assuming I = 0 for the pulley.] A | ^TU FIGURE 10 - 57 Problem 51. I, Atwood's machine. 52. (Ill) A string passing over a pulley has a 3.80-kg mass hanging from one end and a 3.15-kg mass hanging from the other end. The pulley is a uniform solid cylinder of radius 4.0 cm and mass 0.80 kg. (a) If the bearings of the pulley were frictionless, what would be the acceleration of the two masses? (b) In fact, it is found that if the heavier mass is given a downward speed of 0.20 m /s, it comes to rest in .2 s. What is the average frictional torque acting on the pulley? 53. (Ill) A hammer thrower accelerates the hammer (mass = 7.30 kg) from rest within four full turns (revolutions) and releases it at a speed of 26.5 m/s. Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate (a) the angular acceleration, (b) the (linear) tangential acceleration, (c) the centripetal acceleration just before release, (d) the net force being exerted on the hammer by the athlete just before release, and (e) the angle of this force with respect to the radius of the circular motion. Ignore gravity.

54. (Ill) A thin rod of length £ stands vertically on a table. The rod begins to fall, but its lower end does not slide, (a) D eter mine the angular velocity of the rod as a function of the angle cf>it makes with the tabletop. (b) What is the speed of the tip of the rod just before it strikes the table?

10 - 7 Moment of Inertia 55. (I) Use the parallel-axis theorem to show that the moment of inertia of a thin rod about an axis perpendicular to the rod at one end is I = \M l2, given that if the axis passes through the center, I = ^ M f2 (Fig. 10 - 20f and g). 56. (II) Determine the moment of inertia of a 19-kg door that is 2.5 m high and 1.0 m wide and is hinged along one side. Ignore the thickness of the door. 57. (II) Two uniform solid spheres of mass M and radius r0 are connected by a thin (massless) rod of length r0 so that the centers are 3r0 apart, (a) Determine the moment of inertia of this system about an axis perpendicular to the rod at its center, (b) What would be the percentage error if the masses of each sphere were assumed to be concentrated at their centers and a very simple calculation made? 58. (II) A ball of mass M and radius rx on the end of a thin massless rod is rotated in a horizontal circle of radius R 0 about an axis of rotation AB, as shown in Fig. 10 -58. (a) onsidering the mass of the ball to be concentrated at its center of mass, calculate its moment of inertia about AB. (b) Using the parallel-axis theorem and considering the finite radius of the ball, calculate the moment of inertia of the ball about AB. (c) alculate the percentage error intro duced by the point mass approximation for r\ = 9.0 cm and Rn = 1.0 m. U

FIGURE 10 - 58

Problem 58.

59. (II) A thin 7.0-kg wheel of radius 32 cm is weighted to one side by a 1.50-kg weight, small in size, placed 22 cm from the center of the wheel. alculate (a) the position of the center of mass of the weighted wheel and (b) the moment of inertia about an axis through its cm, perpendicular to its face. 60. (Ill) Derive the formula for the moment of inertia of a uniform thin rod of length £ about an axis through its center, perpendicular to the rod (see Fig. 10 -20f). 61. (Ill) (a) Derive the formula given in Fig. 10 - 20h for the moment of inertia of a uniform, flat, rectangular plate of dimensions £ X w about an axis through its center, perpendicular to the plate, (b) What is the moment of inertia about each of the axes through the center that are parallel to the edges of the plate?

10 - 8 Rotational Kinetic Energy 62. (I) An automobile engine develops a torque of 255 m •N at 3750 rpm. What is the horsepower of the engine? 63. (I) A centrifuge rotor has a moment of inertia of 4.25 X 10_2kg - m2. How much energy is required to bring it from rest to 9750 rpm?

Problems

279

64. (II) A rotating uniform cylindrical platform of mass 220 kg and radius 5.5 m slows down from 3.8 rev/s to rest in 16 s when the driving motor is disconnected. Estimate the power output of the motor (hp) required to maintain a steady speed of 3.8 rev/s. 65. (II) A merry -go -round has a mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder. . (II) A uniform thin rod of length £ and mass M is suspended freely from one end. It is pulled to the side an angle 6 and released. If friction can be ignored, what is its angular velocity, and the speed of its free end, at the lowest point? 67. (II) Two masses, raA = 35.0 kg and raB = 38.0 kg, are connected by a rope that hangs over a pulley (as in Fig. 10-59). The pulley is a uniform cylinder of radius 0.381m and mass 3.1kg. Initially m A is on the ground and mB rests 2.5 m above the ground. If the system is released, use conservation of energy to deter mine the speed of mB just before it strikes the ground. Assume the pulley bearing is frictionless. 2.5

73. (II) A sphere of radius r0 = 24.5 cm and mass m = 1.20 kg starts from rest and rolls without slipping down a 30.0° incline that is 10.0 m long, (a) alculate its translational and rotational speeds when it reaches the bottom. (b) What is the ratio of translational to rotational kinetic energy at the bottom? Avoid putting in numbers until the end so you can answer: (c) do your answers in (a) and (b) depend on the radius of the sphere or its mass? 74. (II) A narrow but solid spool of thread has radius R and mass M. If you pull up on the thread so that the cm of the spool remains suspended in the air at the same place as it unwinds, (a) what force must you exert on the thread? (b) How much work have you done by the time the spool turns with angular velocity w? 75. (II) A ball of radius r0rolls on the inside of a track of radius R 0 (see Fig. 10 -61). If the ball starts from rest at the vertical edge of the track, what will be its speed when it reaches the lowest point of the track, rolling without slipping? i \

\ \

I3 5 ' L > \

\ \

FIGURE 10 - 59

Problem 67. 8. (Ill) A 4.00-kg mass and a 3.00-kg mass are attached to opposite ends of a thin 42.0-cm-long horizontal rod (Fig. 10 -60). The system is rotating at angular speed a) = 5.60 rad/s about a vertical axle at the center of the rod. Determine (a) the kinetic energy K of the system, and (b) the net force on each mass, (c) Repeat parts (a) and (b) assuming that the axle passes through the cm of the system.

69. (Ill) A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]

10 - 9 Rotational Plus Translational Motion 70. (I) alculate the translational speed of a cylinder when it reaches the foot of an incline 7.20 m high. Assume it starts from rest and rolls without slipping. 71. (I) A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.7 m /s. alculate its total kinetic energy. 72. (I) Estimate the kinetic energy of the Earth with respect to the Sun as the sum of two terms, (a) that due to its daily rotation about its axis, and (b) that due to its yearly revolu tion about the Sun. [Assume the Earth is a uniform sphere with mass = 6.0 X 1024kg, radius = 6.4 X 10 m, and is 1.5 X 108 km from the Sun.] 28

HAPTER 10

Rotational Motion

FIGURE 10 -61

Problems 75 and 81.

76. (II) A solid rubber ball rests on the floor of a railroad car when the car begins moving with acceleration a. Assuming the ball rolls without slipping, what is its acceleration relative to (a) the car and (b) the ground? 77. (II) A thin, hollow 0.545-kg section of pipe of radius 10.0 cm starts rolling (from rest) down a 17.5° incline 5.60 m long. (a) If the pipe rolls without slipping, what will be its speed at the base of the incline? (b) What will be its total kinetic energy at the base of the incline? (c) What minimum value must the coefficient of static friction have if the pipe is not to slip? *78. (II) In Example 10 -20, (a) how far has the ball moved down the lane when it starts rolling without slipping? (b) What are its final linear and rotational speeds? 79. (Ill) The 1100-kg mass of a car includes four tires, each of mass (including wheels) 35 kg and diameter 0.80 m. Assume each tire and wheel combination acts as a solid cylinder. Determine (a) the total kinetic energy of the car when trav eling 95 km /h and (b) the fraction of the kinetic energy in the tires and wheels, (c) If the car is initially at rest and is then pulled by a tow truck with a force of 1500 N, what is the acceleration of the car? Ignore frictional losses. (d) What percent error would you make in part (c) if you ignored the rotational inertia of the tires and wheels? *80. (Ill) A wheel with rotational inertia I = \M R 2 about its central axle is set spinning with initial angular speed a>0and is then lowered onto the ground so that it touches the ground with no horizontal speed. Initially it slips, but then begins to move forward and eventually rolls without slipping, (a) In what direction does friction act on the slipping wheel? (b) How long does the wheel slip before it begins to roll without slipping? (c) What is the wheel's final translational speed? [Hint Use 2 F = m , 2 t cm = ICMciCM, and recall that only when there is rolling without slipping is -y M = a>i?.]

81. (Ill) A small sphere of radius r0 = 1.5 cm rolls without slipping on the track shown in Fig. 10-61 whose radius is R q = 26.0 cm. The sphere starts rolling at a height R0 above the bottom of the track. When it leaves the track after passing through an angle of 135° as shown, (a) what will be its speed, and (b) at what distance D from the base of the track will the sphere hit the ground?

*10 - 10 Rolling Sphere Slows Down *82. (I) A rolling ball slows down because the normal force does not pass exactly through the c m of the ball, but passes in front of the c m . Using Fig. 10-41, show that the torque resulting from the normal force (tn = £FN in Fig. 10-41) is | of that due to the frictional force, Tfr = r0F where r0 is the ball's radius; that is, show that tn = gTfr.

| Gener l Problems__________ 83. A large spool of rope rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance £, holding onto it, Fig. 10-62. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool's center of mass move?

FIGURE 10 - 62

Problem 83.

;

84. On a 12.0-cm-diameter audio compact disc ( D), digital bits of information are encoded sequentially along an outward spiraling path. The spiral starts at radius Rx = 2.5 cm and winds its way out to radius R2 = 5.8 cm. To read the digital information, a D player rotates the D so that the player's readout laser scans along the spiral's sequence of bits at a constant linear speed of 1.25 m/s. Thus the player must accu rately adjust the rotational frequency / of the D as the laser moves outward. Determine the values for / (in units of rpm) when the laser is located at Ri and when it is at R2. 85. (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo just before it reaches the end of its 1.0-m-long string, if it is released from rest, (b) What fraction of its kinetic energy is rotational? 86. A cyclist accelerates from rest at a rate of 1.00 m/s2. How fast will a point at the top of the rim of the tire (diameter = 8 cm) be moving after 2.5 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest— see Fig. 10 -

88. A 1.4-kg grindstone in the shape of a uniform cylinder of

radius 0.20 m acquires a rotational rate of 1800 rev/s from rest over a .0-s interval at constant angular acceleration. alculate the torque delivered by the motor. Bicycle gears: (a) How is the angular velocity &>R of the rear wheel of a bicycle related to the angular velocity (oF of the front sprocket and pedals? Let NF and NR be the number of teeth on the front and rear sprockets, respectively, Fig. 10-64. The teeth are spaced the same on both sprockets and the rear sprocket is firmly attached to the rear wheel. (b) Evaluate the ratio w r/^ f when the front and rear sprockets r,lR have 52 and 13 teeth, respectively, and (c) when they have 42 R ev sprocket and 28 teeth.

FIGURE 10 - 64

Problem 89. 90. Figure 10-65 illustrates an H 20 molecule. The O — H bond length is 0.96 nm and the H — O — H bonds make an angle of 104°. alculate the moment of inertia for the H 20 molecule about an axis H passing through the center of the oxygen atom (a) perpendicular to the plane of the molecule, and (b) in the plane of the 104" molecule, bisecting the H — O — H bonds. FIGURE 10 - 65

Problem 90. u - L00m/s2

Tnhiii«ipodoutiGinrtiiELe

jit meatimGmairtanily FIGURE 10 - 63

Problem 8 .

87. Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle, accelerating it from rest to a rate of 85 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?

91. One possibility for a low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1100 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel "spinup." (a) Make reasonable assumptions (average frictional retarding force = 450 N, twenty acceleration periods from rest to 95 km/h, equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and estimate what total energy needs to be stored in the flywheel, (b) What is the angular velocity of the flywheel when it has a full "energy charge"? (c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip?

General Problems

281

92. A hollow cylinder (hoop) is rolling on a horizontal surface at speed v = 3.3 m/s when it reaches a 15° incline. (a) How far up the incline will it go? (b) How long will it be on the incline before it arrives back at the bottom? 93. A wheel of mass M has radius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 10-66). The step has height , where < R. What minimum force F is needed?

97. If the coefficient of static friction between tires and pavement is 0.65, calculate the minimum torque that must be applied to the -cm-diameter tire of a 950-kg automobile in order to "lay rubber" (make the wheels spin, slipping as the car accelerates). Assume each wheel supports an equal share of the weight. 98. A cord connected at one end to a block which can slide on an inclined plane has its other end wrapped around a cylinder resting in a depression at the top of the plane as shown in Fig. 10-69. Determine the speed of the block after it has traveled 1.80 m along the plane, starting from rest. Assume (a) there is no friction, (b) the coefficient of friction between all surfaces is {jl = 0.055. [Hint: In part (b) first determine the normal force on the cylinder, and make any reasonable assumptions needed.] M = 33k* R = 0.20 m

FIGURE 10 - 66

Problem 93.

94. A marble of mass m and radius r rolls along the looped rough track of Fig. 10-67. What is the minimum value of the vertical height that the marble must drop if it is to reach the highest point of the loop without leaving the track? (a) Assume r « . R; (b) do not make this assumption. Ignore frictional losses.

FIGURE 10 - 67

Problem 94.

95. The density (mass per unit length) of a thin rod of length £ increases uniformly from A0 at one end to 3A0 at the other end. Determine the moment of inertia about an axis perpendicular to the rod through its geometric center. 96. If a billiard ball is hit in just the right way by a cue stick, the ball will roll without slipping immediately after losing contact with the stick. onsider a billiard ball (radius r, mass M ) at rest on a horizontal pool table. A cue stick exerts a constant horizontal force F on the ball for a time t at a point that is a height above the table's surface (see Fig. 10-68). Assume that the coefficient of kinetic friction between the ball and table is . Determine the value for so that the ball will roll without slipping immediately after losing contact with the stick.

3.0 k£ \J g ' _J__

FIGURE 10 - 69

99. The radius of the roll of paper shown in Fig. 10-70 is 7.6 cm and its moment of inertia is I = 3.3 X 10-3 kg •m2. A force of 2.5 N is exerted on the end of the roll for 1.3 s, but the paper does not tear so it begins to unroll. A constant friction torque of 0.11 m - N is exerted on the roll which gradually brings it to a stop. Assuming that the paper's thickness is negligible, calculate (a) the length of paper that unrolls during the time that the force is applied (1.3 s) and (b) the length of paper that unrolls from the time the force ends to the time when the roll has stopped moving.

FIGURE 10-71

Problem 96. 282

HAPTER 10

Rotational Motion

FIGURE 10 - 70

Problem 99.

100. A solid uniform disk of mass 21.0 kg and radius 85.0 cm is at rest flat on a frictionless surface. Figure 10-71 shows a view from above. A string is wrapped around the rim of the disk and a constant force of 35.0 N is applied to the string. The string does not slip on the rim. (a) In what direction does the cm move? When the disk has moved a distance of 5.5 m, determine (b) how fast it is moving, (c) how fast it is spinning (in radians per second), and (d) how much string has unwrapped from around the rim.

Problem 100, looking down on the disk.

FIGURE 10 - 68

Problem 98.

ft N

101. When bicycle and motorcycle riders "pop a wheelie," a large acceleration causes the bike's front wheel to leave the ground. Let M be the total mass of the bike-plus-rider system; let x and y be the horizontal and vertical distance of this system's c m from the rear wheel's point of contact with the ground (Fig. 10 -72). (a) Determine the horizontal acceleration a required to barely lift the bike's front wheel off of the ground. (b) To minimize the acceleration necessary to pop a wheelie, should x be made as small or as large as possible? How about y l How should a rider position his or her body on the bike in order to achieve these optimal values for x and y l (c) If x = 35 cm and y = 95 cm, find a.

103. A thin uniform stick of mass M and length i is positioned vertically, with its tip on a frictionless table. It is released and allowed to fall. Determine the speed of its c m just before it hits the table (Fig. 10 -74).

-A

FIGURE 10 - 74

Problem 103. * 104. (a) For the yo-yo-like cylinder of Example 10 -19, we saw that the downward acceleration of its c m was a = \g . If it starts from rest, what will be the c m velocity after it has fallen a distance i (b) Now use conservation of energy to determine the cylinder's c m velocity after it has fallen a distance , starting from rest.

*Num rical/Comput r * 105. (II) Determine the torque produced about the support A of the rigid structure, shown in Fig. 10-75, as a function of the leg angle 6 if a force F = 500 N is applied at the point P perpendicular to the leg end. Graph the values of the torque r as a function of 6 from 0 = 0° to 90°, in 1° increments.

FIGURE 10 - 72

Problem 101.

mo c m 102. A crucial part of a piece of machinery starts as a flat uniform cylindrical disk of radius R 0 and mass M. It then has a circular hole of radius R\ drilled into it (Fig. 10 -73). The hole's center is a distance from the center of the disk. Find the moment of inertia of this disk (with off-center hole) when rotated about its center, . [Hint: onsider a solid disk and "subtract" the hole; use the parallel-axis theorem.]

FIGURE 10 - 73

Problem 102.

FIGURE 10 - 75

~ 500 N

Problem 105.

* 106. (II) Use the expression that was derived in Problem 51 for the acceleration of masses on an Atwood's machine to investigate at what point the moment of inertia of the pulley becomes negligible. Assume raA = 0.150 kg, raB = 0.350 kg, and R = 0.040 m. (a) Graph the acceler ation as a function of the moment of inertia. (b) Find the acceleration of the masses when the moment of inertia goes to zero, (c) Using your graph to guide you, at what minimum value of I does the calculated acceleration deviate by 2.0% from the acceleration found in part (b )l (d) If the pulley could be thought of as a uniform disk, find the mass of the pulley using the I found in part (c).

Answers to Exercises A: / = 0.076 Hz; T = 13 s.

D: 4 X 1017 J.

B: Fa .

E: (c).

: Yes; ^ M i 2 + M g £)2 = \M £ 2.

F: a = lg .

General Problems

283

This skater is doing a spin. When her arms are spread outward horizontally, she spins less fast than when her arms are held close to the axis of rotation. This is an example of the conservation of angular momentum. Angular momentum, which we study in this Chapter, is conserved only if no net torque acts on the object or system. Otherwise, the rate of change of angular momentum is proportional to the net applied torque—which, if zero, means the angular momentum is conserved. In this Chapter we also examine more complicated aspects of rotational motion.

P T E R

C H

CONTENTS 11–1 Angular Momentum— Objects Rotating About a Fixed Axis 11–2 Vector Cross Product; Torque as a Vector 11–3 Angular Momentum of a Particle 11–4 Angular Momentum and Torque for a System of Particles; General Motion 11–5 Angular Momentum and Torque for a Rigid Object 11–6 Conservation of Angular Momentum *11–7 The Spinning Top and Gyroscope *11–8 Rotating Frames of Reference; Inertial Forces 11–9 The Coriolis Effect *

284

Angular Momentum; General Rotation CHAPTER-OPENING QUESTION—Guess now! You are standing on a platform at rest, but that is free to rotate. You hold a spinning bicycle wheel by its axle as shown here. You then flip the wheel over so its axle points down. What happens then? B L (a) The platform starts rotating in the direction the bicycle wheel was originally rotating. (b) The platform starts rotating in the direction opposite to the original rotation of the bicycle wheel. (c) The platform stays at rest. (d) The platform turns only while you are flipping the wheel. (e) None of these is correct.

n Chapter 10 we dealt with the kinematics and dynamics of the rotation of a rigid object about an axis whose direction is fixed in an inertial reference frame. We analyzed the motion in terms of the rotational equivalent of Newton's laws (torque plays the role that force does for translational motion), as well as rotational kinetic energy. To keep the axis of a rotating object fixed, the object must usually be constrained by external supports (such as bearings at the end of an axle). The motion of objects that are not constrained to move about a fixed axis is more difficult to describe and analyze. Indeed, the complete analysis of the general rotational motion of an object (or system of objects) is very complicated, and we will only look at some aspects of general rotational motion in this Chapter. We start this Chapter by introducing the concept of angular momentum, which is the rotational analog of linear momentum. We first treat angular momentum and its conservation for an object rotating about a fixed axis. After that, we examine the vector nature of torque and angular momentum. We will derive some general theorems and apply them to some interesting types of motion.

11–1

Angular Momentum — Objects Rotating About a Fixed Axis

In Chapter 10 we saw that if we use the appropriate angular variables, the kinematic and dynamic equations for rotational motion are analogous to those for ordinary linear motion. In like manner, the linear momentum, p = mv, has a rotational analog. It is called angular momentum, L, and for an object rotating about a fixed axis with angular velocity v, it is defined as L = Iv, (11–1) 2 where I is the moment of inertia. The SI units for L are kgm s; there is no special name for this unit. We saw in Chapter 9 (Section 9–1) that Newton's second law can be written not only as ©F = ma, but also more generally in terms of momentum (Eq. 9–2), ©F = dpdt. In a similar way, the rotational equivalent of Newton's second law, which we saw in Eqs. 10–14 and 10–15 can be written as ©t = Ia, can also be written in terms of angular momentum: since the angular acceleration a = dvdt (Eq. 10–3), then Ia = I(dvdt) = d(Iv)dt = dLdt, so dL . ©t = (11–2) dt This derivation assumes that the moment of inertia, I, remains constant. However, Eq. 11–2 is valid even if the moment of inertia changes, and applies also to a system of objects rotating about a fixed axis where ©t is the net external torque (discussed in Section 11–4). Equation 11–2 is Newton's second law for rotational motion about a fixed axis, and is also valid for a moving object if its rotation is about an axis passing through its center of mass (as for Eq. 10–15).

NEWTON'S SECOND LAW FOR ROTATION

Conservation of Angular Momentum Angular momentum is an important concept in physics because, under certain conditions, it is a conserved quantity. What are the conditions for which it is conserved? From Eq. 11–2 we see immediately that if the net external torque ©t on an object (or system of objects) is zero, then dL = 0 and L = Iv = constant. [©t = 0] dt This, then, is the law of conservation of angular momentum for a rotating object: The total angular momentum of a rotating object remains constant if the net external torque acting on it is zero.

CONSERVATION OF ANGULAR MOMENTUM

The law of conservation of angular momentum is one of the great conservation laws of physics, along with those for energy and linear momentum. SECTION 11–1

Angular Momentum—Objects Rotating About a Fixed Axis

285

I large,

I small,

ω small

ω large

When there is zero net torque acting on an object, and the object is rotating about a fixed axis or about an axis through its center of mass whose direction doesn't change, we can write Iv = I0 v0 = constant.

(a)

(b)

FIGURE 11–1 A skater doing a spin on ice, illustrating conservation of angular momentum: (a) I is large and v is small; (b) I is smaller so v is larger. FIGURE 11–2 A diver rotates faster when arms and legs are tucked in than when they are outstretched. Angular momentum is conserved.

I0 and v0 are the moment of inertia and angular velocity, respectively, about the axis at some initial time (t = 0), and I and v are their values at some other time. The parts of the object may alter their positions relative to one another, so that I changes. But then v changes as well and the product Iv remains constant. Many interesting phenomena can be understood on the basis of conservation of angular momentum. Consider a skater doing a spin on the tips of her skates, Fig. 11–1. She rotates at a relatively low speed when her arms are outstretched, but when she brings her arms in close to her body, she suddenly spins much faster. From the definition of moment of inertia, I = ©mR2, it is clear that when she pulls her arms in closer to the axis of rotation, R is reduced for the arms so her moment of inertia is reduced. Since the angular momentum Iv remains constant (we ignore the small torque due to friction), if I decreases, then the angular velocity v must increase. If the skater reduces her moment of inertia by a factor of 2, she will then rotate with twice the angular velocity. A similar example is the diver shown in Fig. 11–2. The push as she leaves the board gives her an initial angular momentum about her center of mass. When she curls herself into the tuck position, she rotates quickly one or more times. She then stretches out again, increasing her moment of inertia which reduces the angular velocity to a small value, and then she enters the water. The change in moment of inertia from the straight position to the tuck position can be a factor of as much as 3 12 . Note that for angular momentum to be conserved, the net torque must be zero, but the net force does not necessarily have to be zero. The net force on the diver in Fig. 11–2, for example, is not zero (gravity is acting), but the net torque about her CM is zero because the force of gravity acts at her center of mass. EXAMPLE 11–1 Object rotating on a string of changing length. A small mass m attached to the end of a string revolves in a circle on a frictionless tabletop. The other end of the string passes through a hole in the table (Fig. 11–3). Initially, the mass revolves with a speed v1 = 2.4 ms in a circle of radius R1 = 0.80 m. The string is then pulled slowly through the hole so that the radius is reduced to R2 = 0.48 m. What is the speed, v2 , of the mass now?

FIGURE 11–3

Example 11–1. R1

APPROACH There is no net torque on the mass m because the force exerted by the string to keep it moving in a circle is exerted toward the axis; hence the lever arm is zero. We can thus apply conservation of angular momentum. SOLUTION Conservation of angular momentum gives I1 v1 = I2 v2 .

Our small mass is essentially a particle whose moment of inertia about the hole is I = mR2 (Eq. 10–11), so we have mR 21 v1 = mR 22 v2 , or v2 = v1 ¢

R 21 ≤. R 22

Then, since v = Rv, we can write v2 = R2 v2 = R2 v1 ¢

R 21 v1 R 21 R1 ≤ = R2 ¢ ≤ = v1 2 R1 R 22 R2 R2 = (2.4 ms) a

The speed increases as the radius decreases.

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CHAPTER 11

Angular Momentum; General Rotation

0.80 m b = 4.0 ms. 0.48 m

EXAMPLE 11–2 Clutch. A simple clutch consists of two cylindrical plates that can be pressed together to connect two sections of an axle, as needed, in a piece of machinery. The two plates have masses MA = 6.0 kg and MB = 9.0 kg, with equal radii R0 = 0.60 m. They are initially separated (Fig. 11–4). Plate MA is accelerated from rest to an angular velocity v1 = 7.2 rads in time ¢ t = 2.0 s. Calculate (a) the angular momentum of MA , and (b) the torque required to have accelerated MA from rest to v1 . (c) Next, plate MB , initially at rest but free to rotate without friction, is placed in firm contact with freely rotating plate MA , and the two plates both rotate at a constant angular velocity v2 , which is considerably less than v1 . Why does this happen, and what is v2 ?

MB MA v1 FIGURE 11–4

Example 11–2.

APPROACH We use the definition of angular momentum L = Iv (Eq. 11–1) plus Newton's second law for rotation, Eq. 11–2. SOLUTION (a) The angular momentum of MA will be LA = IA v1 =

1 2 2 MA R 0 v1

1 2 (6.0

kg)(0.60 m)2(7.2 rads) = 7.8 kgm2s.

(b) The plate started from rest so the torque, assumed constant, was 7.8 kgm2s - 0 ¢L = = 3.9 m N. ¢t 2.0 s (c) Initially, MA is rotating at constant v1 (we ignore friction). When plate B comes in contact, why is their joint rotation speed less? You might think in terms of the torque each exerts on the other upon contact. But quantitatively, it's easier to use conservation of angular momentum, since no external torques are assumed to act. Thus t =

angular momentum before = angular momentum after IA v1 = AIA + IB B v2 . Solving for v2 we find v2 = ¢

6.0 kg IA MA ≤v = ¢ ≤v = a b (7.2 rads) = 2.9 rads. 15.0 kg IA + IB 1 MA + MB 1

EXAMPLE 11–3 ESTIMATE Neutron star. Astronomers detect stars that are rotating extremely rapidly, known as neutron stars. A neutron star is believed to form from the inner core of a larger star that collapsed, under its own gravitation, to a star of very small radius and very high density. Before collapse, suppose the core of such a star is the size of our Sun Ar L 7 * 105 kmB with mass 2.0 times as great as the Sun, and is rotating at a frequency of 1.0 revolution every 100 days. If it were to undergo gravitational collapse to a neutron star of radius 10 km, what would its rotation frequency be? Assume the star is a uniform sphere at all times, and loses no mass.

P H Y S I C S

A P P L I E D

Neutron star

APPROACH We assume the star is isolated (no external forces), so we can use conservation of angular momentum for this process. We use r for the radius of a sphere, as compared to R used for distance from an axis of rotation or cylindrical symmetry: see Fig. 10–2. SOLUTION From conservation of angular momentum, I1 v1 = I2 v2 , where the subscripts 1 and 2 refer to initial (normal star) and final (neutron star), respectively. Then, assuming no mass is lost in the process, v2 = ¢

2 2 I1 r21 5 m1 r 1 ≤ v1 = ¢ 2 ≤ v1 = 2 v1 . 2 I2 r2 5 m2 r 2

The frequency f = v2p, so v2 r 21 f2 = = 2 f1 2p r2 = ¢

7 * 105 km 2 1.0 rev ≤ ¢ ≤ L 6 * 102 revs. 10 km 100 d(24 hd)(3600 sh) SECTION 11–1

Angular Momentum—Objects Rotating About a Fixed Axis

287

Directional Nature of Angular Momentum Angular momentum is a vector, as we shall discuss later in this Chapter. For now we consider the simple case of an object rotating about a fixed axis, and the B direction of L is specified by a plus or minus sign, just as we did for one-dimensional linear motion in Chapter 2. For a symmetrical object rotating about a symmetry axis (such as a cylinder or wheel), the direction of the angular momentum† can be taken as the direction of the angular velocity V. That is,

Axis

R (a) B

L person B

L platform

L = I V.

As a simple example, consider a person standing at rest on a circular platform capable of rotating friction-free about an axis through its center (that is, a simplified merry-go-round). If the person now starts to walk along the edge of the platform, Fig. 11–5a, the platform starts rotating in the opposite direction. Why? One explanation is that the person's foot exerts a force on the platform. Another explanation (and this is the most useful analysis here) is as an example of the conservation of angular momentum. If the person starts walking counterclockwise, the person's angular momentum will be pointed upward along the axis of rotation (remember how we defined the direction of V using the right-hand rule in Section 10–2). The magnitude of the person's angular momentum will be L = Iv = AmR 2 B(vR), where v is the person's speed (relative to the Earth, not the platform), R is his distance from the rotation axis, m is his mass, and mR 2 is his moment of inertia if we consider him a particle (mass concentrated at one point). The platform rotates in the opposite direction, so its angular momentum points downward. If the initial total angular momentum was zero (person and platform at rest), it will remain zero after the person starts walking. That is, the upward angular momentum of the person just balances the oppositely directed downward angular momentum of the platform (Fig. 11–5b), so the total vector angular momentum remains zero. Even though the person exerts a force (and torque) on the platform, the platform exerts an equal and opposite torque on the person. So the net torque on the system of person plus platform is zero (ignoring friction) and the total angular momentum remains constant. B

(b) FIGURE 11–5 (a) A person on a circular platform, both initially at rest, begins walking along the edge at speed v. The platform, assumed to be mounted on frictionfree bearings, begins rotating in the opposite direction, so that the total angular momentum remains zero, as shown in (b).

EXAMPLE 11–4 Running on a circular platform. Suppose a 60-kg person stands at the edge of a 6.0-m-diameter circular platform, which is mounted on frictionless bearings and has a moment of inertia of 1800 kg m2. The platform is at rest initially, but when the person begins running at a speed of 4.2 ms (with respect to the Earth) around its edge, the platform begins to rotate in the opposite direction as in Fig. 11–5. Calculate the angular velocity of the platform. APPROACH We use conservation of angular momentum. The total angular B momentum is zero initially. Since there is no net torque, L is conserved and will remain zero, as in Fig. 11–5. The person's angular momentum is Lper = AmR2 B(vR), and we take this as positive. The angular momentum of the platform is Lplat = –Iv. SOLUTION Conservation of angular momentum gives L = Lper + Lplat 0 = mR 2 a

v b - Iv. R

So v =

(60 kg)(3.0 m)(4.2 ms) mRv = = 0.42 rads. I 1800 kgm2

NOTE The frequency of rotation is f = v2p = 0.067 revs and the period T = 1f = 15 s per revolution. †

For more complicated situations of objects rotating about a fixed axis, there will be a component of L B B along the direction of V and its magnitude will be equal to IV, but there could be other components as B well. If the total angular momentum is conserved, then the component IV will also be conserved. So our results here can be applied to any rotation about a fixed axis.

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CHAPTER 11

Angular Momentum; General Rotation

CONCEPTUAL EXAMPLE 11–5 Spinning bicycle wheel. Your physics teacher is holding a spinning bicycle wheel while he stands on a stationary frictionless turntable (Fig. 11–6).What will happen if the teacher suddenly flips the bicycle wheel over so that it is spinning in the opposite direction?

RESPONSE We consider the system of turntable, teacher, and bicycle wheel. The B total angular momentum initially is L vertically upward. That is also what the B system's angular momentum must be afterward, since L is conserved when there is no net torque. Thus, if the wheel's angular momentum after being flipped over B is –L downward, then the angular momentum of teacher plus turntable will have B to be ±2L upward. We can safely predict that the teacher will begin spinning around in the same direction the wheel was spinning originally.

EXERCISE A In Example 11–5, what if he moves the axis only 90° so it is horizontal? (a) The same direction and speed as above; (b) the same as above, but slower; (c) the opposite result.

FIGURE 11–6

Example 11–5.

EXERCISE B Return to the Chapter-Opening Question, page 284, and answer it again now. Try to explain why you may have answered differently the first time. EXERCISE C Suppose you are standing on the edge of a large freely rotating turntable. If you walk toward the center, (a) the turntable slows down; (b) the turntable speeds up; (c) its rotation speed is unchanged; (d) you need to know the walking speed to answer.

11–2

Vector Cross Product; Torque as a Vector

Vector Cross Product To deal with the vector nature of angular momentum and torque in general, we will need the concept of the vector cross product (often called simply the vector B product or cross product). In general, the vector or cross product of two vectors A B B B B and B is defined as another vector C = A * B whose magnitude is C = @A * B @ = AB sin u, B

(11–3a) B

where u is the angle (6 180°) between A and B, and whose direction is perpendicular B B to both A and B in the sense of the right-hand rule, Fig. 11–7. The angle u is B B measured between A and B when their tails are together at the same point. According to the right-hand rule, as shown in Fig. 11–7, you orient your right hand B B so your fingers point along A, and when you bend your fingers they point along B. When your hand is correctly oriented in this way, your thumb will point along the B B B direction of C = A * B. B B The cross product of two vectors, A = A x ˆi + A y jˆ + A z kˆ , and B = Bx ˆi + By jˆ + Bz kˆ , can be written in component form (see Problem 26) as ˆi A * B = 3 Ax Bx B

jˆ Ay By

kˆ Az 3 Bz

FIGURE 11–7 The vector B B B C = A * B is perpendicular to B B the plane containing A and B; its direction is given by the right-hand rule. B

C=A×B

(11–3b)

= AA y Bz - A z By B iˆ + AA z Bx - A x Bz B jˆ + AA x By - A y Bx B kˆ . (11–3c)

θ B

Equation 11–3b is meant to be evaluated using the rules of determinants (and obtain Eq. 11–3c). SECTION 11–2

Vector Cross Product; Torque as a Vector

289

Some properties of the cross product are the following:

(11–4a)

A * A = 0 θ

A * AB + C B = AA * B B + AA * C B

(11–4b) [distributive law] (11–4c)

d B dA dB . B B B AA * B B = * B + A * dt dt dt

B×A B

FIGURE 11–8 The vector B * A B B equals – A * B; compare to Fig. 11–7.

FIGURE 11–9

A * B = –B * A

Exercise D.

A B

(11–4d)

Equation 11–4a follows from Eqs. 11–3 (since u = 0). So does Eq. 11–4b, since B B B B the magnitude of B * A is the same as that for A * B, but by the right-hand rule the direction is opposite (see Fig. 11–8). Thus the order of the two vectors is crucial. If you change the order, you change the result. That is, the commutative B B B B law does not hold for the cross product AA * B Z B * A B, although it does hold for the dot product of two vectors and for the product of scalars. Note in Eq. 11–4d that the order of quantities in the two products on the right must not be changed (because of Eq. 11–4b). B

EXERCISE D For the vectors A and B in the plane of the page as shown in Fig. 11–9, in B B B B B B what direction is (i) A B, (ii) A * B, (iii) B * A? (a) Into the page; (b) out of the page; B B (c) between A and B; (d) it is a scalar and has no direction; (e) it is zero and has no direction.

The Torque Vector

FIGURE 11–10 The torque due to B the force F (in the plane of the wheel) starts the wheel rotating B B counterclockwise so V and A point out of the page. B

F θ

Torque is an example of a quantity that can be expressed as a cross product. To see this, let us take a simple example: the thin wheel shown in Fig. 11–10 which is free to rotate B about an axis through its center at point O. A force F acts at the edge of the wheel, at a point whose position relative to the center O is given by the position vector r as shown. B The force F tends to rotate the wheel (assumed initially at rest) counterclockwise, so the angular velocity V will point out of the page toward the viewer (remember the B right-hand rule from Section 10–2). The torque due to F will tend to increase V so A also points outward along the rotation axis. The relation between angular acceleration and torque that we developed in Chapter 10 for an object rotating about a fixed axis is ©t = Ia, B

(Eq. 10–14) where I is the moment of inertia. This scalar equation is the rotational equivalent of ©F = ma, and we would like to make it a vector equation just B as © F = ma is a vector equation. To do so in the case of Fig. 11–10 we must have the direction of T point outward along the rotation axis, since A ( = dVdt) has that direction; and the magnitude of the torque must be (see Eqs. 10–10 and Fig. 11–10) t = rF⊥ = rF sin u. We can achieve this by defining the torque vector B to be the cross product of r and F: B T = r * F. (11–5)

O Axis

From the definition of the cross product above (Eq. 11–3a) the magnitude of T will be rF sin u and the direction will be along the axis, as required for this special case. We will see in Sections 11–3 through 11–5 that if we take Eq. 11–5 as the general definition of torque, then the vector relation ©T = IA will hold in general. Thus we state now that Eq. 11–5 is the general definition of torque. It contains both magnitude and direction information. Note that this definition involves the position vector r and thus the torque is being calculated about a point. We can choose that point O as we wish. B For a particle of mass m on which a force F is applied, we define the torque about a point O as B T = r * F where r is the position vector of the particle relative to O (Fig. 11–11). If we have a system of particles (which could be the particles making up a rigid object) the total torque T on the system will be the sum of the torques on the individual particles: B T = ©Ari * Fi B, B where ri is the position vector of the i th particle and Fi is the net force on the i th particle. B

FIGURE 11–11 TB = Br * F, where B r is the position vector.

x B

290

CHAPTER 11

Angular Momentum; General Rotation

EXAMPLE 11–6 Torque Vector. Suppose the vector r is in the xz plane, as in Fig. 11–11, and is given by r = (1.2 m) ˆi + (1.2 m) kˆ . Calculate the torque B vector T if F = (150 N) ˆi. APPROACH We use the determinant form, Eq. 11–3b. ˆi jˆ kˆ B SOLUTION T = r * F = 3 1.2 m 0 1.2 m 3 = 0 ˆi + (180 m N) jˆ + 0 kˆ . B

150 N 0

So t has magnitude 180 mN and points along the positive y axis. EXERCISE E If F = 5.0 N ˆi and r = 2.0 m jˆ, what is T? (a) 10 mN, (b) –10 mN, (c) 10 mN kˆ , (d) – 10 mN jˆ, (e) – 10 mN kˆ . B

11–3

Angular Momentum of a Particle

The most general way of writing Newton's second law for the translational motion of a particle (or system of particles) is in terms of the linear momentum p = mv as given by Eq. 9–2 (or 9–5): dp B . ©F = dt The rotational analog of linear momentum is angular momentum. Just as the rate B of change of p is related to the net force © F, so we might expect the rate of change of angular momentum to be related to the net torque. Indeed, we saw this was true in Section 11–1 for the special case of a rigid object rotating about a fixed axis. Now we will see it is true in general. We first treat a single particle. Suppose a particle of mass m has momentum p and position vector r with respect to the origin O in some chosen inertial reference frame. Then the general B definition of the angular momentum, L, of the particle about point O is the vector cross product of r and p: B [particle] (11–6) L = r * p. † Angular momentum is a vector. Its direction is perpendicular to both r and p as given by the right-hand rule (Fig. 11–12). Its magnitude is given by L = rp sin u or L = rp⊥ = r⊥ p where u is the angle between r and p and p⊥ (= p sin u) and r⊥ (= r sin u) are the components of p and r perpendicular to r and p, respectively. Now let us find the relation between angular momentum and torque for a B particle. If we take the derivative of L with respect to time we have B dp dL d dr . = (r * p) = * p + r * dt dt dt dt But dr * p = v * mv = m(v * v) = 0, dt since sin u = 0 for this case. Thus B dp dL . = r * dt dt B If we let © F represent the resultant force on the particle, then in an inertial reference B frame, © F = dpdt and B dp dL . B r * ©F = r * = dt dt B But r * © F = ©T is the net torque on our particle. Hence B dL . ©T = [particle, inertial frame] (11–7) dt The time rate of change of angular momentum of a particle is equal to the net torque applied to it. Equation 11–7 is the rotational equivalent of Newton's B

L = Br × p

B p

FIGURE 11–12 The angular momentum of a particle of mass m is B B given by L = Br * p = Br * mvB.

†

Actually a pseudovector; see footnote in Section 10–2.

SECTION 11–3

291

second law for a particle, written in its most general form. Equation 11–7 is valid B only in an inertial frame since only then is it true that © F = dpdt, which was used in the proof. B

B = mvB p

m B

CONCEPTUAL EXAMPLE 11–7 A particle's angular momentum. What is the angular momentum of a particle of mass m moving with speed v in a circle of radius r in a counterclockwise direction? RESPONSE The value of the angular momentum depends on the choice of the B point O. Let us calculate L with respect to the center of the circle, Fig. 11–13. Then r is perpendicular to p so L = @ r * p @ = rmv. By the right-hand rule, B the direction of L is perpendicular to the plane of the circle, outward toward the viewer. Since v = vr and I = mr2 for a single particle rotating about an axis a distance r away, we can write B

FIGURE 11–13 The angular momentum of a particle of mass m rotating in a circle of radius Br B with velocity vB is L = Br * mvB (Example 11–7).

L = mvr = mr2v = Iv.

11–4

Angular Momentum and Torque for a System of Particles; General Motion

Relation Between Angular Momentum and Torque B

Consider a system of n particles which have angular momenta L 1 , L 2 , p , L n . The system could be anything from a rigid object to a loose assembly of particles whose B positions are not fixed relative to each other. The total angular momentum L of the system is defined as the vector sum of the angular momenta of all the particles in the system: n

L =

a Li .

(11–8)

i=1

The resultant torque acting on the system is the sum of the net torques acting on all the particles:

Tnet = a Ti . B

This sum includes (1) internal torques due to internal forces that particles of the system exert on other particles of the system, and (2) external torques due to forces exerted by objects outside our system. By Newton's third law, the force each particle exerts on another is equal and opposite (and acts along the same line as) the force that the second particle exerts on the first. Hence the sum of all internal torques adds to zero, and

Tnet = a Ti = a Text . B

Now we take the time derivative of Eq. 11–8 and use Eq. 11–7 for each particle to obtain B

dL = dt

dL i a dt = i

a Text B

or B

NEWTON'S SECOND LAW (rotation, system of particles)

dL = dt

a Text . B

[inertial reference frame] (11–9a)

This fundamental result states that the time rate of change of the total angular momentum of a system of particles (or a rigid object) equals the resultant external B B torque on the system. It is the rotational equivalent of Eq. 9–5, dPdt = © Fext B for translational motion. Note that L and ©T must be calculated about the same origin O. B Equation 11–9a is valid when L and Text are calculated with reference to a point fixed in an inertial reference frame. (In the derivation, we used Eq. 11–7 which is B

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valid only in this case.) It is also valid when Text and L are calculated about a point which is moving uniformly in an inertial reference frame since such a point can be considered the origin of a second inertial reference frame. It is not valid in general B when Text and L are calculated about a point that is accelerating, except for one special (and very important) case—when that point is the center of mass (CM) of the system: B

dL cm = dt

a Tcm . B

[even if accelerating] (11–9b)

NEWTON'S SECOND LAW (for CM even if accelerating)

Equation 11–9b is valid no matter how the CM moves, and ©Tcm is the net external torque calculated about the center of mass. The derivation is in the optional subsection below. It is because of the validity of Eq. 11–9b that we are justified in describing the general motion of a system of particles, as we did in Chapter 10, as translational motion of the center of mass plus rotation about the center of mass. Equations 11–9b B B plus 9–5 AdPcm dt = © Fext B provide the more general statement of this principle. (See also Section 9–8.)

* Derivation of dLcm dt = π Tcm B

The proof of Eq. 11–9b is as follows. Let ri be the position vector of the i th particle in an inertial reference frame, and rcm be the position vector of the center of mass of the system in this reference frame. The position of the i th particle with respect to the CM is r *i where (see Fig. 11–14) ri = rcm + r *i .

r *i

O B

If we multiply each term by mi and take the derivative of this equation, we can write d ri d * pi = mi = mi Ar + rcm B = mi v *i + mi vcm = p*i + mi vcm . dt dt i The angular momentum with respect to the CM is

rCM

* * * a Ar i * pi B = a r i * Api - mi vcm B.

L cm =

Then, taking the time derivative, we have B dp*i dL cm d r *i = a¢ * p*i ≤ + a ¢ r *i * ≤. dt dt dt i i

FIGURE 11–14 The position of mi in the inertial frame is Br i ; with regard to the CM (which could be accelerating) it is Br *i , where B r i = Br *i + Br cm and Br cm is the position of the CM in the inertial frame.

The first term on the right is v*i * mv *i and equals zero because v *i is parallel to itself (sin u = 0). Thus B

d * a r i * dt Api - mi vcm B i dpi dvcm . = a r *i * - a a mi r *i b * dt dt i i

dL cm = dt

The second term on the right is zero since, by Eq. 9–12, ©mi r *i = M r *cm , and r *cm = 0 by definition (the position of the CM is at the origin of the CM reference frame). Furthermore, by Newton's second law we have dpi B = Fi , dt B B where Fi is the net force on mi . (Note that dp*i dt Z Fi because the CM may be accelerating and Newton's second law does not hold in a noninertial reference frame.) Consequently B

dL cm = dt

B * a r i * Fi = a ATi B cm = ©Tcm , B

where ©Tcm is the resultant external torque on the entire system calculated about the CM. (By Newton's third law, the sum over all the Ti eliminates the net torque due to internal forces, as we saw on p. 292.) This last equation is Eq. 11–9b, and this concludes its proof. B

SECTION 11–4

Angular Momentum and Torque for a System of Particles; General Motion

293

Summary To summarize, the relation B dL a Text = dt B is valid only when Text and L are calculated with respect to either (1) the origin of an inertial reference frame or (2) the center of mass of a system of particles (or of a rigid object). B

11–5

z B V

90°

B B Li = ri × pi 90°

B p i

Let us now consider the rotation of a rigid object about an axis that has a fixed direction in space, using the general principles just developed. Let us calculate the component of angular momentum along the rotation axis of the rotating object. We will call this component Lv since the angular velocity V points along the rotation axis. For each particle of the object, B

L i = r i * pi .

Angular Momentum and Torque for a Rigid Object

Let f be the angle between L i and the rotation axis. (See Fig. 11–15; f is not B the angle between ri and pi , which is 90°). Then the component of L i along the rotation axis is Liv = ri pi cos f = mi vi ri cos f, where mi is the mass and vi the velocity of the i th particle. Now vi = Ri v where v is the angular velocity of the object and Ri is the perpendicular distance of mi from the axis of rotation. Furthermore, Ri = ri cos f, as can be seen in Fig. 11–15, so B

Liv = mi vi Ari cos fB = mi R2i v.

FIGURE 11–15 Calculating Lv = Lz = ©Liz . B Note that L i is perpendicular to Br i , B and R i is perpendicular to the z axis, so the three angles marked f are equal.

We sum over all the particles to obtain Lv =

2 a Liv = a a mi Ri b v. i

©mi R 2i

But is the moment of inertia I of the object about the axis of rotation. Therefore the component of the total angular momentum along the rotation axis is given by Lv = Iv.

(11–10)

Note that we would obtain Eq. 11–10 no matter where we choose the point O (for measuring ri) as long as it is on the axis of rotation. Equation 11–10 is the same as Eq. 11–1, which we have now proved from the general definition of angular momentum. If the object rotates about a symmetry axis through the center of mass, then Lv is B the only component of L, as we now show. For each point on one side of the axis there will be a corresponding point on the opposite side. We can see from B Fig. 11–15 that each L i has a component parallel to the axis ALiv B and a component perpendicular to the axis. The components parallel to the axis add together for each pair of opposite points, but the components perpendicular to the axis for opposite points will have the same magnitude but opposite direction and so will cancel. Hence, for an object rotating about a symmetry axis, the angular momentum vector is parallel to the axis and we can write B

L = I V,

[rotation axis = symmetry axis, through CM] (11–11)

where L is measured relative to the center of mass. The general relation between angular momentum and torque is Eq. 11–9: B dL a T = dt B

where ©T and L are calculated either about (1) the origin of an inertial reference frame, or (2) the center of mass of the system. This is a vector relation, and must B

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therefore be valid for each component. Hence, for a rigid object, the component along the rotation axis is dLv d dv ©taxis = = (Iv) = I = Ia, dt dt dt which is valid for a rigid object rotating about an axis fixed relative to the object; also this axis must be either (1) fixed in an inertial system or (2) passing through the CM of the object. This is equivalent to Eqs. 10–14 and 10–15, which we now see B are special cases of Eq. 11–9, ©T = dLdt. B

EXAMPLE 11–8 Atwood's machine. An Atwood machine consists of two masses, mA and mB , which are connected by an inelastic cord of negligible mass that passes over a pulley, Fig. 11–16. If the pulley has radius R0 and moment of inertia I about its axle, determine the acceleration of the masses mA and mB , and compare to the situation where the moment of inertia of the pulley is ignored. APPROACH We first determine the angular momentum of the system, and then apply Newton's second law, t = dLdt. SOLUTION The angular momentum is calculated about an axis along the axle through the center O of the pulley. The pulley has angular momentum Iv, where v = vR0 and v is the velocity of mA and mB at any instant. The angular momentum of mA is R0 mA v and that of mB is R0 mB v. The total angular momentum is v . L = AmA + mB B vR0 + I R0 The external torque on the system, calculated about the axis O (taking clockwise as positive), is t = mB gR0 - mA gR0 . (The force on the pulley exerted by the support on its axle gives rise to no torque because the lever arm is zero.) We apply Eq. 11–9a: dL t = dt dv I dv . AmB - mA B gR0 = AmA + mB B R0 + dt R0 dt Solving for a = dvdt, we get a =

aB mA mB

B mAg

mBgB FIGURE 11–16 Atwood's machine, Example 11–8. We also discussed this in Example 4–13.

AmB - mA B g dv . = dt AmA + mB B + IR 20

If we were to ignore I, a = AmB - mA B gAmB + mA B and we see that the effect of the moment of inertia of the pulley is to slow down the system. This is just what we would expect. CONCEPTUAL EXAMPLE 11–9 Bicycle wheel. Suppose you are holding a bicycle wheel by a handle connected to its axle as in Fig. 11–17a. The wheel is B spinning rapidly so its angular momentum L points horizontally as shown. Now you suddenly try to tilt the axle upward as shown by the dashed line in Fig. 11–17a (so the CM moves vertically). You expect the wheel to go up (and it would if it weren't rotating), but it unexpectedly swerves to the right! Explain. RESPONSE To explain this seemingly odd behavior—you may need to do it to B believe it—we only need to use the relation Tnet = dLdt. In the short time ¢ t, you exert a net torque (about an axis through your wrist) that points along the x axis B B perpendicular to L. Thus the change in L is B

¢ L L Tnet ¢ t; B so ¢ L must also point (approximately) along the x axis, since Tnet does (Fig. 11–17b). B B Thus the new angular momentum, L + ¢ L, points to the right, looking along the axis of the wheel, as shown in Fig. 11–17b. Since the angular momentum is directed along the axle of the wheel, we see that the axle, which now is along B B L + ¢ L, must move sideways to the right, which is what we observe. B

FIGURE 11–17 When you try to tilt a rotating bicycle wheel vertically upward, it swerves to the side instead.

z Rotation axis for spinning wheel Rotation axis for lifting wheel (a)

x z

SECTION 11–5

L B

∆L

L + ∆ LB

x (b)

Angular Momentum and Torque for a Rigid Object

295

C A U T I O N B

L = IV is not always valid

Bearing B

F B V

Although Eq. 11–11, L = IV, is often very useful, it is not valid in general unless the rotation axis is along a symmetry axis through the center of mass. Nonetheless, it can be shown that every rigid object, no matter what its shape, has three "principal axes" about which Eq. 11–11 is valid (we will not go into the details here). As an example of a case where Eq. 11–11 is not valid, consider the nonsymmetrical object shown in Fig. 11–18. It consists of two equal masses, mA and mB , attached to the ends of a rigid (massless) rod which makes an angle f with the axis of rotation. We calculate the angular momentum about the CM at point O. At the moment shown, mA is coming toward the viewer, B B and mB is moving away, so L A = rA * pA and L B = rB * pB are as shown. The B B B total angular momentum is L = L A + L B , which is clearly not along V if f Z 90°. B

L= LA + LB

rA O rB

Axle

mB B

LB B

* Rotational Imbalance Let us go one step further with the system shown in Fig. 11–18, since it is a B fine illustration of ©T = dLdt. If the system rotates with constant angular B velocity, v, the magnitude of L will not change, but its direction will. As the B rod and two masses rotate about the z axis, L also rotates about the axis. At the B moment shown in Fig. 11–18, L is in the plane of the paper. A time dt later, B when the rod has rotated through an angle du = v dt, L will also have rotated B through an angle du (it remains perpendicular to the rod). L will then have a B B component pointing into the page. Thus dL points into the page and so must dLdt. Because B

Bearing B

FIGURE 11–18 In this system L B and V are not parallel. This is an example of rotational imbalance.

dL , a T = dt B

we see that a net torque, directed into the page at the moment shown, must be applied to the axle on which the rod is mounted. The torque is supplied by B bearings (or other constraint) at the ends of the axle. The forces F exerted by the B bearings on the axle are shown in Fig. 11–18. The direction of each force F rotates B as the system does, always being in the plane of L and V for this system. If the torque due to these forces were not present, the system would not rotate about the fixed axis as desired. B The axle tends to move in the direction of F and thus tends to wobble as it rotates. This has many practical applications, such as the vibrations felt in a car whose wheels are not balanced. Consider an automobile wheel that is symmetrical except for an extra mass mA on one rim and an equal mass mB opposite it on the other rim, as shown in Fig. 11–19. Because of the nonsymmetry of mA and mB , the wheel bearings would have to exert a force perpendicular to the axle at all times simply to keep the wheel rotating, just as in Fig. 11–18. The bearings would wear excessively and the wobble of the wheel would be felt by occupants of the car. When the wheels are balanced, they rotate smoothly without wobble. This is why "dynamic balancing" of automobile wheels and tires is important. The wheel of Fig. 11–19 would balance statically just fine. If equal masses mC and mD are added symmetrically, below mA and B above mB , the wheel will be balanced dynamically as well (L will be parallel to V, and Text = 0 ). B

P H Y S I C S

A P P L I E D

Automobile wheel balancing FIGURE 11–19 Unbalanced automobile wheel. Axis

EXAMPLE 11–10 Torque on imbalanced system. Determine the magnitude of the net torque tnet needed to keep the system turning in Fig. 11–18. APPROACH Figure 11–20 is a view of the angular momentum vector, looking down the rotation axis (z axis) of the object depicted in Fig. 11–18, as it rotates. B L cos f is the component of L perpendicular to the axle (points to the right in Fig. 11–18). We find dL from Fig. 11–20 and use tnet = dLdt.

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SOLUTION In a time dt, L changes by an amount (Fig. 11–20 and Eq. 10–2b) dL = (L cos f) du = L cos f v dt, where v = dudt. Hence dL = vL cos f. dt Now L = LA + LB = rA mA vA + rB mB vB = rA mA AvrA sin fB + rB mB AvrB sin fB = AmA rA2 + mB r2B B v sin f. Since I = AmA rA2 + mB r 2B B sin2 f is the moment of inertia about the axis of rotation, then L = Ivsin f. So tnet =

+ dt) (at t f s L co du Axis L cos f (at time t)

dL B

FIGURE 11–20 Angular momentum vector looking down along the rotation axis of the system of Fig. 11–18 as it rotates during a time dt.

tnet = vL cos f = AmA rA2 + mB r2B Bv2 sin f cos f = Iv2tan f.

The situation of Fig. 11–18 illustrates the usefulness of the vector nature of torque and angular momentum. If we had considered only the components of angular momentum and torque along the rotation axis, we could not have calculated B the torque due to the bearings (since the forces F act at the axle and hence produce no torque along that axis). By using the concept of vector angular momentum we have a far more powerful technique for understanding and for attacking problems.

11–6

Conservation of Angular Momentum

In Chapter 9 we saw that the most general form of Newton's second law for the translational motion of a particle or system of particles is B

a Fext

dP , = dt

where P is the (linear) momentum, defined as mv for a particle, or Mvcm for a B system of particles of total mass M whose CM moves with velocity vcm , and © Fext is the net external force acting on the particle or system. This relation is valid only in an inertial reference frame. In this Chapter, we have found a similar relation to describe the general rotation of a system of particles (including rigid objects): B

dL , a T = dt B

where ©T is the net external torque acting on the system, and L is the total angular B momentum. This relation is valid when ©T and L are calculated about a point fixed in an inertial reference frame, or about the CM of the system. B For translational motion, if the net force on the system is zero, dPdt = 0, so the total linear momentum of the system remains constant. This is the law of conservation of linear momentum. For rotational motion, if the net torque on the system is zero, then B

dL = 0 dt

and L = constant.

C ©T = 0D (11–12) B

In words: The total angular momentum of a system remains constant if the net external torque acting on the system is zero.

CONSERVATION OF ANGULAR MOMENTUM

This is the law of conservation of angular momentum in full vector form. It ranks with the laws of conservation of energy and linear momentum (and others to be discussed later) as one of the great laws of physics. In Section 11–1 we saw some Examples of this important law applied to the special case of a rigid object rotating about a fixed axis. Here we have it in general form. We use it now in interesting Examples. SECTION 11–6

Conservation of Angular Momentum

297

Sun

u Planet

dt sin u

FIGURE 11–21 Kepler's second law of planetary motion (Example 11–11).

EXAMPLE 11–11 Kepler's second law derived. Kepler's second law states that each planet moves so that a line from the Sun to the planet sweeps out equal areas in equal times (Section 6–5). Use conservation of angular momentum to show this. APPROACH We determine the angular momentum of a planet in terms of the area swept out with the help of Fig. 11–21. SOLUTION The planet moves in an ellipse as shown in Fig. 11–21. In a time dt, the planet moves a distance v dt and sweeps out an area dA equal to the area of a triangle of base r and height v dt sin u (shown exaggerated in Fig. 11–21). Hence dA =

1 2 (r)(v

dt sin u)

and dA = 12 rv sin u. dt B The magnitude of the angular momentum L about the Sun is L = @ r * mv @ = mrv sin u, B

dA 1 = L. dt 2m B But L = constant, since the gravitational force F is directed toward the Sun so the torque it produces is zero (we ignore the pull of the other planets). Hence dAdt = constant, which is what we set out to prove.

FIGURE 11–22 Bullet strikes and becomes embedded in cylinder at its edge (Example 11–12). m R0 M

EXAMPLE 11–12 Bullet strikes cylinder edge. A bullet of mass m moving with velocity v strikes and becomes embedded at the edge of a cylinder of mass M and radius R0 , as shown in Fig. 11–22. The cylinder, initially at rest, begins to rotate about its symmetry axis, which remains fixed in position. Assuming no frictional torque, what is the angular velocity of the cylinder after this collision? Is kinetic energy conserved? APPROACH We take as our system the bullet and cylinder, on which there is no net external torque. Thus we can use conservation of angular momentum, and we calculate all angular momenta about the center O of the cylinder. SOLUTION Initially, because the cylinder is at rest, the total angular momentum about O is solely that of the bullet: L = @ r * p @ = R0 mv, B

since R0 is the perpendicular distance of p from O. After the collision, the cylinder AIcyl = 12 MR20 B rotates with the bullet AIb = mR 20 B embedded in it at angular velocity v: L = Iv = AIcyl + mR20 B v = A 12 M + mB R20 v.

Hence, because angular momentum is conserved, we find that v is mvR0 L mv . v = 1 = 1 = 1 2 2 A 2 M + mB R 0 A 2 M + mB R0 A 2 M + mB R0 Angular momentum is conserved in this collision, but kinetic energy is not: Kf - Ki = = =

1 1 2 2 Icyl v + 2 1 1 2 2 2 A 2 MR 0 B v 1 1 2 A2 M

AmR20 B v2 -

1 2 2 mv + 12 AmR 20 B v2 - 12 mv2 2 mv mB ¢ 1 ≤ - 12 mv2 2M + m

mM v2, 2M + 4m which is less than zero. Hence Kf 6 Ki . This energy is transformed to thermal energy as a result of the inelastic collision. = –

298

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Angular Momentum; General Rotation

* 11–7

The Spinning Top and Gyroscope

The motion of a rapidly spinning top, or a gyroscope, is an interesting example of rotational motion and of the use of the vector equation B dL . a T = dt Consider a symmetrical top of mass M spinning rapidly about its symmetry axis, as in Fig. 11–23. The top is balanced on its tip at point O in an inertial reference frame. If the axis of the top makes an angle f to the vertical (z axis), when the top is carefully released its axis will move, sweeping out a cone about the vertical as shown by the dashed lines in Fig. 11–23. This type of motion, in which a torque produces a change in the direction of the rotation axis, is called precession. The rate at which the rotation axis moves about the vertical (z) axis is called the angular velocity of precession, (capital Greek omega). Let us now try to understand the reasons for this motion, and calculate . If the top were not spinning, it would immediately fall to the ground when released due to the pull of gravity. The apparent mystery of a top is that when it is spinning, it does not immediately fall to the ground but instead precesses—it moves slowly sideways. But this is not really so mysterious if we examine it from the point of view of angular momentum and torque, which we calculate about the point O. When the top is spinning with angular velocity v about its symmetry axis, B it has an angular momentum L directed along its axis, as shown in Fig. 11–23. (There is also angular momentum due to the precessional motion, so that the total B L is not exactly along the axis of the top; but if V v, which is usually the case, we can ignore this.) To change the angular momentum, a torque is required. If no B torque were applied to the top, L would remain constant in magnitude and direction; the top would neither fall nor precess. But the slightest tip to the side results in a net torque about O, equal to Tnet = r * Mg, where r is the position vector of the top's center of mass with respect to O, and M is the mass of the top. The direction of Tnet is perpendicular to both r and Mg and by the right-hand rule is, as shown in B Fig. 11–23, in the horizontal (xy) plane. The change in L in a time dt is

P H Y S I C S

A P P L I E D

A spinning top

z du dL B

L f r CM

M gB

x FIGURE 11–23

Spinning top.

dL = Tnet dt, B

which is perpendicular to L and horizontal (parallel to Tnet ), as shown in Fig. 11–23. B B B Since dL is perpendicular to L, the magnitude of L does not change. Only the B B direction of L changes. Since L points along the axis of the top, we see that this axis moves to the right in Fig. 11–23. That is, the upper end of the top's axis B moves in a horizontal direction perpendicular to L. This explains why the top B precesses rather than falls. The vector L and the top's axis move together in a B horizontal circle. As they do so, Tnet and dL rotate as well so as to be horizontal B and perpendicular to L. To determine , we see from Fig. 11–23 that the angle du (which is in a horizontal plane) is related to dL by B

dL = L sin f du, B

since L makes an angle f to the z axis. The angular velocity of precession is = dudt, which becomes (since du = dLL sin f ) =

1 dL t . = L sin f dt L sin f

[spinning top] (11–13a)

But tnet = @ r * Mg @ = rMg sin f [because sin(p - f) = sin f ] so we can also write B

Mgr . [spinning top] (11–13b) L Thus the rate of precession does not depend on the angle f; but it is inversely proportional to the top's angular momentum. The faster the top spins, the greater L is and the slower the top precesses. =

*SECTION 11–7

The Spinning Top and Gyroscope

299

B V

FIGURE 11–24

A toy gyroscope.

From Eq. 11–1 (or Eq. 11–11) we can write L = Iv, where I and v are the moment of inertia and angular velocity of the spinning top about its spin axis. Then Eq. 11–13b for the top's precession angular velocity becomes Mgr . (11–13c) = Iv Equations 11–13 apply also to a toy gyroscope, which consists of a rapidly spinning wheel mounted on an axle (Fig. 11–24). One end of the axle rests on a support. The other end of the axle is free and will precess like a top if its "spin" angular velocity v is large compared to the precession rate (v W ). As v decreases due to friction and air resistance, the gyroscope will begin to fall, just as does a top.

* 11–8

Rotating Frames of Reference; Inertial Forces

Inertial and Noninertial Reference Frames FIGURE 11–25 Path of a ball released on a rotating merry-goround (a) in the reference frame of the merry-go-round, and (b) in a reference frame fixed on the ground. People on ground appear to move this way

Path of ball with respect to rotating platform (i.e., as seen by observer on platform) (a) Rotating reference frame Path of ball with respect to ground (i.e., as seen by observers on the ground)

Platform rotating counterclockwise (b) Inertial reference frame

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CHAPTER 11

Up to now, we have examined the motion of objects, including circular and rotational motion, from the outside, as observers fixed on the Earth. Sometimes it is convenient to place ourselves (in theory, if not physically) into a reference frame that is rotating. Let us examine the motion of objects from the point of view, or frame of reference, of persons seated on a rotating platform such as a merry-go-round. It looks to them as if the rest of the world is going around them. But let us focus attention on what they observe when they place a tennis ball on the floor of the rotating platform, which we assume is frictionless. If they put the ball down gently, without giving it any push, they will observe that it accelerates from rest and moves outward as shown in Fig. 11–25a. According to Newton's first law, an object initially at rest should stay at rest if no net force acts on it. But, according to the observers on the rotating platform, the ball starts moving even though there is no net force acting on it. To observers on the ground this is all very clear: the ball has an initial velocity when it is released (because the platform is moving), and it simply continues moving in a straight-line path as shown in Fig. 11–25b, in accordance with Newton's first law. But what shall we do about the frame of reference of the observers on the rotating platform? Since the ball starts moving without any net force on it, Newton's first law, the law of inertia, does not hold in this rotating frame of reference. For this reason, such a frame is called a noninertial reference frame. An inertial reference frame (as we discussed in Chapter 4) is one in which the law of inertia—Newton's first law—does hold, and so do Newton's second and third laws. In a noninertial reference frame, such as our rotating platform, Newton's second law also does not hold. For instance in the situation described above, there is no net force on the ball; yet, with respect to the rotating platform, the ball accelerates.

Fictitious (Inertial) Forces Because Newton's laws do not hold when observations are made with respect to a rotating frame of reference, calculation of motion can be complicated. However, we can still make use of Newton's laws in such a reference frame if we make use of a trick. The ball on the rotating platform of Fig. 11–25a flies outward when released (even though no force is actually acting on it). So the trick we use is to write down the equation ©F = ma as if a force equal to mv2r (or mv2r) were acting radially outward on the object in addition to any other forces that may be acting. This extra force, which might be designated as "centrifugal force" since it seems to act outward, is called a fictitious force or pseudoforce. It is a pseudoforce ("pseudo" means "false") because there is no object that exerts this force. Furthermore, when viewed from an inertial reference frame, the effect doesn't exist at all. We have made up this pseudoforce so that we can make calculations in a noninertial frame using Newton's second law, ©F = ma. Thus the observer in the noninertial frame of Fig. 11–25a uses Newton's second law for the ball's outward motion by assuming that a force equal to mv2r acts on it. Such pseudoforces are also called inertial forces since they arise only because the reference frame is not an inertial one.

Angular Momentum; General Rotation

The Earth itself is rotating on its axis. Thus, strictly speaking, Newton's laws are not valid on the Earth. However, the effect of the Earth's rotation is usually so small that it can be ignored, although it does influence the movement of large air masses and ocean currents. Because of the Earth's rotation, the material of the Earth is concentrated slightly more at the equator. The Earth is thus not a perfect sphere but is slightly fatter at the equator than at the poles.

* 11–9

The Coriolis Effect

In a reference frame that rotates at a constant angular speed v (relative to an inertial frame), there exists another pseudoforce known as the Coriolis force. It appears to act on an object in a rotating reference frame only if the object is moving relative to that rotating reference frame, and it acts to deflect the object sideways. It, too, is an effect of the rotating reference frame being noninertial and hence is referred to as an inertial force. It too affects the weather. To see how the Coriolis force arises, consider two people, A and B, at rest on a platform rotating with angular speed v, as shown in Fig. 11–26a. They are situated at distances rA and rB from the axis of rotation (at O). The woman at A throws a ball with a horizontal velocity v (in her reference frame) radially outward toward the man at B on the outer edge of the platform. In Fig. 11–26a, we view the situation from an inertial reference frame. The ball initially has not only the velocity v radially outward, but also a tangential velocity vA due to the rotation of the platform. Now Eq. 10–4 tells us that vA = rA v, where rA is the woman's radial distance from the axis of rotation at O. If the man at B had this same velocity vA , the ball would reach him perfectly. But his speed is vB = rB v, which is greater than vA because rB 7 rA . Thus, when the ball reaches the outer edge of the platform, it passes a point that the man at B has already gone by because his speed in that direction is greater than the ball's. So the ball passes behind him. Figure 11–26b shows the situation as seen from the rotating platform as frame of reference. Both A and B are at rest, and the ball is thrown with velocity v toward B, but the ball deflects to the right as shown and passes behind B as previously described. This is not a centrifugal-force effect, for the latter acts radially outward. Instead, this effect acts sideways, perpendicular to v, and is called a Coriolis acceleration; it is said to be due to the Coriolis force, which is a fictitious inertial force. Its explanation as seen from an inertial system was given above: it is an effect of being in a rotating system, wherein a point farther from the rotation axis has a higher linear speed. On the other hand, when viewed from the rotating B system, we can describe the motion using Newton's second law, © F = ma, if we add a "pseudoforce" term corresponding to this Coriolis effect. Let us determine the magnitude of the Coriolis acceleration for the simple case described above. (We assume v is large and distances short, so we can ignore gravity.) We do the calculation from the inertial reference frame (Fig. 11–26a). The ball moves radially outward a distance rB - rA at speed v in a short time t given by rB - rA = vt. During this time, the ball moves to the side a distance sA given by sA = vA t. The man at B, in this time t, moves a distance sB = vB t. The ball therefore passes behind him a distance s (Fig. 11–26a) given by s = sB - sA = AvB - vA B t. We saw earlier that vA = rA v and vB = rB v, so s = ArB - rA B vt. We substitute rB - rA = vt (see above) and get s = vvt2. (11–14) This same s equals the sideways displacement as seen from the noninertial rotating system (Fig. 11–26b).

vBA A

vBB vB

Path of ball

Rotation

(a) Inertial reference frame

s Path of ball

(b) Rotating reference frame FIGURE 11–26 The origin of the Coriolis effect. Looking down on a rotating platform, (a) as seen from a nonrotating inertial reference frame, and (b) as seen from the rotating platform as frame of reference.

*SECTION 11–9

The Coriolis Effect

301

FIGURE 11–27 (a) Winds (moving air masses) would flow directly toward a low-pressure area if the Earth did not rotate. (b) and (c): Because of the Earth's rotation, the winds are deflected to the right in the Northern Hemisphere (as in Fig. 11–26) as if a fictitious (Coriolis) force were acting.

We see immediately that Eq. 11–14 corresponds to motion at constant acceleration. For as we saw in Chapter 2 (Eq. 2–12b), y = 12 at2 for a constant acceleration (with zero initial velocity in the y direction). Thus, if we write Eq. 11–14 in the form s = 12 a Cor t2, we see that the Coriolis acceleration a Cor is

Low pressure

a Cor = 2vv.

(a)

Low pressure (b)

(11–15)

This relation is valid for any velocity in the plane of rotation perpendicular to the axis of rotation† (in Fig. 11–26, the axis through point O perpendicular to the page). Because the Earth rotates, the Coriolis effect has some interesting manifestations on the Earth. It affects the movement of air masses and thus has an influence on weather. In the absence of the Coriolis effect, air would rush directly into a region of low pressure, as shown in Fig. 11–27a. But because of the Coriolis effect, the winds are deflected to the right in the Northern Hemisphere (Fig. 11–27b), since the Earth rotates from west to east. So there tends to be a counterclockwise wind pattern around a low-pressure area. The reverse is true in the Southern Hemisphere. Thus cyclones rotate counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere. The same effect explains the easterly trade winds near the equator: any winds heading south toward the equator will be deflected toward the west (that is, as if coming from the east). The Coriolis effect also acts on a falling object. An object released from the top of a high tower will not hit the ground directly below the release point, but will be deflected slightly to the east. Viewed from an inertial frame, this happens because the top of the tower revolves with the Earth at a slightly higher speed than the bottom of the tower. †

The Coriolis acceleration can be written in general in terms of the vector cross product as B B B B a Cor = – 2V * v where V has direction along the rotation axis; its magnitude is aCor = 2wv⊥ where v⊥ is the component of velocity perpendicular to the rotation axis.

(c)

Summary B

The angular momentum L of a rigid object rotating about a fixed axis is given by (11–1) L = Iv. Newton's second law, in terms of angular momentum, is dL . (11–2) ©t = dt If the net torque on an object is zero, dLdt = 0, so L = constant. This is the law of conservation of angular momentum. B B The vector product or cross product of two vectors A and B B B B is another vector C = A * B whose magnitude is AB sin u and B B whose direction is perpendicular to both A and B in the sense of the right-hand rule. B The torque TB due to a force F is a vector quantity and is always calculated about some point O (the origin of a coordinate system) as follows: B

TB = Br * F,

(11–5) B

where r is the position vector of the point at which the force F acts. Angular momentum is also a vector. For a particle having B B momentum p = mvB, the angular momentum L about some point O is B B (11–6) L = Br * p , B where r is the position vector of the particle relative to the point O

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at any instant. The net torque ©TB on a particle is related to its angular momentum by B

©TB =

dL . dt

(11–7) B

For a system of particles, the total angular momentum L = © L i . The total angular momentum of the system is related to the total net torque ©TB on the system by B

©TB =

dL . dt

(11–9)

This last relation is the vector rotational equivalent of Newton's B second law. It is valid when L and ©TB are calculated about an origin (1) fixed in an inertial reference system or (2) situated at the CM of the system. For a rigid object rotating about a fixed axis, the component of angular momentum about the rotation axis is given by Lv = Iv. If an object rotates about an axis of B B symmetry, then the vector relation L = IV holds, but this is not true in general. If the total net torque on a system is zero, then the B total vector angular momentum L remains constant. This is the important law of conservation of angular momentum. B It applies to the vector L and therefore also to each of its components.

Questions 1. If there were a great migration of people toward the Earth's equator, would the length of the day (a) get longer because of conservation of angular momentum; (b) get shorter because of conservation of angular momentum; (c) get shorter because of conservation of energy; (d) get longer because of conservation of energy; or (e) remain unaffected? 2. Can the diver of Fig. 11–2 do a somersault without having any initial rotation when she leaves the board? 3. Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain. 4. When a motorcyclist leaves the ground on a jump and leaves the throttle on (so the rear wheel spins), why does the front of the cycle rise up? 5. Suppose you are standing on the edge of a large freely rotating turntable. What happens if you walk toward the center? 6. A shortstop may leap into the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 11–28). Explain.

FIGURE 11–28 Question 6. A shortstop in the air, throwing the ball. B

7. If all the components of the vectors V1 and V2 were reversed B B in direction, how would this alter V1 * V2 ? 8. Name the four different conditions that could make B B V1 * V2 = 0. B 9. A force F = F jˆ is applied to an object at a position Br = x iˆ + y jˆ + z kˆ where the origin is at the CM. Does the torque about the CM depend on x? On y? On z? 10. A particle moves with constant speed along a straight line. How does its angular momentum, calculated about any point not on its path, change in time? 11. If the net force on a system is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zero? Give examples. 12. Explain how a child "pumps" on a swing to make it go higher.

13. Describe the torque needed if the person in Fig. 11–17 is to tilt the axle of the rotating wheel directly upward without it swerving to the side. 14. An astronaut floats freely in a weightless environment. Describe how the astronaut can move her limbs so as to (a) turn her body upside down and (b) turn her body about-face. 15. On the basis of the law of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate in order to keep the helicopter stable. 16. A wheel is rotating freely about a vertical axis with constant angular velocity. Small parts of the wheel come loose and fly off. How does this affect the rotational speed of the wheel? Is angular momentum conserved? Is kinetic energy conserved? Explain. 17. Consider the following vector quantities: displacement, velocity, acceleration, momentum, angular momentum, torque. (a) Which of these are independent of the choice of origin of coordinates? (Consider different points as origin which are at rest with respect to each other.) (b) Which are independent of the velocity of the coordinate system? 18. How does a car make a right turn? Where does the torque come from that is needed to change the angular momentum? * 19. The axis of the Earth precesses with a period of about 25,000 years. This is much like the precession of a top. Explain how the Earth's equatorial bulge gives rise to a torque exerted by the Sun and Moon on the Earth; see Fig. 11–29, which is drawn for the winter solstice (December 21). About what axis would you expect the Earth's rotation axis to precess as a result 1 of the torque due to the Sun? Does 23 2° the torque exist three months later? Equator Explain. North Pole Orbit plane FIGURE 11–29 Question 19. Sun South Pole (Not to scale.) * 20. Why is it that at most locations on the Earth, a plumb bob does not hang precisely in the direction of the Earth's center? * 21. In a rotating frame of reference, Newton's first and second laws remain useful if we assume that a pseudoforce equal to mv2r is acting. What effect does this assumption have on the validity of Newton's third law? * 22. In the battle of the Falkland Islands in 1914, the shots of British gunners initially fell wide of their marks because their calculations were based on naval battles fought in the Northern Hemisphere. The Falklands are in the Southern Hemisphere. Explain the origin of their problem.

Problems 11–1 Angular Momentum 1. (I) What is the angular momentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.35 m at an angular speed of 10.4 rads?

2. (I) (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1300 rpm? (b) How much torque is required to stop it in 6.0 s?

Problems

303

3. (II) A person stands, hands at his side, on a platform that is rotating at a rate of 0.90 revs. If he raises his arms to a horizontal position, Fig. 11–30, the speed of rotation decreases to 0.70 revs. (a) Why? (b) By what factor has his moment of inertia changed?

FIGURE 11–30 Problem 3. 4. (II) A figure skater can increase her spin rotation rate from an initial rate of 1.0 rev every 1.5 s to a final rate of 2.5 revs. If her initial moment of inertia was 4.6 kgm2, what is her final moment of inertia? How does she physically accomplish this change? 5. (II) A diver (such as the one shown in Fig. 11–2) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed (revs) when in the straight position? 6. (II) A uniform horizontal rod of mass M and length l rotates with angular velocity v about a vertical axis through its center. Attached to each end of the rod is a small mass m. Determine the angular momentum of the system about the axis. 7. (II) Determine the angular momentum of the Earth (a) about its rotation axis (assume the Earth is a uniform sphere), and (b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass = 6.0 * 1024 kg and radius = 6.4 * 106 m, and is 1.5 * 108 km from the Sun. 8. (II) (a) What is the angular momentum of a figure skater spinning at 2.8 revs with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 48 kg? (b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms? 9. (II) A person stands on a platform, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform is IP . The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia IW and angular velocity vW . What will be the angular velocity vP of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60° angle to the vertical, (c) vertically downward? (d) What will vP be if the person reaches up and stops the wheel in part (a)? 10. (II) A uniform disk turns at 3.7 revs around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk's diameter, is dropped onto the freely spinning disk, Fig. 11–31. They then turn together around the spindle with their centers superposed. What is the angular frequency in revs of the combination? FIGURE 11–31 Problem 10.

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11. (II) A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920 kgm2. The platform rotates without friction with angular velocity 0.95 rads. The person walks radially to the edge of the platform. (a) Calculate the angular velocity when the person reaches the edge. (b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person's walk. 12. (II) A potter's wheel is rotating around a vertical axis through its center at a frequency of 1.5 revs. The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 2.6-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it? 13. (II) A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0.80 rads. Its total moment of inertia is 1760 kgm2. Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now? What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)? 14. (II) A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity v0 about a vertical axis through its center, and the woman begins walking with speed v (relative to the platform) toward the center of the platform. (a) Determine the angular velocity of the system as a function of time. (b) What will be the angular velocity when the woman reaches the center? 15. (II) A nonrotating cylindrical disk of moment of inertia I is dropped onto an identical disk rotating at angular speed v. Assuming no external torques, what is the final common angular speed of the two disks? 16. (II) Suppose our Sun eventually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun's new rotation rate be? (Take the Sun's current period to be about 30 days.) What would be its final kinetic energy in terms of its initial kinetic energy of today? 17. (III) Hurricanes can involve winds in excess of 120 kmh at the outer edge. Make a crude estimate of (a) the energy, and (b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 kgm3) of radius 85 km and height 4.5 km. 18. (III) An asteroid of mass 1.0 * 105 kg, traveling at a speed of 35 kms relative to the Earth, hits the Earth at the equator tangentially, in the direction of Earth's rotation, and is embedded there. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision. 19. (III) Suppose a 65-kg person stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1850 kg m2. The turntable is at rest initially, but when the person begins running at a speed of 3.8 ms (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.

11–2 Vector Cross Product and Torque B

20. (I) If vector A points along the negative x axis and vector B B B along the positive z axis, what is the direction of (a) A * B B B B B and (b) B * A? (c) What is the magnitude of A * B and B B B * A? 21. (I) Show that (a) ˆi * ˆi = jˆ * jˆ = kˆ * kˆ = 0, (b) ˆi * jˆ = kˆ , ˆi * kˆ = – jˆ, and jˆ * kˆ = ˆi. B

22. (I) The directions of vectors A and B are given below for B B several cases. For each case, state the direction of A * B. B B B B (a) A points east, B points south. (b) A points east, B points B B straight down. (c) A points straight up, B points north. B B (d) A points straight up, B points straight down. B

23. (II) What is the angle u between two vectors A and B, B B B B if @ A * B @ = A B? 24. (II) A particle is located at Br = A4.0 iˆ + 3.5 jˆ + 6.0 kˆ B m. B A force F = A9.0 jˆ - 4.0 kˆ B N acts on it. What is the torque, calculated about the origin?

25. (II) Consider a particle of a rigid object rotating about a fixed axis. Show that the tangential and radial vector components of the linear acceleration are: B aB tan = A * Br and

B aB R = V * vB.

26. (II) (a) Show that the cross product of two vectors, B B A = A x ˆi + A y jˆ + A z kˆ , and B = Bx ˆi + By jˆ + Bz kˆ is B B A * B = AA y Bz - A z By B ˆi + AA z Bx - A x Bz B jˆ

+ AA x By - A y Bx B kˆ .

29. (II) Use the result of Problem 26 to determine (a) the B B B B vector product A * B and (b) the angle between A and B if B B A = 5.4 ˆi - 3.5 jˆ and B = –8.5 ˆi + 5.6 jˆ + 2.0 kˆ . 30. (III) Show that the velocity vB of any point in an object rotating B with angular velocity V about a fixed axis can be written B vB = V * Br

where Br is the position vector of the point relative to an origin O located on the axis of rotation. Can O be anywhere B on the rotation axis? Will vB = V * Br if O is located at a point not on the axis of rotation? B

31. (III) Let A, B, and C be three vectors, which for generality we assume do not all lie in the same plane. Show that B B B B B B B B B A AB * C B = B AC * A B = C AA * B B.

11–3 Angular Momentum of a Particle 32. (I) What are the x, y, and z components of the angular momentum of a particle located at Br = x ˆi + y jˆ + z kˆ B which has momentum p = px ˆi + py jˆ + pz kˆ ? 33. (I) Show that the kinetic energy K of a particle of mass m, moving in a circular path, is K = L22I, where L is its angular momentum and I is its moment of inertia about the center of the circle. 34. (I) Calculate the angular momentum of a particle of mass m moving with constant velocity v for two cases (see Fig. 11–33): (a) about origin O, and (b) about O¿. m O′ vB d FIGURE 11–33 O Problem 34.

(b) Then show that the cross product can be written 35. (II) Two identical particles have equal but opposite B B momenta, p and –p , but they are not traveling along the same line. Show that the total angular momentum of this system does not depend on the choice of origin.

iˆ jˆ kˆ B B 3 A * B = Ax Ay Az 3 , Bx By Bz where we use the rules for evaluating a determinant. (Note, however, that this is not really a determinant, but a memory aid.) 27. (II) An engineer estimates that under the most adverse expected weather condiy tions, the total force B on the highway sign F in Fig. 11–32 will B be F = A & 2.4 iˆ - 4.1 jˆ B kN, CM acting at the CM. What torque does this force exert 6.0 m about the base O?

36. (II) Determine the angular momentum of a 75-g particle about the origin of coordinates when the particle is at x = 4.4 m, y = –6.0 m, and it has velocity v = A3.2 ˆi - 8.0 kˆ B ms. 37. (II) A particle is at the position (x, y, z) = (1.0, 2.0, 3.0) m. It is traveling with a vector velocity ( –5.0, ±2.8, –3.1) ms. Its mass is 3.8 kg. What is its vector angular momentum about the origin?

11–4 and 11–5 Angular Momentum and Torque: General Motion; Rigid Objects

38. (II) An Atwood machine (Fig. 11–16) consists of two masses, mA = 7.0 kg and mB = 8.2 kg, connected by a cord that passes over a pulley free to rotate about a fixed axis. The pulley is a solid cylinder of radius R0 = 0.40 m and mass 0.80 kg. (a) Determine the acceleration a of each mass. (b) What percentage of error in a would be made if the moment of inertia of the pulley were ignored? Ignore friction in the pulley bearings.

28. (II) The origin of a coordinate system is at the center of a wheel which rotates in the xy plane about its axle which is the z axis. A force F = 215 N acts in the xy plane, at a ±33.0° angle to the x axis, at the point x = 28.0 cm, y = 33.5 cm. Determine the magnitude and direction of the torque produced by this force about the axis.

39. (II) Four identical particles of mass m are mounted at equal intervals on a thin rod of length l and mass M, with one mass at each end of the rod. If the system is rotated with angular velocity v about an axis perpendicular to the rod through one of the end masses, determine (a) the kinetic energy and (b) the angular momentum of the system.

8.0 m FIGURE 11–32 Problem 27.

Problems

305

40. (II) Two lightweight rods 24 cm in length are mounted perpendicular to an axle and at 180° to each other (Fig. 11–34). At the end of each rod is a 480-g mass. The rods are spaced 42 cm apart along the axle. The axle rotates at 4.5 rads. (a) What is the component of the total angular momentum along the axle? (b) What angle does the vector angular momentum make with the axle? [Hint: Remember that the vector angular momentum must be calculated about the same 480 g point for both masses, which 24 cm could be the CM.] 42 cm

11–6 Angular Momentum Conservation 47. (II) A thin rod of mass M and length l is suspended vertically from a frictionless pivot at its upper end. A mass m of putty traveling horizontally with a speed v strikes the rod at its CM and sticks there. How high does the bottom of the rod swing? 48. (II) A uniform stick 1.0 m long with a total mass of 270 g is pivoted at its center. A 3.0-g bullet is shot through the stick midway between the pivot and one 250 m/s 140 m/s end (Fig. 11–36). The bullet approaches at 250 ms and leaves at 140 ms. With what angular speed is the stick spinning after the collision? Pivot

480 g 24 cm

FIGURE 11–34 Problem 40.

41. (II) Figure 11–35 shows two masses connected by a cord passing over a pulley of radius R0 and moment of inertia I. Mass MA slides on a frictionless surface, and MB hangs freely. Determine a formula for (a) the angular momentum of the system about the pulley axis, as a function of the speed v of mass MA or MB , and (b) the acceleraMA tion of the masses.

MB FIGURE 11–35 Problem 41. 42. (III) A thin rod of length l and mass M rotates about a vertical axis through its center with angular velocity v. The rod makes an angle f with the rotation axis. Determine the B magnitude and direction of L. B

B 43. (III) Show that the total angular momentum L = © Br i * p i of a system of particles about the origin of an inertial reference frame can be written as the sum of the angular B momentum about the CM, L* (spin angular momentum), plus the angular momentum of the CM about the origin (orbital B B angular momentum): L = L* + Br cm * MvBcm . [Hint: See the derivation of Eq. 11–9b.]

FIGURE 11–36 Problems 48 and 83. 49. (II) Suppose a 5.8 * 1010 kg meteorite struck the Earth at the equator with a speed v = 2.2 * 104 ms, vB as shown in Fig. 11–37 and remained stuck. By what 45° factor would this affect the North rotational frequency of the Pole Earth (1 revday)? Earth FIGURE 11–37 Problem 49. 50. (III) A 230-kg beam 2.7 m in length slides broadside down the ice with a speed of 18 ms (Fig. 11–38). A 65-kg man at rest grabs one end as it goes past and hangs on as both he and the beam go spinning down the ice. Assume frictionless motion. (a) How fast does the center of mass of the system CM move after the collision? (b) With what angular velocity does the system rotate about its CM?

FIGURE 11–38 Problem 50.

* 44. (III) What is the magnitude of the force F exerted by each bearing in Fig. 11–18 (Example 11–10)? The bearings are a distance d from point O. Ignore the effects of gravity. * 45. (III) Suppose in Fig. 11–18 that mB = 0; that is, only one mass, mA , is actually present. If the bearings are each a distance d from O, determine the forces FA and FB at the upper and lower bearings respectively. [Hint: Choose an B origin—different than O in Fig. 11–18—such that L is B parallel to V. Ignore effects of gravity.] * 46. (III) Suppose in Fig. 11–18 that mA = mB = 0.60 kg, rA = rB = 0.30 m, and the distance between the bearings is 0.23 m. Evaluate the force that each bearing must exert on the axle if f = 34.0°, v = 11.0 rads?

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51. (III) A thin rod of mass M and length l rests on a frictionless table and is struck at a point l4 from its CM by a clay ball of mass m moving at speed v (Fig. 11–39). The ball sticks to the rod. Determine the vB translational and rotational motion 1 of the rod after the collision. 4l CM

FIGURE 11–39 Problems 51 and 84.

52. (III) On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v0 and a "reverse" spin of angular speed v0 (see Fig. 11–40). A kinetic friction force acts on the ball as it initially skids across the table. (a) Explain why the ball's angular momentum is conserved about point O. (b) Using conservation of angular momentum, find the critical angular speed vC such that, if v0 = vC , kinetic friction will bring the ball to a complete (as opposed to momentary) stop. (c) If v0 is 10% smaller than vC , i.e., v0 = 0.90 vC , determine the ball's CM velocity vcm when it starts to roll without slipping. (d) If v0 is 10% larger than vC , i.e., v0 = 1.10 vC , determine the ball's CM velocity vcm when it starts to roll without slipping. [Hint: The ball possesses two types of angular momentum, the first due to the linear speed vcm of its CM relative to point O, the second due to the spin at angular velocity v about its own CM. The ball's total L about O is the sum of these two angular momenta.] ω0 vB0 FIGURE 11–40 Problem 52.

* 11–7 Spinning Top * 53. (II) A 220-g top spinning at 15 revs makes an angle of 25° to the vertical and precesses at a rate of 1.00 rev per 6.5 s. If its CM is 3.5 cm from its tip along its symmetry axis, what is the moment of inertia of the top? * 54. (II) A toy gyroscope consists of a 170-g disk with a radius of 5.5 cm mounted at the center of a thin axle 21 cm long (Fig. 11–41). The gyroscope spins at 45 revs. One end of its axle rests on a stand and the other end precesses horizontally about the stand. (a) How long does it take the gyroscope to precess once around? (b) If all the dimensions of the gyroscope were doubled (radius = 11 cm, axle = 42 cm), how long would it take to precess once? B L FIGURE 11–41 A wheel, rotating about a horizontal axle supported at one end, precesses. Problems 54, 55, and 56. * 55. (II) Suppose the solid wheel of Fig. 11–41 has a mass of 300 g and rotates at 85 rads; it has radius 6.0 cm and is mounted at the center of a horizontal thin axle 25 cm long. At what rate does the axle precess? 56. (II) If a mass equal to half the mass of the wheel in Problem 55 * is placed at the free end of the axle, what will be the precession rate now? Treat the extra mass as insignificant in size. * 57. (II) A bicycle wheel of diameter 65 cm and mass m rotates on its axle; two 20-cm-long wooden handles, one on each side of the wheel, act as the axle. You tie a rope to a small hook on the end of one of the handles, and then spin the bicycle wheel with a flick of the hand. When you release the spinning wheel, it precesses about the vertical axis defined by the rope, instead of falling to the ground (as it would if it were not spinning). Estimate the rate and direction of precession if the wheel rotates counterclockwise at 2.0 revs and its axle remains horizontal.

* 11–8 Rotating Reference Frames * 58. (II) If a plant is allowed to grow from seed on a rotating platform, it will grow at an angle, pointing inward. Calculate what this angle will be (put yourself in the rotating frame) in terms of g, r, and v. Why does it grow inward rather than outward? * 59. (III) Let gB ¿ be the effective acceleration of gravity at a point on the rotating Earth, equal to the vector sum of the "true" value gB plus the effect of the rotating reference frame (mv2r term). See Fig. 11–42. Determine the magnitude and direction of gB ¿ relative to a radial line from the center of the Earth (a) at the North Pole, (b) at a latitude of 45.0° north, and (c) at mv 2r the equator. Assume that N Pole g (if v were zero) is a mgB constant 9.80 ms2. B mg ′

FIGURE 11–42 Problem 59.

S Pole

* 11–9 Coriolis Effect * 60. (II) Suppose the man at B in Fig. 11–26 throws the ball toward the woman at A. (a) In what direction is the ball deflected as seen in the noninertial system? (b) Determine a formula for the amount of deflection and for the (Coriolis) acceleration in this case. * 61. (II) For what directions of velocity would the Coriolis effect on an object moving at the Earth's equator be zero? * 62. (III) We can alter Eqs. 11–14 and 11–15 for use on Earth by considering only the component of vB perpendicular to the axis of rotation. From Fig. 11–43, we see that this is v cos l for a vertically falling object, where l is the latitude of the place on the Earth. If a lead ball is dropped vertically from a 110-m-high tower in Florence, Italy (latitude = 44°), how far from the base of the tower is North Pole cos l it deflected by the Coriolis m force? l vB

FIGURE 11–43 Problem 62. Object of mass m falling vertically to Earth at a latitude l.

South Pole

* 63. (III) An ant crawls with constant speed outward along a radial spoke of a wheel rotating at constant angular velocity v about a vertical axis. Write a vector equation for all the forces (including inertial forces) acting on the ant. Take the x axis along the spoke, y perpendicular to the spoke pointing to the ant's left, and the z axis vertically upward. The wheel rotates counterclockwise as seen from above.

Problems

307

General Problems 64. A thin string is wrapped around a cylindrical hoop of radius R and mass M. One end of the string is fixed, and the hoop is allowed to fall vertically, starting from rest, as the string unwinds. (a) Determine the angular momentum of the hoop about its CM as a function of time. (b) What is the tension in the string as function of time? 65. A particle of mass 1.00 kg is moving with velocity vB = A7.0 ˆi + 6.0 jˆ B ms. (a) Find the angular momentum B L relative to the origin when the particle is at B B r = A2.0 jˆ + 4.0 kˆ B m. (b) At position Br a force of F = 4.0 Nˆi is applied to the particle. Find the torque relative to the origin. 66. A merry-go-round with a moment of inertia equal to 1260 kg m2 and a radius of 2.5 m rotates with negligible friction at 1.70 rads. A child initially standing still next to the merry-go-round jumps onto the edge of the platform straight toward the axis of rotation causing the platform to slow to 1.25 rads. What is her mass? 67. Why might tall narrow SUVs and buses be prone to "rollover"? Consider a vehicle rounding a curve of radius R on a flat road. When just on the verge of rollover, its tires on the inside of the curve are about to leave the ground, so the friction and normal force on these two tires are zero. The total normal force on the outside tires is FN and the total friction force is Ffr . Assume that the vehicle is not skidding. (a) Analysts define a static stability factor SSF = w2h, where a vehicle's "track width" w is the distance between tires on the same axle, and h is the height of the CM above the ground. Show that the critical rollover speed is vC =

Rg a

w b. 2h

[Hint: Take torques about an axis through the center of mass of the SUV, parallel to its direction of motion.] (b) Determine the ratio of highway curve radii (minimum possible) for a typical passenger car with SSF = 1.40 and an SUV with SSF = 1.05 at a speed of 90 kmh. 68. A spherical asteroid with radius r = 123 m and mass M = 2.25 * 1010 kg rotates about an axis at four revolutions per day. A "tug" spaceship attaches itself to the asteroid's south pole (as defined by the axis of rotation) and fires its engine, applying a force F tangentially to the asteroid's surface as shown in Fig. 11–44. If F = 265 N, how long will it take the tug to rotate the asteroid's axis of rotation through an angle of 10.0° by this method?

r 123 m

FIGURE 11–44 Problem 68.

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F 265 N

Angular Momentum; General Rotation

69. The time-dependent position of a point object which moves counterclockwise along the circumference of a circle (radius R) in the xy plane with constant speed v is given by B r = iˆ R cos vt + jˆ R sin vt where the constant v = vR. Determine the velocity vB and B angular velocity V of this object and then show that these B three vectors obey the relation vB = V * Br . 70. The position of a particle with mass m traveling on a helical path (see Fig. 11–45) is given by r = R cos a

2pz ˆ 2pz ˆ b i + R sin a b j + z kˆ d d

where R and d are the radius and pitch of the helix, respectively, and z has time dependence z = vz t where vz is the (constant) component of velocity in the z direction. B Determine the time-dependent angular momentum L of the particle about the origin.

d B

R z

y FIGURE 11–45 Problem 70. 71. A boy rolls a tire along a straight level street. The tire has mass 8.0 kg, radius 0.32 m and moment of inertia about its central axis of symmetry of 0.83 kg m2. The boy pushes the tire forward away from him at a speed of 2.1 ms and sees that the tire leans 12° to the right (Fig. 11–46). (a) How will the resultant torque affect the subsequent motion of the tire? (b) Compare the change in angular momentum caused by this torque 12° in 0.20 s to the original magnitude of angular momentum. CM

mg B

FIGURE 11–46 Problem 71. 72. A 70-kg person stands on a tiny rotating platform with arms outstretched. (a) Estimate the moment of inertia of the person using the following approximations: the body (including head and legs) is a 60-kg cylinder, 12 cm in radius and 1.70 m high; and each arm is a 5.0-kg thin rod, 60 cm long, attached to the cylinder. (b) Using the same approximations, estimate the moment of inertia when the arms are at the person's sides. (c) If one rotation takes 1.5 s when the person's arms are outstretched, what is the time for each rotation with arms at the sides? Ignore the moment of inertia of the lightweight platform. (d) Determine the change in kinetic energy when the arms are lifted from the sides to the horizontal position. (e) From your answer to part (d), would you expect it to be harder or easier to lift your arms when rotating or when at rest?

73. Water drives a waterwheel (or turbine) of radius R = 3.0 m as shown in Fig. 11–47. The water enters at a speed v1 = 7.0 ms and exits from the waterwheel at a speed v2 = 3.8 ms. (a) If 85 kg of water passes through per second, what is the rate at which the water delivers angular momentum to the waterwheel? (b) What is the torque the water applies to the waterwheel? (c) If the water causes the waterwheel to make one revolution every 5.5 s, how much power is delivered to the wheel? R

FIGURE 11–47 Problem 73.

78. A bicyclist traveling with speed v = 9.2 ms on a flat road is making a turn with a radius r = 12 m. The forces acting B on the cyclist and cycle are the normal force AFN B and friction B force AFfr B exerted by the road on the tires and mgB, the total weight of the cyclist and cycle. Ignore the small mass of the wheels. (a) Explain carefully why the angle u the bicycle makes with the vertical (Fig. 11 – 48) must be given by tan u = FfrFN if the cyclist is to maintain balance. (b) Calculate u for the values given. [Hint: Consider the "circular" translational motion of the bicycle and rider.] (c) If the coefficient of static friction between tires and road is ms = 0.65, what is the minimum turning radius?

74. The Moon orbits the Earth such that the same side always faces the Earth. Determine the ratio of the Moon's spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.) 75. A particle of mass m uniformly accelerates as it moves counterclockwise along the circumference of a circle of radius R: B r = ˆi R cos u + jˆ R sin u 1 2

1 r = Avx0 t B ˆi + ¢ vy0 t - gt2 ≤ jˆ 2

where vx0 and vy0 are the initial velocities in the x and y direction, respectively, and g is the acceleration due to gravity. The launch position is defined to be the origin. Determine the torque acting on the projectile about the B B origin using (a) TB = Br * F, (b) TB = dLdt. 77. Most of our Solar System's mass is contained in the Sun, and the planets possess almost all of the Solar System's angular momentum. This observation plays a key role in theories attempting to explain the formation of our Solar System. Estimate the fraction of the Solar System's total angular momentum that is possessed by planets using a simplified model which includes only the large outer planets with the most angular momentum. The central Sun (mass 1.99 * 1030 kg, radius 6.96 * 108 m) spins about its axis once every 25 days and the planets Jupiter, Saturn, Uranus, and Neptune move in nearly circular orbits around the Sun with orbital data given in the Table below. Ignore each planet's spin about its own axis.

Jupiter Saturn Uranus Neptune

FN mgB B

Ffr

with u = v0 t + at , where the constants v0 and a are the initial angular velocity and angular acceleration, respectively. Determine the object's tangential acceleration aB tan and deterB mine the torque acting on the object using (a) TB = Br * F, B (b) TB = IA . 76. A projectile with mass m is launched from the ground and follows a trajectory given by

Planet

(a)

(b)

FIGURE 11–48 Problem 78. 79. Competitive ice skaters commonly perform single, double, and triple axel jumps in which they rotate 1 12 , 2 12 , and 3 12 revolutions, respectively, about a vertical axis while airborne. For all these jumps, a typical skater remains airborne for about 0.70 s. Suppose a skater leaves the ground in an "open" position (e.g., arms outstretched) with moment of inertia I0 and rotational frequency f0 = 1.2 revs, maintaining this position for 0.10 s. The skater then assumes a "closed" position (arms brought closer) with moment of inertia I, acquiring a rotational frequency f, which is maintained for 0.50 s. Finally, the skater immediately returns to the "open" position for 0.10 s until landing (see Fig. 11 – 49). (a) Why is angular momentum conserved during the skater's jump? Neglect air resistance. (b) Determine the minimum rotational frequency f during the flight's middle section for the skater to successfully complete a single and a triple axel. (c) Show that, according to this model, a skater must be able to reduce his or her moment of inertia in midflight by a factor of about 2 and 5 in order to complete a single and triple axel, respectively. f f0

Mean Distance from Orbital Period Mass Sun ( 106 km) (Earth Years) ( 1025 kg) 778 1427 2870 4500

11.9 29.5 84.0 165

190 56.8 8.68 10.2

0.1 s

0.5 s

0.1 s

FIGURE 11–49 Problem 79.

General Problems

309

82. A baseball bat has a "sweet spot" where a ball can be hit with almost effortless transmission of energy. A careful analysis of baseball dynamics shows that this special spot is located at the point where an applied force would result in pure rotation of the bat about the handle grip. Determine the location of the sweet spot of the bat shown in Fig. 11–51. The linear mass density of the bat is given roughly by A0.61 + 3.3x 2 B kgm, where x is in meters measured from the end of the handle. The entire bat is 0.84 m long. The desired rotation point should be 5.0 cm from the end where the bat is held. [Hint: Where is the CM of the bat?]

80. A radio transmission tower has a mass of 80 kg and is 12 m high. The tower is anchored to the ground by a flexible joint at its base, but it is secured by three cables 120° apart (Fig. 11–50). In an analysis of a potential failure, a mechanical engineer needs to determine the behavior of the tower if one of the cables broke. The tower would fall away from the broken cable, rotating about its base. Determine the speed of the top of the tower as a function of the rotation angle u. Start your analysis with the rotational dynamics equation of B motion dLdt = TBnet . Approximate the tower as a tall thin rod.

y = lv = l du dt

xs CM

0.050 m

d dCM

J ∫ F dt

FIGURE 11–51 Problem 82.

mgB

* Numerical/Computer l sin u 2 FIGURE 11–50

Problem 80.

81. Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 9.0 days. If it were to undergo gravitational collapse to a neutron star of radius 12 km, losing 34 of its mass in the process, what would its rotation speed be? Assume the star is a uniform sphere at all times. Assume also that the thrown-off mass carries off either (a) no angular momentum, or (b) its proportional share A 34 B of the initial angular momentum.

* 83. (II) A uniform stick 1.00 m long with a total mass of 330 g is pivoted at its center. A 3.0-g bullet is shot through the stick a distance x from the pivot. The bullet approaches at 250 ms and leaves at 140 ms (Fig. 11–36). (a) Determine a formula for the angular speed of the spinning stick after the collision as a function of x. (b) Graph the angular speed as a function of x, from x = 0 to x = 0.50 m. * 84. (III) Figure 11–39 shows a thin rod of mass M and length l resting on a frictionless table. The rod is struck at a distance x from its CM by a clay ball of mass m moving at speed v. The ball sticks to the rod. (a) Determine a formula for the rotational motion of the system after the collision. (b) Graph the rotational motion of the system as a function of x, from x = 0 to x = l2, with values of M = 450 g, m = 15 g, l = 1.20 m, and v = 12 ms. (c) Does the translational motion depend on x? Explain.

Answers to Exercises A: (b).

D: (i) (d); (ii) (a); (iii) (b).

B: (a).

E: (e).

C: (b).

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Angular Momentum; General Rotation

Our whole built environment, from modern bridges to skyscrapers, has required architects and engineers to determine the forces and stresses within these structures. The object is to keep these structures static—that is, not in motion, especially not falling down.

CONTENTS 12–1 The Conditions for Equilibrium 12–2 Solving Statics Problems 12–3 Stability and Balance 12–4 Elasticity; Stress and Strain 12–5 Fracture

*12–6 Trusses and Bridges *12–7 Arches and Domes

CHAPTER-OPENING QUESTION—Guess Now! The diving board shown here is held by two supports at A and B. Which statement is true about the forces exerted on the diving board at A and B? B B (a) FA is down, FB is up, and FB is larger than FA . A B (b) Both forces are up and FB is larger than FA . B B (c) FA is down, FB is up, and FA is larger than FB . (d) Both forces are down and approximately equal. B B (e) FB is down, FA is up, and they are equal. e now study a special case in mechanics—when the net force and the net torque on an object, or system of objects, are both zero. In this case both the linear acceleration and the angular acceleration of the object or system are zero. The object is either at rest, or its center of mass is moving at constant velocity. We will be concerned mainly with the first situation, in which the object or objects are at rest. We will see how to determine the forces (and torques) that act within a structure. Just how and where these forces act can be very important for buildings, bridges, and other structures, and in the human body. Statics is concerned with the calculation of the forces acting on and within structures that are in equilibrium. Determination of these forces, which occupies us in the first part of this Chapter, then allows a determination of whether the structures can sustain the forces without significant deformation or fracture, since any material can break or buckle if too much force is applied (Fig. 12–1).

C H

Static Equilibrium; Elasticity and Fracture

P T E R

FIGURE 12–1 Elevated walkway collapse in a Kansas City hotel in 1981. How a simple physics calculation could have prevented the tragic loss of over 100 lives is considered in Example 12–9.

311

12–1 Normal force Gravity

FIGURE 12–2 The book is in equilibrium; the net force on it is zero.

The Conditions for Equilibrium

Objects in daily life have at least one force acting on them (gravity). If they are at rest, then there must be other forces acting on them as well so that the net force is zero. A book at rest on a table, for example, has two forces acting on it, the downward force of gravity and the normal force the table exerts upward on it (Fig. 12–2). Because the book is at rest, Newton's second law tells us the net force on it is zero. Thus the upward force exerted by the table on the book must be equal in magnitude to the force of gravity acting downward on the book. Such an object is said to be in equilibrium (Latin for "equal forces" or "balance") under the action of these two forces. Do not confuse the two forces in Fig. 12–2 with the equal and opposite forces of Newton's third law, which act on different objects. Here, both forces act on the same object; and they add up to zero.

The First Condition for Equilibrium For an object to be at rest, Newton's second law tells us that the sum of the forces acting on it must add up to zero. Since force is a vector, the components of the net force must each be zero. Hence, a condition for equilibrium is that ©Fx = 0, ©Fy = 0, ©Fz = 0. (12–1) We will mainly be dealing with forces that act in a plane, so we usually need only the x and y components. We must remember that if a particular force component points along the negative x or y axis, it must have a negative sign. Equations 12–1 are called the first condition for equilibrium. B

FIGURE 12–3 Example 12–1. 60°

1960 N

200 kg

(a) B

(b)

FAy

60° FAx

y x

EXAMPLE 12–1 Chandelier cord tension. Calculate the tensions FA and FB in the two cords that are connected to the vertical cord supporting the 200-kg chandelier in Fig. 12–3. Ignore the mass of the cords. APPROACH We need a free-body diagram, but for which object? If we choose the chandelier, the cord supporting it must exert a force equal to the chandelier's B B weight mg = (200 kg)A9.8 ms2 B = 1960 N. But the forces FA and FB don't get involved. Instead, let us choose as our object the point where the three cords join (it could be a knot). The free-body diagram is then as shown in Fig. 12–3a. B B The three forces— FA , FB , and the tension in the vertical cord equal to the weight of the 200-kg chandelier—act at this point where the three cords join. For this junction point we write ©Fx = 0 and ©Fy = 0, since the problem is laid out in two B B dimensions. The directions of FA and FB are known, since tension in a cord can only be along the cord—any other direction would cause the cord to bend, as already pointed out in Chapter 4. Thus, our unknowns are the magnitudes FA and FB . B SOLUTION We first resolve FA into its horizontal (x) and vertical (y) components. Although we don't know the value of FA , we can write (see Fig. 12–3b) B FAx = –FA cos 60° and FAy = FA sin 60°. FB has only an x component. In the vertical direction, we have the downward force exerted by the vertical cord equal to the B weight of the chandelier = (200 kg)(g), and the vertical component of FA upward: ©Fy = 0 FA sin 60° - (200 kg)(g) = 0 so

(200 kg)g = (231 kg)g = 2260 N. sin 60° In the horizontal direction, with ©Fx = 0, FA =

©Fx = FB - FA cos 60° = 0. Thus FB = FA cos 60° = (231 kg)(g)(0.500) = (115 kg)g = 1130 N. B

The magnitudes of FA and FB determine the strength of cord or wire that must be used. In this case, the cord must be able to hold more than 230 kg.

312

CHAPTER 12

EXERCISE A In Example 12–1, FA has to be greater than the chandelier's weight, mg. Why?

The Second Condition for Equilibrium

Although Eqs. 12–1 are a necessary condition for an object to be in equilibrium, they are not always a sufficient condition. Figure 12–4 shows an object on which B the net force is zero. Although the two forces labeled F add up to give zero net force on the object, they do give rise to a net torque that will rotate the object. Referring to Eq. 10–14, ©t = Ia, we see that if an object is to remain at rest, the net torque applied to it (calculated about any axis) must be zero. Thus we have the second condition for equilibrium: that the sum of the torques acting on an object, as calculated about any axis, must be zero: ©t = 0.

(12–2)

This condition will ensure that the angular acceleration, a, about any axis will be zero. If the object is not rotating initially (v = 0), it will not start rotating. Equations 12–1 and 12–2 are the only requirements for an object to be in equilibrium. We will mainly consider cases in which the forces all act in a plane (we call it the xy plane). In such cases the torque is calculated about an axis that is perpendicular to the xy plane. The choice of this axis is arbitrary. If the object is at rest, then ©t = 0 about any axis whatever. Therefore we can choose any axis that makes our calculation easier. Once the axis is chosen, all torques must be calculated about that axis. CONCEPTUAL EXAMPLE 12–2 A lever. The bar in Fig. 12–5 is being used as a lever to pry up a large rock.The small rock acts as a fulcrum (pivot point).The force FP required at the long end of the bar can be quite a bit smaller than the rock's weight mg, since it is the torques that balance in the rotation about the fulcrum. If, however, the leverage isn't sufficient, and the large rock isn't budged, what are two ways to increase the leverage? RESPONSE One way is to increase the lever arm of the force FP by slipping a pipe over the end of the bar and thereby pushing with a longer lever arm. A second way is to move the fulcrum closer to the large rock. This may change the long lever arm R only a little, but it changes the short lever arm r by a substantial fraction and therefore changes the ratio of Rr dramatically. In order to pry the rock, the torque due to FP must at least balance the torque due to mg; that is, mgr = FP R and FP . r = mg R With r smaller, the weight mg can be balanced with less force FP . The ratio of the load force to your applied force ( mgFP here) is the mechanical advantage of the system, and here equals Rr. A lever is a "simple machine." We discussed another simple machine, the pulley, in Chapter 4, Example 4–14.

FIGURE 12–4 Although the net force on it is zero, the ruler will move (rotate). A pair of equal forces acting in opposite directions but at different points on an object (as shown here) is referred to as a couple. C A U T I O N

Axis choice for ©t = 0 is arbitrary. All torques must be calculated about the same axis. P H Y S I C S

A P P L I E D

The lever FIGURE 12–5 Example 12–2. A lever can "multiply" your force.

mgB

EXERCISE B For simplicity, we wrote the equation in Example 12–2 as if the lever were perpendicular to the forces. Would the equation be valid even for a lever at an angle as shown in Fig. 12–5?

12–2

Solving Statics Problems

The subject of statics is important because it allows us to calculate certain forces on (or within) a structure when some of the forces on it are already known. We will mainly consider situations in which all the forces act in a plane, so we can have two force equations (x and y components) and one torque equation, for a total of three equations. Of course, you do not have to use all three equations if they are not needed. When using a torque equation, a torque that tends to rotate the object counterclockwise is usually considered positive, whereas a torque that tends to rotate it clockwise is considered negative. (But the opposite convention would be okay too.) One of the forces that acts on objects is the force of gravity. As we discussed in Section 9–8, we can consider the force of gravity on an object as acting at its center of gravity (CG) or center of mass (CM), which for practical purposes are the same point. For uniform symmetrically shaped objects, the CG is at the geometric center. For more complicated objects, the CG can be determined as discussed in Section 9–8. There is no single technique for attacking statics problems, but the following procedure may be helpful.

P R O B L E M

S O LV I N G

t 7 0 counterclockwise t 6 0 clockwise

SECTION 12–2

313

L SO VIN OBL E PR

Statics 1. Choose one object at a time for consideration. Make a careful free-body diagram by showing all the forces acting on that object, including gravity, and the points at which these forces act. If you aren't sure of the direction of a force, choose a direction; if the actual direction is opposite, your eventual calculation will give a result with a minus sign. 2. Choose a convenient coordinate system, and resolve the forces into their components. 3. Using letters to represent unknowns, write down the equilibrium equations for the forces: ©Fx = 0

and

©Fy = 0,

assuming all the forces act in a plane.

P H Y S I C S

A P P L I E D

Balancing a seesaw

mA = 30 kg A

2.5 m

(a)

mB = 25 kg

P B

FN 2.5 m B

FA = mAgB (b)

P M gB

x B

FB = mB g

FIGURE 12–6 (a) Two children on a seesaw, Example 12–3. (b) Free-body diagram of the board.

4. For the torque equation, ©t = 0, choose any axis perpendicular to the xy plane that might make the calculation easier. (For example, you can reduce the number of unknowns in the resulting equation by choosing the axis so that one of the unknown forces acts through that axis; then this force will have zero lever arm and produce zero torque, and so won't appear in the torque equation.) Pay careful attention to determining the lever arm for each force correctly. Give each torque a ± or – sign to indicate torque direction. For example, if torques tending to rotate the object counterclockwise are positive, then those tending to rotate it clockwise are negative. 5. Solve these equations for the unknowns. Three equations allow a maximum of three unknowns to be solved for. They can be forces, distances, or even angles.

EXAMPLE 12–3 Balancing a seesaw. A board of mass M = 2.0 kg serves as a seesaw for two children, as shown in Fig. 12–6a. Child A has a mass of 30 kg and sits 2.5 m from the pivot point, P (his center of gravity is 2.5 m from the pivot). At what distance x from the pivot must child B, of mass 25 kg, place herself to balance the seesaw? Assume the board is uniform and centered over the pivot. APPROACH We follow the steps of the Problem Solving Strategy above. SOLUTION 1. Free-body diagram. We choose the board as our object, and assume it is horizontal. Its free-body diagram is shown in Fig. 12–6b. The forces acting on B B the board are the forces exerted downward on it by each child, FA and FB , the B upward force exerted by the pivot FN , and the force of gravity on the board (= Mg) which acts at the center of the uniform board. 2. Coordinate system. We choose y to be vertical, with positive upward, and x horizontal to the right, with origin at the pivot. 3. Force equation. All the forces are in the y (vertical) direction, so B

©Fy = 0 FN - mA g - mB g - Mg = 0, where FA = mA g and FB = mB g because each child is in equilibrium when the seesaw is balanced. 4. Torque equation. Let us calculate the torque about an axis through the board at the pivot point, P. Then the lever arms for FN and for the weight of the board are zero, and they will contribute zero torque about point P. B B Thus the torque equation will involve only the forces FA and FB , which are equal to the weights of the children. The torque exerted by each child will be mg times the appropriate lever arm, which here is the distance of each child from the pivot point. Hence the torque equation is ©t = 0 mA g(2.5 m) - mB gx + Mg(0 m) + FN(0 m) = 0 or mA g(2.5 m) - mB gx = 0, where two terms were dropped because their lever arms were zero.

314

CHAPTER 12

Static Equilibrium; Elasticity and Fracture

5. Solve. We solve the torque equation for x and find 30 kg mA x = (2.5 m) = (2.5 m) = 3.0 m. mB 25 kg To balance the seesaw, child B must sit so that her CM is 3.0 m from the pivot point. This makes sense: since she is lighter, she must sit farther from the pivot than the heavier child in order to provide equal torque. EXERCISE C We did not need to use the force equation to solve Example 12–3 because of our choice of the axis. Use the force equation to find the force exerted by the pivot.

EXERCISE D Return to the Chapter-Opening Question, p. 311, and answer it again now. Try to explain why you may have answered differently the first time.

EXAMPLE 12–4 Force exerted by biceps muscle. How much force must the biceps muscle exert when a 5.0-kg ball is held in the hand (a) with the arm horizontal as in Fig. 12–8a, and (b) when the arm is at a 45° angle as in Fig. 12–8b? The biceps muscle is connected to the forearm by a tendon attached 5.0 cm from the elbow joint. Assume that the mass of forearm and hand together is 2.0 kg and their CG is as shown. APPROACH The free-body diagram for the forearm is shown in Fig. 12–8; the forces B are the weights of the arm and ball, the upward force FM exerted by the muscle, and B a force FJ exerted at the joint by the bone in the upper arm (all assumed to act B vertically). We wish to find the magnitude of FM , which is done most easily by B using the torque equation and by choosing our axis through the joint so that FJ contributes zero torque. B SOLUTION (a) We calculate torques about the point where FJ acts in Fig. 12–8a. The ©t = 0 equation gives (0.050 m)FM - (0.15 m)(2.0 kg)g - (0.35 m)(5.0 kg)g = 0. We solve for FM : (0.15 m)(2.0 kg)g + (0.35 m)(5.0 kg)g FM = = (41 kg)g = 400 N. 0.050 m (b) The lever arm, as calculated about the joint, is reduced by the factor cos 45° for all three forces. Our torque equation will look like the one just above, except that each term will have its lever arm reduced by the same factor, which will cancel out. The same result is obtained, FM = 400 N. NOTE The force required of the muscle (400 N) is quite large compared to the weight of the object lifted (= mg = 49 N). Indeed, the muscles and joints of the body are generally subjected to quite large forces.

P H Y S I C S

A P P L I E D

Cantilever

P R O B L E M

S O LV I N G

If a force comes out negative B

FA 20.0 m

30.0 m CG

FIGURE 12–7 A cantilever.The force vectors shown are hypothetical—one may even have a different direction. P H Y S I C S

A P P L I E D

Forces in muscles and joints FIGURE 12–8 Example 12–4. B

5.0 cm

Figure 12–7 shows a uniform beam that extends beyond its support like a diving board. Such a beam is called a cantilever. The forces acting on the beam in B B Fig. 12–7 are those due to the supports, FA and FB , and the force of gravity which acts at the CG, 5.0 m to the right of the right-hand support. If you follow the procedure of the last Example and calculate FA and FB , assuming they point upward as shown in Fig. 12–7, you will find that FA comes out negative. If the beam has a mass of 1200 kg and a weight mg = 12,000 N, then FB = 15,000 N and FA = –3000 N (see Problem 9). Whenever an unknown force comes out negative, it merely means that the force actually points in the opposite direction B from what you assumed. Thus in Fig. 12–7, FA actually points downward. With a little reflection it should become clear that the left-hand support must indeed pull downward on the beam (by means of bolts, screws, fasteners andor glue) if the beam is to be in equilibrium; otherwise the sum of the torques about the CG (or B about the point where FB acts) could not be zero.

(2.0 kg)gB

(5.0 kg)g B

15 cm 35 cm

(a)

45°

(b)

EXERCISE E How much mass could the person in Example 12–4 hold in the hand with a biceps force of 450 N if the tendon was attached 6.0 cm from the elbow instead of 5.0 cm?

SECTION 12–2

Solving Statics Problems

315

y x B

FHy Hinge

FHx mgB

FTy

FTx M gB

FIGURE 12–9 Example 12–5.

Our next Example involves a beam that is attached to a wall by a hinge and is supported by a cable or cord (Fig. 12–9). It is important to remember that a flexible cable can support a force only along its length. (If there were a component of force perpendicular to the cable, it would bend because it is flexible.) But for a rigid device, such as the hinge in Fig. 12–9, the force can be in any direction and we can know the direction only after solving the problem. The hinge is assumed small and smooth, so it can exert no internal torque (about its center) on the beam. EXAMPLE 12–5 Hinged beam and cable. A uniform beam, 2.20 m long with mass m = 25.0 kg, is mounted by a small hinge on a wall as shown in Fig. 12–9. The beam is held in a horizontal position by a cable that makes an angle u = 30.0°. The beam supports a sign of mass M = 28.0 kg suspended from B its end. Determine the components of the force FH that the (smooth) hinge exerts on the beam, and the tension FT in the supporting cable. APPROACH Figure 12–9 is the free-body diagram for the beam, showing all the B B forces acting on the beam. It also shows the components of FT and a guess for FH . We have three unknowns, FHx , FHy , and FT (we are given u), so we will need all three equations, ©Fx = 0, ©Fy = 0, ©t = 0. SOLUTION The sum of the forces in the vertical (y) direction is ©Fy = 0 FHy + FTy - mg - Mg = 0. (i) In the horizontal (x) direction, the sum of the forces is ©Fx = 0 FHx - FTx = 0. (ii) B For the torque equation, we choose the axis at the point where FT and Mg act (so our equation then contains only one unknown, FHy). We choose torques that tend to rotate the beam counterclockwise as positive. The weight mg of the (uniform) beam acts at its center, so we have ©t = 0 – AFHy B(2.20 m) + mg(1.10 m) = 0. We solve for FHy : 1.10 m FHy = a b mg = (0.500)(25.0 kg)A9.80 ms2 B = 123 N. (iii) 2.20 m B

Next, since the tension FT in the cable acts along the cable (u = 30.0°), we see from Fig. 12–9 that tan u = FTyFTx , or FTy = FTx tan u = FTx(tan 30.0°) = 0.577 FTx . (iv) Equation (i) above gives FTy = (m + M)g - FHy = (53.0 kg)A9.80 ms2 B - 123 N = 396 N; Equations (iv) and (ii) give FTx = FTy0.577 = 687 N; FHx = FTx = 687 N. B

The components of FH are FHy = 123 N and FHx = 687 N. The tension in the wire is FT = 3F 2Tx + F 2Ty = 793 N. Alternate Solution Let us see the effect of choosing a different axis for calculating torques, such as an axis through the hinge. Then the lever arm for FH is zero, and the torque equation (©t = 0) becomes –mg(1.10 m) - Mg(2.20 m) + FTy(2.20 m) = 0.

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CHAPTER 12

We solve this for FTy and find m g + Mg = (12.5 kg + 28.0 kg)A9.80 ms2 B = 397 N. FTy = 2 We get the same result, within the precision of our significant figures. NOTE It doesn't matter which axis we choose for ©t = 0. Using a second axis can serve as a check.

EXAMPLE 12–6 Ladder. A 5.0-m-long ladder leans against a smooth wall at a point 4.0 m above a cement floor as shown in Fig. 12–10. The ladder is uniform and has mass m = 12.0 kg. Assuming the wall is frictionless (but the floor is not), determine the forces exerted on the ladder by the floor and by the wall. APPROACH Figure 12–10 is the free-body diagram for the ladder, showing all the forces acting on the ladder. The wall, since it is frictionless, can exert a force only B perpendicular to the wall, and we label that force FW . The cement floor exerts a force B FC which has both horizontal and vertical force components: FCx is frictional and FCy is the normal force. Finally, gravity exerts a force mg = (12.0 kg)A9.80 ms2 B = 118 N on the ladder at its midpoint, since the ladder is uniform. SOLUTION Again we use the equilibrium conditions, ©Fx = 0, ©Fy = 0, ©t = 0. We will need all three since there are three unknowns: FW , FCx , and FCy . The y component of the force equation is ©Fy = FCy - mg = 0, so immediately we have FCy = mg = 118 N. The x component of the force equation is ©Fx = FCx - FW = 0. To determine both FCx and FW , we need a torque equation. If we choose to calculate torques about an axis through the point where the ladder touches the cement floor, B then FC , which acts at this point, will have a lever arm of zero and so won't enter the equation. The ladder touches the floor a distance x0 = 3(5.0 m)2 - (4.0 m)2 = 3.0 m from the wall (right triangle, c2 = a 2 + b2). The lever arm for mg is half this, or 1.5 m, and the lever arm for FW is 4.0 m, Fig. 12–10. We get ©t = (4.0 m)FW - (1.5 m)mg = 0. Thus (1.5 m)(12.0 kg)A9.8 ms2 B FW = = 44 N. 4.0 m Then, from the x component of the force equation, FCx = FW = 44 N. B Since the components of FC are FCx = 44 N and FCy = 118 N, then

FW x B

4.0 m

FCy

B mg

FCx x0 FIGURE 12–10 A ladder leaning against a wall. Example 12–6.

FC = 3(44 N)2 + (118 N)2 = 126 N L 130 N (rounded off to two significant figures), and it acts at an angle to the floor of u = tan–1(118 N44 N) = 70°. B NOTE The force FC does not have to act along the ladder's direction because the ladder is rigid and not flexible like a cord or cable. EXERCISE F Why is it reasonable to ignore friction along the wall, but not reasonable to ignore it along the floor?

12–3

FIGURE 12–11 (a) Stable equilibrium, and (b) unstable equilibrium.

Stability and Balance

An object in static equilibrium, if left undisturbed, will undergo no translational or rotational acceleration since the sum of all the forces and the sum of all the torques acting on it are zero. However, if the object is displaced slightly, three outcomes are possible: (1) the object returns to its original position, in which case it is said to be in stable equilibrium; (2) the object moves even farther from its original position, and it is said to be in unstable equilibrium; or (3) the object remains in its new position, and it is said to be in neutral equilibrium. Consider the following examples. A ball suspended freely from a string is in stable equilibrium, for if it is displaced to one side, it will return to its original position (Fig. 12–11a) due to the net force and torque exerted on it. On the other hand, a pencil standing on its point is in unstable equilibrium. If its center of gravity is directly over its tip (Fig. 12–11b), the net force and net torque on it will be zero. But if it is displaced ever so slightly as shown—say, by a slight vibration or tiny air current—there will be a torque on it, and this torque acts to make the pencil continue to fall in the direction of the original displacement. Finally, an example of an object in neutral equilibrium is a sphere resting on a horizontal tabletop. If it is placed slightly to one side, it will remain in its new position—no net torque acts on it.

Net force (a)

(b)

SECTION 12–3

317

(a)

(b)

(c)

FIGURE 12–12 Equilibrium of a refrigerator resting on a flat floor.

FIGURE 12–13 Humans adjust their posture to achieve stability when carrying loads.

Total CG

In most situations, such as in the design of structures and in working with the human body, we are interested in maintaining stable equilibrium, or balance, as we sometimes say. In general, an object whose center of gravity (CG) is below its point of support, such as a ball on a string, will be in stable equilibrium. If the CG is above the base of support, we have a more complicated situation. Consider a standing refrigerator (Fig. 12–12a). If it is tipped slightly, it will return to its original position due to the torque on it as shown in Fig. 12–12b. But if it is tipped too far, Fig. 12–12c, it will fall over. The critical point is reached when the CG shifts from one side of the pivot point to the other. When the CG is on one side, the torque pulls the object back onto its original base of support, Fig. 12–12b. If the object is tipped further, the CG goes past the pivot point and the torque causes the object to topple, Fig. 12–12c. In general, an object whose center of gravity is above its base of support will be stable if a vertical line projected downward from the CG falls within the base of support. This is because the normal force upward on the object (which balances out gravity) can be exerted only within the area of contact, so if the force of gravity acts beyond this area, a net torque will act to topple the object. Stability, then, can be relative. A brick lying on its widest face is more stable than a brick standing on its end, for it will take more of an effort to tip it over. In the extreme case of the pencil in Fig. 12–11b, the base is practically a point and the slightest disturbance will topple it. In general, the larger the base and the lower the CG, the more stable the object. In this sense, humans are less stable than four-legged mammals, which have a larger base of support because of their four legs, and most also have a lower center of gravity. When walking and performing other kinds of movement, a person continually shifts the body so that its CG is over the feet, although in the normal adult this requires no conscious thought. Even as simple a movement as bending over requires moving the hips backward so that the CG remains over the feet, and you do this repositioning without thinking about it. To see this, position yourself with your heels and back to a wall and try to touch your toes. You won't be able to do it without falling. People carrying heavy loads automatically adjust their posture so that the CG of the total mass is over their feet, Fig. 12–13.

12–4

Elasticity; Stress and Strain

In the first part of this Chapter we studied how to calculate the forces on objects in equilibrium. In this Section we study the effects of these forces: any object changes shape under the action of applied forces. If the forces are great enough, the object will break, or fracture, as we will discuss in Section 12–5.

Elasticity and Hooke's Law FIGURE 12–14 Hooke's law: ¢l r applied force.

∆l mg B

318

CHAPTER 12

If a force is exerted on an object, such as the vertically suspended metal rod shown in Fig. 12–14, the length of the object changes. If the amount of elongation, ¢l, is small compared to the length of the object, experiment shows that ¢l is proportional to the force exerted on the object. This proportionality, as we saw in Section 7–3, can be written as an equation: (12–3) F = k ¢l. Here F represents the force pulling on the object, ¢l is the change in length, and k is a proportionality constant. Equation 12–3, which is sometimes called Hooke's law† after Robert Hooke (1635–1703), who first noted it, is found to be valid for almost any solid material from iron to bone—but it is valid only up to a point. For if the force is too great, the object stretches excessively and eventually breaks. Figure 12–15 shows a typical graph of applied force versus elongation. Up to a point called the proportional limit, Eq. 12–3 is a good approximation for many †

The term "law" applied to this relation is not really appropriate, since first of all, it is only an approximation, and second, it refers only to a limited set of phenomena. Most physicists prefer to reserve the word "law" for those relations that are deeper and more encompassing and precise, such as Newton's laws of motion or the law of conservation of energy.

Static Equilibrium; Elasticity and Fracture

Young's Modulus The amount of elongation of an object, such as the rod shown in Fig. 12–14, depends not only on the force applied to it, but also on the material of which it is made and on its dimensions. That is, the constant k in Eq. 12–3 can be written in terms of these factors. If we compare rods made of the same material but of different lengths and cross-sectional areas, it is found that for the same applied force, the amount of stretch (again assumed small compared to the total length) is proportional to the original length and inversely proportional to the cross-sectional area. That is, the longer the object, the more it elongates for a given force; and the thicker it is, the less it elongates. These findings can be combined with Eq. 12–3 to yield 1 F ¢l = l , (12–4) EA 0 where l0 is the original length of the object, A is the cross-sectional area, and ¢l is the change in length due to the applied force F. E is a constant of proportionality† known as the elastic modulus, or Young's modulus; its value depends only on the material. The value of Young's modulus for various materials is given in Table 12–1 (the shear modulus and bulk modulus in this Table are discussed later in this Section). Because E is a property only of the material and is independent of the object's size or shape, Eq. 12–4 is far more useful for practical calculation than Eq. 12–3.

Elastic limit

Breaking point

Ela

stic r

egi o

Ultimate strength Proportional limit n e gio Plastic r

Force, F

common materials, and the curve is a straight line. Beyond this point, the graph deviates from a straight line, and no simple relationship exists between F and ¢l. Nonetheless, up to a point farther along the curve called the elastic limit, the object will return to its original length if the applied force is removed. The region from the origin to the elastic limit is called the elastic region. If the object is stretched beyond the elastic limit, it enters the plastic region: it does not return to the original length upon removal of the external force, but remains permanently deformed (such as a bent paper clip). The maximum elongation is reached at the breaking point. The maximum force that can be applied without breaking is called the ultimate strength of the material (actually, force per unit area, as we discuss in Section 12–5).

Elongation, ∆ l FIGURE 12–15 Applied force vs. elongation for a typical metal under tension.

† The fact that E is in the denominator, so 1E is the actual proportionality constant, is merely a convention. When we rewrite Eq. 12–4 to get Eq. 12–5, E is found in the numerator.

TABLE 12–1 Elastic Moduli Material Solids Iron, cast Steel Brass Aluminum Concrete Brick Marble Granite Wood (pine) (parallel to grain) (perpendicular to grain) Nylon Bone (limb) Liquids Water Alcohol (ethyl) Mercury Gases† Air, H 2 , He, CO2 †

Young's Modulus, E (N m2) 100 200 100 70 20 14 50 45 10 1 5 15

* * * * * * * * * * * *

109 109 109 109 109 109 109 109 109 109 109 109

Shear Modulus, G (N m2) 40 80 35 25

* * * *

109 109 109 109

Bulk Modulus, B (N m2) 90 140 80 70

* * * *

109 109 109 109

70 * 109 45 * 109

80 * 109

At normal atmospheric pressure; no variation in temperature during process.

2.0 * 109 1.0 * 109 2.5 * 109 1.01 * 105

SECTION 12–4

319

EXAMPLE 12–7 Tension in piano wire. A 1.60-m-long steel piano wire has a diameter of 0.20 cm. How great is the tension in the wire if it stretches 0.25 cm when tightened? APPROACH We assume Hooke's law holds, and use it in the form of Eq. 12–4, finding E for steel in Table 12–1. SOLUTION We solve for F in Eq. 12–4 and note that the area of the wire is A = pr2 = (3.14)(0.0010 m)2 = 3.14 * 10 –6 m2. Then F = E

¢l 0.0025 m A = A2.0 * 1011 Nm2 B a b A3.14 * 10 –6 m2 B = 980 N. l0 1.60 m

NOTE The large tension in all the wires in a piano must be supported by a strong frame. EXERCISE G Two steel wires have the same length and are under the same tension. But wire A has twice the diameter of wire B. Which of the following is true? (a) Wire B stretches twice as much as wire A. (b) Wire B stretches four times as much as wire A. (c) Wire A stretches twice as much as wire B. (d) Wire A stretches four times as much as wire B. (e) Both wires stretch the same amount.

Stress and Strain From Eq. 12–4, we see that the change in length of an object is directly proportional to the product of the object's length l0 and the force per unit area FA applied to it. It is general practice to define the force per unit area as the stress: stress =

force F, = area A

which has SI units of Nm2. Also, the strain is defined to be the ratio of the change in length to the original length: strain =

change in length ¢l , = original length l0

and is dimensionless (no units). Strain is thus the fractional change in length of the object, and is a measure of how much the rod has been deformed. Stress is applied to the material by external agents, whereas strain is the material's response to the stress. Equation 12–4 can be rewritten as F ¢l = E A l0

(12–5)

or E =

FIGURE 12–16 Stress exists within the material. B

F B

F (a)

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CHAPTER 12

F (b)

FA stress . = ¢ll0 strain

Thus we see that the strain is directly proportional to the stress, in the linear (elastic) region of Fig. 12–15.

Tension, Compression, and Shear Stress The rod shown in Fig. 12–16a is said to be under tension or tensile stress. Not only is there a force pulling down on the rod at its lower end, but since the rod is in equilibrium, we know that the support at the top is exerting an equal† upward force on the rod at its upper end, Fig. 12–16a. In fact, this tensile stress exists throughout the material. Consider, for example, the lower half of a suspended rod as shown in Fig. 12–16b. This lower half is in equilibrium, so there must be an upward force on it to balance the downward force at its lower end. What exerts this upward force? It must be the upper part of the rod. Thus we see that external forces applied to an object give rise to internal forces, or stress, within the material itself. †

Or a greater force if the weight of the rod cannot be ignored compared to F.

Static Equilibrium; Elasticity and Fracture

Strain or deformation due to tensile stress is but one type of stress to which materials can be subjected. There are two other common types of stress: compressive and shear. Compressive stress is the exact opposite of tensile stress. Instead of being stretched, the material is compressed: the forces act inwardly on the object. Columns that support a weight, such as the columns of a Greek temple (Fig. 12–17), are subjected to compressive stress. Equations 12–4 and 12–5 apply equally well to compression and tension, and the values for the modulus E are usually the same. B

F A

∆l

∆l B

FIGURE 12–17 This Greek temple, in Agrigento, Sicily, built 2500 years ago, shows the post-and-beam construction. The columns are under compression.

∆l

F Tension (a)

F Compression (b)

F Shear (c)

FIGURE 12–18 The three types of stress for rigid objects.

Figure 12–18 compares tensile and compressive stresses as well as the third type, shear stress. An object under shear stress has equal and opposite forces applied across its opposite faces. A simple example is a book or brick firmly attached to a tabletop, on which a force is exerted parallel to the top surface. The table exerts an equal and opposite force along the bottom surface. Although the dimensions of the object do not change significantly, the shape of the object does change, Fig. 12–18c. An equation similar to Eq. 12–4 can be applied to calculate shear strain: 1 F ¢l = l , (12–6) GA 0 but ¢l, l0 , and A must be reinterpreted as indicated in Fig. 12–18c. Note that A is the area of the surface parallel to the applied force (and not perpendicular as for tension and compression), and ¢l is perpendicular to l0 . The constant of proportionality G is called the shear modulus and is generally one-half to one-third the value of Young's modulus E (see Table 12–1). Figure 12–19 suggests why ¢l r l0 : the fatter book shifts more for the same shearing force.

FIGURE 12–19 The fatter book (a) shifts more than the thinner book (b) with the same applied shear force.

Volume Change—Bulk Modulus If an object is subjected to inward forces from all sides, its volume will decrease. A common situation is an object submerged in a fluid; in this case, the fluid exerts a pressure on the object in all directions, as we shall see in Chapter 13. Pressure is defined as force per unit area, and thus is the equivalent of stress. For this situation the change in volume, ¢V, is proportional to the original volume, V0 , and to the change in the pressure, ¢P. We thus obtain a relation of the same form as Eq. 12–4 but with a proportionality constant called the bulk modulus B: ¢V 1 = – ¢P (12–7) V0 B or ¢P . B = – ¢VV0 The minus sign means the volume decreases with an increase in pressure. Values for the bulk modulus are given in Table 12–1. Since liquids and gases do not have a fixed shape, only the bulk modulus (not the Young's or shear moduli) applies to them.

(a)

(b)

SECTION 12–4

321

Tension

12–5 Shear

Compression FIGURE 12–20 Fracture as a result of the three types of stress.

Fracture

If the stress on a solid object is too great, the object fractures, or breaks (Fig. 12–20). Table 12–2 lists the ultimate strengths for tension, compression, and shear for a variety of materials. These values give the maximum force per unit area, or stress, that an object can withstand under each of these three types of stress for various types of material. They are, however, representative values only, and the actual value for a given specimen can differ considerably. It is therefore necessary to maintain a safety factor of from 3 to perhaps 10 or more—that is, the actual stresses on a structure should not exceed one-tenth to one-third of the values given in the Table. You may encounter tables of "allowable stresses" in which appropriate safety factors have already been included. TABLE 12–2 Ultimate Strengths of Materials (force/area) Tensile Strength (N m2)

Material Iron, cast Steel Brass Aluminum Concrete Brick Marble Granite Wood (pine) (parallel to grain) (perpendicular to grain) Nylon Bone (limb)

170 500 250 200 2

* * * * *

Compressive Strength (N m2)

106 106 106 106 106

40 * 106 500 * 106 130 * 106

550 500 250 200 20 35 80 170 35 10

* * * * * * * * * *

Shear Strength (N m2)

106 106 106 106 106 106 106 106 106 106

170 250 200 200 2

* * * * *

106 106 106 106 106

5 * 106

170 * 106

EXAMPLE 12–8 ESTIMATE Breaking the piano wire. The steel piano wire we discussed in Example 12–7 was 1.60 m long with a diameter of 0.20 cm. Approximately what tension force would break it? APPROACH We set the tensile stress FA equal to the tensile strength of steel given in Table 12–2. SOLUTION The area of the wire is A = pr2, where r = 0.10 cm = 1.0 * 10–3 m. Table 12–2 tells us F = 500 * 106 Nm2, A so the wire would likely break if the force exceeded F = A500 * 106 Nm2 B(p)A1.0 * 10–3 mB

= 1600 N.

As can be seen in Table 12–2, concrete (like stone and brick) is reasonably strong under compression but extremely weak under tension. Thus concrete can be used as vertical columns placed under compression, but is of little value as a beam because it cannot withstand the tensile forces that result from the inevitable sagging of the lower edge of a beam (see Fig. 12–21). FIGURE 12–21 A beam sags, at least a little (but is exaggerated here), even under its own weight.The beam thus changes shape: the upper edge is compressed, and the lower edge is under tension (elongated). Shearing stress also occurs within the beam.

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Static Equilibrium; Elasticity and Fracture

C o m p r e s sio n

Tension

Reinforced concrete, in which iron rods are embedded in the concrete (Fig. 12–22), is much stronger. Stronger still is prestressed concrete, which also contains iron rods or a wire mesh, but during the pouring of the concrete, the rods or wire are held under tension. After the concrete dries, the tension on the iron is released, putting the concrete under compression. The amount of compressive stress is carefully predetermined so that when loads are applied to the beam, they reduce the compression on the lower edge, but never put the concrete into tension. CONCEPTUAL EXAMPLE 12–9 A tragic substitution. Two walkways, one above the other, are suspended from vertical rods attached to the ceiling of a high hotel lobby, Fig. 12–23a. The original design called for single rods 14 m long, but when such long rods proved to be unwieldy to install, it was decided to replace each long rod with two shorter ones as shown schematically in Fig. 12–23b. Determine the net force exerted by the rods on the supporting pin A (assumed to be the same size) for each design. Assume each vertical rod supports a mass m of each bridge.

FIGURE 12–22 Steel rods around which concrete is poured for strength.

RESPONSE The single long vertical rod in Fig. 12–23a exerts an upward force P H Y S I C S A P P L I E D equal to mg on pin A to support the mass m of the upper bridge. Why? A tragic collapse Because the pin is in equilibrium, and the other force that balances this FIGURE 12–23 Example 12–9. is the downward force mg exerted on it by the upper bridge (Fig. 12–23c). There is thus a shear stress on the pin because the rod pulls up on one side of the pin, and the bridge pulls down on the other side. The situation when two shorter rods support the bridges (Fig. 12–23b) is shown in Fig. 12–23d, in which only the connections at the upper bridge are shown. The lower rod exerts a force mg downward on the lower of the two pins because it supports the A A lower bridge. The upper rod exerts a force 2mg on the upper pin (labelled A) because the upper rod supports both bridges. Thus we see that when the builders substituted two shorter rods for each single long one, the stress in the supporting pin A was doubled. What perhaps seemed like a simple substitution did, in fact, lead to a tragic collapse in 1981 with a loss of life of over 100 people (see Fig. 12–1). Having a feel for physics, and being able to make simple calculations based on (a) (b) physics, can have a great effect, literally, on people's lives. EXAMPLE 12–10 Shear on a beam. A uniform pine beam, 3.6 m long and 9.5 cm * 14 cm in cross section, rests on two supports near its ends, as shown in Fig. 12–24. The beam's mass is 25 kg and two vertical roof supports rest on it, each one-third of the way from the ends. What maximum load force FL can each of the roof supports exert without shearing the pine beam at its supports? Use a safety factor of 5.0. APPROACH The symmetry present simplifies our calculation. We first find the shear strength of pine in Table 12–2 and use the safety factor of 5.0 to get F from FA 15 (shear strength). Then we use ©t = 0 to find FL . SOLUTION Each support exerts an upward force F (there is symmetry) that can be at most (see Table 12–2) 1 1 AA5 * 106 Nm2 B = (0.095 m)(0.14 m)A5 * 106 Nm2 B = 13,000 N. 5 5 To determine the maximum load force FL , we calculate the torque about the left end of the beam (counterclockwise positive): F =

©t = –FL(1.2 m) - (25 kg)A9.8 ms2 B(1.8 m) - FL(2.4 m) + F(3.6 m) = 0 so each of the two roof supports can exert (13,000 N)(3.6 m) - (250 N)(1.8 m) FL = = 13,000 N. (1.2 + 2.4)

2mg

mg (c) Force on pin A exerted by vertical rod

(d) Forces on pins at A exerted by vertical rods

FIGURE 12–24 Example 12–10. B

1.2 m

mgB B

The total mass of roof the beam can support is (2)(13,000 N)A9.8 ms2 B = 2600 kg. SECTION 12–5

Fracture

323

Joints

FIGURE 12–25 A truss bridge.

* 12–6

FIGURE 12–26 A roof truss.

Trusses and Bridges

A beam used to span a wide space, as for a bridge, is subject to strong stresses of all three types as we saw in Fig. 12–21: compression, tension and shear. A basic engineering device to support large spans is the truss, an example of which is shown in Fig. 12–25. Wooden truss bridges were first designed by the great architect Andrea Palladio (1518–1580), famous for his design of public buildings and villas. With the introduction of steel in the nineteenth century, much stronger steel trusses came into use, although wood trusses are still used to support the roofs of houses and mountain lodges (Fig. 12–26). Basically, a truss is a framework of rods or struts joined together at their ends by pins or rivets, always arranged as triangles. (Triangles are relatively stable, as compared to a rectangle, which easily becomes a parallelogram under sideways forces and then collapses.) The place where the struts are joined by a pin is called a joint. It is commonly assumed that the struts of a truss are under pure compression or pure tension—that is, the forces act along the length of each strut, Fig. 12–27a. This is an ideal, valid only if a strut has no mass and supports no weight along its length, in which case a strut has only two forces on it, at the ends, as shown in Fig. 12–27a. If the strut is in equilibrium, these two forces must be equal B and opposite in direction A© F = 0B. But couldn't they be at an angle, as in Fig. 12–27b? No, because then ©T would not be zero. The two forces must act along the strut if the strut is in equilibrium. But in a real case of a strut B with mass, there are three forces on the strut, as shown in Fig. 12–27c, and F1 B and F2 do not act along the strut; the vector diagram in Fig. 12–27d shows B B B B B © F = F1 + F2 + mg = 0. Can you see why F1 and F2 both point above the strut? (Do ©t about each end.) B

F FIGURE 12–27 (a) Each massless strut (or rod) of a truss is assumed to be under tension or compression. (b) The two equal and opposite forces must be along the same line or a net torque would exist. (c) Real struts have mass, so the B B forces F1 and F2 at the joints do not act precisely along the strut. (d) Vector diagram of part (c).

−F B

−F

Tension Compression

F B

mgB

−F (a)

(b)

(c)

F1 (d) B

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Consider again the simple beam in Example 12–5, Fig. 12–9. The force FH at the pin is not along the beam, but acts at an upward angle. If that beam were massB less, we see from Eq. (iii) in Example 12–5 with m = 0, that FHy = 0, and FH would be along the beam. The assumption that the forces in each strut of a truss are purely along the strut is still very useful whenever the loads act only at the joints and are much greater than the weight of the struts themselves.

EXAMPLE 12–11 A truss bridge. Determine the tension or compression in each of the struts of the truss bridge shown in Fig. 12–28a. The bridge is 64 m long and supports a uniform level concrete roadway whose total mass is 1.40 * 106 kg. Use the method of joints, which involves (1) drawing a free-body diagram of the truss as a whole, and (2) drawing a free-body diagram for each of the pins (joints), one by B one, and setting © F = 0 for each pin. Ignore the mass of the struts. Assume all triangles are equilateral.

P H Y S I C S

P R O B L E M

60° 60° l = 64 m (a)

1 2 Mg.

B B

FAB =

F1 = sin 60°

1 2

1 =

F1 60°

M gB

(b)

F1 A

Mg,

120°

FAC

FAB (c) Pin A (different guesses) B

FBA

Mg.

223 Next we look at pin B, and Fig. 12–28d is the free-body diagram. [Convince yourself B B B that if FBD or FBC were in the opposite direction, © F could not be zero; note that B B FBA = –FAB (and FBA = FAB ) because now we are at the opposite end of strut AB.] We see that BC is under tension and BD compression. (Recall that the forces B on the struts are opposite to the forces shown which are on the pin.) We set © F = 0:

FBC

FCD B

FAC

Mg,

FCE

M gB (e) Pin C

and FBD = FAB cos 60° + FBC cos 60° =

FBC

(d) Pin B

Then, because FBA = FAB , we have 1

FBD

©Fx = FBA cos 60° + FBC cos 60° - FBD = 0 ©Fy = FBA sin 60° - FBC sin 60° = 0.

FBC = FAB =

which equals A7.0 * 105 kgBA9.8 ms2 B23 = 4.0 * 106 N; and FAC = FAB cos 60° =

©Fx = FAC - FAB cos 60° = 0 ©Fy = F1 - FAB sin 60° = 0. 1 2 Mg

D 60°

SOLUTION We look at pin A and apply © F = 0 to it. We label the forces on B pin A due to each strut with two subscripts: FAB means the force exerted by the B B B strut AB and FAC is the force exerted by strut AC. FAB and FAC act along their respective struts; but not knowing whether each is compressive or tensile, we could draw four different free-body diagrams, as shown in Fig. 12–28c. Only the one on the B B B left could provide © F = 0, so we immediately know the directions of FAB and FAC .† These forces act on the pin. The force that pin A exerts on strut AB is opposite in B direction to FAB (Newton's third law), so strut AB is under compression and strut B B AC is under tension. Now let's calculate the magnitudes of FAB and FAC . At pin A:

Thus

S O LV I N G

Method of joints

APPROACH Any bridge has two trusses, one on each side of the roadway. Consider only one truss, Fig. 12–28a, and it will support half the weight of the roadway. That is, our truss supports a total mass M = 7.0 * 105 kg. First we draw a free-body diagram for the entire truss as a single unit, which we assume rests on supports at B B either end that exert upward forces F1 and F2 , Fig. 12–28b. We assume the mass of the roadway acts entirely at the center, on pin C, as shown. From symmetry we can see that each of the end supports carries half the weight [or do a torque equation about, say, point A: AF2 B(l) - Mg(l2) = 0], so F1 = F2 =

A P P L I E D

A truss bridge

1 23

MgA 12 B +

1 23

MgA 12 B =

1 23

Mg.

The solution is complete. By symmetry, FDE = FAB , FCE = FAC , and FCD = FBC . NOTE As a check, calculate ©Fx and ©Fy for pin C and see if they equal zero. Figure 12–28e shows the free-body diagram.

FIGURE 12–28 Example 12–11. (a) A truss bridge. Free-body diagrams: (b) for the entire truss, (c) for pin A (different guesses), (d) for pin B and (e) for pin C.

† If we were to choose the direction of a force on a diagram opposite to what it really is, we would get a minus sign.

*SECTION 12–6

Trusses and Bridges

325

F2 C

mgB

(a) B

FAy

FCy

FAx A

FCx

mgB (truck) (b) FIGURE 12–29 (a) Truss with truck of mass m at center of strut AC. (b) Forces on strut AC. P H Y S I C S

Example 12–11 put the roadway load at the center, C. Now consider a heavy load, such as a heavy truck, supported by strut AC at its middle, as shown in Fig. 12–29a. The strut AC sags under this load, telling us there is shear stress in strut AC. Figure 12–29b shows the forces exerted on strut AC: the weight of B B the truck mg, and the forces FA and FC that pins A and C exert on the strut. B [Note that F1 does not appear because it is a force (exerted by external supports) that acts on pin A, not on strut AC.] The forces that pins A and C exert on strut AC will act not only along the strut, but will have vertical components too, perpendicular to the strut to balance the weight of the truck, mg, creating shear stress. The other struts, not bearing weight, remain under pure tension or compression. Problems 53 and 54 deal with this situation, and an early step in B B their solution is to calculate the forces FA and FC by using torque equations for the strut. For very large bridges, truss structures are too heavy. One solution is to build suspension bridges, with the load being carried by relatively light suspension cables under tension, supporting the roadway by means of closely spaced vertical wires, as shown in Fig. 12–30, and in the photo on the first page of this Chapter.

A P P L I E D

Suspension bridge

EXAMPLE 12–12 Suspension bridge. Determine the shape of the cable between the two towers of a suspension bridge (as in Fig. 12–30), assuming the weight of the roadway is supported uniformly along its length. Ignore the weight of the cable. APPROACH We take x = 0, y = 0 at the center of the span, as shown in Fig. 12–31. B Let FT0 be the tension in the cable at x = 0; it acts horizontally as shown. Let FT be the tension in the cable at some other place where the horizontal coordinate is x, as shown. This section of cable supports a portion of the roadway whose weight w is proportional to the distance x, since the roadway is assumed uniform; that is, w = lx where l is the weight per unit length. B SOLUTION We set © F = 0: ©Fx = FT cos u - FT0 = 0 ©Fy = FT sin u - w = 0.

FIGURE 12–30 Suspension bridges (Brooklyn and Manhattan bridges, NY). FIGURE 12–31 Example 12–12.

We divide these two equations, tan u =

The slope of our curve (the cable) at any point is dy = tan u dx

FT θ B

FT0

w lx . = FT0 FT0

or dy l = x. dx FT0 We integrate this:

l FT0

x dx

y = Ax2 + B where we set A = lFT0 and B is a constant of integration. This is just the equation of a parabola. NOTE Real bridges have cables that do have mass, so the cables hang only approximately as a parabola, although often it is quite close.

326

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Static Equilibrium; Elasticity and Fracture

* 12–7

Arches and Domes

There are various ways that engineers and architects can span a space, such as beams, trusses, and suspension bridges. In this Section we discuss arches and domes.

FIGURE 12–32 Round arches in the Roman Forum. The one in the background is the Arch of Titus.

FIGURE 12–33 An arch is used here to good effect in spanning a chasm on the California coast.

The semicircular arch (Figs. 12–32 and 12–33) was introduced by the ancient Romans 2000 years ago. Aside from its aesthetic appeal, it was a tremendous technological innovation. The advantage of the "true" or semicircular arch is that, if well designed, its wedge-shaped stones experience stress which is mainly compressive even when supporting a large load such as the wall and roof of a cathedral. Because the stones are forced to squeeze against one another, they are mainly under compression (see Fig. 12–34). Note, however, that the arch transfers horizontal as well as vertical forces to the supports. A round arch consisting of many well-shaped stones could span a very wide space. However, considerable buttressing on the sides is needed to support the horizontal components of the forces.

P H Y S I C S

A P P L I E D

Architecture: Beams, arches and domes

FIGURE 12–34 Stones in a round arch (see Fig. 12–32) are mainly under compression.

The pointed arch came into use about A.D. 1100 and became the hallmark of the great Gothic cathedrals. It too was an important technical innovation, and was first used to support heavy loads such as the tower and arch of a cathedral. Apparently the builders realized that, because of the steepness of the pointed arch, the forces due to the weight above could be brought down more nearly vertically, so less horizontal buttressing would be needed. The pointed arch reduced the load on the walls, so there could be more openness and light. The smaller buttressing needed was provided on the outside by graceful flying buttresses (Fig. 12–35).

FIGURE 12–35 Flying buttresses (on the cathedral of Notre Dame, in Paris).

*SECTION 12–7

Arches and Domes

327

2.0 m 6.0 × 10 4 N

6.0 × 10 4 N 4.0 m

FV = 6.0 × 10 4 N

8.0 m (a) 2.0 m 6.0 ×

10 4

6.0 × 10 4 N

8.0 m (b) FIGURE 12–36 (a) Forces in a round arch, compared (b) with those in a pointed arch.

FIGURE 12–37 Interior of the Pantheon in Rome, built almost 2000 years ago. This view, showing the great dome and its central opening for light, was painted about 1740 by Panini. Photographs do not capture its grandeur as well as this painting does.

328

CHAPTER 12

(4.0 m)A6.0 * 104 NB - (2.0 m)A6.0 * 104 NB - (4.0 m)AFH B = 0.

Thus FH = 3.0 * 104 N for the round arch. For the pointed arch, the torque equation is (see Fig. 12–36b) (4.0 m)A6.0 * 104 NB - (2.0 m)A6.0 * 104 NB - (8.0 m)AFH B = 0.

8.0 m

FV = 6.0 × 10 4 N

To make an accurate analysis of a stone arch is quite difficult in practice. But if we make some simplifying assumptions, we can show why the horizontal component of the force at the base is less for a pointed arch than for a round one. Figure 12–36 shows a round arch and a pointed arch, each with an 8.0-m span. The height of the round arch is thus 4.0 m, whereas that of the pointed arch is larger and has been chosen to be 8.0 m. Each arch supports a weight of 12.0 * 104 N ( = 12,000 kg * g) which, for simplicity, we have divided into two parts (each 6.0 * 104 N) acting on the two halves of each arch as shown. To be in equilibrium, each of the supports must exert an upward force of 6.0 * 104 N. For rotational equilibrium, each support also exerts a horizontal force, FH , at the base of the arch, and it is this we want to calculate. We focus only on the right half of each arch. We set equal to zero the total torque calculated about the apex of the arch due to the forces exerted on that half arch. For the round arch, the torque equation (©t = 0) is (see Fig. 12–36a)

Solving, we find that FH = 1.5 * 104 N —only half as much as for the round arch! From this calculation we can see that the horizontal buttressing force required for a pointed arch is less because the arch is higher, and there is therefore a longer lever arm for this force. Indeed, the steeper the arch, the less the horizontal component of the force needs to be, and hence the more nearly vertical is the force exerted at the base of the arch. Whereas an arch spans a two-dimensional space, a dome—which is basically an arch rotated about a vertical axis—spans a three-dimensional space. The Romans built the first large domes. Their shape was hemispherical and some still stand, such as that of the Pantheon in Rome (Fig. 12–37), built 2000 years ago.

FIGURE 12–38 The skyline of Florence, showing Brunelleschi's dome on the cathedral.

Fourteen centuries later, a new cathedral was being built in Florence. It was to have a dome 43 m in diameter to rival that of the Pantheon, whose construction has remained a mystery. The new dome was to rest on a "drum" with no external abutments. Filippo Brunelleschi (1377–1446) designed a pointed dome (Fig. 12–38), since a pointed dome, like a pointed arch, exerts a smaller side thrust against its base. A dome, like an arch, is not stable until all the stones are in place. To support smaller domes during construction, wooden frameworks were used. But no trees big enough or strong enough could be found to span the 43-m space required. Brunelleschi decided to try to build the dome in horizontal layers, each bonded to the previous one, holding it in place until the last stone of the circle was placed. Each closed ring was then strong enough to support the next layer. It was an amazing feat. Only in the twentieth century were larger domes built, the largest being that of the Superdome in New Orleans, completed in 1975. Its 200-m diameter dome is made of steel trusses and concrete.

Summary An object at rest is said to be in equilibrium. The subject concerned with the determination of the forces within a structure at rest is called statics. The two necessary conditions for an object to be in equilibrium are that (1) the vector sum of all the forces on it must be zero, and (2) the sum of all the torques (calculated about any arbitrary axis) must also be zero. For a 2-dimensional problem we can write ©Fx = 0, ©Fy = 0, ©t = 0. (12–1, 12–2) It is important when doing statics problems to apply the equilibrium conditions to only one object at a time. An object in static equilibrium is said to be in (a) stable, (b) unstable, or (c) neutral equilibrium, depending on whether a slight displacement leads to (a) a return to the original position, (b) further movement away from the original position, or (c) rest in the new position. An object in stable equilibrium is also said to be in balance.

Hooke's law applies to many elastic solids, and states that the change in length of an object is proportional to the applied force: (12–3)

F = k ¢l.

If the force is too great, the object will exceed its elastic limit, which means it will no longer return to its original shape when the distorting force is removed. If the force is even greater, the ultimate strength of the material can be exceeded, and the object will fracture. The force per unit area acting on an object is the stress, and the resulting fractional change in length is the strain. The stress on an object is present within the object and can be of three types: compression, tension, or shear. The ratio of stress to strain is called the elastic modulus of the material. Young's modulus applies for compression and tension, and the shear modulus for shear. Bulk modulus applies to an object whose volume changes as a result of pressure on all sides. All three moduli are constants for a given material when distorted within the elastic region.

Questions 1. Describe several situations in which an object is not in equilibrium, even though the net force on it is zero. 2. A bungee jumper momentarily comes to rest at the bottom of the dive before he springs back upward. At that moment, is the bungee jumper in equilibrium? Explain. 3. You can find the center of gravity of a meter stick by resting it horizontally on your two index fingers, and then slowly drawing your fingers together. First the meter stick will slip on one finger, and then on the other, but eventually the fingers meet at the CG. Why does this work? 4. Your doctor's scale has arms on which weights slide to counter your weight, Fig. 12–39. These Weights weights are much lighter than you are. How does this work?

6. Can the sum of the torques on an object be zero while the net force on the object is nonzero? Explain. 7. A ladder, leaning against a wall, makes a 60° angle with the ground. When is it more likely to slip: when a person stands on the ladder near the top or near the bottom? Explain. 8. A uniform meter stick supported at the 25-cm mark is in equilibrium when a 1-kg rock is suspended at the 0-cm end (as shown in Fig. 12–41). Is the mass of the meter stick greater than, equal to, or less than the mass of the rock? Explain your reasoning.

FIGURE 12–41 Question 8. 9. Why do you tend to lean backward when carrying a heavy load in your arms? 10. Figure 12–42 shows a cone. Explain how to lay it on a flat table so that it is in (a) stable equilibrium, (b) unstable equilibrium, (c) neutral equilibrium.

FIGURE 12–39 Question 4. 5. A ground retaining wall is shown in Fig. 12–40a. The ground, particularly when wet, can exert a significant force F on the wall. (a) What force produces the torque to keep the wall upright? (b) Explain why the retaining wall in Fig. 12–40b would be much less likely to overturn than that in Fig. 12–40a.

FIGURE 12–42 Question 10. B

(a)

(b)

FIGURE 12–40 Question 5.

11. Place yourself facing the edge of an open door. Position your feet astride the door with your nose and abdomen touching the door's edge. Try to rise on your tiptoes. Why can't this be done? 12. Why is it not possible to sit upright in a chair and rise to your feet without first leaning forward?

Questions

329

13. Why is it more difficult to do sit-ups when your knees are bent than when your legs are stretched out?

15. Name the type of equilibrium for each position of the ball in Fig. 12–44. A

14. Which of the configurations of brick, (a) or (b) of Fig. 12–43, is the more likely to be stable? Why?

C B

1 2

1 4

FIGURE 12–44 Question 15.

1 2

16. Is the Young's modulus for a bungee cord smaller or larger than that for an ordinary rope? 17. Examine how a pair of scissors or shears cuts through a piece of cardboard. Is the name "shears" justified? Explain. 18. Materials such as ordinary concrete and stone are very weak under tension or shear. Would it be wise to use such a material for either of the supports of the cantilever shown in Fig. 12–7? If so, which one(s)? Explain.

(b)

(a)

FIGURE 12–43 Question 14. The dots indicate the CG of each brick. The fractions 14 and 12 indicate what portion of each brick is hanging beyond its support.

Problems 12–1 and 12–2 Equilibrium 1. (I) Three forces are applied to a tree sapling, as shown in B B Fig. 12–45, to stabilize it. If FA = 385 N and FB = 475 N, B find FC in magnitude and direction. B

FB 105° B

FA FIGURE 12–45 Problem 1.

2. (I) Approximately what magnitude force, FM , must the extensor muscle in the upper arm exert on the lower arm to hold a 7.3-kg shot put (Fig. 12–46)? Assume the lower arm has a mass of 2.3 kg and its CG is 12.0 cm from the elbow-joint pivot. 2.5 cm

30.0 cm

Elbow joint

FIGURE 12–46 Problem 2.

3. (I) Calculate the mass m needed in order to suspend the leg shown in Fig. 12–47. Assume the leg (with cast) has a mass of 15.0 kg, and its CG is 35.0 cm from the hip joint; the sling is 78.0 cm from the hip joint.

Hip joint

4. (I) A tower crane (Fig. 12–48a) must always be carefully balanced so that there is no net torque tending to tip it. A particular crane at a building site is about to lift a 2800-kg air-conditioning unit. The crane's dimensions are shown in Fig. 12–48b. (a) Where must the crane's 9500-kg counterweight be placed when the load is lifted from the ground? (Note that the counterweight is usually moved automatically (a) via sensors and motors to precisely Counterweight compensate for the M = 9500 kg load.) (b) Determine the maximum load that can be lifted 7.7 m 3.4 m with this counterweight when it is placed at its full extent. Ignore the mass of the beam. m = 2800 kg FIGURE 12–48 Problem 4.

5. (II) Calculate the forces FA and FB that the supports exert on the diving board of Fig. 12–49 when a 52-kg person stands at its tip. (a) Ignore the weight of the board. (b) Take into account the board's mass of 28 kg. Assume the board's CG is at its center.

FIGURE 12–47 Problem 3.

330

CHAPTER 12

(b)

Static Equilibrium; Elasticity and Fracture

A FIGURE 12–49 Problem 5.

1.0 m

3.0 m

6. (II) Two cords support a chandelier in the manner shown in Fig 12–3 except that the upper cord makes an angle of 45° with the ceiling. If the cords can sustain a force of 1660 N without breaking, what is the maximum chandelier weight that can be supported? 7. (II) The two trees in Fig. 12–50 are 6.6 m apart. A back-packer is trying to lift his pack out of the reach of B bears. Calculate the magnitude of the force F that he must exert downward to hold a 19-kg backpack so that the rope sags at its midpoint by B (a) 1.5 m, (b) 0.15 m. F

FIGURE 12–50 Problems 7 and 83. 8. (II) A 110-kg horizontal beam is supported at each end. A 320-kg piano rests a quarter of the way from one end. What is the vertical force on each of the supports? 9. (II) Calculate FA and FB for the uniform cantilever shown in Fig. 12–7 whose mass is 1200 kg. 10. (II) A 75-kg adult sits at one end of a 9.0-m-long board. His 25-kg child sits on the other end. (a) Where should the pivot be placed so that the board is balanced, ignoring the board's mass? (b) Find the pivot point if the board is uniform and has a mass of 15 kg. 11. (II) Find the tension in the two cords shown in Fig. 12–51. Neglect the mass of the cords, and assume that the angle u is θ 33° and the mass m is 190 kg.

14. (II) The force required to pull the cork out of the top of a wine bottle is in the range of 200 to 400 N. A common B bottle opener is shown F in Fig. 12–54. What 70 mm 9 mm range of forces F is required to open a wine bottle with this device?

FIGURE 12–54 Problem 14. 15. (II) Calculate FA and FB for the beam shown in Fig. 12–55. The downward forces represent the weights of machinery on the beam. Assume B B the beam is uniform FA 4300 N 3100 N 2200 N FB and has a mass of 280 kg. FIGURE 12–55 Problem 15.

2.0 m

4.0 m

3.0 m

1.0 m

16. (II) (a) Calculate the magnitude of the force, FM , required of the "deltoid" muscle to hold up the outstretched arm shown in Fig. 12–56. The total mass of the arm is 3.3 kg. (b) Calculate the magnitude of the force FJ exerted by the shoulder joint on the upper arm and the angle (to the horizontal) at which it acts. 15°

FIGURE 12–51 Problem 11.

12. (II) Find the tension in the two wires supporting the traffic light shown in Fig. 12–52. 53° 37° 33 kg

FIGURE 12–52 Problem 12. 13. (II) How close to the edge of the 24.0-kg table shown in Fig. 12–53 can a 2.20 m 66.0-kg person sit without tipping it over? 0.80 m FIGURE 12–53 Problem 13.

1.20 m

FJ 12 cm

mgB 24 cm

FIGURE 12–56 Problems 16 and 17. 17. (II) Suppose the hand in Problem 16 holds an 8.5-kg mass. What force, FM , is required of the deltoid muscle, assuming the mass is 52 cm from the shoulder joint? 18. (II) Three children are trying to balance on a seesaw, which includes a fulcrum rock acting as a pivot at the center, and a very light board 3.2 m long (Fig. 12–57). Two playmates are already on either end. Boy A has a mass of 45 kg, and boy B a mass of 35 kg. Where should girl C, whose mass is 25 kg, place herself so as to balance the seesaw? x

A m = 45 kg

m = 35 kg

m = 25 kg 0.50 m

FIGURE 12–57 Problem 18.

Problems

331

19. (II) The Achilles tendon is attached to the rear of the foot as shown in Fig. 12–58. When a person elevates himself just barely off the floor on the "ball of one foot," estimate the tension FT in the Achilles tendon (pulling upward), and the (downward) force FB exerted by B the lower leg bone FT Leg bone on the foot.Assume Achilles the person has a tendon Ball of foot mass of 72 kg and D (pivot point) is twice as long as d. B

FIGURE 12–58 Problem 19.

20. (II) A shop sign weighing 215 N is supported by a uniform 155-N beam as shown in Fig. 12–59. Find the tension in the guy wire and the horizontal and vertical forces exerted by 35.0° the hinge on the beam. 1.35 m 1.70 m

FIGURE 12–60 Problem 21. 22. (II) A uniform steel beam has a mass of 940 kg. On it is resting half of an identical beam, as shown in Fig. 12–61. What is the vertical 1– support force at 2M M each end? l FIGURE 12–61 Problem 22.

2.0

2.6 m

23. (II) Two wires run from the top of a pole 2.6 m tall that supports a volleyball net. The two wires are anchored to the ground 2.0 m apart, and each is 2.0 m from the pole (Fig. 12–62). The tension in each wire is 115 N. What is the tension in the net, assumed horizontal and attached at the 2.0 m top of the pole? 2.0 m FIGURE 12–62

332

CHAPTER 12

27. (II) A uniform rod AB of length 5.0 m and mass M = 3.8 kg is hinged at A and held in equilibrium by a light cord, as shown in Fig. 12–64. A load W = 22 N hangs 37° from the rod at a distance x so that the tension in the cord is 85 N. (a) Draw a free-body diagram for the rod. (b) Determine the vertical and horizontal B forces on the rod exerted by the 53° hinge. (c) Determine d from the appropriate torque equation. d Ro

21. (II) A traffic light hangs from a pole as shown in Fig. 12–60. The uniform aluminum pole AB is 7.20 m long and has a mass of 12.0 kg. The mass of the traffic light is 21.5 kg. Determine (a) the B tension in the D C horizontal massless 37° cable CD, and (b) the vertical and hori3.80 m zontal components of the force exerted A by the pivot A on the aluminum pole.

Problem 23.

FIGURE 12–63 Problem 26.

FIGURE 12–59 Problem 20.

24. (II) A large 62.0-kg board is propped at a 45° angle against the edge of a barn door that is 2.6 m wide. How great a horizontal force must a person behind the door exert (at the edge) in order to open it? Assume that there is negligible friction between the door and the board but that the board is firmly set against the ground. 25. (II) Repeat Problem 24 assuming the coefficient of friction between the board and the door is 0.45. 26. (II) A 0.75-kg sheet hangs from a massless clothesline as shown in Fig. 12–63. The clothesline on either side of the sheet makes an angle of 3.5° with the horizontal. Calculate the tension in the clothesline on either side of the sheet. Why is the tension so much greater than the weight of the 3.5° 3.5° sheet?

A FIGURE 12–64 W = 22 N Problem 27. 28. (III) A 56.0-kg person stands 2.0 m from the bottom of the stepladder shown in Fig. 12–65. Determine (a) the tension in the horizontal tie rod, which is halfway up the ladder, (b) the normal force the ground exerts on each side of the ladder, and (c) the force (magnitude and direction) that the left side of the ladder exerts on the right side at the hinge on the top. Ignore the mass of the ladder and assume the ground is frictionless. [Hint: 2.5 m Consider free-body diagrams for each 2.0 m section of the ladder.] FIGURE 12–65 1.8 m Problem 28. 29. (III) A door 2.30 m high and 1.30 m wide has a mass of 13.0 kg. A hinge 0.40 m from the top and another hinge 0.40 m from the bottom each support half the door's weight (Fig. 12–66). 40 cm Assume that the center of gravity is 2.30 m at the geometrical center of the door, and determine the horizontal and vertical force components 1.30 m exerted by each hinge on the door. 40 cm FIGURE 12–66 Problem 29. 30. (III) A cubic crate of side s = 2.0 m is top-heavy: its CG is 18 cm above its true center. How steep an incline can the crate rest on without tipping over? What would your answer be if the crate were to slide at constant speed down the plane without tipping over? [Hint: The normal force would act at the lowest corner.]

Static Equilibrium; Elasticity and Fracture

31. (III) A refrigerator is approximately a uniform rectangular solid 1.9 m tall, 1.0 m wide, and 0.75 m deep. If it sits upright on a truck with its 1.0-m dimension in the direction of travel, and if the refrigerator cannot slide on the truck, how rapidly can the truck accelerate without tipping the refrigerator over? [Hint: The normal force would act at one corner.] 32. (III) A uniform ladder of mass m and length l leans at an angle u against a frictionless wall, Fig. 12–67. If the coefficient of static friction between the ladder and the ground is ms , determine a formula for the minimum angle at which the ladder will not slip. l

FIGURE 12–67 Problem 32.

12–3 Stability and Balance 33. (II) The Leaning Tower of Pisa is 55 m tall and about 7.0 m in diameter. The top is 4.5 m off center. Is the tower in stable equilibrium? If so, how much farther can it lean before it becomes unstable? Assume the tower is of uniform composition.

12–4 Elasticity; Stress and Strain 34. (I) A nylon string on a tennis racket is under a tension of 275 N. If its diameter is 1.00 mm, by how much is it lengthened from its untensioned length of 30.0 cm? 35. (I) A marble column of cross-sectional area 1.4 m2 supports a mass of 25,000 kg. (a) What is the stress within the column? (b) What is the strain? 36. (I) By how much is the column in Problem 35 shortened if it is 8.6 m high? 37. (I) A sign (mass 1700 kg) hangs from the end of a vertical steel girder with a cross-sectional area of 0.012 m2. (a) What is the stress within the girder? (b) What is the strain on the girder? (c) If the girder is 9.50 m long, how much is it lengthened? (Ignore the mass of the girder itself.) 38. (II) How much pressure is needed to compress the volume of an iron block by 0.10%? Express your answer in Nm2, and compare it to atmospheric pressure A1.0 * 105 Nm2 B. 39. (II) A 15-cm-long tendon was found to stretch 3.7 mm by a force of 13.4 N. The tendon was approximately round with an average diameter of 8.5 mm. Calculate Young's modulus of this tendon. 40. (II) At depths of 2000 m in the sea, the pressure is about 200 times atmospheric pressure A1 atm = 1.0 * 105 Nm2 B. By what percentage does the interior space of an iron bathysphere's volume change at this depth? 41. (III) A pole projects horizontally from the front wall of a shop. A 6.1-kg sign hangs from the pole at a point 2.2 m from the wall (Fig. 12–68). (a) What is the torque due to this sign calculated about the point where the pole meets the wall? (b) If the pole is not to fall off, there must be another torque exerted to balance it. What exerts this torque? Use a 2.2 m diagram to show how this torque must act. (c) Discuss whether compression, tension, and/or shear play a role in part (b). FIGURE 12–68 Problem 41.

12–5 Fracture 42. (I) The femur bone in the human leg has a minimum effective cross section of about 3.0 cm2 A = 3.0 * 10–4 m2 B. How much compressive force can it withstand before breaking? 43. (II) (a) What is the maximum tension possible in a 1.00-mm-diameter nylon tennis racket string? (b) If you want tighter strings, what do you do to prevent breakage: use thinner or thicker strings? Why? What causes strings to break when they are hit by the ball? 44. (II) If a compressive force of 3.3 * 104 N is exerted on the end of a 22-cm-long bone of cross-sectional area 3.6 cm2, (a) will the bone break, and (b) if not, by how much does it shorten? 45. (II) (a) What is the minimum cross-sectional area required of a vertical steel cable from which is suspended a 270-kg chandelier? Assume a safety factor of 7.0. (b) If the cable is 7.5 m long, how much does it elongate? 46. (II) Assume the supports of the uniform cantilever shown in Fig. 12–69 (m = 2900 kg) are made of wood. Calculate the B B minimum cross-sectional area FA FB required of each, assuming a 20.0 m safety factor of 9.0. 30.0 m CG

FIGURE 12–69 Problem 46.

47. (II) An iron bolt is used to connect two iron plates together. The bolt must withstand shear forces up to about 3300 N. Calculate the minimum diameter for the bolt, based on a safety factor of 7.0. 48. (III) A steel cable is to support an elevator whose total (loaded) mass is not to exceed 3100 kg. If the maximum acceleration of the elevator is 1.2 ms2, calculate the diameter of cable required. Assume a safety factor of 8.0. * 12–6 Trusses and Bridges * 49. (II) A heavy load Mg = 66.0 kN hangs at point E of the single cantilever truss shown in Fig. 12–70. (a) Use a torque equation for the truss as a whole to determine the tension B FT in the support cable, and then determine the force FA on the truss at pin A. (b) Determine Cable the force in each member D 3.0 m B of the truss. Neglect the weight of the trusses, 3.0 m 3.0 m which is small compared 3.0 m E A to the load. FIGURE 12–70 Problem 49.

3.0 m

Mg = 66.0 kN

* 50. (II) Figure 12–71 shows a simple truss that carries a load at the center (C) of 1.35 * 104 N. B (a) Calculate the force on each strut at the pins, A, B, C, D, and (b) determine which struts (ignore their masses) are under tension and which A 60° 60° D under compression. FIGURE 12–71 C Problem 50. * 51. (II) (a) What minimum cross-sectional area must the trusses have in Example 12–11 if they are of steel (and all the same size for looks), using a safety factor of 7.0? (b) If at any time the bridge may carry as many as 60 trucks with an average mass of 1.3 * 104 kg, estimate again the area needed for the truss members.

Problems

333

* 52. (II) Consider again Example 12–11 but this time assume the roadway is supported uniformly so that 12 its mass M A= 7.0 * 105 kgB acts at the center and 14 M at each end support (think of the bridge as two spans, AC and CE, so the center pin supports two span ends). Calculate the magnitude of the force in each truss member and compare to Example 12–11. * 53. (III) The truss shown in Fig. 12–72 supports a railway bridge. Determine the compressive or tension force in each strut if a 53-ton A1 ton = 103 kgB train locomotive is stopped at the midpoint between the center and one end. Ignore the masses of the rails and truss, and use only 12 the mass of train because there are two trusses (one on each side of B D the train). Assume all triangles are equilateral. [Hint: See Fig. 12–29.] C E A FIGURE 12–72 Problem 53.

* 54. (III) Suppose in Example 12–11, a 23-ton truck Am = 23 * 103 kgB has its CM located 22 m from the left end of the bridge (point A). Determine the magnitude of the force and type of stress in each strut. [Hint: See Fig. 12–29.] * 55. (III) For the "Pratt truss" shown in Fig. 12–73, determine the force on each member and whether it is tensile or compressive. Assume the truss is loaded as shown, and give results in terms of F. The B D G vertical height is a and each of the four a lower horizontal A J spans has length a. C E H FIGURE 12–73 Problem 55.

* 12–7 Arches and Domes * 56. (II) How high must a pointed arch be if it is to span a space 8.0 m wide and exert one-third the horizontal force at its base that a round arch would?

General Problems 57. The mobile in Fig. 12–74 is in equilibrium. Object B has mass of 0.748 kg. Determine the masses of objects A, C, and D. (Neglect the weights of the crossbars.)

30.00 cm

7.50 cm

61. When a wood shelf of mass 6.6 kg is fastened inside a slot in a vertical support as shown in Fig. 12–76, the support exerts a torque on the shelf. (a) Draw a free-body diagram for the shelf, assuming three vertical forces (two exerted by the support slot—explain why). Then calculate (b) the magnitudes of the three forces and (c) the torque exerted by the support (about the left end of the shelf). FIGURE 12–76 Problem 61.

15.00 cm 5.00 cm

17.50 cm 5.00 cm

FIGURE 12–74 Problem 57. 58. A tightly stretched "high wire" is 36 m long. It sags 2.1 m when a 60.0-kg tightrope walker stands at its center. What is the tension in the wire? Is it possible to increase the tension in the wire so that there is no sag? 59. What minimum horizontal force F is needed to pull a wheel of radius R and mass M over a step of height h as shown in Fig. 12–75 (R 7 h)? (a) Assume the force is applied at the top edge as shown. B F (in a) (b) Assume the force is applied instead at the M B wheel's center. F (in b) FIGURE 12–75 Problem 59.

CHAPTER 12

3.0 cm 2.0 cm

62. A 50-story building is being planned. It is to be 180.0 m high with a base 46.0 m by 76.0 m. Its total mass will be about 1.8 * 107 kg, and its weight therefore about 1.8 * 108 N. Suppose a 200-kmh wind exerts a force of 950 Nm2 over the 76.0-m-wide face (Fig. 12–77). Calculate the torque about the potential pivot point, the rear edge of the building B (where FE acts in Fig. 12–77), and determine whether the building will topple. Assume the total force of the wind acts at the midpoint of the building's face, and that the building is not anchored in bedrock. B [Hint: FE in Fig. 12–77 represents the force that the Earth would exert on the building in the B case where the building CG FA would just begin to tip.]

60. A 28-kg round table is supported by three legs equal distances apart on the edge. What minimum mass, placed on the table's edge, will cause the table to overturn?

334

32.0 cm

Static Equilibrium; Elasticity and Fracture

FIGURE 12–77 Forces on a building subjected to B wind AFA B, gravity AmgB B, B and the force FE on the building due to the Earth if the building were just about to tip. Problem 62.

mgB B

63. The center of gravity of a loaded truck depends on how the truck is packed. If it is 4.0 m high and 2.4 m wide, and its CG is 2.2 m above the ground, how steep a slope can the truck be parked on without tipping over (Fig. 12–78)?

68. A uniform flexible steel cable of weight mg is suspended between two points at the same elevation as shown in Fig. 12–82, where u = 56°. Determine the tension in the cable (a) at its lowest point, and (b) at the points of attachment. (c) What is the direction of the tension force in each case? θ θ FIGURE 12–82 Problem 68.

2.4 m CG

4.0 m

2.2 m

FIGURE 12–78 Problem 63.

64. In Fig. 12–79, consider the right-hand (northernmost) section of the Golden Gate Bridge, which has a length d1 = 343 m. Assume the CG of this span is halfway between the tower and anchor. Determine FT1 and FT2 (which act on the northernmost cable) in terms of mg, the weight of the northernmost span, and calculate the tower height h needed for equilibrium. Assume the roadway is supported only by the suspension cables, and neglect the mass of the cables and vertical wires. [Hint: FT3 does not act on this section.] 60°

Suspension cable

F T2

66° B

F T3

h B

F T1 d2

19°

FIGURE 12–79 Problems 64 and 65. 65. Assume that a single-span suspension bridge such as the Golden Gate Bridge has the symmetrical configuration indicated in Fig. 12–79. Assume that the roadway is uniform over the length of the bridge and that each segment of the suspension cable provides the sole support for the roadway directly below it. The ends of the cable are anchored to the ground only, not to the roadway. What must the ratio of d2 to d1 be so that the suspension cable exerts no net horizontal force on the towers? Neglect the mass of the cables and the fact that the roadway isn't precisely horizontal. 66. When a mass of 25 kg is hung from the middle of a fixed straight aluminum wire, the wire sags to make an angle of 12° with the horizontal as shown in Fig. 12–80. (a) Determine the radius of the wire. (b) Would the Al wire break under these conditions? If so, what other material (see Table 12–2) might work? 12°

12°

FIGURE 12–80 25 kg Problem 66. 67. The forces acting on a 77,000-kg aircraft flying at constant velocity are shown in Fig. 12–81. The engine thrust, FT = 5.0 * 105 N, acts on a line 1.6 m below the CM. Determine the drag force FD and the distance above the CM that it B B acts. Assume FD and FT 3.2 B m B are horizontal. (FL is B FL FD the "lift" force on the CM wing.) B FT FIGURE 12–81 Problem 67. MgB

69. A 20.0-m-long uniform beam weighing 650 N rests on walls A and B, as shown in Fig. 12–83. (a) Find the maximum weight of a person who can walk to the extreme end D without tipping the beam. Find the forces that the walls A and B exert on the beam when the person is standing: (b) at D; (c) at a point 2.0 m to the right of B; (d) 2.0 m to the right of A. C

20.0 m

12.0 m

3.0 m FIGURE 12–83 Problem 69.

70. A cube of side l rests on a rough floor. It is subjected to a steady horizontal pull F, exerted a distance h above the floor as shown in Fig. 12–84. As F is increased, the block will either begin to slide, or begin to tip over. Determine the coefficient of static friction ms so that (a) the block begins to slide rather than tip; (b) the block B l F begins to tip. [Hint: Where will the normal force on the block act if it tips?] l h FIGURE 12–84 Problem 70. 71. A 65.0-kg painter is on a uniform 25-kg scaffold supported from above by ropes (Fig. 12–85). There is a 4.0-kg pail of paint to one side, as shown. Can the painter walk safely to both ends of the scaffold? If not, which end(s) is dangerous, and how close to the end can he approach 1.0 m safely? FIGURE 12–85 Problem 71.

4.0 m

1.0 m

72. A man doing push-ups pauses in the position shown in Fig. 12–86. His mass m = 68 kg. Determine the normal force exerted by the floor (a) on each hand; (b) on each foot. 28 cm mgB FIGURE 12–86 Problem 72. 42 cm 95 cm 73. A 23-kg sphere rests between two smooth planes as shown in Fig. 12–87. Determine the magnitude of the force acting on the sphere exerted by each plane. FIGURE 12–87 Problem 73.

67°

23 kg 32°

General Problems

335

74. A 15.0-kg ball is supported from the ceiling by rope A. Rope B pulls downward and to the side on the ball. If the A angle of A to the vertical is 22° and if B makes an angle of 53° to the vertical (Fig. 12–88), 22° find the tensions in ropes A and B. 53°

FIGURE 12–88 Problem 74.

75. Parachutists whose chutes have failed to open have been known to survive if they land in deep snow. Assume that a 75-kg parachutist hits the ground with an area of impact of 0.30 m2 at a velocity of 55 ms, and that the ultimate strength of body tissue is 5 * 105 Nm2. Assume that the person is brought to rest in 1.0 m of snow. Show that the person may escape serious injury. 76. A steel wire 2.3 mm in diameter stretches by 0.030% when a mass is suspended from it. How large is the mass? 77. A 2500-kg trailer is attached to a stationary truck at point B, Fig. 12–89. Determine the normal force exerted by the road on the rear tires at A, and the vertical force exerted on the trailer by the support B.

81. There is a maximum height of a uniform vertical column made of any material that can support itself without buckling, and it is independent of the cross-sectional area (why?). Calculate this height for (a) steel (density 7.8 * 103 kgm3), and (b) granite (density 2.7 * 103 kgm3). 82. A 95,000-kg train locomotive starts across a 280-m-long bridge at time t = 0. The bridge is a uniform beam of mass 23,000 kg and the train travels at a constant 80.0 kmh. What are the magnitudes of the vertical forces, FA(t) and FB(t), on the two end supports, written as a function of time during the train's passage? 83. A 23.0-kg backpack is suspended midway between two trees by a light cord as in Fig. 12–50. A bear grabs the backpack and pulls vertically downward with a constant force, so that each section of cord makes an angle of 27° below the horizontal. Initially, without the bear pulling, the angle was 15°; the tension in the cord with the bear pulling is double what it was when he was not. Calculate the force the bear is exerting on the backpack. 84. A uniform beam of mass M and length l is mounted on a hinge at a wall as shown in Fig. 12–91. It is held in a horizontal position by a wire making an angle u as shown. A mass m is placed on the beam a distance x from the wall, and this distance can be varied. Determine, as a function of x, (a) the tension in the wire and (b) the components of the force exerted by the beam on the hinge. m

Mg B

2.5 m

78. The roof over a 9.0-m * 10.0-m room in a school has a total mass of 13,600 kg. The roof is to be supported by vertical wooden "2 * 4s" (actually about 4.0 cm * 9.0 cm) equally spaced along the 10.0-m sides. How many supports are required on each side, and how far apart must they be? Consider only compression, and assume a safety factor of 12. 79. A 25-kg object is being lifted by pulling on the ends of a 1.15-mm-diameter nylon cord that goes over two 3.00-m-high poles that are 4.0 m apart, as shown in Fig. 12–90. How high above the floor will the object be when the cord breaks?

25 kg

FIGURE 12–90 Problem 79. 80. A uniform 6.0-m-long ladder of mass 16.0 kg leans against a B smooth wall (so the force exerted by the wall, FW , is perpendicular to the wall). The ladder makes an angle of 20.0° with the vertical wall, and the ground is rough. Determine the coefficient of static friction at the base of the ladder if the ladder is not to slip when a 76.0-kg person stands three-fourths of the way up the ladder.

336

CHAPTER 12

FIGURE 12–91 Problem 84.

5.5 m

FIGURE 12–89 Problem 77.

85. Two identical, uniform beams are symmetrically set up against each other (Fig. 12–92) on a floor with which they have a coefficient of friction ms = 0.50. What is the minimum angle the beams can make with the floor and still not fall?

FIGURE 12–92 Problem 85.

86. If 35 kg is the maximum mass m that a person can hold in a hand when the arm is positioned with a 105° angle at the elbow as shown in Fig. 12–93, what is the maximum force Fmax that the biceps muscle exerts on the forearm? Assume the forearm and hand have a total mass of 2.0 kg with a CG that is 15 cm from the elbow, and that the biceps muscle attaches 5.0 cm from the elbow. 105° B Fmax 35 kg

Static Equilibrium; Elasticity and Fracture

5.0 cm 15 cm

(2.0 kg)gB 35 cm

FIGURE 12–93 Problem 86.

87. (a) Estimate the magnitude of the force FM the muscles exert on the back to support the upper body when a person bends forward. Use the model shown in Fig. 12–94b. (b) Estimate the magnitude and B 24 direction of the force FV 12 cm cm acting on the fifth lumbar vertebra (exerted by 36 cm the spine below). 60° 12°

FM Fifth lumbar vertebra

Erector spinae muscles

30°

60°

FV wH = 0.07w (head) wA = 0.12w (arms) w T = 0.46w (trunk)

(a)

B w H

B w B A w T

91. A 2.0-m-high box with a 1.0-m-square base is moved across a rough floor as in Fig. 12–97. The uniform box weighs 250 N and has a coefficient of static friction with the floor of 0.60. What minimum force must be exerted on the box to make it slide? What is the maximum height h above the floor 1.0 m that this force can be applied B without tipping the box over? Note F that as the box tips, the normal force and the friction force will act CG 2.0 m at the lowest corner. h B FN FIGURE 12–97 Problem 91.

y x w = Total weight of person (b)

FIGURE 12–94 Problem 87. 88. One rod of the square frame shown in Fig. 12–95 contains a turnbuckle which, when turned, can put the rod under tension or compression. If the turnbuckle puts rod AB under a compressive force F, deterB mine the forces produced in the other A rods. Ignore the mass of the rods and assume the diagonal rods cross each other freely at the center without friction. [Hint: Use the symmetry of the situation.] FIGURE 12–95 C D Problem 88. 89. A steel rod of radius R = 15 cm and length l0 stands upright on a firm surface. A 65-kg man climbs atop the rod. (a) Determine the percent decrease in the rod's length. (b) When a metal is compressed, each atom throughout its bulk moves closer to its neighboring atom by exactly the same fractional amount. If iron atoms in steel are normally 2.0 * 10–10 m apart, by what distance did this interatomic spacing have to change in order to produce the normal force required to support the man? [Note: Neighboring atoms repel each other, and this repulsion accounts for the observed normal force.] 90. A home mechanic wants to raise the 280-kg engine out of a car. The plan is to stretch a rope vertically from the engine to a branch of a tree 6.0 m above, and back to the bumper (Fig. 12–96). When the mechanic climbs up a stepladder and pulls horizontally on the rope at its midpoint, the engine rises out of the car. (a) How much force must the mechanic exert to hold the engine 0.50 m above its normal position? (b) What is the system's mechanical 6.0 m advantage?

Ffr

92. You are on a pirate ship and being forced to walk the plank (Fig. 12–98). You are standing at the point marked C. The plank is nailed onto the deck at point A, and rests on the support 0.75 m away from A. The center of mass of the uniform plank is located at point B. Your mass is A 3.0 m 65 kg and the mass of the plank is 45 kg. What is the minimum 0.75 m downward force the nails must exert on the plank to hold it in C place? 1.5 m

FIGURE 12–98 Problem 92.

93. A uniform sphere of weight mg and radius r0 is tethered to a wall by a rope of length l. The rope is tied to the wall a distance h above the contact point of the sphere, as shown in Fig. 12–99. The rope makes an angle u with respect to the wall and is not in line with the ball's center. The coefficient of static friction between the wall and sphere is m. (a) Determine the value of θ the frictional force on the sphere due to the wall. [Hint: A wise choice of axis will h make this calculation easy.] (b) Suppose the sphere is just on the verge of r0 slipping. Derive an expression for m in B terms of h and u. mg FIGURE 12–99 Problem 93. * 94. Use the method of joints to determine the force in each member of the truss shown in Fig. 12–100. B State whether each member is in tension or compression. 6.0 m C

A 4.0 m

FIGURE 12–96 Problem 90.

FIGURE 12–100 Problem 94.

D 6.0 m

12,000 N

General Problems

337

95. A uniform ladder of mass m and length l leans at an angle u against a wall, Fig. 12–101. The coefficients of static friction between ladder–ground and ladder–wall are mG and mW , respectively. The ladder will be on the verge of slipping when both the static friction forces due to the ground and due to the wall take on their maximum values. (a) Show that the ladder will be stable if u umin , where the minimum angle umin is given by 1 tan umin = A1 - mG mW B. 2mG

* Numerical/Computer * 97. (III) A metal cylinder has an original diameter of 1.00 cm and a length of 5.00 cm. A tension test was performed on the specimen and the data are listed in the Table. (a) Graph the stress on the specimen vs. the strain. (b) Considering only the elastic region, find the slope of the best-fit straight line and determine the elastic modulus of the metal.

(b) "Leaning ladder problems" are often analyzed under the seemingly unrealistic assumption that the wall is frictionless (see Example 12–6). You wish to investigate the magnitude of error introduced by modeling the wall as frictionless, if in reality it is frictional. Using the relation found in part (a), calculate the true value of umin for a frictional wall, taking mG = mW = 0.40. Then, determine the approximate value of umin for the "frictionmW less wall" model by taking mG = 0.40 and mW = 0. Finally, determine the percent deviation of the approxil mate value of umin from its true value. mgB FIGURE 12–101 Problem 95.

θ mG

96. In a mountain-climbing technique called the "Tyrolean traverse," a rope is anchored on both ends (to rocks or strong trees) across a deep chasm, and then a climber traverses the rope while attached by a sling as in Fig. 12–102. This technique generates tremendous forces in the rope and anchors, so a basic understanding of physics is crucial for safety. A typical climbing rope can undergo a tension force of perhaps 29 kN before breaking, and a "safety factor" of 10 is usually recommended. The length of rope used in the Tyrolean traverse must allow for some "sag" to remain in the recommended safety range. Consider a 75-kg climber at the center of a Tyrolean traverse, spanning a 25-m chasm. (a) To be within its recommended safety range, what minimum distance x must the rope sag? (b) If the Tyrolean traverse is set up incorrectly so that the rope sags by only one-fourth the distance found in (a), determine the 25 m tension in the rope. Will the rope break? x

FIGURE 12–102 Problem 96.

Load (kN)

Elongation (cm)

0 1.50 4.60 8.00 11.00 11.70 11.80 12.00 16.60 20.00 21.50 19.50 18.50

0 0.0005 0.0015 0.0025 0.0035 0.0050 0.0080 0.0200 0.0400 0.1000 0.2800 0.4000 0.4600

* 98. (III) Two springs, attached by a rope, are connected as shown in Fig. 12–103. The length AB is 4.0 m and AC = BC. The spring constant of each spring is k = 20.0 Nm. A force F acts downward at C on the rope. Graph u as a function of F from u = 0 to 75°, assuming the springs are unstretched at u = 0. A

C B

FIGURE 12–103 Problem 98.

75 kg

Answers to Exercises A: FA also has a component to balance the sideways force FB . B: Yes: cos u (angle of bar with ground) appears on both sides and cancels out. C: FN = mA g + mB g + Mg = 560 N. D: (a).

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CHAPTER 12

E: 7.0 kg.

F: Static friction at the cement floor A = FCx B is crucial, or else the ladder would slip. At the top, the ladder can move and adjust, so we wouldn't need or expect a strong static friction force there. G: (b).

Static Equilibrium; Elasticity and Fracture

Underwater divers and sea creatures B experience a buoyant force AFB B that closely balances their weight mg. The buoyant force is equal to the weight of the volume of fluid displaced (Archimedes' principle) and arises because the pressure increases with depth in the fluid. Sea creatures have a density very close to that of water, so their weight very nearly equals the buoyant force. Humans have a density slightly less than water, so they can float. When fluids flow, interesting effects occur because the pressure in the fluid is lower where the fluid velocity is higher (Bernoulli's principle). B

FB2 B

F B1 m2 g B

P T E R

C H

m1 g B

Fluids CHAPTER-OPENING QUESTIONS—Guess now! 1. Which container has the largest pressure at the bottom? Assume each container holds the same volume of water.

CONTENTS 13–1 Phases of Matter 13–2 Density and Specific Gravity

(a)

(b)

(c)

(d)

(e) The pressures are equal.

2. Two balloons are tied and hang with their nearest edges about 3 cm apart. If you blow between the balloons (not at the balloons, but at the opening between them), what will happen? (a) Nothing. (b) The balloons will move closer together. (c) The balloons will move farther apart.

13–3 Pressure in Fluids 13–4 Atmospheric Pressure and Gauge Pressure 13–5 Pascal's Principle 13–6 Measurement of Pressure; Gauges and the Barometer 13–7 Buoyancy and Archimedes' Principle 13–8 Fluids in Motion; Flow Rate and the Equation of Continuity 13–9 Bernoulli's Equation 13–10 Applications of Bernoulli's Principle: Torricelli, Airplanes, Baseballs, TIA

*13–11 Viscosity *13–12 Flow in Tubes: Poiseuille's Equation, Blood Flow

*13–13 Surface Tension and Capillarity *13–14 Pumps, and the Heart

339

n previous Chapters we considered objects that were solid and assumed to maintain their shape except for a small amount of elastic deformation. We sometimes treated objects as point particles. Now we are going to shift our attention to materials that are very deformable and can flow. Such "fluids" include liquids and gases. We will examine fluids both at rest (fluid statics) and in motion (fluid dynamics).

13–1

TABLE 13–1 Densities of Substances† Density, R (kg m3)

Substance Solids Aluminum Iron and steel Copper Lead Gold Concrete Granite Wood (typical) Glass, common Ice AH 2OB Bone Liquids Water (4°C) Blood, plasma Blood, whole Sea water Mercury Alcohol, ethyl Gasoline Gases Air Helium Carbon dioxide Steam (water, 100°C)

2.70 7.8 8.9 11.3 19.3 2.3 2.7 0.3 – 0.9 2.4 – 2.8 0.917 1.7 – 2.0

* * * * * * * * * * *

103 103 103 103 103 103 103 103 103 103 103

1.00 1.03 1.05 1.025 13.6 0.79 0.68

* * * * * * *

103 103 103 103 103 103 103

0.598

Densities are given at 0°C and 1 atm pressure unless otherwise specified.

CHAPTER 13

The three common phases, or states, of matter are solid, liquid, and gas. We can distinguish these three phases as follows. A solid maintains a fixed shape and a fixed size; even if a large force is applied to a solid, it does not readily change in shape or volume. A liquid does not maintain a fixed shape—it takes on the shape of its container—but like a solid it is not readily compressible, and its volume can be changed significantly only by a very large force. A gas has neither a fixed shape nor a fixed volume—it will expand to fill its container. For example, when air is pumped into an automobile tire, the air does not all run to the bottom of the tire as a liquid would; it spreads out to fill the whole volume of the tire. Since liquids and gases do not maintain a fixed shape, they both have ability to flow; they are thus often referred to collectively as fluids. The division of matter into three phases is not always simple. How, for example, should butter be classified? Furthermore, a fourth phase of matter can be distinguished, the plasma phase, which occurs only at very high temperatures and consists of ionized atoms (electrons separated from the nuclei). Some scientists believe that so-called colloids (suspensions of tiny particles in a liquid) should also be considered a separate phase of matter. Liquid crystals, which are used in TV and computer screens, calculators, digital watches, and so on, can be considered a phase of matter intermediate between solids and liquids. However, for our present purposes we will mainly be interested in the three ordinary phases of matter.

13–2

Density and Specific Gravity

It is sometimes said that iron is "heavier" than wood. This cannot really be true since a large log clearly weighs more than an iron nail. What we should say is that iron is more dense than wood. The density, r, of a substance (r is the lowercase Greek letter rho) is defined as its mass per unit volume: m, r = (13–1) V where m is the mass of a sample of the substance and V is its volume. Density is a characteristic property of any pure substance. Objects made of a particular pure substance, such as pure gold, can have any size or mass, but the density will be the same for each. We will sometimes use the concept of density, Eq. 13–1, to write the mass of an object as m = rV, and the weight of an object as

1.29 0.179 1.98

†

340

Phases of Matter

Fluids

mg = rVg. The SI unit for density is kgm3. Sometimes densities are given in gcm3. Note that since 1 kgm3 = 1000 g(100 cm)3 = 103 g106 cm3 = 10 –3 gcm3, then a density given in gcm3 must be multiplied by 1000 to give the result in kgm3. Thus the density of aluminum is r = 2.70 gcm3, which is equal to 2700 kgm3. The densities of a variety of substances are given in Table 13–1. The Table specifies temperature and atmospheric pressure because they affect the density of substances (although the effect is slight for liquids and solids). Note that air is roughly 1000 times less dense than water.

EXAMPLE 13–1 Mass, given volume and density. What is the mass of a solid iron wrecking ball of radius 18 cm? APPROACH First we use the standard formula V = 43 pr 3 (see inside rear cover) to obtain the volume of the sphere. Then Eq. 13–1 and Table 13–1 give us the mass m. SOLUTION The volume of the sphere is V = 43 pr 3 = 43 (3.14)(0.18 m)3 = 0.024 m3. From Table 13–1, the density of iron is r = 7800 kgm3, so Eq. 13–1 gives m = rV = A7800 kgm3 BA0.024 m3 B = 190 kg. The specific gravity of a substance is defined as the ratio of the density of that substance to the density of water at 4.0°C. Because specific gravity (abbreviated SG) is a ratio, it is a simple number without dimensions or units. The density of water is 1.00 gcm3 = 1.00 * 103 kgm3, so the specific gravity of any substance will be equal numerically to its density specified in gcm3, or 10 –3 times its density specified in kgm3. For example (see Table 13–1), the specific gravity of lead is 11.3, and that of alcohol is 0.79. The concepts of density and specific gravity are especially helpful in the study of fluids because we are not always dealing with a fixed volume or mass.

13–3

Pressure in Fluids

Pressure and force are related, but they are not the same thing. Pressure is defined as force per unit area, where the force F is understood to be the magnitude of the force acting perpendicular to the surface area A: F. (13–2) pressure = P = A Although force is a vector, pressure is a scalar. Pressure has magnitude only. The SI unit of pressure is Nm2. This unit has the official name pascal (Pa), in honor of Blaise Pascal (see Section 13–5); that is, 1 Pa = 1 Nm2. However, for simplicity, we will often use Nm2. Other units sometimes used are dynescm2, and lbin.2 (abbreviated "psi"). Several other units for pressure are discussed, along with conversions between them, in Section 13–6 (see also the Table inside the front cover).

C A U T I O N

Pressure is a scalar, not a vector

EXAMPLE 13–2 Calculating pressure. The two feet of a 60-kg person cover an area of 500 cm2. (a) Determine the pressure exerted by the two feet on the ground. (b) If the person stands on one foot, what will the pressure be under that foot? APPROACH Assume the person is at rest. Then the ground pushes up on her with a force equal to her weight mg, and she exerts a force mg on the ground where her feet (or foot) contact it. Because 1 cm2 = A10 –2 mB 2 = 10–4 m2, then 500 cm2 = 0.050 m2. SOLUTION (a) The pressure on the ground exerted by the two feet is

(60 kg)A9.8 ms2 B mg F P = = = = 12 * 103 Nm2. A A A0.050 m2 B (b) If the person stands on one foot, the force is still equal to the person's weight, but the area will be half as much, so the pressure will be twice as much: 24 * 103 Nm2. Pressure is particularly useful for dealing with fluids. It is an experimental observation that a fluid exerts pressure in any direction. This is well known to swimmers and divers who feel the water pressure on all parts of their bodies. At any depth in a fluid at rest, the pressure is the same in all directions at a given depth. To see why, consider a tiny cube of the fluid (Fig. 13–1) which is so small that we can consider it a point and can ignore the force of gravity on it. The pressure on one side of it must equal the pressure on the opposite side. If this weren't true, there would be a net force on the cube and it would start moving. If the fluid is not flowing, then the pressures must be equal.

FIGURE 13–1 Pressure is the same in every direction in a nonmoving fluid at a given depth. If this weren't true, the fluid would be in motion.

SECTION 13–3

341

F F FIGURE 13–2 If there were a component of force parallel to the solid surface of the container, the liquid would move in response to it. For a liquid at rest, F7 = 0. FIGURE 13–3 Calculating the pressure at a depth h in a liquid.

For a fluid at rest, the force due to fluid pressure always acts perpendicular to any solid surface it touches. If there were a component of the force parallel to the surface, as shown in Fig. 13–2, then according to Newton's third law the solid surface would exert a force back on the fluid that also would have a component parallel to the surface. Such a component would cause the fluid to flow, in contradiction to our assumption that the fluid is at rest. Thus the force due to the pressure in a fluid at rest is always perpendicular to the surface. Let us now calculate quantitatively how the pressure in a liquid of uniform density varies with depth. Consider a point at a depth h below the surface of the liquid, as shown in Fig. 13–3 (that is, the surface is a height h above this point). The pressure due to the liquid at this depth h is due to the weight of the column of liquid above it. Thus the force due to the weight of liquid acting on the area A is F = mg = (rV)g = rAhg, where Ah is the volume of the column of liquid, r is the density of the liquid (assumed to be constant), and g is the acceleration of gravity. The pressure P due to the weight of liquid is then rAhg F P = = A A P = rgh.

[liquid] (13–3)

Note that the area A doesn't affect the pressure at a given depth. The fluid pressure is directly proportional to the density of the liquid and to the depth within the liquid. In general, the pressure at equal depths within a uniform liquid is the same.

h A

EXERCISE A Return to Chapter-Opening Question 1, page 339, and answer it again now. Try to explain why you may have answered differently the first time.

FIGURE 13–4 Forces on a flat, slablike volume of fluid for determining the pressure P at a height y in the fluid.

(P + dP)A dy

dFG

Equation 13–3 tells us what the pressure is at a depth h in the liquid, due to the liquid itself. But what if there is additional pressure exerted at the surface of the liquid, such as the pressure of the atmosphere or a piston pushing down? And what if the density of the fluid is not constant? Gases are quite compressible and hence their density can vary significantly with depth. Liquids, too, can be compressed, although we can often ignore the variation in density. (One exception is in the depths of the ocean where the great weight of water above significantly compresses the water and increases its density.) To cover these, and other cases, we now treat the general case of determining how the pressure in a fluid varies with depth. As shown in Fig. 13–4, let us determine the pressure at any height y above some reference point† (such as the ocean floor or the bottom of a tank or swimming pool). Within this fluid, at the height y, we consider a tiny, flat, slablike volume of the fluid whose area is A and whose (infinitesimal) thickness is dy, as shown. Let the pressure acting upward on its lower surface (at height y) be P. The pressure acting downward on the top surface of our tiny slab (at height y + dy) is designated P + dP. The fluid pressure acting on our slab thus exerts a force equal to PA upward on our slab and a force equal to (P + dP)A downward on it. The only other force acting vertically on the slab is the (infinitesimal) force of gravity dFG , which on our slab of mass dm is dFG = (dm)g = rg dV = rgA dy, where r is the density of the fluid at the height y. Since the fluid is assumed to be at rest, our slab is in equilibrium so the net force on it must be zero. Therefore we have PA - (P + dP)A - rgA dy = 0, which when simplified becomes dP = –rg. dy

(13–4)

This relation tells us how the pressure within the fluid varies with height above any reference point. The minus sign indicates that the pressure decreases with an increase in height; or that the pressure increases with depth (reduced height). †

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CHAPTER 13

Fluids

Now we are measuring y positive upwards, the reverse of what we did to get Eq. 13–3 where we measured the depth (i.e. downward as positive).

If the pressure at a height y1 in the fluid is P1 , and at height y2 it is P2 , then we can integrate Eq. 13–4 to obtain

–

dP = –

rg dy

y1 y2

P2 - P1 =

rg dy,

(13–5) P2 = P0

where we assume r is a function of height y: r = r(y). This is a general relation, and we apply it now to two special cases: (1) pressure in liquids of uniform density and (2) pressure variations in the Earth's atmosphere. For liquids in which any variation in density can be ignored, r = constant and Eq. 13–5 is readily integrated: P2 - P1 = –rgAy2 - y1 B. (13–6a) For the everyday situation of a liquid in an open container—such as water in a glass, a swimming pool, a lake, or the ocean—there is a free surface at the top exposed to the atmosphere. It is convenient to measure distances from this top surface. That is, we let h be the depth in the liquid where h = y2 - y1 as shown in Fig. 13–5. If we let y2 be the position of the top surface, then P2 represents the atmospheric pressure, P0 , at the top surface. Then, from Eq. 13–6a, the pressure P A = P1 B at a depth h in the fluid is [h is depth in liquid] (13–6b) P = P0 + rgh. Note that Eq. 13–6b is simply the liquid pressure (Eq. 13–3) plus the pressure P0 due to the atmosphere above. EXAMPLE 13–3 Pressure at a faucet. The surface of the water in a storage tank is 30 m above a water faucet in the kitchen of a house, Fig. 13–6. Calculate the difference in water pressure between the faucet and the surface of the water in the tank. APPROACH Water is practically incompressible, so r is constant even for h = 30 m when used in Eq. 13–6b. Only h matters; we can ignore the "route" of the pipe and its bends. SOLUTION We assume the atmospheric pressure at the surface of the water in the storage tank is the same as at the faucet. So, the water pressure difference between the faucet and the surface of the water in the tank is

h = y2 − y1 y2

P1 = P y1

FIGURE 13–5 Pressure at a depth h = Ay2 - y1 B in a liquid of density r is P = P0 + rgh, where P0 is the external pressure at the liquid's top surface.

FIGURE 13–6 Example 13–3.

h= 30 m

¢P = rgh = A1.0 * 103 kgm3 BA9.8 ms2 B(30 m) = 2.9 * 105 Nm2.

NOTE The height h is sometimes called the pressure head. In this Example, the head of water is 30 m at the faucet. The very different diameters of the tank and faucet don't affect the result—only pressure does. EXAMPLE 13–4 Force on aquarium window. Calculate the force due to water pressure exerted on a 1.0 m * 3.0 m aquarium viewing window whose top edge is 1.0 m below the water surface, Fig. 13–7. APPROACH At a depth h, the pressure due to the water is given by Eq. 13–6b. Divide the window up into thin horizontal strips of length l = 3.0 m and thickness dy, as shown in Fig. 13–7. We choose a coordinate system with y = 0 at the surface of the water and y is positive downward. (With this choice, the minus sign in Eq. 13–6a becomes plus, or we use Eq. 13–6b with y = h.) The force due to water pressure on each strip is dF = PdA = rgy l dy. SOLUTION The total force on the window is given by the integral: y2 = 2.0 m

rgy l dy = y1 = 1.0 m

1 2 2 rglAy 2

y21 B

FIGURE 13–7 Example 13–4. 1.0 m

1.0 m

dy l = 3.0 m

= 12 A1000 kgm3 BA9.8 ms2 B(3.0 m)C(2.0 m)2 - (1.0 m)2 D = 44,000 N.

NOTE To check our answer, we can do an estimate: multiply the area of the window A3.0 m2 B times the pressure at the middle of the window (h = 1.5 m) using Eq. 13–3, P = rgh = A1000 kgm3 BA9.8 ms2 B(1.5 m) L 1.5 * 104 Nm2. So F = PA L A1.5 * 104 Nm2 B(3.0 m)(1.0 m) L 4.5 * 104 N. Good! EXERCISE B A dam holds back a lake that is 85 m deep at the dam. If the lake is 20 km long, how much thicker should the dam be than if the lake were smaller, only 1.0 km long?

SECTION 13–3

343

Now let us apply Eq. 13–4 or 13–5 to gases. The density of gases is normally quite small, so the difference in pressure at different heights can usually be ignored if y2 - y1 is not large (which is why, in Example 13–3, we could ignore the difference in air pressure between the faucet and the top of the storage tank). Indeed, for most ordinary containers of gas, we can assume that the pressure is the same throughout. However, if y2 - y1 is very large, we cannot make this assumption. An interesting example is the air of Earth's atmosphere, whose pressure at sea level is about 1.013 * 105 Nm2 and decreases slowly with altitude. EXAMPLE 13–5 Elevation effect on atmospheric pressure. (a) Determine the variation in pressure in the Earth's atmosphere as a function of height y above sea level, assuming g is constant and that the density of the air is proportional to the pressure. (This last assumption is not terribly accurate, in part because temperature and other weather effects are important.) (b) At what elevation is the air pressure equal to half the pressure at sea level? APPROACH We start with Eq. 13–4 and integrate it from the surface of the Earth where y = 0 and P = P0 , up to height y at pressure P. In (b) we choose P = 12 P0 . SOLUTION (a) We are assuming that r is proportional to P, so we can write r P, = r0 P0 where P0 = 1.013 * 105 Nm2 is atmospheric pressure at sea level and r0 = 1.29 kgm3 is the density of air at sea level at 0°C (Table 13–1). From the differential change in pressure with height, Eq. 13–4, we have r0 dP = –rg = –P ¢ ≤ g, dy P0 so r0 dP = – g dy. P P0 We integrate this from y = 0 (Earth's surface) and P = P0 , to the height y where the pressure is P: P y r0 dP = – g dy P0 0 P0 P r0 P ln = – gy, P0 P0

since ln P - ln P0 = lnAPP0 B. Then

P = P0 e – Ar0 gP0By. So, based on our assumptions, we find that the air pressure in our atmosphere decreases approximately exponentially with height. NOTE The atmosphere does not have a distinct top surface, so there is no natural point from which to measure depth in the atmosphere, as we can do for a liquid. (b) The constant Ar0 gP0 B has the value A1.29 kgm3 BA9.80 ms2 B r0 g = = 1.25 * 10 –4 m–1. P0 A1.013 * 105 Nm2 B

Then, when we set P = 12 P0 in our expression derived in (a), we obtain

= e – A1.25 * 10 m By or, taking natural logarithms of both sides, ln 12 = A –1.25 * 10–4 m–1 By 1 so (recall ln 2 = –ln 2, Appendix A–7, Eq. ii) y = (ln 2.00)A1.25 * 10–4 m–1 B = 5550 m. Thus, at an elevation of about 5500 m (about 18,000 ft), atmospheric pressure drops to half what it is at sea level. It is not surprising that mountain climbers often use oxygen tanks at very high altitudes. 1 2

344

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Fluids

–4

–1

13–4

Atmospheric Pressure and Gauge Pressure

Atmospheric Pressure The pressure of the air at a given place on Earth varies slightly according to the weather. At sea level, the pressure of the atmosphere on average is 1.013 * 105 Nm2 Aor 14.7 lbin.2 B. This value lets us define a commonly used unit of pressure, the atmosphere (abbreviated atm): 1 atm = 1.013 * 105 Nm2 = 101.3 kPa. Another unit of pressure sometimes used (in meteorology and on weather maps) is the bar, which is defined as 1 bar = 1.000 * 105 Nm2. Thus standard atmospheric pressure is slightly more than 1 bar. The pressure due to the weight of the atmosphere is exerted on all objects immersed in this great sea of air, including our bodies. How does a human body withstand the enormous pressure on its surface? The answer is that living cells maintain an internal pressure that closely equals the external pressure, just as the pressure inside a balloon closely matches the outside pressure of the atmosphere. An automobile tire, because of its rigidity, can maintain internal pressures much greater than the external pressure. CONCEPTUAL EXAMPLE 13–6 Finger holds water in a straw. You insert a straw of length l into a tall glass of water. You place your finger over the top of the straw, capturing some air above the water but preventing any additional air from getting in or out, and then you lift the straw from the water. You find that the straw retains most of the water (see Fig. 13–8a). Does the air in the space between your finger and the top of the water have a pressure P that is greater than, equal to, or less than the atmospheric pressure P0 outside the straw? RESPONSE Consider the forces on the column of water (Fig. 13–8b). Atmospheric pressure outside the straw pushes upward on the water at the bottom of the straw, gravity pulls the water downward, and the air pressure inside the top of the straw pushes downward on the water. Since the water is in equilibrium, the upward force due to atmospheric pressure P0 must balance the two downward forces. The only way this is possible is for the air pressure inside the straw to be less than the atmosphere pressure outside the straw. (When you initially remove the straw from the glass of water, a little water may leave the bottom of the straw, thus increasing the volume of trapped air and reducing its density and pressure.)

P H Y S I C S

A P P L I E D

Pressure on living cells

P=?

r h

mg = rgAh

P0 P0 A (a)

(b)

FIGURE 13–8 Example 13–6.

Gauge Pressure It is important to note that tire gauges, and most other pressure gauges, register the pressure above and beyond atmospheric pressure. This is called gauge pressure. Thus, to get the absolute pressure, P, we must add the atmospheric pressure, P0 , to the gauge pressure, PG : P = P0 + PG . If a tire gauge registers 220 kPa, the absolute pressure within the tire is 220 kPa + 101 kPa = 321 kPa, equivalent to about 3.2 atm (2.2 atm gauge pressure). SECTION 13–4

Atmospheric Pressure and Gauge Pressure

345

Fout B

Fin

Pin Ain

Aout

Pout

(a) Master cylinder

Pedal Brake cylinder (b)

Brake pads

Disk, attached to wheel

FIGURE 13–9 Applications of Pascal's principle: (a) hydraulic lift; (b) hydraulic brakes in a car.

P H Y S I C S

A P P L I E D

Hydraulic lift

P H Y S I C S

A P P L I E D

Hydraulic brakes

13–5

The Earth's atmosphere exerts a pressure on all objects with which it is in contact, including other fluids. External pressure acting on a fluid is transmitted throughout that fluid. For instance, according to Eq. 13–3, the pressure due to the water at a depth of 100 m below the surface of a lake is P = rgh = A1000 kgm3 BA9.8 ms2 B(100 m) = 9.8 * 105 Nm2, or 9.7 atm. However, the total pressure at this point is due to the pressure of water plus the pressure of the air above it. Hence the total pressure (if the lake is near sea level) is 9.7 atm + 1.0 atm = 10.7 atm. This is just one example of a general principle attributed to the French philosopher and scientist Blaise Pascal (1623–1662). Pascal's principle states that if an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount. A number of practical devices make use of Pascal's principle. One example is the hydraulic lift, illustrated in Fig. 13–9a, in which a small input force is used to exert a large output force by making the area of the output piston larger than the area of the input piston. To see how this works, we assume the input and output pistons are at the same height (at least approximately). Then the external input force Fin , by Pascal's principle, increases the pressure equally throughout. Therefore, at the same level (see Fig. 13–9a), Pout = Pin where the input quantities are represented by the subscript "in" and the output by "out." Since P = FA, we write the above equality as Fout Fin , = A out A in or Fout A out . = Fin A in The quantity FoutFin is called the mechanical advantage of the hydraulic lift, and it is equal to the ratio of the areas. For example, if the area of the output piston is 20 times that of the input cylinder, the force is multiplied by a factor of 20. Thus a force of 200 lb could lift a 4000-lb car. Figure 13–9b illustrates the brake system of a car. When the driver presses the brake pedal, the pressure in the master cylinder increases. This pressure increase occurs throughout the brake fluid, thus pushing the brake pads against the disk attached to the car's wheel.

13–6

346

CHAPTER 13

Fluids

Pascal's Principle

Measurement of Pressure; Gauges and the Barometer

Many devices have been invented to measure pressure, some of which are shown in Fig. 13–10. The simplest is the open-tube manometer (Fig 13–10a) which is a U-shaped tube partially filled with a liquid, usually mercury or water. The pressure P being measured is related to the difference in height ¢h of the two levels of the liquid by the relation P = P0 + rg ¢h, where P0 is atmospheric pressure (acting on the top of the liquid in the left-hand tube), and r is the density of the liquid. Note that the quantity rg ¢h is the gauge pressure—the amount by which P exceeds atmospheric pressure P0 . If the liquid in the left-hand column were lower than that in the right-hand column, P would have to be less than atmospheric pressure (and ¢h would be negative). Instead of calculating the product rg ¢h, sometimes only the change in height ¢h is specified. In fact, pressures are sometimes specified as so many "millimeters of mercury" (mm-Hg) or "mm of water" Amm-H 2OB. The unit mm-Hg is equivalent to a pressure of 133 Nm2, since rg ¢h for 1 mm = 1.0 * 10–3 m of mercury gives rg ¢h = A13.6 * 103 kgm3 BA9.80 ms2 BA1.00 * 10 –3 mB = 1.33 * 102 Nm2. The unit mm-Hg is also called the torr in honor of Evangelista Torricelli (1608–1647), a student of Galileo's who invented the barometer (see next page).

Scale reading, gauge pressure Atmospheric pressure

Spring

Air pressure P (Pressure being measured)

Δh

Flexible chamber

(b) Aneroid gauge (used mainly for air pressure and then called an aneroid barometer)

(a) Open-tube manometer

Pressure of air in tire (c) Tire gauge

FIGURE 13–10 Pressure gauges: (a) open-tube manometer, (b) aneroid gauge, and (c) common tire-pressure gauge.

Conversion factors among the various units of pressure (an incredible nuisance!) are given in Table 13–2. It is important that only Nm2 = Pa, the proper SI unit, be used in calculations involving other quantities specified in SI units.

P R O B L E M

S O LV I N G

In calculations, use SI units: 1 Pa = 1 Nm2

TABLE 13–2 Conversion Factors Between Different Units of Pressure In Terms of 1 Pa 1 N m2 1 atm = = 1 bar = 1 dynecm2 = 1 lbin.2 = 1 lbft2 = 1 cm-Hg = 1 mm-Hg = 1 torr = 1 mm-H 2O (4°C) =

1 atm in Different Units 2

1.013 * 10 Nm 1.013 * 105 Pa = 101.3 kPa 1.000 * 105 Nm2 0.1 Nm2 6.90 * 103 Nm2 47.9 Nm2 1.33 * 103 Nm2 133 Nm2 133 Nm2 9.80 Nm2

1 atm = 1.013 * 105 Nm2 1 atm 1 atm 1 atm 1 atm 1 atm 1 atm 1 atm 1 atm

= = = = = = = =

1.013 bar 1.013 * 106 dynecm2 14.7 lbin.2 2.12 * 103 lbft2 76.0 cm-Hg 760 mm-Hg 760 torr 1.03 * 104 mm-H 2O (4°C)

Another type of pressure gauge is the aneroid gauge (Fig. 13–10b) in which the pointer is linked to the flexible ends of an evacuated thin metal chamber. In an electronic gauge, the pressure may be applied to a thin metal diaphragm whose resulting distortion is translated into an electrical signal by a transducer. A common tire gauge is shown in Fig. 13–10c. Atmospheric pressure can be measured by a modified kind of mercury manometer with one end closed, called a mercury barometer (Fig. 13–11). The glass tube is completely filled with mercury and then inverted into the bowl of mercury. If the tube is long enough, the level of the mercury will drop, leaving a vacuum at the top of the tube, since atmospheric pressure can support a column of mercury only about 76 cm high (exactly 76.0 cm at standard atmospheric pressure). That is, a column of mercury 76 cm high exerts the same pressure as the atmosphere†: P = rg ¢h = A13.6 * 103 kgm3 BA9.80 ms2 B(0.760 m) = 1.013 * 105 Nm2 = 1.00 atm. †

FIGURE 13–11 A mercury barometer, invented by Torricelli, is shown here when the air pressure is standard atmospheric, 76.0 cm-Hg. P=0

76.0 cm

P = 1 atm

This calculation confirms the entry in Table 13–2, 1 atm = 76.0 cm-Hg.

SECTION 13–6

Measurement of Pressure; Gauges and the Barometer

347

A calculation similar to what we just did will show that atmospheric pressure can maintain a column of water 10.3 m high in a tube whose top is under vacuum (Fig. 13–12). No matter how good a vacuum pump is, water cannot be made to rise more than about 10 m using normal atmospheric pressure. To pump water out of deep mine shafts with a vacuum pump requires multiple stages for depths greater than 10 m. Galileo studied this problem, and his student Torricelli was the first to explain it. The point is that a pump does not really suck water up a tube—it merely reduces the pressure at the top of the tube. Atmospheric air pressure pushes the water up the tube if the top end is at low pressure (under a vacuum), just as it is air pressure that pushes (or maintains) the mercury 76 cm high in a barometer. [Force pumps (Section 13–14) that push up from the bottom can exert higher pressure to push water more than 10 m high.]

FIGURE 13–12 A water barometer: a full tube of water is inserted into a tub of water, keeping the tube's spigot at the top closed. When the bottom end of the tube is unplugged, some water flows out of the tube into the tub, leaving a vacuum between the water's upper surface and the spigot. Why? Because air pressure can not support a column of water more than 10 m high.

FIGURE 13–13 Determination of the buoyant force.

h1 h2

We sometimes mistakenly think of suction as something we actively do. For example, we intuitively think that we pull the soda up through a straw. Instead, what we do is lower the pressure at the top of the straw, and the atmosphere pushes the soda up the straw.

13–7

Buoyancy and Archimedes' Principle

Objects submerged in a fluid appear to weigh less than they do when outside the fluid. For example, a large rock that you would have difficulty lifting off the ground can often be easily lifted from the bottom of a stream. When the rock breaks through the surface of the water, it suddenly seems to be much heavier. Many objects, such as wood, float on the surface of water. These are two examples of buoyancy. In each example, the force of gravity is acting downward. But in addition, an upward buoyant force is exerted by the liquid. The buoyant force on fish and underwater divers (as in the Chapter-Opening photo) almost exactly balances the force of gravity downward, and allows them to "hover" in equilibrium. The buoyant force occurs because the pressure in a fluid increases with depth. Thus the upward pressure on the bottom surface of a submerged object is greater than the downward pressure on its top surface. To see this effect, consider a cylinder of height ¢h whose top and bottom ends have an area A and which is completely submerged in a fluid of density rF , as shown in Fig. 13–13. The fluid exerts a pressure P1 = rF gh1 at the top surface of the cylinder (Eq. 13–3). The force due to this pressure on top of the cylinder is F1 = P1 A = rF gh1 A, and it is directed downward. Similarly, the fluid exerts an upward force on the bottom of the cylinder equal to F2 = P2 A = rF gh2 A. The net force on the cylinder exerted by the fluid B pressure, which is the buoyant force, FB , acts upward and has the magnitude

A ∆h = h2 − h1

348

CONCEPTUAL EXAMPLE 13–7 Suction. A student suggests suction-cup shoes for Space Shuttle astronauts working on the exterior of a spacecraft. Having just studied this Chapter, you gently remind him of the fallacy of this plan. What is it? RESPONSE Suction cups work by pushing out the air underneath the cup. What holds the suction cup in place is the air pressure outside it. (This can be a substantial force when on Earth. For example, a 10-cm-diameter suction cup has an area of 7.9 * 10–3 m2. The force of the atmosphere on it is A7.9 * 10 –3 m2 BA1.0 * 105 Nm2 B L 800 N, about 180 lbs!) But in outer space, there is no air pressure to push the suction cup onto the spacecraft.

CHAPTER 13

Fluids

FB = F2 - F1 = = = =

rF gAAh2 - h1 B rF gA ¢h rF Vg mF g,

where V = A ¢h is the volume of the cylinder, the product rF V is the mass of the fluid displaced, and rF Vg = mF g is the weight of fluid which takes up a volume equal to the volume of the cylinder. Thus the buoyant force on the cylinder is equal to the weight of fluid displaced by the cylinder.

This result is valid no matter what the shape of the object. Its discovery is credited to Archimedes (287?–212 B.C.), and it is called Archimedes' principle: the buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by that object. By "fluid displaced," we mean a volume of fluid equal to the submerged volume of the object (or that part of the object that is submerged). If the object is placed in a glass or tub initially filled to the brim with water, the water that flows over the top represents the water displaced by the object. We can derive Archimedes' principle in general by the following simple but elegant argument. The irregularly shaped object D shown in Fig. 13–14a is acted on B by the force of gravity (its weight, mg, downward) and the buoyant force, FB , upward. We wish to determine FB . To do so, we next consider a body (D¿ in Fig. 13–14b), this time made of the fluid itself, with the same shape and size as the original object, and located at the same depth. You might think of this body of fluid as being separated from the rest of the fluid by an imaginary membrane. The buoyant force FB on this body of fluid will be exactly the same as that on the original object since the surrounding fluid, which exerts FB , is in exactly the same configuration. This body of fluid D¿ is in equilibrium (the fluid as a whole is at rest). Therefore, FB = m¿g, where m¿g is the weight of the body of fluid. Hence the buoyant force FB is equal to the weight of the body of fluid whose volume equals the volume of the original submerged object, which is Archimedes' principle. Archimedes' discovery was made by experiment. What we have done in the last two paragraphs is to show that Archimedes' principle can be derived from Newton's laws. B

mgB

(a)

D′

m′ g B

(b)

FIGURE 13–14 Archimedes' principle.

CONCEPTUAL EXAMPLE 13–8 Two pails of water. Consider two identical pails of water filled to the brim. One pail contains only water, the other has a piece of wood floating in it. Which pail has the greater weight? RESPONSE Both pails weigh the same. Recall Archimedes' principle: the wood displaces a volume of water with weight equal to the weight of the wood. Some water will overflow the pail, but Archimedes' principle tells us the spilled water has weight equal to the weight of the wood object; so the pails have the same weight.

EXAMPLE 13–9 Recovering a submerged statue. A 70-kg ancient statue lies at the bottom of the sea. Its volume is 3.0 * 104 cm3. How much force is needed to lift it? APPROACH The force F needed to lift the statue is equal to the statue's weight mg minus the buoyant force FB . Figure 13–15 is the free-body diagram. SOLUTION The buoyant force on the statue due to the water is equal to the weight of 3.0 * 104 cm3 = 3.0 * 10–2 m3 of water (for seawater, r = 1.025 * 103 kgm3):

FIGURE 13–15 Example 13–9. The B force needed to lift the statue is F. B

FB = mH2Og = rH2OVg = A1.025 * 103 kgm3 BA3.0 * 10 –2 m3 BA9.8 ms2 B = 3.0 * 102 N. The weight of the statue is mg = (70 kg)A9.8 ms2 B = 6.9 * 102 N. Hence the force F needed to lift it is 690 N - 300 N = 390 N. It is as if the statue had a mass of only (390 N)A9.8 ms2 B = 40 kg. NOTE Here F = 390 N is the force needed to lift the statue without acceleration when it is under water. As the statue comes out of the water, the force F increases, reaching 690 N when the statue is fully out of the water. SECTION 13–7

ARCHIMEDES mgB

Buoyancy and Archimedes' Principle

349

Archimedes is said to have discovered his principle in his bath while thinking how he might determine whether the king's new crown was pure gold or a fake. Gold has a specific gravity of 19.3, somewhat higher than that of most metals, but a determination of specific gravity or density is not readily done directly because, even if the mass is known, the volume of an irregularly shaped object is not easily calculated. However, if the object is weighed in air ( = w) and also "weighed" while it is under water ( = w¿), the density can be determined using Archimedes' principle, as the following Example shows. The quantity w¿ is called the apparent weight in water, and is what a scale reads when the object is submerged in water (see Fig. 13–16); w¿ equals the true weight (w = mg) minus the buoyant force. EXAMPLE 13–10 Archimedes: Is the crown gold? When a crown of mass 14.7 kg is submerged in water, an accurate scale reads only 13.4 kg. Is the crown made of gold? APPROACH If the crown is gold, its density and specific gravity must be very high, SG = 19.3 (see Section 13–2 and Table 13–1). We determine the specific gravity using Archimedes' principle and the two free-body diagrams shown in Fig. 13–16. SOLUTION The apparent weight of the submerged object (the crown) is w¿ (what the scale reads), and is the force pulling down on the scale hook. By Newton's third law, w¿ equals the force F Tœ that the scale exerts on the crown in Fig. 13–16b. The sum of the forces on the crown is zero, so w¿ equals the actual weight w ( = mg) minus the buoyant force FB : w¿ = F Tœ = w - FB so w - w¿ = FB . Let V be the volume of the completely submerged object and rO its density (so rO V is its mass), and let rF be the density of the fluid (water). Then ArF VBg is the weight of fluid displaced A = FB B. Now we can write w = mg = rO Vg w - w¿ = FB = rF Vg. We divide these two equations and obtain rO Vg rO w . = = rF w - w¿ rF Vg We see that w(w - w¿) is equal to the specific gravity of the object if the fluid in which it is submerged is water ArF = 1.00 * 103 kgm3 B. Thus (14.7 kg)g 14.7 kg rO w = = = = 11.3. rH2O w - w¿ (14.7 kg - 13.4 kg)g 1.3 kg This corresponds to a density of 11,300 kgm3. The crown is not gold, but seems to be made of lead (see Table 13–1).

FIGURE 13–16 (a) A scale reads the mass of an object in air—in this case the crown of Example 13–10. All objects are at rest, so the tension FT in the connecting cord equals the weight w of the object: FT = mg. We show the free-body diagram of the crown, and FT is what causes the scale reading (it is equal to the net downward force on the scale, by Newton's third law). (b) Submerged, the crown has an additional force on it, the buoyant force FB . The net force is zero, so F Tœ + FB = mg ( = w). The scale now reads m¿ = 13.4 kg, where m¿ is related to the effective weight by w¿ = m¿g. Thus F Tœ = w¿ = w - FB .

CHAPTER 13

Fluids

scale

(FT = −mgB) B

(a)

350

w (13.4 kg) g

w (14.7 kg) g

mgB

F ′T

(b)

B w mgB

Archimedes' principle applies equally well to objects that float, such as wood. In general, an object floats on a fluid if its density (rO) is less than that of the fluid (rF). This is readily seen from Fig. 13–17a, where a submerged log will experience a net upward force and float to the surface if FB 7 mg; that is, if rF Vg 7 rO Vg or rF 7 rO . At equilibrium—that is, when floating—the buoyant force on an object has magnitude equal to the weight of the object. For example, a log whose specific gravity is 0.60 and whose volume is 2.0 m3 has a mass m = rO V = A0.60 * 103 kgm3 BA2.0 m3 B = 1200 kg. If the log is fully submerged, it will displace a mass of water mF = rF V = A1000 kgm3 BA2.0 m3 B = 2000 kg. Hence the buoyant force on the log will be greater than its weight, and it will float upward to the surface (Fig. 13–17). The log will come to equilibrium when it displaces 1200 kg of water, which means that 1.2 m3 of its volume will be submerged. This 1.2 m3 corresponds to 60% of the volume of the log (1.22.0 = 0.60), so 60% of the log is submerged. In general when an object floats, we have FB = mg, which we can write as (see Fig. 13–18) FB = mg rF Vdispl g = rO VO g, where VO is the full volume of the object and Vdispl is the volume of fluid it displaces ( = volume submerged). Thus Vdispl rO . = rF VO

FB = (2000 kg)g mO = 1200 kg V = 2.0 m3 (a)

mg = (1200 kg)g FB = (1200 kg)g

mg (b) FIGURE 13–17 (a) The fully submerged log accelerates upward because FB 7 mg. It comes to equilibrium (b) when ©F = 0, so FB = mg = (1200 kg)g. Thus 1200 kg, or 1.2 m3, of water is displaced. FIGURE 13–18 An object floating in equilibrium: FB = mg.

That is, the fraction of the object submerged is given by the ratio of the object's density to that of the fluid. If the fluid is water, this fraction equals the specific gravity of the object. EXAMPLE 13–11 Hydrometer calibration. A hydrometer is a simple instrument used to measure the specific gravity of a liquid by indicating how deeply the instrument sinks in the liquid. A particular hydrometer (Fig. 13–19) consists of a glass tube, weighted at the bottom, which is 25.0 cm long and 2.00 cm2 in cross-sectional area, and has a mass of 45.0 g. How far from the end should the 1.000 mark be placed? APPROACH The hydrometer will float in water if its density r is less than rw = 1.000 gcm3, the density of water. The fraction of the hydrometer submerged AVdisplacedVtotal B is equal to the density ratio rrw . SOLUTION The hydrometer has an overall density 45.0 g m r = = = 0.900 gcm3. V A2.00 cm2 B(25.0 cm) Thus, when placed in water, it will come to equilibrium when 0.900 of its volume is submerged. Since it is of uniform cross section, (0.900)(25.0 cm) = 22.5 cm of its length will be submerged. The specific gravity of water is defined to be 1.000, so the mark should be placed 22.5 cm from the weighted end.

FB = FVdispl g

mg = OVO g

FIGURE 13–19 A hydrometer. Example 13–11.

1.000

25.0 cm

EXERCISE C On the hydrometer of Example 13–11, will the marks above the 1.000 mark represent higher or lower values of density of the liquid in which it is submerged?

Archimedes' principle is also useful in geology. According to the theories of plate tectonics and continental drift, the continents float on a fluid "sea" of slightly deformable rock (mantle rock). Some interesting calculations can be done using very simple models, which we consider in the Problems at the end of the Chapter. Air is a fluid, and it too exerts a buoyant force. Ordinary objects weigh less in air than they do if weighed in a vacuum. Because the density of air is so small, the effect for ordinary solids is slight. There are objects, however, that float in air—helium-filled balloons, for example, because the density of helium is less than the density of air.

P H Y S I C S

A P P L I E D

Continental drift—plate tectonics

EXERCISE D Which of the following objects, submerged in water, experiences the largest magnitude of the buoyant force? (a) A 1-kg helium balloon; (b) 1 kg of wood; (c) 1 kg of ice; (d) 1 kg of iron; (e) all the same.

SECTION 13–7

Buoyancy and Archimedes' Principle

351

EXERCISE E Which of the following objects, submerged in water, experiences the largest magnitude of the buoyant force? (a) A 1-m3 helium balloon; (b) 1 m3 of wood; (c) 1 m3 of ice; (d) 1 m3 of iron; (e) all the same.

EXAMPLE 13–12 Helium balloon. What volume V of helium is needed if a balloon is to lift a load of 180 kg (including the weight of the empty balloon)? APPROACH The buoyant force on the helium balloon, FB , which is equal to the weight of displaced air, must be at least equal to the weight of the helium plus the weight of the balloon and load (Fig. 13–20). Table 13–1 gives the density of helium as 0.179 kgm3. SOLUTION The buoyant force must have a minimum value of

mHe g B

FB = AmHe + 180 kgBg.

This equation can be written in terms of density using Archimedes' principle: rair Vg = ArHe V + 180 kgBg.

Solving now for V, we find mload g

V =

180 kg 180 kg = = 160 m3. 3 rair - rHe A1.29 kgm - 0.179 kgm3 B

NOTE This is the minimum volume needed near the Earth's surface, where rair = 1.29 kgm3. To reach a high altitude, a greater volume would be needed since the density of air decreases with altitude.

FIGURE 13–20 Example 13–12.

13–8

Fluids in Motion; Flow Rate and the Equation of Continuity

We now turn to the subject of fluids in motion, which is called fluid dynamics, or (especially if the fluid is water) hydrodynamics. We can distinguish two main types of fluid flow. If the flow is smooth, such that neighboring layers of the fluid slide by each other smoothly, the flow is said to be streamline or laminar flow.† In streamline flow, each particle of the fluid follows a smooth path, called a streamline, and these paths do not cross one another (Fig. 13–21a). Above a certain speed, the flow becomes turbulent. Turbulent flow is characterized by erratic, small, whirlpool-like circles called eddy currents or eddies (Fig. 13–21b). Eddies absorb a great deal of energy, and although a certain amount of internal friction called viscosity is present even during streamline flow, it is much greater when the flow is turbulent. A few tiny drops of ink or food coloring dropped into a moving liquid can quickly reveal whether the flow is streamline or turbulent. †

FIGURE 13–21 (a) Streamline, or laminar, flow; (b) turbulent flow. The photos show airflow around an airfoil or airplane wing (more in Section 13–10).

352

CHAPTER 13

Fluids

The word laminar means "in layers."

(a)

(b)

Let us consider the steady laminar flow of a fluid through an enclosed tube or pipe as shown in Fig. 13–22. First we determine how the speed of the fluid changes when the size of the tube changes. The mass flow rate is defined as the mass ¢m of fluid that passes a given point per unit time ¢ t: ¢m . mass flow rate = ¢t In Fig. 13–22, the volume of fluid passing point 1 (that is, through area A 1) in a time ¢ t is A 1 ¢l1 , where ¢l1 is the distance the fluid moves in time ¢ t. Since the velocity† of fluid passing point 1 is v1 = ¢l1¢ t, the mass flow rate through area A 1 is r1 ¢V1 r1 A 1 ¢l1 ¢m1 = = = r1 A 1 v1 , ¢t ¢t ¢t

∆l1

∆l2 vB1

vB2 A2

FIGURE 13–22 Fluid flow through a pipe of varying diameter.

where ¢V1 = A 1 ¢l1 is the volume of mass ¢m1 , and r1 is the fluid density. Similarly, at point 2 (through area A 2), the flow rate is r2 A 2 v2 . Since no fluid flows in or out the sides, the flow rates through A 1 and A 2 must be equal. Thus, since ¢m1 ¢m2 , = ¢t ¢t then r1 A 1 v1 = r2 A 2 v2 . (13–7a) This is called the equation of continuity. If the fluid is incompressible (r doesn't change with pressure), which is an excellent approximation for liquids under most circumstances (and sometimes for gases as well), then r1 = r2 , and the equation of continuity becomes A 1 v1 = A 2 v2 .

[r = constant] (13–7b)

The product Av represents the volume rate of flow (volume of fluid passing a given point per second), since ¢V¢ t = A ¢l¢ t = Av, which in SI units is m3s. Equation 13–7b tells us that where the cross-sectional area is large, the velocity is small, and where the area is small, the velocity is large. That this is reasonable can be seen by looking at a river. A river flows slowly through a meadow where it is broad, but speeds up to torrential speed when passing through a narrow gorge.

v2 Npr2cap = v1pr2aorta so

v1 r2aorta 0.40 ms 1.2 * 10 –2 m 2 = ¢ ≤¢ ≤ L 7 * 109, N = –4 2 v2 r cap 5 * 10 ms 4 * 10 –6 m

P H Y S I C S

A P P L I E D

Blood flow FIGURE 13–23 Human circulatory system. Head Arms

c Lungs v

Aorta

c Body organs c c

Trunk

Kidneys

Legs

Veins

Heart Arteries

EXAMPLE 13–13 ESTIMATE Blood flow. In humans, blood flows from the heart into the aorta, from which it passes into the major arteries. These branch into the small arteries (arterioles), which in turn branch into myriads of tiny capillaries, Fig. 13–23. The blood returns to the heart via the veins. The radius of the aorta is about 1.2 cm, and the blood passing through it has a speed of about 40 cms. A typical capillary has a radius of about 4 * 10 –4 cm, and blood flows through it at a speed of about 5 * 10 –4 ms. Estimate the number of capillaries that are in the body. APPROACH We assume the density of blood doesn't vary significantly from the aorta to the capillaries. By the equation of continuity, the volume flow rate in the aorta must equal the volume flow rate through all the capillaries. The total area of all the capillaries is given by the area of one capillary multiplied by the total number N of capillaries. SOLUTION Let A 1 be the area of the aorta and A 2 be the area of all the capillaries through which blood flows. Then A 2 = Npr2cap , where rcap L 4 * 10 –4 cm is the estimated average radius of one capillary. From the equation of continuity (Eq. 13–7b), we have v2 A 2 = v1 A 1

or on the order of 10 billion capillaries. †

If there were no viscosity, the velocity would be the same across a cross section of the tube. Real fluids have viscosity, and this internal friction causes different layers of the fluid to flow at different speeds. In this case v1 and v2 represent the average speeds at each cross section.

v = valves c = capillaries

SECTION 13–8

353

P H Y S I C S

A P P L I E D

Heating duct Point 1

Point 2

l2 FIGURE 13–24 Example 13–14.

EXAMPLE 13–14 Heating duct to a room. What area must a heating duct have if air moving 3.0 ms along it can replenish the air every 15 minutes in a room of volume 300 m3? Assume the air's density remains constant. APPROACH We apply the equation of continuity at constant density, Eq. 13–7b, to the air that flows through the duct (point 1 in Fig. 13–24) and then into the room (point 2). The volume flow rate in the room equals the volume of the room divided by the 15-min replenishing time. SOLUTION Consider the room as a large section of the duct, Fig. 13–24, and think of air equal to the volume of the room as passing by point 2 in t = 15 min = 900 s. Reasoning in the same way we did to obtain Eq. 13–7a (changing ¢ t to t), we write v2 = l2t so A 2 v2 = A 2 l2t = V2t, where V2 is the volume of the room. Then the equation of continuity becomes A 1 v1 = A 2 v2 = V2t and V2 300 m3 = = 0.11 m2. v1 t (3.0 ms)(900 s) If the duct is square, then each side has length l = 2A = 0.33 m, or 33 cm. A rectangular duct 20 cm * 55 cm will also do. A1 =

13–9

FIGURE 13–25 Fluid flow: for derivation of Bernoulli's equation.

∆l2

∆l1

P2 A2

(a)

At area A 2 , the work done on our cross section of fluid is W2 = –P2 A 2 ¢l2 . The negative sign is present because the force exerted on the fluid is opposite to the motion (thus the fluid shown in color does work on the fluid to the right of point 2). Work is also done on the fluid by the force of gravity. The net effect of the process shown in Fig. 13–25 is to move a mass m of volume A 1 ¢l1 (= A 2 ¢l2 , since the

(b)

354

Have you ever wondered why an airplane can fly, or how a sailboat can move against the wind? These are examples of a principle worked out by Daniel Bernoulli (1700–1782) concerning fluids in motion. In essence, Bernoulli's principle states that where the velocity of a fluid is high, the pressure is low, and where the velocity is low, the pressure is high. For example, if the pressures in the fluid at points 1 and 2 of Fig. 13–22 are measured, it will be found that the pressure is lower at point 2, where the velocity is greater, than it is at point 1, where the velocity is smaller. At first glance, this might seem strange; you might expect that the greater speed at point 2 would imply a higher pressure. But this cannot be the case. For if the pressure in the fluid at point 2 were higher than at point 1, this higher pressure would slow the fluid down, whereas in fact it has sped up in going from point 1 to point 2. Thus the pressure at point 2 must be less than at point 1, to be consistent with the fact that the fluid accelerates. To help clarify any misconceptions, a faster fluid would exert a greater force on an obstacle placed in its path. But that is not what we mean by the pressure in a fluid, and besides we are not considering obstacles that interrupt the flow. We are examining smooth streamline flow. The fluid pressure is exerted on the walls of a tube or pipe, or on the surface of any material the fluid passes over. Bernoulli developed an equation that expresses this principle quantitatively. To derive Bernoulli's equation, we assume the flow is steady and laminar, the fluid is incompressible, and the viscosity is small enough to be ignored. To be general, we assume the fluid is flowing in a tube of nonuniform cross section that varies in height above some reference level, Fig. 13–25. We will consider the volume of fluid shown in color and calculate the work done to move it from the position shown in Fig. 13–25a to that shown in Fig. 13–25b. In this process, fluid entering area A 1 flows a distance ¢l1 and forces the fluid at area A 2 to move a distance ¢l2 . The fluid to the left of area A 1 exerts a pressure P1 on our section of fluid and does an amount of work W1 = F1 ¢l1 = P1 A 1 ¢l1 .

∆l2

∆l1

vB2

Bernoulli's Equation

CHAPTER 13

Fluids

fluid is incompressible) from point 1 to point 2, so the work done by gravity is W3 = –mgAy2 - y1 B,

where y1 and y2 are heights of the center of the tube above some (arbitrary) reference level. In the case shown in Fig. 13–25, this term is negative since the motion is uphill against the force of gravity. The net work W done on the fluid is thus W = W1 + W2 + W3 W = P1 A 1 ¢l1 - P2 A 2 ¢l2 - mgy2 + mgy1 . According to the work-energy principle (Section 7–4), the net work done on a system is equal to its change in kinetic energy. Hence 1 2 2 mv2

1 2 2 mv1

= P1 A 1 ¢l1 - P2 A 2 ¢l2 - mgy2 + mgy1 .

The mass m has volume A 1 ¢l1 = A 2 ¢l2 for an incompressible fluid. Thus we can substitute m = rA 1 ¢l1 = rA 2 ¢l2 , and then divide through by A 1 ¢l1 = A 2 ¢l2 , to obtain 1 1 2 2 2 rv2 - 2 rv1 = P1 - P2 - rgy2 + rgy1 , which we rearrange to get P1 +

1 2 2 rv1

+ rgy1 = P2 +

1 2 2 rv2

(13–8)

+ rgy2 .

Bernoulli's equation

This is Bernoulli's equation. Since points 1 and 2 can be any two points along a tube of flow, Bernoulli's equation can be written as P +

1 2 2 rv

+ rgy = constant

at every point in the fluid, where y is the height of the center of the tube above a fixed reference level. [Note that if there is no flow Av1 = v2 = 0B, then Eq. 13–8 reduces to the hydrostatic equation, Eq. 13–6a: P2 - P1 = –rgAy2 - y1 B.] Bernoulli's equation is an expression of the law of energy conservation, since we derived it from the work-energy principle. EXERCISE F As water in a level pipe passes from a narrow cross section of pipe to a wider cross section, how does the pressure against the walls change?

EXAMPLE 13–15 Flow and pressure in a hot-water heating system. Water circulates throughout a house in a hot-water heating system. If the water is pumped at a speed of 0.50 ms through a 4.0-cm-diameter pipe in the basement under a pressure of 3.0 atm, what will be the flow speed and pressure in a 2.6-cm-diameter pipe on the second floor 5.0 m above? Assume the pipes do not divide into branches. APPROACH We use the equation of continuity at constant density to determine the flow speed on the second floor, and then Bernoulli's equation to find the pressure. SOLUTION We take v2 in the equation of continuity, Eq. 13–7, as the flow speed on the second floor, and v1 as the flow speed in the basement. Noting that the areas are proportional to the radii squared AA = pr 2 B, we obtain v2 =

(0.020 m)2 v1 A 1 v1pr21 = = (0.50 ms) = 1.2 ms. 2 A2 pr2 (0.013 m)2

To find the pressure on the second floor, we use Bernoulli's equation (Eq. 13–8): P2 = P1 + rgAy1 - y2 B +

1 2 2 rAv1

- v22 B

= A3.0 * 105 Nm2 B + A1.0 * 103 kgm3 BA9.8 ms2 B(–5.0 m) +

1 2

A1.0 * 103 kgm3 B C(0.50 ms)2 - (1.2 ms)2 D

= A3.0 * 105 Nm2 B - A4.9 * 104 Nm2 B - A6.0 * 102 Nm2 B = 2.5 * 105 Nm2 = 2.5 atm. NOTE The velocity term contributes very little in this case. SECTION 13–9

Bernoulli's Equation

355

13–10

v2 ≈ 0 y2 − y1 v1

FIGURE 13–26 Torricelli's theorem: v1 = 32gAy2 - y1 B .

Applications of Bernoulli's Principle: Torricelli, Airplanes, Baseballs, TIA

Bernoulli's equation can be applied to many situations. One example is to calculate the velocity, v1 , of a liquid flowing out of a spigot at the bottom of a reservoir, Fig. 13–26. We choose point 2 in Eq. 13–8 to be the top surface of the liquid. Assuming the diameter of the reservoir is large compared to that of the spigot, v2 will be almost zero. Points 1 (the spigot) and 2 (top surface) are open to the atmosphere, so the pressure at both points is equal to atmospheric pressure: P1 = P2 . Then Bernoulli's equation becomes 1 2 2 rv1

FIGURE 13–27 Examples of Bernoulli's principle: (a) atomizer, (b) Ping-Pong ball in jet of air. Low P

(a)

High P (no flow)

(b)

+ rgy1 = rgy2

v1 = 32gAy2 - y1 B .

(13–9)

This result is called Torricelli's theorem. Although it is seen to be a special case of Bernoulli's equation, it was discovered a century earlier by Evangelista Torricelli. Equation 13–9 tells us that the liquid leaves the spigot with the same speed that a freely falling object would attain if falling from the same height. This should not be too surprising since Bernoulli's equation relies on the conservation of energy. Another special case of Bernoulli's equation arises when a fluid is flowing horizontally with no significant change in height; that is, y1 = y2 . Then Eq. 13–8 becomes P1 +

1 2 2 rv1

= P2 +

1 2 2 rv2 ,

(13–10)

which tells us quantitatively that the speed is high where the pressure is low, and vice versa. It explains many common phenomena, some of which are illustrated in Figs. 13–27 to 13–32. The pressure in the air blown at high speed across the top of the vertical tube of a perfume atomizer (Fig. 13–27a) is less than the normal air pressure acting on the surface of the liquid in the bowl. Thus atmospheric pressure in the bowl pushes the perfume up the tube because of the lower pressure at the top. A Ping-Pong ball can be made to float above a blowing jet of air (some vacuum cleaners can blow air), Fig. 13–27b; if the ball begins to leave the jet of air, the higher pressure in the still air outside the jet pushes the ball back in. EXERCISE G Return to Chapter-Opening Question 2, page 339, and answer it again now. Try to explain why you may have answered differently the first time. Try it and see.

Airplane Wings and Dynamic Lift

FIGURE 13–28 Lift on an airplane wing. We are in the reference frame of the wing, seeing the air flow by. Lower pressure

Higher pressure P H Y S I C S

A P P L I E D

Airplanes and dynamic lift

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Airplanes experience a "lift" force on their wings, keeping them up in the air, if they are moving at a sufficiently high speed relative to the air and the wing is tilted upward at a small angle (the "attack angle"), as in Fig. 13–28, where streamlines of air are shown rushing by the wing. (We are in the reference frame of the wing, as if sitting on the wing.) The upward tilt, as well as the rounded upper surface of the wing, causes the streamlines to be forced upward and to be crowded together above the wing. The area for air flow between any two streamlines is reduced as the streamlines get closer together, so from the equation of continuity AA 1 v1 = A 2 v2 B, the air speed increases above the wing where the streamlines are squished together. (Recall also how the crowded streamlines in a pipe constriction, Fig. 13–22, indicate the velocity is higher in the constriction.) Because the air speed is greater above the wing than below it, the pressure above the wing is less than the pressure below the wing (Bernoulli's principle). Hence there is a net upward force on the wing called dynamic lift. Experiments show that the speed of air above the wing can even be double the speed of the air below it. (Friction between the air and wing exerts a drag force, toward the rear, which must be overcome by the plane's engines.) A flat wing, or one with symmetric cross section, will experience lift as long as the front of the wing is tilted upward (attack angle). The wing shown in Fig. 13–28 can experience lift even if the attack angle is zero, because the rounded upper surface deflects air up, squeezing the streamlines together. Airplanes can fly upside down, experiencing lift, if the attack angle is sufficient to deflect streamlines up and closer together.

Our picture considers streamlines; but if the attack angle is larger than about 15°, turbulence sets in (Fig. 13–21b) leading to greater drag and less lift, causing the wing to "stall" and the plane to drop. From another point of view, the upward tilt of a wing means the air moving horizontally in front of the wing is deflected downward; the change in momentum of the rebounding air molecules results in an upward force on the wing (Newton's third law).

Wind

Jib

F water

el Ke

F wind Mainsail

Sailboats A sailboat can move against the wind, with the aid of the Bernoulli effect, by setting the sails at an angle, as shown in Fig. 13–29. The air travels rapidly over the bulging front surface of the sail, and the relatively still air filling the sail exerts a greater B pressure behind the sail, resulting in a net force on the sail, Fwind . This force would tend to make the boat move sideways if it weren't for the keel that extends vertically downB ward beneath the water: the water exerts a force AFwater B on the keel nearly perpendicular B to the keel. The resultant of these two forces AFR B is almost directly forward as shown.

FIGURE 13–29 Sailboat sailing against the wind.

Home plate

Baseball Curve Why a spinning pitched baseball (or tennis ball) curves can also be explained using Bernoulli's principle. It is simplest if we put ourselves in the reference frame of the ball, with the air rushing by, just as we did for the airplane wing. Suppose the ball is rotating counterclockwise as seen from above, Fig. 13–30. A thin layer of air ("boundary layer") is being dragged around by the ball. We are looking down on the ball, and at point A in Fig. 13–30, this boundary layer tends to slow down the oncoming air. At point B, the air rotating with the ball adds its speed to that of the oncoming air, so the air speed is higher at B than at A. The higher speed at B means the pressure is lower at B than at A, resulting in a net force toward B. The ball's path curves toward the left (as seen by the pitcher).

FIGURE 13–30 Looking down on a pitched baseball heading toward home plate.We are in the reference frame of the baseball, with the air flowing by. Basilar artery (to brain)

Lack of Blood to the Brain—TIA In medicine, one of many applications of Bernoulli's principle is to explain a TIA, a transient ischemic attack (meaning a temporary lack of blood supply to the brain). A person suffering a TIA may experience symptoms such as dizziness, double vision, headache, and weakness of the limbs. A TIA can occur as follows. Blood normally flows up to the brain at the back of the head via the two vertebral arteries—one going up each side of the neck—which meet to form the basilar artery just below the brain, as shown in Fig. 13–31. The vertebral arteries issue from the subclavian arteries, as shown, before the latter pass to the arms. When an arm is exercised vigorously, blood flow increases to meet the needs of the arm's muscles. If the subclavian artery on one side of the body is partially blocked, however, as in arteriosclerosis (hardening of the arteries), the blood velocity will have to be higher on that side to supply the needed blood. (Recall the equation of continuity: smaller area means larger velocity for the same flow rate, Eqs. 13–7.) The increased blood velocity past the opening to the vertebral artery results in lower pressure (Bernoulli's principle). Thus blood rising in the vertebral artery on the "good" side at normal pressure can be diverted down into the other vertebral artery because of the low pressure on that side, instead of passing upward to the brain. Hence the blood supply to the brain is reduced.

Other Applications A venturi tube is essentially a pipe with a narrow constriction (the throat). The flowing fluid speeds up as it passes through this constriction, so the pressure is lower in the throat. A venturi meter, Fig. 13–32, is used to measure the flow speed of gases and liquids, including blood velocity in arteries. Why does smoke go up a chimney? It's partly because hot air rises (it's less dense and therefore buoyant). But Bernoulli's principle also plays a role. When wind blows across the top of a chimney, the pressure is less there than inside the house. Hence, air and smoke are pushed up the chimney by the higher indoor pressure. Even on an apparently still night there is usually enough ambient air flow at the top of a chimney to assist upward flow of smoke.

Left vertebral artery

Right vertebral artery

Subclavian artery

Constriction Aorta

FIGURE 13–31 Rear of the head and shoulders showing arteries leading to the brain and to the arms. High blood velocity past the constriction in the left subclavian artery causes low pressure in the left vertebral artery, in which a reverse (downward) blood flow can then occur, resulting in a TIA, a loss of blood to the brain. FIGURE 13–32 Venturi meter. P1

vB1

vB2

SECTION 13–10

357

Bernoulli's equation ignores the effects of friction (viscosity) and the compressibility of the fluid. The energy that is transformed to internal (or potential) energy due to compression and to thermal energy by friction can be taken into account by adding terms to Eq. 13–8. These terms are difficult to calculate theoretically and are normally determined empirically. They do not significantly alter the explanations for the phenomena described above.

* 13–11

Viscosity

Real fluids have a certain amount of internal friction called viscosity, as mentioned in Section 13–8. Viscosity exists in both liquids and gases, and is essentially a Moving plate v F frictional force between adjacent layers of fluid as the layers move past one another. In liquids, viscosity is due to the electrical cohesive forces between the Velocity Fluid gradient l molecules. In gases, it arises from collisions between the molecules. Stationary plate The viscosity of different fluids can be expressed quantitatively by a coefficient of viscosity, h (the Greek lowercase letter eta), which is defined in the following way. A FIGURE 13–33 thin layer of fluid is placed between two flat plates. One plate is stationary and the Determination of viscosity. other is made to move, Fig. 13–33. The fluid directly in contact with each plate is held to the surface by the adhesive force between the molecules of the liquid and those of the plate. Thus the upper surface of the fluid moves with the same speed v as the TABLE 13–3 upper plate, whereas the fluid in contact with the stationary plate remains stationary. Coefficients of Viscosity The stationary layer of fluid retards the flow of the layer just above it, which in turn Fluid Coefficient retards the flow of the next layer, and so on. Thus the velocity varies continuously (temperature of Viscosity, from 0 to v, as shown. The increase in velocity divided by the distance over which this in °C) H (Pa # s)† change is made—equal to vl —is called the velocity gradient. To move the upper Water (0°) 1.8 * 10 –3 plate requires a force, which you can verify by moving a flat plate across a puddle of (20°) 1.0 * 10 –3 syrup on a table. For a given fluid, it is found that the force required, F, is propor(100°) 0.3 * 10 –3 tional to the area of fluid in contact with each plate, A, and to the speed, v, and is Whole blood (37°) L 4 * 10 –3 inversely proportional to the separation, l, of the plates: F r vAl. For different Blood plasma (37°) L 1.5 * 10 –3 fluids, the more viscous the fluid, the greater is the required force. Hence the proporEthyl alcohol (20°) 1.2 * 10 –3 tionality constant for this equation is defined as the coefficient of viscosity, h: v Engine oil (30°) F = hA . (13–11) (SAE 10) 200 * 10 –3 l Glycerine (20°) 1500 * 10 –3 Solving for h, we find h = FlvA. The SI unit for h is Nsm2 = –3 Air (20°) 0.018 * 10 Pas (pascalsecond). In the cgs system, the unit is dynescm2, which is called a –3 poise (P). Viscosities are often given in centipoise A1 cP = 10–2 PB. Table 13–3 lists Hydrogen (0°) 0.009 * 10 –3 the coefficient of viscosity for various fluids. The temperature is also specified, Water vapor (100°) 0.013 * 10 since it has a strong effect; the viscosity of liquids such as motor oil, for example, † 1 Pas = 10 P = 1000 cP. decreases rapidly as temperature increases.† B

* 13–12

Flow in Tubes: Poiseuille's Equation, Blood Flow

If a fluid had no viscosity, it could flow through a level tube or pipe without a force being applied. Viscosity acts like a sort of friction (between fluid layers moving at slightly different speeds), so a pressure difference between the ends of a level tube is necessary for the steady flow of any real fluid, be it water or oil in a pipe, or blood in the circulatory system of a human. The French scientist J. L. Poiseuille (1799–1869), who was interested in the physics of blood circulation (and after whom the poise is named), determined how the variables affect the flow rate of an incompressible fluid undergoing laminar flow in a cylindrical tube. His result, known as Poiseuille's equation, is: pR4 AP1 - P2 B , (13–12) Q = 8hl where R is the inside radius of the tube, l is the tube length, P1 - P2 is the pressure †

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The Society of Automotive Engineers assigns numbers to represent the viscosity of oils: 30 weight (SAE 30) is more viscous than 10 weight. Multigrade oils, such as 20–50, are designed to maintain viscosity as temperature increases; 20–50 means the oil is 20 wt when cool but is like a 50-wt pure oil when it is hot (engine running temperature).

difference between the ends, h is the coefficient of viscosity, and Q is the volume rate of flow (volume of fluid flowing past a given point per unit time which in SI has units of m3s). Equation 13–12 applies only to laminar flow. Poiseuille's equation tells us that the flow rate Q is directly proportional to the "pressure gradient," AP1 - P2 Bl, and it is inversely proportional to the viscosity of the fluid. This is just what we might expect. It may be surprising, however, that Q also depends on the fourth power of the tube's radius. This means that for the same pressure gradient, if the tube radius is halved, the flow rate is decreased by a factor of 16! Thus the rate of flow, or alternately the pressure required to maintain a given flow rate, is greatly affected by only a small change in tube radius. An interesting example of this R 4 dependence is blood flow in the human body. Poiseuille's equation is valid only for the streamline flow of an incompressible fluid. So it cannot be precisely accurate for blood whose flow is not without turbulence and that contains blood cells (whose diameter is almost equal to that of a capillary). Nonetheless, Poiseuille's equation does give a reasonable first approximation. Because the radius of arteries is reduced as a result of arteriosclerosis (thickening and hardening of artery walls) and by cholesterol buildup, the pressure gradient must be increased to maintain the same flow rate. If the radius is reduced by half, the heart would have to increase the pressure by a factor of about 2 4 = 16 in order to maintain the same blood-flow rate. The heart must work much harder under these conditions, but usually cannot maintain the original flow rate. Thus, high blood pressure is an indication both that the heart is working harder and that the blood-flow rate is reduced.

* 13–13

P H Y S I C S

A P P L I E D

Blood flow

Surface Tension and Capillarity

The surface of a liquid at rest behaves in an interesting way, almost as if it were a stretched membrane under tension. For example, a drop of water on the end of a dripping faucet, or hanging from a thin branch in the early morning dew (Fig. 13–34), forms into a nearly spherical shape as if it were a tiny balloon filled with water. A steel needle can be made to float on the surface of water even though it is denser than the water. The surface of a liquid acts like it is under tension, and this tension, acting along the surface, arises from the attractive forces between the molecules. This effect is called surface tension. More specifically, a quantity called the surface tension, g (the Greek letter gamma), is defined as the force F per unit length l that acts perpendicular to any line or cut in a liquid surface, tending to pull the surface closed: g = F. (13–13) l To understand this, consider the U-shaped apparatus shown in Fig. 13–35 which encloses a thin film of liquid. Because of surface tension, a force F is required to pull the movable wire and thus increase the surface area of the liquid. The liquid contained by the wire apparatus is a thin film having both a top and a bottom surface. Hence the total length of the surface being increased is 2l, and the surface tension is g = F2l. A delicate apparatus of this type can be used to measure the surface tension of various liquids. The surface tension of water is 0.072 Nm at 20°C. Table 13–4 gives the values for several substances. Note that temperature has a considerable effect on the surface tension. FIGURE 13–35 U-shaped wire apparatus holding a film of liquid to measure surface tension (g = F2l).

γ F

γ γ

(a) Top view

Liquid Wire (b) Edge view (magnified)

*SECTION 13–13

FIGURE 13–34 Spherical water droplets, dew on a blade of grass.

TABLE 13–4 Surface Tension of Some Substances Substance (temperature in °C)

Surface Tension (Nm)

Mercury (20°) Blood, whole (37°) Blood, plasma (37°) Alcohol, ethyl (20°) Water (0°) (20°) (100°) Benzene (20°) Soap solution (20°) Oxygen ( –193°)

0.44 0.058 0.073 0.023 0.076 0.072 0.059 0.029 L 0.025 0.016

Surface Tension and Capillarity

359

Because of surface tension, some insects (Fig. 13–36) can walk on water, and objects more dense than water, such as a steel needle, can float on the surface. Figure 13–37a shows how the surface tension can support the weight w of an object. Actually, the object sinks slightly into the fluid, so w is the "effective weight" of that object—its true weight less the buoyant force. g = F/l

FIGURE 13–36 A water strider.

FIGURE 13–37 Surface tension acting on (a) a sphere, and (b) an insect leg. Example 13–16.

(a)

r r

g = F/l

(b)

EXAMPLE 13–16 ESTIMATE Insect walks on water. The base of an insect's leg is approximately spherical in shape, with a radius of about 2.0 * 10–5 m. The 0.0030-g mass of the insect is supported equally by its six legs. Estimate the angle u (see Fig. 13–37) for an insect on the surface of water. Assume the water temperature is 20°C. APPROACH Since the insect is in equilibrium, the upward surface tension force is equal to the pull of gravity downward on each leg. We ignore the buoyant force for this estimate. SOLUTION For each leg, we assume the surface tension force acts all around a circle of radius r, at an angle u, as shown in Fig. 13–37a. Only the vertical component, g cos u, acts to balance the weight mg. So we set the length l in Eq. 13–13 equal to the circumference of the circle, l L 2pr. Then the net upward force due to surface tension is Fy L (g cos u)l L 2prg cos u. We set this surface tension force equal to one-sixth the weight of the insect since it has six legs: 2prg cos u L 16 mg

A3.0 * 10 –6 kgBA9.8 ms2 B 0.49 cos u L = 0.54. 0.90 So u L 57°. If cos u had come out greater than 1, the surface tension would not be great enough to support the insect's weight. NOTE Our estimate ignored the buoyant force and ignored any difference between the radius of the insect's "foot" and the radius of the surface depression. (6.28)A2.0 * 10 –5 mB(0.072 Nm) cos u L

Water

Mercury

(a)

(b)

FIGURE 13–38 (a) Water "wets" the surface of glass, whereas (b) mercury does not "wet" the glass. FIGURE 13–39 Capillarity.

(a) Glass tube in water

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(b) Glass tube in mercury

Fluids

1 6

Soaps and detergents lower the surface tension of water. This is desirable for washing and cleaning since the high surface tension of pure water prevents it from penetrating easily between the fibers of material and into tiny crevices. Substances that reduce the surface tension of a liquid are called surfactants. Surface tension plays a role in another interesting phenomenon, capillarity. It is a common observation that water in a glass container rises up slightly where it touches the glass, Fig. 13–38a. The water is said to "wet" the glass. Mercury, on the other hand, is depressed when it touches the glass, Fig. 13–38b; the mercury does not wet the glass. Whether a liquid wets a solid surface is determined by the relative strength of the cohesive forces between the molecules of the liquid compared to the adhesive forces between the molecules of the liquid and those of the container. Cohesion refers to the force between molecules of the same type, whereas adhesion refers to the force between molecules of different types. Water wets glass because the water molecules are more strongly attracted to the glass molecules than they are to other water molecules. The opposite is true for mercury: the cohesive forces are stronger than the adhesive forces. In tubes having very small diameters, liquids are observed to rise or fall relative to the level of the surrounding liquid. This phenomenon is called capillarity, and such thin tubes are called capillaries. Whether the liquid rises or falls (Fig. 13–39) depends on the relative strengths of the adhesive and cohesive forces. Thus water rises in a glass tube, whereas mercury falls. The actual amount of rise (or fall) depends on the surface tension—which is what keeps the liquid surface from breaking apart.

Outlet

Pumps, and the Heart

We conclude this Chapter with a brief discussion of pumps, including the heart. Pumps can be classified into categories according to their function. A vacuum pump is designed to reduce the pressure (usually of air) in a given vessel. A force pump, on the other hand, is a pump that is intended to increase the pressure—for example, to lift a liquid (such as water from a well) or to push a fluid through a pipe. Figure 13–40 illustrates the principle behind a simple reciprocating pump. It could be a vacuum pump, in which case the intake is connected to the vessel to be evacuated. A similar mechanism is used in some force pumps, and in this case the fluid is forced under increased pressure through the outlet. A centrifugal pump (Fig. 13–41), or any force pump, can be used as a circulating pump—that is, to circulate a fluid around a closed path, such as the cooling water or lubricating oil in an automobile. The heart of a human (and of other animals as well) is essentially a circulating pump. The action of a human heart is shown in Fig. 13–42. There are actually two separate paths for blood flow. The longer path takes blood to the parts of the body, via the arteries, bringing oxygen to body tissues and picking up carbon dioxide, which it carries back to the heart via veins. This blood is then pumped to the lungs (the second path), where the carbon dioxide is released and oxygen is taken up. The oxygen-laden blood is returned to the heart, where it is again pumped to the tissues of the body.

Piston Intake

* 13–14

FIGURE 13–40 One kind of pump: the intake valve opens and air (or fluid that is being pumped) fills the empty space when the piston moves to the left. When the piston moves to the right (not shown), the outlet valve opens and fluid is forced out. FIGURE 13–41 Centrifugal pump: the rotating blades force fluid through the outlet pipe; this kind of pump is used in vacuum cleaners and as a water pump in automobiles. Intake

Outlet

FIGURE 13–42 (a) In the diastole phase, the heart relaxes between beats. Blood moves into the heart; both atria fill rapidly. (b) When the atria contract, the systole or pumping phase begins. The contraction pushes the blood through the mitral and tricuspid valves into the ventricles. (c) The contraction of the ventricles forces the blood through the semilunar valves into the pulmonary artery, which leads to the lungs, and to the aorta (the body's largest artery), which leads to the arteries serving all the body. (d) When the heart relaxes, the semilunar valves close; blood fills the atria, beginning the cycle again. Right atrium (from the body)

Tricuspid valve

Pulmonary artery (to lungs)

Left atrium (from the lungs)

Mitral valve (a)

Aorta (to the body)

Semilunar valves Right ventricle

(b)

Left ventricle

Rotor

Semilunar valves (c)

Right atrium Left atrium Tricuspid valve

Mitral valve (d)

Summary The three common phases of matter are solid, liquid, and gas. Liquids and gases are collectively called fluids, meaning they have the ability to flow. The density of a material is defined as its mass per unit volume: m. (13–1) r = V Specific gravity is the ratio of the density of the material to the density of water (at 4°C). Pressure is defined as force per unit area: F. P = (13–2) A The pressure P at a depth h in a liquid is given by P = rgh,

(13–3)

where r is the density of the liquid and g is the acceleration due

to gravity. If the density of a fluid is not uniform, the pressure P varies with height y as dP = –rg. dy

(13–4)

Pascal's principle says that an external pressure applied to a confined fluid is transmitted throughout the fluid. Pressure is measured using a manometer or other type of gauge. A barometer is used to measure atmospheric pressure. Standard atmospheric pressure (average at sea level) is 1.013 * 105 Nm2. Gauge pressure is the total (absolute) pressure less atmospheric pressure. Archimedes' principle states that an object submerged wholly or partially in a fluid is buoyed up by a force equal to the weight of fluid it displaces AFB = mF g = rF Vdispl gB.

Summary

361

Fluid flow can be characterized either as streamline (sometimes called laminar), in which the layers of fluid move smoothly and regularly along paths called streamlines, or as turbulent, in which case the flow is not smooth and regular but is characterized by irregularly shaped whirlpools. Fluid flow rate is the mass or volume of fluid that passes a given point per unit time. The equation of continuity states that for an incompressible fluid flowing in an enclosed tube, the product of the velocity of flow and the cross-sectional area of the tube remains constant: Av = constant. (13–7b) Bernoulli's principle tells us that where the velocity of a

fluid is high, the pressure in it is low, and where the velocity is low, the pressure is high. For steady laminar flow of an incompressible and nonviscous fluid, Bernoulli's equation, which is based on the law of conservation of energy, is P1 + 12 rv21 + rgy1 = P2 + 12 rv22 + rgy2 , (13–8) for two points along the flow. [*Viscosity refers to friction within a fluid and is essentially a frictional force between adjacent layers of fluid as they move past one another.] [*Liquid surfaces hold together as if under tension (surface tension), allowing drops to form and objects like needles and insects to stay on the surface.]

Questions 1. If one material has a higher density than another, must the molecules of the first be heavier than those of the second? Explain. 2. Airplane travelers sometimes note that their cosmetics bottles and other containers have leaked during a flight. What might cause this? 3. The three containers in Fig. 13–43 are filled with water to the same height and have the same surface area at the base; hence the water pressure, and the total force on the base of each, is the same. Yet the total weight of water is different for each. Explain this "hydrostatic paradox."

6. 7.

8. 9. 10. 11.

FIGURE 13–43 Question 3. Consider what happens when you push both a pin and the blunt end of a pen against your skin with the same force. Decide what determines whether your skin is cut—the net force applied to it or the pressure. A small amount of water is boiled in a 1-gallon metal can. The can is removed from the heat and the lid put on. As the can cools, it collapses. Explain. When blood pressure is measured, why must the cuff be held at the level of the heart? An ice cube floats in a glass of water filled to the brim. What can you say about the density of ice? As the ice melts, will the water overflow? Explain. Will an ice cube float in a glass of alcohol? Why or why not? A submerged can of Coke® will sink, but a can of Diet Coke® will float. (Try it!) Explain. Why don't ships made of iron sink? Explain how the tube in Fig. 13–44, known as a siphon, can transfer liquid from one container to a lower one even though the liquid must flow uphill for part of its journey. (Note that the tube must be filled with liquid to start with.) FIGURE 13–44 Question 11. A siphon.

12. A barge filled high with sand approaches a low bridge over the river and cannot quite pass under it. Should sand be added to, or removed from, the barge? [Hint: Consider Archimedes' principle.] 13. Explain why helium weather balloons, which are used to measure atmospheric conditions at high altitudes, are normally released while filled to only 10–20% of their maximum volume.

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14. A row boat floats in a swimming pool, and the level of the water at the edge of the pool is marked. Consider the following situations and explain whether the level of the water will rise, fall, or stay the same. (a) The boat is removed from the water. (b) The boat in the water holds an iron anchor which is removed from the boat and placed on the shore. (c) The iron anchor is removed from the boat and dropped in the pool. 15. Will an empty balloon have precisely the same apparent weight on a scale as a balloon filled with air? Explain. 16. Why do you float higher in salt water than in fresh water? 17. If you dangle two pieces of paper vertically, a few inches apart (Fig. 13–45), and blow between them, how do you think the papers will move? Try it and see. Explain.

FIGURE 13–45 Question 17.

FIGURE 13–46 Question 18. Water coming from a faucet.

18. Why does the stream of water from a faucet become narrower as it falls (Fig. 13–46)? 19. Children are told to avoid standing too close to a rapidly moving train because they might get sucked under it. Is this possible? Explain. 20. A tall Styrofoam cup is filled with water. Two holes are punched in the cup near the bottom, and water begins rushing out. If the cup is dropped so it falls freely, will the water continue to flow from the holes? Explain. 21. Why do airplanes normally take off into the wind? 22. Two ships moving in parallel paths close to one another risk colliding. Why? 23. Why does the canvas top of a convertible bulge out when the car is traveling at high speed? [Hint: The windshield deflects air upward, pushing streamlines closer together.] 24. Roofs of houses are sometimes "blown" off (or are they pushed off?) during a tornado or hurricane. Explain using Bernoulli's principle.

Problems 13–2 Density and Specific Gravity 1. (I) The approximate volume of the granite monolith known as El Capitan in Yosemite National Park (Fig. 13–47) is about 108 m3. What is its approximate mass?

FIGURE 13–47 Problem 1. 2. (I) What is the approximate mass of air in a living room 5.6 m * 3.8 m * 2.8 m? 3. (I) If you tried to smuggle gold bricks by filling your backpack, whose dimensions are 56 cm * 28 cm * 22 cm, what would its mass be? 4. (I) State your mass and then estimate your volume. [Hint: Because you can swim on or just under the surface of the water in a swimming pool, you have a pretty good idea of your density.] 5. (II) A bottle has a mass of 35.00 g when empty and 98.44 g when filled with water. When filled with another fluid, the mass is 89.22 g. What is the specific gravity of this other fluid? 6. (II) If 5.0 L of antifreeze solution Aspecific gravity = 0.80B is added to 4.0 L of water to make a 9.0-L mixture, what is the specific gravity of the mixture? 7. (III) The Earth is not a uniform sphere, but has regions of varying density. Consider a simple model of the Earth divided into three regions—inner core, outer core, and mantle. Each region is taken to have a unique constant density (the average density of that region in the real Earth): Region

Radius (km)

Density (kg m3)

Inner Core Outer Core Mantle

0–1220 1220–3480 3480–6371

13,000 11,100 4,400

(a) Use this model to predict the average density of the entire Earth. (b) The measured radius of the Earth is 6371 km and its mass is 5.98 * 1024 kg. Use these data to determine the actual average density of the Earth and compare it (as a percent difference) with the one you determined in (a).

13–3 to 13–6 Pressure; Pascal's Principle 8. (I) Estimate the pressure needed to raise a column of water to the same height as a 35-m-tall oak tree. 9. (I) Estimate the pressure exerted on a floor by (a) one pointed chair leg (66 kg on all four legs) of area = 0.020 cm2, and (b) a 1300-kg elephant standing on one foot Aarea = 800 cm2 B.

10. (I) What is the difference in blood pressure (mm-Hg) between the top of the head and bottom of the feet of a 1.70-m-tall person standing vertically? 11. (II) How high would the level be in an alcohol barometer at normal atmospheric pressure? 12. (II) In a movie, Tarzan evades his captors by hiding underwater for many minutes while breathing through a long, thin reed. Assuming the maximum pressure difference his lungs can manage and still breathe is –85 mm-Hg, calculate the deepest he could have been. 13. (II) The maximum gauge pressure in a hydraulic lift is 17.0 atm. What is the largest-size vehicle (kg) it can lift if the diameter of the output line is 22.5 cm? 14. (II) The gauge pressure in each of the four tires of an automobile is 240 kPa. If each tire has a "footprint" of 220 cm2, estimate the mass of the car. 15. (II) (a) Determine the total force and the absolute pressure on the bottom of a swimming pool 28.0 m by 8.5 m whose uniform depth is 1.8 m. (b) What will be the pressure against the side of the pool near the bottom? 16. (II) A house at the bottom of a hill is fed by a full tank of water 5.0 m deep and connected to the house by a pipe that is 110 m long at an angle of 58° from the horizontal (Fig. 13–48). (a) Determine the water gauge pressure at the house. (b) How high could the 5.0 m water shoot if it came vertically out of a broken 110 m pipe in front of the house? FIGURE 13–48 Problem 16.

58°

17. (II) Water and then oil (which don't mix) are poured into a U-shaped tube, open at both ends. They come to equilibrium as shown 8.62 cm in Fig. 13–49. What is the density of the oil? [Hint: Pressures at points a Oil 27.2 cm and b are equal. Why?] a b Water FIGURE 13–49 Problem 17. 18. (II) In working out his principle, Pascal showed dramatically how force can be multiplied with fluid pressure. He placed a long, thin tube of radius r = 0.30 cm r = 0.30 cm vertically into a wine barrel of radius R = 21 cm, Fig. 13–50. He found that when the barrel was filled with water and the tube filled to a height 12 m of 12 m, the barrel burst. Calculate (a) the mass of water in the tube, and (b) R = 21 cm the net force exerted by the water in the barrel on the lid just before rupture. FIGURE 13–50 Problem 18 (not to scale).

Problems

363

19. (II) What is the normal pressure of the atmosphere at the summit of Mt. Everest, 8850 m above sea level? 20. (II) A hydraulic press for compacting powdered samples has a large cylinder which is 10.0 cm in diameter, and a small cylinder with a diameter of 2.0 cm (Fig. 13–51). A lever is attached to the small cylinder as shown. The sample, which is placed on the large cylinder, has an area of 4.0 cm2. What is the pressure on the sample if 350 N is applied to the lever? 350 N

Sample

Hydraulic fluid

10.0 cm

Small cylinder 2.0 cm

FIGURE 13–51 Problem 20. 21. (II) An open-tube mercury manometer is used to measure the pressure in an oxygen tank. When the atmospheric pressure is 1040 mbar, what is the absolute pressure (in Pa) in the tank if the height of the mercury in the open tube is (a) 21.0 cm higher, (b) 5.2 cm lower, than the mercury in the tube connected to the tank? 22. (III) A beaker of liquid accelerates from rest, on a horizontal surface, with acceleration a to the right. (a) Show that the surface of the liquid makes an angle u = tan–1 (ag) with the horizontal. (b) Which edge of the water surface is higher? (c) How does the pressure vary with depth below the surface? 23. (III) Water stands at a height h behind a vertical dam of uniform width b. (a) Use integration to show that the total force of the water on the dam is F = 12 rgh2b. (b) Show that the torque about the base of the dam due to this force can be considered to act with a lever arm equal to h3. (c) For a freestanding concrete dam of uniform thickness t and height h, what minimum thickness is needed to prevent overturning? Do you need to add in atmospheric pressure for this last part? Explain. 24. (III) Estimate the density of the water 5.4 km deep in the sea. (See Table 12–1 and Section 12–4 regarding bulk modulus.) By what fraction does it differ from the density at the surface? 25. (III) A cylindrical bucket of liquid (density r) is rotated about its symmetry axis, which is vertical. If the angular velocity is v, show that the pressure at a distance r from the rotation axis is P = P0 +

1 2 2 2 rv r ,

where P0 is the pressure at r = 0.

13–7 Buoyancy and Archimedes' Principle 26. (I) What fraction of a piece of iron will be submerged when it floats in mercury? 27. (I) A geologist finds that a Moon rock whose mass is 9.28 kg has an apparent mass of 6.18 kg when submerged in water. What is the density of the rock?

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28. (II) A crane lifts the 16,000-kg steel hull of a sunken ship out of the water. Determine (a) the tension in the crane's cable when the hull is fully submerged in the water, and (b) the tension when the hull is completely out of the water. 29. (II) A spherical balloon has a radius of 7.35 m and is filled with helium. How large a cargo can it lift, assuming that the skin and structure of the balloon have a mass of 930 kg? Neglect the buoyant force on the cargo volume itself. 30. (II) A 74-kg person has an apparent mass of 54 kg (because of buoyancy) when standing in water that comes up to the hips. Estimate the mass of each leg. Assume the body has SG = 1.00. 31. (II) What is the likely identity of a metal (see Table 13–1) if a sample has a mass of 63.5 g when measured in air and an apparent mass of 55.4 g when submerged in water? 32. (II) Calculate the true mass (in vacuum) of a piece of aluminum whose apparent mass is 3.0000 kg when weighed in air. 33. (II) Because gasoline is less dense than water, drums containing gasoline will float in water. Suppose a 230-L steel drum is completely full of gasoline. What total volume of steel can be used in making the drum if the gasoline-filled drum is to float in fresh water? 34. (II) A scuba diver and her gear displace a volume of 65.0 L and have a total mass of 68.0 kg. (a) What is the buoyant force on the diver in seawater? (b) Will the diver sink or float? 35. (II) The specific gravity of ice is 0.917, whereas that of seawater is 1.025. What percent of an iceberg is above the surface of the water? 36. (II) Archimedes' principle can be used not only to determine the specific gravity of a solid using a known liquid (Example 13–10); the reverse can be done as well. (a) As an example, a 3.80-kg aluminum ball has an apparent mass of 2.10 kg when submerged in a particular liquid: calculate the density of the liquid. (b) Derive a formula for determining the density of a liquid using this procedure. 37. (II) (a) Show that the buoyant force FB on a partially submerged object such as a ship acts at the center of gravity of the fluid before it is displaced. This point is called the center of buoyancy. (b) To ensure that a ship is in stable equilibrium, would it be better if its center of buoyancy was above, below, or at the same point B FB as, its center of gravity? Explain. (See Fig. 13–52.)

FIGURE 13–52 Problem 37.

mgB

38. (II) A cube of side length 10.0 cm and made of unknown material floats at the surface between water and oil. The oil has a density of 810 kgm3. If the cube floats so that it is 72% in the water and 28% in the oil, what is the mass of the cube and what is the buoyant force on the cube? 39. (II) How many helium-filled balloons would it take to lift a person? Assume the person has a mass of 75 kg and that each helium-filled balloon is spherical with a diameter of 33 cm.

40. (II) A scuba tank, when fully submerged, displaces 15.7 L of seawater. The tank itself has a mass of 14.0 kg and, when "full," contains 3.00 kg of air. Assuming only a weight and buoyant force act, determine the net force (magnitude and direction) on the fully submerged tank at the beginning of a dive (when it is full of air) and at the end of a dive (when it no longer contains any air). 41. (III) If an object floats in water, its density can be determined by tying a sinker to it so that both the object and the sinker are submerged. Show that the specific gravity is given by wAw1 - w2 B, where w is the weight of the object alone in air, w1 is the apparent weight when a sinker is tied to it and the sinker only is submerged, and w2 is the apparent weight when both the object and the sinker are submerged. 42. (III) A 3.25-kg piece of wood (SG = 0.50) floats on water. What minimum mass of lead, hung from the wood by a string, will cause it to sink?

13–8 to 13–10 Fluid Flow, Bernoulli's Equation 43. (I) A 15-cm-radius air duct is used to replenish the air of a room 8.2 m * 5.0 m * 3.5 m every 12 min. How fast does the air flow in the duct? 44. (I) Using the data of Example 13–13, calculate the average speed of blood flow in the major arteries of the body which have a total cross-sectional area of about 2.0 cm2. 45. (I) How fast does water flow from a hole at the bottom of a very wide, 5.3-m-deep storage tank filled with water? Ignore viscosity. 46. (II) A fish tank has dimensions 36 cm wide by 1.0 m long by 0.60 m high. If the filter should process all the water in the tank once every 4.0 h, what should the flow speed be in the 3.0-cm-diameter input tube for the filter? 47. (II) What gauge pressure in the water mains is necessary if a firehose is to spray water to a height of 18 m? 48. (II) A 58-in. (inside) diameter garden hose is used to fill a round swimming pool 6.1 m in diameter. How long will it take to fill the pool to a depth of 1.2 m if water flows from the hose at a speed of 0.40 ms? 49. (II) A 180-kmh wind blowing over the flat roof of a house causes the roof to lift off the house. If the house is 6.2 m * 12.4 m in size, estimate the weight of the roof. Assume the roof is not nailed down. 50. (II) A 6.0-cm-diameter horizontal pipe gradually narrows to 4.5 cm. When water flows through this pipe at a certain rate, the gauge pressure in these two sections is 32.0 kPa and 24.0 kPa, respectively. What is the volume rate of flow? 51. (II) Estimate the air pressure inside a category 5 hurricane, where the wind speed is 300 kmh (Fig. 13–53).

52. (II) What is the lift (in newtons) due to Bernoulli's principle on a wing of area 88 m2 if the air passes over the top and bottom surfaces at speeds of 280 ms and 150 ms, respectively? 53. (II) Show that the power needed to drive a fluid through a pipe with uniform cross-section is equal to the volume rate of flow, Q, times the pressure difference, P1 - P2 . 54. (II) Water at a gauge pressure of 3.8 atm at street level flows into an office building at a speed of 0.68 ms through a pipe 5.0 cm Faucet in diameter. The pipe tapers down to 2.8 cm in diameter by the top floor, 18 m above (Fig. 13–54), where the faucet has been left open. Calculate the flow velocity and the 18 m gauge pressure in the pipe on the top floor. Assume no branch pipes and ignore viscosity.

FIGURE 13–54 Problem 54.

P =3.8 atm

55. (II) In Fig. 13–55, take into account the speed of the top surface of the tank and show that the speed of fluid leaving the opening at the bottom is v1 =

2gh , B A1 - A 21A 22 B

where h = y2 - y1 , and A 1 and A 2 are the areas of the opening and of the top surface, respectively. Assume A 1 V A 2 so that the flow remains nearly steady and laminar. vB2 y2 − y1 vB1 FIGURE 13–55 Problems 55, 56, 58, and 59. 56. (II) Suppose the top surface of the vessel in Fig. 13–55 is subjected to an external gauge pressure P2 . (a) Derive a formula for the speed, v1 , at which the liquid flows from the opening at the bottom into atmospheric pressure, P0 . Assume the velocity of the liquid surface, v2 , is approximately zero. (b) If P2 = 0.85 atm and y2 - y1 = 2.4 m, determine v1 for water.

FIGURE 13–53 Problem 51.

57. (II) You are watering your lawn with a hose when you put your finger over the hose opening to increase the distance the water reaches. If you are pointing the hose at the same angle, and the distance the water reaches increases by a factor of 4, what fraction of the hose opening did you block?

Problems

365

58. (III) Suppose the opening in the tank of Fig. 13–55 is a height h1 above the base and the liquid surface is a height h2 above the base. The tank rests on level ground. (a) At what horizontal distance from the base of the tank will the fluid strike the vB2 ground? (b) At what other height, h1œ , can a hole be y2 − y1 placed so that the emerging vB1 liquid will have the same "range"? Assume v2 L 0. FIGURE 13–55 (repeated) Problems 55, 56, 58, and 59. 59. (III) (a) In Fig. 13–55, show that Bernoulli's principle predicts that the level of the liquid, h = y2 - y1 , drops at a rate 2ghA 21 dh , = – dt C A 22 - A 21 where A 1 and A 2 are the areas of the opening and the top surface, respectively, assuming A 1 V A 2 , and viscosity is ignored. (b) Determine h as a function of time by integrating. Let h = h0 at t = 0. (c) How long would it take to empty a 10.6-cm-tall cylinder filled with 1.3 L of water if the opening is at the bottom and has a 0.50-cm diameter? 60. (III) (a) Show that the flow speed measured by a venturi meter (see Fig. 13–32) is given by the relation

2AP1 - P2 B . v1 = A 2 C rAA 21 - A 22 B (b) A venturi meter is measuring the flow of water; it has a main diameter of 3.0 cm tapering down to a throat diameter of 1.0 cm. If the pressure difference is measured to be 18 mm-Hg, what is the speed of the water entering the venturi throat? 61. (III) Thrust of a rocket. (a) Use Bernoulli's equation and the equation of continuity to show that the emission speed of the propelling gases of a rocket is v = 32AP - P0 Br , where r is the density of the gas, P is the pressure of the gas inside the rocket, and P0 is atmospheric pressure just outside the exit orifice. Assume that the gas density stays approximately constant, and that the area of the exit orifice, A 0 , is much smaller than the cross-sectional area, A, of the inside of the rocket (take it to be a large cylinder). Assume also that the gas speed is not so high that significant turbulence or nonsteady flow sets in. (b) Show that the thrust force on the rocket due to the emitted gases is F = 2A 0 AP - P0 B. 62. (III) A fire hose exerts a force on the person holding it. This is because the water accelerates as it goes from the hose through the nozzle. How much force is required to hold a 7.0-cm-diameter hose delivering 450 Lmin through a 0.75-cm-diameter nozzle? * 13–11 Viscosity * 63. (II) A viscometer consists of two concentric cylinders, 10.20 cm and 10.60 cm in diameter. A liquid fills the space between them to a depth of 12.0 cm. The outer cylinder is fixed, and a torque of 0.024 m N keeps the inner cylinder turning at a steady rotational speed of 57 revmin. What is the viscosity of the liquid? * 64. (III) A long vertical hollow tube with an inner diameter of 1.00 cm is filled with SAE 10 motor oil. A 0.900-cm-diameter, 30.0-cm-long 150-g rod is dropped vertically through the oil in the tube. What is the maximum speed attained by the rod as it falls?

366

CHAPTER 13

Fluids

* 13–12 Flow in Tubes; Poiseuille's Equation * 65. (I) Engine oil (assume SAE 10, Table 13–3) passes through a fine 1.80-mm-diameter tube that is 8.6 cm long. What pressure difference is needed to maintain a flow rate of 6.2 mLmin? * 66. (I) A gardener feels it is taking too long to water a garden with a 38-in.-diameter hose. By what factor will the time be cut using a 5 8 -in.-diameter hose instead? Assume nothing else is changed. * 67. (II) What diameter must a 15.5-m-long air duct have if the ventilation and heating system is to replenish the air in a room 8.0 m * 14.0 m * 4.0 m every 12.0 min? Assume the pump can exert a gauge pressure of 0.710 * 10–3 atm. * 68. (II) What must be the pressure difference between the two ends of a 1.9-km section of pipe, 29 cm in diameter, if it is to transport oil Ar = 950 kgm3, h = 0.20 PasB at a rate of 650 cm3s? * 69. (II) Poiseuille's equation does not hold if the flow velocity is high enough that turbulence sets in. The onset of turbulence occurs when the Reynolds number, Re, exceeds approximately 2000. Re is defined as 2vrr , Re = h where v is the average speed of the fluid, r is its density, h is its viscosity, and r is the radius of the tube in which the fluid is flowing. (a) Determine if blood flow through the aorta is laminar or turbulent when the average speed of blood in the aorta (r = 0.80 cm) during the resting part of the heart's cycle is about 35 cms. (b) During exercise, the blood-flow speed approximately doubles. Calculate the Reynolds number in this case, and determine if the flow is laminar or turbulent. * 70. (II) Assuming a constant pressure gradient, if blood flow is reduced by 85%, by what factor is the radius of a blood vessel decreased? * 71. (III) A patient is to be given a blood transfusion. The blood is to flow through a tube from a raised bottle to a needle inserted in the vein (Fig. 13–56). The inside diameter of the 25-mm-long needle is 0.80 mm, and the required flow rate is 2.0 cm3 of blood per minute. How high h should the bottle be placed above the needle? Obtain r and h from the Tables. Assume the blood pressure is 78 torr above atmospheric pressure. h

FIGURE 13–56 Problems 71 and 79. * 13–13 Surface Tension and Capillarity * 72. (I) If the force F needed to move the wire in Fig. 13–35 is 3.4 * 10 –3 N, calculate the surface tension g of the enclosed fluid. Assume l = 0.070 m. * 73. (I) Calculate the force needed to move the wire in Fig. 13–35 if it is immersed in a soapy solution and the wire is 24.5 cm long. * 74. (II) The surface tension of a liquid can be determined by measuring the force F needed to just lift a circular platinum ring of radius r from the surface of the liquid. (a) Find a formula for g in terms of F and r. (b) At 30°C, if F = 5.80 * 10–3 N and r = 2.8 cm, calculate g for the tested liquid. * 75. (III) Estimate the diameter of a steel needle that can just "float" on water due to surface tension.

* 76. (III) Show that inside a soap bubble, there must be a pressure ¢P in excess of that outside equal to ¢P = 4gr, where r is the radius of the bubble and g is the surface tension. [Hint: Think of the bubble as two hemispheres in contact with each other; and remember that there are two surfaces to the bubble. Note that this result applies to any kind of membrane, where 2g is the tension per unit length in that membrane.]

* 77. (III) A common effect of surface tension is the ability of a liquid to rise up a narrow tube due to what is called capillary action. Show that for a narrow tube of radius r placed in a liquid of density r and surface tension g, the liquid in the tube will reach a height h = 2grgr above the level of the liquid outside the tube, where g is the gravitational acceleration. Assume that the liquid "wets" the capillary (the liquid surface is vertical at the contact with the inside of the tube).

General Problems 78. A 2.8-N force is applied to the plunger of a hypodermic needle. If the diameter of the plunger is 1.3 cm and that of the needle 0.20 mm, (a) with what force does the fluid leave the needle? (b) What force on the plunger would be needed to push fluid into a vein where the gauge pressure is 75 mm-Hg? Answer for the instant just before the fluid starts to move. 79. Intravenous infusions are often made under gravity, as shown in Fig. 13–56. Assuming the fluid has a density of 1.00 gcm3, at what height h should the bottle be placed so the liquid pressure is (a) 55 mm-Hg, and (b) 650 mm-H 2O? (c) If the blood pressure is 78 mm-Hg above atmospheric pressure, how high should the bottle be placed so that the fluid just barely enters the vein? 80. A beaker of water rests on an electronic balance that reads 998.0 g. A 2.6-cm-diameter solid copper ball attached to a string is submerged in the water, but does not touch the bottom. What are the tension in the string and the new balance reading? 81. Estimate the difference in air pressure between the top and the bottom of the Empire State building in New York City. It is 380 m tall and is located at sea level. Express as a fraction of atmospheric pressure at sea level. 82. A hydraulic lift is used to jack a 920-kg car 42 cm off the floor. The diameter of the output piston is 18 cm, and the input force is 350 N. (a) What is the area of the input piston? (b) What is the work done in lifting the car 42 cm? (c) If the input piston moves 13 cm in each stroke, how high does the car move up for each stroke? (d) How many strokes are required to jack the car up 42 cm? (e) Show that energy is conserved. 83. When you ascend or descend a great deal when driving in a car, your ears "pop," which means that the pressure behind the eardrum is being equalized to that outside. If this did not happen, what would be the approximate force on an eardrum of area 0.20 cm2 if a change in altitude of 950 m takes place? 84. Giraffes are a wonder of cardiovascular engineering. Calculate the difference in pressure (in atmospheres) that the blood vessels in a giraffe's head must accommodate as the head is lowered from a full upright position to ground level for a drink. The height of an average giraffe is about 6 m. 85. Suppose a person can reduce the pressure in his lungs to –75 mm-Hg gauge pressure. How high can water then be "sucked" up a straw? 86. Airlines are allowed to maintain a minimum air pressure within the passenger cabin equivalent to that at an altitude of 8000 ft (2400 m) to avoid adverse health effects among passengers due to oxygen deprivation. Estimate this minimum pressure (in atm).

87. A simple model (Fig. 13–57) considers a continent as a block Adensity L 2800 kgm3 B floating in the mantle rock around it Adensity L 3300 kgm3 B. Assuming the continent is 35 km thick (the average thickness of the Earth's continental crust), estiB mate the height of FB the continent above the surrounding rock. Continent (density ≈ 2800 kg/m3) FIGURE 13–57 Problem 87.

mgB

Mantle rock (density ≈ 3300 kg/m3)

88. A ship, carrying fresh water to a desert island in the Caribbean, has a horizontal cross-sectional area of 2240 m2 at the waterline. When unloaded, the ship rises 8.50 m higher in the sea. How many cubic meters of water was delivered? 89. During ascent, and especially during descent, volume changes of trapped air in the middle ear can cause ear discomfort until the middle-ear pressure and exterior pressure are equalized. (a) If a rapid descent at a rate of 7.0 ms or faster commonly causes ear discomfort, what is the maximum rate of increase in atmospheric pressure (that is, dPdt) tolerable to most people? (b) In a 350-m-tall building, what will be the fastest possible descent time for an elevator traveling from the top to ground floor, assuming the elevator is properly designed to account for human physiology? 90. A raft is made of 12 logs lashed together. Each is 45 cm in diameter and has a length of 6.1 m. How many people can the raft hold before they start getting their feet wet, assuming the average person has a mass of 68 kg? Do not neglect the weight of the logs. Assume the specific gravity of wood is 0.60. 91. Estimate the total mass of the Earth's atmosphere, using the known value of atmospheric pressure at sea level. 92. During each heartbeat, approximately 70 cm3 of blood is pushed from the heart at an average pressure of 105 mm-Hg. Calculate the power output of the heart, in watts, assuming 70 beats per minute. 93. Four lawn sprinkler heads are fed by a 1.9-cm-diameter pipe. The water comes out of the heads at an angle of 35° to the horizontal and covers a radius of 7.0 m. (a) What is the velocity of the water coming out of each sprinkler head? (Assume zero air resistance.) (b) If the output diameter of each head is 3.0 mm, how many liters of water do the four heads deliver per second? (c) How fast is the water flowing inside the 1.9-cm-diameter pipe?

General Problems

367

94. A bucket of water is accelerated upward at 1.8 g. What is the buoyant force on a 3.0-kg granite rock (SG = 2.7) submerged in the water? Will the rock float? Why or why not? 95. The stream of water from a faucet decreases in diameter as it falls (Fig. 13–58). Derive an equation for the diameter of the stream as a function of the distance y below the faucet, given that the water has speed v0 when it leaves the faucet, whose diameter is d.

FIGURE 13–58 Problem 95. Water coming from a faucet. 96. You need to siphon water from a clogged sink. The sink has an area of 0.38 m2 and is filled to a height of 4.0 cm. Your siphon tube rises 45 cm above the bottom of the sink and then descends 85 cm to a pail as shown in Fig. 13–59. The siphon tube has a diameter of 2.0 cm. 45 cm (a) Assuming that the water level in the sink has almost 85 cm zero velocity, estimate the water 4.0 cm velocity when it enters the pail. (b) Estimate how long it will take to empty the sink. FIGURE 13–59 Problem 96.

101. Three forces act significantly on a freely floating heliumfilled balloon: gravity, air resistance (or drag force), and a buoyant force. Consider a spherical helium-filled balloon of radius r = 15 cm rising upward through 0°C air, and m = 2.8 g is the mass of the (deflated) balloon itself. For all speeds v, except the very slowest ones, the flow of air past a rising balloon is turbulent, and the drag force FD is given by the relation FD =

97. An airplane has a mass of 1.7 * 10 kg, and the air flows past the lower surface of the wings at 95 ms. If the wings have a surface area of 1200 m2, how fast must the air flow over the upper surface of the wing if the plane is to stay in the air? 98. A drinking fountain shoots water about 14 cm up in the air from a nozzle of diameter 0.60 cm. The pump at the base of the unit (1.1 m below the nozzle) pushes water into a 1.2cm-diameter supply pipe that goes up to the nozzle. What gauge pressure does the pump have to provide? Ignore the viscosity; your answer will therefore be an underestimate. 99. A hurricane-force wind of 200 kmh blows across the face of a storefront window. Estimate the force on the 2.0 m * 3.0 m window due to the difference in air pressure inside and outside the window. Assume the store is airtight so the inside pressure remains at 1.0 atm. (This is why you should not tightly seal a building in preparation for a hurricane). 100. Blood from an animal is placed in a bottle 1.30 m above a 3.8-cm-long needle, of inside diameter 0.40 mm, from which it flows at a rate of 4.1 cm3min. What is the viscosity of this blood?

where the constant CD = 0.47 is the "drag coefficient" for a smooth sphere of radius r. If this balloon is released from rest, it will accelerate very quickly (in a few tenths of a second) to its terminal velocity vT , where the buoyant force is cancelled by the drag force and the balloon's total weight. Assuming the balloon's acceleration takes place over a negligible time and distance, how long does it take the released balloon to rise a distance h = 12 m? * 102. If cholesterol buildup reduces the diameter of an artery by 15%, by what % will the blood flow rate be reduced, assuming the same pressure difference? 103. A two-component model used to determine percent body fat in a human body assumes that a fraction f (6 1) of the body's total mass m is composed of fat with a density of 0.90 gcm3, and that the remaining mass of the body is composed of fat-free tissue with a density of 1.10 gcm3. If the specific gravity of the entire body's density is X, show that the percent body fat ( = f * 100) is given by 495 % Body fat = - 450. X

* 104. (III) Air pressure decreases with altitude. The following data show the air pressure at different altitudes.

A: (d).

E: (e).

B: The same. Pressure depends on depth, not on length.

F: Increases.

C: Lower.

G: (b).

CHAPTER 13

Fluids

Altitude (m)

Pressure (kPa)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000

101.3 89.88 79.50 70.12 61.66 54.05 47.22 41.11 35.65 30.80 26.50

(a) Determine the best-fit quadratic equation that shows how the air pressure changes with altitude. (b) Determine the best-fit exponential equation that describes the change of air pressure with altitude. (c) Use each fit to find the air pressure at the summit of the mountain K2 at 8611 m, and give the % difference.

Answers to Exercises

368

CD rair pr 2v2

* Numerical/Computer 6

D: (a).

1 2

An object attached to a coil spring can exhibit oscillatory motion. Many kinds of oscillatory motion are sinusoidal in time, or nearly so, and are referred to as being simple harmonic motion. Real systems generally have at least some friction, causing the motion to be damped. The automobile spring shown here has a shock absorber (yellow) that purposefully dampens the oscillation to make for a smooth ride. When an external sinusoidal force is exerted on a system able to oscillate, resonance occurs if the driving force is at or near the natural frequency of oscillation.

P T E R

C H

Oscillations CHAPTER-OPENING QUESTION—Guess now! A simple pendulum consists of a mass m (the "bob") hanging on the end of a thin string of length l and negligible mass. The bob is pulled sideways so the string makes a 5.0° angle to the vertical; when released, it oscillates back and forth at a frequency f. If the pendulum was raised to a 10.0° angle instead, its frequency would be (a) twice as great. 10.0° (b) half as great. l (c) the same, or very close to it. (d) not quite twice as great. (e) a bit more than half as great. m any objects vibrate or oscillate—an object on the end of a spring, a tuning fork, the balance wheel of an old watch, a pendulum, a plastic ruler held firmly over the edge of a table and gently struck, the strings of a guitar or piano. Spiders detect prey by the vibrations of their webs; cars oscillate up and down when they hit a bump; buildings and bridges vibrate when heavy trucks pass or the wind is fierce. Indeed, because most solids are elastic (see Chapter 12), they vibrate (at least briefly) when given an impulse. Electrical oscillations are necessary in radio and television sets. At the atomic level, atoms vibrate within a molecule, and the atoms of a solid vibrate about their relatively fixed positions. Because it is so common in everyday life and occurs in so many areas of physics, oscillatory motion is of great importance. Mechanical oscillations are fully described on the basis of Newtonian mechanics.

CONTENTS 14–1 Oscillations of a Spring 14–2 Simple Harmonic Motion 14–3 Energy in the Simple Harmonic Oscillator 14–4 Simple Harmonic Motion Related to Uniform Circular Motion 14–5 The Simple Pendulum

*14–6 The Physical Pendulum

and the Torsion Pendulum

14–7 Damped Harmonic Motion 14–8 Forced Oscillations; Resonance

369

14–1 m x=0

(a)

m x

(b)

(> 0) B

(c)

(< 0)

FIGURE 14–2 Force on, and velocity of, a mass at different positions of its oscillation cycle on a frictionless surface. B

F v=0 x = −A

x=0 F=0 v = +vmax B

(max. in positive direction)

x=0

(b)

F v=0 x=0

(c)

x=A

F=0 v = −vmax B

(max. in negative direction)

x=0

(d)

When an object vibrates or oscillates back and forth, over the same path, each oscillation taking the same amount of time, the motion is periodic. The simplest form of periodic motion is represented by an object oscillating on the end of a uniform coil spring. Because many other types of oscillatory motion closely resemble this system, we will look at it in detail. We assume that the mass of the spring can be ignored, and that the spring is mounted horizontally, as shown in Fig. 14–1a, so that the object of mass m slides without friction on the horizontal surface. Any spring has a natural length at which it exerts no force on the mass m. The position of the mass at this point is called the equilibrium position. If the mass is moved either to the left, which compresses the spring, or to the right, which stretches it, the spring exerts a force on the mass that acts in the direction of returning the mass to the equilibrium position; hence it is called a restoring force. We consider the common situation where we can assume the restoring force F is directly proportional to the displacement x the spring has been stretched (Fig. 14–1b) or compressed (Fig. 14–1c) from the equilibrium position: F = –kx.

FIGURE 14–1 A mass oscillating at the end of a uniform spring.

(a)

Oscillations of a Spring

v=0 x = −A

370

CHAPTER 14

Note that the equilibrium position has been chosen at x = 0 and the minus sign in Eq. 14–1 indicates that the restoring force is always in the direction opposite to the displacement x. For example, if we choose the positive direction to the right in Fig. 14–1, x is positive when the spring is stretched (Fig. 14–1b), but the direction of the restoring force is to the left (negative direction). If the spring is compressed, x is negative (to the left) but the force F acts toward the right (Fig. 14–1c). Equation 14–1 is often referred to as Hooke's law (Sections 7–3, 8–2 and 12–4), and is accurate only if the spring is not compressed to where the coils are close to touching, or stretched beyond the elastic region (see Fig. 12–15). Hooke's law works not only for springs but for other oscillating solids as well; it thus has wide applicability, even though it is valid only over a certain range of F and x values. The proportionality constant k in Eq. 14–1 is called the spring constant for that particular spring, or its spring stiffness constant. To stretch the spring a distance x, one has to exert an (external) force on the free end of the spring with a magnitude at least equal to Fext = ±kx. [external force on spring] The greater the value of k, the greater the force needed to stretch a spring a given distance. That is, the stiffer the spring, the greater the spring constant k. Note that the force F in Eq. 14–1 is not a constant, but varies with position. Therefore the acceleration of the mass m is not constant, so we cannot use the equations for constant acceleration developed in Chapter 2. Let us examine what happens when our uniform spring is initially compressed a distance x = –A, as shown in Fig. 14–2a, and then released on the frictionless surface. The spring exerts a force on the mass that pushes it toward the equilibrium position. But because the mass has inertia, it passes the equilibrium position with considerable speed. Indeed, as the mass reaches the equilibrium position, the force on it decreases to zero, but its speed at this point is a maximum, vmax (Fig. 14–2b). As the mass moves farther to the right, the force on it acts to slow it down, and it stops for an instant at x = A (Fig. 14–2c). It then begins moving back in the opposite direction, accelerating until it passes the equilibrium point (Fig. 14–2d), and then slows down until it reaches zero speed at the original starting point, x = –A (Fig. 14–2e). It then repeats the motion, moving back and forth symmetrically between x = A and x = –A. EXERCISE A An object is oscillating back and forth. Which of the following statements are true at some time during the course of the motion? (a) The object can have zero velocity and, simultaneously, nonzero acceleration. (b) The object can have zero velocity and, simultaneously, zero acceleration. (c) The object can have zero acceleration and, simultaneously, nonzero velocity. (d) The object can have nonzero velocity and nonzero acceleration simultaneously.

(e)

[force exerted by spring] (14–1)

x=0

Oscillations

EXERCISE B A mass is oscillating on a frictionless surface at the end of a horizontal spring. Where, if anywhere, is the acceleration of the mass zero (see Fig. 14–2)? (a) At x = –A; (b) at x = 0; (c) at x = ±A; (d) at both x = –A and x = ±A; (e) nowhere.

To discuss oscillatory motion, we need to define a few terms. The distance x of the mass from the equilibrium point at any moment is called the displacement. The maximum displacement—the greatest distance from the equilibrium point—is called the amplitude, A. One cycle refers to the complete to-and-fro motion from some initial point back to that same point—say, from x = –A to x = A and back to x = –A. The period, T, is defined as the time required to complete one cycle. Finally, the frequency, f, is the number of complete cycles per second. Frequency is generally specified in hertz (Hz), where 1 Hz = 1 cycle per second As–1 B. It is easy to see, from their definitions, that frequency and period are inversely related, as we saw earlier (Eqs. 5–2 and 10–8): 1 1 f = and T = ; (14–2) T f for example, if the frequency is 5 cycles per second, then each cycle takes 15 s. The oscillation of a spring hung vertically is essentially the same as that of a horizontal spring. Because of gravity, the length of a vertical spring with a mass m on the end will be longer at equilibrium than when that same spring is horizontal, as shown in Fig. 14–3. The spring is in equilibrium when ©F = 0 = mg - kx0 , so the spring stretches an extra amount x0 = mgk to be in equilibrium. If x is measured from this new equilibrium position, Eq. 14–1 can be used directly with the same value of k.

C A U T I O N

For vertical spring, measure displacement (x or y) from the vertical equilibrium position

EXERCISE C If an oscillating mass has a frequency of 1.25 Hz, it makes 100 oscillations in (a) 12.5 s, (b) 125 s, (c) 80 s, (d) 8.0 s.

F = −kx0

m x

x now measured from here

FIGURE 14–3 (a) Free spring, hung vertically. (b) Mass m attached to spring in new equilibrium position, which occurs when ©F = 0 = mg - kx0 .

m gB (a)

(b)

EXAMPLE 14–1 Car springs. When a family of four with a total mass of 200 kg step into their 1200-kg car, the car's springs compress 3.0 cm. (a) What is the spring constant of the car's springs (Fig. 14–4), assuming they act as a single spring? (b) How far will the car lower if loaded with 300 kg rather than 200 kg? APPROACH We use Hooke's law: the weight of the people, mg, causes a 3.0-cm displacement. SOLUTION (a) The added force of (200 kg)A9.8 ms2 B = 1960 N causes the springs to compress 3.0 * 10 –2 m. Therefore (Eq. 14–1), the spring constant is F 1960 N k = = = 6.5 * 104 Nm. x 3.0 * 10–2 m (b) If the car is loaded with 300 kg, Hooke's law gives (300 kg)A9.8 ms2 B F x = = = 4.5 * 10–2 m, k A6.5 * 104 NmB or 4.5 cm. NOTE In (b), we could have obtained x without solving for k: since x is proportional to F, if 200 kg compresses the spring 3.0 cm, then 1.5 times the force will compress the spring 1.5 times as much, or 4.5 cm. SECTION 14–1

FIGURE 14–4 Photo of a car's spring. (Also visible is the shock absorber, in blue—see Section 14–7.)

Oscillations of a Spring

371

14–2

Simple Harmonic Motion

Any oscillating system for which the net restoring force is directly proportional to the negative of the displacement (as in Eq. 14–1, F = –kx) is said to exhibit simple harmonic motion (SHM). Such a system is often called a simple harmonic oscillator (SHO). We saw in Chapter 12 (Section 12–4) that most solid materials stretch or compress according to Eq. 14–1 as long as the displacement is not too great. Because of this, many natural oscillations are simple harmonic or close to it. EXERCISE D Which of the following represents a simple harmonic oscillator: (a) F = –0.5x2, (b) F = – 2.3y, (c) F = 8.6x, (d) F = –4u?

Let us now determine the position x as a function of time for a mass attached to the end of a simple spring with spring constant k. To do so, we make use of Newton's second law, F = ma. Since the acceleration a = d2xdt2, we have ma = ©F d2x m 2 = –kx, dt

Paper motion

where m is the mass† which is oscillating. We rearrange this to obtain

x A 0 −A

1 4T

1 2T

3 4T

3 2T

d 2x k + x = 0, 2 m dt

FIGURE 14–5 Sinusoidal nature of SHM as a function of time. In this case, x = A cos(2ptT).

[SHM] (14–3)

which is known as the equation of motion for the simple harmonic oscillator. Mathematically it is called a differential equation, since it involves derivatives. We want to determine what function of time, x(t), satisfies this equation. We might guess the form of the solution by noting that if a pen were attached to an oscillating mass (Fig. 14–5) and a sheet of paper moved at a steady rate beneath it, the pen would trace the curve shown. The shape of this curve looks a lot like it might be sinusoidal (such as cosine or sine) as a function of time, and its height is the amplitude A. Let us then guess that the general solution to Eq. 14–3 can be written in a form such as x = A cos(vt + f),

(14–4) ‡

where we include the constant f in the argument to be general. Let us now put this trial solution into Eq. 14–3 and see if it really works. We need to differentiate the x = x(t) twice: dx d = [A cos(vt + f)] = –vA sin(vt + f) dt dt d 2x = –v2A cos(vt + f). dt2 We now put the latter into Eq. 14–3, along with Eq. 14–4 for x: d2x k + x = 0 m dt2 –v2A cos(vt + f) +

k A cos(vt + f) = 0 m

or a

k - v2 b A cos(vt + f) = 0. m

Our solution, Eq. 14–4, does indeed satisfy the equation of motion (Eq. 14–3) for †

In the case of a mass, m¿, on the end of a spring, the spring itself also oscillates and at least a part of its mass must be included. It can be shown—see the Problems—that approximately one-third the mass of the spring, ms , must be included, so then m = m¿ + 13 ms in our equation. Often ms is small enough to be ignored. ‡ Another possible way to write the solution is the combination x = a cos vt + b sin vt, where a and b are constants. This is equivalent to Eq. 14–4 as can be seen using the trigonometric identity cos(A6B) = cos A cos B7sin A sin B.

372

CHAPTER 14

Oscillations

any time t, but only if Akm - v2 B = 0. Hence k. v2 = (14–5) m Equation 14–4 is the general solution to Eq. 14–3, and it contains two arbitrary constants A and f, which we should expect because the second derivative in Eq. 14–3 implies that two integrations are needed, each yielding a constant. They are "arbitrary" only in a calculus sense, in that they can be anything and still satisfy the differential equation, Eq. 14–3. In real physical situations, however, A and f are determined by the initial conditions. Suppose, for example, that the mass is started at its maximum displacement and is released from rest. This is, in fact, what is shown in Fig. 14–5, and for this case x = A cos vt. Let us confirm it: we are given v = 0 at t = 0, where v =

dx d = [A cos(vt + f)] = –vA sin(vt + f) = 0. dt dt

[at t = 0]

For v to be zero at t = 0, then sin(vt + f) = sin(0 + f) is zero if f = 0 (f could also be p, 2p, etc.), and when f = 0, then x = A cos vt, as we expected. We see immediately that A is the amplitude of the motion, and it is determined initially by how far you pulled the mass m from equilibrium before releasing it. Consider another interesting case: at t = 0, the mass m is at x = 0 and is struck, giving it an initial velocity toward increasing values of x. Then at t = 0, x = 0, so we can write x = A cos(vt + f) = A cos f = 0, which can happen only if f = & p2 (or &90°). Whether f = ±p2 or –p2 depends on v = dxdt = –vA sin(vt + f) = –vA sin f at t = 0, which we are given as positive (v 7 0 at t = 0); hence f = –p2 because sin( –90°) = –1. Thus our solution for this case is x = A cos avt -

p b 2

where we used cos(u - p2) = sin u. The solution in this case is a pure sine wave, Fig. 14–6, where A is still the amplitude. Many other situations are possible, such as that shown in Fig. 14–7. The constant f is called the phase angle, and it tells us how long after (or before) t = 0 the peak at x = A is reached. Notice that the value of f does not affect the shape of the x(t) curve, but only affects the displacement at some arbitrary time, t = 0. Simple harmonic motion is thus always sinusoidal. Indeed, simple harmonic motion is defined as motion that is purely sinusoidal. Since our oscillating mass repeats its motion after a time equal to its period T, it must be at the same position and moving in the same direction at t = T as it was at t = 0. Since a sine or cosine function repeats itself after every 2p radians, then from Eq. 14–4, we must have vT = 2p. v =

A 0

1 1 4 T 2T

−A

= A sin vt,

Hence

x(t)

3 4T

3 2T

FIGURE 14–6 Special case of SHM where the mass m starts, at t = 0, at the equilibrium position x = 0 and has initial velocity toward positive values of x (v 7 0 at t = 0). FIGURE 14–7 A plot of x = A cos(vt + f) when f 6 0.

∆t = −

A 0

φ ω

x = A cos (ω t + φ)

A A

−A

2p = 2pf, T

where f is the frequency of the motion. Strictly speaking, we call v the angular frequency (units are rads) to distinguish it from the frequency f (units are s–1 = Hz); sometimes the word "angular" is dropped, so the symbol v or f needs to be specified. Because v = 2pf = 2pT, we can write Eq. 14–4 as x = A cos a

2pt + fb T

(14–6a)

or x = A cos(2pft + f).

(14–6b) SECTION 14–2

Simple Harmonic Motion

373

Because v = 2pf = 1km (Eq. 14–5), then 1 k, (14–7a) 2p B m m. (14–7b) T = 2p Ak Note that the frequency and period do not depend on the amplitude. Changing the amplitude of a simple harmonic oscillator does not affect its frequency. Equation 14–7a tells us that the greater the mass, the lower the frequency; and the stiffer the spring, the higher the frequency. This makes sense since a greater mass means more inertia and therefore a slower response (or acceleration); and larger k means greater force and therefore quicker response. The frequency f (Eq. 14–7a) at which a SHO oscillates naturally is called its natural frequency (to distinguish it from a frequency at which it might be forced to oscillate by an outside force, as discussed in Section 14–8). The simple harmonic oscillator is important in physics because whenever we have a net restoring force proportional to the displacement (F = –kx), which is at least a good approximation for a variety of systems, then the motion is simple harmonic—that is, sinusoidal. f =

P H Y S I C S

A P P L I E D

Car springs

EXAMPLE 14–2 Car springs again. Determine the period and frequency of the car in Example 14–1a after hitting a bump. Assume the shock absorbers are poor, so the car really oscillates up and down. APPROACH We put m = 1400 kg and k = 6.5 * 104 Nm from Example 14–1a into Eqs. 14–7. SOLUTION From Eq. 14–7b, 1400 kg m = 2p = 0.92 s, Ak B 6.5 * 104 Nm or slightly less than a second. The frequency f = 1T = 1.09 Hz. T = 2p

EXERCISE E By how much should the mass on the end of a spring be changed to halve the frequency of its oscillations? (a) No change; (b) doubled; (c) quadrupled; (d) halved; (e) quartered. FIGURE 14–8 Displacement, x, velocity, dxdt, and acceleration, d2xdt2, of a simple harmonic oscillator when f = 0.

EXERCISE F The position of a SHO is given by x = (0.80 m) cos(3.14t - 0.25). The frequency is (a) 3.14 Hz, (b) 1.0 Hz, (c) 0.50 Hz, (d) 9.88 Hz, (e) 19.8 Hz.

Displacement x

A 0 −A

1 4T

1 2T

3 4T

3 2T

Velocity v

(a) vmax 0 −vmax

1 4T

1 3 2T 4T

3 2T

Acceleration a

0 − mk A

3 T 4 1 1 4T 2T

3 2T

(c)

374

CHAPTER 14

k (14–9a) A Bm when the oscillating object is passing through its equilibrium point, x = 0. And the speed is zero at points of maximum displacement, x = &A. This is in accord with our discussion of Fig. 14–2. Similarly, the acceleration has its maximum value k a max = v2A = (14–9b) A m which occurs where x = &A; and a is zero at x = 0, as we expect, since ma = F = –kx. vmax = vA =

(b) k mA

Let us continue our analysis of a simple harmonic oscillator. The velocity and acceleration of the oscillating mass can be obtained by differentiation of Eq. 14–4, x = A cos(vt + f): dx v = = –vA sin(vt + f) (14–8a) dt d 2x dv = –v2A cos(vt + f). (14–8b) a = = 2 dt dt The velocity and acceleration of a SHO also vary sinusoidally. In Fig. 14–8 we plot the displacement, velocity, and acceleration of a SHO as a function of time for the case when f = 0. As can be seen, the speed reaches its maximum

Oscillations

For the general case when f Z 0, we can relate the constants A and f to the initial values of x, v, and a by setting t = 0 in Eqs. 14–4, 14–8, and 14–9: x0 = x(0) = A cos f v0 = v(0) = –vA sin f = –vmax sin f a0 = a(0) = –v2A cos f = –a max cos f. EXAMPLE 14–3 ESTIMATE A vibrating floor. A large motor in a factory causes the floor to vibrate at a frequency of 10 Hz. The amplitude of the floor's motion near the motor is about 3.0 mm. Estimate the maximum acceleration of the floor near the motor. APPROACH Assuming the motion of the floor is roughly SHM we can make an estimate for the maximum acceleration using Eq. 14–9b. SOLUTION Given v = 2pf = (2p)A10 s–1 B = 62.8 rads, then Eq. 14–9b gives a max = v2A = (62.8 rads)2(0.0030 m) = 12 ms2. NOTE The maximum acceleration is a little over g, so when the floor accelerates down, objects sitting on the floor will actually lose contact momentarily, which will cause noise and serious wear.

P H Y S I C S

A P P L I E D

Unwanted floor vibrations

EXAMPLE 14–4 Loudspeaker. The cone of a loudspeaker (Fig. 14–9) oscillates in SHM at a frequency of 262 Hz ("middle C"). The amplitude at the center of the cone is A = 1.5 * 10–4 m, and at t = 0, x = A. (a) What equation describes the motion of the center of the cone? (b) What are the velocity and acceleration as a function of time? (c) What is the position of the cone at t = 1.00 ms A = 1.00 * 10–3 sB? APPROACH The motion begins (t = 0) with the cone at its maximum displacement (x = A at t = 0). So we use the cosine function, x = A cos vt, with f = 0. SOLUTION (a) The amplitude A = 1.5 * 10–4 m and v = 2pf = (6.28 rad)A262 s–1 B = 1650 rads. The motion is described as x = A cos vt = A1.5 * 10–4 mB cos(1650 t),

FIGURE 14–9 Example 14–4. A loudspeaker cone.

where t is in seconds. (b) The maximum velocity, from Eq. 14–9a, is vmax = vA = (1650 rads)A1.5 * 10–4 mB = 0.25 ms, so v = –(0.25 ms) sin(1650 t). From Eq. 14–9b the maximum acceleration is amax = v2A = 2 –4 2 (1650 rads) A1.5 * 10 mB = 410 ms , which is more than 40 g's. Then a = – A410 ms2 B cos(1650 t).

= A1.5 * 10–4 mB cos(1.65 rad) = –1.2 * 10–5 m.

NOTE Be sure your calculator is set in RAD mode, not DEG mode, for these cos vt calculations. SECTION 14–2

C A U T I O N

Always be sure your calculator is in the correct mode for angles

Simple Harmonic Motion

375

EXAMPLE 14–5 Spring calculations. A spring stretches 0.150 m when a 0.300-kg mass is gently attached to it as in Fig. 14–3b. The spring is then set up horizontally with the 0.300-kg mass resting on a frictionless table as in Fig. 14–2. The mass is pushed so that the spring is compressed 0.100 m from the equilibrium point, and released from rest. Determine: (a) the spring stiffness constant k and angular frequency v; (b) the amplitude of the horizontal oscillation A; (c) the magnitude of the maximum velocity vmax ; (d) the magnitude of the maximum acceleration amax of the mass; (e) the period T and frequency f; (f) the displacement x as a function of time; and (g) the velocity at t = 0.150 s. APPROACH When the 0.300-kg mass hangs at rest from the spring as in Fig. 14–3b, we apply Newton's second law for the vertical forces: ©F = 0 = mg - kx0 , so k = mgx0 . For the horizontal oscillations, the amplitude is given, and the other quantities can be found from Eqs. 14–4, 14–5, 14–7, and 14–9. We choose x positive to the right. SOLUTION (a) The spring stretches 0.150 m due to the 0.300-kg load, so k =

(0.300 kg)A9.80 ms2 B mg F = = = 19.6 Nm. x0 x0 0.150 m

From Eq. 14–5, 19.6 Nm k = = 8.08 s–1. Bm B 0.300 kg (b) The spring is now horizontal (on a table). It is compressed 0.100 m from equilibrium and is given no initial speed, so A = 0.100 m. (c) From Eq. 14–9a, the maximum velocity has magnitude v =

vmax = vA = A8.08 s–1 B(0.100 m) = 0.808 ms.

(d) Since F = ma, the maximum acceleration occurs where the force is greatest—that is, when x = &A = &0.100 m. Thus its magnitude is (19.6 Nm)(0.100 m) F kA amax = = = = 6.53 ms2. m m 0.300 kg [This result could also have been obtained directly from Eq. 14–9b, but it is often useful to go back to basics as we did here.] (e) Equations 14–7b and 14–2 give T = 2p

0.300 kg m = 2p = 0.777 s Ak B 19.6 Nm

1 = 1.29 Hz. T (f) The motion begins at a point of maximum compression. If we take x positive to the right in Fig. 14–2, then at t = 0, x = –A = –0.100 m. So we need a sinusoidal curve that has its maximum negative value at t = 0; this is just a negative cosine: x = –A cos vt. To write this in the form of Eq. 14–4 (no minus sign), recall that cos u = –cos(u - p). Then, putting in numbers, and recalling –cos u = cos(p - u) = cos(u - p), we have x = –(0.100 m) cos 8.08 t = (0.100 m) cos(8.08 t - p), f =

where t is in seconds and x is in meters. Note that the phase angle (Eq. 14–4) is f = –p or –180°. (g) The velocity at any time t is dxdt (see also part c): dx v = = vA sin vt = (0.808 ms) sin 8.08 t. dt At t = 0.150 s, v = (0.808 ms) sin (1.21 rad) = 0.756 ms, and is to the right (±).

376

CHAPTER 14

Oscillations

EXAMPLE 14–6 Spring is started with a push. Suppose the spring of Example 14–5 is compressed 0.100 m from equilibrium Ax0 = –0.100 mB but is given a shove to create a velocity in the ±x direction of v0 = 0.400 ms. Determine (a) the phase angle f, (b) the amplitude A, and (c) the displacement x as a function of time, x(t). APPROACH We use Eq. 14–8a, at t = 0, to write v0 = –vA sin f, and Eq. 14–4 to write x0 = A cos f. Combining these, we can obtain f. We obtain A by using Eq. 14–4 again at t = 0. From Example 14–5, v = 8.08 s–1. SOLUTION (a) We combine Eqs. 14–8a and 14–4 at t = 0 and solve for the tangent: Av0–vAB v0 sin f 0.400 ms tan f = = = – = – = 0.495. vx0 cos f Ax0AB A8.08 s–1 B(–0.100 m) A calculator gives the angle as 26.3°, but we note from this equation that both the sine and cosine are negative, so our angle is in the third quadrant. Hence f = 26.3° + 180° = 206.3° = 3.60 rad. (b) Again using Eq. 14–4 at t = 0, as given in the Approach above, (–0.100 m) x0 A = = = 0.112 m. cos f cos(3.60 rad) (c) x = A cos(vt + f) = (0.112 m) cos(8.08 t + 3.60).

14–3

Energy in the Simple Harmonic Oscillator

When forces are not constant, as is the case here with simple harmonic motion, it is often convenient and useful to use the energy approach, as we saw in Chapters 7 and 8. For a simple harmonic oscillator, such as a mass m oscillating on the end of a massless spring, the restoring force is given by F = –kx. The potential energy function, as we saw in Chapter 8, is given by

U = – F dx =

E = 2 kA2

(a)

x = −A x = 0 x = A (v = 0) 1

2 E = 2 mvmax

1 2 2 kx ,

where we set the constant of integration equal to zero so U = 0 at x = 0 (the equilibrium position). The total mechanical energy is the sum of the kinetic and potential energies, E = 12 mv2 + 12 kx2, where v is the velocity of the mass m when it is a distance x from the equilibrium position. SHM can occur only if there is no friction, so the total mechanical energy E remains constant.As the mass oscillates back and forth, the energy continuously changes from potential energy to kinetic energy, and back again (Fig. 14–10). At the extreme points, x = A and x = –A, all the energy is stored in the spring as potential energy (and is the same whether the spring is compressed or stretched to the full amplitude).At these extreme points, the mass stops for an instant as it changes direction, so v = 0 and: E = 12 m(0)2 + 12 kA2 = 12 kA2. (14–10a) Thus, the total mechanical energy of a simple harmonic oscillator is proportional to the square of the amplitude. At the equilibrium point, x = 0, all the energy is kinetic: E = 12 mv2 + 12 k(0)2 = 12 mv2max , (14–10b) where vmax is the maximum velocity during the motion. At intermediate points the energy is part kinetic and part potential, and because energy is conserved (14–10c) E = 12 mv2 + 12 kx2 . We can confirm Eqs. 14–10a and b explicitly by inserting Eqs. 14–4 and 14–8a into this last relation: E = 12 mv2A2 sin2(vt + f) + 12 kA2 cos2(vt + f). Substituting with v2 = km, or kA2 = mv2A2 = mv2max , and recalling the important trigonometric identity sin2(vt + f) + cos2(vt + f) = 1, we obtain Eqs. 14–10a and b: E = 12 kA2 = 12 mv2max . [Note that we can check for consistency by solving this equation for vmax , and we do get Eq. 14–9a.]

U K

m (b)

x = −A x = 0 x = A 1

E = 2 kA2

U K

m (c)

x = −A x = 0 x = A (v = 0) 1

E = 2 mv 2 + 2 kx 2 m (d)

U K

x = −A x = 0 x = A x

FIGURE 14–10 Energy changes from potential energy to kinetic energy and back again as the spring oscillates. Energy bar graphs (on the right) are described in Section 8–4.

SECTION 14–3

377

Energy

We can now obtain an equation for the velocity v as a function of x by solving for v in Eq. 14–10c:

U(x)

v = &

U(x)

1 kx2 2

x2 . (14–11b) B A2 Again we see that v is a maximum at x = 0, and is zero at x = &A. The potential energy, U = 12 kx2, is plotted in Fig. 14–11 (see also Section 8–9). The upper horizontal line represents a particular value of the total energy E = 12 kA2. The distance between the E line and the U curve represents the kinetic energy, K, and the motion is restricted to x values between –A and ± A. These results are, of course, consistent with our full solution of the previous Section. Energy conservation is a convenient way to obtain v, for example, if x is given (or vice versa), without having to deal with time t. v = &vmax

U −A

(14–11a)

or, since vmax = A 2km ,

E = K + U = 12 kA2 U=

k 2 AA - x2 B Bm

FIGURE 14–11 Graph of potential energy, U = 12 kx2. K + U = E = constant for any point x where – A x A. Values of K and U are indicated for an arbitrary position x.

1 -

EXAMPLE 14–7 Energy calculations. For the simple harmonic oscillation of Example 14–5, determine (a) the total energy, (b) the kinetic and potential energies as a function of time, (c) the velocity when the mass is 0.050 m from equilibrium, (d) the kinetic and potential energies at half amplitude (x = & A2). APPROACH We use conservation of energy for a spring–mass system, Eqs. 14–10 and 14–11. SOLUTION (a) From Example 14–5, k = 19.6 Nm and A = 0.100 m, so the total energy E from Eq. 14–10a is E = 12 kA2 = 12 (19.6 Nm)(0.100 m)2 = 9.80 * 10 –2 J. (b) We have, from parts (f) and (g) of Example 14–5, x = –(0.100 m) cos 8.08 t and v = (0.808 ms) sin 8.08 t, so U =

1 2 2 kx

K =

1 2 2 mv

= =

1 2 (19.6

Nm)(0.100 m)2 cos2 8.08 t = A9.80 * 10 –2 JB cos2 8.08 t

1 2 (0.300

kg)(0.808 ms)2 sin2 8.08 t = A9.80 * 10–2 JB sin2 8.08 t.

v = vmax31 - x2A2 = (0.808 ms)31 - A 12 B 2 = 0.70 ms.

(d) At x = A2 = 0.050 m, we have U = 12 kx2 = 12 (19.6 Nm)(0.050 m)2 = 2.5 * 10 –2 J K = E - U = 7.3 * 10 –2 J. CONCEPTUAL EXAMPLE 14–8 Doubling the amplitude. Suppose the spring in Fig. 14–10 is stretched twice as far (to x = 2A). What happens to (a) the energy of the system, (b) the maximum velocity of the oscillating mass, (c) the maximum acceleration of the mass? RESPONSE (a) From Eq. 14–10a, the total energy is proportional to the square of the amplitude A, so stretching it twice as far quadruples the energy A2 2 = 4B. You may protest, "I did work stretching the spring from x = 0 to x = A. Don't I do the same work stretching it from A to 2A?" No. The force you exert is proportional to the displacement x, so for the second displacement, from x = A to 2A, you do more work than for the first displacement (x = 0 to A). (b) From Eq. 14–10b, we can see that when the energy is quadrupled, the maximum velocity must be doubled. Cvmax r 1E r A. D (c) Since the force is twice as great when we stretch the spring twice as far, the acceleration is also twice as great: a r F r x. EXERCISE G Suppose the spring in Fig. 14–10 is compressed to x = –A, but is given a push to the right so that the initial speed of the mass m is v0 . What effect does this push have on (a) the energy of the system, (b) the maximum velocity, (c) the maximum acceleration?

378

CHAPTER 14

Oscillations

14–4

Simple Harmonic Motion Related to Uniform Circular Motion

v √ A2 − x2

A θ

Simple harmonic motion has a simple relationship to a particle rotating in a circle with uniform speed. Consider a mass m rotating in a circle of radius A with speed vM on top of a table as shown in Fig. 14–12. As viewed from above, the motion is a circle. But a person who looks at the motion from the edge of the table sees an oscillatory motion back and forth, and this corresponds precisely to SHM as we shall now see. What the person sees, and what we are interested in, is the projection of the circular motion onto the x axis, Fig. 14–12. To see that this motion is analogous to SHM, let us calculate the x component of the velocity vM which is labeled v in Fig. 14–12. The two right triangles indicated in Fig. 14–12 are similar, so

y x

2 2 v 3A - x = vM A

(a)

or v = vM

1 -

x2 . A2

This is exactly the equation for the speed of a mass oscillating with SHM, Eq. 14–11b, where vM = vmax . Furthermore, we can see from Fig. 14–12 that if the angular displacement at t = 0 is f, then after a time t the particle will have rotated through an angle u = vt, and so

x A (b)

x = A cos(u + f) = A cos(vt + f). But what is v here? The linear velocity vM of our particle undergoing rotational motion is related to v by vM = vA, where A is the radius of the circle (see Eq. 10–4, v = Rv). To make one revolution requires a time T, so we also have vM = 2pAT where 2pA is the circle's circumference. Hence v =

vM 2pAT = = 2pT = 2pf A A

where T is the time required for one rotation and f is the frequency. This corresponds precisely to the back-and-forth motion of a simple harmonic oscillator. Thus, the projection on the x axis of a particle rotating in a circle has the same motion as a mass undergoing SHM. Indeed, we can say that the projection of circular motion onto a straight line is simple harmonic motion. The projection of uniform circular motion onto the y axis is also simple harmonic. Thus uniform circular motion can be thought of as two simple harmonic motions operating at right angles.

14–5

FIGURE 14–12 Analysis of simple harmonic motion as a side view (b) of circular motion (a).

FIGURE 14–13 Strobe-light photo of an oscillating pendulum photographed at equal time intervals.

The Simple Pendulum

A simple pendulum consists of a small object (the pendulum bob) suspended from the end of a lightweight cord, Fig. 14–13. We assume that the cord does not stretch and that its mass can be ignored relative to that of the bob. The motion of a simple pendulum moving back and forth with negligible friction resembles simple harmonic motion: the pendulum oscillates along the arc of a circle with equal amplitude on either side of its equilibrium point and as it passes through the equilibrium point (where it would hang vertically) it has its maximum speed. But is it really undergoing SHM? That is, is the restoring force proportional to its displacement? Let us find out. SECTION 14–5

The Simple Pendulum

379

The displacement of the pendulum along the arc is given by x = lu, where u is the angle (in radians) that the cord makes with the vertical and l is the length of the cord (Fig. 14–14). If the restoring force is proportional to x or to u, the motion will be simple harmonic. The restoring force is the net force on the bob, equal to the component of the weight, mg, tangent to the arc:

l B

l sin θ

mg cos θ

mg sin θ mg B

FIGURE 14–14 Simple pendulum.

FIGURE 14–15 The swinging motion of this lamp, hanging by a very long cord from the ceiling of the cathedral at Pisa, is said to have been observed by Galileo and to have inspired him to the conclusion that the period of a pendulum does not depend on amplitude.

P H Y S I C S

A P P L I E D

Pendulum clock

F = –mg sin u, where g is the acceleration of gravity. The minus sign here, as in Eq. 14–1, means that the force is in the direction opposite to the angular displacement u. Since F is proportional to the sine of u and not to u itself, the motion is not SHM. However, if u is small, then sin u is very nearly equal to u when the latter is specified in radians. This can be seen by looking at the series expansion† of sin u (or by looking at the trigonometry Table in Appendix A), or simply by noting in Fig. 14–14 that the arc length x ( = lu) is nearly the same length as the chord ( = l sin u) indicated by the straight dashed line, if u is small. For angles less than 15°, the difference between u (in radians) and sin u is less than 1%. Thus, to a very good approximation for small angles, F = –mg sin u L –mgu. Substituting x = lu or u = xl, we have mg F L – x. l Thus, for small displacements, the motion is essentially simple harmonic, since this equation fits Hooke's law, F = –kx. The effective force constant is k = mgl. Thus we can write u = umax cos(vt + f) where umax is the maximum angular displacement and v = 2pf = 2pT. To obtain v we use Eq. 14–5, where for k we substitute mgl: that is,‡ v = 1km = 1(mgl)m , or g . Al Then the frequency f is g v 1 , f = = 2p 2p A l and the period T is 1 l. T = = 2p f Bg v =

CHAPTER 14

Oscillations

[u small] (14–12b)

[u small] (14–12c)

The mass m of the pendulum bob does not appear in these formulas for T and f. Thus we have the surprising result that the period and frequency of a simple pendulum do not depend on the mass of the pendulum bob. You may have noticed this if you pushed a small child and then a large one on the same swing. We also see from Eq. 14–12c that the period of a pendulum does not depend on the amplitude (like any SHM, Section 14–2), as long as the amplitude u is small. Galileo is said to have first noted this fact while watching a swinging lamp in the cathedral at Pisa (Fig. 14–15). This discovery led to the invention of the pendulum clock, the first really precise timepiece, which became the standard for centuries. Because a pendulum does not undergo precisely SHM, the period does depend slightly on the amplitude, the more so for large amplitudes. The accuracy of a pendulum clock would be affected, after many swings, by the decrease in amplitude due to friction. But the mainspring in a pendulum clock (or the falling weight in a grandfather clock) supplies energy to compensate for the friction and to maintain the amplitude constant, so that the timing remains precise. u3 u5 u7 + + p. 3! 5! 7! ‡ Be careful not to think that v = dudt as in rotational motion. Here u is the angle of the pendulum at any instant (Fig. 14–14), but we use v now to represent not the rate this angle u changes, but rather a constant related to the period, v = 2pf = 1gl . †

380

[u small] (14–12a)

sin u = u -

EXERCISE H If a simple pendulum is taken from sea level to the top of a high mountain and started at the same angle of 5°, it would oscillate at the top of the mountain (a) slightly slower, (b) slightly faster, (c) at exactly the same frequency, (d) not at all—it would stop, (e) none of these. EXERCISE I Return to the Chapter-Opening Question, p. 369, and answer it again now. Try to explain why you may have answered differently the first time.

EXAMPLE 14–9 Measuring g. A geologist uses a simple pendulum that has a length of 37.10 cm and a frequency of 0.8190 Hz at a particular location on the Earth. What is the acceleration of gravity at this location? APPROACH We can use the length l and frequency f of the pendulum in Eq. 14-12b, which contains our unknown, g. SOLUTION We solve Eq. 14–12b for g and obtain g = (2pf)2l = A6.283 * 0.8190 s–1 B (0.3710 m) = 9.824 ms2. 2

EXERCISE J (a) Estimate the length of a simple pendulum that makes one swing back and forth per second. (b) What would be the period of a 1.0-m-long pendulum?

* 14–6

The Physical Pendulum and the Torsion Pendulum

Physical Pendulum The term physical pendulum refers to any real extended object which oscillates back and forth, in contrast to the rather idealized simple pendulum where all the mass is assumed concentrated in the tiny pendulum bob. An example of a physical pendulum is a baseball bat suspended from the point O, as shown in Fig. 14–16. The force of gravity acts at the center of gravity (CG) of the object located a distance h from the pivot point O. The physical pendulum is best analyzed using the equations of rotational motion. The torque on a physical pendulum, calculated about point O, is t = –mgh sin u. Newton's second law for rotational motion, Eq. 10–14, states that d 2u ©t = Ia = I 2 , dt where I is the moment of inertia of the object about the pivot point and a = d2udt2 is the angular acceleration. Thus we have d2u I 2 = –mgh sin u dt or mgh d 2u + sin u = 0, 2 I dt where I is calculated about an axis through point O. For small angular amplitude, sin u L u, so we have mgh d 2u + a b u = 0. [small angular displacement] (14–13) 2 I dt This is just the equation for SHM, Eq. 14–3, except that u replaces x and mghI replaces km. Thus, for small angular displacements, a physical pendulum undergoes SHM, given by u = umax cos(vt + f), where umax is the maximum angular displacement and v = 2pT. The period, T, is (see Eq. 14–7b, replacing mk with Imgh): T = 2p

I . B mgh

O h θ

mgB

d⊥(= h sin θ ) FIGURE 14–16 A physical pendulum suspended from point O.

[small angular displacement] (14–14) *SECTION 14–6

The Physical Pendulum and the Torsion Pendulum

381

Pivot 42 cm

EXAMPLE 14–10 Moment of inertia measurement. An easy way to measure the moment of inertia of an object about any axis is to measure the period of oscillation about that axis. (a) Suppose a nonuniform 1.0-kg stick can be balanced at a point 42 cm from one end. If it is pivoted about that end (Fig. 14–17), it oscillates with a period of 1.6 s. What is its moment of inertia about this end? (b) What is its moment of inertia about an axis perpendicular to the stick through its center of mass? APPROACH We put the given values into Eq. 14–14 and solve for I. For (b) we use the parallel-axis theorem (Section 10–7). SOLUTION (a) Given T = 1.6 s, and h = 0.42 m, Eq. 14–14 gives

FIGURE 14–17 Example 14–10.

I = mghT 24p2 = 0.27 kg m2. (b) We use the parallel-axis theorem, Eq. 10–17. The CM is where the stick balanced, 42 cm from the end, so Icm = I - mh2 = 0.27 kgm2 - (1.0 kg)(0.42 m)2 = 0.09 kgm2. NOTE Since an object does not oscillate about its CM, we cannot measure Icm directly, but the parallel-axis theorem provides a convenient method to determine Icm .

Torsion Pendulum Wire

+θmax 0 Equilibrium − θmax FIGURE 14–18 A torsion pendulum. The disc oscillates in SHM between umax and – umax .

Another type of oscillatory motion is a torsion pendulum, in which a disc (Fig. 14–18) or a bar (as in Cavendish's apparatus, Fig. 6–3) is suspended from a wire. The twisting (torsion) of the wire serves as the elastic force. The motion here will be SHM since the restoring torque is very closely proportional to the negative of the angular displacement, t = –Ku, where K is a constant that depends on the wire stiffness. Then v = 2KI. There is no small angle restriction here, as there is for the physical pendulum (where gravity acts), as long as the wire responds linearly in accordance with Hooke's law.

14–7 FIGURE 14–19 Damped harmonic motion. The solid red curve represents a cosine times a decreasing exponential (the dashed curves).

x 0

The amplitude of any real oscillating spring or swinging pendulum slowly decreases in time until the oscillations stop altogether. Figure 14–19 shows a typical graph of the displacement as a function of time. This is called damped harmonic motion. The damping† is generally due to the resistance of air and to internal friction within the oscillating system. The energy that is dissipated to thermal energy is reflected in a decreased amplitude of oscillation. Since natural oscillating systems are damped in general, why do we even talk about (undamped) simple harmonic motion? The answer is that SHM is much t easier to deal with mathematically. And if the damping is not large, the oscillations can be thought of as simple harmonic motion on which the damping is superposed, as represented by the dashed curves in Fig. 14–19. Although damping does alter the frequency of vibration, the effect is usually small if the damping is small. Let us look at this in more detail. The damping force depends on the speed of the oscillating object, and opposes the motion. In some simple cases the damping force can be approximated as being directly proportional to the speed: Fdamping = –bv, where b is a constant.‡ For a mass oscillating on the end of a spring, the restoring force of the spring is F = –kx; so Newton's second law (ma = ©F) becomes ma = –kx - bv. We bring all terms to the left side of the equation and substitute v = dxdt and † ‡

382

CHAPTER 14

Oscillations

Damped Harmonic Motion

To "damp" means to diminish, restrain, or extinguish, as to "dampen one's spirits." Such velocity-dependent forces were discussed in Section 5–6.

a = d2xdt2 to obtain d 2x dx + b + kx = 0, (14–15) dt dt2 which is the equation of motion. To solve this equation, we guess at a solution and then check to see if it works. If the damping constant b is small, x as a function of t is as plotted in Fig. 14–19, which looks like a cosine function times a factor (represented by the dashed lines) that decreases in time. A simple function that g does this is the exponential, e – t, and the solution that satisfies Eq. 14–15 is m

x = Ae – t cos v¿ t,

(14–16)

where A, g, and v¿ are assumed to be constants, and x = A at t = 0. We have called the angular frequency v¿ (and not v) because it is not the same as the v for SHM without damping Av = 1kmB. If we substitute Eq. 14–16 into Eq. 14–15 (we do this in the optional subsection below), we find that Eq. 14–16 is indeed a solution if g and v¿ have the values g = b (14–17) 2m k b2 . (14–18) Bm 4m2 Thus x as a function of time t for a (lightly) damped harmonic oscillator is v¿ =

x = Ae(–b2m)t cos v¿ t.

(14–19)

Of course a phase constant, f, can be added to the argument of the cosine in Eq. 14–19. As it stands with f = 0, it is clear that the constant A in Eq. 14–19 is simply the initial displacement, x = A at t = 0. The frequency f¿ is f¿ =

v¿ 1 k b2 . = 2p 2p B m 4m2

(14–20)

The frequency is lower, and the period longer, than for undamped SHM. (In many practical cases of light damping, however, v¿ differs only slightly from v = 1km . ) This makes sense since we expect damping to slow down the motion. Equation 14–20 reduces to Eq. 14–7a, as it should, when there is no damping (b = 0). The constant g = b2m is a measure of how quickly the oscillations decrease toward zero (Fig. 14–19). The time tL = 2mb is the time taken for the oscillations to drop to 1e of the original amplitude; tL is called the "mean lifetime" of the oscillations. Note that the larger b is, the more quickly the oscillations die away. The solution, Eq. 14–19, is not valid if b is so large that 2

b 7 4mk since then v¿ (Eq. 14–18) would become imaginary. In this case the system does not oscillate at all but returns directly to its equilibrium position, as we now discuss. Three common cases of heavily damped systems are shown in Fig. 14–20. Curve C represents the situation when the damping is so large Ab2 W 4mkB that it takes a long time to reach equilibrium; the system is overdamped. Curve A represents an underdamped situation in which the system makes several swings before coming to rest Ab2 6 4mkB and corresponds to a more heavily damped version of Eq. 14–19. Curve B represents critical damping: b2 = 4mk; in this case equilibrium is reached in the shortest time. These terms all derive from the use of practical damped systems such as door-closing mechanisms and shock absorbers in a car (Fig. 14–21), which are usually designed to give critical damping. But as they wear out, underdamping occurs: a door slams and a car bounces up and down several times whenever it hits a bump. In many systems, the oscillatory motion is what counts, as in clocks and watches, and damping needs to be minimized. In other systems, oscillations are the problem, such as a car's springs, so a proper amount of damping (i.e., critical) is desired. Well-designed damping is needed for all kinds of applications. Large buildings, especially in California, are now built (or retrofitted) with huge dampers to reduce earthquake damage.

x C A

FIGURE 14–20 Underdamped (A), critically damped (B), and overdamped (C) motion. FIGURE 14–21 Automobile spring and shock absorber provide damping so that a car won't bounce up and down endlessly. Attached to car frame

Piston Viscous fluid Attached to car axle

SECTION 14–7

383

EXAMPLE 14–11 Simple pendulum with damping. A simple pendulum has a length of 1.0 m (Fig. 14–22). It is set swinging with small-amplitude oscillations. After 5.0 minutes, the amplitude is only 50% of what it was initially. (a) What is the value of g for the motion? (b) By what factor does the frequency, f¿, differ from f, the undamped frequency? 1.0 m

FIGURE 14–22 Example 14–11.

APPROACH We assume the damping force is proportional to angular speed, dudt. The equation of motion for damped harmonic motion is b k b2 , g x = Ae – t cos v¿ t, where g = and v¿ = 2m Bm 4m2 for motion of a mass on the end of a spring. For the simple pendulum without damping, we saw in Section 14–5 that F = –mgu for small u. Since F = ma, where a can be written in terms of the angular acceleration a = d2udt2 as a = la = l d 2udt2, then F = ml d 2udt2, and d 2u + gu = 0. dt2 Introducing a damping term, b(dudt), we have l

d2u du + b + gu = 0, 2 dt dt which is the same as Eq. 14–15 with u replacing x, and l and g replacing m and k. SOLUTION (a) We compare Eq. 14–15 with our equation just above and see that g our equation x = Ae – t cos v¿ t becomes an equation for u with l

g b b2 and v¿ = - 2. 2l Bl 4l At t = 0, we rewrite Eq. 14–16 with u replacing x as g# u0 = Ae – 0 cos v¿ # 0 = A. g =

Then at t = 5.0 min = 300 s, the amplitude given by Eq. 14–16 has fallen to 0.50 A, so g 0.50A = Ae – (300 s). We solve this for g and obtain g = ln 2.0(300 s) = 2.3 * 10–3 s–1. (b) We have l = 1.0 m, so b = 2gl = 2 A2.3 * 10–3 s–1 B(1.0 m) = 4.6 * 10–3 ms. Thus Ab24l2 B is very much less than gl A = 9.8 s–2 B, and the angular frequency of the motion remains almost the same as that of the undamped motion. Specifically (see Eq. 14–20), f¿ =

g 1 l b2 c1 - ¢ 2 ≤ g 4l 2p A l

1 2

g 1 1 l b2 c1 ¢ ≤ 2p A l 2 g 4l2

where we used the binomial expansion. Then, with f = (12 p)2gl (Eq. 14–12b), f - f¿ 1 l b2 L ¢ ≤ = 2.7 * 10 –7. f 2 g 4l2 So f¿ differs from f by less than one part in a million. G * Showing x Ae t cos Vt is a Solution

We start with Eq. 14–16, to see if it is a solution to Eq. 14–15. First we take the first and second derivatives dx g g = – gAe – t cos v¿ t - v¿Ae – t sin v¿ t dt d 2x g g g g = g2Ae – t cos v¿ t + gAv¿e – t sin v¿ t + v¿ gAe – t sin v¿ t - v¿ 2Ae – t cos v¿ t. 2 dt We next substitute these relations back into Eq. 14–15 and reorganize to

384

CHAPTER 14

Oscillations

obtain g Ae – t C Amg2 - mv¿ 2 - bg + kB cos v¿ t + (2v¿ gm - bv¿) sin v¿ t D = 0.

(i)

The left side of this equation must equal zero for all times t, but this can only be so for certain values of g and v¿. To determine g and v¿, we choose two values of t that will make their evaluation easy. At t = 0, sin v¿ t = 0, so the above relation reduces to AAmg2 - mv¿ 2 - bg + kB = 0, which means† that mg2 - mv¿ 2 - bg + k = 0.

(ii)

Then at t = p2v¿, cos v¿ t = 0 so Eq. (i) can be valid only if 2gm - b = 0.

(iii)

From Eq. (iii) we have g =

b 2m

and from Eq. (ii) v¿ =

g2 -

bg k b2 . k + = m m Bm 4m2

Thus we see that Eq. 14–16 is a solution to the equation of motion for the damped harmonic oscillator as long as g and v¿ have these specific values, as already given in Eqs. 14–17 and 14–18.

Forced Oscillations; Resonance

When an oscillating system is set into motion, it oscillates at its natural frequency (Eqs. 14–7a and 14–12b). However, a system may have an external force applied to it that has its own particular frequency and then we have a forced oscillation. For example, we might pull the mass on the spring of Fig. 14–1 back and forth at a frequency f. The mass then oscillates at the frequency f of the external force, even if this frequency is different from the natural frequency of the spring, which we will now denote by f0 where (see Eqs. 14–5 and 14–7a) v0 = 2pf0 =

k. Bm

In a forced oscillation the amplitude of oscillation, and hence the energy transferred to the oscillating system, is found to depend on the difference between f and f0 as well as on the amount of damping, reaching a maximum when the frequency of the external force equals the natural frequency of the system—that is, when f = f0 . The amplitude is plotted in Fig. 14–23 as a function of the external frequency f. Curve A represents light damping and curve B heavy damping. The amplitude can become large when the driving frequency f is near the natural frequency, f L f0 , as long as the damping is not too large. When the damping is small, the increase in amplitude near f = f0 is very large (and often dramatic). This effect is known as resonance. The natural frequency f0 of a system is called its resonant frequency. A simple illustration of resonance is pushing a child on a swing. A swing, like any pendulum, has a natural frequency of oscillation that depends on its length l. If you push on the swing at a random frequency, the swing bounces around and reaches no great amplitude. But if you push with a frequency equal to the natural frequency of the swing, the amplitude increases greatly. At resonance, relatively little effort is required to obtain a large amplitude.

Amplitude of oscillating system

14–8

B f0

External frequency f FIGURE 14–23 Resonance for lightly damped (A) and heavily damped (B) systems. (See Fig. 14–26 for a more detailed graph.)

† It would also be satisfied by A = 0, but this gives the trivial and uninteresting solution x = 0 for all t —that is, no oscillation.

SECTION 14–8

Forced Oscillations; Resonance

385

FIGURE 14–24 This goblet breaks as it vibrates in resonance to a trumpet call. FIGURE 14–25 (a) Large-amplitude oscillations of the Tacoma Narrows Bridge, due to gusty winds, led to its collapse (1940). (b) Collapse of a freeway in California, due to the 1989 earthquake.

The great tenor Enrico Caruso was said to be able to shatter a crystal goblet by singing a note of just the right frequency at full voice. This is an example of resonance, for the sound waves emitted by the voice act as a forced oscillation on the glass. At resonance, the resulting oscillation of the goblet may be large enough in amplitude that the glass exceeds its elastic limit and breaks (Fig. 14–24). Since material objects are, in general, elastic, resonance is an important phenomenon in a variety of situations. It is particularly important in structural engineering, although the effects are not always foreseen. For example, it has been reported that a railway bridge collapsed because a nick in one of the wheels of a crossing train set up a resonant oscillation in the bridge. Indeed, marching soldiers break step when crossing a bridge to avoid the possibility that their normal rhythmic march might match a resonant frequency of the bridge. The famous collapse of the Tacoma Narrows Bridge (Fig. 14–25a) in 1940 occurred as a result of strong gusting winds driving the span into large-amplitude oscillatory motion. The Oakland freeway collapse in the 1989 California earthquake (Fig. 14–25b) involved resonant oscillation of a section built on mudfill that readily transmitted that frequency. We will meet important examples of resonance later. We will also see that vibrating objects often have not one, but many resonant frequencies.

* Equation of Motion and Its Solution We now look at the equation of motion for a forced oscillation and its solution. Suppose the external force is sinusoidal and can be represented by Fext = F0 cos vt, where v = 2pf is the angular frequency applied externally to the oscillator. Then the equation of motion (with damping) is ma = –kx - bv + F0 cos vt. This can be written as m

(a)

d2x dx + kx = F0 cos vt. + b dt dt2

(14–21)

The external force, on the right of the equation, is the only term that does not involve x or one of its derivatives. Problem 68 asks you to show that x = A 0 sin Avt + f0 B

(14–22)

is a solution to Eq. 14–21, by direct substitution, where A0 = and (b)

m 3 Av - v20 B 2 + b2v2m2

f0 = tan–1

v20 - v2 . v(bm)

(14–23)

(14–24)

Actually, the general solution to Eq. 14–21 is Eq. 14–22 plus another term of the form of Eq. 14–19 for the natural damped motion of the oscillator; this second term approaches zero in time, so in many cases we need to be concerned only with Eq. 14–22. The amplitude of forced harmonic motion, A 0 , depends strongly on the difference between the applied and the natural frequency. A plot of A 0 (Eq. 14–23) as a function of the applied frequency, v, is shown in Fig. 14–26 (a more detailed version of Fig. 14–23) for three specific values of the damping constant b. Curve A Ab = 16 mv0 B represents light damping, curve B Ab = 12 mv0 B fairly heavy damping, and curve C Ab = 12mv0 B overdamped motion. The amplitude can become large when the driving frequency is near the natural frequency, v L v0 , as long as the damping is not too large. When the damping is small, the increase in amplitude near v = v0 is very large and, as we saw, is known as resonance.

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The natural oscillating frequency f0 ( v02p) of a system is its resonant frequency.† If b = 0, resonance occurs at v = v0 and the resonant peak (of A 0) becomes infinite; in such a case, energy is being continuously transferred into the system and none is dissipated. For real systems, b is never precisely zero, and the resonant peak is finite. The peak does not occur precisely at v = v0 (because of the term b2v2m2 in the denominator of Eq. 14–23), although it is quite close to v0 unless the damping is very large. If the damping is large, there is little or no peak (curve C in Fig. 14–26).

F0 k

* Q value The height and narrowness of a resonant peak is often specified by its quality factor or Q value, defined as mv0 . Q = (14–25) b

F0 k

In Fig. 14–26, curve A has Q = 6, curve B has Q = 2, and curve C has Q = 112 . The smaller the damping constant b, the larger the Q value becomes, and the higher the resonance peak. The Q value is also a measure of the width of the peak. To see why, let v1 and v2 be the frequencies where the square of the amplitude A 0 has half its maximum value (we use the square because the power transferred to the system is proportional to A 20 ); then ¢v = v1 - v2 , which is called the width of the resonance peak, is related to Q by ¢v 1. = v0 Q

B C 2v0

v0 v

FIGURE 14–26 Amplitude of a forced harmonic oscillator as a function of v. Curves A, B, and C correspond to light, heavy, and overdamped systems, respectively AQ = mv0b = 6, 2, 0.71B.

(14–26)

This relation is accurate only for weak damping. The larger the Q value, the narrower will be the resonance peak relative to its height. Thus a large Q value, representing a system of high quality, has a high, narrow resonance peak. † Sometimes the resonant frequency is defined as the actual value of v at which the amplitude has its maximum value, and this depends somewhat on the damping constant. Except for very heavy damping, this value is quite close to v0 .

Summary An oscillating object undergoes simple harmonic motion (SHM) if the restoring force is proportional to the displacement, F = –kx. (14–1) The maximum displacement from equilibrium is called the amplitude. The period, T, is the time required for one complete cycle (back and forth), and the frequency, f, is the number of cycles per second; they are related by 1. (14–2) f = T The period of oscillation for a mass m on the end of an ideal massless spring is given by m. (14–7b) Ak SHM is sinusoidal, which means that the displacement as a function of time follows a sine or cosine curve. The general solution can be written (14–4) x = A cos(vt + f) where A is the amplitude, f is the phase angle, and T = 2p

k. (14–5) Bm The values of A and f depend on the initial conditions (x and v at t = 0 ). v = 2pf =

During SHM, the total energy E = 12 mv2 + 12 kx2 is continually changing from potential to kinetic and back again. A simple pendulum of length l approximates SHM if its amplitude is small and friction can be ignored. For small amplitudes, its period is given by T = 2p

l, Bg

(14–12c)

where g is the acceleration of gravity. When friction is present (for all real springs and pendulums), the motion is said to be damped. The maximum displacement decreases in time, and the mechanical energy is eventually all transformed to thermal energy. If the friction is very large, so no oscillations occur, the system is said to be overdamped. If the friction is small enough that oscillations occur, the system is underdamped, and the displacement is given by g

x = Ae – t cos v¿ t,

(14–16)

where g and v¿ are constants. For a critically damped system, no oscillations occur and equilibrium is reached in the shortest time. If an oscillating force is applied to a system capable of vibrating, the amplitude of vibration can be very large if the frequency of the applied force is near the natural (or resonant) frequency of the oscillator; this is called resonance.

Summary

387

Questions 1. Give some examples of everyday vibrating objects. Which exhibit SHM, at least approximately? 2. Is the acceleration of a simple harmonic oscillator ever zero? If so, where? 3. Explain why the motion of a piston in an automobile engine is approximately simple harmonic. 4. Real springs have mass. Will the true period and frequency be larger or smaller than given by the equations for a mass oscillating on the end of an idealized massless spring? Explain. 5. How could you double the maximum speed of a simple harmonic oscillator (SHO)? 6. A 5.0-kg trout is attached to the hook of a vertical spring scale, and then is released. Describe the scale reading as a function of time. 7. If a pendulum clock is accurate at sea level, will it gain or lose time when taken to high altitude? Why? 8. A tire swing hanging from a branch reaches nearly to the ground (Fig. 14–27). How could you estimate the height of the branch using only a stopwatch?

9. For a simple harmonic oscillator, when (if ever) are the displacement and velocity vectors in the same direction? When are the displacement and acceleration vectors in the same direction? 10. A 100-g mass hangs from a long cord forming a pendulum. The mass is pulled a short distance to one side and released from rest. The time to swing over and back is carefully measured to be 2.0 s. If the 100-g mass is replaced by a 200-g mass, which is then pulled over the same distance and released from rest, the time will be (a) 1.0 s, (b) 1.41 s, (c) 2.0 s, (d) 2.82 s, (e) 4.0 s. 11. Two equal masses are attached to separate identical springs next to one another. One mass is pulled so its spring stretches 20 cm and the other is pulled so its spring stretches only 10 cm. The masses are released simultaneously. Which mass reaches the equilibrium point first? 12. Does a car bounce on its springs faster when it is empty or when it is fully loaded? 13. What is the approximate period of your walking step? 14. What happens to the period of a playground swing if you rise up from sitting to a standing position? * 15. A thin uniform rod of mass m is suspended from one end and oscillates with a frequency f. If a small sphere of mass 2m is attached to the other end, does the frequency increase or decrease? Explain. 16. A tuning fork of natural frequency 264 Hz sits on a table at the front of a room. At the back of the room, two tuning forks, one of natural frequency 260 Hz and one of 420 Hz are initially silent, but when the tuning fork at the front of the room is set into vibration, the 260-Hz fork spontaneously begins to vibrate but the 420-Hz fork does not. Explain. 17. Why can you make water slosh back and forth in a pan only if you shake the pan at a certain frequency? 18. Give several everyday examples of resonance. 19. Is a rattle in a car ever a resonance phenomenon? Explain.

FIGURE 14–27 Question 8.

20. Over the years, buildings have been able to be built out of lighter and lighter materials. How has this affected the natural oscillation frequencies of buildings and the problems of resonance due to passing trucks, airplanes, or by wind and other natural sources of vibration?

Problems 14–1 and 14–2 Simple Harmonic Motion 1. (I) If a particle undergoes SHM with amplitude 0.18 m, what is the total distance it travels in one period? 2. (I) An elastic cord is 65 cm long when a weight of 75 N hangs from it but is 85 cm long when a weight of 180 N hangs from it. What is the "spring" constant k of this elastic cord? 3. (I) The springs of a 1500-kg car compress 5.0 mm when its 68-kg driver gets into the driver's seat. If the car goes over a bump, what will be the frequency of oscillations? Ignore damping.

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4. (I) (a) What is the equation describing the motion of a mass on the end of a spring which is stretched 8.8 cm from equilibrium and then released from rest, and whose period is 0.66 s? (b) What will be its displacement after 1.8 s? 5. (II) Estimate the stiffness of the spring in a child's pogo stick if the child has a mass of 35 kg and bounces once every 2.0 seconds. 6. (II) A fisherman's scale stretches 3.6 cm when a 2.4-kg fish hangs from it. (a) What is the spring stiffness constant and (b) what will be the amplitude and frequency of oscillation if the fish is pulled down 2.5 cm more and released so that it oscillates up and down?

7. (II) Tall buildings are designed to sway in the wind. In a 100-kmh wind, for example, the top of the 110-story Sears Tower oscillates horizontally with an amplitude of 15 cm. The building oscillates at its natural frequency, which has a period of 7.0 s. Assuming SHM, find the maximum horizontal velocity and acceleration experienced by a Sears employee as she sits working at her desk located on the top floor. Compare the maximum acceleration (as a percentage) with the acceleration due to gravity. 8. (II) Construct a Table indicating the position x of the mass in Fig. 14–2 at times t = 0, 14 T, 12 T, 34 T, T, and 54 T, where T is the period of oscillation. On a graph of x vs. t, plot these six points. Now connect these points with a smooth curve. Based on these simple considerations, does your curve resemble that of a cosine or sine wave? 9. (II) A small fly of mass 0.25 g is caught in a spider's web. The web oscillates predominately with a frequency of 4.0 Hz. (a) What is the value of the effective spring stiffness constant k for the web? (b) At what frequency would you expect the web to oscillate if an insect of mass 0.50 g were trapped? 10. (II) A mass m at the end of a spring oscillates with a frequency of 0.83 Hz. When an additional 680-g mass is added to m, the frequency is 0.60 Hz. What is the value of m? 11. (II) A uniform meter stick of mass M is pivoted on a hinge at one end and held horizontal by a spring with spring constant k attached at the other end (Fig. 14–28). If the stick oscillates up and down slightly, what is its frequency? [Hint: Write a torque equation about the hinge.]

1.25 m FIGURE 14–28 Problem 11. 12. (II) A balsa wood block of mass 55 g floats on a lake, bobbing up and down at a frequency of 3.0 Hz. (a) What is the value of the effective spring constant of the water? (b) A partially filled water bottle of mass 0.25 kg and almost the same size and shape of the balsa block is tossed into the water. At what frequency would you expect the bottle to bob up and down? Assume SHM. 13. (II) Figure 14–29 shows two examples of SHM, labeled A and B. For each, what is (a) the amplitude, (b) the frequency, and (c) the period? (d) Write the equations for both A and B in the form of a sine or cosine. x (m) 4 3 2 1 0 −1 −2 −3 −4

1.0

2.0

3.0

4.0

FIGURE 14–29 Problem 13.

t (s)

14. (II) Determine the phase constant f in Eq. 14–4 if, at t = 0, the oscillating mass is at (a) x = –A, (b) x = 0, (c) x = A, (d) x = 12 A, (e) x = – 12 A, (f ) x = A 22 . 15. (II) A vertical spring with spring stiffness constant 305 Nm oscillates with an amplitude of 28.0 cm when 0.260 kg hangs from it. The mass passes through the equilibrium point (y = 0) with positive velocity at t = 0. (a) What equation describes this motion as a function of time? (b) At what times will the spring be longest and shortest? 16. (II) The graph of displacement vs. time for a small mass m at the end of a spring is shown in Fig. 14–30. At t = 0, x = 0.43 cm. (a) If m = 9.5 g, find the spring constant, k. (b) Write the equation for displacement x as a function of time.

x 0.82 cm 0.43 cm

0.82 cm

0.69 s FIGURE 14–30 Problem 16. 17. (II) The position of a SHO as a function of time is given by x = 3.8 cos(5pt4 + p6) where t is in seconds and x in meters. Find (a) the period and frequency, (b) the position and velocity at t = 0, and (c) the velocity and acceleration at t = 2.0 s. 18. (II) A tuning fork oscillates at a frequency of 441 Hz and the tip of each prong moves 1.5 mm to either side of center. Calculate (a) the maximum speed and (b) the maximum acceleration of the tip of a prong. 19. (II) An object of unknown mass m is hung from a vertical spring of unknown spring constant k, and the object is observed to be at rest when the spring has extended by 14 cm. The object is then given a slight push and executes SHM. Determine the period T of this oscillation. 20. (II) A 1.25-kg mass stretches a vertical spring 0.215 m. If the spring is stretched an additional 0.130 m and released, how long does it take to reach the (new) equilibrium position again? 21. (II) Consider two objects, A and B, both undergoing SHM, but with different frequencies, as described by the equations xA = (2.0 m) sin(2.0 t) and xB = (5.0 m) sin(3.0 t), where t is in seconds. After t = 0, find the next three times t at which both objects simultaneously pass through the origin. 22. (II) A 1.60-kg object oscillates from a vertically hanging light spring once every 0.55 s. (a) Write down the equation giving its position y (± upward) as a function of time t, assuming it started by being compressed 16 cm from the equilibrium position (where y = 0 ), and released. (b) How long will it take to get to the equilibrium position for the first time? (c) What will be its maximum speed? (d) What will be its maximum acceleration, and where will it first be attained?

Problems

389

23. (II) A bungee jumper with mass 65.0 kg jumps from a high bridge. After reaching his lowest point, he oscillates up and down, hitting a low point eight more times in 43.0 s. He finally comes to rest 25.0 m below the level of the bridge. Estimate the spring stiffness constant and the unstretched length of the bungee cord assuming SHM. 24. (II) A block of mass m is supported by two identical parallel vertical springs, each with spring stiffness constant k (Fig. 14–31). What will be the frequency of vertical oscillation? k k m

FIGURE 14–31 Problem 24.

25. (III) A mass m is connected to two springs, with spring constants k1 and k2 , in two different ways as shown in Fig. 14–32a and b. Show that the period for the configuration shown in part (a) is given by T = 2p

m¢

1 1 + ≤ k1 k2

and for that in part (b) is given by T = 2p

m . A k1 + k2

Ignore friction. k2

(a) k1 FIGURE 14–32 Problem 25.

(b)

26. (III) A mass m is at rest on the end of a spring of spring constant k. At t = 0 it is given an impulse J by a hammer. Write the formula for the subsequent motion in terms of m, k, J, and t.

30. (II) A 0.35-kg mass at the end of a spring oscillates 2.5 times per second with an amplitude of 0.15 m. Determine (a) the velocity when it passes the equilibrium point, (b) the velocity when it is 0.10 m from equilibrium, (c) the total energy of the system, and (d) the equation describing the motion of the mass, assuming that at t = 0, x was a maximum. 31. (II) It takes a force of 95.0 N to compress the spring of a toy popgun 0.175 m to "load" a 0.160-kg ball. With what speed will the ball leave the gun if fired horizontally? 32. (II) A 0.0125-kg bullet strikes a 0.240-kg block attached to a fixed horizontal spring whose spring constant is 2.25 * 103 Nm and sets it into oscillation with an amplitude of 12.4 cm. What was the initial speed of the bullet if the two objects move together after impact? 33. (II) If one oscillation has 5.0 times the energy of a second one of equal frequency and mass, what is the ratio of their amplitudes? 34. (II) A mass of 240 g oscillates on a horizontal frictionless surface at a frequency of 3.0 Hz and with amplitude of 4.5 cm. (a) What is the effective spring constant for this motion? (b) How much energy is involved in this motion? 35. (II) A mass resting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. It takes 3.6 J of work to compress the spring by 0.13 m. If the spring is compressed, and the mass is released from rest, it experiences a maximum acceleration of 15 ms2. Find the value of (a) the spring constant and (b) the mass. 36. (II) An object with mass 2.7 kg is executing simple harmonic motion, attached to a spring with spring constant k = 280 Nm. When the object is 0.020 m from its equilibrium position, it is moving with a speed of 0.55 ms. (a) Calculate the amplitude of the motion. (b) Calculate the maximum speed attained by the object. 37. (II) Agent Arlene devised the following method of measuring the muzzle velocity of a rifle (Fig. 14–33). She fires a bullet into a 4.648-kg wooden block resting on a smooth surface, and attached to a spring of spring constant k = 142.7 Nm. The bullet, whose mass is 7.870 g, remains embedded in the wooden block. She measures the maximum distance that the block compresses the spring to be 9.460 cm. What is the speed v of the bullet?

14–3 Energy in SHM 27. (I) A 1.15-kg mass oscillates according to the equation x = 0.650 cos 7.40t where x is in meters and t in seconds. Determine (a) the amplitude, (b) the frequency, (c) the total energy, and (d) the kinetic energy and potential energy when x = 0.260 m. 28. (I) (a) At what displacement of a SHO is the energy half kinetic and half potential? (b) What fraction of the total energy of a SHO is kinetic and what fraction potential when the displacement is one third the amplitude? 29. (II) Draw a graph like Fig. 14–11 for a horizontal spring whose spring constant is 95 Nm and which has a mass of 55 g on the end of it. Assume the spring was started with an initial amplitude of 2.0 cm. Neglect the mass of the spring and any friction with the horizontal surface. Use your graph to estimate (a) the potential energy, (b) the kinetic energy, and (c) the speed of the mass, for x = 1.5 cm.

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vB m

k M

9.460 cm

M+m

FIGURE 14–33 Problem 37. 38. (II) Obtain the displacement x as a function of time for the simple harmonic oscillator using the conservation of energy, Eqs. 14–10. [Hint: Integrate Eq. 14–11a with v = dxdt. ]

39. (II) At t = 0, a 785-g mass at rest on the end of a horizontal spring (k = 184 Nm) is struck by a hammer which gives it an initial speed of 2.26 ms. Determine (a) the period and frequency of the motion, (b) the amplitude, (c) the maximum acceleration, (d) the position as a function of time, (e) the total energy, and (f) the kinetic energy when x = 0.40A where A is the amplitude. 40. (II) A pinball machine uses a spring launcher that is compressed 6.0 cm to launch a ball up a 15° ramp. Assume that the pinball is a solid uniform sphere of radius r = 1.0 cm and mass m = 25 g. If it is rolling without slipping at a speed of 3.0 ms when it leaves the launcher, what is the spring constant of the spring launcher?

14–5 Simple Pendulum 41. (I) A pendulum has a period of 1.35 s on Earth. What is its period on Mars, where the acceleration of gravity is about 0.37 that on Earth? 42. (I) A pendulum makes 32 oscillations in exactly 50 s. What is its (a) period and (b) frequency? 43. (II) A simple pendulum is 0.30 m long. At t = 0 it is released from rest starting at an angle of 13°. Ignoring friction, what will be the angular position of the pendulum at (a) t = 0.35 s, (b) t = 3.45 s, and (c) t = 6.00 s? 44. (II) What is the period of a simple pendulum 53 cm long (a) on the Earth, and (b) when it is in a freely falling elevator? 45. (II) A simple pendulum oscillates with an amplitude of 10.0°. What fraction of the time does it spend between ±5.0° and – 5.0°? Assume SHM. 46. (II) Your grandfather clock's pendulum has a length of 0.9930 m. If the clock loses 26 s per day, how should you adjust the length of the pendulum? 47. (II) Derive a formula for the maximum speed vmax of a simple pendulum bob in terms of g, the length l, and the maximum angle of swing umax . * 14–6 Physical Pendulum and Torsion Pendulum * 48. (II) A pendulum consists of a tiny bob of mass M and a uniform cord of mass m and length l. (a) Determine a formula for the period using the small angle approximation. (b) What would be the fractional error if you use the formula for a simple pendulum, Eq. 14–12c? * 49. (II) The balance wheel of a watch is a thin ring of radius 0.95 cm and oscillates with a frequency of 3.10 Hz. If a torque of 1.1 * 10–5 m N causes the wheel to rotate 45°, calculate the mass of the balance wheel. * 50. (II) The human leg can be compared to a physical pendulum, with a "natural" swinging period at which walking is easiest. Consider the leg as two rods joined rigidly together at the knee; the axis for the leg is the hip joint. The length of each rod is about the same, 55 cm. The upper rod has a mass of 7.0 kg and the lower rod has a mass of 4.0 kg. (a) Calculate the natural swinging period of the system. (b) Check your answer by standing on a chair and measuring the time for one or more complete back-andforth swings. The effect of a shorter leg is a shorter swinging period, enabling a faster "natural" stride. * 51. (II) (a) Determine the equation of motion (for u as a function of time) for a torsion pendulum, Fig. 14–18, and show that the motion is simple harmonic. (b) Show that the period T is T = 2p 1IK . [The balance wheel of a mechanical watch is an example of a torsion pendulum in which the restoring torque is applied by a coil spring.]

* 52. (II) A student wants to use a meter stick as a pendulum. She plans to drill a small hole through the meter stick and suspend it from a smooth pin attached to the wall (Fig. 14–34). x Where in the meter stick should Pin she drill the hole to obtain the shortest possible period? How 1.00 m short an oscillation period can she obtain with a meter stick in this way? FIGURE 14–34 Problem 52.

* 53. (II) A meter stick is hung at its center from a thin wire (Fig. 14–35a). It is twisted and oscillates with a period of 5.0 s. The meter stick is sawed off to a length of 70.0 cm. This piece is again balanced at its center and set in oscillation (Fig. 14–35b). With what period does it oscillate?

(a)

(b)

FIGURE 14–35 Problem 53. * 54. (II) An aluminum disk, 12.5 cm in diameter and 375 g in mass, is mounted on a vertical shaft with very low friction (Fig. 14–36). One end of a flat coil spring is attached to the disk, the other end to the base of the apparatus. The disk is set into rotational oscillation and the frequency is 0.331 Hz. What is the torsional spring constant K (t = –Ku)? Coil spring

Aluminum disk

Air bearing

FIGURE 14–36 Problem 54. * 55. (II) A plywood disk of radius 20.0 cm and mass 2.20 kg has a small hole drilled through Pin it, 2.00 cm from its edge 2.00 cm (Fig. 14–37). The disk is hung from the wall by means of a metal pin through the hole, and is used as a pendulum. What is R = 20.0 cm the period of this pendulum for M = 2.20 kg small oscillations?

FIGURE 14–37 Problem 55.

Problems

391

14–7 Damping 56. (II) A 0.835-kg block oscillates on the end of a spring whose spring constant is k = 41.0 Nm. The mass moves in a fluid which offers a resistive force F = – bv, where b = 0.662 N sm. (a) What is the period of the motion? (b) What is the fractional decrease in amplitude per cycle? (c) Write the displacement as a function of time if at t = 0, x = 0, and at t = 1.00 s, x = 0.120 m. 57. (II) Estimate how the damping constant changes when a car's shock absorbers get old and the car bounces three times after going over a speed bump. 58. (II) A physical pendulum consists of an 85-cm-long, 240-g-mass, uniform wooden rod hung from a nail near one end (Fig. 14–38). The motion is damped because of friction in the pivot; the damping force is approximately proportional to dudt. The rod is set in oscillation by displacing it 15° from its equilibrium position and releasing it. After 8.0 s, the amplitude of the oscillation has been reduced to 5.5°. If the angular displacement 85 cm g can be written as u = Ae – t cos v¿ t, find (a) g, (b) the approximate period of the motion, and (c) how long it takes for the θ0 amplitude to be reduced to 12 of its original = 15° value. FIGURE 14–38 Problem 58. 59. (II) A damped harmonic oscillator loses 6.0% of its mechanical energy per cycle. (a) By what percentage does its frequency differ from the natural frequency f0 = (12p)1km ? (b) After how many periods will the amplitude have decreased to 1e of its original value? 60. (II) A vertical spring of spring constant 115 Nm supports a mass of 75 g. The mass oscillates in a tube of liquid. If the mass is initially given an amplitude of 5.0 cm, the mass is observed to have an amplitude of 2.0 cm after 3.5 s. Estimate the damping constant b. Neglect buoyant forces. 61. (III) (a) Show that the total mechanical energy, E = 12 mv2 + 12 kx2, as a function of time for a lightly damped harmonic oscillator is E = 12 kA2e –(bm) t = E0 e –(bm) t, where E0 is the total mechanical energy at t = 0. (Assume v¿ W b2m.) (b) Show that the fractional energy lost per period is 2p , ¢E 2pb = = mv0 Q E where v0 = 2km and Q = mv0b is called the quality factor or Q value of the system. A larger Q value means the system can undergo oscillations for a longer time.

62. (III) A glider on an air track is connected by springs to either end of the track (Fig. 14–39). Both springs have the same spring constant, k, and the glider has mass M. (a) Determine the frequency of the oscillation, assuming no damping, if k = 125 Nm and M = 215 g. (b) It is observed that after 55 oscillations, the amplitude of the oscillation has dropped to one-half of its initial value. Estimate the value of g, using Eq. 14–16. (c) How long does it take the amplitude to decrease to one-quarter of its initial value? k k FIGURE 14–39 Problem 62.

14–8 Forced Oscillations; Resonance

63. (II) (a) For a forced oscillation at resonance Av = v0 B, what is the value of the phase angle f0 in Eq. 14–22? (b) What, then, is the displacement at a time when the driving force Fext is a maximum, and at a time when Fext = 0? (c) What is the phase difference (in degrees) between the driving force and the displacement in this case? 64. (II) Differentiate Eq. 14–23 to show that the resonant amplitude peaks at b2 . B 2m2 65. (II) An 1150 kg automobile has springs with k = 16,000 Nm. One of the tires is not properly balanced; it has a little extra mass on one side compared to the other, causing the car to shake at certain speeds. If the tire radius is 42 cm, at what speed will the wheel shake most? * 66. (II) Construct an accurate resonance curve, from v = 0 to v = 2v0 , for Q = 6.0. * 67. (II) The amplitude of a driven harmonic oscillator reaches a value of 23.7F0 k at a resonant frequency of 382 Hz. What is the Q value of this system? 68. (III) By direct substitution, show that Eq. 14–22, with Eqs. 14–23 and 14–24, is a solution of the equation of motion (Eq. 14–21) for the forced oscillator. [Hint: To find sin f0 and cos f0 from tan f0 , draw a right triangle.] * 69. (III) Consider a simple pendulum (point mass bob) 0.50 m long with a Q of 350. (a) How long does it take for the amplitude (assumed small) to decrease by two-thirds? (b) If the amplitude is 2.0 cm and the bob has mass 0.27 kg, what is the initial energy loss rate of the pendulum in watts? (c) If we are to stimulate resonance with a sinusoidal driving force, how close must the driving frequency be to the natural frequency of the pendulum (give ¢f = f - f0)? v =

v20 -

General Problems 70. A 62-kg person jumps from a window to a fire net 20.0 m below, which stretches the net 1.1 m. Assume that the net behaves like a simple spring. (a) Calculate how much it would stretch if the same person were lying in it. (b) How much would it stretch if the person jumped from 38 m? 71. An energy-absorbing car bumper has a spring constant of 430 kNm. Find the maximum compression of the bumper if the car, with mass 1300 kg, collides with a wall at a speed of 2.0 ms (approximately 5 mih).

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Oscillations

72. The length of a simple pendulum is 0.63 m, the pendulum bob has a mass of 295 g, and it is released at an angle of 15° to the vertical. (a) With what frequency does it oscillate? (b) What is the pendulum bob's speed when it passes through the lowest point of the swing? Assume SHM. (c) What is the total energy stored in this oscillation assuming no losses? 73. A simple pendulum oscillates with frequency f. What is its frequency if the entire pendulum accelerates at 0.50 g (a) upward, and (b) downward?

74. A 0.650-kg mass oscillates according to the equation x = 0.25 sin(5.50t) where x is in meters and t is in seconds. Determine (a) the amplitude, (b) the frequency, (c) the period, (d) the total energy, and (e) the kinetic energy and potential energy when x is 15 cm. 75. (a) A crane has hoisted a 1350-kg car at the junkyard. The crane's steel cable is 20.0 m long and has a diameter of 6.4 mm. If the car starts bouncing at the end of the cable, what is the period of the bouncing? [Hint: Refer to Table 12–1.] (b) What amplitude of bouncing will likely cause the cable to snap? (See Table 12–2, and assume Hooke's law holds all the way up to the breaking point.) 76. An oxygen atom at a particular site within a DNA molecule can be made to execute simple harmonic motion when illuminated by infrared light. The oxygen atom is bound with a spring-like chemical bond to a phosphorus atom, which is rigidly attached to the DNA backbone. The oscillation of the oxygen atom occurs with frequency f = 3.7 * 1013 Hz. If the oxygen atom at this site is chemically replaced with a sulfur atom, the spring constant of the bond is unchanged (sulfur is just below oxygen in the Periodic Table). Predict the frequency for a DNA molecule after the sulfur substitution. 77. A "seconds" pendulum has a period of exactly 2.000 s. That is, each one-way swing takes 1.000 s. What is the length of a seconds pendulum in Austin, Texas, where g = 9.793 ms2 ? If the pendulum is moved to Paris, where g = 9.809 ms2, by how many millimeters must we lengthen the pendulum? What is the length of a seconds pendulum on the Moon, where g = 1.62 ms2 ? 78. A 320-kg wooden raft floats on a lake. When a 75-kg man stands on the raft, it sinks 3.5 cm deeper into the water. When he steps off, the raft oscillates for a while. (a) What is the frequency of oscillation? (b) What is the total energy of oscillation (ignoring damping)? 79. At what displacement from equilibrium is the speed of a SHO half the maximum value? 80. A diving board oscillates with simple harmonic motion of frequency 2.5 cycles per second. What is the maximum amplitude with which the end of the board can oscillate in order that a pebble placed there (Fig. 14–40) does not lose contact with the board during the oscillation?

FIGURE 14–42 Problem 83. 84. In some diatomic molecules, the force each atom exerts on the other can be approximated by F = –Cr2 + Dr3, where r is the atomic separation and C and D are positive constants. (a) Graph F vs. r from r = 0.8DC to r = 4DC. (b) Show that equilibrium occurs at r = r0 = DC. (c) Let ¢r = r - r0 be a small displacement from equilibrium, where ¢r V r0 . Show that for such small displacements, the motion is approximately simple harmonic, and (d) determine the force constant. (e) What is the period of such motion? [Hint: Assume one atom is kept at rest.] 85. A mass attached to the end of a spring is stretched a distance x0 from equilibrium and released. At what distance from equilibrium will it have (a) velocity equal to half its maximum velocity, and (b) acceleration equal to half its maximum acceleration? 86. Carbon dioxide is a linear molecule. The carbon–oxygen bonds in this molecule act very much like springs. Figure 14–43 shows one possible way the oxygen atoms in this molecule can oscillate: the oxygen atoms oscillate symmetrically in and out, while the central carbon atom remains at rest. Hence each oxygen atom acts like a simple harmonic oscillator with a mass equal to the mass of an oxygen atom. It is observed that this oscillation occurs with a frequency of f = 2.83 * 1013 Hz. What is the spring constant of the C ¬ O bond? O FIGURE 14–43 Problem 86, the CO2 molecule.

FIGURE 14–40 Problem 80. 81. A rectangular block of wood floats in a calm lake. Show that, if friction is ignored, when the block is pushed gently down into the water and then released, it will then oscillate with SHM. Also, determine an equation for the force constant. 82. A 950-kg car strikes a huge spring at a speed of 25 ms (Fig. 14–41), compressing the spring 5.0 m. (a) What is the spring stiffness constant of the spring? (b) How long is the car in contact with the spring before it bounces off in the opposite direction? 950 kg FIGURE 14–41 Problem 82.

83. A 1.60-kg table is supported on four springs. A 0.80-kg chunk of modeling clay is held above the table and dropped so that it hits the table with a speed of 1.65 ms (Fig. 14–42). The clay makes an inelastic collision with the table, and the table and clay oscillate up and down. After a long time the table comes to rest 6.0 cm below its original position. (a) What is the effective spring constant of all four springs taken together? (b) With what maximum amplitude does the platform oscillate?

87. Imagine that a 10-cm-diameter circular hole was drilled all the way through the center of the Earth (Fig. 14–44). At one end of the hole, you drop an apple into the hole. Show that, if you assume that the Earth has a constant density, the subsequent motion of the apple is simple harmonic. How long will the apple take to return? Assume C that we can ignore all frictional effects. [Hint: See Appendix D.] FIGURE 14–44 Problem 87.

General Problems

393

88. A thin, straight, uniform rod of length l = 1.00 m and mass m = 215 g hangs from a pivot at one end. (a) What is its period for small-amplitude oscillations? (b) What is the length of a simple pendulum that will have the same period? 89. A mass m is gently placed on the end of a freely hanging spring. The mass then falls 32.0 cm before it stops and begins to rise. What is the frequency of the oscillation? 90. A child of mass m sits on top of a rectangular slab of mass M = 35 kg, which in turn rests on the frictionless horizontal floor at a pizza shop. The slab is attached to a horizontal spring with spring constant k = 430 Nm (the other end is attached to an immovable wall, Fig. 14–45). The coefficient of static friction between the child and the top of the slab is m = 0.40. The shop owner's intention is that, when displaced from the equilibrium position and released, the slab and child (with no slippage between the two) execute SHM with amplitude A = 0.50 m. Should there be a weight restriction for this ride? If so, A 0 A what is it? x

* Numerical/Computer * 93. (II) A mass m on a frictionless surface is attached to a spring with spring constant k as shown in Fig. 14–47. This mass–spring system is then observed to execute simple harmonic motion with a period T. The mass m is changed several times and the associated period T is measured in each case, generating the following data Table:

k 450 N/m

91. Estimate the effective spring constant of a trampoline. 92. In Section 14–5, the oscillation of a simple pendulum (Fig. 14–46) is viewed as linear motion along the arc length x and analyzed via F = ma. Alternatively, the pendulum's movement can be regarded as rotational motion about its point of support and analyzed using t = Ia. Carry out this alternative analysis and show that u(t) = umax cos a

FIGURE 14–46 Problem 92.

0.5 1.0 2.0 3.0 4.5

0.445 0.520 0.630 0.723 0.844

k m

g t + fb, Al

where u(t) is the angular displacement of the pendulum from the vertical at time t, as long as its maximum value is less than about 15°.

Period T (s)

(a) Starting with Eq. 14–7b, show why a graph of T2 vs. m is expected to yield a straight line. How can k be determined from the straight line's slope? What is the line's y-intercept expected to be? (b) Using the data in the Table, plot T2 vs. m and show that this graph yields a straight line. Determine the slope and (nonzero) y-intercept. (c) Show that a nonzero y-intercept can be expected in our plot theoretically if, rather than simply using m for the mass in Eq. 14–7b, we use m + m0 , where m0 is a constant. That is, repeat part (a) using m + m0 for the mass in Eq. 14–7b. Then use the result of this analysis to determine k and m0 from your graph's slope and y-intercept. (d) Offer a physical interpretation for m0 , a mass that appears to be oscillating in addition to the attached mass m.

µ 0.40 FIGURE 14–45 Problem 90.

Mass m (kg)

FIGURE 14–47 Problem 93.

* 94. (III) Damping proportional to v2. Suppose the oscillator of Example 14–5 is damped by a force proportional to the square of the velocity, Fdamping = –cv2, where c = 0.275 kgm is a constant. Numerically integrate† the differential equation from t = 0 to t = 2.00 s to an accuracy of 2%, and plot your results.

†

See Section 2–9.

Answers to Exercises A: B: C: D: E: F:

(a), (c), (d). (b). (c). (b), (d). (c). (c).

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G: All are larger. H: (a). I: (c). J: (a) 25 cm; (b) 2.0 s.

Oscillations

Wave Motion CHAPTER-OPENING QUESTION—Guess now! You drop a rock into a pond, and water waves spread out in circles. (a) The waves carry water outward, away from where the rock hit. That moving water carries energy outward. (b) The waves only make the water move up and down. No energy is carried outward from where the rock hit. (c) The waves only make the water move up and down, but the waves do carry energy outward, away from where the rock hit. hen you throw a stone into a lake or pool of water, circular waves form and move outward, as shown in the photos above. Waves will also travel along a cord that is stretched out flat on a table if you vibrate one end back and forth as shown in Fig. 15–1. Water waves and waves on a cord are two common examples of mechanical waves, which propagate as oscillations of matter. We will discuss other kinds of waves in later Chapters, including electromagnetic waves and light.

FIGURE 15–1 Wave traveling on a cord. The wave travels to the right along the cord. Particles of the cord oscillate back and forth on the tabletop. Velocity of cord particle Velocity of wave

P T E R

C H

Waves—such as these water waves—spread outward from a source. The source in this case is a small spot of water oscillating up and down briefly where a rock was thrown in (left photo). Other kinds of waves include waves on a cord or string, which also are produced by a vibration. Waves move away from their source, but we also study waves that seem to stand still ("standing waves"). Waves reflect, and they can interfere with each other when they pass through any point at the same time.

CONTENTS 15–1 Characteristics of Wave Motion 15–2 Types of Waves: Transverse and Longitudinal 15–3 Energy Transported by Waves 15–4 Mathematical Representation of a Traveling Wave

*15–5 The Wave Equation 15–6 The Principle of Superposition 15–7 Reflection and Transmission 15–8 Interference 15–9 Standing Waves; Resonance

*15–10 Refraction *15–11 Diffraction

395

If you have ever watched ocean waves moving toward shore before they break, you may have wondered if the waves were carrying water from far out at sea onto the beach. They don't.† Water waves move with a recognizable velocity. But each particle (or molecule) of the water itself merely oscillates about an equilibrium point. This is clearly demonstrated by observing leaves on a pond as waves move by. The leaves (or a cork) are not carried forward by the waves, but simply oscillate about an equilibrium point because this is the motion of the water itself. CONCEPTUAL EXAMPLE 15–1 Wave vs. particle velocity. Is the velocity of a wave moving along a cord the same as the velocity of a particle of the cord? See Fig. 15–1. RESPONSE No. The two velocities are different, both in magnitude and direction. The wave on the cord of Fig. 15–1 moves to the right along the tabletop, but each piece of the cord only vibrates to and fro. (The cord clearly does not travel in the direction that the wave on it does.)

FIGURE 15–1 (repeated) Wave traveling on a cord. The wave travels to the right along the cord. Particles of the cord oscillate back and forth on the tabletop.

Velocity of cord particle Velocity of wave

Waves can move over large distances, but the medium (the water or the cord) itself has only a limited movement, oscillating about an equilibrium point as in simple harmonic motion. Thus, although a wave is not matter, the wave pattern can travel in matter. A wave consists of oscillations that move without carrying matter with them. Waves carry energy from one place to another. Energy is given to a water wave, for example, by a rock thrown into the water, or by wind far out at sea. The energy is transported by waves to the shore. The oscillating hand in Fig. 15–1 transfers energy to the cord, and that energy is transported down the cord and can be transferred to an object at the other end. All forms of traveling waves transport energy. FIGURE 15–2 Motion of a wave pulse to the right. Arrows indicate velocity of cord particles.

EXERCISE A Return to the Chapter-Opening Question, page 395, and answer it again now. Try to explain why you may have answered differently the first time.

15–1 (a)

Let us look a little more closely at how a wave is formed and how it comes to "travel." We first look at a single wave bump, or pulse. A single pulse can be formed on a cord by a quick up-and-down motion of the hand, Fig. 15–2. The hand pulls up on one end of the cord. Because the end section is attached to adjacent sections, these also feel an upward force and they too begin to move upward. As each succeeding section of cord moves upward, the wave crest moves outward along the cord. Meanwhile, the end section of cord has been returned to its original position by the hand. As each succeeding section of cord reaches its peak position, it too is pulled back down again by tension from the adjacent section of cord. Thus the source of a traveling wave pulse is a disturbance, and cohesive forces between adjacent sections of cord cause the pulse to travel. Waves in other media are created

(b)

(c)

†

Do not be confused by the "breaking" of ocean waves, which occurs when a wave interacts with the ground in shallow water and hence is no longer a simple wave.

(d)

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Characteristics of Wave Motion

Wave Motion

and propagate outward in a similar fashion. A dramatic example of a wave pulse is a tsunami or tidal wave that is created by an earthquake in the Earth's crust under the ocean. The bang you hear when a door slams is a sound wave pulse. A continuous or periodic wave, such as that shown in Fig. 15–1, has as its source a disturbance that is continuous and oscillating; that is, the source is a vibration or oscillation. In Fig. 15–1, a hand oscillates one end of the cord. Water waves may be produced by any vibrating object at the surface, such as your hand; or the water itself is made to vibrate when wind blows across it or a rock is thrown into it. A vibrating tuning fork or drum membrane gives rise to sound waves in air. And we will see later that oscillating electric charges give rise to light waves. Indeed, almost any vibrating object sends out waves. The source of any wave, then, is a vibration. And it is a vibration that propagates outward and thus constitutes the wave. If the source vibrates sinusoidally in SHM, then the wave itself—if the medium is perfectly elastic—will have a sinusoidal shape both in space and in time. (1) In space: if you take a picture of the wave in space at a given instant of time, the wave will have the shape of a sine or cosine as a function of position. (2) In time: if you look at the motion of the medium at one place over a long period of time—for example, if you look between two closely spaced posts of a pier or out of a ship's porthole as water waves pass by—the up-and-down motion of that small segment of water will be simple harmonic motion. The water moves up and down sinusoidally in time. Some of the important quantities used to describe a periodic sinusoidal wave are shown in Fig. 15–3. The high points on a wave are called crests; the low points, troughs. The amplitude, A, is the maximum height of a crest, or depth of a trough, relative to the normal (or equilibrium) level. The total swing from a crest to a trough is twice the amplitude. The distance between two successive crests is called the wavelength, l (the Greek letter lambda). The wavelength is also equal to the distance between any two successive identical points on the wave. The frequency, f, is the number of crests—or complete cycles—that pass a given point per unit time. The period, T, equals 1f and is the time elapsed between two successive crests passing by the same point in space. Crest

Amplitude

FIGURE 15–3 Characteristics of a single-frequency continuous wave moving through space.

Amplitude Trough

The wave velocity, v, is the velocity at which wave crests (or any other part of the waveform) move forward. The wave velocity must be distinguished from the velocity of a particle of the medium itself as we saw in Example 15–1. A wave crest travels a distance of one wavelength, l, in a time equal to one period, T. Thus the wave velocity is v = lT. Then, since 1T = f, v = lf.

(15–1)

For example, suppose a wave has a wavelength of 5 m and a frequency of 3 Hz. Since three crests pass a given point per second, and the crests are 5 m apart, the first crest (or any other part of the wave) must travel a distance of 15 m during the 1 s. So the wave velocity is 15 ms. EXERCISE B You notice a water wave pass by the end of a pier with about 0.5 s between crests. Therefore (a) the frequency is 0.5 Hz; (b) the velocity is 0.5 ms; (c) the wavelength is 0.5 m; (d) the period is 0.5 s.

SECTION 15–1

Characteristics of Wave Motion

397

15–2

Types of Waves: Transverse and Longitudinal

When a wave travels down a cord—say, from left to right as in Fig. 15–1—the particles of the cord vibrate up and down in a direction transverse (that is, perpendicular) to the motion of the wave itself. Such a wave is called a transverse wave (Fig. 15–4a). There exists another type of wave known as a longitudinal wave. In a longitudinal wave, the vibration of the particles of the medium is along the direction of the wave's motion. Longitudinal waves are readily formed on a stretched spring or Slinky by alternately compressing and expanding one end. This is shown in Fig. 15–4b, and can be compared to the transverse wave in Fig. 15–4a.

FIGURE 15–4 (a) Transverse wave; (b) longitudinal wave.

(a)

Wavelength Compression

(b)

FIGURE 15–5 Production of a sound wave, which is longitudinal, shown at two moments in time about a half period A 12 TB apart.

Drum membrane

Compression

Expansion

Wavelength

A series of compressions and expansions propagate along the spring. The compressions are those areas where the coils are momentarily close together. Expansions (sometimes called rarefactions) are regions where the coils are momentarily far apart. Compressions and expansions correspond to the crests and troughs of a transverse wave. An important example of a longitudinal wave is a sound wave in air. A vibrating drumhead, for instance, alternately compresses and rarefies the air in contact with it, producing a longitudinal wave that travels outward in the air, as shown in Fig. 15–5. As in the case of transverse waves, each section of the medium in which a longitudinal wave passes oscillates over a very small distance, whereas the wave itself can travel large distances. Wavelength, frequency, and wave velocity all have meaning for a longitudinal wave. The wavelength is the distance between successive compressions (or between successive expansions), and frequency is the number of compressions that pass a given point per second. The wave velocity is the velocity with which each compression appears to move; it is equal to the product of wavelength and frequency, v = lf (Eq. 15–1). A longitudinal wave can be represented graphically by plotting the density of air molecules (or coils of a Slinky) versus position at a given instant, as shown in Fig. 15–6. Such a graphical representation makes it easy to illustrate what is happening. Note that the graph looks much like a transverse wave.

(a)

(b)

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Wave Motion

Density of air

FIGURE 15–6 (a) A longitudinal wave with (b) its graphical representation at a particular instant in time.

High Normal Low

Velocity of Transverse Waves The velocity of a wave depends on the properties of the medium in which it travels. The velocity of a transverse wave on a stretched string or cord, for example, depends on the tension in the cord, FT , and on the mass per unit length of the cord, m (the Greek letter mu, where here m = ml). For waves of small amplitude, the relationship is v =

FT . Bm

transverse d (15–2) wave on a cord

Before giving a derivation of this formula, it is worth noting that at least qualitatively it makes sense on the basis of Newtonian mechanics. That is, we do expect the tension to be in the numerator and the mass per unit length in the denominator. Why? Because when the tension is greater, we expect the velocity to be greater since each segment of cord is in tighter contact with its neighbor. And, the greater the mass per unit length, the more inertia the cord has and the more slowly the wave would be expected to propagate. EXERCISE C A wave starts at the left end of a long cord (see Fig. 15–1) when someone shakes the cord back and forth at the rate of 2.0 Hz. The wave is observed to move to the right at 4.0 ms. If the frequency is increased from 2.0 to 3.0 Hz, the new speed of the wave is (a) 1.0 ms, (b) 2.0 ms, (c) 4.0 ms, (d) 8.0 ms, (e) 16.0 ms.

FT (a) B

vB′

v ′t

FIGURE 15–7 Diagram of simple wave pulse on a cord for derivation of Eq. 15–2. The vector shown in B B (b) as the resultant of FT + Fy has to be directed along the cord because the cord is flexible. (Diagram is not to scale: we assume v¿ V v; the upward angle of the cord is exaggerated for visibility.)

C vt

(b)

We can make a simple derivation of Eq. 15–2 using a simple model of a cord under a tension FT as shown in Fig. 15–7a. The cord is pulled upward at a speed v¿ by the force Fy . As shown in Fig. 15–7b all points of the cord to the left of point C move upward at the speed v¿, and those to the right are still at rest. The speed of propagation, v, of this wave pulse is the speed of point C, the leading edge of the pulse. Point C moves to the right a distance vt in a time t, whereas the end of the cord moves upward a distance v¿ t. By similar triangles we have the approximate relation FT vt v, = = Fy v¿ t v¿ which is accurate for small displacements (v¿ t V vt) so that FT does not change appreciably. As we saw in Chapter 9, the impulse given to an object is equal to its change in momentum. During the time t the total upward impulse is Fy t = (v¿v)FT t. The change in momentum of the cord, ¢p, is the mass of cord moving upward times its velocity. Since the upward moving segment of cord has mass equal to the mass per unit length m times its length vt we have Fy t = ¢p v¿ F t = (mvt)v¿. v T Solving for v we find v = 1FTm which is Eq. 15–2. Although it was derived for a special case, it is valid for any wave shape since other shapes can be considered to be made up of many tiny such lengths. But it is valid only for small displacements (as was our derivation). Experiment is in accord with this result derived from Newtonian mechanics.

SECTION 15–2

399

EXAMPLE 15–2 Pulse on a wire. An 80.0-m-long, 2.10-mm-diameter copper wire is stretched between two poles. A bird lands at the center point of the wire, sending a small wave pulse out in both directions. The pulses reflect at the ends and arrive back at the bird's location 0.750 seconds after it landed. Determine the tension in the wire. APPROACH From Eq. 15–2, the tension is given by FT = mv2. The speed v is distance divided by the time. The mass per unit length m is calculated from the density of copper and the dimensions of the wire. SOLUTION Each wave pulse travels 40.0 m to the pole and back again (= 80.0 m) in 0.750 s. Thus their speed is v = (80.0 m)(0.750 s) = 107 ms. We take (Table 13–1) the density of copper as 8.90 * 103 kgm3. The volume of copper in the wire is the cross-sectional area Apr 2 B times the length l, and the mass of the wire is the volume times the density: m = rApr2 B l for a wire of radius r and length l. Then m = ml is m = rpr2ll = rpr 2 = A8.90 * 103 kgm3 BpA1.05 * 10–3 mB = 0.0308 kgm. 2

Thus, the tension is FT = mv2 = A0.0308 kgmB(107 ms)2 = 353 N.

Velocity of Longitudinal Waves The velocity of a longitudinal wave has a form similar to that for a transverse wave on a cord (Eq. 15–2); that is, elastic force factor . B inertia factor In particular, for a longitudinal wave traveling down a long solid rod, v =

v =

E, Br

longitudinal d (15–3) wave in a long rod

where E is the elastic modulus (Section 12–4) of the material and r is its density. For a longitudinal wave traveling in a liquid or gas, B, longitudinal wave c d (15–4) in a fluid Br where B is the bulk modulus (Section 12–4) and r again is the density. v =

P H Y S I C S

A P P L I E D

Space perception by animals, using sound waves

EXAMPLE 15–3 Echolocation. Echolocation is a form of sensory perception used by animals such as bats, toothed whales, and dolphins. The animal emits a pulse of sound (a longitudinal wave) which, after reflection from objects, returns and is detected by the animal. Echolocation waves can have frequencies of about 100,000 Hz. (a) Estimate the wavelength of a sea animal's echolocation wave. (b) If an obstacle is 100 m from the animal, how long after the animal emits a wave is its reflection detected? APPROACH We first compute the speed of longitudinal (sound) waves in sea water, using Eq. 15–4 and Tables 12–1 and 13–1. The wavelength is l = vf. SOLUTION (a) The speed of longitudinal waves in sea water, which is slightly more dense than pure water, is 2.0 * 109 Nm2 B = = 1.4 * 103 ms. Br B 1.025 * 103 kgm3 Then, using Eq. 15–1, we find A1.4 * 103 msB v l = = = 14 mm. f A1.0 * 105 HzB v =

(b) The time required for the round-trip between the animal and the object is 2(100 m) distance t = = = 0.14 s. speed 1.4 * 103 ms NOTE We shall see later that waves can be used to "resolve" (or detect) objects only if the wavelength is comparable to or smaller than the object. Thus, a dolphin can resolve objects on the order of a centimeter or larger in size.

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Wave Motion

* Deriving Velocity of Wave in a Fluid We now derive Eq. 15–4. Consider a wave pulse traveling in a fluid in a long tube, so that the wave motion is one dimensional. The tube is fitted with a piston at the end and is filled with a fluid which, at t = 0, is of uniform density r and at uniform pressure P0 , Fig. 15–8a. At this moment the piston is abruptly made to start moving to the right with speed v¿, compressing the fluid in front of it. In the (short) time t the piston moves a distance v¿t. The compressed fluid itself also moves with speed v¿, but the leading edge of the compressed region moves to the right at the characteristic speed v of compression waves in that fluid; we assume the wave speed v is much larger than the piston speed v¿. The leading edge of the compression (which at t = 0 was at the piston face) thus moves a distance vt in time t as shown in Fig. 15–8b. Let the pressure in the compression be P0 + ¢P, which is ¢P higher than in the uncompressed fluid. To move the piston to the right requires an external force AP0 + ¢PBS acting to the right, where S is the cross-sectional area of the tube. (S for "surface area"; we save A for amplitude.) The net force on the compressed region of the fluid is Fnet = AP0 + ¢PBS - P0 S = S ¢P

r, P0

(a) t = 0

v ′t

vB ′

Leading edge of compression

vB′ P0 + ∆P

(b) after time t FIGURE 15–8 Determining the speed of a one-dimensional longitudinal wave in a fluid contained in a long narrow tube.

since the uncompressed fluid exerts a force P0 S to the left at the leading edge. Hence the impulse given to the compressed fluid, which equals its change in momentum, is Fnet t = ¢mv¿ S ¢P t = (rSvt)v¿, where (rSvt) represents the mass of fluid which is given the speed v¿ (the compressed fluid of area S moves a distance vt, Fig. 15–8, so the volume moved is Svt). Hence we have ¢P = rvv¿. From the definition of the bulk modulus, B (Eq. 12–7): B = –

rvv¿ ¢P , = – ¢VV0 ¢VV0

where ¢VV0 is the fractional change in volume due to compression. The original volume of the compressed fluid is V0 = Svt (see Fig. 15–8), and it has been compressed by an amount ¢V = –Sv¿ t (Fig. 15–8b). Thus B = –

rvv¿ Svt = –rvv¿ a b = rv2, ¢VV0 –Sv¿ t

and so v =

B, Br

which is what we set out to show, Eq. 15–4. The derivation of Eq. 15–3 follows similar lines, but takes into account the expansion of the sides of a rod when the end of the rod is compressed.

Other Waves Both transverse and longitudinal waves are produced when an earthquake occurs. The transverse waves that travel through the body of the Earth are called S waves (S for shear), and the longitudinal waves are called P waves (P for pressure) or compression waves. Both longitudinal and transverse waves can travel through a solid since the atoms or molecules can vibrate about their relatively fixed positions in any direction. But only longitudinal waves can propagate through a fluid, because any transverse motion would not experience any restoring force since a fluid is readily deformable. This fact was used by geophysicists to infer that a portion of the Earth's core must be liquid: after an earthquake, longitudinal waves are detected diametrically across the Earth, but not transverse waves. SECTION 15–2

P H Y S I C S

A P P L I E D

Earthquake waves

Types of Waves: Transverse and Longitudinal

401

FIGURE 15–9 A water wave is an example of a surface wave, which is a combination of transverse and longitudinal wave motions. FIGURE 15–10 How a wave breaks. The green arrows represent the local velocity of water molecules. vB

Besides these two types of waves that can pass through the body of the Earth (or other substance), there can also be surface waves that travel along the boundary between two materials. A wave on water is actually a surface wave that moves on the boundary between water and air. The motion of each particle of water at the surface is circular or elliptical (Fig. 15–9), so it is a combination of transverse and longitudinal motions. Below the surface, there is also transverse plus longitudinal wave motion, as shown. At the bottom, the motion is only longitudinal. (When a wave approaches shore, the water drags at the bottom and is slowed down, while the crests move ahead at higher speed (Fig. 15–10) and "spill" over the top.) Surface waves are also set up on the Earth when an earthquake occurs. The waves that travel along the surface are mainly responsible for the damage caused by earthquakes. Waves which travel along a line in one dimension, such as transverse waves on a stretched string, or longitudinal waves in a rod or fluid-filled tube, are linear or one-dimensional waves. Surface waves, such as the water waves pictured at the start of this Chapter, are two-dimensional waves. Finally, waves that move out from a source in all directions, such as sound from a loudspeaker or earthquake waves through the Earth, are three-dimensional waves.

15–3

Energy Transported by Waves

Waves transport energy from one place to another. As waves travel through a medium, the energy is transferred as vibrational energy from particle to particle of the medium. For a sinusoidal wave of frequency f, the particles move in simple harmonic motion (Chapter 14) as a wave passes, and each particle has energy E = 12 kA2 where A is the maximum displacement (amplitude) of its motion, either transversely or longitudinally (Eq. 14–10a). Using Eq. 14–7a we can write k = 4p2mf2, where m is the mass of a particle (or small volume) of the medium. Then in terms of the frequency f and amplitude A, E =

FIGURE 15–11 Calculating the energy carried by a wave moving with velocity v. S

1 2 2 kA

= 2p2mf2A2.

For three-dimensional waves traveling in an elastic medium, the mass m = rV, where r is the density of the medium and V is the volume of a small slice of the medium. The volume V = Sl where S is the cross-sectional area through which the wave travels (Fig. 15–11), and we can write l as the distance the wave travels in a time t as l = vt, where v is the speed of the wave. Thus m = rV = rSl = rSvt and E = 2p2rSvtf2A2.

(15–5)

From this equation we have the important result that the energy transported by a wave is proportional to the square of the amplitude, and to the square of the frequency. The average rate of energy transferred is the average power P:

l = vt

g =

= 2p2rSvf2A2.

(15–6)

Finally, the intensity, I, of a wave is defined as the average power transferred across unit area perpendicular to the direction of energy flow: I =

g = 2p2vrf2A2. S

(15–7)

If a wave flows out from the source in all directions, it is a three-dimensional wave. Examples are sound traveling in the open air, earthquake waves, and light waves.

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CHAPTER 15

Wave Motion

If the medium is isotropic (same in all directions), the wave from a point source is a spherical wave (Fig. 15–12). As the wave moves outward, the energy it carries is spread over a larger and larger area since the surface area of a sphere of radius r is 4pr 2. Thus the intensity of a wave is g g . I = = S 4pr2

Source

If the power output g is constant, then the intensity decreases as the inverse square of the distance from the source: I r

1. r2

[spherical wave] (15–8a)

If we consider two points at distances r1 and r2 from the source, as in Fig. 15–12, then I1 = g4pr 21 and I2 = g4pr22, so I2 g4pr 22 r21 . = = I1 r22 g4pr 21

(15–8b)

FIGURE 15–12 Wave traveling outward from a point source has spherical shape. Two different crests (or compressions) are shown, of radius r1 and r2 .

Thus, for example, when the distance doubles Ar2r1 = 2B, then the intensity is 2 reduced to 14 of its earlier value: I2I1 = A 12 B = 14 . The amplitude of a wave also decreases with distance. Since the intensity is proportional to the square of the amplitude (Eq. 15–7), I r A2, the amplitude A must decrease as 1r, so that I can be proportional to 1r 2 (Eq. 15–8a). Hence A r

1. r

To see this directly from Eq. 15–6, consider again two different distances from the source, r1 and r2 . For constant power output, S1 A 21 = S2 A 22 where A 1 and A 2 are the amplitudes of the wave at r1 and r2 , respectively. Since S1 = 4pr21 and S2 = 4pr 22 , we have AA 21 r21 B = AA 22 r 22 B, or A2 r1 . = r A1 2 When the wave is twice as far from the source, the amplitude is half as large, and so on (ignoring damping due to friction). EXAMPLE 15–4 Earthquake intensity. The intensity of an earthquake P wave traveling through the Earth and detected 100 km from the source is 1.0 * 106 Wm2. What is the intensity of that wave if detected 400 km from the source? APPROACH We assume the wave is spherical, so the intensity decreases as the square of the distance from the source. SOLUTION At 400 km the distance is 4 times greater than at 100 km, so the 2 intensity will be A 14 B = 161 of its value at 100 km, or A1.0 * 106 Wm2 B16 = 4 2 6.3 * 10 Wm . NOTE Using Eq. 15–8b directly gives I2 = I1 r 21r22 = A1.0 * 106 Wm2 B(100 km)2(400 km)2 = 6.3 * 104 Wm2.

The situation is different for a one-dimensional wave, such as a transverse wave on a string or a longitudinal wave pulse traveling down a thin uniform metal rod. The area remains constant, so the amplitude A also remains constant (ignoring friction). Thus the amplitude and the intensity do not decrease with distance. In practice, frictional damping is generally present, and some of the energy is transformed into thermal energy. Thus the amplitude and intensity of a onedimensional wave decrease with distance from the source. For a three-dimensional wave, the decrease will be greater than that discussed above, although the effect may often be small. SECTION 15–3

Energy Transported by Waves

403

15–4

Mathematical Representation of a Traveling Wave

Let us now consider a one-dimensional wave traveling along the x axis. It could be, for example, a transverse wave on a cord or a longitudinal wave traveling in a rod or in a fluid-filled tube. Let us assume the wave shape is sinusoidal and has a particular wavelength l and frequency f. At t = 0, suppose the wave shape is given by D(x) = A sin y

wave at t=0

vt A

wave at time t x

−A l FIGURE 15–13 A traveling wave. In time t, the wave moves a distance vt.

2p x, l

(15–9)

as shown by the solid curve in Fig. 15–13: D(x) is the displacement† of the wave (be it a longitudinal or transverse wave) at position x, and A is the amplitude (maximum displacement) of the wave. This relation gives a shape that repeats itself every wavelength, which is needed so that the displacement is the same at x = 0, x = l, x = 2l, and so on (since sin 4p = sin 2p = sin 0). Now suppose the wave is moving to the right with velocity v. Then, after a time t, each part of the wave (indeed, the whole wave "shape") has moved to the right a distance vt; see the dashed curve in Fig. 15–13. Consider any point on the wave at t = 0: say, a crest which is at some position x. After a time t, that crest will have traveled a distance vt so its new position is a distance vt greater than its old position. To describe this same point on the wave shape, the argument of the sine function must be the same, so we replace x in Eq. 15–9 by (x - vt): D(x, t) = A sin c

1-D wave moving in positive x direction

2p (x - vt) d . l

(15–10a)

Said another way, if you are riding on a crest, the argument of the sine function, (2pl)(x - vt), remains the same ( p2, 5p2, and so on); as t increases, x must increase at the same rate so that (x - vt) remains constant. Equation 15–10a is the mathematical representation of a sinusoidal wave traveling along the x axis to the right (increasing x). It gives the displacement D(x, t) of the wave at any chosen point x at any time t. The function D(x, t) describes a curve that represents the actual shape of the wave in space at time t. Since v = lf (Eq. 15–1) we can write Eq. 15–10a in other ways that are often convenient: D(x, t) = A sin a

1-D wave moving in

2px 2pt b, l T

(15–10b)

where T = 1f = lv is the period; and

positive x

D(x, t) = A sin(kx - vt),

direction

(15–10c)

where v = 2pf = 2pT is the angular frequency and 2p (15–11) l is called the wave number. (Do not confuse the wave number k with the spring constant k; they are very different quantities.) All three forms, Eqs. 15–10a, b, and c, are equivalent; Eq. 15–10c is the simplest to write and is perhaps the most common. The quantity (kx - vt), and its equivalent in the other two equations, is called the phase of the wave. The velocity v of the wave is often called the phase velocity, since it describes the velocity of the phase (or shape) of the wave and it can be written in terms of v and k: k =

C A U T I O N

Don't confuse wave number k with spring constant k

v = lf = a †

2p v v. ba b = k 2p k

(15–12)

Some books use y(x) in place of D(x). To avoid confusion, we reserve y (and z) for the coordinate positions of waves in two or three dimensions. Our D(x) can stand for pressure (in longitudinal waves), position displacement (transverse mechanical waves), or—as we will see later—electric or magnetic fields (for electromagnetic waves).

404

CHAPTER 15

Wave Motion

For a wave traveling along the x axis to the left (decreasing values of x), we start again with Eq. 15–9 and note that the velocity is now –v. A particular point on the wave changes position by – vt in a time t, so x in Eq. 15–9 must be replaced by (x + vt). Thus, for a wave traveling to the left with velocity v, D(x, t) = A sin c

2p (x + vt) d l

(15–13a)

= A sin a

2px 2pt + b l T

(15–13b)

= A sin(kx + vt).

1-D wave moving in negative x direction

(15–13c)

In other words, we simply replace v in Eqs. 15–10 by –v. Let us look at Eq. 15–13c (or, just as well, at Eq. 15–10c). At t = 0 we have D(x, 0) = A sin kx, which is what we started with, a sinusoidal wave shape. If we look at the wave shape in space at a particular later time t1 , then we have DAx, t1 B = A sin Akx + vt1 B.

That is, if we took a picture of the wave at t = t1 , we would see a sine wave with a phase constant vt1 . Thus, for fixed t = t1 , the wave has a sinusoidal shape in space. On the other hand, if we consider a fixed point in space, say x = 0, we can see how the wave varies in time: D(0, t) = A sin vt where we used Eq. 15–13c. This is just the equation for simple harmonic motion (Section 14–2). For any other fixed value of x, say x = x1 , D = A sin Avt + kx1 B which differs only by a phase constant kx1 . Thus, at any fixed point in space, the displacement undergoes the oscillations of simple harmonic motion in time. Equations 15–10 and 15–13 combine both these aspects to give us the representation for a traveling sinusoidal wave (also called a harmonic wave). The argument of the sine in Eqs. 15–10 and 15–13 can in general contain a phase angle f, which for Eq. 15–10c is D(x, t) = A sin(kx - vt + f), to adjust for the position of the wave at t = 0, x = 0, just as in Section 14–2 (see Fig. 14–7). If the displacement is zero at t = 0, x = 0, as in Fig. 14–6 (or Fig. 15–13), then f = 0. Now let us consider a general wave (or wave pulse) of any shape. If frictional losses are small, experiment shows that the wave maintains its shape as it travels. Thus we can make the same arguments as we did right after Eq. 15–9. Suppose our wave has some shape at t = 0, given by D(x, 0) = D(x) where D(x) is the displacement of the wave at x and is not necessarily sinusoidal. At some later time, if the wave is traveling to the right along the x axis, the wave will have the same shape but all parts will have moved a distance vt where v is the phase velocity of the wave. Hence we must replace x by x - vt to obtain the amplitude at time t: D(x, t) = D(x - vt).

(15–14)

Similarly, if the wave moves to the left, we must replace x by x + vt, so D(x, t) = D(x + vt).

(15–15)

Thus, any wave traveling along the x axis must have the form of Eq. 15–14 or 15–15. EXERCISE D A wave is given by D(x, t) = (5.0 mm) sin(2.0x - 20.0t) where x is in meters and t is in seconds. What is the speed of the wave? (a) 10 ms, (b) 0.10 ms, (c) 40 ms, (d) 0.005 ms, (e) 2.5 * 10 –4 ms.

SECTION 15–4

Mathematical Representation of a Traveling Wave

405

y x

EXAMPLE 15–5 A traveling wave. The left-hand end of a long horizontal stretched cord oscillates transversely in SHM with frequency f = 250 Hz and amplitude 2.6 cm. The cord is under a tension of 140 N and has a linear density m = 0.12 kgm. At t = 0, the end of the cord has an upward displacement of 1.6 cm and is falling (Fig. 15–14). Determine (a) the wavelength of waves produced and (b) the equation for the traveling wave.

FIGURE 15–14 Example 15–5. The wave at t = 0 (the hand is falling). Not to scale.

APPROACH We first find the phase velocity of the transverse wave from Eq. 15–2; then l = vf. In (b), we need to find the phase f using the initial conditions. SOLUTION (a) The wave velocity is

Wave velocity

Hand

v =

FT 140 N = = 34 ms. Bm B 0.12 kgm

l =

34 ms v = = 0.14 m f 250 Hz

Then or

14 cm.

(b) Let x = 0 at the left-hand end of the cord. The phase of the wave at t = 0 is not zero in general as was assumed in Eqs. 15–9, 10, and 13. The general form for a wave traveling to the right is D(x, t) = A sin(kx - vt + f), where f is the phase angle. In our case, the amplitude A = 2.6 cm; and at t = 0, x = 0, we are given D = 1.6 cm. Thus 1.6 = 2.6 sin f, –1

so f = sin (1.62.6) = 38° = 0.66 rad. We also have v = 2pf = 1570 s–1 and k = 2pl = 2p0.14 m = 45 m–1. Hence D = (0.026 m) sin C A45 m–1 Bx - (1570 s) t + 0.66D which we can write more simply as D = 0.026 sin(45x - 1570t + 0.66), and we specify clearly that D and x are in meters and t in seconds.

* 15–5

FIGURE 15–15 Deriving the wave equation from Newton's second law: a segment of string under tension FT .

y ∆x θ1

FT θ2 m = m∆ x

The Wave Equation

Many types of waves satisfy an important general equation that is the equivalent of Newton's second law of motion for particles. This "equation of motion for a wave" is called the wave equation, and we derive it now for waves traveling on a stretched horizontal string. We assume the amplitude of the wave is small compared to the wavelength so that each point on the string can be assumed to move only vertically and the tension in the string, FT , does not vary during a vibration. We apply Newton's second law, ©F = ma, to the vertical motion of a tiny section of the string as shown in Fig. 15–15. The amplitude of the wave is small, so the angles u1 and u2 that the string makes with the horizontal are small. The length of this section is then approximately ¢x, and its mass is m ¢x, where m is the mass per unit length of the string. The net vertical force on this section of string is FT sin u2 - FT sin u1 . So Newton's second law applied to the vertical (y) direction gives ©Fy = ma y FT sin u2 - FT sin u1 = (m ¢x)

0 2D . 0 t2

(i)

We have written the acceleration as a y = 0 2D0 t2 since the motion is only vertical, and we use the partial derivative notation because the displacement D is a function of both x and t.

406

CHAPTER 15

Wave Motion

Because the angles u1 and u2 are assumed small, sin u L tan u and tan u is equal to the slope s of the string at each point: sin u L tan u =

0D = s. 0x

Thus our equation (i) at the bottom of the previous page becomes FT As2 - s1 B = m ¢x

0 2D 0 t2

or FT

¢s 0 2D = m 2 , ¢x 0t

(ii)

where ¢s = s2 - s1 is the difference in the slope between the two ends of our tiny section. Now we take the limit of ¢x S 0, so that FT lim

¢x S 0

¢s 0s = FT ¢x 0x 0 0D 0 2D = FT a b = FT 2 0x 0x 0x

since the slope s = 0D0x, as we wrote above. Substituting this into the equation labeled (ii) above gives FT

0 2D 0 2D = m 0x2 0 t2

or m 0 2D 0 2D . = FT 0 t2 0x2 We saw earlier in this Chapter (Eq. 15–2) that the velocity of waves on a string is given by v = 1FTm , so we can write this last equation as 0 2D 1 0 2D . = 0x2 v2 0 t 2

(15–16)

This is the one-dimensional wave equation, and it can describe not only small amplitude waves on a stretched string, but also small amplitude longitudinal waves (such as sound waves) in gases, liquids, and elastic solids, in which case D can refer to the pressure variations. In this case, the wave equation is a direct consequence of Newton's second law applied to a continuous elastic medium. The wave equation also describes electromagnetic waves for which D refers to the electric or magnetic field, as we shall see in Chapter 31. Equation 15–16 applies to waves traveling in one dimension only. For waves spreading out in three dimensions, the wave equation is the same, with the addition of 0 2D0y 2 and 0 2D0z2 to the left side of Eq. 15–16. The wave equation is a linear equation: the displacement D appears singly in each term. There are no terms that contain D2, or D(0D0x), or the like in which D appears more than once. Thus, if D1(x, t) and D2(x, t) are two different solutions of the wave equation, then the linear combination D3(x, t) = aD1(x, t) + bD2(x, t), where a and b are constants, is also a solution. This is readily seen by direct substitution into the wave equation. This is the essence of the superposition principle, which we discuss in the next Section. Basically it says that if two waves pass through the same region of space at the same time, the actual displacement is the sum of the separate displacements. For waves on a string, or for sound waves, this is valid only for small-amplitude waves. If the amplitude is not small enough, the equations for wave propagation may become nonlinear and the principle of superposition would not hold and more complicated effects may occur. *SECTION 15–5

The Wave Equation

407

EXAMPLE 15–6 Wave equation solution. Verify that the sinusoidal wave of Eq. 15–10c, D(x, t) = A sin(kx - vt), satisfies the wave equation. APPROACH We substitute Eq. 15–10c into the wave equation, Eq. 15–16. SOLUTION We take the derivative of Eq. 15–10c twice with respect to t: 0D = –vA cos(kx - vt) 0t 0 2D = –v2A sin(kx - vt). 0 t2 With respect to x, the derivatives are 0D = kA cos(kx - vt) 0x 0 2D = –k2A sin(kx - vt). 0x2 If we now divide the second derivatives we get –v2A sin(kx - vt) 0 2D0 t2 v2 = = 2. 2 2 2 0 D0x –k A sin(kx - vt) k FIGURE 15–16 The superposition principle for one-dimensional waves. The composite wave formed from three sinusoidal waves of different amplitudes and frequencies Af0 , 2f0 , 3f0 B is shown at the bottom at a certain instant in time. The displacement of the composite wave at each point in space, at any time, is the algebraic sum of the displacements of the component waves. Amplitudes are shown exaggerated; for the superposition principle to hold, they must be small compared to the wavelengths. D1(x, t)

D2(x, t)

D3(x, t)

D(x, t) = D1(x, t) + D2(x, t) + D3(x, t)

Sum of all three

From Eq. 15–12 we have v2k2 = v2, so we see that Eq. 15–10 does satisfy the wave equation (Eq. 15–16).

15–6

The Principle of Superposition

When two or more waves pass through the same region of space at the same time, it is found that for many waves the actual displacement is the vector (or algebraic) sum of the separate displacements. This is called the principle of superposition. It is valid for mechanical waves as long as the displacements are not too large and there is a linear relationship between the displacement and the restoring force of the oscillating medium.† If the amplitude of a mechanical wave, for example, is so large that it goes beyond the elastic region of the medium, and Hooke's law is no longer operative, the superposition principle is no longer accurate.‡ For the most part, we will consider systems for which the superposition principle can be assumed to hold. One result of the superposition principle is that if two waves pass through the same region of space, they continue to move independently of one another. You may have noticed, for example, that the ripples on the surface of water (two-dimensional waves) that form from two rocks striking the water at different places will pass through each other. Figure 15–16 shows an example of the superposition principle. In this case there are three waves present on a stretched string, each of different amplitude and frequency. At any time, such as at the instant shown, the actual displacement at any position x is the algebraic sum of the displacements of the three waves at that position. The actual wave is not a simple sinusoidal wave and is called a composite (or complex) wave. (Amplitudes are exaggerated in Fig. 15–16.) It can be shown that any complex wave can be considered as being composed of many simple sinusoidal waves of different amplitudes, wavelengths, and frequencies. This is known as Fourier's theorem. A complex periodic wave of period T can be represented as a sum of pure sinusoidal terms whose frequencies are integral multiples of f = 1T. If the wave is not periodic, the sum becomes an integral (called a Fourier integral). Although we will not go into the details here, we see the importance of considering sinusoidal waves (and simple harmonic motion): because any other wave shape can be considered a sum of such pure sinusoidal waves. †

For electromagnetic waves in vacuum, Chapter 31, the superposition principle always holds. Intermodulation distortion in high-fidelity equipment is an example of the superposition principle not holding when two frequencies do not combine linearly in the electronics. ‡

408

CHAPTER 15

Wave Motion

CONCEPTUAL EXAMPLE 15–7 Making a square wave. At t = 0, three waves are given by D1 = A cos kx, D2 = – 13 A cos 3kx, and D3 = 15 A cos 5kx, where A = 1.0 m and k = 10 m–1. Plot the sum of the three waves from x = –0.4 m to ±0.4 m. (These three waves are the first three Fourier components of a "square wave.") RESPONSE The first wave, D1 , has amplitude of 1.0 m and wavelength l = 2pk = (2p10) m = 0.628 m. The second wave, D2 , has amplitude of 0.33 m and wavelength l = 2p3k = (2p30) m = 0.209 m. The third wave, D3 , has amplitude of 0.20 m and wavelength l = 2p5k = (2p50) m = 0.126 m. Each wave is plotted in Fig. 15–17a. The sum of the three waves is shown in Fig. 15–17b.The sum begins to resemble a "square wave," shown in blue in Fig. 15–17b.

P H Y S I C S

y (m)

1.0 0.5

D1 D3

0.0 D2

−0.5

When the restoring force is not precisely proportional to the displacement for mechanical waves in some continuous medium, the speed of sinusoidal waves depends on the frequency. The variation of speed with frequency is called dispersion. The different sinusoidal waves that compose a complex wave will travel with slightly different speeds in such a case. Consequently, a complex wave will change shape as it travels if the medium is "dispersive." A pure sine wave will not change shape under these conditions, however, except by the influence of friction or dissipative forces. If there is no dispersion (or friction), even a complex linear wave does not change shape.

15–7

Reflection and Transmission

When a wave strikes an obstacle, or comes to the end of the medium in which it is traveling, at least a part of the wave is reflected. You have probably seen water waves reflect off a rock or the side of a swimming pool. And you may have heard a shout reflected from a distant cliff—which we call an "echo." A wave pulse traveling down a cord is reflected as shown in Fig. 15–18. The reflected pulse returns inverted as in Fig. 15–18a if the end of the cord is fixed; it returns right side up if the end is free as in Fig. 15–18b. When the end is fixed to a support, as in Fig. 15–18a, the pulse reaching that fixed end exerts a force (upward) on the support. The support exerts an equal but opposite force downward on the cord (Newton's third law). This downward force on the cord is what "generates" the inverted reflected pulse.

A P P L I E D

Square wave

−1.0 −0.4

x(m)

−0.2

0.0 (a)

0.2

0.4

−0.2

0.0 (b)

0.2

0.4

y (m)

1.0 0.5 0.0 −0.5

−1.0 −0.4

x(m)

FIGURE 15–17 Example 15–7. Making a square wave.

FIGURE 15–18 Reflection of a wave pulse on a cord lying on a table. (a) The end of the cord is fixed to a peg. (b) The end of the cord is free to move.

FIGURE 15–19 When a wave pulse traveling to the right along a thin cord (a) reaches a discontinuity where the cord becomes thicker and heavier, then part is reflected and part is transmitted (b). (a)

(b)

Consider next a pulse that travels down a cord which consists of a light section and a heavy section, as shown in Fig. 15–19. When the wave pulse reaches the boundary between the two sections, part of the pulse is reflected and part is transmitted, as shown. The heavier the second section of the cord, the less the energy that is transmitted. (When the second section is a wall or rigid support, very little is transmitted and most is reflected, as in Fig. 15–18a.) For a periodic wave, the frequency of the transmitted wave does not change across the boundary since the boundary point oscillates at that frequency. Thus if the transmitted wave has a lower speed, its wavelength is also less (l = vf).

Light section

Heavy section (a)

Reflected pulse

Transmitted pulse

(b)

SECTION 15–7

409

Ray Wave fronts

Wave fronts

Ray

Ray (a)

(b)

FIGURE 15–20 Rays, signifying the direction of motion, are always perpendicular to the wave fronts (wave crests). (a) Circular or spherical waves near the source. (b) Far from the source, the wave fronts are nearly straight or flat, and are called plane waves.

For a two- or three-dimensional wave, such as a water wave, we are concerned with wave fronts, by which we mean all the points along the wave forming the wave crest (what we usually refer to simply as a "wave" at the seashore). A line drawn in the direction of motion, perpendicular to the wave front, is called a ray, as shown in Fig. 15–20. Wave fronts far from the source have lost almost all their curvature (Fig. 15–20b) and are nearly straight, as ocean waves often are; they are then called plane waves. For reflection of a two- or three-dimensional plane wave, as shown in Fig. 15–21, the angle that the incoming or incident wave makes with the reflecting surface is equal to the angle made by the reflected wave. This is the law of reflection: the angle of reflection equals the angle of incidence. The "angle of incidence" is defined as the angle (ui) the incident ray makes with the perpendicular to the reflecting surface (or the wave front makes with a tangent to the surface), and the "angle of reflection" is the corresponding angle (ur) for the reflected wave. Incident ray

Reflected ray θi

θr

FIGURE 15–21 Law of reflection: ur = ui .

Re wa flect ve ed fro nt θr

nt t ide ron Incave f w θi

15–8

Interference

Interference refers to what happens when two waves pass through the same region of space at the same time. Consider, for example, the two wave pulses on a cord traveling toward each other as shown in Fig. 15–22. In Fig. 15–22a the two pulses have the same amplitude, but one is a crest and the other a trough; in Fig. 15–22b they are both crests. In both cases, the waves meet and pass right by each other. However, in the region where they overlap, the resultant displacement is the algebraic sum of their separate displacements (a crest is considered positive and a trough negative). This is another example of the principle of superposition. In Fig. 15–22a, the two waves have opposite displacements at the instant they pass one another, and they add to zero. The result is called destructive interference. In Fig. 15–22b, at the instant the two pulses overlap, they produce a resultant displacement that is greater than the displacement of either separate pulse, and the result is constructive interference. (a)

(b)

Pulses far apart, approaching FIGURE 15–22 Two wave pulses pass each other. Where they overlap, interference occurs: (a) destructive, and (b) constructive.

Pulses overlap precisely (for an instant)

Time

Pulses far apart, receding

410

CHAPTER 15

[You may wonder where the energy is at the moment of destructive interference in Fig. 15–22a; the cord may be straight at this instant, but the central parts of it are still moving up or down.]

Constructive interference

Destructive interference (b)

(a)

FIGURE 15–23 (a) Interference of water waves. (b) Constructive interference occurs where one wave's maximum (a crest) meets the other's maximum. Destructive interference ("flat water") occurs where one wave's maximum (a crest) meets the other's miminum (a trough).

When two rocks are thrown into a pond simultaneously, the two sets of circular waves interfere with one another as shown in Fig. 15–23a. In some areas of overlap, crests of one wave repeatedly meet crests of the other (and troughs meet troughs), Fig. 15–23b. Constructive interference is occurring at these points, and the water continuously oscillates up and down with greater amplitude than either wave separately. In other areas, destructive interference occurs where the water does not move up and down at all over time. This is where crests of one wave meet troughs of the other, and vice versa. Figure 15–24a shows the displacement of two identical waves graphically as a function of time, as well as their sum, for the case of constructive interference. For constructive interference (Fig. 15–24a), the two waves are in phase. At points where destructive interference occurs (Fig. 15–24b) crests of one wave repeatedly meet troughs of the other wave and the two waves are out of phase by one-half wavelength or 180°. The crests of one wave occur a half wavelength behind the crests of the other wave. The relative phase of the two water waves in Fig. 15–23 in most areas is intermediate between these two extremes, resulting in partially destructive interference, as illustrated in Fig. 15–24c. If the amplitudes of two interfering waves are not equal, fully destructive interference (as in Fig. 15–24b) does not occur.

FIGURE 15–24 Graphs showing two identical waves, and their sum, as a function of time at three locations. In (a) the two waves interfere constructively, in (b) destructively, and in (c) partially destructively.

(a)

(b)

(c)

SECTION 15–8

Interference

411

15–9 Antinode

If you shake one end of a cord and the other end is kept fixed, a continuous wave will travel down to the fixed end and be reflected back, inverted, as we saw in Fig. 15–18a. As you continue to vibrate the cord, waves will travel in both directions, and the wave traveling along the cord, away from your hand, will interfere with the reflected wave coming back. Usually there will be quite a jumble. But if you vibrate the cord at just the right frequency, the two traveling waves will interfere in such a way that a large-amplitude standing wave will be produced, Fig. 15–25. It is called a "standing wave" because it does not appear to be traveling. The cord simply appears to have segments that oscillate up and down in a fixed pattern. The points of destructive interference, where the cord remains still at all times, are called nodes. Points of constructive interference, where the cord oscillates with maximum amplitude, are called antinodes. The nodes and antinodes remain in fixed positions for a particular frequency. Standing waves can occur at more than one frequency. The lowest frequency of vibration that produces a standing wave gives rise to the pattern shown in Fig. 15–25a. The standing waves shown in Figs. 15–25b and 15–25c are produced at precisely twice and three times the lowest frequency, respectively, assuming the tension in the cord is the same. The cord can also vibrate with four loops (four antinodes) at four times the lowest frequency, and so on. The frequencies at which standing waves are produced are the natural frequencies or resonant frequencies of the cord, and the different standing wave patterns shown in Fig. 15–25 are different "resonant modes of vibration." A standing wave on a cord is the result of the interference of two waves traveling in opposite directions. A standing wave can also be considered a vibrating object at resonance. Standing waves represent the same phenomenon as the resonance of a vibrating spring or pendulum, which we discussed in Chapter 14. However, a spring or pendulum has only one resonant frequency, whereas the cord has an infinite number of resonant frequencies, each of which is a whole-number multiple of the lowest resonant frequency. Consider a string stretched between two supports that is plucked like a guitar or violin string, Fig. 15–26a. Waves of a great variety of frequencies will travel in both directions along the string, will be reflected at the ends, and will travel back in the opposite direction. Most of these waves interfere with each other and quickly die out. However, those waves that correspond to the resonant frequencies of the string will persist. The ends of the string, since they are fixed, will be nodes. There may be other nodes as well. Some of the possible resonant modes of vibration (standing waves) are shown in Fig. 15–26b. Generally, the motion will be a combination of these different resonant modes, but only those frequencies that correspond to a resonant frequency will be present.

Node

(a) Node

Standing Waves; Resonance

Antinode

(b) Node Antinode

FIGURE 15–26 (a) A string is plucked. (b) Only standing waves corresponding to resonant frequencies persist for long. (a)

(b)

l= l

1 λ 2 1

Fundamental or first harmonic, f1

l = λ2 First overtone or second harmonic, f2 = 2 f1

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Second overtone or third harmonic, f3 = 3f1

3 λ 2 3

To determine the resonant frequencies, we first note that the wavelengths of the standing waves bear a simple relationship to the length l of the string. The lowest frequency, called the fundamental frequency, corresponds to one antinode (or loop). And as can be seen in Fig. 15–26b, the whole length corresponds to one-half wavelength. Thus l = 12 l1 , where l1 stands for the wavelength of the fundamental frequency. The other natural frequencies are called overtones; for a vibrating string they are whole-number (integral) multiples of the fundamental, and then are also called harmonics, with the fundamental being referred to as the first harmonic.† The next mode of vibration after the fundamental has two loops and is called the second harmonic (or first overtone), Fig. 15–26b. The length of the string l at the second harmonic corresponds to one complete wavelength: l = l2 . For the third and fourth harmonics, l = 32 l3 , and l = 2l4 , respectively, and so on. In general, we can write nln , l = where n = 1, 2, 3, p . 2 The integer n labels the number of the harmonic: n = 1 for the fundamental, n = 2 for the second harmonic, and so on. We solve for ln and find 2l , string fixed ln = n = 1, 2, 3, p . c d (15–17a) at both ends n To find the frequency f of each vibration we use Eq. 15–1, f = vl, and we see that v v fn = = n = nf1 , n = 1, 2, 3, p , (15–17b) ln 2l where f1 = vl1 = v2l is the fundamental frequency. We see that each resonant frequency is an integer multiple of the fundamental frequency. Because a standing wave is equivalent to two traveling waves moving in opposite directions, the concept of wave velocity still makes sense and is given by Eq. 15–2 in terms of the tension FT in the string and its mass per unit length (m = ml). That is, v = 1FTm for waves traveling in both directions. EXAMPLE 15–8 Piano string. A piano string is 1.10 m long and has a mass of 9.00 g. (a) How much tension must the string be under if it is to vibrate at a fundamental frequency of 131 Hz? (b) What are the frequencies of the first four harmonics? APPROACH To determine the tension, we need to find the wave speed using Eq. 15–1 (v = lf), and then use Eq. 15–2, solving it for FT . SOLUTION (a) The wavelength of the fundamental is l = 2l = 2.20 m (Eq. 15–17a with n = 1). The speed of the wave on the string is v = lf = (2.20 m)A131 s–1 B = 288 ms. Then we have (Eq. 15–2) FT = mv2 =

9.00 * 10 –3 kg m 2 v = ¢ ≤ (288 ms)2 = 679 N. l 1.10 m

(b) The frequencies of the second, third, and fourth harmonics are two, three, and four times the fundamental frequency: 262, 393, and 524 Hz, respectively. NOTE The speed of the wave on the string is not the same as the speed of the sound wave that the piano string produces in the air (as we shall see in Chapter 16). A standing wave does appear to be standing in place (and a traveling wave appears to move). The term "standing" wave is also meaningful from the point of view of energy. Since the string is at rest at the nodes, no energy flows past these points. Hence the energy is not transmitted down the string but "stands" in place in the string. Standing waves are produced not only on strings, but on any object that is struck, such as a drum membrane or an object made of metal or wood. The resonant frequencies depend on the dimensions of the object, just as for a string they depend on its length. Large objects have lower resonant frequencies than small objects. All musical instruments, from stringed instruments to wind instruments (in which a column of air vibrates as a standing wave) to drums and other percussion instruments, depend on standing waves to produce their particular musical sounds, as we shall see in Chapter 16. †

The term "harmonic" comes from music, because such integral multiples of frequencies "harmonize."

SECTION 15–9

413

Mathematical Representation of a Standing Wave As we saw, a standing wave can be considered to consist of two traveling waves that move in opposite directions. These can be written (see Eqs. 15–10c and 15–13c) D1(x, t) = A sin(kx - vt)

and

D2(x, t) = A sin(kx + vt)

since, assuming no damping, the amplitudes are equal as are the frequencies and wavelengths. The sum of these two traveling waves produces a standing wave which can be written mathematically as D = D1 + D2 = A Csin(kx - vt) + sin(kx + vt)D.

From the trigonometric identity sin u1 + sin u2 = 2 sin 12 Au1 + u2 B cos 12 Au1 - u2 B, we can rewrite this as D = 2A sin kx cos vt.

(15–18)

If we let x = 0 at the left-hand end of the string, then the right-hand end is at x = l where l is the length of the string. Since the string is fixed at its two ends (Fig. 15–26), D(x, t) must be zero at x = 0 and at x = l. Equation 15–18 already satisfies the first condition (D = 0 at x = 0) and satisfies the second condition if sin kl = 0 which means kl = p, 2p, 3p, p , np, p where n = an integer. Since k = 2pl, then l = 2ln, which is just Eq. 15–17a. Equation 15–18, with the condition l = 2ln, is the mathematical representation of a standing wave. We see that a particle at any position x vibrates in simple harmonic motion (because of the factor cos vt). All particles of the string vibrate with the same frequency f = v2p, but the amplitude depends on x and equals 2A sin kx. (Compare this to a traveling wave for which all particles vibrate with the same amplitude.) The amplitude has a maximum, equal to 2A, when kx = p2, 3p2, 5p2, and so on—that is, at l , 3l , 5l , p . 4 4 4 These are, of course, the positions of the antinodes (see Fig. 15–26). x =

FIGURE 15–27 Example 15–9: possible lengths for the string.

x l = 1.57 m

x l = 3.14 m

x l = (np/2.0) m = n(1.57 m)

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EXAMPLE 15–9 Wave forms. Two waves traveling in opposite directions on a string fixed at x = 0 are described by the functions D1 = (0.20 m) sin(2.0x - 4.0t) and D2 = (0.20 m) sin(2.0x + 4.0t) (where x is in m, t is in s), and they produce a standing wave pattern. Determine (a) the function for the standing wave, (b) the maximum amplitude at x = 0.45 m, (c) where the other end is fixed (x 7 0), (d) the maximum amplitude, and where it occurs. APPROACH We use the principle of superposition to add the two waves. The given waves have the form we used to obtain Eq. 15–18, which we thus can use. SOLUTION (a) The two waves are of the form D = A sin(kx6vt), so k = 2.0 m–1 and v = 4.0 s–1. These combine to form a standing wave of the form of Eq. 15–18: D = 2A sin kx cos vt = (0.40 m) sin(2.0x) cos(4.0t), where x is in meters and t in seconds. (b) At x = 0.45 m, D = (0.40 m) sin(0.90) cos(4.0t) = (0.31 m) cos(4.0t). The maximum amplitude at this point is D = 0.31 m and occurs when cos(4.0t) = 1. (c) These waves make a standing wave pattern, so both ends of the string must be nodes. Nodes occur every half wavelength, which for our string is l 1 2p p = = m = 1.57 m. 2 2 k 2.0 If the string includes only one loop, its length is l = 1.57 m. But without more information, it could be twice as long, l = 3.14 m, or any integral number times 1.57 m, and still provide a standing wave pattern, Fig. 15–27.

Ray

(d) The nodes occur at x = 0, x = 1.57 m, and, if the string is longer than l = 1.57 m, at x = 3.14 m, 4.71 m, and so on. The maximum amplitude (antinode) is 0.40 m [from part (b) above] and occurs midway between the nodes. For l = 1.57 m, there is only one antinode, at x = 0.79 m.

* 15–10

Refraction†

θr

θi

ve Wa

nts

fro

1 2 Ray

When any wave strikes a boundary, some of the energy is reflected and some is transmitted or absorbed. When a two- or three-dimensional wave traveling in one medium crosses a boundary into a medium where its speed is different, the transmitted wave may move in a different direction than the incident wave, as shown in Fig. 15–28. This phenomenon is known as refraction. One example is a water wave; the velocity decreases in shallow water and the waves refract, as shown in Fig. 15–29 below. [When the wave velocity changes gradually, as in Fig. 15–29, without a sharp boundary, the waves change direction (refract) gradually.] In Fig. 15–28, the velocity of the wave in medium 2 is less than in medium 1. In this case, the wave front bends so that it travels more nearly parallel to the boundary. That is, the angle of refraction, ur , is less than the angle of incidence, ui . To see why this is so, and to help us get a quantitative relation between ur and ui , let us think of each wave front as a row of soldiers. The soldiers are marching from firm ground (medium 1) into mud (medium 2) and hence are slowed down after the boundary. The soldiers that reach the mud first are slowed down first, and the row bends as shown in Fig. 15–30a. Let us consider the wave front (or row of soldiers) labeled A in Fig. 15–30b. In the same time t that A 1 moves a distance l1 = v1 t, we see that A 2 moves a distance l2 = v2 t. The two right triangles in Fig. 15–30b, shaded yellow and green, have the side labeled a in common. Thus l1 v1 t sin u1 = = a a since a is the hypotenuse, and l2 v2 t . sin u2 = = a a Dividing these two equations, we obtain the law of refraction: sin u2 v2 . = (15–19) v1 sin u1

θi

θr

FIGURE 15–28 Refraction of waves passing a boundary. FIGURE 15–29 Water waves refract gradually as they approach the shore, as their velocity decreases. There is no distinct boundary, as in Fig. 15–28, because the wave velocity changes gradually.

Since u1 is the angle of incidence Aui B, and u2 is the angle of refraction Aur B, Eq. 15–19 gives the quantitative relation between the two. If the wave were going in the opposite direction, the geometry would not change; only u1 and u2 would change roles: u1 would be the angle of refraction and u2 the angle of incidence. Clearly then, if the wave travels into a medium where it can move faster, it will bend the opposite way, ur 7 ui . We see from Eq. 15–19 that if the velocity increases, the angle increases, and vice versa. Earthquake waves refract within the Earth as they travel through rock layers of different densities (and therefore the velocity is different) just as water waves do. Light waves refract as well, and when we discuss light, we shall find Eq. 15–19 very useful. †

This Section and the next are covered in more detail in Chapters 32 to 35 on optics.

(a) Firm ground w Ro

(b)

θ1

rs die l o s

A2 A

Mud

θ2

θ1

FIGURE 15–30 (a) Soldier analogy to derive (b) law of refraction for waves.

θ2

*SECTION 15–10

415

P H Y S I C S

EXAMPLE 15–10 Refraction of an earthquake wave. An earthquake P wave passes across a boundary in rock where its velocity increases from 6.5 kms to 8.0 kms. If it strikes this boundary at 30°, what is the angle of refraction? APPROACH We apply the law of refraction, Eq. 15–19, sin u2 sin u1 = v2 v1 . SOLUTION Since sin 30° = 0.50, Eq. 15–19 yields (8.0 ms) sin u2 = (0.50) = 0.62. (6.5 ms)

A P P L I E D

Earthquake wave refraction

So u2 = sin–1(0.62) = 38°. NOTE Be careful with angles of incidence and refraction. As we discussed in Section 15–7 (Fig. 15–21), these angles are between the wave front and the boundary line, or—equivalently—between the ray (direction of wave motion) and the line perpendicular to the boundary. Inspect Fig. 15–30b carefully.

* 15–11

FIGURE 15–31 Wave diffraction. The waves are coming from the upper left. Note how the waves, as they pass the obstacle, bend around it, into the "shadow region" behind the obstacle.

Diffraction

Waves spread as they travel. When they encounter an obstacle, they bend around it somewhat and pass into the region behind it, as shown in Fig. 15–31 for water waves. This phenomenon is called diffraction. The amount of diffraction depends on the wavelength of the wave and on the size of the obstacle, as shown in Fig. 15–32. If the wavelength is much larger than the object, as with the grass blades of Fig. 15–32a, the wave bends around them almost as if they are not there. For larger objects, parts (b) and (c), there is more of a "shadow" region behind the obstacle where we might not expect the waves to penetrate—but they do, at least a little. Then notice in part (d), where the obstacle is the same as in part (c) but the wavelength is longer, that there is more diffraction into the shadow region. As a rule of thumb, only if the wavelength is smaller than the size of the object will there be a significant shadow region. This rule applies to reflection from an obstacle as well. Very little of a wave is reflected unless the wavelength is smaller than the size of the obstacle. A rough guide to the amount of diffraction is l, u(radians) L l where u is roughly the angular spread of waves after they have passed through an opening of width l or around an obstacle of width l.

FIGURE 15–32 Water waves passing objects of various sizes. Note that the longer the wavelength compared to the size of the object, the more diffraction there is into the "shadow region."

(a) Water waves passing blades of grass

(b) Stick in water

(d) Long-wavelength waves passing log

That waves can bend around obstacles, and thus can carry energy to areas behind obstacles, is very different from energy carried by material particles. A clear example is the following: if you are standing around a corner on one side of a building, you cannot be hit by a baseball thrown from the other side, but you can hear a shout or other sound because the sound waves diffract around the edges of the building.

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Summary Vibrating objects act as sources of waves that travel outward from the source. Waves on water and on a string are examples. The wave may be a pulse (a single crest) or it may be continuous (many crests and troughs). The wavelength of a continuous wave is the distance between two successive crests (or any two identical points on the wave shape). The frequency is the number of full wavelengths (or crests) that pass a given point per unit time. The wave velocity (how fast a crest moves) is equal to the product of wavelength and frequency, v = lf.

(15–1)

The amplitude of a wave is the maximum height of a crest, or depth of a trough, relative to the normal (or equilibrium) level. In a transverse wave, the oscillations are perpendicular to the direction in which the wave travels. An example is a wave on a string. In a longitudinal wave, the oscillations are along (parallel to) the line of travel; sound is an example. The velocity of both longitudinal and transverse waves in matter is proportional to the square root of an elastic force factor divided by an inertia factor (or density). Waves carry energy from place to place without matter being carried. The intensity of a wave (energy transported across unit area per unit time) is proportional to the square of the amplitude of the wave. For a wave traveling outward in three dimensions from a point source, the intensity (ignoring damping) decreases with the square of the distance from the source, I r

1. r2

(15–8a)

The amplitude decreases linearly with distance from the source. A one-dimensional transverse wave traveling in a medium to the right along the x axis (x increasing) can be represented by a formula for the displacement of the medium from equilibrium at any point x as a function of time as 2p b (x - vt) d D(x, t) = A sin c a l = A sin(kx - vt)

(15–10a) (15–10c)

where k =

2p l

(15–11)

and v = 2pf. If a wave is traveling toward decreasing values of x, D(x, t) = A sin(kx + vt).

(15–13c)

[*Waves can be described by the wave equation, which in one dimension is 0 2D0x2 = (1v2) 0 2D0 t2, Eq. 15–16.] When two or more waves pass through the same region of space at the same time, the displacement at any given point will be the vector sum of the displacements of the separate waves. This is the principle of superposition. It is valid for mechanical waves if the amplitudes are small enough that the restoring force of the medium is proportional to displacement. Waves reflect off objects in their path. When the wave front of a two- or three-dimensional wave strikes an object, the angle of reflection equals the angle of incidence, which is the law of reflection. When a wave strikes a boundary between two materials in which it can travel, part of the wave is reflected and part is transmitted. When two waves pass through the same region of space at the same time, they interfere. From the superposition principle, the resultant displacement at any point and time is the sum of their separate displacements. This can result in constructive interference, destructive interference, or something in between depending on the amplitudes and relative phases of the waves. Waves traveling on a cord of fixed length interfere with waves that have reflected off the end and are traveling back in the opposite direction. At certain frequencies, standing waves can be produced in which the waves seem to be standing still rather than traveling. The cord (or other medium) is vibrating as a whole. This is a resonance phenomenon, and the frequencies at which standing waves occur are called resonant frequencies. The points of destructive interference (no vibration) are called nodes. Points of constructive interference (maximum amplitude of vibration) are called antinodes. On a cord of length l fixed at both ends, the wavelengths of standing waves are given by ln = 2ln

(15–17a)

where n is an integer. [*Waves change direction, or refract, when traveling from one medium into a second medium where their speed is different. Waves spread, or diffract, as they travel and encounter obstacles. A rough guide to the amount of diffraction is u L ll, where l is the wavelength and l the width of an opening or obstacle. There is a significant "shadow region" only if the wavelength l is smaller than the size of the obstacle.]

Questions 1. Is the frequency of a simple periodic wave equal to the frequency of its source? Why or why not? 2. Explain the difference between the speed of a transverse wave traveling down a cord and the speed of a tiny piece of the cord. 3. You are finding it a challenge to climb from one boat up onto a higher boat in heavy waves. If the climb varies from 2.5 m to 4.3 m, what is the amplitude of the wave? Assume the centers of the two boats are a half wavelength apart. 4. What kind of waves do you think will travel down a horizontal metal rod if you strike its end (a) vertically from above and (b) horizontally parallel to its length?

5. Since the density of air decreases with an increase in temperature, but the bulk modulus B is nearly independent of temperature, how would you expect the speed of sound waves in air to vary with temperature? 6. Describe how you could estimate the speed of water waves across the surface of a pond. 7. The speed of sound in most solids is somewhat greater than in air, yet the density of solids is much greater (103 to 104 times). Explain. 8. Give two reasons why circular water waves decrease in amplitude as they travel away from the source.

Questions

417

9. Two linear waves have the same amplitude and speed, and otherwise are identical, except one has half the wavelength of the other. Which transmits more energy? By what factor? 10. Will any function of (x - vt) —see Eq. 15–14—represent a wave motion? Why or why not? If not, give an example. 11. When a sinusoidal wave crosses the boundary between two sections of cord as in Fig. 15–19, the frequency does not change (although the wavelength and velocity do change). Explain why. 12. If a sinusoidal wave on a two-section cord (Fig. 15–19) is inverted upon reflection, does the transmitted wave have a longer or shorter wavelength? 13. Is energy always conserved when two waves interfere? Explain. 14. If a string is vibrating as a standing wave in three segments, are there any places you could touch it with a knife blade without disturbing the motion?

15. When a standing wave exists on a string, the vibrations of incident and reflected waves cancel at the nodes. Does this mean that energy was destroyed? Explain. 16. Can the amplitude of the standing waves in Fig. 15–25 be greater than the amplitude of the vibrations that cause them (up and down motion of the hand)? 17. When a cord is vibrated as in Fig. 15–25 by hand or by a mechanical oscillator, the "nodes" are not quite true nodes (at rest). Explain. [Hint: Consider damping and energy flow from hand or oscillator.] * 18. AM radio signals can usually be heard behind a hill, but FM often cannot. That is, AM signals bend more than FM. Explain. (Radio signals, as we shall see, are carried by electromagnetic waves whose wavelength for AM is typically 200 to 600 m and for FM about 3 m.) * 19. If we knew that energy was being transmitted from one place to another, how might we determine whether the energy was being carried by particles (material objects) or by waves?

Problems 15–1 and 15–2 Characteristics of Waves 1. (I) A fisherman notices that wave crests pass the bow of his anchored boat every 3.0 s. He measures the distance between two crests to be 8.0 m. How fast are the waves traveling? 2. (I) A sound wave in air has a frequency of 262 Hz and travels with a speed of 343 ms. How far apart are the wave crests (compressions)? 3. (I) Calculate the speed of longitudinal waves in (a) water, (b) granite, and (c) steel. 4. (I) AM radio signals have frequencies between 550 kHz and 1600 kHz (kilohertz) and travel with a speed of 3.0 * 108 ms. What are the wavelengths of these signals? On FM the frequencies range from 88 MHz to 108 MHz (megahertz) and travel at the same speed. What are their wavelengths? 5. (I) Determine the wavelength of a 5800-Hz sound wave traveling along an iron rod. 6. (II) A cord of mass 0.65 kg is stretched between two supports 8.0 m apart. If the tension in the cord is 140 N, how long will it take a pulse to travel from one support to the other? 7. (II) A 0.40-kg cord is stretched between two supports, 7.8 m apart. When one support is struck by a hammer, a transverse wave travels down the cord and reaches the other support in 0.85 s. What is the tension in the cord? 8. (II) A sailor strikes the side of his ship just below the surface of the sea. He hears the echo of the wave reflected from the ocean floor directly below 2.8 s later. How deep is the ocean at this point? 9. (II) A ski gondola is connected to the top of a hill by a steel cable of length 660 m and diameter 1.5 cm. As the gondola comes to the end of its run, it bumps into the terminal and sends a wave pulse along the cable. It is observed that it took 17 s for the pulse to return. (a) What is the speed of the pulse? (b) What is the tension in the cable? 10. (II) P and S waves from an earthquake travel at different speeds, and this difference helps locate the earthquake "epicenter" (where the disturbance took place). (a) Assuming typical speeds of 8.5 kms and 5.5 kms for P and S waves, respectively, how far away did the earthquake occur if a particular seismic station detects the arrival of these two types of waves 1.7 min apart? (b) Is one seismic station sufficient to determine the position of the epicenter? Explain.

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11. (II) The wave on a string shown in Fig. 15–33 is moving to the right with a speed of 1.10 ms. (a) Draw the shape of the string 1.00 s later and indicate which parts of the string are moving up and which down at that instant. (b) Estimate the vertical speed of point A on the string at the instant shown in the Figure.

A 0

2 cm 1 cm 0 −1 cm −2 cm

FIGURE 15–33 Problem 11. 12. (II) A 5.0 kg ball hangs from a steel wire 1.00 mm in diameter and 5.00 m long. What would be the speed of a wave in the steel wire? 13. (II) Two children are sending signals along a cord of total mass 0.50 kg tied between tin cans with a tension of 35 N. It takes the vibrations in the string 0.50 s to go from one child to the other. How far apart are the children? * 14. (II) Dimensional analysis. Waves on the surface of the ocean do not depend significantly on the properties of water such as density or surface tension. The primary "return force" for water piled up in the wave crests is due to the gravitational attraction of the Earth. Thus the speed v (ms) of ocean waves depends on the acceleration due to gravity g. It is reasonable to expect that v might also depend on water depth h and the wave's wavelength l. Assume the wave speed is given by the functional form g v = Cgahbl , where a, b, g, and C are numbers without dimension. (a) In deep water, the water deep below does not affect the motion of waves at the surface. Thus v should be independent of depth h (i.e., b = 0). Using only dimensional analysis (Section 1–7), determine the formula for the speed of surface waves in deep water. (b) In shallow water, the speed of surface waves is found experimentally to be independent of the wavelength (i.e., g = 0). Using only dimensional analysis, determine the formula for the speed of waves in shallow water.

15–3 Energy Transported by Waves 15. (I) Two earthquake waves of the same frequency travel through the same portion of the Earth, but one is carrying 3.0 times the energy. What is the ratio of the amplitudes of the two waves? 16. (II) What is the ratio of (a) the intensities, and (b) the amplitudes, of an earthquake P wave passing through the Earth and detected at two points 15 km and 45 km from the source? 17. (II) Show that if damping is ignored, the amplitude A of circular water waves decreases as the square root of the distance r from the source: A r 11r . 18. (II) The intensity of an earthquake wave passing through the Earth is measured to be 3.0 * 106 Jm2 s at a distance of 48 km from the source. (a) What was its intensity when it passed a point only 1.0 km from the source? (b) At what rate did energy pass through an area of 2.0 m2 at 1.0 km? 19. (II) A small steel wire of diameter 1.0 mm is connected to an oscillator and is under a tension of 7.5 N. The frequency of the oscillator is 60.0 Hz and it is observed that the amplitude of the wave on the steel wire is 0.50 cm. (a) What is the power output of the oscillator, assuming that the wave is not reflected back? (b) If the power output stays constant but the frequency is doubled, what is the amplitude of the wave? 20. (II) Show that the intensity of a wave is equal to the energy density (energy per unit volume) in the wave times the wave speed. 21. (II) (a) Show that the average rate with which energy is transported along a cord by a mechanical wave of frequency f and amplitude A is g = 2p2mvf2A2, where v is the speed of the wave and m is the mass per unit length of the cord. (b) If the cord is under a tension FT = 135 N and has mass per unit length 0.10 kgm, what power is required to transmit 120-Hz transverse waves of amplitude 2.0 cm?

15–4 Mathematical Representation of Traveling Wave 22. (I) A transverse wave on a wire is given by D(x, t) = 0.015 sin(25x - 1200 t) where D and x are in meters and t is

23.

24.

25.

26.

in seconds. (a) Write an expression for a wave with the same amplitude, wavelength, and frequency but traveling in the opposite direction. (b) What is the speed of either wave? (I) Suppose at t = 0, a wave shape is represented by D = A sin(2pxl + f); that is, it differs from Eq. 15–9 by a constant phase factor f. What then will be the equation for a wave traveling to the left along the x axis as a function of x and t? (II) A transverse traveling wave on a cord is represented by D = 0.22 sin(5.6x + 34t) where D and x are in meters and t is in seconds. For this wave determine (a) the wavelength, (b) frequency, (c) velocity (magnitude and direction), (d) amplitude, and (e) maximum and minimum speeds of particles of the cord. (II) Consider the point x = 1.00 m on the cord of Example 15–5. Determine (a) the maximum velocity of this point, and (b) its maximum acceleration. (c) What is its velocity and acceleration at t = 2.50 s? (II) A transverse wave on a cord is given by D(x, t) = 0.12 sin(3.0x - 15.0 t), where D and x are in m and t is in s. At t = 0.20 s, what are the displacement and velocity of the point on the cord where x = 0.60 m?

27. (II) A transverse wave pulse travels to the right along a string with a speed v = 2.0 ms. At t = 0 the shape of the pulse is given by the function D = 0.45 cos(2.6x + 1.2), where D and x are in meters. (a) Plot D vs. x at t = 0. (b) Determine a formula for the wave pulse at any time t assuming there are no frictional losses. (c) Plot D(x, t) vs. x at t = 1.0 s. (d) Repeat parts (b) and (c) assuming the pulse is traveling to the left. Plot all 3 graphs on the same axes for easy comparison. 28. (II) A 524-Hz longitudinal wave in air has a speed of 345 ms. (a) What is the wavelength? (b) How much time is required for the phase to change by 90° at a given point in space? (c) At a particular instant, what is the phase difference (in degrees) between two points 4.4 cm apart? 29. (II) Write the equation for the wave in Problem 28 traveling to the right, if its amplitude is 0.020 cm, and D = –0.020 cm, at t = 0 and x = 0. 30. (II) A sinusoidal wave traveling on a string in the negative x direction has amplitude 1.00 cm, wavelength 3.00 cm, and frequency 245 Hz. At t = 0, the particle of string at x = 0 is displaced a distance D = 0.80 cm above the origin and is moving upward. (a) Sketch the shape of the wave at t = 0 and (b) determine the function of x and t that describes the wave. * 15–5 The Wave Equation * 31. (II) Determine if the function D = A sin kx cos vt is a solution of the wave equation. * 32. (II) Show by direct substitution that the following functions satisfy the wave equation: (a) D(x, t) = A ln(x + vt); (b) D(x, t) = (x - vt)4. * 33. (II) Show that the wave forms of Eqs. 15–13 and 15–15 satisfy the wave equation, Eq. 15–16. * 34. (II) Let two linear waves be represented by D1 = f1(x, t) and D2 = f2(x, t). If both these waves satisfy the wave equation (Eq. 15–16), show that any combination D = C1 D1 + C2 D2 does as well, where C1 and C2 are constants. 2 * 35. (II) Does the function D(x, t) = e –(kx - vt) satisfy the wave equation? Why or why not? * 36. (II) In deriving Eq. 15–2, v = 2FT m, for the speed of a transverse wave on a string, it was assumed that the wave's amplitude A is much less than its wavelength l. Assuming a sinusoidal wave shape D = A sin(kx - vt), show via the partial derivative v¿ = 0D0 t that the assumption A V l implies that the maximum transverse œ speed vmax of the string itself is much less than the wave œ velocity. If A = l100 determine the ratio vmax v.

15–7 Reflection and Transmission 37. (II) A cord has two sections with linear densities of 0.10 kgm and 0.20 kgm, Fig. 15–34. An incident wave, given by D = (0.050 m) sin(7.5x - 12.0t), where x is in meters and t in seconds, travels along the lighter cord. (a) What is the wavelength on the lighter section of the cord? (b) What is the tension in the cord? (c) What is the wavelength when the wave travels on the heavier section? m1 = 0.10 kg/m FIGURE 15–34 Problem 37.

m2 = 0.20 kg/m

D = (0.050 m) sin(7.5 x − 12.0 t)

Problems

419

38. (II) Consider a sine wave traveling down the stretched twopart cord of Fig. 15–19. Determine a formula (a) for the ratio of the speeds of the wave in the two sections, vHvL , and (b) for the ratio of the wavelengths in the two sections. (The frequency is the same in both sections. Why?) (c) Is the wavelength larger in the heavier cord or the lighter? 39. (II) Seismic reflection prospecting is commonly used to map deeply buried formations containing oil. In this technique, a seismic wave generated on the Earth's surface (for example, by an explosion or falling weight) reflects from the subsurface formation and is detected upon its return to ground level. By placing ground-level detectors at a variety of locations relative to the source, and observing the variation in the source-to-detector travel times, the depth of the subsurface formation can be determined. (a) Assume a ground-level detector is placed a distance x away from a seismic-wave source and that a horizontal boundary between overlying rock and a subsurface formation exists at depth D (Fig. 15–35a). Determine an x expression for the time t Detector Source taken by the reflected wave to travel from source to detector, assuming the D seismic wave propagates at constant speed v. (b) Suppose several detecOil tors are placed along a line at different distances x (a) from the source as in Fig. 15–35b. Then, when a x3 seismic wave is generated, x2 the different travel times t Source x1 for each detector are measured. Starting with 1 2 3 4 your result from part (a), Detectors explain how a graph of t2 vs. x2 can be used to determine D. Oil FIGURE 15–35 (b) Problem 39. 40. (III) A cord stretched to a tension FT consists of two sections (as in Fig. 15–19) whose linear densities are m1 and m2 . Take x = 0 to be the point (a knot) where they are joined, with m1 referring to that section of cord to the left and m2 that to the right. A sinusoidal wave, D = A sin Ck1 Ax - v1 t B D, starts at the left end of the cord. When it reaches the knot, part of it is reflected and part is transmitted. Let the equation of the reflected wave be DR = A R sin Ck1 Ax + v1 t B D and that for the transmitted wave be DT = A T sin Ck2 Ax - v2 t B D. Since the frequency must be the same in both sections, we have v1 = v2 or k1 v1 = k2 v2 . (a) Because the cord is continuous, a point an infinitesimal distance to the left of the knot has the same displacement at any moment (due to incident plus reflected waves) as a point just to the right of the knot (due to the transmitted wave). Thus show that A = A T + A R . (b) Assuming that the slope (0D0x) of the cord just to the left of the knot is the same as the slope just to the right of the knot, show that the amplitude of the reflected wave is given by v1 - v2 k2 - k1 AR = ¢ ≤A = ¢ ≤ A. v1 + v2 k2 + k1 (c) What is A T in terms of A?

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15–8 Interference 41. (I) The two pulses shown in Fig. 15–36 are moving toward each other. (a) Sketch the shape of the string at the moment they directly overlap. (b) Sketch the shape of the string a few moments later. (c) In Fig. 15–22a, at the moment the pulses pass each other, the string is straight. What has happened to the energy at this moment?

FIGURE 15–36 Problem 41. 42. (II) Suppose two linear waves of equal amplitude and frequency have a phase difference f as they travel in the same medium. They can be represented by D1 = A sin(kx - vt) D2 = A sin(kx - vt + f). (a) Use the trigonometric identity sin u1 + sin u2 = 2 sin 12 Au1 + u2 B cos 12 Au1 - u2 B to show that the resultant wave is given by D = a 2A cos

f f b sin a kx - vt + b . 2 2

(b) What is the amplitude of this resultant wave? Is the wave purely sinusoidal, or not? (c) Show that constructive interference occurs if f = 0, 2p, 4p, and so on, and destructive interference occurs if f = p, 3p, 5p, etc. (d) Describe the resultant wave, by equation and in words, if f = p2.

15–9 Standing Waves; Resonance 43. (I) A violin string vibrates at 441 Hz when unfingered. At what frequency will it vibrate if it is fingered one-third of the way down from the end? (That is, only two-thirds of the string vibrates as a standing wave.) 44. (I) If a violin string vibrates at 294 Hz as its fundamental frequency, what are the frequencies of the first four harmonics? 45. (I) In an earthquake, it is noted that a footbridge oscillated up and down in a one-loop (fundamental standing wave) pattern once every 1.5 s. What other possible resonant periods of motion are there for this bridge? What frequencies do they correspond to? 46. (I) A particular string resonates in four loops at a frequency of 280 Hz. Name at least three other frequencies at which it will resonate. 47. (II) A cord of length 1.0 m has two equal-length sections with linear densities of 0.50 kgm and 1.00 kgm. The tension in the entire cord is constant. The ends of the cord are oscillated so that a standing wave is set up in the cord with a single node where the two sections meet. What is the ratio of the oscillatory frequencies? 48. (II) The velocity of waves on a string is 96 ms. If the frequency of standing waves is 445 Hz, how far apart are the two adjacent nodes? 49. (II) If two successive harmonics of a vibrating string are 240 Hz and 320 Hz, what is the frequency of the fundamental? 50. (II) A guitar string is 90.0 cm long and has a mass of 3.16 g. From the bridge to the support post ( = l) is 60.0 cm and the string is under a tension of 520 N. What are the frequencies of the fundamental and first two overtones?

51. (II) Show that the frequency of standing waves on a cord of length l and linear density m, which is stretched to a tension FT , is given by f =

FT n 2l B m

where n is an integer. 52. (II) One end of a horizontal string of linear density 6.6 * 10–4 kgm is attached to a small-amplitude mechanical 120-Hz oscillator. The string passes over a pulley, a distance l = 1.50 m away, and weights are hung from this end, Fig. 15–37. What mass m must be hung from this end of the string to produce (a) one loop, (b) two loops, and (c) five loops of a standing wave? Assume the string at the oscillator is a node, which is 1.50 m nearly true.

Oscillator FIGURE 15–37 Problems 52 and 53.

53. (II) In Problem 52, Fig. 15–37, the length of the string may be adjusted by moving the pulley. If the hanging mass m is fixed at 0.070 kg, how many different standing wave patterns may be achieved by varying l between 10 cm and 1.5 m? 54. (II) The displacement of a standing wave on a string is given by D = 2.4 sin(0.60x) cos(42 t), where x and D are in centimeters and t is in seconds. (a) What is the distance (cm) between nodes? (b) Give the amplitude, frequency, and speed of each of the component waves. (c) Find the speed of a particle of the string at x = 3.20 cm when t = 2.5 s. 55. (II) The displacement of a transverse wave traveling on a string is represented by D1 = 4.2 sin(0.84x - 47t + 2.1), where D1 and x are in cm and t in s. (a) Find an equation that represents a wave which, when traveling in the opposite direction, will produce a standing wave when added to this one. (b) What is the equation describing the standing wave? 56. (II) When you slosh the water back and forth in a tub at just the right frequency, the water alternately rises and falls at each end, remaining relatively calm at the center. Suppose the frequency to produce such a standing wave in a 45-cm-wide tub is 0.85 Hz. What is the speed of the water wave? 57. (II) A particular violin string plays at a frequency of 294 Hz. If the tension is increased 15%, what will the new frequency be? 58. (II) Two traveling waves are described by the functions D1 = A sin(kx - vt) D2 = A sin(kx + vt), where A = 0.15 m, k = 3.5 m–1, and v = 1.8 s–1. (a) Plot these two waves, from x = 0 to a point x( 7 0) that includes one full wavelength. Choose t = 1.0 s. (b) Plot the sum of the two waves and identify the nodes and antinodes in the plot, and compare to the analytic (mathematical) representation. 59. (II) Plot the two waves given in Problem 58 and their sum, as a function of time from t = 0 to t = T (one period). Choose (a) x = 0 and (b) x = l4. Interpret your results.

60. (II) A standing wave on a 1.64-m-long horizontal string displays three loops when the string vibrates at 120 Hz. The maximum swing of the string (top to bottom) at the center of each loop is 8.00 cm. (a) What is the function describing the standing wave? (b) What are the functions describing the two equal-amplitude waves traveling in opposite directions that make up the standing wave? 61. (II) On an electric guitar, a "pickup" under each string transforms the string's vibrations directly into an electrical signal. If a pickup is placed 16.25 cm from one of the fixed ends of a 65.00-cm-long string, which of the harmonics from n = 1 to n = 12 will not be "picked up" by this pickup? 62. (II) A 65-cm guitar string is fixed at both ends. In the frequency range between 1.0 and 2.0 kHz, the string is found to resonate only at frequencies 1.2, 1.5, and 1.8 kHz. What is the speed of traveling waves on this string? 63. (II) Two oppositely directed traveling waves given by D1 = (5.0 mm) cos C A2.0 m–1 Bx - (3.0 rads) t D and D2 = (5.0 mm) cos C A2.0 m–1 Bx + (3.0 rads) t D form a standing wave. Determine the position of nodes along the x axis. 64. (II) A wire is composed of aluminum with length l1 = 0.600 m and mass per unit length m1 = 2.70 gm joined to a steel section with length l2 = 0.882 m and mass per unit length m2 = 7.80 gm. This composite wire is fixed at both ends and held at a uniform tension of 135 N. Find the lowest frequency standing wave that can exist on this wire, assuming there is a node at the joint between aluminum and steel. How many nodes (including the two at the ends) does this standing wave possess? * 15–10 Refraction * 65. (I) An earthquake P wave traveling 8.0 kms strikes a boundary within the Earth between two kinds of material. If it approaches the boundary at an incident angle of 52° and the angle of refraction is 31°, what is the speed in the second medium? * 66. (I) Water waves approach an underwater "shelf" where the velocity changes from 2.8 ms to 2.5 ms. If the incident wave crests make a 35° angle with the shelf, what will be the angle of refraction? * 67. (II) A sound wave is traveling in warm air (25°C) when it hits a layer of cold ( –15°C) denser air. If the sound wave hits the cold air interface at an angle of 33°, what is the angle of refraction? The speed of sound as a function of temperature can be approximated by v = (331 + 0.60 T) ms, where T is in °C. 68. (II) Any type of wave that reaches a boundary beyond which * its speed is increased, there is a maximum incident angle if there is to be a transmitted refracted wave. This maximum incident angle uiM corresponds to an angle of refraction equal to 90°. If ui 7 uiM , all the wave is reflected at the boundary and none is refracted, because this would correspond to sin ur 7 1 (where ur is the angle of refraction), which is impossible. This phenomenon is referred to as total internal reflection. (a) Find a formula for uiM using the law of refraction, Eq. 15–19. (b) How far from the bank should a trout fisherman stand (Fig. 15–38) so trout won't be frightened by his voice (1.8 m above the ground)? The speed of sound is about θi 343 ms in air and 1440 ms in water. FIGURE 15–38 x Problem 68b.

Problems

421

* 69. (II) A longitudinal earthquake wave strikes a boundary between two types of rock at a 38° angle. As the wave crosses the boundary, the specific gravity of the rock changes from 3.6 to 2.8. Assuming that the elastic modulus is the same for both types of rock, determine the angle of refraction.

* 15–11 Diffraction * 70. (II) A satellite dish is about 0.5 m in diameter. According to the user's manual, the dish has to be pointed in the direction of the satellite, but an error of about 2° to either side is allowed without loss of reception. Estimate the wavelength of the electromagnetic waves (speed = 3 * 108 ms) received by the dish.

General Problems 71. A sinusoidal traveling wave has frequency 880 Hz and phase velocity 440 ms. (a) At a given time, find the distance between any two locations that correspond to a difference in phase of p6 rad. (b) At a fixed location, by how much does the phase change during a time interval of 1.0 * 10–4 s? 72. When you walk with a cup of coffee (diameter 8 cm) at just the right pace of about one step per second, the coffee sloshes higher and higher in your cup until eventually it starts to spill over the top, Fig 15–39. Estimate the speed of the waves in the coffee.

79. A transverse wave pulse travels to the right along a string with a speed v = 2.4 ms. At t = 0 the shape of the pulse is given by the function D =

4.0 m3 x2 + 2.0 m2

where D and x are in meters. (a) Plot D vs. x at t = 0 from x = –10 m to x = ±10 m. (b) Determine a formula for the wave pulse at any time t assuming there are no frictional losses. (c) Plot D(x, t) vs. x at t = 1.00 s. (d) Repeat parts (b) and (c) assuming the pulse is traveling to the left. 80. (a) Show that if the tension in a stretched string is changed by a small amount ¢FT , the frequency of the fundamental is changed by an amount ¢f = 12 A¢FTFT Bf. (b) By what percent must the tension in a piano string be increased or decreased to raise the frequency from 436 Hz to 442 Hz. (c) Does the formula in part (a) apply to the overtones as well?

FIGURE 15–39 Problem 72. 73. Two solid rods have the same bulk modulus but one is 2.5 times as dense as the other. In which rod will the speed of longitudinal waves be greater, and by what factor? 74. Two waves traveling along a stretched string have the same frequency, but one transports 2.5 times the power of the other. What is the ratio of the amplitudes of the two waves? 75. A bug on the surface of a pond is observed to move up and down a total vertical distance of 0.10 m, lowest to highest point, as a wave passes. (a) What is the amplitude of the wave? (b) If the amplitude increases to 0.15 m, by what factor does the bug's maximum kinetic energy change? 76. A guitar string is supposed to vibrate at 247 Hz, but is measured to actually vibrate at 255 Hz. By what percentage should the tension in the string be changed to get the frequency to the correct value? 77. An earthquake-produced surface wave can be approximated by a sinusoidal transverse wave. Assuming a frequency of 0.60 Hz (typical of earthquakes, which actually include a mixture of frequencies), what amplitude is needed so that objects begin to leave contact with the ground? [Hint: Set the acceleration a 7 g.] 78. A uniform cord of length l and mass m is hung vertically from a support. (a) Show that the speed of transverse waves in this cord is 1gh , where h is the height above the lower end. (b) How long does it take for a pulse to travel upward from one end to the other?

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CHAPTER 15

Wave Motion

81. Two strings on a musical instrument are tuned to play at 392 Hz (G) and 494 Hz (B). (a) What are the frequencies of the first two overtones for each string? (b) If the two strings have the same length and are under the same tension, what must be the ratio of their masses AmGmB B? (c) If the strings, instead, have the same mass per unit length and are under the same tension, what is the ratio of their lengths AlGlB B? (d) If their masses and lengths are the same, what must be the ratio of the tensions in the two strings? 82. The ripples in a certain groove 10.8 cm from the center of a 33-rpm phonograph record have a wavelength of 1.55 mm. What will be the frequency of the sound emitted? 83. A 10.0-m-long wire of mass 152 g is stretched under a tension of 255 N. A pulse is generated at one end, and 20.0 ms later a second pulse is generated at the opposite end. Where will the two pulses first meet? 84. A wave with a frequency of 220 Hz and a wavelength of 10.0 cm is traveling along a cord. The maximum speed of particles on the cord is the same as the wave speed. What is the amplitude of the wave? 85. A string can have a "free" end if that end is attached to a ring that can slide without friction on a vertical pole (Fig. 15–40). Determine the wavelengths of the resonant vibrations of such a string with one end fixed and the other free. Free end

Fixed end FIGURE 15–40 Problem 85.

86. A highway overpass was observed to resonate as one full loop A 12 lB when a small earthquake shook the ground vertically at 3.0 Hz. The highway department put a support at the center of the overpass, anchoring it to the ground as shown in Fig. 15–41. What resonant frequency would you now expect for the overpass? It is noted that earthquakes rarely do significant shaking above 5 or 6 Hz. Did the modifications do any good? Explain.

90. Two wave pulses are traveling in opposite directions with the same speed of 7.0 cms as shown in Fig. 15–43. At t = 0, the leading edges of the two pulses are 15 cm apart. Sketch the wave pulses at t = 1.0, 2.0 and 3.0 s. 7.0 cm/s

7.0 cm/s

Distance (cm)

FIGURE 15–43 Problem 90. Before modification

91. For a spherical wave traveling uniformly away from a point source, show that the displacement can be represented by D = a

Added support

A b sin(kr - vt), r

where r is the radial distance from the source and A is a constant. After modification FIGURE 15–41 Problem 86. 87. Figure 15–42 shows the wave shape at two instants of time for a sinusoidal wave traveling to the right. What is the mathematical representation of this wave?

y (cm)

at t = 0

at t = 0.80 s

3.5 0

x (cm)

FIGURE 15–42 Problem 87. 88. Estimate the average power of a water wave when it hits the chest of an adult standing in the water at the seashore. Assume that the amplitude of the wave is 0.50 m, the wavelength is 2.5 m, and the period is 4.0 s. 89. A tsunami of wavelength 215 km and velocity 550 kmh travels across the Pacific Ocean. As it approaches Hawaii, people observe an unusual decrease of sea level in the harbors. Approximately how much time do they have to run to safety? (In the absence of knowledge and warning, people have died during tsunamis, some of them attracted to the shore to see stranded fishes and boats.)

92. What frequency of sound would have a wavelength the same size as a 1.0-m-wide window? (The speed of sound is 344 ms at 20°C.) What frequencies would diffract through the window? * Numerical/Computer * 93. (II) Consider a wave generated by the periodic vibration of a source and given by the expression D(x, t) = A sin2 k(x - ct), where x represents position (in meters), t represents time (in seconds), and c is a positive constant. We choose A = 5.0 m, k = 1.0 m - 1, and c = 0.50 ms. Use a spreadsheet to make a graph with three curves of D(x, t) from x = –5.0 m to ±5.0 m in steps of 0.050 m at times t = 0.0, 1.0, and 2.0 s. Determine the speed, direction of motion, period, and wavelength of the wave. * 94. (II) The displacement of a bell-shaped wave pulse is described by a relation that involves the exponential function: D(x, t) = Ae –a(x - vt)

where the constants A = 10.0 m, a = 2.0 m–2, and v = 3.0 ms. (a) Over the range –10.0 m x ±10.0 m, use a graphing calculator or computer program to plot D(x, t) at each of the three times t = 0, t = 1.0, and t = 2.0 s. Do these three plots demonstrate the wave-pulse shape shifting along the x axis by the expected amount over the span of each 1.0-s interval? (b) Repeat part (a) but assume 2 D(x, t) = Ae –a(x + vt) .

Answers to Exercises A: (c). B: (d).

C: (c). D: (a).

General Problems

423

"If music be the food of physics, play on." [See Shakespeare, Twelfth Night, line 1.] Stringed instruments depend on transverse standing waves on strings to produce their harmonious sounds. The sound of wind instruments originates in longitudinal standing waves of an air column. Percussion instruments create more complicated standing waves. Besides examining sources of sound, we also study the decibel scale of sound level, sound wave interference and beats, the Doppler effect, shock waves and sonic booms, and ultrasound imaging.

P T E R

C H

CONTENTS

16–1 Characteristics of Sound 16–2 Mathematical Representation of Longitudinal Waves 16–3 Intensity of Sound: Decibels 16–4 Sources of Sound:Vibrating Strings and Air Columns

*16–5 Quality of Sound, and Noise; Superposition

16–6 Interference of Sound Waves; Beats 16–7 Doppler Effect

*16–8 Shock Waves and the Sonic Boom

*16–9 Applications: Sonar, Ultrasound, and Medical Imaging

424

Sound CHAPTER-OPENING QUESTION—Guess now! A pianist plays the note "middle C." The sound is made by the vibration of the piano string and is propagated outward as a vibration of the air (which can reach your ear). Comparing the vibration on the string to the vibration in the air, which of the following is true? (a) The vibration on the string and the vibration in the air have the same wavelength. (b) They have the same frequency. (c) They have the same speed. (d) Neither wavelength, frequency, nor speed are the same in the air as on the string. ound is associated with our sense of hearing and, therefore, with the physiology of our ears and the psychology of our brain, which interprets the sensations that reach our ears. The term sound also refers to the physical sensation that stimulates our ears: namely, longitudinal waves. We can distinguish three aspects of any sound. First, there must be a source for a sound; as with any mechanical wave, the source of a sound wave is a vibrating object. Second, the energy is transferred from the source in the form of longitudinal sound waves. And third, the sound is detected by an ear or by a microphone. We start this Chapter by looking at some aspects of sound waves themselves.

16–1

Characteristics of Sound

We saw in Chapter 15, Fig. 15–5, how a vibrating drumhead produces a sound wave in air. Indeed, we usually think of sound waves traveling in the air, for normally it is the vibrations of the air that force our eardrums to vibrate. But sound waves can also travel in other materials. Two stones struck together under water can be heard by a swimmer beneath the surface, for the vibrations are carried to the ear by the water. When you put your ear flat against the ground, you can hear an approaching train or truck. In this case the ground does not actually touch your eardrum, but the longitudinal wave transmitted by the ground is called a sound wave just the same, for its vibrations cause the outer ear and the air within it to vibrate. Sound cannot travel in the absence of matter. For example, a bell ringing inside an evacuated jar cannot be heard, nor does sound travel through the empty reaches of outer space. The speed of sound is different in different materials. In air at 0°C and 1 atm, sound travels at a speed of 331 ms. We saw in Eq. 15–4 Av = 1BrB that the speed depends on the elastic modulus, B, and the density, r, of the material. Thus for helium, whose density is much less than that of air but whose elastic modulus is not greatly different, the speed is about three times as great as in air. In liquids and solids, which are much less compressible and therefore have much greater elastic moduli, the speed is larger still. The speed of sound in various materials is given in Table 16–1. The values depend somewhat on temperature, but this is significant mainly for gases. For example, in air at normal (ambient) temperatures, the speed increases approximately 0.60 ms for each Celsius degree increase in temperature: v L (331 + 0.60 T) ms,

TABLE 16–1 Speed of Sound in Various Materials (20°C and 1 atm)

Speed (m s)

Material Air Air (0°C) Helium Hydrogen Water Sea water Iron and steel Glass Aluminum Hardwood Concrete

343 331 1005 1300 1440 1560 L 5000 L 4500 L 5100 L 4000 L 3000

[speed of sound in air]

where T is the temperature in °C. Unless stated otherwise, we will assume in this Chapter that T = 20°C, so that† v = C331 + (0.60)(20)D ms = 343 ms. CONCEPTUAL EXAMPLE 16–1 Distance from a lightning strike. A rule of thumb that tells how close lightning has struck is, "one mile for every five seconds before the thunder is heard." Explain why this works, noting that the speed of light is so high (3 * 108 ms, almost a million times faster than sound) that the time for light to travel to us is negligible compared to the time for the sound.

P H Y S I C S

A P P L I E D

How far away is the lightning?

RESPONSE The speed of sound in air is about 340 ms, so to travel 1 km = 1000 m takes about 3 seconds. One mile is about 1.6 kilometers, so the time for the thunder to travel a mile is about (1.6)(3) L 5 seconds. EXERCISE A What would be the rule used in Example 16–1 in terms of kilometers?

Two aspects of any sound are immediately evident to a human listener: "loudness" and "pitch." Each refers to a sensation in the consciousness of the listener. But to each of these subjective sensations there corresponds a physically measurable quantity. Loudness is related to the intensity (energy per unit time crossing unit area) in the sound wave, and we shall discuss it in Section 16–3. The pitch of a sound refers to whether it is high, like the sound of a piccolo or violin, or low, like the sound of a bass drum or string bass. The physical quantity that determines pitch is the frequency, as was first noted by Galileo. The lower the frequency, the lower the pitch; the higher the frequency, the higher the pitch.‡ The best human ears can respond to frequencies from about 20 Hz to almost 20,000 Hz. (Recall that 1 Hz is 1 cycle per second.) This frequency range is called the audible range. These limits vary somewhat from one individual to another. One general trend is that as people age, they are less able to hear high frequencies, so the high-frequency limit may be 10,000 Hz or less. †

We treat the 20°C ("room temperature") as accurate to 2 significant figures. Although pitch is determined mainly by frequency, it also depends to a slight extent on loudness. For example, a very loud sound may seem slightly lower in pitch than a quiet sound of the same frequency. ‡

SECTION 16–1

425

C A U T I O N

Do not confuse ultrasonic (high frequency) with supersonic (high speed)

P H Y S I C S

A P P L I E D

Autofocusing camera

FIGURE 16–1 Example 16–2. Autofocusing camera emits an ultrasonic pulse. Solid lines represent the wave front of the outgoing wave pulse moving to the right; dashed lines represent the wave front of the pulse reflected off the person's face, returning to the camera. The time information allows the camera mechanism to adjust the lens to focus at the proper distance.

Sound waves whose frequencies are outside the audible range may reach the ear, but we are not generally aware of them. Frequencies above 20,000 Hz are called ultrasonic (do not confuse with supersonic, which is used for an object moving with a speed faster than the speed of sound). Many animals can hear ultrasonic frequencies; dogs, for example, can hear sounds as high as 50,000 Hz, and bats can detect frequencies as high as 100,000 Hz. Ultrasonic waves have many useful applications in medicine and other fields, which we discuss later in this Chapter. EXAMPLE 16–2 Autofocusing with sound waves. Older autofocusing cameras determine the distance by emitting a pulse of very high frequency (ultrasonic) sound that travels to the object being photographed, and include a sensor that detects the returning reflected sound, as shown in Fig. 16–1. To get an idea of the time sensitivity of the detector, calculate the travel time of the pulse for an object (a) 1.0 m away, and (b) 20 m away. APPROACH If we assume the temperature is about 20°C, then the speed of sound is 343 ms. Using this speed v and the total distance d back and forth in each case, we can obtain the time (v = dt). SOLUTION (a) The pulse travels 1.0 m to the object and 1.0 m back, for a total of 2.0 m. We solve for t in v = dt: d 2.0 m t = = = 0.0058 s = 5.8 ms. v 343 ms (b) The total distance now is 2 * 20 m = 40 m, so 40 m t = = 0.12 s = 120 ms. 343 ms NOTE Newer autofocus cameras use infrared light (v = 3 * 108 ms) instead of ultrasound, andor a digital sensor array that detects light intensity differences between adjacent receptors as the lens is automatically moved back and forth, choosing the lens position that provides maximum intensity differences (sharpest focus). Sound waves whose frequencies are below the audible range (that is, less than 20 Hz) are called infrasonic. Sources of infrasonic waves include earthquakes, thunder, volcanoes, and waves produced by vibrating heavy machinery. This last source can be particularly troublesome to workers, for infrasonic waves—even though inaudible—can cause damage to the human body. These low-frequency waves act in a resonant fashion, causing motion and irritation of the body's organs.

FIGURE 16–2 Longitudinal sound wave traveling to the right, and its graphical representation in terms of pressure. Expansion (pressure lower)

Compression (pressure higher)

16–2

Mathematical Representation of Longitudinal Waves

In Section 15–4, we saw that a one-dimensional sinusoidal wave traveling along the x axis can be represented by the relation (Eq. 15–10c) D = A sin(kx - vt),

Pressure

∆PM Patm −∆PM

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(16–1)

where D is the displacement of the wave at position x and time t, and A is its amplitude (maximum value). The wave number k is related to the wavelength l by k = 2pl, and v = 2pf where f is the frequency. For a transverse wave—such as a wave on a string—the displacement D is perpendicular to the direction of wave propagation along the x axis. But for a longitudinal wave the displacement D is along the direction of wave propagation. That is, D is parallel to x and represents the displacement of a tiny volume element of the medium from its equilibrium position. Longitudinal (sound) waves can also be considered from the point of view of variations in pressure rather than displacement. Indeed, longitudinal waves are often called pressure waves. The pressure variation is usually easier to measure than the displacement (see Example 16–7). As can be seen in Fig. 16–2, in a wave "compression" (where molecules are closest together), the pressure is higher than normal, whereas in an expansion (or rarefaction) the pressure is less than normal.

Let us now derive the mathematical representation of the pressure variation in a traveling longitudinal wave. From the definition of the bulk modulus, B (Eq. 12–7), ¢P = –B(¢VV), where ¢P represents the pressure difference from the normal pressure P0 (no wave present) and ¢VV is the fractional change in volume of the medium due to the pressure change ¢P. The negative sign reflects the fact that the volume decreases (¢V 6 0) if the pressure is increased. Consider now a layer of fluid through which the longitudinal wave is passing (Fig. 16–4). If this layer has thickness ¢x and area S, then its volume is V = S ¢x. As a result of pressure variation in the wave, the volume will change by an amount ¢V = S ¢D, where ¢D is the change in thickness of this layer as it compresses or expands. (Remember that D represents the displacement of the medium.) Thus we have S ¢D . ¢P = –B S ¢x To be precise, we take the limit of ¢x S 0, so we obtain 0D , ¢P = –B (16–2) 0x where we use the partial derivative notation since D is a function of both x and t. If the displacement D is sinusoidal as given by Eq. 16–1, then we have from Eq. 16–2 that ¢P = –(BAk) cos(kx - vt).

(16–3)

(Here A is the displacement amplitude, not area which is S.) Thus the pressure varies sinusoidally as well, but is out of phase from the displacement by 90° or a quarter wavelength, as in Fig. 16–3. The quantity BAk is called the pressure amplitude, ¢PM . It represents the maximum and minimum amounts by which the pressure varies from the normal ambient pressure. We can thus write ¢P = – ¢PM cos(kx - vt), (16–4)

Displacement (cm)

Pressure Wave Derivation

+ 0.0005 0 − 0.0005

x (cm)

(a)

Pressure (atm)

Figure 16–3 shows a graphical representation of a sound wave in air in terms of (a) displacement and (b) pressure. Note that the displacement wave is a quarter wavelength, or 90° (p2 rad), out of phase with the pressure wave: where the pressure is a maximum or minimum, the displacement from equilibrium is zero; and where the pressure variation is zero, the displacement is a maximum or minimum.

1.0002

x (cm)

1.0000 0.9998 (b)

FIGURE 16–3 Representation of a sound wave in space at a given instant in terms of (a) displacement, and (b) pressure. FIGURE 16–4 Longitudinal wave in a fluid moves to the right. A thin layer of fluid, in a thin cylinder of area S and thickness ¢x, changes in volume as a result of pressure variation as the wave passes. At the moment shown, the pressure will increase as the wave moves to the right, so the thickness of our layer will decrease, by an amount ¢D. ∆x ∆D S

where, using v = 1Br (Eq. 15–4), and k = vv = 2pfv (Eq. 15–12), then ¢PM = BAk = rv2 Ak = 2prv Af. (16–5)

16–3

Intensity of Sound: Decibels

Loudness is a sensation in the consciousness of a human being and is related to a physically measurable quantity, the intensity of the wave. Intensity is defined as the energy transported by a wave per unit time across a unit area perpendicular to the energy flow. As we saw in Chapter 15, intensity is proportional to the square of the wave amplitude. Intensity has units of power per unit area, or wattsmeter 2 AWm2 B. The human ear can detect sounds with an intensity as low as 10 –12 Wm2 and as high as 1 Wm2 (and even higher, although above this it is painful). This is an incredibly wide range of intensity, spanning a factor of 1012 from lowest to highest. Presumably because of this wide range, what we perceive as loudness is not directly proportional to the intensity. To produce a sound that sounds about twice as loud requires a sound wave that has about 10 times the intensity. This is roughly valid at any sound level for frequencies near the middle of the audible range. For example, a sound wave of intensity 10–2 Wm2 sounds to an average human being like it is about twice as loud as one whose intensity is 10 –3 Wm2, and four times as loud as 10 –4 Wm2. SECTION 16–3

P H Y S I C S

A P P L I E D

Wide range of human hearing

Intensity of Sound: Decibels

427

Sound Level Because of this relationship between the subjective sensation of loudness and the physically measurable quantity "intensity," sound intensity levels are usually specified on a logarithmic scale. The unit on this scale is a bel, after the inventor Alexander Graham Bell, or much more commonly, the decibel (dB), which is 1 10 bel (10 dB = 1 bel). The sound level, b, of any sound is defined in terms of its intensity, I, as b (in dB) = 10 log

I, I0

(16–6)

where I0 is the intensity of a chosen reference level, and the logarithm is to the base 10. I0 is usually taken as the minimum intensity audible to a good ear—the "threshold of hearing," which is I0 = 1.0 * 10–12 Wm2. Thus, for example, the sound level of a sound whose intensity I = 1.0 * 10–10 Wm2 will be b = 10 log ¢ C A U T I O N

0 dB does not mean zero intensity

TABLE 16–2 Intensity of Various Sounds Source of the Sound

Sound Level Intensity (dB) (W m2)

Jet plane at 30 m 140 Threshold of pain 120 Loud rock concert 120 Siren at 30 m 100 Truck traffic 90 Busy street traffic 80 Noisy restaurant 70 Talk, at 50 cm 65 Quiet radio 40 Whisper 30 Rustle of leaves 10 Threshold of hearing 0

1 1 1 1 3 1 1 1 1

* * * * * * * * *

100 1 1 10 –2 10 –3 10 –4 10 –5 10 –6 10 –8 10 –9 10 –11 10–12

1.0 * 10–10 Wm2 ≤ = 10 log 100 = 20 dB, 1.0 * 10–12 Wm2

since log 100 is equal to 2.0. (Appendix A has a brief review of logarithms.) Notice that the sound level at the threshold of hearing is 0 dB. That is, b = 10 log 10 –1210 –12 = 10 log 1 = 0 since log 1 = 0. Notice too that an increase in intensity by a factor of 10 corresponds to a sound level increase of 10 dB. An increase in intensity by a factor of 100 corresponds to a sound level increase of 20 dB. Thus a 50-dB sound is 100 times more intense than a 30-dB sound, and so on. Intensities and sound levels for a number of common sounds are listed in Table 16–2. EXAMPLE 16–3 Sound intensity on the street. At a busy street corner, the sound level is 75 dB. What is the intensity of sound there? APPROACH We have to solve Eq. 16–6 for intensity I, remembering that I0 = 1.0 * 10–12 Wm2. SOLUTION From Eq. 16–6 log

b I , = I0 10

so I = 10b10. I0 With b = 75, then I = I0 10b10 = A1.0 * 10 –12 Wm2 BA107.5 B = 3.2 * 10–5 Wm2. NOTE Recall that x = log y is the same as y = 10x (Appendix A).

P H Y S I C S

A P P L I E D

Loudspeaker response (&3 dB)

EXAMPLE 16–4 Loudspeaker response. A high-quality loudspeaker is advertised to reproduce, at full volume, frequencies from 30 Hz to 18,000 Hz with uniform sound level &3 dB. That is, over this frequency range, the sound level output does not vary by more than 3 dB for a given input level. By what factor does the intensity change for the maximum change of 3 dB in output sound level? APPROACH Let us call the average intensity I1 and the average sound level b1 . Then the maximum intensity, I2 , corresponds to a level b2 = b1 + 3 dB. We then use the relation between intensity and sound level, Eq. 16–6.

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SOLUTION Equation 16–6 gives b2 - b1 = 10 log

I2 I1 - 10 log I0 I0

3 dB = 10 ¢ log = 10 log

I2 I1 - log ≤ I0 I0

I2 I1

because (log a - log b) = log ab (see Appendix A). This last equation gives log

I2 = 0.30, I1

or I2 = 100.30 = 2.0. I1 So & 3 dB corresponds to a doubling or halving of the intensity. It is worth noting that a sound-level difference of 3 dB (which corresponds to a doubled intensity, as we just saw) corresponds to only a very small change in the subjective sensation of apparent loudness. Indeed, the average human can distinguish a difference in sound level of only about 1 or 2 dB. EXERCISE B If an increase of 3 dB means "twice as intense," what does an increase of 6 dB mean?

CONCEPTUAL EXAMPLE 16–5 Trumpet players. A trumpeter plays at a sound level of 75 dB. Three equally loud trumpet players join in. What is the new sound level? RESPONSE The intensity of four trumpets is four times the intensity of one trumpet A = I1 B or 4I1 . The sound level of the four trumpets would be b = 10 log

4I1 I1 = 10 log 4 + 10 log I0 I0 =

6.0 dB + 75 dB = 81 dB.

EXERCISE C From Table 16–2, we see that ordinary conversation corresponds to a sound level of about 65 dB. If two people are talking at once, the sound level is (a) 65 dB, (b) 68 dB, (c) 75 dB, (d) 130 dB, (e) 62 dB.

Normally, the loudness or intensity of a sound decreases as you get farther from the source of the sound. In interior rooms, this effect is altered because of reflections from the walls. However, if a source is in the open so that sound can radiate out freely in all directions, the intensity decreases as the inverse square of the distance, I r

1, r2

as we saw in Section 15–3. Over large distances, the intensity decreases faster than 1r2 because some of the energy is transferred into irregular motion of air molecules. This loss happens more for higher frequencies, so any sound of mixed frequencies will be less "bright" at a distance. SECTION 16–3

Intensity of Sound: Decibels

429

P H Y S I C S

A P P L I E D

Jet plane noise

EXAMPLE 16–6 Airplane roar. The sound level measured 30 m from a jet plane is 140 dB. What is the sound level at 300 m? (Ignore reflections from the ground.) APPROACH Given the sound level, we can determine the intensity at 30 m using Eq. 16–6. Because intensity decreases as the square of the distance, ignoring reflections, we can find I at 300 m and again apply Eq. 16–6 to obtain the sound level. SOLUTION The intensity I at 30 m is I 140 dB = 10 log ¢ –12 ≤ 10 Wm2 or I 14 = log ¢ –12 ≤. 10 Wm2 We raise both sides of this equation to the power 10 (recall 10log x = x) and have I , 1014 = –12 10 Wm2 so I = A1014 BA10–12 Wm2 B = 102 Wm2. At 300 m, 10 times as far, the intensity will 2 be A 101 B = 1100 as much, or 1 Wm2. Hence, the sound level is

FIGURE 16–5 Example 16–6. Airport worker with soundintensity-reducing ear covers (headphones).

b = 10 log ¢

1 Wm2 ≤ = 120 dB. 10–12 Wm2

Even at 300 m, the sound is at the threshold of pain. This is why workers at airports wear ear covers to protect their ears from damage (Fig. 16–5). NOTE Here is a simpler approach that avoids Eq. 16–6. Because the intensity decreases as the square of the distance, at 10 times the distance the intensity 2 1 . decreases by A 101 B = 100 We can use the result that 10 dB corresponds to an intensity change by a factor of 10 (see text just before Example 16–3). Then an intensity change by a factor of 100 corresponds to a sound-level change of (2)(10 dB) = 20 dB. This confirms our result above: 140 dB - 20 dB = 120 dB.

Intensity Related to Amplitude The intensity I of a wave is proportional to the square of the wave amplitude, as we saw in Chapter 15. We can therefore relate the amplitude quantitatively to the intensity I or level b, as the following Example shows. P H Y S I C S

A P P L I E D

Incredible sensitivity of the ear

EXAMPLE 16–7 How tiny the displacement is. (a) Calculate the displacement of air molecules for a sound having a frequency of 1000 Hz at the threshold of hearing. (b) Determine the maximum pressure variation in such a sound wave. APPROACH In Section 15–3 we found a relation between intensity I and displacement amplitude A of a wave, Eq. 15–7. The amplitude of oscillation of air molecules is what we want to solve for, given the intensity. The pressure is found from Eq. 16–5. SOLUTION (a) At the threshold of hearing, I = 1.0 * 10–12 Wm2 (Table 16–2). We solve for the amplitude A in Eq. 15–7: A =

1 I pf B 2rv

1.0 * 10 –12 Wm2 1 3 –1 B (3.14)A1.0 * 10 s B (2)A1.29 kgm3 B(343 ms) = 1.1 * 10 –11 m,

where we have taken the density of air to be 1.29 kgm3 and the speed of sound in air (assumed 20°C) as 343 ms. NOTE We see how incredibly sensitive the human ear is: it can detect displacements of air molecules which are actually less than the diameter of atoms (about 10 –10 m). (b) Now we are dealing with sound as a pressure wave (Section 16–2). From Eq. 16–5, ¢PM = 2prvAf = 2pA1.29 kgm3 BA343 msBA1.1 * 10 –11 mBA1.0 * 103 s–1 B = 3.1 * 10 –5 Pa

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or 3.1 * 10 –10 atm. Again we see that the human ear is incredibly sensitive.

By combining Eqs. 15–7 and 16–5, we can write the intensity in terms of the pressure amplitude, ¢PM : ¢PM 2 I = 2p2vrf2A2 = 2p2vrf2 a b 2prvf 2 A¢PM B . I = (16–7) 2vr The intensity, when given in terms of pressure amplitude, thus does not depend on frequency.

The Ear's Response The ear is not equally sensitive to all frequencies. To hear the same loudness for sounds of different frequencies requires different intensities. Studies averaged over large numbers of people have produced the curves shown in Fig. 16–6. On this graph, each curve represents sounds that seemed to be equally loud. The number labeling each curve represents the loudness level (the units are called phons), which is numerically equal to the sound level in dB at 1000 Hz. For example, the curve labeled 40 represents sounds that are heard by an average person to have the same loudness as a 1000-Hz sound with a sound level of 40 dB. From this 40-phon curve, we see that a 100-Hz tone must be at a level of about 62 dB to be perceived as loud as a 1000-Hz tone of only 40 dB. The lowest curve in Fig. 16–6 (labeled 0) represents the sound level, as a function of frequency, for the threshold of hearing, the softest sound that is just audible by a very good ear. Note that the ear is most sensitive to sounds of frequency between 2000 and 4000 Hz, which are common in speech and music. Note too that whereas a 1000-Hz sound is audible at a level of 0 dB, a 100-Hz sound must be nearly 40 dB to be heard. The top curve in Fig. 16–6, labeled 120 phons, represents the threshold of pain. Sounds above this level can actually be felt and cause pain. Figure 16–6 shows that at lower sound levels, our ears are less sensitive to the high and low frequencies relative to middle frequencies. The "loudness" control on some stereo systems is intended to compensate for this low-volume insensitivity. As the volume is turned down, the loudness control boosts the high and low frequencies relative to the middle frequencies so that the sound will have a more "normal-sounding" frequency balance. Many listeners, however, find the sound more pleasing or natural without the loudness control.

16–4

Sources of Sound: Vibrating Strings and Air Columns

The source of any sound is a vibrating object. Almost any object can vibrate and hence be a source of sound. We now discuss some simple sources of sound, particularly musical instruments. In musical instruments, the source is set into vibration by striking, plucking, bowing, or blowing. Standing waves are produced and the source vibrates at its natural resonant frequencies. The vibrating source is in contact with the air (or other medium) and pushes on it to produce sound waves that travel outward. The frequencies of the waves are the same as those of the source, but the speed and wavelengths can be different. A drum has a stretched membrane that vibrates. Xylophones and marimbas have metal or wood bars that can be set into vibration. Bells, cymbals, and gongs also make use of a vibrating metal. Many instruments make use of vibrating strings, such as the violin, guitar, and piano, or make use of vibrating columns of air, such as the flute, trumpet, and pipe organ. We have already seen that the pitch of a pure sound is determined by the frequency. Typical frequencies for musical notes on the "equally tempered chromatic scale" are given in Table 16–3 for the octave beginning with middle C. Note that one octave corresponds to a doubling of frequency. For example, middle C has frequency of 262 Hz whereas C¿ (C above middle C) has twice that frequency, 524 Hz. [Middle C is the C or "do" note at the middle of a piano keyboard.] SECTION 16–4

Sound level (dB)

Intensity (W/m2)

Threshold of pain

120

120 phons

100

10 −2

10 − 4

40 20

re sh old

0 20

50 100

10 −6

40 of h

ear

10 −8

20 in g

10 −10

10 −12

500 1000

5000 10,000

Frequency (Hz)

FIGURE 16–6 Sensitivity of the human ear as a function of frequency (see text). Note that the frequency scale is "logarithmic" in order to cover a wide range of frequencies.

TABLE 16–3 Equally Tempered Chromatic Scale† Note C C or D D D or E E F F or G G G or A A A or B B C¿ †

Frequency (Hz) 262 277 294 311 330 349 370 392 415 440 466 494 524

Only one octave is included.

Sources of Sound: Vibrating Strings and Air Columns

431

1 l = 2l1

Fundamental or first harmonic, f1 FIGURE 16–7 Standing waves on a string—only the lowest three frequencies are shown.

l = l2 First overtone or second harmonic, f2 = 2 f1 l = 3l3 2 Second overtone or third harmonic, f3 = 3f1

Stringed Instruments P H Y S I C S

A P P L I E D

Stringed instruments

(a)

(b) FIGURE 16–8 The wavelength of (a) an unfingered string is longer than that of (b) a fingered string. Hence, the frequency of the fingered string is higher. Only one string is shown on this guitar, and only the simplest standing wave, the fundamental, is shown.

We saw in Chapter 15, Fig. 15–26b, how standing waves are established on a string, and we show this again here in Fig. 16–7. Such standing waves are the basis for all stringed instruments. The pitch is normally determined by the lowest resonant frequency, the fundamental, which corresponds to nodes occurring only at the ends. The string vibrating up and down as a whole corresponds to a half wavelength as shown at the top of Fig. 16–7; so the wavelength of the fundamental on the string is equal to twice the length of the string. Therefore, the fundamental frequency is f1 = vl = v2l, where v is the velocity of the wave on the string (not in the air). The possible frequencies for standing waves on a stretched string are whole-number multiples of the fundamental frequency: v fn = nf1 = n , n = 1, 2, 3, p 2l where n = 1 refers to the fundamental and n = 2, 3, p are the overtones. All of the standing waves, n = 1, 2, 3, p , are called harmonics,† as we saw in Section 15–9. When a finger is placed on the string of a guitar or violin, the effective length of the string is shortened. So its fundamental frequency, and pitch, is higher since the wavelength of the fundamental is shorter (Fig. 16–8). The strings on a guitar or violin are all the same length. They sound at a different pitch because the strings have different mass per unit length, m, which affects the velocity on the string, Eq. 15–2, v = 3FTm . [stretched string] Thus the velocity on a heavier string is lower and the frequency will be lower for the same wavelength. The tension FT may also be different. Adjusting the tension is the means for tuning the pitch of each string. In pianos and harps the strings are of different lengths. For the lower notes the strings are not only longer, but heavier as well, and the reason is illustrated in the following Example. EXAMPLE 16–8 Piano strings. The highest key on a piano corresponds to a frequency about 150 times that of the lowest key. If the string for the highest note is 5.0 cm long, how long would the string for the lowest note have to be if it had the same mass per unit length and was under the same tension? APPROACH Since v = 3FTm , the velocity would be the same on each string. So the frequency is inversely proportional to the length l of the string (f = vl = v2l). SOLUTION We can write, for the fundamental frequencies of each string, the ratio fH lL , = lH fL where the subscripts L and H refer to the lowest and highest notes, respectively. Thus lL = lH AfHfL B = (5.0 cm)(150) = 750 cm, or 7.5 m. This would be ridiculously long (L 25 ft) for a piano. NOTE The longer strings of lower frequency are made heavier, of higher mass per unit length, so even on grand pianos the strings are less than 3 m long. †

432

CHAPTER 16

Sound

When the resonant frequencies above the fundamental (that is, the overtones) are integral multiples of the fundamental, as here, they are called harmonics. But if the overtones are not integral multiples of the fundamental, as is the case for a vibrating drumhead, for example, they are not harmonics.

EXERCISE D Two strings have the same length and tension, but one is more massive than the other. Which plays the higher note?

EXAMPLE 16–9 Frequencies and wavelengths in the violin. A 0.32-m-long violin string is tuned to play A above middle C at 440 Hz. (a) What is the wavelength of the fundamental string vibration, and (b) what are the frequency and wavelength of the sound wave produced? (c) Why is there a difference?

C A U T I O N

Speed of standing wave on string Z speed of sound wave in air

APPROACH The wavelength of the fundamental string vibration equals twice the length of the string (Fig. 16–7). As the string vibrates, it pushes on the air, which is thus forced to oscillate at the same frequency as the string. SOLUTION (a) From Fig. 16–7 the wavelength of the fundamental is l = 2l = 2(0.32 m) = 0.64 m = 64 cm. This is the wavelength of the standing wave on the string. (b) The sound wave that travels outward in the air (to reach our ears) has the same frequency, 440 Hz. Its wavelength is l =

343 ms v = = 0.78 m = 78 cm, f 440 Hz

(a)

where v is the speed of sound in air (assumed at 20°C), Section 16–1. (c) The wavelength of the sound wave is different from that of the standing wave on the string because the speed of sound in air (343 ms at 20°C) is different from the speed of the wave on the string ( fl = 440 Hz * 0.64 m = 280 ms) which depends on the tension in the string and its mass per unit length. NOTE The frequencies on the string and in the air are the same: the string and air are in contact, and the string "forces" the air to vibrate at the same frequency. But the wavelengths are different because the wave speed on the string is different than that in air. Stringed instruments would not be very loud if they relied on their vibrating strings to produce the sound waves since the strings are too thin to compress and expand much air. Stringed instruments therefore make use of a kind of mechanical amplifier known as a sounding board (piano) or sounding box (guitar, violin), which acts to amplify the sound by putting a greater surface area in contact with the air (Fig. 16–9). When the strings are set into vibration, the sounding board or box is set into vibration as well. Since it has much greater area in contact with the air, it can produce a more intense sound wave. On an electric guitar, the sounding box is not so important since the vibrations of the strings are amplified electronically.

(b) FIGURE 16–9 (a) Piano, showing sounding board to which the strings are attached; (b) sounding box (guitar).

Wind Instruments Instruments such as woodwinds, the brasses, and the pipe organ produce sound from the vibrations of standing waves in a column of air within a tube (Fig. 16–10). Standing waves can occur in the air of any cavity, but the frequencies present are complicated for any but very simple shapes such as the uniform, narrow tube of a flute or an organ pipe. In some instruments, a vibrating reed or the vibrating lip of the player helps to set up vibrations of the air column. In others, a stream of air is directed against one edge of the opening or mouthpiece, leading to turbulence which sets up the vibrations. Because of the disturbance, whatever its source, the air within the tube vibrates with a variety of frequencies, but only frequencies that correspond to standing waves will persist. For a string fixed at both ends, Fig. 16–7, we saw that the standing waves have nodes (no movement) at the two ends, and one or more antinodes (large amplitude of vibration) in between. A node separates successive antinodes. The lowestfrequency standing wave, the fundamental, corresponds to a single antinode. The higher-frequency standing waves are called overtones or harmonics, as we saw in Section 15–9. Specifically, the first harmonic is the fundamental, the second harmonic ( first overtone) has twice the frequency of the fundamental, and so on. SECTION 16–4

FIGURE 16–10 Wind instruments: flute (left) and clarinet.

Sources of Sound: Vibrating Strings and Air Columns

433

TUBE OPEN AT BOTH ENDS (a) Displacement of air l A

node

antinode

1 2

v f1 = 2l

[motion of air molecules]

Second harmonic l = l2

f2 = vl = 2 f1

Third harmonic l = 32 l 3

3v f3 = 2l = 3 f1

B FIGURE 16–11 Graphs of the three simplest modes of vibration (standing waves) for a uniform tube open at both ends ("open tube").These simplest modes of vibration are graphed in (a), on the left, in terms of the motion of the air (displacement), and in (b), on the right, in terms of air pressure. Each graph shows the wave format at two times,A and B, a half period apart.The actual motion of molecules for one case, the fundamental, is shown just below the tube at top left.

P H Y S I C S

A P P L I E D

Wind instruments

434

First harmonic = fundamental

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Sound

(b) Pressure variation in the air l node

antinode

node

A B

A B Overtones A B

The situation is similar for a column of air in a tube of uniform diameter, but we must remember that it is now air itself that is vibrating. We can describe the waves either in terms of the flow of the air—that is, in terms of the displacement of air—or in terms of the pressure in the air (see Figs. 16–2 and 16–3). In terms of displacement, the air at the closed end of a tube is a displacement node since the air is not free to move there, whereas near the open end of a tube there will be an antinode because the air can move freely in and out. The air within the tube vibrates in the form of longitudinal standing waves. The possible modes of vibration for a tube open at both ends (called an open tube) are shown graphically in Fig. 16–11. They are shown for a tube that is open at one end but closed at the other (called a closed tube) in Fig. 16–12. [A tube closed at both ends, having no connection to the outside air, would be useless as an instrument.] The graphs in part (a) of each Figure (left-hand sides) represent the displacement amplitude of the vibrating air in the tube. Note that these are graphs, and that the air molecules themselves oscillate horizontally, parallel to the tube length, as shown by the small arrows in the top diagram of Fig. 16–11a (on the left). The exact position of the antinode near the open end of a tube depends on the diameter of the tube, but if the diameter is small compared to the length, which is the usual case, the antinode occurs very close to the end as shown. We assume this is the case in what follows. (The position of the antinode may also depend slightly on the wavelength and other factors.) Let us look in detail at the open tube, in Fig. 16–11a, which might be an organ pipe or a flute. An open tube has displacement antinodes at both ends since the air is free to move at open ends. There must be at least one node within an open tube if there is to be a standing wave at all. A single node corresponds to the fundamental frequency of the tube. Since the distance between two successive nodes, or between two successive antinodes, is 12 l, there is one-half of a wavelength within the length of the tube for the simplest case of the fundamental (top diagram in Fig. 16–11a): l = 12 l, or l = 2l. So the fundamental frequency is f1 = vl = v2l, where v is the velocity of sound in air (the air in the tube). The standing wave with two nodes is the first overtone or second harmonic and has half the wavelength (l = l) and twice the frequency of the fundamental. Indeed, in a uniform tube open at both ends, the frequency of each overtone is an integral multiple of the fundamental frequency, as shown in Fig. 16–11a. This is just what is found for a string. For a closed tube, shown in Fig. 16–12a, which could be an organ pipe, there is always a displacement node at the closed end (because the air is not free to move) and an antinode at the open end (where the air can move freely). Since the distance between a node and the nearest antinode is 14 l, we see that the fundamental in a closed tube corresponds to only one-fourth of a wavelength within the length of the tube: l = l4, and l = 4l. The fundamental frequency is thus f1 = v4l, or half that for an open pipe of the same length. There is another

(a) Displacement of air l A

TUBE CLOSED AT ONE END

(b) Pressure variation in the air

First harmonic = fundamental l=

1 4

v f1 = 4l

Third harmonic l = 34 l 3

f3 = 34lv = 3 f1

Fifth harmonic l = 54 l 5

A B

A B Overtones

f5 = 54lv = 5 f1

difference, for as we can see from Fig. 16–12a, only the odd harmonics are present in a closed tube: the overtones have frequencies equal to 3, 5, 7, p times the fundamental frequency. There is no way for waves with 2, 4, 6, p times the fundamental frequency to have a node at one end and an antinode at the other, and thus they cannot exist as standing waves in a closed tube. Another way to analyze the vibrations in a uniform tube is to consider a description in terms of the pressure in the air, shown in part (b) of Figs. 16–11 and 16–12 (right-hand sides). Where the air in a wave is compressed, the pressure is higher, whereas in a wave expansion (or rarefaction), the pressure is less than normal. The open end of a tube is open to the atmosphere. Hence the pressure variation at an open end must be a node: the pressure does not alternate, but remains at the outside atmospheric pressure. If a tube has a closed end, the pressure at that closed end can readily alternate to be above or below atmospheric pressure. Hence there is a pressure antinode at a closed end of a tube. There can be pressure nodes and antinodes within the tube. Some of the possible vibrational modes in terms of pressure for an open tube are shown in Fig. 16–11b, and for a closed tube are shown in Fig. 16–12b.

A B FIGURE 16–12 Modes of vibration (standing waves) for a tube closed at one end ("closed tube"). See caption for Fig. 16–11.

EXAMPLE 16–10 Organ pipes. What will be the fundamental frequency and first three overtones for a 26-cm-long organ pipe at 20°C if it is (a) open and (b) closed? APPROACH All our calculations can be based on Figs. 16–11a and 16–12a. SOLUTION (a) For the open pipe, Fig. 16–11a, the fundamental frequency is f1 =

343 ms v = = 660 Hz. 2l 2(0.26 m)

The speed v is the speed of sound in air (the air vibrating in the pipe). The overtones include all harmonics: 1320 Hz, 1980 Hz, 2640 Hz, and so on. (b) For a closed pipe, Fig. 16–12a, the fundamental frequency is 343 ms v f1 = = = 330 Hz. 4l 4(0.26 m) Only odd harmonics are present: the first three overtones are 990 Hz, 1650 Hz, and 2310 Hz. NOTE The closed pipe plays 330 Hz, which, from Table 16–3, is E above middle C, whereas the open pipe of the same length plays 660 Hz, an octave higher. Pipe organs use both open and closed pipes, with lengths from a few centimeters to 5 m or more. A flute acts as an open tube, for it is open not only where you blow into it, but also at the opposite end. The different notes on a flute are obtained by shortening the length of the vibrating air column, by uncovering holes along the tube (so a displacement antinode can occur at the hole). The shorter the length of the vibrating air column, the higher the fundamental frequency.

SECTION 16–4

435

EXAMPLE 16–11 Flute. A flute is designed to play middle C (262 Hz) as the fundamental frequency when all the holes are covered. Approximately how long should the distance be from the mouthpiece to the far end of the flute? (This is only approximate since the antinode does not occur precisely at the mouthpiece.) Assume the temperature is 20°C. APPROACH When all holes are covered, the length of the vibrating air column is the full length. The speed of sound in air at 20°C is 343 ms. Because a flute is open at both ends, we use Fig. 16–11: the fundamental frequency f1 is related to the length l of the vibrating air column by f = v2l. SOLUTION Solving for l, we find 343 ms v l = = = 0.655 m L 0.66 m. 2f 2A262 s–1 B

EXERCISE E To see why players of wind instruments "warm up" their instruments (so they will be in tune), determine the fundamental frequency of the flute of Example 16–11 when all holes are covered and the temperature is 10°C instead of 20°C.

EXERCISE F Return to the Chapter-Opening Question, page 424, and answer it again now. Try to explain why you may have answered differently the first time.

* 16–5 Sum of all three

FIGURE 16–13 The displacements of the fundamental and first two overtones are added at each point to get the "sum," or composite waveform.

Relative amplitude

FIGURE 16–14 Sound spectra for different instruments. The spectra change when the instruments play different notes. The clarinet is a bit complicated: it acts like a closed tube at lower frequencies, having only odd harmonics, but at higher frequencies all harmonics occur as for an open tube.

436

1.0

Clarinet

0.5 0

1000 2000 3000 Frequency (Hz)

1.0

Piano

0.5 0

1000 2000 3000 Frequency (Hz)

1.0

Violin

0.5 0

1000 2000 3000 Frequency (Hz)

CHAPTER 16

Sound

Quality of Sound, and Noise; Superposition

Whenever we hear a sound, particularly a musical sound, we are aware of its loudness, its pitch, and also of a third aspect called its timbre or "quality." For example, when a piano and then a flute play a note of the same loudness and pitch (say, middle C), there is a clear difference in the overall sound. We would never mistake a piano for a flute. This is what is meant by the timbre or quality of a sound. For musical instruments, the term tone color is also used. Just as loudness and pitch can be related to physically measurable quantities, so too can quality. The quality of a sound depends on the presence of overtones—their number and their relative amplitudes. Generally, when a note is played on a musical instrument, the fundamental as well as overtones are present simultaneously. Figure 16–13 illustrates how the principle of superposition (Section 15–6) applies to three wave forms, in this case the fundamental and first two overtones (with particular amplitudes): they add together at each point to give a composite waveform (by which we mean the displacement as a function of position at a given time). Normally, more than two overtones are present. [Any complex wave can be analyzed into a superposition of sinusoidal waves of appropriate amplitudes, wavelengths, and frequencies—see Section 15–6. Such an analysis is called a Fourier analysis.] The relative amplitudes of the overtones for a given note are different for different musical instruments, which is what gives each instrument its characteristic quality or timbre. A bar graph showing the relative amplitudes of the harmonics for a given note produced by an instrument is called a sound spectrum. Several typical examples for different musical instruments are shown in Fig. 16–14. The fundamental usually has the greatest amplitude, and its frequency is what is heard as the pitch. The manner in which an instrument is played strongly influences the sound quality. Plucking a violin string, for example, makes a very different sound than pulling a bow across it. The sound spectrum at the very start (or end) of a note (as when a hammer strikes a piano string) can be very different from the subsequent sustained tone. This too affects the subjective tone quality of an instrument. An ordinary sound, like that made by striking two stones together, is a noise that has a certain quality, but a clear pitch is not discernible. Such a noise is a mixture of many frequencies which bear little relation to one another. A sound spectrum made of that noise would not show discrete lines like those of Fig. 16–14. Instead it would show a continuous, or nearly continuous, spectrum of frequencies. Such a sound we call "noise" in comparison with the more harmonious sounds which contain frequencies that are simple multiples of the fundamental.

16–6

Interference of Sound Waves; Beats

Interference in Space We saw in Section 15–8 that when two waves simultaneously pass through the same region of space, they interfere with one another. Interference also occurs with sound waves. Consider two large loudspeakers, A and B, a distance d apart on the stage of an auditorium as shown in Fig. 16–15. Let us assume the two speakers are emitting sound waves of the same single frequency and that they are in phase: that is, when one speaker is forming a compression, so is the other. (We ignore reflections from walls, floor, etc.) The curved lines in the diagram represent the crests of sound waves from each speaker at one instant in time.We must remember that for a sound wave, a crest is a compression in the air whereas a trough—which falls between two crests—is a rarefaction. A human ear or detector at a point such as C, which is the same distance from each speaker, will experience a loud sound because the interference will be constructive—two crests reach it at one moment, two troughs reach it a moment later. On the other hand, at a point such as D in the diagram, little if any sound will be heard because destructive interference occurs—compressions of one wave meet rarefactions of the other and vice versa (see Fig. 15–24 and the related discussion on water waves in Section 15–8). An analysis of this situation is perhaps clearer if we graphically represent the waveforms as in Fig. 16–16. In Fig. 16–16a, it can be seen that at point C, constructive interference occurs since both waves simultaneously have crests or simultaneously have troughs when they arrive at C. In Fig. 16–16b we see that, to reach point D, the wave from speaker B must travel a greater distance than the wave from A. Thus the wave from B lags behind that from A. In this diagram, point E is chosen so that the distance ED is equal to AD. Thus we see that if the distance BE is equal to precisely one-half the wavelength of the sound, the two waves will be exactly out of phase when they reach D, and destructive interference occurs. This then is the criterion for determining at what points destructive interference occurs: destructive interference occurs at any point whose distance from one speaker is one-half wavelength greater than its distance from the other speaker. Notice that if this extra distance (BE in Fig. 16–16b) is equal to a whole wavelength (or 2, 3, p wavelengths), then the two waves will be in phase and constructive interference occurs. If the distance BE equals 12 , 1 12 , 2 12 , p wavelengths, destructive interference occurs. It is important to realize that a person at point D in Fig. 16–15 or 16–16 hears nothing at all (or nearly so), yet sound is coming from both speakers. Indeed, if one of the speakers is turned off, the sound from the other speaker will be clearly heard. If a loudspeaker emits a whole range of frequencies, only specific wavelengths will destructively interfere completely at a given point.

FIGURE 16–15 Sound waves from two loudspeakers interfere. FIGURE 16–16 Sound waves of a single frequency from loudspeakers A and B (see Fig. 16–15) constructively interfere at C and destructively interfere at D. [Shown here are graphical representations, not the actual longitudinal sound waves.] A

EXAMPLE 16–12 Loudspeakers' interference. Two loudspeakers are 1.00 m apart. A person stands 4.00 m from one speaker. How far must this person be from the second speaker to detect destructive interference when the speakers emit an 1150-Hz sound? Assume the temperature is 20°C. APPROACH To sense destructive interference, the person must be one-half wavelength closer to or farther from one speaker than from the other—that is, at a distance = 4.00 m6l2. We can determine l since we know f and v. SOLUTION The speed of sound at 20°C is 343 ms, so the wavelength of this sound is (Eq. 15–1) 343 ms v l = = = 0.30 m. f 1150 Hz For destructive interference to occur, the person must be one-half wavelength farther from one loudspeaker than from the other, or 0.15 m. Thus the person must be 3.85 m or 4.15 m from the second speaker. NOTE If the speakers are less than 0.15 m apart, there will be no point that is 0.15 m farther from one speaker than the other, and there will be no point where destructive interference could occur.

(a)

B A

D d E B

(b)

SECTION 16–6

437

fA = 50 Hz fB = 60 Hz t

t=0

sum

t = 0.05 s

t = 0.10 s

FIGURE 16–17 Beats occur as a result of the superposition of two sound waves of slightly different frequency.

t = 0.15 s

beat period (0.10 s)

Beats—Interference in Time We have been discussing interference of sound waves that takes place in space. An interesting and important example of interference that occurs in time is the phenomenon known as beats: If two sources of sound—say, two tuning forks—are close in frequency but not exactly the same, sound waves from the two sources interfere with each other. The sound level at a given position alternately rises and falls in time, because the two waves are sometimes in phase and sometimes out of phase due to their different wavelengths. The regularly spaced intensity changes are called beats. To see how beats arise, consider two equal-amplitude sound waves of frequency fA = 50 Hz and fB = 60 Hz, respectively. In 1.00 s, the first source makes 50 vibrations whereas the second makes 60. We now examine the waves at one point in space equidistant from the two sources. The waveforms for each wave as a function of time, at a fixed position, are shown on the top graph of Fig. 16–17; the red line represents the 50-Hz wave, and the blue line represents the 60-Hz wave. The lower graph in Fig. 16–17 shows the sum of the two waves as a function of time. At time t = 0 the two waves are shown to be in phase and interfere constructively. Because the two waves vibrate at different rates, at time t = 0.05 s they are completely out of phase and interfere destructively. At t = 0.10 s, they are again in phase and the resultant amplitude again is large. Thus the resultant amplitude is large every 0.10 s and drops drastically in between. This rising and falling of the intensity is what is heard as beats.† In this case the beats are 0.10 s apart. That is, the beat frequency is ten per second, or 10 Hz. This result, that the beat frequency equals the difference in frequency of the two waves, is valid in general, as we now show. Let the two waves, of frequencies f1 and f2 , be represented at a fixed point in space by D1 = A sin 2pf1t and D2 = A sin 2pf2t. The resultant displacement, by the principle of superposition, is D = D1 + D2 = A Asin 2pf1t + sin 2pf2t B.

Using the trigonometric identity sin u1 + sin u2 = 2 sin 12 Au1 + u2 B cos 12 Au1 - u2 B , we have f1 - f2 f1 + f2 D = c2A cos 2p ¢ ≤ t d sin 2p ¢ ≤ t. (16–8) 2 2 We can interpret Eq. 16–8 as follows. The superposition of the two waves results in a wave that vibrates at the average frequency of the two components, Af1 + f2 B2. This vibration has an amplitude given by the expression in brackets, and this amplitude varies in time, from zero to a maximum of 2A (the sum of the separate amplitudes), with a frequency of Af1 - f2 B2. A beat occurs whenever cos 2pC Af1 - f2 B2D t equals ±1 or – 1 (see Fig. 16–17); that is, two beats occur per cycle, so the beat frequency is twice Af1 - f2 B2 which is just f1 - f2 , the difference in frequency of the component waves.

438

CHAPTER 16

Sound

†

Beats will be heard even if the amplitudes are not equal, as long as the difference in amplitude is not great.

The phenomenon of beats can occur with any kind of wave and is a very sensitive method for comparing frequencies. For example, to tune a piano, a piano tuner listens for beats produced between his standard tuning fork and that of a particular string on the piano, and knows it is in tune when the beats disappear. The members of an orchestra tune up by listening for beats between their instruments and that of a standard tone (usually A above middle C at 440 Hz) produced by a piano or an oboe. A beat frequency is perceived as an intensity modulation (a wavering between loud and soft) for beat frequencies below 20 Hz or so, and as a separate low tone for higher beat frequencies (audible if the tones are strong enough).

P H Y S I C S

A P P L I E D

Tuning a piano

EXAMPLE 16–13 Beats. A tuning fork produces a steady 400-Hz tone. When this tuning fork is struck and held near a vibrating guitar string, twenty beats are counted in five seconds. What are the possible frequencies produced by the guitar string? APPROACH For beats to occur, the string must vibrate at a frequency different from 400 Hz by whatever the beat frequency is. SOLUTION The beat frequency is fbeat = 20 vibrations5 s = 4 Hz. This is the difference of the frequencies of the two waves. Because one wave is known to be 400 Hz, the other must be either 404 Hz or 396 Hz.

16–7

Doppler Effect

You may have noticed that you hear the pitch of the siren on a speeding fire truck drop abruptly as it passes you. Or you may have noticed the change in pitch of a blaring horn on a fast-moving car as it passes by you. The pitch of the engine noise of a racecar changes as the car passes an observer. When a source of sound is moving toward an observer, the pitch the observer hears is higher than when the source is at rest; and when the source is traveling away from the observer, the pitch is lower. This phenomenon is known as the Doppler effect† and occurs for all types of waves. Let us now see why it occurs, and calculate the difference between the perceived and source frequencies when there is relative motion between source and observer. Consider the siren of a fire truck at rest, which is emitting sound of a particular frequency in all directions as shown in Fig. 16–18a. The sound waves are moving at the speed of sound in air, vsnd , which is independent of the velocity of the source or observer. If our source, the fire truck, is moving, the siren emits sound at the same frequency as it does at rest. But the sound wavefronts it emits forward, in front of it, are closer together than when the fire truck is at rest, as shown in Fig. 16–18b. This is because the fire truck, as it moves, is "chasing" the previously emitted wavefronts, and emits each crest closer to the previous one. Thus an observer on the sidewalk in front of the truck will detect more wave crests passing per second, so the frequency heard is higher. The wavefronts emitted behind the truck, on the other hand, are farther apart than when the truck is at rest because the truck is speeding away from them. Hence, fewer wave crests per second pass by an observer behind the moving truck (Fig. 16–18b) and the perceived pitch is lower.

(a) At rest

FIGURE 16–18 (a) Both observers on the sidewalk hear the same frequency from a fire truck at rest. (b) Doppler effect: observer toward whom the fire truck moves hears a higher-frequency sound, and observer behind the fire truck hears a lower-frequency sound.

(b) Fire truck moving †

After J. C. Doppler (1803–1853).

SECTION 16–7

Doppler Effect

439

Source

We can calculate the frequency shift perceived by making use of Fig. 16–19, and we assume the air (or other medium) is at rest in our reference frame. (The stationary observer is off to the right.) In Fig. 16–19a, the source of the sound is shown as a red dot, and is at rest. Two successive wave crests are shown, the second of which has just been emitted and so is still near the source. The distance between these crests is l, the wavelength. If the frequency of the source is f, then the time between emissions of wave crests is 1 l . T = = vsnd f

d=l

(a) Source fixed

In Fig. 16–19b, the source is moving with a velocity vsource toward the observer. In a time T (as just defined), the first wave crest has moved a distance d = vsnd T = l, where vsnd is the velocity of the sound wave in air (which is the same whether the source is moving or not). In this same time, the source has moved a distance dsource = vsource T. Then the distance between successive wave crests, which is the wavelength l¿ the observer will perceive, is

Crest emitted when source was at point 1. Crest emitted when source was at point 2. 1

vBsource

l′

dsource = vsourceT (b) Source moving FIGURE 16–19 Determination of the frequency shift in the Doppler effect (see text). The red dot is the source.

l¿ = d - dsource = l - vsource T l = l - vsource vsnd vsource = l¢1 ≤. vsnd We subtract l from both sides of this equation and find that the shift in wavelength, ¢l, is vsource . ¢l = l¿ - l = –l vsnd So the shift in wavelength is directly proportional to the source speed vsource . The frequency f¿ that will be perceived by our stationary observer on the ground is given by vsnd vsnd . f¿ = = vsource l¿ l¢1 ≤ vsnd Since vsndl = f, then f¿ =

f . vsource ¢1 ≤ vsnd

source moving toward d (16–9a) stationary observer

Because the denominator is less than 1, the observed frequency f¿ is greater than the source frequency f. That is, f¿ 7 f. For example, if a source emits a sound of frequency 400 Hz when at rest, then when the source moves toward a fixed observer with a speed of 30 ms, the observer hears a frequency (at 20°C) of 400 Hz = 438 Hz. f¿ = 30 ms 1 343 ms Now consider a source moving away from the stationary observer at a speed vsource . Using the same arguments as above, the wavelength l¿ perceived by our observer will have the minus sign on dsource (second equation on this page) changed to plus: l¿ = d + dsource vsource = l¢1 + ≤. vsnd The difference between the observed and emitted wavelengths will be ¢l = l¿ - l = ±lAvsourcevsnd B. The observed frequency of the wave, f¿ = vsndl¿, will be f source moving away from . f¿ = c d (16–9b) vsource stationary observer ¢1 + ≤ vsnd

440

CHAPTER 16

Sound

If a source emitting at 400 Hz is moving away from a fixed observer at 30 ms, the observer hears a frequency f¿ = (400 Hz)C1 + (30 ms)(343 ms)D = 368 Hz.

FIGURE 16–20 Observer moving with speed vobs toward a stationary source detects wave crests passing at speed v¿ = vsnd + vobs where vsnd is the speed of the sound waves in air.

Observer

Source l

vsource = 0

vBobs

vBsnd

The Doppler effect also occurs when the source is at rest and the observer is in motion. If the observer is traveling toward the source, the pitch heard is higher than that of the emitted source frequency. If the observer is traveling away from the source, the pitch heard is lower. Quantitatively the change in frequency is different than for the case of a moving source. With a fixed source and a moving observer, the distance between wave crests, the wavelength l, is not changed. But the velocity of the crests with respect to the observer is changed. If the observer is moving toward the source, Fig. 16–20, the speed v¿ of the waves relative to the observer is a simple addition of velocities: v¿ = vsnd + vobs , where vsnd is the velocity of sound in air (we assume the air is still) and vobs is the velocity of the observer. Hence, the frequency heard is f¿ =

vsnd + vobs v¿ . = l l

Because l = vsndf, then f¿ =

Avsnd + vobs Bf vsnd

or f¿ = ¢ 1 +

vobs ≤ f. vsnd

observer moving toward d (16–10a) stationary source

If the observer is moving away from the source, the relative velocity is v¿ = vsnd - vobs , so f¿ = ¢ 1 -

vobs ≤ f. vsnd

observer moving away d (16–10b) from stationary source

EXAMPLE 16–14 A moving siren. The siren of a police car at rest emits at a predominant frequency of 1600 Hz. What frequency will you hear if you are at rest and the police car moves at 25.0 ms (a) toward you, and (b) away from you? APPROACH The observer is fixed, and the source moves, so we use Eqs. 16–9. The frequency you (the observer) hear is the emitted frequency f divided by the factor A16vsourcevsnd B where vsource is the speed of the police car. Use the minus sign when the car moves toward you (giving a higher frequency); use the plus sign when the car moves away from you (lower frequency). SOLUTION (a) The car is moving toward you, so (Eq. 16–9a) f¿ =

f 1600 Hz = = 1726 Hz L 1730 Hz. vsource 25.0 ms ¢1 ≤ ¢1 ≤ vsnd 343 ms

(b) The car is moving away from you, so (Eq. 16–9b) f 1600 Hz f¿ = = = 1491 Hz L 1490 Hz. vsource 25.0 ms ¢1 + ≤ ¢1 + ≤ vsnd 343 ms EXERCISE G Suppose the police car of Example 16–14 is at rest and emits at 1600 Hz. What frequency would you hear if you were moving at 25.0 ms (a) toward it, and (b) away from it?

SECTION 16–7

Doppler Effect

441

When a sound wave is reflected from a moving obstacle, the frequency of the reflected wave will, because of the Doppler effect, be different from that of the incident wave. This is illustrated in the following Example.

Object

Original

Wave

source

velocity

vBobs = 3.50 m/s (a)

Source and detector

Object Wave velocity

vBsource = 3.50 m/s

EXAMPLE 16–15 Two Doppler shifts. A 5000-Hz sound wave is emitted by a stationary source. This sound wave reflects from an object moving 3.50 ms toward the source (Fig. 16–21). What is the frequency of the wave reflected by the moving object as detected by a detector at rest near the source? APPROACH There are actually two Doppler shifts in this situation. First, the moving object acts like an observer moving toward the source with speed vobs = 3.50 ms (Fig. 16–21a) and so "detects" a sound wave of frequency (Eq. 16–10a) f¿ = fC1 + Avobsvsnd B D. Second, reflection of the wave from the moving object is equivalent to the object reemitting the wave, acting effectively as a moving source with speed vsource = 3.50 ms (Fig. 16–21b). The final frequency detected, f–, is given by f– = f¿C 1 - vsourcevsnd D, Eq. 16–9a. SOLUTION The frequency f¿ that is "detected" by the moving object is (Eq. 16–10a): f¿ = ¢ 1 +

(b) FIGURE 16–21 Example 16–15.

vobs 3.50 ms ≤f = ¢1 + ≤ (5000 Hz) = 5051 Hz. vsnd 343 ms

The moving object now "emits" (reflects) a sound of frequency (Eq. 16–9a) f– =

f¿ 5051 Hz = = 5103 Hz. vsource 3.50 ms ¢1 ≤ ¢1 ≤ vsnd 343 ms

Thus the frequency shifts by 103 Hz. NOTE Bats use this technique to be aware of their surroundings. This is also the principle behind Doppler radar as speed-measuring devices for vehicles and other objects.

P H Y S I C S

A P P L I E D

Doppler blood-flow meter and other medical uses

The incident wave and the reflected wave in Example 16–15, when mixed together (say, electronically), interfere with one another and beats are produced. The beat frequency is equal to the difference in the two frequencies, 103 Hz. This Doppler technique is used in a variety of medical applications, usually with ultrasonic waves in the megahertz frequency range. For example, ultrasonic waves reflected from red blood cells can be used to determine the velocity of blood flow. Similarly, the technique can be used to detect the movement of the chest of a young fetus and to monitor its heartbeat. For convenience, we can write Eqs. 16–9 and 16–10 as a single equation that covers all cases of both source and observer in motion: f¿ = f ¢

P R O B L E M

S O LV I N G

Getting the signs right

vsnd6vobs ≤. vsnd7vsource

source and d (16–11) observer moving

To get the signs right, recall from your own experience that the frequency is higher when observer and source approach each other, and lower when they move apart. Thus the upper signs in numerator and denominator apply if source and/or observer move toward each other; the lower signs apply if they are moving apart. EXERCISE H How fast would a source have to approach an observer for the observed frequency to be one octave above (twice) the produced frequency? (a) 12 vsnd , (b) vsnd , (c) 2vsnd , (d) 4vsnd .

442

CHAPTER 16

Sound

Doppler Effect for Light The Doppler effect occurs for other types of waves as well. Light and other types of electromagnetic waves (such as radar) exhibit the Doppler effect: although the formulas for the frequency shift are not identical to Eqs. 16–9 and 16–10, as discussed in Chapter 44, the effect is similar. One important application is for weather forecasting using radar. The time delay between the emission of radar pulses and their reception after being reflected off raindrops gives the position of precipitation. Measuring the Doppler shift in frequency (as in Example 16–15) tells how fast the storm is moving and in which direction. Another important application is to astronomy, where the velocities of distant galaxies can be determined from the Doppler shift. Light from distant galaxies is shifted toward lower frequencies, indicating that the galaxies are moving away from us. This is called the redshift since red has the lowest frequency of visible light. The greater the frequency shift, the greater the velocity of recession. It is found that the farther the galaxies are from us, the faster they move away. This observation is the basis for the idea that the universe is expanding, and is one basis for the idea that the universe began as a great explosion, affectionately called the "Big Bang" (Chapter 44).

* 16–8

P H Y S I C S

A P P L I E D

Doppler effect for EM waves and weather forecasting

P H Y S I C S

A P P L I E D

Redshift in cosmology

Shock Waves and the Sonic Boom

An object such as an airplane traveling faster than the speed of sound is said to have a supersonic speed. Such a speed is often given as a Mach† number, which is defined as the ratio of the speed of the object to the speed of sound in the surrounding medium. For example, a plane traveling 600 ms high in the atmosphere, where the speed of sound is only 300 ms, has a speed of Mach 2. Sh

ock

wa ve

ock

(a) vobj = 0

(b) vobj < v snd

ve wa

Sh (d) vobj > vsnd

FIGURE 16–22 Sound waves emitted by an object (a) at rest or (b, c, and d) moving. (b) If the object's velocity is less than the velocity of sound, the Doppler effect occurs; (d) if its velocity is greater than the velocity of sound, a shock wave is produced.

When a source of sound moves at subsonic speeds (less than the speed of sound), the pitch of the sound is altered as we have seen (the Doppler effect); see also Fig. 16–22a and b. But if a source of sound moves faster than the speed of sound, a more dramatic effect known as a shock wave occurs. In this case, the source is actually "outrunning" the waves it produces. As shown in Fig. 16–22c, when the source is traveling at the speed of sound, the wave fronts it emits in the forward direction "pile up" directly in front of it. When the object moves faster, at a supersonic speed, the wave fronts pile up on one another along the sides, as shown in Fig. 16–22d. The different wave crests overlap one another and form a single very large crest which is the shock wave. Behind this very large crest there is usually a very large trough. A shock wave is essentially the result of constructive interference of a large number of wave fronts. A shock wave in air is analogous to the bow wave of a boat traveling faster than the speed of the water waves it produces, Fig. 16–23. †

FIGURE 16–23 Bow waves produced by a boat.

After the Austrian physicist Ernst Mach (1838–1916).

*SECTION 16–8

Shock Waves and the Sonic Boom

443

l Tai

k hoc

ve wa

ck sho t n Fro

ve wa

(a)

(b)

FIGURE 16–24 (a) The (double) sonic boom has already been heard by person A on the left. The front shock wave is just being heard by person B in the center. And it will shortly be heard by person C on the right. (b) Special photo of supersonic aircraft showing shock waves produced in the air. (Several closely spaced shock waves are produced by different parts of the aircraft.)

P H Y S I C S

A P P L I E D

Sonic boom

When an airplane travels at supersonic speeds, the noise it makes and its disturbance of the air form into a shock wave containing a tremendous amount of sound energy. When the shock wave passes a listener, it is heard as a loud sonic boom. A sonic boom lasts only a fraction of a second, but the energy it contains is often sufficient to break windows and cause other damage. Actually, a sonic boom is made up of two or more booms since major shock waves can form at the front and the rear of the aircraft, as well as at the wings, etc. (Fig. 16–24). Bow waves of a boat are also multiple, as can be seen in Fig. 16–23. When an aircraft approaches the speed of sound, it encounters a barrier of sound waves in front of it (see Fig. 16–22c). To exceed the speed of sound, the aircraft needs extra thrust to pass through this "sound barrier." This is called "breaking the sound barrier." Once a supersonic speed is attained, this barrier no longer impedes the motion. It is sometimes erroneously thought that a sonic boom is produced only at the moment an aircraft is breaking through the sound barrier. Actually, a shock wave follows the aircraft at all times it is traveling at supersonic speeds. A series of observers on the ground will each hear a loud "boom" as the shock wave passes, Fig. 16–24. The shock wave consists of a cone whose apex is at the aircraft. The angle of this cone, u (see Fig. 16–22d), is given by sin u =

vsnd , vobj

(16–12)

where vobj is the velocity of the object (the aircraft) and vsnd is the velocity of sound in the medium. (The proof is left as Problem 75.)

* 16–9

Applications: Sonar, Ultrasound, and Medical Imaging

* Sonar P H Y S I C S

A P P L I E D

Sonar: depth finding, Earth soundings

The reflection of sound is used in many applications to determine distance. The sonar† or pulse-echo technique is used to locate underwater objects. A transmitter sends out a sound pulse through the water, and a detector receives its reflection, or echo, a short time later. This time interval is carefully measured, and from it the distance to the reflecting object can be determined since the speed of sound in water is known. The depth of the sea and the location of reefs, sunken ships, submarines, or schools of fish can be determined in this way. The interior structure of the Earth is studied in a similar way by detecting reflections of waves traveling through the Earth whose source was a deliberate explosion (called "soundings"). An analysis of waves reflected from various structures and boundaries within the Earth reveals characteristic patterns that are also useful in the exploration for oil and minerals. †

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Sonar stands for "sound navigation ranging."

Sonar generally makes use of ultrasonic frequencies: that is, waves whose frequencies are above 20 kHz, beyond the range of human detection. For sonar, the frequencies are typically in the range 20 kHz to 100 kHz. One reason for using ultrasound waves, other than the fact that they are inaudible, is that for shorter wavelengths there is less diffraction (Section 15–11) so the beam spreads less and smaller objects can be detected.

* Ultrasound Medical Imaging The diagnostic use of ultrasound in medicine, in the form of images (sometimes called sonograms), is an important and interesting application of physical principles. A pulse-echo technique is used, much like sonar, except that the frequencies used are in the range of 1 to 10 MHz A1 MHz = 106 HzB. A high-frequency sound pulse is directed into the body, and its reflections from boundaries or interfaces between organs and other structures and lesions in the body are then detected. Tumors and other abnormal growths, or pockets of fluid, can be distinguished; the action of heart valves and the development of a fetus can be examined; and information about various organs of the body, such as the brain, heart, liver, and kidneys, can be obtained. Although ultrasound does not replace X-rays, for certain kinds of diagnosis it is more helpful. Some kinds of tissue or fluid are not detected in X-ray photographs, but ultrasound waves are reflected from their boundaries. "Real-time" ultrasound images are like a movie of a section of the interior of the body. The pulse-echo technique for medical imaging works as follows. A brief pulse of ultrasound is emitted by a transducer that transforms an electrical pulse into a sound-wave pulse. Part of the pulse is reflected as echoes at each interface in the body, and most of the pulse (usually) continues on, Fig. 16–25a. The detection of reflected pulses by the same transducer can then be displayed on the screen of a display terminal or monitor. The time elapsed from when the pulse is emitted to when each reflection (echo) is received is proportional to the distance to the reflecting surface. For example, if the distance from transducer to the vertebra is 25 cm, the pulse travels a round-trip distance of 2 * 25 cm = 0.50 m. The speed of sound in human tissue is about 1540 ms (close to that of sea water), so the time taken is

t =

P H Y S I C S

A P P L I E D

Ultrasound medical imaging

(0.50 m) d = = 320 ms. v (1540 ms)

The strength of a reflected pulse depends mainly on the difference in density of the two materials on either side of the interface and can be displayed as a pulse or as a dot (Figs. 16–25b and c). Each echo dot (Fig. 16–25c) can be represented as a point whose position is given by the time delay and whose brightness depends on the strength of the echo.

Transducer

Abdominal wall (a)

Vertebra Pulse Echoes

Echoes

Strength of reflected pulse

Organs inside the body

(b)

FIGURE 16–25 (a) Ultrasound pulse passes through the abdomen, reflecting from surfaces in its path. (b) Reflected pulses plotted as a function of time when received by transducer. The vertical dashed lines point out which reflected pulse goes with which surface. (c) Dot display for the same echoes: brightness of each dot is related to signal strength.

Time

(c)

*SECTION 16–9

Applications: Sonar, Ultrasound, and Medical Imaging

445

FIGURE 16–26 (a) Ten traces are made across the abdomen by moving the transducer, or by using an array of transducers. (b) The echoes are plotted as dots to produce the image. More closely spaced traces would give a more detailed image.

FIGURE 16–27 Ultrasound image of a human fetus within the uterus.

(a)

Body organs

(b)

A two-dimensional image can then be formed out of these dots from a series of scans. The transducer is moved, or an array of transducers is used, each of which sends out a pulse at each position and receives echoes as shown in Fig. 16–26a. Each trace can be plotted, spaced appropriately one below the other, to form an image on a monitor screen as shown in Fig. 16–26b. Only 10 lines are shown in Fig. 16–26, so the image is crude. More lines give a more precise image.† An ultrasound image is shown in Fig. 16–27. †

Radar used for aircraft involves a similar pulse-echo technique except that it uses electromagnetic (EM) waves, which, like light, travel with a speed of 3 * 108 ms.

Summary Sound travels as a longitudinal wave in air and other materials. In air, the speed of sound increases with temperature; at 20°C, it is about 343 ms. The pitch of a sound is determined by the frequency; the higher the frequency, the higher the pitch. The audible range of frequencies for humans is roughly 20 Hz to 20,000 Hz (1 Hz = 1 cycle per second). The loudness or intensity of a sound is related to the amplitude squared of the wave. Because the human ear can detect sound intensities from 10 –12 Wm2 to over 1 Wm2, sound levels are specified on a logarithmic scale. The sound level b, specified in decibels, is defined in terms of intensity I as b (in dB) = 10 log ¢

I ≤, I0

(16–6)

where the reference intensity I0 is usually taken to be 10 –12 Wm2. Musical instruments are simple sources of sound in which standing waves are produced. The strings of a stringed instrument may vibrate as a whole with nodes only at the ends; the frequency at which this standing wave occurs is called the fundamental. The fundamental frequency corresponds to a wavelength equal to twice the length of the string, l1 = 2l. The string can also vibrate at higher frequencies, called overtones or harmonics, in which there are

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one or more additional nodes. The frequency of each harmonic is a whole-number multiple of the fundamental. In wind instruments, standing waves are set up in the column of air within the tube. The vibrating air in an open tube (open at both ends) has displacement antinodes at both ends. The fundamental frequency corresponds to a wavelength equal to twice the tube length: l1 = 2l. The harmonics have frequencies that are 1, 2, 3, 4, p times the fundamental frequency, just as for strings. For a closed tube (closed at one end), the fundamental corresponds to a wavelength four times the length of the tube: l1 = 4l. Only the odd harmonics are present, equal to 1, 3, 5, 7, p times the fundamental frequency. Sound waves from different sources can interfere with each other. If two sounds are at slightly different frequencies, beats can be heard at a frequency equal to the difference in frequency of the two sources. The Doppler effect refers to the change in pitch of a sound due to the motion either of the source or of the listener. If source and listener are approaching each other, the perceived pitch is higher; if they are moving apart, the perceived pitch is lower. [*Shock waves and a sonic boom occur when an object moves at a supersonic speed—faster than the speed of sound. Ultrasonic-frequency (higher than 20 kHz) sound waves are used in many applications, including sonar and medical imaging.]

Questions 1. What is the evidence that sound travels as a wave? 2. What is the evidence that sound is a form of energy? 3. Children sometimes play with a homemade "telephone" by attaching a string to the bottoms of two paper cups. When the string is stretched and a child speaks into one cup, the sound can be heard at the other cup (Fig. 16–28). Explain clearly how the sound wave travels from one cup to the other.

16. Consider the two waves shown in Fig. 16–30. Each wave can be thought of as a superposition of two sound waves with slightly different frequencies, as in Fig. 16–17. In which of the waves, (a) or (b), are the two component frequencies farther apart? Explain. 2 1 0

FIGURE 16–28 Question 3. 4. When a sound wave passes from air into water, do you expect the frequency or wavelength to change? 5. What evidence can you give that the speed of sound in air does not depend significantly on frequency? 6. The voice of a person who has inhaled helium sounds very high-pitched. Why? 7. What is the main reason the speed of sound in hydrogen is greater than the speed of sound in air? 8. Two tuning forks oscillate with the same amplitude, but one has twice the frequency. Which (if either) produces the more intense sound? 9. How will the air temperature in a room affect the pitch of organ pipes? 10. Explain how a tube might be used as a filter to reduce the amplitude of sounds in various frequency ranges. (An example is a car muffler.) 11. Why are the frets on a guitar (Fig. 16–29) spaced closer together as you move up Bridge the fingerboard toward Frets the bridge?

FIGURE 16–29 Question 11. 12. A noisy truck approaches you from behind a building. Initially you hear it but cannot see it. When it emerges and you do see it, its sound is suddenly "brighter"—you hear more of the high-frequency noise. Explain. [Hint: See Section 15–11 on diffraction.] 13. Standing waves can be said to be due to "interference in space," whereas beats can be said to be due to "interference in time." Explain. 14. In Fig. 16–15, if the frequency of the speakers is lowered, would the points D and C (where destructive and constructive interference occur) move farther apart or closer together? 15. Traditional methods of protecting the hearing of people who work in areas with very high noise levels have consisted mainly of efforts to block or reduce noise levels. With a relatively new technology, headphones are worn that do not block the ambient noise. Instead, a device is used which detects the noise, inverts it electronically, then feeds it to the headphones in addition to the ambient noise. How could adding more noise reduce the sound levels reaching the ears?

0.05

0.1

0.15

0.2

0.15

0.2

t (s)

1 2 (a) 2 1 0

0.05

0.1

t (s)

1 2 (b) FIGURE 16–30 Question 16. 17. Is there a Doppler shift if the source and observer move in the same direction, with the same velocity? Explain. 18. If a wind is blowing, will this alter the frequency of the sound heard by a person at rest with respect to the source? Is the wavelength or velocity changed? 19. Figure 16–31 shows various positions of a child on a swing moving toward a person on the ground who is blowing a whistle. At which position, A through E, will the child hear the highest frequency Whistle for the sound of the whistle? Explain your reasoning.

Whistle A FIGURE 16–31 Question 19.

20. Approximately how many octaves are there in the human audible range? 21. At a race track, you can estimate the speed of cars just by listening to the difference in pitch of the engine noise between approaching and receding cars. Suppose the sound of a certain car drops by a full octave (frequency halved) as it goes by on the straightaway. How fast is it going?

Questions

447

Problems [Unless stated otherwise, assume T = 20°C and vsound = 343 ms in air.]

16–1 Characteristics of Sound 1. (I) A hiker determines the length of a lake by listening for the echo of her shout reflected by a cliff at the far end of the lake. She hears the echo 2.0 s after shouting. Estimate the length of the lake. 2. (I) A sailor strikes the side of his ship just below the waterline. He hears the echo of the sound reflected from the ocean floor directly below 2.5 s later. How deep is the ocean at this point? Assume the speed of sound in sea water is 1560 ms (Table 16–1) and does not vary significantly with depth. 3. (I) (a) Calculate the wavelengths in air at 20°C for sounds in the maximum range of human hearing, 20 Hz to 20,000 Hz. (b) What is the wavelength of a 15-MHz ultrasonic wave? 4. (I) On a warm summer day (27°C), it takes 4.70 s for an echo to return from a cliff across a lake. On a winter day, it takes 5.20 s. What is the temperature on the winter day? 5. (II) A motion sensor can accurately measure the distance d to an object repeatedly via the sonar technique used in Example 16–2. A short ultrasonic pulse is emitted and reflects from any objects it encounters, creating echo pulses upon their arrival back at the sensor. The sensor measures the time interval t between the emission of the original pulse and the arrival of the first echo. (a) The smallest time interval t that can be measured with high precision is 1.0 ms. What is the smallest distance (at 20°C) that can be measured with the motion sensor? (b) If the motion sensor makes 15 distance measurements every second (that is, it emits 15 sound pulses per second at evenly spaced time intervals), the measurement of t must be completed within the time interval between the emissions of successive pulses. What is the largest distance (at 20°C) that can be measured with the motion sensor? (c) Assume that during a lab period the room's temperature increases from 20°C to 23°C. What percent error will this introduce into the motion sensor's distance measurements? 6. (II) An ocean fishing boat is drifting just above a school of tuna on a foggy day. Without warning, an engine backfire occurs on another boat 1.35 km away (Fig. 16–32). How much time elapses before the backfire is heard (a) by the fish, and (b) by the (b) fishermen? 1.35 km FIGURE 16–32 Problem 6.

(a)

7. (II) A stone is dropped from the top of a cliff. The splash it makes when striking the water below is heard 3.0 s later. How high is the cliff? 8. (II) A person, with his ear to the ground, sees a huge stone strike the concrete pavement. A moment later two sounds are heard from the impact: one travels in the air and the other in the concrete, and they are 0.75 s apart. How far away did the impact occur? See Table 16–1. 9. (II) Calculate the percent error made over one mile of distance by the "5-second rule" for estimating the distance from a lightning strike if the temperature is (a) 30°C, and (b) 10°C.

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16–2 Mathematical Representation of Waves 10. (I) The pressure amplitude of a sound wave in air Ar = 1.29 kgm3 B at 0°C is 3.0 * 10 –3 Pa. What is the displacement amplitude if the frequency is (a) 150 Hz and (b) 15 kHz? 11. (I) What must be the pressure amplitude in a sound wave in air (0°C) if the air molecules undergo a maximum displacement equal to the diameter of an oxygen molecule, about 3.0 * 10–10 m? Assume a sound-wave frequency of (a) 55 Hz and (b) 5.5 kHz. 12. (II) Write an expression that describes the pressure variation as a function of x and t for the waves described in Problem 11. 13. (II) The pressure variation in a sound wave is given by ¢P = 0.0035 sin(0.38px - 1350pt), where ¢P is in pascals, x in meters, and t in seconds. Determine (a) the wavelength, (b) the frequency, (c) the speed, and (d) the displacement amplitude of the wave. Assume the density of the medium to be r = 2.3 * 103 kgm3.

16–3 Intensity of Sound; Decibels 14. (I) What is the intensity of a sound at the pain level of 120 dB? Compare it to that of a whisper at 20 dB. 15. (I) What is the sound level of a sound whose intensity is 2.0 * 10 –6 Wm2? 16. (I) What are the lowest and highest frequencies that an ear can detect when the sound level is 40 dB? (See Fig. 16–6.) 17. (II) Your auditory system can accommodate a huge range of sound levels. What is the ratio of highest to lowest intensity at (a) 100 Hz, (b) 5000 Hz? (See Fig. 16–6.) 18. (II) You are trying to decide between two new stereo amplifiers. One is rated at 100 W per channel and the other is rated at 150 W per channel. In terms of dB, how much louder will the more powerful amplifier be when both are producing sound at their maximum levels? 19. (II) At a painfully loud concert, a 120-dB sound wave travels away from a loudspeaker at 343 ms. How much sound wave energy is contained in each 1.0-cm3 volume of air in the region near this loudspeaker? 20. (II) If two firecrackers produce a sound level of 95 dB when fired simultaneously at a certain place, what will be the sound level if only one is exploded? 21. (II) A person standing a certain distance from an airplane with four equally noisy jet engines is experiencing a sound level of 130 dB. What sound level would this person experience if the captain shut down all but one engine? 22. (II) A cassette player is said to have a signal-to-noise ratio of 62 dB, whereas for a CD player it is 98 dB. What is the ratio of intensities of the signal and the background noise for each device? 23. (II) (a) Estimate the power output of sound from a person speaking in normal conversation. Use Table 16–2. Assume the sound spreads roughly uniformly over a sphere centered on the mouth. (b) How many people would it take to produce a total sound output of 75 W of ordinary conversation? [Hint: Add intensities, not dBs.] 24. (II) A 50-dB sound wave strikes an eardrum whose area is 5.0 * 10–5 m2. (a) How much energy is received by the eardrum per second? (b) At this rate, how long would it take your eardrum to receive a total energy of 1.0 J?

25. (II) Expensive amplifier A is rated at 250 W, while the more modest amplifier B is rated at 45 W. (a) Estimate the sound level in decibels you would expect at a point 3.5 m from a loudspeaker connected in turn to each amp. (b) Will the expensive amp sound twice as loud as the cheaper one? 26. (II) At a rock concert, a dB meter registered 130 dB when placed 2.2 m in front of a loudspeaker on the stage. (a) What was the power output of the speaker, assuming uniform spherical spreading of the sound and neglecting absorption in the air? (b) How far away would the sound level be a somewhat reasonable 85 dB? 27. (II) A fireworks shell explodes 100 m above the ground, creating a colorful display of sparks. How much greater is the sound level of the explosion for a person standing at a point directly below the explosion than for a person a horizontal distance of 200 m 100 m away (Fig. 16–33)? FIGURE 16–33 Problem 27.

200 m

28. (II) If the amplitude of a sound wave is made 2.5 times greater, (a) by what factor will the intensity increase? (b) By how many dB will the sound level increase? 29. (II) Two sound waves have equal displacement amplitudes, but one has 2.6 times the frequency of the other. (a) Which has the greater pressure amplitude and by what factor is it greater? (b) What is the ratio of their intensities? 30. (II) What would be the sound level (in dB) of a sound wave in air that corresponds to a displacement amplitude of vibrating air molecules of 0.13 mm at 380 Hz? 31. (II) (a) Calculate the maximum displacement of air molecules when a 330-Hz sound wave passes whose intensity is at the threshold of pain (120 dB). (b) What is the pressure amplitude in this wave? 32. (II) A jet plane emits 5.0 * 105 J of sound energy per second. (a) What is the sound level 25 m away? Air absorbs sound at a rate of about 7.0 dBkm; calculate what the sound level will be (b) 1.00 km and (c) 7.50 km away from this jet plane, taking into account air absorption.

16–4 Sources of Sound: Strings and Air Columns 33. (I) What would you estimate for the length of a bass clarinet, assuming that it is modeled as a closed tube and that the lowest note that it can play is a D whose frequency is 69.3 Hz? 34. (I) The A string on a violin has a fundamental frequency of 440 Hz. The length of the vibrating portion is 32 cm, and it has a mass of 0.35 g. Under what tension must the string be placed? 35. (I) An organ pipe is 124 cm long. Determine the fundamental and first three audible overtones if the pipe is (a) closed at one end, and (b) open at both ends. 36. (I) (a) What resonant frequency would you expect from blowing across the top of an empty soda bottle that is 21 cm deep, if you assumed it was a closed tube? (b) How would that change if it was one-third full of soda? 37. (I) If you were to build a pipe organ with open-tube pipes spanning the range of human hearing (20 Hz to 20 kHz), what would be the range of the lengths of pipes required?

38. (II) Estimate the frequency of the "sound of the ocean" when you put your ear very near a 20-cm-diameter seashell (Fig. 16–34).

FIGURE 16–34 Problem 38. 39. (II) An unfingered guitar string is 0.73 m long and is tuned to play E above middle C (330 Hz). (a) How far from the end of this string must a fret (and your finger) be placed to play A above middle C (440 Hz)? (b) What is the wavelength on the string of this 440-Hz wave? (c) What are the frequency and wavelength of the sound wave produced in air at 25°C by this fingered string? 40. (II) (a) Determine the length of an open organ pipe that emits middle C (262 Hz) when the temperature is 15°C. (b) What are the wavelength and frequency of the fundamental standing wave in the tube? (c) What are l and f in the traveling sound wave produced in the outside air? 41. (II) An organ is in tune at 22.0°C. By what percent will the frequency be off at 5.0°C? 42. (II) How far from the mouthpiece of the flute in Example 16–11 should the hole be that must be uncovered to play F above middle C at 349 Hz? 43. (II) A bugle is simply a tube of fixed length that behaves as if it is open at both ends. A bugler, by adjusting his lips correctly and blowing with proper air pressure, can cause a harmonic (usually other than the fundamental) of the air column within the tube to sound loudly. Standard military tunes like Taps and Reveille require only four musical notes: G4 (392 Hz), C5 (523 Hz), E5 (659 Hz), and G5 (784 Hz). (a) For a certain length l, a bugle will have a sequence of four consecutive harmonics whose frequencies very nearly equal those associated with the notes G4, C5, E5, and G5. Determine this l. (b) Which harmonic is each of the (approximate) notes G4, C5, E5, and G5 for the bugle? 44. (II) A particular organ pipe can resonate at 264 Hz, 440 Hz, and 616 Hz, but not at any other frequencies in between. (a) Show why this is an open or a closed pipe. (b) What is the fundamental frequency of this pipe? 45. (II) When a player's finger presses a guitar string down onto a fret, the length of the vibrating portion of the string is shortened, thereby increasing the string's fundamental frequency (see Fig. 16–35). The string's tension and mass per unit length remain unchanged. If the unfingered length of the string is l = 65.0 cm, determine the positions x of the first six frets, if each fret raises the pitch of the fundamental by one musical note in comparison to the neighboring fret. On the equally tempered chromatic scale, the ratio of frequencies of neighboring notes is 2 112.

x FIGURE 16–35 Problem 45.

l 65.0 cm

Problems

449

46. (II) A uniform narrow tube 1.80 m long is open at both ends. It resonates at two successive harmonics of frequencies 275 Hz and 330 Hz. What is (a) the fundamental frequency, and (b) the speed of sound in the gas in the tube? 47. (II) A pipe in air at 23.0°C is to be designed to produce two successive harmonics at 240 Hz and 280 Hz. How long must the pipe be, and is it open or closed? 48. (II) How many overtones are present within the audible range for a 2.48-m-long organ pipe at 20°C (a) if it is open, and (b) if it is closed? 49. (II) Determine the fundamental and first overtone frequencies for an 8.0-m-long hallway with all doors closed. Model the hallway as a tube closed at both ends. 50. (II) In a quartz oscillator, used as a stable clock in electronic devices, a transverse (shear) standing sound wave is excited across the thickness d of a quartz disk and its frequency f is detected electronically. The parallel faces of the disk are unsupported and so behave as "free ends" when the sound wave reflects from them (see Fig. 16–36). If the oscillator is designed to operate with the first harmonic, determine the required disk thickness if f = 12.0 MHz. The density and shear modulus of quartz are r = 2650 kgm3 and G = 2.95 * 1010 Nm2. f3

Air Quartz Air

FIGURE 16–36 Problem 50. 51. (III) The human ear canal is approximately 2.5 cm long. It is open to the outside and is closed at the other end by the eardrum. Estimate the frequencies (in the audible range) of the standing waves in the ear canal. What is the relationship of your answer to the information in the graph of Fig. 16–6? * 16–5 Quality of Sound, Superposition * 52. (II) Approximately what are the intensities of the first two overtones of a violin compared to the fundamental? How many decibels softer than the fundamental are the first and second overtones? (See Fig. 16–14.)

16–6 Interference; Beats 53. (I) A piano tuner hears one beat every 2.0 s when trying to adjust two strings, one of which is sounding 370 Hz. How far off in frequency is the other string? 54. (I) What is the beat frequency if middle C (262 Hz) and C (277 Hz) are played together? What if each is played two octaves lower (each frequency reduced by a factor of 4)? 55. (II) A guitar string produces 4 beatss when sounded with a 350-Hz tuning fork and 9 beatss when sounded with a 355-Hz tuning fork. What is the vibrational frequency of the string? Explain your reasoning. 56. (II) The two sources of sound in Fig. 16–15 face each other and emit sounds of equal amplitude and equal frequency (294 Hz) but 180° out of phase. For what minimum separation of the two speakers will there be some point at which (a) complete constructive interference occurs and (b) complete destructive interference occurs. (Assume T = 20°C.)

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Sound

57. (II) How many beats will be heard if two identical flutes, each 0.66 m long, try to play middle C (262 Hz), but one is at 5.0°C and the other at 28°C? 58. (II) Two loudspeakers are placed 3.00 m apart, as shown in Fig. 16–37. They emit 494-Hz sounds, in phase. A microphone is placed 3.20 m distant from a point midway between the two speakers, where an intensity maximum is recorded. (a) How far must the microphone be moved to the right to find the first intensity minimum? (b) Suppose the speakers are reconnected 3.00 m so that the 494-Hz sounds they emit are exactly out of 3.20 m phase. At what positions are S2 S1 the intensity maximum and minimum now? FIGURE 16–37 Problem 58. 59. (II) Two piano strings are supposed to be vibrating at 220 Hz, but a piano tuner hears three beats every 2.0 s when they are played together. (a) If one is vibrating at 220.0 Hz, what must be the frequency of the other (is there only one answer)? (b) By how much (in percent) must the tension be increased or decreased to bring them in tune? 60. (II) A source emits sound of wavelengths 2.64 m and 2.72 m in air. (a) How many beats per second will be heard? (Assume T = 20°C.) (b) How far apart in space are the regions of maximum intensity?

16–7 Doppler Effect 61. (I) The predominant frequency of a certain fire truck's siren is 1350 Hz when at rest. What frequency do you detect if you move with a speed of 30.0 ms (a) toward the fire truck, and (b) away from it? 62. (I) A bat at rest sends out ultrasonic sound waves at 50.0 kHz and receives them returned from an object moving directly away from it at 30.0 ms. What is the received sound frequency? 63. (II) (a) Compare the shift in frequency if a 2300-Hz source is moving toward you at 18 ms, versus you moving toward it at 18 ms. Are the two frequencies exactly the same? Are they close? (b) Repeat the calculation for 160 ms and then again (c) for 320 ms. What can you conclude about the asymmetry of the Doppler formulas? (d) Show that at low speeds (relative to the speed of sound), the two formulas— source approaching and detector approaching—yield the same result. 64. (II) Two automobiles are equipped with the same singlefrequency horn. When one is at rest and the other is moving toward the first at 15 ms, the driver at rest hears a beat frequency of 4.5 Hz. What is the frequency the horns emit? Assume T = 20°C. 65. (II) A police car sounding a siren with a frequency of 1280 Hz is traveling at 120.0 kmh. (a) What frequencies does an observer standing next to the road hear as the car approaches and as it recedes? (b) What frequencies are heard in a car traveling at 90.0 kmh in the opposite direction before and after passing the police car? (c) The police car passes a car traveling in the same direction at 80.0 kmh. What two frequencies are heard in this car?

66. (II) A bat flies toward a wall at a speed of 7.0 ms. As it flies, the bat emits an ultrasonic sound wave with frequency 30.0 kHz. What frequency does the bat hear in the reflected wave? 67. (II) In one of the original Doppler experiments, a tuba was played on a moving flat train car at a frequency of 75 Hz, and a second identical tuba played the same tone while at rest in the railway station. What beat frequency was heard in the station if the train car approached the station at a speed of 12.0 ms? 68. (II) If a speaker mounted on an automobile broadcasts a song, with what speed (kmh) does the automobile have to move toward a stationary listener so that the listener hears the song with each musical note shifted up by one note in comparison to the song heard by the automobile's driver? On the equally tempered chromatic scale, the ratio of frequencies of neighboring notes is 2 112. 69. (II) A wave on the surface of the ocean with wavelength 44 m is traveling east at a speed of 18 ms relative to the ocean floor. If, on this stretch of ocean surface, a powerboat is moving at 15 ms (relative to the ocean floor), how often does the boat encounter a wave crest, if the boat is traveling (a) west, and (b) east? 70. (III) A factory whistle emits sound of frequency 720 Hz. When the wind velocity is 15.0 ms from the north, what frequency will observers hear who are located, at rest, (a) due north, (b) due south, (c) due east, and (d) due west, of the whistle? What frequency is heard by a cyclist heading (e) north or (f) west, toward the whistle at 12.0 ms? Assume T = 20°C. 71. (III) The Doppler effect using ultrasonic waves of frequency 2.25 * 106 Hz is used to monitor the heartbeat of a fetus. A (maximum) beat frequency of 260 Hz is observed. Assuming that the speed of sound in tissue is 1.54 * 103 ms, calculate the maximum velocity of the surface of the beating heart. * 16–8 Shock Waves; Sonic Boom * 72. (II) An airplane travels at Mach 2.0 where the speed of sound is 310 ms. (a) What is the angle the shock wave makes with the direction of the airplane's motion? (b) If the plane is flying at a height of 6500 m, how long after it is directly overhead will a person on the ground hear the shock wave?

* 73. (II) A space probe enters the thin atmosphere of a planet where the speed of sound is only about 45 ms. (a) What is the probe's Mach number if its initial speed is 15,000 kmh? (b) What is the angle of the shock wave relative to the direction of motion? * 74. (II) A meteorite traveling 8800 ms strikes the ocean. Determine the shock wave angle it produces (a) in the air just before entering the ocean, and (b) in the water just after entering. Assume T = 20°C. * 75. (II) Show that the angle u a sonic boom makes with the path of a supersonic object is given by Eq. 16–12. * 76. (II) You look directly overhead and see a plane exactly 1.25 km above the ground flying faster than the speed of sound. By the time you hear the sonic boom, the plane has traveled a horizontal distance of 2.0 km. See Fig. 16–38. Determine (a) the angle of the shock cone, u, and (b) the speed of the plane (the Mach number). Assume the speed of sound is 330 ms. θ FIGURE 16–38 Problem 76. * 77. (II) A supersonic jet traveling at Mach 2.2 at an altitude of 9500 m passes directly over an observer on the ground. Where will the plane be relative to the observer when the latter hears the sonic boom? (See Fig. 16–39.)

θ θ (90° − θ )

9500 m Observer

Sound direction

(a)

Observer (b)

FIGURE 16–39 Problem 77.

General Problems 78. A fish finder uses a sonar device that sends 20,000-Hz sound pulses downward from the bottom of the boat, and then detects echoes. If the maximum depth for which it is designed to work is 75 m, what is the minimum time between pulses (in fresh water)? 79. A science museum has a display called a sewer pipe symphony. It consists of many plastic pipes of various lengths, which are open on both ends. (a) If the pipes have lengths of 3.0 m, 2.5 m, 2.0 m, 1.5 m and 1.0 m, what frequencies will be heard by a visitor's ear placed near the ends of the pipes? (b) Why does this display work better on a noisy day than on a quiet day? 80. A single mosquito 5.0 m from a person makes a sound close to the threshold of human hearing (0 dB). What will be the sound level of 100 such mosquitoes? 81. What is the resultant sound level when an 82-dB sound and an 89-dB sound are heard simultaneously? 82. The sound level 9.00 m from a loudspeaker, placed in the open, is 115 dB. What is the acoustic power output (W) of the speaker, assuming it radiates equally in all directions?

83. A stereo amplifier is rated at 175 W output at 1000 Hz. The power output drops by 12 dB at 15 kHz. What is the power output in watts at 15 kHz? 84. Workers around jet aircraft typically wear protective devices over their ears. Assume that the sound level of a jet airplane engine, at a distance of 30 m, is 130 dB, and that the average human ear has an effective radius of 2.0 cm. What would be the power intercepted by an unprotected ear at a distance of 30 m from a jet airplane engine? 85. In audio and communications systems, the gain, b, in decibels is defined as b = 10 log ¢

Pout ≤, Pin

where Pin is the power input to the system and Pout is the power output. A particular stereo amplifier puts out 125 W of power for an input of 1.0 mW. What is its gain in dB?

General Problems

451

86. For large concerts, loudspeakers are sometimes used to amplify a singer's sound. The human brain interprets sounds that arrive within 50 ms of the original sound as if they came from the same source. Thus if the sound from a loudspeaker reaches a listener first, it would sound as if the loudspeaker is the source of the sound. Conversely, if the singer is heard first and the loudspeaker adds to the sound within 50 ms, the sound would seem to come from the singer, who would now seem to be singing louder. The second situation is desired. Because the signal to the loudspeaker travels at the speed of light A3 * 108 msB, which is much faster than the speed of sound, a delay is added to the signal sent to the loudspeaker. How much delay must be added if the loudspeaker is 3.0 m behind the singer and we want its sound to arrive 30 ms after the singer's? 87. Manufacturers typically offer a particular guitar string in a choice of diameters so that players can tune their instruments with a preferred string tension. For example, a nylon high-E string is available in a low- and high-tension model with diameter 0.699 mm and 0.724 mm, respectively. Assuming the density r of nylon is the same for each model, compare (as a ratio) the tension in a tuned high- and low-tension string. 88. The high-E string on a guitar is fixed at both ends with length l = 65.0 cm and fundamental frequency f1 = 329.6 Hz. On an acoustic guitar, this string typically has a diameter of 0.33 mm and is commonly made of brass A7760 kgm3 B, while on an electric guitar it has a diameter of 0.25 mm and is made of nickel-coated steel A7990 kgm3 B. Compare (as a ratio) the high-E string tension on an acoustic versus an electric guitar. 89. The A string of a violin is 32 cm long between fixed points with a fundamental frequency of 440 Hz and a mass per unit length of 7.2 * 10–4 kgm. (a) What are the wave speed and tension in the string? (b) What is the length of the tube of a simple wind instrument (say, an organ pipe) closed at one end whose fundamental is also 440 Hz if the speed of sound is 343 ms in air? (c) What is the frequency of the first overtone of each instrument? 90. A tuning fork is set into vibration above a vertical open tube filled with water (Fig. 16–40). The water level is allowed to drop slowly. As it does so, the air in the tube above the water level is heard to resonate with the tuning fork when the distance from the tube opening to the water level is 0.125 m and again at 0.395 m. What is the 0.125 m frequency of the tuning fork? 0.395 m FIGURE 16–40 Problem 90. 91. Two identical tubes, each closed at one end, have a fundamental frequency of 349 Hz at 25.0°C. The air temperature is increased to 30.0°C in one tube. If the two pipes are sounded together now, what beat frequency results? 92. Each string on a violin is tuned to a frequency 1 12 times that of its neighbor. The four equal-length strings are to be placed under the same tension; what must be the mass per unit length of each string relative to that of the lowest string?

452

CHAPTER 16

Sound

93. The diameter D of a tube does affect the node at the open end of a tube. The end correction can be roughly approximated as adding D3 to the effective length of the tube. For a closed tube of length 0.60 m and diameter 3.0 cm, what are the first four harmonics, taking the end correction into consideration? 94. A person hears a pure tone in the 500 to 1000-Hz range coming from two sources. The sound is loudest at points equidistant from the two sources. To determine exactly what the frequency is, the person moves about and finds that the sound level is minimal at a point 0.28 m farther from one source than the other. What is the frequency of the sound? 95. The frequency of a steam train whistle as it approaches you is 552 Hz. After it passes you, its frequency is measured as 486 Hz. How fast was the train moving (assume constant velocity)? 96. Two trains emit 516-Hz whistles. One train is stationary. The conductor on the stationary train hears a 3.5-Hz beat frequency when the other train approaches. What is the speed of the moving train? 97. Two loudspeakers are at opposite ends of a railroad car as it moves past a stationary observer at 10.0 ms, as shown in Fig. 16–41. If the speakers have identical sound frequencies of 348 Hz, what is the beat frequency heard by the observer when (a) he listens from the position A, in front of the car, (b) he is between the speakers, at B, and (c) he hears the speakers after they have passed him, at C? v = 10.0 m/s

FIGURE 16–41 Problem 97. 98. Two open organ pipes, sounding together, produce a beat frequency of 8.0 Hz. The shorter one is 2.40 m long. How long is the other? 99. A bat flies toward a moth at speed 7.5 ms while the moth is flying toward the bat at speed 5.0 ms. The bat emits a sound wave of 51.35 kHz. What is the frequency of the wave detected by the bat after that wave reflects off the moth? 100. If the velocity of blood flow in the aorta is normally about 0.32 ms, what beat frequency would you expect if 3.80-MHz ultrasound waves were directed along the flow and reflected from the red blood cells? Assume that the waves travel with a speed of 1.54 * 103 ms. 101. A bat emits a series of high-frequency sound pulses as it approaches a moth. The pulses are approximately 70.0 ms apart, and each is about 3.0 ms long. How far away can the moth be detected by the bat so that the echo from one pulse returns before the next pulse is emitted? 102. (a) Use the binomial expansion to show that Eqs. 16–9a and 16–10a become essentially the same for small relative velocity between source and observer. (b) What percent error would result if Eq. 16–10a were used instead of Eq. 16–9a for a relative velocity of 18.0 ms? 103. Two loudspeakers face each other at opposite ends of a long corridor. They are connected to the same source which produces a pure tone of 282 Hz. A person walks from one speaker toward the other at a speed of 1.4 ms. What "beat" frequency does the person hear?

104. A Doppler flow meter is used to measure the speed of blood flow. Transmitting and receiving elements are placed on the skin, as shown in Fig. 16–42. Typical sound-wave frequencies of about 5.0 MHz are used, which have a reasonable chance of being reflected from red blood cells. By measuring the frequency of the reflected waves, which are Doppler-shifted because the red blood cells are moving, the speed of the blood flow can be deduced. "Normal" blood flow speed is about 0.1 ms. Suppose that an artery is partly constricted, so that the speed of the blood flow is increased, and the flow meter measures a Doppler shift of 780 Hz. What is the speed of blood flow in the constricted region? The effective angle between the sound waves (both transmitted and reflected) and the direction Receiver of blood flow Transmitter is 45°. Assume the velocity of sound in tissue is 1540 ms. FIGURE 16–42 Problem 104.

Red blood cells

105. The wake of a speedboat is 15° in a lake where the speed of the water wave is 2.2 kmh. What is the speed of the boat? 106. A source of sound waves (wavelength l) is a distance l from a detector. Sound reaches the detector directly, and also by reflecting off an obstacle, as shown in Fig. 16–43. The obstacle is equidistant from source and detector. When the obstacle is a distance d to the right of the line of sight between source and detector, as shown, the two waves arrive in phase. Detector How much farther to the right must the obstacle be moved if the two waves are to be out of phase by 12 wavel d Obstacle length, so destructive interference occurs? (Assume l V l, d.) FIGURE 16–43 Problem 106.

Source

107. A dramatic demonstration, called "singing rods," involves a long, slender aluminum rod held in the hand near the rod's midpoint. The rod is stroked with the other hand. With a little practice, the rod can be made to "sing," or emit a clear, loud, ringing sound. For a 75-cm-long rod, (a) what is the fundamental frequency of the sound? (b) What is its wavelength in the rod, and (c) what is the wavelength of the sound in air at 20°C?

108. Assuming that the maximum displacement of the air molecules in a sound wave is about the same as that of the speaker cone that produces the sound (Fig. 16–44), estimate by how Rarefaction Compression much a loudspeaker cone moves for a fairly loud (105 dB) sound of (a) 8.0 kHz, l and (b) 35 Hz. Motion of speaker cone (exaggerated)

FIGURE 16–44 Problem 108. * Numerical/Computer

* 109. (III) The manner in which a string is plucked determines the mixture of harmonic amplitudes in the resulting wave. Consider a string exactly 12-m long that is fixed at both its ends located at x = 0.0 and x = 12 m. The first five harmonics of this string have wavelengths l1 = 1.0 m, l2 = 12 m, l3 = 13 m, l4 = 14 m, and l5 = 15 m. According to Fourier's theorem, any shape of this string can be formed by a sum of its harmonics, with each harmonic having its own unique amplitude A. We limit the sum to the first five harmonics in the expression 2p 2p D(x) = A 1 sin ¢ x≤ + A 2 sin ¢ x≤ l1 l2 + A 3 sin ¢

2p 2p 2p x≤ + A 4 sin ¢ x≤ + A 5 sin ¢ x≤ , l3 l4 l5

and D is the displacement of the string at a time t = 0. Imagine plucking this string at its midpoint (Fig. 16–45a) or at a point two-thirds from the left end (Fig. 16–45b). Using a graphing calculator or computer program, show that the above expression can fairly accurately represent the shape in: (a) Fig. 16–45a, if A 1 = 1.00, A 2 = 0.00, A 3 = –0.11, 1m 1m 4 4 A 4 = 0.00, and (a) A 5 = 0.040; and in (b) Fig. 16–45b, if A 1 = 0.87, A 2 = –0.22, A 3 = 0.00, A 4 = 0.054, and A 5 = –0.035. FIGURE 16–45 Problem 109.

(b)

1 3

1 6

Answers to Exercises A: 1 km for every 3 s before the thunder is heard.

E: 257 Hz.

B: 4 times as intense.

F: (b).

C: (b).

G: (a) 1717 Hz, (b) 1483 Hz.

D: The less-massive one.

H: (a).

General Problems

453

Heating the air inside a "hot-air" balloon raises the air's temperature, causing it to expand, and forces air out the opening at the bottom. The reduced amount of air inside means its density is lower than the outside air, so there is a net buoyant force upward on the balloon. In this Chapter we study temperature and its effects on matter: thermal expansion and the gas laws.

P T E R

C H

CONTENTS 17–1 Atomic Theory of Matter 17–2 Temperature and Thermometers 17–3 Thermal Equilibrium and the Zeroth Law of Thermodynamics

Temperature, Thermal Expansion, and the Ideal Gas Law CHAPTER-OPENING QUESTION—Guess now! A hot-air balloon, open at one end (see photos above), rises when the air inside is heated by a flame. For the following properties, is the air inside the balloon higher, lower, or the same as for the air outside the balloon? (i) Temperature, (ii) Pressure, (iii) Density.

17–4 Thermal Expansion

*17–5 Thermal Stresses 17–6 The Gas Laws and Absolute Temperature 17–7 The Ideal Gas Law 17–8 Problem Solving with the Ideal Gas Law 17–9 Ideal Gas Law in Terms of Molecules: Avogadro's Number

*17–10 Ideal Gas Temperature Scale—a Standard

454

n the next four Chapters, Chapters 17 through 20, we study temperature, heat and thermodynamics, and the kinetic theory of gases. We will often consider a particular system, by which we mean a particular object or set of objects; everything else in the universe is called the "environment." We can describe the state (or condition) of a particular system— such as a gas in a container—from either a microscopic or macroscopic point of view. A microscopic description would involve details of the motion of all the atoms or molecules making up the system, which could be very complicated. A macroscopic description is given in terms of quantities that are detectable directly by our senses and instruments, such as volume, mass, pressure, and temperature.

The description of processes in terms of macroscopic quantities is the field of thermodynamics. Quantities that can be used to describe the state of a system are called state variables. To describe the state of a pure gas in a container, for example, requires only three state variables, which are typically the volume, the pressure, and the temperature. More complex systems require more than three state variables to describe them. The emphasis in this Chapter is on the concept of temperature. We begin, however, with a brief discussion of the theory that matter is made up of atoms and that these atoms are in continual random motion. This theory is called kinetic theory ("kinetic," you may recall, is Greek for "moving"), and we discuss it in more detail in Chapter 18.

17–1

Atomic Theory of Matter

The idea that matter is made up of atoms dates back to the ancient Greeks. According to the Greek philosopher Democritus, if a pure substance—say, a piece of iron—were cut into smaller and smaller bits, eventually a smallest piece of that substance would be obtained which could not be divided further. This smallest piece was called an atom, which in Greek means "indivisible."† Today the atomic theory is universally accepted. The experimental evidence in its favor, however, came mainly in the eighteenth, nineteenth, and twentieth centuries, and much of it was obtained from the analysis of chemical reactions. We will often speak of the relative masses of individual atoms and molecules—what we call the atomic mass or molecular mass, respectively.‡ These are based on arbitrarily assigning the abundant carbon atom, 12C, the atomic mass of exactly 12.0000 unified atomic mass units (u). In terms of kilograms, 1 u = 1.6605 * 10–27 kg. The atomic mass of hydrogen is then 1.0078 u, and the values for other atoms are as listed in the Periodic Table inside the back cover of this book, and also in Appendix F. The molecular mass of a compound is the sum of atomic masses of the atoms making up the molecules of that compound.§ An important piece of evidence for the atomic theory is called Brownian motion, named after the biologist Robert Brown, who is credited with its discovery in 1827. While he was observing tiny pollen grains suspended in water under his microscope, Brown noticed that the tiny grains moved about in tortuous paths (Fig. 17–1), even though the water appeared to be perfectly still. The atomic theory easily explains Brownian motion if the further reasonable assumption is made that the atoms of any substance are continually in motion. Then Brown's tiny pollen grains are jostled about by the vigorous barrage of rapidly moving molecules of water. In 1905, Albert Einstein examined Brownian motion from a theoretical point of view and was able to calculate from the experimental data the approximate size and mass of atoms and molecules. His calculations showed that the diameter of a typical atom is about 10–10 m.

FIGURE 17–1 Path of a tiny particle (pollen grain, for example) suspended in water. The straight lines connect observed positions of the particle at equal time intervals.

† Today we do not consider the atom as indivisible, but rather as consisting of a nucleus (containing protons and neutrons) and electrons. ‡ The terms atomic weight and molecular weight are sometimes used for these quantities, but properly speaking we are comparing masses. § An element is a substance, such as gold, iron, or copper, that cannot be broken down into simpler substances by chemical means. Compounds are substances made up of elements, and can be broken down into them; examples are carbon dioxide and water. The smallest piece of an element is an atom; the smallest piece of a compound is a molecule. Molecules are made up of atoms; a molecule of water, for example, is made up of two atoms of hydrogen and one of oxygen; its chemical formula is H 2O.

SECTION 17–1

Atomic Theory of Matter

455

At the start of Chapter 13, we distinguished the three common phases (or states) of matter—solid, liquid, gas—based on macroscopic, or "large-scale," properties. Now let us see how these three phases of matter differ, from the atomic or microscopic point of view. Clearly, atoms and molecules must exert attractive forces on each other. For how else could a brick or a block of aluminum hold together in one piece? The attractive forces between molecules are of an electrical nature (more on this in later Chapters). When molecules come too close together, the force between them must become repulsive (electric repulsion between their outer electrons), for how else could matter take up space? Thus molecules maintain a minimum distance from each other. In a solid material, the attractive forces are strong enough that the atoms or molecules move only slightly (oscillate) about relatively fixed positions, often in an array known as a crystal lattice, as shown in Fig. 17–2a. In a liquid, the atoms or molecules are moving more rapidly, or the forces between them are weaker, so that they are sufficiently free to pass around one another, as in Fig. 17–2b. In a gas, the forces are so weak, or the speeds so high, that the molecules do not even stay close together. They move rapidly every which way, Fig. 17–2c, filling any container and occasionally colliding with one another. On average, the speeds are sufficiently high in a gas that when two molecules collide, the force of attraction is not strong enough to keep them close together and they fly off in new directions.

(a)

(b)

EXAMPLE 17–1 ESTIMATE Distance between atoms. The density of copper is 8.9 * 103 kgm3, and each copper atom has a mass of 63 u. Estimate the average distance between the centers of neighboring copper atoms. APPROACH We consider a cube of copper 1 m on a side. From the given density r we can calculate the mass m of a cube of volume V = 1 m3 (m = rV). We divide this by the mass of one atom (63 u) to obtain the number of atoms in 1 m3. We assume the atoms are in a uniform array, and we let N be the number of atoms in a 1-m length; then (N)(N)(N) = N 3 equals the total number of atoms in 1 m3. SOLUTION The mass of 1 copper atom is 63 u = 63 * 1.66 * 10–27 kg = 1.05 * 10 –25 kg. This means that in a cube of copper 1 m on a side Avolume = 1 m3 B, there are 8.9 * 103 kgm3 = 8.5 * 1028 atomsm3. 1.05 * 10 –25 kgatom The volume of a cube 1of side l is V = l3, so on one edge of the 1-m-long cube there are A8.5 * 1028 B 3 atoms = 4.4 * 109 atoms. Hence the distance between neighboring atoms is 1m = 2.3 * 10 –10 m. 4.4 * 109 atoms NOTE Watch out for units. Even though "atoms" is not a unit, it is helpful to include it to make sure you calculate correctly.

FIGURE 17–3 Expansion joint on a bridge.

17–2

Temperature and Thermometers

In everyday life, temperature is a measure of how hot or cold something is. A hot oven is said to have a high temperature, whereas the ice of a frozen lake is said to have a low temperature. Many properties of matter change with temperature. For example, most materials expand when their temperature is increased.† An iron beam is longer when hot than when cold. Concrete roads and sidewalks expand and contract slightly according to temperature, which is why compressible spacers or expansion joints (Fig. 17–3) are placed at regular intervals. The electrical resistance of matter changes with temperature (Chapter 25). So too does the color radiated by objects, at least at high temperatures: you may have noticed that the heating element of an electric stove glows with a red color when hot. †

Most materials expand when their temperature is raised, but not all. Water, for example, in the range 0°C to 4°C contracts with an increase in temperature (see Section 17–4).

456

CHAPTER 17

Temperature, Thermal Expansion, and the Ideal Gas Law

At higher temperatures, solids such as iron glow orange or even white. The white light from an ordinary incandescent lightbulb comes from an extremely hot tungsten wire. The surface temperatures of the Sun and other stars can be measured by the predominant color (more precisely, wavelengths) of light they emit. Instruments designed to measure temperature are called thermometers. There are many kinds of thermometers, but their operation always depends on some property of matter that changes with temperature. Many common thermometers rely on the expansion of a material with an increase in temperature. The first idea for a thermometer, by Galileo, made use of the expansion of a gas. Common thermometers today consist of a hollow glass tube filled with mercury or with alcohol colored with a red dye, as were the earliest usable thermometers (Fig. 17–4). Inside a common liquid-in-glass thermometer, the liquid expands more than the glass when the temperature is increased, so the liquid level rises in the tube (Fig. 17–5a). Although metals also expand with temperature, the change in length of a metal rod, say, is generally too small to measure accurately for ordinary changes in temperature. However, a useful thermometer can be made by bonding together two dissimilar metals whose rates of expansion are different (Fig. 17–5b). When the temperature is increased, the different amounts of expansion cause the bimetallic strip to bend. Often the bimetallic strip is in the form of a coil, one end of which is fixed while the other is attached to a pointer, Fig. 17–6. This kind of thermometer is used as ordinary air thermometers, oven thermometers, automatic off switches in electric coffeepots, and in room thermostats for determining when the heater or air conditioner should go on or off. Very precise thermometers make use of electrical properties (Chapter 25), such as resistance thermometers, thermocouples, and thermistors, often with a digital readout.

Temperature Scales In order to measure temperature quantitatively, some sort of numerical scale must be defined. The most common scale today is the Celsius scale, sometimes called the centigrade scale. In the United States, the Fahrenheit scale is also common. The most important scale in scientific work is the absolute, or Kelvin, scale, and it will be discussed later in this Chapter. One way to define a temperature scale is to assign arbitrary values to two readily reproducible temperatures. For both the Celsius and Fahrenheit scales these two fixed points are chosen to be the freezing point and the boiling point† of water, both taken at standard atmospheric pressure. On the Celsius scale, the freezing point of water is chosen to be 0°C ("zero degrees Celsius") and the boiling point 100°C. On the Fahrenheit scale, the freezing point is defined as 32°F and the boiling point 212°F. A practical thermometer is calibrated by placing it in carefully prepared environments at each of the two temperatures and marking the position of the liquid or pointer. For a Celsius scale, the distance between the two marks is divided into one hundred equal intervals representing each degree between 0°C and 100°C (hence the name "centigrade scale" meaning "hundred steps"). For a Fahrenheit scale, the two points are labeled 32°F and 212°F and the distance between them is divided into 180 equal intervals. For temperatures below the freezing point of water and above the boiling point of water, the scales may be extended using the same equally spaced intervals. However, thermometers can be used only over a limited temperature range because of their own limitations—for example, the liquid mercury in a mercury-in-glass thermometer solidifies at some point, below which the thermometer will be useless. It is also rendered useless above temperatures where the fluid, such as alcohol, vaporizes. For very low or very high temperatures, specialized thermometers are required, some of which we will mention later.

FIGURE 17–4 Thermometers built by the Accademia del Cimento (1657–1667) in Florence, Italy, are among the earliest known.These sensitive and exquisite instruments contained alcohol, sometimes colored, like many thermometers today.

FIGURE 17–5 (a) Mercury- or alcohol-in-glass thermometer; (b) bimetallic strip.

Tube

Bulb (acts as a reservoir) (a)

(b)

FIGURE 17–6 Photograph of a thermometer using a coiled bimetallic strip.

†

The freezing point of a substance is defined as that temperature at which the solid and liquid phases coexist in equilibrium—that is, without any net liquid changing into the solid or vice versa. Experimentally, this is found to occur at only one definite temperature, for a given pressure. Similarly, the boiling point is defined as that temperature at which the liquid and gas coexist in equilibrium. Since these points vary with pressure, the pressure must be specified (usually it is 1 atm).

SECTION 17–2

Temperature and Thermometers

457

100 C

212 F 200 F

150 F 50 C

Every temperature on the Celsius scale corresponds to a particular temperature on the Fahrenheit scale, Fig. 17–7. It is easy to convert from one to the other if you remember that 0°C corresponds to 32°F and that a range of 100° on the Celsius scale corresponds to a range of 180° on the Fahrenheit scale. Thus, one Fahrenheit degree (1 F°) corresponds to 100180 = 59 of a Celsius degree (1 C°). That is, 1 F° = 59 C°. (Notice that when we refer to a specific temperature, we say "degrees Celsius," as in 20°C; but when we refer to a change in temperature or a temperature interval, we say "Celsius degrees," as in "2 C°.") The conversion between the two temperature scales can be written

100 F

CT(°F) - 32D

T(°C) =

5 9

T(°F) =

9 5 T(°C)

or 50 F 32 F

0 C

Celsius (centigrade)

+ 32.

Rather than memorizing these relations (it would be easy to confuse them), it is usually easier simply to remember that 0°C = 32°F and that a change of 5 C° = a change of 9 F°.

0 F Fahrenheit

EXAMPLE 17–2 Taking your temperature. Normal body temperature is 98.6°F. What is this on the Celsius scale?

FIGURE 17–7 Celsius and Fahrenheit scales compared. C A U T I O N

Convert temperature by remembering 0°C = 32°F and a change of 5 C° = 9 F°

APPROACH We recall that 0°C = 32°F and 5 C° = 9 F°. SOLUTION First we relate the given temperature to the freezing point of water (0°C). That is, 98.6°F is 98.6 - 32.0 = 66.6 F° above the freezing point of water. Since each F° is equal to 59 C°, this corresponds to 66.6 * 59 = 37.0 Celsius degrees above the freezing point. The freezing point is 0°C, so the temperature is 37.0°C. EXERCISE A Determine the temperature at which both scales give the same numerical reading ATC = TF B.

FIGURE 17–8 Constant-volume gas thermometer.

h Gas Bulb

458

Reference mark Mercury Connecting hose

CHAPTER 17

Different materials do not expand in quite the same way over a wide temperature range. Consequently, if we calibrate different kinds of thermometers exactly as described above, they will not usually agree precisely. Because of how we calibrate them, they will agree at 0°C and at 100°C. But because of different expansion properties, they may not agree precisely at intermediate temperatures (remember we arbitrarily divided the thermometer scale into 100 equal divisions between 0°C and 100°C). Thus a carefully calibrated mercury-in-glass thermometer might register 52.0°C, whereas a carefully calibrated thermometer of another type might read 52.6°C. Discrepancies below 0°C and above 100°C can also be significant. Because of such discrepancies, some standard kind of thermometer must be chosen so that all temperatures can be precisely defined. The chosen standard for this purpose is the constant-volume gas thermometer. As shown in the simplified diagram of Fig. 17–8, this thermometer consists of a bulb filled with a dilute gas connected by a thin tube to a mercury manometer (Section 13–6). The volume of the gas is kept constant by raising or lowering the right-hand tube of the manometer so that the mercury in the left-hand tube coincides with the reference mark. An increase in temperature causes a proportional increase in pressure in the bulb. Thus the tube must be lifted higher to keep the gas volume constant. The height of the mercury in the right-hand column is then a measure of the temperature. This thermometer gives the same results for all gases in the limit of reducing the gas pressure in the bulb toward zero. The resulting scale serves as a basis for the standard temperature scale (Section 17–10).

Temperature, Thermal Expansion, and the Ideal Gas Law

17–3

Thermal Equilibrium and the Zeroth Law of Thermodynamics

We are all familiar with the fact that if two objects at different temperatures are placed in thermal contact (meaning thermal energy can transfer from one to the other), the two objects will eventually reach the same temperature. They are then said to be in thermal equilibrium. For example, you leave a fever thermometer in your mouth until it comes into thermal equilibrium with that environment, and then you read it. Two objects are defined to be in thermal equilibrium if, when placed in thermal contact, no net energy flows from one to the other, and their temperatures don't change. Experiments indicate that if two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. This postulate is called the zeroth law of thermodynamics. It has this unusual name because it was not until after the great first and second laws of thermodynamics (Chapters 19 and 20) were worked out that scientists realized that this apparently obvious postulate needed to be stated first. Temperature is a property of a system that determines whether the system will be in thermal equilibrium with other systems. When two systems are in thermal equilibrium, their temperatures are, by definition, equal, and no net thermal energy will be exchanged between them. This is consistent with our everyday notion of temperature, since when a hot object and a cold one are put into contact, they eventually come to the same temperature. Thus the importance of the zeroth law is that it allows a useful definition of temperature.

17–4

Thermal Expansion

Most substances expand when heated and contract when cooled. However, the amount of expansion or contraction varies, depending on the material.

Linear Expansion Experiments indicate that the change in length ¢l of almost all solids is, to a good approximation, directly proportional to the change in temperature ¢T, as long as ¢T is not too large. The change in length is also proportional to the original length of the object, l0 . That is, for the same temperature increase, a 4-m-long iron rod will increase in length twice as much as a 2-m-long iron rod. We can write this proportionality as an equation: (17–1a)

¢l = al0 ¢T,

where a, the proportionality constant, is called the coefficient of linear expansion for the particular material and has units of (C°)–1. We write l = l0 + ¢l, Fig. 17–9, and rewrite this equation as l = l0 + ¢l = l0 + al0 ¢T, or l = l0(1 + a ¢T),

(17–1b)

where l0 is the length initially, at temperature T0 , and l is the length after heating or cooling to a temperature T. If the temperature change ¢T = T - T0 is negative, then ¢l = l - l0 is also negative; the length shortens as the temperature decreases. l0 at T0

∆l at T

FIGURE 17–9 A thin rod of length l0 at temperature T0 is heated to a new uniform temperature T and acquires length l, where l = l0 + ¢l.

SECTION 17–4

Thermal Expansion

459

TABLE 17–1 Coefficients of Expansion, near 20°C Coefficient of Linear Expansion, A (C°)1

Material Solids Aluminum Brass Copper Gold Iron or steel Lead Glass (Pyrex®) Glass (ordinary) Quartz Concrete and brick Marble Liquids Gasoline Mercury Ethyl alcohol Glycerin Water Gases Air (and most other gases at atmospheric pressure)

25 19 17 14 12 29 3 9 0.4 L 12 1.4– 3.5

* * * * * * * * * * *

10–6 10–6 10–6 10–6 10–6 10 –6 10 –6 10 –6 10 –6 10 –6 10 –6

Coefficient of Volume Expansion, B (C°)1 75 56 50 42 35 87 9 27 1 L 36 4– 10

* * * * * * * * * * *

10–6 10–6 10–6 10–6 10–6 10–6 10–6 10–6 10–6 10–6 10–6

950 180 1100 500 210

* * * * *

10–6 10–6 10–6 10–6 10–6

3400 * 10 –6

The values of a for various materials at 20°C are listed in Table 17–1. Actually, a does vary slightly with temperature (which is why thermometers made of different materials do not agree precisely). However, if the temperature range is not too great, the variation can usually be ignored. P H Y S I C S

A P P L I E D

Expansion in structures

EXAMPLE 17–3 Bridge expansion. The steel bed of a suspension bridge is 200 m long at 20°C. If the extremes of temperature to which it might be exposed are –30°C to ±40°C, how much will it contract and expand? APPROACH We assume the bridge bed will expand and contract linearly with temperature, as given by Eq. 17–1a. SOLUTION From Table 17–1, we find that a = 12 * 10–6(C°)–1 for steel. The increase in length when it is at 40°C will be ¢l = al0 ¢T = A12 * 10 –6C°B(200 m)(40°C - 20°C) = 4.8 * 10–2 m,

or 4.8 cm. When the temperature decreases to –30°C, ¢T = –50 C°. Then ¢l = A12 * 10 –6C°B(200 m)(–50 C°) = –12.0 * 10 –2 m,

or a decrease in length of 12 cm. The total range the expansion joints must accommodate is 12 cm + 4.8 cm L 17 cm (Fig. 17–3).

FIGURE 17–10 Example 17–4.

(a)

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CHAPTER 17

(b)

CONCEPTUAL EXAMPLE 17–4 Do holes expand or contract? If you heat a thin, circular ring (Fig. 17–10a) in the oven, does the ring's hole get larger or smaller? RESPONSE You might guess that the metal expands into the hole, making the hole smaller. But it is not so. Imagine the ring is solid, like a coin (Fig. 17–10b). Draw a circle on it with a pen as shown. When the metal expands, the material inside the circle will expand along with the rest of the metal; so the circle expands. Cutting the metal where the circle is makes clear to us that the hole in Fig. 17–10a increases in diameter.

Temperature, Thermal Expansion, and the Ideal Gas Law

EXAMPLE 17–5 Ring on a rod. An iron ring is to fit snugly on a cylindrical iron rod. At 20°C, the diameter of the rod is 6.445 cm and the inside diameter of the ring is 6.420 cm. To slip over the rod, the ring must be slightly larger than the rod diameter by about 0.008 cm. To what temperature must the ring be brought if its hole is to be large enough so it will slip over the rod? APPROACH The hole in the ring must be increased from a diameter of 6.420 cm to 6.445 cm + 0.008 cm = 6.453 cm. The ring must be heated since the hole diameter will increase linearly with temperature (Example 17–4). SOLUTION We solve for ¢T in Eq. 17–1a and find ¢l 6.453 cm - 6.420 cm ¢T = = = 430 C°. al0 A12 * 10–6C°B(6.420 cm) So it must be raised at least to T = (20°C + 430 C°) = 450°C. NOTE In doing Problems, do not forget the last step, adding in the initial temperature (20°C here). CONCEPTUAL EXAMPLE 17–6 Opening a tight jar lid. When the lid of a glass jar is tight, holding the lid under hot water for a short time will often make it easier to open (Fig. 17–11). Why? RESPONSE The lid may be struck by the hot water more directly than the glass and so expand sooner. But even if not, metals generally expand more than glass for the same temperature change (a is greater—see Table 17–1). NOTE If you put a hard-boiled egg in cold water immediately after cooking it, it is easier to peel: the different thermal expansions of the shell and egg cause the egg to separate from the shell.

P H Y S I C S

A P P L I E D

Opening a tight lid

P H Y S I C S

A P P L I E D

Peeling a hard-boiled egg

Volume Expansion The change in volume of a material which undergoes a temperature change is given by a relation similar to Eq. 17–1a, namely, ¢V = bV0 ¢T, (17–2) where ¢T is the change in temperature, V0 is the original volume, ¢V is the change in volume, and b is the coefficient of volume expansion. The units of b are (C°)–1. Values of b for various materials are given in Table 17–1. Notice that for solids, b is normally equal to approximately 3a. To see why, consider a rectangular solid of length l0 , width W0 , and height H0 . When its temperature is changed by ¢T, its volume changes from V0 = l0 W0 H0 to V = l0(1 + a ¢T)W0(1 + a ¢T)H0(1 + a ¢T), using Eq. 17–1b and assuming a is the same in all directions. Thus, ¢V = V - V0 = V0(1 + a ¢T)3 - V0 = V0 C3a ¢T + 3(a ¢T)2 + (a ¢T)3 D. If the amount of expansion is much smaller than the original size of the object, then a ¢T V 1 and we can ignore all but the first term and obtain ¢V L (3a)V0 ¢T. This is Eq. 17–2 with b L 3a. For solids that are not isotropic (having the same properties in all directions), however, the relation b L 3a is not valid. Note also that linear expansion has no meaning for liquids and gases since they do not have fixed shapes.

FIGURE 17–11 Example 17–6.

EXERCISE B A long thin bar of aluminum at 0°C is 1.0 m long and has a volume of 1.0000 * 10 –3 m3. When heated to 100°C, the length of the bar becomes 1.0025 m. What is the approximate volume of the bar at 100°C? (a) 1.0000 * 10–3 m3; (b) 1.0025 * 10–3 m3; (c) 1.0050 * 10 –3 m3; (d) 1.0075 * 10–3 m3; (e) 2.5625 * 10–3 m3.

Equations 17–1 and 17–2 are accurate only if ¢l (or ¢V ) is small compared to l0 (or V0). This is of particular concern for liquids and even more so for gases because of the large values of b. Furthermore, b itself varies substantially with temperature for gases. Therefore, a more convenient way of dealing with gases is needed, and will be discussed starting in Section 17–6. SECTION 17–4

Thermal Expansion

461

P H Y S I C S

A P P L I E D

Gas tank overflow

EXAMPLE 17–7 Gas tank in the Sun. The 70-liter (L) steel gas tank of a car is filled to the top with gasoline at 20°C. The car sits in the Sun and the tank reaches a temperature of 40°C (104°F). How much gasoline do you expect to overflow from the tank? APPROACH Both the gasoline and the tank expand as the temperature increases, and we assume they do so linearly as described by Eq. 17–2. The volume of overflowing gasoline equals the volume increase of the gasoline minus the increase in volume of the tank. SOLUTION The gasoline expands by ¢V = bV0 ¢T = A950 * 10–6C°B(70 L)(40°C - 20°C) = 1.3 L. The tank also expands. We can think of it as a steel shell that undergoes volume expansion Ab L 3a = 36 * 10–6C°B. If the tank were solid, the surface layer (the shell) would expand just the same. Thus the tank increases in volume by ¢V = A36 * 10–6C°B(70 L)(40°C - 20°C) = 0.050 L, so the tank expansion has little effect. More than a liter of gas could spill out. NOTE Want to save a few pennies? You pay for gas by volume, so fill your gas tank when it is cool and the gas is denser—more molecules for the same price. But don't fill the tank quite all the way.

Anomalous Behavior of Water Below 4°C

P H Y S I C S

A P P L I E D

Life under ice

Most substances expand more or less uniformly with an increase in temperature, as long as no phase change occurs. Water, however, does not follow the usual pattern. If water at 0°C is heated, it actually decreases in volume until it reaches 4°C. Above 4°C water behaves normally and expands in volume as the temperature is increased, Fig. 17–12. Water thus has its greatest density at 4°C. This anomalous behavior of water is of great importance for the survival of aquatic life during cold winters. When the water in a lake or river is above 4°C and begins to cool by contact with cold air, the water at the surface sinks because of its greater density. It is replaced by warmer water from below. This mixing continues until the temperature reaches 4°C. As the surface water cools further, it remains on the surface because it is less dense than the 4°C water below. Water then freezes first at the surface, and the ice remains on the surface since ice (specific gravity = 0.917) is less dense than water. The water at the bottom remains liquid unless it is so cold that the whole body of water freezes. If water were like most substances, becoming more dense as it cools, the water at the bottom of a lake would be frozen first. Lakes would freeze solid more easily since circulation would bring the warmer water to the surface to be efficiently cooled. The complete freezing of a lake would cause severe damage to its plant and animal life. Because of the unusual behavior of water below 4°C, it is rare for any large body of water to freeze completely, and this is helped by the layer of ice on the surface which acts as an insulator to reduce the flow of heat out of the water into the cold air above. Without this peculiar but wonderful property of water, life on this planet as we know it might not have been possible. Not only does water expand as it cools from 4°C to 0°C, it expands even more as it freezes to ice. This is why ice cubes float in water and pipes break when water inside them freezes. (a)

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CHAPTER 17

5°

(b) 10° 100°

1.04343

1.00013 1.00000

ρ (g/cm3)

FIGURE 17–12 Behavior of water as a function of temperature near 4°C. (a) Volume of 1.00000 g of water, as a function of temperature. (b) Density vs. temperature. [Note the break in each axis.]

Volume (cm3) of 1.00000 g water

0°

0° 5° 10° 100° Temperature (°C)

1.0000 0.9999 0.9998

0.96

0° 2° 4° 6°

100° 1.00000 0.9999 0.9998

0.96 0° 2° 4° 6° 100° Temperature (°C)

* 17–5

Thermal Stresses

In many situations, such as in buildings and roads, the ends of a beam or slab of material are rigidly fixed, which greatly limits expansion or contraction. If the temperature should change, large compressive or tensile stresses, called thermal stresses, will occur. The magnitude of such stresses can be calculated using the concept of elastic modulus developed in Chapter 12. To calculate the internal stress, we can think of this process as occurring in two steps: (1) the beam tries to expand (or contract) by an amount ¢l given by Eq. 17–1; (2) the solid in contact with the beam exerts a force to compress (or expand) it, keeping it at its original length. The force F required is given by Eq. 12–4: 1 F ¢l = l , EA 0 where E is Young's modulus for the material. To calculate the internal stress, FA, we then set ¢l in Eq. 17–1a equal to ¢l in the equation above and find 1 F al0 ¢T = l . EA 0 Hence, the stress F = aE ¢T. A EXAMPLE 17–8 Stress in concrete on a hot day. A highway is to be made of blocks of concrete 10 m long placed end to end with no space between them to allow for expansion. If the blocks were placed at a temperature of 10°C, what compressive stress would occur if the temperature reached 40°C? The contact area between each block is 0.20 m2. Will fracture occur? APPROACH We use the expression for the stress FA we just derived, and find the value of E from Table 12–1. To see if fracture occurs, we compare this stress to the ultimate strength of concrete in Table 12–2. SOLUTION F = aE ¢T = A12 * 10–6C°BA20 * 109 Nm2 B(30 C°) = 7.2 * 106 Nm2. A This stress is not far from the ultimate strength of concrete under compression (Table 12–2) and exceeds it for tension and shear. If the concrete is not perfectly aligned, part of the force will act in shear, and fracture is likely. This is why soft spacers or expansion joints (Fig. 17–3) are used in concrete sidewalks, highways, and bridges.

P H Y S I C S

A P P L I E D

Highway buckling

EXERCISE C How much space would you allow between the 10-m-long concrete blocks if you expected a temperature range of 0°F to 110°F?

17–6

The Gas Laws and Absolute Temperature

Equation 17–2 is not very useful for describing the expansion of a gas, partly because the expansion can be so great, and partly because gases generally expand to fill whatever container they are in. Indeed, Eq. 17–2 is meaningful only if the pressure is kept constant. The volume of a gas depends very much on the pressure as well as on the temperature. It is therefore valuable to determine a relation between the volume, the pressure, the temperature, and the mass of a gas. Such a relation is called an equation of state. (By the word state, we mean the physical condition of the system.) If the state of a system is changed, we will always wait until the pressure and temperature have reached the same values throughout. We thus consider only equilibrium states of a system—when the variables that describe it (such as temperature and pressure) are the same throughout the system and are not changing in time. We also note that the results of this Section are accurate only for gases that are not too dense (the pressure is not too high, on the order of an atmosphere or less) and not close to the liquefaction (boiling) point. SECTION 17–6

The Gas Laws and Absolute Temperature

463

For a given quantity of gas it is found experimentally that, to a good approximation, the volume of a gas is inversely proportional to the absolute pressure applied to it when the temperature is kept constant. That is, 1, [constant T] P where P is the absolute pressure (not "gauge pressure"—see Section 13–4). For example, if the pressure on a gas is doubled, the volume is reduced to half its original volume. This relation is known as Boyle's law, after Robert Boyle (1627–1691), who first stated it on the basis of his own experiments. A graph of P vs. V for a fixed temperature is shown in Fig. 17–13. Boyle's law can also be written V r

V FIGURE 17–13 Pressure vs. volume of a fixed amount of gas at a constant temperature, showing the inverse relationship as given by Boyle's law: as the pressure decreases, the volume increases.

Volume

FIGURE 17–14 Volume of a fixed amount of gas as a function of (a) Celsius temperature, and (b) Kelvin temperature, when the pressure is kept constant.

0°C

−273°C

100°C

200°C

Volume

Temperature (°C) (a)

100 K

200 K

300 K

400 K

Temperature (kelvins, or K) (b)

500 K

PV = constant.

[constant T ]

That is, at constant temperature, if either the pressure or volume of a fixed amount of gas is allowed to vary, the other variable also changes so that the product PV remains constant. Temperature also affects the volume of a gas, but a quantitative relationship between V and T was not found until more than a century after Boyle's work. The Frenchman Jacques Charles (1746–1823) found that when the pressure is not too high and is kept constant, the volume of a gas increases with temperature at a nearly linear rate, as shown in Fig. 17–14a. However, all gases liquefy at low temperatures (for example, oxygen liquefies at –183°C), so the graph cannot be extended below the liquefaction point. Nonetheless, the graph is essentially a straight line and if projected to lower temperatures, as shown by the dashed line, it crosses the axis at about –273°C. Such a graph can be drawn for any gas, and the straight line always projects back to –273°C at zero volume. This seems to imply that if a gas could be cooled to –273°C, it would have zero volume, and at lower temperatures a negative volume, which makes no sense. It could be argued that –273°C is the lowest temperature possible; indeed, many other more recent experiments indicate that this is so. This temperature is called the absolute zero of temperature. Its value has been determined to be –273.15°C. Absolute zero forms the basis of a temperature scale known as the absolute scale or Kelvin scale, and it is used extensively in scientific work. On this scale the temperature is specified as degrees Kelvin or, preferably, simply as kelvins (K) without the degree sign. The intervals are the same as for the Celsius scale, but the zero on this scale (0 K) is chosen as absolute zero. Thus the freezing point of water (0°C) is 273.15 K, and the boiling point of water is 373.15 K. Indeed, any temperature on the Celsius scale can be changed to kelvins by adding 273.15 to it: T(K) = T(°C) + 273.15. Now let us look at Fig. 17–14b, where the graph of the volume of a gas versus absolute temperature is a straight line that passes through the origin. Thus, to a good approximation, the volume of a given amount of gas is directly proportional to the absolute temperature when the pressure is kept constant. This is known as Charles's law, and is written V r T.

[constant P]

A third gas law, known as Gay-Lussac's law, after Joseph Gay-Lussac (1778–1850), states that at constant volume, the absolute pressure of a gas is directly proportional to the absolute temperature: P r T.

[constant V]

The laws of Boyle, Charles, and Gay-Lussac are not really laws in the sense that we use this term today (precise, deep, wide-ranging validity). They are really only approximations that are accurate for real gases only as long as the pressure

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Temperature, Thermal Expansion, and the Ideal Gas Law

and density of the gas are not too high, and the gas is not too close to liquefaction (condensation). The term law applied to these three relationships has become traditional, however, so we have stuck with that usage. CONCEPTUAL EXAMPLE 17–9 Why you should not throw a closed glass jar into a campfire. What can happen if you did throw an empty glass jar, with the lid on tight, into a fire, and why? RESPONSE The inside of the jar is not empty. It is filled with air. As the fire heats the air inside, its temperature rises. The volume of the glass jar changes only slightly due to the heating. According to Gay-Lussac's law the pressure P of the air inside the jar can increase dramatically, enough to cause the jar to explode, throwing glass pieces outward.

17–7

The Ideal Gas Law

The gas laws of Boyle, Charles, and Gay-Lussac were obtained by means of a very useful scientific technique: namely, holding one or more variables constant to see clearly the effects on one variable due to changing one other variable. These laws can now be combined into a single more general relation between the absolute pressure, volume, and absolute temperature of a fixed quantity of gas: PV r T. This relation indicates how any of the quantities P, V, or T will vary when the other two quantities change. This relation reduces to Boyle's, Charles's, or Gay-Lussac's law when either T, P, or V, respectively, is held constant. Finally, we must incorporate the effect of the amount of gas present. Anyone who has blown up a balloon knows that the more air forced into the balloon, the bigger it gets (Fig. 17–15). Indeed, careful experiments show that at constant temperature and pressure, the volume V of an enclosed gas increases in direct proportion to the mass m of gas present. Hence we write PV r mT. This proportion can be made into an equation by inserting a constant of proportionality. Experiment shows that this constant has a different value for different gases. However, the constant of proportionality turns out to be the same for all gases if, instead of the mass m, we use the number of moles. One mole (abbreviated mol) is defined as the amount of substance that contains as many atoms or molecules as there are in precisely 12 grams of carbon 12 (whose atomic mass is exactly 12 u). A simpler but equivalent definition is this: 1 mol is that quantity of substance whose mass in grams is numerically equal to the molecular mass (Section 17–1) of the substance. For example, the molecular mass of hydrogen gas AH 2 B is 2.0 u (since each molecule contains two atoms of hydrogen and each atom has an atomic mass of 1.0 u). Thus 1 mol of H 2 has a mass of 2.0 g. Similarly, 1 mol of neon gas has a mass of 20 g, and 1 mol of CO2 has a mass of [12 + (2 * 16)] = 44 g since oxygen has atomic mass of 16 (see Periodic Table inside the rear cover). The mole is the official unit of amount of substance in the SI system. In general, the number of moles, n, in a given sample of a pure substance is equal to the mass of the sample in grams divided by the molecular mass specified as grams per mole: n (mole) =

FIGURE 17–15 Blowing up a balloon means putting more air (more air molecules) into the balloon, which increases its volume. The pressure is nearly constant (atmospheric) except for the small effect of the balloon's elasticity.

mass (grams) . molecular mass (gmol)

For example, the number of moles in 132 g of CO2 (molecular mass 44 u) is n =

132 g = 3.0 mole. 44 gmol SECTION 17–7

The Ideal Gas Law

465

We can now write the proportion discussed above (PV r mT) as an equation: (17–3)

PV = nRT,

IDEAL GAS LAW

where n represents the number of moles and R is the constant of proportionality. R is called the universal gas constant because its value is found experimentally to be the same for all gases. The value of R, in several sets of units (only the first is the proper SI unit), is R = 8.314 J(mol K) = 0.0821 (Latm)(molK) = 1.99 calories(molK).†

C A U T I O N

Always give T in kelvins and P as absolute (not gauge) pressure

[SI units]

Equation 17–3 is called the ideal gas law, or the equation of state for an ideal gas. We use the term "ideal" because real gases do not follow Eq. 17–3 precisely, particularly at high pressure (and density) or when the gas is near the liquefaction point ( = boiling point). However, at pressures less than an atmosphere or so, and when T is not close to the liquefaction point of the gas, Eq. 17–3 is quite accurate and useful for real gases. Always remember, when using the ideal gas law, that temperatures must be given in kelvins (K) and that the pressure P must always be absolute pressure, not gauge pressure (Section 13–4). EXERCISE D Return to the Chapter-Opening Question, page 454, and answer it again now. Try to explain why you may have answered differently the first time. EXERCISE E An ideal gas is contained in a steel sphere at 27.0°C and 1.00 atm absolute pressure. If no gas is allowed to escape and the temperature is raised to 127°C, what will be the new pressure? (a) 1.33 atm; (b) 0.75 atm; (c) 4.7 atm; (d) 0.21 atm; (e) 1.00 atm.

17–8

Problem Solving with the Ideal Gas Law

The ideal gas law is an extremely useful tool, and we now consider some Examples. We will often refer to "standard conditions" or standard temperature and pressure (STP), which means: T = 273 K (0°C)

STP

and

P = 1.00 atm = 1.013 * 105 Nm2 = 101.3 kPa.

EXAMPLE 17–10 Volume of one mole at STP. Determine the volume of 1.00 mol of any gas, assuming it behaves like an ideal gas, at STP. APPROACH We use the ideal gas law, solving for V. SOLUTION We solve for V in Eq. 17–3: V = P R O B L E M

(1.00 mol)(8.314 JmolK)(273 K) nRT = = 22.4 * 10–3 m3. P A1.013 * 105 Nm2 B

Since 1 liter (L) is 1000 cm3 = 1.00 * 10 –3 m3, 1.00 mol of any (ideal) gas has volume V = 22.4 L at STP.

S O LV I N G

1 mol of gas at STP has V = 22.4 L

The value of 22.4 L for the volume of 1 mol of an ideal gas at STP is worth remembering, for it sometimes makes calculation simpler. EXERCISE F What is the volume of 1.00 mol of ideal gas at 546 K ( = 2 * 273 K) and 2.0 atm absolute pressure? (a) 11.2 L, (b) 22.4 L, (c) 44.8 L, (d) 67.2 L, (e) 89.6 L. †

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CHAPTER 17

Calories will be defined in Section 19–1; sometimes it is useful to use R as given in terms of calories.

Temperature, Thermal Expansion, and the Ideal Gas Law

EXAMPLE 17–11 Helium balloon. A helium party balloon, assumed to be a perfect sphere, has a radius of 18.0 cm. At room temperature (20°C), its internal pressure is 1.05 atm. Find the number of moles of helium in the balloon and the mass of helium needed to inflate the balloon to these values. APPROACH We can use the ideal gas law to find n, since we are given P and T, and can find V from the given radius. SOLUTION We get the volume V from the formula for a sphere: V = =

4 3 3 pr 4 3 p (0.180

m)3 = 0.0244 m3.

The pressure is given as 1.05 atm = 1.064 * 105 Nm2. The temperature must be expressed in kelvins, so we change 20°C to (20 + 273)K = 293 K. Finally, we use the value R = 8.314 J(molK) because we are using SI units. Thus A1.064 * 105 Nm2 BA0.0244 m3 B PV = = 1.066 mol. n = (8.314 JmolK)(293 K) RT The mass of helium (atomic mass = 4.00 gmol as given in the Periodic Table or Appendix F) can be obtained from mass = n * molecular mass = (1.066 mol)(4.00 gmol) = 4.26 g or 4.26 * 10 –3 kg. EXAMPLE 17–12 ESTIMATE Mass of air in a room. Estimate the mass of air in a room whose dimensions are 5 m * 3 m * 2.5 m high, at STP. APPROACH First we determine the number of moles n using the given volume. Then we can multiply by the mass of one mole to get the total mass. SOLUTION Example 17–10 told us that 1 mol of a gas at 0°C has a volume of 22.4 L. The room's volume is 5 m * 3 m * 2.5 m, so (5 m)(3 m)(2.5 m) n = L 1700 mol. 22.4 * 10–3 m3 Air is a mixture of about 20% oxygen AO2 B and 80% nitrogen AN2 B. The molecular masses are 2 * 16 u = 32 u and 2 * 14 u = 28 u, respectively, for an average of about 29 u. Thus, 1 mol of air has a mass of about 29 g = 0.029 kg, so our room has a mass of air m L (1700 mol)(0.029 kgmol) L 50 kg. NOTE That is roughly 100 lb of air!

P H Y S I C S

A P P L I E D

Mass (and weight) of the air in a room

EXERCISE G At 20°C, would there be (a) more, (b) less, or (c) the same air mass in a room than at 0°C?

Frequently, volume is specified in liters and pressure in atmospheres. Rather than convert these to SI units, we can instead use the value of R given in Section 17–7 as 0.0821 L atmmolK. In many situations it is not necessary to use the value of R at all. For example, many problems involve a change in the pressure, temperature, and volume of a fixed amount of gas. In this case, PVT = nR = constant, since n and R remain constant. If we now let P1 , V1 , and T1 represent the appropriate variables initially, and P2 , V2 , T2 represent the variables after the change is made, then we can write P1 V1 P2 V2 . = T1 T2 If we know any five of the quantities in this equation, we can solve for the sixth. Or, if one of the three variables is constant (V1 = V2 , or P1 = P2 , or T1 = T2) then we can use this equation to solve for one unknown when given the other three quantities. SECTION 17–8

P R O B L E M

S O LV I N G

Using the ideal gas law as a ratio

Problem Solving with the Ideal Gas Law

467

P H Y S I C S

EXAMPLE 17–13 Check tires cold. An automobile tire is filled (Fig. 17–16) to a gauge pressure of 200 kPa at 10°C. After a drive of 100 km, the temperature within the tire rises to 40°C. What is the pressure within the tire now? APPROACH We do not know the number of moles of gas, or the volume of the tire, but we assume they are constant. We use the ratio form of the ideal gas law. SOLUTION Since V1 = V2 , then P1 P2 . = T1 T2 This is, incidentally, a statement of Gay-Lussac's law. Since the pressure given is the gauge pressure (Section 13–4), we must add atmospheric pressure ( = 101 kPa) to get the absolute pressure P1 = (200 kPa + 101 kPa) = 301 kPa. We convert temperatures to kelvins by adding 273 and solve for P2 : T2 313 K P2 = P1 ¢ ≤ = A3.01 * 105 PaB a b = 333 kPa. T1 283 K Subtracting atmospheric pressure, we find the resulting gauge pressure to be 232 kPa, which is a 16% increase. This Example shows why car manuals suggest checking tire pressure when the tires are cold.

A P P L I E D

Pressure in a hot tire

FIGURE 17–16 Example 17–13.

17–9

Avogadro's number

IDEAL GAS LAW (in terms of molecules)

Ideal Gas Law in Terms of Molecules: Avogadro's Number

The fact that the gas constant, R, has the same value for all gases is a remarkable reflection of simplicity in nature. It was first recognized, although in a slightly different form, by the Italian scientist Amedeo Avogadro (1776–1856). Avogadro stated that equal volumes of gas at the same pressure and temperature contain equal numbers of molecules. This is sometimes called Avogadro's hypothesis. That this is consistent with R being the same for all gases can be seen as follows. From Eq. 17–3, PV = nRT, we see that for the same number of moles, n, and the same pressure and temperature, the volume will be the same for all gases as long as R is the same. Second, the number of molecules in 1 mole is the same for all gases.† Thus Avogadro's hypothesis is equivalent to R being the same for all gases. The number of molecules in one mole of any pure substance is known as Avogadro's number, NA . Although Avogadro conceived the notion, he was not able to actually determine the value of NA . Indeed, precise measurements were not done until the twentieth century. A number of methods have been devised to measure NA , and the accepted value today is NA = 6.02 * 1023. [molecules/mole] Since the total number of molecules, N, in a gas is equal to the number per mole times the number of moles AN = nNA B, the ideal gas law, Eq. 17–3, can be written in terms of the number of molecules present: N PV = nRT = RT, NA or PV = NkT, (17–4) where k = RNA is called the Boltzmann constant and has the value 8.314 JmolK R k = = = 1.38 * 10–23 JK. NA 6.02 * 1023mol †

For example, the molecular mass of H 2 gas is 2.0 atomic mass units (u), whereas that of O2 gas is 32.0 u. Thus 1 mol of H 2 has a mass of 0.0020 kg and 1 mol of O2 gas, 0.0320 kg. The number of molecules in a mole is equal to the total mass M of a mole divided by the mass m of one molecule; since this ratio (Mm) is the same for all gases by definition of the mole, a mole of any gas must contain the same number of molecules.

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Temperature, Thermal Expansion, and the Ideal Gas Law

EXAMPLE 17–14 Hydrogen atom mass. Use Avogadro's number to determine the mass of a hydrogen atom. APPROACH The mass of one atom equals the mass of 1 mol divided by the number of atoms in 1 mol, NA . SOLUTION One mole of hydrogen atoms (atomic mass = 1.008 u, Section 17–1 or Appendix F) has a mass of 1.008 * 10 –3 kg and contains 6.02 * 1023 atoms. Thus one atom has a mass m =

1.008 * 10–3 kg 6.02 * 1023

= 1.67 * 10–27 kg.

EXAMPLE 17–15 ESTIMATE How many molecules in one breath? Estimate how many molecules you breathe in with a 1.0-L breath of air.

P H Y S I C S

A P P L I E D

Molecules in a breath

APPROACH We determine what fraction of a mole 1.0 L is by using the result of Example 17–10 that 1 mole has a volume of 22.4 L at STP, and then multiply that by NA to get the number of molecules in this number of moles. SOLUTION One mole corresponds to 22.4 L at STP, so 1.0 L of air is (1.0 L)(22.4 Lmol) = 0.045 mol. Then 1.0 L of air contains (0.045 mol)A6.02 * 1023 moleculesmolB L 3 * 1022 molecules.

* 17–10

Ideal Gas Temperature Scale— a Standard

It is important to have a very precisely defined temperature scale so that measurements of temperature made at different laboratories around the world can be accurately compared. We now discuss such a scale that has been accepted by the general scientific community. The standard thermometer for this scale is the constant-volume gas thermometer discussed in Section 17–2. The scale itself is called the ideal gas temperature scale, since it is based on the property of an ideal gas that the pressure is directly proportional to the absolute temperature (Gay-Lussac's law). A real gas, which would need to be used in any real constant-volume gas thermometer, approaches this ideal at low density. In other words, the temperature at any point in space is defined as being proportional to the pressure in the (nearly) ideal gas used in the thermometer. To set up a scale we need two fixed points. One fixed point will be P = 0 at T = 0 K. The second fixed point is chosen to be the triple point of water, which is that point where water in the solid, liquid, and gas states can coexist in equilibrium. This occurs only at a unique temperature and pressure,† and can be reproduced at different laboratories with great precision. The pressure at the triple point of water is 4.58 torr and the temperature is 0.01°C. This temperature corresponds to 273.16 K, since absolute zero is about –273.15°C. In fact, the triple point is now defined to be exactly 273.16 K.

† Liquid water and steam can coexist (the boiling point) at a range of temperatures depending on the pressure. Water boils at a lower temperature when the pressure is less, such as high in the mountains. The triple point represents a more precisely reproducible fixed point than does either the freezing point or boiling point of water at, say, 1 atm. See Section 18–3 for further discussion.

*SECTION 17–10

Ideal Gas Temperature Scale—a Standard

469

374.00

The absolute or Kelvin temperature T at any point is then defined, using a constant-volume gas thermometer for an ideal gas, as

T (K)

373.80

373.60

T = (273.16 K) ¢

373.40

373.20

373.00

H2 0

200 400 600 800 Ptp (torr)

FIGURE 17–17 Temperature readings of a constant-volume gas thermometer for the boiling point of water at 1.00 atm are plotted, for different gases, as a function of the gas pressure in the thermometer at the triple point APtp B. Note that as the amount of gas in the thermometer is reduced, so that Ptp S 0, all gases give the same reading, 373.15 K. For pressure less than 0.10 atm (76 torr), the variation shown is less than 0.07 K.

P ≤. Ptp

[ideal gas; constant volume] (17–5a)

In this relation, Ptp is the pressure of the gas in the thermometer at the triple point temperature of water, and P is the pressure in the thermometer when it is at the point where T is being determined. Note that if we let P = Ptp in this relation, then T = 273.16 K, as it must. The definition of temperature, Eq. 17–5a, with a constant-volume gas thermometer filled with a real gas is only approximate because we find that we get different results for the temperature depending on the type of gas that is used in the thermometer. Temperatures determined in this way also vary depending on the amount of gas in the bulb of the thermometer: for example, the boiling point of water at 1.00 atm is found from Eq. 17–5a to be 373.87 K when the gas is O2 and Ptp = 1000 torr. If the amount of O2 in the bulb is reduced so that at the triple point Ptp = 500 torr, the boiling point of water from Eq. 17–5a is then found to be 373.51 K. If H 2 gas is used instead, the corresponding values are 373.07 K and 373.11 K (see Fig. 17–17). But now suppose we use a particular real gas and make a series of measurements in which the amount of gas in the thermometer bulb is reduced to smaller and smaller amounts, so that Ptp becomes smaller and smaller. It is found experimentally that an extrapolation of such data to Ptp = 0 always gives the same value for the temperature of a given system (such as T = 373.15 K for the boiling point of water at 1.00 atm) as shown in Fig. 17–17. Thus the temperature T at any point in space, determined using a constantvolume gas thermometer containing a real gas, is defined using this limiting process: T = (273.16 K) lim ¢ Ptp S 0

P ≤. Ptp

[constant volume] (17–5b)

This defines the ideal gas temperature scale. One of the great advantages of this scale is that the value for T does not depend on the kind of gas used. But the scale does depend on the properties of gases in general. Helium has the lowest condensation point of all gases; at very low pressures it liquefies at about 1 K, so temperatures below this cannot be defined on this scale.

Summary The atomic theory of matter postulates that all matter is made up of tiny entities called atoms, which are typically 10–10 m in diameter. Atomic and molecular masses are specified on a scale where ordinary carbon A 12CB is arbitrarily given the value 12.0000 u (atomic mass units). The distinction between solids, liquids, and gases can be attributed to the strength of the attractive forces between the atoms or molecules and to their average speed. Temperature is a measure of how hot or cold something is. Thermometers are used to measure temperature on the Celsius (°C), Fahrenheit (°F), and Kelvin (K) scales. Two standard points on each scale are the freezing point of water (0°C, 32°F, 273.15 K) and the boiling point of water (100°C, 212°F, 373.15 K). A one-kelvin change in temperature equals a change of one Celsius degree or 95 Fahrenheit degrees. Kelvins are related to °C by T(K) = T(°C) + 273.15. The change in length, ¢l, of a solid, when its temperature changes by an amount ¢T, is directly proportional to the temperature change and to its original length l0 . That is, ¢l = al0 ¢T,

(17–1a)

where a is the coefficient of linear expansion. The change in volume of most solids, liquids, and gases is proportional to the temperature change and to the original

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volume V0 : (17–2) ¢V = bV0 ¢T. The coefficient of volume expansion, b, is approximately equal to 3a for uniform solids. Water is unusual because, unlike most materials whose volume increases with temperature, its volume actually decreases as the temperature increases in the range from 0°C to 4°C. The ideal gas law, or equation of state for an ideal gas, relates the pressure P, volume V, and temperature T (in kelvins) of n moles of gas by the equation (17–3) PV = nRT, where R = 8.314 JmolK for all gases. Real gases obey the ideal gas law quite accurately if they are not at too high a pressure or near their liquefaction point. One mole is that amount of a substance whose mass in grams is numerically equal to the atomic or molecular mass of that substance. Avogadro's number, NA = 6.02 * 1023, is the number of atoms or molecules in 1 mol of any pure substance. The ideal gas law can be written in terms of the number of molecules N in the gas as (17–4) PV = NkT, where k = RNA = 1.38 * 10 –23 JK is Boltzmann's constant.

Temperature, Thermal Expansion, and the Ideal Gas Law

Questions 1. Which has more atoms: 1 kg of iron or 1 kg of aluminum? See the Periodic Table or Appendix F. 2. Name several properties of materials that could be exploited to make a thermometer. 3. Which is larger, 1C° or 1F°? 4. If system A is in equilibrium with system B, but B is not in equilibrium with system C, what can you say about the temperatures of A, B, and C? 5. Suppose system C is not in equilibrium with system A nor in equilibrium with system B. Does this imply that A and B are not in equilibrium? What can you infer regarding the temperatures of A, B, and C? 6. In the relation ¢l = al0 ¢T, should l0 be the initial length, the final length, or does it matter? 7. A flat bimetallic strip consists of a strip of aluminum riveted to a strip of iron. When heated, the strip will bend. Which metal will be on the outside of the curve? Why? 8. Long steam pipes that are fixed at the ends often have a section in the shape of a ´. Why? 9. A flat uniform cylinder of lead floats in mercury at 0°C. Will the lead float higher or lower if the temperature is raised? 10. Figure 17–18 shows a diagram of a simple thermostat used to control a furnace (or other heating or cooling system). The bimetallic strip consists of two strips of different metals bonded together. The electric switch (attached to the bimetallic strip) is a glass vessel containing liquid mercury that conducts electricity when it can Temperature flow to touch both setting lever contact wires. Explain Bimetallic strip how this device controls the furnace and how it Liquid mercury switch can be set at different temperatures. FIGURE 17–18 A thermostat (Question 10).

Liquid mercury

Wires to heater

11. Explain why it is advisable to add water to an overheated automobile engine only slowly, and only with the engine running. 12. The units for the coefficients of expansion a are (C°)–1, and there is no mention of a length unit such as meters. Would the expansion coefficient change if we used feet or millimeters instead of meters? 13. When a cold mercury-in-glass thermometer is first placed in a hot tub of water, the mercury initially descends a bit and then rises. Explain. 14. The principal virtue of Pyrex glass is that its coefficient of linear expansion is much smaller than that for ordinary glass (Table 17–1). Explain why this gives rise to the higher resistance to heat of Pyrex. 15. Will a grandfather clock, accurate at 20°C, run fast or slow on a hot day (30°C)? The clock uses a pendulum supported on a long thin brass rod. 16. Freezing a can of soda will cause its bottom and top to bulge so badly the can will not stand up. What has happened? 17. Why might you expect an alcohol-in-glass thermometer to be more precise than a mercury-in-glass thermometer? 18. Will the buoyant force on an aluminum sphere submerged in water increase, decrease, or remain the same, if the temperature is increased from 20°C to 40°C? 19. If an atom is measured to have a mass of 6.7 * 10–27 kg, what atom do you think it is? 20. From a practical point of view, does it really matter what gas is used in a constant-volume gas thermometer? If so, explain. [Hint: See Fig. 17–17.] 21. A ship loaded in sea water at 4°C later sailed up a river into fresh water where it sank in a storm. Explain why a ship might be more likely to sink in fresh water than on the open sea. [Hint: Consider the buoyant force due to water.]

Problems 17–1 Atomic Theory 1. (I) How does the number of atoms in a 21.5-g gold ring compare to the number in a silver ring of the same mass? 2. (I) How many atoms are there in a 3.4-g copper penny?

17–2 Temperature and Thermometers 3. (I) (a) "Room temperature" is often taken to be 68°F. What is this on the Celsius scale? (b) The temperature of the filament in a lightbulb is about 1900°C. What is this on the Fahrenheit scale? 4. (I) Among the highest and lowest natural air temperatures recorded are 136°F in the Libyan desert and –129°F in Antarctica. What are these temperatures on the Celsius scale? 5. (I) A thermometer tells you that you have a fever of 39.4°C. What is this in Fahrenheit? 6. (II) In an alcohol-in-glass thermometer, the alcohol column has length 11.82 cm at 0.0°C and length 21.85 cm at 100.0°C. What is the temperature if the column has length (a) 18.70 cm, and (b) 14.60 cm?

17–4 Thermal Expansion 7. (I) The Eiffel Tower (Fig. 17–19) is built of wrought iron approximately 300 m tall. Estimate how much its height changes between January (average temperature of 2°C) and July (average temperature of 25°C). Ignore the angles of the iron beams and treat the tower as a vertical beam.

FIGURE 17–19 Problem 7. The Eiffel Tower in Paris.

Problems

471

8. (I) A concrete highway is built of slabs 12 m long (15°C). How wide should the expansion cracks between the slabs be (at 15°C) to prevent buckling if the range of temperature is –30°C to ± 50°C? 9. (I) Super Invar™, an alloy of iron and nickel, is a strong material with a very low coefficient of thermal expansion A0.20 * 10 –6C°B. A 1.6-m-long tabletop made of this alloy is used for sensitive laser measurements where extremely high tolerances are required. How much will this alloy table expand along its length if the temperature increases 5.0 C°? Compare to tabletops made of steel. 10. (II) To what temperature would you have to heat a brass rod for it to be 1.0% longer than it is at 25°C? 11. (II) The density of water at 4°C is 1.00 * 103 kgm3. What is water's density at 94°C? Assume a constant coefficient of volume expansion. 12. (II) At a given latitude, ocean water in the so-called "mixed layer" (from the surface to a depth of about 50 m) is at approximately the same temperature due to the mixing action of waves. Assume that because of global warming, the temperature of the mixed layer is everywhere increased by 0.5°C, while the temperature of the deeper portions of the ocean remains unchanged. Estimate the resulting rise in sea level. The ocean covers about 70% of the Earth's surface. 13. (II) To make a secure fit, rivets that are larger than the rivet hole are often used and the rivet is cooled (usually in dry ice) before it is placed in the hole. A steel rivet 1.872 cm in diameter is to be placed in a hole 1.870 cm in diameter in a metal at 20°C. To what temperature must the rivet be cooled if it is to fit in the hole? 14. (II) A uniform rectangular plate of length l and width w has a coefficient of linear expansion a. Show that, if we neglect very small quantities, the change in area of the plate due to a temperature change ¢T l ∆l is ¢A = 2alw ¢T. See Fig. 17–20. FIGURE 17–20 Problem 14. A rectangular plate is heated.

∆w

15. (II) An aluminum sphere is 8.75 cm in diameter. What will be its change in volume if it is heated from 30°C to 180°C? 16. (II) A typical car has 17 L of liquid coolant circulating at a temperature of 93°C through the engine's cooling system. Assume that, in this normal condition, the coolant completely fills the 3.5-L volume of the aluminum radiator and the 13.5-L internal cavities within the steel engine. When a car overheats, the radiator, engine, and coolant expand and a small reservoir connected to the radiator catches any resultant coolant overflow. Estimate how much coolant overflows to the reservoir if the system is heated from 93°C to 105°C. Model the radiator and engine as hollow shells of aluminum and steel, respectively. The coefficient of volume expansion for coolant is b = 410 * 10–6C°. 17. (II) It is observed that 55.50 mL of water at 20°C completely fills a container to the brim. When the container and the water are heated to 60°C, 0.35 g of water is lost. (a) What is the coefficient of volume expansion of the container? (b) What is the most likely material of the container? Density of water at 60°C is 0.98324 gmL.

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CHAPTER 17

18. (II) (a) A brass plug is to be placed in a ring made of iron. At 15°C, the diameter of the plug is 8.753 cm and that of the inside of the ring is 8.743 cm. They must both be brought to what common temperature in order to fit? (b) What if the plug were iron and the ring brass? 19. (II) If a fluid is contained in a long narrow vessel so it can expand in essentially one direction only, show that the effective coefficient of linear expansion a is approximately equal to the coefficient of volume expansion b. 20. (II) (a) Show that the change in the density r of a substance, when the temperature changes by ¢T, is given by ¢r = –br ¢T. (b) What is the fractional change in density of a lead sphere whose temperature decreases from 25°C to –55°C? 21. (II) Wine bottles are never completely filled: a small volume of air is left in the glass bottle's cylindrically shaped neck (inner diameter d = 18.5 mm) to allow for wine's fairly large coefficient of thermal expansion. The distance H between the surface of the liquid contents and the bottom of the cork is called the "headspace height" (Fig. 17–21), and is typically H = 1.5 cm d for a 750-mL bottle Cork filled at 20°C. Due to its alcoholic content, Air wine's coefficient of (headspace) volume expansion is H about double that of water; in comparison, Glass the thermal expansion bottle of glass can be neglected. Estimate H if the bottle is kept (a) at 10°C, (b) at 30°C. Liquid wine FIGURE 17–21 Problem 21. 22. (III) (a) Determine a formula for the change in surface area of a uniform solid sphere of radius r if its coefficient of linear expansion is a (assumed constant) and its temperature is changed by ¢T. (b) What is the increase in area of a solid iron sphere of radius 60.0 cm if its temperature is raised from 15°C to 275°C? 23. (III) The pendulum in a grandfather clock is made of brass and keeps perfect time at 17°C. How much time is gained or lost in a year if the clock is kept at 28°C? (Assume the frequency dependence on length for a simple pendulum applies.) 24. (III) A 28.4-kg solid aluminum cylindrical wheel of radius 0.41 m is rotating about its axle in frictionless bearings with angular velocity v = 32.8 rads. If its temperature is then raised from 20.0°C to 95.0°C, what is the fractional change in v? * 17–5 Thermal Stresses * 25. (I) An aluminum bar has the desired length when at 18°C. How much stress is required to keep it at this length if the temperature increases to 35°C? * 26. (II) (a) A horizontal steel I-beam of cross-sectional area 0.041 m2 is rigidly connected to two vertical steel girders. If the beam was installed when the temperature was 25°C, what stress is developed in the beam when the temperature drops to –25°C? (b) Is the ultimate strength of the steel exceeded? (c) What stress is developed if the beam is concrete and has a cross-sectional area of 0.13 m2? Will it fracture?

Temperature, Thermal Expansion, and the Ideal Gas Law

* 27. (III) A barrel of diameter 134.122 cm at 20°C is to be enclosed by an iron band. The circular band has an inside diameter of 134.110 cm at 20°C. It is 9.4 cm wide and 0.65 cm thick. (a) To what temperature must the band be heated so that it will fit over the barrel? (b) What will be the tension in the band when it cools to 20°C?

17–6 Gas Laws; Absolute Temperature 28. (I) What are the following temperatures on the Kelvin scale: (a) 66°C, (b) 92°F, (c) – 55°C, (d) 5500°C? 29. (I) Absolute zero is what temperature on the Fahrenheit scale? 30. (II) Typical temperatures in the interior of the Earth and Sun are about 4000°C and 15 * 106 °C, respectively. (a) What are these temperatures in kelvins? (b) What percent error is made in each case if a person forgets to change °C to K?

17–7 and 17–8 Ideal Gas Law 31. (I) If 3.80 m3 of a gas initially at STP is placed under a pressure of 3.20 atm, the temperature of the gas rises to 38.0°C. What is the volume? 32. (I) In an internal combustion engine, air at atmospheric pressure and a temperature of about 20°C is compressed in the cylinder by a piston to 18 of its original volume (compression ratio = 8.0). Estimate the temperature of the compressed air, assuming the pressure reaches 40 atm. 33. (II) Calculate the density of nitrogen at STP using the ideal gas law. 34. (II) If 14.00 mol of helium gas is at 10.0°C and a gauge pressure of 0.350 atm, calculate (a) the volume of the helium gas under these conditions, and (b) the temperature if the gas is compressed to precisely half the volume at a gauge pressure of 1.00 atm. 35. (II) A stoppered test tube traps 25.0 cm3 of air at a pressure of 1.00 atm and temperature of 18°C. The cylindrically shaped stopper at the test tube's mouth has a diameter of 1.50 cm and will "pop off" the test tube if a net upward force of 10.0 N is applied to it. To what temperature would one have to heat the trapped air in order to "pop off" the stopper? Assume the air surrounding the test tube is always at a pressure of 1.00 atm.

36. (II) A storage tank contains 21.6 kg of nitrogen AN2 B at an absolute pressure of 3.85 atm. What will the pressure be if the nitrogen is replaced by an equal mass of CO2 at the same temperature?

37. (II) A storage tank at STP contains 28.5 kg of nitrogen AN2 B. (a) What is the volume of the tank? (b) What is the pressure if an additional 25.0 kg of nitrogen is added without changing the temperature? 38. (II) A scuba tank is filled with air to a pressure of 204 atm when the air temperature is 29°C. A diver then jumps into the ocean and, after a short time treading water on the ocean surface, checks the tank's pressure and finds that it is only 194 atm. Assuming the diver has inhaled a negligible amount of air from the tank, what is the temperature of the ocean water? 39. (II) What is the pressure inside a 38.0-L container holding 105.0 kg of argon gas at 20.0°C? 40. (II) A tank contains 30.0 kg of O2 gas at a gauge pressure of 8.20 atm. If the oxygen is replaced by helium at the same temperature, how many kilograms of the latter will be needed to produce a gauge pressure of 7.00 atm?

41. (II) A sealed metal container contains a gas at 20.0°C and 1.00 atm. To what temperature must the gas be heated for the pressure to double to 2.00 atm? (Ignore expansion of the container.) 42. (II) A tire is filled with air at 15°C to a gauge pressure of 250 kPa. If the tire reaches a temperature of 38°C, what fraction of the original air must be removed if the original pressure of 250 kPa is to be maintained? 43. (II) If 61.5 L of oxygen at 18.0°C and an absolute pressure of 2.45 atm are compressed to 48.8 L and at the same time the temperature is raised to 56.0°C, what will the new pressure be? 44. (II) A helium-filled balloon escapes a child's hand at sea level and 20.0°C. When it reaches an altitude of 3600 m, where the temperature is 5.0°C and the pressure only 0.68 atm, how will its volume compare to that at sea level? 45. (II) A sealed metal container can withstand a pressure difference of 0.50 atm. The container initially is filled with an ideal gas at 18°C and 1.0 atm. To what temperature can you cool the container before it collapses? (Ignore any changes in the container's volume due to thermal expansion.) 46. (II) You buy an "airtight" bag of potato chips packaged at sea level, and take the chips on an airplane flight. When you take the potato chips out of your luggage, you notice it has noticeably "puffed up." Airplane cabins are typically pressurized at 0.75 atm, and assuming the temperature inside an airplane is about the same as inside a potato chip processing plant, by what percentage has the bag "puffed up" in comparison to when it was packaged? 47. (II) A typical scuba tank, when fully charged, contains 12 L of air at 204 atm. Assume an "empty" tank contains air at 34 atm and is connected to an air compressor at sea level. The air compressor intakes air from the atmosphere, compresses it to high pressure, and then inputs this highpressure air into the scuba tank. If the (average) flow rate of air from the atmosphere into the intake port of the air compressor is 290 Lmin, how long will it take to fully charge the scuba tank? Assume the tank remains at the same temperature as the surrounding air during the filling process. 48. (III) A sealed container containing 4.0 mol of gas is squeezed, changing its volume from 0.020 m3 to 0.018 m3. During this process, the temperature decreases by 9.0 K while the pressure increases by 450 Pa. What was the original pressure and temperature of the gas in the container? 49. (III) Compare the value for the density of water vapor at exactly 100°C and 1 atm (Table 13–1) with the value predicted from the ideal gas law. Why would you expect a difference? 50. (III) An air bubble at the bottom of a lake 37.0 m deep has a volume of 1.00 cm3. If the temperature at the bottom is 5.5°C and at the top 18.5°C, what is the volume of the bubble just before it reaches the surface?

17–9 Ideal Gas Law in Terms of Molecules; Avogadro's Number 51. (I) Calculate the number of moleculesm3 in an ideal gas at STP. 52. (I) How many moles of water are there in 1.000 L at STP? How many molecules? 53. (II) What is the pressure in a region of outer space where there is 1 moleculecm3 and the temperature is 3 K? 54. (II) Estimate the number of (a) moles and (b) molecules of water in all the Earth's oceans. Assume water covers 75% of the Earth to an average depth of 3 km.

Problems

473

55. (II) The lowest pressure attainable using the best available vacuum techniques is about 10 –12 Nm2. At such a pressure, how many molecules are there per cm3 at 0°C? 56. (II) Is a gas mostly empty space? Check by assuming that the spatial extent of common gas molecules is about l0 = 0.3 nm so one gas molecule occupies an approximate volume equal to l30 . Assume STP. 57. (III) Estimate how many molecules of air are in each 2.0-L breath you inhale that were also in the last breath Galileo took. [Hint: Assume the atmosphere is about 10 km high and of constant density.] * 17–10 Ideal Gas Temperature Scale * 58. (I) In a constant-volume gas thermometer, what is the limiting ratio of the pressure at the boiling point of water at 1 atm to that at the triple point? (Keep five significant figures.)

* 59. (I) At the boiling point of sulfur (444.6°C) the pressure in a constant-volume gas thermometer is 187 torr. Estimate (a) the pressure at the triple point of water, (b) the temperature when the pressure in the thermometer is 118 torr. * 60. (II) Use Fig. 17–17 to determine the inaccuracy of a constant-volume gas thermometer using oxygen if it reads a pressure P = 268 torr at the boiling point of water at 1 atm. Express answer (a) in kelvins and (b) as a percentage. * 61. (III) A constant-volume gas thermometer is being used to determine the temperature of the melting point of a substance. The pressure in the thermometer at this temperature is 218 torr; at the triple point of water, the pressure is 286 torr. Some gas is now released from the thermometer bulb so that the pressure at the triple point of water becomes 163 torr. At the temperature of the melting substance, the pressure is 128 torr. Estimate, as accurately as possible, the melting-point temperature of the substance.

General Problems 62. A Pyrex measuring cup was calibrated at normal room temperature. How much error will be made in a recipe calling for 350 mL of cool water, if the water and the cup are hot, at 95°C, instead of at room temperature? Neglect the glass expansion. 63. A precise steel tape measure has been calibrated at 15°C. At 36°C, (a) will it read high or low, and (b) what will be the percentage error? 64. A cubic box of volume 6.15 * 10 –2 m3 is filled with air at atmospheric pressure at 15°C. The box is closed and heated to 185°C. What is the net force on each side of the box? 65. The gauge pressure in a helium gas cylinder is initially 32 atm. After many balloons have been blown up, the gauge pressure has decreased to 5 atm. What fraction of the original gas remains in the cylinder? 66. If a rod of original length l1 has its temperature changed from T1 to T2 , determine a formula for its new length l2 in terms of T1 , T2 , and a. Assume (a) a = constant, (b) a = a(T) is some function of temperature, and (c) a = a0 + bT where a0 and b are constants. 67. If a scuba diver fills his lungs to full capacity of 5.5 L when 8.0 m below the surface, to what volume would his lungs expand if he quickly rose to the surface? Is this advisable? 68. (a) Use the ideal gas law to show that, for an ideal gas at constant pressure, the coefficient of volume expansion is equal to b = 1T, where T is the kelvin temperature. Compare to Table 17–1 for gases at T = 293 K. (b) Show that the bulk modulus (Section 12–4) for an ideal gas held at constant temperature is B = P, where P is the pressure. 69. A house has a volume of 870 m3. (a) What is the total mass of air inside the house at 15°C? (b) If the temperature drops to – 15°C, what mass of air enters or leaves the house? 70. Assume that in an alternate universe, the laws of physics are very different from ours and that "ideal" gases behave as follows: (i) At constant temperature, pressure is inversely proportional to the square of the volume. (ii) At constant pressure, the volume varies directly with the 23 power of the temperature. (iii) At 273.15 K and 1.00 atm pressure, 1.00 mole of an ideal gas is found to occupy 22.4 L. Obtain the form of the ideal gas law in this alternate universe, including the value of the gas constant R.

474

CHAPTER 17

71. An iron cube floats in a bowl of liquid mercury at 0°C. (a) If the temperature is raised to 25°C, will the cube float higher or lower in the mercury? (b) By what percent will the fraction of volume submerged change? 72. (a) The tube of a mercury thermometer has an inside diameter of 0.140 mm. The bulb has a volume of 0.275 cm3. How far will the thread of mercury move when the temperature changes from 10.5°C to 33.0°C? Take into account expansion of the Pyrex glass. (b) Determine a formula for the change in length of the mercury column in terms of relevant variables. Ignore tube volume compared to bulb volume. 73. From the known value of atmospheric pressure at the surface of the Earth, estimate the total number of air molecules in the Earth's atmosphere. 74. Estimate the percent difference in the density of iron at STP, and when it is a solid deep in the Earth where the temperature is 2000°C and under 5000 atm of pressure. Assume the bulk modulus A90 * 109 Nm2 B and the coefficient of volume expansion do not vary with temperature and are the same as at STP. 75. What is the average distance between nitrogen molecules at STP? 76. A helium balloon, assumed to be a perfect sphere, has a radius of 22.0 cm. At room temperature (20°C), its internal pressure is 1.06 atm. Determine the number of moles of helium in the balloon, and the mass of helium needed to inflate the balloon to these values. 77. A standard cylinder of oxygen used in a hospital has gauge pressure = 2000 psi (13,800 kPa) and volume = 14 L A0.014 m3 B at T = 295 K. How long will the cylinder last if the flow rate, measured at atmospheric pressure, is constant at 2.4 Lmin? 78. A brass lid screws tightly onto a glass jar at 15°C. To help open the jar, it can be placed into a bath of hot water. After this treatment, the temperatures of the lid and the jar are both 75°C. The inside diameter of the lid is 8.0 cm. Find the size of the gap (difference in radius) that develops by this procedure. 79. The density of gasoline at 0°C is 0.68 * 103 kgm3. (a) What is the density on a hot day, when the temperature is 35°C? (b) What is the percent change in density?

Temperature, Thermal Expansion, and the Ideal Gas Law

80. A helium balloon has volume V0 and temperature T0 at sea level where the pressure is P0 and the air density is r0 . The balloon is allowed to float up in the air to altitude y where the temperature is T1 . (a) Show that the volume occupied by the balloon is then V = V0 AT1T0 Be ±cy where c = r0 gP0 = 1.25 * 10–4 m–1. (b) Show that the buoyant force does not depend on altitude y. Assume that the skin of the balloon maintains the helium pressure at a constant factor of 1.05 times greater than the outside pressure. [Hint: Assume that the pressure change with altitude is P = P0 e –cy, as in Example 13–5, Chapter 13.] 81. The first real length standard, adopted more than 200 years ago, was a platinum bar with two very fine marks separated by what was defined to be exactly one meter. If this standard bar was to be accurate to within &1.0 mm, how carefully would the trustees have needed to control the temperature? The coefficient of linear expansion is 9 * 10 –6C°. 82. A scuba tank when fully charged has a pressure of 180 atm at 20°C. The volume of the tank is 11.3 L. (a) What would the volume of the air be at 1.00 atm and at the same temperature? (b) Before entering the water, a person consumes 2.0 L of air in each breath, and breathes 12 times a minute. At this rate, how long would the tank last? (c) At a depth of 20.0 m in sea water at a temperature of 10°C, how long would the same tank last assuming the breathing rate does not change? 83. A temperature controller, designed to work in a steam environment, involves a bimetallic strip constructed of brass and steel, connected at their ends by rivets. Each of the metals is 2.0 mm thick. At 20°C, the strip is 10.0 cm long and straight. Find the radius of curvature r of the assembly at 100°C. See Fig. 17–22.

Steel

Brass

l0 10.0 cm l0 + ∆ l2

r ∆θ

85. Snorkelers breathe through short tubular "snorkels" while swimming under water very near the surface. One end of the snorkel attaches to the snorkeler's mouth while the other end protrudes above the water's surface. Unfortunately, snorkels cannot support breathing to any great depth: it is said that a typical snorkeler below a water depth of only about 30 cm cannot draw a breath through a snorkel. Based on this claim, what is the approximate fractional change in a typical person's lung volume when drawing a breath? Assume, in equilibrium, the air pressure in a snorkeler's lungs matches that of the surrounding water pressure. * Numerical/Computer * 86. (II) A thermocouple consists of a junction of two different types of materials that produces a voltage depending on its temperature. A thermocouple's voltages were recorded when at different temperatures as follows: Temperature (°C) Voltage (mV)

100 2.96

200 5.90

300 8.92

Use a spreadsheet to fit these data to a cubic equation and determine the temperature when the thermocouple produces 3.21 mV. Get a second value of the temperature by fitting the data to a quadratic equation. * 87. (III) You have a vial of an unknown liquid which might be octane (gasoline), water, glycerin, or ethyl alcohol. You are trying to determine its identity by studying how its volume changes with temperature changes. You fill a Pyrex graduated cylinder to 100.00 mL with the liquid when the liquid and the cylinder are at 0.000°C. You raise the temperature in five-degree increments, allowing the graduated cylinder and liquid to come to equilibrium at each temperature. You read the volumes listed below off the graduated cylinder at each temperature. Take into account the expansion of the Pyrex glass cylinder. Graph the data, possibly using a spreadsheet program, and determine the slope of the line to find the effective (combined) coefficient of volume expansion b. Then determine b for the liquid and which liquid is in the vial. Temperature (°C)

Volume Reading (apparent mL)

0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000 40.000 45.000 50.000

100.00 100.24 100.50 100.72 100.96 101.26 101.48 101.71 101.97 102.20 102.46

l0 + ∆ l1

(a)

50 1.41

(b) FIGURE 17–22 Problem 83.

84. A copper wire sags 50.0 cm between two utility poles 30.0 m apart when the temperature is – 15°C. Estimate the amount of sag when the temperature is ± 35°C. [Hint: An estimate can be made by assuming the shape of the wire is approximately an arc of a circle; hard equations can sometimes be solved by guessing values.]

Answers to Exercises A: B: C: D:

–40°. (d). 8 mm. (i) Higher, (ii) same, (iii) lower.

E: (a). F: (b). G: (b) Less.

General Problems

475

In this winter scene in Yellowstone Park, we recognize the three states of matter for water: as a liquid, as a solid (snow and ice), and as a gas (steam). In this Chapter we examine the microscopic theory of matter as atoms or molecules that are always in motion, which we call kinetic theory. We will see that the temperature of a gas is directly related to the average translational kinetic energy of its molecules. We will consider ideal gases, but we will also look at real gases and how they change phase, including evaporation, vapor pressure, and humidity.

P T E R

C H

CONTENTS 18–1 The Ideal Gas Law and the Molecular Interpretation of Temperature 18–2 Distribution of Molecular Speeds 18–3 Real Gases and Changes of Phase 18–4 Vapor Pressure and Humidity 18–5 Van der Waals Equation * of State *18–6 Mean Free Path *18–7 Diffusion

Kinetic Theory of Gases CHAPTER-OPENING QUESTION—Guess now! The typical speed of an air molecule at room temperature (20°C) is (a) nearly at rest (6 10 kmh). (b) on the order of 10 kmh. (c) on the order of 100 kmh. (d) on the order of 1000 kmh. (e) nearly the speed of light. he analysis of matter in terms of atoms in continuous random motion is called kinetic theory. We now investigate the properties of a gas from the point of view of kinetic theory, which is based on the laws of classical mechanics. But to apply Newton's laws to each one of the vast number of molecules in a gas (7 1025m3 at STP) is far beyond the capability of any present computer. Instead we take a statistical approach and determine averages of certain quantities, and these averages correspond to macroscopic variables. We will, of course, demand that our microscopic description correspond to the macroscopic properties of gases; otherwise our theory would be of little value. Most importantly, we will arrive at an important relation between the average kinetic energy of molecules in a gas and the absolute temperature.

18–1

The Ideal Gas Law and the Molecular Interpretation of Temperature

We make the following assumptions about the molecules in a gas. These assumptions reflect a simple view of a gas, but nonetheless the results they predict correspond well to the essential features of real gases that are at low pressures and far from the liquefaction point. Under these conditions real gases follow the ideal gas law quite closely, and indeed the gas we now describe is referred to as an ideal gas.

476

The assumptions, which represent the basic postulates of the kinetic theory for an ideal gas, are 1. There are a large number of molecules, N, each of mass m, moving in random directions with a variety of speeds. This assumption is in accord with our observation that a gas fills its container and, in the case of air on Earth, is kept from escaping only by the force of gravity. 2. The molecules are, on average, far apart from one another. That is, their average separation is much greater than the diameter of each molecule. 3. The molecules are assumed to obey the laws of classical mechanics, and are assumed to interact with one another only when they collide. Although molecules exert weak attractive forces on each other between collisions, the potential energy associated with these forces is small compared to the kinetic energy, and we ignore it for now. 4. Collisions with another molecule or the wall of the vessel are assumed to be perfectly elastic, like the collisions of perfectly elastic billiard balls (Chapter 9). We assume the collisions are of very short duration compared to the time between collisions. Then we can ignore the potential energy associated with collisions in comparison to the kinetic energy between collisions. We can see immediately how this kinetic view of a gas can explain Boyle's law (Section 17–6).The pressure exerted on a wall of a container of gas is due to the constant bombardment of molecules. If the volume is reduced by (say) half, the molecules are closer together and twice as many will be striking a given area of the wall per second. Hence we expect the pressure to be twice as great, in agreement with Boyle's law. Now let us calculate quantitatively the pressure a gas exerts on its container as based on kinetic theory. We imagine that the molecules are inside a rectangular container (at rest) whose ends have area A and whose length is l, as shown in Fig. 18–1a. The pressure exerted by the gas on the walls of its container is, according to our model, due to the collisions of the molecules with the walls. Let us focus our attention on the wall, of area A, at the left end of the container and examine what happens when one molecule strikes this wall, as shown in Fig. 18–1b. This molecule exerts a force on the wall, and according to Newton's third law the wall exerts an equal and opposite force back on the molecule. The magnitude of this force on the molecule, according to Newton's second law, is equal to the molecule's rate of change of momentum, F = dpdt (Eq. 9–2). Assuming the collision is elastic, only the x component of the molecule's momentum changes, and it changes from –mvx (it is moving in the negative x direction) to ±mvx . Thus the change in the molecule's momentum, ¢(mv), which is the final momentum minus the initial momentum, is

y l

(a)

y l

¢(mv) = mvx - A –mvx B = 2mvx

for one collision. This molecule will make many collisions with the wall, each separated by a time ¢ t, which is the time it takes the molecule to travel across the container and back again, a distance (x component) equal to 2l. Thus 2l = vx ¢ t, or 2l . ¢t = vx The time ¢ t between collisions is very small, so the number of collisions per second is very large. Thus the average force—averaged over many collisions—will be equal to the momentum change during one collision divided by the time between collisions (Newton's second law): ¢(mv) 2mvx mv2x . F = = = [due to one molecule] ¢t 2lvx l During its passage back and forth across the container, the molecule may collide with the tops and sides of the container, but this does not alter its x component of momentum and thus does not alter our result. It may also collide with other molecules, which may change its vx . However, any loss (or gain) of momentum is acquired by other molecules, and because we will eventually sum over all the molecules, this effect will be included. So our result above is not altered. SECTION 18–1

(b)

FIGURE 18–1 (a) Molecules of a gas moving about in a rectangular container. (b) Arrows indicate the momentum of one molecule as it rebounds from the end wall.

The Ideal Gas Law and the Molecular Interpretation of Temperature

477

The actual force due to one molecule is intermittent, but because a huge number of molecules are striking the wall per second, the force is, on average, nearly constant. To calculate the force due to all the molecules in the container, we have to add the contributions of each. Thus the net force on the wall is m 2 F = Av + vx2 2 + p + vx2 N B, l x1 where vx 1 means vx for molecule number 1 (we arbitrarily assign each molecule a number) and the sum extends over the total number of molecules N in the container. The average value of the square of the x component of velocity is vx2 1 + vx2 2 + p + vx2 N , (18–1) Ox = N where the overbar ( ) means "average." Thus we can write the force as m F = NOx . l We know that the square of any vector is equal to the sum of the squares of its components (theorem of Pythagoras). Thus v2 = v2x + v2y + v2z for any velocity v. Taking averages, we obtain O = Ox + Oy + Oz . Since the velocities of the molecules in our gas are assumed to be random, there is no preference to one direction or another. Hence Ox = Oy = Oz . Combining this relation with the one just above, we get O = 3Ox . We substitute this into the equation for net force F: F =

m O. N l 3

The pressure on the wall is then P =

F 1 NmO = A 3 Al

P =

1 NmO , 3 V

or (18–2)

where V = lA is the volume of the container. This is the result we were seeking, the pressure exerted by a gas on its container expressed in terms of molecular properties. Equation 18–2 can be rewritten in a clearer form by multiplying both sides by V and rearranging the right-hand side: 2 1 3 NA 2 mOB.

PV =

(18–3)

1 2 mO

The quantity is the average translational kinetic energy † of the molecules in the gas. If we compare Eq. 18–3 with Eq. 17–4, the ideal gas law PV = NkT, we see that the two agree if TEMPERATURE RELATED TO AVERAGE KINETIC ENERGY OF MOLECULES

2 1 3 A 2 mOB

† =

= kT,

1 2 mO

3 2 kT.

[ideal gas] (18–4)

This equation tells us that the average translational kinetic energy of molecules in random motion in an ideal gas is directly proportional to the absolute temperature of the gas. The higher the temperature, according to kinetic theory, the faster the molecules are moving on the average. This relation is one of the triumphs of the kinetic theory.

478

CHAPTER 18

Kinetic Theory of Gases

EXAMPLE 18–1 Molecular kinetic energy. What is the average translational kinetic energy of molecules in an ideal gas at 37°C? APPROACH We use the absolute temperature in Eq. 18–4. SOLUTION We change 37°C to 310 K and insert into Eq. 18–4: † =

3 2 kT

3 2

A1.38 * 10–23 JKB(310 K) = 6.42 * 10 –21 J.

NOTE A mole of molecules would have a total translational kinetic energy equal to A6.42 * 10–21 JBA6.02 * 1023 B = 3860 J, which equals the kinetic energy of a 1-kg stone traveling almost 90 ms. EXERCISE A In a mixture of the gases oxygen and helium, which statement is valid: (a) the helium molecules will be moving faster than the oxygen molecules, on average; (b) both kinds of molecules will be moving at the same speed; (c) the oxygen molecules will, on average, be moving more rapidly than the helium molecules; (d) the kinetic energy of the helium will exceed that of the oxygen; (e) none of the above.

Equation 18–4 holds not only for gases, but also applies reasonably accurately to liquids and solids. Thus the result of Example 18–1 would apply to molecules within living cells at body temperature (37°C). We can use Eq. 18–4 to calculate how fast molecules are moving on the average. Notice that the average in Eqs. 18–1 through 18–4 is over the square of the speed. The square root of O is called the root-mean-square speed, vrms (since we are taking the square root of the mean of the square of the speed): vrms = 3O =

3kT . B m

(18–5)

EXAMPLE 18–2 Speeds of air molecules. What is the rms speed of air molecules (O2 and N2) at room temperature (20°C)? APPROACH To obtain vrms , we need the masses of O2 and N2 molecules and then apply Eq. 18–5 to oxygen and nitrogen separately, since they have different masses. SOLUTION The masses of one molecule of O2 (molecular mass = 32 u) and N2 (molecular mass = 28 u) are (where 1 u = 1.66 * 10–27 kg) m(O2) = (32)A1.66 * 10–27 kgB = 5.3 * 10–26 kg, m(N2) = (28)A1.66 * 10–27 kgB = 4.6 * 10–26 kg. Thus, for oxygen vrms =

(3)A1.38 * 10–23 JKB(293 K) 3kT = = 480 ms, B m C A5.3 * 10–26 kgB

and for nitrogen the result is vrms = 510 ms. These speeds are more than 1700 kmh or 1000 mih, and are greater than the speed of sound L 340 ms at 20°C (Chapter 16). NOTE The speed vrms is a magnitude only. The velocity of molecules averages to zero: the velocity has direction, and as many molecules move to the right as to the left, as many up as down, as many inward as outward. EXERCISE B Now you can return to the Chapter-Opening Question, page 476, and answer it correctly. Try to explain why you may have answered differently the first time. EXERCISE C If you double the volume of a gas while keeping the pressure and number of moles constant, the average (rms) speed of the molecules (a) doubles, (b) quadruples, (c) increases by 12 , (d) is half, (e) is 14 . EXERCISE D By what factor must the absolute temperature change to double vrms? (a) 12 ; (b) 2; (c) 2 12 ; (d) 4; (e) 16.

SECTION 18–1

The Ideal Gas Law and the Molecular Interpretation of Temperature

479

CONCEPTUAL EXAMPLE 18–3 Less gas in the tank. A tank of helium is used to fill balloons. As each balloon is filled, the number of helium atoms remaining in the tank decreases. How does this affect the rms speed of molecules remaining in the tank? RESPONSE The rms speed is given by Eq. 18–5: vrms = 13kTm. So only the temperature matters, not pressure P or number of moles n. If the tank remains at a constant (ambient) temperature, then the rms speed remains constant even though the pressure of helium in the tank decreases. In a collection of molecules, the average speed, v, is the average of the magnitudes of the speeds themselves; v is generally not equal to vrms . To see the difference between the average speed and the rms speed, consider the following Example. EXAMPLE 18–4 Average speed and rms speed. Eight particles have the following speeds, given in ms: 1.0, 6.0, 4.0, 2.0, 6.0, 3.0, 2.0, 5.0. Calculate (a) the average speed and (b) the rms speed. APPROACH In (a) we sum the speeds and divide by N = 8. In (b) we square each speed, sum the squares, divide by N = 8, and take the square root. SOLUTION (a) The average speed is 1.0 + 6.0 + 4.0 + 2.0 + 6.0 + 3.0 + 2.0 + 5.0 v = = 3.6 ms. 8 (b) The rms speed is (Eq. 18–1): (1.0)2 + (6.0)2 + (4.0)2 + (2.0)2 + (6.0)2 + (3.0)2 + (2.0)2 + (5.0)2 ms C 8 = 4.0 ms.

vrms =

We see in this Example that v and vrms are not necessarily equal. In fact, for an ideal gas they differ by about 8%. We will see in the next Section how to calculate v for an ideal gas. We already have the tool to calculate vrms (Eq. 18–5).

* Kinetic Energy Near Absolute Zero Equation 18–4, † = 32 kT, implies that as the temperature approaches absolute zero, the kinetic energy of molecules approaches zero. Modern quantum theory, however, tells us this is not quite so. Instead, as absolute zero is approached, the kinetic energy approaches a very small nonzero minimum value. Even though all real gases become liquid or solid near 0 K, molecular motion does not cease, even at absolute zero.

18–2

Relative number of molecules f(v)

FIGURE 18–2 Distribution of speeds of molecules in an ideal gas. Note that v and vrms are not at the peak of the curve. This is because the curve is skewed to the right: it is not symmetrical. The speed at the peak of the curve is the "most probable speed," vp .

480

vp vrms v

CHAPTER 18

Speed v

Distribution of Molecular Speeds

The Maxwell Distribution The molecules in a gas are assumed to be in random motion, which means that many molecules have speeds less than the average speed and others have speeds greater than the average. In 1859, James Clerk Maxwell (1831–1879) worked out a formula for the most probable distribution of speeds in a gas containing N molecules. We will not give a derivation here but merely quote his result: 3

mv2 2 m –1 ≤ v2 e 2 kT f(v) = 4pN ¢ 2pkT

(18–6)

where f(v) is called the Maxwell distribution of speeds, and is plotted in Fig. 18–2. The quantity f(v) dv represents the number of molecules that have speed between v and v + dv. Notice that f(v) does not give the number of molecules with speed v; f(v) must be multiplied by dv to give the number of molecules (the number of molecules depends on the "width" or "range" of velocities included, dv). In the formula for f(v), m is the mass of a single molecule, T is the absolute temperature, and k is the Boltzmann constant. Since N is the total number of molecules in the gas,

Kinetic Theory of Gases

when we sum over all the molecules in the gas we must get N; thus we must have

f(v) dv q

= N.

(Problem 22 is an exercise to show that this is true.) Experiments to determine the distribution of speeds in real gases, starting in the 1920s, confirmed with considerable accuracy the Maxwell distribution (for gases at not too high a pressure) and the direct proportion between average kinetic energy and absolute temperature, Eq. 18–4. The Maxwell distribution for a given gas depends only on the absolute temperature. Figure 18–3 shows the distributions for two different temperatures. Just as vrms increases with temperature, so the whole distribution curve shifts to the right at higher temperatures. Figure 18–3 illustrates how kinetic theory can be used to explain why many chemical reactions, including those in biological cells, take place more rapidly as the temperature increases. Most chemical reactions take place in a liquid solution, and the molecules in a liquid have a distribution of speeds close to the Maxwell distribution. Two molecules may chemically react only if their kinetic energy is great enough so that when they collide, they partially penetrate into each other. The minimum energy required is called the activation energy, EA , and it has a specific value for each chemical reaction. The molecular speed corresponding to a kinetic energy of EA for a particular reaction is indicated in Fig. 18–3. The relative number of molecules with energy greater than this value is given by the area under the curve to the right of v AEA B, shown in Fig. 18–3 by the two different shadings. We see that the number of molecules that have kinetic energies in excess of EA increases greatly for only a small increase in temperature. The rate at which a chemical reaction occurs is proportional to the number of molecules with energy greater than EA , and thus we see why reaction rates increase rapidly with increased temperature.

Relative number of molecules

T = 273 K (0°C) T = 310 K (37°C)

Speed

v (EA)

FIGURE 18–3 Distribution of molecular speeds for two different temperatures.

P H Y S I C S

A P P L I E D

How chemical reactions depend on temperature

* Calculations Using the Maxwell Distribution Let us see how the Maxwell distribution can be used to obtain some interesting results. EXAMPLE 18–5 Determining £ and £p . Determine formulas for (a) the average speed, v, and (b) the most probable speed, vp , of molecules in an ideal gas at temperature T. APPROACH (a) The average value of any quantity is found by multiplying each possible value of the quantity (here, speed) by the number of molecules that have that value, and then summing all these numbers and dividing by N (the total number). For (b), we want to find where the curve of Fig. 18–2 has zero slope; so we set dfdv = 0. SOLUTION (a) We are given a continuous distribution of speeds (Eq. 18–6), so the sum over the speeds becomes an integral over the product of v and the number f(v) dv that have speed v:

v f(v) dv q

= 4p a

q mv2 2 m –1 b v3e 2 kT dv. 2pkT N 0 We can integrate by parts or look up the definite integral in a Table, and obtain

v =

2 2 2 2k T m 8 kT kT . v = 4p a b ¢ ≤ = L 1.60 2 2pkT Bp m Bm m

(b) The most probable speed is that speed which occurs more than any others, and thus is that speed where f(v) has its maximum value. At the maximum of the curve, the slope is zero: df(v)dv = 0. Taking the derivative of Eq. 18–6 gives 3

mv2 mv2 df(v) 2 m 2mv3 – 2kT – = 4pN a b ¢ 2ve 2kT e ≤ = 0. dv 2pkT 2kT Solving for v, we find

2kT kT . L 1.41 B m Bm (Another solution is v = 0, but this corresponds to a minimum, not a maximum.) vp =

SECTION 18–2

481

In summary, Most probable speed, vp

vp =

kT kT L 1.41 B m Bm

(18–7a)

Average speed, v

v =

8 kT kT L 1.60 p m B Bm

(18–7b)

and from Eq. 18–5 vrms =

rms speed, vrms

kT kT . L 1.73 B m Bm 3

These are all indicated in Fig. 18–2. From Eq. 18–6 and Fig. 18–2, it is clear that the speeds of molecules in a gas vary from zero up to many times the average speed, but as can be seen from the graph, most molecules have speeds that are not far from the average. Less than 1% of the molecules exceed four times vrms .

18–3 B′ A′

Real Gases and Changes of Phase

The ideal gas law PV = NkT

Gas

Liquid

c a

A B C D

Vapor Liquidvapor region

V FIGURE 18–4 PV diagram for a real substance. Curves A, B, C, and D represent the same substance at different temperatures ATA 7 TB 7 TC 7 TD B.

482

CHAPTER 18

is an accurate description of the behavior of a real gas as long as the pressure is not too high and as long as the temperature is far from the liquefaction point. But what happens when these two criteria are not satisfied? First we discuss real gas behavior, and then we examine how kinetic theory can help us understand this behavior. Let us look at a graph of pressure plotted against volume for a given amount of gas. On such a "PV diagram," Fig. 18–4, each point represents an equilibrium state of the given substance. The various curves (labeled A, B, C, and D) show how the pressure varies, as the volume is changed at constant temperature, for several different values of the temperature. The dashed curve A¿ represents the behavior of a gas as predicted by the ideal gas law; that is, PV = constant. The solid curve A represents the behavior of a real gas at the same temperature. Notice that at high pressure, the volume of a real gas is less than that predicted by the ideal gas law. The curves B and C in Fig. 18–4 represent the gas at successively lower temperatures, and we see that the behavior deviates even more from the curves predicted by the ideal gas law (for example, B¿ ), and the deviation is greater the closer the gas is to liquefying. To explain this, we note that at higher pressure we expect the molecules to be closer together. And, particularly at lower temperatures, the potential energy associated with the attractive forces between the molecules (which we ignored before) is no longer negligible compared to the now reduced kinetic energy of the molecules. These attractive forces tend to pull the molecules closer together so that at a given pressure, the volume is less than expected from the ideal gas law, as in Fig. 18–4. At still lower temperatures, these forces cause liquefaction, and the molecules become very close together. Section 18–5 discusses in more detail the effect of these attractive molecular forces, as well as the effect of the volume which the molecules themselves occupy. Curve D represents the situation when liquefaction occurs. At low pressure on curve D (on the right in Fig. 18–4), the substance is a gas and occupies a large volume. As the pressure is increased, the volume decreases until point b is reached. Beyond b, the volume decreases with no change in pressure; the substance is gradually changing from the gas to the liquid phase. At point a, all of the substance has changed to

Kinetic Theory of Gases

SECTION 18–3

TABLE 18–1 Critical Temperatures and Pressures Critical Temperature Substance

°C

Critical Pressure (atm)

Water CO2 Oxygen Nitrogen Hydrogen Helium

374 31 – 118 – 147 – 239.9 – 267.9

647 304 155 126 33.3 5.3

218 72.8 50 33.5 12.8 2.3

218 1.0

0.006

Critical point s-l

P (atm)

Solid

Liquid

Gas l -v Vapor Triple v s point 0.00 0.01 100 374

T (°C)

FIGURE 18–5 Phase diagram for water (note that the scales are not linear). FIGURE 18–6 Phase diagram for carbon dioxide. Critical point

73 Solid

P (atm)

liquid. Further increase in pressure reduces the volume only slightly—liquids are nearly incompressible—so on the left the curve is very steep as shown. The colored area under the dashed line represents the region where the gas and liquid phases exist together in equilibrium. Curve C in Fig. 18–4 represents the behavior of the substance at its critical temperature; the point c (the one point where curve C is horizontal) is called the critical point. At temperatures less than the critical temperature (and this is the definition of the term), a gas will change to the liquid phase if sufficient pressure is applied. Above the critical temperature, no amount of pressure can cause a gas to change phase and become a liquid. The critical temperatures for various gases are given in Table 18–1. Scientists tried for many years to liquefy oxygen without success. Only after the discovery of the behavior of substances associated with the critical point was it realized that oxygen can be liquefied only if first cooled below its critical temperature of – 118°C. Often a distinction is made between the terms "gas" and "vapor": a substance below its critical temperature in the gaseous state is called a vapor; above the critical temperature, it is called a gas. The behavior of a substance can be diagrammed not only on a PV diagram but also on a PT diagram. A PT diagram, often called a phase diagram, is particularly convenient for comparing the different phases of a substance. Figure 18–5 is the phase diagram for water. The curve labeled l-v represents those points where the liquid and vapor phases are in equilibrium—it is thus a graph of the boiling point versus pressure. Note that the curve correctly shows that at a pressure of 1 atm the boiling point is 100°C and that the boiling point is lowered for a decreased pressure. The curve s-l represents points where solid and liquid exist in equilibrium and thus is a graph of the freezing point versus pressure. At 1 atm, the freezing point of water is 0°C, as shown. Notice also in Fig. 18–5 that at a pressure of 1 atm, the substance is in the liquid phase if the temperature is between 0°C and 100°C, but is in the solid or vapor phase if the temperature is below 0°C or above 100°C. The curve labeled s-v is the sublimation point versus pressure curve. Sublimation refers to the process whereby at low pressures a solid changes directly into the vapor phase without passing through the liquid phase. For water, sublimation occurs if the pressure of the water vapor is less than 0.0060 atm. Carbon dioxide, which in the solid phase is called dry ice, sublimates even at atmospheric pressure (Fig. 18–6). The intersection of the three curves (in Fig. 18–5) is the triple point. For water this occurs at T = 273.16 K and P = 6.03 * 10–3 atm. It is only at the triple point that the three phases can exist together in equilibrium. Because the triple point corresponds to a unique value of temperature and pressure, it is precisely reproducible and is often used as a point of reference. For example, the standard of temperature is usually specified as exactly 273.16 K at the triple point of water, rather than 273.15 K at the freezing point of water at 1 atm. Notice that the solid liquid (s-l) curve for water slopes upward to the left. This is true only of substances that expand upon freezing: at a higher pressure, a lower temperature is needed to cause the liquid to freeze. More commonly, substances contract upon freezing and the s-l curve slopes upward to the right, as shown for carbon dioxide ACO2 B in Fig. 18–6. The phase transitions we have been discussing are the common ones. Some substances, however, can exist in several forms in the solid phase. A transition from one phase to another occurs at a particular temperature and pressure, just like ordinary phase changes. For example, ice has been observed in at least eight forms at very high pressure. Ordinary helium has two distinct liquid phases, called helium I and II. They exist only at temperatures within a few degrees of absolute zero. Helium II exhibits very unusual properties referred to as superfluidity. It has essentially zero viscosity and exhibits strange properties such as climbing up the sides of an open container. Also interesting are liquid crystals (used for TV and computer monitors, Section 35–11) which can be considered to be in a phase between liquid and solid.

5.11

Liquid

Triple point Vapor

−56.6

T (°C)

P H Y S I C S

A P P L I E D

Liquid crystals

Real Gases and Changes of Phase

483

18–4

Vapor Pressure and Humidity

Evaporation

P H Y S I C S

A P P L I E D

Evaporation cools

FIGURE 18–7 Vapor appears above a liquid in a closed container.

TABLE 18–2 Saturated Vapor Pressure of Water Saturated Vapor Pressure Temperature torr Pa (°C) ( = mm-Hg) ( = N m2) – 50 – 10 0 5 10 15 20 25 30 40 50 60 70† 80 90 100‡ 120 150 † ‡

0.030 1.95 4.58 6.54 9.21 12.8 17.5 23.8 31.8 55.3 92.5 149 234 355 526 760 1489 3570

2.60 6.11 8.72 1.23 1.71 2.33 3.17 4.24 7.37 1.23 1.99 3.12 4.73 7.01 1.01 1.99 4.76

4.0 * 102 * 102 * 102 * 103 * 103 * 103 * 103 * 103 * 103 * 104 * 104 * 104 * 104 * 104 * 105 * 105 * 105

Boiling point on summit of Mt. Everest. Boiling point at sea level.

484

CHAPTER 18

If a glass of water is left out overnight, the water level will have dropped by morning. We say the water has evaporated, meaning that some of the water has changed to the vapor or gas phase. This process of evaporation can be explained on the basis of kinetic theory. The molecules in a liquid move past one another with a variety of speeds that follow, approximately, the Maxwell distribution. There are strong attractive forces between these molecules, which is what keeps them close together in the liquid phase. A molecule near the surface of the liquid may, because of its speed, leave the liquid momentarily. But just as a rock thrown into the air returns to the Earth, so the attractive forces of the other molecules can pull the vagabond molecule back to the liquid surface—that is, if its velocity is not too large. A molecule with a high enough velocity, however, will escape the liquid entirely (like an object leaving Earth with a high enough speed, Section 8–7), and become part of the gas phase. Only those molecules that have kinetic energy above a particular value can escape to the gas phase. We have already seen that kinetic theory predicts that the relative number of molecules with kinetic energy above a particular value (such as EA in Fig. 18–3) increases with temperature. This is in accord with the well-known observation that the evaporation rate is greater at higher temperatures. Because it is the fastest molecules that escape from the surface, the average speed of those remaining is less. When the average speed is less, the absolute temperature is less. Thus kinetic theory predicts that evaporation is a cooling process. You have no doubt noticed this effect when you stepped out of a warm shower and felt cold as the water on your body began to evaporate; and after working up a sweat on a hot day, even a slight breeze makes you feel cool through evaporation.

Vapor Pressure Air normally contains water vapor (water in the gas phase), and it comes mainly from evaporation. To look at this process in a little more detail, consider a closed container that is partially filled with water (or another liquid) and from which the air has been removed (Fig. 18–7). The fastest moving molecules quickly evaporate into the empty space above the liquid's surface. As they move about, some of these molecules strike the liquid surface and again become part of the liquid phase: this is called condensation. The number of molecules in the vapor increases until a point is reached when the number of molecules returning to the liquid equals the number leaving in the same time interval. Equilibrium then exists, and the space above the liquid surface is said to be saturated. The pressure of the vapor when it is saturated is called the saturated vapor pressure (or sometimes simply the vapor pressure). The saturated vapor pressure does not depend on the volume of the container. If the volume above the liquid were reduced suddenly, the density of molecules in the vapor phase would be increased temporarily. More molecules would then be striking the liquid surface per second. There would be a net flow of molecules back to the liquid phase until equilibrium was again reached, and this would occur at the same value of the saturated vapor pressure, as long as the temperature had not changed. The saturated vapor pressure of any substance depends on the temperature. At higher temperatures, more molecules have sufficient kinetic energy to break from the liquid surface into the vapor phase. Hence equilibrium will be reached at a higher pressure. The saturated vapor pressure of water at various temperatures is given in Table 18–2. Notice that even solids—for example, ice—have a measurable saturated vapor pressure. In everyday situations, evaporation from a liquid takes place into the air above it rather than into a vacuum. This does not materially alter the discussion above relating to Fig. 18–7. Equilibrium will still be reached when there are sufficient molecules in the gas phase that the number reentering the liquid equals the number leaving. The concentration of particular molecules (such as water) in the gas phase

Kinetic Theory of Gases

is not affected by the presence of air, although collisions with air molecules may lengthen the time needed to reach equilibrium. Thus equilibrium occurs at the same value of the saturated vapor pressure as if air were not there. If the container is large or is not closed, all the liquid may evaporate before saturation is reached. And if the container is not sealed—as, for example, a room in your house—it is not likely that the air will become saturated with water vapor (unless it is raining outside).

Boiling The saturated vapor pressure of a liquid increases with temperature. When the temperature is raised to the point where the saturated vapor pressure at that temperature equals the external pressure, boiling occurs (Fig. 18–8). As the boiling point is approached, tiny bubbles tend to form in the liquid, which indicate a change from the liquid to the gas phase. However, if the vapor pressure inside the bubbles is less than the external pressure, the bubbles immediately are crushed. As the temperature is increased, the saturated vapor pressure inside a bubble eventually becomes equal to or exceeds the external pressure. The bubble will then not collapse but can rise to the surface. Boiling has then begun. A liquid boils when its saturated vapor pressure equals the external pressure. This occurs for water at a pressure of 1 atm (760 torr) at 100°C, as can be seen from Table 18–2. The boiling point of a liquid clearly depends on the external pressure. At high elevations, the boiling point of water is somewhat less than at sea level since the air pressure is less up there. For example, on the summit of Mt. Everest (8850 m) the air pressure is about one-third of what it is at sea level, and from Table 18–2 we can see that water will boil at about 70°C. Cooking food by boiling takes longer at high elevations, since the temperature is less. Pressure cookers, however, reduce cooking time, because they build up a pressure as high as 2 atm, allowing higher boiling temperatures to be attained.

FIGURE 18–8 Boiling: bubbles of water vapor float upward from the bottom (where the temperature is highest).

Partial Pressure and Humidity When we refer to the weather as being dry or humid, we are referring to the water vapor content of the air. In a gas such as air, which is a mixture of several types of gases, the total pressure is the sum of the partial pressures of each gas present.† By partial pressure, we mean the pressure each gas would exert if it alone were present. The partial pressure of water in the air can be as low as zero and can vary up to a maximum equal to the saturated vapor pressure of water at the given temperature. Thus, at 20°C, the partial pressure of water cannot exceed 17.5 torr (see Table 18–2). The relative humidity is defined as the ratio of the partial pressure of water vapor to the saturated vapor pressure at a given temperature. It is usually expressed as a percentage: Relative humidity =

partial pressure of H 2O * 100%. saturated vapor pressure of H 2O

Thus, when the humidity is close to 100%, the air holds nearly all the water vapor it can. EXAMPLE 18–6 Relative humidity. On a particular hot day, the temperature is 30°C and the partial pressure of water vapor in the air is 21.0 torr. What is the relative humidity? APPROACH From Table 18–2, we see that the saturated vapor pressure of water at 30°C is 31.8 torr. SOLUTION The relative humidity is thus 21.0 torr * 100% = 66%. 31.8 torr † For example, 78% (by volume) of air molecules are nitrogen and 21% oxygen, with much smaller amounts of water vapor, argon, and other gases. At an air pressure of 1 atm, oxygen exerts a partial pressure of 0.21 atm and nitrogen 0.78 atm.

SECTION 18–4

Vapor Pressure and Humidity

485

P H Y S I C S

A P P L I E D

Humidity and comfort

P H Y S I C S

A P P L I E D

Weather

FIGURE 18–9 Fog or mist settling around a castle where the temperature has dropped below the dew point.

Humans are sensitive to humidity. A relative humidity of 40–50% is generally optimum for both health and comfort. High humidity, particularly on a hot day, reduces the evaporation of moisture from the skin, which is one of the body's vital mechanisms for regulating body temperature. Very low humidity, on the other hand, can dry the skin and mucous membranes. Air is saturated with water vapor when the partial pressure of water in the air is equal to the saturated vapor pressure at that temperature. If the partial pressure of water exceeds the saturated vapor pressure, the air is said to be supersaturated. This situation can occur when a temperature decrease occurs. For example, suppose the temperature is 30°C and the partial pressure of water is 21 torr, which represents a humidity of 66% as we saw in Example 18–6. Suppose now that the temperature falls to, say, 20°C, as might happen at nightfall. From Table 18–2 we see that the saturated vapor pressure of water at 20°C is 17.5 torr. Hence the relative humidity would be greater than 100%, and the supersaturated air cannot hold this much water. The excess water may condense and appear as dew, or as fog or rain (Fig. 18–9). When air containing a given amount of water is cooled, a temperature is reached where the partial pressure of water equals the saturated vapor pressure. This is called the dew point. Measurement of the dew point is the most accurate means of determining the relative humidity. One method uses a polished metal surface in contact with air, which is gradually cooled down. The temperature at which moisture begins to appear on the surface is the dew point, and the partial pressure of water can then be obtained from saturated vapor pressure Tables. If, for example, on a given day the temperature is 20°C and the dew point is 5°C, then the partial pressure of water (Table 18–2) in the 20°C air is 6.54 torr, whereas its saturated vapor pressure is 17.5 torr; hence the relative humidity is 6.5417.5 = 37%. EXERCISE E As the air warms up in the afternoon, how would the relative humidity change if there were no further evaporation? It would (a) increase, (b) decrease, (c) stay the same.

CONCEPTUAL EXAMPLE 18–7 Dryness in winter. Why does the air inside heated buildings seem very dry on a cold winter day? RESPONSE Suppose the relative humidity outside on a –10°C day is 50%. Table 18–2 tells us the partial pressure of water in the air is about 1.0 torr. If this air is brought indoors and heated to ±20°C, the relative humidity is (1.0 torr)(17.5 torr) = 5.7%. Even if the outside air were saturated at a partial pressure of 1.95 torr, the inside relative humidity would be at a low 11%.

* 18–5

FIGURE 18–10 Molecules, of radius r, colliding.

486

CHAPTER 18

Van der Waals Equation of State

In Section 18–3, we discussed how real gases deviate from ideal gas behavior, particularly at high densities or when near condensing to a liquid. We would like to understand these deviations using a microscopic (molecular) point of view. J. D. van der Waals (1837–1923) analyzed this problem and in 1873 arrived at an equation of state which fits real gases more accurately than the ideal gas law. His analysis is based on kinetic theory but takes into account: (1) the finite size of molecules (we previously neglected the actual volume of the molecules themselves, compared to the total volume of the container, and this assumption becomes poorer as the density increases and molecules become closer together); (2) the range of the forces between molecules may be greater than the size of the molecules (we previously assumed that intermolecular forces act only during collisions, when the molecules are "in contact"). Let us now look at this analysis and derive the van der Waals equation of state. Assume the molecules in a gas are spherical with radius r. If we assume these molecules behave like hard spheres, then two molecules collide and bounce off one another if the distance between their centers (Fig. 18–10) gets as small as 2r. Thus the actual volume in which the molecules can move about is somewhat less than the volume V of the container holding the gas. The amount of "unavailable volume" depends on the number of molecules and on their size. Let b represent the "unavailable volume per mole" of gas.

Then in the ideal gas law we replace V by (V - nb), where n is the number of moles, and we obtain P(V - nb) = nRT. If we divide through by n, we have V - b b = RT. Pa (18–8) n This relation (sometimes called the Clausius equation of state) predicts that for a given temperature T and volume V, the pressure P will be greater than for an ideal gas. This makes sense since the reduced "available" volume means the number of collisions with the walls is increased. Next we consider the effects of attractive forces between molecules, which are responsible for holding molecules in the liquid and solid states at lower temperatures. These forces are electrical in nature and although they act even when molecules are not touching, we assume their range is small—that is, they act mainly between nearest neighbors. Molecules at the edge of the gas, headed toward a wall of the container, are slowed down by a net force pulling them back into the gas. Thus these molecules will exert less force and less pressure on the wall than if there were no attractive forces. The reduced pressure will be proportional to the density of molecules in the layer of gas at the surface, and also to the density in the next layer, which exerts the inward force.† Therefore we expect the pressure to be reduced by a factor proportional to the density squared (nV)2, here written as moles per volume. If the pressure P is given by Eq. 18–8, then we should reduce this by an amount a(nV)2 where a is a proportionality constant. Thus we have RT a P = (Vn) - b (Vn)2 or a V - b b = RT, ¢P + ≤a (18–9) 2 n (Vn) which is the van der Waals equation of state. The constants a and b in the van der Waals equation are different for different gases and are determined by fitting to experimental data for each gas. For CO2 gas, the best fit is obtained for a = 0.36 N m4mol2 and b = 4.3 * 10 –5 m3mol. Figure 18–11 shows a typical PV diagram for Eq. 18–9 (a "van der Waals gas") for four different temperatures, with detailed caption, and it should be compared to Fig. 18–4 for real gases. Neither the van der Waals equation of state nor the many other equations of state that have been proposed are accurate for all gases under all conditions. Yet Eq. 18–9 is a very useful relation. And because it is quite accurate for many situations, its derivation gives us further insight into the nature of gases at the microscopic level. Note that at low densities, a(Vn)2 V P and b V Vn, so that the van der Waals equation reduces to the equation of state for an ideal gas, PV = nRT.

* 18–6

Critical point

TA TB TC = Tcr TD

FIGURE 18–11 PV diagram for a van der Waals gas, shown for four different temperatures. For TA , TB , and TC (TC is chosen equal to the critical temperature), the curves fit experimental data very well for most gases. The curve labeled TD , a temperature below the critical point, passes through the liquid–vapor region. The maximum (point b) and minimum (point d) would seem to be artifacts, since we usually see constant pressure, as indicated by the horizontal dashed line (and Fig. 18–4). However, for very pure supersaturated vapors or supercooled liquids, the sections ab and ed, respectively, have been observed. (The section bd would be unstable and has not been observed.)

Mean Free Path

If gas molecules were truly point particles, they would have zero cross-section and never collide with one another. If you opened a perfume bottle, you would be able to smell it almost instantaneously across the room, since molecules travel hundreds of meters per second. In reality, it takes time before you detect an odor and, according to kinetic theory, this must be due to collisions between molecules of nonzero size. If we were to follow the path of a particular molecule, we would expect to see it follow a zigzag path as shown in Fig. 18–12. Between each collision the molecule would move in a straight-line path. (Not quite true if we take account of the small intermolecular forces that act between collisions.) An important parameter for a given situation is the mean free path, which is defined as the average distance a molecule travels between collisions. We would expect that the greater the gas density, and the larger the molecules, the shorter the mean free path would be. We now determine the nature of this relationship for an ideal gas. † This is similar to the gravitational force in which the force on mass m1 due to mass m2 is proportional to the product of their masses (Newton's law of universal gravitation, Chapter 6).

FIGURE 18–12 Zigzag path of a molecule colliding with other molecules.

*SECTION 18–6

487

2r r

v ∆t FIGURE 18–13 Molecule at left moves to the right with speed v. It collides with any molecule whose center is within the cylinder of radius 2r.

Suppose our gas is made up of molecules which are hard spheres of radius r. A collision will occur whenever the centers of two molecules come within a distance 2r of one another. Let us follow a molecule as it traces a straight-line path. In Fig. 18–13, the dashed line represents the path of our particle if it makes no collisions. Also shown is a cylinder of radius 2r. If the center of another molecule lies within this cylinder, a collision will occur. (Of course, when a collision occurs the particle's path would change direction, as would our imagined cylinder, but our result won't be altered by unbending a zigzag cylinder into a straight one for purposes of calculation.) Assume our molecule is an average one, moving at the mean speed v in the gas. For the moment, let us assume that the other molecules are not moving, and that the concentration of molecules (number per unit volume) is NV. Then the number of molecules whose centers lie within the cylinder of Fig. 18–13 is NV times the volume of this cylinder, and this also represents the number of collisions that will occur. In a time ¢ t, our molecule travels a distance v ¢ t, so the length of the cylinder is v ¢ t and its volume is p(2r)2 v ¢ t. Hence the number of collisions that occur in a time ¢ t is (NV)p(2r)2 v ¢ t. We define the mean free path, lM , as the average distance between collisions. This distance is equal to the distance traveled (v ¢ t) in a time ¢ t divided by the number of collisions made in time ¢ t: lM =

v ¢t 1 . = 2 2 (NV)p(2r) v ¢ t 4pr (NV)

(18–10a)

Thus we see that lM is inversely proportional to the cross-sectional area A = pr 2 B of the molecules and to their concentration (number/volume), NV. However, Eq. 18–10a is not fully correct since we assumed the other molecules are all at rest. In fact, they are moving, and the number of collisions in a time ¢ t must depend on the relative speed of the colliding molecules, rather than on v. Hence the number of collisions per second is (NV)p(2r)2vrel ¢ t (rather than (NV)p(2r)2 v ¢ t), where vrel is the average relative speed of colliding molecules. A careful calculation shows that for a Maxwellian distribution of speeds vrel = 12v. Hence the mean free path is lM =

Mean free path

1 . 4p12r2(NV)

(18–10b)

EXAMPLE 18–8 ESTIMATE Mean free path of air molecules at STP. Estimate the mean free path of air molecules at STP, standard temperature and pressure (0°C, 1 atm). The diameter of O2 and N2 molecules is about 3 * 10 –10 m. APPROACH We saw in Example 17–10 that 1 mol of an ideal gas occupies a volume of 22.4 * 10 –3 m3 at STP. We can thus determine NV and apply Eq. 18–10b. SOLUTION N 6.02 * 1023 molecules = 2.69 * 1025 moleculesm3. = V 22.4 * 10 –3 m3 Then lM =

4p 22 A1.5 * 10

1 –10

mB 2 A2.7 * 1025 m–3 B

L 9 * 10–8 m.

NOTE This is about 300 times the diameter of an air molecule. At very low densities, such as in an evacuated vessel, the concept of mean free path loses meaning since collisions with the container walls may occur more frequently than collisions with other molecules. For example, in a cubical box that is (say) 20 cm on a side containing air at 10–7 torr AL 10–10 atmB, the mean free path would be about 900 m, which means many more collisions are made with the walls than with other molecules. (Note, nonetheless, that the box contains over 1012 molecules.) If the concept of mean free path included also collision with the walls, it would be closer to 0.2 m than to the 900 m calculated from Eq. 18–10b.

488

CHAPTER 18

Kinetic Theory of Gases

* 18–7

Diffusion

If you carefully place a few drops of food coloring in a container of water as in Fig. 18–14, you will find that the color spreads throughout the water. The process may take some time (assuming you do not shake the glass), but eventually the color will become uniform. This mixing, known as diffusion, takes place because of the random movement of the molecules. Diffusion occurs in gases too. Common examples include perfume or smoke (or the odor of something cooking on the stove) diffusing in air, although convection (moving air currents) often plays a greater role in spreading odors than does diffusion. Diffusion depends on concentration, by which we mean the number of molecules or moles per unit volume. In general, the diffusing substance moves from a region where its concentration is high to one where its concentration is low.

FIGURE 18–14 A few drops of food coloring (a) dropped into water, (b) spreads slowly throughout the water, eventually (c) becoming uniform.

(a)

(b)

(c)

Diffusion can be readily understood on the basis of kinetic theory and the random motion of molecules. Consider a tube of cross-sectional area A containing molecules in a higher concentration on the left than on the right, Fig. 18–15. We assume the molecules are in random motion. Yet there will be a net flow of molecules to the right. To see why this is true, let us consider the small section of tube of length ¢x as shown. Molecules from both regions 1 and 2 cross into this central section as a result of their random motion. The more molecules there are in a region, the more will strike a given area or cross a boundary. Since there is a greater concentration of molecules in region 1 than in region 2, more molecules cross into the central section from region 1 than from region 2. There is, then, a net flow of molecules from left to right, from high concentration toward low concentration. The net flow becomes zero only when the concentrations become equal. You might expect that the greater the difference in concentration, the greater the flow rate. Indeed, the rate of diffusion, J (number of molecules or moles or kg per second), is directly proportional to the difference in concentration per unit distance, AC1 - C2 B¢x (which is called the concentration gradient), and to the cross-sectional area A (see Fig. 18–15): C1 - C2 , ¢x or, in terms of derivatives, J = DA

J = DA

dC . dx

FIGURE 18–15 Diffusion occurs from a region of high concentration to one of lower concentration (only one type of molecule is shown).

Region 1; concentration = C1

D is a constant of proportionality called the diffusion constant. Equation 18–11 is known as the diffusion equation, or Fick's law. If the concentrations are given in molm3, then J is the number of moles passing a given point per second. If the concentrations are given in kgm3, then J is the mass movement per second (kgs). The length ¢x is given in meters. The values of D for a variety of substances are given in Table 18–3.

Region 2; concentration = C2

TABLE 18–3 Diffusion Constants, D (20°C, 1 atm) Diffusing Molecules

(18–11)

∆x

H2 O2 O2 Blood hemoglobin Glycine (an amino acid) DNA (mass 6 * 106 u)

Medium

D (m2 s)

Air Air Water

6.3 * 10–5 1.8 * 10–5 100 * 10 –11

Water

6.9 * 10–11

Water

95 * 10 –11

Water

0.13 * 10–11

*SECTION 18–7

Diffusion

489

EXAMPLE 18–9 ESTIMATE Diffusion of ammonia in air. To get an idea of the time required for diffusion, estimate how long it might take for ammonia ANH 3 B to be detected 10 cm from a bottle after it is opened, assuming only diffusion is occurring. P H Y S I C S

A P P L I E D

Diffusion time

APPROACH This will be an order-of-magnitude calculation. The rate of diffusion J can be set equal to the number of molecules N diffusing across area A in a time t: J = Nt. Then the time t = NJ, where J is given by Eq. 18–11. We will have to make some assumptions and rough approximations about concentrations to use Eq. 18–11. SOLUTION Using Eq. 18–11, we find

t =

N N ¢x . = J DA ¢C

The average concentration (midway between bottle and nose) can be approximated by K L NV, where V is the volume over which the molecules move and is roughly of the order of V L A ¢x, where ¢x is 10 cm = 0.10 m. We substitute N = KV = KA ¢x into the above equation:

t L

(KA ¢x) ¢x K (¢x)2 . = DA ¢C ¢C D

The concentration of ammonia is high near the bottle (C) and low near the detecting nose (L 0), so K L C2 L ¢C2, or (K¢C) L 12 . Since NH 3 molecules have a size somewhere between H 2 and O2 , from Table 18–3 we can estimate D L 4 * 10–5 m2s. Then (0.10 m)2 t L 12 L 100 s, A4 * 10 –5 m2sB or about a minute or two. NOTE This result seems rather long from experience, suggesting that air currents (convection) are more important than diffusion for transmitting odors. CONCEPTUAL EXAMPLE 18–10 Colored rings on a paper towel. A child colors a small spot on a wet paper towel with a brown marker. Later, she discovers that instead of a brown spot, there are concentric colored rings around the marked spot. What happened? P H Y S I C S

A P P L I E D

Chromatography

RESPONSE The ink in a brown marker is composed of several different inks that mix to make brown. These inks each diffuse at different rates through the wet paper towel. After a period of time the inks have diffused far enough that the differences in distances traveled is sufficient to separate the different colors. Chemists and biochemists use a similar technique, called chromatography, to separate substances based on their diffusion rates through a medium.

Summary According to the kinetic theory of gases, which is based on the idea that a gas is made up of molecules that are moving rapidly and at random, the average translational kinetic energy of the molecules is proportional to the Kelvin temperature T: † =

1 2 mO

3 2 kT

(18–4)

where k is Boltzmann's constant. At any moment, there exists a wide distribution of molecular speeds within a gas. The Maxwell distribution of speeds is derived from simple kinetic theory assumptions, and is in good accord with experiment for gases at not too high a pressure.

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The behavior of real gases at high pressure, andor when near their liquefaction point, deviates from the ideal gas law due to the finite size of molecules and to the attractive forces between molecules. Below the critical temperature, a gas can change to a liquid if sufficient pressure is applied; but if the temperature is higher than the critical temperature, no amount of pressure will cause a liquid surface to form. The triple point of a substance is that unique temperature and pressure at which all three phases—solid, liquid, and gas—can coexist in equilibrium. Because of its precise

reproducibility, the triple point of water is often taken as a standard reference point. Evaporation of a liquid is the result of the fastest moving molecules escaping from the surface. Because the average molecular velocity is less after the fastest molecules escape, the temperature decreases when evaporation takes place. Saturated vapor pressure refers to the pressure of the vapor above a liquid when the two phases are in equilibrium. The vapor pressure of a substance (such as water) depends strongly on temperature and is equal to atmospheric pressure at the boiling point.

Relative humidity of air at a given place is the ratio of the partial pressure of water vapor in the air to the saturated vapor pressure at that temperature; it is usually expressed as a percentage. [*The van der Waals equation of state takes into account the finite volume of molecules, and the attractive forces between molecules, to better approximate the behavior of real gases.] [*The mean free path is the average distance a molecule moves between collisions with other molecules.] [*Diffusion is the process whereby molecules of a substance move (on average) from one area to another because of a difference in that substance's concentration.]

Questions 1. Why doesn't the size of different molecules enter into the ideal gas law?

16. Explain why a hot humid day is far more uncomfortable than a hot dry day at the same temperature.

2. When a gas is rapidly compressed (say, by pushing down a piston) its temperature increases. When a gas expands against a piston, it cools. Explain these changes in temperature using the kinetic theory, in particular noting what happens to the momentum of molecules when they strike the moving piston.

17. Is it possible to boil water at room temperature (20°C) without heating it? Explain. 18. What exactly does it mean when we say that oxygen boils at –183°C?

3. In Section 18–1 we assumed the gas molecules made perfectly elastic collisions with the walls of the container. This assumption is not necessary as long as the walls are at the same temperature as the gas. Why?

19. A length of thin wire is placed over a block of ice (or an ice cube) at 0°C and weights are hung from the ends of the wire. It is found that the wire cuts its way through the ice cube, but leaves a solid block of ice behind it. This process is called regelation. Explain how this happens by inferring how the freezing point of water depends on pressure.

4. Explain in words how Charles's law follows from kinetic theory and the relation between average kinetic energy and the absolute temperature.

20. Consider two days when the air temperature is the same but the humidity is different. Which is more dense, the dry air or the humid air at the same T? Explain.

5. Explain in words how Gay-Lussac's law follows from kinetic theory.

21. (a) Why does food cook faster in a pressure cooker? (b) Why does pasta or rice need to boil longer at high altitudes? (c) Is it harder to boil water at high altitudes?

6. As you go higher in the Earth's atmosphere, the ratio of N2 molecules to O2 molecules increases. Why? 7. Can you determine the temperature of a vacuum? 8. Is temperature a macroscopic or microscopic variable? 9. Explain why the peak of the curve for 310 K in Fig. 18–3 is not as high as for 273 K. (Assume the total number of molecules is the same for both.) 10. Escape velocity for the Earth refers to the minimum speed an object must have to leave the Earth and never return. (a) The escape velocity for the Moon is about one-fifth what it is for the Earth due to the Moon's smaller mass; explain why the Moon has practically no atmosphere. (b) If hydrogen was once in the Earth's atmosphere, why would it have probably escaped?

22. How do a gas and a vapor differ? 23. (a) At suitable temperatures and pressures, can ice be melted by applying pressure? (b) At suitable temperatures and pressures, can carbon dioxide be melted by applying pressure? 24. Why does dry ice not last long at room temperature? 25. Under what conditions can liquid CO2 exist? Be specific. Can it exist as a liquid at normal room temperature? 26. Why does exhaled air appear as a little white cloud in the winter (Fig. 18–16)?

11. If a container of gas is at rest, the average velocity of molecules must be zero. Yet the average speed is not zero. Explain. 12. If the pressure in a gas is doubled while its volume is held constant, by what factor do (a) vrms and (b) v change? 13. What everyday observation would tell you that not all molecules in a material have the same speed? 14. We saw that the saturated vapor pressure of a liquid (say, water) does not depend on the external pressure. Yet the temperature of boiling does depend on the external pressure. Is there a contradiction? Explain. 15. Alcohol evaporates more quickly than water at room temperature. What can you infer about the molecular properties of one relative to the other?

FIGURE 18–16 Question 26. * 27. Discuss why sound waves can travel in a gas only if their wavelength is somewhat larger than the mean free path. * 28. Name several ways to reduce the mean free path in a gas.

Questions

491

Problems 18–1 Molecular Interpretation of Temperature 1. (I) (a) What is the average translational kinetic energy of an oxygen molecule at STP? (b) What is the total translational kinetic energy of 1.0 mol of O2 molecules at 25°C? 2. (I) Calculate the rms speed of helium atoms near the surface of the Sun at a temperature of about 6000 K. 3. (I) By what factor will the rms speed of gas molecules increase if the temperature is increased from 0°C to 180°C? 4. (I) A gas is at 20°C. To what temperature must it be raised to triple the rms speed of its molecules? 5. (I) What speed would a 1.0-g paper clip have if it had the same kinetic energy as a molecule at 15°C? 6. (I) A 1.0-mol sample of helium gas has a temperature of 27°C. (a) What is the total kinetic energy of all the gas atoms in the sample? (b) How fast would a 65-kg person have to run to have the same kinetic energy? 7. (I) Twelve molecules have the following speeds, given in arbitrary units: 6.0, 2.0, 4.0, 6.0, 0.0, 4.0, 1.0, 8.0, 5.0, 3.0, 7.0, and 8.0. Calculate (a) the mean speed, and (b) the rms speed. 8. (II) The rms speed of molecules in a gas at 20.0°C is to be increased by 2.0%. To what temperature must it be raised? 9. (II) If the pressure in a gas is tripled while its volume is held constant, by what factor does vrms change? 10. (II) Show that the rms speed of molecules in a gas is given by vrms = 13Pr , where P is the pressure in the gas, and r is the gas density. 11. (II) Show that for a mixture of two gases at the same temperature, the ratio of their rms speeds is equal to the inverse ratio of the square roots of their molecular masses. 12. (II) What is the rms speed of nitrogen molecules contained in an 8.5-m3 volume at 3.1 atm if the total amount of nitrogen is 1800 mol? 13. (II) (a) For an ideal gas at temperature T show that

17. (II) Calculate (a) the rms speed of a nitrogen molecule at 0°C and (b) determine how many times per second it would move back and forth across a 5.0-m-long room on the average, assuming it made very few collisions with other molecules. 18. (III) Estimate how many air molecules rebound from a wall in a typical room per second, assuming an ideal gas of N molecules contained in a cubic room with sides of length l at temperature T and pressure P. (a) Show that the frequency f with which gas molecules strike a wall is vx P 2 f = l 2 kT where vx is the average x component of the molecule's velocity. (b) Show that the equation can then be written as Pl2 f L 24mkT where m is the mass of a gas molecule. (c) Assume a cubic air-filled room is at sea level, has a temperature 20°C, and has sides of length l = 3 m. Determine f.

18–2 Distribution of Molecular Speeds 19. (I) If you double the mass of the molecules in a gas, is it possible to change the temperature to keep the velocity distribution from changing? If so, what do you need to do to the temperature? 20. (I) A group of 25 particles have the following speeds: two have speed 10 ms, seven have 15 ms, four have 20 ms, three have 25 ms, six have 30 ms, one has 35 ms, and two have 40 ms. Determine (a) the average speed, (b) the rms speed, and (c) the most probable speed. 21. (II) A gas consisting of 15,200 molecules, each of mass 2.00 * 10–26 kg, has the following distribution of speeds, which crudely mimics the Maxwell distribution: Number of Molecules

Speed (m s)

1600 4100 4700 3100 1300 400

220 440 660 880 1100 1320

dvrms 1 vrms = , dT 2 T and using the approximation ¢vrms L

dvrms ¢T, show that dT

¢vrms 1 ¢T . L vrms 2 T (b) If the average air temperature changes from –5°C in winter to 25°C in summer, estimate the percent change in the rms speed of air molecules between these seasons. 14. (II) What is the average distance between oxygen molecules at STP? 15. (II) Two isotopes of uranium, 235U and 238U (the superscripts refer to their atomic masses), can be separated by a gas diffusion process by combining them with fluorine to make the gaseous compound UF6 . Calculate the ratio of the rms speeds of these molecules for the two isotopes, at constant T. Use Appendix F for masses. 16. (II) Can pockets of vacuum persist in an ideal gas? Assume that a room is filled with air at 20°C and that somehow a small spherical region of radius 1 cm within the room becomes devoid of air molecules. Estimate how long it will take for air to refill this region of vacuum. Assume the atomic mass of air is 29 u.

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(a) Determine vrms for this distribution of speeds. (b) Given your value for vrms , what (effective) temperature would you assign to this gas? (c) Determine the mean speed v of this distribution and use this value to assign an (effective) temperature to the gas. Is the temperature you find here consistent with the one you determined in part (b)? 22. (III) Starting from the Maxwell distribution of speeds, q Eq. 18–6, show (a) 0 f(v) dv = N, and (b)

v q

f(v) dvN = 3kTm.

18–3 Real Gases 23. (I) CO2 exists in what phase when the pressure is 30 atm and the temperature is 30°C (Fig. 18–6)? 24. (I) (a) At atmospheric pressure, in what phases can CO2 exist? (b) For what range of pressures and temperatures can CO2 be a liquid? Refer to Fig. 18–6. 25. (I) Water is in which phase when the pressure is 0.01 atm and the temperature is (a) 90°C, (b) –20°C?

26. (II) You have a sample of water and are able to control temperature and pressure arbitrarily. (a) Using Fig. 18–5, describe the phase changes you would see if you started at a temperature of 85°C, a pressure of 180 atm, and decreased the pressure down to 0.004 atm while keeping the temperature fixed. (b) Repeat part (a) with the temperature at 0.0°C. Assume that you held the system at the starting conditions long enough for the system to stabilize before making further changes.

18–4 Vapor Pressure and Humidity 27. (I) What is the partial pressure of water vapor at 30°C if the humidity is 85%? 28. (I) What is the partial pressure of water on a day when the temperature is 25°C and the relative humidity is 55%? 29. (I) What is the air pressure at a place where water boils at 80°C? 30. (II) What is the dew point if the humidity is 75% on a day when the temperature is 25°C? 31. (II) If the air pressure at a particular place in the mountains is 0.75 atm, estimate the temperature at which water boils. 32. (II) What is the mass of water in a closed room 5.0 m * 6.0 m * 2.4 m when the temperature is 24.0°C and the relative humidity is 65%? 33. (II) What is the approximate pressure inside a pressure cooker if the water is boiling at a temperature of 120°C? Assume no air escaped during the heating process, which started at 12°C. 34. (II) If the humidity in a room of volume 440 m3 at 25°C is 65%, what mass of water can still evaporate from an open pan? 35. (II) A pressure cooker is a sealed pot designed to cook food with the steam produced by boiling water somewhat above 100°C. The pressure cooker in Fig. 18–17 uses a weight of mass m to allow steam to escape at a certain pressure through a small hole (diameter d) in the cooker's lid. If d = 3.0 mm, what should m be in order to cook food at 120°C? Assume that atmospheric pressure outside the cooker is 1.01 * 105 Pa. Weight (mass m)

m Diameter d

38. (III) Air that is at its dew point of 5°C is drawn into a building where it is heated to 20°C. What will be the relative humidity at this temperature? Assume constant pressure of 1.0 atm. Take into account the expansion of the air. 39. (III) What is the mathematical relation between water's boiling temperature and atmospheric pressure? (a) Using the data from Table 18–2, in the temperature range from 50°C to 150°C, plot ln P versus (1T), where P is water's saturated vapor pressure (Pa) and T is temperature on the Kelvin scale. Show that a straight-line plot results and determine the slope and y-intercept of the line. (b) Show that your result implies P = Be –AT where A and B are constants. Use the slope and y-intercept from your plot to show that A L 5000 K and B L 7 * 1010 Pa. * 18–5 Van der Waals Equation of State * 40. (II) In the van der Waals equation of state, the constant b represents the amount of "unavailable volume" occupied by the molecules themselves. Thus V is replaced by (V - nb), where n is the number of moles. For oxygen, b is about 3.2 * 10–5 m3mol. Estimate the diameter of an oxygen molecule. * 41. (II) For oxygen gas, the van der Waals equation of state achieves its best fit for a = 0.14 N m4mol2 and b = 3.2 * 10 –5 m3mol. Determine the pressure in 1.0 mol of the gas at 0°C if its volume is 0.70 L, calculated using (a) the van der Waals equation, (b) the ideal gas law. * 42. (III) A 0.5-mol sample of O2 gas is in a large cylinder with a movable piston on one end so it can be compressed. The initial volume is large enough that there is not a significant difference between the pressure given by the ideal gas law and that given by the van der Waals equation. As the gas is slowly compressed at constant temperature (use 300 K), at what volume does the van der Waals equation give a pressure that is 5% different than the ideal gas law pressure? Let a = 0.14 Nm4mol2 and b = 3.2 * 10–5 m3mol. * 43. (III) (a) From the van der Waals equation of state, show that the critical temperature and pressure are given by Tcr =

Steam

FIGURE 18–17 Problem 35.

Water Water

36. (II) When using a mercury barometer (Section 13–6), the vapor pressure of mercury is usually assumed to be zero. At room temperature mercury's vapor pressure is about 0.0015 mm-Hg. At sea level, the height h of mercury in a barometer is about 760 mm. (a) If the vapor pressure of mercury is neglected, is the true atmospheric pressure greater or less than the value read from the barometer? (b) What is the percent error? (c) What is the percent error if you use a water barometer and ignore water's saturated vapor pressure at STP? 37. (II) If the humidity is 45% at 30.0°C, what is the dew point? Use linear interpolation to find the temperature of the dew point to the nearest degree.

8a , 27bR

Pcr =

a .

27b2

[Hint: Use the fact that the P versus V curve has an inflection point at the critical point so that the first and second derivatives are zero.] (b) Determine a and b for CO2 from the measured values of Tcr = 304 K and Pcr = 72.8 atm. * 44. (III) How well does the ideal gas law describe the pressurized air in a scuba tank? (a) To fill a typical scuba tank, an air compressor intakes about 2300 L of air at 1.0 atm and compresses this gas into the tank's 12-L internal volume. If the filling process occurs at 20°C, show that a tank holds about 96 mol of air. (b) Assume the tank has 96 mol of air at 20°C. Use the ideal gas law to predict the air's pressure within the tank. (c) Use the van der Waals equation of state to predict the air's pressure within the tank. For air, the van der Waals constants are a = 0.1373 N m4mol2 and b = 3.72 * 10 –5 m3mol. (d) Taking the van der Waals pressure as the true air pressure, show that the ideal gas law predicts a pressure that is in error by only about 3%.

Problems

493

* 18–6 Mean Free Path * 45. (II) At about what pressure would the mean free path of air molecules be (a) 0.10 m and (b) equal to the diameter of air molecules, L 3 * 10–10 m? Assume T = 20°C. * 46. (II) Below a certain threshold pressure, the air molecules (0.3-nm diameter) within a research vacuum chamber are in the "collision-free regime," meaning that a particular air molecule is as likely to cross the container and collide first with the opposite wall, as it is to collide with another air molecule. Estimate the threshold pressure for a vacuum chamber of side 1.0 m at 20°C. * 47. (II) A very small amount of hydrogen gas is released into the air. If the air is at 1.0 atm and 15°C, estimate the mean free path for a H 2 molecule. What assumptions did you make? * 48. (II) (a) The mean free path of CO2 molecules at STP is measured to be about 5.6 * 10 –8 m. Estimate the diameter of a CO2 molecule. (b) Do the same for He gas for which lM L 25 * 10–8 m at STP. * 49. (II) (a) Show that the number of collisions a molecule makes per second, called the collision frequency, f, is given by f = vlM , and thus f = 412 pr2 vNV. (b) What is the collision frequency for N2 molecules in air at T = 20°C and P = 1.0 * 10–2 atm? * 50. (II) We saw in Example 18–8 that the mean free path of air molecules at STP, lM , is about 9 * 10–8 m. Estimate the collision frequency f, the number of collisions per unit time.

* 51. (II) A cubic box 1.80 m on a side is evacuated so the pressure of air inside is 10–6 torr. Estimate how many collisions molecules make with each other for each collision with a wall (0°C). * 52. (III) Estimate the maximum allowable pressure in a 32-cmlong cathode ray tube if 98% of all electrons must hit the screen without first striking an air molecule. * 18–7 Diffusion * 53. (I) Approximately how long would it take for the ammonia of Example 18–9 to be detected 1.0 m from the bottle after it is opened? What does this suggest about the relative importance of diffusion and convection for carrying odors? * 54. (II) Estimate the time needed for a glycine molecule (see Table 18–3) to diffuse a distance of 15 mm in water at 20°C if its concentration varies over that distance from 1.00 molm3 to 0.50 molm3? Compare this "speed" to its rms (thermal) speed. The molecular mass of glycine is about 75 u. * 55. (II) Oxygen diffuses from the surface of insects to the interior through tiny tubes called tracheae. An average trachea is about 2 mm long and has cross-sectional area of 2 * 10–9 m2. Assuming the concentration of oxygen inside is half what it is outside in the atmosphere, (a) show that the concentration of oxygen in the air (assume 21% is oxygen) at 20°C is about 8.7 molm3, then (b) calculate the diffusion rate J, and (c) estimate the average time for a molecule to diffuse in. Assume the diffusion constant is 1 * 10 –5 m2s.

General Problems 56. A sample of ideal gas must contain at least N = 106 molecules in order for the Maxwell distribution to be a valid description of the gas, and to assign it a meaningful temperature. For an ideal gas at STP, what is the smallest length scale l (volume V = l3) for which a valid temperature can be assigned? 57. In outer space the density of matter is about one atom per cm3, mainly hydrogen atoms, and the temperature is about 2.7 K. Calculate the rms speed of these hydrogen atoms, and the pressure (in atmospheres). 58. Calculate approximately the total translational kinetic energy of all the molecules in an E. coli bacterium of mass 2.0 * 10 –15 kg at 37°C. Assume 70% of the cell, by weight, is water, and the other molecules have an average molecular mass on the order of 105 u. 59. (a) Estimate the rms speed of an amino acid, whose molecular mass is 89 u, in a living cell at 37°C. (b) What would be the rms speed of a protein of molecular mass 85,000 u at 37°C? 60. The escape speed from the Earth is 1.12 * 104 ms, so that a gas molecule travelling away from Earth near the outer boundary of the Earth's atmosphere would, at this speed, be able to escape from the Earth's gravitational field and be lost to the atmosphere. At what temperature is the average speed of (a) oxygen molecules, and (b) helium atoms equal to 1.12 * 104 ms? (c) Can you explain why our atmosphere contains oxygen but not helium?

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61. The second postulate of kinetic theory is that the molecules are, on the average, far apart from one another. That is, their average separation is much greater than the diameter of each molecule. Is this assumption reasonable? To check, calculate the average distance between molecules of a gas at STP, and compare it to the diameter of a typical gas molecule, about 0.3 nm. If the molecules were the diameter of ping-pong balls, say 4 cm, how far away would the next ping-pong ball be on average? 62. A sample of liquid cesium is heated in an oven to 400°C and the resulting vapor is used to produce an atomic beam. The volume of the oven is 55 cm3, the vapor pressure of Cs at 400°C is 17 mm-Hg, and the diameter of cesium atoms in the vapor is 0.33 nm. (a) Calculate the mean speed of cesium atoms in the vapor. (b) Determine the number of collisions a single Cs atom undergoes with other cesium atoms per second. (c) Determine the total number of collisions per second between all of the cesium atoms in the vapor. Note that a collision involves two Cs atoms and assume the ideal gas law holds. 63. Consider a container of oxygen gas at a temperature of 20°C that is 1.00 m tall. Compare the gravitational potential energy of a molecule at the top of the container (assuming the potential energy is zero at the bottom) with the average kinetic energy of the molecules. Is it reasonable to neglect the potential energy?

64. In humid climates, people constantly dehumidify their cellars to prevent rot and mildew. If the cellar in a house (kept at 20°C) has 115 m2 of floor space and a ceiling height of 2.8 m, what is the mass of water that must be removed from it in order to drop the humidity from 95% to a more reasonable 40%? 65. Assuming a typical nitrogen or oxygen molecule is about 0.3 nm in diameter, what percent of the room you are sitting in is taken up by the volume of the molecules themselves? 66. A scuba tank has a volume of 3100 cm3. For very deep dives, the tank is filled with 50% (by volume) pure oxygen and 50% pure helium. (a) How many molecules are there of each type in the tank if at 20°C the gauge pressure reads 12 atm? (b) What is the ratio of the average kinetic energies of the two types of molecule? (c) What is the ratio of the rms speeds of the two types of molecule? 67. A space vehicle returning from the Moon enters the atmosphere at a speed of about 42,000 kmh. Molecules (assume nitrogen) striking the nose of the vehicle with this speed correspond to what temperature? (Because of this high temperature, the nose of a space vehicle must be made of special materials; indeed, part of it does vaporize, and this is seen as a bright blaze upon reentry.) 68. At room temperature, it takes approximately 2.45 * 103 J to evaporate 1.00 g of water. Estimate the average speed of evaporating molecules. What multiple of vrms (at 20°C) for water molecules is this? (Assume Eq. 18–4 holds.) 69. Calculate the total water vapor pressure in the air on the following two days: (a) a hot summer day, with the temperature 30°C and the relative humidity at 65%; (b) a cold winter day, with the temperature 5°C and the relative humidity at 75%. * 70. At 300 K, an 8.50-mol sample of carbon dioxide occupies a volume of 0.220 m3. Calculate the gas pressure, first by assuming the ideal gas law, and then by using the van der Waals equation of state. (The values for a and b are given in Section 18–5.) In this range of pressure and volume, the van der Waals equation is very accurate. What percent error did you make in assuming ideal-gas-law behavior? * 71. The density of atoms, mostly hydrogen, in interstellar space is about one per cubic centimeter. Estimate the mean free path of the hydrogen atoms, assuming an atomic diameter of 10 –10 m.

* 72. Using the ideal gas law, find an expression for the mean free path lM that involves pressure and temperature instead of (NV). Use this expression to find the mean free path for nitrogen molecules at a pressure of 7.5 atm and 300 K. 73. A sauna has 8.5 m3 of air volume, and the temperature is 90°C. The air is perfectly dry. How much water (in kg) should be evaporated if we want to increase the relative humidity from 0% to 10%? (See Table 18–2.) 74. A 0.50-kg trash-can lid is suspended against gravity by tennis balls thrown vertically upward at it. How many tennis balls per second must rebound from the lid elastically, assuming they have a mass of 0.060 kg and are thrown at 12 ms? * 75. Sound waves in a gas can only propagate if the gas molecules collide with each other on the time scale of the sound wave's period. Thus the highest possible frequency fmax for a sound wave in a gas is approximately equal to the inverse of the average collision time between molecules. Assume a gas, composed of molecules with mass m and radius r, is at pressure P and temperature T. (a) Show that fmax L 16Pr 2

p . B mkT

(b) Determine fmax for 20°C air at sea level. How many times larger is fmax than the highest frequency in the human audio range (20 kHz)? * Numerical/Computer * 76. (II) Use a spreadsheet to calculate and graph the fraction of molecules in each 50-ms speed interval from 100 ms to 5000 ms if T = 300 K. * 77. (II) Use numerical integration [Section 2–9] to estimate (within 2%) the fraction of molecules in air at 1.00 atm and 20°C that have a speed greater than 1.5 times the most probable speed. * 78. (II) For oxygen gas the van der Waals constants are a = 0.14 Nm4mol2 and b = 3.2 * 10 –5 m3mol. Using these values, graph six curves of pressure vs. volume between V = 2 * 10–5 m3 to 2.0 * 10–4 m3, for 1 mol of oxygen gas at temperatures of 80 K, 100 K, 120 K, 130 K, 150 K, and 170 K. From the graphs determine approximately the critical temperature for oxygen.

Answers to Exercises A: (a).

D: (d).

B: (d).

E: (b).

C: (c).

General Problems

495

When it is cold, warm clothes act as insulators to reduce heat loss from the body to the environment by conduction and convection. Heat radiation from a campfire can warm you and your clothes. The fire can also transfer energy directly by heat convection and conduction to what you are cooking. Heat, like work, represents a transfer of energy. Heat is defined as a transfer of energy due to a difference of temperature. Work is a transfer of energy by mechanical means, not due to a temperature difference. The first law of thermodynamics links the two in a general statement of energy conservation: the heat Q added to a system minus the net work W done by the system equals the change in internal energy ¢Eint of the system: ¢Eint = Q - W. Internal energy Eint is the sum total of all the energy of the molecules of the system.

P T E R

C H

CONTENTS 19–1 Heat as Energy Transfer 19–2 Internal Energy 19–3 Specific Heat 19–4 Calorimetry—Solving Problems 19–5 Latent Heat 19–6 The First Law of Thermodynamics 19–7 The First Law of Thermodynamics Applied; Calculating the Work 19–8 Molar Specific Heats for Gases, and the Equipartition of Energy 19–9 Adiabatic Expansion of a Gas 19–10 Heat Transfer: Conduction, Convection, Radiation

496

Heat and the First Law of Thermodynamics CHAPTER-OPENING QUESTION—Guess now! A 5-kg cube of warm iron (60°C) is put in thermal contact with a 10-kg cube of cold iron (15°C). Which statement is valid: (a) Heat flows spontaneously from the warm cube to the cold cube until both cubes have the same heat content. (b) Heat flows spontaneously from the warm cube to the cold cube until both cubes have the same temperature. (c) Heat can flow spontaneously from the warm cube to the cold cube, but can also flow spontaneously from the cold cube to the warm cube. (d) Heat can never flow from a cold object or area to a hot object or area. (e) Heat flows from the larger cube to the smaller one because the larger one has more internal energy. hen a pot of cold water is placed on a hot burner of a stove, the temperature of the water increases. We say that heat "flows" from the hot burner to the cold water. When two objects at different temperatures are put in contact, heat spontaneously flows from the hotter one to the colder one. The spontaneous flow of heat is in the direction tending to equalize the temperature. If the two objects are kept in contact long enough for their temperatures to become equal, the objects are said to be in thermal equilibrium, and there is no further heat flow between them. For example, when a fever thermometer is first placed in your mouth, heat flows from your mouth to the thermometer. When the thermometer reaches the same temperature as the inside of your mouth, the thermometer and your mouth are then in equilibrium, and no more heat flows.

Heat and temperature are often confused. They are very different concepts and we will make the clear distinction between them. We start this Chapter by defining and using the concept of heat. We also begin our discussion of thermodynamics, which is the name we give to the study of processes in which energy is transferred as heat and as work.

19–1

Heat as Energy Transfer

We use the term "heat" in everyday life as if we knew what we meant. But the term is often used inconsistently, so it is important for us to define heat clearly, and to clarify the phenomena and concepts related to heat. We commonly speak of the flow of heat—heat flows from a stove burner to a pot of soup, from the Sun to the Earth, from a person's mouth into a fever thermometer. Heat flows spontaneously from an object at higher temperature to one at lower temperature. Indeed, an eighteenth-century model of heat pictured heat flow as movement of a fluid substance called caloric. However, the caloric fluid could never be detected. In the nineteenth century, it was found that the various phenomena associated with heat could be described consistently using a new model that views heat as akin to work, as we will discuss in a moment. First we note that a common unit for heat, still in use today, is named after caloric. It is called the calorie (cal) and is defined as the amount of heat necessary to raise the temperature of 1 gram of water by 1 Celsius degree. [To be precise, the particular temperature range from 14.5°C to 15.5°C is specified because the heat required is very slightly different at different temperatures. The difference is less than 1% over the range 0 to 100°C, and we will ignore it for most purposes.] More often used than the calorie is the kilocalorie (kcal), which is 1000 calories. Thus 1 kcal is the heat needed to raise the temperature of 1 kg of water by 1 C°. Often a kilocalorie is called a Calorie (with a capital C), and this Calorie (or the kJ) is used to specify the energy value of food. In the British system of units, heat is measured in British thermal units (Btu). One Btu is defined as the heat needed to raise the temperature of 1 lb of water by 1 F°. It can be shown (Problem 4) that 1 Btu = 0.252 kcal = 1056 J. The idea that heat is related to energy transfer was pursued by a number of scientists in the 1800s, particularly by an English brewer, James Prescott Joule (1818–1889). One of Joule's experiments is shown (simplified) in Fig. 19–1. The falling weight causes the paddle wheel to turn. The friction between the water and the paddle wheel causes the temperature of the water to rise slightly (barely measurable, in fact, by Joule). In this and many other experiments (some involving electrical energy), Joule determined that a given amount of work done was always equivalent to a particular amount of heat input. Quantitatively, 4.186 joules (J) of work was found to be equivalent to 1 calorie (cal) of heat. This is known as the mechanical equivalent of heat:

C A U T I O N

Heat is not a fluid

Weight

FIGURE 19–1 Joule's experiment on the mechanical equivalent of heat.

4.186 J = 1 cal; 4.186 kJ = 1 kcal. As a result of these and other experiments, scientists came to interpret heat not as a substance, and not exactly as a form of energy. Rather, heat refers to a transfer of energy: when heat flows from a hot object to a cooler one, it is energy that is being transferred from the hot to the cold object. Thus, heat is energy transferred from one object to another because of a difference in temperature. In SI units, the unit for heat, as for any form of energy, is the joule. Nonetheless, calories and kcal are still sometimes used. Today the calorie is defined in terms of the joule (via the mechanical equivalent of heat, above), rather than in terms of the properties of water, as given previously. The latter is still handy to remember: 1 cal raises 1 g of water by 1 C°, or 1 kcal raises 1 kg of water by 1 C°. Joule's result was crucial because it extended the work-energy principle to include processes involving heat. It also led to the establishment of the law of conservation of energy, which we shall discuss more fully later in this Chapter. SECTION 19–1

C A U T I O N

Heat is energy transferred because of a ¢T

Heat as Energy Transfer

497

P H Y S I C S

EXAMPLE 19–1 ESTIMATE Working off the extra calories. Suppose you throw caution to the wind and eat too much ice cream and cake on the order of 500 Calories. To compensate, you want to do an equivalent amount of work climbing stairs or a mountain. How much total height must you climb?

A P P L I E D

Working off Calories

APPROACH The work W you need to do in climbing stairs equals the change in gravitational potential energy: W = ¢PE = mgh, where h is the vertical height climbed. For this estimate, approximate your mass as m L 60 kg. SOLUTION 500 food Calories are 500 kcal, which in joules is (500 kcal)A4.186 * 103 JkcalB = 2.1 * 106 J. The work done to climb a vertical height h is W = mgh. We solve for h: h =

W 2.1 * 106 J = = 3600 m. mg (60 kg)A9.80 ms2 B

This is a huge elevation change (over 11,000 ft). NOTE The human body does not transform food energy with 100% efficiency—it is more like 20% efficient. As we shall discuss in the next Chapter, some energy is always "wasted," so you would actually have to climb only about (0.2)(3600 m) L 700 m, which is more reasonable (about 2300 ft of elevation gain).

19–2

Internal Energy

The sum total of all the energy of all the molecules in an object is called its internal energy. (Sometimes thermal energy is used to mean the same thing.) We introduce the concept of internal energy now since it will help clarify ideas about heat.

Distinguishing Temperature, Heat, and Internal Energy C A U T I O N

Distinguish heat from internal energy and from temperature

C A U T I O N

Direction of heat flow depends on temperature (not on amount of internal energy)

Using the kinetic theory, we can make a clear distinction between temperature, heat, and internal energy. Temperature (in kelvins) is a measure of the average kinetic energy of individual molecules. Internal energy refers to the total energy of all the molecules within the object. (Thus two equal-mass hot ingots of iron may have the same temperature, but two of them have twice as much internal energy as one does.) Heat, finally, refers to a transfer of energy from one object to another because of a difference in temperature. Notice that the direction of heat flow between two objects depends on their temperatures, not on how much internal energy each has. Thus, if 50 g of water at 30°C is placed in contact (or mixed) with 200 g of water at 25°C, heat flows from the water at 30°C to the water at 25°C even though the internal energy of the 25°C water is much greater because there is so much more of it. EXERCISE A Return to the Chapter-Opening Question, page 496, and answer it again now. Try to explain why you may have answered differently the first time.

Internal Energy of an Ideal Gas Let us calculate the internal energy of n moles of an ideal monatomic (one atom per molecule) gas. The internal energy, Eint , is the sum of the translational kinetic energies of all the atoms.† This sum is equal to the average kinetic energy per molecule times the total number of molecules, N: Eint = NA 12 mOB. Using Eq. 18–4, † = 12 mO = 32 kT, we can write this as Eint =

3 2 NkT

†

(19–1a)

The symbol U is used in some books for internal energy. The use of Eint avoids confusion with U which stands for potential energy (Chapter 8).

498

CHAPTER 19

Heat and the First Law of Thermodynamics

or (recall Section 17–9) 3 2 nRT,

Eint =

[ideal monatomic gas]

(19–1b)

where n is the number of moles. Thus, the internal energy of an ideal gas depends only on temperature and the number of moles of gas. If the gas molecules contain more than one atom, then the rotational and vibrational energy of the molecules (Fig. 19–2) must also be taken into account. The internal energy will be greater at a given temperature than for a monatomic gas, but it will still be a function only of temperature for an ideal gas. The internal energy of real gases also depends mainly on temperature, but where real gases deviate from ideal gas behavior, their internal energy depends also somewhat on pressure and volume (due to atomic potential energy). The internal energy of liquids and solids is quite complicated, for it includes electrical potential energy associated with the forces (or "chemical" bonds) between atoms and molecules.

19–3

Specific Heat

If heat flows into an object, the object's temperature rises (assuming no phase change). But how much does the temperature rise? That depends. As early as the eighteenth century, experimenters had recognized that the amount of heat Q required to change the temperature of a given material is proportional to the mass m of the material present and to the temperature change ¢T. This remarkable simplicity in nature can be expressed in the equation Q = mc ¢T, (19–2) where c is a quantity characteristic of the material called its specific heat. Because c = Qm ¢T, specific heat is specified in units† of Jkg C° (the proper SI unit) or kcalkgC°. For water at 15°C and a constant pressure of 1 atm, c = 4.19 * 103 JkgC° or 1.00 kcalkg C°, since, by definition of the cal and the joule, it takes 1 kcal of heat to raise the temperature of 1 kg of water by 1 C°. Table 19–1 gives the values of specific heat for other solids and liquids at 20°C. The values of c for solids and liquids depend to some extent on temperature (as well as slightly on pressure), but for temperature changes that are not too great, c can often be considered constant.‡ Gases are more complicated and are treated in Section 19–8. EXAMPLE 19–2 How heat transferred depends on specific heat. (a) How much heat input is needed to raise the temperature of an empty 20-kg vat made of iron from 10°C to 90°C? (b) What if the vat is filled with 20 kg of water? APPROACH We apply Eq. 19–2 to the different materials involved. SOLUTION (a) Our system is the iron vat alone. From Table 19–1, the specific heat of iron is 450 JkgC°. The change in temperature is (90°C - 10°C) = 80 C°. Thus, Q = mc ¢T = (20 kg)(450 JkgC°)(80 C°) = 7.2 * 105 J = 720 kJ. (b) Our system is the vat plus the water. The water alone would require Q = mc ¢T = (20 kg)(4186 JkgC°)(80 C°) = 6.7 * 106 J = 6700 kJ, or almost 10 times what an equal mass of iron requires. The total, for the vat plus the water, is 720 kJ + 6700 kJ = 7400 kJ. NOTE In (b), the iron vat and the water underwent the same temperature change, ¢T = 80 C°, but their specific heats are different. J Note that JkgC° means and not (Jkg) C° = J C°kg (otherwise we would have written it kg C° that way). ‡ To take into account the dependence of c on T, we can write Eq. 19–2 in differential form: dQ = mc(T) dT, where c(T) means c is a function of temperature T. Then the heat Q required to change the temperature from T1 to T2 is †

Q =

mc(T) dT.

(a)

(b) FIGURE 19–2 Besides translational kinetic energy, molecules can have (a) rotational kinetic energy, and (b) vibrational energy (both kinetic and potential).

TABLE 19–1 Specific Heats (at 1 atm constant pressure and 20°C unless otherwise stated)

Specific Heat, c Substance

kcal kg C° J kg C° ( cal g C°)

Aluminum Alcohol (ethyl) Copper Glass Iron or steel Lead Marble Mercury Silver Wood Water Ice (–5°C) Liquid (15°C) Steam (110°C) Human body (average) Protein

0.22

900

0.58 0.093 0.20 0.11 0.031 0.21 0.033 0.056 0.4

2400 390 840 450 130 860 140 230 1700

0.50 1.00 0.48

2100 4186 2010

0.83 0.4

3470 1700

SECTION 19–3

Specific Heat

499

P H Y S I C S

A P P L I E D

Practical effects of water's high specific heat

If the iron vat in part (a) of Example 19–2 had been cooled from 90°C to 10°C, 720 kJ of heat would have flowed out of the iron. In other words, Eq. 19–2 is valid for heat flow either in or out, with a corresponding increase or decrease in temperature. We saw in part (b) that water requires almost 10 times as much heat as an equal mass of iron to make the same temperature change. Water has one of the highest specific heats of all substances, which makes it an ideal substance for hot-water space-heating systems and other uses that require a minimal drop in temperature for a given amount of heat transfer. It is the water content, too, that causes the apples rather than the crust in hot apple pie to burn our tongues, through heat transfer.

19–4

Calorimetry—Solving Problems

In discussing heat and thermodynamics, we shall often refer to particular systems. As already mentioned in earlier Chapters, a system is any object or set of objects that we wish to consider. Everything else in the universe we will refer to as its "environment" or the "surroundings." There are several categories of systems. A closed system is one for which no mass enters or leaves (but energy may be exchanged with the environment). In an open system, mass may enter or leave (as may energy). Many (idealized) systems we study in physics are closed systems. But many systems, including plants and animals, are open systems since they exchange materials (food, oxygen, waste products) with the environment. A closed system is said to be isolated if no energy in any form passes across its boundaries; otherwise it is not isolated. When different parts of an isolated system are at different temperatures, heat will flow (energy is transferred) from the part at higher temperature to the part at lower temperature—that is, within the system. If the system is truely isolated, no energy is transferred into or out of it. So the conservation of energy again plays an important role for us: the heat lost by one part of the system is equal to the heat gained by the other part: heatl ost = heat gained or energy out of one part = energy into another part. These simple relations are very useful, but depend on the (often very good) approximation that the whole system is isolated (no other energy transfers occur). Let us take an Example.

95°C

25°C (a) FIGURE 19–3 Example 19–3.

500

CHAPTER 19

T =? (b)

EXAMPLE 19–3 The cup cools the tea. If 200 cm3 of tea at 95°C is poured into a 150-g glass cup initially at 25°C (Fig. 19–3), what will be the common final temperature T of the tea and cup when equilibrium is reached, assuming no heat flows to the surroundings? APPROACH We apply conservation of energy to our system of tea plus cup, which we are assuming is isolated: all of the heat that leaves the tea flows into the cup. We use the specific heat equation, Eq. 19–2, to determine how the heat flow is related to the temperature changes. SOLUTION Because tea is mainly water, its specific heat is 4186 Jkg C° (Table 19–1), and its mass m is its density times its volume AV = 200 cm3 = 200 * 10–6 m3 B: m = rV = A1.0 * 103 kgm3 BA200 * 10–6 m3 B = 0.20 kg. We use Eq. 19–2, apply conservation of energy, and let T be the as yet unknown final temperature: heat lost by tea = heat gained by cup mtea ctea(95°C - T) = mcup ccup(T - 25°C). Putting in numbers and using Table 19–1 (ccup = 840 Jkg C° for glass), we solve for T, and find (0.20 kg)(4186 JkgC°)(95°C - T) = (0.15 kg)(840 Jkg C°)(T - 25°C) 79,500 J - (837 JC°)T = (126 JC°)T - 3150 J T = 86°C. The tea drops in temperature by 9 C° by coming into equilibrium with the cup. NOTE The cup increases in temperature by 86°C - 25°C = 61 C°. Its much greater change in temperature (compared with that of the tea water) is due to its much smaller specific heat compared to that of water.

NOTE In this calculation, the ¢T (of Eq. 19–2, Q = mc ¢T ) is a positive quantity on both sides of our conservation of energy equation. On the left is "heat lost" and ¢T is the initial minus the final temperature (95°C - T), whereas on the right is "heat gained" and ¢T is the final minus the initial temperature. But consider the following alternate approach. Alternate Solution We can set up this Example (and others) by a different approach. We can write that the total heat transferred into or out of the isolated system is zero: ©Q = 0. Then each term is written as Q = mcATf - Ti B, and ¢T = Tf - Ti is always the final minus the initial temperature, and each ¢T can be positive or negative. In the present Example: ©Q = mcup ccup(T - 25°C) + mtea ctea(T - 95°C) = 0. The second term is negative because T will be less than 95°C. Solving the algebra gives the same result. The exchange of energy, as exemplified in Example 19–3, is the basis for a technique known as calorimetry, which is the quantitative measurement of heat exchange. To make such measurements, a calorimeter is used; a simple water calorimeter is shown in Fig. 19–4. It is very important that the calorimeter be well insulated so that almost no heat is exchanged with the surroundings. One important use of the calorimeter is in the determination of specific heats of substances. In the technique known as the "method of mixtures," a sample of a substance is heated to a high temperature, which is accurately measured, and then quickly placed in the cool water of the calorimeter. The heat lost by the sample will be gained by the water and the calorimeter cup. By measuring the final temperature of the mixture, the specific heat can be calculated, as illustrated in the following Example.

C A U T I O N

When using heat lost = heat gained, ¢T is positive on both sides

P R O B L E M

S O LV I N G

Alternate approach: ©Q = 0

Thermometer

Stirrer Insulating lid

Water

Air (insulation) Insulating Calorimeter jacket cup FIGURE 19–4 Simple water calorimeter.

EXAMPLE 19–4 Unknown specific heat determined by calorimetry. An engineer wishes to determine the specific heat of a new metal alloy. A 0.150-kg sample of the alloy is heated to 540°C. It is then quickly placed in 0.400 kg of water at 10.0°C, which is contained in a 0.200-kg aluminum calorimeter cup. (We do not need to know the mass of the insulating jacket since we assume the air space between it and the cup insulates it well, so that its temperature does not change significantly.) The final temperature of the system is 30.5°C. Calculate the specific heat of the alloy. APPROACH We apply conservation of energy to our system, which we take to be the alloy sample, the water, and the calorimeter cup. We assume this system is isolated, so the energy lost by the hot alloy equals the energy gained by the water and calorimeter cup. SOLUTION The heat lost equals the heat gained: heat lost heat gained heat gained by ≤ = ¢ ≤ + ¢ ≤ by alloy by water calorimeter cup ma ca ¢Ta = mw cw ¢Tw + mcal ccal ¢Tcal where the subscripts a, w, and cal refer to the alloy, water, and calorimeter, respectively, and each ¢T 7 0. When we put in values and use Table 19–1, this equation becomes (0.150 kg)(ca)(540°C - 30.5°C) = (0.400 kg)(4186 JkgC°)(30.5°C - 10.0°C) ± (0.200 kg)(900 JkgC°)(30.5°C - 10.0°C) A76.4 kgC°B ca = (34,300 + 3690) J ca = 497 JkgC°. In making this calculation, we have ignored any heat transferred to the thermometer and the stirrer (which is used to quicken the heat transfer process and thus reduce heat loss to the outside). It can be taken into account by adding additional terms to the right side of the above equation and will result in a slight correction to the value of ca . ¢

In all Examples and Problems of this sort, be sure to include all objects that gain or lose heat (within reason). On the "heat loss" side here, it is only the hot metal alloy. On the "heat gain" side, it is both the water and the aluminum calorimeter cup. For simplicity, we have ignored very small masses, such as the thermometer and the stirrer, which will affect the energy balance only very slightly.

P R O B L E M

S O LV I N G

Be sure to consider all possible sources of energy transfer

SECTION 19–4

501

120 Water vapor (steam)

80 60 40

Water and steam

20 0

−20 − 40

Ice

Temperature (°C)

100

0 20

Water and ice

Water (all liquid)

100

200

300

400

Heat added (kcal)

500

600

700

740

FIGURE 19–5 Temperature as a function of the heat added to bring 1.0 kg of ice at – 40°C to steam above 100°C.

19–5

502

CHAPTER 19

Latent Heat

When a material changes phase from solid to liquid, or from liquid to gas (see also Section 18–3), a certain amount of energy is involved in this change of phase. For example, let us trace what happens when a 1.0-kg block of ice at –40°C is heated at a slow steady rate until all the ice has changed to water, then the (liquid) water is heated to 100°C and changed to steam, and heated further above 100°C, all at 1 atm pressure. As shown in the graph of Fig. 19–5, as the ice is heated starting at –40°C, its temperature rises at a rate of about 2 C°kcal of heat added (for ice, c L 0.50 kcalkg C°). However, when 0°C is reached, the temperature stops increasing even though heat is still being added. The ice gradually changes to water in the liquid state, with no change in temperature. After about 40 kcal has been added at 0°C, half the ice remains and half has changed to water. After about 80 kcal, or 330 kJ, has been added, all the ice has changed to water, still at 0°C. Continued addition of heat causes the water's temperature to again increase, now at a rate of 1 C°kcal. When 100°C is reached, the temperature again remains constant as the heat added changes the liquid water to vapor (steam). About 540 kcal (2260 kJ) is required to change the 1.0 kg of water completely to steam, after which the graph rises again, indicating that the temperature of the steam rises as heat is added. The heat required to change 1.0 kg of a substance from the solid to the liquid state is called the heat of fusion; it is denoted by LF . The heat of fusion of water is 79.7 kcalkg or, in proper SI units, 333 kJkg A = 3.33 * 105 JkgB. The heat required to change a substance from the liquid to the vapor phase is called the heat of vaporization, LV . For water it is 539 kcalkg or 2260 kJkg. Other substances follow graphs similar to Fig. 19–5, although the melting-point and boiling-point temperatures are different, as are the specific heats and heats of fusion and vaporization. Values for the heats of fusion and vaporization, which are also called the latent heats, are given in Table 19–2 for a number of substances. The heats of vaporization and fusion also refer to the amount of heat released by a substance when it changes from a gas to a liquid, or from a liquid to a solid. Thus, steam releases 2260 kJkg when it changes to water, and water releases 333 kJkg when it becomes ice. The heat involved in a change of phase depends not only on the latent heat but also on the total mass of the substance. That is, Q = mL, (19–3) where L is the latent heat of the particular process and substance, m is the mass of the substance, and Q is the heat added or released during the phase change. For example, when 5.00 kg of water freezes at 0°C, (5.00 kg)A3.33 * 105 JkgB = 1.67 * 106 J of energy is released.

TABLE 19–2 Latent Heats (at 1 atm) Substance Oxygen Nitrogen Ethyl alcohol Ammonia Water Lead Silver Iron Tungsten †

Heat of Fusion

Melting Point (°C)

kcal kg†

kJ kg

Boiling Point (°C)

– 218.8 – 210.0 – 114 – 77.8 0 327 961 1808 3410

3.3 6.1 25 8.0 79.7 5.9 21 69.1 44

14 26 104 33 333 25 88 289 184

– 183 – 195.8 78 – 33.4 100 1750 2193 3023 5900

Heat of Vaporization kcal kg†

kJ kg

51 48 204 33 539 208 558 1520 1150

210 200 850 137 2260 870 2300 6340 4800

Numerical values in kcalkg are the same in calg.

EXERCISE B A pot of water is boiling on a gas stove, and then you turn up the heat. What happens? (a) The temperature of the water starts increasing. (b) There is a tiny decrease in the rate of water loss by evaporation. (c) The rate of water loss by boiling increases. (d) There is an appreciable increase in both the rate of boiling and the temperature of the water. (e) None of these.

Calorimetry sometimes involves a change of state, as the following Examples show. Indeed, latent heats are often measured using calorimetry. EXAMPLE 19–5 Will all the ice melt? A 0.50-kg chunk of ice at –10°C is placed in 3.0 kg of "iced" tea at 20°C. At what temperature and in what phase will the final mixture be? The tea can be considered as water. Ignore any heat flow to the surroundings, including the container. APPROACH Before we can write down an equation applying conservation of energy, we must first check to see if the final state will be all ice, a mixture of ice and water at 0°C, or all water. To bring the 3.0 kg of water at 20°C down to 0°C would require an energy release of (Eq. 19–2)

P R O B L E M

S O LV I N G

First determine (or estimate) the final state

mw cw(20°C - 0°C) = (3.0 kg)(4186 JkgC°)(20 C°) = 250 kJ. On the other hand, to raise the ice from –10°C to 0°C would require

mice cice C0°C - (–10°C)D = (0.50 kg)(2100 JkgC°)(10 C°) = 10.5 kJ,

and to change the ice to water at 0°C would require (Eq. 19–3) mice LF = (0.50 kg)(333 kJkg) = 167 kJ, for a total of 10.5 kJ + 167 kJ = 177 kJ. This is not enough energy to bring the 3.0 kg of water at 20°C down to 0°C, so we see that the mixture must end up all water, somewhere between 0°C and 20°C. SOLUTION To determine the final temperature T, we apply conservation of energy and write: heat gain = heat loss,

P R O B L E M

S O LV I N G

Then determine the final temperature

heat to raise heat to change heat to raise heat lost by 0.50 kg of ice 0.50 kg 0.50 kg of water 3.0 kg of § ¥ + § ¥ + § ¥ = § ¥. from –10°C of ice from 0°C water cooling to 0°C to water to T from 20°C to T Using some of the results from above, we obtain 10.5 kJ + 167 kJ + (0.50 kg)(4186 JkgC°)(T - 0°C) = (3.0 kg)(4186 JkgC°)(20°C - T). Solving for T we obtain T = 5.0°C. EXERCISE C How much more ice at – 10°C would be needed in Example 19–5 to bring the tea down to 0°C, while just melting all the ice?

SECTION 19–5

503

L SO VIN OBL E PR

Calorimetry 1. Be sure you have sufficient information to apply energy conservation. Ask yourself: is the system isolated (or very nearly so, enough to get a good estimate)? Do we know or can we calculate all significant sources of energy transfer? 2. Apply conservation of energy: heat gained = heat lost. For each substance in the system, a heat (energy) term will appear on either the left or right side of this equation. [Alternatively, use ©Q = 0.] 3. If no phase changes occur, each term in the energy conservation equation (above) will have the form Q(gain) = mcATf - Ti B or Q(lost) = mcATi - Tf B where Ti and Tf are the initial and final temperatures

of the substance, and m and c are its mass and specific heat, respectively. If phase changes do or might occur, there may be terms in the energy conservation equation of the form Q = mL, where L is the latent heat. But before applying energy conservation, determine (or estimate) in which phase the final state will be, as we did in Example 19–5 by calculating the different contributing values for heat Q. Be sure each term appears on the correct side of the energy equation (heat gained or heat lost) and that each ¢T is positive. Note that when the system reaches thermal equilibrium, the final temperature of each substance will have the same value. There is only one Tf . Solve your energy equation for the unknown.

EXAMPLE 19–6 Determining a latent heat. The specific heat of liquid mercury is 140 JkgC°. When 1.0 kg of solid mercury at its melting point of –39°C is placed in a 0.50-kg aluminum calorimeter filled with 1.2 kg of water at 20.0°C, the mercury melts and the final temperature of the combination is found to be 16.5°C. What is the heat of fusion of mercury in Jkg? APPROACH We follow the Problem Solving Strategy above. SOLUTION 1. Is the system isolated? The mercury is placed in a calorimeter, which we assume is well insulated. Our isolated system is the calorimeter, the water, and the mercury. 2. Conservation of energy. The heat gained by the mercury = the heat lost by the water and calorimeter. 3. and 4. Phase changes. There is a phase change (of mercury), plus we use specific heat equations. The heat gained by the mercury (Hg) includes a term representing the melting of the Hg, Q(melt solid Hg) = mHg LHg , plus a term representing the heating of the liquid Hg from –39°C to ±16.5°C: Q(heat liquid Hg) = mHg cHg[16.5°C - (–39°C)] = (1.0 kg)(140 JkgC°)(55.5 C°) = 7770 J. All of this heat gained by the mercury is obtained from the water and calorimeter, which cool down: Qcal + Qw = mcal ccal(20.0°C - 16.5°C) + mw cw(20.0°C - 16.5°C) = (0.50 kg)(900 JkgC°)(3.5 C°) + (1.2 kg)(4186 Jkg C°)(3.5 C°) = 19,200 J. 5. Energy equation. The conservation of energy tells us the heat lost by the water and calorimeter cup must equal the heat gained by the mercury: Qcal + Qw = Q(melt solid Hg) + Q(heat liquid Hg) or 19,200 J = mHg LHg + 7770 J. 6. Equilibrium temperature. It is given as 16.5°C, and we already used it.

504

CHAPTER 19

Heat and the First Law of Thermodynamics

7. Solve. The only unknown in our energy equation (point 5) is LHg , the latent heat of fusion (or melting) of mercury. We solve for it, putting in mHg = 1.0 kg: LHg =

19,200 J - 7770 J = 11,400 Jkg L 11 kJkg, 1.0 kg

where we rounded off to 2 significant figures.

Evaporation The latent heat to change a liquid to a gas is needed not only at the boiling point. Water can change from the liquid to the gas phase even at room temperature. This process is called evaporation (see also Section 18–4). The value of the heat of vaporization of water increases slightly with a decrease in temperature: at 20°C, for example, it is 2450 kJkg (585 kcalkg) compared to 2260 kJkg ( = 539 kcalkg) at 100°C. When water evaporates, the remaining liquid cools, because the energy required (the latent heat of vaporization) comes from the water itself; so its internal energy, and therefore its temperature, must drop.† Evaporation of water from the skin is one of the most important methods the body uses to control its temperature. When the temperature of the blood rises slightly above normal, the hypothalamus region of the brain detects this temperature increase and sends a signal to the sweat glands to increase their production. The energy (latent heat) required to vaporize this water comes from the body, and hence the body cools.

P H Y S I C S

A P P L I E D

Body temperature

Kinetic Theory of Latent Heats We can make use of kinetic theory to see why energy is needed to melt or vaporize a substance. At the melting point, the latent heat of fusion does not act to increase the average kinetic energy (and the temperature) of the molecules in the solid, but instead is used to overcome the potential energy associated with the forces between the molecules. That is, work must be done against these attractive forces to break the molecules loose from their relatively fixed positions in the solid so they can freely roll over one another in the liquid phase. Similarly, energy is required for molecules held close together in the liquid phase to escape into the gaseous phase. This process is a more violent reorganization of the molecules than is melting (the average distance between the molecules is greatly increased), and hence the heat of vaporization is generally much greater than the heat of fusion for a given substance.

19–6

The First Law of Thermodynamics

Up to now in this Chapter we have discussed internal energy and heat. But work too is often involved in thermodynamic processes. In Chapter 8 we saw that work is done when energy is transferred from one object to another by mechanical means. In Section 19–1 we saw that heat is a transfer of energy from one object to a second one at a lower temperature. Thus, heat is much like work. To distinguish them, heat is defined as a transfer of energy due to a difference in temperature, whereas work is a transfer of energy that is not due to a temperature difference. In Section 19–2, we defined the internal energy of a system as the sum total of all the energy of the molecules within the system. We would expect that the internal energy of a system would be increased if work was done on the system, or if heat were added to it. Similarly the internal energy would be decreased if heat flowed out of the system or if work were done by the system on something in the surroundings. †

According to kinetic theory, evaporation is a cooling process because it is the fastest-moving molecules that escape from the surface. Hence the average speed of the remaining molecules is less, so by Eq. 18–4 the temperature is less.

SECTION 19–6

The First Law of Thermodynamics

505

Thus it is reasonable to extend conservation of energy and propose an important law: the change in internal energy of a closed system, ¢Eint , will be equal to the energy added to the system by heating minus the work done by the system on the surroundings. In equation form we write FIRST LAW OF THERMODYNAMICS C A U T I O N

Heat added is Heat lost is Work on system is Work by system is

± – – ±

¢Eint = Q - W

(19–4)

where Q is the net heat added to the system and W is the net work done by the system.† We must be careful and consistent in following the sign conventions for Q and W. Because W in Eq. 19–4 is the work done by the system, then if work is done on the system, W will be negative and Eint will increase. Similarly, Q is positive for heat added to the system, so if heat leaves the system, Q is negative. Equation 19–4 is known as the first law of thermodynamics. It is one of the great laws of physics, and its validity rests on experiments (such as Joule's) to which no exceptions have been seen. Since Q and W represent energy transferred into or out of the system, the internal energy changes accordingly. Thus, the first law of thermodynamics is a great and broad statement of the law of conservation of energy. It is worth noting that the conservation of energy law was not formulated until the nineteenth century, for it depended on the interpretation of heat as a transfer of energy. Equation 19–4 applies to a closed system. It also applies to an open system (Section 19–4) if we take into account the change in internal energy due to the increase or decrease in the amount of matter. For an isolated system (p. 500), no work is done and no heat enters or leaves the system, so W = Q = 0, and hence ¢Eint = 0. A given system at any moment is in a particular state and can be said to have a certain amount of internal energy, Eint . But a system does not "have" a certain amount of heat or work. Rather, when work is done on a system (such as compressing a gas), or when heat is added or removed from a system, the state of the system changes. Thus, work and heat are involved in thermodynamic processes that can change the system from one state to another; they are not characteristic of the state itself. Quantities which describe the state of a system, such as internal energy Eint , pressure P, volume V, temperature T, and mass m or number of moles n, are called state variables. Q and W are not state variables. Because Eint is a state variable, which depends only on the state of the system and not on how the system arrived in that state, we can write ¢Eint = Eint, 2 - Eint, 1 = Q - W where Eint, 1 and Eint, 2 represent the internal energy of the system in states 1 and 2, and Q and W are the heat added to the system and work done by the system in going from state 1 to state 2. It is sometimes useful to write the first law of thermodynamics in differential form: dEint = dQ - dW. Here, dEint represents an infinitesimal change in internal energy when an infinitesimal amount of heat dQ is added to the system, and the system does an infinitesimal amount of work dW.‡

†

This convention relates historically to steam engines: the interest was in the heat input and the work output, both regarded as positive. In other books you may see the first law of thermodynamics written as ¢Eint = Q + W, in which case W refers to the work done on the system.

‡

The differential form of the first law is often written dEint = ÎQ - ÎW, where the bars on the differential sign (Î) are used to remind us that W and Q are not functions of the state variables (such as P, V, T, n). Internal energy, Eint , is a function of the state variables, and so dEint represents the differential (called an exact differential) of some function Eint . The differentials ÎW and ÎQ are not exact differentials (they are not the differential of some mathematical function); they thus only represent infinitesimal amounts. This issue won't really be of concern in this book.

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CHAPTER 19

Heat and the First Law of Thermodynamics

EXAMPLE 19–7 Using the first law. 2500 J of heat is added to a system, and 1800 J of work is done on the system. What is the change in internal energy of the system? APPROACH We apply the first law of thermodynamics, Eq. 19–4, to our system. SOLUTION The heat added to the system is Q = 2500 J. The work W done by the system is –1800 J. Why the minus sign? Because 1800 J done on the system (as given) equals –1800 J done by the system, and it is the latter we need for the sign conventions we used in Eq. 19–4. Hence ¢Eint = 2500 J - (–1800 J) = 2500 J + 1800 J = 4300 J. You may have intuitively thought that the 2500 J and the 1800 J would need to be added together, since both refer to energy added to the system. You would have been right. EXERCISE D What would be the internal energy change in Example 19–7 if 2500 J of heat is added to the system and 1800 J of work is done by the system (i.e., as output)?

* The First Law of Thermodynamics Extended To write the first law of thermodynamics in its complete form, consider a system that has kinetic energy K (there is motion) as well as potential energy U. Then the first law of thermodynamics would have to include these terms and would be written as ¢K + ¢U + ¢Eint = Q - W. (19–5) EXAMPLE 19–8 Kinetic energy transformed to thermal energy. A 3.0-g bullet traveling at a speed of 400 ms enters a tree and exits the other side with a speed of 200 ms. Where did the bullet's lost kinetic energy go, and what was the energy transferred? APPROACH Take the bullet and tree as our system. No potential energy is involved. No work is done on (or by) the system by outside forces, nor is any heat added because no energy was transferred to or from the system due to a temperature difference. Thus the kinetic energy gets transformed into internal energy of the bullet and tree. SOLUTION From the first law of thermodynamics as given in Eq. 19–5, we are given Q = W = ¢U = 0, so we have ¢K + ¢Eint = 0 or, using subscripts i and f for initial and final velocities ¢Eint = – ¢K = – AKf - Ki B = 12 mAv2i - v2f B = 12 A3.0 * 10 –3 kgB C(400 ms)2 - (200 ms)2 D = 180 J. NOTE The internal energy of the bullet and tree both increase, as both experience a rise in temperature. If we had chosen the bullet alone as our system, work would be done on it and heat transfer would occur.

19–7

The First Law of Thermodynamics Applied; Calculating the Work

Let us analyze some simple processes in the light of the first law of thermodynamics.

Isothermal Processes (T 0) First we consider an idealized process that is carried out at constant temperature. Such a process is called an isothermal process (from the Greek meaning "same temperature"). If the system is an ideal gas, then PV = nRT (Eq. 17–3), so for a fixed amount of gas kept at constant temperature, PV = constant. Thus the process follows a curve like AB on the PV diagram shown in Fig. 19–6, which is a curve for PV = constant. Each point on the curve, such as point A, represents the state of the system at a given moment—that is, its pressure P and volume V. At a lower temperature, another isothermal process would be represented by a curve like A¿B¿ in Fig. 19–6 (the product PV = nRT = constant is less when T is less). The curves shown in Fig. 19–6 are referred to as isotherms. SECTION 19–7

FIGURE 19–6 PV diagram for an ideal gas undergoing isothermal processes at two different temperatures.

P A′

A B

B′

Higher T Lower T

The First Law of Thermodynamics Applied; Calculating the Work

507

Movable piston

Ideal gas

FIGURE 19–7 An ideal gas in a cylinder fitted with a movable piston.

Let us assume that the gas is enclosed in a container fitted with a movable piston, Fig. 19–7, and that the gas is in contact with a heat reservoir (a body whose mass is so large that, ideally, its temperature does not change significantly when heat is exchanged with our system). We also assume that the process of compression (volume decreases) or expansion (volume increases) is done quasistatically ("almost statically"), by which we mean extremely slowly, so that all of the gas moves between a series of equilibrium states each of which are at the same constant temperature. If an amount of heat Q is added to the system and temperature is to remain constant, the gas will expand and do an amount of work W on the environment (it exerts a force on the piston and moves it through a distance). The temperature and mass are kept constant so, from Eq. 19–1, the internal energy does not change: ¢Eint = 32 nR ¢T = 0. Hence, by the first law of thermodynamics, Eq. 19–4, ¢Eint = Q - W = 0, so W = Q: the work done by the gas in an isothermal process equals the heat added to the gas.

Adiabatic Processes (Q 0)

Isothermal B

Adiabatic

FIGURE 19–8 PV diagram for adiabatic (AC) and isothermal (AB) processes on an ideal gas. FIGURE 19–9 (a) Isobaric ("same pressure") process. (b) Isovolumetric ("same volume") process.

P 0

Isobaric and Isovolumetric Processes

Isothermal and adiabatic processes are just two possible processes that can occur. Two other simple thermodynamic processes are illustrated on the PV diagrams of Fig. 19–9: (a) an isobaric process is one in which the pressure is kept constant, so P A the process is represented by a horizontal straight line on the PV diagram, B Fig. 19–9a; (b) an isovolumetric (or isochoric) process is one in which the volume 0 V does not change (Fig. 19–9b). In these, and in all other processes, the first law of (b) Isovolumetric thermodynamics holds.

An adiabatic process is one in which no heat is allowed to flow into or out of the system: Q = 0. This situation can occur if the system is extremely well insulated, or the process happens so quickly that heat—which flows slowly—has no time to flow in or out. The very rapid expansion of gases in an internal combustion engine is one example of a process that is very nearly adiabatic. A slow adiabatic expansion of an ideal gas follows a curve like that labeled AC in Fig. 19–8. Since Q = 0, we have from Eq. 19–4 that ¢Eint = –W. That is, the internal energy decreases if the gas expands; hence the temperature decreases as well (because ¢Eint = 32 nR ¢T ). This is evident in Fig. 19–8 where the product PV ( = nRT) is less at point C than at point B (curve AB is for an isothermal process, for which ¢Eint = 0 and ¢T = 0). In the reverse operation, an adiabatic compression (going from C to A, for example), work is done on the gas, and hence the internal energy increases and the temperature rises. In a diesel engine, the fuel–air mixture is rapidly compressed adiabatically by a factor of 15 or more; the temperature rise is so great that the mixture ignites spontaneously.

(a) Isobaric

Work Done in Volume Changes

FIGURE 19–10 The work done by a gas when its volume increases by dV = A dl is dW = P dV.

area A

We often want to calculate the work done in a process. Suppose we have a gas confined to a cylindrical container fitted with a movable piston (Fig. 19–10). We must always be careful to define exactly what our system is. In this case we choose our system to be the gas; so the container's walls and the piston are parts of the environment. Now let us calculate the work done by the gas when it expands quasistatically, so that P and T are defined for the system at all instants.† The gas expands against the piston, whose area is A. The gas exerts a force F = PA on the piston, where P is the pressure in the gas. The work done by the gas to move the piston an infinitesimal displacement dL is B

dW = F dL = PA dl = P dV

(19–6)

since the infinitesimal increase in volume is dV = A dl. If the gas was compressed so that dL pointed into the gas, the volume would decrease and dV 6 0. The work done by the gas in this case would then be negative, which is equivalent to saying that positive work was done on the gas, not by it. For a finite change in volume B

†

508

CHAPTER 19

If the gas expands or is compressed quickly, there would be turbulence and different parts would be at different pressure (and temperature).

from VA to VB , the work W done by the gas will be

dW =

(19–7)

P dV.

Equations 19–6 and 19–7 are valid for the work done in any volume change—by a gas, a liquid, or a solid—as long as it is done quasistatically. In order to integrate Eq. 19–7, we need to know how the pressure varies during the process, and this depends on the type of process. Let us first consider a quasistatic isothermal expansion of an ideal gas. This process is represented by the curve between points A and B on the PV diagram of Fig. 19–11. The work done by the gas in this process, according to Eq. 19–7, is just the area between the PV curve and the V axis, and is shown shaded in Fig. 19–11. We can do the integral in Eq. 19–7 for an ideal gas by using the ideal gas law, P = nRTV. The work done at constant T is W =

P dV = nRT

VB . dV = nRT ln V VA

isothermal process; d (19–8) ideal gas

Let us next consider a different way of taking an ideal gas between the same states A and B. This time, let us lower the pressure in the gas from PA to PB , as indicated by the line AD in Fig. 19–12. (In this isovolumetric process, heat must be allowed to flow out of the gas so its temperature drops.) Then let the gas expand from VA to VB at constant pressure A = PB B, which is indicated by the line DB in Fig. 19–12. (In this isobaric process, heat is added to the gas to raise its temperature.) No work is done in the isovolumetric process AD, since dV = 0: W = 0.

[isovolumetric process]

FIGURE 19–11 Work done by an ideal gas in an isothermal process equals the area under the PV curve. Shaded area equals the work done by the gas when it expands from VA to VB . FIGURE 19–12 Process ADB consists of an isovolumetric (AD) and an isobaric (DB) process.

P PA

[isobaric process] (19–9a)

The work done is again represented on the PV diagram by the area between the curve (ADB) and the V axis, as indicated by the shading in Fig. 19–12. Using the ideal gas law, we can also write W = PB AVB - VA B = nRTB ¢ 1 -

VA ≤. VB

P dV = PB AVB - VA B = P ¢V.

Iso

In the isobaric process DB the pressure remains constant, so W =

Isovolumetric

W =

Isobaric

isobaric process; d (19–9b) ideal gas

As can be seen from the shaded areas in Figs. 19–11 and 19–12, or by putting in numbers in Eqs. 19–8 and 19–9 (try it for VB = 2VA ), the work done in these two processes is different. This is a general result. The work done in taking a system from one state to another depends not only on the initial and final states but also on the type of process (or "path"). This result reemphasizes the fact that work cannot be considered a property of a system. The same is true of heat. The heat input required to change the gas from state A to state B depends on the process; for the isothermal process of Fig. 19–11, the heat input turns out to be greater than for the process ADB of Fig. 19–12. In general, the amount of heat added or removed in taking a system from one state to another depends not only on the initial and final states but also on the path or process.

CONCEPTUAL EXAMPLE 19–9 Work in isothermal and adiabatic processes. In Fig. 19–8 we saw the PV diagrams for a gas expanding in two ways, isothermally and adiabatically. The initial volume VA was the same in each case, and the final volumes were the same AVB = VC B. In which process was more work done by the gas? RESPONSE Our system is the gas. More work was done by the gas in the isothermal process, which we can see in two simple ways by looking at Fig. 19–8. First, the "average" pressure was higher during the isothermal process AB, so W = g ¢V was greater ( ¢V is the same for both processes). Second, we can look at the area under each curve: the area under curve AB, which represents the work done, was greater (since curve AB is higher) than that under AC. EXERCISE E Is the work done by the gas in process ADB of Fig. 19–12 greater than, less than, or equal to the work done in the isothermal process AB?

SECTION 19–7

509

EXAMPLE 19–10 First law in isobaric and isovolumetric processes. An ideal gas is slowly compressed at a constant pressure of 2.0 atm from 10.0 L to 2.0 L. This process is represented in Fig. 19–13 as the path B to D. (In this process, some heat flows out of the gas and the temperature drops.) Heat is then added to the gas, holding the volume constant, and the pressure and temperature are allowed to rise (line DA) until the temperature reaches its original value ATA = TB B. Calculate (a) the total work done by the gas in the process BDA, and (b) the total heat flow into the gas.

P PA Isovolumetric

D 0

Isothermal B Isobaric 4

10 V (L)

FIGURE 19–13 Example 19–10.

APPROACH (a) Work is done only in the compression process BD. In process DA, the volume is constant so ¢V = 0 and no work is done. (b) We use the first law of thermodynamics, Eq. 19–4. SOLUTION (a) During the compression BD, the pressure is 2.0 atm = 2A1.01 * 105 Nm2 B and the work done is (since 1 L = 103 cm3 = 10–3 m3) W = P ¢V = A2.02 * 105 Nm2 BA2.0 * 10–3 m3 - 10.0 * 10–3 m3 B = –1.6 * 103 J.

The total work done by the gas is –1.6 * 103 J, where the minus sign means that ±1.6 * 103 J of work is done on the gas. (b) Because the temperature at the beginning and at the end of process BDA is the same, there is no change in internal energy: ¢Eint = 0. From the first law of thermodynamics we have 0 = ¢Eint = Q - W so Q = W = –1.6 * 103 J. Because Q is negative, 1600 J of heat flows out of the gas for the whole process, BDA. EXERCISE F In Example 19–10, if the heat lost from the gas in the process BD is 8.4 * 103 J, what is the change in internal energy of the gas during process BD?

EXAMPLE 19–11 Work done in an engine. In an engine, 0.25 mol of an ideal monatomic gas in the cylinder expands rapidly and adiabatically against the piston. In the process, the temperature of the gas drops from 1150 K to 400 K. How much work does the gas do? APPROACH We take the gas as our system (the piston is part of the surroundings). The pressure is not constant, and its varying value is not given. Instead, we can use the first law of thermodynamics because we can determine ¢Eint given Q = 0 (the process is adiabatic). SOLUTION We determine ¢Eint from Eq. 19–1 for the internal energy of an ideal monatomic gas, using subscripts f and i for final and initial states: ¢Eint = Eint, f - Eint, i = 32 nRATf - Ti B = 32 (0.25 mol)(8.314 JmolK)(400 K - 1150 K) = –2300 J.

Then, from the first law of thermodynamics, Eq. 19–4, W = Q - ¢Eint = 0 - (–2300 J) = 2300 J. Table 19–3 gives a brief summary of the processes we have discussed.

Free Expansion One type of adiabatic process is a so-called free expansion in which a gas is allowed to expand in volume adiabatically without doing any work. The apparatus to TABLE 19–3 Simple Thermodynamic Processes and the First Law

510

CHAPTER 19

Process

What is constant:

Isothermal Isobaric Isovolumetric Adiabatic

T P V Q

= = = =

constant constant constant 0

The first law predicts: ¢T = 0 makes ¢Eint = 0, so Q = W Q = ¢Eint + W = ¢Eint + P ¢V ¢V = 0 makes W = 0, so Q = ¢Eint ¢Eint = –W

accomplish a free expansion is shown in Fig. 19–14. It consists of two well-insulated compartments (to ensure no heat flow in or out) connected by a valve or stopcock. One compartment is filled with gas, the other is empty. When the valve is opened, the gas expands to fill both containers. No heat flows in or out (Q = 0), and no work is done because the gas does not move any other object. Thus Q = W = 0 and by the first law of thermodynamics, ¢Eint = 0. The internal energy of a gas does not change in a free expansion. For an ideal gas, ¢T = 0 also, since Eint depends only on T (Section 19–2). Experimentally, the free expansion has been used to determine if the internal energy of real gases depends only on T. The experiments are very difficult to do accurately, but it has been found that the temperature of a real gas drops very slightly in a free expansion. Thus the internal energy of real gases does depend, a little, on pressure or volume as well as on temperature. A free expansion can not be plotted on a PV diagram, because the process is rapid, not quasistatic. The intermediate states are not equilibrium states, and hence the pressure (and even the volume at some instants) is not clearly defined.

19–8

FIGURE 19–14 Free expansion.

Molar Specific Heats for Gases, and the Equipartition of Energy

In Section 19–3 we discussed the concept of specific heat and applied it to solids and liquids. Much more than for solids and liquids, the values of the specific heat for gases depends on how the process is carried out. Two important processes are those in which either the volume or the pressure is kept constant. Although for solids and liquids it matters little, Table 19–4 shows that the specific heats of gases at constant volume AcV B and at constant pressure AcP B are quite different.

Molar Specific Heats for Gases The difference in specific heats for gases is nicely explained in terms of the first law of thermodynamics and kinetic theory. Our discussion is simplified if we use molar specific heats, CV and CP , which are defined as the heat required to raise 1 mol of the gas by 1 C° at constant volume and at constant pressure, respectively. That is, in analogy to Eq. 19–2, the heat Q needed to raise the temperature of n moles of gas by ¢T is Q = nCV ¢T

[volume constant] (19–10a)

Q = nCP ¢T.

[pressure constant] (19–10b)

It is clear from the definition of molar specific heat (or by comparing Eqs. 19–2 and 19–10) that CV = McV CP = McP , where M is the molecular mass of the gas (M = mn in gramsmol). The values for molar specific heats are included in Table 19–4, and we see that the values are nearly the same for different gases that have the same number of atoms per molecule. TABLE 19–4 Specific Heats of Gases at 15°C Specific heats (kcal kg K) Gas Monatomic He Ne Diatomic N2 O2 Triatomic CO2 H 2O (100°C)

Molar specific heats (cal mol K)

CP CV (cal mol K)

0.75 0.148

1.15 0.246

2.98 2.98

4.97 4.97

1.99 1.99

1.67 1.67

0.177 0.155

0.248 0.218

4.96 5.03

6.95 7.03

1.99 2.00

1.40 1.40

0.153 0.350

0.199 0.482

6.80 6.20

8.83 8.20

2.03 2.00

1.30 1.32

CP CV

SECTION 19–8

511

Now we use kinetic theory and imagine that an ideal gas is slowly heated via two different processes—first at constant volume, and then at constant pressure. In both of these processes, we let the temperature increase by the same amount, ¢T. In the process done at constant volume, no work is done since ¢V = 0. Thus, according to the first law of thermodynamics, the heat added (which we denote by QV) all goes into increasing the internal energy of the gas: QV = ¢Eint . In the process carried out at constant pressure, work is done, and hence the heat added, QP , must not only increase the internal energy but also is used to do the work W = P ¢V. Thus, more heat must be added in this process at constant pressure than in the first process at constant volume. For the process at constant pressure, we have from the first law of thermodynamics QP = ¢Eint + P ¢V. Since ¢Eint is the same in the two processes ( ¢T was chosen to be the same), we can combine the two above equations: QP - QV = P ¢V. From the ideal gas law, V = nRTP, so for a process at constant pressure we have ¢V = nR ¢TP. Putting this into the above equation and using Eqs. 19–10, we find nCP ¢T - nCV ¢T = P ¢

nR ¢T ≤ P

or, after cancellations, (19–11) CP - CV = R. Since the gas constant R = 8.314 Jmol K = 1.99 calmol K, our prediction is that CP will be larger than CV by about 1.99 calmolK. Indeed, this is very close to what is obtained experimentally, as shown in the next to last column of Table 19–4. Now we calculate the molar specific heat of a monatomic gas using kinetic theory. In a process carried out at constant volume, no work is done; so the first law of thermodynamics tells us that if heat Q is added to the gas, the internal energy of the gas changes by ¢Eint = Q. For an ideal monatomic gas, the internal energy Eint is the total kinetic energy of all the molecules, Eint = NA 12 mOB = 32 nRT as we saw in Section 19–2. Then, using Eq. 19–10a, we can write ¢Eint = Q in the form (19–12) ¢Eint = 32 nR ¢T = nCV ¢T or (19–13) CV = 32 R. Since R = 8.314 JmolK = 1.99 calmolK, kinetic theory predicts that CV = 2.98 calmolK for an ideal monatomic gas. This is very close to the experimental values for monatomic gases such as helium and neon (Table 19–4). From Eq. 19–11, CP is predicted to be about 4.97 calmolK, also in agreement with experiment.

Equipartition of Energy

FIGURE 19–15 A diatomic molecule can rotate about two different axes. Axis

Axis (a)

512

CHAPTER 19

(b)

The measured molar specific heats for more complex gases (Table 19–4), such as diatomic (two atoms) and triatomic (three atoms) gases, increase with the increased number of atoms per molecule. We can explain this by assuming that the internal energy includes not only translational kinetic energy but other forms of energy as well. Take, for example, a diatomic gas. As shown in Fig. 19–15 the two atoms can rotate about two different axes (but rotation about a third axis passing through the two atoms would give rise to very little energy since the moment of inertia is so small). The molecules can have rotational as well as translational kinetic energy. It is useful to introduce the idea of degrees of freedom, by which we mean the number of independent ways molecules can possess energy. For example, a monatomic gas is said to have three degrees of freedom, since an atom can have velocity along the x axis, the y axis, and the z axis. These are considered to be three independent motions because a change in any one of the components would not affect any of the others. A diatomic molecule has the same three degrees of freedom associated with translational kinetic

energy plus two more degrees of freedom associated with rotational kinetic energy, for a total of five degrees of freedom. A quick look at Table 19–4 indicates that the CV for diatomic gases is about 53 times as great as for a monatomic gas—that is, in the same ratio as their degrees of freedom. This result led nineteenth-century physicists to an important idea, the principle of equipartition of energy. This principle states that energy is shared equally among the active degrees of freedom, and in particular each active degree of freedom of a molecule has on the average an energy equal to 1 3 2 kT. Thus, the average energy for a molecule of a monatomic gas would be 2 kT 5 (which we already knew) and of a diatomic gas 2 kT. Hence the internal energy of a diatomic gas would be Eint = NA 52 kTB = 52 nRT, where n is the number of moles. Using the same argument we did for monatomic gases, we see that for diatomic gases the molar specific heat at constant volume would be 52 R = 4.97 calmolK, in accordance with measured values. More complex molecules have even more degrees of freedom and thus greater molar specific heats. The situation was complicated, however, by measurements that showed that for diatomic gases at very low temperatures, CV has a value of only 32 R, as if it had only three degrees of freedom. And at very high temperatures, CV was about 72 R, as if there were seven degrees of freedom. The explanation is that at low temperatures, nearly all molecules have only translational kinetic energy. That is, no energy goes into rotational energy, so only three degrees of freedom are "active." At very high temperatures, on the other hand, all five degrees of freedom are active plus two additional ones. We can interpret the two new degrees of freedom as being associated with the two atoms vibrating as if they were connected by a spring, as shown in Fig. 19–16. One degree of freedom comes from the kinetic energy of the vibrational motion, and the second comes from the potential energy of vibrational motion A 12 kx 2 B. At room temperature, these two degrees of freedom are apparently not active. See Fig. 19–17.

3R Rotational kinetic energy

2R R 0

7 2R Vibrational kinetic energy 5 R 2 3 R 2

Translational kinetic energy 20

100

200

500

1000

2000

5000

10,000

T (K)

FIGURE 19–17 Molar specific heat CV as a function of temperature for hydrogen molecules AH 2 B. As the temperature is increased, some of the translational kinetic energy can be transferred in collisions into rotational kinetic energy and, at still higher temperature, into vibrational kinetic energy. [Note: H 2 dissociates into two atoms at about 3200 K, so the last part of the curve is shown dashed.]

Just why fewer degrees of freedom are "active" at lower temperatures was eventually explained by Einstein using the quantum theory. [According to quantum theory, energy does not take on continuous values but is quantized—it can have only certain values, and there is a certain minimum energy. The minimum rotational and vibrational energies are higher than for simple translational kinetic energy, so at lower temperatures and lower translational kinetic energy, there is not enough energy to excite the rotational or vibrational kinetic energy.] Calculations based on kinetic theory and the principle of equipartition of energy (as modified by the quantum theory) give numerical results in accord with experiment.

* Solids The principle of equipartition of energy can be applied to solids as well. The molar specific heat of any solid, at high temperature, is close to 3R (6.0 calmolK), Fig. 19–18. This is called the Dulong and Petit value after the scientists who first measured it in 1819. (Note that Table 19–1 gives the specific heats per kilogram, not per mole.) At high temperatures, each atom apparently has six degrees of freedom, although some are not active at low temperatures. Each atom in a crystalline solid can vibrate about its equilibrium position as if it were connected by springs to each of its neighbors. Thus it can have three degrees of freedom for kinetic energy and three more associated with potential energy of vibration in each of the x, y, and z directions, which is in accord with the measured values.

FIGURE 19–18 Molar specific heats of solids as a function of temperature. Molar specific heat at constant volume (cal/mol · K)

FIGURE 19–16 A diatomic molecule can vibrate, as if the two atoms were connected by a spring. Of course they are not connected by a spring; rather they exert forces on each other that are electrical in nature, but of a form that resembles a spring force.

6 5 4 3 2 1 0

Lead

Copper Beryllium Diamond

0 100

300 500 700 Temperature (K)

SECTION 19–8

900

513

19–9

Adiabatic Expansion of a Gas

The PV curve for the quasistatic (slow) adiabatic expansion (Q = 0) of an ideal gas was shown in Fig. 19–8 (curve AC). It is somewhat steeper than for an isothermal process (¢T = 0), which indicates that for the same change in volume the change in pressure will be greater. Hence the temperature of the gas must drop during an adiabatic expansion. Conversely, the temperature rises during an adiabatic compression. We can derive the relation between the pressure P and the volume V of an ideal gas that is allowed to slowly expand adiabatically. We begin with the first law of thermodynamics, written in differential form: dEint = dQ - dW = –dW = –P dV, since dQ = 0 for an adiabatic process. Equation 19–12 gives us a relation between ¢Eint and CV , which is valid for any ideal gas process since Eint is a function only of T for an ideal gas. We write this in differential form: dEint = nCV dT. When we combine these last two equations, we obtain nCV dT + P dV = 0. We next take the differential of the ideal gas law, PV = nRT, allowing P, V, and T to vary: P dV + V dP = nR dT. We solve for dT in this relation and substitute it into the previous relation and get P dV + V dP nCV ¢ ≤ + P dV = 0 nR or, multiplying through by R and rearranging, ACV + RBP dV + CV V dP = 0. We note from Eq. 19–11 that CV + R = CP , so we have CP P dV + CV V dP = 0, or CP P dV + V dP = 0. CV We define C g = P (19–14) CV so that our last equation becomes dP dV + g = 0. P V This is integrated to become ln P + g ln V = constant. This simplifies (using the rules for addition and multiplication of logarithms) to PVg = constant.

quasistatic adiabatic d (19–15) process; ideal gas

This is the relation between P and V for a quasistatic adiabatic expansion or contraction. We will find it very useful when we discuss heat engines in the next Chapter. Table 19–4 (p. 511) gives values of g for some real gases. Figure 19–8 compares an adiabatic expansion (Eq. 19–15) in curve AC to an isothermal expansion (PV = constant) in curve AB. It is important to remember that the ideal gas law, PV = nRT, continues to hold even for an adiabatic expansion APVg = constantB; clearly PV is not constant, meaning T is not constant. EXAMPLE 19–12 Compressing an ideal gas. An ideal monatomic gas is compressed starting at point A in the PV diagram of Fig. 19–19, where PA = 100 kPa, VA = 1.00 m3, and TA = 300 K. The gas is first compressed adiabatically to state B (PB = 200 kPa). The gas is then further compressed from point B to point C AVC = 0.50 m3 B in an isothermal process. (a) Determine VB . (b) Calculate the work done on the gas for the whole process.

514

CHAPTER 19

APPROACH Volume VB is obtained using Eq. 19–15. The work done by a gas is given by Eq. 19–7, W = P dV. The work on the gas is the negative of this: Won = – P dV. SOLUTION In the adiabatic process, Eq. 19–15 tells us PVg = constant. Therefore, g g PVg = PA VA = PB VB where for a monatomic gas g = CPCV = (52)(32) = 53 . 1 Bg

(a) Eq. 19–15 gives VB = VA APA PB = A1.00 m BA100 kPa200 = 0.66 m . (b) The pressure P at any instant during the adiabatic process is given by g g P = PA VAV – . The work done on the gas in going from VA to VB is WAB = –

P dV = –PA VA

3 kPaB 5

V – dV = –PA VA ¢

WAB = – aPA VA3 b a– 5

VB 3 –2 b aVA 3 b c ¢ ≤ 2 VA

1 g g ≤ AV1B- - V1A- B. g – + 1

- 1d = ±

Isothermal B

200

Adiabatic

Since g = 53 , then – g + 1 = 1 - g = – 23 , so –2 3

P (kPa)

100

0.50

V (m3)

1.00

FIGURE 19–19 Example 19–12.

VB 3 PA VA c ¢ ≤ 2 VA

–2 3

- 1d

3 –2 (100 kPa)(1.00 m3) c (0.66) 3 - 1 d = ±48 kJ. 2 For the isothermal process from B to C, the work is done at constant temperature, so the pressure at any instant during the process is P = nRTBV and = ±

WBC = –

P dV = –nRTB

VC VC dV = –nRTB ln = –PB VB ln = ±37 kJ. V VB VB

The total work done on the gas is 48 kJ + 37 kJ = 85 kJ.

19–10

Heat Transfer: Conduction, Convection, Radiation

Heat is transferred from one place or body to another in three different ways: by conduction, convection, and radiation. We now discuss each of these in turn; but in practical situations, any two or all three may be operating at the same time. We start with conduction.

Conduction When a metal poker is put in a hot fire, or a silver spoon is placed in a hot bowl of soup, the exposed end of the poker or spoon soon becomes hot as well, even though it is not directly in contact with the source of heat. We say that heat has been conducted from the hot end to the cold end. Heat conduction in many materials can be visualized as being carried out via molecular collisions. As one end of an object is heated, the molecules there move faster and faster. As they collide with slower-moving neighbors, they transfer some of their kinetic energy to these other molecules, which in turn transfer energy by collision with molecules still farther along the object. In metals, collisions of free electrons are mainly responsible for conduction. Heat conduction from one point to another takes place only if there is a difference in temperature between the two points. Indeed, it is found experimentally that the rate of heat flow through a substance is proportional to the difference in temperature between its ends. The rate of heat flow also depends on the size and shape of the object. To investigate this quantitatively, let us consider the heat flow through a uniform cylinder, as illustrated in Fig. 19–20. It is found experimentally that the heat flow ¢Q over a time interval ¢ t is given by the relation T1 - T2 ¢Q = kA (19–16a) ¢t l where A is the cross-sectional area of the object, l is the distance between the two ends, which are at temperatures T1 and T2 , and k is a proportionality constant called the thermal conductivity which is characteristic of the material. From Eq. 19–16a, we see that the rate of heat flow (units of Js) is directly proportional to the cross-sectional area and to the temperature gradient AT1 - T2 Bl.

Hotter

Cooler A

Heat

flow

l FIGURE 19–20 Heat conduction between areas at temperatures T1 and T2 . If T1 is greater than T2 , the heat flows to the right; the rate is given by Eq. 19–16a.

SECTION 19–10

515

TABLE 19–5 Thermal Conductivities Thermal conductivity, k Substance

kcal J (s m C°) (s m C°)

Silver 10 Copper 9.2 Aluminum 5.0 Steel 1.1 Ice 5 Glass 2.0 Brick 2.0 Concrete 2.0 Water 1.4 Human tissue 0.5 Wood 0.3 Fiberglass 0.12 Cork 0.1 Wool 0.1 Goose down 0.06 Polyurethane 0.06 Air 0.055

* * * * * * * * * * * * * * * * *

10 –2 10 –2 10 –2 10 –2 10 –4 10 –4 10 –4 10 –4 10 –4 10 –4 10 –4 10 –4 10 –4 10 –4 10 –4 10 –4 10 –4

420 380 200 40 2 0.84 0.84 0.84 0.56 0.2 0.1 0.048 0.042 0.040 0.025 0.024 0.023

In some cases (such as when k or A cannot be considered constant) we need to consider the limit of an infinitesimally thin slab of thickness dx. Then Eq. 19–16a becomes dQ dT , (19–16b) = –kA dt dx where dTdx is the temperature gradient† and the negative sign is included since the heat flow is in the direction opposite to the temperature gradient. The thermal conductivities, k, for a variety of substances are given in Table 19–5. Substances for which k is large conduct heat rapidly and are said to be good thermal conductors. Most metals fall in this category, although there is a wide range even among them, as you may observe by holding the ends of a silver spoon and a stainless-steel spoon immersed in the same hot cup of soup. Substances for which k is small, such as wool, fiberglass, polyurethane, and goose down, are poor conductors of heat and are therefore good thermal insulators. The relative magnitudes of k can explain simple phenomena such as why a tile floor is much colder on the feet than a rug-covered floor at the same temperature. Tile is a better conductor of heat than the rug. Heat that flows from your foot to the rug is not conducted away rapidly, so the rug's surface quickly warms up to the temperature of your foot and feels good. But the tile conducts the heat away rapidly and thus can take more heat from your foot quickly, so your foot's surface temperature drops. EXAMPLE 19–13 Heat loss through windows. A major source of heat loss from a house is through the windows. Calculate the rate of heat flow through a glass window 2.0 m * 1.5 m in area and 3.2 mm thick, if the temperatures at the inner and outer surfaces are 15.0°C and 14.0°C, respectively (Fig. 19–21). APPROACH Heat flows by conduction through the glass from the higher inside temperature to the lower outside temperature. We use the heat conduction equation, Eq. 19–16a. SOLUTION Here A = (2.0 m)(1.5 m) = 3.0 m2 and l = 3.2 * 10–3 m. Using Table 19–5 to get k, we have (0.84 Js mC°)A3.0 m2 B(15.0°C - 14.0°C) T1 - T2 ¢Q = kA = ¢t l A3.2 * 10–3 mB = 790 Js. NOTE This rate of heat flow is equivalent to (790 Js)A4.19 * 103 JkcalB = 0.19 kcals, or (0.19 kcals) * (3600 sh) = 680 kcalh.

FIGURE 19–21 Example 19–13.

A= 3.0 m2 14.0°C

15.0°C

l = 3.2 ×

P H Y S I C S

10−3 m

A P P L I E D

Thermal windows

You might notice in Example 19–13 that 15°C is not very warm for the living room of a house. The room itself may indeed be much warmer, and the outside might be colder than 14°C. But the temperatures of 15°C and 14°C were specified as those at the window surfaces, and there is usually a considerable drop in temperature of the air in the vicinity of the window both on the inside and the outside. That is, the layer of air on either side of the window acts as an insulator, and normally the major part of the temperature drop between the inside and outside of the house takes place across the air layer. If there is a heavy wind, the air outside a window will constantly be replaced with cold air; the temperature gradient across the glass will be greater and there will be a much greater rate of heat loss. Increasing the width of the air layer, such as using two panes of glass separated by an air gap, will reduce the heat loss more than simply increasing the glass thickness, since the thermal conductivity of air is much less than that for glass. The insulating properties of clothing come from the insulating properties of air. Without clothes, our bodies in still air would heat the air in contact with the skin and would soon become reasonably comfortable because air is a very good insulator. †

516

CHAPTER 19

Equations 19–16 are quite similar to the relations describing diffusion (Section 18–7) and the flow of fluids through a pipe (Section 13–12). In those cases, the flow of matter was found to be proportional to the concentration gradient dCdx, or to the pressure gradient AP1 - P2 Bl. This close similarity is one reason we speak of the "flow" of heat. Yet we must keep in mind that no substance is flowing in the case of heat—it is energy that is being transferred.

But since air moves—there are breezes and drafts, and people move about—the warm air would be replaced by cold air, thus increasing the temperature difference and the heat loss from the body. Clothes keep us warm by trapping air so it cannot move readily. It is not the cloth that insulates us, but the air that the cloth traps. Goose down is a very good insulator because even a small amount of it fluffs up and traps a great amount of air. [For practical purposes the thermal properties of building materials, particularly when considered as insulation, are usually specified by R-values (or "thermal resistance"), defined for a given thickness l of material as: R = lk. The R-value of a given piece of material combines the thickness l and the thermal conductivity k in one number. In the United States, R-values are given in British units as ft2 hF°Btu (for example, R-19 means R = 19 ft2 h F°Btu ). Table 19–6 gives R-values for some common building materials. R-values increase directly with material thickness: for example, 2 inches of fiberglass is R-6, whereas 4 inches is R-12.]

Convection Although liquids and gases are generally not very good conductors of heat, they can transfer heat quite rapidly by convection. Convection is the process whereby heat flows by the mass movement of molecules from one place to another. Whereas conduction involves molecules (andor electrons) moving only over small distances and colliding, convection involves the movement of large numbers of molecules over large distances. A forced-air furnace, in which air is heated and then blown by a fan into a room, is an example of forced convection. Natural convection occurs as well, and one familiar example is that hot air rises. For instance, the air above a radiator (or other type of heater) expands as it is heated (Chapter 17), and hence its density decreases. Because its density is less than that of the surrounding cooler air, it rises, just as a log submerged in water floats upward because its density is less than that of water. Warm or cold ocean currents, such as the balmy Gulf Stream, represent natural convection on a global scale. Wind is another example of convection, and weather in general is strongly influenced by convective air currents. When a pot of water is heated (Fig. 19–22), convection currents are set up as the heated water at the bottom of the pot rises because of its reduced density. That heated water is replaced by cooler water from above. This principle is used in many heating systems, such as the hot-water radiator system shown in Fig. 19–23. Water is heated in the furnace, and as its temperature increases, it expands and rises as shown. This causes the water to circulate in the heating system. Hot water then enters the radiators, heat is transferred by conduction to the air, and the cooled water returns to the furnace. Thus, the water circulates because of convection; pumps are sometimes used to improve circulation. The air throughout the room also becomes heated as a result of convection. The air heated by the radiators rises and is replaced by cooler air, resulting in convective air currents, as shown by the green arrows in Fig. 19–23. Other types of furnaces also depend on convection. Hot-air furnaces with registers (openings) near the floor often do not have fans but depend on natural convection, which can be appreciable. In other systems, a fan is used. In either case, it is important that cold air can return to the furnace so that convective currents circulate throughout the room if the room is to be uniformly heated. Convection is not always favorable. Much of the heat from a fireplace, for example, goes up the chimney and not out into the room.

P H Y S I C S

A P P L I E D

Clothes insulate by trapping an air layer

P H Y S I C S

A P P L I E D

R-values of thermal insulation

TABLE 19–6 R-values Material

Thickness

R-value (ft2 h F°Btu)

1 Glass 8 inch Brick 3 12 inches 1 Plywood 2 inch Fiberglass insulation 4 inches

1 0.6–1 0.6 12

Cooler water

Hotter water FIGURE 19–22 Convection currents in a pot of water being heated on a stove. P H Y S I C S

A P P L I E D

Convective home heating FIGURE 19–23 Convection plays a role in heating a house. The circular arrows show the convective air currents in the rooms.

Radiator

Cool

Hot Furnace water

Radiation Convection and conduction require the presence of matter as a medium to carry the heat from the hotter to the colder region. But a third type of heat transfer occurs without any medium at all. All life on Earth depends on the transfer of energy from the Sun, and this energy is transferred to the Earth over empty (or nearly empty) space. This form of energy transfer is heat—since the Sun's surface temperature is much higher (6000 K) than Earth's—and is referred to as radiation. The warmth we receive from a fire is mainly radiant energy.

SECTION 19–10

517

As we shall see in later Chapters, radiation consists essentially of electromagnetic waves. Suffice it to say for now that radiation from the Sun consists of visible light plus many other wavelengths that the eye is not sensitive to, including infrared (IR) radiation. The rate at which an object radiates energy has been found to be proportional to the fourth power of the Kelvin temperature, T. That is, an object at 2000 K, as compared to one at 1000 K, radiates energy at a rate 2 4 = 16 times as much. The rate of radiation is also proportional to the area A of the emitting object, so the rate at which energy leaves the object, ¢Q¢ t, is ¢Q (19–17) = sAT4. ¢t This is called the Stefan-Boltzmann equation, and s is a universal constant called the Stefan-Boltzmann constant which has the value s = 5.67 * 10–8 Wm2 K4.

P H Y S I C S

A P P L I E D

Dark vs. light clothing

P H Y S I C S

A P P L I E D

The body's radiative heat loss

P R O B L E M

S O LV I N G

Must use the Kelvin temperature

The factor (Greek letter epsilon), called the emissivity, is a number between 0 and 1 that is characteristic of the surface of the radiating material. Very black surfaces, such as charcoal, have emissivity close to 1, whereas shiny metal surfaces have close to zero and thus emit correspondingly less radiation. The value depends somewhat on the temperature of the material. Not only do shiny surfaces emit less radiation, but they absorb little of the radiation that falls upon them (most is reflected). Black and very dark objects are good emitters ( L 1), and they also absorb nearly all the radiation that falls on them—which is why light-colored clothing is usually preferable to dark clothing on a hot day. Thus, a good absorber is also a good emitter. Any object not only emits energy by radiation but also absorbs energy radiated by other objects. If an object of emissivity and area A is at a temperature T1 , it radiates energy at a rate sAT41 . If the object is surrounded by an environment at temperature T2 , the rate at which the surroundings radiate energy is proportional to T42 , and the rate that energy is absorbed by the object is proportional to T42 . The net rate of radiant heat flow from the object is given by ¢Q (19–18) = sAAT41 - T42 B, ¢t where A is the surface area of the object, T1 its temperature and its emissivity (at temperature T1), and T2 is the temperature of the surroundings. This equation is consistent with the experimental fact that equilibrium between the object and its surroundings is reached when they come to the same temperature. That is, ¢Q¢ t must equal zero when T1 = T2 , so must be the same for emission and absorption. This confirms the idea that a good emitter is a good absorber. Because both the object and its surroundings radiate energy, there is a net transfer of energy from one to the other unless everything is at the same temperature. EXAMPLE 19–14 ESTIMATE Cooling by radiation. An athlete is sitting unclothed in a locker room whose dark walls are at a temperature of 15°C. Estimate his rate of heat loss by radiation, assuming a skin temperature of 34°C and = 0.70. Take the surface area of the body not in contact with the chair to be 1.5 m2. APPROACH We use Eq. 19–18, with Kelvin temperatures. SOLUTION We have ¢Q = sAAT41 - T42 B ¢t = (0.70)A5.67 * 10–8 Wm2 K4 BA1.5 m2 B C(307 K)4 - (288 K)4 D = 120 W. NOTE The "output" of this resting person is a bit more than what a 100-W lightbulb uses.

518

CHAPTER 19

Heat and the First Law of Thermodynamics

A resting person naturally produces heat internally at a rate of about 100 W, which is less than the heat loss by radiation as calculated in Example 19–14. Hence, the person's temperature would drop, causing considerable discomfort. The body responds to excessive heat loss by increasing its metabolic rate, and shivering is one method by which the body increases its metabolism. Naturally, clothes help a lot. Example 19–14 illustrates that a person may be uncomfortable even if the temperature of the air is, say, 25°C, which is quite a warm room. If the walls or floor are cold, radiation to them occurs no matter how warm the air is. Indeed, it is estimated that radiation accounts for about 50% of the heat loss from a sedentary person in a normal room. Rooms are most comfortable when the walls and floor are warm and the air is not so warm. Floors and walls can be heated by means of hot-water conduits or electric heating elements. Such first-rate heating systems are becoming more common today, and it is interesting to note that 2000 years ago the Romans, even in houses in the remote province of Great Britain, made use of hot-water and steam conduits in the floor to heat their houses. Heating of an object by radiation from the Sun cannot be calculated using Eq. 19–18 since this equation assumes a uniform temperature, T2 , of the environment surrounding the object, whereas the Sun is essentially a point source. Hence the Sun must be treated as a separate source of energy. Heating by the Sun is calculated using the fact that about 1350 J of energy strikes the atmosphere of the Earth from the Sun per second per square meter of area at right angles to the Sun's rays. This number, 1350 Wm2, is called the solar constant. The atmosphere may absorb as much as 70% of this energy before it reaches the ground, depending on the cloud cover. On a clear day, about 1000 Wm2 reaches the Earth's surface. An object of emissivity with area A facing the Sun absorbs energy from the Sun at a rate, in watts, of about ¢Q = A1000 Wm2 B A cos u, ¢t

(19–19)

where u is the angle between the Sun's rays and a line perpendicular to the area A (Fig. 19–24). That is, A cos u is the "effective" area, at right angles to the Sun's rays. The explanation for the seasons and the polar ice caps (see Fig. 19–25) depends on this cos u factor in Eq. 19–19. The seasons are not a result of how close the Earth is to the Sun—in fact, in the Northern Hemisphere, summer occurs when the Earth is farthest from the Sun. It is the angle (i.e., cos u) that really matters. Furthermore, the reason the Sun heats the Earth more at midday than at sunrise or sunset is also related to this cos u factor. An interesting application of thermal radiation to diagnostic medicine is thermography. A special instrument, the thermograph, scans the body, measuring the intensity of radiation from many points and forming a picture that resembles an X-ray (Fig. 19–26). Areas where metabolic activity is high, such as in tumors, can often be detected on a thermogram as a result of their higher temperature and consequent increased radiation.

FIGURE 19–24 Radiant energy striking a body at an angle u. FIGURE 19–25 (a) Earth's seasons arise from the 23 12° angle Earth's axis makes with its orbit around the Sun. (b) June sunlight makes an angle of about 23° with the equator. Thus u in the southern United States (A) is near 0° (direct summer sunlight), whereas in the Southern Hemisphere (B), u is 50° or 60°, and less heat can be absorbed—hence it is winter. Near the poles (C), there is never strong direct sunlight; cos u varies from about 12 in summer to 0 in winter; so with little heating, ice can form. Axis

Equator Earth (June)

Sun

Earth (December)

(a) Axis

June

(A) θ ≈ 0° (Summer)

Equ

ator

Sun's rays (June) 23°

FIGURE 19–26 Thermograms of a healthy person's arms and hands (a) before and (b) after smoking a cigarette, showing a temperature decrease due to impaired blood circulation associated with smoking. The thermograms have been color-coded according to temperature; the scale on the right goes from blue (cold) to white (hot). (a)

(b)

SECTION 19–10

Heat Transfer: Conduction, Convection, Radiation

519

P H Y S I C S

A P P L I E D

Astronomy—size of a star

EXAMPLE 19–15 ESTIMATE Star radius. The giant star Betelgeuse emits radiant energy at a rate 104 times greater than our Sun, whereas its surface temperature is only half (2900 K) that of our Sun. Estimate the radius of Betelgeuse, assuming = 1 for both. The Sun's radius is rS = 7 * 108 m. APPROACH We assume both Betelgeuse and the Sun are spherical, with surface area 4pr 2. SOLUTION We solve Eq. 19–17 for A: 4pr 2 = A =

(¢Q¢ t) sT4

Then (¢Q¢ t)B T4S r 2B = = A104 BA2 4 B = 16 * 104. (¢Q¢ t)S T4B r2S Hence rB = 316 * 104 rS = (400)A7 * 108 mB L 3 * 1011 m. NOTE If Betelgeuse were our Sun, it would envelop us (Earth is 1.5 * 1011 m from the Sun). EXERCISE G Fanning yourself on a hot day cools you by (a) increasing the radiation rate of the skin; (b) increasing conductivity; (c) decreasing the mean free path of air; (d) increasing the evaporation of perspiration; (e) none of these.

Summary Internal energy, Eint , refers to the total energy of all the molecules in an object. For an ideal monatomic gas, Eint =

3 2 NkT

= 32 nRT

(19–1)

where N is the number of molecules or n is the number of moles. Heat refers to the transfer of energy from one object to another because of a difference of temperature. Heat is thus measured in energy units, such as joules. Heat and internal energy are also sometimes specified in calories or kilocalories (kcal), where 1 kcal = 4.186 kJ is the amount of heat needed to raise the temperature of 1 kg of water by 1 C°. The specific heat, c, of a substance is defined as the energy (or heat) required to change the temperature of unit mass of substance by 1 degree; as an equation, Q = mc ¢T,

CHAPTER 19

¢Eint = Q - W.

(19–4)

This important law is a broad restatement of the conservation of energy and is found to hold for all processes. Two simple thermodynamic processes are isothermal, which is a process carried out at constant temperature, and adiabatic, a process in which no heat is exchanged. Two more are isobaric (a process carried out at constant pressure) and isovolumetric (a process at constant volume). The work done by (or on) a gas to change its volume by dV is dW = P dV, where P is the pressure. Work and heat are not functions of the state of a system (as are P, V, T, n, and Eint) but depend on the type of process that takes a system from one state to another. The molar specific heat of an ideal gas at constant volume, CV , and at constant pressure, CP , are related by

(19–2)

where Q is the heat absorbed or given off, ¢T is the temperature increase or decrease, and m is the mass of the substance. When heat flows between parts of an isolated system, conservation of energy tells us that the heat gained by one part of the system is equal to the heat lost by the other part of the system. This is the basis of calorimetry, which is the quantitative measurement of heat exchange. Exchange of energy occurs, without a change in temperature, whenever a substance changes phase. The heat of fusion is the heat required to melt 1 kg of a solid into the liquid phase; it is also equal to the heat given off when the substance changes from liquid to solid. The heat of vaporization is the energy required to change 1 kg of a substance from the liquid to the vapor phase; it is also the energy given off when the substance changes from vapor to liquid.

520

The first law of thermodynamics states that the change in internal energy ¢Eint of a system is equal to the heat added to the system, Q, minus the work, W, done by the system:

CP - CV = R,

(19–11)

where R is the gas constant. For a monatomic ideal gas, CV = 32 R. For ideal gases made up of diatomic or more complex molecules, CV is equal to 12 R times the number of degrees of freedom of the molecule. Unless the temperature is very high, some of the degrees of freedom may not be active and so do not contribute. According to the principle of equipartition of energy, energy is shared equally among the active degrees of freedom in an amount 12 kT per molecule on average. When an ideal gas expands (or contracts) adiabatically g (Q = 0), the relation PV = constant holds, where g =

CP . CV

(19–14)

Heat is transferred from one place (or object) to another in three different ways: conduction, convection, and radiation.

Heat and the First Law of Thermodynamics

In conduction, energy is transferred by collisions between molecules or electrons with higher kinetic energy to slowermoving neighbors. Convection is the transfer of energy by the mass movement of molecules over considerable distances. Radiation, which does not require the presence of matter, is energy transfer by electromagnetic waves, such as from the Sun.

All objects radiate energy in an amount that is proportional to the fourth power of their Kelvin temperature AT4 B and to their surface area. The energy radiated (or absorbed) also depends on the nature of the surface, which is characterized by the emissivity, (dark surfaces absorb and radiate more than do bright shiny ones). Radiation from the Sun arrives at the surface of the Earth on a clear day at a rate of about 1000 Wm2.

Questions 1. What happens to the work done on a jar of orange juice when it is vigorously shaken? 2. When a hot object warms a cooler object, does temperature flow between them? Are the temperature changes of the two objects equal? Explain. 3. (a) If two objects of different temperature are placed in contact, will heat naturally flow from the object with higher internal energy to the object with lower internal energy? (b) Is it possible for heat to flow even if the internal energies of the two objects are the same? Explain. 4. In warm regions where tropical plants grow but the temperature may drop below freezing a few times in the winter, the destruction of sensitive plants due to freezing can be reduced by watering them in the evening. Explain. 5. The specific heat of water is quite large. Explain why this fact makes water particularly good for heating systems (that is, hot-water radiators). 6. Why does water in a canteen stay cooler if the cloth jacket surrounding the canteen is kept moist? 7. Explain why burns caused by steam at 100°C on the skin are often more severe than burns caused by water at 100°C. 8. Explain why water cools (its temperature drops) when it evaporates, using the concepts of latent heat and internal energy. 9. Will potatoes cook faster if the water is boiling more vigorously? 10. Very high in the Earth's atmosphere the temperature can be 700°C. Yet an animal there would freeze to death rather than roast. Explain. 11. What happens to the internal energy of water vapor in the air that condenses on the outside of a cold glass of water? Is work done or heat exchanged? Explain. 12. Use the conservation of energy to explain why the temperature of a well-insulated gas increases when it is compressed—say, by pushing down on a piston—whereas the temperature decreases when the gas expands. 13. In an isothermal process, 3700 J of work is done by an ideal gas. Is this enough information to tell how much heat has been added to the system? If so, how much? 14. Explorers on failed Arctic expeditions have survived by covering themselves with snow. Why would they do that? 15. Why is wet sand at the beach cooler to walk on than dry sand? 16. When hot-air furnaces are used to heat a house, why is it important that there be a vent for air to return to the furnace? What happens if this vent is blocked by a bookcase? 17. Is it possible for the temperature of a system to remain constant even though heat flows into or out of it? If so, give examples.

18. Discuss how the first law of thermodynamics can apply to metabolism in humans. In particular, note that a person does work W, but very little heat Q is added to the body (rather, it tends to flow out). Why then doesn't the internal energy drop drastically in time? 19. Explain in words why CP is greater than CV . 20. Explain why the temperature of a gas increases when it is adiabatically compressed. 21. An ideal monatomic gas is allowed to expand slowly to twice its volume (1) isothermally; (2) adiabatically; (3) isobarically. Plot each on a PV diagram. In which process is ¢Eint the greatest, and in which is ¢Eint the least? In which is W the greatest and the least? In which is Q the greatest and the least? 22. Ceiling fans are sometimes reversible, so that they drive the air down in one season and pull it up in another season. Which way should you set the fan for summer? For winter? 23. Goose down sleeping bags and parkas are often specified as so many inches or centimeters of loft, the actual thickness of the garment when it is fluffed up. Explain. 24. Microprocessor chips nowadays have a "heat sink" glued on top that looks like a series of fins. Why is it shaped like that? 25. Sea breezes are often encountered on sunny days at the shore of a large body of water. Explain, assuming the temperature of the land rises more rapidly than that of the nearby water. 26. The Earth cools off at night much more quickly when the weather is clear than when cloudy. Why? 27. Explain why air-temperature readings are always taken with the thermometer in the shade. 28. A premature baby in an incubator can be dangerously cooled even when the air temperature in the incubator is warm. Explain. 29. The floor of a house on a foundation under which the air can flow is often cooler than a floor that rests directly on the ground (such as a concrete slab foundation). Explain. 30. Why is the liner of a thermos bottle silvered (Fig. 19–27), and why does it have a Stopper vacuum between its two walls? Outer casing Air Liner Liquid, hot or cold Vacuum FIGURE 19–27 Question 30.

Insulating support

Questions

521

31. A 22°C day is warm, while a swimming pool at 22°C feels cool. Why? 32. In the Northern Hemisphere the amount of heat required to heat a room where the windows face north is much higher than that required where the windows face south. Explain. 33. Heat loss occurs through windows by the following processes: (1) ventilation around edges; (2) through the frame, particularly if it is metal; (3) through the glass panes; and (4) radiation. (a) For the first three, what is (are) the mechanism(s): conduction, convection, or radiation? (b) Heavy curtains reduce which of these heat losses? Explain in detail.

34. Early in the day, after the Sun has reached the slope of a mountain, there tends to be a gentle upward movement of air. Later, after a slope goes into shadow, there is a gentle downdraft. Explain. 35. A piece of wood lying in the Sun absorbs more heat than a piece of shiny metal. Yet the metal feels hotter than the wood when you pick it up. Explain. 36. An "emergency blanket" is a thin shiny (metal-coated) plastic foil. Explain how it can help to keep an immobile person warm. 37. Explain why cities situated by the ocean tend to have less extreme temperatures than inland cities at the same latitude.

Problems 19–1 Heat as Energy Transfer 1. (I) To what temperature will 8700 J of heat raise 3.0 kg of water that is initially at 10.0°C? 2. (II) When a diver jumps into the ocean, water leaks into the gap region between the diver's skin and her wetsuit, forming a water layer about 0.5 mm thick. Assuming the total surface area of the wetsuit covering the diver is about 1.0 m2, and that ocean water enters the suit at 10°C and is warmed by the diver to skin temperature of 35°C, estimate how much energy (in units of candy bars = 300 kcal) is required by this heating process. 3. (II) An average active person consumes about 2500 Cal a day. (a) What is this in joules? (b) What is this in kilowatthours? (c) If your power company charges about 10 ¢ per kilowatt-hour, how much would your energy cost per day if you bought it from the power company? Could you feed yourself on this much money per day? 4. (II) A British thermal unit (Btu) is a unit of heat in the British system of units. One Btu is defined as the heat needed to raise 1 lb of water by 1 F°. Show that 1 Btu = 0.252 kcal = 1056 J. 5. (II) How many joules and kilocalories are generated when the brakes are used to bring a 1200-kg car to rest from a speed of 95 kmh? 6. (II) A small immersion heater is rated at 350 W. Estimate how long it will take to heat a cup of soup (assume this is 250 mL of water) from 15°C to 75°C.

19–3 and 19–4 Specific Heat; Calorimetry 7. (I) An automobile cooling system holds 18 L of water. How much heat does it absorb if its temperature rises from 15°C to 95°C? 8. (I) What is the specific heat of a metal substance if 135 kJ of heat is needed to raise 5.1 kg of the metal from 18.0°C to 37.2°C? 9. (II) (a) How much energy is required to bring a 1.0-L pot of water at 20°C to 100°C? (b) For how long could this amount of energy run a 100-W lightbulb? 10. (II) Samples of copper, aluminum, and water experience the same temperature rise when they absorb the same amount of heat. What is the ratio of their masses? 11. (II) How long does it take a 750-W coffeepot to bring to a boil 0.75 L of water initially at 8.0°C? Assume that the part of the pot which is heated with the water is made of 280 g of aluminum, and that no water boils away.

522

CHAPTER 19

12. (II) A hot iron horseshoe (mass = 0.40 kg), just forged (Fig. 19–28), is dropped into 1.05 L of water in a 0.30-kg iron pot initially at 20.0°C. If the final equilibrium temperature is 25.0°C, estimate the initial temperature of the hot horseshoe. FIGURE 19–28 Problem 12. 13. (II) A 31.5-g glass thermometer reads 23.6°C before it is placed in 135 mL of water. When the water and thermometer come to equilibrium, the thermometer reads 39.2°C. What was the original temperature of the water? [Hint: Ignore the mass of fluid inside the glass thermometer.] 14. (II) Estimate the Calorie content of 65 g of candy from the following measurements.A 15-g sample of the candy is placed in a small aluminum container of mass 0.325 kg filled with oxygen. This container is placed in 2.00 kg of water in an aluminum calorimeter cup of mass 0.624 kg at an initial temperature of 15.0°C. The oxygen-candy mixture in the small container is ignited, and the final temperature of the whole system is 53.5°C. 15. (II) When a 290-g piece of iron at 180°C is placed in a 95-g aluminum calorimeter cup containing 250 g of glycerin at 10°C, the final temperature is observed to be 38°C. Estimate the specific heat of glycerin. 16. (II) The heat capacity, C, of an object is defined as the amount of heat needed to raise its temperature by 1 C°. Thus, to raise the temperature by ¢T requires heat Q given by Q = C ¢T. (a) Write the heat capacity C in terms of the specific heat, c, of the material. (b) What is the heat capacity of 1.0 kg of water? (c) Of 35 kg of water? 17. (II) The 1.20-kg head of a hammer has a speed of 7.5 ms just before it strikes a nail (Fig. 19–29) and is brought to rest. Estimate the temperature rise of a 14-g iron nail generated by 10 such hammer blows done in quick succession. Assume the nail absorbs all the energy.

Heat and the First Law of Thermodynamics

FIGURE 19–29 Problem 17.

19–5 Latent Heat 18. (I) How much heat is needed to melt 26.50 kg of silver that is initially at 25°C? 19. (I) During exercise, a person may give off 180 kcal of heat in 25 min by evaporation of water from the skin. How much water has been lost? 20. (II) A 35-g ice cube at its melting point is dropped into an insulated container of liquid nitrogen. How much nitrogen evaporates if it is at its boiling point of 77 K and has a latent heat of vaporization of 200 kJkg? Assume for simplicity that the specific heat of ice is a constant and is equal to its value near its melting point. 21. (II) High-altitude mountain climbers do not eat snow, but always melt it first with a stove. To see why, calculate the energy absorbed from your body if you (a) eat 1.0 kg of – 10°C snow which your body warms to body temperature of 37°C. (b) You melt 1.0 kg of – 10°C snow using a stove and drink the resulting 1.0 kg of water at 2°C, which your body has to warm to 37°C. 22. (II) An iron boiler of mass 180 kg contains 730 kg of water at 18°C. A heater supplies energy at the rate of 52,000 kJh. How long does it take for the water (a) to reach the boiling point, and (b) to all have changed to steam? 23. (II) In a hot day's race, a bicyclist consumes 8.0 L of water over the span of 3.5 hours. Making the approximation that all of the cyclist's energy goes into evaporating this water as sweat, how much energy in kcal did the rider use during the ride? (Since the efficiency of the rider is only about 20%, most of the energy consumed does go to heat, so our approximation is not far off.) 24. (II) The specific heat of mercury is 138 Jkg C°. Determine the latent heat of fusion of mercury using the following calorimeter data: 1.00 kg of solid Hg at its melting point of – 39.0°C is placed in a 0.620-kg aluminum calorimeter with 0.400 kg of water at 12.80°C; the resulting equilibrium temperature is 5.06°C. 25. (II) At a crime scene, the forensic investigator notes that the 7.2-g lead bullet that was stopped in a doorframe apparently melted completely on impact. Assuming the bullet was shot at room temperature (20°C), what does the investigator calculate as the minimum muzzle velocity of the gun? 26. (II) A 58-kg ice-skater moving at 7.5 ms glides to a stop. Assuming the ice is at 0°C and that 50% of the heat generated by friction is absorbed by the ice, how much ice melts?

19–6 and 19–7 First Law of Thermodynamics 27. (I) Sketch a PV diagram of the following process: 2.0 L of ideal gas at atmospheric pressure are cooled at constant pressure to a volume of 1.0 L, and then expanded isothermally back to 2.0 L, whereupon the pressure is increased at constant volume until the original pressure is reached. 28. (I) A gas is enclosed in a cylinder fitted with a light frictionless piston and maintained at atmospheric pressure. When 1250 kcal of heat is added to the gas, the volume is observed to increase slowly from 12.0 m3 to 18.2 m3. Calculate (a) the work done by the gas and (b) the change in internal energy of the gas. 29. (II) The pressure in an ideal gas is cut in half slowly, while being kept in a container with rigid walls. In the process, 365 kJ of heat left the gas. (a) How much work was done during this process? (b) What was the change in internal energy of the gas during this process?

30. (II) A 1.0-L volume of air initially at 3.5 atm of (absolute) pressure is allowed to expand isothermally until the pressure is 1.0 atm. It is then compressed at constant pressure to its initial volume, and lastly is brought back to its original pressure by heating at constant volume. Draw the process on a PV diagram, including numbers and labels for the axes. 31. (II) Consider the following two-step process. Heat is allowed to flow out of an ideal gas at constant volume so that its pressure drops from 2.2 atm to 1.4 atm. Then the gas expands at constant pressure, from a volume of 5.9 L to 9.3 L, where the temperature reaches its original value. See Fig. 19–30. Calculate (a) the total work done by the gas in the process, P (b) the change in internal energy of a the gas in the 2.2 atm process, and (c) the total heat flow into or out of the gas.

FIGURE 19–30 Problem 31.

1.4 atm

5.9 L 9.3 L

32. (II) The PV diagram in Fig. 19–31 shows two possible states of a system containing 1.55 moles of a monatomic ideal gas. AP1 = P2 = 455 Nm2, V1 = 2.00 m3, V2 = 8.00 m3.B (a) Draw the process which depicts an isobaric expansion from state 1 to state 2, and label this process A. (b) Find the work done by the gas and the change in internal energy of the gas in process A. (c) Draw the two-step process which depicts an isothermal expansion from state 1 to the volume V2 , followed by an isovolumetric increase in temperature to state 2, and label this process B. (d) Find the change in internal energy of P the gas for the (N/m2) two-step process B. 500 1 2 400 300 200 100 FIGURE 19–31 0 2 4 6 8 10 V (m3) Problem 32. 33. (II) Suppose 2.60 mol of an ideal gas of volume V1 = 3.50 m3 at T1 = 290 K is allowed to expand isothermally to V2 = 7.00 m3 at T2 = 290 K. Determine (a) the work done by the gas, (b) the heat added to the gas, and (c) the change in internal energy of the gas. 34. (II) In an engine, an almost ideal gas is compressed adiabatically to half its volume. In doing so, 2850 J of work is done on the gas. (a) How much heat flows into or out of the gas? (b) What is the change in internal energy of the gas? (c) Does its temperature rise or fall? 35. (II) One and one-half moles of an ideal monatomic gas expand adiabatically, performing 7500 J of work in the process. What is the change in temperature of the gas during this expansion? 36. (II) Determine (a) the work done and (b) the change in internal energy of 1.00 kg of water when it is all boiled to steam at 100°C. Assume a constant pressure of 1.00 atm.

Problems

523

37. (II) How much work is done by a pump to slowly compress, isothermally, 3.50 L of nitrogen at 0°C and 1.00 atm to 1.80 L at 0°C? 38. (II) When a gas is taken from a to c along the curved path in Fig. 19–32, the work done by the gas is W = – 35 J and the heat added to the gas is Q = – 63 J. Along path abc, the work done is W = – 54 J. (a) What is Q for path abc? (b) If Pc = 12 Pb , what is W for path cda? (c) What is Q for path cda? (d) What is Eint, a - Eint, c? (e) If Eint, d - Eint, c = 12 J, what is Q for path da?

P b

FIGURE 19–32 Problems 38, 39, and 40.

46. (II) Show that the work done by n moles of an ideal gas when it expands adiabatically is W = nCV AT1 - T2 B, where T1 and T2 are the initial and final temperatures, and CV is the molar specific heat at constant volume.

48. (II) The specific heat at constant volume of a particular gas is 0.182 kcalkgK at room temperature, and its molecular mass is 34. (a) What is its specific heat at constant pressure? (b) What do you think is the molecular structure of this gas?

Qa S c = – 85 J Wa S c = – 55 J Wcda = 38 J Eint, a - Eint, b = 15 J Pa = 2.2Pd . 40. (III) Suppose a gas is taken clockwise around the rectangular cycle shown in Fig. 19–32, starting at b, then to a, to d, to c, and returning to b. Using the values given in Problem 39, (a) describe each leg of the process, and then calculate (b) the net work done during the cycle, (c) the total internal energy change during the cycle, and (d) the net heat flow during the cycle. (e) What percentage of the intake heat was turned into usable work: i.e., how efficient is this "rectangular" cycle (give as a percentage)? * 41. (III) Determine the work done by 1.00 mol of a van der Waals gas (Section 18–5) when it expands from volume V1 to V2 isothermally.

19–8 Molecular Specific Heat for Gases; Equipartition of Energy 42. (I) What is the internal energy of 4.50 mol of an ideal diatomic gas at 645 K, assuming all degrees of freedom are active? 43. (I) If a heater supplies 1.8 * 106 Jh to a room 3.5 m * 4.6 m * 3.0 m containing air at 20°C and 1.0 atm, by how much will the temperature rise in one hour, assuming no losses of heat or air mass to the outside? Assume air is an ideal diatomic gas with molecular mass 29.

CHAPTER 19

45. (II) A certain monatomic gas has specific heat cV = 0.0356 kcalkgC°, which changes little over a wide temperature range. What is the atomic mass of this gas? What gas is it?

47. (II) An audience of 1800 fills a concert hall of volume 22,000 m3. If there were no ventilation, by how much would the temperature of the air rise over a period of 2.0 h due to the metabolism of the people (70 Wperson)?

39. (III) In the process of taking a gas from state a to state c along the curved path shown in Fig. 19–32, 85 J of heat leaves the system and 55 J of work is done on the system. (a) Determine the change in internal energy, Eint, a - Eint, c . (b) When the gas is taken along the path cda, the work done by the gas is W = 38 J. How much heat Q is added to the gas in the process cda? (c) If Pa = 2.2Pd , how much work is done by the gas in the process abc? (d) What is Q for path abc? (e) If Eint, a - Eint, b = 15 J, what is Q for the process bc? Here is a summary of what is given:

524

44. (I) Show that if the molecules of a gas have n degrees of freedom, then theory predicts and CV = 12 nR 1 CP = 2 (n + 2)R.

49. (II) A 2.00 mole sample of N2 gas at 0°C is heated to 150°C at constant pressure (1.00 atm). Determine (a) the change in internal energy, (b) the work the gas does, and (c) the heat added to it. 50. (III) A 1.00-mol sample of an ideal diatomic gas at a pressure of 1.00 atm and temperature of 420 K undergoes a process in which its pressure increases linearly with temperature. The final temperature and pressure are 720 K and 1.60 atm. Determine (a) the change in internal energy, (b) the work done by the gas, and (c) the heat added to the gas. (Assume five active degrees of freedom.)

19–9 Adiabatic Expansion of a Gas 51. (I) A 1.00-mol sample of an ideal diatomic gas, originally at 1.00 atm and 20°C, expands adiabatically to 1.75 times its initial volume. What are the final pressure and temperature for the gas? (Assume no molecular vibration.) 52. (II) Show, using Eqs. 19–6 and 19–15, that the work done by a gas that slowly expands adiabatically from pressure P1 and volume V1 , to P2 and V2 , is given by W = AP1 V1 - P2 V2 B(g - 1). 53. (II) A 3.65-mol sample of an ideal diatomic gas expands adiabatically from a volume of 0.1210 m3 to 0.750 m3. Initially the pressure was 1.00 atm. Determine: (a) the initial and final temperatures; (b) the change in internal energy; (c) the heat lost by the gas; (d) the work done on the gas. (Assume no molecular vibration.) 54. (II) An ideal monatomic gas, consisting of 2.8 mol of volume 0.086 m3, expands adiabatically. The initial and final temperatures are 25°C and –68°C. What is the final volume of the gas? 55. (III) A 1.00-mol sample of an ideal monatomic gas, originally at a pressure of 1.00 atm, undergoes a three-step process: (1) it is expanded adiabatically from T1 = 588 K to T2 = 389 K; (2) it is compressed at constant pressure until its temperature reaches T3 ; (3) it then returns to its original pressure and temperature by a constant-volume process. (a) Plot these processes on a PV diagram. (b) Determine T3 . (c) Calculate the change in internal energy, the work done by the gas, and the heat added to the gas for each process, and (d) for the complete cycle.

Heat and the First Law of Thermodynamics

56. (III) Consider a parcel of air moving to a different altitude y in the Earth's atmosphere (Fig. 19–33). As the parcel changes altitude it acquires the pressure P of the surrounding air. From Eq. 13–4 we have dP = – rg dy where r is the parcel's altitude-dependent mass density.

59. (II) How long does it take the Sun to melt a block of ice at 0°C with a flat horizontal area 1.0 m2 and thickness 1.0 cm? Assume that the Sun's rays make an angle of 35° with the vertical and that the emissivity of ice is 0.050. 60. (II) Heat conduction to skin. Suppose 150 W of heat flows by conduction from the blood capillaries beneath the skin to the body's surface area of 1.5 m2. If the temperature difference is 0.50 C°, estimate the average distance of capillaries below the skin surface.

"Parcel" of N air molecules y0 FIGURE 19–33 Problem 56. During this motion, the parcel's volume will change and, because air is a poor heat conductor, we assume this expansion or contraction will take place adiabatically. (a) Starting with g Eq. 19–15, PV = constant, show that for an ideal g g gas undergoing an adiabatic process, P1- T = constant. Then show that the parcel's pressure and temperature are related by (1 - g)

58. (I) One end of a 45-cm-long copper rod with a diameter of 2.0 cm is kept at 460°C, and the other is immersed in water at 22°C. Calculate the heat conduction rate along the rod.

P dT dP + g = 0 dy T dy

61. (II) A ceramic teapot ( = 0.70) and a shiny one ( = 0.10) each hold 0.55 L of tea at 95°C. (a) Estimate the rate of heat loss from each, and (b) estimate the temperature drop after 30 min for each. Consider only radiation, and assume the surroundings are at 20°C. 62. (II) A copper rod and an aluminum rod of the same length and cross-sectional area are attached end to end (Fig. 19–34). The copper end is placed in a furnace maintained at a constant temperature of 225°C. The aluminum end is placed in an ice bath held at constant temperature of 0.0°C. Calculate the temperature at the point where the two rods are joined. Cu 225°C

Al T=?

0.0°C

FIGURE 19–34 Problem 62.

and thus (1 - g)(– rg) + g

P dT = 0. T dy

(b) Use the ideal gas law with the result from part (a) to show that the change in the parcel's temperature with change in altitude is given by 1 - g mg dT = g dy k where m is the average mass of an air molecule and k is the Boltzmann constant. (c) Given that air is a diatomic gas with an average molecular mass of 29, show that dTdy = – 9.8 C°km. This value is called the adiabatic lapse rate for dry air. (d) In California, the prevailing westerly winds descend from one of the highest elevations (the 4000-m Sierra Nevada mountains) to one of the lowest elevations (Death Valley, – 100 m) in the continental United States. If a dry wind has a temperature of – 5°C at the top of the Sierra Nevada, what is the wind's temperature after it has descended to Death Valley?

19–10 Conduction, Convection, Radiation 57. (I) (a) How much power is radiated by a tungsten sphere (emissivity = 0.35 ) of radius 16 cm at a temperature of 25°C? (b) If the sphere is enclosed in a room whose walls are kept at – 5°C, what is the net flow rate of energy out of the sphere?

63. (II) (a) Using the solar constant, estimate the rate at which the whole Earth receives energy from the Sun. (b) Assume the Earth radiates an equal amount back into space (that is, the Earth is in equilibrium). Then, assuming the Earth is a perfect emitter ( = 1.0), estimate its average surface temperature. [Hint: Use area A = 4pr2E , and state why.] 64. (II) A 100-W lightbulb generates 95 W of heat, which is dissipated through a glass bulb that has a radius of 3.0 cm and is 0.50 mm thick. What is the difference in temperature between the inner and outer surfaces of the glass? 65. (III) A house thermostat is normally set to 22°C, but at night it is turned down to 12°C for 9.0 h. Estimate how much more heat would be needed (state as a percentage of daily usage) if the thermostat were not turned down at night. Assume that the outside temperature averages 0°C for the 9.0 h at night and 8°C for the remainder of the day, and that the heat loss from the house is proportional to the difference in temperature inside and out. To obtain an estimate from the data, you will have to make other simplifying assumptions; state what these are. 66. (III) Approximately how long should it take 9.5 kg of ice at 0°C to melt when it is placed in a carefully sealed Styrofoam ice chest of dimensions 25 cm * 35 cm * 55 cm whose walls are 1.5 cm thick? Assume that the conductivity of Styrofoam is double that of air and that the outside temperature is 34°C.

Problems

525

67. (III) A cylindrical pipe has inner radius R1 and outer radius R2 . The interior of the pipe carries hot water at temperature T1 . The temperature outside is T2 A6 T1 B. (a) Show that the rate of heat loss for a length l of pipe is 2pkAT1 - T2 B l dQ , = dt lnAR2R1 B where k is the thermal conductivity of the pipe. (b) Suppose the pipe is steel with R1 = 3.3 cm, R2 = 4.0 cm, and T2 = 18°C. If the pipe holds still water at T1 = 71°C, what will be the initial rate of change of its temperature? (c) Suppose water at 71°C enters the pipe and moves at a speed of 8.0 cms. What will be its temperature drop per centimeter of travel?

68. (III) Suppose the insulating qualities of the wall of a house come mainly from a 4.0-in. Brick Insulation layer of brick and an R-19 (R1) (R2) layer of insulation, as shown in Fig. 19–35. What is the total rate of heat loss through such T1 a wall, if its total area is 195 ft2 T2 and the temperature difference across it is 12 F°? Heat flow

FIGURE 19–35 Problem 68. Two layers insulating a wall.

General Problems 69. A soft-drink can contains about 0.20 kg of liquid at 5°C. Drinking this liquid can actually consume some of the fat in the body, since energy is needed to warm the liquid to body temperature (37°C). How many food Calories should the drink have so that it is in perfect balance with the heat needed to warm the liquid (essentially water)? 70. (a) Find the total power radiated into space by the Sun, assuming it to be a perfect emitter at T = 5500 K. The Sun's radius is 7.0 * 108 m. (b) From this, determine the power per unit area arriving at the Earth, 1.5 * 1011 m away. 71. To get an idea of how much thermal energy is contained in the world's oceans, estimate the heat liberated when a cube of ocean water, 1 km on each side, is cooled by 1 K. (Approximate the ocean water as pure water for this estimate.) 72. A mountain climber wears a goose-down jacket 3.5 cm thick with total surface area 0.95 m2. The temperature at the surface of the clothing is – 18°C and at the skin is 34°C. Determine the rate of heat flow by conduction through the jacket (a) assuming it is dry and the thermal conductivity k is that of goose down, and (b) assuming the jacket is wet, so k is that of water and the jacket has matted to 0.50 cm thickness. 73. During light activity, a 70-kg person may generate 200 kcalh. Assuming that 20% of this goes into useful work and the other 80% is converted to heat, estimate the temperature rise of the body after 30 min if none of this heat is transferred to the environment.

76. A house has well-insulated walls 19.5 cm thick (assume conductivity of air) and area 410 m2, a roof of wood 5.5 cm thick and area 280 m2, and uncovered windows 0.65 cm thick and total area 33 m2. (a) Assuming that heat is lost only by conduction, calculate the rate at which heat must be supplied to this house to maintain its inside temperature at 23°C if the outside temperature is –15°C. (b) If the house is initially at 12°C, estimate how much heat must be supplied to raise the temperature to 23°C within 30 min. Assume that only the air needs to be heated and that its volume is 750 m3. (c) If natural gas costs $0.080 per kilogram and its heat of combustion is 5.4 * 107 Jkg, how much is the monthly cost to maintain the house as in part (a) for 24 h each day, assuming 90% of the heat produced is used to heat the house? Take the specific heat of air to be 0.24 kcalkgC°. 77. In a typical game of squash (Fig. 19–36), two people hit a soft rubber ball at a wall until they are about to drop due to dehydration and exhaustion. Assume that the ball hits the wall at a velocity of 22 ms and bounces back with a velocity of 12 ms, and that the kinetic energy lost in the process heats the ball. What will be the temperature increase of the ball after one bounce? (The specific heat of rubber is about 1200 Jkg C°.)

74. Estimate the rate at which heat can be conducted from the interior of the body to the surface. Assume that the thickness of tissue is 4.0 cm, that the skin is at 34°C and the interior at 37°C, and that the surface area is 1.5 m2. Compare this to the measured value of about 230 W that must be dissipated by a person working lightly. This clearly shows the necessity of convective cooling by the blood. 75. A marathon runner has an average metabolism rate of about 950 kcalh during a race. If the runner has a mass of 55 kg, estimate how much water she would lose to evaporation from the skin for a race that lasts 2.2 h.

526

CHAPTER 19

Heat and the First Law of Thermodynamics

FIGURE 19–36 Problem 77.

78. A bicycle pump is a cylinder 22 cm long and 3.0 cm in diameter. The pump contains air at 20.0°C and 1.0 atm. If the outlet at the base of the pump is blocked and the handle is pushed in very quickly, compressing the air to half its original volume, how hot does the air in the pump become? 79. A microwave oven is used to heat 250 g of water. On its maximum setting, the oven can raise the temperature of the liquid water from 20°C to 100°C in 1 min 45 s A = 105 sB. (a) At what rate does the oven input energy to the liquid water? (b) If the power input from the oven to the water remains constant, determine how many grams of water will boil away if the oven is operated for 2 min (rather than just 1 min 45 s). 80. The temperature within the Earth's crust increases about 1.0 C° for each 30 m of depth. The thermal conductivity of the crust is 0.80 WC° m. (a) Determine the heat transferred from the interior to the surface for the entire Earth in 1.0 h. (b) Compare this heat to the amount of energy incident on the Earth in 1.0 h due to radiation from the Sun. 81. An ice sheet forms on a lake. The air above the sheet is at – 18°C, whereas the water is at 0°C. Assume that the heat of fusion of the water freezing on the lower surface is conducted through the sheet to the air above. How much time will it take to form a sheet of ice 15 cm thick? 82. An iron meteorite melts when it enters the Earth's atmosphere. If its initial temperature was – 105°C outside of Earth's atmosphere, calculate the minimum velocity the meteorite must have had before it entered Earth's atmosphere. 83. A scuba diver releases a 3.60-cm-diameter (spherical) bubble of air from a depth of 14.0 m. Assume the temperature is constant at 298 K, and that the air behaves as an ideal gas. (a) How large is the bubble when it reaches the surface? (b) Sketch a PV diagram for the process. (c) Apply the first law of thermodynamics to the bubble, and find the work done by the air in rising to the surface, the change in its internal energy, and the heat added or removed from the air in the bubble as it rises. Take the density of water to be 1000 kgm3. 84. A reciprocating compressor is a device that compresses air by a back-and-forth straight-line motion, like a piston in a cylinder. Consider a reciprocating compressor running at 150 rpm. During a compression stroke, 1.00 mol of air is compressed. The initial temperature of the air is 390 K, the engine of the compressor is supplying 7.5 kW of power to compress the air, and heat is being removed at the rate of 1.5 kW. Calculate the temperature change per compression stroke.

85. The temperature of the glass surface of a 75-W lightbulb is 75°C when the room temperature is 18°C. Estimate the temperature of a 150-W lightbulb with a glass bulb the same size. Consider only radiation, and assume that 90% of the energy is emitted as heat. 86. Suppose 3.0 mol of neon (an ideal monatomic gas) at STP are compressed slowly and isothermally to 0.22 the original volume. The gas is then allowed to expand quickly and adiabatically back to its original volume. Find the highest and lowest temperatures and pressures attained by the gas, and show on a PV diagram where these values occur. 87. At very low temperatures, the molar specific heat of many substances varies as the cube of the absolute temperature: C = k

T3 ,

T30

which is sometimes called Debye's law. For rock salt, T0 = 281 K and k = 1940 JmolK. Determine the heat needed to raise 2.75 mol of salt from 22.0 K to 48.0 K. 88. A diesel engine accomplishes ignition without a spark plug by an adiabatic compression of air to a temperature above the ignition temperature of the diesel fuel, which is injected into the cylinder at the peak of the compression. Suppose air is taken into the cylinder at 280 K and volume V1 and is compressed adiabatically to 560°C (L 1000°F) and volume V2 . Assuming that the air behaves as an ideal gas whose ratio of CP to CV is 1.4, calculate the compression ratio V1V2 of the engine. 89. When 6.30 * 105 J of heat is added to a gas enclosed in a cylinder fitted with a light frictionless piston maintained at atmospheric pressure, the volume is observed to increase from 2.2 m3 to 4.1 m3. Calculate (a) the work done by the gas, and (b) the change in internal energy of the gas. (c) Graph this process on a PV diagram. 90. In a cold environment, a person can lose heat by conduction and radiation at a rate of about 200 W. Estimate how long it would take for the body temperature to drop from 36.6°C to 35.6°C if metabolism were nearly to stop. Assume a mass of 70 kg. (See Table 19–1.) * Numerical/Computer * 91. (II) Suppose 1.0 mol of steam at 100°C of volume 0.50 m3 is expanded isothermally to volume 1.00 m3. Assume steam obeys the van der Waals equation Eq. 18–9, with AP + n2aV2 B(Vn - b) = RT, a = 0.55 Nm4mol2 and b = 3.0 * 10 –5 m3mol. Using the expression dW = P dV, determine numerically the total work done W. Your result should agree within 2% of the result obtained by integrating the expression for dW.

Answers to Exercises A: (b).

E: Less.

B: (c).

F: –6.8 * 103 J.

C: 0.21 kg.

G: (d).

D: 700 J.

General Problems

527

There are many uses for a heat engine, such as old steam trains and modern coal-burning power plants. Steam engines produce steam which does work: on turbines to generate electricity, and on a piston that moves linkage to turn locomotive wheels. The efficiency of any engine—no matter how carefully engineered—is limited by nature as described in the second law of thermodynamics. This great law is best stated in terms of a quantity called entropy, which is unlike any other. Entropy is not conserved, but instead is constrained always to increase in any real process. Entropy is a measure of disorder. The second law of thermodynamics tells us that as time moves forward, the disorder in the universe increases. We discuss many practical matters including heat engines, heat pumps, and refrigeration.

P T E R

C H

CONTENTS 20–1 The Second Law of Thermodynamics—Introduction 20–2 Heat Engines 20–3 Reversible and Irreversible Processes; the Carnot Engine 20–4 Refrigerators, Air Conditioners, and Heat Pumps 20–5 Entropy 20–6 Entropy and the Second Law of Thermodynamics 20–7 Order to Disorder 20–8 Unavailability of Energy; Heat Death *20–9 Statistical Interpretation of Entropy and the Second Law 20–10 Thermodynamic Temperature; * Third Law of Thermodynamics *20–11 Thermal Pollution, Global Warming, and Energy Resources

528

Second Law of Thermodynamics CHAPTER-OPENING QUESTION—Guess now! Fossil-fuel electric generating plants produce "thermal pollution." Part of the heat produced by the burning fuel is not converted to electric energy. The reason for this waste is (a) The efficiency is higher if some heat is allowed to escape. (b) Engineering technology has not yet reached the point where 100% waste heat recovery is possible. (c) Some waste heat must be produced: this is a fundamental property of nature when converting heat to useful work. (d) The plants rely on fossil fuels, not nuclear fuel. (e) None of the above. n this final Chapter on heat and thermodynamics, we discuss the famous second law of thermodynamics, and the quantity "entropy" that arose from this fundamental law and is its quintessential expression. We also discuss heat engines—the engines that transform heat into work in power plants, trains, and motor vehicles—because they first showed us that a new law was needed. Finally, we briefly discuss the third law of thermodynamics.

20–1

The Second Law of Thermodynamics—Introduction

The first law of thermodynamics states that energy is conserved. There are, however, many processes we can imagine that conserve energy but are not observed to occur in nature. For example, when a hot object is placed in contact with a cold object, heat flows from the hotter one to the colder one, never spontaneously the reverse. If heat were to leave the colder object and pass to the hotter one, energy could still be conserved. Yet it does not happen spontaneously.† As a second example, consider what happens when you drop a rock and it hits the ground. The initial potential energy of the rock changes to kinetic energy as the rock falls. When the rock hits the ground, this energy in turn is transformed into internal energy of the rock and the ground in the vicinity of the impact; the molecules move faster and the temperature rises slightly. But have you seen the reverse happen—a rock at rest on the ground suddenly rise up in the air because the thermal energy of molecules is transformed into kinetic energy of the rock as a whole? Energy could be conserved in this process, yet we never see it happen. There are many other examples of processes that occur in nature but whose reverse does not. Here are two more. (1) If you put a layer of salt in a jar and cover it with a layer of similar-sized grains of pepper, when you shake it you get a thorough mixture. But no matter how long you shake it, the mixture does not separate into two layers again. (2) Coffee cups and glasses break spontaneously if you drop them. But they do not go back together spontaneously (Fig. 20–1).

(a) Initial state.

FIGURE 20–1 Have you ever observed this process, a broken cup spontaneously reassembling and rising up onto a table? This process could conserve energy and other laws of mechanics.

(b) Later: cup reassembles and rises up.

The first law of thermodynamics (conservation of energy) would not be violated if any of these processes occurred in reverse. To explain this lack of reversibility, scientists in the latter half of the nineteenth century formulated a new principle known as the second law of thermodynamics. The second law of thermodynamics is a statement about which processes occur in nature and which do not. It can be stated in a variety of ways, all of which are equivalent. One statement, due to R. J. E. Clausius (1822–1888), is that heat can flow spontaneously from a hot object to a cold object; heat will not flow spontaneously from a cold object to a hot object.

SECOND LAW OF THERMODYNAMICS (Clausius statement)

Since this statement applies to one particular process, it is not obvious how it applies to other processes. A more general statement is needed that will include other possible processes in a more obvious way. The development of a general statement of the second law of thermodynamics was based partly on the study of heat engines. A heat engine is any device that changes thermal energy into mechanical work, such as a steam engine or automobile engine. We now examine heat engines, both from a practical point of view and to show their importance in developing the second law of thermodynamics. † By spontaneously, we mean by itself without input of work of some sort. (A refrigerator does move heat from a cold environment to a warmer one, but only because its motor does work—Section 20–4.)

SECTION 20–1

The Second Law of Thermodynamics—Introduction

529

20–2

High temperature, TH QH Engine W QL Low temperature, TL FIGURE 20–2 Schematic diagram of energy transfers for a heat engine.

C A U T I O N

New sign convention: QH 7 0, QL 7 0, W 7 0

Heat Engines

It is easy to produce thermal energy by doing work—for example, by simply rubbing your hands together briskly, or indeed by any frictional process. But to get work from thermal energy is more difficult, and a practical device to do this was invented only about 1700 with the development of the steam engine. The basic idea behind any heat engine is that mechanical energy can be obtained from thermal energy only when heat is allowed to flow from a high temperature to a low temperature. In the process, some of the heat can then be transformed to mechanical work, as diagrammed schematically in Fig. 20–2. We will be interested only in engines that run in a repeating cycle (that is, the system returns repeatedly to its starting point) and thus can run continuously. In each cycle the change in internal energy of the system is ¢Eint = 0 because it returns to the starting state. Thus a heat input QH at a high temperature TH is partly transformed into work W and partly exhausted as heat QL at a lower temperature TL (Fig. 20–2). By conservation of energy, QH = W + QL . The high and low temperatures, TH and TL , are called the operating temperatures of the engine. Note carefully that we are now using a new sign convention: we take QH , QL , and W as always positive. The direction of each energy transfer is shown by the arrow on the applicable diagram, such as Fig. 20–2.

Steam Engine and Internal Combustion Engine P H Y S I C S

A P P L I E D

Engines

The operation of a steam engine is illustrated in Fig. 20–3. Steam engines are of two main types, each making use of steam heated by combustion of coal, oil or gas, or by nuclear energy. In a reciprocating engine, Fig. 20–3a, the heated steam passes through the intake valve and expands against a piston, forcing it to move. As the piston returns to its original position, it forces the gases out the exhaust valve. A steam turbine, Fig. 20–3b, is very similar except that the reciprocating piston is replaced by a rotating turbine that resembles a paddlewheel with many sets of blades. Most of our electricity today is generated using steam turbines.† The material that is heated and cooled, steam in this case, is called the working substance. †

Even nuclear power plants utilize steam turbines; the nuclear fuel—uranium—merely serves as fuel to heat the steam.

(b) Turbine (boiler and condenser not shown)

(a) Reciprocating type

High-pressure steam, from boiler

High temperature Boiler Steam Water FIGURE 20–3 Steam engines.

Intake valve (open during expansion)

Heat input

Exhaust valve (closed during expansion) Pump Water

Condenser

Low temperature

530

CHAPTER 20

Second Law of Thermodynamics

Piston

Condenser

Low-pressure steam, exhausted to condenser

Exhaust valve (closed)

Intake valve (open)

Both valves closed

Both valves closed Spark plug (firing)

Cylinder

Gas–air mixture from carburetor

Piston

Both valves closed

Exhaust valve (open) To exhaust pipe

Rings

Connecting rod Crankshaft (a) Intake

(b) Compression

(d) Expansion (power stroke)

In an internal combustion engine (used in most automobiles), the high temperature is achieved by burning the gasoline–air mixture in the cylinder itself (ignited by the spark plug), as described in Fig. 20–4.

Why a T Is Needed to Drive a Heat Engine To see why a temperature difference is required to run an engine, let us examine the steam engine. In the reciprocating engine, for example, suppose there were no condenser or pump (Fig. 20–3a), and that the steam was at the same temperature throughout the system. This would mean that the pressure of the gas being exhausted would be the same as that on intake. Thus, although work would be done by the gas on the piston when it expanded, an equal amount of work would have to be done by the piston to force the steam out the exhaust; hence, no net work would be done. In a real engine, the exhausted gas is cooled to a lower temperature and condensed so that the exhaust pressure is less than the intake pressure. Thus, although the piston must do work on the gas to expel it on the exhaust stroke, it is less than the work done by the gas on the piston during the intake. So a net amount of work can be obtained—but only if there is a difference of temperature. Similarly, in the gas turbine if the gas isn't cooled, the pressure on each side of the blades would be the same. By cooling the gas on the exhaust side, the pressure on the back side of the blade is less and hence the turbine turns.

(e) Exhaust FIGURE 20–4 Four-stroke-cycle internal combustion engine: (a) the gasoline–air mixture flows into the cylinder as the piston moves down; (b) the piston moves upward and compresses the gas; (c) the brief instant when firing of the spark plug ignites the highly compressed gasoline–air mixture, raising it to a high temperature; (d) the gases, now at high temperature and pressure, expand against the piston in this, the power stroke; (e) the burned gases are pushed out to the exhaust pipe; when the piston reaches the top, the exhaust valve closes and the intake valve opens, and the whole cycle repeats. (a), (b), (d), and (e) are the four strokes of the cycle.

Efficiency and the Second Law The efficiency, e, of any heat engine can be defined as the ratio of the work it does, W, to the heat input at the high temperature, QH (Fig. 20–2): W. e = QH This is a sensible definition since W is the output (what you get from the engine), whereas QH is what you put in and pay for in burned fuel. Since energy is conserved, the heat input QH must equal the work done plus the heat that flows out at the low temperature AQL B: QH = W + QL . Thus W = QH - QL , and the efficiency of an engine is W QH QH - QL QL . = = 1 QH QH

e =

(20–1a) (20–1b)

To give the efficiency as a percent, we multiply Eqs. 20–1 by 100. Note that e could be 1.0 (or 100%) only if QL were zero—that is, only if no heat were exhausted to the environment. SECTION 20–2

Heat Engines

531

EXAMPLE 20–1 Car efficiency. An automobile engine has an efficiency of 20% and produces an average of 23,000 J of mechanical work per second during operation. (a) How much heat input is required, and (b) how much heat is discharged as waste heat from this engine, per second? APPROACH We want to find the heat input QH as well as the heat output QL , given W = 23,000 J each second and an efficiency e = 0.20. We can use the definition of efficiency, Eq. 20–1 in its various forms, to find first QH and then QL . SOLUTION (a) From Eq. 20–1a, e = WQH , we solve for QH : QH =

23,000 J W = e 0.20 = 1.15 * 105 J = 115 kJ.

The engine requires 115 kJs = 115 kW of heat input. (b) Now we use Eq. 20–1b Ae = 1 - QLQH B and solve for QL : QL = (1 - e)QH = (0.80)115 kJ = 92 kJ. P

C 0

FIGURE 20–5 Adiabatic process, Exercise A.

The engine discharges heat to the environment at a rate of 92 kJs = 92 kW. NOTE Of the 115 kJ that enters the engine per second, only 23 kJ does useful work whereas 92 kJ is wasted as heat output. NOTE The problem was stated in terms of energy per unit time. We could just as well have stated it in terms of power, since 1 Js = 1 W. EXERCISE A An adiabatic process is defined as one in which no heat flows in or out of the system. If an ideal gas expands as shown in Fig. 20–5 (see also Fig. 19–8), the work W done in this expansion equals the area under the graph, shown shaded. The efficiency of this process would be e = WQ, much greater than 100% ( = q since Q = 0). Is this a violation of the second law?

It is clear from Eq. 20–1b, e = 1 - QLQH , that the efficiency of an engine will be greater if QL can be made small. However, from experience with a wide variety of systems, it has not been found possible to reduce QL to zero. If QL could be reduced to zero we would have a 100% efficient engine, as diagrammed in Fig. 20–6. High temperature FIGURE 20–6 Schematic diagram of a hypothetical perfect heat engine in which all the heat input would be used to do work.

Heat

T NO IBLE Work S S PO Engine

That such a perfect engine (running continuously in a cycle) is not possible is another way of expressing the second law of thermodynamics: SECOND LAW OF THERMODYNAMICS (Kelvin-Planck statement)

No device is possible whose sole effect is to transform a given amount of heat completely into work. This is known as the Kelvin-Planck statement of the second law of thermodynamics. Said another way, there can be no perfect (100% efficient) heat engine such as that diagrammed in Fig. 20–6. If the second law were not true, so that a perfect engine could be built, some rather remarkable things could happen. For example, if the engine of a ship did not need a low-temperature reservoir to exhaust heat into, the ship could sail across the ocean using the vast resources of the internal energy of the ocean water. Indeed, we would have no fuel problems at all!

532

CHAPTER 20

Second Law of Thermodynamics

20–3

Reversible and Irreversible Processes; the Carnot Engine

In the early nineteenth century, the French scientist N. L. Sadi Carnot (1796–1832) studied in detail the process of transforming heat into mechanical energy. His aim had been to determine how to increase the efficiency of heat engines, but his studies soon led him to investigate the foundations of thermodynamics itself. In 1824, Carnot invented (on paper) an idealized type of engine which we now call the Carnot engine. No Carnot engine actually exists, but as a theoretical idea it played an important role in the establishment and understanding of the second law of thermodynamics.

Reversible and Irreversible Processes The Carnot engine involves reversible processes, so before we discuss it we must discuss what is meant by reversible and irreversible processes. A reversible process is one that is carried out infinitely slowly, so that the process can be considered as a series of equilibrium states, and the whole process could be done in reverse with no change in magnitude of the work done or heat exchanged. For example, a gas contained in a cylinder fitted with a tight, movable, but frictionless piston could be compressed isothermally in a reversible way if done infinitely slowly. Not all very slow (quasistatic) processes are reversible, however. If there is friction present, for example (as between the movable piston and cylinder just mentioned), the work done in one direction (going from some state A to state B) will not be the negative of the work done in the reverse direction (state B to state A). Such a process would not be considered reversible. A perfectly reversible process is not possible in reality because it would require an infinite time; reversible processes can be approached arbitrarily closely, however, and they are very important theoretically. All real processes are irreversible: they are not done infinitely slowly. There could be turbulence in the gas, friction would be present, and so on. Any process could not be done precisely in reverse since the heat lost to friction would not reverse itself, the turbulence would be different, and so on. For any given volume there would not be a well-defined pressure P and temperature T since the system would not always be in an equilibrium state. Thus a real, irreversible, process cannot be plotted on a PV diagram, except insofar as it may approach an ideal reversible process. But a reversible process (since it is a quasistatic series of equilibrium states) always can be plotted on a PV diagram; and a reversible process, when done in reverse, retraces the same path on a PV diagram. Although all real processes are irreversible, reversible processes are conceptually important, just as the concept of an ideal gas is.

Carnot's Engine Now let us look at Carnot's idealized engine. The Carnot engine makes use of a reversible cycle, by which we mean a series of reversible processes that take a given substance (the working substance) from an initial equilibrium state through many other equilibrium states and returns it again to the same initial state. In particular, the Carnot engine utilizes the Carnot cycle, which is illustrated in Fig. 20–7, with the working substance assumed to be an ideal gas. Let us take point a as the initial state. The gas is first expanded isothermally and reversibly, path ab, at temperature TH , as heat QH is added to it. Next the gas is expanded adiabatically and reversibly, path bc; no heat is exchanged and the temperature of the gas is reduced to TL . The third step is a reversible isothermal compression, path cd, during which heat QL flows out of the working substance. Finally, the gas is compressed adiabatically, path da, back to its original state. Thus a Carnot cycle consists of two isothermal and two adiabatic processes. The net work done in one cycle by a Carnot engine (or any other type of engine using a reversible cycle) is equal to the area enclosed by the curve representing the cycle on the PV diagram, the curve abcd in Fig. 20–7. (See Section 19–7.) SECTION 20–3

TL TH

P a 4 1 QH b d

3 QL

2 c

TH TL

FIGURE 20–7 The Carnot cycle. Heat engines work in a cycle, and the cycle for the Carnot engine begins at point a on this PV diagram. (1) The gas is first expanded isothermally, with the addition of heat QH , along the path ab at temperature TH . (2) Next the gas expands adiabatically from b to c—no heat is exchanged, but the temperature drops to TL . (3) The gas is then compressed at constant temperature TL , path cd, and heat QL flows out. (4) Finally, the gas is compressed adiabatically, path da, back to its original state. No Carnot engine actually exists, but as a theoretical idea it played an important role in the development of thermodynamics.

Reversible and Irreversible Processes; the Carnot Engine

533

Carnot Efficiency and the Second Law of Thermodynamics The efficiency of a Carnot engine, like any heat engine, is given by Eq. 20–1b: QL . QH

e = 1 -

For a Carnot engine using an ideal gas, however, we can show that the efficiency depends only on the temperatures of the heat reservoirs, TH and TL . In the first isothermal process ab in Fig. 20–7, the work done by the gas is (see Eq. 19–8) Wab = nRTH ln

Vb , Va

where n is the number of moles of the ideal gas used as working substance. Because the internal energy of an ideal gas does not change when the temperature remains constant, the first law of thermodynamics tells us that the heat added to the gas equals the work done by the gas: QH = nRTH ln

Vb . Va

Similarly, the heat lost by the gas in the isothermal process cd is QL = nRTL ln

Vc . Vd

The paths bc and da are adiabatic, so we have from Eq. 19–15: g

and

Pb Vb = Pc Vc

Pd Vd = Pa Va ,

where g = CPCV is the ratio of molar specific heats (Eq. 19–14). Also, from the ideal gas law, Pb Vb Pc Vc = TH TL

and

Pa Va Pd Vd . = TL TH

When we divide these last equations, term by term, into the corresponding set of equations on the line above, we obtain g-1

TH Vb

g-1

= TL Vc

and

g-1

TL Vd

g-1

= TH Va

Next we divide the equation on the left by the one on the right and obtain Vb ¢ ≤ Va

g-1

Vc = ¢ ≤ Vd

g-1

or Vb Vc . = Va Vd Inserting this result in our equations for QH and QL above, we obtain QL TL . = QH TH

[Carnot cycle] (20–2)

Hence the efficiency of a reversible Carnot engine can now be written eideal = 1 -

QL QH

eideal = 1 -

TL . TH

or c

Carnot efficiency; d (20–3) Kelvin temperatures

The temperatures TL and TH are the absolute or Kelvin temperatures as measured on the ideal gas temperature scale. Thus the efficiency of a Carnot engine depends only on the temperatures TL and TH .

534

CHAPTER 20

Second Law of Thermodynamics

We could imagine other possible reversible cycles that could be used for an ideal reversible engine. According to a theorem stated by Carnot: All reversible engines operating between the same two constant temperatures TH and TL have the same efficiency. Any irreversible engine operating between the same two fixed temperatures will have an efficiency less than this.

This is known as Carnot's theorem.† It tells us that Eq. 20–3, e = 1 - ATLTH B, applies to any ideal reversible engine with fixed input and exhaust temperatures, TH and TL , and that this equation represents a maximum possible efficiency for a real (i.e., irreversible) engine. In practice, the efficiency of real engines is always less than the Carnot efficiency. Well-designed engines reach perhaps 60% to 80% of Carnot efficiency. EXAMPLE 20–2 A phony claim? An engine manufacturer makes the following claims: An engine's heat input per second is 9.0 kJ at 435 K. The heat output per second is 4.0 kJ at 285 K. Do you believe these claims? APPROACH The engine's efficiency can be calculated from the definition, Eqs. 20–1. It must be less than the maximum possible, Eq. 20–3. SOLUTION The claimed efficiency of the engine is (Eq. 20–1b) QL 4.0 kJ e = 1 = 1 = 0.56, QH 9.0 kJ or 56%. However, the maximum possible efficiency is given by the Carnot efficiency, Eq. 20–3: TL 285 K eideal = 1 = 1 = 0.34, TH 435 K or 34%. The manufacturer's claims violate the second law of thermodynamics and cannot be believed. EXERCISE B A motor is running with an intake temperature TH = 400 K and an exhaust temperature TL = 300 K. Which of the following is not a possible efficiency for the engine? (a) 0.10; (b) 0.16; (c) 0.24; (d) 0.30.

It is clear from Eq. 20–3 that a 100% efficient engine is not possible. Only if the exhaust temperature, TL , were at absolute zero would 100% efficiency be obtainable. But reaching absolute zero is a practical (as well as theoretical) impossibility.‡ Thus we can state, as we already did in Section 20–2, that no device is possible whose sole effect is to transform a given amount of heat completely into work. As we saw in Section 20–2, this is known as the Kelvin-Planck statement of the second law of thermodynamics. It tells us that there can be no perfect (100% efficient) heat engine such as the one diagrammed in Fig. 20–6. EXERCISE C Return to the Chapter-Opening Question, page 528, and answer it again now. Try to explain why you may have answered differently the first time.

* The Otto Cycle The operation of an automobile internal combustion engine (Fig. 20–4) can be approximated by a reversible cycle known as the Otto cycle, whose PV diagram is shown in Fig. 20–8. Unlike the Carnot cycle, the input and exhaust temperatures of the Otto cycle are not constant. Paths ab and cd are adiabatic, and paths bc and da are at constant volume. The gas (gasoline-air mixture) enters the cylinder at point a and is compressed adiabatically (compression stroke) to point b. At b ignition occurs (spark plug) and the burning of the gas adds heat QH to the system at constant volume (approximately in a real engine). The temperature and pressure rise, and then in the power stroke, cd, the gas expands adiabatically. In the exhaust stroke, da, heat QL is ejected to the environment (in a real engine, the gas leaves the engine and is replaced by a new mixture of air and fuel). † Carnot's theorem can be shown to follow directly from either the Clausius or Kelvin-Planck statements of the second law of thermodynamics. ‡

This result is known as the third law of thermodynamics, as discussed in Section 20–10.

FIGURE 20–8 The Otto cycle. c QH

b d

a 0

SECTION 20–3

535

EXAMPLE 20–3 The Otto cycle. (a) Show that for an ideal gas as working substance, the efficiency of an Otto cycle engine is

c QH

e = 1 - a

b d

a 0

Va 1 - g b Vb

where g is the ratio of specific heats (g = CPCV , Eq. 19–14) and VaVb is the compression ratio. (b) Calculate the efficiency for a compression ratio VaVb = 8.0 assuming a diatomic gas like O2 and N2 . APPROACH We use the original definition of efficiency and the results from Chapter 19 for constant volume and adiabatic processes (Sections 19–8 and 19–9). SOLUTION The heat exchanges take place at constant volume in the ideal Otto cycle, so from Eq. 19–10a:

FIGURE 20–8 (repeated for Example 20–3) The Otto cycle.

QH = nCV ATc - Tb B

QL = nCV ATd - Ta B.

and

Then from Eq. 20–1b, e = 1 -

Td - Ta QL = 1 - c d. QH Tc - Tb

To get this in terms of the compression ratio, VaVb , we use the result from g Section 19–9, Eq. 19–15, PV = constant during the adiabatic processes ab and cd. Thus g

and

Pa Va = Pb Vb

Pc Vc = Pd Vd .

We use the ideal gas law, P = nRTV, and substitute P into these two equations g-1

Ta Va

g-1

and

= Tb Vb

g-1

Tc Vc

g-1

= Td Vd

Then the efficiency (see above) is Tc AVcVd B - Tb AVbVa B Td - Ta e = 1 - c d = 1 - c Tc - Tb Tc - Tb g-1

g-1

But processes bc and da are at constant volume, so Vc = Vb and Vd = Va . Hence VcVd = VbVa and e = 1 - c

AVbVa B

g-1

ATc - Tb B

Tc - Tb

d = 1 - a

Vb g - 1 Va 1 - g b = 1 - a b . Va Vb

(b) For diatomic molecules (Section 19–8), g = CPCV = 1.4 so e = 1 - (8.0)1 FIGURE 20–9 Schematic diagram of energy transfers for a refrigerator or air conditioner.

= 1 - (8.0)–0.4 = 0.56.

Real engines do not reach this high efficiency because they do not follow perfectly the Otto cycle, plus there is friction, turbulence, heat loss and incomplete combustion of the gases.

TH QH

20–4 W Refrigerator or air conditioner

QL TL

536

CHAPTER 20

Refrigerators, Air Conditioners, and Heat Pumps

The operating principle of refrigerators, air conditioners, and heat pumps is just the reverse of a heat engine. Each operates to transfer heat out of a cool environment into a warm environment. As diagrammed in Fig. 20–9, by doing work W, heat is taken from a low-temperature region, TL (such as inside a refrigerator), and a greater amount of heat is exhausted at a high temperature, TH (the room). You can often feel this heated air blowing out beneath a refrigerator. The work W is usually done by an electric motor which compresses a fluid, as illustrated in Fig. 20–10.

Second Law of Thermodynamics

Low pressure High pressure

Expansion valve

Condenser coils (outside refrigerator)

Cooling coils (inside refrigerator) Sensor

QH (to outside)

QL (from inside refrigerator to coils) Low pressure vapor

(a)

Compressor motor

High pressure W vapor

Plug

(b)

Motor

FIGURE 20–10 (a) Typical refrigerator system. The electric compressor motor forces a gas at high pressure through a heat exchanger (condenser) on the rear outside wall of the refrigerator where QH is given off and the gas cools to become liquid. The liquid passes from a high-pressure region, via a valve, to low-pressure tubes on the inside walls of the refrigerator; the liquid evaporates at this lower pressure and thus absorbs heat AQL B from the inside of the refrigerator. The fluid returns to the compressor where the cycle begins again. (b) Schematic diagram, like Fig. 20–9.

Plug

A perfect refrigerator—one in which no work is required to take heat from the low-temperature region to the high-temperature region—is not possible. This is the Clausius statement of the second law of thermodynamics, already mentioned in Section 20–1, which can be stated formally as No device is possible whose sole effect is to transfer heat from one system at a temperature TL into a second system at a higher temperature TH . To make heat flow from a low-temperature object (or system) to one at a higher temperature, work must be done. Thus, there can be no perfect refrigerator. The coefficient of performance (COP) of a refrigerator is defined as the heat QL removed from the low-temperature area (inside a refrigerator) divided by the work W done to remove the heat (Fig. 20–9 or 20–10b): QL . refrigerator and COP = c d (20–4a) air conditioner W

SECOND LAW OF THERMODYNAMICS (Clausius statement)

P H Y S I C S

A P P L I E D

Refrigerator

This makes sense because the more heat QL that can be removed from the inside of the refrigerator for a given amount of work, the better (more efficient) the refrigerator is. Energy is conserved, so from the first law of thermodynamics we can write (see Fig. 20–9 or 20–10b) QL + W = QH , or W = QH - QL . Then Eq. 20–4a becomes QL QL refrigerator . COP = = c d (20–4b) air conditioner W QH - QL For an ideal refrigerator (not a perfect one, which is impossible), the best we could do would be COPideal =

TL , TH - TL

refrigerator d (20–4c) air conditioner

analagous to an ideal (Carnot) engine (Eqs. 20–2 and 20–3). An air conditioner works very much like a refrigerator, although the actual construction details are different: an air conditioner takes heat QL from inside a room or building at a low temperature, and deposits heat QH outside to the environment at a higher temperature. Equations 20–4 also describe the coefficient of performance for an air conditioner. SECTION 20–4

P H Y S I C S

A P P L I E D

Air conditioner

Refrigerators, Air Conditioners, and Heat Pumps

537

EXAMPLE 20–4 Making ice. A freezer has a COP of 3.8 and uses 200 W of power. How long would it take this otherwise empty freezer to freeze an ice-cube tray that contains 600 g of water at 0°C? APPROACH In Eq. 20–4b, QL is the heat that must be transferred out of the water so it will become ice. To determine QL , we use the latent heat of fusion L of water and Eq. 19–3, Q = mL. SOLUTION From Table 19–2, L = 333 kJkg. Hence Q = mL = (0.600 kg)A3.33 * 105 JkgB = 2.0 * 105 J is the total energy that needs to be removed from the water. The freezer does work at the rate of 200 W = 200 Js = Wt, which is the work W it can do in t seconds. We solve for t: t = W(200 Js). For W, we use Eq. 20–4a: W = QLCOP. Thus

t =

QLCOP W = 200 Js 200 Js =

A2.0 * 105 JBA3.8B 200 Js

= 260 s,

or about 4 12 min.

P H Y S I C S

A P P L I E D

Heat pump Warm inside of house

Cold outside QL

QH Heat pump W

FIGURE 20–11 A heat pump uses an electric motor to "pump" heat from the cold outside to the warm inside of a house.

Heat naturally flows from high temperature to low temperature. Refrigerators and air conditioners do work to accomplish the opposite: to make heat flow from cold to hot. We might say they "pump" heat from cold areas to hotter areas, against the natural tendency of heat to flow from hot to cold, just as water can be pumped uphill, against the natural tendency to flow downhill. The term heat pump is usually reserved for a device that can heat a house in winter by using an electric motor that does work W to take heat QL from the outside at low temperature and delivers heat QH to the warmer inside of the house; see Fig. 20–11. As in a refrigerator, there is an indoor and an outdoor heat exchanger (coils of the refrigerator) and an electric compressor motor. The operating principle is like that for a refrigerator or air conditioner; but the objective of a heat pump is to heat (deliver QH), rather than to cool (remove QL). Thus, the coefficient of performance of a heat pump is defined differently than for an air conditioner because it is the heat QH delivered to the inside of the house that is important now: COP =

QH . W

[heat pump] (20–5)

C A U T I O N

Heat pumps and air conditioners have different COP definitions

The COP is necessarily greater than 1. Most heat pumps can be "turned around" and used as air conditioners in the summer. EXAMPLE 20–5 Heat pump. A heat pump has a coefficient of performance of 3.0 and is rated to do work at 1500 W. (a) How much heat can it add to a room per second? (b) If the heat pump were turned around to act as an air conditioner in the summer, what would you expect its coefficient of performance to be, assuming all else stays the same? APPROACH We use the definitions of coefficient of performance, which are different for the two devices in (a) and (b). SOLUTION (a) We use Eq. 20–5 for the heat pump, and, since our device does 1500 J of work per second, it can pour heat into the room at a rate of QH = COP * W = 3.0 * 1500 J = 4500 J per second, or at a rate of 4500 W.

538

CHAPTER 20

Second Law of Thermodynamics

(b) If our device is turned around in summer, it can take heat QL from inside the house, doing 1500 J of work per second to then dump QH = 4500 J per second to the hot outside. Energy is conserved, so QL + W = QH (see Fig. 20–11, but reverse the inside and outside of the house). Then QL = QH - W = 4500 J - 1500 J = 3000 J. The coefficient of performance as an air conditioner would thus be (Eq. 20–4a) COP =

20–5

QL 3000 J = = 2.0. W 1500 J

Entropy

Thus far we have stated the second law of thermodynamics for specific situations. What we really need is a general statement of the second law of thermodynamics that will cover all situations, including those discussed earlier in this Chapter that are not observed in nature even though they would not violate the first law of thermodynamics. It was not until the latter half of the nineteenth century that the second law of thermodynamics was finally stated in a general way—namely, in terms of a quantity called entropy, introduced by Clausius in the 1860s. In Section 20–7 we will see that entropy can be interpreted as a measure of the order or disorder of a system. When we deal with entropy—as with potential energy—it is the change in entropy during a process that is important, not the absolute amount. According to Clausius, the change in entropy S of a system, when an amount of heat Q is added to it by a reversible process at constant temperature, is given by ¢S =

Q, T

(20–6)

where T is the kelvin temperature. If the temperature is not constant, we define entropy S by the relation dS =

dQ . T

(20–7)

Then the change in entropy of a system taken reversibly between two states a and b is given by† b

¢S = Sb - Sa =

dS a

dQ . T

[reversible process] (20–8)

A careful analysis (see next page) shows that the change in entropy when a system moves by a reversible process from any state a to another state b does not depend on the process. That is, ¢S = Sb - Sa depends only on the states a and b of the system. Thus entropy (unlike heat) is a state variable. Any system in a given state has a temperature, a volume, a pressure, and also has a particular value of entropy. It is easy to see why entropy is a state variable for a Carnot cycle. In Eq. 20–2 we saw that QLQH = TLTH , which we rewrite as QL QH . = TL TH In the PV diagram for a Carnot cycle, Fig. 20–7, the entropy change ¢S = QT in going from state a to state c along path abc ( QHTH + 0) is thus the same as going along the path adc. That is, the change in entropy is path independent—it depends only on the initial and final states of the system. †

Equation 20–8 says nothing about the absolute value of S; it only gives the change in S. This is much like potential energy (Chapter 8). However, one form of the so-called third law of thermodynamics (see also Section 20–10) states that as T S 0, S S 0.

SECTION 20–5

Entropy

539

* Showing Entropy Is a State Variable

a 0

FIGURE 20–12 Any reversible cycle can be approximated as a series of Carnot cycles. (The dashed lines represent isotherms.)

In our study of the Carnot cycle we found (Eq. 20–2) that QLQH = TLTH . We rewrite this as QH QL . = TH TL In this relation, both QH and QL are positive. But now let us recall our original convention as used in the first law (Section 19–6), that Q is positive when it represents a heat flow into the system (as QH) and negative for a heat flow out of the system (as –QL). Then this relation becomes QH QL + = 0. [Carnot cycle] (20–9) TH TL Now consider any reversible cycle, as represented by the smooth (oval-shaped) curve in Fig. 20–12. Any reversible cycle can be approximated as a series of Carnot cycles. Figure 20–12 shows only six—the isotherms (dashed lines) are connected by adiabatic paths for each—and the approximation becomes better and better if we increase the number of Carnot cycles. Equation 20–9 is valid for each of these cycles, so we can write Q © = 0 [Carnot cycles] (20–10) T for the sum of all these cycles. But note that the heat output QL of one cycle crosses the boundary below it and is approximately equal to the negative of the heat input, QH , of the cycle below it (actual equality occurs in the limit of an infinite number of infinitely thin Carnot cycles). Hence the heat flows on the inner paths of all these Carnot cycles cancel out, so the net heat transferred, and the work done, is the same for the series of Carnot cycles as for the original cycle. Hence, in the limit of infinitely many Carnot cycles, Eq. 20–10 applies to any reversible cycle. In this case Eq. 20–10 becomes dQ = 0, T

§ FIGURE 20–13 The integral, dS, of the entropy for a reversible cycle is zero. Hence the difference in entropy between states a and b, b Sb - Sa = a dS, is the same for path I as for path II.

dQ + T

dQ = 0. T

The first term is the integral from point a to point b along path I in Fig. 20–13, and the second term is the integral from b back to a along path II. If path II is taken in reverse, dQ at each point becomes –dQ, since the path is reversible. Therefore b

dQ = T

where dQ represents an infinitesimal heat flow.† The symbol means that the integral is taken around a closed path; the integral can be started at any point on the path such as at a or b in Fig. 20–12, and proceed in either direction. If we divide the cycle of Fig. 20–12 into two parts as indicated in Fig. 20–13, then

I II

[reversible cycle] (20–11)

dQ . T

[reversible paths] (20–12)

The integral of dQT between any two equilibrium states, a and b, does not depend on the path of the process. By defining entropy as dS = dQT (Eq. 20–7), we see from Eq. 20–12 that the change in entropy between any two states along a reversible path is independent of the path between two points a and b. Thus entropy is a state variable—its value depends only on the state of the system, and not on the process or the past history of how it got there.‡ This is in clear distinction to Q and W which are not state variables; their values do depend on the processes undertaken. dQ is often written d–Q: see footnote at the end of Section 19–6. Real processes are irreversible. Because entropy is a state variable, the change in entropy ¢S for an irreversible process can be determined by calculating ¢S for a reversible process between the same two states. † ‡

540

CHAPTER 20

Second Law of Thermodynamics

20–6

Entropy and the Second Law of Thermodynamics

We have defined a new quantity, S, the entropy, which can be used to describe the state of the system, along with P, T, V, Eint , and n. But what does this rather abstract quantity have to do with the second law of thermodynamics? To answer this, let us take some examples in which we calculate the entropy changes during particular processes. But note first that Eq. 20–8 can be applied only to reversible processes. How then do we calculate ¢S = Sb - Sa for a real process that is irreversible? What we can do is this: we figure out some other reversible process that takes the system between the same two states, and calculate ¢S for this reversible process. This will equal ¢S for the irreversible process since ¢S depends only on the initial and final states of the system. If the temperature varies during a process, a summation of the heat flow over the changing temperature can often be calculated using calculus or a computer. However, if the temperature change is not too great, a reasonable approximation can be made using the average value of the temperature, as indicated in the next Example. EXAMPLE 20–6 ESTIMATE Entropy change when mixing water. A sample of 50.0 kg of water at 20.00°C is mixed with 50.0 kg of water at 24.00°C. Estimate the change in entropy. APPROACH The final temperature of the mixture will be 22.00°C, since we started with equal amounts of water. We use the specific heat of water and the methods of calorimetry (Sections 19–3 and 19–4) to determine the heat transferred. Then we use the average temperature of each sample of water to estimate the entropy change (¢QT). SOLUTION A quantity of heat, Q = mc ¢T = (50.0 kg)(4186 JkgC°)(2.00 C°) = 4.186 * 105 J, flows out of the hot water as it cools down from 24°C to 22°C, and this heat flows into the cold water as it warms from 20°C to 22°C. The total change in entropy, ¢S, will be the sum of the changes in entropy of the hot water, ¢SH , and that of the cold water, ¢SC : ¢S = ¢SH + ¢SC . We estimate entropy changes by writing ¢S = QT, where T is an "average" temperature for each process, which ought to give a reasonable estimate since the temperature change is small. For the hot water we use an average temperature of 23°C (296 K), and for the cold water an average temperature of 21°C (294 K). Thus ¢SH L –

4.186 * 105 J = –1414 JK 296 K

which is negative because this heat flows out, whereas heat is added to the cold water: ¢SC L

4.186 * 105 J = 1424 JK. 294 K

The entropy of the hot water ASH B decreases since heat flows out of the hot water. But the entropy of the cold water ASC B increases by a greater amount. The total change in entropy is ¢S = ¢SH + ¢SC L –1414 JK + 1424 JK L 10 JK. We see that although the entropy of one part of the system decreased, the entropy of the other part increased by a greater amount so that the net change in entropy of the whole system is positive. SECTION 20–6

Entropy and the Second Law of Thermodynamics

541

We can now show in general that for an isolated system of two objects, the flow of heat from the higher-temperature ATH B object to the lower-temperature ATL B object always results in an increase in the total entropy. The two objects eventually come to some intermediate temperature, TM . The heat lost by the hotter object AQH = –Q, where Q is positive B is equal to the heat gained by the colder one AQL = QB, so the total change in entropy is ¢S = ¢SH + ¢SL = –

Q Q , + THM TLM

where THM is some intermediate temperature between TH and TM for the hot object as it cools from TH to TM , and TLM is the counterpart for the cold object. Since the temperature of the hot object is, at all times during the process, greater than that of the cold object, then THM 7 TLM . Hence ¢S = Q ¢

1 1 ≤ 7 0. TLM THM

One object decreases in entropy, while the other gains in entropy, but the total change is positive. EXAMPLE 20–7 Entropy changes in a free expansion. Consider the adiabatic free expansion of n moles of an ideal gas from volume V1 to volume V2 , where V2 7 V1 as was discussed in Section 19–7, Fig. 19–14. Calculate the change in entropy (a) of the gas and (b) of the surrounding environment. (c) Evaluate ¢S for 1.00 mole, with V2 = 2.00 V1 . APPROACH We saw in Section 19–7 that the gas is initially in a closed container of volume V1 , and, with the opening of a valve, it expands adiabatically into a previously empty container. The total volume of the two containers is V2 . The whole apparatus is thermally insulated from the surroundings, so no heat flows into the gas, Q = 0. The gas does no work, W = 0, so there is no change in internal energy, ¢Eint = 0, and the temperature of the initial and final states is the same, T2 = T1 = T. The process takes place very quickly, and so is irreversible. Thus we cannot apply Eq. 20–8 to this process. Instead we must think of a reversible process that will take the gas from volume V1 to V2 at the same temperature, and use Eq. 20–8 on this reversible process to get ¢S. A reversible isothermal process will do the trick; in such a process, the internal energy does not change, so from the first law, dQ = dW = P dV. SOLUTION (a) For the gas, ¢Sgas =

dQ 1 = T T

P dV.

The ideal gas law tells us P = nRTV, so ¢Sgas =

nRT T

V2 dV = nR ln . V V1

Since V2 7 V1 , ¢Sgas 7 0. (b) Since no heat is transferred to the surrounding environment, there is no change of the state of the environment due to this process. Hence ¢Senv = 0. Note that the total change in entropy, ¢Sgas + ¢Senv , is greater than zero. (c) Since n = 1.00 and V2 = 2.00 V1 , then ¢Sgas = R ln 2.00 = 5.76 JK.

542

CHAPTER 20

Second Law of Thermodynamics

EXAMPLE 20–8 Heat transfer. A red-hot 2.00-kg piece of iron at temperature T1 = 880 K is thrown into a huge lake whose temperature is T2 = 280 K. Assume the lake is so large that its temperature rise is insignificant. Determine the change in entropy (a) of the iron and (b) of the surrounding environment (the lake). APPROACH The process is irreversible, but the same entropy change will occur for a reversible process, and we use the concept of specific heat, Eq. 19–2. SOLUTION (a) We assume the specific heat of the iron is constant at c = 450 Jkg K. Then dQ = mc dT and in a quasistatic reversible process

dQ = mc T Putting in numbers, we find ¢Siron =

T2 T1 dT = mc ln = –mc ln . T T1 T2

880 K = –1030 JK. 280 K (b) The initial and final temperatures of the lake are the same, T = 280 K. The lake receives from the iron an amount of heat ¢Siron = –(2.00 kg)(450 JkgK) ln

Q = mcAT2 - T1 B = (2.00 kg)(450 JkgK)(880 K - 280 K) = 540 kJ.

Strictly speaking, this is an irreversible process (the lake heats up locally before equilibrium is reached), but is equivalent to a reversible isothermal transfer of heat Q = 540 kJ at T = 280 K. Hence 540 kJ ¢Senv = = 1930 JK. 280 K Thus, although the entropy of the iron actually decreases, the total change in entropy of iron plus environment is positive: 1930 JK - 1030 JK = 900 JK. EXERCISE D A 1.00-kg piece of ice at 0°C melts very slowly to water at 0°C. Assume the ice is in contact with a heat reservoir whose temperature is only infinitesimally greater than 0°C. Determine the entropy change of (a) the ice cube and (b) the heat reservoir.

In each of these Examples, the entropy of our system plus that of the environment (or surroundings) either stayed constant or increased. For any reversible process, such as that in Exercise D, the total entropy change is zero. This can be seen in general as follows: any reversible process can be considered as a series of quasistatic isothermal transfers of heat ¢Q between a system and the environment, which differ in temperature only by an infinitesimal amount. Hence the change in entropy of either the system or environment is ¢QT and that of the other is – ¢QT, so the total is ¢S = ¢Ssyst + ¢Senv = 0.

[any reversible process]

In Examples 20–6, 20–7, and 20–8, we found that the total entropy of system plus environment increases. Indeed, it has been found that for all real (irreversible) processes, the total entropy increases. No exceptions have been found. We can thus make the general statement of the second law of thermodynamics as follows: The entropy of an isolated system never decreases. It either stays constant (reversible processes) or increases (irreversible processes). Since all real processes are irreversible, we can equally well state the second law as: The total entropy of any system plus that of its environment increases as a result of any natural process: ¢S = ¢Ssyst + ¢Senv 7 0.

(20–13)

SECOND LAW OF THERMODYNAMICS (general statement)

Although the entropy of one part of the universe may decrease in any process (see the Examples above), the entropy of some other part of the universe always increases by a greater amount, so the total entropy always increases. SECTION 20–6

Entropy and the Second Law of Thermodynamics

543

Now that we finally have a quantitative general statement of the second law of thermodynamics, we can see that it is an unusual law. It differs considerably from other laws of physics, which are typically equalities (such as F = ma) or conservation laws (such as for energy and momentum). The second law of thermodynamics introduces a new quantity, the entropy, S, but does not tell us it is conserved. Quite the opposite. Entropy is not conserved in natural processes. Entropy always increases in time.

"Time's Arrow" The second law of thermodynamics summarizes which processes are observed in nature, and which are not. Or, said another way, it tells us about the direction processes go. For the reverse of any of the processes in the last few Examples, the entropy would decrease; and we never observe them. For example, we never observe heat flowing spontaneously from a cold object to a hot object, the reverse of Example 20–8. Nor do we ever observe a gas spontaneously compressing itself into a smaller volume, the reverse of Example 20–7 (gases always expand to fill their containers). Nor do we see thermal energy transform into kinetic energy of a rock so the rock rises spontaneously from the ground. Any of these processes would be consistent with the first law of thermodynamics (conservation of energy). But they are not consistent with the second law of thermodynamics, and this is why we need the second law. If you were to see a movie run backward, you would probably realize it immediately because you would see odd occurrences—such as rocks rising spontaneously from the ground, or air rushing in from the atmosphere to fill an empty balloon (the reverse of free expansion). When watching a movie or video, we are tipped off to a faked reversal of time by observing whether entropy is increasing or decreasing. Hence entropy has been called time's arrow, for it can tell us in which direction time is going.

20–7

Order to Disorder

The concept of entropy, as we have discussed it so far, may seem rather abstract. But we can relate it to the more ordinary concepts of order and disorder. In fact, the entropy of a system can be considered a measure of the disorder of the system. Then the second law of thermodynamics can be stated simply as: SECOND LAW OF THERMODYNAMICS (general statement)

Natural processes tend to move toward a state of greater disorder. Exactly what we mean by disorder may not always be clear, so we now consider a few examples. Some of these will show us how this very general statement of the second law applies beyond what we usually consider as thermodynamics. Let us look at the simple processes mentioned in Section 20–1. First, a jar containing separate layers of salt and pepper is more orderly than when the salt and pepper are all mixed up. Shaking a jar containing separate layers results in a mixture, and no amount of shaking brings the layers back again. The natural process is from a state of relative order (layers) to one of relative disorder (a mixture), not the reverse. That is, disorder increases. Next, a solid coffee cup is a more "orderly" and useful object than the pieces of a broken cup. Cups break when they fall, but they do not spontaneously mend themselves (as faked in Fig. 20–1). Again, the normal course of events is an increase of disorder. Let us consider some processes for which we have actually calculated the entropy change, and see that an increase in entropy results in an increase in disorder (or vice versa). When ice melts to water at 0°C, the entropy of the water increases (Exercise D). Intuitively, we can think of solid water, ice, as being more ordered than the less orderly fluid state which can flow all over the place. This change from order to disorder can be seen more clearly from the molecular point of view: the orderly arrangement of water molecules in an ice crystal has changed to the disorderly and somewhat random motion of the molecules in the fluid state. When a hot object is put in contact with a cold object, heat flows from the high temperature to the low until the two objects reach the same intermediate temperature. At the beginning of the process we can distinguish two classes of

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molecules: those with a high average kinetic energy (the hot object), and those with a low average kinetic energy (the cooler object). After the process in which heat flows, all the molecules are in one class with the same average kinetic energy. We no longer have the more orderly arrangement of molecules in two classes. Order has gone to disorder. Furthermore, the separate hot and cold objects could serve as the hot- and cold-temperature regions of a heat engine, and thus could be used to obtain useful work. But once the two objects are put in contact and reach the same temperature, no work can be obtained. Disorder has increased, since a system that has the ability to perform work must surely be considered to have a higher order than a system no longer able to do work. When a stone falls to the ground, its macroscopic kinetic energy is transformed to thermal energy. Thermal energy is associated with the disorderly random motion of molecules, but the molecules in the falling stone all have the same velocity downward in addition to their own random velocities. Thus, the more orderly kinetic energy of the stone as a whole is changed to disordered thermal energy when the stone strikes the ground. Disorder increases in this process, as it does in all processes that occur in nature.

* Biological Evolution An interesting example of the increase in entropy relates to biological evolution and to growth of organisms. Clearly, a human being is a highly ordered organism. The theory of evolution describes the process from the early macromolecules and simple forms of life to Homo sapiens, which is a process of increasing order. So, too, the development of an individual from a single cell to a grown person is a process of increasing order. Do these processes violate the second law of thermodynamics? No, they do not. In the processes of evolution and growth, and even during the mature life of an individual, waste products are eliminated. These small molecules that remain as a result of metabolism are simple molecules without much order. Thus they represent relatively higher disorder or entropy. Indeed, the total entropy of the molecules cast aside by organisms during the processes of evolution and growth is greater than the decrease in entropy associated with the order of the growing individual or evolving species.

20–8

P H Y S I C S

A P P L I E D

Biological Evolution and Development

Unavailability of Energy; Heat Death

In the process of heat conduction from a hot object to a cold one, we have seen that entropy increases and that order goes to disorder. The separate hot and cold objects could serve as the high- and low-temperature regions for a heat engine and thus could be used to obtain useful work. But after the two objects are put in contact with each other and reach the same uniform temperature, no work can be obtained from them. With regard to being able to do useful work, order has gone to disorder in this process. The same can be said about a falling rock that comes to rest upon striking the ground. Before hitting the ground, all the kinetic energy of the rock could have been used to do useful work. But once the rock's mechanical kinetic energy becomes thermal energy, doing useful work is no longer possible. Both these examples illustrate another important aspect of the second law of thermodynamics: in any natural process, some energy becomes unavailable to do useful work. In any process, no energy is ever lost (it is always conserved). Rather, energy becomes less useful—it can do less useful work. As time goes on, energy is degraded, in a sense; it goes from more orderly forms (such as mechanical) eventually to the least orderly form, internal, or thermal, energy. Entropy is a factor here because the amount of energy that becomes unavailable to do work is proportional to the change in entropy during any process.† † It can be shown that the amount of energy that becomes unavailable to do useful work is equal to TL ¢S, where TL is the lowest available temperature and ¢S is the total increase in entropy during the process.

SECTION 20–8

Unavailability of Energy; Heat Death

545

A natural outcome of the degradation of energy is the prediction that as time goes on, the universe should approach a state of maximum disorder. Matter would become a uniform mixture, and heat will have flowed from high-temperature regions to low-temperature regions until the whole universe is at one temperature. No work could then be done. All the energy of the universe would have degraded to thermal energy. This prediction, called the heat death of the universe, has been much discussed, but would lie very far in the future. It is a complicated subject, and some scientists question whether thermodynamic modeling of the universe is possible or appropriate.

* 20–9

Statistical Interpretation of Entropy and the Second Law

The ideas of entropy and disorder are made clearer with the use of a statistical or probabilistic analysis of the molecular state of a system. This statistical approach, which was first applied toward the end of the nineteenth century by Ludwig Boltzmann (1844–1906), makes a clear distinction between the "macrostate" and the "microstate" of a system. The microstate of a system would be specified in giving the position and velocity of every particle (or molecule). The macrostate of a system is specified by giving the macroscopic properties of the system—the temperature, pressure, number of moles, and so on. In reality, we can know only the macrostate of a system. We could not possibily know the velocity and position of every one of the huge number of molecules in a system at a given moment. Nonetheless, we can hypothesize a great many different microstates that can correspond to the same macrostate. Let us take a very simple example. Suppose you repeatedly shake four coins in your hand and drop them on a table. Specifying the number of heads and the number of tails that appear on a given throw is the macrostate of this system. Specifying each coin as being a head or a tail is the microstate of the system. In the following Table we see how many microstates correspond to each macrostate: Possible Microstates (H heads, T tails)

Macrostate 4 heads 3 heads, 1 tail 2 heads, 2 tails 1 head, 3 tails 4 tails

HHHH H H H T, H H T H, H T H H, T H H H H H T T, H T H T, T H H T, H T T H, T H T H, T T H H T T T H, T T H T, T H T T, H T T T TTTT

Number of Microstates 1 4 6 4 1

A basic assumption behind the statistical approach is that each microstate is equally probable. Thus the number of microstates that give the same macrostate corresponds to the relative probability of that macrostate occurring. The macrostate of two heads and two tails is the most probable one in our case of tossing four coins; out of the total of 16 possible microstates, six correspond to two heads and two tails, so the probability of throwing two heads and two tails is 6 out of 16, or 38%. The probability of throwing one head and three tails is 4 out of 16, or 25%. The probability of four heads is only 1 in 16, or 6%. If you threw the coins 16 times, you might not find that two heads and two tails appear exactly 6 times, or four tails exactly once. These are only probabilities or averages. But if you made 1600 throws, very nearly 38% of them would be two heads and two tails. The greater the number of tries, the closer the percentages are to the calculated probabilities. EXERCISE E In the Table above, what is the probability that there will be at least two heads? (a) 12 ; (b) 161 ; (c) 18 ; (d) 38 ; (e) 11 16 .

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If we toss more coins—say, 100 all at the same time—the relative probability of throwing all heads (or all tails) is greatly reduced. There is only one microstate corresponding to all heads. For 99 heads and 1 tail, there are 100 microstates since each of the coins could be the one tail. The relative probabilities for other macrostates are given in Table 20–1. About 1.3 * 1030 microstates are possible.† Thus the relative probability of finding all heads is 1 in 1030, an incredibly unlikely event! The probability of obtaining 50 heads and 50 tails (see Table 20–1) is A1.0 * 1029 B1.3 * 1030 = 0.08 or 8%. The probability of obtaining anything between 45 and 55 heads is over 70%. TABLE 20–1 Probabilities of Various Macrostates for 100 Coin Tosses Macrostate Number of Microstates

Probability

100 99 90 80 60 55 50 45 40 20 10 1 0

0 1 10 20 40 45 50 55 60 80 90 99 100

1 * * * * * * * * * * * 1

7.9 * 10–31 7.9 * 10–29 1.4 * 10–17 4.2 * 10–10 0.01 0.05 0.08 0.05 0.01 4.2 * 10–10 1.4 * 10–17 7.9 * 10–29 7.9 * 10–31

1.0 1.7 5.4 1.4 6.1 1.0 6.1 1.4 5.4 1.7 1.0

10 1013 1020 1028 1028 1029 1028 1028 1020 1013 102

Thus we see that as the number of coins increases, the probability of obtaining the most orderly arrangement (all heads or all tails) becomes extremely unlikely. The least orderly arrangement (half heads, half tails) is the most probable, and the probability of being within, say, 5% of the most probable arrangement greatly increases as the number of coins increases. These same ideas can be applied to the molecules of a system. For example, the most probable state of a gas (say, the air in a room) is one in which the molecules take up the whole space and move about randomly; this corresponds to the Maxwellian distribution, Fig. 20–14a (and see Section 18–2). On the other hand, the very orderly arrangement of all the molecules located in one corner of the room and all moving with the same velocity (Fig. 20–14b) is extremely unlikely. From these examples, it is clear that probability is directly related to disorder and hence to entropy. That is, the most probable state is the one with greatest entropy or greatest disorder and randomness. Boltzmann showed, consistent with Clausius's definition (dS = dQT), that the entropy of a system in a given (macro) state can be written (20–14)

S = k ln w,

where k is Boltzmann's constant Ak = RNA = 1.38 * 10 –23 JKB and w is the number of microstates corresponding to the given macrostate. That is, w is proportional to the probability of occurrence of that state. w is called the thermodynamic probability, or, sometimes, the disorder parameter. †

Each coin has two possibilities, heads or tails. Then the possible number of microstates is 2 * 2 * 2 * p = 2 100 = 1.27 * 1030 (using a calculator or logarithms).

*SECTION 20–9

FIGURE 20–14 (a) Most probable distribution of molecular speeds in a gas (Maxwellian, or random); (b) orderly, but highly unlikely, distribution of speeds in which all molecules have nearly the same speed.

Fraction of molecules

Tails

Speed, (a)

Fraction of molecules

Heads

Speed, (b)

Statistical Interpretation of Entropy and the Second Law

547

EXAMPLE 20–9 Free expansion—statistical determination of entropy. Use Eq. 20–14 to determine the change in entropy for the adiabatic free expansion of a gas, a calculation we did macroscopically in Example 20–7. Assume w, the number of microstates for each macrostate, is the number of possible positions. APPROACH We assume the number of moles is n = 1, and then the number of molecules is N = nNA = NA . We let the volume double, just as in Example 20–7. Because the volume doubles, the number of possible positions for each molecule doubles. SOLUTION When the volume doubles, each molecule has two times as many positions (microstates) available. For two molecules, the number of total microstates increases by 2 * 2 = 2 2. For NA molecules, the total number of microstates increases by a factor of 2 * 2 * 2 * p = 2 NA. That is w2 = 2 NA. w1 The change in entropy is, from Eq. 20–14, ¢S = S2 - S1 = kAln w2 - ln w1 B = k ln

w2 = k ln 2 NA = kNA ln 2 = R ln 2 w1 which is the same result we obtained in Example 20–7. In terms of probability, the second law of thermodynamics—which tells us that entropy increases in any process—reduces to the statement that those processes occur which are most probable. The second law thus becomes a trivial statement. However there is an additional element now. The second law in terms of probability does not forbid a decrease in entropy. Rather, it says the probability is extremely low. It is not impossible that salt and pepper could separate spontaneously into layers, or that a broken tea cup could mend itself. It is even possible that a lake could freeze over on a hot summer day (that is, heat flow out of the cold lake into the warmer surroundings). But the probability for such events occurring is extremely small. In our coin examples, we saw that increasing the number of coins from 4 to 100 drastically reduced the probability of large deviations from the average, or most probable, arrangement. In ordinary systems we are not dealing with 100 molecules, but with incredibly large numbers of molecules: in 1 mol alone there are 6 * 1023 molecules. Hence the probability of deviation far from the average is incredibly tiny. For example, it has been calculated that the probability that a stone resting on the ground should transform 1 cal of thermal energy into mechanical energy and rise up into the air is much less likely than the probability that a group of monkeys typing randomly would by chance produce the complete works of Shakespeare.

* 20–10

Thermodynamic Temperature; Third Law of Thermodynamics

In Section 20–3 we saw for a Carnot cycle that the ratio of the heat absorbed QH from the high-temperature reservoir and the heat exhausted QL to the low-temperature reservoir is directly related to the ratio of the temperatures of the two reservoirs (Eq. 20–2): QL TL . = QH TH This result is valid for any reversible engine and does not depend on the working substance. It can thus serve as the basis for the Kelvin or thermodynamic temperature scale. We use this relation and the ideal gas temperature scale (Section 17–10) to complete the definition of the thermodynamic scale: we assign the value

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Ttp = 273.16 K to the triple point of water so that Q ≤, Qtp where Q and Qtp are the magnitudes of the heats exchanged by a Carnot engine with reservoirs at temperatures T and Ttp . Then the thermodynamic scale is identical to the ideal gas scale over the latter's range of validity. Very low temperatures are difficult to obtain experimentally. The closer the temperature is to absolute zero, the more difficult it is to reduce the temperature further, and it is generally accepted that it is not possible to reach absolute zero in any finite number of processes. This last statement is one way to state† the third law of thermodynamics. Since the maximum efficiency that any heat engine can have is the Carnot efficiency TL , e = 1 TH and since TL can never be zero, we see that a 100% efficient heat engine is not possible. T = (273.16 K) ¢

FIGURE 20–15 (a) An array of mirrors focuses sunlight on a boiler to produce steam at a solar energy installation. (b) A fossil-fuel steam plant (this one uses forest waste products, biomass). (c) Large cooling towers at an electric generating plant.

(a)

* 20–11

(b)

Thermal Pollution, Global Warming, and Energy Resources

Much of the energy we utilize in everyday life—from motor vehicles to most of the electricity produced by power plants—makes use of a heat engine. Electricity produced by falling water at dams, by windmills, or by solar cells (Fig. 20–15a) does not involve a heat engine. But over 90% of the electric energy produced in the U.S. is generated at fossil-fuel steam plants (coal, oil, or gas—see Fig. 20–15b), and they make use of a heat engine (essentially steam engines). In electric power plants, the steam drives the turbines and generators (Fig. 20–16) whose output is electric energy. The various means to turn the turbine are discussed briefly in Table 20–2 (next page), along with some of the advantages and disadvantages of each. Even nuclear power plants use nuclear fuel to run a steam engine. The heat output QL from every heat engine, from power plants to cars, is referred to as thermal pollution because this heat AQL B must be absorbed by the environment—such as by water from rivers or lakes, or by the air using large cooling towers (Fig. 20–15c). When water is the coolant, this heat raises the temperature of the water, altering the natural ecology of aquatic life (largely because warmer water holds less oxygen). In the case of air cooling towers, the output heat QL raises the temperature of the atmosphere, which affects the weather. †

(c) FIGURE 20–16 Mechanical energy is transformed to electric energy with a turbine and generator. Source of energy: water, steam, or wind Electric generator

Turbine Electric energy P H Y S I C S

A P P L I E D

Heat engines and thermal pollution

See also the statement in the footnote on p. 539.

*SECTION 20–11

Thermal Pollution, Global Warming, and Energy Resources

549

TABLE 20–2 Electric Energy Resources Form of Electric Energy Production

% of Production (approx.) U.S.

World

Advantages

We know how to build them; for now relatively inexpensive.

Air pollution; thermal pollution; limited efficiency; land devastation from extraction of raw materials (mining); global warming; accidents such as oil spills at sea; limited fuel supply (estimates range from a couple of decades to a few centuries).

Normally almost no air pollution; less contribution to global warming; relatively inexpensive. Relatively "clean"; vast fuel supply (hydrogen in water molecules in oceans); less contribution to global warming.

Thermal pollution; accidents can release damaging radioactivity; difficult disposal of radioactive by-products; possible diversion of nuclear material by terrorists; limited fuel supply. Not yet workable.

No heat engine needed; no air, water, or thermal pollution; relatively inexpensive; high efficiency; dams can control flooding.

Reservoirs behind dams inundate scenic or inhabited land; dams block upstream migration of salmon and other fish for reproduction; few locations remain for new dams; drought.

Geothermal: natural steam from inside the Earth comes to the surface (hot springs, geysers, steam vents); or cold water passed down into contact with hot, dry rock is heated to steam. Wind power: 3-kW to 5-MW windmills (vanes up to 50 m wide) turn a generator.

No heat engine needed; little air pollution; good efficiency; relatively inexpensive and "clean."

Few appropriate sites; small production; mineral content of spent hot water can pollute.

No heat engine; no air, water, or thermal pollution; relatively inexpensive.

Large array of big windmills might affect weather and be eyesores; hazardous to migratory birds; winds not always strong.

Solar energy: Active solar heating: rooftop solar panels absorb the Sun's rays, which heat water in tubes for space heating and hot water supply.

61 No heat engine needed; no air or thermal pollution; unlimited fuel supply. No heat engine needed; no air or thermal pollution; relatively inexpensive.

Space limitations; may require back-up; relatively expensive; less effective when cloudy.

Fossil-fuel steam plants: burn coal, oil, or natural gas to boil water, producing high-pressure steam that turns a turbine of a generator (Figs. 20–3b, 20–16); uses heat engine.

Nuclear energy: Fission: nuclei of uranium or plutonium atoms split ("fission") with release of energy (Chapter 42) that heats steam; uses heat engine.

Fusion: energy released when isotopes of hydrogen (or other small nuclei) combine or "fuse" (Chapter 42). Hydroelectric: falling water turns turbines at the base of a dam.

Passive solar heating: architectural devices—windows along southern exposure, sunshade over windows to keep Sun's rays out in summer.

Disadvantages

Almost none, but other methods needed too.

Summer sun shield

Window

Summer sun

Winter sun

Energy-absorbing floor

Solar cells (photovoltaic cells): convert sunlight directly into electricity without use of heat engine.

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CHAPTER 20

No heat engine; thermal, air, and water pollution very low; good efficiency (7 30% and improving).

Second Law of Thermodynamics

Expensive; chemical pollution at manufacture; large land area needed as Sun's energy not concentrated.

Air pollution—by which we mean the chemicals released in the burning of fossil fuels in cars, power plants, and industrial furnaces—gives rise to smog and other problems. Another much talked about issue is the buildup of CO2 in the Earth's atmosphere due to the burning of fossil fuels. CO2 absorbs some of the infrared radiation that the Earth naturally emits (Section 19–10) and thus can contribute to global warming. Limiting the burning of fossil fuels can help these problems. Thermal pollution, however, is unavoidable. Engineers can try to design and build engines that are more efficient, but they cannot surpass the Carnot efficiency and must live with TL being at best the ambient temperature of water or air. The second law of thermodynamics tells us the limit imposed by nature. What we can do, in the light of the second law of thermodynamics, is use less energy and conserve our fuel resources.

L SO VIN OBL E PR

Thermodynamics 1. Define the system you are dealing with; distinguish the system under study from its surroundings. 2. Be careful of signs associated with work and heat. In the first law, work done by the system is positive; work done on the system is negative. Heat added to the system is positive, but heat removed from it is negative. With heat engines, we usually consider the heat intake, the heat exhausted, and the work done as positive. 3. Watch the units used for work and heat; work is most often expressed in joules, and heat can be in calories, kilocalories, or joules. Be consistent: choose only one unit for use throughout a given problem.

4. Temperatures must generally be expressed in kelvins; temperature differences may be expressed in C° or K. 5. Efficiency (or coefficient of performance) is a ratio of two energy transfers: useful output divided by required input. Efficiency (but not coefficient of performance) is always less than 1 in value, and hence is often stated as a percentage. 6. The entropy of a system increases when heat is added to the system, and decreases when heat is removed. If heat is transferred from system A to system B, the change in entropy of A is negative and the change in entropy of B is positive.

Summary A heat engine is a device for changing thermal energy, by means of heat flow, into useful work. The efficiency of a heat engine is defined as the ratio of the work W done by the engine to the heat input QH . Because of conservation of energy, the work output equals QH - QL , where QL is the heat exhausted to the environment; hence the efficiency e =

QL W . = 1 QH QH

(20–1)

Carnot's (idealized) engine consists of two isothermal and two adiabatic processes in a reversible cycle. For a Carnot engine, or any reversible engine operating between two temperatures, TH and TL (in kelvins), the efficiency is eideal = 1 -

TL . TH

(20–3)

Irreversible (real) engines always have an efficiency less than this. The operation of refrigerators and air conditioners is the reverse of that of a heat engine: work is done to extract heat from a cool region and exhaust it to a region at a higher temperature. The coefficient of performance (COP) for either is COP =

QL , W

refrigerator or d (20–4a) air conditioner

where W is the work needed to remove heat QL from the area with the low temperature.

A heat pump does work W to bring heat QL from the cold outside and deliver heat QH to warm the interior. The coefficient of performance of a heat pump is QH . [heat pump] (20–5) COP = W The second law of thermodynamics can be stated in several equivalent ways: (a) heat flows spontaneously from a hot object to a cold one, but not the reverse; (b) there can be no 100% efficient heat engine—that is, one that can change a given amount of heat completely into work; (c) natural processes tend to move toward a state of greater disorder or greater entropy. Statement (c) is the most general statement of the second law of thermodynamics, and can be restated as: the total entropy, S, of any system plus that of its environment increases as a result of any natural process: (20–13) ¢S 7 0. Entropy, which is a state variable, is a quantitative measure of the disorder of a system. The change in entropy of a system during a reversible process is given by ¢S = dQT. The second law of thermodynamics tells us in which direction processes tend to proceed; hence entropy is called "time's arrow." As time goes on, energy is degraded to less useful forms— that is, it is less available to do useful work. [*All heat engines give rise to thermal pollution because they exhaust heat to the environment.]

Summary

551

Questions 1. Can mechanical energy ever be transformed completely into heat or internal energy? Can the reverse happen? In each case, if your answer is no, explain why not; if yes, give one or two examples. 2. Can you warm a kitchen in winter by leaving the oven door open? Can you cool the kitchen on a hot summer day by leaving the refrigerator door open? Explain. 3. Would a definition of heat engine efficiency as e = WQL be useful? Explain. 4. What plays the role of high-temperature and low-temperature areas in (a) an internal combustion engine, and (b) a steam engine? Are they, strictly speaking, heat reservoirs? 5. Which will give the greater improvement in the efficiency of a Carnot engine, a 10 C° increase in the high-temperature reservoir, or a 10 C° decrease in the low-temperature reservoir? Explain. 6. The oceans contain a tremendous amount of thermal (internal) energy. Why, in general, is it not possible to put this energy to useful work? 7. Discuss the factors that keep real engines from reaching Carnot efficiency. 8. The expansion valve in a refrigeration system, Fig. 20–10, is crucial for cooling the fluid. Explain how the cooling occurs. 9. Describe a process in nature that is nearly reversible. 10. (a) Describe how heat could be added to a system reversibly. (b) Could you use a stove burner to add heat to a system reversibly? Explain. 11. Suppose a gas expands to twice its original volume (a) adiabatically, (b) isothermally. Which process would result in a greater change in entropy? Explain. 12. Give three examples, other than those mentioned in this Chapter, of naturally occurring processes in which order goes to disorder. Discuss the observability of the reverse process. 13. Which do you think has the greater entropy, 1 kg of solid iron or 1 kg of liquid iron? Why?

14. (a) What happens if you remove the lid of a bottle containing chlorine gas? (b) Does the reverse process ever happen? Why or why not? (c) Can you think of two other examples of irreversibility? 15. You are asked to test a machine that the inventor calls an "in-room air conditioner": a big box, standing in the middle of the room, with a cable that plugs into a power outlet. When the machine is switched on, you feel a stream of cold air coming out of it. How do you know that this machine cannot cool the room? 16. Think up several processes (other than those already mentioned) that would obey the first law of thermodynamics, but, if they actually occurred, would violate the second law. 17. Suppose a lot of papers are strewn all over the floor; then you stack them neatly. Does this violate the second law of thermodynamics? Explain. 18. The first law of thermodynamics is sometimes whimsically stated as, "You can't get something for nothing," and the second law as, "You can't even break even." Explain how these statements could be equivalent to the formal statements. 19. Powdered milk is very slowly (quasistatically) added to water while being stirred. Is this a reversible process? Explain. 20. Two identical systems are taken from state a to state b by two different irreversible processes. Will the change in entropy for the system be the same for each process? For the environment? Answer carefully and completely. 21. It can be said that the total change in entropy during a process is a measure of the irreversibility of the process. Discuss why this is valid, starting with the fact that ¢Stotal = ¢Ssystem + ¢Senvironment = 0 for a reversible process. 22. Use arguments, other than the principle of entropy increase, to show that for an adiabatic process, ¢S = 0 if it is done reversibly and ¢S 7 0 if done irreversibly.

Problems 20–2 Heat Engines 1. (I) A heat engine exhausts 7800 J of heat while performing 2600 J of useful work. What is the efficiency of this engine? 2. (I) A certain power plant puts out 580 MW of electric power. Estimate the heat discharged per second, assuming that the plant has an efficiency of 35%.

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4. (II) A four-cylinder gasoline engine has an efficiency of 0.22 and delivers 180 J of work per cycle per cylinder. The engine fires at 25 cycles per second. (a) Determine the work done per second. (b) What is the total heat input per second from the gasoline? (c) If the energy content of gasoline is 130 MJ per gallon, how long does one gallon last?

3. (II) A typical compact car experiences a total drag force at 55 mih of about 350 N. If this car gets 35 miles per gallon of gasoline at this speed, and a liter of gasoline (1 gal = 3.8 L) releases about 3.2 * 107 J when burned, what is the car's efficiency?

5. (II) The burning of gasoline in a car releases about 3.0 * 104 kcalgal. If a car averages 38 kmgal when driving 95 kmh, which requires 25 hp, what is the efficiency of the engine under those conditions? 6. (II) Figure 20–17 is a PV diagram for a reversible heat engine in which 1.0 mol of argon, a nearly ideal monatomic gas, is initially at STP (point a). Points b and c are on an isotherm at b T = 423 K. Process ab is at constant volume, process ac at constant pressure. (a) Is the path of the P cycle carried out clockwise or 3 counterclockwise? (b) What is K the efficiency of this engine? a c 0

FIGURE 20–17 Problem 6.

7. (III) The operation of a diesel engine can be idealized by the cycle shown in Fig. 20–18. Air is drawn into the cylinder during the intake stroke (not part of the idealized cycle). The air is compressed adiabatically, path ab. At point b diesel fuel is injected into the cylinder which immediately burns since the temperature is very high. Combustion is slow, and during the first part of the power stroke, the gas expands at (nearly) constant pressure, path bc. After burning, the rest of the power stroke is adiabatic, path cd. Path da corresponds to the exhaust stroke. (a) Show that, for a quasistatic reversible engine undergoing this cycle using an ideal gas, the ideal efficiency is e = 1 -

AVaVc B

–g

- AVaVb B

–g

g C AVaVc B –1 - AVaVb B –1 D

where VaVb is the "compression ratio", VaVc is the "expansion ratio", and g is defined by Eq. 19–14. (b) If VaVb = 16 and VaVc = 4.5, QH calculate the efficiency assuming b c the gas is diatomic (like N2 and P O2) and ideal. d

a 0

FIGURE 20–18 Problem 7.

20–3 Carnot Engine 8. (I) What is the maximum efficiency of a heat engine whose operating temperatures are 550°C and 365°C? 9. (I) It is not necessary that a heat engine's hot environment be hotter than ambient temperature. Liquid nitrogen (77 K) is about as cheap as bottled water. What would be the efficiency of an engine that made use of heat transferred from air at room temperature (293 K) to the liquid nitrogen "fuel" (Fig. 20–19)?

FIGURE 20–19 Problem 9. 10. (II) A heat engine exhausts its heat at 340°C and has a Carnot efficiency of 38%. What exhaust temperature would enable it to achieve a Carnot efficiency of 45%? 11. (II) (a) Show that the work done by a Carnot engine is equal to the area enclosed by the Carnot cycle on a PV diagram, Fig. 20–7. (See Section 19–7.) (b) Generalize this to any reversible cycle. 12. (II) A Carnot engine's operating temperatures are 210°C and 45°C. The engine's power output is 950 W. Calculate the rate of heat output. 13. (II) A nuclear power plant operates at 65% of its maximum theoretical (Carnot) efficiency between temperatures of 660°C and 330°C. If the plant produces electric energy at the rate of 1.2 GW, how much exhaust heat is discharged per hour?

14. (II) A Carnot engine performs work at the rate of 520 kW with an input of 950 kcal of heat per second. If the temperature of the heat source is 560°C, at what temperature is the waste heat exhausted? 15. (II) Assume that a 65 kg hiker needs 4.0 * 103 kcal of energy to supply a day's worth of metabolism. Estimate the maximum height the person can climb in one day, using only this amount of energy. As a rough prediction, treat the person as an isolated heat engine, operating between the internal temperature of 37°C (98.6°F) and the ambient air temperature of 20°C. 16. (II) A particular car does work at the rate of about 7.0 kJs when traveling at a steady 20.0 ms along a level road. This is the work done against friction. The car can travel 17 km on 1 L of gasoline at this speed (about 40 migal). What is the minimum value for TH if TL is 25°C? The energy available from 1 L of gas is 3.2 * 107 J. 17. (II) A heat engine utilizes a heat source at 580°C and has a Carnot efficiency of 32%. To increase the efficiency to 38%, what must be the temperature of the heat source? 18. (II) The working substance of a certain Carnot engine is 1.0 mol of an ideal monatomic gas. During the isothermal expansion portion of this engine's cycle, the volume of the gas doubles, while during the adiabatic expansion the volume increases by a factor of 5.7. The work output of the engine is 920 J in each cycle. Compute the temperatures of the two reservoirs between which this engine operates. 19. (III) A Carnot cycle, shown in Fig. 20–7, has the following conditions: Va = 7.5 L, Vb = 15.0 L, TH = 470°C, and TL = 260°C. The gas used in the cycle is 0.50 mol of a diatomic gas, g = 1.4. Calculate (a) the pressures at a and b; (b) the volumes at c and d. (c) What is the work done along process ab? (d) What is the heat lost along process cd? (e) Calculate the net work done for the whole cycle. (f) What is the efficiency of the cycle, using the definition e = WQH? Show that this is the same as given by Eq. 20–3. 20. (III) One mole of monatomic gas undergoes a Carnot cycle with TH = 350°C and TL = 210°C. The initial pressure is 8.8 atm. During the isothermal expansion, the volume doubles. (a) Find the values of the pressure and volume at the points a, b, c, and d (see Fig. 20–7). (b) Determine Q, W, and ¢Eint for each segment of the cycle. (c) Calculate the efficiency of the cycle using Eqs. 20–1 and 20–3. * 21. (III) In an engine that approximates the Otto cycle (Fig. 20–8), gasoline vapor must be ignited at the end of the cylinder's adiabatic compression by the spark from a spark plug. The ignition temperature of 87-octane gasoline vapor is about 430°C and, assuming that the working gas is diatomic and enters the cylinder at 25°C, determine the maximum compression ratio of the engine.

20–4 Refrigerators, Air Conditioners, Heat Pumps 22. (I) If an ideal refrigerator keeps its contents at 3.0°C when the house temperature is 22°C, what is its coefficient of performance? 23. (I) The low temperature of a freezer cooling coil is –15°C and the discharge temperature is 33°C. What is the maximum theoretical coefficient of performance? 24. (II) An ideal (Carnot) engine has an efficiency of 38%. If it were possible to run it backward as a heat pump, what would be its coefficient of performance?

Problems

553

25. (II) An ideal heat pump is used to maintain the inside temperature of a house at Tin = 22°C when the outside temperature is Tout . Assume that when it is operating, the heat pump does work at a rate of 1500 W. Also assume that the house loses heat via conduction through its walls and other surfaces at a rate given by (650 WC°) (Tin - Tout). (a) For what outside temperature would the heat pump have to operate at all times in order to maintain the house at an inside temperature of 22°C? (b) If the outside temperature is 8°C, what percentage of the time does the heat pump have to operate in order to maintain the house at an inside temperature of 22°C? 26. (II) A restaurant refrigerator has a coefficient of performance of 5.0. If the temperature in the kitchen outside the refrigerator is 32°C, what is the lowest temperature that could be obtained inside the refrigerator if it were ideal? 27. (II) A heat pump is used to keep a house warm at 22°C. How much work is required of the pump to deliver 3100 J of heat into the house if the outdoor temperature is (a) 0°C, (b) – 15°C? Assume ideal (Carnot) behavior. 28. (II) (a) Given that the coefficient of performance of a refrigerator is defined (Eq. 20–4a) as COP =

QL , W

show that for an ideal (Carnot) refrigerator, COPideal =

TL . TH - TL

(b) Write the COP in terms of the efficiency e of the reversible heat engine obtained by running the refrigerator backward. (c) What is the coefficient of performance for an ideal refrigerator that maintains a freezer compartment at –18°C when the condenser's temperature is 24°C? 29. (II) A "Carnot" refrigerator (reverse of a Carnot engine) absorbs heat from the freezer compartment at a temperature of – 17°C and exhausts it into the room at 25°C. (a) How much work must be done by the refrigerator to change 0.40 kg of water at 25°C into ice at – 17°C? (b) If the compressor output is 180 W, what minimum time is needed to take 0.40 kg of 25°C water and freeze it at 0°C? 30. (II) A central heat pump operating as an air conditioner draws 33,000 Btu per hour from a building and operates between the temperatures of 24°C and 38°C. (a) If its coefficient of performance is 0.20 that of a Carnot air conditioner, what is the effective coefficient of performance? (b) What is the power (kW) required of the compressor motor? (c) What is the power in terms of hp? 31. (II) What volume of water at 0°C can a freezer make into ice cubes in 1.0 h, if the coefficient of performance of the cooling unit is 7.0 and the power input is 1.2 kilowatt?

34. (I) What is the change in entropy of 1.00 m3 of water at 0°C when it is frozen to ice at 0°C? 35. (II) If 1.00 m3 of water at 0°C is frozen and cooled to –10°C by being in contact with a great deal of ice at –10°C, estimate the total change in entropy of the process. 36. (II) If 0.45 kg of water at 100°C is changed by a reversible process to steam at 100°C, determine the change in entropy of (a) the water, (b) the surroundings, and (c) the universe as a whole. (d) How would your answers differ if the process were irreversible? 37. (II) An aluminum rod conducts 9.50 cals from a heat source maintained at 225°C to a large body of water at 22°C. Calculate the rate at which entropy increases in this process. 38. (II) A 2.8-kg piece of aluminum at 43.0°C is placed in 1.0 kg of water in a Styrofoam container at room temperature (20°C). Estimate the net change in entropy of the system. 39. (II) An ideal gas expands isothermally (T = 410 K) from a volume of 2.50 L and a pressure of 7.5 atm to a pressure of 1.0 atm. What is the entropy change for this process? 40. (II) When 2.0 kg of water at 12.0°C is mixed with 3.0 kg of water at 38.0°C in a well-insulated container, what is the change in entropy of the system? (a) Make an estimate; (b) use the integral ¢S = dQT. 41. (II) (a) An ice cube of mass m at 0°C is placed in a large 20°C room. Heat flows (from the room to the ice cube) such that the ice cube melts and the liquid water warms to 20°C. The room is so large that its temperature remains nearly 20°C at all times. Calculate the change in entropy for the (water + room) system due to this process. Will this process occur naturally? (b) A mass m of liquid water at 20°C is placed in a large 20°C room. Heat flows (from the water to the room) such that the liquid water cools to 0°C and then freezes into a 0°C ice cube. The room is so large that its temperature remains 20°C at all times. Calculate the change in entropy for the (water + room) system due to this process. Will this process occur naturally? 42. (II) The temperature of 2.0 mol of an ideal diatomic gas goes from 25°C to 55°C at a constant volume. What is the change in entropy? Use ¢S = dQT. 43. (II) Calculate the change in entropy of 1.00 kg of water when it is heated from 0°C to 75°C. (a) Make an estimate; (b) use the integral ¢S = dQT. (c) Does the entropy of the surroundings change? If so, by how much? 44. (II) An ideal gas of n moles undergoes the reversible process ab shown in the PV diagram of Fig. 20–20. The temperature T of the gas is the same at points a and b. Determine the change in entropy of the gas due to this process.

P a

20–5 and 20–6 Entropy 32. (I) What is the change in entropy of 250 g of steam at 100°C when it is condensed to water at 100°C? 33. (I) A 7.5-kg box having an initial speed of 4.0 ms slides along a rough table and comes to rest. Estimate the total change in entropy of the universe. Assume all objects are at room temperature (293 K).

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FIGURE 20–20 Problem 44.

45. (II) Two samples of an ideal gas are initially at the same temperature and pressure. They are each compressed reversibly from a volume V to volume V/2, one isothermally, the other adiabatically. (a) In which sample is the final pressure greater? (b) Determine the change in entropy of the gas for each process by integration. (c) What is the entropy change of the environment for each process? 46. (II) A 150-g insulated aluminum cup at 15°C is filled with 215 g of water at 100°C. Determine (a) the final temperature of the mixture, and (b) the total change in entropy as a result of the mixing process (use ¢S = dQT). 47. (II) (a) Why would you expect the total entropy change in a Carnot cycle to be zero? (b) Do a calculation to show that it is zero. 48. (II) 1.00 mole of nitrogen AN2 B gas and 1.00 mole of argon (Ar) gas are in separate, equal-sized, insulated containers at the same temperature. The containers are then connected and the gases (assumed ideal) allowed to mix. What is the change in entropy (a) of the system and (b) of the environment? (c) Repeat part (a) but assume one container is twice as large as the other. 49. (II) Thermodynamic processes are sometimes represented on TS (temperature–entropy) diagrams, rather than PV diagrams. Determine the slope of a constant-volume process on a TS diagram when a system with n moles of an ideal gas with constant-volume molar specific heat CV is at temperature T. 50. (III) The specific heat per mole of potassium at low temperatures is given by CV = aT + bT3, where a = 2.08 mJmolK2 and b = 2.57 mJmolK4. Determine (by integration) the entropy change of 0.15 mol of potassium when its temperature is lowered from 3.0 K to 1.0 K. 51. (III) Consider an ideal gas of n moles with molar specific heats CV and CP . (a) Starting with the first law, show that when the temperature and volume of this gas are changed by a reversible process, its change in entropy is given by dS = nCV

dT dV . + nR T V

(b) Show that the expression in part (a) can be written as dS = nCV

* 20–9 Statistical Interpretation of Entropy * 54. (I) Use Eq. 20–14 to determine the entropy of each of the five macrostates listed in the Table on page 546. * 55. (II) Suppose that you repeatedly shake six coins in your hand and drop them on the floor. Construct a table showing the number of microstates that correspond to each macrostate. What is the probability of obtaining (a) three heads and three tails and (b) six heads? * 56. (II) Calculate the relative probabilities, when you throw two dice, of obtaining (a) a 7, (b) an 11, (c) a 4. * 57. (II) (a) Suppose you have four coins, all with tails up. You now rearrange them so two heads and two tails are up. What was the change in entropy of the coins? (b) Suppose your system is the 100 coins of Table 20–1; what is the change in entropy of the coins if they are mixed randomly initially, 50 heads and 50 tails, and you arrange them so all 100 are heads? (c) Compare these entropy changes to ordinary thermodynamic entropy changes, such as Examples 20–6, 20–7, and 20–8. 58. (III) Consider an isolated gas-like system consisting of a box * that contains N = 10 distinguishable atoms, each moving at the same speed v. The number of unique ways that these atoms can be arranged so that NL atoms are within the left-hand half of the box and NR atoms are within the right-hand half of the box is given by N!NL!NR!, where, for example, the factorial 4! = 4 32 1 (the only exception is that 0! = 1). Define each unique arrangement of atoms within the box to be a microstate of this system. Now imagine the following two possible macrostates: state A where all of the atoms are within the left-hand half of the box and none are within the right-hand half; and state B where the distribution is uniform (that is, there is the same number in each half). See Fig. 20–21. (a) Assume the system is initially in state A and, at a later time, is found to be in state B. Determine the system's change in entropy. Can this process occur naturally? (b) Assume the system is initially in state B and, at a later time, is found to be in state A. Determine the system's change in entropy. Can this process occur naturally? State A (NL 10, NR 0) v

dP dV . + nCP P V

(c) Using the expression from part (b), show that if dS = 0 for the reversible process (that is, the process is adiabatic), g then PV = constant, where g = CPCV .

State B (NL 5, NR 5)

20–8 Unavailability of Energy 52. (III) A general theorem states that the amount of energy that becomes unavailable to do useful work in any process is equal to TL ¢S, where TL is the lowest temperature available and ¢S is the total change in entropy during the process. Show that this is valid in the specific cases of (a) a falling rock that comes to rest when it hits the ground; (b) the free adiabatic expansion of an ideal gas; and (c) the conduction of heat, Q, from a high-temperature ATH B reservoir to a low-temperature ATL B reservoir. [Hint: In part (c) compare to a Carnot engine.] 53. (III) Determine the work available in a 3.5-kg block of copper at 490 K if the surroundings are at 290 K. Use results of Problem 52.

FIGURE 20–21 Problem 58. * 20–11 Energy Resources * 59. (II) Energy may be stored for use during peak demand by pumping water to a high reservoir when demand is low and then releasing it to drive turbines when needed. Suppose water is pumped to a lake 135 m above the turbines at a rate of 1.35 * 105 kgs for 10.0 h at night. (a) How much energy (kWh) is needed to do this each night? (b) If all this energy is released during a 14-h day, at 75% efficiency, what is the average power output?

Problems

555

* 60. (II) If solar cells (Fig. 20–22) can produce about 40 W of electricity per square meter of surface area when directly facing the Sun, how large an area is required to supply the needs of a house that requires 22 kWhday? Would this fit on the roof of an average house? (Assume the Sun shines about 9 hday.)

* 61. (II) Water is stored in an artificial lake created by a dam (Fig. 20–23). The water depth is 38 m at the dam, and a steady flow rate of 32 m3s is maintained through hydroelectric turbines installed near the base of the dam. How much electrical power can be produced?

FIGURE 20–23 Problem 61.

FIGURE 20–22 Problem 60.

General Problems 62. It has been suggested that a heat engine could be developed that made use of the temperature difference between water at the surface of the ocean and water several hundred meters deep. In the tropics, the temperatures may be 27°C and 4°C, respectively. (a) What is the maximum efficiency such an engine could have? (b) Why might such an engine be feasible in spite of the low efficiency? (c) Can you imagine any adverse environmental effects that might occur? 63. A heat engine takes a diatomic gas around the cycle shown in Fig. 20–24. (a) Using the ideal gas law, determine how many moles of gas are in this engine. (b) Determine the temperature at point c. (c) Calculate the heat input into the gas during the constant volume process from points b to c. (d) Calculate the work done by the gas during the isothermal process from points a to b. (e) Calculate the work done by the gas during the adiabatic process from points c to a. (f) Determine the engine's efficiency. (g) What is the maximum efficiency possible for an engine working between Ta and Tc?

P c

the

1.0 atm

rma

0.0050 m3

ti c

a Ta 300 K

0.010 m3

FIGURE 20–24 Problem 63.

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64. A 126.5-g insulated aluminum cup at 18.00°C is filled with 132.5 g of water at 46.25°C. After a few minutes, equilibrium is reached. Determine (a) the final temperature, and (b) the total change in entropy. 65. (a) At a steam power plant, steam engines work in pairs, the heat output of the first one being the approximate heat input of the second. The operating temperatures of the first are 710°C and 430°C, and of the second 415°C and 270°C. If the heat of combustion of coal is 2.8 * 107 Jkg, at what rate must coal be burned if the plant is to put out 950 MW of power? Assume the efficiency of the engines is 65% of the ideal (Carnot) efficiency. (b) Water is used to cool the power plant. If the water temperature is allowed to increase by no more than 5.5 C°, estimate how much water must pass through the plant per hour. 66. Refrigeration units can be rated in "tons." A 1-ton air conditioning system can remove sufficient energy to freeze 1 British ton (2000 pounds = 909 kg) of 0°C water into 0°C ice in one 24-h day. If, on a 35°C day, the interior of a house is maintained at 22°C by the continuous operation of a 5-ton air conditioning system, how much does this cooling cost the homeowner per hour? Assume the work done by the refrigeration unit is powered by electricity that costs $0.10 per kWh and that the unit's coefficient of performance is 15% that of an ideal refrigerator. 1 kWh = 3.60 * 106 J. 67. A 35% efficient power plant puts out 920 MW of electrical power. Cooling towers are used to take away the exhaust heat. (a) If the air temperature (15°C) is allowed to rise 7.0 C°, estimate what volume of air Akm3 B is heated per day. Will the local climate be heated significantly? (b) If the heated air were to form a layer 150 m thick, estimate how large an area it would cover for 24 h of operation. Assume the air has density 1.2 kgm3 and that its specific heat is about 1.0 kJkgC° at constant pressure.

68. (a) What is the coefficient of performance of an ideal heat pump that extracts heat from 11°C air outside and deposits heat inside your house at 24°C? (b) If this heat pump operates on 1400 W of electrical power, what is the maximum heat it can deliver into your house each hour? 69. The operation of a certain heat engine takes an ideal monatomic gas through a cycle shown as the rectangle on the PV diagram of Fig. 20–25. (a) Determine the efficiency of this engine. Let QH and QL be the total heat input and total heat exhausted during one cycle of this engine. (b) Compare (as a ratio) the efficiency of this engine to that of P a Carnot engine operating between TH and TL , where d c 3Po TH and TL are the highest and lowest temperatures achieved.

2Vo

FIGURE 20–25 Problem 69.

70. A car engine whose output power is 155 hp operates at about 15% efficiency. Assume the engine's water temperature of 95°C is its cold-temperature (exhaust) reservoir and 495°C is its thermal "intake" temperature (the temperature of the exploding gas–air mixture). (a) What is the ratio of its efficiency relative to its maximum possible (Carnot) efficiency? (b) Estimate how much power (in watts) goes into moving the car, and how much heat, in joules and in kcal, is exhausted to the air in 1.0 h. 71. Suppose a power plant delivers energy at 850 MW using steam turbines. The steam goes into the turbines superheated at 625 K and deposits its unused heat in river water at 285 K. Assume that the turbine operates as an ideal Carnot engine. (a) If the river's flow rate is 34 m3s, estimate the average temperature increase of the river water immediately downstream from the power plant. (b) What is the entropy increase per kilogram of the downstream river water in Jkg K? 72. 1.00 mole of an ideal monatomic gas at STP first undergoes an isothermal expansion so that the volume at b is 2.5 times the volume at a (Fig. 20–26). Next, heat is extracted at a constant volume so that the pressure drops. The gas is then compressed adiabatically back to the original state. (a) Calculate the pressures at b and c. (b) Determine the temperature at c. (c) Determine the work done, heat input or extracted, and the change in entropy for each process. (d) What is the efficiency of this cycle?

74. Metabolizing 1.0 kg of fat results in about 3.7 * 107 J of internal energy in the body. (a) In one day, how much fat does the body burn to maintain the body temperature of a person staying in bed and metabolizing at an average rate of 95 W? (b) How long would it take to burn 1.0-kg of fat this way assuming there is no food intake? 75. A cooling unit for a new freezer has an inner surface area of 6.0 m2, and is bounded by walls 12 cm thick with a thermal conductivity of 0.050 WmK. The inside must be kept at –10°C in a room that is at 20°C. The motor for the cooling unit must run no more than 15% of the time. What is the minimum power requirement of the cooling motor? 76. An ideal air conditioner keeps the temperature inside a room at 21°C when the outside temperature is 32°C. If 3.3 kW of power enters a room through the windows in the form of direct radiation from the Sun, how much electrical power would be saved if the windows were shaded so only 500 W came through them? 77. The Stirling cycle, shown in Fig. 20–27, is useful to describe external combustion engines as well as solar-power systems. Find the efficiency of the cycle in terms of the parameters shown, assuming a monatomic gas as the working substance. The P processes ab and cd are a isothermal whereas bc and da are at constant TH volume. How does it b compare to the Carnot efficiency? d

FIGURE 20–27 Problem 77.

iab

ati

78. A gas turbine operates under the Brayton cycle, which is depicted in the PV diagram of Fig. 20–28. In process ab the air–fuel mixture undergoes an adiabatic compression. This is followed, in process bc, with an isobaric (constant pressure) heating, by combustion. Process cd is an adiabatic expansion with expulsion of the products to the atmosphere. The return step, da, takes place at constant pressure. If the working gas behaves like an ideal gas, show that the efficiency of the Brayton cycle is c 1-g b Pb g e = 1 - ¢ ≤ . Pa P tic d a iab

73. Two 1100-kg cars are traveling 75 kmh in opposite directions when they collide and are brought to rest. Estimate the change in entropy of the universe as a result of this collision. Assume T = 15°C.

ia b

a ti

FIGURE 20–28 Problem 78.

c 22.4 L

56.0 L

FIGURE 20–26 Problem 72.

79. Thermodynamic processes can be represented not only on PV and PT diagrams; another useful one is a TS (temperature–entropy) diagram. (a) Draw a TS diagram for a Carnot cycle. (b) What does the area within the curve represent?

General Problems

557

80. An aluminum can, with negligible heat capacity, is filled with 450 g of water at 0°C and then is brought into thermal contact with a similar can filled with 450 g of water at 50°C. Find the change in entropy of the system if no heat is allowed to exchange with the surroundings. Use ¢S = dQT. 81. A dehumidifier is essentially a "refrigerator with an open door." The humid air is pulled in by a fan and guided to a cold coil, whose temperature is less than the dew point, and some of the air's water condenses. After this water is extracted, the air is warmed back to its original temperature and sent into the room. In a well-designed dehumidifier, the heat is exchanged between the incoming and outgoing air. Thus the heat that is removed by the refrigerator coil mostly comes from the condensation of water vapor to liquid. Estimate how much water is removed in 1.0 h by an ideal dehumidifier, if the temperature of the room is 25°C, the water condenses at 8°C, and the dehumidifier does work at the rate of 650 W of electrical power.

* 82. A bowl contains a large number of red, orange, and green jelly beans. You are to make a line of three jelly beans. (a) Construct a table showing the number of microstates that correspond to each macrostate. Then determine the probability of (b) all 3 beans red, and (c) 2 greens, 1 orange.

* Numerical/Computer * 83. (II) At low temperature the specific heat of diamond varies with absolute temperature T according to the Debye 3 equation CV = 1.88 * 103 ATTD B J mol–1 K–1 where the Debye temperature for diamond is TD = 2230 K. Use a spreadsheet and numerical integration to determine the entropy change of 1.00 mol of diamond when it is heated at constant volume from 4 K to 40 K. Your result should agree within 2% of the result obtained by integrating the expression for dS. [Hint: dS = nCV dTT, where n is the number of moles.]

Answers to Exercises A: No. Efficiency makes no sense for a single process. It is defined (Eqs. 20–1) only for cyclic processes that return to the initial state. B: (d). C: (c).

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D: 1220 JK; –1220 JK. (Note that the total entropy change, ¢Sice + ¢Sres , is zero.) E: (e).

This comb has acquired a static electric charge, either from passing through hair, or being rubbed by a cloth or paper towel. The electrical charge on the comb induces a polarization (separation of charge) in scraps of paper, and thus attracts them. Our introduction to electricity in this Chapter covers conductors and insulators, and Coulomb's law which relates the force between two point charges as a function of their distance apart. We also introduce the powerful concept of electric field.

CHAPTER-OPENING QUESTION—Guess now! Two identical tiny spheres have the same electric charge. If the electric charge on each of them is doubled, and their separation is also doubled, the force each exerts on the other will be (a) half. (b) double. (c) four times larger. (d) one-quarter as large. (e) unchanged. he word "electricity" may evoke an image of complex modern technology: lights, motors, electronics, and computers. But the electric force plays an even deeper role in our lives. According to atomic theory, electric forces between atoms and molecules hold them together to form liquids and solids, and electric forces are also involved in the metabolic processes that occur within our bodies. Many of the forces we have dealt with so far, such as elastic forces, the normal force, and friction and other contact forces (pushes and pulls), are now considered to result from electric forces acting at the atomic level. Gravity, on the other hand, is a separate force.†

T †

As we discussed in Section 6–7, physicists in the twentieth century came to recognize four different fundamental forces in nature: (1) gravitational force, (2) electromagnetic force (we will see later that electric and magnetic forces are intimately related), (3) strong nuclear force, and (4) weak nuclear force. The last two forces operate at the level of the nucleus of an atom. Recent theory has combined the electromagnetic and weak nuclear forces so they are now considered to have a common origin known as the electroweak force. We will discuss these forces in later Chapters.

C H

Electric Charge and Electric Field

P T E R

CONTENTS 21–1

Static Electricity; Electric Charge and Its Conservation

21–2

Electric Charge in the Atom

21–3

Insulators and Conductors

21–4

Induced Charge; the Electroscope

21–5

Coulomb's Law

21–6

The Electric Field

21–7

Electric Field Calculations for Continuous Charge Distributions

21–8

Field Lines

21–9

Electric Fields and Conductors

21–10 Motion of a Charged Particle in an Electric Field 21–11 Electric Dipoles

*21–12 Electric Forces in Molecular Biology; DNA

*21–13 Photocopy Machines and Computer Printers Use Electrostatics

559

The earliest studies on electricity date back to the ancients, but only in the past two centuries has electricity been studied in detail. We will discuss the development of ideas about electricity, including practical devices, as well as its relation to magnetism, in the next eleven Chapters.

21–1 (a)

(b) FIGURE 21–1 (a) Rub a plastic ruler and (b) bring it close to some tiny pieces of paper. FIGURE 21–2 Like charges repel one another; unlike charges attract. (Note color coding: positive and negative charged objects are often colored pink and blue-green, respectively, when we want to emphasize them. We use these colors especially for point charges, but not often for real objects.)

(a) Two charged plastic rulers repel

(b) Two charged glass rods repel

LAW OF CONSERVATION OF ELECTRIC CHARGE

560

CHAPTER 21

Static Electricity; Electric Charge and Its Conservation

The word electricity comes from the Greek word elektron, which means "amber." Amber is petrified tree resin, and the ancients knew that if you rub a piece of amber with a cloth, the amber attracts small pieces of leaves or dust. A piece of hard rubber, a glass rod, or a plastic ruler rubbed with a cloth will also display this "amber effect," or static electricity as we call it today. You can readily pick up small pieces of paper with a plastic comb or ruler that you have just vigorously rubbed with even a paper towel. See the photo on the previous page and Fig. 21–1. You have probably experienced static electricity when combing your hair or when taking a synthetic blouse or shirt from a clothes dryer. And you may have felt a shock when you touched a metal doorknob after sliding across a car seat or walking across a nylon carpet. In each case, an object becomes "charged" as a result of rubbing, and is said to possess a net electric charge. Is all electric charge the same, or is there more than one type? In fact, there are two types of electric charge, as the following simple experiments show. A plastic ruler suspended by a thread is vigorously rubbed with a cloth to charge it. When a second plastic ruler, which has been charged in the same way, is brought close to the first, it is found that one ruler repels the other. This is shown in Fig. 21–2a. Similarly, if a rubbed glass rod is brought close to a second charged glass rod, again a repulsive force is seen to act, Fig. 21–2b. However, if the charged glass rod is brought close to the charged plastic ruler, it is found that they attract each other, Fig. 21–2c. The charge on the glass must therefore be different from that on the plastic. Indeed, it is found experimentally that all charged objects fall into one of two categories. Either they are attracted to the plastic and repelled by the glass; or they are repelled by the plastic and attracted to the glass. Thus there seem to be two, and only two, types of electric charge. Each type of charge repels the same type but attracts the opposite type. That is: unlike charges attract; like charges repel. The two types of electric charge were referred to as positive and negative by the American statesman, philosopher, and scientist Benjamin Franklin (1706–1790). The choice of which name went with which type of charge was arbitrary. Franklin's choice set the charge on the rubbed glass rod to be positive charge, so the charge on a rubbed plastic ruler (or amber) is called negative charge. We still follow this convention today. Franklin argued that whenever a certain amount of charge is produced on one object, an equal amount of the opposite type of charge is produced on another object. The positive and negative are to be treated algebraically, so during any process, the net change in the amount of charge produced is zero. For example, when a plastic ruler is rubbed with a paper towel, the plastic acquires a negative charge and the towel acquires an equal amount of positive charge. The charges are separated, but the sum of the two is zero. This is an example of a law that is now well established: the law of conservation of electric charge, which states that the net amount of electric charge produced in any process is zero; or, said another way, no net electric charge can be created or destroyed. If one object (or a region of space) acquires a positive charge, then an equal amount of negative charge will be found in neighboring areas or objects. No violations have ever been found, and this conservation law is as firmly established as those for energy and momentum.

Electric Charge and Electric Field

21–2

Electric Charge in the Atom

Only within the past century has it become clear that an understanding of electricity originates inside the atom itself. In later Chapters we will discuss atomic structure and the ideas that led to our present view of the atom in more detail. But it will help our understanding of electricity if we discuss it briefly now. A simplified model of an atom shows it as having a tiny but heavy, positively charged nucleus surrounded by one or more negatively charged electrons (Fig. 21–3). The nucleus contains protons, which are positively charged, and neutrons, which have no net electric charge. All protons and all electrons have exactly the same magnitude of electric charge; but their signs are opposite. Hence neutral atoms, having no net charge, contain equal numbers of protons and electrons. Sometimes an atom may lose one or more of its electrons, or may gain extra electrons, in which case it will have a net positive or negative charge and is called an ion. In solid materials the nuclei tend to remain close to fixed positions, whereas some of the electrons may move quite freely. When an object is neutral, it contains equal amounts of positive and negative charge. The charging of a solid object by rubbing can be explained by the transfer of electrons from one object to the other. When a plastic ruler becomes negatively charged by rubbing with a paper towel, the transfer of electrons from the towel to the plastic leaves the towel with a positive charge equal in magnitude to the negative charge acquired by the plastic. In liquids and gases, nuclei or ions can move as well as electrons. Normally when objects are charged by rubbing, they hold their charge only for a limited time and eventually return to the neutral state. Where does the charge go? Usually the charge "leaks off" onto water molecules in the air. This is because water molecules are polar—that is, even though they are neutral, their charge is not distributed uniformly, Fig. 21–4. Thus the extra electrons on, say, a charged plastic ruler can "leak off" into the air because they are attracted to the positive end of water molecules. A positively charged object, on the other hand, can be neutralized by transfer of loosely held electrons from water molecules in the air. On dry days, static electricity is much more noticeable since the air contains fewer water molecules to allow leakage. On humid or rainy days, it is difficult to make any object hold a net charge for long.

21–3

Insulators and Conductors

Suppose we have two metal spheres, one highly charged and the other electrically neutral (Fig. 21–5a). If we now place a metal object, such as a nail, so that it touches both spheres (Fig. 21–5b), the previously uncharged sphere quickly becomes charged. If, instead, we had connected the two spheres by a wooden rod or a piece of rubber (Fig. 21–5c), the uncharged ball would not become noticeably charged. Materials like the iron nail are said to be conductors of electricity, whereas wood and rubber are nonconductors or insulators. Metals are generally good conductors, whereas most other materials are insulators (although even insulators conduct electricity very slightly). Nearly all natural materials fall into one or the other of these two very distinct categories. However, a few materials (notably silicon and germanium) fall into an intermediate category known as semiconductors. From the atomic point of view, the electrons in an insulating material are bound very tightly to the nuclei. In a good conductor, on the other hand, some of the electrons are bound very loosely and can move about freely within the material (although they cannot leave the object easily) and are often referred to as free electrons or conduction electrons. When a positively charged object is brought close to or touches a conductor, the free electrons in the conductor are attracted by this positively charged object and move quickly toward it. On the other hand, the free electrons move swiftly away from a negatively charged object that is brought close to the conductor. In a semiconductor, there are many fewer free electrons, and in an insulator, almost none. SECTION 21–3

FIGURE 21–3 atom.

Simple model of the

O H

FIGURE 21–4 Diagram of a water molecule. Because it has opposite charges on different ends, it is called a "polar" molecule. FIGURE 21–5 (a) A charged metal sphere and a neutral metal sphere. (b) The two spheres connected by a conductor (a metal nail), which conducts charge from one sphere to the other. (c) The original two spheres connected by an insulator (wood); almost no charge is conducted. Charged Neutral +++ +++

(a) Metal + + +

+ + +

(b) Wood +++ +++

(c)

Insulators and Conductors

561

21–4 (a) Neutral metal rod Charged

metal object

e− passage

(b) Metal rod acquires charge by contact FIGURE 21–6 A neutral metal rod in (a) will acquire a positive charge if placed in contact (b) with a positively charged metal object. (Electrons move as shown by the orange arrow.) This is called charging by conduction.

Induced Charge; the Electroscope

Suppose a positively charged metal object is brought close to an uncharged metal object. If the two touch, the free electrons in the neutral one are attracted to the positively charged object and some will pass over to it, Fig. 21–6. Since the second object, originally neutral, is now missing some of its negative electrons, it will have a net positive charge. This process is called "charging by conduction," or "by contact," and the two objects end up with the same sign of charge. Now suppose a positively charged object is brought close to a neutral metal rod, but does not touch it. Although the free electrons of the metal rod do not leave the rod, they still move within the metal toward the external positive charge, leaving a positive charge at the opposite end of the rod (Fig. 21–7). A charge is said to have been induced at the two ends of the metal rod. No net charge has been created in the rod: charges have merely been separated. The net charge on the metal rod is still zero. However, if the metal is separated into two pieces, we would have two charged objects: one charged positively and one charged negatively.

(a) (a)

Neutral metal rod

–– –

–+ –+ –+ –+ –+ Nonconductor

FIGURE 21–9 A charged object brought near an insulator causes a charge separation within the insulator's molecules. FIGURE 21–10 Electroscope. Insulator

Gold leaves

562

CHAPTER 21

Metal

Glass

++ +

Metal rod still neutral, but with a separation of charge

FIGURE 21–7 Charging by induction.

+++++ +

(b)

++++ (b)

Another way to induce a net charge on a metal object is to first connect it with a conducting wire to the ground (or a conducting pipe leading into the ground) as shown in Fig. 21–8a (the symbol means connected to "ground"). The object is then said to be "grounded" or "earthed." The Earth, because it is so large and can conduct, easily accepts or gives up electrons; hence it acts like a reservoir for charge. If a charged object—say negative this time—is brought up close to the metal object, free electrons in the metal are repelled and many of them move down the wire into the Earth, Fig. 21–8b. This leaves the metal positively charged. If the wire is now cut, the metal object will have a positive induced charge on it (Fig. 21–8c). If the wire were cut after the negative object was moved away, the electrons would all have moved back into the metal object and it would be neutral. Charge separation can also be done in nonconductors. If you bring a positively charged object close to a neutral nonconductor as shown in Fig. 21–9, almost no electrons can move about freely within the nonconductor. But they can move slightly within their own atoms and molecules. Each oval in Fig. 21–9 represents a molecule (not to scale); the negatively charged electrons, attracted to the external positive charge, tend to move in its direction within their molecules. Because the negative charges in the nonconductor are nearer to the external positive charge, the nonconductor as a whole is attracted to the external positive charge (see the Chapter-Opening Photo, page 559). An electroscope is a device that can be used for detecting charge. As shown in Fig. 21–10, inside of a case are two movable metal leaves, often made of gold, connected to a metal knob on the outside. (Sometimes only one leaf is movable.)

Electric Charge and Electric Field

If a positively charged object is brought close to the knob, a separation of charge is induced: electrons are attracted up into the knob, leaving the leaves positively charged, Fig. 21–11a. The two leaves repel each other as shown, because they are both positively charged. If, instead, the knob is charged by conduction, the whole apparatus acquires a net charge as shown in Fig. 21–11b. In either case, the greater the amount of charge, the greater the separation of the leaves. Note that you cannot tell the sign of the charge in this way, since negative charge will cause the leaves to separate just as much as an equal amount of positive charge; in either case, the two leaves repel each other. An electroscope can, however, be used to determine the sign of the charge if it is first charged by conduction, say, negatively, as in Fig. 21–12a. Now if a negative object is brought close, as in Fig. 21–12b, more electrons are induced to move down into the leaves and they separate further. If a positive charge is brought close instead, the electrons are induced to flow upward, leaving the leaves less negative and their separation is reduced, Fig. 21–12c. The electroscope was used in the early studies of electricity. The same principle, aided by some electronics, is used in much more sensitive modern electrometers.

21–5

Coulomb's Law

We have seen that an electric charge exerts a force of attraction or repulsion on other electric charges. What factors affect the magnitude of this force? To find an answer, the French physicist Charles Coulomb (1736–1806) investigated electric forces in the 1780s using a torsion balance (Fig. 21–13) much like that used by Cavendish for his studies of the gravitational force (Chapter 6). Precise instruments for the measurement of electric charge were not available in Coulomb's time. Nonetheless, Coulomb was able to prepare small spheres with different magnitudes of charge in which the ratio of the charges was known.† Although he had some difficulty with induced charges, Coulomb was able to argue that the force one tiny charged object exerts on a second tiny charged object is directly proportional to the charge on each of them. That is, if the charge on either one of the objects is doubled, the force is doubled; and if the charge on both of the objects is doubled, the force increases to four times the original value. This was the case when the distance between the two charges remained the same. If the distance between them was allowed to increase, he found that the force decreased with the square of the distance between them. That is, if the distance was doubled, the force fell to one-fourth of its original value. Thus, Coulomb concluded, the force one small charged object exerts on a second one is proportional to the product of the magnitude of the charge on one, Q1 , times the magnitude of the charge on the other, Q2 , and inversely proportional to the square of the distance r between them (Fig. 21–14). As an equation, we can write Coulomb's law as F = k

Q1 Q2 , r2

[magnitudes] (21–1)

where k is a proportionality constant.‡ †

Coulomb reasoned that if a charged conducting sphere is placed in contact with an identical uncharged sphere, the charge on the first would be shared equally by the two of them because of symmetry. He thus had a way to produce charges equal to 12 , 14 , and so on, of the original charge. ‡ The validity of Coulomb's law today rests on precision measurements that are much more sophisticated than Coulomb's original experiment. The exponent 2 in Coulomb's law has been shown to be accurate to 1 part in 1016 [that is, 26A1 * 10 –16 B ].

(a)

(b)

FIGURE 21–11 Electroscope charged (a) by induction, (b) by conduction. FIGURE 21–12 A previously charged electroscope can be used to determine the sign of a charged object. (a)

(b)

(c)

FIGURE 21–13 (below) Coulomb used a torsion balance to investigate how the electric force varies as a function of the magnitude of the charges and of the distance between them. When an external charged sphere is placed close to the charged one on the suspended bar, the bar rotates slightly. The suspending fiber resists the twisting motion, and the angle of twist is proportional to the electric force. Fiber

Rod

COULOMB'S LAW FIGURE 21–14 Coulomb's law, Eq. 21–1, gives the force between two point charges, Q1 and Q2 , a distance r apart.

SECTION 21–5

Coulomb's Law

563

F12 = force on 1 due to 2 B

F12 B

F12

F21 = force on 2 due to 1

(a)

F21

(b) B

F 21

F12 (c)

As we just saw, Coulomb's law,

FIGURE 21–15 The direction of the static electric force one point charge exerts on another is always along the line joining the two charges, and depends on whether the charges have the same sign as in (a) and (b), or opposite signs (c).

F = k

Q1 Q2 , r2

[magnitudes] (21–1)

gives the magnitude of the electric force that either charge exerts on the other. The direction of the electric force is always along the line joining the two charges. If the two charges have the same sign, the force on either charge is directed away from the other (they repel each other). If the two charges have opposite signs, the force on one is directed toward the other (they attract). See Fig. 21–15. Notice that the force one charge exerts on the second is equal but opposite to that exerted by the second on the first, in accord with Newton's third law. The SI unit of charge is the coulomb (C).† The precise definition of the coulomb today is in terms of electric current and magnetic field, and will be discussed later (Section 28–3). In SI units, the constant k in Coulomb's law has the value k = 8.99 * 109 N m2C 2 or, when we only need two significant figures, k L 9.0 * 109 N m2C 2. Thus, 1 C is that amount of charge which, if placed on each of two point objects that are 1.0 m apart, will result in each object exerting a force of A9.0 * 109 N m2C 2 B(1.0 C)(1.0 C)(1.0 m)2 = 9.0 * 109 N on the other. This would be an enormous force, equal to the weight of almost a million tons. We rarely encounter charges as large as a coulomb. Charges produced by rubbing ordinary objects (such as a comb or plastic ruler) are typically around a microcoulomb A1 mC = 10–6 CB or less. Objects that carry a positive charge have a deficit of electrons, whereas negatively charged objects have an excess of electrons. The charge on one electron has been determined to have a magnitude of about 1.602 * 10–19 C, and is negative. This is the smallest charge found in nature,‡ and because it is fundamental, it is given the symbol e and is often referred to as the elementary charge: e = 1.602 * 10 –19 C. Note that e is defined as a positive number, so the charge on the electron is –e. (The charge on a proton, on the other hand, is ±e.) Since an object cannot gain or lose a fraction of an electron, the net charge on any object must be an integral multiple of this charge. Electric charge is thus said to be quantized (existing only in discrete amounts: 1e, 2e, 3e, etc.). Because e is so small, however, we normally do not notice this discreteness in macroscopic charges (1 mC requires about 1013 electrons), which thus seem continuous. Coulomb's law looks a lot like the law of universal gravitation, F = Gm1 m2r2, which expresses the gravitational force a mass m1 exerts on a mass m2 (Eq. 6–1). Both are inverse square laws AF r 1r 2 B. Both also have a proportionality to a property of each object—mass for gravity, electric charge for electricity. And both act over a distance (that is, there is no need for contact). A major difference between the two laws is that gravity is always an attractive force, whereas the electric force can be either attractive or repulsive. Electric charge comes in two types, positive and negative; gravitational mass is only positive. †

In the once common cgs system of units, k is set equal to 1, and the unit of electric charge is called the electrostatic unit (esu) or the statcoulomb. One esu is defined as that charge, on each of two point objects 1 cm apart, that gives rise to a force of 1 dyne. ‡ According to the standard model of elementary particle physics, subnuclear particles called quarks have a smaller charge than that on the electron, equal to 13 e or 23 e. Quarks have not been detected directly as isolated objects, and theory indicates that free quarks may not be detectable.

564

CHAPTER 21

Electric Charge and Electric Field

The constant k in Eq. 21–1 is often written in terms of another constant, 0 , called the permittivity of free space. It is related to k by k = 14p 0 . Coulomb's law can then be written F =

1 Q1 Q2 , 4p 0 r 2

0 =

1 = 8.85 * 10–12 C 2N m2. 4pk

(21–2)

COULOMB'S LAW (in terms of 0)

where

Equation 21–2 looks more complicated than Eq. 21–1, but other fundamental equations we haven't seen yet are simpler in terms of 0 rather than k. It doesn't matter which form we use since Eqs. 21–1 and 21–2 are equivalent. (The latest precise values of e and 0 are given inside the front cover.) [Our convention for units, such as C 2N m2 for 0 , means m2 is in the denominator. That is, C 2Nm2 does not mean AC 2NB m2 = C 2 m2N.] Equations 21–1 and 21–2 apply to objects whose size is much smaller than the distance between them. Ideally, it is precise for point charges (spatial size negligible compared to other distances). For finite-sized objects, it is not always clear what value to use for r, particularly since the charge may not be distributed uniformly on the objects. If the two objects are spheres and the charge is known to be distributed uniformly on each, then r is the distance between their centers. Coulomb's law describes the force between two charges when they are at rest. Additional forces come into play when charges are in motion, and will be discussed in later Chapters. In this Chapter we discuss only charges at rest, the study of which is called electrostatics, and Coulomb's law gives the electrostatic force. When calculating with Coulomb's law, we usually ignore the signs of the charges and determine the direction of a force separately based on whether the force is attractive or repulsive.

P R O B L E M

S O LV I N G

Use magnitudes in Coulomb's law; find force direction from signs of charges

EXERCISE A Return to the Chapter-Opening Question, page 559, and answer it again now. Try to explain why you may have answered differently the first time.

CONCEPTUAL EXAMPLE 21–1 Which charge exerts the greater force? Two positive point charges, Q1 = 50 mC and Q2 = 1 mC, are separated by a distance l, Fig. 21–16. Which is larger in magnitude, the force that Q1 exerts on Q2 , or the force that Q2 exerts on Q1?

Q1 = 50 mC

FIGURE 21–16

Q2 = 1 mC Example 21–1.

RESPONSE From Coulomb's law, the force on Q1 exerted by Q2 is F12 = k

Q1 Q2 . l2

The force on Q2 exerted by Q1 is F21 = k

Q2 Q1 l2

which is the same magnitude. The equation is symmetric with respect to the two charges, so F21 = F12 . NOTE Newton's third law also tells us that these two forces must have equal magnitude. EXERCISE B What is the magnitude of F12 (and F21) in Example 21–1 if l = 30 cm?

Keep in mind that Eq. 21–2 (or 21–1) gives the force on a charge due to only one other charge. If several (or many) charges are present, the net force on any one of them will be the vector sum of the forces due to each of the others. This principle of superposition is based on experiment, and tells us that electric force vectors add like any other vector. For continuous distributions of charge, the sum becomes an integral. SECTION 21–5

Coulomb's Law

565

EXAMPLE 21–2 Three charges in a line. Three charged particles are arranged in a line, as shown in Fig. 21–17. Calculate the net electrostatic force on particle 3 (the –4.0 mC on the right) due to the other two charges. B

APPROACH The net force on particle 3 is the vector sum of the force F31 exerted B B B B on 3 by particle 1 and the force F32 exerted on 3 by particle 2: F = F31 + F32 . SOLUTION The magnitudes of these two forces are obtained using Coulomb's law, Eq. 21–1: F31 = k

0.30 m Q1 = −8.0 m C

Q2 = Q3 = +3.0 m C −4.0 m C

= B

F32

F31

Q3 (b)

FIGURE 21–17

(0.50 m)2

F32 = k

(a)

A9.0 * 109 N m2C 2 BA4.0 * 10–6 CBA8.0 * 10 –6 CB

= 1.2 N,

where r31 = 0.50 m is the distance from Q3 to Q1 . Similarly,

0.20 m

Q3 Q1 r 231

Example 21–2.

C A U T I O N

Each charge exerts its own force. No charge blocks the effect of the others

Q3 Q2 r 232

A9.0 * 109 N m2C 2 BA4.0 * 10–6 CBA3.0 * 10 –6 CB (0.20 m)2

= 2.7 N.

Since we were calculating the magnitudes of the forces, we omitted the signs of the charges. But we must be aware of them to get the direction of each force. Let the line joining the particles be the x axis, and we take it positive to the right. B B Then, because F31 is repulsive and F32 is attractive, the directions of the forces are as shown in Fig. 21–17b: F31 points in the positive x direction and F32 points in the negative x direction. The net force on particle 3 is then F = –F32 + F31 = –2.7 N + 1.2 N = –1.5 N. The magnitude of the net force is 1.5 N, and it points to the left. NOTE Charge Q1 acts on charge Q3 just as if Q2 were not there (this is the principle of superposition). That is, the charge in the middle, Q2 , in no way blocks the effect of charge Q1 acting on Q3 . Naturally, Q2 exerts its own force on Q3 . EXERCISE C Determine the magnitude and direction of the net force on Q1 in Fig. 21–17a.

EXAMPLE 21–3 Electric force using vector components. Calculate the net electrostatic force on charge Q3 shown in Fig. 21–18a due to the charges Q1 and Q2 . APPROACH We use Coulomb's law to find the magnitudes of the individual forces. The direction of each force will be along the line connecting Q3 to B B Q1 or Q2 . The forces F31 and F32 have the directions shown in Fig. 21–18a, since Q1 exerts an attractive force on Q3 , and Q2 exerts a repulsive force. The forces B B F31 and F32 are not along the same line, so to find the resultant force on Q3 we B B resolve F31 and F32 into x and y components and perform the vector addition. B B SOLUTION The magnitudes of F31 and F32 are (ignoring signs of the charges since we know the directions) F31 = k

A9.0 * 109 Nm2C2 BA6.5 * 10–5 CBA8.6 * 10–5 CB Q3 Q1 = = 140 N, r231 (0.60 m)2

F32 = k

A9.0 * 109 Nm2C2 BA6.5 * 10–5 CBA5.0 * 10–5 CB Q3 Q2 = = 330 N. r232 (0.30 m)2

We resolve F31 into its components along the x and y axes, as shown in Fig. 21–18a: F31x = F31 cos 30° = (140 N) cos 30° = 120 N, F31y = –F31 sin 30° = –(140 N) sin 30° = –70 N.

566

CHAPTER 21

Electric Charge and Electric Field

F 32

F31x

Q3 = +65 mC

30°

F31y

F 32

30 cm

F 31

90°

Q2 = +50 mC

52 cm (a)

30°

Q1 = −86 m C

FIGURE 21–18 Determining the forces for Example 21–3. (a) The directions of the individual forces are as shown because B F32 is repulsive (the force on Q3 is in the direction away from Q2 because Q3 and Q2 B are both positive) whereas F31 is attractive B (Q3 and Q1 have opposite signs), so F31 points B B toward Q1 . (b) Adding F32 to F31 to obtain the B net force F.

F 31 (b) B

The force F32 has only a y component. So the net force F on Q3 has components Fx = F31x = 120 N, Fy = F32 + F31y = 330 N - 70 N = 260 N. The magnitude of the net force is F = 3F 2x + F 2y = 3(120 N)2 + (260 N)2 = 290 N; and it acts at an angle u (see Fig. 21–18b) given by Fy 260 N tan u = = = 2.2, Fx 120 N so u = tan–1(2.2) = 65°. B B B NOTE Because F31 and F32 are not along the same line, the magnitude of F3 is not equal to the sum (or difference as in Example 21–2) of the separate magnitudes.

y B

CONCEPTUAL EXAMPLE 21–4 Make the force on Q 3 zero. In Fig. 21–18, where could you place a fourth charge, Q4 = –50 mC , so that the net force on Q3 would be zero? RESPONSE By the principle of superposition, we need a force in exactly the opposite B direction to the resultant F due to Q2 and Q1 that we calculated in Example 21–3, Fig. 21–18b. Our force must have magnitude 290 N, and must point down and to the left of Q3 in Fig. 21–18b. So Q4 must be along this line. See Fig. 21–19. EXERCISE D (a) Consider two point charges of the same magnitude but opposite sign ( ± Q and – Q), which are fixed a distance d apart. Can you find a location where a third positive charge Q could be placed so that the net electric force on this third charge is zero? (b) What if the first two charges were both ±Q?

* Vector Form of Coulomb's Law

F = F31 + F32 u = 65°

115°

F 34

Q4 FIGURE 21–19 Example 21–4: B Q4 exerts force AF34 B that makes the net force on Q3 zero.

Coulomb's law can be written in vector form (as we did for Newton's law of universal gravitation in Chapter 6, Section 6–2) as B

F12 = k

Q1 Q2 rˆ 21 , r 221

where F12 is the vector force on charge Q1 due to Q2 and rˆ 21 is the unit vector pointing from Q2 toward Q1 . That is, r21 points from the "source" charge AQ2 B toward the charge on which we want to know the force AQ1 B. See Fig. 21–20. The charges Q1 and Q2 can be either positive or negative, and this will affect the direction of the electric force. If Q1 and Q2 have the same sign, the product Q1 Q2 7 0 and the force on Q1 points away from Q2 —that is, it is repulsive. If Q1 and Q2 have B opposite signs, Q1 Q2 6 0 and F12 points toward Q2 —that is, it is attractive. B

FIGURE 21–20 Determining the force on Q1 due to Q2 , showing the direction of the unit vector rˆ 21 . Q1 r21

r21

SECTION 21–5

Coulomb's Law

567

21–6 P Q

FIGURE 21–21 An electric field surrounds every charge. P is an arbitrary point.

FIGURE 21–22 Force exerted by charge ± Q on a small test charge, q, placed at points A, B, and C. B

FA A +Q

FIGURE 21–23 (a) Electric field at a given point in space. (b) Force on a positive charge at that point. (c) Force on a negative charge at that point. B

E (a) E B

F +q

(b) B

E –q

568

CHAPTER 21

(c)

(21–3) B

More precisely, E is defined as the limit of Fq as q is taken smaller and smaller, approaching zero. That is, q is so tiny that it exerts essentially no force on the other charges which created the field. From this definition (Eq. 21–3), we see that the electric field at any point in space is a vector whose direction is the direction of the force on a tiny positive test charge at that point, and whose magnitude is the force B per unit charge. Thus E has SI units of newtons per coulomb (NC). B B B The reason for defining E as Fq (with q S 0) is so that E does not depend B on the magnitude of the test charge q. This means that E describes only the effect of the charges creating the electric field at that point. The electric field at any point in space can be measured, based on the definition, Eq. 21–3. For simple situations involving one or several point charges, we can B calculate E. For example, the electric field at a distance r from a single point charge Q would have magnitude kqQr 2 F = q q Q E = k 2; r or, in terms of 0 as in Eq. 21–2 Ak = 14p 0 B: E =

[single point charge] (21–4a)

1 Q. [single point charge] (21–4b) 4p 0 r2 Notice that E is independent of the test charge q—that is, E depends only on the charge Q which produces the field, and not on the value of the test charge q. Equations 21–4 are referred to as the electric field form of Coulomb's law. B If we are given the electric field E at a given point in space, then we can B calculate the force F on any charge q placed at that point by writing (see Eq. 21–3): B B F = qE. (21–5) This is valid even if q is not small as long as q does not cause the charges creating B B B B E to move. If q is positive, F and E point in the same direction. If q is negative, F B and E point in opposite directions. See Fig. 21–23. E =

F. E = q B

Many common forces might be referred to as "contact forces," such as your hands pushing or pulling a cart, or a tennis racket hitting a tennis ball. In contrast, both the gravitational force and the electrical force act over a distance: there is a force between two objects even when the objects are not touching. The idea of a force acting at a distance was a difficult one for early thinkers. Newton himself felt uneasy with this idea when he published his law of universal gravitation. A helpful way to look at the situation uses the idea of the field, developed by the British scientist Michael Faraday (1791–1867). In the electrical case, according to Faraday, an electric field extends outward from every charge and permeates all of space (Fig. 21–21). If a second charge (call it Q2) is placed near the first charge, it feels a force exerted by the electric field that is there (say, at point P in Fig. 21–21). The electric field at point P is considered to interact directly with charge Q2 to produce the force on Q2 . We can in principle investigate the electric field surrounding a charge or group of charges by measuring the force on a small positive test charge at rest. By a test charge we mean a charge so small that the force it exerts does not significantly affect the charges that create the field. If a tiny positive test charge q is placed at various locations in the vicinity of a single positive charge Q as shown in Fig. 21–22 (points A, B, C), the force exerted on q is as shown. The force at B is less than at A because B's distance from Q is greater (Coulomb's law); and the force at C is smaller still. In each case, the force on q is directed radially away from Q. The electric field is defined in terms of the force on such a positive test charge. In particular, B B the electric field, E, at any point in space is defined as the force F exerted on a tiny positive test charge placed at that point divided by the magnitude of the test charge q: B

The Electric Field

EXAMPLE 21–5 Photocopy machine. A photocopy machine works by arranging positive charges (in the pattern to be copied) on the surface of a drum, then gently sprinkling negatively charged dry toner (ink) particles onto the drum. The toner particles temporarily stick to the pattern on the drum (Fig. 21–24) and are later transferred to paper and "melted" to produce the copy. Suppose each toner particle has a mass of 9.0 * 10–16 kg and carries an average of 20 extra electrons to provide an electric charge. Assuming that the electric force on a toner particle must exceed twice its weight in order to ensure sufficient attraction, compute the required electric field strength near the surface of the drum.

P H Y S I C S

A P P L I E D

Photocopier

APPROACH The electric force on a toner particle of charge q = 20e is F = qE, where E is the needed electric field. This force needs to be at least as great as twice the weight (mg) of the particle. SOLUTION The minimum value of electric field satisfies the relation

Surface of drum

qE = 2mg where q = 20e. Hence 2A9.0 * 10–16 kgBA9.8 ms2 B 2mg E = = = 5.5 * 103 NC. q 20A1.6 * 10 –19 CB

E Toner particles held to drum surface B by electric field E

EXAMPLE 21–6 Electric field of a single point charge. Calculate the magnitude and direction of the electric field at a point P which is 30 cm to the right of a point charge Q = –3.0 * 10–6 C. APPROACH The magnitude of the electric field due to a single point charge is given by Eq. 21–4. The direction is found using the sign of the charge Q. SOLUTION The magnitude of the electric field is: E = k

A9.0 * 109 N m2C 2 BA3.0 * 10 –6 CB Q = = 3.0 * 105 NC. r2 (0.30 m)2

The direction of the electric field is toward the charge Q, to the left as shown in Fig. 21–25a, since we defined the direction as that of the force on a positive test charge which here would be attractive. If Q had been positive, the electric field would have pointed away, as in Fig. 21–25b. NOTE There is no electric charge at point P. But there is an electric field there. The only real charge is Q.

FIGURE 21–24 Example 21–5.

FIGURE 21–25 Example 21–6. Electric field at point P (a) due to a negative charge Q, and (b) due to a positive charge Q, each 30 cm from P. 30 cm P Q = −3.0 × 10–6 C E = 3.0 × 105 N/C (a) Q = +3.0 × 10–6 C (b)

P E = 3.0 × 105 N/C

This Example illustrates a general result: The electric field E due to a positive B charge points away from the charge, whereas E due to a negative charge points toward that charge. EXERCISE E Four charges of equal magnitude, but possibly different sign, are placed on the corners of a square. What arrangement of charges will produce an electric field with the greatest magnitude at the center of the square? (a) All four positive charges; (b) all four negative charges; (c) three positive and one negative; (d) two positive and two negative; (e) three negative and one positive.

If the electric field at a given point in space is due to more than one charge, B B the individual fields (call them E1 , E 2 , etc.) due to each charge are added vectorially to get the total field at that point: B B B E = E1 + E2 + p .

The validity of this superposition principle for electric fields is fully confirmed by experiment. SECTION 21–6

The Electric Field

569

Q1 = –25 m C

Q2 = +50 m C

r1 = 2.0 cm

FIGURE 21–26 Example 21–7. In (b), we B B don't know the relative lengths of E1 and E 2 until we do the calculation.

r2 = 8.0 cm

(a)

E1 B

(b)

EXAMPLE 21–7 E at a point between two charges. Two point charges are separated by a distance of 10.0 cm. One has a charge of –25 mC and the other ±50 mC. (a) Determine the direction and magnitude of the electric field at a point P between the two charges that is 2.0 cm from the negative charge (Fig. 21–26a). (b) If an electron Amass = 9.11 * 10 –31 kgB is placed at rest at P and then released, what will be its initial acceleration (direction and magnitude)? APPROACH The electric field at P will be the vector sum of the fields created separately by Q1 and Q2 . The field due to the negative charge Q1 points toward Q1 , and the field due to the positive charge Q2 points away from Q2 . Thus both fields point to the left as shown in Fig. 21–26b and we can add the magnitudes of the two fields together algebraically, ignoring the signs of the charges. In (b) we use Newton's second law (F = ma) to determine the acceleration, where F = qE (Eq. 21–5). SOLUTION (a) Each field is due to a point charge as given by Eq. 21–4, E = kQr2. The total field is E = k

Q1 Q2 Q1 Q2 + k 2 = k¢ 2 + 2 ≤ 2 r1 r2 r1 r2

= A9.0 * 109 N m2C 2 B ¢ = 6.3 * 108 NC.

25 * 10 –6 C

A2.0 * 10–2 mB

50 * 10 –6 C

A8.0 * 10–2 mB

≤

(b) The electric field points to the left, so the electron will feel a force to the right since it is negatively charged. Therefore the acceleration a = Fm (Newton's second law) will be to the right. The force on a charge q in an electric field E is F = qE (Eq. 21–5). Hence the magnitude of the acceleration is a =

A1.60 * 10 –19 CBA6.3 * 108 NCB qE F = 1.1 * 1020 ms2. = = m m 9.11 * 10–31 kg B

NOTE By carefully considering the directions of each field (E1 and E 2) before doing any calculations, we made sure our calculation could be done simply and correctly. B

EXAMPLE 21–8 E above two point charges. Calculate the total electric field (a) at point A and (b) at point B in Fig. 21–27 due to both charges, Q1 and Q2 . APPROACH The calculation is much like that of Example 21–3, except now we are dealing with electric fields instead of force. The electric field at point A is B B the vector sum of the fields EA1 due to Q1 , and EA2 due to Q2 . We find the magnitude of the field produced by each point charge, then we add their components to find the total field at point A. We do the same for point B. SOLUTION (a) The magnitude of the electric field produced at point A by each of the charges Q1 and Q2 is given by E = kQr2, so P R O B L E M

S O LV I N G

Ignore signs of charges and determine direction physically, showing directions on diagram

EA1 =

A9.0 * 109 N m2C 2 BA50 * 10–6 CB

= 1.25 * 106 NC,

EA2 =

A9.0 * 109 N m2C 2 BA50 * 10–6 CB

= 5.0 * 106 NC.

(0.60 m)2 (0.30 m)2

The direction of EA1 points from A toward Q1 (negative charge), whereas EA2 points

570

CHAPTER 21

Electric Charge and Electric Field

EA2

A B

EA1

30° 60

FIGURE 21–27 Calculation of the electric field at points A and B for Example 21–8.

EB1

30 cm

EB2

26 cm

Q2 = +50 m C

Q1 = −50 mC B

from A away from Q2 , as shown; so the total electric field at A, EA , has components EAx = EA1 cos 30° = 1.1 * 106 NC, EAy = EA2 - EA1 sin 30° = 4.4 * 106 NC. B Thus the magnitude of EA is EA = 3(1.1)2 + (4.4)2 * 106 NC = 4.5 * 106 NC, and its direction is f given by tan f = EAyEAx = 4.41.1 = 4.0, so f = 76°. (b) Because B is equidistant from the two equal charges (40 cm by the Pythagorean theorem), the magnitudes of EB1 and EB2 are the same; that is, A9.0 * 109 N m2C 2 BA50 * 10 –6 CB kQ EB1 = EB2 = = r2 (0.40 m)2 6 = 2.8 * 10 NC. Also, because of the symmetry, the y components are equal and opposite, and so cancel out. Hence the total field EB is horizontal and equals EB1 cos u + EB2 cos u = 2EB1 cos u. From the diagram, cos u = 26 cm40 cm = 0.65. Then EB = 2EB1 cos u = 2A2.8 * 106 NCB(0.65) = 3.6 * 106 NC, B and the direction of E B is along the ±x direction. NOTE We could have done part (b) in the same way we did part (a). But symmetry allowed us to solve the problem with less effort.

P R O B L E M

S O LV I N G

Use symmetry to save work, when possible

L SO VIN OBL E PR

Electrostatics: Electric Forces and Electric Fields Solving electrostatics problems follows, to a large extent, the general problem-solving procedure discussed in Section 4–8. Whether you use electric field or electrostatic forces, the procedure is similar: 1. Draw a careful diagram—namely, a free-body diagram for each object, showing all the forces acting on that object, or showing the electric field at a point due to all significant charges present. Determine the direction of each force or electric field physically: like charges repel each other, unlike charges attract; fields point away from a ± charge, and toward

a – charge. Show and label each vector force or field on your diagram. 2. Apply Coulomb's law to calculate the magnitude of the force that each contributing charge exerts on a charged object, or the magnitude of the electric field each charge produces at a given point. Deal only with magnitudes of charges (leaving out minus signs), and obtain the magnitude of each force or electric field. 3. Add vectorially all the forces on an object, or the contributing fields at a point, to get the resultant. Use symmetry (say, in the geometry) whenever possible. 4. Check your answer. Is it reasonable? If a function of distance, does it give reasonable results in limiting cases? SECTION 21–6

The Electric Field

571

21–7

Electric Field Calculations for Continuous Charge Distributions

In many cases we can treat charge as being distributed continuously.† We can divide up a charge distribution into infinitesimal charges dQ, each of which will act as a tiny point charge. The contribution to the electric field at a distance r from each dQ is 1 dQ . dE = (21–6a) 4p 0 r2 B Then the electric field, E, at any point is obtained by summing over all the infinitesimal contributions, which is the integral B

E =

dE. B

(21–6b)

Note that dE is a vector (Eq. 21–6a gives its magnitude). [In situations where Eq. 21–6b is difficult to evaluate, other techniques (discussed in the next two Chapters) can often B be used instead to determine E. Numerical integration can also be used in many cases.] EXAMPLE 21–9 A ring of charge. A thin, ring-shaped object of radius a holds a total charge ±Q distributed uniformly around it. Determine the electric field at a point P on its axis, a distance x from the center. See Fig. 21–28. Let l be the charge per unit length (Cm). dl a

APPROACH AND SOLUTION We explicitly follow the steps of the Problem Solving Strategy on page 571. 1. Draw a careful diagram. The direction of the electric field due to one infinitesimal length dl of the charged ring is shown in Fig. 21–28. B 2. Apply Coulomb's law. The electric field, dE, due to this particular segment of the ring of length dl has magnitude 1 dQ . dE = 4p 0 r2 The whole ring has length (circumference) of 2pa, so the charge on a length dl is

y x

r x

P dEx dE cos u

dE⊥

FIGURE 21–28

Example 21–9.

dl ≤ = l dl 2pa where l = Q2pa is the charge per unit length. Now we write dE as 1 l dl . dE = 4p 0 r 2 B 3. Add vectorially and use symmetry: The vector dE has components dEx along the x axis and dE⊥ perpendicular to the x axis (Fig. 21–28). We are going to sum (integrate) around the entire ring. We note that an equal-length segment B diametrically opposite the dl shown will produce a dE whose component perpendicular to the x axis will just cancel the dE⊥ shown. This is true for all B segments of the ring, so by symmetry E will have zero y component, and so we need only sum the x components, dEx . The total field is then 1 dl E = Ex = dEx = dE cos u = l cos u. 4p 0 r2 dQ = Q ¢

P R O B L E M

S O LV I N G

Use symmetry when possible

Since cos u = xr, where r = Ax 2 + a B , we have

2 12

E = P R O B L E M

†

CHAPTER 21

2pa

dl 0

lx(2pa) Qx 1 1 . 3 = 3 2 2 2 4p 0 Ax + a B 2 4p 0 Ax + a 2 B 2

4. To check reasonableness, note that at great distances, x W a, this result reduces to E = Q(4p 0 x2). We would expect this result because at great distances the ring would appear to be a point charge (1r 2 dependence). Also note that our result gives E = 0 at x = 0, as we might expect because all components will cancel at the center of the circle.

S O LV I N G

Check result by noting that at a great distance the ring looks like a point charge

572

l x 3 2 A4p 0 B Ax + a 2 B 2

Because we believe there is a minimum charge (e), the treatment here is only for convenience; it is nonetheless useful and accurate since e is usually very much smaller than macroscopic charges.

Note in this Example three important problem-solving techniques that can be used elsewhere: (1) using symmetry to reduce the complexity of the problem; (2) expressing the charge dQ in terms of a charge density (here linear, l = Q2pa); and (3) checking the answer at the limit of large r, which serves as an indication (but not proof) of the correctness of the answer—if the result does not check at large r, your result has to be wrong.

P R O B L E M

S O LV I N G

Use symmetry, charge density, and values at r = 0 and q

CONCEPTUAL EXAMPLE 21–10 Charge at the center of a ring. Imagine a small positive charge placed at the center of a nonconducting ring carrying a uniformly distributed negative charge. Is the positive charge in equilibrium if it is displaced slightly from the center along the axis of the ring, and if so is it stable? What if the small charge is negative? Neglect gravity, as it is much smaller than the electrostatic forces. RESPONSE The positive charge is in equilibrium because there is no net force on it, by symmetry. If the positive charge moves away from the center of the ring along the axis in either direction, the net force will be back towards the center of the ring and so the charge is in stable equilibrium. A negative charge at the center of the ring would feel no net force, but is in unstable equilibrium because if it moved along the ring's axis, the net force would be away from the ring and the charge would be pushed farther away. EXAMPLE 21–11 Long line of charge. Determine the magnitude of the electric field at any point P a distance x from the midpoint 0 of a very long line (a wire, say) of uniformly distributed positive charge, Fig. 21–29. Assume x is much smaller than the length of the wire, and let l be the charge per unit length (Cm). APPROACH We set up a coordinate system so the wire is on the y axis with B origin 0 as shown. A segment of wire dy has charge dQ = l dy. The field dE at point P due to this length dy of wire (at y) has magnitude l dy 1 dQ 1 , dE = = 2 2 4p 0 r 4p 0 Ax + y2 B

where r = Ax 2 + y 2 B 2 as shown in Fig. 21–29. The vector dE has components dEx and dEy as shown where dEx = dE cos u and dEy = dE sin u. B SOLUTION Because 0 is at the midpoint of the wire, the y component of E will be zero since there will be equal contributions to Ey = dEy from above and below point 0: 1

Ey =

r = (x2 + y2) 2 y 0

dEy

dEx B

dE sin u

FIGURE 21–29 Example 21–11.

= 0.

Thus we have E = Ex =

dE cos u

l 4p 0

cos u dy 2

+ y2

The integration here is over y, along the wire, with x treated as constant. We must now write u as a function of y, or y as a function of u. We do the latter: since y = x tan u, then dy = x ducos2 u. Furthermore, because cos u = x 2x2 + y2, then 1Ax 2 + y 2 B = cos2 ux2 and our integrand above is Acos uB Ax ducos2 uBAcos2 ux2 B = cos u dux. Hence E =

l 1 4p 0 x

cos u du =

–p2

p2 l 1 l, (sin u) 2 = 4p 0 x 2p 0 x –p2

where we have assumed the wire is extremely long in both directions (y S &q) which corresponds to the limits u = &p2. Thus the field near a long straight wire of uniform charge decreases inversely as the first power of the distance from the wire. NOTE This result, obtained for an infinite wire, is a good approximation for a wire of finite length as long as x is small compared to the distance of P from the ends of the wire. SECTION 21–7

Electric Field Calculations for Continuous Charge Distributions

573

EXAMPLE 21–12 Uniformly charged disk. Charge is distributed uniformly over a thin circular disk of radius R. The charge per unit area ACm2 B is s. Calculate the electric field at a point P on the axis of the disk, a distance z above its center, Fig. 21–30.

E P z R

r dQ

FIGURE 21–30 Example 21–12; a uniformly charged flat disk of radius R.

APPROACH We can think of the disk as a set of concentric rings. We can then apply the result of Example 21–9 to each of these rings, and then sum over all the rings. SOLUTION For the ring of radius r shown in Fig. 21–30, the electric field has magnitude (see result of Example 21–9) z dQ 1 dE = 3 4p 0 Az2 + r 2 B 2 where we have written dE (instead of E) for this thin ring of total charge dQ. The ring has area (dr)(2pr) and charge per unit area s = dQ(2pr dr). We solve this for dQ ( s 2pr dr) and insert it in the equation above for dE: dE =

1 zs2pr dr zsr dr 3 = 3 . 4p 0 Az2 + r 2 B 2 2 0 Az2 + r2 B 2

Now we sum over all the rings, starting at r = 0 out to the largest with r = R: E =

zs 2 0

R zs 1 – 1 2 0 c Az2 + r 2 B 2 d 0

s z 1 1 . c 2 2 0 Az + R 2 B 2 d

r dr

Az2 + r B

2 3 2

This gives the magnitude of E at any point z along the axis of the disk. The B direction of each dE due to each ring is along the z axis (as in Example 21–9), B B and therefore the direction of E is along z. If Q (and s) are positive, E points B away from the disk; if Q (and s) are negative, E points toward the disk. If the radius of the disk in Example 21–12 is much greater than the distance of our point P from the disk (i.e., z V R) then we can obtain a very useful result: the second term in the solution above becomes very small, so E =

s . 2 0

[infinite plane] (21–7)

This result is valid for any point above (or below) an infinite plane of any shape holding a uniform charge density s. It is also valid for points close to a finite plane, as long as the point is close to the plane compared to the distance to the edges of the plane. Thus the field near a large uniformly charged plane is uniform, and directed outward if the plane is positively charged. It is interesting to compare here the distance dependence of the electric field due to a point charge AE ' 1r2 B, due to a very long uniform line of charge (E ' 1r), and due to a very large uniform plane of charge (E does not depend on r). EXAMPLE 21–13 Two parallel plates. Determine the electric field between two large parallel plates or sheets, which are very thin and are separated by a distance d which is small compared to their height and width. One plate carries a uniform surface charge density s and the other carries a uniform surface charge density –s, as shown in Fig. 21–31 (the plates extend upward and downward beyond the part shown). APPROACH From Eq. 21–7, each plate sets up an electric field of magnitude s2 0 . The field due to the positive plate points away from that plate whereas the field due to the negative plate points toward that plate.

574

CHAPTER 21

Electric Charge and Electric Field

E+ B

E = E+ + E − =

B E− 0

E+ B

E− B E = E+ + E − s s = 22e = e 0 0 B

B E − B E

E+ B

= E+ + E− = 0

FIGURE 21–31 Example 21–13. (Only the center portion of these large plates is shown: their dimensions are large compared to their separation d.)

SOLUTION In the region between the plates, the fields add together as shown: E = E± + E– =

s s s. + = 0 2 0 2 0

The field is uniform, since the plates are very large compared to their separation, so this result is valid for any point, whether near one or the other of the plates, or midway between them as long as the point is far from the ends. Outside the plates, the fields cancel, E = E± + E– =

s s = 0, 2 0 2 0

as shown in Fig. 21–31. These results are valid ideally for infinitely large plates; they are a good approximation for finite plates if the separation is much less than the dimensions of the plate and for points not too close to the edge. NOTE: These useful and extraordinary results illustrate the principle of superposition and its great power.

21–8

Field Lines

Since the electric field is a vector, it is sometimes referred to as a vector field. We could indicate the electric field with arrows at various points in a given B B B situation, such as at A, B, and C in Fig. 21–32. The directions of EA , E B , and E C are the same as that of the forces shown earlier in Fig. 21–22, but the B magnitudes (arrow lengths) are different since we divide F in Fig. 21–22 by q to B B B B get E . However, the relative lengths of E A , EB , and EC are the same as for the forces since we divide by the same q each time. To indicate the electric field in such a way at many points, however, would result in many arrows, which might appear complicated or confusing. To avoid this, we use another technique, that of field lines. To visualize the electric field, we draw a series of lines to indicate the direction of the electric field at various points in space. These electric field lines (sometimes called lines of force) are drawn so that they indicate the direction of the force due to the given field on a positive test charge. The lines of force due to a single isolated positive charge are shown in Fig. 21–33a, and for a single isolated negative charge in Fig. 21–33b. In part (a) the lines point radially outward from the charge, and in part (b) they point radially inward toward the charge because that is the direction the force would be on a positive test charge in each case (as in Fig. 21–25). Only a few representative lines are shown. We could just as well draw lines in between those shown since the electric field exists there as well. We can draw the lines so that the number of lines starting on a positive charge, or ending on a negative charge, is proportional to the magnitude of the charge. Notice that nearer the charge, where the electric field is greater AF r 1r2 B, the lines are closer together. This is a general property of electric field lines: the closer together the lines are, the stronger the electric field in that region. In fact, field lines can be drawn B so that the number of lines crossing unit area perpendicular to E is proportional to the magnitude of the electric field.

EA A +Q

B B

EB FIGURE 21–32 Electric field vector shown at three points, due to a single point charge Q. (Compare to Fig. 21–22.)

FIGURE 21–33 Electric field lines (a) near a single positive point charge, (b) near a single negative point charge.

–

(a)

(b)

SECTION 21–8

Field Lines

575

(a)

Figure 21–34a shows the electric field lines due to two equal charges of opposite sign, a combination known as an electric dipole. The electric field lines are curved in this case and are directed from the positive charge to the negative charge. The direction of the electric field at any point is tangent to the field line at that point as B shown by the vector arrow E at point P. To satisfy yourself that this is the correct pattern for the electric field lines, you can make a few calculations such as those done in Example 21–8 for just this case (see Fig. 21–27). Figure 21–34b shows the electric field lines for two equal positive charges, and Fig. 21–34c for unequal charges, –Q and ±2Q . Note that twice as many lines leave ±2Q , as enter –Q (number of lines is proportional to magnitude of Q). Finally, in Fig. 21–34d, we see the field lines between two parallel plates carrying equal but opposite charges. Notice that the electric field lines between the two plates start out perpendicular to the surface of the metal plates (we will see why this is true in the next Section) and go directly from one plate to the other, as we expect because a positive test charge placed between the plates would feel a strong repulsion from the positive plate and a strong attraction to the negative plate. The field lines between two close plates are parallel and equally spaced in the central region, but fringe outward near the edges. Thus, in the central region, the electric field has the same magnitude at all points, and we can write (see Example 21–13)

(b)

E = constant =

+2Q

–Q

−

FIGURE 21–34 Electric field lines for four arrangements of charges.

between two closely spaced, d (21–8) oppositely charged, parallel plates

The fringing of the field near the edges can often be ignored, particularly if the separation of the plates is small compared to their height and width. We summarize the properties of field lines as follows: 1. Electric field lines indicate the direction of the electric field; the field points in the direction tangent to the field line at any point. 2. The lines are drawn so that the magnitude of the electric field, E, is proportional to the number of lines crossing unit area perpendicular to the lines. The closer together the lines, the stronger the field. 3. Electric field lines start on positive charges and end on negative charges; and the number starting or ending is proportional to the magnitude of the charge. Also note that field lines never cross. Why not? Because the electric field can not have two directions at the same point, nor exert more than one force on a test charge.

The field concept can also be applied to the gravitational force as mentioned in Chapter 6. Thus we can say that a gravitational field exists for every object that has mass. One object attracts another by means of the gravitational field. The Earth, for example, can be said to possess a gravitational field (Fig. 21–35) which is responsible for the gravitational force on objects. The gravitational field is defined as the force per unit mass. The magnitude of the Earth's gravitational field at any point above the Earth's surface is thus AGMEr2 B, where ME is the mass of the Earth, r is the distance of the point from the Earth's center, and G is the gravitational constant (Chapter 6). At the Earth's surface, r is the radius of the Earth and the gravitational field is equal to g , the acceleration due to gravity. Beyond the Earth, the gravitational field can be calculated at any point as a sum of terms due to Earth, Sun, Moon, and other bodies that contribute significantly.

FIGURE 21–35 The Earth's gravitational field.

CHAPTER 21

Gravitational Field

(d)

576

s. 0

Electric Charge and Electric Field

21–9

Electric Fields and Conductors

We now discuss some properties of conductors. First, the electric field inside a conductor is zero in the static situation—that is, when the charges are at rest. If there were an electric field within a conductor, there would be a force on the free electrons. The electrons would move until they reached positions where the electric field, and therefore the electric force on them, did become zero. This reasoning has some interesting consequences. For one, any net charge on a conductor distributes itself on the surface. (If there were charges inside, there would be an electric field.) For a negatively charged conductor, you can imagine that the negative charges repel one another and race to the surface to get as far from one another as possible. Another consequence is the following. Suppose that a positive charge Q is surrounded by an isolated uncharged metal conductor whose shape is a spherical shell, Fig. 21–36. Because there can be no field within the metal, the lines leaving the central positive charge must end on negative charges on the inner surface of the metal. Thus an equal amount of negative charge, –Q , is induced on the inner surface of the spherical shell. Then, since the shell is neutral, a positive charge of the same magnitude, ±Q , must exist on the outer surface of the shell. Thus, although no field exists in the metal itself, an electric field exists outside of it, as shown in Fig. 21–36, as if the metal were not even there. A related property of static electric fields and conductors is that the electric field is always perpendicular to the surface outside of a conductor. If there were a B component of E parallel to the surface (Fig. 21–37), it would exert a force on free electrons at the surface, causing the electrons to move along the surface until they reached positions where no net force was exerted on them parallel to the surface—that is, until the electric field was perpendicular to the surface. These properties apply only to conductors. Inside a nonconductor, which does not have free electrons, a static electric field can exist as we will see in the next Chapter. Also, the electric field outside a nonconductor does not necessarily make an angle of 90° to the surface. + + + + + + + + + + + + + +

− − − − − − − − − − − − − −

(a)

+ + + + + + + + + + + + + +

− − − −

+ + + +

− − − − − − − − − − − − − −

Conductor

− +−

−+

−

FIGURE 21–36 A charge inside a neutral spherical metal shell induces charge on its surfaces. The electric field exists even beyond the shell, but not within the conductor itself. FIGURE 21–37 If the electric B field E at the surface of a conductor had a component parallel to the B surface, E ∑ ∑ , the latter would accelerate electrons into motion. In the static B case, E ∑ ∑ must be zero, and the electric field must be perpendicular to the B B conductor's surface: E = E⊥ . B

E Good conductor

FIGURE 21–38 Example 21–14. FIGURE 21–39 A strong electric field exists in the vicinity of this "Faraday cage," so strong that stray electrons in the atmosphere are accelerated to the kinetic energy needed to knock electrons out of air atoms, causing an avalanche of charge which flows to (or from) the metal cage. Yet the person inside the cage is not affected.

(b)

CONCEPTUAL EXAMPLE 21–14 Shielding, and safety in a storm. A neutral hollow metal box is placed between two parallel charged plates as shown in Fig. 21–38a. What is the field like inside the box? RESPONSE If our metal box had been solid, and not hollow, free electrons in the box would have redistributed themselves along the surface until all their individual fields would have canceled each other inside the box. The net field inside the box would have been zero. For a hollow box, the external field is not changed since the electrons in the metal can move just as freely as before to the surface. Hence the field inside the hollow metal box is also zero, and the field lines are shown in Fig. 21–38b. A conducting box used in this way is an effective device for shielding delicate instruments and electronic circuits from unwanted external electric fields. We also can see that a relatively safe place to be during a lightning storm is inside a parked car, surrounded by metal. See also Fig. 21–39, where a person inside a porous "cage" is protected from a strong electric discharge. SECTION 21–9

P H Y S I C S

A P P L I E D

Electrical shielding

Electric Fields and Conductors

577

21–10

Motion of a Charged Particle in an Electric Field

If an object having an electric charge q is at a point in space where the electric field B is E, the force on the object is given by B B F = qE B (see Eq. 21–5). In the past few Sections we have seen how to determine E for some B particular situations. Now let us suppose we know E and we want to find the force on a charged object and the object's subsequent motion. (We assume no other forces act.) B E − e

FIGURE 21–40 Example 21–15.

EXAMPLE 21–15 Electron accelerated by electric field. An electron (mass B m = 9.1 * 10 –31 kg) is accelerated in the uniform field E AE = 2.0 * 104 NCB between two parallel charged plates. The separation of the plates is 1.5 cm. The electron is accelerated from rest near the negative plate and passes through a tiny hole in the positive plate, Fig. 21–40. (a) With what speed does it leave the hole? (b) Show that the gravitational force can be ignored. Assume the hole is so small that it does not affect the uniform field between the plates. APPROACH We can obtain the electron's velocity using the kinematic equations of Chapter 2, after first finding its acceleration from Newton's second law, F = ma. The magnitude of the force on the electron is F = qE and is directed to the right. SOLUTION (a) The magnitude of the electron's acceleration is qE F . a = = m m B

Between the plates E is uniform so the electron undergoes uniformly accelerated motion with acceleration a =

A1.6 * 10 –19 CBA2.0 * 104 NCB A9.1 * 10 –31 kgB

= 3.5 * 1015 ms2.

It travels a distance x = 1.5 * 10 –2 m before reaching the hole, and since its initial speed was zero, we can use the kinematic equation, v2 = v20 + 2ax (Eq. 2–12c), with v0 = 0:

v = 22ax = 32A3.5 * 1015 ms2 BA1.5 * 10–2 mB = 1.0 * 107 ms. There is no electric field outside the plates, so after passing through the hole, the electron moves with this speed, which is now constant. (b) The magnitude of the electric force on the electron is qE = A1.6 * 10–19 CBA2.0 * 104 NCB = 3.2 * 10–15 N.

The gravitational force is

mg = A9.1 * 10 –31 kgBA9.8 ms2 B = 8.9 * 10–30 N,

which is 1014 times smaller! Note that the electric field due to the electron does not enter the problem (since a particle cannot exert a force on itself). B

FIGURE 21–41

Example 21–16.

y B E x B P 0 v0

578

CHAPTER 21

EXAMPLE 21–16 Electron moving perpendicular to E. Suppose an B electron traveling with speed v0 enters a uniform electric field E, which is at right angles to v0 as shown in Fig. 21–41. Describe its motion by giving the equation of its path while in the electric field. Ignore gravity. B

APPROACH Again we use Newton's second law, with F = qE, and the kinematic equations from Chapter 2. SOLUTION When the electron enters the electric field (at x = y = 0) it has B velocity v0 = v0 ˆi in the x direction. The electric field E, pointing vertically upward, imparts a uniform vertical acceleration to the electron of qE F eE , ay = = = – m m m where we set q = –e for the electron. B

The electron's vertical position is given by Eq. 2–12b, 1 eE 2 y = a t2 = – t 2 y 2m since the motion is at constant acceleration. The horizontal position is given by x = v0 t since ax = 0. We eliminate t between these two equations and obtain eE 2 y = – x, 2mv20 which is the equation of a parabola (just as in projectile motion, Section 3–7).

21–11

Electric Dipoles

−Q

The combination of two equal charges of opposite sign, ±Q and –Q, separated by a distance l, is referred to as an electric dipole. The quantity Ql is called the dipole moment and is represented† by the symbol p. The dipole moment can be considered to be a vector p, of magnitude Ql, that points from the negative to the positive charge as shown in Fig. 21–42. Many molecules, such as the diatomic molecule CO, have a dipole moment (C has a small positive charge and O a small negative charge of equal magnitude), and are referred to as polar molecules. Even though the molecule as a whole is neutral, there is a separation of charge that results from an uneven sharing of electrons by the two atoms.‡ (Symmetric diatomic molecules, like O2 , have no dipole moment.) The water molecule, with its uneven sharing of electrons (O is negative, the two H are positive), also has a dipole moment—see Fig. 21–43. B

Dipole in an External Field First let us consider a dipole, of dipole moment p = Ql, that is placed in a B uniform electric field E, as shown in Fig. 21–44. If the field is uniform, the force B B QE on the positive charge and the force –QE on the negative charge result in no net force on the dipole. There will, however, be a torque on the dipole (Fig. 21–44) which has magnitude (calculated about the center, 0, of the dipole) l l (21–9a) t = QE sin u + QE sin u = pE sin u. 2 2 This can be written in vector notation as B

T = p * E. B

FIGURE 21–42 A dipole consists of equal but opposite charges, ±Q and –Q, separated by a distance l. The dipole B B moment is p = Q L and points from the negative to the positive charge. FIGURE 21–43 In the water molecule AH 2OB, the electrons spend more time around the oxygen atom than around the two hydrogen B atoms. The net dipole moment p can be considered as the vector sum of B B two dipole moments p 1 and p2 that point from the O toward each H as B B B shown: p = p 1 + p2 . B p 1

B p

104°

(21–9b)

B p 2

The effect of the torque is to try to turn the dipole so p is parallel to E. The work done on the dipole by the electric field to change the angle u from u1 to u2 is (see Eq. 10–22) W =

t du.

We need to write the torque as t = –pE sin u because its direction is opposite to the direction of increasing u (right-hand rule). Then W =

t du = –pE

sin u du = pE cos u 2

u2 u1

= pEAcos u2 - cos u1 B.

FIGURE 21–44 (below) An electric dipole in a uniform electric field.

Positive work done by the field decreases the potential energy, U, of the dipole in this field. (Recall the relation between work and potential energy, Eq. 8–4, B ¢U = –W.) If we choose U = 0 when p is perpendicular to E (that is, choosing u1 = 90° so cos u1 = 0), and setting u2 = u, then

U = –W = –pE cos u = –p E. B

B p

(21–10)

If the electric field is not uniform, the force on the ±Q of the dipole may not have the same magnitude as on the –Q, so there may be a net force as well as a torque.

F− = −QE

F+ = QE

u 0

−Q

†

Be careful not to confuse this p for dipole moment with p for momentum. ‡ The value of the separated charges may be a fraction of e (say & 0.2e or & 0.4e) but note that such charges do not violate what we said about e being the smallest charge. These charges less than e cannot be isolated and merely represent how much time electrons spend around one atom or the other.

SECTION 21–11

579

EXAMPLE 21–17 Dipole in a field. The dipole moment of a water molecule is 6.1 * 10–30 Cm. A water molecule is placed in a uniform electric field with magnitude 2.0 * 105 NC. (a) What is the magnitude of the maximum torque that the field can exert on the molecule? (b) What is the potential energy when the torque is at its maximum? (c) In what position will the potential energy take on its greatest value? Why is this different than the position where the torque is maximum? APPROACH The torque is given by Eq. 21–9 and the potential energy by Eq. 21–10. SOLUTION (a) From Eq. 21–9 we see that t is maximized when u is 90°. Then t = pE = A6.1 * 10–30 CmBA2.0 * 105 NCB = 1.2 * 10–24 N m. (b) The potential energy for u = 90° is zero (Eq. 21–10). Note that the potential energy is negative for smaller values of u, so U is not a minimum for u = 90°. (c) The potential energy U will be a maximum when cos u = –1 in Eq. 21–10, B so u = 180°, meaning E and p are antiparallel. The potential energy is maximized when the dipole is oriented so that it has to rotate through the largest angle, 180°, to reach the equilibrium position at u = 0°. The torque on the other hand is maximized when the electric forces are perpendicular to p. B

Electric Field Produced by a Dipole

E+ f

y P

E− r f −Q

l 2

We have just seen how an external electric field affects an electric dipole. Now let us suppose that there is no external field, and we want to determine the electric field produced by the dipole. For brevity, we restrict ourselves to points that are on the perpendicular bisector of the dipole, such as point P in Fig. 21–45 which is a distance r above the midpoint of the dipole. Note that r in Fig. 21–45 is not the 1 distance from either charge to point P; the latter distance is Ar 2 + l24B 2 and this is what must be used in Eq. 21–4. The total field at P is

f 0

E = E± + E– ,

l +Q 2

FIGURE 21–45 Electric field due to an electric dipole.

where E ± and E – are the fields due to the ± and – charges respectively. The magnitudes E± and E– are equal: E± = E– =

1 4p 0

Q . + l24

Their y components cancel at point P (symmetry again), so the magnitude of the B total field E is E = 2E± cos f =

Q 1 l ¢ ≤ 1 2p 0 r 2 + l24 2Ar 2 + l24B 2

or, setting Ql = p, E =

p 1 . 3 4p 0 Ar2 + l24B 2

on perpendicular bisector d (21–11) of dipole

Far from the dipole, r W l, this reduces to E =

1 p. 4p 0 r3

on perpendicular bisector d (21–12) of dipole; r W l

So the field decreases more rapidly for a dipole than for a single point charge A1r 3 versus 1r2 B, which we expect since at large distances the two opposite charges appear so close together as to neutralize each other. This 1r 3 dependence also applies for points not on the perpendicular bisector (see Problem 67).

580

CHAPTER 21

Electric Charge and Electric Field

* 21–12

Electric Forces in Molecular Biology; DNA

The interior of every biological cell is mainly water. We can imagine a cell as a vast sea of molecules continually in motion (kinetic theory, Chapter 18), colliding with one another with various amounts of kinetic energy. These molecules interact with one another because of electrostatic attraction between molecules. Indeed, cellular processes are now considered to be the result of random ("thermal") molecular motion plus the ordering effect of the electrostatic force. As an example, we look at DNA structure and replication. The picture we present has not been seen "in action." Rather, it is a model of what happens based on physical theories and experiment. The genetic information that is passed on from generation to generation in all living cells is contained in the chromosomes, which are made up of genes. Each gene contains the information needed to produce a particular type of protein molecule, and that information is built into the principal molecule of a chromosome, DNA (deoxyribonucleic acid), Fig. 21–46. DNA molecules are made up of many small molecules known as nucleotide bases which are each polar due to unequal sharing of electrons. There are four types of nucleotide bases in DNA: adenine (A), cytosine (C), guanine (G), and thymine (T). The DNA of a chromosome generally consists of two long DNA strands wrapped about one another in the shape of a "double helix." The genetic information is contained in the specific order of the four bases (A, C, G, T) along the strand. As shown in Fig. 21–47, the two strands are attracted by electrostatic forces—that is, by the attraction of positive charges to negative charges that exist on parts of the molecules. We see in Fig. 21–47a that an A (adenine) on one strand is always opposite a T on the other strand; similarly, a G is always opposite a C. This important ordering effect occurs because the shapes of A, T, C, and G are such that a T fits closely only into an A, and a G into a C; and only in the case of this close proximity of the charged portions is the electrostatic force great enough to hold them together even for a short time (Fig. 21–47b), forming what are referred to as "weak bonds."

T G C A

0.280 n

in ha To c

A A

C G T T A

O –

C A

FIGURE 21–46 DNA replicating in a human HeLa cancer cell. This is a false-color image made by a transmission electron microscope (TEM; discussed in Chapter 37).

m H Adenine (A) H N C N + – H C 0.300 n C C C m N H N C N N C – + – C N O H 1.11 nm

+ C

H T C G T G C

A P P L I E D

c To

A C G T A G

Thymine (T) H H H C

P H Y S I C S

Inside a cell: kinetic theory plus electrostatic force

T (a)

To c

in ha

c To

FIGURE 21–47 (a) Section of a DNA double helix. (b) "Close-up" view of the helix, showing how A and T attract each other and how G and C attract each other through electrostatic forces. The ± and – signs represent net charges, usually a fraction of e, due to uneven sharing of electrons. The red dots indicate the electrostatic attraction (often called a "weak bond" or "hydrogen bond"—Section 40–3). Note that there are two weak bonds between A and T, and three between C and G.

Cytosine (C) H 0.290 nm H Guanine (G) N H – O + C C – + N H C 0.300 nm C C C N H N – C N N C + – + 0.290 nm C N O H N – + – H 1.08 nm (b)

*SECTION 21–12

Electric Forces in Molecular Biology; DNA

581

Old T

A A G

T T

T G

A Old

New

T A T 3 G T T A A T

A A

New New

Old

New

Old A T

C C

A A C

A T C

Old

CHAPTER 21

G C

T A

A T

FIGURE 21–48 Replication of DNA.

Old

G G C

When the DNA replicates (duplicates) itself just before cell division, the arrangement of A opposite T and G opposite C is crucial for ensuring that the genetic information is passed on accurately to the next generation, Fig. 21–48. The two strands of DNA separate (with the help of enzymes, which also operate via the electrostatic force), leaving the charged parts of the bases exposed. Once replication starts, let us see how the correct order of bases occurs by looking at the G molecule indicated by the red arrow in Fig. 21–48. Many unattached nucleotide bases of all four kinds are bouncing around in the cellular fluid, and the only type that will experience attraction to our G, if it bounces close to it, will be a C. The charges on the other three bases can not get close enough to those on the G to provide a significant attractive force—remember that the force decreases rapidly with distance Ar 1r 2 B. Because the G does not attract an A, T, or G appreciably, an A, T, or G will be knocked away by collisions with other molecules before enzymes can attach it to the growing chain (number 3). But the electrostatic force will often hold a C opposite our G long enough so that an enzyme can attach the C to the growing end of the new chain. Thus we see that electrostatic forces are responsible for selecting the bases in the proper order during replication. This process of DNA replication is often presented as if it occurred in clockwork fashion—as if each molecule knew its role and went to its assigned place. But this is not the case. The forces of attraction are rather weak, and if the molecular shapes are not just right, there is almost no electrostatic attraction, which is why there are few mistakes. Thus, out of the random motion of the molecules, the electrostatic force acts to bring order out of chaos. The random (thermal) velocities of molecules in a cell affect cloning. When a bacterial cell divides, the two new bacteria have nearly identical DNA. Even if the DNA were perfectly identical, the two new bacteria would not end up behaving in the same way. Long protein, DNA, and RNA molecules get bumped into different shapes, and even the expression of genes can thus be different. Loosely held parts of large molecules such as a methyl group (CH3) can also be knocked off by a strong collision with another molecule. Hence, cloned organisms are not identical, even if their DNA were identical. Indeed, there can not really be genetic determinism.

* 21–13

582

Photocopy Machines and Computer Printers Use Electrostatics

Photocopy machines and laser printers use electrostatic attraction to print an image. They each use a different technique to project an image onto a special cylindrical drum. The drum is typically made of aluminum, a good conductor; its surface is coated with a thin layer of selenium, which has the interesting property (called "photoconductivity") of being an electrical nonconductor in the dark, but a conductor when exposed to light. In a photocopier, lenses and mirrors focus an image of the original sheet of paper onto the drum, much like a camera lens focuses an image on film. Step 1 is

(2) Lens focuses image of original (3) Toner hopper

(4) Paper

Charging rod

(1) Charging rod or roller

FIGURE 21–49 Inside a photocopy machine: (1) the selenium drum is given a + charge; (2) the lens focuses image on drum—only dark spots stay charged; (3) toner particles (negatively charged) are attracted to positive areas on drum; (4) the image is transferred to paper; (5) heat binds the image to the paper.

(5) Heater rollers

the placing of a uniform positive charge on the drum's selenium layer by a charged rod or roller, done in the dark. In step 2, the image to be copied or printed is projected onto the drum. For simplicity, let us assume the image is a dark letter A on a white background (as on the page of a book) as shown in Fig. 21–49. The letter A on the drum is dark, but all around it is light. At all these light places, the selenium becomes conducting and electrons flow in from the aluminum beneath, neutralizing those positive areas. In the dark areas of the letter A, the selenium is nonconducting and so retains a positive charge, Fig. 21–49. In step 3, a fine dark powder known as toner is given a negative charge, and brushed on the drum as it rotates. The negatively charged toner particles are attracted to the positive areas on the drum (the A in our case) and stick only there. In step 4, the rotating drum presses against a piece of paper which has been positively charged more strongly than the selenium, so the toner particles are transferred to the paper, forming the final image. Finally, step 5, the paper is heated to fix the toner particles firmly on the paper. In a color copier (or printer), this process is repeated for each color—black, cyan (blue), magenta (red), and yellow. Combining these four colors in different proportions produces any desired color. A laser printer, on the other hand, uses a computer output to program the intensity of a laser beam onto the selenium-coated drum. The thin beam of light from the laser is scanned (by a movable mirror) from side to side across the drum in a series of horizontal lines, each line just below the previous line. As the beam sweeps across the drum, the intensity of the beam is varied by the computer output, being strong for a point that is meant to be white or bright, and weak or zero for points that are meant to come out dark. After each sweep, the drum rotates very slightly for additional sweeps, Fig. 21–50, until a complete image is formed on it. The light parts of the selenium become conducting and lose their electric charge, and the toner sticks only to the dark, electrically charged areas. The drum then transfers the image to paper, as in a photocopier. An inkjet printer does not use a drum. Instead nozzles spray tiny droplets of ink directly at the paper. The nozzles are swept across the paper, each sweep just above the previous one as the paper moves down. On each sweep, the ink makes dots on the paper, except for those points where no ink is desired, as directed by the computer. The image consists of a huge number of very tiny dots. The quality or resolution of a printer is usually specified in dots per inch (dpi) in each (linear) direction. Laser beam scans

P H Y S I C S

A P P L I E D

Photocopy machines

P H Y S I C S

A P P L I E D

Laser printer

P H Y S I C S

A P P L I E D

Inkjet printer

Laser

Toner hopper

Movable mirror Paper

FIGURE 21–50 Inside a laser printer: A movable mirror sweeps the laser beam in horizontal lines across the drum.

Heater rollers

*SECTION 21–13

Photocopy Machines and Computer Printers Use Electrostatics

583

Summary There are two kinds of electric charge, positive and negative. These designations are to be taken algebraically—that is, any charge is plus or minus so many coulombs (C), in SI units. Electric charge is conserved: if a certain amount of one type of charge is produced in a process, an equal amount of the opposite type is also produced; thus the net charge produced is zero. According to atomic theory, electricity originates in the atom, each consisting of a positively charged nucleus surrounded by negatively charged electrons. Each electron has a charge –e = – 1.6 * 10 –19 C. Electric conductors are those materials in which many electrons are relatively free to move, whereas electric insulators are those in which very few electrons are free to move. An object is negatively charged when it has an excess of electrons, and positively charged when it has less than its normal amount of electrons. The charge on any object is thus a whole number times ± e or – e. That is, charge is quantized. An object can become charged by rubbing (in which electrons are transferred from one material to another), by conduction (which is transfer of charge from one charged object to another by touching), or by induction (the separation of charge within an object because of the close approach of another charged object but without touching). Electric charges exert a force on each other. If two charges are of opposite types, one positive and one negative, they each exert an attractive force on the other. If the two charges are the same type, each repels the other. The magnitude of the force one point charge exerts on another is proportional to the product of their charges, and inversely proportional to the square of the distance between them: F = k

Q1 Q2 r

1 Q1 Q2 ; 4p 0 r2

(21–1, 21–2)

this is Coulomb's law. In SI units, k is often written as 14p 0 .

We think of an electric field as existing in space around any charge or group of charges. The force on another charged object is then said to be due to the electric field present at its location. B The electric field, E, at any point in space due to one or more charges, is defined as the force per unit charge that would act on a positive test charge q placed at that point: B

E =

F. q

(21–3)

The magnitude of the electric field a distance r from a point charge Q is Q (21–4a) E = k 2. r The total electric field at a point in space is equal to the vector sum of the individual fields due to each contributing charge (principle of superposition). Electric fields are represented by electric field lines that start on positive charges and end on negative charges. Their direction indicates the direction the force would be on a tiny positive test charge placed at each point. The lines can be drawn so that the number per unit area is proportional to the magnitude of E. The static electric field inside a conductor is zero, and the electric field lines just outside a charged conductor are perpendicular to its surface. An electric dipole is a combination of two equal but opposite charges, ±Q and –Q, separated by a distance l. The dipole moment is p = Ql. A dipole placed in a uniform electric B field feels no net force but does feel a net torque (unless p is B parallel to E). The electric field produced by a dipole decreases as the third power of the distance r from the dipole AE r 1r3 B for r large compared to l. [*In the replication of DNA, the electrostatic force plays a crucial role in selecting the proper molecules so the genetic information is passed on accurately from generation to generation.]

Questions 1. If you charge a pocket comb by rubbing it with a silk scarf, how can you determine if the comb is positively or negatively charged? 2. Why does a shirt or blouse taken from a clothes dryer sometimes cling to your body? 3. Explain why fog or rain droplets tend to form around ions or electrons in the air. 4. A positively charged rod is brought close to a neutral piece of paper, which it attracts. Draw a diagram showing the separation of charge in the paper, and explain why attraction occurs. 5. Why does a plastic ruler that has been rubbed with a cloth have the ability to pick up small pieces of paper? Why is this difficult to do on a humid day? 6. Contrast the net charge on a conductor to the "free charges" in the conductor. 7. Figures 21–7 and 21–8 show how a charged rod placed near an uncharged metal object can attract (or repel) electrons. There are a great many electrons in the metal, yet only some of them move as shown. Why not all of them? 8. When an electroscope is charged, the two leaves repel each other and remain at an angle. What balances the electric force of repulsion so that the leaves don't separate further?

584

CHAPTER 21

Electric Charge and Electric Field

9. The form of Coulomb's law is very similar to that for Newton's law of universal gravitation. What are the differences between these two laws? Compare also gravitational mass and electric charge. 10. We are not normally aware of the gravitational or electric force between two ordinary objects. What is the reason in each case? Give an example where we are aware of each one and why. 11. Is the electric force a conservative force? Why or why not? (See Chapter 8.) 12. What experimental observations mentioned in the text rule out the possibility that the numerator in Coulomb's law contains the sum AQ1 + Q2 B rather than the product Q1 Q2? 13. When a charged ruler attracts small pieces of paper, sometimes a piece jumps quickly away after touching the ruler. Explain. 14. Explain why the test charges we use when measuring electric fields must be small. 15. When determining an electric field, must we use a positive test charge, or would a negative one do as well? Explain. 16. Draw the electric field lines surrounding two negative electric charges a distance l apart.

17. Assume that the two opposite charges in Fig. 21–34a are 12.0 cm apart. Consider the magnitude of the electric field 2.5 cm from the positive charge. On which side of this charge—top, bottom, left, or right—is the electric field the strongest? The weakest? 18. Consider the electric field at the three points indicated by the letters A, B, and C in Fig. 21–51. First draw an arrow at each point indicating the direction of the net force that a positive test charge would experience if placed at that point, then list B the letters in order of decreasing field strength (strongest C first). A FIGURE 21–51 Question 18. 19. Why can electric field lines never cross? 20. Show, using the three rules for field lines given in Section 21–8, that the electric field lines starting or ending on a single point charge must be symmetrically spaced around the charge.

21. Given two point charges, Q and 2Q, a distance l apart, is there a point along the straight line that passes through them where E = 0 when their signs are (a) opposite, (b) the same? If yes, state roughly where this point will be. 22. Suppose the ring of Fig. 21–28 has a uniformly distributed negative charge Q. What is the magnitude and direction of B E at point P? 23. Consider a small positive test charge located on an electric field line at some point, such as point P in Fig. 21–34a. Is the direction of the velocity andor acceleration of the test charge along this line? Discuss. 24. We wish to determine the electric field at a point near a positively charged metal sphere (a good conductor). We do so by bringing a small test charge, q0 , to this point and measure the force F0 on it. Will F0q0 be greater than, less B than, or equal to the electric field E as it was at that point before the test charge was present? 25. In what ways does the electron motion in Example 21–16 resemble projectile motion (Section 3–7)? In which ways not? 26. Describe the motion of the dipole shown in Fig. 21–44 if it is released from rest at the position shown. 27. Explain why there can be a net force on an electric dipole placed in a nonuniform electric field.

Problems 1. (I) What is the magnitude of the electric force of attraction between an iron nucleus (q = ± 26e) and its innermost electron if the distance between them is 1.5 * 10 –12 m? 2. (I) How many electrons make up a charge of –38.0 mC? 3. (I) What is the magnitude of the force a ± 25 mC charge exerts on a ± 2.5 mC charge 28 cm away? 4. (I) What is the repulsive electrical force between two protons 4.0 * 10 –15 m apart from each other in an atomic nucleus? 5. (II) When an object such as a plastic comb is charged by rubbing it with a cloth, the net charge is typically a few microcoulombs. If that charge is 3.0 mC, by what percentage does the mass of a 35-g comb change during charging? 6. (II) Two charged dust particles exert a force of 3.2 * 10 –2 N on each other. What will be the force if they are moved so they are only one-eighth as far apart?

9. (II) What is the total charge of all the electrons in a 15-kg bar of gold? What is the net charge of the bar? (Gold has 79 electrons per atom and an atomic mass of 197 u.) 10. (II) Compare the electric force holding the electron in orbit Ar = 0.53 * 10–10 mB around the proton nucleus of the hydrogen atom, with the gravitational force between the same electron and proton. What is the ratio of these two forces?

FIGURE 21–52 Problem 12.

75 C

85 C

48 C

0.35 m

13. (II) Three charged particles are placed at the corners of an equilateral triangle of side 1.20 m (Fig. 21–53). The charges are ±7.0 mC, –8.0 mC, and –6.0 mC. Calculate the magnitude and direction of the net force on each due to the other Q1 = +7.0 mC two.

FIGURE 21–53 Problem 13.

Q2 = −8.0 mC

8. (II) A person scuffing her feet on a wool rug on a dry day accumulates a net charge of – 46 mC. How many excess electrons does she get, and by how much does her mass increase?

12. (II) Particles of charge ±75, ± 48, and –85 mC are placed in a line (Fig. 21–52). The center one is 0.35 m from each of the others. Calculate the net force on each charge due to the other two.

0 1.2

7. (II) Two charged spheres are 8.45 cm apart. They are moved, and the force on each of them is found to have been tripled. How far apart are they now?

11. (II) Two positive point charges are a fixed distance apart. The sum of their charges is QT . What charge must each have in order to (a) maximize the electric force between them, and (b) minimize it?

C1 mC = 10 –3 C, 1 mC = 10 –6 C, 1 nC = 10 –9 C. D

1.2 0

21–5 Coulomb's Law

1.20 m Q3 = −6.0 mC

14. (II) Two small nonconducting spheres have a total charge of 90.0 mC. (a) When placed 1.16 m apart, the force each exerts on the other is 12.0 N and is repulsive. What is the charge on each? (b) What if the force were attractive? 15. (II) A charge of 4.15 mC is placed at each corner of a square 0.100 m on a side. Determine the magnitude and direction of the force on each charge.

Problems

585

16. (II) Two negative and two positive point charges (magnitude Q = 4.15 mC) are placed on opposite corners of a square as shown in Fig. 21–54. Determine the −4.15 mC 0.100 m 4.15 mC magnitude and direction of the force on each charge. 0.100 m 0.100 m FIGURE 21–54 Problem 16.

4.15 mC 0.100 m −4.15 mC

17. (II) A charge Q is transferred from an initially uncharged plastic ball to an identical ball 12 cm away. The force of attraction is then 17 mN. How many electrons were transferred from one ball to the other? 18. (III) Two charges, – Q0 and – 4Q0 , are a distance l apart. These two charges are free to move but do not because there is a third charge nearby. What must be the magnitude of the third charge and its placement in order for the first two to be in equilibrium? 19. (III) Two positive charges ± Q are affixed rigidly to the x axis, one at x = ± d and the other at x = – d. A third charge ±q of mass m, which is constrained to move only along the x axis, is displaced from the origin by a small distance s V d and then released from rest. (a) Show that (to a good approximation) ± q will execute simple harmonic motion and determine an expression for its oscillation period T. (b) If these three charges are each singly ionized sodium atoms (q = Q = ±e) at the equilibrium spacing d = 3 * 10–10 m typical of the atomic spacing in a solid, find T in picoseconds. 20. (III) Two small charged spheres hang from cords of equal length l as shown in Fig. 21–55 and make small angles u1 and u2 with the vertical. (a) If Q1 = Q, Q2 = 2Q, and m1 = m2 = m, determine the ratio u1u2 . (b) If Q1 = Q, Q2 = 2Q, m1 = m, and m2 = 2m, deterl l mine the ratio u1u2 . (c) Estimate the u2 distance between the spheres for each case. u1 FIGURE 21–55 Problem 20.

21–6 to 21–8 Electric Field, Field Lines 21. (I) What are the magnitude and direction of the electric force on an electron in a uniform electric field of strength 1920 NC that points due east? 22. (I) A proton is released in a uniform electric field, and it experiences an electric force of 2.18 * 10 –14 N toward the south. What are the magnitude and direction of the electric field? 23. (I) Determine the magnitude and direction of the electric field 16.4 cm directly above an isolated 33.0 * 10–6 C charge. 24. (I) A downward electric force of 8.4 N is exerted on a –8.8 mC charge. What are the magnitude and direction of the electric field at the position of this charge? 25. (I) The electric force on a ± 4.20-mC charge is B F = A7.22 * 10–4 NB jˆ. What is the electric field at the position of the charge? 26. (I) What is the electric field at a point when the force on a 1.25-mC charge placed at that point is B F = (3.0iˆ - 3.9jˆ) * 10 –3 N?

586

CHAPTER 21

Electric Charge and Electric Field

27. (II) Determine the magnitude of the acceleration experienced by an electron in an electric field of 576 NC. How does the direction of the acceleration depend on the direction of the field at that point? 28. (II) Determine the magnitude and direction of the electric field at a point midway between a –8.0 mC and a ±5.8 mC charge 8.0 cm apart. Assume no other charges are nearby. 29. (II) Draw, approximately, the electric field lines about two point charges, ±Q and –3Q, which are a distance l apart. 30. (II) What is the electric field strength at a point in space where a proton experiences an acceleration of 1.8 million "g's"? 31. (II) A long uniformly charged thread (linear charge density l = 2.5 Cm) lies along the x axis in Fig. 21–56. A small charged sphere (Q = –2.0 C) is at the point x = 0 cm, y = –5.0 cm. What y is the electric field at the B point x = 7.0 cm, y = 7.0 cm? E thread B B 7.0 cm E thread and EQ represent fields due to the long u thread and the charge 7.0 cm Q, respectively. B EQ thread

λ = 2.5 C/m 5.0 cm FIGURE 21–56 Problem 31.

Q = −2.0 C

32. (II) The electric field midway between two equal but opposite point charges is 586 NC, and the distance between the charges is 16.0 cm. What is the magnitude of the charge on each? 33. (II) Calculate the electric field at one corner of a square 1.22 m on a side if the other three corners are occupied by 2.25 * 10 –6 C charges. 34. (II) Calculate the electric field at the center of a square 52.5 cm on a side if one corner is occupied by a –38.6 mC charge and the other three are occupied by –27.0 mC charges. 35. (II) Determine the direction and magnitude of the electric field at the point P in Fig. 21–57. The charges are separated by a distance 2a, and point P is a distance x from the midpoint between the two charges. Express your answer in terms of Q, x, a, and k. FIGURE 21–57 Problem 35.

Q a

a Q x

36. (II) Two point charges, Q1 = –25 mC and Q2 = ±45 mC, are separated by a distance of 12 cm. The electric field at the point P (see Fig. 21–58) is zero. How far from Q1 is P? FIGURE 21–58 P Problem 36.

Q1 25 mC

12 cm

Q2 45 mC

37. (II) A very thin line of charge lies along the x axis from x = – q to x = ± q. Another similar line of charge lies along the y axis from y = – q to y = ± q. Both lines have a uniform charge per length l. Determine the resulting electric field magnitude and direction (relative to the x axis) at a point (x, y) in the first quadrant of the xy plane.

38. (II) (a) Determine the electric field E at the origin 0 in Fig. 21–59 due to the two charges at A and B. (b) Repeat, but let the charge at B be reversed y in sign. Q A l Q l B FIGURE 21–59 l x Problem 38. 0 39. (II) Draw, approximately, the electric field lines emanating from a uniformly charged straight wire whose length l is not great. The spacing between lines near the wire should be much less than l. [Hint: Also consider points very far from the wire.] 40. (II) Two parallel circular rings of radius R have their centers on the x axis separated by a distance l as shown in Fig. 21–60. If each ring carries a uniformly y distributed charge Q, find the B electric field, E(x), at points l l R R along the x axis. 2 2 0

Q2 l

42. (II) Use Coulomb's law to determine the magnitude and direction of the electric field at points A and B in Fig. 21–62 due to the two positive charges (Q = 5.7 mC) shown. Are your results consistent B A with Fig. 21–34b? 5.0 cm +Q FIGURE 21–62 Problem 42.

+Q 5.0 cm 5.0 cm

10.0 cm

43. (II) (a) Two equal charges Q are positioned at points and (x = – l, y = 0). Determine the (x = l, y = 0) electric field as a function of y for points along the y axis. (b) Show that the field is a maximum at y = &l 12 . 44. (II) At what position, x = xM , is the magnitude of the electric field along the axis of the ring of Example 21–9 a maximum? 45. (II) Estimate the electric field at a point 2.40 cm perpendicular to the midpoint of a uniformly charged 2.00-m-long thin wire carrying a total charge of 4.75 mC. 46. (II) The uniformly charged straight wire in Fig. 21–29 has the length l, where point 0 is at the midpoint. Show that the field at point P, a perpendicular distance x from 0, is given by E =

l l , 2 2p 0 x(l + 4x2) 12

where l is the charge per unit length.

48. (II) Determine the direction and magnitude of the electric field at the point P shown in Fig. 21–64. The two charges are separated by a distance of 2a. Point P is on the perpendicular bisector of the line joining the charges, a distance x from the midpoint between them. Express your answers in terms of Q, x, a, and k. +Q FIGURE 21–64 Problem 48.

41. (II) You are given two unknown point charges, Q1 and Q2 . At a point on the line joining them, one-third of the way from Q1 to Q2 , the electric field is zero (Fig. 21–61). What is the ratio Q1Q2? E=0 FIGURE 21–61 Problem 41.

FIGURE 21–63 Problem 47.

FIGURE 21–60 Problem 40.

l Q1 3

47. (II) Use your result from Problem 46 to find the electric field (magnitude and direction) a distance z above the center of a square loop of wire, each of whose sides has length l and uniform charge per length l z (Fig. 21–63).

–Q

a a

49. (III) A thin rod bent into the shape of an arc of a circle of radius R carries a uniform charge per unit length l. The arc subtends a total angle 2u0 , symmetric about the x axis, as R shown in Fig. 21–65. Determine the B electric field E at the origin 0. u 0

u0 R

FIGURE 21–65 Problem 49.

50. (III) A thin glass rod is a semicircle of radius R, Fig. 21–66. A charge is nonuniformly distributed along the rod with a linear charge density given by l = l0 sin u, where l0 is a positive constant. Point P is at the center of the semicircle. B (a) Find the electric field E y (magnitude and direction) at point P. [Hint: Remember sin(–u) = –sin u, so the two halves of the rod are oppositely charged.] (b) Determine the R acceleration (magnitude and u direction) of an electron placed at P point P, assuming R = 1.0 cm x and l0 = 1.0 mCm. FIGURE 21–66 Problem 50.

51. (III) Suppose a uniformly charged wire starts at point 0 and rises vertically along the positive y axis to a length l. (a) Determine the components of the electric field Ex and B Ey at point (x, 0). That is, calculate E near one end of a long wire, in the plane perpendicular to the wire. (b) If the wire extends from y = 0 to y = q, so that l = q, show B that E makes a 45° angle to the horizontal for any x. [Hint: See Example 21–11 and Fig. 21–29.]

Problems

587

52. (III) Suppose in Example 21–11 that x = 0.250 m, Q = 3.15 mC, and that the uniformly charged wire is only 6.50 m long and extends along the y axis from y = –4.00 m to y = ± 2.50 m. (a) Calculate Ex and Ey at point P. (b) Determine what the error would be if you simply used the result of Example 21–11, E = l2p 0 x. Express this error as AEx - EBE and EyE. 53. (III) A thin rod of length l carries a total charge Q distributed uniformly along its length. See Fig. 21–67. Determine the electric field along the axis of the rod starting y at one end—that is, find E(x) l for x 0 in Fig. 21–67. 0

FIGURE 21–67 Problem 53.

54. (III) Uniform plane of charge. Charge is distributed uniformly over a large square plane of side l, as shown in Fig. 21–68. The charge per unit area ACm2 B is s. Determine the electric field at a point P a distance z above the center of the plane, in the limit z l S q. [Hint: Divide the B dE plane into long narrow dEz strips of width dy, u and use the result of Example 21–11; then dEy P sum the fields due to dy each strip to get the z total field at P.] 0 y

FIGURE 21–68 Problem 54.

55. (III) Suppose the charge Q on the ring of Fig. 21–28 was all distributed uniformly on only the upper half of the ring, and no charge was on the lower half. Determine the electric B field E at P. (Take y vertically upward.)

21–10 Motion of Charges in an Electric Field 56. (II) An electron with speed v0 = 27.5 * 106 ms is traveling parallel to a uniform electric field of magnitude E = 11.4 * 103 NC. (a) How far will the electron travel before it stops? (b) How much time will elapse before it returns to its starting point? 57. (II) An electron has an initial velocity vB0 = A8.0 * 104 msB jˆ. B It enters a region where E = (2.0iˆ + 8.0jˆ) * 104 NC. (a) Determine the vector acceleration of the electron as a function of time. (b) At what angle u is it moving (relative to its initial direction) at t = 1.0 ns? 58. (II) An electron moving to the right at 7.5 * 105 ms enters a uniform electric field parallel to its direction of motion. If the electron is to be brought to rest in the space of 4.0 cm, (a) what direction is required for the electric field, and (b) what is the strength of the field? 59. (II) At what angle will the electrons in Example 21–16 leave the uniform electric field at the end of the parallel plates (point P in Fig. 21–41)? Assume the plates are 4.9 cm long, E = 5.0 * 103 NC, and v0 = 1.00 * 107 ms. Ignore fringing of the field.

588

CHAPTER 21

Electric Charge and Electric Field

60. (II) An electron is traveling through a uniform electric B field. The field is constant and given by E = A2.00 * 10 –11 NCB ˆi - A1.20 * 10–11 NCB jˆ. At t = 0, the electron is at the origin and traveling in the x direction with a speed of 1.90 ms. What is its position 2.00 s later? 61. (II) A positive charge q is placed at the center of a circular ring of radius R. The ring carries a uniformly distributed negative charge of total magnitude –Q. (a) If R the charge q is displaced from the center a small distance x as shown in Fig. 21–69, show that it will undergo simple harmonic motion when released. (b) If its mass is m, what is its period?

x 0q

FIGURE 21–69 Problem 61.

21–11 Electric Dipoles 62. (II) A dipole consists of charges ±e and –e separated by 0.68 nm. It is in an electric field E = 2.2 * 104 NC. (a) What is the value of the dipole moment? (b) What is the torque on the dipole when it is perpendicular to the field? (c) What is the torque on the dipole when it is at an angle of 45° to the field? (d) What is the work required to rotate the dipole from being oriented parallel to the field to being antiparallel to the field? 63. (II) The HCl molecule has a dipole moment of about 3.4 * 10–30 C m. The two atoms are separated by about 1.0 * 10–10 m. (a) What is the net charge on each atom? (b) Is this equal to an integral multiple of e? If not, explain. (c) What maximum torque would this dipole experience in a 2.5 * 104 NC electric field? (d) How much energy would be needed to rotate one molecule 45° from its equilibrium position of lowest potential energy? 64. (II) Suppose both charges in Fig. 21–45 (for a dipole) were positive. (a) Show that the field on the perpendicular bisector, for r W l, is given by A14p 0 B A2Qr2 B. (b) Explain why the field decreases as 1r 2 here whereas for a dipole it decreases as 1r3. 65. (II) An electric dipole, of dipole moment p and moment of B inertia I, is placed in a uniform electric field E. (a) If displaced by an angle u as shown in Fig. 21–44 and released, under what conditions will it oscillate in simple harmonic motion? (b) What will be its frequency? B 66. (III) Suppose a dipole p is placed in a nonuniform electric B B ˆ field E = E i that points along the x axis. If E depends only on x, show that the net force on the dipole is B

B F = ¢p

dE ˆ ≤ i, dx

where dEdx is the gradient of the field in the x direction. 67. (III) (a) Show that at points along the axis of a dipole (along the same line that contains ±Q and –Q), the electric field has magnitude 1 2p E = 4p 0 r 3 for r W l (Fig. 21–45), where r is the distance from a point to the center of the dipole. (b) In what direction does B E point?

General Problems 68. How close must two electrons be if the electric force between them is equal to the weight of either at the Earth's surface? 69. Given that the human body is mostly made of water, estimate the total amount of positive charge in a 65-kg person. 70. A 3.0-g copper penny has a positive charge of 38 mC. What fraction of its electrons has it lost? 71. Measurements indicate that there is an electric field surrounding the Earth. Its magnitude is about 150 NC at the Earth's surface and points inward toward the Earth's center. What is the magnitude of the electric charge on the Earth? Is it positive or negative? [Hint: The electric field outside a uniformly charged sphere is the same as if all the charge were concentrated at its center.] 72. (a) The electric field near the Earth's surface has magnitude of about 150 NC. What is the acceleration experienced by an electron near the surface of the Earth? (b) What about a proton? (c) Calculate the ratio of each acceleration to g = 9.8 ms2. 73. A water droplet of radius 0.018 mm remains stationary in the air. If the downward-directed electric field of the Earth is 150 NC, how many excess electron charges must the water droplet have? 74. Estimate the net force between the CO group and the HN group shown in Fig. 21–70. The C and O have charges & 0.40e, and the H and N have charges & 0.20e, where e = 1.6 * 10–19 C. [Hint: Do not include the "internal" forces between C and O, or between H and N.]

C FIGURE 21–70 Problem 74.

0.12 nm

−

H 0.28 nm

−

0.10 nm

75. Suppose that electrical attraction, rather than gravity, were responsible for holding the Moon in orbit around the Earth. If equal and opposite charges Q were placed on the Earth and the Moon, what should be the value of Q to maintain the present orbit? Use data given on the inside front cover of this book. Treat the Earth and Moon as point particles. 76. In a simple model of the hydrogen atom, the electron revolves in a circular orbit around the proton with a speed of 2.2 * 106 ms. Determine the radius of the electron's orbit. [Hint: See Chapter 5 on circular motion.] 77. A positive point charge Q1 = 2.5 * 10–5 C is fixed at the origin of coordinates, and a negative point charge Q2 = – 5.0 * 10–6 C is fixed to the x axis at x = ±2.0 m. Find the location of the place(s) along the x axis where the electric field due to these two charges is zero. 78. When clothes are removed from a dryer, a 40-g sock is stuck to a sweater, even with the sock clinging to the sweater's underside. Estimate the minimum attractive force between the sock and the sweater. Then estimate the minimum charge on the sock and the sweater. Assume the charging came entirely from the sock rubbing against the sweater so that they have equal and opposite charges, and approximate the sweater as a flat sheet of uniform charge.

79. A small lead sphere is encased in insulating plastic and suspended vertically from an ideal spring (spring constant k = 126 Nm) as in Fig. 21–71. The total mass of the coated sphere is 0.650 kg, and its center lies 15.0 cm above a tabletop when in equilibrium. The sphere is pulled down 5.00 cm below equilibrium, an electric charge Q = –3.00 * 10–6 C is deposited on it, and then it is released. Using what you know about harmonic oscillation, write 10.0 cm an expression for the electric field strength as a function of time that 15.0 cm would be measured at the point on P the tabletop (P) directly below the sphere. FIGURE 21–71 Problem 79. 80. A large electroscope is made with "leaves" that are 78-cmlong wires with tiny 24-g spheres at the ends. When charged, nearly all the charge resides on the spheres. If the wires each make a 26° angle with the vertical (Fig. 21–72), what total charge Q 26° 26° must have been applied to the 78 cm 78 cm electroscope? Ignore the mass of the wires. FIGURE 21–72 Q Q Problem 80. 2 2 81. Dry air will break down and generate a spark if the electric field exceeds about 3 * 106 NC. How much charge could be packed onto a green pea (diameter 0.75 cm) before the pea spontaneously discharges? [Hint: Eqs. 21–4 work outside a sphere if r is measured from its center.] 82. Two point charges, Q1 = –6.7 mC and Q2 = 1.8 mC, are located between two oppositely charged parallel plates, as shown in Fig. 21–73. The two charges are separated by a distance of x = 0.34 m. Assume that the − + electric field produced by the charged − + plates is uniform and equal to + Q Q2 − 1 E = 73,000 NC. Calculate the net electro− + − + static force on Q1 and give its direction. −

− FIGURE 21–73 x Problem 82. 83. Packing material made of pieces of foamed polystyrene can easily become charged and stick to each other. Given that the density of this material is about 35 kgm3, estimate how much charge might be on a 2.0-cm-diameter foamed polystyrene sphere, assuming the electric force between two spheres stuck together is equal to the weight of one sphere. 84. One type of electric quadrupole consists of two dipoles placed end to end with their negative charges (say) overlapping; that is, in the center is –2Q flanked (on a line) by a ±Q to either side (Fig. 21–74). Determine the B electric field E at points along the perpendicular bisector and show that E decreases as 1r4. Measure r from the –2Q charge and assume r W l. +

FIGURE 21–74 Problem 84.

l +Q

l −2Q

General Problems

589

85. Suppose electrons enter a uniform electric field midway between two plates at an angle u0 to the horizontal, as shown in Fig. 21–75. The path is symmetrical, so they leave at the same angle u0 and just barely miss the top plate. What is u0? Ignore fringing of the field. 6.0 cm 1.0 cm u0 u0 E = 3.8 × 103 N/C FIGURE 21–75 Problem 85. 86. An electron moves in a circle of radius r around a very long uniformly charged wire in a vacuum chamber, as shown in Fig. 21–76. The charge density on the wire is l = 0.14 mCm. (a) What is the electric field at the electron (magnitude and direction in terms of r and l)? (b) What is the speed of the electron? l = 0.14 mC/m r FIGURE 21–76 Problem 86. 87. Three very large square planes of charge are arranged as shown (on edge) in Fig. 21–77. From left to right, the planes have charge densities per unit area of – 0.50 mCm2, ± 0.25 mCm2, and – 0.35 mCm2. Find the total electric field (direction and magnitude) at the points A, B, C, and D. Assume the plates are much larger than A B C D the distance AD. FIGURE 21–77 Problem 87. 88. A point charge (m = 1.0 g) at the end of an insulating cord of length 55 cm is observed to be in equilibrium in a uniform horizontal electric field of 15,000 NC, when the pendulum's position is as shown in Fig. 21–78, with the charge 12 cm above the lowest (vertical) position. If the field l = 55 cm points to the right in Fig. 21–78, u determine the magnitude and sign B Q E of the point charge. m FIGURE 21–78 Problem 88.

12 cm

89. Four equal positive point charges, each of charge 8.0 mC, are at the corners of a square of side 9.2 cm. What charge should be placed at the center of the square so that all charges are at equilibrium? Is this a stable or unstable equilibrium (Section 12–3) in the plane? 90. Two small, identical conducting spheres A and B are a distance R apart; each carries the same charge Q. (a) What is the force sphere B exerts on sphere A? (b) An identical sphere with zero charge, sphere C, makes contact with sphere B and is then moved very far away. What is the net force now acting on sphere A? (c) Sphere C is brought back and now makes contact with sphere A and is then moved far away. What is the force on sphere A in this third case?

91. A point charge of mass 0.210 kg, and net charge ±0.340 mC, hangs at rest at the end of an insulating cord above a large sheet of charge. The horizontal sheet of fixed uniform charge creates a uniform vertical electric field in the vicinity of the point charge. The tension in the cord is measured to be 5.18 N. (a) Calculate the magnitude and direction of the electric field due to the sheet of charge (Fig. 21–79). (b) What is the surface charge density s (Cm2) on gB the sheet? Q = 0.340 µC m = 0.210 kg FIGURE 21–79 Uniform sheet of charge Problem 91. 92. A one-dimensional row of positive ions, each with charge ±Q and separated from its neighbors by a distance d, occupies the right-hand half of the x axis. That is, there is a ±Q charge at x = 0, x = ±d, x = ±2d, x = ±3d, and so on out to q. (a) If an electron is placed at the position x = –d, determine F, the magnitude of force that this row of charges exerts on the electron. (b) If the electron is instead placed at x = –3d, what is the value of F? [Hint: The n=q 1 p2 infinite sum a 2 = , where n is a positive integer.] 6 n=1 n

* Numerical/Computer * 93. (III) A thin ring-shaped object of radius a contains a total charge Q uniformly distributed over its length. The electric field at a point on its axis a distance x from its center is given in Example 21–9 as Qx 1 . E = 3 4p 0 Ax2 + a2 B 2 (a) Take the derivative to find where on the x axis (x 7 0) Ex is a maximum. Assume Q = 6.00 mC and a = 10.0 cm. (b) Calculate the electric field for x = 0 to x = ±12.0 cm in steps of 0.1 cm, and make a graph of the electric field. Does the maximum of the graph coincide with the maximum of the electric field you obtained analytically? Also, calculate and graph the electric field (c) due to the ring, and (d) due to a point charge Q = 6.00 mC at the center of the ring. Make a single graph, from x = 0 (or x = 1.0 cm) out to x = 50.0 cm in 1.0 cm steps, with two curves of the electric fields, and show that both fields converge at large distances from the center. (e) At what distance does the electric field of the ring differ from that of the point charge by 10%? * 94. (III) An 8.00 mC charge is on the x axis of a coordinate system at x = ±5.00 cm. A –2.00 mC charge is at x = –5.00 cm. (a) Plot the x component of the electric field for points on the x axis from x = –30.0 cm to x = ±30.0 cm. The sign B of Ex is positive when E points to the right and negative when it points to the left. (b) Make a plot of Ex and Ey for points on the y axis from y = –30.0 to ±30.0 cm.

Answers to Exercises D: (a) No; (b) yes, midway between them. E: (d), if the two ± charges are not at opposite corners (use symmetry).

A: (e). B: 5 N. C: 1.2 N, to the right.

590

CHAPTER 21

Electric Charge and Electric Field

E # dA = Q B

encl 0

Q A2

Gauss's law is an elegant relation between electric charge and electric field. It is more general than Coulomb's law. Gauss's law involves B an integral of the electric field E at each point on a closed surface. The surface is only imaginary, and we choose the shape and placement of the surface so that we can evaluate the integral. In this drawing, two different 3-D surfaces are shown (one green, one blue), both enclosing a point charge Q. Gauss's law states B B B that the product E dA, where dA is an infinitesimal area of the surface, integrated over the entire surface, equals the charge enclosed by the surface Qencl divided by 0 . Both surfaces here enclose the same B B charge Q. Hence E dA will give the same result for both surfaces.

P T E R

C H

Gauss's Law CHAPTER-OPENING QUESTION—Guess now! A nonconducting sphere has a uniform charge density throughout. How does the magnitude of the electric field vary inside with distance from the center? (a) The electric field is zero throughout. (b) The electric field is constant but nonzero throughout. (c) The electric field is linearly increasing from the center to the outer edge. (d) The electric field is exponentially increasing from the center to the outer edge. (e) The electric field increases quadratically from the center to the outer edge. he great mathematician Karl Friedrich Gauss (1777–1855) developed an important relation, now known as Gauss's law, which we develop and discuss in this Chapter. It is a statement of the relation between electric charge and electric field and is a more general and elegant form of Coulomb's law. We can, in principle, determine the electric field due to any given distribution of electric charge using Coulomb's law. The total electric field at any point will be the vector sum (or integral) of contributions from all charges present (see Eq. 21–6). Except for some simple cases, the sum or integral can be quite complicated to evaluate. For situations in which an analytic solution (such as we carried out in the Examples of Sections 21–6 and 21–7) is not possible, a computer can be used. In some cases, however, the electric field due to a given charge distribution can be calculated more easily or more elegantly using Gauss's law, as we shall see in this Chapter. But the major importance of Gauss's law is that it gives us additional insight into the nature of electrostatic fields, and a more general relationship between charge and field. Before discussing Gauss's law itself, we first discuss the concept of flux.

CONTENTS 22–1 Electric Flux 22–2 Gauss's Law 22–3 Applications of Gauss's Law

*22–4 Experimental Basis of Gauss's and Coulomb's Laws

591

Area A

d rpen

o ar

ar t icul

22–1

Gauss's law involves the concept of electric flux, which refers to the electric field B passing through a given area. For a uniform electric field E passing through an area A, as shown in Fig. 22–1a, the electric flux £ E is defined as £ E = EA cos u, where u is the angle between the electric field direction and a line drawn perpendicular to the area. The flux can be written equivalently as

(a)

E⊥

[E uniform] (22–1a)

where E⊥ = E cos u is the component of E along the perpendicular to the area (Fig. 22–1b) and, similarly, A ⊥ = A cos u is the projection of the area A perpenB dicular to the field E (Fig. 22–1c). B The area A of a surface can be represented by a vector A whose magnitude is A and whose direction is perpendicular to the surface, as shown in Fig. 22–1c. B B The angle u is the angle between E and A, so the electric flux can also be written

(b) B

£ E = E A.

A⊥

£ E = E⊥ A = EA⊥ = EA cos u,

(c)

Electric Flux

Area A

FIGURE 22–1 (a) A uniform electric B field E passing through a flat area A. (b) E⊥ = E cos u is the component B of E perpendicular to the plane of area A. (c) A ⊥ = A cos u is the projection (dashed) of the area A B perpendicular to the field E. FIGURE 22–2 Electric flux through a curved surface. One small area of B the surface, ¢A i , is indicated.

[E uniform] (22–1b)

Electric flux has a simple intuitive interpretation in terms of field lines. We mentioned in Section 21–8 that field lines can always be drawn so that the number (N) passing through unit area perpendicular to the field AA ⊥ B is proportional to the magnitude of the field (E): that is, E r NA ⊥ . Hence, N r EA ⊥ = £ E , so the flux through an area is proportional to the number of lines passing through that area. EXAMPLE 22–1 Electric flux. Calculate the electric flux through the rectangle shown in Fig. 22–1a. The rectangle is 10 cm by 20 cm, the electric field is uniform at 200 NC, and the angle u is 30°. B B APPROACH We use the definition of flux, £ E = E A = EA cos u. SOLUTION The electric flux is £ E = (200 NC)(0.10 m * 0.20 m) cos 30° = 3.5 N m2C. EXERCISE A Which of the following would cause a change in the electric flux through a circle lying in the xz plane where the electric field is (10 N) jˆ? (a) Changing the magnitude of the electric field. (b) Changing the size of the circle. (c) Tipping the circle so that it is lying in the xy plane. (d) All of the above. (e) None of the above.

In the more general case, when the electric field E is not uniform and the surface is not flat, Fig. 22–2, we divide up the chosen surface into n small elements of surface whose areas are ¢A1 , ¢A2 , p , ¢An . We choose the division so that each ¢Ai is small enough that (1) it can be considered flat, and (2) the electric field varies so little over this small area that it can be considered uniform. Then the electric flux through the entire surface is approximately

E B

∆ Ai

£E L

# a E i ¢A i ,

i=1 B

E d A. B

£E = B

E B

§ E d A, B

CHAPTER 22

(22–2)

Gauss's law involves the total flux through a closed surface—a surface of any shape that completely encloses a volume of space, such as that shown in Fig. 22–3. In this case, the net flux through the enclosing surface is given by £E =

592

where Ei is the field passing through ¢A i . In the limit as we let ¢A i S 0, the sum becomes an integral over the entire surface and the relation becomes mathematically exact:

FIGURE 22–3 Electric flux through a closed surface.

(22–3)

where the integral sign is written to indicate that the integral is over the value B of E at every point on an enclosing surface.

Up to now we have not been concerned with an ambiguity in the direction B B of the vector A or dA that represents a surface. For example, in Fig. 22–1c, the B vector A could point upward and to the right (as shown) or downward to the left and still be perpendicular to the surface. For a closed surface, we define (arbitrarily) B B the direction of A, or of dA, to point outward from the enclosed volume, Fig. 22–4. For an electric field line leaving the enclosed volume (on the right in Fig. 22–4), B B the angle u between E and dA must be less than p2 ( = 90°); hence cos u 7 0. For a line entering the volume (on the left in Fig. 22–4) u 7 p2; hence cos u 6 0. Hence, flux entering the enclosed volume is negative A E cos u dA 6 0B, whereas flux leaving the volume is positive. Consequently, Eq. 22–3 gives the net flux out of the volume. If £ E is negative, there is a net flux into the volume. In Figs. 22–3 and 22–4, each field line that enters the volume also leaves the B B volume. Hence £ E = E dA = 0. There is no net flux into or out of this enclosed B B surface. The flux, E dA, will be nonzero only if one or more lines start or end within the surface. Since electric field lines start and stop only on electric charges, the flux will be nonzero only if the surface encloses a net charge. For example, the surface labeled A 1 in Fig. 22–5 encloses a positive charge and there is a net outward flux through this surface A £ E 7 0B. The surface A 2 encloses an equal magnitude negative charge and there is a net inward flux A£ E 6 0B. For the configuration shown in Fig. 22–6, the flux through the surface shown is negative (count the lines). The value of £ E depends on the charge enclosed by the surface, and this is what Gauss's law is all about.

22–2

dA B

p u (> ) 2

p u (< ) 2 B

E B

FIGURE 22–4 The direction of an B element of area dA is taken to point outward from an enclosed surface. FIGURE 22–5 An electric dipole. Flux through surface A 1 is positive. Flux through A 2 is negative.

Gauss's Law

The precise relation between the electric flux through a closed surface and the net charge Qencl enclosed within that surface is given by Gauss's law: Qencl B B , E dA = (22–4) 0 where 0 is the same constant (permittivity of free space) that appears in Coulomb's B law. The integral on the left is over the value of E on any closed surface, and we choose that surface for our convenience in any given situation. The charge Qencl is the net charge enclosed by that surface. It doesn't matter where or how the charge is distributed within the surface. Any charge outside this surface must not be included. A charge outside the chosen surface may affect the position of the electric field lines, but will not affect the net number of lines entering or leaving the surface. For example, Qencl for the gaussian surface A 1 in Fig. 22–5 would be the positive charge enclosed by A 1 ; the negative charge does contribute to the electric field at A 1 but it is not enclosed by surface A 1 and so is not included in Qencl . Now let us see how Gauss's law is related to Coulomb's law. First, we show that Coulomb's law follows from Gauss's law. In Fig. 22–7 we have a single isolated charge Q. For our "gaussian surface," we choose an imaginary sphere of radius r centered on the charge. Because Gauss's law is supposed to be valid for any surface, we have chosen one that will make our calculation easy. Because of the symmetry B of this (imaginary) sphere about the charge at its center, we know that E must have B the same magnitude at any point on the surface, and that E points radially outward B (inward for a negative charge) parallel to dA, an element of the surface area. Hence, we write the integral in Gauss's law as

FIGURE 22–6 Net flux through surface A is negative.

§ E dA B

§ E dA

A −2Q

FIGURE 22–7 A single point charge Q at the center of an imaginary sphere of radius r (our "gaussian surface"—that is, the closed surface we choose to use for applying Gauss's law in this case). B

= E dA = EA4pr B

dA B

since the surface area of a sphere of radius r is 4pr 2, and the magnitude of E is the same at all points on this spherical surface. Then Gauss's law becomes, with Qencl = Q, Q B B = E dA = EA4pr2 B 0

because E and dA are both perpendicular to the surface at each point, and cos u = 1. Solving for E we obtain Q , E = 4p 0 r2 which is the electric field form of Coulomb's law, Eq. 21–4b.

SECTION 22–2

593

Now let us do the reverse, and derive Gauss's law from Coulomb's law for static electric charges†. First we consider a single point charge Q surrounded by an imaginary spherical surface as in Fig. 22–7(and shown again, green, in Fig. 22-8). Coulomb's law tells us that the electric field at the spherical surface is E = A14p 0 BAQr2 B. Reversing the argument we just used, we have Q Q. 1 Q B B E dA = dA = A4pr 2 B = 2 2 4p 0 r 4p 0 r 0

Q A2

E FIGURE 22–8 A single point charge surrounded by a spherical surface, A 1 , and an irregular surface, A 2 .

This is Gauss's law, with Qencl = Q, and we derived it for the special case of a spherical surface enclosing a point charge at its center. But what about some other surface, such as the irregular surface labeled A 2 in Fig. 22–8? The same number of field lines (due to our charge Q) pass through surface A 2 , as pass through the spherical surface, A 1 . Therefore, because the flux through a surface is proportional to the number of lines through it as we saw in Section 22–1, the flux through A 2 is the same as through A 1 : Q. B B B B E dA = E dA = 0 A2 A1

Hence, we can expect that

§ E dA B

Q 0

would be valid for any surface surrounding a single point charge Q. Finally, let us look at the case of more than one charge. For each charge, Qi , enclosed by the chosen surface, Qi B B , Ei dA = 0

where Ei refers to the electric field produced by Qi alone. By the superposition B principle for electric fields (Section 21–6), the total field E is equal to the sum of B B the fields due to each separate charge, E = ©Ei . Hence Qi Qencl B B B B , E dA = A©Ei B # dA = © = 0 0

FIGURE 22–9 Electric flux through a closed surface. (Same as Fig. 22–3.) No electric charge is enclosed by this surface AQencl = 0B.

where Qencl = ©Qi is the total net charge enclosed within the surface. Thus we see, based on this simple argument, that Gauss's law follows from Coulomb's law for any distribution of static electric charge enclosed within a closed surface of any shape. The derivation of Gauss's law from Coulomb's law is valid for electric fields produced by static electric charges. We will see later that electric fields can also be produced by changing magnetic fields. Coulomb's law cannot be used to describe such electric fields. But Gauss's law is found to hold also for electric fields produced in any of these ways. Hence Gauss's law is a more general law than Coulomb's law. It holds for any electric field whatsoever. Even for the case of static electric fields that we are considering in this Chapter, B it is important to recognize that E on the left side of Gauss's law is not necessarily due only to the charge Qencl that appears on the right. For example, in Fig. 22–9 B there is an electric field E at all points on the imaginary gaussian surface, but it is not due to the charge enclosed by the surface (which is Qencl = 0 in this case). The B electric field E which appears on the left side of Gauss's law is the total electric field at each point, on the gaussian surface chosen, not just that due to the charge Qencl , which appears on the right side. Gauss's law has been found to be valid for the total field at any surface. It tells us that any difference between the input and output flux of the electric field over any surface is due to charge within that surface. †

Note that Gauss's law would look more complicated in terms of the constant k = 14p 0 that we originally used in Coulomb's law (Eqs. 21–1 or 21–4a): Coulomb's law Gauss's law Q B B E = k 2 E dA = 4pkQ r Q B B 1 Q . E = E dA = 4p 0 r2 0

§ §

594

CHAPTER 22

Gauss's law has a simpler form using 0 ; Coulomb's law is simpler using k. The normal convention is to use 0 rather than k because Gauss's law is considered more general and therefore it is preferable to have it in simpler form.

CONCEPTUAL EXAMPLE 22–2 Flux from Gauss's law. Consider the two gaussian surfaces, A 1 and A 2 , shown in Fig. 22–10.The only charge present is the charge Q at the center of surface A 1 . What is the net flux through each surface, A 1 and A 2 ? RESPONSE The surface A 1 encloses the charge ±Q. By Gauss's law, the net flux through A 1 is then Q 0 . For surface A 2 , the charge ±Q is outside the surface. Surface A 2 encloses zero net charge, so the net electric flux through A 2 is zero, by Gauss's law. Note that all field lines that enter the volume enclosed by surface A 2 also leave it. EXERCISE B A point charge Q is at the center of a spherical gaussian surface A. When a second charge Q is placed just outside A, the total flux through this spherical surface A is (a) unchanged, (b) doubled, (c) halved, (d) none of these. EXERCISE C Three 2.95 mC charges are in a small box. What is the net flux leaving the box? (a) 3.3 * 1012 N m2C, (b) 3.3 * 105 Nm2C, (c) 1.0 * 1012 Nm2C, (d) 1.0 * 106 Nm2C, (e) 6.7 * 106 Nm2C.

FIGURE 22–10 Example 22–2. Two gaussian surfaces.

We note that the integral in Gauss's law is often rather difficult to carry out in practice. We rarely need to do it except for some fairly simple situations that we now discuss.

22–3

Applications of Gauss's Law

Gauss's law is a very compact and elegant way to write the relation between electric charge and electric field. It also offers a simple way to determine the electric field when the charge distribution is simple andor possesses a high degree of symmetry. In order to apply Gauss's law, however, we must choose the "gaussian" surface very carefully (for the integral on the left side of Gauss's law) B so we can determine E. We normally try to think of a surface that has just the symmetry needed so that E will be constant on all or on parts of its surface. Sometimes we choose a surface so the flux through part of the surface is zero. EXAMPLE 22–3 Spherical conductor. A thin spherical shell of radius r0 possesses a total net charge Q that is uniformly distributed on it (Fig. 22–11). Determine the electric field at points (a) outside the shell, and (b) inside the shell. (c) What if the conductor were a solid sphere? APPROACH Because the charge is distributed symmetrically, the electric field must also be symmetric. Thus the field outside the sphere must be directed radially outward (inward if Q 6 0) and must depend only on r, not on angle (spherical coordinates). SOLUTION (a) The electric field will have the same magnitude at all points on an imaginary gaussian surface, if we choose that surface as a sphere of radius r Ar 7 r0 B concentric with the shell, and shown in Fig. 22–11 as the dashed circle A 1 . B Because E is perpendicular to this surface, the cosine of the angle between B B E and dA is always 1. Gauss's law then gives (with Qencl = Q in Eq. 22–4) Q, B B E dA = EA4pr2 B = 0 where 4pr 2 is the surface area of our sphere (gaussian surface) of radius r. Thus 1 Q. E = Cr 7 r0 D 4p 0 r2 Thus the field outside a uniformly charged spherical shell is the same as if all the charge were concentrated at the center as a point charge. (b) Inside the shell, the electric field must also be symmetric. So E must again have the same value at all points on a spherical gaussian surface (A 2 in Fig. 22–11) concentric with the shell. Thus E can be factored out of the integral and, with Qencl = 0 because the charge enclosed within the sphere A 2 is zero, we have

§ E dA B

FIGURE 22–11 Cross-sectional drawing of a thin spherical shell of radius r0 , carrying a net charge Q uniformly distributed. A 1 and A 2 represent two gaussian surfaces we B use to determine E. Example 22–3.

+ +

A1 +

E +

+ +

= EA4pr2 B = 0.

Hence

E = 0 Cr 6 r0 D inside a uniform spherical shell of charge. (c) These same results also apply to a uniformly charged solid spherical conductor, since all the charge would lie in a thin layer at the surface.

SECTION 22–3

595

EXERCISE D A charge Q is placed on a hollow metal ball. We saw in Chapter 21 that the charge is all on the surface of the ball because metal is a conductor. How does the charge distribute itself on the ball? (a) Half on the inside surface and half on the outside surface. (b) Part on each surface in inverse proportion to the two radii. (c) Part on each surface but with a more complicated dependence on the radii than in answer (b). (d) All on the inside surface. (e) All on the outside surface.

FIGURE 22–12 A solid sphere of uniform charge density. Example 22–4.

EXAMPLE 22–4 Solid sphere of charge. An electric charge Q is distributed uniformly throughout a nonconducting sphere of radius r0 , Fig. 22–12. Determine the electric field (a) outside the sphere Ar 7 r0 B and (b) inside the sphere Ar 6 r0 B.

APPROACH Since the charge is distributed symmetrically in the sphere, the B electric field at all points must again be symmetric. E depends only on r and is directed radially outward (or inward if Q 6 0). SOLUTION (a) For our gaussian surface we choose a sphere of radius r Ar 7 r0 B, labeled A 1 in Fig. 22–12. Since E depends only on r, Gauss's law gives, with Qencl = Q, Q B B E dA = EA4pr2 B = 0 or 1 Q. E = 4p 0 r2 Again, the field outside a spherically symmetric distribution of charge is the same as that for a point charge of the same magnitude located at the center of the sphere. (b) Inside the sphere, we choose for our gaussian surface a concentric sphere of B radius r Ar 6 r0 B, labeled A 2 in Fig. 22–12. From symmetry, the magnitude of E is B the same at all points on A 2 , and E is perpendicular to the surface, so

§ E dA B

= EA4pr2 B.

We must equate this to Qencl 0 where Qencl is the charge enclosed by A 2 . Qencl is not the total charge Q but only a portion of it. We define the charge density, rE , as the charge per unit volume ArE = dQdVB, and here we are given that rE = constant. So the charge enclosed by the gaussian surface A 2 , a sphere of radius r, is FIGURE 22–13 Magnitude of the electric field as a function of the distance r from the center of a uniformly charged solid sphere. Q E= 1 4pe0 r02

4 3 3 pr rE 3 3 pr 0 rE

Qencl = ¢ 4

EA4pr2 B = or

r3 Q. r 30

Hence, from Gauss's law,

E = 0

≤ Q =

Qencl r3 Q = 3 0 r0 0

1 Q r. 4p 0 r30

Cr 6 r0 D

Thus the field increases linearly with r, until r = r0 . It then decreases as 1r 2, as plotted in Fig. 22–13. EXERCISE E Return to the Chapter-Opening Question, page 591, and answer it again now. Try to explain why you may have answered differently the first time.

The results in Example 22–4 would have been difficult to obtain from Coulomb's law by integrating over the sphere. Using Gauss's law and the symmetry of the situation, this result is obtained rather easily, and shows the great power of Gauss's law. However, its use in this way is limited mainly to cases where the charge distribution has a high degree of symmetry. In such cases, we choose a simple surface on which E = constant, so the integration is simple. Gauss's law holds, of course, for any surface.

596

CHAPTER 22

Gauss's Law

EXAMPLE 22–5 Nonuniformly charged solid sphere. Suppose the charge density of the solid sphere in Fig. 22–12, Example 22–4, is given by rE = ar 2, where a is a constant. (a) Find a in terms of the total charge Q on the sphere and its radius r0 . (b) Find the electric field as a function of r inside the sphere.

APPROACH We divide the sphere up into concentric thin shells of thickness dr as shown in Fig. 22–14, and integrate (a) setting Q = rE dV and (b) using Gauss's law. SOLUTION (a) A thin shell of radius r and thickness dr (Fig. 22–14) has volume dV = 4pr2 dr. The total charge is given by

Q =

rE dV =

Aar 2 BA4pr 2 drB = 4pa

r 4 dr =

4pa 5 r . 5 0

5Q4pr 50 .

Thus a = (b) To find E inside the sphere at distance r from its center, we apply Gauss's law to an imaginary sphere of radius r which will enclose a charge

Qencl =

r E dV =

Aar 2 B 4pr2 dr =

FIGURE 22–14 Example 22–5.

5Q 2 r5 . 2 r ≤ 4pr dr = Q 4pr50 r 50

By symmetry, E will be the same at all points on the surface of a sphere of radius r, so Gauss's law gives Qencl 0 r5 , (E)A4pr2 B = Q 0 r50

§ E dA B

E =

Qr3 . 4p 0 r 50

EXAMPLE 22–6 Long uniform line of charge. A very long straight wire possesses a uniform positive charge per unit length, l. Calculate the electric field at points near (but outside) the wire, far from the ends. APPROACH Because of the symmetry, we expect the field to be directed radially outward and to depend only on the perpendicular distance, R, from the wire. Because of the cylindrical symmetry, the field will be the same at all points on a B gaussian surface that is a cylinder with the wire along its axis, Fig. 22–15. E is perpendicular to this surface at all points. For Gauss's law, we need a closed B surface, so we include the flat ends of the cylinder. Since E is parallel to the ends, B B there is no flux through the ends (the cosine of the angle between E and dA on the ends is cos 90° = 0). SOLUTION For our chosen gaussian surface Gauss's law gives Qencl ll , B B E dA = E(2pRl) = = 0 0

where l is the length of our chosen gaussian surface (l V length of wire), and 2pR is its circumference. Hence 1 l. E = 2p 0 R NOTE This is the same result we found in Example 21–11 using Coulomb's law (we used x there instead of R), but here it took much less effort. Again we see the great power of Gauss's law.† NOTE Recall from Chapter 10, Fig. 10–2, that we use R for the distance of a particle from an axis (cylindrical symmetry), but lower case r for the distance from a point (usually the origin 0).

FIGURE 22–15 Calculation of E due to a very long line of charge. Example 22–6. B

E R

+ + + + + + + + + + + + + + + +

†

But note that the method of Example 21–11 allows calculation of E also for a short line of charge by using the appropriate limits for the integral, whereas Gauss's law is not readily adapted due to lack of symmetry.

SECTION 22–3

Applications of Gauss's Law

597

EXAMPLE 22–7 Infinite plane of charge. Charge is distributed uniformly, with a surface charge density s (s = charge per unit area = dQdA), over a very large but very thin nonconducting flat plane surface. Determine the electric field at points near the plane.

FIGURE 22–16 Calculation of the electric field outside a large uniformly charged nonconducting plane surface. Example 22–7.

APPROACH We choose as our gaussian surface a small closed cylinder whose axis is perpendicular to the plane and which extends through the plane as shown B in Fig. 22–16. Because of the symmetry, we expect E to be directed perpendicular to the plane on both sides as shown, and to be uniform over the end caps of the cylinder, each of whose area is A. SOLUTION Since no flux passes through the curved sides of our chosen cylindrical surface, all the flux is through the two end caps. So Gauss's law gives

§ E dA B

= 2EA =

Qencl sA , = 0 0

where Qencl = sA is the charge enclosed by our gaussian cylinder. The electric field is then s . E = 2 0 NOTE This is the same result we obtained much more laboriously in Chapter 21, Eq. 21–7. The field is uniform for points far from the ends of the plane, and close to its surface. EXAMPLE 22–8 Electric field near any conducting surface. Show that the electric field just outside the surface of any good conductor of arbitrary shape is given by s, E = 0 where s is the surface charge density on the conductor's surface at that point.

FIGURE 22–17 Electric field near surface of a conductor. Example 22–8.

so that

E =

C A U T I O N

When is E = s 0 and when is E = s2 0

598

APPROACH We choose as our gaussian surface a small cylindrical box, as we did in the previous Example. We choose the cylinder to be very small in height, so that one of its circular ends is just above the conductor (Fig. 22–17). The other end is just below the conductor's surface, and the sides are perpendicular to it. SOLUTION The electric field is zero inside a conductor and is perpendicular to the surface just outside it (Section 21–9), so electric flux passes only through the outside end of our cylindrical box; no flux passes through the short sides or inside end. We choose the area A (of the flat cylinder end) small enough so that E is essentially uniform over it. Then Gauss's law gives Qencl sA , B B E dA = EA = = 0 0

CHAPTER 22

Gauss's Law

s. 0

[at surface of conductor] (22–5)

NOTE This useful result applies for a conductor of any shape. Why is it that the field outside a large plane nonconductor is E = s2 0 (Example 22–7) whereas outside a conductor it is E = s 0 (Example 22–8)? The reason for the factor of 2 comes not from conductor verses nonconductor. It comes instead from how we define charge per unit area s. For a thin flat nonconductor, Fig. 22–16, the charge may be distributed throughout the volume (not only on the surface, as for a conductor). The charge per unit area s represents all the charge throughout the thickness of the thin nonconductor. Also our gaussian surface has its ends outside the nonconductor on each side, so as to include all this charge.

For a conductor, on the other hand, the charge accumulates on the outer surfaces only. For a thin flat conductor, as shown in Fig. 22–18, the charge accumulates on both surfaces, and using the same small gaussian surface we did in Fig. 22–17, with one end inside and the other end outside the conductor, we came up with the result, E = s 0 . If we defined s for a conductor, as we did for a nonconductor, s would represent the charge per area for the entire conductor. Then Fig. 22–18 would show s2 as the surface charge on each surface, and Gauss's law would B B give E dA = EA = (s2)A 0 = sA2 0 so E = s2 0 , just as for a nonconductor. We need to be careful about how we define charge per unit area s. We saw in Section 21–9 that in the static situation, the electric field inside any conductor must be zero even if it has a net charge. (Otherwise, the free charges in the conductor would move—until the net force on each, B and hence E, were zero.) We also mentioned there that any net electric charge on a conductor must all reside on its outer surface. This is readily shown using Gauss's law. Consider any charged conductor of any shape, such as that shown in Fig. 22–19, which carries a net charge Q. We choose the gaussian surface, shown dashed in the diagram, so that it all lies just below the surface of the conductor and encloses essentially the whole volume of the conductor. Our gaussian surface can be arbitrarily close to the surface, but still inside the conductor. The electric field is zero at all points on this gaussian surface since it is inside the conductor. Hence, from Gauss's law, Eq. 22–4, the net charge within the surface must be zero. Thus, there can be no net charge within the conductor. Any net charge must lie on the surface of the conductor. If there is an empty cavity inside a conductor, can charge accumulate on that (inner) surface too? As shown in Fig. 22–20, if we imagine a gaussian surface (shown dashed) just inside the conductor above the cavity, we B know that E must be zero everywhere on this surface since it is inside the conductor. Hence, by Gauss's law, there can be no net charge at the surface of the cavity. But what if the cavity is not empty and there is a charge inside it?

FIGURE 22–18 Thin flat charged conductor with surface charge density s at each surface. For the conductor as a whole, the charge density is s¿ = 2s. FIGURE 22–19 An insulated charged conductor of arbitrary shape, showing a gaussian surface (dashed) just below the surface of the conductor. Surface of conductor

Gaussian surface

Q Empty cavity

Conductor (net charge Q)

Gaussian surface

FIGURE 22–20 An empty cavity inside a charged conductor carries zero net charge.

FIGURE 22–21 Example 22–9.

CONCEPTUAL EXAMPLE 22–9 Conductor with charge inside a cavity. Suppose a conductor carries a net charge ±Q and contains a cavity, inside of which resides a point charge ±q. What can you say about the charges on the inner and outer surfaces of the conductor? RESPONSE As shown in Fig. 22–21, a gaussian surface just inside the conductor surrounding the cavity must contain zero net charge (E = 0 in a conductor). Thus a net charge of –q must exist on the cavity surface. The conductor itself carries a net charge ±Q, so its outer surface must carry a charge equal to Q + q. These results apply to a cavity of any shape. SECTION 22–3

Point charge + + + + + + Cavity + − − + + − +q − + − − + Gaussian + surface + + + + Conductor + + + + +

Applications of Gauss's Law

599

EXERCISE F Which of the following statements about Gauss's law is correct? (a) If we know the charge enclosed by a surface, we always know the electric field everywhere at the surface. (b) When finding the electric field with Gauss's law, we always use a sphere for the gaussian surface. (c) If we know the total flux through a surface, we also know the total charge inside the surface. (d) We can only use Gauss's law if the electric field is constant in space.

L SO VIN OBL E PR

Gauss's Law for Symmetric Charge Distributions 1. First identify the symmetry of the charge distribution: spherical, cylindrical, planar. This identification B should suggest a gaussian surface for which E will be constant andor zero on all or on parts of the surface: a sphere for spherical symmetry, a cylinder for cylindrical symmetry and a small cylinder or "pillbox" for planar symmetry. 2. Draw the appropriate gaussian surface making sure it passes through the point where you want to know the electric field.

FIGURE 22–22 (a) A charged conductor (metal ball) is lowered into an insulated metal can (a good conductor) carrying zero net charge. (b) The charged ball is touched to the can and all of its charge quickly flows to the outer surface of the can. (c) When the ball is then removed, it is found to carry zero net charge.

(a)

Insulator

(b)

(c)

600

CHAPTER 22

* 22–4

3. Use the symmetry of the charge distribution to B determine the direction of E at points on the gaussian surface. B B 4. Evaluate the flux, E dA. With an appropriate B B gaussian surface, the dot product E dA should be zero or equal to &E dA, with the magnitude of E being constant over all or parts of the surface. 5. Calculate the charge enclosed by the gaussian surface. Remember it's the enclosed charge that matters. Ignore all the charge outside the gaussian surface. 6. Equate the flux to the enclosed charge and solve for E.

Experimental Basis of Gauss's and Coulomb's Laws

Gauss's law predicts that any net charge on a conductor must lie only on its surface. But is this true in real life? Let us see how it can be verified experimentally. And in confirming this prediction of Gauss's law, Coulomb's law is also confirmed since the latter follows from Gauss's law, as we saw in Section 22–2. Indeed, the earliest observation that charge resides only on the outside of a conductor was recorded by Benjamin Franklin some 30 years before Coulomb stated his law. A simple experiment is illustrated in Fig. 22–22. A metal can with a small opening at the top rests on an insulator. The can, a conductor, is initially uncharged (Fig. 22–22a). A charged metal ball (also a conductor) is lowered by an insulating thread into the can, and is allowed to touch the can (Fig. 22–22b). The ball and can now form a single conductor. Gauss's law, as discussed above, predicts that all the charge will flow to the outer surface of the can. (The flow of charge in such situations does not occur instantaneously, but the time involved is usually negligible). These predictions are confirmed in experiments by (1) connecting an electroscope to the can, which will show that the can is charged, and (2) connecting an electroscope to the ball after it has been withdrawn from the can (Fig. 22–22c), which will show that the ball carries zero charge. The precision with which Coulomb's and Gauss's laws hold can be stated quantitatively by writing Coulomb's law as F = k

Q1 Q2 . r2 + d

For a perfect inverse-square law, d = 0. The most recent and precise experiments (1971) give d = (2.763.1) * 10 –16. Thus Coulomb's and Gauss's laws are found to be valid to an extremely high precision!

Gauss's Law

Summary The electric flux passing through a flat area A for a uniform B electric field E is B

£ E = E A.

(22–1b)

If the field is not uniform, the flux is determined from the integral

E dA. B

£E =

(22–2) B

The direction of the vector A or dA is chosen to be perpendicular to the surface whose area is A or dA, and points outward from an enclosed surface. The flux through a surface is proportional to the number of field lines passing through it.

Gauss's law states that the net flux passing through any closed surface is equal to the net charge Qencl enclosed by the surface divided by 0 :

§ E dA B

Qencl . 0

(22–4)

Gauss's law can in principle be used to determine the electric field due to a given charge distribution, but its usefulness is mainly limited to a small number of cases, usually where the charge distribution displays much symmetry. The real importance of Gauss's law is that it is a more general and elegant statement (than Coulomb's law) for the relation between electric charge and electric field. It is one of the basic equations of electromagnetism.

Questions 1. If the electric flux through a closed surface is zero, is the electric field necessarily zero at all points on the surface? B Explain. What about the converse: If E = 0 at all points on the surface is the flux through the surface zero? B B B 2. Is the electric field E in Gauss's law, E dA = Qencl 0 , created only by the charge Qencl ? 3. A point charge is surrounded by a spherical gaussian surface of radius r. If the sphere is replaced by a cube of side r, will £ E be larger, smaller, or the same? Explain. 4. What can you say about the flux through a closed surface that encloses an electric dipole? B 5. The electric field E is zero at all points on a closed surface; is there necessarily no net charge within the surface? If a surface encloses zero net charge, is the electric field necessarily zero at all points on the surface? 6. Define gravitational flux in analogy to electric flux. Are there "sources" and "sinks" for the gravitational field as there are for the electric field? Discuss. 7. Would Gauss's law be helpful in determining the electric field due to an electric dipole? 8. A spherical basketball (a nonconductor) is given a charge Q distributed uniformly over its surface. What can you say about the electric field inside the ball? A person now steps on the ball, collapsing it, and forcing most of the air out without altering the charge. What can you say about the field inside now? 9. In Example 22–6, it may seem that the electric field calculated is due only to the charge on the wire that is enclosed by the cylinder chosen as our gaussian surface. In fact, the entire charge along the whole length of the wire contributes to the field. Explain how the charge outside the cylindrical gaussian surface of Fig. 22–15 contributes to E at the gaussian surface. [Hint: Compare to what the field would be due to a short wire.]

10. Suppose the line of charge in Example 22–6 extended only a short way beyond the ends of the cylinder shown in Fig. 22–15. How would the result of Example 22–6 be altered? 11. A point charge Q is surrounded by a spherical surface of radius r0 , whose center is at C. Later, the charge is moved to the right a distance 12 r0 , but the sphere remains where it was, Fig. 22–23. How is the electric flux £ E through the sphere changed? Is the electric field at the surface of the r0 1 r 0 sphere changed? For each "yes" answer, describe the change.

FIGURE 22–23 Question 11. 12. A solid conductor carries a net positive charge Q. There is a hollow cavity within the conductor, at whose center is a negative point charge Q –q (Fig. 22–24). What is the charge on (a) the outer surface of the conductor and (b) the inner surface of the conductor's cavity? −q FIGURE 22–24 Question 12. 13. A point charge q is placed at the center of the cavity of a thin metal shell which is neutral. Will a charge Q placed outside the shell feel an electric force? Explain. 14. A small charged ball is inserted into a balloon. The balloon is then blown up slowly. Describe how the flux through the balloon's surface changes as the balloon is blown up. Consider both the total flux and the flux per unit surface area of the balloon.

Problems 22–1 Electric Flux 1. (I) A uniform electric field of magnitude 5.8 * 102 NC passes through a circle of radius 13 cm. What is the electric flux through the circle when its face is (a) perpendicular to the field lines, (b) at 45° to the field lines, and (c) parallel to the field lines?

2. (I) The Earth possesses an electric field of (average) magnitude 150 NC near its surface. The field points radially inward. Calculate the net electric flux outward through a spherical surface surrounding, and just beyond, the Earth's surface.

Problems

601

3. (II) A cube of side l is placed in a uniform field E0 with edges parallel to the field lines. (a) What is the net flux through the cube? (b) What is the flux through each of its six faces? B 4. (II) A uniform field E is parallel to the axis of a hollow hemisphere of radius r, Fig. 22–25. (a) What is the electric flux through the hemispherical surface? (b) What is the result B if E is instead perpendicular to the Axis B axis? E FIGURE 22–25 Problem 4.

5. (I) The total electric flux from a cubical box 28.0 cm on a side is 1.84 * 103 N m2C. What charge is enclosed by the box? 6. (I) Figure 22–26 shows five closed surfaces that surround various charges in a plane, as indicated. Determine the electric flux through each surface, S4 S1 , S2 , S3 , S4 , and S5 . The surfaces are flat "pillbox" surfaces that extend only S5 slightly above and below +2Q +Q the plane in which the S1 charges lie. −3Q

7. (II) In Fig. 22–27, two objects, O1 and O2 , have charges ±1.0 mC and – 2.0 mC respectively, and a third object, O3 , is electrically neutral. (a) What is the electric flux through the surface A 1 that encloses all the three objects? (b) What is the electric flux through the surface A 2 that encloses the third object only? A1 O3

O1 +1.0 mC FIGURE 22–27 Problem 7.

O2 −2.0 mC

8. (II) A ring of charge with uniform charge density is completely enclosed in a hollow donut shape. An exact copy of the ring is completely enclosed in a hollow sphere. What is the ratio of the flux out of the donut shape to that out of the sphere? 9. (II) In a certain region of space, the electric field is constant in direction (say horizontal, in the x direction), but its magnitude decreases from E = 560 NC at x = 0 to E = 410 NC at x = 25 m. Determine the charge within a cubical box of side l = 25 m, where the box is oriented so that four of its sides are parallel x=0 x = 25 m to the field lines (Fig. 22–28). B

E FIGURE 22–28 Problem 9.

E 25 m

10. (II) A point charge Q is placed at the center of a cube of side l. What is the flux through one face of the cube?

602

CHAPTER 22

Gauss's Law

22–3 Applications of Gauss's Law 12. (I) Draw the electric field lines around a negatively charged metal egg. 13. (I) The field just outside a 3.50-cm-radius metal ball is 6.25 * 102 NC and points toward the ball. What charge resides on the ball? 14. (I) Starting from the result of Example 22–3, show that the electric field just outside a uniformly charged spherical conductor is E = s 0 , consistent with Example 22–8.

22–2 Gauss's Law

FIGURE 22–26 Problem 6.

11. (II) A 15.0-cm-long uniformly charged plastic rod is sealed inside a plastic bag. The total electric flux leaving the bag is 7.3 * 105 Nm2C. What is the linear charge density on the rod?

15. (I) A long thin wire, hundreds of meters long, carries a uniformly distributed charge of –7.2 mC per meter of length. Estimate the magnitude and direction of the electric field at points (a) 5.0 m and (b) 1.5 m perpendicular from the center of the wire. 16. (I) A metal globe has 1.50 mC of charge put on it at the north pole. Then –3.00 mC of charge is applied to the south pole. Draw the field lines for this system after it has come to equilibrium. 17. (II) A nonconducting sphere is made of two layers. The innermost section has a radius of 6.0 cm and a uniform charge density of –5.0 Cm3. The outer layer has a uniform charge density of ±8.0 Cm3 and extends from an inner radius of 6.0 cm to an outer radius of 12.0 cm. Determine the electric field for (a) 0 6 r 6 6.0 cm, (b) 6.0 cm 6 r 6 12.0 cm, and (c) 12.0 cm 6 r 6 50.0 cm. (d) Plot the magnitude of the electric field for 0 6 r 6 50.0 cm. Is the field continuous at the edges of the layers? 18. (II) A solid metal sphere of radius 3.00 m carries a total charge of –5.50 mC. What is the magnitude of the electric field at a distance from the sphere's center of (a) 0.250 m, (b) 2.90 m, (c) 3.10 m, and (d) 8.00 m? How would the answers differ if the sphere were (e) a thin shell, or (f) a solid nonconductor uniformly charged throughout? 19. (II) A 15.0-cm-diameter nonconducting sphere carries a total charge of 2.25 mC distributed uniformly throughout its volume. Graph the electric field E as a function of the distance r from the center of the sphere from r = 0 to r = 30.0 cm. 20. (II) A flat square sheet of thin aluminum foil, 25 cm on a side, carries a uniformly distributed 275 nC charge. What, approximately, is the electric field (a) 1.0 cm above the center of the sheet and (b) 15 m above the center of the sheet? 21. (II) A spherical cavity of radius 4.50 cm is at the center of a metal sphere of radius 18.0 cm. A point charge Q = 5.50 mC rests at the very center of the cavity, whereas the metal conductor carries no net charge. Determine the electric field at a point (a) 3.00 cm from the center of the cavity, (b) 6.00 cm from the center of the cavity, (c) 30.0 cm from the center. 22. (II) A point charge Q rests at the center of an uncharged thin spherical conducting shell. What is the electric field E as a function of r (a) for r less than the radius of the shell, (b) inside the shell, and (c) beyond the shell? (d) Does the shell affect the field due to Q alone? Does the charge Q affect the shell?

23. (II) A solid metal cube has a spherical cavity at its center as shown in Fig. 22–29. At the center of the cavity there is a point charge Q = ± 8.00 mC. The metal cube carries a net charge q = – 6.10 mC (not including Q). Determine (a) the total charge on the surface of the spherical cavity and Q (b) the total charge on the outer surface of the cube. FIGURE 22–29 Problem 23. 24. (II) Two large, flat metal plates are separated by a distance that is very small compared to their height and width. The conductors are given equal but opposite uniform surface charge densities &s. Ignore edge effects and use Gauss's law to show (a) that for points far from the edges, the electric field between the plates is E = s 0 and (b) that outside the plates on either side the field is zero. (c) How would your results be altered if the two plates were nonconductors? (See Fig. 22–30). FIGURE 22–30 Problems 24, 25, and 26.

+s −s

25. (II) Suppose the two conducting plates in Problem 24 have the same sign and magnitude of charge. What then will be the electric field (a) between them and (b) outside them on either side? (c) What if the plates are nonconducting? 26. (II) The electric field between two square metal plates is 160 NC. The plates are 1.0 m on a side and are separated by 3.0 cm, as in Fig. 22–30. What is the charge on each plate? Neglect edge effects. 27. (II) Two thin concentric spherical shells of radii r1 and r2 Ar1 6 r2 B contain uniform surface charge densities s1 and s2 , respectively (see Fig. 22–31). Determine the electric field for (a) 0 6 r 6 r1 , (b) r1 6 r 6 r2 , and (c) r 7 r2 . (d) Under what conditions r2 will E = 0 for r 7 r2 ? (e) Under what conditions will E = 0 for r1 6 r 6 r2 ? r1 r Neglect the thickness of the shells. s1 FIGURE 22–31 Two spherical s2 shells (Problem 27).

28. (II) A spherical rubber balloon carries a total charge Q uniformly distributed on its surface. At t = 0 the nonconducting balloon has radius r0 and the balloon is then slowly blown up so that r increases linearly to 2r0 in a time T. Determine the electric field as a function of time (a) just outside the balloon surface and (b) at r = 3.2r0 . 29. (II) Suppose the nonconducting sphere of Example 22–4 has a spherical cavity of radius r1 centered at the sphere's center (Fig. 22–32). Assuming the charge Q is distributed uniformly in the "shell" (between r = r1 and r = r0), determine the electric field as a function of r for (a) 0 6 r 6 r1 , (b) r1 6 r 6 r0 , Q and (c) r 7 r0 . r0 r1 FIGURE 22–32 Problems 29, 30, 31, and 44. 30. (II) Suppose in Fig. 22–32, Problem 29, there is also a charge q at the center of the cavity. Determine the electric field for (a) 0 6 r 6 r1 , (b) r1 6 r 6 r0 , and (c) r 7 r0 .

31. (II) Suppose the thick spherical shell of Problem 29 is a conductor. It carries a total net charge Q and at its center there is a point charge q. What total charge is found on (a) the inner surface of the shell and (b) the outer surface of the shell? Determine the electric field for (c) 0 6 r 6 r1 , (d) r1 6 r 6 r0 , and (e) r 7 r0 . 32. (II) Suppose that at the center of the cavity inside the shell (charge Q) of Fig. 22–11 (and Example 22–3), there is a point charge q (Z &Q). Determine the electric field for (a) 0 6 r 6 r0 , and for (b) r 7 r0 . What are your answers if (c) q = Q and (d) q = –Q? 33. (II) A long cylindrical shell of radius R0 and length l AR0 V lB possesses a uniform surface charge density (charge per unit area) s (Fig. 22–33). Determine the electric field at points (a) outside the cylinder AR 7 R0 B and (b) inside the cylinder A0 6 R 6 R0 B; assume the points are far from the ends and not too far from the shell (R V l). R0 (c) Compare to the result for a long line of charge, Example 22–6. Neglect the thickness of shell. FIGURE 22–33 Problem 33. 34. (II) A very long solid nonconducting cylinder of radius R0 and length l AR0 V lB possesses a uniform volume charge density rE ACm3 B, Fig. 22–34. Determine the electric field at points (a) outside the cylinder AR 7 R0 B and (b) inside the cylinder AR 6 R0 B. Do only for points far from the R0 ends and for which R V l. FIGURE 22–34 Problem 34. 35. (II) A thin cylindrical shell of radius R1 is surrounded by a second concentric cylindrical shell of radius R2 (Fig. 22–35). The inner shell has a total charge ±Q and the outer shell –Q. Assuming the length l of the shells is much greater than R1 or R2 , determine the electric field as a function of R (the perpendicular distance from the common axis of the cylinders) for (a) 0 6 R 6 R1 , (b) R1 6 R 6 R2 , and (c) R 7 R2 . (d) What is the kinetic energy of an electron if it moves between (and concentric with) the shells in a circular orbit of radius AR1 + R2 B2? Neglect R2 thickness of shells. R R1 FIGURE 22–35 Problems 35, 36, and 37. 36. (II) A thin cylindrical shell of radius R1 = 6.5 cm is surrounded by a second cylindrical shell of radius R2 = 9.0 cm, as in Fig. 22–35. Both cylinders are 5.0 m long and the inner one carries a total charge Q1 = –0.88 mC and the outer one Q2 = ±1.56 mC. For points far from the ends of the cylinders, determine the electric field at a radial distance R from the central axis of (a) 3.0 cm, (b) 7.0 cm, and (c) 12.0 cm. 37. (II) (a) If an electron Am = 9.1 * 10–31 kgB escaped from the surface of the inner cylinder in Problem 36 (Fig. 22–35) with negligible speed, what would be its speed when it reached the outer cylinder? (b) If a proton Am = 1.67 * 10–27 kgB revolves in a circular orbit of radius R = 7.0 cm about the axis (i.e., between the cylinders), what must be its speed?

Problems

603

Gauss's Law

CHAPTER 22

604

FIGURE 22–38 Problem 42.

44. (III) Suppose the density of charge between r1 and r0 of the hollow sphere of Problem 29 (Fig. 22–32) varies as rE = r0 r1r. Determine the electric field as a function of r for (a) 0 6 r 6 r1 , (b) r1 6 r 6 r0 , and (c) r 7 r0 . (d) Plot E versus r from r = 0 to r = 2r0 . 45. (III) Suppose two thin flat plates measure 1.0 m * 1.0 m and are separated by 5.0 mm. They are oppositely charged with & 15 mC. (a) Estimate the total force exerted by one plate on the other (ignore edge effects). (b) How much work would be required to move the plates from 5.0 mm apart to 1.00 cm apart? 46. (III) A flat slab of nonconducting material (Fig. 22–40) carries a uniform charge per unit volume, rE . The slab has thickness d which is small compared to the y height and breadth of the slab. Determine the electric field as a function of x (a) inside the slab and (b) outside the slab (at distances much less than the slab's height or breadth). x 0 Take the origin at the center of the slab.

FIGURE 22–37 Problem 40. 41. (II) A flat ring (inner radius R0 , outer radius 4R0) is uniformly charged. In terms of the total charge Q, determine the electric field on the axis at points (a) 0.25R0 and (b) 75R0 from the center of the ring. [Hint: The ring can be replaced with two oppositely charged superposed disks.] 42. (II) An uncharged solid conducting sphere of radius r0 contains two spherical cavities of radii r1 and r2 , respectively. Point charge Q1 is then placed within the cavity of radius r1 and point charge r Q2 is placed within the cavity of radius r2 (Fig. 22–38). Determine the r0 resulting electric field Q1 r (magnitude and direc1 tion) at locations outside the solid sphere (r 7 r0) , where r is the distance r Q2 2 from its center.

FIGURE 22–39 Problem 43.

39. (II) A nonconducting sphere of radius r0 is uniformly charged with volume charge density rE . It is surrounded by a concentric metal (conducting) spherical shell of inner radius r1 and outer radius r2 , which carries a net charge ±Q. Determine the resulting electric field in the regions (a) 0 6 r 6 r0 , (b) r0 6 r 6 r1 , (c) r1 6 r 6 r2 , and (d) r 7 r2 where the radial distance r is measured from the center of the nonconducting sphere. 40. (II) A very long solid nonconducting cylinder of radius R1 is uniformly charged with charge density rE. It is surrounded by a cylindrical metal (conducting) tube of inner radius R2 and outer radius R3 , which has no net charge (cross-sectional view shown in Fig. 22–37). If the axes of the two cylinders are parallel, but displaced from each other by a distance d, determine the resulting electric field in the region R 7 R3 , where the R radial distance R is R2 measured from the R3 metal cylinder's axis. Assume d 6 AR2 - R1 B. R1 d

FIGURE 22–36 Problem 38.

43. (III) A very large (i.e., assume infinite) flat slab of nonconducting material has thickness d and a uniform volume charge density ±rE . (a) Show that a uniform electric field exists outside of this slab. Determine its magnitude E and its direction (relative to the slab's surface). (b) As shown in Fig. 22–39, the slab is now aligned so that one of its surfaces lies on the line y = x. At time t = 0, a pointlike particle (mass m, charge ±q) is located at y yx position Br = ±y0 jˆ and has B d velocity v = v0 iˆ. Show q, m y 0 that the particle will v0 collide with the slab if v0 212qy0 rEd 0 m. Ignore gravity. 0 x

38. (II) A very long solid nonconducting cylinder of radius R1 is uniformly charged with a charge density rE . It is surrounded by a concentric cylindrical tube of inner radius R2 and outer radius R3 as shown in Fig. 22–36, and it too carries a uniform charge density rE . Determine the electric field as a function of the distance R from the center of the cylinders for (a) 0 6 R 6 R1 , (b) R1 6 R 6 R2 , (c) R2 6 R 6 R3 , and (d) R 7 R3 . (e) If rE = 15 mCm3 and R1 = 12 R2 = 13 R3 = 5.0 cm, plot E as a function of R from R = 0 to R = 20.0 cm. Assume R1 the cylinders are very long R2 compared to R3 .

FIGURE 22–40 Problem 46.

47. (III) A flat slab of nonconducting material has thickness 2d, which is small compared to its height and breadth. Define the x axis to be along the direction of the slab's thickness with the origin at the center of the slab (Fig. 22–41). If the slab carries a volume y charge density rE(x) = –r0 in the region –d x 6 0, and rE(x) = ±r0 in the region 0 6 x ±d, determine the B electric field E as a function of x in the regions (a) outside the slab, (b) 0 6 x ± d, and (c) –d x 6 0. Let r0 be a x d 0 d positive constant.

FIGURE 22–41 Problem 47.

48. (III) An extremely long, solid nonconducting cylinder has a radius R0 . The charge density within the cylinder is a function of the distance R from the axis, given by rE (R) = r0(RR0)2. What is the electric field everywhere inside and outside the cylinder (far away from the ends) in terms of r0 and R0?

49. (III) Charge is distributed within a solid sphere of radius r0 in such a way that the charge density is a function of the radial position within the sphere of the form: rE (r) = r0(rr0). If the total charge within the sphere is Q (and positive), what is the electric field everywhere within the sphere in terms of Q, r0 , and the radial position r?

General Problems 50. A point charge Q is on the axis of a short cylinder at its center. The diameter of the cylinder is equal to its length l (Fig. 22–42). What is the total flux through the curved sides of the cylinder? [Hint: First calculate l = 2R0 the flux through the ends.] R0

I II III

FIGURE 22–44 Problem 58.

Q FIGURE 22–42 Problem 50. B

51. Write Gauss's law for the gravitational field g (see Section 6–6). 52. The Earth is surrounded by an electric field, pointing inward at every point, of magnitude E L 150 NC near the surface. (a) What is the net charge on the Earth? (b) How many excess electrons per square meter on the Earth's surface does this correspond to? 53. A cube of side l has one corner at the origin of coordinates, and extends along the positive x, y, and z axes. Suppose the B electric field in this region is given by E = (ay + b)jˆ. Determine the charge inside the cube. 54. A solid nonconducting sphere of radius r0 has a total charge Q which is distributed according to rE = br, where rE is the charge per unit volume, or charge density ACm3 B, and b is a constant. Determine (a) b in terms of Q, (b) the electric field at points inside the sphere, and (c) the electric field at points outside the sphere. 55. A point charge of 9.20 nC is located at the origin and a second charge of – 5.00 nC is located on the x axis at x = 2.75 cm. Calculate the electric flux through a sphere centered at the origin with radius 1.00 m. Repeat the calculation for a sphere of radius 2.00 m. 56. A point charge produces an electric flux of ±235 Nm2C through a gaussian sphere of radius 15.0 cm centered on the charge. (a) What is the flux through a gaussian sphere with a radius 27.5 cm? (b) What is the magnitude and sign of the charge? 57. A point charge Q is placed a distance r02 above the surface of an imaginary spherical surface of radius r0 (Fig. 22–43). (a) What is the electric flux through the sphere? (b) What range of values does E have at the surface of the sphere? B (c) Is E perpendicular to the sphere at all points? (d) Is Gauss's law useful for obtaining E at the Q 1r surface of the sphere? 0 2

r0 FIGURE 22–43 Problem 57.

58. Three large but thin charged sheets are parallel to each other as shown in Fig. 22–44. Sheet I has a total surface charge density of 6.5 nCm2, sheet II a charge of –2.0 nCm2, and sheet III a charge of 5.0 nCm2. Estimate the force per unit area on each sheet, in Nm2.

59. Neutral hydrogen can be modeled as a positive point charge ±1.6 * 10–19 C surrounded by a distribution of negative charge with volume density given by rE(r) = –Ae –2ra0 where a0 = 0.53 * 10 –10 m is called the Bohr radius, A is a constant such that the total amount of negative charge is –1.6 * 10–19 C, and e = 2.718 p is the base of the natural log. (a) What is the net charge inside a sphere of radius a0 ? (b) What is the strength of the electric field at a distance a 0 from the nucleus? [Hint: Do not confuse the exponential number e with the elementary charge e which uses the same symbol but has a completely different meaning and value Ae = 1.6 * 10 –19 CB.] 60. A very large thin plane has uniform surface charge density s. Touching it on the right (Fig. 22–45) is a long wide slab of thickness d with uniform volume charge density rE . Determine the electric field (a) to the left of the plane, (b) to the right of the slab, and (c) everywhere inside the slab. rE s d

FIGURE 22–45 Problem 60.

61. A sphere of radius r0 carries a volume charge density rE (Fig. 22–46). A spherical cavity of radius r02 is then scooped out and left empty, as shown. (a) What is the magnitude and direction of the electric field at point A? (b) What is the direction and magnitude of the electric field at point B? Points A and C are at the centers of the respective spheres. rE B

FIGURE 22–46 Problem 61. 62. Dry air will break down and generate a spark if the electric field exceeds about 3 * 106 NC. How much charge could be packed onto the surface of a green pea (diameter 0.75 cm) before the pea spontaneously discharges?

General Problems

605

63. Three very large sheets are separated by equal distances of 15.0 cm (Fig. 22–47). The first and third sheets are very thin and nonconducting and have charge per unit area s of ± 5.00 mCm2 and – 5.00 mCm2 respectively. The middle sheet is conducting but has no net charge. (a) What is the electric field inside the middle sheet? What is the electric field (b) between the left and middle sheets, and (c) between the middle and right sheets? (d) What is the charge density on the surface of the middle sheet facing the left sheet, and (e) on the surface facing the right sheet?

65. A conducting spherical shell (Fig. 22–49) has inner radius = 10.0 cm, outer radius = 15.0 cm, and has a ±3.0 mC point charge at the center. A charge of –3.0 mC is put on the conductor. (a) Where on the conductor does the 10.0 cm –3.0 mC end up? (b) What is the Q 15.0 cm electric field both inside and outside the shell? FIGURE 22–49 Problem 65.

s1 = +5.00 mC/m2 Qnet = 0 s3 = −5.00 mC/m2

15.0 cm

66. A hemisphere of radius R is placed in a charge-free region of space where a uniform electric field exists of magnitude E directed perpendicular to the hemisphere's circular base (Fig. 22–50). (a) Using the definition of £ E through an "open" surface, calculate (via explicit integration) the electric flux through the hemisphere. [Hint: In Fig. 22–50 you can see that, on the surface of a sphere, the infinitesimal area located between the angles u and u + du is dA = (2pR sin u)(R du) = 2pR2 sin u du.] (b) Choose an appropriate gaussian B E surface and use Gauss's law to much more easily obtain the same result dA 2pR2 sin u du for the electric flux through the hemisphere. u du

15.0 cm

FIGURE 22–47 Problem 63.

64. In a cubical volume, 0.70 m on a side, the electric field is

z z E = E0 a 1 + b ˆi + E0 a b jˆ a a B

where E0 = 0.125 NC and a = 0.70 m. The cube has its sides parallel to the coordinate axes, Fig. 22–48. Determine the net charge within the cube.

a dz

* Numerical/Computer * 67. (III) An electric field is given by

E = Ex0 e

a y

FIGURE 22–48 Problem 64.

FIGURE 22–50 Problem 66.

A: (d).

D: (e).

B: (a).

E: (c).

C: (d).

F: (c).

CHAPTER 22

Gauss's Law

i ± Ey0 e

x+y 2 b a

jˆ ,

where Ex0 = 50 NC, Ey0 = 25 NC, and a = 1.0 m. Given a cube with sides parallel to the coordinate axes, with one corner at the origin (as in Fig. 22–48), and with sides of length 1.0 m, estimate the flux out of the cube using a spreadsheet or other numerical method. How much total charge is enclosed by the cube?

Answers to Exercises

606

x+y 2 -a a b ˆ

We are used to voltage in our lives—a 12-volt car battery, 110 V or 220 V at home, 1.5 volt flashlight batteries, and so on. Here we see a Van de Graaff generator, whose voltage may reach 50,000 V or more. Voltage is the same as electric potential difference between two points. Electric potential is defined as the potential energy per unit charge. The children here, whose hair stands on end because each hair has received the same sign of charge, are not harmed by the voltage because the Van de Graaff cannot provide much current before the voltage drops. (It is current through the body that is harmful, as we will see later.)

P T E R

C H

Electric Potential CHAPTER-OPENING QUESTION—Guess now! Consider a pair of parallel plates with equal and opposite charge densities, s. Which of the following actions will increase the voltage between the plates (assuming fixed charge density)? (a) Moving the plates closer together. (b) Moving the plates apart. (c) Doubling the area of the plates. (d) Halving the area of the plates. e saw in Chapters 7 and 8 that the concept of energy was extremely useful in dealing with the subject of mechanics. The energy point of view is especially useful for electricity. It not only extends the law of conservation of energy, but it gives us another way to view electrical phenomena. Energy is also a powerful tool for solving Problems more easily in many cases than by using forces and electric fields.

W 23–1

CONTENTS

23–1 Electric Potential Energy and Potential Difference 23–2 Relation between Electric Potential and Electric Field 23–3 Electric Potential Due to Point Charges 23–4 Potential Due to Any Charge Distribution 23–5 Equipotential Surfaces 23–6 Electric Dipole Potential B 23–7 E Determined from V 23–8 Electrostatic Potential Energy; the Electron Volt 23–9 Cathode Ray Tube: TV and * Computer Monitors, Oscilloscope

Electric Potential Energy and Potential Difference

Electric Potential Energy To apply conservation of energy, we need to define electric potential energy as we did for other types of potential energy. As we saw in Chapter 8, potential energy can be defined only for a conservative force. The work done by a conservative force in moving an object between any two positions is independent of the path taken. The electrostatic force between any two charges (Eq. 21–1, F = kQ1 Q2r 2) is conservative since the dependence on position is just like the gravitational force, 1r2, which we saw in Section 8–7 is conservative. Hence we can define potential energy U for the electrostatic force.

607

−

High potential energy High potential

−

+ + +

− − −

−

Low potential energy Low potential

We saw in Chapter 8 that the change in potential energy between two points, a and b, equals the negative of the work done by the conservative force as an object moves from a to b: ¢U = –W. Thus we define the change in electric potential energy, Ub - Ua , when a point charge q moves from some point a to another point b, as the negative of the work done by the electric force as the charge moves from a to b. For example, consider the electric field between two equally but oppositely charged parallel plates; we B assume their separation is small compared to their width and height, so the field E will be uniform over most of the region, Fig. 23–1. Now consider a tiny positive point charge q placed at point a very near the positive plate as shown. This B charge q is so small it has no effect on E. If this charge q at point a is released, the electric force will do work on the charge and accelerate it toward the negative plate. The work W done by the electric field E to move the charge a distance d is W = Fd = qEd where we used Eq. 21–5, F = qE. The change in electric potential energy equals the negative of the work done by the electric force:

−

Ub - Ua = –W = –qEd

[uniform E] (23–1)

FIGURE 23–1 Work is done by the electric field in moving the positive charge q from position a to position b.

for this case of uniform electric field E. In the case illustrated, the potential energy decreases ( ¢U is negative); and as the charged particle accelerates from point a to point b in Fig. 23–1, the particle's kinetic energy K increases—by an equal amount. In accord with the conservation of energy, electric potential energy is transformed into kinetic energy, and the total energy is conserved. Note that the positive charge q has its greatest potential energy at point a, near the positive plate.† The reverse is true for a negative charge: its potential energy is greatest near the negative plate.

Electric Potential and Potential Difference In Chapter 21, we found it useful to define the electric field as the force per unit charge. Similarly, it is useful to define the electric potential (or simply the potential when "electric" is understood) as the electric potential energy per unit charge. Electric potential is given the symbol V. If a positive test charge q in an electric field has electric potential energy Ua at some point a (relative to some zero potential energy), the electric potential Va at this point is Ua . Va = (23–2a) q As we discussed in Chapter 8, only differences in potential energy are physically meaningful. Hence only the difference in potential, or the potential difference, between two points a and b (such as between a and b in Fig. 23–1) is measurable. When the electric force does positive work on a charge, the kinetic energy increases and the potential energy decreases. The difference in potential energy, Ub - Ua , is equal to the negative of the work, Wba , done by the electric field as the charge moves from a to b; so the potential difference Vba is Ub - Ua Wba . (23–2b) Vba = ¢V = Vb - Va = = – q q Note that electric potential, like electric field, does not depend on our test charge q. V depends on the other charges that create the field, not on q; q acquires potential energy by being in the potential V due to the other charges. We can see from our definition that the positive plate in Fig. 23–1 is at a higher potential than the negative plate. Thus a positively charged object moves naturally from a high potential to a low potential. A negative charge does the reverse. The unit of electric potential, and of potential difference, is joulescoulomb and is given a special name, the volt, in honor of Alessandro Volta (1745–1827) who is best known for inventing the electric battery. The volt is abbreviated V, so 1 V = 1 JC. Potential difference, since it is measured in volts, is often referred to as voltage. †

608

CHAPTER 23

At this point the charge has its greatest ability to do work (on some other object or system).

Electric Potential

If we wish to speak of the potential Va at some point a, we must be aware that Va depends on where the potential is chosen to be zero. The zero for electric potential in a given situation can be chosen arbitrarily, just as for potential energy, because only differences in potential energy can be measured. Often the ground, or a conductor connected directly to the ground (the Earth), is taken as zero potential, and other potentials are given with respect to ground. (Thus, a point where the voltage is 50 V is one where the difference of potential between it and ground is 50 V.) In other cases, as we shall see, we may choose the potential to be zero at an infinite distance (r = q). CONCEPTUAL EXAMPLE 23–1 A negative charge. Suppose a negative charge, such as an electron, is placed near the negative plate in Fig. 23–1, at point b, shown here in Fig. 23–2. If the electron is free to move, will its electric potential energy increase or decrease? How will the electric potential change? RESPONSE An electron released at point b will move toward the positive plate. As the electron moves toward the positive plate, its potential energy decreases as its kinetic energy gets larger, so Ua 6 Ub and ¢U = Ua - Ub 6 0. But note that the electron moves from point b at low potential to point a at higher potential: Vab = Va - Vb 7 0. (Potentials Va and Vb are due to the charges on the plates, not due to the electron.) The sign of ¢U and ¢V are opposite because of the negative charge.

High potential Va

−

+ +a +

Low potential Vb

−

b−

High PE for negative charge here

−

FIGURE 23–2 Central part of Fig. 23–1, showing a negative point charge near the negative plate, where its potential energy (pe) is high. Example 23–1. C A U T I O N

A negative charge has high potential energy when potential V is low

Because the electric potential difference is defined as the potential energy difference per unit charge, then the change in potential energy of a charge q when moved between two points a and b is ¢U = Ub - Ua = q AVb - Va B = qVba .

(23–3)

That is, if an object with charge q moves through a potential difference Vba , its potential energy changes by an amount qVba . For example, if the potential difference between the two plates in Fig. 23–1 is 6 V, then a ±1 C charge moved (say by an external force) from point b to point a will gain (1 C)(6 V) = 6 J of electric potential energy. (And it will lose 6 J of electric potential energy if it moves from a to b.) Similarly, a ±2 C charge will gain 12 J, and so on. Thus, electric potential difference is a measure of how much energy an electric charge can acquire in a given situation. And, since energy is the ability to do work, the electric potential difference is also a measure of how much work a given charge can do. The exact amount depends both on the potential difference and on the charge. To better understand electric potential, let's make a comparison to the gravitational case when a rock falls from the top of a cliff. The greater the height, h, of a cliff, the more potential energy ( = mgh) the rock has at the top of the cliff, relative to the bottom, and the more kinetic energy it will have when it reaches the bottom. The actual amount of kinetic energy it will acquire, and the amount of work it can do, depends both on the height of the cliff and the mass m of the rock. A large rock and a small rock can be at the same height h (Fig. 23–3a) and thus have the same "gravitational potential," but the larger rock has the greater potential energy (it has more mass). The electrical case is similar (Fig. 23–3b): the potential energy change, or the work that can be done, depends both on the potential difference (corresponding to the height of the cliff) and on the charge (corresponding to mass), Eq. 23–3. But note a significant difference: electric charge comes in two types, ± and –, whereas gravitational mass is always ±. Sources of electrical energy such as batteries and electric generators are meant to maintain a potential difference. The actual amount of energy transformed by such a device depends on how much charge flows, as well as the potential difference (Eq. 23–3). For example, consider an automobile headlight connected to a 12.0-V battery. The amount of energy transformed (into light and thermal energy) is proportional to how much charge flows, which depends on how long the light is on. If over a given period of time 5.0 C of charge flows through the light, the total energy transformed is (5.0 C)(12.0 V) = 60 J. If the headlight is left on twice as long, 10.0 C of charge will flow and the energy transformed is (10.0 C)(12.0 V) = 120 J. Table 23–1 presents some typical voltages. SECTION 23–1

−

Vba

−

(a)

(b)

FIGURE 23–3 (a) Two rocks are at the same height.The larger rock has more potential energy. (b) Two charges have the same electric potential.The 2Q charge has more potential energy.

TABLE 23–1 Some Typical Potential Differences (Voltages) Source Thundercloud to ground High-voltage power line Power supply for TV tube Automobile ignition Household outlet Automobile battery Flashlight battery Resting potential across nerve membrane Potential changes on skin (EKG and EEG)

Voltage (approx.) 108 V 105 – 106 V 104 V 104 V 102 V 12 V 1.5 V 10 –1 V 10 –4 V

Electric Potential Energy and Potential Difference

609

− − − −

Vba 5000 V

− − −

EXAMPLE 23–2 Electron in CRT. Suppose an electron in a cathode ray tube (Section 23–9) is accelerated from rest through a potential difference Vb - Va = Vba = ±5000 V (Fig. 23–4). (a) What is the change in electric potential energy of the electron? (b) What is the speed of the electron Am = 9.1 * 10 –31 kgB as a result of this acceleration?

+ ++

e−

++ ++

−

b High voltage

FIGURE 23–4 Electron accelerated in CRT. Example 23–2.

APPROACH The electron, accelerated toward the positive plate, will decrease in potential energy by an amount ¢U = qVba (Eq. 23–3). The loss in potential energy will equal its gain in kinetic energy (energy conservation). SOLUTION (a) The charge on an electron is q = –e = –1.6 * 10–19 C. Therefore its change in potential energy is ¢U = qVba = A –1.6 * 10–19 CB(±5000 V) = –8.0 * 10–16 J.

The minus sign indicates that the potential energy decreases. The potential difference Vba has a positive sign since the final potential Vb is higher than the initial potential Va . Negative electrons are attracted toward a positive electrode and repelled away from a negative electrode. (b) The potential energy lost by the electron becomes kinetic energy K. From conservation of energy (Eq. 8–9a), ¢K + ¢U = 0, so 1 2 2 mv

¢K = – ¢U - 0 = –qAVb - Va B = –qVba ,

where the initial kinetic energy is zero since we are given that the electron started from rest. We solve for v: v =

–

2A –1.6 * 10–19 CB(5000 V) 2qVba = – = 4.2 * 107 ms. m C 9.1 * 10–31 kg

NOTE The electric potential energy does not depend on the mass, only on the charge and voltage. The speed does depend on m.

23–2

Relation between Electric Potential and Electric Field

The effects of any charge distribution can be described either in terms of electric field or in terms of electric potential. Electric potential is often easier to use because it is a scalar, as compared to electric field which is a vector. There is a crucial connection between the electric potential produced by a given arrangement of charges and the electric field due to those charges, which we now examine. B We start by recalling the relation between a conservative force F and the potential energy U associated with that force. As discussed in Section 8–2, the difference in potential energy between any two points in space, a and b, is given by Eq. 8–4: b

Ub - Ua = –

F dL, B

FIGURE 23–5 To find Vba in B a nonuniform electric field E, we B B integrate E dL from point a to point b. field c tri c Ele

li n e s

where dL is an infinitesimal increment of displacement, and the integral is taken along any path in space from point a to point b. For the electrical case, we are more interested in the potential difference, given by Eq. 23–2b, Vba = Vb - Va = AUb - Ua Bq, rather than in the potential energy itself. Also, the B electric field E at any point in space is defined as the force per unit charge B B (Eq. 21–3): E = Fq. Putting these two relations in the above equation gives us

Vba = Vb - Va = –

610

CHAPTER 23

(23–4a)

b E

E dL.

This is the general relation between electric field and potential difference. See Fig. 23–5. If we are given the electric field due to some arrangement of electric charge, we can use Eq. 23–4a to determine Vba .

Electric Potential

A simple special case is a uniform field. In Fig. 23–1, for example, a path parallel to the electric field lines from point a at the positive plate to point b at the B negative plate gives (since E and dL are in the same direction at each point),

Vba 50 V

−

Vba = Vb - Va = –

E dL B

= –E

= –Ed

[only if E is uniform] (23–4b)

Vba = –Ed

−

where d is the distance, parallel to the field lines, between points a and b. Be careful not to use Eq. 23–4b unless you are sure the electric field is uniform. From either of Eqs. 23–4 we can see that the units for electric field intensity can be written as volts per meter (Vm) as well as newtons per coulomb (NC). These are equivalent in general, since 1 NC = 1 N mCm = 1 JCm = 1 Vm. EXERCISE A Return to the Chapter-Opening Question, page 607, and answer it again now. Try to explain why you may have answered differently the first time.

−

d= 5.0 cm FIGURE 23–6 Example 23–3.

EXAMPLE 23–4 Charged conducting sphere. Determine the potential at a distance r from the center of a charged conducting sphere of radius r0 for (a) r 7 r0 , (b) r = r0 , (c) r 6 r0 . The total charge on the sphere is Q.

and we set dl = dr. If we let V = 0 for r = q (let's choose Vb = 0 at rb = q ), then at any other point r (for r 7 r0) we have 1 Q. V = Cr 7 r0 D 4p 0 r We will see in the next Section that this same equation applies for the potential a distance r from a single point charge. Thus the electric potential outside a spherical conductor with a uniformly distributed charge is the same as if all the charge were at its center. (b) As r approaches r0 , we see that 1 Q V = Cr = r0 D 4p 0 r0 at the surface of the conductor. B (c) For points within the conductor, E = 0. Thus the integral, E dL , between r = r0 and any point within the conductor gives zero change in V. Hence V is constant within the conductor: 1 Q. V = Cr r0 D 4p 0 r0 The whole conductor, not just its surface, is at this same potential. Plots of both E and V as a function of r are shown in Fig. 23–8 for a positively charged conducting sphere. B

−

EXAMPLE 23–3 Electric field obtained from voltage. Two parallel plates are charged to produce a potential difference of 50 V. If the separation between the plates is 0.050 m, calculate the magnitude of the electric field in the space between the plates (Fig. 23–6). APPROACH We apply Eq. 23–4b to obtain the magnitude of E, assumed uniform. SOLUTION The electric field magnitude is E = Vbad = (50 V0.050 m) = 1000 Vm.

APPROACH The charge Q is distributed over the surface of the sphere since it is a conductor. We saw in Example 22–3 that the electric field outside a conducting sphere is 1 Q E = Cr 7 r0 D 4p 0 r2 B and points radially outward (inward if Q 6 0). Since we know E, we can start by using Eq. 23–4a. SOLUTION (a) We use Eq. 23–4a and integrate along a radial line with dL B parallel to E (Fig. 23–7) between two points which are distances ra and rb from the sphere's center: rb rb Q Q dr 1 1 B Vb - Va = – E dL = – = ¢ ≤ 2 ra 4p 0 ra r 4p 0 rb ra

−

E=?

ra d L

FIGURE 23–7 Example 23–4. B B Integrating E dL for the field outside a spherical conductor. FIGURE 23–8 (a) E versus r, and (b) V versus r, for a positively charged solid conducting sphere of radius r0 (the charge distributes itself evenly on the surface); r is the distance from the center of the sphere. E 1 r2

2r0 (a)

3r0

V 1r

2r0 (b)

3r0

SECTION 23–2

611

EXAMPLE 23–5 Breakdown voltage. In many kinds of equipment, very high voltages are used. A problem with high voltage is that the air can become ionized due to the high electric fields: free electrons in the air (produced by cosmic rays, for example) can be accelerated by such high fields to speeds sufficient to ionize O2 and N2 molecules by collision, knocking out one or more of their electrons. The air then becomes conducting and the high voltage cannot be maintained as charge flows. The breakdown of air occurs for electric fields of about 3 * 106 Vm. (a) Show that the breakdown voltage for a spherical conductor in air is proportional to the radius of the sphere, and (b) estimate the breakdown voltage in air for a sphere of diameter 1.0 cm. APPROACH The electric potential at the surface of a spherical conductor of radius r0 (Example 23–4), and the electric field just outside its surface, are V =

1 Q 4p 0 r0

and

E =

1 Q. 4p 0 r 20

SOLUTION (a) We combine these two equations and obtain [at surface of spherical conductor]

V = r0 E.

(b) For r0 = 5 * 10–3 m, the breakdown voltage in air is

V = A5 * 10–3 mBA3 * 106 VmB L 15,000 V.

When high voltages are present, a glow may be seen around sharp points, known as a corona discharge, due to the high electric fields at these points which ionize air molecules. The light we see is due to electrons jumping down to empty lower states. Lightning rods, with their sharp tips, are intended to ionize the surrounding air when a storm cloud is near, and to provide a conduction path to discharge a dangerous high-voltage cloud slowly, over a period of time. Thus lightning rods, connected to the ground, are intended to draw electric charge off threatening clouds before a large buildup of charge results in a swift destructive lightning bolt. EXERCISE B On a dry day, a person can become electrically charged by rubbing against rugs and other ordinary objects. Suppose you notice a small shock as you reach for a metal doorknob, noting that the shock occurs along with a tiny spark when your hand is about 3.0 mm from the doorknob. As a rough estimate, use Eq. 23–4b to estimate the potential difference between your hand and the doorknob. (a) 9 V; (b) 90 V; (c) 900 V; (d) 9000 V; (e) none of these.

23–3 FIGURE 23–9 We integrate Eq. 23–4a along the straight line (shown in black) from point a to point b. The line ab is parallel to a field line.

Electric Potential Due to Point Charges

The electric potential at a distance r from a single point charge Q can be derived B directly from Eq. 23–4a, Vb - Va = – E dL . The electric field due to a single point charge has magnitude (Eq. 21–4) B

E = B

E +Q

rb a

1 Q 4p 0 r2

E = k

Q r2

(where k = 14p 0 = 8.99 * 109 N m2C 2), and is directed radially outward from a positive charge (inward if Q 6 0). We take the integral in Eq. 23–4a along a (straight) field line (Fig. 23–9) from point a, a distance ra from Q, to point b, a B distance rb from Q. Then dL will be parallel to E and dl = dr. Thus B

Vb - Va = –

E dL = –

Q 4p 0

Q Q 1 1 dr = ¢ ≤. 2 ra 4p 0 rb r

As mentioned earlier, only differences in potential have physical meaning. We are free, therefore, to choose the value of the potential at some one point to

612

CHAPTER 23

Electric Potential

be whatever we please. It is common to choose the potential to be zero at infinity (let Vb = 0 at rb = q ). Then the electric potential V at a distance r from a single point charge is

1 Q when Q > 0 4pe0 r

single point charge; c d V = 0 at r = q

1 Q. V = 4p 0 r

(23–5)

We can think of V here as representing the absolute potential, where V = 0 at r = q, or we can think of V as the potential difference between r and infinity. Notice that the potential V decreases with the first power of the distance, whereas the electric field (Eq. 21–4) decreases as the square of the distance. The potential near a positive charge is large, and it decreases toward zero at very large distances (Fig. 23–10). For a negative charge, the potential is negative and increases toward zero at large distances (Fig. 23–11). Equation 23–5 is sometimes called the Coulomb potential (it has its origin in Coulomb's law). In Example 23–4 we found that the potential due to a uniformly charged sphere is given by the same relation, Eq. 23–5, for points outside the sphere. Thus we see that the potential outside a uniformly charged sphere is the same as if all the charge were concentrated at its center. EXERCISE C What is the potential at a distance of 3.0 cm from a point charge Q = – 2.0 * 10 –9 C? (a) 600 V; (b) 60 V; (c) 6 V; (d) –600 V; (e) –60 V; (f) –6 V.

FIGURE 23–10 Potential V as a function of distance r from a single point charge Q when the charge is positive. FIGURE 23–11 Potential V as a function of distance r from a single point charge Q when the charge is negative.

V r

0 V=

1 Q when Q < 0 4pe0 r

EXAMPLE 23–6 Work required to bring two positive charges close together. What minimum work must be done by an external force to bring a charge q = 3.00 mC from a great distance away (take r = q ) to a point 0.500 m from a charge Q = 20.0 mC? APPROACH To find the work we cannot simply multiply the force times distance because the force is not constant. Instead we can set the change in potential energy equal to the (positive of the) work required of an external force (Chapter 8), and Eq. 23–3: W = ¢U = q AVb - Va B. We get the potentials Vb and Va using Eq. 23–5.

C A U T I O N

We cannot use W = Fd when F is not constant

SOLUTION The work required is equal to the change in potential energy: W = q AVb - Va B = q¢

kQ kQ ≤, ra rb

where rb = 0.500 m and ra = q. The right-hand term within the parentheses is zero (1q = 0) so W = A3.00 * 10

–6

A8.99 * 109 N m2C 2 BA2.00 * 10 –5 CB (0.500 m)

= 1.08 J.

NOTE We could not use Eq. 23–4b here because it applies only to uniform fields. But we did use Eq. 23–3 because it is always valid. To determine the electric field at points near a collection of two or more point charges requires adding up the electric fields due to each charge. Since the electric field is a vector, this can be time consuming or complicated. To find the electric potential at a point due to a collection of point charges is far easier, since the electric potential is a scalar, and hence you only need to add numbers (with appropriate signs) without concern for direction. This is a major advantage in using electric potential for solving Problems. SECTION 23–3

613

y A

EXAMPLE 23–7 Potential above two charges. Calculate the electric potential (a) at point A in Fig. 23–12 due to the two charges shown, and (b) at point B. [This is the same situation as Example 21–8, Fig. 21–27, where we calculated the electric field at these points.]

B cm

40 cm

30 cm

26 cm x Q1 = −50 mC

26 cm Q2 = +50 mC

FIGURE 23–12 Example 23–7. (See also Example 21–8, Fig. 21–27.)

C A U T I O N

Potential is a scalar and has no components

APPROACH The total potential at point A (or at point B) is the sum of the potentials at that point due to each of the two charges Q1 and Q2 . The potential due to each single charge is given by Eq. 23–5. We do not have to worry about directions because electric potential is a scalar quantity. But we do have to keep track of the signs of charges. SOLUTION (a) We add the potentials at point A due to each charge Q1 and Q2 , and we use Eq. 23–5 for each: VA = VA2 + VA1 Q2 Q1 = k + k r2A r1A where r1A = 60 cm and r2A = 30 cm. Then VA =

A9.0 * 109 N m2C 2 BA5.0 * 10 –5 CB 0.30 m A9.0 * 10 N m2C 2 BA –5.0 * 10 –5 CB 9

0.60 m = 1.50 * 106 V - 0.75 * 106 V = 7.5 * 105 V. (b) At point B, r1B = r2B = 0.40 m, so VB = VB2 + VB1 =

A9.0 * 109 N m2C 2 BA5.0 * 10 –5 CB 0.40 m A9.0 * 10 N m2C 2 BA –5.0 * 10–5 CB 9

0.40 m = 0 V. NOTE The two terms in the sum in (b) cancel for any point equidistant from Q1 and Q2 Ar1B = r2B B. Thus the potential will be zero everywhere on the plane equidistant between the two opposite charges. This plane where V is constant is called an equipotential surface.

–

(i) –

Simple summations like these can easily be performed for any number of point charges.

(ii) +

(iii) FIGURE 23–13 Exercise D.

EXERCISE D Consider the three pairs of charges, Q1 and Q2 , in Fig. 23–13. (a) Which set has a positive potential energy? (b) Which set has the most negative potential energy? (c) Which set requires the most work to separate the charges to infinity? Assume the charges all have the same magnitude.

23–4

Potential Due to Any Charge Distribution

If we know the electric field in a region of space due to any distribution of electric charge, we can determine the difference in potential between two points in the B b B region using Eq. 23–4a, Vba = – a E dL . In many cases we don't know E as a function of position, and it may be difficult to calculate. We can calculate the potential V due to a given charge distribution in another way, using the potential due to a single point charge, Eq. 23–5: 1 Q, V = 4p 0 r where we choose V = 0 at r = q. Then we can sum over all the charges. B

614

CHAPTER 23

Electric Potential

If we have n individual point charges, the potential at some point a (relative to V = 0 at r = q ) is n

a Vi =

Va =

i=1

n Qi 1 , a 4p 0 i = 1 ria

(23–6a)

where ria is the distance from the i th charge AQi B to the point a. (We already used this approach in Example 23–7.) If the charge distribution can be considered continuous, then 1 4p 0

V =

dq , r

(23–6b)

where r is the distance from a tiny element of charge, dq, to the point where V is being determined. EXAMPLE 23–8 Potential due to a ring of charge. A thin circular ring of radius R has a uniformly distributed charge Q. Determine the electric potential at a point P on the axis of the ring a distance x from its center, Fig. 23–14. APPROACH We integrate over the ring using Eq. 23–6b. SOLUTION Each point on the ring is equidistant from point P, and this distance is 1 Ax2 + R 2 B 2 . So the potential at P is: 1 4p 0

V =

dq 1 1 = 1 r 4p 0 Ax2 + R2 B 2

Q 1 . 1 4p 0 Ax2 + R2 B 2

NOTE For points very far away from the ring, x W R, this result reduces to A14p 0 B(Qx), the potential of a point charge, as we should expect. EXAMPLE 23–9 Potential due to a charged disk. A thin flat disk, of radius R0 , has a uniformly distributed charge Q, Fig. 23–15. Determine the potential at a point P on the axis of the disk, a distance x from its center. APPROACH We divide the disk into thin rings of radius R and thickness dR and use the result of Example 23–8 to sum over the disk. SOLUTION The charge Q is distributed uniformly, so the charge contained in each ring is proportional to its area. The disk has area pR 20 and each thin ring has area dA = (2pR)(dR). Hence

dq (x 2

P x

FIGURE 23–14 Example 23–8. Calculating the potential at point P, a distance x from the center of a uniform ring of charge. FIGURE 23–15 Example 23–9. Calculating the electric potential at point P on the axis of a uniformly charged thin disk.

dR R

dq 2pR dR = Q pR 20

+ R2 1 )2

(x 2

+ R2 1 )2

(2pR)(dR)

2QR dR . = pR 20 R 20 1 Then the potential at P, using Eq. 23–6b in which r is replaced by Ax2 + R 2 B 2 , is dq = Q

V = =

1 4p 0

dq 2

+ R B

2 1 2

2Q 4p 0 R 20

Q 1 C Ax 2 + R 20 B 2 - xD. 2p 0 R 20

R dR

Ax2 + R B

2 1 2

1 R = R0 Q 2 2 22 Ax + R B 2p 0 R 20 R=0

NOTE For x W R0, this formula reduces to V L

Q Q . 1 R 20 cx¢1 + ≤ - xd = 2 2 x2 4p 0 x 2p 0 R0

This is the formula for a point charge, as we expect. SECTION 23–4

Potential Due to Any Charge Distribution

615

23–5

−

0V 20 V 15 V 5V 10 V FIGURE 23–16 Equipotential lines (the green dashed lines) between two oppositely charged parallel plates. Note that they are perpendicular to the electric field lines (solid red lines).

FIGURE 23–17 Example 23–10. Electric field lines and equipotential surfaces for a point charge. B

30 V 20 V 10 V

FIGURE 23–18 Equipotential lines (green, dashed) are always perpendicular to the electric field lines (solid red) shown here for two equal but oppositely charged particles.

616

CHAPTER 23

The electric potential can be represented graphically by drawing equipotential lines or, in three dimensions, equipotential surfaces. An equipotential surface is one on which all points are at the same potential. That is, the potential difference between any two points on the surface is zero, and no work is required to move a charge from one point to the other. An equipotential surface must be perpendicular to the electric field at any point. If this were not so—that is, if there were a component B of E parallel to the surface—it would require work to move the charge along the B surface against this component of E; and this would contradict the idea that it is an B equipotential surface. This can also be seen from Eq. 23–4a, ¢V = – E dL . B On a surface where V is constant, ¢V = 0, so we must have either E = 0, dL = 0, B B or cos u = 0 where u is the angle between E and dL . Thus in a region where E is not zero, the path dL along an equipotential must have cos u = 0, meaning B u = 90° and E is perpendicular to the equipotential. The fact that the electric field lines and equipotential surfaces are mutually perpendicular helps us locate the equipotentials when the electric field lines are known. In a normal two-dimensional drawing, we show equipotential lines, which are the intersections of equipotential surfaces with the plane of the drawing. In Fig. 23–16, a few of the equipotential lines are drawn (dashed green lines) for the electric field (red lines) between two parallel plates at a potential difference of 20 V. The negative plate is arbitrarily chosen to be zero volts and B the potential of each equipotential line is indicated. Note that E points toward lower values of V. B

−

Equipotential Surfaces

–

EXAMPLE 23–10 Point charge equipotential surfaces. For a single point charge with Q = 4.0 * 10 –9 C, sketch the equipotential surfaces (or lines in a plane containing the charge) corresponding to V1 = 10 V, V2 = 20 V, and V3 = 30 V. APPROACH The electric potential V depends on the distance r from the charge (Eq. 23–5). SOLUTION The electric field for a positive point charge is directed radially outward. Since the equipotential surfaces must be perpendicular to the lines of electric field, they will be spherical in shape, centered on the point charge, Fig. 23–17. From Eq. 23–5 we have r = A14p 0 B(QV), so that for V1 = 10 V, r1 = A9.0 * 109 N m2C 2 B A4.0 * 10 –9 CB(10 V) = 3.6 m, for V2 = 20 V, r2 = 1.8 m, and for V3 = 30 V, r3 = 1.2 m, as shown. NOTE The equipotential surface with the largest potential is closest to the positive charge. How would this change if Q were negative?

The equipotential lines for the case of two equal but oppositely charged particles are shown in Fig. 23–18 as green dashed lines. Equipotential lines and surfaces, unlike field lines, are always continuous and never end, and so continue beyond the borders of Figs. 23–16 and 23–18. We saw in Section 21–9 that there can be no electric field within a conductor in the static case, for otherwise the free electrons would feel a force and would move. Indeed, the entire volume of a conductor must be entirely at the same potential in the static case, and the surface of a conductor is then an equipotential surface. (If it weren't, the free electrons at the surface would move, since whenever there is a potential difference between two points, free charges will move.) This is fully consistent with our result, discussed earlier, that the electric field at the surface of a conductor must be perpendicular to the surface.

Electric Potential

A useful analogy for equipotential lines is a topographic map: the contour lines are essentially gravitational equipotential lines (Fig. 23–19).

FIGURE 23–19 A topographic map (here, a portion of the Sierra Nevada in California) shows continuous contour lines, each of which is at a fixed height above sea level. Here they are at 80 ft (25 m) intervals. If you walk along one contour line, you neither climb nor descend. If you cross lines, and especially if you climb perpendicular to the lines, you will be changing your gravitational potential (rapidly, if the lines are close together).

23–6

Electric Dipole Potential P

Two equal point charges Q, of opposite sign, separated by a distance l, are called an electric dipole, as we saw in Section 21–11. Also, the two charges we saw in Figs. 23–12 and 23–18 constitute an electric dipole, and the latter shows the electric field lines and equipotential surfaces for a dipole. Because electric dipoles occur often in physics, as well as in other fields, it is useful to examine them more closely. The electric potential at an arbitrary point P due to a dipole, Fig. 23–20, is the sum of the potentials due to each of the two charges (we take V = 0 at r = q ): (–Q) Q 1 Q 1 1 1 1 ¢r , V = + = Q¢ ≤ = r 4p 0 r 4p 0 (r + ¢r) 4p 0 r + ¢r 4p 0 r(r + ¢r)

where r is the distance from P to the positive charge and r + ¢r is the distance to the negative charge. This equation becomes simpler if we consider points P whose distance from the dipole is much larger than the separation of the two charges— that is, for r W l. From Fig. 23–20 we see that ¢r L l cos u; since r W ¢r = l cos u, we can neglect ¢r in the denominator as compared to r. Therefore, we obtain V =

1 Ql cos u 1 p cos u = 2 4p 0 4p 0 r2 r

[dipole; r W l] (23–7)

∆r −Q

We can use Eq. 23–4a, Vb - Va = – a E dL , to determine the difference in potential between two points if the electric field is known in the region between those two points. By inverting Eq. 23–4a, we can write the electric field in terms of the potential. Then the electric field can be determined from a knowledge of V. Let us see how to do this. B

TABLE 23–2 Dipole Moments of Selected Molecules Molecule

Dipole Moment (C m)

H 2(±)O (–) H (±)Cl(–) N (–)H 3(±) »N (–) ¬ H (±) »C (±) " O (–)

6.1 3.4 5.0 L 3.0† L 8.0†

* * * * *

10 –30 10 –30 10 –30 10–30 10–30

†

E Determined from V b

FIGURE 23–20 Electric dipole. Calculation of potential V at point P.

where p = Ql is called the dipole moment. We see that the potential decreases as the square of the distance from the dipole, whereas for a single point charge the potential decreases with the first power of the distance (Eq. 23–5). It is not surprising that the potential should fall off faster for a dipole; for when you are far from a dipole, the two equal but opposite charges appear so close together as to tend to neutralize each other. Table 23–2 gives the dipole moments for several molecules. The ± and – signs indicate on which atoms these charges lie. The last two entries are a part of many organic molecules and play an important role in molecular biology. A dipole moment has units of coulomb-meters (Cm), although for molecules a smaller unit called a debye is sometimes used: 1 debye = 3.33 * 10–30 Cm.

23–7

SECTION 23–7

These groups often appear on larger molecules; hence the value for the dipole moment will vary somewhat, depending on the rest of the molecule.

E Determined from V

617

We write Eq. 23–4a in differential form as B dV = –E dL = –El dl, where dV is the infinitesimal difference in potential between two points a distance dl apart, and El is the component of the electric field in the direction of the infinitesimal displacement dL . We can then write dV . (23–8) El = – dl Thus the component of the electric field in any direction is equal to the negative of the rate of change of the electric potential with distance in that direction. The quantity dVdl is called the gradient of V in a particular direction. If the direction is not specified, the term gradient refers to that direction in which V changes most B rapidly; this would be the direction of E at that point, so we can write dV . B [if dL 7 E] E = – dl B If E is written as a function of x, y, and z, and we let l refer to the x, y, and z axes, then Eq. 23–8 becomes 0V , 0V , 0V . (23–9) Ex = – Ey = – Ez = – 0x 0y 0z Here, 0V0x is the "partial derivative" of V with respect to x, with y and z held constant.† For example, if V(x, y, z) = A2 Vm2 B x 2 + A8 Vm3 B y2z + A2 Vm2 B z2, then B

Ex = – 0V0x = – A4 Vm2 B x,

and

Ey = – 0V0y = – A16 Vm3 B yz, Ez = – 0V0z = – A8 Vm3 B y 2 - A4 Vm2 B z. B

EXAMPLE 23–11 E for ring and disk. Use electric potential to determine the electric field at point P on the axis of (a) a circular ring of charge (Fig. 23–14) and (b) a uniformly charged disk (Fig. 23–15). APPROACH We obtained V as a function of x in Examples 23–8 and 23–9, so we find E by taking derivatives (Eqs. 23–9). SOLUTION (a) From Example 23–8, Q 1 . V = 1 4p 0 Ax2 + R 2 B 2 Then

Ex = –

Qx 0V 1 . = 0x 4p 0 Ax2 + R 2 B 32

This is the same result we obtained in Example 21–9. (b) From Example 23–9, Q 1 V = C Ax2 + R 20 B 2 - xD, 2 2p 0 R0 so Q 0V x . Ex = – = 1 1d 2 c 2 0x 2p 0 R 0 Ax + R 20 B 2 For points very close to the disk, x V R0, this can be approximated by Q s Ex L = 2 2 0 2p 0 R 0 where s = QpR20 is the surface charge density. We also obtained these results in Chapter 21, Example 21–12 and Eq. 21–7. †

618

CHAPTER 23

Equation 23–9 can be written as a vector equation,

B B 0 0 0 E = – grad V = – V = – ¢ ˆi + jˆ + kˆ ≤V 0x 0y 0z B B 0 0 0 . where the symbol is called the del or gradient operator: = ˆi + jˆ + kˆ 0x 0y 0z

If we compare this last Example with Examples 21–9 and 21–12, we see that here, B as for many charge distributions, it is easier to calculate V first, and then E from B Eq. 23–9, rather than to calculate E due to each charge from Coulomb's law. This is B because V due to many charges is a scalar sum, whereas E is a vector sum.

23–8

Electrostatic Potential Energy; the Electron Volt

Suppose a point charge q is moved between two points in space, a and b, where the electric potential due to other charges is Va and Vb , respectively. The change in electrostatic potential energy of q in the field of these other charges is, according to Eq. 23–2b, ¢U = Ub - Ua = q AVb - Va B.

Now suppose we have a system of several point charges. What is the electrostatic potential energy of the system? It is most convenient to choose the electric potential energy to be zero when the charges are very far (ideally infinitely far) apart. A single point charge, Q1 , in isolation, has no potential energy, because if there are no other charges around, no electric force can be exerted on it. If a second point charge Q2 is brought close to Q1 , the potential due to Q1 at the position of this second charge is 1 Q1 , V = 4p 0 r1 2 where r1 2 is the distance between the two. The potential energy of the two charges, relative to V = 0 at r = q, is 1 Q1 Q2 . U = Q2 V = (23–10) 4p 0 r1 2 This represents the work that needs to be done by an external force to bring Q2 from infinity (V = 0) to a distance r1 2 from Q1 . It is also the negative of the work needed to separate them to infinity. If the system consists of three charges, the total potential energy will be the work needed to bring all three together. Equation 23–10 represents the work needed to bring Q2 close to Q1 ; to bring a third charge Q3 so that it is a distance r1 3 from Q1 and r2 3 from Q2 requires work equal to 1 Q1 Q3 1 Q2 Q3 . + 4p 0 r1 3 4p 0 r2 3 So the potential energy of a system of three point charges is U =

Q1 Q3 Q2 Q3 Q1 Q2 1 ¢ + + ≤. r1 2 r1 3 r2 3 4p 0

[V = 0 at r = q]

For a system of four charges, the potential energy would contain six such terms, and so on. (Caution must be used when making such sums to avoid double counting of the different pairs.)

The Electron Volt Unit The joule is a very large unit for dealing with energies of electrons, atoms, or molecules (see Example 23–2), and for this purpose, the unit electron volt (eV) is used. One electron volt is defined as the energy acquired by a particle carrying a charge whose magnitude equals that on the electron (q = e) as a result of moving through a potential difference of 1 V. Since e = 1.6 * 10 –19 C, and since the change in potential energy equals qV, 1 eV is equal to A1.6 * 10–19 CB(1.0 V) = 1.6 * 10–19 J: 1 eV = 1.6 * 10 –19 J. An electron that accelerates through a potential difference of 1000 V will lose 1000 eV of potential energy and will thus gain 1000 eV or 1 keV (kiloelectron volt) of kinetic energy. On the other hand, if a particle with a charge equal to twice the magnitude of the charge on the electron A = 2e = 3.2 * 10 –19 CB moves through a potential difference of 1000 V, its energy will change by 2000 eV = 2 keV. SECTION 23–8

Electrostatic Potential Energy; the Electron Volt

619

Although the electron volt is handy for stating the energies of molecules and elementary particles, it is not a proper SI unit. For calculations it should be converted to joules using the conversion factor given above. In Example 23–2, for example, the electron acquired a kinetic energy of 8.0 * 10–16 J. We normally would quote this energy as 5000 eV A = 8.0 * 10–16 J1.6 * 10–19 JeVB. But when determining the speed of a particle in SI units, we must use the kinetic energy in J. EXERCISE E What is the kinetic energy of a He2± ion released from rest and accelerated through a potential difference of 1.0 kV? (a) 1000 eV, (b) 500 eV, (c) 2000 eV, (d) 4000 eV, (e) 250 eV.

EXAMPLE 23–12 Disassembling a hydrogen atom. Calculate the work needed to "disassemble" a hydrogen atom. Assume that the proton and electron are initially separated by a distance equal to the "average" radius of the hydrogen atom in its ground state, 0.529 * 10–10 m, and that they end up an infinite distance apart from each other. APPROACH The work necessary will be equal to the total energy, kinetic plus potential, of the electron and proton as an atom, compared to their total energy when infinitely far apart. SOLUTION From Eq. 23–10 we have initially – A8.99 * 109 N m2C 2 BA1.60 * 10 –19 CB 2 1 Q1 Q2 1 (e)(–e) U = = = r 4p 0 r 4p 0 A0.529 * 10–10 mB = –27.2A1.60 * 10 –19 B J = –27.2 eV.

This represents the potential energy. The total energy must include also the kinetic energy of the electron moving in an orbit of radius r = 0.529 * 10 –10 m. From F = ma for centripetal acceleration, we have

Then

1 e2 mv2 . a 2b = r 4p 0 r K =

1 2 2 mv

1 2

1 e2 ≤ 4p 0 r

which equals – 12 U (as calculated above), so K = ±13.6 eV. The total energy initially is E = K + U = 13.6 eV - 27.2 eV = –13.6 eV. To separate a stable hydrogen atom into a proton and an electron at rest very far apart (U = 0 at r = q, K = 0 because v = 0) requires ±13.6 eV. This is, in fact, the measured ionization energy for hydrogen. NOTE To treat atoms properly, we need to use quantum theory (Chapters 37 to 39). But our "classical" calculation does give the correct answer here. EXERCISE F The kinetic energy of a 1000-kg automobile traveling 20 ms (70 kmh) would be about (a) 100 GeV, (b) 1000 TeV, (c) 106 TeV, (d) 1012 TeV, (e) 1018 TeV. FIGURE 23–21 If the cathode inside the evacuated glass tube is heated to glowing, negatively charged "cathode rays" (electrons) are "boiled off" and flow across to the anode (± ) to which they are attracted. Cathode

Anode –– ––

+ + + +

– +

Battery

620

CHAPTER 23

* 23–9

Cathode Ray Tube: TV and Computer Monitors, Oscilloscope

An important device that makes use of voltage, and that allows us to "visualize" how a voltage changes in time, is the cathode ray tube (CRT). A CRT used in this way is an oscilloscope. The CRT has also been used for many years as the picture tube of television sets and computer monitors, but LCD (Chapter 35) and other screens are now common. The operation of a CRT depends on the phenomenon of thermionic emission discovered by Thomas Edison (1847–1931). Consider two small plates (electrodes) inside an evacuated "bulb" or "tube" as shown in Fig. 23–21, to which is applied a potential difference. The negative electrode is called the cathode, the positive one the anode. If the negative cathode is heated (usually by an electric current, as in a lightbulb) so that it becomes hot and glowing, it is found that negative charge leaves the cathode and flows to the positive anode. These negative charges are now called electrons, but originally they were called cathode rays since they seemed to come from the cathode (see Section 27–7 on the discovery of the electron).

Cathode

Anode

Bright spot on screen where electrons hit

Horizontal deflection plates

Heater current Fluorescent screen

Grid Vertical deflection plates

Path of electrons

FIGURE 23–22 A cathode ray tube. Magnetic deflection coils are often used in place of the electric deflection plates shown here. The relative positions of the elements have been exaggerated for clarity.

The cathode ray tube (CRT) derives its name from the fact that inside an evacuated glass tube, a beam of cathode rays (electrons) is directed to various parts of a screen to produce a "picture." A simple CRT is diagrammed in Fig. 23–22. Electrons emitted by the heated cathode are accelerated by a high voltage (5000–50,000 V) applied between the anode and cathode. The electrons pass out of this "electron gun" through a small hole in the anode. The inside of the tube face is coated with a fluorescent material that glows when struck by electrons. A tiny bright spot is thus visible where the electron beam strikes the screen. Two horizontal and two vertical plates can deflect the beam of electrons when a voltage is applied to them. The electrons are deflected toward whichever plate is positive. By varying the voltage on the deflection plates, the bright spot can be placed at any point on the screen. Many CRTs use magnetic deflection coils (Chapter 27) instead of electric plates. In the picture tube or monitor for a computer or television set, the electron beam is made to sweep over the screen in the manner shown in Fig. 23–23 by changing voltages applied to the deflection plates. For standard television in the United States, 525 lines constitutes a complete sweep in 301 s, over the entire screen. High-definition TV provides more than double this number of lines (1080), giving greater picture sharpness. We see a picture because the image is retained by the fluorescent screen and by our eyes for about 201 s. The picture we see consists of the varied brightness of the spots on the screen, controlled by the grid (a "porous" electrode, such as a wire grid, that allows passage of electrons). The grid limits the flow of electrons by means of the voltage (the "video signal") applied to it: the more negative this voltage, the more electrons are repelled and the fewer pass through. This video signal sent out by the TV station, and received by the TV set, is accompanied by signals that synchronize the grid voltage to the horizontal and vertical sweeps. (More in Chapter 31.) An oscilloscope is a device for amplifying, measuring, and visually observing an electrical signal as a function of time on the screen of a CRT (a "signal" is usually a time-varying voltage). The electron beam is swept horizontally at a uniform rate in time by the horizontal deflection plates. The signal to be displayed is applied (after amplification) to the vertical deflection plates. The visible "trace" on the screen, which could be an electrocardiogram (Fig. 23–24), or a signal from an experiment on nerve conduction, is a plot of the signal voltage (vertically) versus time (horizontally).

P H Y S I C S

A P P L I E D

CRT, TV and computer monitors

FIGURE 23–23 Electron beam sweeps across a television screen in a succession of horizontal lines. Each horizontal sweep is made by varying the voltage on the horizontal deflection plates. Then the electron beam is moved down a short distance by a change in voltage on the vertical deflection plates, and the process is repeated.

P H Y S I C S

A P P L I E D

Oscilloscope

FIGURE 23–24 An electrocardiogram (ECG) trace displayed on a CRT.

*SECTION 23–9

621

Summary Electric potential is defined as electric potential energy per unit charge. That is, the electric potential difference between any two points in space is defined as the difference in potential energy of a test charge q placed at those two points, divided by the charge q: Ub - Ua . (23–2b) Vba = q Potential difference is measured in volts (1 V = 1 JC) and is sometimes referred to as voltage. The change in potential energy of a charge q when it moves through a potential difference Vba is (23–3)

¢U = qVba .

The potential difference Vba between two points, a and b, is given by the relation Vba = Vb - Va = –

E dL .

(23–4a)

a B

Thus Vba can be found in any region where E is known. If the electric field is uniform, the integral is easy: Vba = –Ed,

where d is the distance (parallel to the field lines) between the two points. An equipotential line or surface is all at the same potential, and is perpendicular to the electric field at all points. The electric potential due to a single point charge Q, relative to zero potential at infinity, is given by 1 Q. (23–5) V = 4p 0 r The potential due to any charge distribution can be obtained by summing (or integrating) over the potentials for all the charges. The potential due to an electric dipole drops off as 1r 2. The dipole moment is p = Ql, where l is the distance between the two equal but opposite charges of magnitude Q. B When V is known, the components of E can be found from the inverse of Eq. 23–4a, namely 0V , 0V , 0V . Ex = – (23–9) Ey = – Ez = – 0x 0y 0z [*Television and computer monitors traditionally use a cathode ray tube (CRT) that accelerates electrons by high voltage, and sweeps them across the screen in a regular way using deflection plates.]

Questions 1. If two points are at the same potential, does this mean that no work is done in moving a test charge from one point to the other? Does this imply that no force must be exerted? Explain. 2. If a negative charge is initially at rest in an electric field, will it move toward a region of higher potential or lower potential? What about a positive charge? How does the potential energy of the charge change in each instance? 3. State clearly the difference (a) between electric potential and electric field, (b) between electric potential and electric potential energy. 4. An electron is accelerated from rest by a potential difference of, say, 0.10 V. How much greater would its final speed be if it is accelerated with four times as much voltage? Explain. 5. Can a particle ever move from a region of low electric potential to one of high potential and yet have its electric potential energy decrease? Explain. B

6. If V = 0 at a point in space, must E = 0? If E = 0 at some point, must V = 0 at that point? Explain. Give examples for each. 7. When dealing with practical devices, we often take the ground (the Earth) to be 0 V. (a) If instead we said the ground was – 10 V, how would this affect V and E at other points? (b) Does the fact that the Earth carries a net charge affect the choice of V at its surface? 8. Can two equipotential lines cross? Explain. 9. Draw in a few equipotential lines in Fig. 21–34b and c. 10. What can you say about the electric field in a region of space that has the same potential throughout? 11. A satellite orbits the Earth along a gravitational equipotential line. What shape must the orbit be?

622

CHAPTER 23

Electric Potential

12. Suppose the charged ring of Example 23–8 was not uniformly charged, so that the density of charge was twice as great near the top as near the bottom. Assuming the total charge Q is unchanged, would this affect the potential at point P on the axis (Fig. 23–14)? Would it B affect the value of E at that point? Is there a discrepancy here? Explain. 13. Consider a metal conductor in the shape of a football. If it carries a total charge Q, where would you expect the charge density s to be greatest, at the ends or along the flatter sides? Explain. [Hint: Near the surface of a conductor, E = s 0 .] B 14. If you know V at a point in space, can you calculate E B at that point? If you know E at a point can you calculate V at that point? If not, what else must be known in each case? 15. A conducting sphere carries a charge Q and a second identical conducting sphere is neutral. The two are initially isolated, but then they are placed in contact. (a) What can you say about the potential of each when they are in contact? (b) Will charge flow from one to the other? If so, how much? (c) If the spheres do not have the same radius, how are your answers to parts (a) and (b) altered? 16. At a particular location, the electric field points due north. In what direction(s) will the rate of change of potential be (a) greatest, (b) least, and (c) zero? 17. Equipotential lines are spaced 1.00 V apart. Does the distance between the lines in different regions of space tell B you anything about the relative strengths of E in those regions? If so, what? B 18. If the electric field E is uniform in a region, what can you infer about the electric potential V? If V is uniform in a B region of space, what can you infer about E ? 19. Is the electric potential energy of two unlike charges positive or negative? What about two like charges? What is the significance of the sign of the potential energy in each case?

Problems 23–1 Electric Potential 1. (I) What potential difference is needed to stop an electron that has an initial velocity v = 5.0 * 105 ms? 2. (I) How much work does the electric field do in moving a proton from a point with a potential of ± 185 V to a point where it is – 55 V? 3. (I) An electron acquires 5.25 * 10–16 J of kinetic energy when it is accelerated by an electric field from plate A to plate B. What is the potential difference between the plates, and which plate is at the higher potential? 4. (II) The work done by an external force to move a – 9.10 mC charge from point a to point b is 7.00 * 10–4 J. If the charge was started from rest and had 2.10 * 10 –4 J of kinetic energy when it reached point b, what must be the potential difference between a and b?

23–2 Potential Related to Electric Field 5. (I) Thunderclouds typically develop voltage differences of about 1 * 108 V. Given that an electric field of 3 * 106 Vm is required to produce an electrical spark within a volume of air, estimate the length of a thundercloud lightning bolt. [Can you see why, when lightning strikes from a cloud to the ground, the bolt actually has to propagate as a sequence of steps?] 6. (I) The electric field between two parallel plates connected to a 45-V battery is 1300 Vm. How far apart are the plates? 7. (I) What is the maximum amount of charge that a spherical conductor of radius 6.5 cm can hold in air? 8. (I) What is the magnitude of the electric field between two parallel plates 4.0 mm apart if the potential difference between them is 110 V? 9. (I) What minimum radius must a large conducting sphere of an electrostatic generating machine have if it is to be at 35,000 V without discharge into the air? How much charge will it carry? 10. (II) A manufacturer claims that a carpet will not generate more than 5.0 kV of static electricity. What magnitude of charge would have to be transferred between a carpet and a shoe for there to be a 5.0-kV potential difference between the shoe and the carpet. Approximate the shoe and the carpet as large sheets of charge separated by a distance d = 1.0 mm. B 11. (II) A uniform electric field E = – 4.20 NC ˆi points in the negative x direction as shown in Fig. 23–25. The x and y coordinates of points A, B, and C are given on the diagram (in meters). Determine y the differences in C(−3.00, 4.00) potential (a) VBA , B B(4.00, 4.00) E (b) VCB , and (c) VCA . A(4.00, 1.00) FIGURE 23–25 Problem 11.

12. (II) The electric potential of a very large isolated flat metal plate is V0 . It carries a uniform distribution of charge of surface density s ACm2 B, or s2 on each surface. Determine V at a distance x from the plate. Consider the point x to be far from the edges and assume x is much smaller than the plate dimensions.

13. (II) The Earth produces an inwardly directed electric field of magnitude 150 Vm near its surface. (a) What is the potential of the Earth's surface relative to V = 0 at r = q? (b) If the potential of the Earth is chosen to be zero, what is the potential at infinity? (Ignore the fact that positive charge in the ionosphere approximately cancels the Earth's net charge; how would this affect your answer?) 14. (II) A 32-cm-diameter conducting sphere is charged to 680 V relative to V = 0 at r = q. (a) What is the surface charge density s? (b) At what distance will the potential due to the sphere be only 25 V? 15. (II) An insulated spherical conductor of radius r1 carries a charge Q. A second conducting sphere of radius r2 and initially uncharged is then connected to the first by a long conducting wire. (a) After the connection, what can you say about the electric potential of each sphere? (b) How much charge is transferred to the second sphere? Assume the connected spheres are far apart compared to their radii. (Why make this assumption?) 16. (II) Determine the difference in potential between two points that are distances Ra and Rb from a very long ( W Ra or Rb) straight wire carrying a uniform charge per unit length l. 17. (II) Suppose the end of your finger is charged. (a) Estimate the breakdown voltage in air for your finger. (b) About what surface charge density would have to be on your finger at this voltage? 18. (II) Estimate the electric field in the membrane wall of a living cell. Assume the wall is 10 nm thick and has a potential of 0.10 V across it. 19. (II) A nonconducting sphere of radius r0 carries a total charge Q distributed uniformly throughout its volume. Determine the electric potential as a function of the distance r from the center of the sphere for (a) r 7 r0 and (b) r 6 r0 . Take V = 0 at r = q. (c) Plot V versus r and E versus r. 20. (III) Repeat Problem 19 assuming the charge density rE increases as the square of the distance from the center of the sphere, and rE = 0 at the center. 21. (III) The volume charge density rE within a sphere of radius r0 is distributed in accordance with the following spherically symmetric relation rE(r) = r0 c 1 -

d r 20 where r is measured from the center of the sphere and r0 is a constant. For a point P inside the sphere (r 6 r0), determine the electric potential V. Let V = 0 at infinity. 22. (III) A hollow spherical conductor, carrying a net charge ±Q, has inner radius r1 and outer radius r2 = 2r1 (Fig. 23–26). At the center of the sphere is a point charge ±Q2. (a) Write the electric field strength E in all three regions as a function of r. Then determine the potential as a function of r, the distance from the center, r2 for (b) r 7 r2 , (c) r1 6 r 6 r2 , and (d) 0 6 r 6 r1 . (e) Plot both V and E Q Q r1 as a function of r from r = 0 to 2 r = 2r2 . FIGURE 23–26 Problem 22.

Problems

623

23. (III) A very long conducting cylinder (length l) of radius R0 AR0 V lB carries a uniform surface charge density s ACm2 B. The cylinder is at an electric potential V0 . What is the potential, at points far from the end, at a distance R from the center of the cylinder? Determine for (a) R 7 R0 and (b) R 6 R0 . (c) Is V = 0 at R = q (assume l = q )? Explain.

23–3 Potential Due to Point Charges 24. (I) A point charge Q creates an electric potential of ±185 V at a distance of 15 cm. What is Q (let V = 0 at r = q) ? 25. (I) (a) What is the electric potential 0.50 * 10–10 m from a proton (charge ± e)? Let V = 0 at r = q. (b) What is the potential energy of an electron at this point? 26. (II) Two point charges, 3.4 mC and – 2.0 mC, are placed 5.0 cm apart on the x axis. At what points along the x axis is (a) the electric field zero and (b) the potential zero? Let V = 0 at r = q. 27. (II) A ± 25 mC point charge is placed 6.0 cm from an identical ± 25 mC point charge. How much work would be required by an external force to move a ± 0.18 mC test charge from a point midway between them to a point 1.0 cm closer to either of the charges? 28. (II) Point a is 26 cm north of a – 3.8 mC point charge, and point b is 36 cm west of the charge (Fig. 23–27). Determine B B (a) Vb - Va , and (b) Eb - Ea a (magnitude and direction). FIGURE 23–27 Problem 28.

26 cm Q = –3.8 mC

36 cm

29. (II) How much voltage must be used to accelerate a proton (radius 1.2 * 10 –15 m) so that it has sufficient energy to just "touch" a silicon nucleus? A silicon nucleus has a charge of ± 14e and its radius is about 3.6 * 10–15 m. Assume the potential is that for point charges. 30. (II) Two identical ± 5.5 mC point charges are initially spaced 6.5 cm from each other. If they are released at the same instant from rest, how fast will they be moving when they are very far away from each other? Assume they have identical masses of 1.0 mg. 31. (II) An electron starts from rest 42.5 cm from a fixed point charge with Q = – 0.125 nC. How fast will the electron be moving when it is very far away? 32. (II) Two equal but opposite charges are separated by a distance d, as shown in Fig. 23–28. Determine a formula for VBA = VB - VA for points B and A on the line between the charges situated as shown. d FIGURE 23–28 Problem 32.

−q

23–4 Potential Due to Charge Distribution 33. (II) A thin circular ring of radius R (as in Fig. 23–14) has charge ± Q2 uniformly distributed on the top half, and –Q2 on the bottom half. (a) What is the value of the electric potential at a point a distance x along the axis through the center of the circle? (b) What can you say about B the electric field E at a distance x along the axis? Let V = 0 at r = q.

624

CHAPTER 23

Electric Potential

34. (II) Three point charges are arranged at the corners of a square of side l as shown in Fig. 23–29. l What is the potential at the fourth +Q −2Q corner (point A), taking V = 0 at a great distance? l l FIGURE 23–29 Problem 34.

+3Q

35. (II) A flat ring of inner radius R1 and outer radius R2 , Fig. 23–30, carries a uniform surface R2 charge density s. Determine the electric potential at points along the axis R1 (the x axis). [Hint: Try substituting variables.] x 0 FIGURE 23–30 Problem 35. 36. (II) A total charge Q is uniformly distributed on a thread of length l. The thread forms a semicircle. What is the potential at the center? (Assume V = 0 at large distances.) 37. (II) A 12.0-cm-radius thin ring carries a uniformly distributed 15.0 mC charge. A small 7.5-g sphere with a charge of 3.0 mC is placed exactly at the center of the ring and given a very small push so it moves along the ring axis ( ±x axis). How fast will the sphere be moving when it is 2.0 m from the center of the ring (ignore gravity)? 38. (II) A thin rod of length 2l is centered on the x axis as shown in Fig. 23–31. The rod carries a uniformly distributed charge Q. Determine the potential V as y a function of y for points along the y axis. Let V = 0 at infinity. FIGURE 23–31 Problems 38, 39, 40, and 53.

l 2l

39. (II) Determine the potential V(x) for points along the x axis outside the rod of Fig. 23–31 (Problem 38). 40. (III) The charge on the rod of Fig. 23–31 has a nonuniform linear charge distribution, l = ax. Determine the potential V for (a) points along the y axis and (b) points along the x axis outside the rod. 41. (III) Suppose the flat circular disk of Fig. 23–15 (Example 23–9) has a nonuniform surface charge density s = aR2, where R is measured from the center of the disk. Find the potential V(x) at points along the x axis, relative to V = 0 at x = q.

23–5 Equipotentials 42. (I) Draw a conductor in the shape of a football. This conductor carries a net negative charge, –Q. Draw in a dozen or so electric field lines and equipotential lines. 43. (II) Equipotential surfaces are to be drawn 100 V apart near a very large uniformly charged metal plate carrying a surface charge density s = 0.75 mCm2. How far apart (in space) are the equipotential surfaces? 44. (II) A metal sphere of radius r0 = 0.44 m carries a charge Q = 0.50 mC. Equipotential surfaces are to be drawn for 100-V intervals outside the sphere. Determine the radius r of (a) the first, (b) the tenth, and (c) the 100th equipotential from the surface.

23–6 Dipoles 45. (II) Calculate the electric potential due to a tiny dipole whose dipole moment is 4.8 * 10–30 C m at a point 4.1 * 10–9 m away if this point is (a) along the axis of the dipole nearer the positive charge; (b) 45° above the axis but nearer the positive charge; (c) 45° above the axis but nearer the negative charge. Let V = 0 at r = q. 46. (III) The dipole moment, considered as a vector, points from the negative to the positive charge. The water B molecule, Fig. 23–32, has a dipole moment p which can be B considered as the vector sum of the two dipole moments p 1 B and p2 as shown. The distance between each H and the O is about 0.96 * 10 –10 m; the lines joining the center of the O atom with each H atom make an angle of 104° as shown, and the net dipole moment has been measured to be p = 6.1 * 10–30 C m. (a) Determine the effective charge q on each H atom. (b) Determine the electric potenB B tial, far from the molecule, due to each dipole, p 1 and p2 , and show that V =

1 p cos u , 4p 0 r2

where p is the magnitude of the net dipole moment, B B B p = p 1 + p2 , and V is the total potential due to both B B p 1 and p2 . Take V = 0 at r = q. FIGURE 23–32 Problem 46.

B p 1

B p

104°

H B p 2

23–7 E Determined from V 47. (I) Show that the electric field of a single point charge (Eq. 21–4) follows from Eq. 23–5, V = A14p 0 B(Qr). 48. (I) What is the potential gradient just outside the surface of a uranium nucleus (Q = ± 92e) whose diameter is about 15 * 10 –15 m? 49. (II) The electric potential between two parallel plates is given by V(x) = (8.0 Vm) x + 5.0 V, with x = 0 taken at one of the plates and x positive in the direction toward the other plate. What is the charge density on the plates? 50. (II) The electric potential in a region of space varies as B V = byAa2 + y 2 B. Determine E. 51. (II) In a certain region of space, the electric potential is given by V = y 2 + 2.5xy - 3.5xyz. Determine the electric B field vector, E, in this region. 52. (II) A dust particle with mass of 0.050 g and a charge of 2.0 * 10 –6 C is in a region of space where the potential is given by V(x) = A2.0 Vm2 B x 2 - A3.0 Vm3 B x3. If the particle starts at x = 2.0 m, what is the initial acceleration of the charge? 53. (III) Use the results of Problems 38 and 39 to determine the electric field due to the uniformly charged rod of Fig. 23–31 for points (a) along the y axis and (b) along the x axis.

23–8 Electrostatic Potential Energy; Electron Volt 54. (I) How much work must be done to bring three electrons from a great distance apart to within 1.0 * 10 –10 m from one another (at the corners of an equilateral triangle)? 55. (I) What potential difference is needed to give a helium nucleus AQ = 3.2 * 10–19 CB 125 keV of kinetic energy?

56. (I) What is the speed of (a) a 1.5-keV (kinetic energy) electron and (b) a 1.5-keV proton? 57. (II) Many chemical reactions release energy. Suppose that at the beginning of a reaction, an electron and proton are separated by 0.110 nm, and their final separation is 0.100 nm. How much electric potential energy was lost in this reaction (in units of eV)? 58. (II) An alpha particle (which is a helium nucleus, Q = ±2e, m = 6.64 * 10–27 kg) is emitted in a radioactive decay with kinetic energy 5.53 MeV. What is its speed? 59. (II) Write the total electrostatic potential energy, U, for (a) four point charges and (b) five point charges. Draw a diagram defining all quantities. 60. (II) Four equal point charges, Q, are fixed at the corners of a square of side b. (a) What is their total electrostatic potential energy? (b) How much potential energy will a fifth charge, Q, have at the center of the square (relative to V = 0 at r = q )? (c) If constrained to remain in that plane, is the fifth charge in stable or unstable equilibrium? (d) If a negative (–Q) charge is at the center, is it in stable equilibrium? 61. (II) An electron starting from rest acquires 1.33 keV of kinetic energy in moving from point A to point B. (a) How much kinetic energy would a proton acquire, starting from rest at B and moving to point A? (b) Determine the ratio of their speeds at the end of their respective trajectories. 62. (II) Determine the total electrostatic potential energy of a conducting sphere of radius r0 that carries a total charge Q distributed uniformly on its surface. 63. (II) The liquid-drop model of the nucleus suggests that highenergy oscillations of certain nuclei can split ("fission") a large nucleus into two unequal fragments plus a few neutrons. Using this model, consider the case of a uranium nucleus fissioning into two spherical fragments, one with a charge q1 = ±38e and radius r1 = 5.5 * 10–15 m, the other with q2 = ±54e and r2 = 6.2 * 10–15 m. Calculate the electric potential energy (MeV) of these fragments, assuming that the charge is uniformly distributed throughout the volume of each spherical nucleus and that their surfaces are initially in contact at rest. The electrons surrounding the nuclei can be neglected. This electric potential energy will then be entirely converted to kinetic energy as the fragments repel each other. How does your predicted kinetic energy of the fragments agree with the observed value associated with uranium fission (approximately 200 MeV total)? C1 MeV = 106 eV. D 64. (III) Determine the total electrostatic potential energy of a nonconducting sphere of radius r0 carrying a total charge Q distributed uniformly throughout its volume. * 23–9 CRT * 65. (I) Use the ideal gas as a model to estimate the rms speed of a free electron in a metal at 273 K, and at 2700 K (a typical temperature of the cathode in a CRT). * 66. (III) Electrons are accelerated by 6.0 kV in a CRT. The screen is 28 cm wide and is 34 cm from the 2.6-cm-long deflection plates. Over what range must the horizontally deflecting electric field vary to sweep the beam fully across the screen? * 67. (III) In a given CRT, electrons are accelerated horizontally by 7.2 kV. They then pass through a uniform electric field E for a distance of 2.8 cm which deflects them upward so they reach the top of the screen 22 cm away, 11 cm above the center. Estimate the value of E.

Problems

625

General Problems 68. If the electrons in a single raindrop, 3.5 mm in diameter, could be removed from the Earth (without removing the atomic nuclei), by how much would the potential of the Earth increase? 69. By rubbing a nonconducting material, a charge of 10–8 C can readily be produced. If this is done to a sphere of radius 15 cm, estimate the potential produced at the surface. Let V = 0 at r = q. 70. Sketch the electric field and equipotential lines for two charges of the same sign and magnitude separated by a distance d. 71. A ± 33 mC point charge is placed 36 cm from an identical ±33 mC charge. A – 1.5 mC charge is moved from point a to point b, Fig. 23–33. b What is the change in potential energy? FIGURE 23–33 Problem 71.

14 cm 24 cm 12 cm 33 mC a

33 mC

72. At each corner of a cube of side l there is a point charge Q, Fig. 23–34. (a) What is the potential z at the center of the cube (V = 0 at Q Q r = q)? (b) What is the potential at each corner due to the other Q Q seven charges? (c) What is the total potential energy of this system? y Q Q FIGURE 23–34 Q x Q Problem 72. 73. In a television picture tube (CRT), electrons are accelerated by thousands of volts through a vacuum. If a television set is laid on its back, would electrons be able to move upward against the force of gravity? What potential difference, acting over a distance of 3.5 cm, would be needed to balance the downward force of gravity so that an electron would remain stationary? Assume that the electric field is uniform. 74. Four point charges are located at the corners of a square that is 8.0 cm on a side. The charges, going in rotation around the square, are Q, 2Q, – 3Q, and 2Q, where Q = 3.1 mC (Fig. 23–35). What is the Q 8.0 cm 2Q total electric potential energy stored in the system, relative to U = 0 at infinite separation? FIGURE 23–35 Problem 74.

CHAPTER 23

Electric Potential

+ + + + + + + +

vB FIGURE 23–37 Problem 76.

− − − − − − − −

77. Three charges are at the corners of an equilateral triangle (side l) as shown in Fig. 23–38. Determine the potential at the y midpoint of each of the sides. Let V = 0 at r = q. –Q l +Q l

FIGURE 23–38 Problem 77.

–3Q

78. Near the surface of the Earth there is an electric field of about 150 Vm which points downward. Two identical balls with mass m = 0.340 kg are dropped from a height of 2.00 m, but one of the balls is positively charged with q1 = 450 mC, and the second is negatively charged with q2 = –450 mC. Use conservation of energy to determine the difference in the speeds of the two balls when they hit the ground. (Neglect air resistance.) 79. A lightning flash transfers 4.0 C of charge and 4.8 MJ of energy to the Earth. (a) Between what potential difference did it travel? (b) How much water could this energy boil, starting from room temperature? [Hint: See Chapter 19.] 80. Determine the components of the electric field, Ex and Ey , as a function of x and y in the xy plane due to a y dipole, Fig. 23–39, starting P(x, y) with Eq. 23–7. Assume 2 2 12 r = Ax + y B W l. B

−3Q

75. In a photocell, ultraviolet (UV) light provides enough energy to some electrons in barium metal to eject them from the surface at high speed. See Fig. 23–36. To measure the maximum energy of the electrons, another plate above the barium surface is kept at a negative enough potential that the emitted electrons are slowed down and stopped, and return to the barium surface. If the plate voltage is –3.02 V (compared to the barium) when the fastest electrons are stopped, what Plate was the speed of these electrons when they were V = −3.02 V UV – emitted? – light + V=0 FIGURE 23–36 Barium Problem 75.

626

76. An electron is accelerated horizontally from rest in a television picture tube by a potential difference of 5500 V. It then passes between two horizontal plates 6.5 cm long and 1.3 cm apart that have a potential difference of 250 V (Fig. 23–37). At what angle u will the electron be traveling after it passes between the plates?

FIGURE 23–39 Problem 80.

u −Q

81. A nonconducting sphere of radius r2 contains a concentric spherical cavity of radius r1 . The material between r1 and r2 carries a uniform charge density rE ACm3 B. Determine the electric potential V, relative to V = 0 at r = q, as a function of the distance r from the center for (a) r 7 r2 , (b) r1 6 r 6 r2 , and (c) 0 6 r 6 r1 . Is V continuous at r1 and r2? 82. A thin flat nonconducting disk, with radius R0 and charge Q, has a hole with a radius R02 in its center. Find the electric potential V(x) at points along the symmetry (x) axis of the disk (a line perpendicular to the disk, passing through its center). Let V = 0 at x = q.

83. A Geiger counter is used to detect charged particles emitted by radioactive nuclei. It consists of a thin, positively charged central wire of radius Ra surrounded by a concentric conducting cylinder of radius Rb with an equal negative charge (Fig. 23–40). The charge per unit length on the inner wire is l (units Cm). The interior space between wire and cylinder is filled with low-pressure inert gas. Charged particles ionize some of these gas atoms; the resulting free electrons are attracted toward the positive central wire. If the radial electric field is strong enough, the freed electrons gain enough energy to ionize other atoms, causing an "avalanche" of electrons to strike the central wire, generating an electric "signal." Find the expression for the electric field between the wire and the cylinder, and show that the potential difference between Ra and Rb is Va - Vb = ¢

Rb l ≤ ln ¢ ≤. Ra 2p 0

86. A charge –q1 of mass m rests on the y axis at a distance b above the x axis. Two positive charges of magnitude ±q2 are fixed on the x axis at x = ± a and x = –a, respectively (Fig. 23–42). If the –q1 charge is given an initial velocity v0 in the positive y direction, what is the minimum value of v0 such that the charge y escapes to a point infinitely far away from +b –q1, m the two positive charges?

+q2

FIGURE 23–42 Problem 86.

–a

+q2

* Numerical/Computer Rb

– +

Central wire, radius Ra

FIGURE 23–40 Problem 83.

84. A Van de Graaff generator (Fig. 23–41) can develop a very large potential difference, even millions of volts. Electrons are pulled off the belt by the high voltage pointed electrode at A, leaving the belt positively charged. (Recall Example 23–5 where we saw that near sharp points the electric field is high and ionization can occur.) The belt carries the positive charge up inside the spherical shell where electrons from the large conducting sphere are attracted over to the pointed conductor at B, leaving the outer surface of the conducting sphere positively charged. As more charge is brought up, the sphere reaches extremely high voltage. Consider a Van de Graaff generator with a GENERATOR sphere of radius 0.20 m. + + + Conductor (a) What is the electric + + potential on the surface + + of the sphere when B electrical breakdown + + Pulley + occurs? (Assume V = 0 + + + at r = q.) (b) What is + + + the charge on the + + sphere for the poten+ + tial found in part (a)? + Insulator + + + Motor-driven pulley A FIGURE 23–41 50 kV Problem 84. +

Belt

85. The potential in a region of space is given by V = BAx2 + R2 B 2 where B = 150 Vm4 and R = 0.20 m. B (a) Find V at x = 0.20 m. (b) Find E as a function of x. B (c) Find E at x = 0.20 m.

* 87. (II) A dipole is composed of a –1.0 nC charge at x = –1.0 cm and a ±1.0 nC charge at x = ±1.0 cm. (a) Make a plot of V along the x axis from x = 2.0 cm to x = 15 cm. (b) On the same graph, plot the approximate V using Eq. 23–7 from x = 2.0 cm to x = 15 cm. Let V = 0 at x = q. * 88. (II) A thin flat disk of radius R0 carries a total charge Q that is distributed uniformly over its surface. The electric potential at a distance x on the x axis is given by V(x) =

Q 2p 0 R 20

C Ax 2 + R20 B 2 - x D. 1

(See Example 23–9.) Show that the electric field at a distance x on the x axis is given by E(x) =

Q 2p 0 R20

x . ¢ 1 - Ax 2 + R 2 B 12 ≤ 0

Make graphs of V(x) and E(x) as a function of xR0 for xR0 = 0 to 4. (Do the calculations in steps of 0.1.) Use Q = 5.0 mC and R0 = 10 cm for the calculation and graphs. * 89. (III) You are trying to determine an unknown amount of charge using only a voltmeter and a ruler, knowing that it is either a single sheet of charge or a point charge that is creating it. You determine the direction of greatest change of potential, and then measure potentials along a line in that direction. The potential versus position (note that the zero of position is arbitrary, and the potential is measured relative to ground) is measured as follows: x (cm) V (volts)

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 3.9 3.0 2.5 2.0 1.7 1.5 1.4 1.4 1.2 1.1

(a) Graph V versus position. Do you think the field is caused by a sheet or a point charge? (b) Graph the data in such a way that you can determine the magnitude of the charge and determine that value. (c) Is it possible to determine where the charge is from this data? If so, give the position of the charge.

Answers to Exercises A: (b).

D: (a) iii, (b) i, (c) i.

B: (d).

E: (c).

C: (d).

F: (d).

General Problems

627

Capacitors come in a wide range of sizes and shapes, only a few of which are shown here. A capacitor is basically two conductors that do not touch, and which therefore can store charge of opposite sign on its two conductors. Capacitors are used in a wide variety of circuits, as we shall see in this and later Chapters.

P T E R

C H

CONTENTS 24–1 Capacitors 24–2 Determination of Capacitance 24–3 Capacitors in Series and Parallel 24–4 Electric Energy Storage

Capacitance, Dielectrics, Electric Energy Storage CHAPTER-OPENING QUESTION—Guess now! A fixed potential difference V exists between a pair of close parallel plates carrying opposite charges ±Q and –Q. Which of the following would not increase the magnitude of charge that you could put on the plates? (a) Increase the size of the plates. (b) Move the plates farther apart. (c) Fill the space between the plates with paper. (d) Increase the fixed potential difference V. (e) None of the above.

24–5 Dielectrics

*24–6 Molecular Description of Dielectrics

his Chapter will complete our study of electrostatics. It deals first of all with an important device, the capacitor, which is used in many electronic circuits. We will also discuss electric energy storage and the effects of an insulator, or dielectric, on electric fields and potential differences.

24–1 P H Y S I C S

A P P L I E D

Uses of capacitors

628

Capacitors

A capacitor is a device that can store electric charge, and normally consists of two conducting objects (usually plates or sheets) placed near each other but not touching. Capacitors are widely used in electronic circuits. They store charge for later use, such as in a camera flash, and as energy backup in computers if the power fails. Capacitors also block surges of charge and energy to protect circuits.

Insulator

(a)

FIGURE 24–1 Capacitors: diagrams of (a) parallel plate, (b) cylindrical (rolled up parallel plate).

(b)

Very tiny capacitors serve as memory for the "ones" and "zeros" of the binary code in the random access memory (RAM) of computers. Capacitors serve many other applications, some of which we will discuss. A simple capacitor consists of a pair of parallel plates of area A separated by a small distance d (Fig. 24–1a). Often the two plates are rolled into the form of a cylinder with plastic, paper, or other insulator separating the plates, Fig. 24–1b. In a diagram, the symbol or

[capacitor symbol]

represents a capacitor. A battery, which is a source of voltage, is indicated by the symbol:

[battery symbol]

with unequal arms. If a voltage is applied across a capacitor by connecting the capacitor to a battery with conducting wires as in Fig. 24–2, the two plates quickly become charged: one plate acquires a negative charge, the other an equal amount of positive charge. Each battery terminal and the plate of the capacitor connected to it are at the same potential; hence the full battery voltage appears across the capacitor. For a given capacitor, it is found that the amount of charge Q acquired by each plate is proportional to the magnitude of the potential difference V between them: Q = CV.

Q Q

(24–1)

The constant of proportionality, C, in the above relation is called the capacitance of the capacitor. The unit of capacitance is coulombs per volt and this unit is called a farad (F). Common capacitors have capacitance in the range of 1 pF Apicofarad = 10 –12 FB to 103 mF Amicrofarad = 10–6 FB. The relation, Eq. 24–1, was first suggested by Volta in the late eighteenth century. The capacitance C does not in general depend on Q or V. Its value depends only on the size, shape, and relative position of the two conductors, and also on the material that separates them. In Eq. 24–1, and from now on, we use simply V (in italics) to represent a potential difference, rather than Vba , ¢V, or Vb - Va , as previously. (Be sure not to confuse italic V and C which stand for voltage and capacitance, with non-italic V and C which stand for the units volts and coulombs).

−

12 V

(a)

(b)

V FIGURE 24–2 (a) Parallel-plate capacitor connected to a battery. (b) Same circuit shown using symbols. FIGURE 24–3 Exercise A. A B

V C A U T I O N

V = potential difference from here on

EXERCISE A Graphs for charge versus voltage are shown in Fig. 24–3 for three capacitors, A, B, and C. Which has the greatest capacitance?

SECTION 24–1

Capacitors

629

24–2 d

B E a b

FIGURE 24–4 Parallel-plate capacitor, each of whose plates has area A. Fringing of the field is ignored.

Determination of Capacitance

The capacitance of a given capacitor can be determined experimentally directly from Eq. 24–1, by measuring the charge Q on either conductor for a given potential difference V. For capacitors whose geometry is simple, we can determine C analytically, and in this Section we assume the conductors are separated by a vacuum or air. First, we determine C for a parallel-plate capacitor, Fig. 24–4. Each plate has area A and the two plates are separated by a distance d. We assume d is small compared to the B dimensions of each plate so that the electric field E is uniform between them and B we can ignore fringing (lines of E not straight) at the edges. We saw earlier (Example 21–13) that the electric field between two closely spaced parallel plates has magnitude E = s 0 and its direction is perpendicular to the plates. Since s is the charge per unit area, s = QA, then the field between the plates is Q . E = 0 A The relation between electric field and electric potential, as given by Eq. 23–4a, is b

V = Vba = Vb - Va = –

E dL. B

We can take the line integral along a path antiparallel to the field lines, from plate a to plate b; then u = 180° and cos 180° = –1, so b b b Q Qd . V = Vb - Va = – E dl cos 180° = ± E dl = dl = 0 A a 0 A a a This relates Q to V, and from it we can get the capacitance C in terms of the geometry of the plates: Q A C = = 0 . [parallel-plate capacitor] (24–2) V d Note from Eq. 24–2 that the value of C does not depend on Q or V, so Q is predicted to be proportional to V as is found experimentally.

EXAMPLE 24–1 Capacitor calculations. (a) Calculate the capacitance of a parallel-plate capacitor whose plates are 20 cm * 3.0 cm and are separated by a 1.0-mm air gap. (b) What is the charge on each plate if a 12-V battery is connected across the two plates? (c) What is the electric field between the plates? (d) Estimate the area of the plates needed to achieve a capacitance of 1 F, given the same air gap d. APPROACH The capacitance is found by using Eq. 24–2, C = 0 Ad. The charge on each plate is obtained from the definition of capacitance, Eq. 24–1, Q = CV. The electric field is uniform, so we can use Eq. 23–4b for the magnitude E = Vd. In (d) we use Eq. 24–2 again. SOLUTION (a) The area A = A20 * 10–2 mBA3.0 * 10 –2 mB = 6.0 * 10–3 m2. The capacitance C is then A 6.0 * 10 –3 m2 C = 0 = A8.85 * 10–12 C 2N m2 B = 53 pF. d 1.0 * 10 –3 m (b) The charge on each plate is Q = CV = A53 * 10–12 FB(12 V) = 6.4 * 10–10 C. (c) From Eq. 23–4b for a uniform electric field, the magnitude of E is V 12 V E = = = 1.2 * 104 Vm. d 1.0 * 10–3 m (d) We solve for A in Eq. 24–2 and substitute C = 1.0 F and d = 1.0 mm to find that we need plates with an area

(1 F)A1.0 * 10–3 mB Cd L L 108 m2. 0 A9 * 10 –12 C 2N m2 B NOTE This is the area of a square 104 m or 10 km on a side.That is the size of a city like San Francisco or Boston! Large-capacitance capacitors will not be simple parallel plates. A =

630

CHAPTER 24

EXERCISE B Two circular plates of radius 5.0 cm are separated by a 0.10-mm air gap. What is the magnitude of the charge on each plate when connected to a 12-V battery?

Not long ago, a capacitance greater than a few mF was unusual. Today capacitors are available that are 1 or 2 F, yet they are just a few cm on a side. Such capacitors are used as power backups, for example, in computer memory and electronics where the time and date can be maintained through tiny charge flow. [Capacitors are superior to rechargable batteries for this purpose because they can be recharged more than 105 times with no degradation.] Such high-capacitance capacitors can be made of "activated" carbon which has very high porosity, so that the surface area is very large; one tenth of a gram of activated carbon can have a surface area of 100 m2. Furthermore, the equal and opposite charges exist in an electric "double layer" about 10–9 m thick. Thus, the capacitance of 0.1 g of activated carbon, whose internal area can be 102 m2, is equivalent to a parallel-plate capacitor with C L 0 Ad = A8.85 * 10–12 C 2N m2 BA102 m2 BA10–9 mB L 1 F. One type of computer keyboard operates by capacitance. As shown in Fig. 24–5, each key is connected to the upper plate of a capacitor. The upper plate moves down when the key is pressed, reducing the spacing between the capacitor plates, and increasing the capacitance (Eq. 24–2: smaller d, larger C). The change in capacitance results in an electric signal that is detected by an electronic circuit. The proportionality, C r Ad in Eq. 24–2, is valid also for a parallel-plate capacitor that is rolled up into a spiral cylinder, as in Fig. 24–1b. However, the constant factor, 0 , must be replaced if an insulator such as paper separates the plates, as is usual, and this is discussed in Section 24–5. For a true cylindrical capacitor—consisting of two long coaxial cylinders—the result is somewhat different as the next Example shows. EXAMPLE 24–2 Cylindrical capacitor. A cylindrical capacitor consists of a cylinder (or wire) of radius Rb surrounded by a coaxial cylindrical shell of inner radius Ra , Fig. 24–6a. Both cylinders have length l which we assume is much greater than the separation of the cylinders, Ra - Rb , so we can neglect end effects. The capacitor is charged (by connecting it to a battery) so that one cylinder has a charge ±Q (say, the inner one) and the other one a charge –Q. Determine a formula for the capacitance. APPROACH To obtain C = QV, we need to determine the potential difference V between the cylinders in terms of Q. We can use our earlier result (Example 21–11 or 22–6) that the electric field outside a long wire is directed radially outward and has magnitude E = A12p 0 B(lR), where R is the distance from the axis and l is the charge per unit length, Ql. Then E = A12p 0 B(QlR) for points between the cylinders. SOLUTION To obtain the potential difference V in terms of Q, we use this result bB for E in Eq. 23–4a, V = Vb - Va = – a E dL , and write the line integral from the outer cylinder to the inner one (so V 7 0) along a radial line:† b Rb Q dR B V = Vb - Va = – E dL = – 2p 0 l Ra R a Rb Ra Q Q . ln = ln = – 2p 0 l Ra 2p 0 l Rb Q and V are proportional, and the capacitance C is 2p 0 l Q . [cylindrical capacitor] C = = V ln ARaRb B

P H Y S I C S

A P P L I E D

Very high capacitance

Key Movable plate

Capacitor

Insulator (flexible)

Fixed plate

FIGURE 24–5 Key on a computer keyboard. Pressing the key reduces the capacitor spacing thus increasing the capacitance which can be detected electronically. P H Y S I C S

A P P L I E D

Computer key

FIGURE 24–6 (a) Cylindrical capacitor consists of two coaxial cylindrical conductors. (b) The electric field lines are shown in cross-sectional view.

Ra l

EXERCISE C What is the capacitance per unit length of a cylindrical capacitor with radii Ra = 2.5 mm and Rb = 0.40 mm? (a) 30 pFm; (b) –30 pFm; (c) 56 pFm; (d) –56 pFm; (e) 100 pFm; (f ) – 100 pFm. B

(a)

NOTE If the space between cylinders, Ra - Rb = ¢R is small, we have ln ARaRb B = ln C ARb + ¢RBRb D = ln C1 + ¢RRb D L ¢RRb (see Appendix A–3) so C L 2p 0 lRb¢R = 0 A¢R because the area of cylinder b is A = 2pRbl. This is just Eq. 24–2 (d = ¢R), a nice check.

†

Rb +Q

(b)

Note that E points outward in Fig. 24–6b, but dL points inward for our chosen direction of integration; B B the angle between E and dL is 180° and cos 180° = – 1. Also, dl = – dr because dr increases outward. These two minus signs cancel.

SECTION 24–2

631

B Q E r rb

EXAMPLE 24–3 Spherical capacitor. A spherical capacitor consists of two thin concentric spherical conducting shells, of radius ra and rb as shown in Fig. 24–7. The inner shell carries a uniformly distributed charge Q on its surface, and the outer shell an equal but opposite charge –Q. Determine the capacitance of the two shells.

Vba = –

FIGURE 24–7 Cross section through the center of a spherical capacitor. The thin inner shell has radius rb and the thin outer shell has radius ra .

P R O B L E M

APPROACH In Example 22–3 we used Gauss's law to show that the electric field outside a uniformly charged conducting sphere is E = Q4p 0 r2 as if all the bB charge were concentrated at the center. Now we use Eq. 23–4a, V = – a E dL . SOLUTION We integrate Eq. 23–4a along a radial line to obtain the potential difference between the two conducting shells:

S O LV I N G

Checking with a limiting case

E dL = –

Q 4p 0

1 dr r2

ra - rb Q Q 1 1 ¢ ≤ = ¢ ≤. ra ra rb 4p 0 rb 4p 0

Finally, C =

ra rb Q = 4p 0 ¢ ≤. ra - rb Vba

NOTE If the separation ¢r = ra - rb is very small, then C = 4p 0 r2¢r L 0 A¢r (since A = 4pr2), which is the parallel-plate formula, Eq. 24–2. A single isolated conductor can also be said to have a capacitance, C. In this case, C can still be defined as the ratio of the charge to absolute potential V on the conductor (relative to V = 0 at r = q ), so that the relation Q = CV remains valid. For example, the potential of a single conducting sphere of radius rb can be obtained from our results in Example 24–3 by letting ra become infinitely large. As ra S q, then V =

Q 1 1 1 Q ¢ ≤ = ; ra 4p 0 rb 4p 0 rb

so its capacitance is C =

Q = 4p 0 rb . V

In practical cases, a single conductor may be near other conductors or the Earth (which can be thought of as the other "plate" of a capacitor), and these will affect the value of the capacitance.

FIGURE 24–8 Example 24–4. a

x –Q

EXAMPLE 24–4 Capacitance of two long parallel wires. Estimate the capacitance per unit length of two very long straight parallel wires, each of radius R, carrying uniform charges ±Q and –Q, and separated by a distance d which is large compared to R (d W R), Fig. 24–8. APPROACH We calculate the potential difference between the wires by treating the electric field at any point between them as the superposition of the two fields created by each wire. (The electric field inside each wire conductor is zero.) SOLUTION The electric field outside of a long straight conductor was found in Examples 21–11 and 22–6 to be radial and given by E = l(2p 0 x) where l is the charge per unit length, l = Ql. The total electric field at distance x from the left-hand wire in Fig. 24–8 has magnitude E =

l l , + 2p 0 x 2p 0(d - x)

and points to the left (from ± to – ). Now we find the potential difference

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between the two wires using Eq. 23–4a and integrating along the straight line from the surface of the negative wire to the surface of the positive wire, noting B B that E and dL point in opposite directions AE dL 6 0B: B

d-R

l V = Vb - Va = – E dL = ¢ ≤ 2p 0 a B

c x + (d - x) d dx 1

= ¢

d-R l ≤ Cln(x) - ln(d - x)D 2 2p 0 R

= ¢

l ≤ Cln(d - R) - ln R - ln R + ln(d - R)D 2p 0

l l ≤ Cln(d - R) - ln(R)D L ¢ ≤ Cln(d) - ln(R)D. p 0 p 0 We are given that d W R, so Q d V L ¢ ≤ B ln ¢ ≤ R . p 0 l R = ¢

The capacitance from Eq. 24–1 is C = QV L Ap 0 lBln(dR), capacitance per unit length is given approximately by

so the

p 0 C . L l ln(dR)

24–3

Capacitors in Series and Parallel

Capacitors are found in many electric circuits. By electric circuit we mean a closed path of conductors, usually wires connecting capacitors and/or other devices, in which charge can flow and which includes a source of voltage such as a battery. The battery voltage is usually given the symbol V, which means that V represents a potential difference. Capacitors can be connected together in various ways. Two common ways are in series, or in parallel, and we now discuss both. A circuit containing three capacitors connected in parallel is shown in Fig. 24–9. They are in "parallel" because when a battery of voltage V is connected to points a and b, this voltage V = Vab exists across each of the capacitors. That is, since the left-hand plates of all the capacitors are connected by conductors, they all reach the same potential Va when connected to the battery; and the right-hand plates each reach potential Vb . Each capacitor plate acquires a charge given by Q1 = C1 V, Q2 = C2 V, and Q3 = C3 V. The total charge Q that must leave the battery is then

Q1 a

V = Vab FIGURE 24–9 Capacitors in parallel: Ceq = C1 + C2 + C3 .

Q = Q1 + Q2 + Q3 = C1 V + C2 V + C3 V. Let us try to find a single equivalent capacitor that will hold the same charge Q at the same voltage V = Vab . It will have a capacitance Ceq given by Q = Ceq V. Combining the two previous equations, we have or

Ceq V = C1 V + C2 V + C3 V = AC1 + C2 + C3 BV Ceq = C1 + C2 + C3 .

[parallel] (24–3)

The net effect of connecting capacitors in parallel is thus to increase the capacitance. This makes sense because we are essentially increasing the area of the plates where charge can accumulate (see, for example, Eq. 24–2). SECTION 24–3

Capacitors in Series and Parallel

633

Q Q Q Q Q Q

V = Vab FIGURE 24–10 Capacitors in series:

Capacitors can also be connected in series: that is, end to end as shown in Fig. 24–10. A charge ±Q flows from the battery to one plate of C1 , and –Q flows to one plate of C3 . The regions A and B between the capacitors were originally neutral; so the net charge there must still be zero. The ±Q on the left plate of C1 attracts a charge of –Q on the opposite plate. Because region A must have a zero net charge, there is thus ±Q on the left plate of C2 . The same considerations apply to the other capacitors, so we see the charge on each capacitor is the same value Q. A single capacitor that could replace these three in series without affecting the circuit (that is, Q and V the same) would have a capacitance Ceq where Q = Ceq V.

1 1 1 1 . = + + Ceq C1 C2 C3

Now the total voltage V across the three capacitors in series must equal the sum of the voltages across each capacitor: V = V1 + V2 + V3 . We also have for each capacitor Q = C1 V1 , Q = C2 V2 , and we substitute for V, V1 , V2 , and V3 into the last equation and get

Q = C3 V3 , so

Q Q Q Q 1 1 1 = + + = Q¢ + + ≤ Ceq C1 C2 C3 C1 C2 C3 or 1 1 1 1 . = + + Ceq C1 C2 C3

[series] (24–4)

Notice that the equivalent capacitance Ceq is smaller than the smallest contributing capacitance. EXERCISE D Consider two identical capacitors C1 = C2 = 10 mF. What are the minimum and maximum capacitances that can be obtained by connecting these in series or parallel combinations? (a) 0.2 mF, 5 mF; (b) 0.2 mF, 10 mF; (c) 0.2 mF, 20 mF; (d) 5 mF, 10 mF; (e) 5 mF, 20 mF; ( f ) 10 mF, 20 mF.

FIGURE 24–11 Examples 24–5 and 24–6.

C2 C1

(a) a

V (b)

EXAMPLE 24–5 Equivalent capacitance. Determine the capacitance of a single capacitor that will have the same effect as the combination shown in Fig. 24–11a. Take C1 = C2 = C3 = C. APPROACH First we find the equivalent capacitance of C2 and C3 in parallel, and then consider that capacitance in series with C1 . SOLUTION Capacitors C2 and C3 are connected in parallel, so they are equivalent to a single capacitor having capacitance

C23

Other connections of capacitors can be analyzed similarly using charge conservation, and often simply in terms of series and parallel connections.

C23 = C2 + C3 = 2C. b

This C23 is in series with C1 , Fig. 24–11b, so the equivalent capacitance of the entire circuit, Ceq , is given by 1 1 1 1 1 3 . = + = + = Ceq C1 C23 C 2C 2C Hence the equivalent capacitance of the entire combination is Ceq = 23 C, and it is smaller than any of the contributing capacitors, C1 = C2 = C3 = C.

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EXAMPLE 24–6 Charge and voltage on capacitors. Determine the charge on each capacitor in Fig. 24–11a of Example 24–5 and the voltage across each, assuming C = 3.0 mF and the battery voltage is V = 4.0 V. APPROACH We have to work "backward" through Example 24–5. That is, we find the charge Q that leaves the battery, using the equivalent capacitance. Then we find the charge on each separate capacitor and the voltage across each. Each step uses Eq. 24–1, Q = CV. SOLUTION The 4.0-V battery behaves as if it is connected to a capacitance Ceq = 23 C = 23 (3.0 mF) = 2.0 mF. Therefore the charge Q that leaves the battery, by Eq. 24–1, is Q = CV = (2.0 mF)(4.0 V) = 8.0 mC. From Fig. 24–11a, this charge arrives at the negative plate of C1 , so Q1 = 8.0 mC. The charge Q that leaves the positive plate of the battery is split evenly between C2 and C3 (symmetry: C2 = C3 ) and is Q2 = Q3 = 12 Q = 4.0 mC. Next, the voltages across C2 and C3 have to be the same. The voltage across each capacitor is obtained using V = QC. So V1 = Q1C1 = (8.0 mC)(3.0 mF) = 2.7 V V2 = Q2C2 = (4.0 mC)(3.0 mF) = 1.3 V V3 = Q3C3 = (4.0 mC)(3.0 mF) = 1.3 V. EXAMPLE 24–7 Capacitors reconnected. Two capacitors, C1 = 2.2 mF and C2 = 1.2 mF, are connected in parallel to a 24-V source as shown in Fig. 24–12a. After they are charged they are disconnected from the source and from each other, and then reconnected directly to each other with plates of opposite sign connected together (see Fig. 24–12b). Find the charge on each capacitor and the potential across each after equilibrium is established. APPROACH We find the charge Q = CV on each capacitor initially. Charge is conserved, although rearranged after the switch. The two new voltages will have to be equal. SOLUTION First we calculate how much charge has been placed on each capacitor after the power source has charged them fully, using Eq. 24–1: Q1 = C1 V = (2.2 mF)(24 V) = 52.8 mC, Q2 = C2 V = (1.2 mF)(24 V) = 28.8 mC.

and

(a) Initial configuration.

(b) At the instant of reconnection only. q1

Next the capacitors are connected in parallel, Fig. 24–12b, and the potential difference across each must quickly equalize. Thus, the charge cannot remain as shown in Fig. 24–12b, but the charge must rearrange itself so that the upper plates at least have the same sign of charge, with the lower plates having the opposite charge as shown in Fig. 24–12c. Equation 24–1 applies for each: q1 = C1 V¿

24 V

q2 q2

q2 = C2 V¿,

where V¿ is the voltage across each capacitor after the charges have rearranged themselves. We don't know q1 , q2 , or V¿, so we need a third equation. This is provided by charge conservation. The charges have rearranged themselves between Figs. 24–12b and c. The total charge on the upper plates in those two Figures must be the same, so we have q1 + q2 = Q1 - Q2 = 24.0 mC. Combining the last three equations we find:

V¿ = Aq1 + q2 BAC1 + C2 B = 24.0 mC3.4 mF = 7.06 V L 7.1 V q1 = C1 V¿ = (2.2 mF)(7.06 V) = 15.5 mC L 16 mC q2 = C2 V¿ = (1.2 mF)(7.06 V) = 8.5 mC

where we have kept only two significant figures in our final answers. SECTION 24–3

Capacitors in Series and Parallel

635

24–4

Electric Energy Storage

A charged capacitor stores electrical energy. The energy stored in a capacitor will be equal to the work done to charge it. The net effect of charging a capacitor is to remove charge from one plate and add it to the other plate. This is what a battery does when it is connected to a capacitor. A capacitor does not become charged instantly. It takes time (Section 26–4). Initially, when the capacitor is uncharged, it requires no work to move the first bit of charge over. When some charge is on each plate, it requires work to add more charge of the same sign because of the electric repulsion. The more charge already on a plate, the more work required to add additional charge. The work needed to add a small amount of charge dq, when a potential difference V is across the plates, is dW = V dq. Since V = qC at any moment (Eq. 24–1), where C is the capacitance, the work needed to store a total charge Q is W =

V dq =

1 C

Q2 . q dq = 12 C 0

Thus we can say that the energy "stored" in a capacitor is Q2 U = 12 C when the capacitor C carries charges ±Q and –Q on its two conductors. Since Q = CV, where V is the potential difference across the capacitor, we can also write Q2 U = 12 = 12 CV 2 = 12 QV. C P H Y S I C S

A P P L I E D

Camera flash

(24–5)

EXAMPLE 24–8 Energy stored in a capacitor. A camera flash unit (Fig. 24–13) stores energy in a 150-mF capacitor at 200 V. (a) How much electric energy can be stored? (b) What is the power output if nearly all this energy is released in 1.0 ms? APPROACH We use Eq. 24–5 in the form U = 12 CV 2 because we are given C and V. SOLUTION The energy stored is U =

FIGURE 24–13 A camera flash unit.

1 2 2 CV

1 2

A150 * 10 –6 FB(200 V)2 = 3.0 J.

1 If this energy is released in 1000 of a second, the power output is –3 P = Ut = (3.0 J)A1.0 * 10 sB = 3000 W.

CONCEPTUAL EXAMPLE 24–9 Capacitor plate separation increased. A parallel-plate capacitor carries charge Q and is then disconnected from a battery. The two plates are initially separated by a distance d. Suppose the plates are pulled apart until the separation is 2d. How has the energy stored in this capacitor changed? RESPONSE If we increase the plate separation d, we decrease the capacitance according to Eq. 24–2, C = 0 Ad, by a factor of 2. The charge Q hasn't changed. So according to Eq. 24–5, where we choose the form U = 12 Q2C because we know Q is the same and C has been halved, the reduced C means the potential energy stored increases by a factor of 2. NOTE We can see why the energy stored increases from a physical point of view: the two plates are charged equal and opposite, so they attract each other. If we pull them apart, we must do work, so we raise their potential energy.

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EXAMPLE 24–10 Moving parallel capacitor plates. The plates of a parallelplate capacitor have area A, separation x, and are connected to a battery with voltage V. While connected to the battery, the plates are pulled apart until they are separated by 3x. (a) What are the initial and final energies stored in the capacitor? (b) How much work is required to pull the plates apart (assume constant speed)? (c) How much energy is exchanged with the battery? APPROACH The stored energy is given by Eq. 24–5: U = 12 CV 2, where C = 0 Ax. Unlike Example 24–9, here the capacitor remains connected to the battery. Hence charge and energy can flow to or from the battery, and we can not set B the work W = ¢U. Instead, the work can be calculated from Eq. 7–7, W = F dL . SOLUTION (a) When the separation is x, the capacitance is C1 = 0 Ax and the energy stored is B

0 A 2 U1 = 12 C1 V2 = 12 V . x When the separation is 3x, C2 = 0 A3x and 0 A 2 U2 = 12 V. 3x Then ¢Ucap = U2 - U1 = –

0 AV2 . 3x

The potential energy decreases as the oppositely charged plates are pulled apart, which makes sense. The plates remain connected to the battery, so V does not change and C decreases; hence some charge leaves each plate (Q = CV), causing U to decrease. 3x 3x (b) The work done in pulling the plates apart is W = x F dl = x QE dl, where Q is the charge on one plate at a given moment when the plates are a distance l apart, and E is the field due to the other plate at that instant. You might think we could use E = Vl where l is the separation of the plates (Eq. 23–4b). But we want the force on one plate (of charge Q) due to the electric field of the other plate only—which is half by symmetry: so we take E = V2l. The charge at any separation l is given by Q = CV, where C = 0 Al. Substituting, the work is l = 3x

QEdl =

l=x

0 AV2 2

0 AV2 l = 3x 0 AV2 –1 0 AV2 dl 1 . 2 = – = ¢ + ≤ = x 2l 2 3x 3x l2 l=x

As you might expect, the work required to pull these oppositely charged plates apart is positive. (c) Even though the work done is positive, the potential energy decreased, which tells us that energy must have gone into the battery (as if charging it). Conservation of energy tells us that the work W done on the system must equal the change in potential energy of the capacitor plus that of the battery (kinetic energy can be assumed to be essentially zero): W = ¢Ucap + ¢Ubatt . Thus the battery experiences a change in energy of ¢Ubatt = W - ¢Ucap =

0 AV2 0 AV2 2 0 AV2 . + = 3x 3x 3x

Thus charge flows back into the battery, raising its stored energy. In fact, the battery energy increase is double the work we do. It is useful to think of the energy stored in a capacitor as being stored in the electric field between the plates. As an example let us calculate the energy stored in a parallel-plate capacitor in terms of the electric field. SECTION 24–4

637

We have seen (Eq. 23–4b) that the electric field E between two close parallel plates is (approximately) uniform and its magnitude is related to the potential difference by V = Ed where d is the plate separation. Also, Eq. 24–2 tells us C = 0 Ad for a parallel-plate capacitor. Thus 0 A U = 12 CV 2 = 12 ¢ ≤ AE 2d2 B d = 12 0 E 2Ad. The quantity Ad is the volume between the plates in which the electric field E exists. If we divide both sides by the volume, we obtain an expression for the energy per unit volume or energy density, u: (24–6) u = energy density = 12 0 E 2. The electric energy stored per unit volume in any region of space is proportional to the square of the electric field in that region. We derived Eq. 24–6 for the special case of a parallel-plate capacitor. But it can be shown to be true for any region of space where there is an electric field. Note that the units check: for A 0 E 2 B we have AC 2N m2 B(NC)2 = Nm2 = (N m)m3 = Jm3. P H Y S I C S

A P P L I E D

Shocks, burns, defibrillators

FIGURE 24–14 Heart defibrillator.

TABLE 24–1 Dielectric Constants (at 20°C)

Material

Dielectric Dielectric constant strength K (V m)

Vacuum 1.0000 Air (1 atm) 1.0006 Paraffin 2.2 Polystyrene 2.6 Vinyl (plastic) 2–4 Paper 3.7 Quartz 4.3 Oil 4 Glass, Pyrex 5 Porcelain 6–8 Mica 7 Water (liquid) 80 Strontium titanate 300

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3 10 24 50 15 8 12 14 5 150

* * * * * * * * * *

106 106 106 106 106 106 106 106 106 106

8 * 106

Health Effects The energy stored in a large capacitance can do harm, giving you a burn or a shock. One reason you are warned not to touch a circuit, or the inside of electronic devices, is because capacitors may still be carrying charge even if the external power has been turned off. On the other hand, the basis of a heart defibrillator is a capacitor charged to a high voltage. A heart attack can be characterized by fast irregular beating of the heart, known as ventricular (or cardiac) fibrillation. The heart then does not pump blood to the rest of the body properly, and if it lasts for long, death results. A sudden, brief jolt of charge through the heart from a defibrillator can cause complete heart stoppage, sometimes followed by a resumption of normal beating. The defibrillator capacitor is charged to a high voltage, typically a few thousand volts, and is allowed to discharge very rapidly through the heart via a pair of wide contacts known as "paddles" that spread out the current over the chest (Fig. 24–14).

24–5

Dielectrics

In most capacitors there is an insulating sheet of material, such as paper or plastic, called a dielectric between the plates. This serves several purposes. First of all, dielectrics break down (allowing electric charge to flow) less readily than air, so higher voltages can be applied without charge passing across the gap. Furthermore, a dielectric allows the plates to be placed closer together without touching, thus allowing an increased capacitance because d is smaller in Eq. 24–2. Finally, it is found experimentally that if the dielectric fills the space between the two conductors, it increases the capacitance by a factor K which is known as the dielectric constant. Thus C = KC0 , (24–7) where C0 is the capacitance when the space between the two conductors of the capacitor is a vacuum, and C is the capacitance when the space is filled with a material whose dielectric constant is K. The values of the dielectric constant for various materials are given in Table 24–1. Also shown in Table 24–1 is the dielectric strength, the maximum electric field before breakdown (charge flow) occurs. For a parallel-plate capacitor (see Eq. 24–2), A C = K 0 [parallel-plate capacitor] (24–8) d when the space between the plates is completely filled with a dielectric whose dielectric constant is K. (The situation when the dielectric only partially fills the space will be discussed shortly in Example 24–11.) The quantity K 0 appears so

often in formulas that we define a new quantity = K 0 (24–9) called the permittivity of a material. Then the capacitance of a parallel-plate capacitor becomes A C = . d Note that 0 represents the permittivity of free space (a vacuum), as in Section 21–5. The energy density stored in an electric field E (Section 24–4) in a dielectric is given by (see Eq. 24–6) u =

1 2 2 K 0 E

1 2 2 E .

[E in a dielectric]

EXERCISE E Return to the Chapter-Opening Question, page 628, and answer it again now. Try to explain why you may have answered differently the first time.

Two simple experiments illustrate the effect of a dielectric. In the first, Fig. 24–15a, a battery of voltage V0 is kept connected to a capacitor as a dielectric is inserted between the plates. If the charge on the plates without dielectric is Q0 , then when the dielectric is inserted, it is found experimentally (first by Faraday) that the charge Q on the plates is increased by a factor K, Q = KQ0 . [voltage constant] The capacitance has increased to C = QV0 = KQ0V0 = KC0 , which is Eq. 24–7. In a second experiment, Fig. 24–15b, a battery V0 is connected to a capacitor C0 which then holds a charge Q0 = C0 V0 . The battery is then disconnected, leaving the capacitor isolated with charge Q0 and still at voltage V0 . Next a dielectric is inserted between the plates of the capacitor. The charge remains Q0 (there is nowhere for the charge to go) but the voltage is found experimentally to drop by a factor K: V0 . V = [charge constant] K Note that the capacitance changes to C = Q0V = Q0AV0KB = KQ0V0 = KC0 , so this experiment too confirms Eq. 24–7. Q0 Q C = 0 Q0 0 V0

V0 no dielectric

no dielectric

(b) Charge constant

Q = KQ0 Q = KQ0

Q = KC0 V0

with dielectric

(a) Voltage constant Q0 Q C = 0 Q0 0 V0

Q0 Q0

Q V0, C0 = 0 V0

battery disconnected

Q0 V = V0 K Q0 C = KC

FIGURE 24–15 Two experiments with a capacitor. Dielectric inserted with (a) voltage held constant, (b) charge held constant. 0

dielectric inserted

The electric field when a dielectric is inserted is also altered. When no dielectric is present, the electric field between the plates of a parallel-plate capacitor is given by Eq. 23–4b: V0 , E0 = d where V0 is the potential difference between the plates and d is their separation. If the capacitor is isolated so that the charge remains fixed on the plates when a dielectric is inserted, filling the space between the plates, the potential difference drops to V = V0K. So the electric field in the dielectric is V0 V E = ED = = d Kd or E0 . ED = [in a dielectric] (24–10) K The electric field in a dielectric is reduced by a factor equal to the dielectric constant. The field in a dielectric (or insulator) is not reduced all the way to zero as in a conductor. Equation 24–10 is valid even if the dielectric's width is smaller than the gap between the capacitor plates.

SECTION 24–5

639

A K = 3.4

4.0 mm (a)

100 V 0V

A d = 4.0 mm (b) FIGURE 24–16 Example 24–11.

EXAMPLE 24–11 Dielectric removal. A parallel-plate capacitor, filled with a dielectric with K = 3.4, is connected to a 100-V battery (Fig. 24–16a). After the capacitor is fully charged, the battery is disconnected. The plates have area A = 4.0 m2, and are separated by d = 4.0 mm. (a) Find the capacitance, the charge on the capacitor, the electric field strength, and the energy stored in the capacitor. (b) The dielectric is carefully removed, without changing the plate separation nor does any charge leave the capacitor (Fig. 24–16b). Find the new values of capacitance, electric field strength, voltage between the plates, and the energy stored in the capacitor. APPROACH We use the formulas for parallel-plate capacitance and electric field with and without a dielectric. SOLUTION (a) First we find the capacitance, with dielectric: C =

3.4A8.85 * 10–12 C 2N m2 BA4.0 m2 B K 0 A = d 4.0 * 10–3 m = 3.0 * 10–8 F.

The charge Q on the plates is Q = CV = A3.0 * 10–8 FB(100 V) = 3.0 * 10–6 C. The electric field between the plates is E =

V 100 V = = 25 kVm. d 4.0 * 10–3 m

Finally, the total energy stored in the capacitor is U =

1 2 2 CV

1 2

A3.0 * 10 –8 FB(100 V)2 = 1.5 * 10 –4 J.

(b) The capacitance without dielectric decreases by a factor K = 3.4: C0 =

A3.0 * 10–8 FB C = = 8.8 * 10–9 F. K 3.4

Because the battery has been disconnected, the charge Q can not change; when the dielectric is removed, V = QC increases by a factor K = 3.4 to 340 V. The electric field is E =

V 340 V = = 85 kVm. d 4.0 * 10–3 m

The energy stored is U =

1 2 2 CV

1 2

A8.8 * 10 –9 FB(340 V)2

= 5.1 * 10 –4 J. NOTE Where did all this extra energy come from? The energy increased because work had to be done to remove the dielectric. The work required was W = A5.1 * 10–4 JB - A1.5 * 10–4 JB = 3.6 * 10–4 J. (We will see in the next Section that work is required because of the force of attraction between induced charge on the dielectric and the charges on the plates, Fig. 24–17c.)

* 24–6

Molecular Description of Dielectrics

Let us examine, from the molecular point of view, why the capacitance of a capacitor should be larger when a dielectric is between the plates. A capacitor whose plates are separated by an air gap has a charge ±Q on one plate and –Q on

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Capacitance, Dielectrics, Electric Energy Storage

the other (Fig. 24–17a). Assume it is isolated (not connected to a battery) so charge cannot flow to or from the plates. The potential difference between the plates, V0 , is given by Eq. 24–1: Q = C0 V0 , where the subscripts refer to air between the plates. Now we insert a dielectric between the plates (Fig. 24–17b). Because of the electric field between the capacitor plates, the dielectric molecules will tend to become oriented as shown in Fig. 24–17b. If the dielectric molecules are polar, the positive end is attracted to the negative plate and vice versa. Even if the dielectric molecules are not polar, electrons within them will tend to move slightly toward the positive capacitor plate, so the effect is the same. The net effect of the aligned dipoles is a net negative charge on the outer edge of the dielectric facing the positive plate, and a net positive charge on the opposite side, as shown in Fig. 24–17c. Some of the electric field lines, then, do not pass through the dielectric but instead end on charges induced on the surface of the dielectric as shown in Fig. 24–17c. Hence the electric field within the dielectric is less than in air. That is, the electric field between the capacitor plates, assumed filled by the dielectric, has been reduced by some factor K. The voltage across the capacitor is reduced by the same factor K because V = Ed (Eq. 23–4b) and hence, by Eq. 24–1, Q = CV, the capacitance C must increase by that same factor K to keep Q constant. As shown in Fig. 24–17d, the electric field within the dielectric ED can be B considered as the vector sum of the electric field E0 due to the "free" charges on B the conducting plates, and the field Eind due to the induced charge on the surfaces of the dielectric. Since these two fields are in opposite directions, the net field within the dielectric, E0 - Eind , is less than E0 . The precise relationship is given by Eq. 24–10, even if the dielectric does not fill the gap between the plates: ED = E0 - Eind =

E0 , K

(a)

(b)

so Eind = E0 ¢ 1 -

1 ≤. K

The electric field between two parallel plates is related to the surface charge density, s, by E = s 0 (Example 21–13 or 22–8). Thus E0 = s 0 where s = QA is the surface charge density on the conductor; Q is the net charge on the conductor and is often called the free charge (since charge is free to move in a conductor). Similarly, we define an equivalent induced surface charge density sind on the dielectric; then

B E 0

(c)

Eind

(d) FIGURE 24–17 Molecular view of the effects of a dielectric.

Eind = sind 0 where Eind is the electric field due to the induced charge Qind = sind A on the surface of the dielectric, Fig. 24–17d. Qind is often called the bound charge, since it is on an insulator and is not free to move. Since Eind = E0(1 - 1K) as shown above, we now have sind = s ¢ 1 -

1 ≤ K

(24–11a)

Qind = Q ¢ 1 -

1 ≤. K

(24–11b)

and

Since K is always greater than 1, we see that the charge induced on the dielectric is always less than the free charge on each of the capacitor plates. *SECTION 24–6

Molecular Description of Dielectrics

641

d = 2.00 mm Q Q A

K = 3.50

EXAMPLE 24–12 Dielectric partially fills capacitor. A parallel-plate capacitor has plates of area A = 250 cm2 and separation d = 2.00 mm. The capacitor is charged to a potential difference V0 = 150 V. Then the battery is disconnected (the charge Q on the plates then won't change), and a dielectric sheet (K = 3.50) of the same area A but thickness l = 1.00 mm is placed between the plates as shown in Fig. 24–18. Determine (a) the initial capacitance of the air-filled capacitor, (b) the charge on each plate before the dielectric is inserted, (c) the charge induced on each face of the dielectric after it is inserted, (d) the electric field in the space between each plate and the dielectric, (e) the electric field in the dielectric, (f) the potential difference between the plates after the dielectric is added, and (g) the capacitance after the dielectric is in place. APPROACH We use the expressions for capacitance and charge developed in B this Section plus (part e), Eq. 23–4a, V = – E dL . SOLUTION (a) Before the dielectric is in place, the capacitance is B

l = 1.00 mm FIGURE 24–18 Example 24–12.

C0 = 0

A 2.50 * 10–2 m2 = A8.85 * 10–12 C 2N m2 B ¢ ≤ = 111 pF. d 2.00 * 10 –3 m

(b) The charge on each plate is

Q = C0 V0 = A1.11 * 10–10 FB(150 V) = 1.66 * 10 –8 C.

1 1 ≤ = A1.66 * 10–8 CB ¢ 1 ≤ = 1.19 * 10–8 C. K 3.50

(d) The electric field in the gaps between the plates and the dielectric (see Fig. 24–17c) is the same as in the absence of the dielectric since the charge on the plates has not been altered. The result of Example 21–13 can be used here, which gives E0 = s 0 . [Or we can note that, in the absence of the dielectric, E0 = V0d = QC0 d Asince V0 = QC0 B = Q 0 A Asince C0 = 0 AdB which is the same result.] Thus E0 =

Q 1.66 * 10 –8 C = = 7.50 * 104 Vm. 0 A A8.85 * 10–12 C 2N m2 BA2.50 * 10 –2 m2 B

(e) In the dielectric the electric field is (Eq. 24–10) E0 7.50 * 104 Vm = = 2.14 * 104 Vm. K 3.50 (f) To obtain the potential difference in the presence of the dielectric we use Eq. 23–4a, and integrate from the surface of one plate to the other along a straight line parallel to the field lines: ED =

V = – E dL = E0(d - l) + ED l, which can be simplified to V = E0 ¢ d - l +

l ≤ K

= A7.50 * 104 VmB ¢ 1.00 * 10 –3 m +

1.00 * 10–3 m ≤ 3.50

= 96.4 V. (g) In the presence of the dielectric, the capacitance is Q 1.66 * 10–8 C = = 172 pF. 96.4 V V NOTE If the dielectric filled the space between the plates, the answers to (f) and (g) would be 42.9 V and 387 pF, respectively. C =

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CHAPTER 24

Capacitance, Dielectrics, Electric Energy Storage

Summary A capacitor is a device used to store charge (and electric energy), and consists of two nontouching conductors. The two conductors generally hold equal and opposite charges of magnitude Q. The ratio of this charge Q to the potential difference V between the conductors is called the capacitance, C: C =

Q V

Q = CV.

(24–1)

The capacitance of a parallel-plate capacitor is proportional to the area A of each plate and inversely proportional to their separation d: C = 0

A. d

(24–2)

When capacitors are connected in parallel, the equivalent capacitance is the sum of the individual capacitances: Ceq = C1 + C2 + p .

U =

1 2 QV

(24–4)

1 2 2 CV

1 2

Q2 . C

(24–5)

This energy can be thought of as stored in the electric field B between the plates. In any electric field E in free space the energy density u (energy per unit volume) is u =

1 2 2 0 E .

(24–6)

The space between the conductors contains a nonconducting material such as air, paper, or plastic. These materials are referred to as dielectrics, and the capacitance is proportional to a property of dielectrics called the dielectric constant, K (nearly equal to 1 for air). For a parallel-plate capacitor

(24–3)

When capacitors are connected in series, the reciprocal of the equivalent capacitance equals the sum of the reciprocals of the individual capacitances: 1 1 1 = + + p. Ceq C1 C2

A charged capacitor stores an amount of electric energy given by

C = K 0

A A = d d

(24–8)

where = K 0 is called the permittivity of the dielectric material. When a dielectric is present, the energy density is u =

1 2 2 K 0 E

1 2 2 E .

Questions 1. Suppose two nearby conductors carry the same negative charge. Can there be a potential difference between them? If so, can the definition of capacitance, C = QV, be used here? 2. Suppose the separation of plates d in a parallel-plate capacitor is not very small compared to the dimensions of the plates. Would you expect Eq. 24–2 to give an overestimate or underestimate of the true capacitance? Explain. 3. Suppose one of the plates of a parallel-plate capacitor was moved so that the area of overlap was reduced by half, but they are still parallel. How would this affect the capacitance? 4. When a battery is connected to a capacitor, why do the two plates acquire charges of the same magnitude? Will this be true if the two conductors are different sizes or shapes? 5. Describe a simple method of measuring 0 using a capacitor. 6. Suppose three identical capacitors are connected to a battery. Will they store more energy if connected in series or in parallel? 7. A large copper sheet of thickness l is placed between the parallel plates of a capacitor, but does not touch the plates. How will this affect the capacitance? 8. The parallel plates of an isolated capacitor carry opposite charges, Q. If the separation of the plates is increased, is a force required to do so? Is the potential difference changed? What happens to the work done in the pulling process? 9. How does the energy in a capacitor change if (a) the potential difference is doubled, (b) the charge on each plate is doubled, and (c) the separation of the plates is doubled, as the capacitor remains connected to a battery in each case?

10. If the voltage across a capacitor is doubled, the amount of energy it can store (a) doubles; (b) is halved; (c) is quadrupled; (d) is unaffected; (e) none of these. 11. An isolated charged capacitor has horizontal plates. If a thin dielectric is inserted a short way between the plates, Fig. 24–19, will it move left or right when it is released? Dielectric FIGURE 24–19 Question 11. 12. Suppose a battery remains connected to the capacitor in Question 11. What then will happen when the dielectric is released? 13. How does the energy stored in a capacitor change when a dielectric is inserted if (a) the capacitor is isolated so Q does not change; (b) the capacitor remains connected to a battery so V does not change? 14. For dielectrics consisting of polar molecules, how would you expect the dielectric constant to change with temperature? 15. A dielectric is pulled out from between the plates of a capacitor which remains connected to a battery. What changes occur to the capacitance, charge on the plates, potential difference, energy stored in the capacitor, and electric field? 16. We have seen that the capacitance C depends on the size, shape, and position of the two conductors, as well as on the dielectric constant K. What then did we mean when we said that C is a constant in Eq. 24–1? 17. What value might we assign to the dielectric constant for a good conductor? Explain.

Questions

643

Problems 24–1 Capacitors 1. (I) The two plates of a capacitor hold ± 2800 mC and –2800 mC of charge, respectively, when the potential difference is 930 V. What is the capacitance? 2. (I) How much charge flows from a 12.0-V battery when it is connected to a 12.6-mF capacitor? 3. (I) The potential difference between two short sections of parallel wire in air is 24.0 V. They carry equal and opposite charge of magnitude 75 pC. What is the capacitance of the two wires? 4. (I) The charge on a capacitor increases by 26 mC when the voltage across it increases from 28 V to 78 V. What is the capacitance of the capacitor? 5. (II) A 7.7-mF capacitor is charged by a 125-V battery (Fig. 24–20a) and then is disconnected from the battery. When this capacitor AC1 B is then connected (Fig. 24–20b) to a second (initially uncharged) capacitor, C2 , the final voltage on each capacitor is 15 V. What is the value of C2? [Hint: Charge is conserved.]

(a)

(b)

FIGURE 24–20 Problems 5 and 48. 6. (II) An isolated capacitor C1 carries a charge Q0 . Its wires are then connected to those of a second capacitor C2 , previously uncharged. What charge will each carry now? What will be the potential difference across each? 7. (II) It takes 15 J of energy to move a 0.20-mC charge from one plate of a 15-mF capacitor to the other. How much charge is on each plate? 8. (II) A 2.70-mF capacitor is charged to 475 V and a 4.00-mF capacitor is charged to 525 V. (a) These capacitors are then disconnected from their batteries, and the positive plates are now connected to each other and the negative plates are connected to each other. What will be the potential difference across each capacitor and the charge on each? (b) What is the voltage and charge for each capacitor if plates of opposite sign are connected? 9. (II) Compact "ultracapacitors" with capacitance values up to several thousand farads are now commercially available. One application for ultracapacitors is in providing power for electrical circuits when other sources (such as a battery) are turned off. To get an idea of how much charge can be stored in such a component, assume a 1200-F ultracapacitor is initially charged to 12.0 V by a battery and is then disconnected from the battery. If charge is then drawn off the plates of this capacitor at a rate of 1.0 mCs, say, to power the backup memory of some electrical gadget, how long (in days) will it take for the potential difference across this capacitor to drop to 6.0 V?

644

CHAPTER 24

10. (II) In a dynamic random access memory (DRAM) computer chip, each memory cell chiefly consists of a capacitor for charge storage. Each of these cells represents a single binary-bit value of 1 when its 35-fF capacitor A1 fF = 10–15 FB is charged at 1.5 V, or 0 when uncharged at 0 V. (a) When it is fully charged, how many excess electrons are on a cell capacitor's negative plate? (b) After charge has been placed on a cell capacitor's plate, it slowly "leaks" off (through a variety of mechanisms) at a constant rate of 0.30 fCs. How long does it take for the potential difference across this capacitor to decrease by 1.0% from its fully charged value? (Because of this leakage effect, the charge on a DRAM capacitor is "refreshed" many times per second.)

24–2 Determination of Capacitance 11. (I) To make a 0.40-mF capacitor, what area must the plates have if they are to be separated by a 2.8-mm air gap? 12. (I) What is the capacitance per unit length (Fm) of a coaxial cable whose inner conductor has a 1.0-mm diameter and the outer cylindrical sheath has a 5.0-mm diameter? Assume the space between is filled with air. 13. (I) Determine the capacitance of the Earth, assuming it to be a spherical conductor. B

14. (II) Use Gauss's law to show that E = 0 inside the inner conductor of a cylindrical capacitor (see Fig. 24–6 and Example 24–2) as well as outside the outer cylinder. 15. (II) Dry air will break down if the electric field exceeds about 3.0 * 106 Vm. What amount of charge can be placed on a capacitor if the area of each plate is 6.8 cm2? 16. (II) An electric field of 4.80 * 105 Vm is desired between two parallel plates, each of area 21.0 cm2 and separated by 0.250 cm of air. What charge must be on each plate? 17. (II) How strong is the electric field between the plates of a 0.80-mF air-gap capacitor if they are 2.0 mm apart and each has a charge of 92 mC? 18. (II) A large metal sheet of thickness l is placed between, and parallel to, the plates of the parallel-plate capacitor of Fig. 24–4. It does not touch the plates, and extends beyond their edges. (a) What is now the net capacitance in terms of A, d, and l? (b) If l = 0.40 d, by what factor does the capacitance change when the sheet is inserted? 19. (III) Small distances are commonly measured capacitively. Consider an air-filled parallel-plate capacitor with fixed plate area A = 25 mm2 and a variable plate-separation distance x. Assume this capacitor is attached to a capacitance-measuring instrument which can measure capacitance C in the range 1.0 pF to 1000.0 pF with an accuracy of ¢C = 0.1 pF. (a) If C is measured while x is varied, over what range Axmin x xmax B can the plate-separation distance (in mm) be determined by this setup? (b) Define ¢x to be the accuracy (magnitude) to which x can be determined, and determine a formula for ¢x. (c) Determine the percent accuracy to which xmin and xmax can be measured.

Capacitance, Dielectrics, Electric Energy Storage

20. (III) In an electrostatic air cleaner ("precipitator"), the strong nonuniform electric field in the central region of a cylindrical capacitor (with outer and inner cylindrical radii Ra and Rb) is used to create ionized air molecules for use in charging dust and soot particles (Fig. 24–21). Under standard atmospheric conditions, if air is subjected to an electric field magnitude that exceeds its dielectric strength ES = 2.7 * 106 NC, air molecules will dissociate into positively charged ions and free electrons. In a precipitator, the region within which air is ionized (the corona discharge region) occupies a cylindrical volume of radius R that is typically five times that of the inner cylinder. Assume a particular precipitator is constructed with Rb = 0.10 mm and Ra = 10.0 cm. In order to create a corona discharge region with radius R = 5.0 Rb , what potential difference V should be applied between the precipitator's inner and outer conducting cylinders? [Besides dissociating air, the charged inner cylinder repels the resulting positive ions from the corona discharge region, where they are put to use in charging dust particles, which are then "collected" on the negatively charged outer cylinder.]

Corona discharge region (ionized air)

V−

+ − + − + − + _ + _

− + − + − + _ + _ +

+ + + + + + + + + + + + +

− + − + − + _ + _ +

+ − + − + − + _ + _

Air R5.0 R b

26. (II) Three conducting plates, each of area A, are connected as shown in Fig. 24–22. (a) Are the two capacitors thus formed connected in series or in parallel? (b) Determine C as a function of d1 , d2 , and A. Assume d1 + d2 is much less than the dimensions of the plates. (c) The middle plate can be moved (changing the values of d1 and d2), so as to vary the capacitance. What are the minimum and maximum values of the net capacitance? d1 d2 FIGURE 24–22 Problem 26.

27. (II) Consider three capacitors, of capacitance 3600 pF, 5800 pF, and 0.0100 mF. What maximum and minimum capacitance can you form from these? How do you make the connection in each case? 28. (II) A 0.50-mF and a 0.80-mF capacitor are connected in series to a 9.0-V battery. Calculate (a) the potential difference across each capacitor and (b) the charge on each. (c) Repeat parts (a) and (b) assuming the two capacitors are in parallel. 29. (II) In Fig. 24–23, suppose C1 = C2 = C3 = C4 = C. (a) Determine the equivalent capacitance between points a C1 C2 and b. (b) Determine the charge on each capacitor and the potential difference across each in terms of V. C3

Ra FIGURE 24–21 Problem 20.

FIGURE 24–23 Problems 29 and 30.

24–3 Capacitors in Series and Parallel 21. (I) The capacitance of a portion of a circuit is to be reduced from 2900 pF to 1600 pF. What capacitance can be added to the circuit to produce this effect without removing existing circuit elements? Must any existing connections be broken to accomplish this? 22. (I) (a) Six 3.8-mF capacitors are connected in parallel. What is the equivalent capacitance? (b) What is their equivalent capacitance if connected in series? 23. (II) Given three capacitors, C1 = 2.0 mF, C2 = 1.5 mF, and C3 = 3.0 mF, what arrangement of parallel and series connections with a 12-V battery will give the minimum voltage drop across the 2.0-mF capacitor? What is the minimum voltage drop? 24. (II) Suppose three parallel-plate capacitors, whose plates have areas A 1 , A 2 , and A 3 and separations d1 , d2 , and d3 , are connected in parallel. Show, using only Eq. 24–2, that Eq. 24–3 is valid. 25. (II) An electric circuit was accidentally constructed using a 5.0-mF capacitor instead of the required 16-mF value. Without removing the 5.0-mF capacitor, what can a technician add to correct this circuit?

30. (II) Suppose in Fig. 24–23 that C1 = C2 = C3 = 16.0 mF and C4 = 28.5 mF. If the charge on C2 is Q2 = 12.4 mC, determine the charge on each of the other capacitors, the voltage across each capacitor, and the voltage Vab across the entire combination. 31. (II) The switch S in Fig. 24–24 is connected downward so that capacitor C2 becomes fully charged by the battery of voltage V0 . If the switch is then C1 connected upward, determine the charge on each capacitor after the switching. S C2

FIGURE 24–24 Problem 31.

Problems

645

32. (II) (a) Determine the equivalent capacitance between points a and b for the combination of capacitors shown in Fig. 24–25. (b) Determine the charge on each capacitor and the voltage across each if Vba = V. C1

FIGURE 24–25 Problems 32 and 33. 33. (II) Suppose in Problem 32, Fig. 24–25, that C1 = C3 = 8.0 mF, C2 = C4 = 16 mF, and Q3 = 23 mC. Determine (a) the charge on each of the other capacitors, (b) the voltage across each capacitor, and (c) the voltage Vba across the combination. 34. (II) Two capacitors connected in parallel produce an equivalent capacitance of 35.0 mF but when connected in series the equivalent capacitance is only 5.5 mF. What is the individual capacitance of each capacitor? 35. (II) In the capacitance bridge shown in Fig. 24–26, a voltage V0 is applied and the variable capacitor C1 is adjusted until there is zero voltage between points a and b as measured on the voltmeter ( ). Determine the unknown capaciV tance Cx if C1 = 8.9 mF and the fixed capacitors have C2 = 18.0 mF and C3 = 4.8 mF. a Assume no charge flows C1 Cx through the voltmeter. V C3

C2 b FIGURE 24–26 Problem 35.

36. (II) Two capacitors, C1 = 3200 pF and C2 = 1800 pF, are connected in series to a 12.0-V battery. The capacitors are later disconnected from the battery and connected directly to each other, positive plate to positive plate, and negative plate to negative plate. What then will be the charge on each capacitor? 37. (II) (a) Determine the equivalent capacitance of the circuit shown in Fig. 24–27. (b) If C1 = C2 = 2C3 = 24.0 mF, how much charge is stored on each C1 capacitor when V = 35.0 V? C2 FIGURE 24–27 Problems 37, 38, and 45.

C2 = 3.00 mF, 38. (II) In Fig. 24–27, let C1 = 2.00 mF, C3 = 4.00 mF, and V = 24.0 V. What is the potential difference across each capacitor?

646

CHAPTER 24

39. (III) Suppose one plate of a parallel-plate capacitor is tilted so it makes a small angle u with the other plate, as shown in Fig. 24–28. Determine a formula for the capacitance C in terms of A, d, and u, where A is the area of each plate and u is small. Assume the plates are square. [Hint: Imagine the capacitor as many infinitesimal capacitors in θ parallel.] FIGURE 24–28 Problem 39. d 40. (III) A voltage V is applied to the capacitor network shown in Fig. 24–29. (a) What is the equivalent capacitance? [Hint: Assume a potential difference Vab exists across the network as shown; write potential differences for various pathways through the network from a to b in terms of the charges on the capacitors and the capac- a itances.] (b) Determine C2 the equivalent capacitance if C2 = C4 = 8.0 mF and C5 C1 = C3 = C5 = 4.5 mF. C3 V C1 FIGURE 24–29 Problem 40.

C4 b

24–4 Electric Energy Storage 41. (I) 2200 V is applied to a 2800-pF capacitor. How much electric energy is stored? 42. (I) There is an electric field near the Earth's surface whose intensity is about 150 Vm. How much energy is stored per cubic meter in this field? 43. (I) How much energy is stored by the electric field between two square plates, 8.0 cm on a side, separated by a 1.3-mm air gap? The charges on the plates are equal and opposite and of magnitude 420 mC. 44. (II) A parallel-plate capacitor has fixed charges ±Q and –Q. The separation of the plates is then tripled. (a) By what factor does the energy stored in the electric field change? (b) How much work must be done to increase the separation of the plates from d to 3.0d? The area of each plate is A. 45. (II) In Fig. 24–27, let V = 10.0 V and C1 = C2 = C3 = 22.6 mF. How much energy is stored in the capacitor network? 46. (II) How much energy must a 28-V battery expend to charge a 0.45-mF and a 0.20-mF capacitor fully when they are placed (a) in parallel, (b) in series? (c) How much charge flowed from the battery in each case? 47. (II) (a) Suppose the outer radius Ra of a cylindrical capacitor was tripled, but the charge was kept constant. By what factor would the stored energy change? Where would the energy come from? (b) Repeat part (a), assuming the voltage remains constant. 48. (II) A 2.20-mF capacitor is charged by a 12.0-V battery. It is disconnected from the battery and then connected to an uncharged 3.50-mF capacitor (Fig. 24–20). Determine the total stored energy (a) before the two capacitors are connected, and (b) after they are connected. (c) What is the change in energy?

Capacitance, Dielectrics, Electric Energy Storage

49. (II) How much work would be required to remove a metal sheet from between the plates of a capacitor (as in Problem 18a), assuming: (a) the battery remains connected so the voltage remains constant; (b) the battery is disconnected so the charge remains constant? 50. (II) (a) Show that each plate of a parallel-plate capacitor exerts a force F =

1 Q2 2 0 A

on the other, by calculating dW/dx where dW is the work needed to increase the separation by dx. (b) Why does using F = QE, with E being the electric field between the plates, give the wrong answer? 51. (II) Show that the electrostatic energy stored in the electric field outside an isolated spherical conductor of radius r0 carrying a net charge Q is U =

1 Q2 . 8p 0 r0

Do this in three ways: (a) Use Eq. 24–6 for the energy density in an electric field [Hint: Consider spherical shells of thickness dr]; (b) use Eq. 24–5 together with the capacitance of an isolated sphere (Section 24–2); (c) by calculating the work needed to bring all the charge Q up from infinity in infinitesimal bits dq. 52. (II) When two capacitors are connected in parallel and then connected to a battery, the total stored energy is 5.0 times greater than when they are connected in series and then connected to the same battery. What is the ratio of the two capacitances? (Before the battery is connected in each case, the capacitors are fully discharged.) 53. (II) For commonly used CMOS (complementary metal oxide semiconductor) digital circuits, the charging of the component capacitors C to their working potential difference V accounts for the major contribution of its energy input requirements. Thus, if a given logical operation requires such circuitry to charge its capacitors N times, we can assume that the operation requires an energy of NA 12 CV 2 B. In the past 20 years, the capacitance in digital circuits has been reduced by a factor of about 20 and the voltage to which these capacitors are charged has been reduced from 5.0 V to 1.5 V. Also, present-day alkaline batteries hold about five times the energy of older batteries. Two present-day AA alkaline cells, each of which measures 1 cm diameter by 4 cm long, can power the logic circuitry of a hand-held personal digital assistant (PDA) with its display turned off for about two months. If an attempt was made to construct a similar PDA (i.e., same digital capabilities so N remains constant) 20 years ago, how many (older) AA batteries would have been required to power its digital circuitry for two months? Would this PDA fit in a pocket or purse?

24–5 Dielectrics 54. (I) What is the capacitance of two square parallel plates 4.2 cm on a side that are separated by 1.8 mm of paraffin? 55. (II) Suppose the capacitor in Example 24–11 remains connected to the battery as the dielectric is removed. What will be the work required to remove the dielectric in this case?

56. (II) How much energy would be stored in the capacitor of Problem 43 if a mica dielectric is placed between the plates? Assume the mica is 1.3 mm thick (and therefore fills the space between the plates). 57. (II) In the DRAM computer chip of Problem 10, the cell capacitor's two conducting parallel plates are separated by a 2.0-nm thick insulating material with dielectric constant K = 25. (a) Determine the area A Ain mm2 B of the cell capacitor's plates. (b) In (older) "planar" designs, the capacitor was mounted on a silicon-wafer surface with its plates parallel to the plane of the wafer. Assuming the plate area A accounts for half of the area of each cell, estimate how many megabytes of memory can be placed on a 3.0-cm2 silicon wafer with the planar design? (1 byte = 8 bits.) 58. (II) A 3500-pF air-gap capacitor is connected to a 32-V battery. If a piece of mica fills the space between the plates, how much charge will flow from the battery? 59. (II) Two different dielectrics each fill half the space between the plates of a parallel-plate capacitor as shown in Fig. 24–30. Determine a formula for the capacitance in terms of K1 , K2 , the area A of the plates, and the separation d. [Hint: Can you consider this capacitor as two capacitors in series or in parallel?] K1

FIGURE 24–30 Problem 59.

60. (II) Two different dielectrics fill the space between the plates of a parallel-plate capacitor as shown in Fig. 24–31. Determine a formula for the capacitance in terms of K1 , K2 , the area A, of the plates, and the separation d1 = d2 = d2. [Hint: Can you consider this capacitor as two capacitors in series or in parallel?]

FIGURE 24–31 Problems 60 and 61.

61. (II) Repeat Problem 60 (Fig. 24–31) but assume the separation d1 Z d2 . 62. (II) Two identical capacitors are connected in parallel and each acquires a charge Q0 when connected to a source of voltage V0 . The voltage source is disconnected and then a dielectric (K = 3.2) is inserted to fill the space between the plates of one of the capacitors. Determine (a) the charge now on each capacitor, and (b) the voltage now across each capacitor. 63. (III) A slab of width d and dielectric constant K is inserted a distance x into the space between the square parallel plates (of side l) of a capacitor as shown in Fig. 24–32. Determine, as a function of x, (a) the capacitance, (b) the energy stored if the potential difference is V0 , and (c) the magnitude and direction of the force exerted on the slab (assume V0 is constant). l FIGURE 24–32 Problem 63.

d x

Problems

647

64. (III) The quantity of liquid (such as cryogenic liquid nitrogen) available in its storage tank is often monitored by a capacitive level sensor. This sensor is a vertically aligned cylindrical capacitor with outer and inner conductor radii Ra and Rb , whose length l spans the height of the tank. When a nonconducting liquid fills the tank to a height h ( l) from the tank's bottom, the dielectric in the lower and upper region between the cylindrical conductors is the liquid AKliq B and its vapor AKV B, respectively (Fig. 24–33). (a) Determine a formula for the fraction F of the tank filled by liquid in terms of the level-sensor capacitance C. [Hint: Consider the sensor as a combination of two capacitors.] (b) By connecting a capacitance-measuring instrument to the level sensor, F can be monitored. Assume the sensor dimensions are l = 2.0 m, Ra = 5.0 mm, and Rb = 4.5 mm. For liquid nitrogen AKliq = 1.4, KV = 1.0B, what values of C (in pF) will correspond to the tank being completely full and completely empty?

KV l h

FIGURE 24–33 Problem 64.

* 24–6 Molecular Description of Dielectrics * 65. (II) Show that the capacitor in Example 24–12 with dielectric inserted can be considered as equivalent to three capacitors in series, and using this assumption show that the same value for the capacitance is obtained as was obtained in part (g) of the Example. * 66. (II) Repeat Example 24–12 assuming the battery remains connected when the dielectric is inserted. Also, what is the free charge on the plates after the dielectric is added (let this be part (h) of this Problem)? 67. (II) Using Example 24–12 as a model, derive a formula * for the capacitance of a parallel-plate capacitor whose plates have area A, separation d, with a dielectric of dielectric constant K and thickness l (l 6 d) placed between the plates. * 68. (II) In Example 24–12 what percent of the stored energy is stored in the electric field in the dielectric? * 69. (III) The capacitor shown in Fig. 24–34 is connected to a 90.0-V battery. Calculate (and sketch) the electric field everywhere between the capacitor plates. Find both the free charge on the capacitor plate and the induced charge on the faces of the glass dielectric plate.

Kliq R Ra b

90.0 V a = 3.00 mm FIGURE 24–34 Problem 69.

b = 2.00 mm

A = 1.45 m2 Air Glass

K = 5.80

General Problems 70. (a) A general rule for estimating the capacitance C of an isolated conducting sphere with radius r is C (in pF) L r (in cm). That is, the numerical value of C in pF is about the same as the numerical value of the sphere's radius in cm. Justify this rule. (b) Modeling the human body as a 1-m-radius conducting sphere, use the given rule to estimate your body's capacitance. (c) While walking across a carpet, you acquire an excess "static electricity" charge Q and produce a 0.5-cm spark when reaching out to touch a metallic doorknob. The dielectric strength of air is 30 kVcm. Use this information to estimate Q (in mC). 71. A cardiac defibrillator is used to shock a heart that is beating erratically. A capacitor in this device is charged to 7.5 kV and stores 1200 J of energy. What is its capacitance? 72. A homemade capacitor is assembled by placing two 9-in. pie pans 5.0 cm apart and connecting them to the opposite terminals of a 9-V battery. Estimate (a) the capacitance, (b) the charge on each plate, (c) the electric field halfway between the plates, and (d) the work done by the battery to charge the plates. (e) Which of the above values change if a dielectric is inserted? 73. An uncharged capacitor is connected to a 34.0-V battery until it is fully charged, after which it is disconnected from the battery. A slab of paraffin is then inserted between the plates. What will now be the voltage between the plates? 74. It takes 18.5 J of energy to move a 13.0-mC charge from one plate of a 17.0-mF capacitor to the other. How much charge is on each plate?

648

CHAPTER 24

75. A huge 3.0-F capacitor has enough stored energy to heat 3.5 kg of water from 22°C to 95°C. What is the potential difference across the plates? 76. A coaxial cable, Fig. 24–35, consists of an inner cylindrical conducting wire of radius Rb surrounded by a dielectric insulator. Surrounding the dielectric insulator is an outer conducting sheath of radius Ra , which is usually "grounded." (a) Determine an expression for the capacitance per unit length of a cable whose insulator has dielectric constant K. (b) For a given cable, Rb = 2.5 mm and Ra = 9.0 mm. The dielectric constant of the dielectric insulator is K = 2.6. Suppose that there is a potential of 1.0 kV between the inner conducting wire and the outer conducting sheath. Find the capacitance per meter of the cable. Outer conducting sheath, radius Ra Dielectric Outer covering

FIGURE 24–35 Problem 76.

Radius Rb inner conductor

77. The electric field between the plates of a paper-separated (K = 3.75) capacitor is 9.21 * 104 Vm. The plates are 1.95 mm apart and the charge on each plate is 0.675 mC. Determine the capacitance of this capacitor and the area of each plate.

Capacitance, Dielectrics, Electric Energy Storage

78. Capacitors can be used as "electric charge counters." Consider an initially uncharged capacitor of capacitance C with its bottom plate grounded and its top plate connected to a source of electrons. (a) If N electrons flow onto the capacitor's top plate, show that the resulting potential difference V across the capacitor is directly proportional to N. (b) Assume the voltage-measuring device can accurately resolve voltage changes of about 1 mV. What value of C would be necessary to detect each new collected electron? (c) Using modern semiconductor technology, a micron-size capacitor can be constructed with parallel conducting plates separated by an insulating oxide of dielectric constant K = 3 and thickness d = 100 nm. To resolve the arrival of an individual electron on the plate of such a capacitor, determine the required value of l (in mm) assuming square plates of side length l.

85. The variable capacitance of an old radio tuner consists of four plates connected together placed alternately between four other plates, also connected together (Fig. 24–36). Each plate is separated from its neighbor by 1.6 mm of air. One set of plates can move so that the area of overlap of each plate varies from 2.0 cm2 to 9.0 cm2. (a) Are these seven capacitors connected in series or in parallel? (b) Determine the range of capacitance values.

79. A parallel-plate capacitor is isolated with a charge & Q on each plate. If the separation of the plates is halved and a dielectric (constant K) is inserted in place of air, by what factor does the energy storage change? To what do you attribute the change in stored potential energy? How does the new value of the electric field between the plates compare with the original value?

86. A high-voltage supply can be constructed from a variable capacitor with interleaving plates which can be rotated as in Fig. 24–36. A version of this type of capacitor with more plates has a capacitance which can be varied from 10 pF to 1 pF. (a) Initially, this capacitor is charged by a 7500-V power supply when the capacitance is 8.0 pF. It is then disconnected from the power supply and the capacitance reduced to 1.0 pF by rotating the plates. What is the voltage across the capacitor now? (b) What is a major disadvantage of this as a high-voltage power supply?

80. In lightning storms, the potential difference between the Earth and the bottom of the thunderclouds can be as high as 35,000,000 V. The bottoms of thunderclouds are typically 1500 m above the Earth, and may have an area of 120 km2. Modeling the Earth–cloud system as a huge capacitor, calculate (a) the capacitance of the Earth–cloud system, (b) the charge stored in the "capacitor," and (c) the energy stored in the "capacitor." 81. A multilayer film capacitor has a maximum voltage rating of 100 V and a capacitance of 1.0 mF. It is made from alternating sheets of metal foil connected together, separated by films of polyester dielectric. The sheets are 12.0 mm by 14.0 mm and the total thickness of the capacitor is 6.0 mm (not counting the thickness of the insulator on the outside). The metal foil is actually a very thin layer of metal deposited directly on the dielectric, so most of the thickness of the capacitor is due to the dielectric. The dielectric strength of the polyester is about 30 * 106 Vm. Estimate the dielectric constant of the polyester material in the capacitor. 82. A 3.5-mF capacitor is charged by a 12.4-V battery and then is disconnected from the battery. When this capacitor AC1 B is then connected to a second (initially uncharged) capacitor, C2 , the voltage on the first drops to 5.9 V. What is the value of C2 ? 83. The power supply for a pulsed nitrogen laser has a 0.080-mF capacitor with a maximum voltage rating of 25 kV. (a) Estimate how much energy could be stored in this capacitor. (b) If 15% of this stored electrical energy is converted to light energy in a pulse that is 4.0-ms long, what is the power of the laser pulse? 84. A parallel-plate capacitor has square plates 12 cm on a side separated by 0.10 mm of plastic with a dielectric constant of K = 3.1. The plates are connected to a battery, causing them to become oppositely charged. Since the oppositely charged plates attract each other, they exert a pressure on the dielectric. If this pressure is 40.0 Pa, what is the battery voltage?

FIGURE 24–36 Problems 85 and 86.

87. A 175-pF capacitor is connected in series with an unknown capacitor, and as a series combination they are connected to a 25.0-V battery. If the 175-pF capacitor stores 125 pC of charge on its plates, what is the unknown capacitance? 88. A parallel-plate capacitor with plate area 2.0 cm2 and airgap separation 0.50 mm is connected to a 12-V battery, and fully charged. The battery is then disconnected. (a) What is the charge on the capacitor? (b) The plates are now pulled to a separation of 0.75 mm. What is the charge on the capacitor now? (c) What is the potential difference across the plates now? (d) How much work was required to pull the plates to their new separation? 89. In the circuit shown in Fig. 24–37, C1 = 1.0 mF, C2 = 2.0 mF, C3 = 2.4 mF, and a voltage Vab = 24 V is applied across points a and b. After C1 is fully charged the switch is thrown to the right. What is the final charge and potential difference on each capacitor? S

a FIGURE 24–37 Problem 89.

C2 C3

90. The long cylindrical capacitor shown in Fig. 24–38 consists of four concentric cylinders, with respective radii Ra , Rb , Rc , and Rd . The cylinders b and c are joined by metal strips. Determine the capacitance per unit length of this arrangement. (Assume equal and opposite charges are placed on the innermost and outermost cylinders.)

Rd FIGURE 24–38 Problem 90.

Ra Rb

General Problems

649

91. A parallel-plate capacitor has plate area A, plate separation x, and has a charge Q stored on its plates (Fig. 24–39). Find the amount of work required to double the plate separation to 2x, assuming the charge remains constant at Q. Show that your answer is consistent with the change in energy stored by the capacitor. (Hint: See Example 24–10.) A Q A Q 2x FIGURE 24–39 x Q Q Problem 91. 92. Consider the use of capacitors as memory cells. A charged capacitor would represent a one and an uncharged capacitor a zero. Suppose these capacitors were fabricated on a silicon chip and each has a capacitance of 30 femto-farads A1 fF = 10–15 F.B The dielectric filling the space between the parallel plates has dielectric constant K = 25 and a dielectric strength of 1.0 * 109 Vm. (a) If the operating voltage is 1.5 V, how many electrons would be stored on one of these capacitors when charged? (b) If no safety factor is allowed, how thin a dielectric layer could we use for operation at 1.5 V? (c) Using the layer thickness from your answer to part (b), what would be the area of the capacitor plates? 93. To get an idea how big a farad is, suppose you want to make a 1-F air-filled parallel-plate capacitor for a circuit you are building. To make it a reasonable size, suppose you limit the plate area to 1.0 cm2. What would the gap have to be between the plates? Is this practically achievable? 94. A student wearing shoes with thin insulating soles is standing on a grounded metal floor when he puts his hand flat against the screen of a CRT computer monitor. The voltage inside the monitor screen, 6.3 mm from his hand, is 25,000 V. The student's hand and the monitor form a capacitor; the student is a conductor, and there is another capacitor between the floor and his feet. Using reasonable numbers for hand and foot areas, estimate the student's voltage relative to the floor. Assume vinyl-soled shoes 1 cm thick. 95. A parallel-plate capacitor with plate area A = 2.0 m2 and plate separation d = 3.0 mm is connected to a 45-V battery (Fig. 24–40a). (a) Determine the charge on the capacitor, the electric field, the capacitance, and the energy stored in the capacitor. (b) With the capacitor still connected to the battery, a slab of plastic with dielectric strength K = 3.2 is placed between the plates of the capacitor, so that the gap is completely filled with the dielectric. What are the new values of charge, electric field, capacitance, and the energy U stored in the capacitor? A = 2.0 m2 45 V d = 3.0 mm (a) FIGURE 24–40 Problem 95.

45 V

96. Let us try to estimate the maximum "static electricity" charge that might result during each walking step across an insulating floor. Assume the sole of a person's shoe has area A L 150 cm2, and when the foot is lifted from the ground during each step, the sole acquires an excess charge Q from rubbing contact with the floor. (a) Model the sole as a plane conducting surface with Q uniformly distributed across it as the foot is lifted from the ground. If the dielectric strength of the air between the sole and floor as the foot is lifted is ES = 3 * 106 NC, determine Qmax , the maximum possible excess charge that can be transferred to the sole during each step. (b) Modeling a person as an isolated conducting sphere of radius r L 1 m, estimate a person's capacitance. (c) After lifting the foot from the floor, assume the excess charge Q quickly redistributes itself over the entire surface area of the person. Estimate the maximum potential difference that the person can develop with respect to the floor. 97. Paper has a dielectric constant K = 3.7 and a dielectric strength of 15 * 106 Vm. Suppose that a typical sheet of paper has a thickness of 0.030 mm. You make a "homemade" capacitor by placing a sheet of 21 * 14 cm paper between two aluminum foil sheets (Fig. 24–41). The thickness of the aluminum foil is 0.040 mm. (a) What is the capacitance C0 of your device? (b) About how much charge could you store on your capacitor before it would break down? (c) Show in a sketch how you could overlay sheets of paper and aluminum for a parallel combination. If you made 100 such capacitors, and connected the edges of the sheets in parallel so that you have a single large capacitor of capacitance 100 C0 , how thick would your new large capacitor be? (d) What is the maximum voltage you can apply to this 100 C0 capacitor without breakdown? Aluminum Paper FIGURE 24–41 Problem 97. Aluminum * Numerical/Computer * 98. (II) Six physics students were each given an air filled capacitor. Although the areas were different, the spacing between the plates, d, was the same for all six capacitors, but was unknown. Each student made a measurement of the area A and capacitance C of their capacitor. Below is a Table for their data. Using the combined data and a graphing program or spreadsheet, determine the spacing d between the plates.

K = 3.2 3.0 mm

(b)

Answers to Exercises D: (e).

A: A. B: 8.3 * 10

–9

E: (b).

C: (a).

650

CHAPTER 24

Capacitance, Dielectrics, Electric Energy Storage

Area (m2)

Capacitance (pF)

0.01 0.03 0.04 0.06 0.09 0.12

90 250 340 450 800 1050

The glow of the thin wire filament of a lightbulb is caused by the electric current passing through it. Electric energy is transformed to thermal energy (via collisions between moving electrons and atoms of the wire), which causes the wire's temperature to become so high that it glows. Electric current and electric power in electric circuits are of basic importance in everyday life. We examine both dc and ac in this Chapter, and include the microscopic analysis of electric current.

CHAPTER-OPENING QUESTION—Guess now! The conductors shown are all made of copper and are at the same temperature. Which conductor would have the greatest resistance to the flow of charge entering from the left? Which would offer the least resistance? Current

Current

(a)

C H

Electric Currents and Resistance

P T E R

CONTENTS 25–1 The Electric Battery 25–2 Electric Current

(b)

25–3 Ohm's Law: Resistance and Resistors 25–4 Resistivity

Current

(c)

25–5 Electric Power

(d)

n the previous four Chapters we have been studying static electricity: electric charges at rest. In this Chapter we begin our study of charges in motion, and we call a flow of charge an electric current. In everyday life we are familiar with electric currents in wires and other conductors. Indeed, most practical electrical devices depend on electric current: current through a lightbulb, current in the heating element of a stove or electric heater, and currents in electronic devices. Electric currents can exist in conductors such as wires, and also in other devices such as the CRT of a television or computer monitor whose charged electrons flow through space (Section 23–9).

25–6 Power in Household Circuits 25–7 Alternating Current 25–8 Microscopic View of Electric Current: Current Density and Drift Velocity

*25–9 Superconductivity *25–10 Electrical Conduction in the Nervous System

651

FIGURE 25–1 Alessandro Volta. In this portrait, Volta exhibits his battery to Napoleon in 1801.

In electrostatic situations, we saw in Sections 21–9 and 22–3 that the electric field must be zero inside a conductor (if it weren't, the charges would move). But when charges are moving in a conductor, there usually is an electric field in the conductor. Indeed, an electric field is needed to set charges into motion, and to keep them in motion in any normal conductor. We can control the flow of charge using electric fields and electric potential (voltage), concepts we have just been discussing. In order to have a current in a wire, a potential difference is needed, which can be provided by a battery. We first look at electric current from a macroscopic point of view: that is, current as measured in a laboratory. Later in the Chapter we look at currents from a microscopic (theoretical) point of view as a flow of electrons in a wire. Until the year 1800, the technical development of electricity consisted mainly of producing a static charge by friction. It all changed in 1800 when Alessandro Volta (1745–1827; Fig. 25–1) invented the electric battery, and with it produced the first steady flow of electric charge—that is, a steady electric current.

25–1

FIGURE 25–2 A voltaic battery, from Volta's original publication.

The Electric Battery

The events that led to the discovery of the battery are interesting. For not only was this an important discovery, but it also gave rise to a famous scientific debate. In the 1780s, Luigi Galvani (1737–1798), professor at the University of Bologna, carried out a series of experiments on the contraction of a frog's leg muscle through electricity produced by static electricity. Galvani found that the muscle also contracted when dissimilar metals were inserted into the frog. Galvani believed that the source of the electric charge was in the frog muscle or nerve itself, and that the metal merely transmitted the charge to the proper points. When he published his work in 1791, he termed this charge "animal electricity." Many wondered, including Galvani himself, if he had discovered the long-sought "life-force." Volta, at the University of Pavia 200 km away, was skeptical of Galvani's results, and came to believe that the source of the electricity was not in the animal itself, but rather in the contact between the dissimilar metals. Volta realized that a moist conductor, such as a frog muscle or moisture at the contact point of two dissimilar metals, was necessary in the circuit if it was to be effective. He also saw that the contracting frog muscle was a sensitive instrument for detecting electric "tension" or "electromotive force" (his words for what we now call potential), in fact more sensitive than the best available electroscopes (Section 21–4) that he and others had developed.† Volta's research found that certain combinations of metals produced a greater effect than others, and, using his measurements, he listed them in order of effectiveness. (This "electrochemical series" is still used by chemists today.) He also found that carbon could be used in place of one of the metals. Volta then conceived his greatest contribution to science. Between a disc of zinc and one of silver, he placed a piece of cloth or paper soaked in salt solution or dilute acid and piled a "battery" of such couplings, one on top of another, as shown in Fig. 25–2. This "pile" or "battery" produced a much increased potential difference. Indeed, when strips of metal connected to the two ends of the pile were brought close, a spark was produced. Volta had designed and built the first electric battery; he published his discovery in 1800. †

Volta's most sensitive electroscope measured about 40 V per degree (angle of leaf separation). Nonetheless, he was able to estimate the potential differences produced by dissimilar metals in contact: for a silver–zinc contact he got about 0.7 V, remarkably close to today's value of 0.78 V.

652

CHAPTER 25

Electric Currents and Resistance

Electric Cells and Batteries A battery produces electricity by transforming chemical energy into electrical energy. Today a great variety of electric cells and batteries are available, from flashlight batteries to the storage battery of a car. The simplest batteries contain two plates or rods made of dissimilar metals (one can be carbon) called electrodes. The electrodes are immersed in a solution, such as a dilute acid, called the electrolyte. Such a device is properly called an electric cell, and several cells connected together is a battery, although today even a single cell is called a battery. The chemical reactions involved in most electric cells are quite complicated. Here we describe how one very simple cell works, emphasizing the physical aspects. The cell shown in Fig. 25–3 uses dilute sulfuric acid as the electrolyte. One of the electrodes is made of carbon, the other of zinc. That part of each electrode outside the solution is called the terminal, and connections to wires and circuits are made here. The acid tends to dissolve the zinc electrode. Each zinc atom leaves two electrons behind on the electrode and enters the solution as a positive ion. The zinc electrode thus acquires a negative charge. As the electrolyte becomes positively charged, electrons are pulled off the carbon electrode by the electrolyte. Thus the carbon electrode becomes positively charged. Because there is an opposite charge on the two electrodes, there is a potential difference between the two terminals. In a cell whose terminals are not connected, only a small amount of the zinc is dissolved, for as the zinc electrode becomes increasingly negative, any new positive zinc ions produced are attracted back to the electrode. Thus, a particular potential difference (or voltage) is maintained between the two terminals. If charge is allowed to flow between the terminals, say, through a wire (or a lightbulb), then more zinc can be dissolved. After a time, one or the other electrode is used up and the cell becomes "dead." The voltage that exists between the terminals of a battery depends on what the electrodes are made of and their relative ability to be dissolved or give up electrons. When two or more cells are connected so that the positive terminal of one is connected to the negative terminal of the next, they are said to be connected in series and their voltages add up. Thus, the voltage between the ends of two 1.5-V flashlight batteries connected in series is 3.0 V, whereas the six 2-V cells of an automobile storage battery give 12 V. Figure 25–4a shows a diagram of a common "dry cell" or "flashlight battery" used in portable radios and CD players, flashlights, etc., and Fig. 25–4b shows two smaller ones in series, connected to a flashlight bulb. A lightbulb consists of a thin, coiled wire (filament) inside an evacuated glass bulb, as shown in Fig. 25–5 and in the large photo opening this Chapter, page 651. The filament gets very hot (3000 K) and glows when charge passes through it. FIGURE 25–4 (a) Diagram of an ordinary dry cell (like a D-cell or AA). The cylindrical zinc cup is covered on the sides; its flat bottom is the negative terminal. (b) Two dry cells (AA type) connected in series. Note that the positive terminal of one cell pushes against the negative terminal of the other. Insulation

+ Terminal (top of carbon electrode)

+ – Terminal Terminal Carbon electrode + (+)

Sulfuric acid FIGURE 25–3 Simple electric cell.

FIGURE 25–5 A lightbulb: the fine wire of the filament becomes so hot that it glows. This type of lightbulb is called an incandescent bulb (as compared, say, to a fluorescent bulb). Filament

Connecting wires

Electrolyte paste

Zinc – electrode (–)

Insulator

External connections

− Terminal

Negative electrode (zinc cup) (a) One D-cell

(b) Two AA batteries

SECTION 25–1

The Electric Battery

653

25–2

Electric Current

The purpose of a battery is to produce a potential difference, which can then make charges move. When a continuous conducting path is connected between the terminals of a battery, we have an electric circuit, Fig. 25–6a. On any diagram of a circuit, as in Fig. 25–6b, we use the symbol [battery symbol]

(a) Electric current Device (bulb) A

–

(b) FIGURE 25–6 (a) A simple electric circuit. (b) Schematic drawing of the same circuit, consisting of a battery, connecting wires (thick gray lines), and a lightbulb or other device.

C A U T I O N

A battery does not create charge; a lightbulb does not destroy charge

to represent a battery. The device connected to the battery could be a lightbulb, a heater, a radio, or whatever. When such a circuit is formed, charge can flow through the wires of the circuit, from one terminal of the battery to the other, as long as the conducting path is continuous. Any flow of charge such as this is called an electric current. More precisely, the electric current in a wire is defined as the net amount of charge that passes through the wire's full cross section at any point per unit time. Thus, the average current I is defined as I =

¢Q , ¢t

(25–1a)

where ¢Q is the amount of charge that passes through the conductor at any location during the time interval ¢ t. The instantaneous current is defined by the derivative limit I =

dQ . dt

(25–1b)

Electric current is measured in coulombs per second; this is given a special name, the ampere (abbreviated amp or A), after the French physicist André Ampère (1775–1836). Thus, 1 A = 1 Cs. Smaller units of current are often used, such as the milliampere A1 mA = 10–3 AB and microampere A1 mA = 10–6 AB. A current can flow in a circuit only if there is a continuous conducting path. We then have a complete circuit. If there is a break in the circuit, say, a cut wire, we call it an open circuit and no current flows. In any single circuit, with only a single path for current to follow such as in Fig. 25–6b, a steady current at any instant is the same at one point (say, point A) as at any other point (such as B). This follows from the conservation of electric charge: charge doesn't disappear. A battery does not create (or destroy) any net charge, nor does a lightbulb absorb or destroy charge. EXAMPLE 25–1 Current is flow of charge. A steady current of 2.5 A exists in a wire for 4.0 min. (a) How much total charge passed by a given point in the circuit during those 4.0 min? (b) How many electrons would this be? APPROACH Current is flow of charge per unit time, Eqs. 25–1, so the amount of charge passing a point is the product of the current and the time interval. To get the number of electrons (b), we divide the total charge by the charge on one electron. SOLUTION (a) Since the current was 2.5 A, or 2.5 Cs, then in 4.0 min ( = 240 s) the total charge that flowed past a given point in the wire was, from Eq. 25–1a, ¢Q = I ¢ t = (2.5 Cs)(240 s) = 600 C. (b) The charge on one electron is 1.60 * 10–19 C, so 600 C would consist of 600 C = 3.8 * 1021 electrons. 1.60 * 10 –19 Celectron EXERCISE A If 1 million electrons per second pass a point in a wire, what is the current in amps?

654

CHAPTER 25

Electric Currents and Resistance

CONCEPTUAL EXAMPLE 25–2 How to connect a battery. What is wrong with each of the schemes shown in Fig. 25–7 for lighting a flashlight bulb with a flashlight battery and a single wire? RESPONSE (a) There is no closed path for charge to flow around. Charges might briefly start to flow from the battery toward the lightbulb, but there they run into a "dead end," and the flow would immediately come to a stop. (b) Now there is a closed path passing to and from the lightbulb; but the wire touches only one battery terminal, so there is no potential difference in the circuit to make the charge move. (c) Nothing is wrong here. This is a complete circuit: charge can flow out from one terminal of the battery, through the wire and the bulb, and into the other terminal. This scheme will light the bulb. In many real circuits, wires are connected to a common conductor that provides continuity. This common conductor is called ground, usually represented as or , and really is connected to the ground in a building or house. In a car, one terminal of the battery is called "ground," but is not connected to the ground—it is connected to the frame of the car, as is one connection to each lightbulb and other devices. Thus the car frame is a conductor in each circuit, ensuring a continuous path for charge flow. We saw in Chapter 21 that conductors contain many free electrons. Thus, if a continuous conducting wire is connected to the terminals of a battery, negatively charged electrons flow in the wire. When the wire is first connected, the potential difference between the terminals of the battery sets up an electric field inside the wire† and parallel to it. Free electrons at one end of the wire are attracted into the positive terminal, and at the same time other electrons leave the negative terminal of the battery and enter the wire at the other end. There is a continuous flow of electrons throughout the wire that begins as soon as the wire is connected to both terminals. However, when the conventions of positive and negative charge were invented two centuries ago, it was assumed that positive charge flowed in a wire. For nearly all purposes, positive charge flowing in one direction is exactly equivalent to negative charge flowing in the opposite direction,‡ as shown in Fig. 25–8. Today, we still use the historical convention of positive charge flow when discussing the direction of a current. So when we speak of the current direction in a circuit, we mean the direction positive charge would flow. This is sometimes referred to as conventional current. When we want to speak of the direction of electron flow, we will specifically state it is the electron current. In liquids and gases, both positive and negative charges (ions) can move.

25–3

Ohm's Law: Resistance and Resistors

(a)

–

(b)

–

(c)

–

FIGURE 25–7 Example 25–2. FIGURE 25–8 Conventional current from ± to – is equivalent to a negative electron flow from – to ±. Conventional current

Electron flow

Device Conventional current Electron current

–

To produce an electric current in a circuit, a difference in potential is required. One way of producing a potential difference along a wire is to connect its ends to the opposite terminals of a battery. It was Georg Simon Ohm (1787–1854) who established experimentally that the current in a metal wire is proportional to the potential difference V applied to its two ends: I r V. If, for example, we connect a wire to the two terminals of a 6-V battery, the current flow will be twice what it would be if the wire were connected to a 3-V battery. It is also found that reversing the sign of the voltage does not affect the magnitude of the current. † This does not contradict what was said in Section 21–9 that in the static case, there can be no electric field within a conductor since otherwise the charges would move. Indeed, when there is an electric field in a conductor, charges do move, and we get an electric current. ‡ An exception is discussed in Section 27–8.

SECTION 25–3

Ohm's Law: Resistance and Resistors

655

A useful analogy compares the flow of electric charge in a wire to the flow of water in a river, or in a pipe, acted on by gravity. If the river or pipe is nearly level, the flow rate is small. But if one end is somewhat higher than the other, the flow rate—or current—is greater. The greater the difference in height, the swifter the current. We saw in Chapter 23 that electric potential is analogous, in the gravitational case, to the height of a cliff. This applies in the present case to the height through which the fluid flows. Just as an increase in height can cause a greater flow of water, so a greater electric potential difference, or voltage, causes a greater electric current. Exactly how large the current is in a wire depends not only on the voltage but also on the resistance the wire offers to the flow of electrons. The walls of a pipe, or the banks of a river and rocks in the middle, offer resistance to the water current. Similarly, electron flow is impeded because of interactions with the atoms of the wire. The higher this resistance, the less the current for a given voltage V. We then define electrical resistance so that the current is inversely proportional to the resistance: that is, I =

V R

(25–2a)

where R is the resistance of a wire or other device, V is the potential difference applied across the wire or device, and I is the current through it. Equation 25–2a is often written as OHM'S "LAW" I

(a)

∆I 1 ∆V = R

(b)

FIGURE 25–9 Graphs of current vs. voltage for (a) a metal conductor which obeys Ohm's law, and (b) for a nonohmic device, in this case a semiconductor diode.

FIGURE 25–10 Example 25–3.

I A

V = IR.

(25–2b)

As mentioned above, Ohm found experimentally that in metal conductors R is a constant independent of V, a result known as Ohm's law. Equation 25–2b, V = IR, is itself sometimes called Ohm's law, but only when referring to materials or devices for which R is a constant independent of V. But R is not a constant for many substances other than metals, nor for devices such as diodes, vacuum tubes, transistors, and so on. Even for metals, R is not constant if the temperature changes much: for a lightbulb filament the measured resistance is low for small currents, but is much higher at its normal large operating current that puts it at the high temperature needed to make it glow (3000 K). Thus Ohm's "law" is not a fundamental law, but rather a description of a certain class of materials: metal conductors, whose temperature does not change much. Materials or devices that do not follow Ohm's law are said to be nonohmic. See Fig. 25–9. The unit for resistance is called the ohm and is abbreviated (Greek capital letter omega). Because R = VI, we see that 1.0 is equivalent to 1.0 VA. CONCEPTUAL EXAMPLE 25–3 Current and potential. Current I enters a resistor R as shown in Fig. 25–10. (a) Is the potential higher at point A or at point B? (b) Is the current greater at point A or at point B? RESPONSE (a) Positive charge always flows from ± to –, from high potential to low potential. Think again of the gravitational analogy: a mass will fall down from high gravitational potential to low. So for positive current I, point A is at a higher potential than point B. (b) Conservation of charge requires that whatever charge flows into the resistor at point A, an equal amount of charge emerges at point B. Charge or current does not get "used up" by a resistor, just as an object that falls through a gravitational potential difference does not gain or lose mass. So the current is the same at A and B. An electric potential decrease, as from point A to point B in Example 25–3, is often called a potential drop or a voltage drop.

656

CHAPTER 25

Electric Currents and Resistance

EXAMPLE 25–4 Flashlight bulb resistance. A small flashlight bulb (Fig. 25–11) draws 300 mA from its 1.5-V battery. (a) What is the resistance of the bulb? (b) If the battery becomes weak and the voltage drops to 1.2 V, how would the current change? APPROACH We can apply Ohm's law to the bulb, where the voltage applied across it is the battery voltage. SOLUTION (a) We change 300 mA to 0.30 A and use Eq. 25–2a or b: 1.5 V on

V 1.5 V = = 5.0 . I 0.30 A (b) If the resistance stays the same, the current would be R =

off

V 1.2 V = = 0.24 A = 240 mA, R 5.0 or a decrease of 60 mA. NOTE With the smaller current in part (b), the bulb filament's temperature would be lower and the bulb less bright. Also, resistance does depend on temperature (Section 25–4), so our calculation is only a rough approximation. I =

FIGURE 25–11 Flashlight (Example 25–4). Note how the circuit is completed along the side strip.

EXERCISE B What resistance should be connected across a 9.0-V battery to make a 10-mA current? (a) 9 , (b) 0.9 , (c) 900 , (d) 1.1 , (e) 0.11 .

All electric devices, from heaters to lightbulbs to stereo amplifiers, offer resistance to the flow of current. The filaments of lightbulbs (Fig. 25–5) and electric heaters are special types of wires whose resistance results in their becoming very hot. Generally, the connecting wires have very low resistance in comparison to the resistance of the wire filaments or coils, so the connecting wires usually have a minimal effect on the magnitude of the current. In many circuits, particularly in electronic devices, resistors are used to control the amount of current. Resistors have resistances ranging from less than an ohm to millions of ohms (see Figs. 25–12 and 25–13). The main types are "wire-wound" resistors which consist of a coil of fine wire, "composition" resistors which are usually made of carbon, and thin carbon or metal films. When we draw a diagram of a circuit, we use the symbol

FIGURE 25–12 Photo of resistors (striped), plus other devices on a circuit board.

[resistor symbol] to indicate a resistance. Wires whose resistance is negligible, however, are shown simply as straight lines.

First digit Second digit Multiplier Tolerance

Resistor Color Code Color Black Brown Red Orange Yellow Green Blue Violet Gray White Gold Silver No color

Number

Multiplier

0 1 2 3 4 5 6 7 8 9

1 101 102 103 104 105 106 107 108 109 10–1 10 –2

Tolerance 1% 2%

5% 10% 20%

FIGURE 25–13 The resistance value of a given resistor is written on the exterior, or may be given as a color code as shown above and in the Table: the first two colors represent the first two digits in the value of the resistance, the third color represents the power of ten that it must be multiplied by, and the fourth is the manufactured tolerance. For example, a resistor whose four colors are red, green, yellow, and silver has a resistance of 25 * 104 = 250,000 = 250 k, plus or minus 10%. An alternate example of a simple code is a number such as 104, which means R = 1.0 * 104 .

SECTION 25–3

Ohm's Law: Resistance and Resistors

657

Some Helpful Clarifications C A U T I O N

Voltage is applied across a device; current passes through a device

C A U T I O N

Current is not consumed

Here we briefly summarize some possible misunderstandings and clarifications. Batteries do not put out a constant current. Instead, batteries are intended to maintain a constant potential difference, or very nearly so. (Details in the next Chapter.) Thus a battery should be considered a source of voltage. The voltage is applied across a wire or device. Electric current passes through a wire or device (connected to a battery), and its magnitude depends on that device's resistance. The resistance is a property of the wire or device. The voltage, on the other hand, is external to the wire or device, and is applied across the two ends of the wire or device. The current through the device might be called the "response": the current increases if the voltage increases or the resistance decreases, as I = VR. In a wire, the direction of the current is always parallel to the wire, no matter how the wire curves, just like water in a pipe. The direction of conventional (positive) current is from high potential (±) toward lower potential (–). Current and charge do not increase or decrease or get "used up" when going through a wire or other device. The amount of charge that goes in at one end comes out at the other end.

25–4

Resistivity

It is found experimentally that the resistance R of any wire is directly proportional to its length l and inversely proportional to its cross-sectional area A. That is, l R = r , (25–3) A where r, the constant of proportionality, is called the resistivity and depends on the material used. Typical values of r, whose units are m (see Eq. 25–3), are given for various materials in the middle column of Table 25–1, which is divided into the categories conductors, insulators, and semiconductors (see Section 21–3). The values depend somewhat on purity, heat treatment, temperature, and other factors. Notice that silver has the lowest resistivity and is thus the best conductor (although it is expensive). Copper is close, and much less expensive, which is why most wires are made of copper. Aluminum, although it has a higher resistivity, is much less dense than copper; it is thus preferable to copper in some situations, such as for transmission lines, because its resistance for the same weight is less than that for copper. TABLE 25–1 Resistivity and Temperature Coefficients (at 20°C) Material Conductors Silver Copper Gold Aluminum Tungsten Iron Platinum Mercury Nichrome (Ni, Fe, Cr alloy) Semiconductors† Carbon (graphite) Germanium Silicon Insulators Glass Hard rubber

658

CHAPTER 25

†

Resistivity, R ( m) 1.59 1.68 2.44 2.65 5.6 9.71 10.6 98 100

* * * * * * * * *

Temperature Coefficient, A (C°)1

10–8 10–8 10–8 10–8 10–8 10–8 10–8 10–8 10–8

(3 – 60) * 10 –5 (1 – 500) * 10 –3 0.1 – 60 109 – 1012 1013 – 1015

Values depend strongly on the presence of even slight amounts of impurities.

0.0061 0.0068 0.0034 0.00429 0.0045 0.00651 0.003927 0.0009 0.0004 – 0.0005 – 0.05 – 0.07

The reciprocal of the resistivity, called the conductivity s, is 1 s = r

(25–4)

and has units of (m)–1. EXERCISE C Return to the Chapter-Opening Question, page 651, and answer it again now. Try to explain why you may have answered differently the first time.

APPROACH We solve Eq. 25–3 to get the area A, from which we can calculate the wire's radius using A = pr2. The diameter is 2r. In (b) we can use Ohm's law, V = IR. SOLUTION (a) We solve Eq. 25–3 for the area A and find r for copper in Table 25–1:

A1.68 * 10–8 mB(20 m) l = = 3.4 * 10–6 m2. R (0.10 ) The cross-sectional area A of a circular wire is A = pr 2. The radius must then be at least A r = = 1.04 * 10 –3 m = 1.04 mm. Bp The diameter is twice the radius and so must be at least 2r = 2.1 mm. (b) From V = IR we find that the voltage drop across each wire is

20 m

EXAMPLE 25–5 Speaker wires. Suppose you want to connect your stereo to remote speakers (Fig. 25–14). (a) If each wire must be 20 m long, what diameter copper wire should you use to keep the resistance less than 0.10 per wire? (b) If the current to each speaker is 4.0 A, what is the potential difference, or voltage drop, across each wire?

20 m

EXERCISE D A copper wire has a resistance of 10 . What will its resistance be if it is only half as long? (a) 20 , (b) 10 , (c) 5 , (d) 1 , (e) none of these.

A = r

FIGURE 25–14 Example 25–5.

V = IR = (4.0 A)(0.10 ) = 0.40 V. NOTE The voltage drop across the wires reduces the voltage that reaches the speakers from the stereo amplifier, thus reducing the sound level a bit. CONCEPTUAL EXAMPLE 25–6 Stretching changes resistance. Suppose a wire of resistance R could be stretched uniformly until it was twice its original length. What would happen to its resistance? RESPONSE If the length l doubles, then the cross-sectional area A is halved, because the volume (V = Al) of the wire remains the same. From Eq. 25–3 we see that the resistance would increase by a factor of four A212 = 4B. EXERCISE E Copper wires in houses typically have a diameter of about 1.5 mm. How long a wire would have a 1.0- resistance?

Temperature Dependence of Resistivity The resistivity of a material depends somewhat on temperature. The resistance of metals generally increases with temperature. This is not surprising, for at higher temperatures, the atoms are moving more rapidly and are arranged in a less orderly fashion. So they might be expected to interfere more with the flow of electrons. If the temperature change is not too great, the resistivity of metals usually increases nearly linearly with temperature. That is, rT = r0 C1 + a AT - T0 B D (25–5) where r0 is the resistivity at some reference temperature T0 (such as 0°C or 20°C), rT is the resistivity at a temperature T, and a is the temperature coefficient of resistivity. Values for a are given in Table 25–1. Note that the temperature coefficient for semiconductors can be negative. Why? It seems that at higher temperatures, some of the electrons that are normally not free in a semiconductor become free and can contribute to the current. Thus, the resistance of a semiconductor can decrease with an increase in temperature. SECTION 25–4

Resistivity

659

P H Y S I C S

A P P L I E D

Resistance thermometer

EXAMPLE 25–7 Resistance thermometer. The variation in electrical resistance with temperature can be used to make precise temperature measurements. Platinum is commonly used since it is relatively free from corrosive effects and has a high melting point. Suppose at 20.0°C the resistance of a platinum resistance thermometer is 164.2 . When placed in a particular solution, the resistance is 187.4 . What is the temperature of this solution? APPROACH Since the resistance R is directly proportional to the resistivity r, we can combine Eq. 25–3 with Eq. 25–5 to find R as a function of temperature T, and then solve that equation for T. SOLUTION We multiply Eq. 25–5 by (lA) to obtain (see also Eq. 25–3) R = R0 C1 + a AT - T0 B D.

Here R0 = r0 lA is the resistance of the wire at T0 = 20.0°C. We solve this equation for T and find (see Table 25–1 for a) T = T0 +

FIGURE 25–15 A thermistor shown next to a millimeter ruler for scale.

R - R0 187.4 - 164.2 = 20.0°C + = 56.0°C. aR0 A3.927 * 10–3(C°)–1 B(164.2 )

NOTE Resistance thermometers have the advantage that they can be used at very high or low temperatures where gas or liquid thermometers would be useless. NOTE More convenient for some applications is a thermistor (Fig. 25–15), which consists of a metal oxide or semiconductor whose resistance also varies in a repeatable way with temperature. Thermistors can be made quite small and respond very quickly to temperature changes. EXERCISE F The resistance of the tungsten filament of a common incandescent lightbulb is how many times greater at its operating temperature of 3000 K than its resistance at room temperature? (a) Less than 1% greater; (b) roughly 10% greater; (c) about 2 times greater; (d) roughly 10 times greater; (e) more than 100 times greater.

The value of a in Eq. 25–5 itself can depend on temperature, so it is important to check the temperature range of validity of any value (say, in a handbook of physical data). If the temperature range is wide, Eq. 25–5 is not adequate and terms proportional to the square and cube of the temperature are needed, but they are generally very small except when T - T0 is large.

25–5

FIGURE 25–16 Hot electric stove burner glows because of energy transformed by electric current.

660

CHAPTER 25

Electric Power

Electric energy is useful to us because it can be easily transformed into other forms of energy. Motors transform electric energy into mechanical energy, and are examined in Chapter 27. In other devices such as electric heaters, stoves, toasters, and hair dryers, electric energy is transformed into thermal energy in a wire resistance known as a "heating element." And in an ordinary lightbulb, the tiny wire filament (Fig. 25–5 and Chapter-opening photo) becomes so hot it glows; only a few percent of the energy is transformed into visible light, and the rest, over 90%, into thermal energy. Lightbulb filaments and heating elements (Fig. 25–16) in household appliances have resistances typically of a few ohms to a few hundred ohms. Electric energy is transformed into thermal energy or light in such devices, and there are many collisions between the moving electrons and the atoms of the wire. In each collision, part of the electron's kinetic energy is transferred to the atom with which it collides. As a result, the kinetic energy of the wire's atoms increases and hence the temperature of the wire element increases. The increased thermal energy can be transferred as heat by conduction and convection to the air in a heater or to food in a pan, by radiation to bread in a toaster, or radiated as light. To find the power transformed by an electric device, recall that the energy transformed when an infinitesimal charge dq moves through a potential difference V is dU = V dq (Eq. 23–3). Let dt be the time required for an amount of charge dq

Electric Currents and Resistance

to move through a potential difference V. Then the power P, which is the rate energy is transformed, is P =

dq dU = V. dt dt

The charge that flows per second, dqdt, is the electric current I. Thus we have (25–6)

P = IV.

This general relation gives us the power transformed by any device, where I is the current passing through it and V is the potential difference across it. It also gives the power delivered by a source such as a battery. The SI unit of electric power is the same as for any kind of power, the watt (1 W = 1 Js). The rate of energy transformation in a resistance R can be written in two other ways, starting with the general relation P = IV and substituting in V = IR: P = IV = I(IR) = I 2R P = IV = ¢

V V2 . ≤V = R R

(25–7a) (25–7b)

Equations 25–7a and b apply only to resistors, whereas Eq. 25–6, P = IV, is more general and applies to any device, including a resistor. EXAMPLE 25–8 Headlights. Calculate the resistance of a 40-W automobile headlight designed for 12 V (Fig. 25–17). APPROACH We solve Eq. 25–7b for R. SOLUTION From Eq. 25–7b,

40-W Headlight

(12 V)2 V2 R = = = 3.6 . P (40 W)

FIGURE 25–17 Example 25–8.

NOTE This is the resistance when the bulb is burning brightly at 40 W. When the bulb is cold, the resistance is much lower, as we saw in Eq. 25–5. Since the current is high when the resistance is low, lightbulbs burn out most often when first turned on. It is energy, not power, that you pay for on your electric bill. Since power is the rate energy is transformed, the total energy used by any device is simply its power consumption multiplied by the time it is on. If the power is in watts and the time is in seconds, the energy will be in joules since 1 W = 1 Js. Electric companies usually specify the energy with a much larger unit, the kilowatt-hour (kWh). One kWh = (1000 W)(3600 s) = 3.60 * 106 J.

P H Y S I C S

A P P L I E D

Why lightbulbs burn out when first turned on

C A U T I O N

You pay for energy, which is power * time, not for power

EXAMPLE 25–9 Electric heater. An electric heater draws a steady 15.0 A on a 120-V line. How much power does it require and how much does it cost per month (30 days) if it operates 3.0 h per day and the electric company charges 9.2 cents per kWh? APPROACH We use Eq. 25–6, P = IV, to find the power. We multiply the power (in kW) by the time (h) used in a month and by the cost per energy unit, $0.092 per kWh, to get the cost per month. SOLUTION The power is P = IV = (15.0 A)(120 V) = 1800 W or 1.80 kW.The time (in hours) the heater is used per month is (3.0 hd)(30 d) = 90 h, which at 9.2¢kWh would cost (1.80 kW)(90 h)($0.092kWh) = $15. NOTE Household current is actually alternating (ac), but our solution is still valid assuming the given values for V and I are the proper averages (rms) as we discuss in Section 25–7. SECTION 25–5

Electric Power

661

P H Y S I C S

A P P L I E D

Lightning

EXAMPLE 25–10 ESTIMATE Lightning bolt. Lightning is a spectacular example of electric current in a natural phenomenon (Fig. 25–18). There is much variability to lightning bolts, but a typical event can transfer 109 J of energy across a potential difference of perhaps 5 * 107 V during a time interval of about 0.2 s. Use this information to estimate (a) the total amount of charge transferred between cloud and ground, (b) the current in the lightning bolt, and (c) the average power delivered over the 0.2 s. APPROACH We estimate the charge Q, recalling that potential energy change equals the potential difference ¢V times the charge Q, Eq. 23–3. We equate ¢U with the energy transferred, ¢U L 109 J. Next, the current I is Qt (Eq. 25–1a) and the power P is energy/time. SOLUTION (a) From Eq. 23–3, the energy transformed is ¢U = Q ¢V. We solve for Q: Q =

¢U 109 J L = 20 coulombs. ¢V 5 * 107 V

(b) The current during the 0.2 s is about FIGURE 25–18 Example 25–10. A lightning bolt.

I =

20 C = 100 A. 0.2 s

energy 109 J = = 5 * 109 W = 5 GW. time 0.2 s

We can also use Eq. 25–6: P = IV = (100 A)A5 * 107 VB = 5 GW. NOTE Since most lightning bolts consist of several stages, it is possible that individual parts could carry currents much higher than the 100 A calculated above.

25–6 P H Y S I C S

A P P L I E D

Safety—wires getting hot FIGURE 25–19 (a) Fuses. When the current exceeds a certain value, the metallic ribbon melts and the circuit opens. Then the fuse must be replaced. (b) One type of circuit breaker. The electric current passes through a bimetallic strip. When the current exceeds a safe level, the heating of the bimetallic strip causes the strip to bend so far to the left that the notch in the spring-loaded metal strip drops down over the end of the bimetallic strip; (c) the circuit then opens at the contact points (one is attached to the metal strip) and the outside switch is also flipped. As soon as the bimetallic strip cools down, it can be reset using the outside switch. Magnetic-type circuit breakers are discussed in Chapters 28 and 29.

662

CHAPTER 25

Power in Household Circuits

The electric wires that carry electricity to lights and other electric appliances have some resistance, although usually it is quite small. Nonetheless, if the current is large enough, the wires will heat up and produce thermal energy at a rate equal to I 2R, where R is the wire's resistance. One possible hazard is that the current-carrying wires in the wall of a building may become so hot as to start a fire. Thicker wires have less resistance (see Eq. 25–3) and thus can carry more current without becoming too hot. When a wire carries more current than is safe, it is said to be "overloaded." To prevent overloading, fuses or circuit breakers are installed in circuits. They are basically switches (Fig. 25–19)

Compressed spring

Contact points

Outside switch Fuse ribbon

(a) Types of fuses

Electric Currents and Resistance

Contacts open Metal strip

Bimetallic strip

To electric circuit

(b) Circuit breaker (closed)

that open the circuit when the current exceeds some particular value. A 20-A fuse or circuit breaker, for example, opens when the current passing through it exceeds 20 A. If a circuit repeatedly burns out a fuse or opens a circuit breaker, there are two possibilities: there may be too many devices drawing current in that circuit; or there is a fault somewhere, such as a "short." A short, or "short circuit," means that two wires have touched that should not have (perhaps because the insulation has worn through) so the resistance is much reduced and the current becomes very large. Short circuits should be remedied immediately. Household circuits are designed with the various devices connected so that each receives the standard voltage (usually 120 V in the United States) from the electric company (Fig. 25–20). Circuits with the devices arranged as in Fig. 25–20 are called parallel circuits, as we will discuss in the next Chapter. When a fuse blows or circuit breaker opens, it is important to check the total current being drawn on that circuit, which is the sum of the currents in each device.

P H Y S I C S

Switch

0.8 A + 15.0 A + 2.9 A + 10.0 A = 28.7 A. NOTE The heater draws as much current as 18 100-W lightbulbs. For safety, the heater should probably be on a circuit by itself. If the circuit in Fig. 25–20 is designed for a 20-A fuse, the fuse should blow, and we hope it will, to prevent overloaded wires from getting hot enough to start a fire. Something will have to be turned off to get this circuit below 20 A. (Houses and apartments usually have several circuits, each with its own fuse or circuit breaker; try moving one of the devices to another circuit.) If the circuit is designed with heavier wire and a 30-A fuse, the fuse shouldn't blow—if it does, a short may be the problem. (The most likely place for a short is in the cord of one of the devices.) Proper fuse size is selected according to the wire used to supply the current. A properly rated fuse should never be replaced by a higher-rated one. A fuse blowing or a circuit breaker opening is acting like a switch, making an "open circuit." By an open circuit, we mean that there is no longer a complete conducting path, so no current can flow; it is as if R = q. CONCEPTUAL EXAMPLE 25–12 A dangerous extension cord. Your 1800-W portable electric heater is too far from your desk to warm your feet. Its cord is too short, so you plug it into an extension cord rated at 11 A. Why is this dangerous?

Lightbulb 100 W

Electric heater 1800 W

EXAMPLE 25–11 Will a fuse blow? Determine the total current drawn by all the devices in the circuit of Fig. 25–20. APPROACH Each device has the same 120-V voltage across it. The current each draws from the source is found from I = PV, Eq. 25–6. SOLUTION The circuit in Fig. 25–20 draws the following currents: the lightbulb draws I = PV = 100 W120 V = 0.8 A; the heater draws 1800 W120 V = 15.0 A; the stereo draws a maximum of 350 W120 V = 2.9 A; and the hair dryer draws 1200 W120 V = 10.0 A. The total current drawn, if all devices are used at the same time, is

A P P L I E D

Fuses and circuit breakers

Stereo receiver 350 W

Fuse or circuit breaker

Hair dryer 1200 W

120 V (from electric company)

FIGURE 25–20 Connection of household appliances.

P H Y S I C S

A P P L I E D

Proper fuses and shorts

P H Y S I C S

A P P L I E D

Extension cords and possible danger

RESPONSE 1800 W at 120 V draws a 15-A current. The wires in the extension cord rated at 11 A could become hot enough to melt the insulation and cause a fire. EXERCISE G How many 60-W 120-V lightbulbs can operate on a 20-A line? (a) 2; (b) 3; (c) 6; (d) 20; (e) 40.

SECTION 25–6

Power in Household Circuits

663

Current

25–7 t

Time (a) DC

Current

I0 t

Time

Alternating Current

When a battery is connected to a circuit, the current moves steadily in one direction. This is called a direct current, or dc. Electric generators at electric power plants, however, produce alternating current, or ac. (Sometimes capital letters are used, DC and AC.) An alternating current reverses direction many times per second and is commonly sinusoidal, as shown in Fig. 25–21. The electrons in a wire first move in one direction and then in the other. The current supplied to homes and businesses by electric companies is ac throughout virtually the entire world. We will discuss and analyze ac circuits in detail in Chapter 30. But because ac circuits are so common in real life, we will discuss some of their basic aspects here. The voltage produced by an ac electric generator is sinusoidal, as we shall see later. The current it produces is thus sinusoidal (Fig. 25–21b). We can write the voltage as a function of time as

−I0

V = V0 sin 2pft = V0 sin vt. (b) AC

FIGURE 25–21 (a) Direct current. (b) Alternating current.

The potential V oscillates between ±V0 and –V0 , and V0 is referred to as the peak voltage. The frequency f is the number of complete oscillations made per second, and v = 2pf. In most areas of the United States and Canada, f is 60 Hz (the unit "hertz," as we saw in Chapters 10 and 14, means cycles per second). In many other countries, 50 Hz is used. Equation 25–2, V = IR, works also for ac: if a voltage V exists across a resistance R, then the current I through the resistance is V0 V (25–8) I = = sin vt = I0 sin vt. R R The quantity I0 = V0R is the peak current. The current is considered positive when the electrons flow in one direction and negative when they flow in the opposite direction. It is clear from Fig. 25–21b that an alternating current is as often positive as it is negative. Thus, the average current is zero. This does not mean, however, that no power is needed or that no heat is produced in a resistor. Electrons do move back and forth, and do produce heat. Indeed, the power transformed in a resistance R at any instant is P = I 2R = I 20 R sin2 vt.

FIGURE 25–22 Power transformed in a resistor in an ac circuit. I 2R = 12 I 20R

I 20 R

Because the current is squared, we see that the power is always positive, as graphed in Fig. 25–22. The quantity sin2 vt varies between 0 and 1; and it is not too difficult to show† that its average value is 12 , as indicated in Fig. 25–22. Thus, the average power transformed, P, is P =

1 2 2 I 0R.

Power

Since power can also be written P = V2R = AV20RB sin2 vt, we also have that the average power is Time

V20 R The average or mean value of the square of the current or voltage is thus what is important for calculating average power: Y = 12 I 20 and Z = 12 V20 . The square root of each of these is the rms (root-mean-square) value of the current or voltage: I0 Irms = 3Y = = 0.707 I0 , (25–9a) 22 V0 Vrms = 3Z = = 0.707 V0 . (25–9b) 22 The rms values of V and I are sometimes called the effective values. P =

1 2

A graph of cos2 vt versus t is identical to that for sin2 vt in Fig. 25–22, except that the points are shifted (by 14 cycle) on the time axis. Hence the average value of sin2 and cos2, averaged over one or more full cycles, will be the same: sin2 vt = cos2 vt. From the trigonometric identity sin2 u + cos2 u = 1, we can write †

664

CHAPTER 25

Asin2 vt B + Acos2 vt B = 2Asin2 vt B = 1. Hence the average value of sin2 vt is 12 .

They are useful because they can be substituted directly into the power formulas, Eqs. 25–6 and 25–7, to get the average power: P = Irms Vrms P =

1 2 2 I0 R

P =

1 2

= I 2rms R

V2rms V20 . = R R

(25–10a) (25–10b) (25–10c)

Thus, a direct current whose values of I and V equal the rms values of I and V for an alternating current will produce the same power. Hence it is usually the rms value of current and voltage that is specified or measured. For example, in the United States and Canada, standard line voltage† is 120-V ac. The 120 V is Vrms ; the peak voltage V0 is V0 = 22 Vrms = 170 V. In much of the world (Europe, Australia, Asia) the rms voltage is 240 V, so the peak voltage is 340 V. EXAMPLE 25–13 Hair dryer. (a) Calculate the resistance and the peak current in a 1000-W hair dryer (Fig. 25–23) connected to a 120-V line. (b) What happens if it is connected to a 240-V line in Britain?

Motor Fan

APPROACH We are given P and Vrms , so Irms = PVrms (Eq. 25–10a or 25–6), and I0 = 12 Irms . Then we find R from V = IR. SOLUTION (a) We solve Eq. 25–10a for the rms current: Irms =

Heating coils

P 1000 W = = 8.33 A. Vrms 120 V

Switch

Then I0 = 22 Irms = 11.8 A.

Cord

The resistance is

FIGURE 25–23 A hair dryer. Most of the current goes through the heating coils, a pure resistance; a small part goes to the motor to turn the fan. Example 25–13.

Vrms 120 V R = = = 14.4 . Irms 8.33 A The resistance could equally well be calculated using peak values: R =

V0 170 V = = 14.4 . I0 11.8 A

(b) When connected to a 240-V line, more current would flow and the resistance would change with the increased temperature (Section 25–4). But let us make an estimate of the power transformed based on the same 14.4- resistance. The average power would be P =

(240 V)2 V2rms = = 4000 W. R (14.4 )

This is four times the dryer's power rating and would undoubtedly melt the heating element or the wire coils of the motor. EXERCISE H Each channel of a stereo receiver is capable of an average power output of 100 W into an 8- loudspeaker (see Fig. 25–14). What are the rms voltage and the rms current fed to the speaker (a) at the maximum power of 100 W, and (b) at 1.0 W when the volume is turned down? † The line voltage can vary, depending on the total load; the frequency of 60 Hz or 50 Hz, however, remains extremely steady.

SECTION 25–7

Alternating Current

665

25–8 A B

–

FIGURE 25–24 Electric field E in a uniform wire of cross-sectional area A carrying a current I. The current density j = IA.

Microscopic View of Electric Current: Current Density and Drift Velocity

Up to now in this Chapter we have dealt mainly with a macroscopic view of electric current. We saw, however, that according to atomic theory, the electric current in metal wires is carried by negatively charged electrons, and that in liquid solutions current can also be carried by positive and/or negative ions. Let us now look at this microscopic picture in more detail. When a potential difference is applied to the two ends of a wire of uniform B cross section, the direction of the electric field E is parallel to the walls of the wire B (Fig. 25–24). The existence of E within the conducting wire does not contradict our B earlier result that E = 0 inside a conductor in the electrostatic case, as we are no longer dealing with the static case. Charges are free to move in a conductor, and hence can move under the action of the electric field. If all the charges are at rest, B then E must be zero (electrostatics). We now define a new microscopic quantity, the current density, j . It is defined as the electric current per unit cross-sectional area at any point in space. If the current density j in a wire of cross-sectional area A is uniform over the cross section, then j is related to the electric current by I j = or I = jA. (25–11) A If the current density is not uniform, then the general relation is B

I = B

j dA, B

+ E

– vBd

FIGURE 25–26 Electrons in the volume Al will all pass through the cross section indicated in a time ¢ t, where l = vd ¢ t. vBd

l = vd ∆t

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CHAPTER 25

(25–12)

where dA is an element of surface and I is the current through the surface over which the integration is taken. The direction of the current density at any point is the direction that a positive charge would move when placed at that point—that B is, the direction of j at any point is generally the same as the direction of E, Fig. 25–24. The current density exists for any point in space. The current I, on the other hand, refers to a conductor as a whole, and hence is a macroscopic quantity. The direction of j is chosen to represent the direction of net flow of positive charge. In a conductor, it is negatively charged electrons that move, so they move in the B direction of – j , or –E (to the left in Fig. 25–24). We can imagine the free electrons as moving about randomly at high speeds, bouncing off the atoms of the wire (somewhat like the molecules of a gas—Chapter 18). When an electric field exists in the wire, Fig. 25–25, the electrons feel a force and initially begin to accelerate. But they soon B reach a more or less steady average velocity in the direction of E, known as their drift velocity, £d (collisions with atoms in the wire keep them from accelerating further). The drift velocity is normally very much smaller than the electrons' average random speed. We can relate the drift velocity vd to the macroscopic current I in the wire. In a time ¢ t, the electrons will travel a distance l = vd ¢ t on average. Suppose the wire has cross-sectional area A. Then in time ¢ t, electrons in a volume V = Al = Avd ¢ t will pass through the cross section A of wire, as shown in Fig. 25–26. If there are n free electrons (each of charge –e) per unit volume (n = NV), then the total charge ¢Q that passes through the area A in a time ¢ t is ¢Q = (no. of charges, N) * (charge per particle) = (nV)(–e) = – AnAvd ¢ t B(e). The current I in the wire is thus ¢Q I = = –neAvd . (25–13) ¢t The current density, j = IA, is j = –nevd . (25–14) In vector form, this is written j = –neTd , (25–15) where the minus sign indicates that the direction of (positive) current flow is opposite to the drift velocity of electrons. B

FIGURE 25–25 Electric field E in a wire gives electrons in random motion a drift velocity vd .

We can generalize Eq. 25–15 to any type of charge flow, such as flow of ions in an electrolyte. If there are several types of ions (which can include free electrons), each of density ni (number per unit volume), charge qi (qi = –e for electrons) and drift velocity Tdi , then the net current density at any point is B

j =

a ni qi Tdi .

(25–16)

The total current I passing through an area A perpendicular to a uniform j is then I =

a ni qi vdi A. i

EXAMPLE 25–14 Electron speeds in a wire. A copper wire 3.2 mm in diameter carries a 5.0-A current. Determine (a) the current density in the wire, and (b) the drift velocity of the free electrons. (c) Estimate the rms speed of electrons assuming they behave like an ideal gas at 20°C. Assume that one electron per Cu atom is free to move (the others remain bound to the atom). APPROACH For (a) j = IA = Ipr2. For (b) we can apply Eq. 25–14 to find vd if we can determine the number n of free electrons per unit volume. Since we assume there is one free electron per atom, the density of free electrons, n, is the same as the density of Cu atoms. The atomic mass of Cu is 63.5 u (see Periodic Table inside the back cover), so 63.5 g of Cu contains one mole or 6.02 * 1023 free electrons. The mass density of copper (Table 13–1) is rD = 8.9 * 103 kgm3, where rD = mV. (We use rD to distinguish it here from r for resistivity.) In (c) we use K = 32kT, Eq. 18–4. (Do not confuse V for volume with V for voltage.) SOLUTION (a) The current density is (with r = 12 A3.2 mmB = 1.6 * 10–3 mB I I 5.0 A j = = = = 6.2 * 105 Am2. 2 2 A pr p A1.6 * 10 –3 mB (b) The number of free electrons per unit volume, n = NV (where V = mrD), is N(1 mole) N N = r n = = m(1 mole) D V mrD n = ¢

6.02 * 1023 electrons ≤ A8.9 * 103 kgm3 B = 8.4 * 1028 m–3. 63.5 * 10 –3 kg

Then, by Eq. 25–14, the drift velocity has magnitude vd =

j 6.2 * 105 Am2 = = 4.6 * 10 –5 ms L 0.05 mms. ne A8.4 * 1028 m–3 BA1.6 * 10 –19 CB

(c) If we model the free electrons as an ideal gas (a rather rough approximation), we use Eq. 18–5 to estimate the random rms speed of an electron as it darts around: vrms =

3 A1.38 * 10–23 JKB(293 K) 3kT = = 1.2 * 105 ms. B m C 9.11 * 10–31 kg

The drift velocity (average speed in the direction of the current) is very much less than the rms thermal speed of the electrons, by a factor of about 109. NOTE The result in (c) is an underestimate. Quantum theory calculations, and experiments, give the rms speed in copper to be about 1.6 * 106 ms. The drift velocity of electrons in a wire is very slow, only about 0.05 mms (Example 25–14 above), which means it takes an electron 20 * 103 s, or 5 12 h, to travel only 1 m. This is not, of course, how fast "electricity travels": when you flip a light switch, the light—even if many meters away—goes on nearly instantaneously. Why? Because electric fields travel essentially at the speed of light A3 * 108 msB. We can think of electrons in a wire as being like a pipe full of water: when a little water enters one end of the pipe, almost immediately some water comes out at the other end. SECTION 25–8

Microscopic View of Electric Current: Current Density and Drift Velocity

667

* Electric Field Inside a Wire Equation 25–2b, V = IR, can be written in terms of microscopic quantities as follows. We write the resistance R in terms of the resistivity r: l R = r ; A and we write V and I as I = jA and V = El. The last relation follows from Eqs. 23–4, where we assume the electric field is uniform within the wire and l is the length of the wire (or a portion of the wire) between whose ends the potential difference is V. Thus, from V = IR, we have El = (jA) ¢ r so j =

l ≤ = jrl A

1 E = sE, r

(25–17)

where s = 1r is the conductivity (Eq. 25–4). For a metal conductor, r and s do not depend on V (and hence not on E). Therefore the current density j is B proportional to the electrical field E in the conductor. This is the "microscopic" statement of Ohm's law. Equation 25–17, which can be written in vector form as 1 B B j = sE = E, r is sometimes taken as the definition of conductivity s and resistivity r. B

EXAMPLE 25–15 Electric field inside a wire. What is the electric field inside the wire of Example 25–14? APPROACH We use Eq. 25–17 and r = 1.68 * 10 –8 m for copper. SOLUTION Example 25–14 gives j = 6.2 * 105 Am2, so

E = rj = A1.68 * 10 –8 mBA6.2 * 105 Am2 B = 1.0 * 10 –2 Vm.

NOTE For comparison, the electric field between the plates of a capacitor is often much larger; in Example 24–1, for example, E is on the order of 104 Vm. Thus we see that only a modest electric field is needed for current flow in practical cases.

* 25–9

Resistivity, r

FIGURE 25–27 A superconducting material has zero resistivity when its temperature is below TC , its "critical temperature." At TC , the resistivity jumps to a "normal" nonzero value and increases with temperature as most materials do (Eq. 25–5).

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CHAPTER 25

Superconductivity

At very low temperatures, well below 0°C, the resistivity (Section 25–4) of certain metals and certain compounds or alloys becomes zero as measured by the highestprecision techniques. Materials in such a state are said to be superconducting. It was first observed by H. K. Onnes (1853–1926) in 1911 when he cooled mercury below 4.2 K (–269°C) and found that the resistance of mercury suddenly dropped to zero. In general, superconductors become superconducting only below a certain transition temperature or critical temperature, TC , which is usually within a few degrees of absolute zero. Current in a ring-shaped superconducting material has been observed to flow for years in the absence of a potential difference, with no measurable decrease. Measurements show that the resistivity r of superconductors is less than 4 * 10–25 m, which is over 1016 times smaller than that for copper, and is considered to be zero in practice. See Fig. 25–27. Before 1986 the highest temperature at which a material was found to superconduct was 23 K, which required liquid helium to keep the material cold. In 1987, a compound of yttrium, barium, copper, and oxygen (YBCO) was developed that can be superconducting at 90 K. This was an important breakthrough since liquid nitrogen, which boils at 77 K (sufficiently cold to keep the material superconducting), is more easily and cheaply obtained than the liquid helium needed for conventional superconductors. Superconductivity at temperatures as high as 160 K has been reported, though in fragile compounds.

Signal from another neuron

Most applications today use a bismuth-strontium-calcium-copper oxide, known (for short) as BSCCO. A major challenge is how to make a useable, bendable wire out of the BSCCO, which is very brittle. (One solution is to embed tiny filaments of the high-TC superconductor in a metal alloy, which is not resistanceless, but the resistance is much less than that of a conventional copper cable.)

* 25–10

Synapse Dendrites

Electrical Conduction in the Nervous System

The flow of electric charge in the human nervous system provides us the means for being aware of the world. Although the detailed functioning is not well understood, we do have a reasonable understanding of how messages are transmitted within the nervous system: they are electrical signals passing along the basic element of the nervous system, the neuron. Neurons are living cells of unusual shape (Fig. 25–28). Attached to the main cell body are several small appendages known as dendrites and a long tail called the axon. Signals are received by the dendrites and are propagated along the axon. When a signal reaches the nerve endings, it is transmitted to the next neuron or to a muscle at a connection called a synapse. A neuron, before transmitting an electrical signal, is in the so-called "resting state." Like nearly all living cells, neurons have a net positive charge on the outer surface of the cell membrane and a negative charge on the inner surface. This difference in charge, or "dipole layer," means that a potential difference exists across the cell membrane. When a neuron is not transmitting a signal, this "resting potential," normally stated as Vinside - Voutside , is typically –60 mV to –90 mV, depending on the type of organism. The most common ions in a cell are K±, Na±, and Cl–. There are large differences in the concentrations of these ions inside and outside a cell, as indicated by the typical values given in Table 25–2. Other ions are also present, so the fluids both inside and outside the axon are electrically neutral. Because of the differences in concentration, there is a tendency for ions to diffuse across the membrane (see Section 18–7 on diffusion). However, in the resting state the cell membrane prevents any net flow of Na± (through a mechanism of "active pumping" of Na± out of the cell). But it does allow the flow of Cl– ions, and less so of K± ions, and it is these two ions that produce the dipole charge layer on the membrane. Because there is a greater concentration of K± inside the cell than outside, more K± ions tend to diffuse outward across the membrane than diffuse inward. A K± ion that passes through the membrane becomes attached to the outer surface of the membrane, and leaves behind an equal negative charge that lies on the inner surface of the membrane (Fig. 25–29). Independently, Cl– ions tend to diffuse into the cell since their concentration outside is higher. Both K± and Cl– diffusion tends to charge the interior surface of the membrane negatively and the outside positively. As charge accumulates on the membrane surface, it becomes increasingly difficult for more ions to diffuse: K± ions trying to move outward, for example, are repelled by the positive charge already there. Equilibrium is reached when the tendency to diffuse because of the concentration difference is just balanced by the electrical potential difference across the membrane. The greater the concentration difference, the greater the potential difference across the membrane ( –60 mV to –90 mV). The most important aspect of a neuron is not that it has a resting potential (most cells do), but rather that it can respond to a stimulus and conduct an electrical signal along its length. The stimulus could be thermal (when you touch a hot stove) or chemical (as in taste buds); it could be pressure (as on the skin or at the eardrum), or light (as in the eye); or it could be the electric stimulus of a signal coming from the brain or another neuron. In the laboratory, the stimulus is usually electrical and is applied by a tiny probe at some point on the neuron. If the stimulus exceeds some threshold, a voltage pulse will travel down the axon. This voltage pulse can be detected at a point on the axon using a voltmeter or an oscilloscope connected as in Fig. 25–30.

Cell body

Nucleus

Axon Myelin sheath Node of Ranvier

Synapse

Nerve endings

Another neuron or a muscle FIGURE 25–28 A simplified sketch of a typical neuron.

TABLE 25–2 Concentrations of Ions Inside and Outside a Typical Axon Concentration Concentration inside axon outside axon (mol m3) (mol m3) K± Na± Cl–

140 15 9

Na+

K+ – +

+ –

5 140 125

Extracellular fluid Cl– + – Axon

Membrane

– +

FIGURE 25–29 How a dipole layer of charge forms on a cell membrane. FIGURE 25–30 Measuring the potential difference between the inside and outside of a nerve cell.

Voutside

Vinside

Axon

*SECTION 25–10

669

Potential V (mV)

40 20

Time (ms)

− 20

− 40 − 60 − 80

Resting potential

Action potential

FIGURE 25–31 Action potential. FIGURE 25–32 Propagation of an action potential along an axon membrane. Point of stimulation Membrane Exterior + – – ++++++++++ – ++ – – – – – – – – – – Interior (a) + – – ++++++++++ – ++ – – – – – – – – – – (b) – ++ – +++++++++ + – – + – – – – – – – – – (c) ++++ – ++++++++ – – – – + – – – – – – – – Action potential moving to the right (d)

This voltage pulse has the shape shown in Fig. 25–31, and is called an action potential. As can be seen, the potential increases from a resting potential of about –70 mV and becomes a positive 30 mV or 40 mV. The action potential lasts for about 1 ms and travels down an axon with a speed of 30 ms to 150 ms. When an action potential is stimulated, the nerve is said to have "fired." What causes the action potential? Apparently, the cell membrane has the ability to alter its permeability properties. At the point where the stimulus occurs, the membrane suddenly becomes much more permeable to Na± than to K± and Cl– ions. Thus, Na± ions rush into the cell and the inner surface of the wall becomes positively charged, and the potential difference quickly swings positive (L ±35 mV in Fig. 25–31). Just as suddenly, the membrane returns to its original characteristics: it becomes impermeable to Na± and in fact pumps out Na± ions. The diffusion of Cl– and K± ions again predominates and the original resting potential is restored ( –70 mV in Fig. 25–31). What causes the action potential to travel along the axon? The action potential occurs at the point of stimulation, as shown in Fig. 25–32a. The membrane momentarily is positive on the inside and negative on the outside at this point. Nearby charges are attracted toward this region, as shown in Fig. 25–32b. The potential in these adjacent regions then drops, causing an action potential there. Thus, as the membrane returns to normal at the original point, nearby it experiences an action potential, so the action potential moves down the axon (Figs. 25–32c and d). You may wonder if the number of ions that pass through the membrane would significantly alter the concentrations. The answer is no; and we can show why by treating the axon as a capacitor in the following Example. EXAMPLE 25–16 ESTIMATE Capacitance of an axon. (a) Do an order-ofmagnitude estimate for the capacitance of an axon 10 cm long of radius 10 mm. The thickness of the membrane is about 10–8 m, and the dielectric constant is about 3. (b) By what factor does the concentration (number of ions per volume) of Na± ions in the cell change as a result of one action potential? APPROACH We model the membrane of an axon as a cylindrically shaped parallel-plate capacitor, with opposite charges on each side. The separation of the "plates" is the thickness of the membrane, d L 10–8 m. We first calculate the area of the cylinder and then can use Eq. 24–8, C = K 0 Ad, to find the capacitance. In (b), we use the voltage change during one action potential to find the amount of charge moved across the membrane. SOLUTION (a) The area A is the area of a cylinder of radius r and length l: A = 2prl L (6.28)A10 –5 mB(0.1 m) L 6 * 10–6 m2. From Eq. 24–8, we have A 6 * 10–6 m2 L (3)A8.85 * 10 –12 C 2N m2 B C = K 0 L 10–8 F. d 10–8 m (b) Since the voltage changes from –70 mV to about ±30 mV, the total change is about 100 mV. The amount of charge that moves is then Q = CV L A10–8 FB(0.1 V) = 10–9 C.

Each ion carries a charge e = 1.6 * 10–19 C, so the number of ions that flow per action potential is Qe = A10 –9 CBA1.6 * 10–19 CB L 1010. The volume of our cylindrical axon is 2

V = pr 2l L (3)A10 –5 mB (0.1 m) = 3 * 10–11 m3, and the concentration of Na± ions inside the cell (Table 25–2) is 15 molm3 = 15 * 6.02 * 1023 ionsm3 L 1025 ionsm3. Thus, the cell contains A1025 ionsm3 B * A3 * 10–11 m3 B L 3 * 1014 Na± ions. One action potential, then, will change the concentration of Na± ions by about 1010A3 * 1014 B = 13 * 10–4, or 1 part in 30,000. This tiny change would not be measurable.

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CHAPTER 25

Thus, even 1000 action potentials will not alter the concentration significantly. The sodium pump does not, therefore, have to remove Na± ions quickly after an action potential, but can operate slowly over time to maintain a relatively constant concentration.

Summary An electric battery serves as a source of nearly constant potential difference by transforming chemical energy into electric energy. A simple battery consists of two electrodes made of different metals immersed in a solution or paste known as an electrolyte. Electric current, I, refers to the rate of flow of electric charge and is measured in amperes (A): 1 A equals a flow of 1 Cs past a given point. The direction of conventional current is that of positive charge flow. In a wire, it is actually negatively charged electrons that move, so they flow in a direction opposite to the conventional current. A positive charge flow in one direction is almost always equivalent to a negative charge flow in the opposite direction. Positive conventional current always flows from a high potential to a low potential. The resistance R of a device is defined by the relation (25–2)

V = IR,

where I is the current in the device when a potential difference V is applied across it. For materials such as metals, R is a constant independent of V (thus I r V), a result known as Ohm's law. Thus, the current I coming from a battery of voltage V depends on the resistance R of the circuit connected to it. Voltage is applied across a device or between the ends of a wire. Current passes through a wire or device. Resistance is a property of the wire or device. The unit of resistance is the ohm (), where 1 = 1 VA. See Table 25–3.

TABLE 25–3 Summary of Units Current Potential difference Power Resistance

1 A = 1 Cs 1 V = 1 JC 1 W = 1 Js 1 = 1 VA

The rate at which energy is transformed in a resistance R from electric to other forms of energy (such as heat and light) is equal to the product of current and voltage. That is, the power transformed, measured in watts, is given by (25–6)

P = IV, which for resistors can be written as

V2 . (25–7) R The SI unit of power is the watt (1 W = 1 Js). The total electric energy transformed in any device equals the product of the power and the time during which the device is operated. In SI units, energy is given in joules (1 J = 1 W s), but electric companies use a larger unit, the kilowatt-hour A1 kWh = 3.6 * 106 JB. Electric current can be direct current (dc), in which the current is steady in one direction; or it can be alternating current (ac), in which the current reverses direction at a particular frequency f, typically 60 Hz. Alternating currents are typically sinusoidal in time, P = I 2R =

(25–8)

I = I0 sin vt,

where v = 2pf, and are produced by an alternating voltage. The rms values of sinusoidally alternating currents and voltages are given by Irms =

I0 22

and

Vrms =

(25–9)

respectively, where I0 and V0 are the peak values. The power relationship, P = IV = I 2R = V2R, is valid for the average power in alternating currents when the rms values of V and I are used. B Current density j is the current per cross-sectional area. From a microscopic point of view, the current density is related to the number of charge carriers per unit volume, n, their charge, q, and their drift velocity, Td , by B

(25–16)

j = nqTd . B

The resistance R of a wire is inversely proportional to its cross-sectional area A, and directly proportional to its length l and to a property of the material called its resistivity: rl . R = (25–3) A The resistivity, r, increases with temperature for metals, but for semiconductors it may decrease.

The electric field within a wire is related to j by j = sE where s = 1r is the conductivity. [*At very low temperatures certain materials become superconducting, which means their electrical resistance becomes zero.] [*The human nervous system operates via electrical conduction: when a nerve "fires," an electrical signal travels as a voltage pulse known as an action potential.]

Questions 1. What quantity is measured by a battery rating given in ampere-hours (A h)? 2. When an electric cell is connected to a circuit, electrons flow away from the negative terminal in the circuit. But within the cell, electrons flow to the negative terminal. Explain. 3. When a flashlight is operated, what is being used up: battery current, battery voltage, battery energy, battery power, or battery resistance? Explain. 4. One terminal of a car battery is said to be connected to "ground." Since it is not really connected to the ground, what is meant by this expression?

5. When you turn on a water faucet, the water usually flows immediately. You don't have to wait for water to flow from the faucet valve to the spout. Why not? Is the same thing true when you connect a wire to the terminals of a battery? 6. Can a copper wire and an aluminum wire of the same length have the same resistance? Explain. 7. The equation P = V2R indicates that the power dissipated in a resistor decreases if the resistance is increased, whereas the equation P = I 2R implies the opposite. Is there a contradiction here? Explain. 8. What happens when a lightbulb burns out?

Questions

671

9. If the resistance of a small immersion heater (to heat water for tea or soup, Fig. 25–33) was increased, would it speed up or slow down the heating process? Explain.

FIGURE 25–33 Question 9. 10. If a rectangular solid made of carbon has sides of lengths a, 2a, and 3a, how would you connect the wires from a battery so as to obtain (a) the least resistance, (b) the greatest resistance? 11. Explain why lightbulbs almost always burn out just as they are turned on and not after they have been on for some time. 12. Which draws more current, a 100-W lightbulb or a 75-W bulb? Which has the higher resistance? 13. Electric power is transferred over large distances at very high voltages. Explain how the high voltage reduces power losses in the transmission lines.

14. A 15-A fuse blows repeatedly. Why is it dangerous to replace this fuse with a 25-A fuse? 15. When electric lights are operated on low-frequency ac (say, 5 Hz), they flicker noticeably. Why? 16. Driven by ac power, the same electrons pass back and forth through your reading lamp over and over again. Explain why the light stays lit instead of going out after the first pass of electrons. 17. The heating element in a toaster is made of Nichrome wire. Immediately after the toaster is turned on, is the current AIrms B in the wire increasing, decreasing, or staying constant? Explain. 18. Is current used up in a resistor? Explain. 19. Compare the drift velocities and electric currents in two wires that are geometrically identical and the density of atoms is similar, but the number of free electrons per atom in the material of one wire is twice that in the other. 20. A voltage V is connected across a wire of length l and radius r. How is the electron drift velocity affected if (a) l is doubled, (b) r is doubled, (c) V is doubled? 21. Why is it more dangerous to turn on an electric appliance when you are standing outside in bare feet than when you are inside wearing shoes with thick soles?

Problems 25–2 and 25–3 Electric Current, Resistance, Ohm's Law (Note: The charge on one electron is 1.60 * 10 –19 C.)

1. (I) A current of 1.30 A flows in a wire. How many electrons are flowing past any point in the wire per second? 2. (I) A service station charges a battery using a current of 6.7-A for 5.0 h. How much charge passes through the battery? 3. (I) What is the current in amperes if 1200 Na± ions flow across a cell membrane in 3.5 ms? The charge on the sodium is the same as on an electron, but positive. 4. (I) What is the resistance of a toaster if 120 V produces a current of 4.2 A? 5. (II) An electric clothes dryer has a heating element with a resistance of 8.6 . (a) What is the current in the element when it is connected to 240 V? (b) How much charge passes through the element in 50 min? (Assume direct current.) 6. (II) A hair dryer draws 9.5 A when plugged into a 120-V line. (a) What is its resistance? (b) How much charge passes through it in 15 min? (Assume direct current.) 7. (II) A 4.5-V battery is connected to a bulb whose resistance is 1.6 . How many electrons leave the battery per minute? 8. (II) A bird stands on a dc electric transmission line carrying 3100 A (Fig. 25–34). The line has 2.5 * 10–5 resistance per meter, and the bird's feet are 4.0 cm apart. What is the potential difference between the bird's feet?

FIGURE 25–34 Problem 8.

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CHAPTER 25

Electric Currents and Resistance

9. (II) A 12-V battery causes a current of 0.60 A through a resistor. (a) What is its resistance, and (b) how many joules of energy does the battery lose in a minute? 10. (II) An electric device draws 6.50 A at 240 V. (a) If the voltage drops by 15%, what will be the current, assuming nothing else changes? (b) If the resistance of the device were reduced by 15%, what current would be drawn at 240 V?

25–4 Resistivity 11. (I) What is the diameter of a 1.00-m length of tungsten wire whose resistance is 0.32 ? 12. (I) What is the resistance of a 4.5-m length of copper wire 1.5 mm in diameter? 13. (II) Calculate the ratio of the resistance of 10.0 m of aluminum wire 2.0 mm in diameter, to 20.0 m of copper wire 1.8 mm in diameter. 14. (II) Can a 2.2-mm-diameter copper wire have the same resistance as a tungsten wire of the same length? Give numerical details. 15. (II) A sequence of potential differences V is applied across a wire (diameter = 0.32 mm, length = 11 cm) and the resulting currents I are measured as follows: V (V) I (mA)

0.100 72

0.200 144

0.300 216

0.400 288

0.500 360

(a) If this wire obeys Ohm's law, graphing I vs. V will result in a straight-line plot. Explain why this is so and determine the theoretical predictions for the straight line's slope and y-intercept. (b) Plot I vs. V. Based on this plot, can you conclude that the wire obeys Ohm's law (i.e., did you obtain a straight line with the expected y-intercept)? If so, determine the wire's resistance R. (c) Calculate the wire's resistivity and use Table 25–1 to identify the solid material from which it is composed.

4. 0

16. (II) How much would you have to raise the temperature of a copper wire (originally at 20°C) to increase its resistance by 15%? 17. (II) A certain copper wire has a resistance of 10.0 . At what point along its length must the wire be cut so that the resistance of one piece is 4.0 times the resistance of the other? What is the resistance of each piece? 18. (II) Determine at what temperature aluminum will have the same resistivity as tungsten does at 20°C. 19. (II) A 100-W lightbulb has a resistance of about 12 when cold (20°C) and 140 when on (hot). Estimate the temperature of the filament when hot assuming an average temperature coefficient of resistivity a = 0.0045 (C°)–1. 20. (II) Compute the voltage drop along a 26-m length of household no. 14 copper wire (used in 15-A circuits). The wire has diameter 1.628 mm and carries a 12-A current. 21. (II) Two aluminum wires have the same resistance. If one has twice the length of the other, what is the ratio of the diameter of the longer wire to the diameter of the shorter wire? 22. (II) A rectangular solid made of carbon has sides of lengths 1.0 cm, 2.0 cm, and 4.0 cm, lying y along the x, y, and z axes, respectively (Fig. 25–35). Determine the resistance for current that passes through the solid in (a) the x direc2.0 cm tion, (b) the y direction, and (c) the z direction. x Assume the resistivity is –5 r = 3.0 * 10 m. FIGURE 25–35 Problem 22.

1.0 cm

23. (II) A length of aluminum wire is connected to a precision 10.00-V power supply, and a current of 0.4212 A is precisely measured at 20.0°C. The wire is placed in a new environment of unknown temperature where the measured current is 0.3818 A. What is the unknown temperature? 24. (II) Small changes in the length of an object can be measured using a strain gauge sensor, which is a wire with undeformed length l0 , cross-sectional area A 0 , and resistance R0 . This sensor is rigidly affixed to the object's surface, aligning its length in the direction in which length changes are to be measured. As the object deforms, the length of the wire sensor changes by ¢l, and the resulting change ¢R in the sensor's resistance is measured. Assuming that as the solid wire is deformed to a length l, its density (and volume) remains constant (only approximately valid), show that the strain A = ¢ll0 B of the wire sensor, and thus of the object to which it is attached, is ¢R2R0 . 25. (II) A length of wire is cut in half and the two lengths are wrapped together side by side to make a thicker wire. How does the resistance of this new combination compare to the resistance of the original wire? 26. (III) For some applications, it is important that the value of a resistance not change with temperature. For example, suppose you made a 3.70-k resistor from a carbon resistor and a Nichrome wire-wound resistor connected together so the total resistance is the sum of their separate resistances. What value should each of these resistors have (at 0°C) so that the combination is temperature independent?

27. (III) Determine a formula for the total resistance of a spherical shell made of material whose conductivity is s and whose inner and outer radii are r1 and r2 . Assume the current flows radially outward. 28. (III) The filament of a lightbulb has a resistance of 12 at 20°C and 140 when hot (as in Problem 19). (a) Calculate the temperature of the filament when it is hot, and take into account the change in length and area of the filament due to thermal expansion (assume tungsten for which the thermal expansion coefficient is L 5.5 * 10–6 C°–1). (b) In this temperature range, what is the percentage change in resistance due to thermal expansion, and what is the percentage change in resistance due solely to the change in r? Use Eq. 25–5. 29. (III) A 10.0-m length of wire consists of 5.0 m of copper followed by 5.0 m of aluminum, both of diameter 1.4 mm. A voltage difference of 85 mV is placed across the composite wire. (a) What is the total resistance (sum) of the two wires? (b) What is the current through the wire? (c) What are the voltages across the aluminum part and across the copper part? 30. (III) A hollow cylindrical resistor with inner radius r1 and outer radius r2 , and length l, is made of a material whose resistivity is r (Fig. 25–36). (a) Show that the resistance is given by r r2 R = ln 2pl r1 for current that flows radially outward. [Hint: Divide the resistor into concentric cylindrical shells and integrate.] (b) Evaluate the resistance R for such a resistor made of carbon whose inner and outer radii are 1.0 mm and 1.8 mm and whose length is 2.4 cm. (Choose r = 15 * 10–5 m.) (c) What is the resistance in l part (b) for current flowing r2 parallel to the axis? FIGURE 25–36 Problem 30.

25–5 and 25–6 Electric Power 31. (I) What is the maximum power consumption of a 3.0-V portable CD player that draws a maximum of 270 mA of current? 32. (I) The heating element of an electric oven is designed to produce 3.3 kW of heat when connected to a 240-V source. What must be the resistance of the element? 33. (I) What is the maximum voltage that can be applied across a 3.3-k resistor rated at 14 watt? 34. (I) (a) Determine the resistance of, and current through, a 75-W lightbulb connected to its proper source voltage of 110 V. (b) Repeat for a 440-W bulb. 35. (II) An electric power plant can produce electricity at a fixed power P, but the plant operator is free to choose the voltage V at which it is produced. This electricity is carried as an electric current I through a transmission line (resistance R) from the plant to the user, where it provides the user with electric power P¿. (a) Show that the reduction in power ¢P = P - P¿ due to transmission losses is given by ¢P = P2RV2. (b) In order to reduce power losses during transmission, should the operator choose V to be as large or as small as possible? 36. (II) A 120-V hair dryer has two settings: 850 W and 1250 W. (a) At which setting do you expect the resistance to be higher? After making a guess, determine the resistance at (b) the lower setting; and (c) the higher setting.

Problems

673

37. (II) A 115-V fish-tank heater is rated at 95 W. Calculate (a) the current through the heater when it is operating, and (b) its resistance. 38. (II) You buy a 75-W lightbulb in Europe, where electricity is delivered to homes at 240 V. If you use the lightbulb in the United States at 120 V (assume its resistance does not change), how bright will it be relative to 75-W 120-V bulbs? [Hint: Assume roughly that brightness is proportional to power consumed.] 39. (II) How many kWh of energy does a 550-W toaster use in the morning if it is in operation for a total of 6.0 min? At a cost of 9.0 centskWh, estimate how much this would add to your monthly electric energy bill if you made toast four mornings per week. 40. (II) At $0.095kWh, what does it cost to leave a 25-W porch light on day and night for a year? 41. (II) What is the total amount of energy stored in a 12-V, 75-Ah car battery when it is fully charged? 42. (II) An ordinary flashlight uses two D-cell 1.5-V batteries connected in series as in Fig. 25–4b (Fig. 25–37). The bulb draws 380 mA when turned on. (a) Calculate the resistance of the bulb and the power dissipated. (b) By what factor would the power increase if four D-cells in series were used with the same bulb? (Neglect heating effects of the filament.) Why shouldn't you try this?

25–7 Alternating Current 49. (I) Calculate the peak current in a 2.7-k resistor connected to a 220-V rms ac source. 50. (I) An ac voltage, whose peak value is 180 V, is across a 380- resistor. What are the rms and peak currents in the resistor? 51. (II) Estimate the resistance of the 120-Vrms circuits in your house as seen by the power company, when (a) everything electrical is unplugged, and (b) there are two 75-W lightbulbs burning. 52. (II) The peak value of an alternating current in a 1500-W device is 5.4 A. What is the rms voltage across it? 53. (II) An 1800-W arc welder is connected to a 660-Vrms ac line. Calculate (a) the peak voltage and (b) the peak current. 54. (II) (a) What is the maximum instantaneous power dissipated by a 2.5-hp pump connected to a 240-Vrms ac power source? (b) What is the maximum current passing through the pump? 55. (II) A heater coil connected to a 240-Vrms ac line has a resistance of 44 . (a) What is the average power used? (b) What are the maximum and minimum values of the instantaneous power? 56. (II) For a time-dependent voltage V(t), which is periodic with period T, the rms voltage is defined to be 1 T Vrms = C T1 0 V2 dt D 2 . Use this definition to determine Vrms (in terms of the peak voltage V0) for (a) a sinusoidal voltage, i.e., V(t) = V0 sin(2ptT) for 0 t T; and (b) a positive square-wave voltage, i.e., T 2 .

T tT 2

V(t) = d

25–8 Microscopic View of Electric Current

FIGURE 25–37 Problem 42. 43. (II) How many 75-W lightbulbs, connected to 120 V as in Fig. 25–20, can be used without blowing a 15-A fuse? 44. (II) An extension cord made of two wires of diameter 0.129 cm (no. 16 copper wire) and of length 2.7 m (9 ft) is connected to an electric heater which draws 15.0 A on a 120-V line. How much power is dissipated in the cord? 45. (II) A power station delivers 750 kW of power at 12,000 V to a factory through wires with total resistance 3.0 . How much less power is wasted if the electricity is delivered at 50,000 V rather than 12,000 V? 46. (III) A small immersion heater can be used in a car to heat a cup of water for coffee or tea. If the heater can heat 120 mL of water from 25°C to 95°C in 8.0 min, (a) approximately how much current does it draw from the car's 12-V battery, and (b) what is its resistance? Assume the manufacturer's claim of 75% efficiency. 47. (III) The current in an electromagnet connected to a 240-V line is 17.5 A. At what rate must cooling water pass over the coils if the water temperature is to rise by no more than 6.50 C°? 48. (III) A 1.0-m-long round tungsten wire is to reach a temperature of 3100 K when a current of 15.0 A flows through it. What diameter should the wire be? Assume the wire loses energy only by radiation (emissivity = 1.0, Section 19–10) and the surrounding temperature is 20°C.

674

CHAPTER 25

Electric Currents and Resistance

57. (II) A 0.65-mm-diameter copper wire carries a tiny current of 2.3 mA. Estimate (a) the electron drift velocity, (b) the current density, and (c) the electric field in the wire. 58. (II) A 5.80-m length of 2.0-mm-diameter wire carries a 750-mA current when 22.0 mV is applied to its ends. If the drift velocity is 1.7 * 10–5 ms, determine (a) the resistance R of the wire, (b) the resistivity r, (c) the current density j, (d) the electric field inside the wire, and (e) the number n of free electrons per unit volume. 59. (II) At a point high in the Earth's atmosphere, He2± ions in a concentration of 2.8 * 1012m3 are moving due north at a speed of 2.0 * 106 ms. Also, a 7.0 * 1011m3 concentration of O2– ions is moving due south at a speed of 6.2 * 106 ms. Determine the magnitude and direction of the current B density j at this point. * 25–10 Nerve Conduction * 60. (I) What is the magnitude of the electric field across an axon membrane 1.0 * 10 –8 m thick if the resting potential is –70 mV? * 61. (II) A neuron is stimulated with an electric pulse. The action potential is detected at a point 3.40 cm down the axon 0.0052 s later. When the action potential is detected 7.20 cm from the point of stimulation, the time required is 0.0063 s. What is the speed of the electric pulse along the axon? (Why are two measurements needed instead of only one?) * 62. (III) During an action potential, Na± ions move into the cell at a rate of about 3 * 10–7 molm2 s. How much power must be produced by the "active Na± pumping" system to produce this flow against a ±30-mV potential difference? Assume that the axon is 10 cm long and 20 mm in diameter.

General Problems 63. A person accidentally leaves a car with the lights on. If each of the two headlights uses 40 W and each of the two taillights 6 W, for a total of 92 W, how long will a fresh 12-V battery last if it is rated at 85 A h? Assume the full 12 V appears across each bulb. 64. How many coulombs are there in 1.00 ampere-hour? 65. You want to design a portable electric blanket that runs on a 1.5-V battery. If you use copper wire with a 0.50-mm diameter as the heating element, how long should the wire be if you want to generate 15 W of heating power? What happens if you accidentally connect the blanket to a 9.0-V battery? 66. What is the average current drawn by a 1.0-hp 120-V motor? (1 hp = 746 W.) 67. The conductance G of an object is defined as the reciprocal of the resistance R; that is, G = 1R. The unit of conductance is a mho A = ohm–1 B, which is also called the siemens (S). What is the conductance (in siemens) of an object that draws 480 mA of current at 3.0 V? 68. The heating element of a 110-V, 1500-W heater is 3.5 m long. If it is made of iron, what must its diameter be? 69. (a) A particular household uses a 1.8-kW heater 2.0 hday ("on" time), four 100-W lightbulbs 6.0 hday, a 3.0-kW electric stove element for a total of 1.0 hday, and miscellaneous power amounting to 2.0 kWhday. If electricity costs $0.105 per kWh, what will be their monthly bill (30 d)? (b) How much coal (which produces 7500 kcalkg) must be burned by a 35%-efficient power plant to provide the yearly needs of this household? 70. A small city requires about 15 MW of power. Suppose that instead of using high-voltage lines to supply the power, the power is delivered at 120 V. Assuming a two-wire line of 0.50-cm-diameter copper wire, estimate the cost of the energy lost to heat per hour per meter. Assume the cost of electricity is about 9.0 cents per kWh. 71. A 1400-W hair dryer is designed for 117 V. (a) What will be the percentage change in power output if the voltage drops to 105 V? Assume no change in resistance. (b) How would the actual change in resistivity with temperature affect your answer? 72. The wiring in a house must be thick enough so it does not become so hot as to start a fire. What diameter must a copper wire be if it is to carry a maximum current of 35 A and produce no more than 1.5 W of heat per meter of length? 73. Determine the resistance of the tungsten filament in a 75-W 120-V incandescent lightbulb (a) at its operating temperature of about 3000 K, (b) at room temperature. 74. Suppose a current is given by the equation I = 1.80 sin 210t, where I is in amperes and t in seconds. (a) What is the frequency? (b) What is the rms value of the current? (c) If this is the current through a 24.0- resistor, write the equation that describes the voltage as a function of time. 75. A microwave oven running at 65% efficiency delivers 950 W to the interior. Find (a) the power drawn from the source, and (b) the current drawn. Assume a source voltage of 120 V. 76. A 1.00- wire is stretched uniformly to 1.20 times its original length. What is its resistance now? 77. 220 V is applied to two different conductors made of the same material. One conductor is twice as long and twice the diameter of the second. What is the ratio of the power transformed in the first relative to the second?

78. An electric heater is used to heat a room of volume 54 m3. Air is brought into the room at 5°C and is completely replaced twice per hour. Heat loss through the walls amounts to approximately 850 kcalh. If the air is to be maintained at 20°C, what minimum wattage must the heater have? (The specific heat of air is about 0.17 kcalkg C°.) 79. A 2800-W oven is connected to a 240-V source. (a) What is the resistance of the oven? (b) How long will it take to bring 120 mL of 15°C water to 100°C assuming 75% efficiency? (c) How much will this cost at 11 centskWh? 80. A proposed electric vehicle makes use of storage batteries as its source of energy. Its mass is 1560 kg and it is powered by 24 batteries, each 12 V, 95 A h. Assume that the car is driven on level roads at an average speed of 45 kmh, and the average friction force is 240 N. Assume 100% efficiency and neglect energy used for acceleration. No energy is consumed when the vehicle is stopped, since the engine doesn't need to idle. (a) Determine the horsepower required. (b) After approximately how many kilometers must the batteries be recharged? 81. A 12.5- resistor is made from a coil of copper wire whose total mass is 15.5 g. What is the diameter of the wire, and how long is it? 82. A fish-tank heater is rated at 95 W when connected to 120 V. The heating element is a coil of Nichrome wire. When uncoiled, the wire has a total length of 3.8 m. What is the diameter of the wire? 83. A 100-W, 120-V lightbulb has a resistance of 12 when cold (20°C) and 140 when on (hot). Calculate its power consumption (a) at the instant it is turned on, and (b) after a few moments when it is hot. 84. In an automobile, the system voltage varies from about 12 V when the car is off to about 13.8 V when the car is on and the charging system is in operation, a difference of 15%. By what percentage does the power delivered to the headlights vary as the voltage changes from 12 V to 13.8 V? Assume the headlight resistance remains constant. 85. The Tevatron accelerator at Fermilab (Illinois) is designed to carry an 11-mA beam of protons traveling at very nearly the speed of light A3.0 * 108 msB around a ring 6300 m in circumference. How many protons are in the beam? 86. Lightbulb A is rated at 120 V and 40 W for household applications. Lightbulb B is rated at 12 V and 40 W for automotive applications. (a) What is the current through each bulb? (b) What is the resistance of each bulb? (c) In one hour, how much charge passes through each bulb? (d) In one hour, how much energy does each bulb use? (e) Which bulb requires larger diameter wires to connect its power source and the bulb? 87. An air conditioner draws 14 A at 220-V ac. The connecting cord is copper wire with a diameter of 1.628 mm. (a) How much power does the air conditioner draw? (b) If the total length of wire is 15 m, how much power is dissipated in the wiring? (c) If no. 12 wire, with a diameter of 2.053 mm, was used instead, how much power would be dissipated in the wiring? (d) Assuming that the air conditioner is run 12 h per day, how much money per month (30 days) would be saved by using no. 12 wire? Assume that the cost of electricity is 12 cents per kWh.

General Problems

675

Normal

95. The level of liquid helium (temperature 4 K ) in its storage tank can be monitored using a vertically aligned niobium–titanium (NbTi) wire, whose length l spans the height of the tank. In this level-sensing setup, an electronic circuit maintains a constant electrical current I at all times in the NbTi wire and a voltmeter monitors the voltage difference V across this wire. Since the superconducting transition temperature for NbTi is 10 K, the portion of the wire immersed in the liquid helium is in the superconducting state, while the portion above the liquid (in helium vapor with temperature above 10 K) is in the normal state. Define f = xl to be the fraction of the tank filled with liquid helium (Fig. 25–40) and V0 to be the value Constant I of V when the tank is empty (f = 0). DeterHelium mine the relation vapor between f and V (10 K) (in terms of V0).

l Superconducting

88. Copper wire of diameter 0.259 cm is used to connect a set of appliances at 120 V, which draw 1750 W of power total. (a) What power is wasted in 25.0 m of this wire? (b) What is your answer if wire of diameter 0.412 cm is used? 89. Battery-powered electricity is very expensive compared with that available from a wall receptacle. Estimate the cost per kWh of (a) an alkaline D-cell (cost $1.70) and (b) an alkaline AA-cell (cost $1.25). These batteries can provide a continuous current of 25 mA for 820 h and 120 h, respectively, at 1.5 V. Compare to normal 120-V ac house current at $0.10kWh. 90. How far does an average electron move along the wires of a 550-W toaster during an alternating current cycle? The power cord has copper wires of diameter 1.7 mm and is plugged into a standard 60-Hz 120-V ac outlet. [Hint: The maximum current in the cycle is related to the maximum drift velocity. The maximum velocity in an oscillation is related to the maximum displacement; see Chapter 14.] 91. A copper pipe has an inside diameter of 3.00 cm and an outside diameter of 5.00 cm (Fig. 25–38). What is the resistance of a 10.0-m length of this pipe? 3.00 cm

5.00 cm FIGURE 25–38 Problem 91. 92. For the wire in Fig. 25–39, whose diameter varies uniformly from a to b as shown, suppose a current I = 2.0 A enters at a. If a = 2.5 mm and b = 4.0 mm, what is the current density (assume uniform) at each end? l

Liquid helium (4 K)

FIGURE 25–40 Problem 95. * Numerical/Computer

* 96. (II) The resistance, R, of a particular thermistor as a function of temperature T is shown in this Table:

T (°C)

R ()

T (°C)

R ()

20 22 24 26 28 30 32 34

126,740 115,190 104,800 95,447 87,022 79,422 72,560 66,356

36 38 40 42 44 46 48 50

60,743 55,658 51,048 46,863 43,602 39,605 36,458 33,591

FIGURE 25–39 Problems 92 and 93. 93. The cross section of a portion of wire increases uniformly as shown in Fig. 25–39 so it has the shape of a truncated cone. The diameter at one end is a and at the other it is b, and the total length along the axis is l. If the material has resistivity r, determine the resistance R between the two ends in terms of a, b, l, and r. Assume that the current flows uniformly through each section, and that the taper is small, i.e., (b - a) V l. 94. A tungsten filament used in a flashlight bulb operates at 0.20 A and 3.2 V. If its resistance at 20°C is 1.5 , what is the temperature of the filament when the flashlight is on?

Determine what type of best-fit equation (linear, quadratic, exponential, other) describes the variation of R with T. The resistance of the thermistor is 57,641 when embedded in a substance whose temperature is unknown. Based on your equation, what is the unknown temperature?

Answers to Exercises A: 1.6 * 10 –13 A.

E: 110 m.

B: (c).

F: (d).

C: (b), (c).

G: (e).

D: (c).

H: (a) 28 V, 3.5 A; (b) 2.8 V, 0.35 A.

676

CHAPTER 25

Electric Currents and Resistance

These MP3 players contain circuits that are dc, at least in part. (The audio signal is ac.) The circuit diagram below shows a possible amplifier circuit for each stereo channel. We have already met two of the circuit elements shown: resistors and capacitors, and we discuss them in circuits in this Chapter. (The large triangle is an amplifier chip containing transistors.) We also discuss voltmeters and ammeters, and how they are built and used to make measurements.

15 V

470 mF

2.2 mF

IC3

470 mF 0.1 mF

50 k

47 k

2.2 mF

Headset 8

2.7 k

DC Circuits CHAPTER-OPENING QUESTION—Guess now! The automobile headlight bulbs shown in the circuits here are identical. The connection which produces more light is (a) circuit 1. (b) circuit 2. (c) both the same. (d) not enough information. + –

P T E R

C H

2.2 mF Input to each channel

CONTENTS

26–1 EMF and Terminal Voltage 26–2 Resistors in Series and in Parallel 26–3 Kirchhoff's Rules 26–4 Series and Parallel EMFs; Battery Charging +

–

26–5 Circuits Containing Resistor and Capacitor (RC Circuits) 26–6 Electric Hazards

Circuit 1

Circuit 2

* 26–7 Ammeters and Voltmeters

lectric circuits are basic parts of all electronic devices from radio and TV sets to computers and automobiles. Scientific measurements, from physics to biology and medicine, make use of electric circuits. In Chapter 25, we discussed the basic principles of electric current. Now we will apply these principles to analyze dc circuits involving combinations of batteries, resistors, and capacitors. We also study the operation of some useful instruments.†

E †

AC circuits that contain only a voltage source and resistors can be analyzed like the dc circuits in this Chapter. However, ac circuits that contain capacitors and other circuit elements are more complicated, and we discuss them in Chapter 30.

677

TABLE 26–1 Symbols for Circuit Elements Symbol

Device

Battery

ƒ or

Capacitor Resistor Wire with negligible resistance Switch

Ground

C A U T I O N

Why battery voltage isn't perfectly constant

–

b Terminal voltage

Vab

FIGURE 26–1 Diagram for an electric cell or battery.

When we draw a diagram for a circuit, we represent batteries, capacitors, and resistors by the symbols shown in Table 26–1. Wires whose resistance is negligible compared with other resistance in the circuit are drawn simply as straight lines. Some circuit diagrams show a ground symbol ( or ) which may mean a real connection to the ground, perhaps via a metal pipe, or it may simply mean a common connection, such as the frame of a car. For the most part in this Chapter, except in Section 26–5 on RC circuits, we will be interested in circuits operating in their steady state.That is, we won't be looking at a circuit at the moment a change is made in it, such as when a battery or resistor is connected or disconnected, but rather later when the currents have reached their steady values.

26–1

To have current in an electric circuit, we need a device such as a battery or an electric generator that transforms one type of energy (chemical, mechanical, or light, for example) into electric energy. Such a device is called a source of electromotive force or of emf. (The term "electromotive force" is a misnomer since it does not refer to a "force" that is measured in newtons. Hence, to avoid confusion, we prefer to use the abbreviation, emf.) The potential difference between the terminals of such a source, when no current flows to an external circuit, is called the emf of the source. The symbol e is usually used for emf (don't confuse it with E for electric field), and its unit is volts. A battery is not a source of constant current—the current out of a battery varies according to the resistance in the circuit. A battery is, however, a nearly constant voltage source, but not perfectly constant as we now discuss. You may have noticed in your own experience that when a current is drawn from a battery, the potential difference (voltage) across its terminals drops below its rated emf. For example, if you start a car with the headlights on, you may notice the headlights dim. This happens because the starter draws a large current, and the battery voltage drops as a result. The voltage drop occurs because the chemical reactions in a battery (Section 25–1) cannot supply charge fast enough to maintain the full emf. For one thing, charge must move (within the electrolyte) between the electrodes of the battery, and there is always some hindrance to completely free flow. Thus, a battery itself has some resistance, which is called its internal resistance; it is usually designated r. A real battery is modeled as if it were a perfect emf e in series with a resistor r, as shown in Fig. 26–1. Since this resistance r is inside the battery, we can never separate it from the battery. The two points a and b in the diagram represent the two terminals of the battery. What we measure is the terminal voltage Vab = Va - Vb . When no current is drawn from the battery, the terminal voltage equals the emf, which is determined by the chemical reactions in the battery: Vab = e. However, when a current I flows naturally from the battery there is an internal drop in voltage equal to Ir. Thus the terminal voltage (the actual voltage) is† Vab = e - Ir. (26–1) For example, if a 12-V battery has an internal resistance of 0.1 , then when 10 A flows from the battery, the terminal voltage is 12 V - (10 A)(0.1 ) = 11 V. The internal resistance of a battery is usually small. For example, an ordinary flashlight battery when fresh may have an internal resistance of perhaps 0.05 . (However, as it ages and the electrolyte dries out, the internal resistance increases to many ohms.) Car batteries have lower internal resistance. EXAMPLE 26–1 Battery with internal resistance. A 65.0- resistor is connected to the terminals of a battery whose emf is 12.0 V and whose internal resistance is 0.5 , Fig. 26–2. Calculate (a) the current in the circuit, (b) the terminal voltage of the battery, Vab , and (c) the power dissipated in the resistor R and in the battery's internal resistance r. APPROACH We first consider the battery as a whole, which is shown in Fig. 26–2 as an emf e and internal resistance r between points a and b. Then we apply V = IR to the circuit itself.

FIGURE 26–2 Example 26–1.

R = 65.0 Ω

0.5 Ω

678

CHAPTER 26

= 12.0 V

EMF and Terminal Voltage

†

When a battery is being charged, a current is forced to pass through it; we then have to write Vab = e + Ir. See Section 26–4 or Problem 28 and Fig. 26–46.

SOLUTION (a) From Eq. 26–1, we have Vab = e - Ir. We apply Ohm's law (Eqs. 25–2) to this battery and the resistance R of the circuit: Vab = IR. Hence IR = e - Ir or e = I(R + r), and so e 12.0 V 12.0 V I = = = = 0.183 A. R + r 65.0 + 0.5 65.5 (b) The terminal voltage is Vab = e - Ir = 12.0 V - (0.183 A)(0.5 ) = 11.9 V. (c) The power dissipated (Eq. 25–7) in R is PR = I 2R = (0.183 A)2(65.0 ) = 2.18 W, and in r is Pr = I 2r = (0.183 A)2(0.5 ) = 0.02 W. EXERCISE A Repeat Example 26–1 assuming now that the resistance R = 10.0 , whereas e and r remain as before.

In much of what follows, unless stated otherwise, we assume that the battery's internal resistance is negligible, and that the battery voltage given is its terminal voltage, which we will usually write simply as V rather than Vab . Be careful not to confuse V (italic) for voltage and V (not italic) for the volt unit.

26–2

Resistors in Series and in Parallel

When two or more resistors are connected end to end along a single path as shown in Fig. 26–3a, they are said to be connected in series. The resistors could be simple resistors as were pictured in Fig. 25–12, or they could be lightbulbs (Fig. 26–3b), or heating elements, or other resistive devices. Any charge that passes through R1 in Fig. 26–3a will also pass through R2 and then R3 . Hence the same current I passes through each resistor. (If it did not, this would imply that either charge was not conserved, or that charge was accumulating at some point in the circuit, which does not happen in the steady state.) We let V represent the potential difference (voltage) across all three resistors in Fig. 26–3a. We assume all other resistance in the circuit can be ignored, so V equals the terminal voltage supplied by the battery. We let V1 , V2 , and V3 be the potential differences across each of the resistors, R1 , R2 , and R3 , respectively. From Ohm's law, V = IR, we can write V1 = IR1 , V2 = IR2 , and V3 = IR3 . Because the resistors are connected end to end, energy conservation tells us that the total voltage V is equal to the sum of the voltages† across each resistor: V = V1 + V2 + V3 = IR1 + IR2 + IR3 .

FIGURE 26–3 (a) Resistances connected in series. (b) Resistances could be lightbulbs, or any other type of resistance. (c) Equivalent single resistance Req that draws the same current: Req = R1 + R2 + R3 .

–

(a)

[series] (26–2)

Now let us determine the equivalent single resistance Req that would draw the same current I as our combination of three resistors in series; see Fig. 26–3c. Such a single resistance Req would be related to V by V = IReq .

We equate this expression with Eq. 26–2, V = IAR1 + R2 + R3 B, and find Req = R1 + R2 + R3 .

[series] (26–3)

This is, in fact, what we expect. When we put several resistances in series, the total or equivalent resistance is the sum of the separate resistances. (Sometimes we may also call it the "net resistance.") This sum applies to any number of resistances in series. Note that when you add more resistance to the circuit, the current through the circuit will decrease. For example, if a 12-V battery is connected to a 4- resistor, the current will be 3 A. But if the 12-V battery is connected to three 4- resistors in series, the total resistance is 12 and the current through the entire circuit will be only 1 A. †

To see in more detail why this is true, note that an electric charge q passing through R1 loses an amount of potential energy equal to qV1 . In passing through R2 and R3 , the potential energy U decreases by qV2 and qV3 , for a total ¢U = qV1 + qV2 + qV3 ; this sum must equal the energy given to q by the battery, qV, so that energy is conserved. Hence qV = qAV1 + V2 + V3 B, and so V = V1 + V2 + V3 , which is Eq. 26–2.

Battery (b)

Req I

(c)

–

SECTION 26–2

679

–

(a)

–

(b)

Req

(c)

–

FIGURE 26–4 (a) Resistances connected in parallel. (b) The resistances could be lightbulbs. (c) The equivalent circuit with Req obtained from Eq. 26–4: 1 1 1 1 . = + + R1 R2 R3 Req

FIGURE 26–5 Water pipes in parallel—analogy to electric currents in parallel.

680

CHAPTER 26

DC Circuits

Another simple way to connect resistors is in parallel so that the current from the source splits into separate branches or paths, as shown in Fig. 26–4a and b. The wiring in houses and buildings is arranged so all electric devices are in parallel, as we already saw in Chapter 25, Fig. 25–20. With parallel wiring, if you disconnect one device (say, R1 in Fig. 26–4a), the current to the other devices is not interrupted. Compare to a series circuit, where if one device (say, R1 in Fig. 26–3a) is disconnected, the current is stopped to all the others. In a parallel circuit, Fig. 26–4a, the total current I that leaves the battery splits into three separate paths. We let I1 , I2 , and I3 be the currents through each of the resistors, R1 , R2 , and R3 , respectively. Because electric charge is conserved, the current I flowing into junction A (where the different wires or conductors meet, Fig. 26–4a) must equal the current flowing out of the junction. Thus [parallel] I = I1 + I2 + I3 . When resistors are connected in parallel, each has the same voltage across it. (Indeed, any two points in a circuit connected by a wire of negligible resistance are at the same potential.) Hence the full voltage of the battery is applied to each resistor in Fig. 26–4a. Applying Ohm's law to each resistor, we have V, V, V. I1 = I2 = and I3 = R1 R2 R3 Let us now determine what single resistor Req (Fig. 26–4c) will draw the same current I as these three resistances in parallel. This equivalent resistance Req must satisfy Ohm's law too: V . I = Req We now combine the equations above: I = I1 + I2 + I3 , V V V V. = + + Req R1 R2 R3 When we divide out the V from each term, we have 1 1 1 1 . = + + Req R1 R2 R3

[parallel] (26–4)

For example, suppose you connect two 4- loudspeakers to a single set of output terminals of your stereo amplifier or receiver. (Ignore the other channel for a moment—our two speakers are both connected to the left channel, say.) The equivalent resistance of the two 4- "resistors" in parallel is 1 1 1 2 1 , = + = = Req 4 4 4 2 and so Req = 2 . Thus the net (or equivalent) resistance is less than each single resistance. This may at first seem surprising. But remember that when you connect resistors in parallel, you are giving the current additional paths to follow. Hence the net resistance will be less. Equations 26–3 and 26–4 make good sense. Recalling Eq. 25–3 for resistivity, R = rlA, we see that placing resistors in series increases the length and therefore the resistance; putting resistors in parallel increases the area through which current flows, thus reducing the overall resistance. An analogy may help here. Consider two identical pipes taking in water near the top of a dam and releasing it below as shown in Fig. 26–5. The gravitational potential difference, proportional to the height h, is the same for both pipes, just as the voltage is the same for parallel resistors. If both pipes are open, rather than only one, twice as much water will flow through. That is, with two equal pipes open, the net resistance to the flow of water will be reduced, by half, just as for electrical resistors in parallel. Note that if both pipes are closed, the dam offers infinite resistance to the flow of water. This corresponds in the electrical case to an open circuit—when the path is not continuous and no current flows—so the electrical resistance is infinite. EXERCISE B You have a 10- and a 15- resistor. What is the smallest and largest equivalent resistance that you can make with these two resistors?

CONCEPTUAL EXAMPLE 26–2 Series or parallel? (a) The lightbulbs in Fig. 26–6 are identical.Which configuration produces more light? (b) Which way do you think the headlights of a car are wired? Ignore change of filament resistance R with current. RESPONSE (a) The equivalent resistance of the parallel circuit is found from Eq. 26–4, 1Req = 1R + 1R = 2R. Thus Req = R2. The parallel combination then has lower resistance ( = R2) than the series combination AReq = R + R = 2RB. There will be more total current in the parallel configuration (2), since I = VReq and V is the same for both circuits. The total power transformed, which is related to the light produced, is P = IV, so the greater current in (2) means more light produced. (b) Headlights are wired in parallel (2), because if one bulb goes out, the other bulb can stay lit. If they were in series (1), when one bulb burned out (the filament broke), the circuit would be open and no current would flow, so neither bulb would light. NOTE When you answered the Chapter-Opening Question on page 677, was your answer circuit 2? Can you express any misconceptions you might have had? CONCEPTUAL EXAMPLE 26–3 An illuminating surprise. A 100-W, 120-V lightbulb and a 60-W, 120-V lightbulb are connected in two different ways as shown in Fig. 26–7. In each case, which bulb glows more brightly? Ignore change of filament resistance with current (and temperature). RESPONSE (a) These are normal lightbulbs with their power rating given for 120 V. They both receive 120 V, so the 100-W bulb is naturally brighter. (b) The resistance of the 100-W bulb is less than that of the 60-W bulb (calculated from P = V2R at constant 120 V). Here they are connected in series and receive the same current. Hence, from P = I 2R (I = constant) the higher-resistance "60-W" bulb will transform more power and thus be brighter. NOTE When connected in series as in (b), the two bulbs do not dissipate 60 W and 100 W because neither bulb receives 120 V.

–

(1) Series

–

(2) Parallel

FIGURE 26–6 Example 26–2. FIGURE 26–7 Example 26–3. 60 W 100 W (a) 120 V

60 W (b)

100 W

120 V

Note that whenever a group of resistors is replaced by the equivalent resistance, current and voltage and power in the rest of the circuit are unaffected. EXAMPLE 26–4 Circuit with series and parallel resistors. How much current is drawn from the battery shown in Fig. 26–8a? APPROACH The current I that flows out of the battery all passes through the 400- resistor, but then it splits into I1 and I2 passing through the 500- and 700- resistors. The latter two resistors are in parallel with each other. We look for something that we already know how to treat. So let's start by finding the equivalent resistance, Rp , of the parallel resistors, 500 and 700 . Then we can consider this Rp to be in series with the 400- resistor. SOLUTION The equivalent resistance, Rp , of the 500- and 700- resistors in parallel is given by 1 1 1 = + = 0.0020 –1 + 0.0014 –1 = 0.0034 –1. RP 500 700 This is 1RP , so we take the reciprocal to find RP . It is a common mistake to forget to take this reciprocal. Notice that the units of reciprocal ohms, –1, are a reminder. Thus 1 RP = = 290 . 0.0034 –1 This 290 is the equivalent resistance of the two parallel resistors, and is in series with the 400- resistor as shown in the equivalent circuit of Fig. 26–8b. To find the total equivalent resistance Req , we add the 400- and 290- resistances together, since they are in series, and find Req = 400 + 290 = 690 . The total current flowing from the battery is then V 12.0 V I = = = 0.0174 A L 17 mA. Req 690 NOTE This I is also the current flowing through the 400- resistor, but not through the 500- and 700- resistors (both currents are less—see the next Example). NOTE Complex resistor circuits can often be analyzed in this way, considering the circuit as a combination of series and parallel resistances.

FIGURE 26–8 (a) Circuit for Examples 26–4 and 26–5. (b) Equivalent circuit, showing the equivalent resistance of 290 for the two parallel resistors in (a). 500 Ω I1

a 400 Ω b

700 Ω I2

c I

I 12.0 V (a)

400 Ω

RP = 290 Ω

12.0 V (b) C A U T I O N

Remember to take the reciprocal

SECTION 26–2

681

500 Ω I1

a 400 Ω b

700 Ω I2

EXAMPLE 26–5 Current in one branch. What is the current through the 500- resistor in Fig. 26–8a?

c I

I 12.0 V (a)

400 Ω

RP = 290 Ω

12.0 V (b) FIGURE 26–8 (repeated) (a) Circuit for Examples 26–4 and 26–5. (b) Equivalent circuit, showing the equivalent resistance of 290 for the two parallel resistors in (a).

A C S

B FIGURE 26–9 Example 26–6, three identical lightbulbs. Each yellow circle with inside represents a lightbulb and its resistance.

APPROACH We need to find the voltage across the 500- resistor, which is the voltage between points b and c in Fig. 26–8a, and we call it Vbc . Once Vbc is known, we can apply Ohm's law, V = IR, to get the current. First we find the voltage across the 400- resistor, Vab , since we know that 17.4 mA passes through it (Example 26–4). SOLUTION Vab can be found using V = IR: Vab = (0.0174 A)(400 ) = 7.0 V. Since the total voltage across the network of resistors is Vac = 12.0 V, then Vbc must be 12.0 V - 7.0 V = 5.0 V. Then Ohm's law applied to the 500- resistor tells us that the current I1 through that resistor is 5.0 V I1 = = 1.0 * 10–2 A = 10 mA. 500 This is the answer we wanted. We can also calculate the current I2 through the 700- resistor since the voltage across it is also 5.0 V: 5.0 V I2 = = 7 mA. 700 NOTE When I1 combines with I2 to form the total current I (at point c in Fig. 26–8a), their sum is 10 mA + 7 mA = 17 mA. This equals the total current I as calculated in Example 26–4, as it should. CONCEPTUAL EXAMPLE 26–6 Bulb brightness in a circuit. The circuit shown in Fig. 26–9 has three identical lightbulbs, each of resistance R. (a) When switch S is closed, how will the brightness of bulbs A and B compare with that of bulb C? (b) What happens when switch S is opened? Use a minimum of mathematics in your answers. RESPONSE (a) With switch S closed, the current that passes through bulb C must split into two equal parts when it reaches the junction leading to bulbs A and B. It splits into equal parts because the resistance of bulb A equals that of B. Thus, bulbs A and B each receive half of C's current; A and B will be equally bright, but they will be less bright than bulb C (P = I 2R). (b) When the switch S is open, no current can flow through bulb A, so it will be dark. We now have a simple one-loop series circuit, and we expect bulbs B and C to be equally bright. However, the equivalent resistance of this circuit ( = R + R) is greater than that of the circuit with the switch closed. When we open the switch, we increase the resistance and reduce the current leaving the battery. Thus, bulb C will be dimmer when we open the switch. Bulb B gets more current when the switch is open (you may have to use some mathematics here), and so it will be brighter than with the switch closed; and B will be as bright as C. EXAMPLE 26–7 ESTIMATE A two-speed fan. One way a multiple-speed ventilation fan for a car can be designed is to put resistors in series with the fan motor. The resistors reduce the current through the motor and make it run more slowly. Suppose the current in the motor is 5.0 A when it is connected directly across a 12-V battery. (a) What series resistor should be used to reduce the current to 2.0 A for low-speed operation? (b) What power rating should the resistor have? APPROACH An electric motor in series with a resistor can be treated as two resistors in series. The power comes from P = IV. SOLUTION (a) When the motor is connected to 12 V and drawing 5.0 A, its resistance is R = VI = (12 V)(5.0 A) = 2.4 . We will assume that this is the motor's resistance for all speeds. (This is an approximation because the current through the motor depends on its speed.) Then, when a current of 2.0 A is flowing, the voltage across the motor is (2.0 A)(2.4 ) = 4.8 V. The remaining 12.0 V - 4.8 V = 7.2 V must appear across the series resistor. When 2.0 A flows through the resistor, its resistance must be R = (7.2 V)(2.0 A) = 3.6 . (b) The power dissipated by the resistor is P = (7.2 V)(2.0 A) = 14.4 W. To be safe, a power rating of 20 W would be appropriate.

682

CHAPTER 26

DC Circuits

10.0 Ω

EXAMPLE 26–8 Analyzing a circuit. A 9.0-V battery whose internal resistance r is 0.50 is connected in the circuit shown in Fig. 26–10a. (a) How much current is drawn from the battery? (b) What is the terminal voltage of the battery? (c) What is the current in the 6.0- resistor?

8.0 Ω 6.0 Ω

APPROACH To find the current out of the battery, we first need to determine the equivalent resistance Req of the entire circuit, including r, which we do by identifying and isolating simple series or parallel combinations of resistors. Once we find I from Ohm's law, I = eReq , we get the terminal voltage using Vab = e - Ir. For (c) we apply Ohm's law to the 6.0- resistor. SOLUTION (a) We want to determine the equivalent resistance of the circuit. But where do we start? We note that the 4.0- and 8.0- resistors are in parallel, and so have an equivalent resistance Req1 given by

4.0 Ω 5.0 Ω

r = 0.50 Ω (a)

= 9.0 V 10.0 Ω

1 1 1 3 = + = ; Req1 8.0 4.0 8.0 so Req1 = 2.7 . This 2.7 is in series with the 6.0- resistor, as shown in the equivalent circuit of Fig. 26–10b. The net resistance of the lower arm of the circuit is then

5.0 Ω

r = 0.50 Ω

Req2 = 6.0 + 2.7 = 8.7 , (b)

as shown in Fig. 26–10c. The equivalent resistance Req3 of the 8.7- and 10.0- resistances in parallel is given by

= 9.0 V 10.0 Ω

1 1 1 = + = 0.21 –1, Req3 10.0 8.7

Req2 = 8.7 Ω

so Req3 = A10.21 –1 B = 4.8 . This 4.8 is in series with the 5.0- resistor and the 0.50- internal resistance of the battery (Fig. 26–10d), so the total equivalent resistance Req of the circuit is Req = 4.8 + 5.0 + 0.50 = 10.3 . Hence the current drawn is e 9.0 V I = = = 0.87 A. Req 10.3

5.0 Ω

r = 0.50 Ω (c)

= 9.0 V

(b) The terminal voltage of the battery is

Req3 = 4.8 Ω

Vab = e - Ir = 9.0 V - (0.87 A)(0.50 ) = 8.6 V. (c) Now we can work back and get the current in the 6.0- resistor. It must be the same as the current through the 8.7 shown in Fig. 26–10c (why?). The voltage across that 8.7 will be the emf of the battery minus the voltage drops across r and the 5.0- resistor: V8.7 = 9.0 V - (0.87 A)(0.50 + 5.0 ). Applying Ohm's law, we get the current (call it I¿ ) 9.0 V - (0.87 A)(0.50 + 5.0 ) I¿ = = 0.48 A. 8.7 This is the current through the 6.0- resistor.

26–3

Req1 = 2.7 Ω

6.0 Ω

5.0 Ω

r = 0.50 Ω (d)

= 9.0 V

FIGURE 26–10 Circuit for Example 26–8, where r is the internal resistance of the battery.

Kirchhoff's Rules

In the last few Examples we have been able to find the currents in circuits by combining resistances in series and parallel, and using Ohm's law. This technique can be used for many circuits. However, some circuits are too complicated for that analysis. For example, we cannot find the currents in each part of the circuit shown in Fig. 26–11 simply by combining resistances as we did before. To deal with such complicated circuits, we use Kirchhoff's rules, devised by G. R. Kirchhoff (1824–1887) in the mid-nineteenth century. There are two rules, and they are simply convenient applications of the laws of conservation of charge and energy.

FIGURE 26–11 Currents can be calculated using Kirchhoff's rules. 30 Ω

I1 a

40 Ω

r= 2= 1 Ω 45 V

b 1= 80 V

I2 g

SECTION 26–3

d 20 Ω

r= 1Ω e

Kirchhoff's Rules

683

Kirchhoff's first rule or junction rule is based on the conservation of electric charge that we already used to derive the rule for parallel resistors. It states that Junction rule (conservation of charge)

at any junction point, the sum of all currents entering the junction must equal the sum of all currents leaving the junction. That is, whatever charge goes in must come out. We already saw an instance of this in the NOTE at the end of Example 26–5. Kirchhoff's second rule or loop rule is based on the conservation of energy. It states that

Loop rule (conservation of energy)

400 Ω

290 Ω

–

12.0 V d (a) a

7.0 V 12.0 V

5.0 V

12.0 V

400 Ω 290 Ω (b)

FIGURE 26–12 Changes in potential around the circuit in (a) are plotted in (b).

the sum of the changes in potential around any closed loop of a circuit must be zero. To see why this rule should hold, consider a rough analogy with the potential energy of a roller coaster on its track. When the roller coaster starts from the station, it has a particular potential energy. As it climbs the first hill, its potential energy increases and reaches a maximum at the top. As it descends the other side, its potential energy decreases and reaches a local minimum at the bottom of the hill. As the roller coaster continues on its path, its potential energy goes through more changes. But when it arrives back at the starting point, it has exactly as much potential energy as it had when it started at this point. Another way of saying this is that there was as much uphill as there was downhill. Similar reasoning can be applied to an electric circuit. We will analyze the circuit of Fig. 26–11 shortly but first we consider the simpler circuit in Fig. 26–12. We have chosen it to be the same as the equivalent circuit of Fig. 26–8b already discussed. The current in this circuit is I = (12.0 V)(690 ) = 0.0174 A, as we calculated in Example 26–4. (We keep an extra digit in I to reduce rounding errors.) The positive side of the battery, point e in Fig. 26–12a, is at a high potential compared to point d at the negative side of the battery. That is, point e is like the top of a hill for a roller coaster. We follow the current around the circuit starting at any point. We choose to start at point d and follow a positive test charge completely around this circuit. As we go, we note all changes in potential. When the test charge returns to point d, the potential will be the same as when we started (total change in potential around the circuit is zero). We plot the changes in potential around the circuit in Fig. 26–12b; point d is arbitrarily taken as zero. As our positive test charge goes from point d, which is the negative or low potential side of the battery, to point e, which is the positive terminal (high potential side) of the battery, the potential increases by 12.0 V. (This is like the roller coaster being pulled up the first hill.) That is, Ved = ±12.0 V.

P R O B L E M

S O LV I N G

Be consistent with signs when applying the loop rule

When our test charge moves from point e to point a, there is no change in potential since there is no source of emf and we assume negligible resistance in the connecting wires. Next, as the charge passes through the 400- resistor to get to point b, there is a decrease in potential of V = IR = (0.0174 A)(400 ) = 7.0 V. The positive test charge is flowing "downhill" since it is heading toward the negative terminal of the battery, as indicated in the graph of Fig. 26–12b. Because this is a decrease in potential, we use a negative sign: Vba = Vb - Va = –7.0 V. As the charge proceeds from b to c there is another potential decrease (a "voltage drop") of (0.0174 A) * (290 ) = 5.0 V, and this too is a decrease in potential: Vcb = –5.0 V. There is no change in potential as our test charge moves from c to d as we assume negligible resistance in the wires. The sum of all the changes in potential around the circuit of Fig. 26–12 is +12.0 V – 7.0 V - 5.0 V = 0.

684

CHAPTER 26

This is exactly what Kirchhoff's loop rule said it would be.

L SO VIN OBL E PR

Kirchhoff's Rules 1. Label the current in each separate branch of the given circuit with a different subscript, such as I1 , I2 , I3 (see Fig. 26–11 or 26–13). Each current refers to a segment between two junctions. Choose the direction of each current, using an arrow. The direction can be chosen arbitrarily: if the current is actually in the opposite direction, it will come out with a minus sign in the solution. 2. Identify the unknowns. You will need as many independent equations as there are unknowns. You may write down more equations than this, but you will find that some of the equations will be redundant (that is, not be independent in the sense of providing new information). You may use V = IR for each resistor, which sometimes will reduce the number of unknowns. 3. Apply Kirchhoff's junction rule at one or more junctions.

4. Apply Kirchhoff's loop rule for one or more loops: follow each loop in one direction only. Pay careful attention to subscripts, and to signs: (a) For a resistor, apply Ohm's law; the potential difference is negative (a decrease) if your chosen loop direction is the same as the chosen current direction through that resistor; the potential difference is positive (an increase) if your chosen loop direction is opposite to the chosen current direction. (b) For a battery, the potential difference is positive if your chosen loop direction is from the negative terminal toward the positive terminal; the potential difference is negative if the loop direction is from the positive terminal toward the negative terminal. 5. Solve the equations algebraically for the unknowns. Be careful when manipulating equations not to err with signs. At the end, check your answers by plugging them into the original equations, or even by using any additional loop or junction rule equations not used previously.

EXAMPLE 26–9 Using Kirchhoff's rules. Calculate the currents I1 , I2 , and I3 in the three branches of the circuit in Fig. 26–13 (which is the same as Fig. 26–11). APPROACH AND SOLUTION 1. Label the currents and their directions. Figure 26–13 uses the labels I1 , I2 , and I3 for the current in the three separate branches. Since (positive) current tends to move away from the positive terminal of a battery, we choose I2 and I3 to have the directions shown in Fig. 26–13. The direction of I1 is not obvious in advance, so we arbitrarily chose the direction indicated. If the current actually flows in the opposite direction, our answer will have a negative sign. 2. Identify the unknowns. We have three unknowns and therefore we need three equations, which we get by applying Kirchhoff's junction and loop rules. 3. Junction rule: We apply Kirchhoff's junction rule to the currents at point a, where I3 enters and I2 and I1 leave: (a) I3 = I1 + I2 . This same equation holds at point d, so we get no new information by writing an equation for point d. 4. Loop rule: We apply Kirchhoff's loop rule to two different closed loops. First we apply it to the upper loop ahdcba. We start (and end) at point a. From a to h we have a potential decrease Vha = – AI1 B(30 ). From h to d there is no change, but from d to c the potential increases by 45 V: that is, Vcd = ±45 V. From c to a the potential decreases through the two resistances by an amount Vac = – AI3 B(40 + 1 ) = –(41 )I3 . Thus we have Vha + Vcd + Vac = 0, or –30I1 + 45 - 41I3 = 0, (b) where we have omitted the units (volts and amps) so we can more easily do the algebra. For our second loop, we take the outer loop ahdefga. (We could have chosen the lower loop abcdefga instead.) Again we start at point a and have Vha = – AI1 B(30 ), and Vdh = 0. But when we take our positive test charge from d to e, it actually is going uphill, against the current—or at least against the assumed direction of the current, which is what counts in this calculation. Thus Ved = I2 (20 ) has a positive sign. Similarly, Vfe = I2 (1 ). From f to g there is a decrease in potential of 80 V since we go from the high potential terminal of the battery to the low. Thus Vgf = –80 V. Finally, Vag = 0, and the sum of the potential changes around this loop is –30I1 + (20 + 1)I2 - 80 = 0. (c) Our major work is done. The rest is algebra.

P R O B L E M

S O LV I N G

Choose current directions arbitrarily 30 Ω

I1 a

40 Ω

h r= 2= 1 Ω 45 V

b 1= 80 V

I2 g

d 20 Ω

r= 1Ω e

FIGURE 26–13 Currents can be calculated using Kirchhoff's rules. See Example 26–9.

SECTION 26–3

685

5. Solve the equations. We have three equations—labeled (a), (b), and (c)—and three unknowns. From Eq. (c) we have 80 + 30I1 (d) I2 = = 3.8 + 1.4I1 . 21 From Eq. (b) we have 45 - 30I1 (e) I3 = = 1.1 - 0.73I1 . 41 We substitute Eqs. (d) and (e) into Eq. (a): I1 = I3 - I2 = 1.1 - 0.73I1 - 3.8 - 1.4I1 . We solve for I1 , collecting terms: 3.1I1 = –2.7 I1 = –0.87 A. The negative sign indicates that the direction of I1 is actually opposite to that initially assumed and shown in Fig. 26–13. The answer automatically comes out in amperes because all values were in volts and ohms. From Eq. (d) we have I2 = 3.8 + 1.4I1 = 3.8 + 1.4(–0.87) = 2.6 A, and from Eq. (e) I3 = 1.1 - 0.73I1 = 1.1 - 0.73(–0.87) = 1.7 A. This completes the solution. NOTE The unknowns in different situations are not necessarily currents. It might be that the currents are given and we have to solve for unknown resistance or voltage. The variables are then different, but the technique is the same. EXERCISE C Write the equation for the lower loop abcdefga of Example 26–9 and show, assuming the currents calculated in this Example, that the potentials add to zero for this lower loop.

26–4 R I (a)

1.5 V

– +

1.5 V

– +

R I (b)

20 V

12 V c + –

– +

R 12 V – +

686

CHAPTER 26

DC Circuits

Series and Parallel EMFs; Battery Charging

When two or more sources of emf, such as batteries, are arranged in series as in Fig. 26–14a, the total voltage is the algebraic sum of their respective voltages. On the other hand, when a 20-V and a 12-V battery are connected oppositely, as shown in Fig. 26–14b, the net voltage Vca is 8 V (ignoring voltage drop across internal resistances). That is, a positive test charge moved from a to b gains in potential by 20 V, but when it passes from b to c it drops by 12 V. So the net change is 20 V - 12 V = 8 V. You might think that connecting batteries in reverse like this would be wasteful. For most purposes that would be true. But such a reverse arrangement is precisely how a battery charger works. In Fig. 26–14b, the 20-V source is charging up the 12-V battery. Because of its greater voltage, the 20-V source is forcing charge back into the 12-V battery: electrons are being forced into its negative terminal and removed from its positive terminal. An automobile alternator keeps the car battery charged in the same way. A voltmeter placed across the terminals of a (12-V) car battery with the engine running fairly fast can tell you whether or not the alternator is charging the battery. If it is, the voltmeter reads 13 or 14 V. If the battery is not being charged, the voltage will be 12 V, or less if the battery is discharging. Car batteries can be recharged, but other batteries may not be rechargeable, since the chemical reactions in many cannot be reversed. In such cases, the arrangement of Fig. 26–14b would simply waste energy. Sources of emf can also be arranged in parallel, Fig. 26–14c. With equal emfs, a parallel arrangement can provide more energy when large currents are needed. Each of the cells in parallel has to produce only a fraction of the total current, so the energy loss due to internal resistance is less than for a single cell; and the batteries will go dead less quickly.

EXAMPLE 26–10 Jump starting a car. A good car battery is being used to jump start a car with a weak battery. The good battery has an emf of 12.5 V and internal resistance 0.020 . Suppose the weak battery has an emf of 10.1 V and internal resistance 0.10 . Each copper jumper cable is 3.0 m long and 0.50 cm in diameter, and can be attached as shown in Fig. 26–15. Assume the starter motor can be represented as a resistor Rs = 0.15 . Determine the current through the starter motor (a) if only the weak battery is connected to it, and (b) if the good battery is also connected, as shown in Fig. 26–15. APPROACH We apply Kirchhoff's rules, but in (b) we will first need to determine the resistance of the jumper cables using their dimensions and the resistivity (r = 1.68 * 10–8 m for copper) as discussed in Section 25–4. SOLUTION (a) The circuit with only the weak battery and no jumper cables is simple: an emf of 10.1 V connected to two resistances in series, 0.10 + 0.15 = 0.25 . Hence the current is I = VR = (10.1 V)(0.25 ) = 40 A. (b) We need to find the resistance of the jumper cables that connect the good battery. From Eq. 25–3, each has resistance RJ = rlA = A1.68 * 10 –8 mB(3.0 m)(p)A0.25 * 10 –2 mB 2 = 0.0026 . Kirchhoff's loop rule for the full outside loop gives

r1 = 0.020 Ω

10.1 V - I3 (0.15 ) - I2 (0.10 ) = 0.

(b)

RJ Weak battery

Starter switch S (closed)

12.5 V - AI3 - I2 B(0.025 ) - I3 (0.15 ) = 0,

Rs = 0.15 Ω

Starter motor

(c)

We have three equations in three unknowns. From Eq. (c), I1 = I3 - I2 and we substitute this into Eq. (a):

B = 10.1 V

0.10 Ω

The junction rule at point B gives I1 + I2 = I3 .

Jumper cables

(a)

since A2RJ + rB = (0.0052 + 0.020 ) = 0.025 . The loop rule for the lower loop, including the weak battery and the starter, gives

= 12.5 V

12.5 V - I1 A2RJ + r1 B - I3 RS = 0

12.5 V - I1 (0.025 ) - I3 (0.15 ) = 0

Good battery

FIGURE 26–15 Example 26–10, a jump start.

12.5 V - I3 (0.175 ) + I2 (0.025 ) = 0.

Combining this last equation with (b) gives I3 = 71 A, quite a bit better than in (a). The other currents are I2 = –5 A and I1 = 76 A. Note that I2 = –5 A is in the opposite direction from that assumed in Fig. 26–15. The terminal voltage of the weak 10.1-V battery is now VBA = 10.1 V - (–5 A)(0.10 ) = 10.6 V. NOTE The circuit shown in Fig. 26–15, without the starter motor, is how a battery can be charged. The stronger battery pushes charge back into the weaker battery. EXERCISE D If the jumper cables of Example 26–10 were mistakenly connected in reverse, the positive terminal of each battery would be connected to the negative terminal of the other battery (Fig. 26–16). What would be the current I even before the starter motor is engaged (the switch S in Fig. 26–16 is open)? Why could this cause the batteries to explode?

26–5

Circuits Containing Resistor and Capacitor (RC Circuits)

FIGURE 26–16 Exercise D. 0.020 Ω

12.5 V

RJ I C S

DON'T TRY THIS

0.10 Ω

10.1 V

Rs Starter motor

Our study of circuits in this Chapter has, until now, dealt with steady currents that do not change in time. Now we examine circuits that contain both resistance and capacitance. Such a circuit is called an RC circuit. RC circuits are common in everyday life: they are used to control the speed of a car's windshield wiper, and the timing of the change of traffic lights. They are used in camera flashes, in heart pacemakers, and in many other electronic devices. In RC circuits, we are not so interested in the final "steady state" voltage and charge on the capacitor, but rather in how these variables change in time. SECTION 26–5

Circuits Containing Resistor and Capacitor (RC Circuits)

687

–

(a)

VC 0.63

t = 0 t = RC (b)

2RC Time

3RC

Current I

0 (c)

Current 0.37 /R RC

2RC Time

FIGURE 26–17 After the switch S closes in the RC circuit shown in (a), the voltage across the capacitor increases with time as shown in (b), and the current through the resistor decreases with time as shown in (c).

Let us now examine the RC circuit shown in Fig. 26–17a. When the switch S is closed, current immediately begins to flow through the circuit. Electrons will flow out from the negative terminal of the battery, through the resistor R, and accumulate on the upper plate of the capacitor. And electrons will flow into the positive terminal of the battery, leaving a positive charge on the other plate of the capacitor. As charge accumulates on the capacitor, the potential difference across it increases AVC = QCB, and the current is reduced until eventually the voltage across the capacitor equals the emf of the battery, e. There is then no potential difference across the resistor, and no further current flows. Potential difference VC across the capacitor thus increases in time as shown in Fig. 26–17b. The mathematical form of this curve—that is, VC as a function of time—can be derived using conservation of energy (or Kirchhoff's loop rule). The emf e of the battery will equal the sum of the voltage drops across the resistor (IR) and the capacitor (QC): Q. (26–5) e = IR + C The resistance R includes all resistance in the circuit, including the internal resistance of the battery; I is the current in the circuit at any instant, and Q is the charge on the capacitor at that same instant. Although e, R, and C are constants, both Q and I are functions of time. The rate at which charge flows through the resistor (I = dQdt) is equal to the rate at which charge accumulates on the capacitor. Thus we can write dQ 1 e = R + Q. dt C This equation can be solved by rearranging it: dQ dt . = Ce - Q RC We now integrate from t = 0, when Q = 0, to time t when a charge Q is on the capacitor: Q t dQ 1 = dt RC 0 0 Ce - Q

–ln(Ce - Q) ` = –ln(Ce - Q) - (–ln Ce) =

t 0

RC t ln(Ce - Q) - ln(Ce) = – RC

Q t . ≤ = – Ce RC We take the exponential† of both sides Q 1 = e –tRC Ce or Q = CeA1 - e –tRC B. ln ¢ 1 -

(26–6a)

The potential difference across the capacitor is VC = QC, so

(26–6b) VC = eA1 - e –tRC B. From Eqs. 26–6 we see that the charge Q on the capacitor, and the voltage VC across it, increase from zero at t = 0 to maximum values Qmax = Ce and VC = e after a very long time. The quantity RC that appears in the exponent is called the time constant t of the circuit: (26–7) t = RC. †† –1 It represents the time required for the capacitor to reach A1 - e B = 0.63 or 63% of its full charge and voltage. Thus the product RC is a measure of how quickly the †

688

CHAPTER 26

The constant e, known as the base for natural logarithms, has the value e = 2.718 p . Do not confuse this e with e for the charge on the electron. † The units of RC are F = (VA)(CV) = C(Cs) = s.

capacitor gets charged. In a circuit, for example, where R = 200 k and C = 3.0 mF, the time constant is A2.0 * 105 BA3.0 * 10–6 FB = 0.60 s. If the resistance is much lower, the time constant is much smaller. This makes sense, since a lower resistance will retard the flow of charge less. All circuits contain some resistance (if only in the connecting wires), so a capacitor never can be charged instantaneously when connected to a battery. From Eqs. 26–6, it appears that Q and VC never quite reach their maximum values within a finite time. However, they reach 86% of maximum in 2RC, 95% in 3RC, 98% in 4RC, and so on. Q and VC approach their maximum values asymptotically. For example, if R = 20 k and C = 0.30 mF, the time constant is A2.0 * 104 B A3.0 * 10–7 FB = 6.0 * 10 –3 s. So the capacitor is more than 98% charged in less than 401 of a second. The current I through the circuit of Fig. 26–17a at any time t can be obtained by differentiating Eq. 26–6a: dQ e –tRC I = = e . (26–8) dt R Thus, at t = 0, the current is I = eR, as expected for a circuit containing only a resistor (there is not yet a potential difference across the capacitor). The current then drops exponentially in time with a time constant equal to RC, as the voltage across the capacitor increases. This is shown in Fig. 26–17c. The time constant RC represents the time required for the current to drop to 1e L 0.37 of its initial value. EXAMPLE 26–11 RC circuit, with emf. The capacitance in the circuit of Fig. 26–17a is C = 0.30 mF, the total resistance is 20 k, and the battery emf is 12 V. Determine (a) the time constant, (b) the maximum charge the capacitor could acquire, (c) the time it takes for the charge to reach 99% of this value, (d) the current I when the charge Q is half its maximum value, (e) the maximum current, and (f) the charge Q when the current I is 0.20 its maximum value. APPROACH We use Fig. 26–17 and Eqs. 26–5, 6, 7, and 8. SOLUTION (a) The time constant is RC = A2.0 * 104 BA3.0 * 10 –7 FB = 6.0 * 10 –3 s. (b) The maximum charge would be Q = Ce = A3.0 * 10–7 FB(12 V) = 3.6 mC. (c) In Eq. 26–6a, we set Q = 0.99Ce: 0.99Ce = CeA1 - e –tRC B,

e –tRC = 1 - 0.99 = 0.01.

Then

t RC

= – ln(0.01) = 4.6

t = 4.6RC = 28 * 10–3 s or 28 ms (less than 301 s). (d) From part (b) the maximum charge is 3.6 mC. When the charge is half this value, 1.8 mC, the current I in the circuit can be found using the original differential equation, or Eq. 26–5: I =

Q 1 1 1.8 * 10–6 C ¢e ≤ = ¢ 12 V ≤ = 300 mA. 4 R C 2.0 * 10 0.30 * 10 –6 F

(e) The current is a maximum when there is no charge on the capacitor (Q = 0): e 12 V Imax = = = 600 mA. R 2.0 * 104 (f) Again using Eq. 26–5, now with I = 0.20Imax = 120 mA, we have Q = C(e - IR) = A3.0 * 10–7 FB C12 V - A1.2 * 10 –4 ABA2.0 * 104 B D = 2.9 mC. SECTION 26–5

Circuits Containing Resistor and Capacitor (RC Circuits)

689

– V + 0

The circuit just discussed involved the charging of a capacitor by a battery through a resistance. Now let us look at another situation: when a capacitor is already charged (say to a voltage V0), and it is then allowed to discharge through a resistance R as shown in Fig. 26–18a. (In this case there is no battery.) When the switch S is closed, charge begins to flow through resistor R from one side of the capacitor toward the other side, until the capacitor is fully discharged. The voltage across the resistor at any instant equals that across the capacitor:

S (a)

IR =

VC V0

The rate at which charge leaves the capacitor equals the negative of the current in the resistor, I = –dQdt, because the capacitor is discharging (Q is decreasing). So we write the above equation as

0.37V0 0

Q. C

t = RC

2RC Time (b)

3RC

–

dQ Q. R = dt C

We rearrange this to

FIGURE 26–18 For the RC circuit shown in (a), the voltage VC across the capacitor decreases with time, as shown in (b), after the switch S is closed at t = 0. The charge on the capacitor follows the same curve since VC r Q.

dQ dt = – Q RC and integrate it from t = 0 when the charge on the capacitor is Q0 , to some time t later when the charge is Q: ln or

Q t = – Q0 RC

Q = Q0 e –tRC.

(26–9a)

The voltage across the capacitor (VC = QC) as a function of time is VC = V0 e –tRC,

(26–9b)

where the initial voltage V0 = Q0C. Thus the charge on the capacitor, and the voltage across it, decrease exponentially in time with a time constant RC. This is shown in Fig. 26–18b. The current is I = –

Q0 –tRC dQ = e = I0 e –tRC, dt RC

(26–10)

and it too is seen to decrease exponentially in time with the same time constant RC. The charge on the capacitor, the voltage across it, and the current in the resistor all decrease to 37% of their original value in one time constant t = t = RC. EXERCISE E In 10 time constants, the charge on the capacitor in Fig. 26–18 will be about (a) Q020,000, (b) Q05000, (c) Q01000, (d) Q010, (e) Q03?

FIGURE 26–19 Example 26–12. a =+ 20.0 V –

R C= 1.02 mF

EXAMPLE 26–12 Discharging RC circuit. In the RC circuit shown in Fig. 26–19, the battery has fully charged the capacitor, so Q0 = Ce. Then at t = 0 the switch is thrown from position a to b. The battery emf is 20.0 V, and the capacitance C = 1.02 mF. The current I is observed to decrease to 0.50 of its initial value in 40 ms. (a) What is the value of Q, the charge on the capacitor, at t = 0? (b) What is the value of R? (c) What is Q at t = 60 ms? APPROACH At t = 0, the capacitor has charge Q0 = Ce, and then the battery is removed from the circuit and the capacitor begins discharging through the resistor, as in Fig. 26–18. At any time t later (Eq. 26–9a) we have Q = Q0 e –tRC = Cee –tRC.

690

CHAPTER 26

DC Circuits

SOLUTION (a) At t = 0, Q = Q0 = Ce = A1.02 * 10–6 FB(20.0 V) = 2.04 * 10 –5 C = 20.4 mC. (b) To find R, we are given that at t = 40 ms, I = 0.50I0 . Hence 0.50I0 = I0 e –tRC. Taking natural logs on both sides (ln 0.50 = –0.693): 0.693 =

t RC

so R =

t (0.693)C

A40 * 10–6 sB

(0.693)A1.02 * 10 –6 FB

= 57 .

Q = Q0 e

–6

= (20.4 * 10 C)e

–

60 * 10 – 6 s (57 )A1.02 * 10 – 6 FB

= 7.3 mC. S

CONCEPTUAL EXAMPLE 26–13 Bulb in RC circuit. In the circuit of Fig. 26–20, the capacitor is originally uncharged. Describe the behavior of the lightbulb from the instant switch S is closed until a long time later. RESPONSE When the switch is first closed, the current in the circuit is high and the lightbulb burns brightly. As the capacitor charges, the voltage across the capacitor increases causing the current to be reduced, and the lightbulb dims. As the potential difference across the capacitor approaches the same voltage as the battery, the current decreases toward zero and the lightbulb goes out.

FIGURE 26–20 Example 26–13.

* Applications of RC Circuits The charging and discharging in an RC circuit can be used to produce voltage pulses at a regular frequency. The charge on the capacitor increases to a particular voltage, and then discharges. One way of initiating the discharge of the capacitor is by the use of a gas-filled tube which has an electrical breakdown when the voltage across it reaches a certain value V0 . After the discharge is finished, the tube no longer conducts current and the recharging process repeats itself, starting at a lower voltage V0œ . Figure 26–21 shows a possible circuit, and the "sawtooth" voltage it produces. A simple blinking light can be an application of a sawtooth oscillator circuit. Here the emf is supplied by a battery; the neon bulb flashes on at a rate of perhaps 1 cycle per second. The main component of a "flasher unit" is a moderately large capacitor. The intermittent windshield wipers of a car can also use an RC circuit. The RC time constant, which can be changed using a multi-positioned switch for different values of R with fixed C, determines the rate at which the wipers come on.

P H Y S I C S

FIGURE 26–21 (a) An RC circuit, coupled with a gas-filled tube as a switch, can produce a repeating "sawtooth" voltage, as shown in (b).

R + –

R =

t 0.5 s = L 500 k. C 1 * 10–6 F SECTION 26–5

R' C Gas-filled tube (a)

EXAMPLE 26–14 ESTIMATE Resistor in a turn signal. Estimate the order of magnitude of the resistor in a turn-signal circuit. APPROACH A typical turn signal flashes perhaps twice per second, so the time constant is on the order of 0.5 s. A moderate capacitor might have C = 1 mF. SOLUTION Setting t = RC = 0.5 s, we find

A P P L I E D

Sawtooth, blinkers, windshield wipers

V0 V0' 0

Time (b)

Circuits Containing Resistor and Capacitor (RC Circuits)

691

P H Y S I C S

A P P L I E D

Heart pacemaker

FIGURE 26–22 Electronic batterypowered pacemaker can be seen on the rib cage in this X-ray.

P H Y S I C S

A P P L I E D

Dangers of electricity

FIGURE 26–23 A person receives an electric shock when the circuit is completed.

I 120 V

(a)

An interesting medical use of an RC circuit is the electronic heart pacemaker, which can make a stopped heart start beating again by applying an electric stimulus through electrodes attached to the chest. The stimulus can be repeated at the normal heartbeat rate if necessary. The heart itself contains pacemaker cells, which send out tiny electric pulses at a rate of 60 to 80 per minute. These signals induce the start of each heartbeat. In some forms of heart disease, the natural pacemaker fails to function properly, and the heart loses its beat. Such patients use electronic pacemakers which produce a regular voltage pulse that starts and controls the frequency of the heartbeat. The electrodes are implanted in or near the heart (Fig. 26–22), and the circuit contains a capacitor and a resistor. The charge on the capacitor increases to a certain point and then discharges a pulse to the heart. Then it starts charging again. The pulsing rate depends on the values of R and C.

26–6

Electric Hazards

Excess electric current can heat wires in buildings and cause fires, as discussed in Section 25–6. Electric current can also damage the human body or even be fatal. Electric current through the human body can cause damage in two ways: (1) Electric current heats tissue and can cause burns; (2) electric current stimulates nerves and muscles, and we feel a "shock." The severity of a shock depends on the magnitude of the current, how long it acts, and through what part of the body it passes. A current passing through vital organs such as the heart or brain is especially serious for it can interfere with their operation. Most people can "feel" a current of about 1 mA. Currents of a few mA cause pain but rarely cause much damage in a healthy person. Currents above 10 mA cause severe contraction of the muscles, and a person may not be able to let go of the source of the current (say, a faulty appliance or wire). Death from paralysis of the respiratory system can occur. Artificial respiration, however, can sometimes revive a victim. If a current above about 80 to 100 mA passes across the torso, so that a portion passes through the heart for more than a second or two, the heart muscles will begin to contract irregularly and blood will not be properly pumped. This condition is called ventricular fibrillation. If it lasts for long, death results. Strangely enough, if the current is much larger, on the order of 1 A, death by heart failure may be less likely,† but such currents can cause serious burns, especially if concentrated through a small area of the body. The seriousness of a shock depends on the applied voltage and on the effective resistance of the body. Living tissue has low resistance since the fluid of cells contains ions that can conduct quite well. However, the outer layer of skin, when dry, offers high resistance and is thus protective. The effective resistance between two points on opposite sides of the body when the skin is dry is in the range of 104 to 106 . But when the skin is wet, the resistance may be 103 or less. A person who is barefoot or wearing thin-soled shoes will be in good contact with the ground, and touching a 120-V line with a wet hand can result in a current I =

"Yikes!"

120 V = 120 mA. 1000

As we saw, this could be lethal. A person who has received a shock has become part of a complete circuit. Figure 26–23 shows two ways the circuit might be completed when a person †

Larger currents apparently bring the entire heart to a standstill. Upon release of the current, the heart returns to its normal rhythm. This may not happen when fibrillation occurs because, once started, it can be hard to stop. Fibrillation may also occur as a result of a heart attack or during heart surgery. A device known as a defibrillator (described in Section 24–4) can apply a brief high current to the heart, causing complete heart stoppage which is often followed by resumption of normal beating.

(b)

692

CHAPTER 26

DC Circuits

Current

I I

120 V

Current

I (a)

(c)

(b)

FIGURE 26–24 (a) An electric oven operating normally with a 2-prong plug. (b) Short to the case with ungrounded case: shock. (c) Short to the case with the case grounded by a 3-prong plug.

accidentally touches a "hot" electric wire—"hot" meaning a high potential such as 120 V (normal U.S. household voltage) relative to ground. The other wire of building wiring is connected to ground—either by a wire connected to a buried conductor, or via a metal water pipe into the ground. In Fig. 26–23a, the current passes from the high-voltage wire, through the person, to the ground through his bare feet, and back along the ground (a fair conductor) to the ground terminal of the source. If the person stands on a good insulator—thick rubber-soled shoes or a dry wood floor—there will be much more resistance in the circuit and consequently much less current through the person. If the person stands with bare feet on the ground, or is in a bathtub, there is lethal danger because the resistance is much less and the current greater. In a bathtub (or swimming pool), not only are you wet, which reduces your resistance, but the water is in contact with the drain pipe (typically metal) that leads to the ground. It is strongly recommended that you not touch anything electrical when wet or in bare feet. Building codes that require the use of non-metal pipes would be protective. In Fig. 26–23b, a person touches a faulty "hot" wire with one hand, and the other hand touches a sink faucet (connected to ground via the pipe). The current is particularly dangerous because it passes across the chest, through the heart and lungs. A useful rule: if one hand is touching something electrical, keep your other hand in your pocket (don't use it!), and wear thick rubber-soled shoes. It is also a good idea to remove metal jewelry, especially rings (your finger is usually moist under a ring). You can come into contact with a hot wire by touching a bare wire whose insulation has worn off, or from a bare wire inside an appliance when you're tinkering with it. (Always unplug an electrical device before investigating † its insides!) Another possibility is that a wire inside a device may break or lose its insulation and come in contact with the case. If the case is metal, it will conduct electricity. A person could then suffer a severe shock merely by touching the case, as shown in Fig. 26–24b. To prevent an accident, metal cases are supposed to be connected directly to ground by a separate ground wire. Then if a "hot" wire touches the grounded case, a short circuit to ground immediately occurs internally, as shown in Fig. 26–24c, and most of the current passes through the low-resistance ground wire rather than through the person. Furthermore, the high current should open the fuse or circuit breaker. Grounding a metal case is done by a separate ground wire connected to the third (round) prong of a 3-prong plug. Never cut off the third prong of a plug—it could save your life.

C A U T I O N

Keep one hand in your pocket when other touches electricity

P H Y S I C S

A P P L I E D

Grounding and shocks

† Even then you can get a bad shock from a capacitor that hasn't been discharged until you touch it.

SECTION 26–6

Electric Hazards

693

(a)

(b)

(c) FIGURE 26–25 (a) A 3-prong plug, and (b) an adapter (gray) for old-fashioned 2-prong outlets— be sure to screw down the ground tab. (c) A polarized 2-prong plug. FIGURE 26–26 Four wires entering a typical house. The color codes for wires are not always as shown here—be careful! 120 V From electric company 120 V

Black (or red)

120 V

120 Vrms 240 Vrms

White

120 Vrms

Black

Ground at electric pole

694

Hot

CHAPTER 26

Neutral

Hot 120 V Green 0 Ground at house

DC Circuits

A three-prong plug, and an adapter, are shown in Figs. 26–25a and b. Why is a third wire needed? The 120 V is carried by the other two wires—one hot (120 V ac), the other neutral, which is itself grounded. The third "dedicated" ground wire with the round prong may seem redundant. But it is protection for two reasons: (1) it protects against internal wiring that may have been done incorrectly; (2) the neutral wire carries normal current ("return" current from the 120 V) and it does have resistance; so there can be a voltage drop along it—normally small, but if connections are poor or corroded, or the plug is loose, the resistance could be large enough that you might feel that voltage if you touched the neutral wire some distance from its grounding point. Some electrical devices come with only two wires, and the plug's two prongs are of different widths; the plug can be inserted only one way into the outlet so that the intended neutral (wider prong) in the device is connected to neutral in the wiring (Fig. 26–25c). For example, the screw threads of a lightbulb are meant to be connected to neutral (and the base contact to hot), to avoid shocks when changing a bulb in a possibly protruding socket. Devices with 2-prong plugs do not have their cases grounded; they are supposed to have double electric insulation. Take extra care anyway. The insulation on a wire may be color coded. Hand-held meters may have red (hot) and black (ground) lead wires. But in a house, black is usually hot (or it may be red), whereas white is neutral and green is the dedicated ground, Fig. 26–26. But beware: these color codes cannot always be trusted. [In the U.S., three wires normally enter a house: two hot wires at 120 V each (which add together to 240 V for appliances or devices that run on 240 V) plus the grounded neutral (carrying return current for the two hots). See Fig. 26–26. The "dedicated" ground wire (non-current carrying) is a fourth wire that does not come from the electric company but enters the house from a nearby heavy stake in the ground or a buried metal pipe. The two hot wires can feed separate 120-V circuits in the house, so each 120-V circuit inside the house has only three wires, including ground.] Normal circuit breakers (Sections 25–6 and 28–8) protect equipment and buildings from overload and fires. They protect humans only in some circumstances, such as the very high currents that result from a short, if they respond quickly enough. Ground fault circuit interrupters (GFCI), described in Section 29–8, are designed to protect people from the much lower currents (10 mA to 100 mA) that are lethal but would not throw a 15-A circuit breaker or blow a 20-A fuse. It is current that harms, but it is voltage that drives the current. 30 volts is sometimes said to be the threshhold for danger. But even a 12-V car battery (which can supply large currents) can cause nasty burns and shock. Another danger is leakage current, by which we mean a current along an unintended path. Leakage currents are often "capacitively coupled." For example, a wire in a lamp forms a capacitor with the metal case; charges moving in one conductor attract or repel charge in the other, so there is a current. Typical electrical codes limit leakage currents to 1 mA for any device. A 1-mA leakage current is usually harmless. It can be very dangerous, however, to a hospital patient with implanted electrodes connected to ground through the apparatus. This is due to the absence of the protective skin layer and because the current can pass directly through the heart as compared to the usual situation where the current enters at the hands and spreads out through the body. Although 100 mA may be needed to cause heart fibrillation when entering through the hands (very little of it actually passes through the heart), as little as 0.02 mA has been known to cause fibrillation when passing directly to the heart. Thus, a "wired" patient is in considerable danger from leakage current even from as simple an act as touching a lamp. Finally, don't touch a downed power line (lethal!) or even get near it. A hot power line is at thousands of volts. A huge current can flow along the ground or pavement, from where the high-voltage wire touches the ground along its path to the grounding point of the neutral line, enough that the voltage between your two feet could be large. Tip: stand on one foot or run (only one foot touching the ground at a time).

* 26–7

Ammeters and Voltmeters

An ammeter is used to measure current, and a voltmeter measures potential difference or voltage. Measurements of current and voltage are made with meters that are of two types: (1) analog meters, which display numerical values by the position of a pointer that can move across a scale (Fig. 26–27a); and (2) digital meters, which display the numerical value in numbers (Fig. 26–27b). We now discuss the meters themselves and how they work, then how they are connected to circuits to make measurements. Finally we will discuss how using meters affects the circuit being measured, possibly causing erroneous results—and what to do about it.

P H Y S I C S

A P P L I E D

DC meters

* Analog Ammeters and Voltmeters The crucial part of an analog ammeter or voltmeter, in which the reading is by a pointer on a scale (Fig. 26–27a), is a galvanometer. The galvanometer works on the principle of the force between a magnetic field and a current-carrying coil of wire, and will be discussed in Chapter 27. For now, we merely need to know that the deflection of the needle of a galvanometer is proportional to the current flowing through it. The full-scale current sensitivity of a galvanometer, Im , is the electric current needed to make the needle deflect full scale. A galvanometer can be used directly to measure small dc currents. For example, a galvanometer whose sensitivity Im is 50 mA can measure currents from about 1 mA (currents smaller than this would be hard to read on the scale) up to 50 mA. To measure larger currents, a resistor is placed in parallel with the galvanometer. Thus, an analog ammeter, represented by the symbol A , consists of a galvanometer ( G ) in parallel with a resistor called the shunt resistor, as shown in Fig. 26–28. ("Shunt" is a synonym for "in parallel.") The shunt resistance is Rsh , and the resistance of the galvanometer coil, through which current passes, is r. The value of Rsh is chosen according to the full-scale deflection desired; Rsh is normally very small—giving an ammeter a very small net resistance—so most of the current passes through Rsh and very little (f 50 mA) passes through the galvanometer to deflect the needle. r I

Ammeter A

IG IR Rsh

(a)

(b) FIGURE 26–27 (a) An analog multimeter being used as a voltmeter. (b) An electronic digital meter.

FIGURE 26–28 An ammeter is a galvanometer in parallel with a (shunt) resistor with low resistance, Rsh .

EXAMPLE 26–15 Ammeter design. Design an ammeter to read 1.0 A at full scale using a galvanometer with a full-scale sensitivity of 50 mA and a resistance r = 30 . Check if the scale is linear.

APPROACH Only 50 mA A = IG = 0.000050 AB of the 1.0-A current must pass through the galvanometer to give full-scale deflection. The rest of the current AIR = 0.999950 AB passes through the small shunt resistor, Rsh , Fig. 26–28. The potential difference across the galvanometer equals that across the shunt resistor (they are in parallel). We apply Ohm's law to find Rsh . SOLUTION Because I = IG + IR , when I = 1.0 A flows into the meter, we want IR through the shunt resistor to be IR = 0.999950 A. The potential difference across the shunt is the same as across the galvanometer, so Ohm's law tells us IR Rsh = IG r; then A5.0 * 10–5 AB(30 ) IG r Rsh = = = 1.5 * 10 –3 , (0.999950 A) IR or 0.0015 . The shunt resistor must thus have a very low resistance and most of the current passes through it. Because IG = IR ARshrB and ARshrB is constant, we see that the scale is linear. *SECTION 26–7

Ammeters and Voltmeters

695

An analog voltmeter ( V ) also consists of a galvanometer and a resistor. But the resistor Rser is connected in series, Fig. 26–29, and it is usually large, giving a voltmeter a high internal resistance. FIGURE 26–29 A voltmeter is a galvanometer in series with a resistor with high resistance, Rser .

Voltmeter V

Rser

EXAMPLE 26–16 Voltmeter design. Using a galvanometer with internal resistance r = 30 and full-scale current sensitivity of 50 mA, design a voltmeter that reads from 0 to 15 V. Is the scale linear? APPROACH When a potential difference of 15 V exists across the terminals of our voltmeter, we want 50 mA to be passing through it so as to give a full-scale deflection. SOLUTION From Ohm's law, V = IR, we have (see Fig. 26–29) so

15 V = (50 mA)Ar + Rser B,

15 V - r = 300 k - 30 = 300 k. 5.0 * 10–5 A Notice that r = 30 is so small compared to the value of Rser that it doesn't influence the calculation significantly. The scale will again be linear: if the voltage to be measured is 6.0 V, the current passing through the voltmeter will be (6.0 V)A3.0 * 105 B = 2.0 * 10 –5 A, or 20 mA. This will produce two-fifths of full-scale deflection, as required (6.0 V15.0 V = 25). Rser =

Rser

Rsh

V R to be measured FIGURE 26–30 An ohmmeter.

FIGURE 26–31 Measuring current and voltage. a

R2 V (a)

R1 A

* How to Connect Meters

V (b)

V a

R2 V (c)

696

CHAPTER 26

The meters just described are for direct current. A dc meter can be modified to measure ac (alternating current, Section 25–7) with the addition of diodes (Chapter 40), which allow current to flow in one direction only. An ac meter can be calibrated to read rms or peak values. Voltmeters and ammeters can have several series or shunt resistors to offer a choice of range. Multimeters can measure voltage, current, and resistance. Sometimes a multimeter is called a VOM (Volt-Ohm-Meter or Volt-Ohm-Milliammeter). An ohmmeter measures resistance, and must contain a battery of known voltage connected in series to a resistor ARser B and to an ammeter (Fig. 26–30). The resistor whose resistance is to be measured completes the circuit. The needle deflection is inversely proportional to the resistance. The scale calibration depends on the value of the series resistor. Because an ohmmeter sends a current through the device whose resistance is to be measured, it should not be used on very delicate devices that could be damaged by the current. The sensitivity of a meter is generally specified on the face. It may be given as so many ohms per volt, which indicates how many ohms of resistance there are in the meter per volt of full-scale reading. For example, if the sensitivity is 30,000 V, this means that on the 10-V scale the meter has a resistance of 300,000 , whereas on a 100-V scale the meter resistance is 3 M. The full-scale current sensitivity, Im , discussed earlier, is just the reciprocal of the sensitivity in V.

DC Circuits

Suppose you wish to determine the current I in the circuit shown in Fig. 26–31a, and the voltage V across the resistor R1 . How exactly are ammeters and voltmeters connected to the circuit being measured? Because an ammeter is used to measure the current flowing in the circuit, it must be inserted directly into the circuit, in series with the other elements, as shown in Fig. 26–31b. The smaller its internal resistance, the less it affects the circuit. A voltmeter, on the other hand, is connected "externally," in parallel with the circuit element across which the voltage is to be measured. It is used to measure the potential difference between two points. Its two wire leads (connecting wires) are connected to the two points, as shown in Fig. 26–31c, where the voltage across R1 is being measured. The larger its internal resistance, ARser + rB in Fig. 26–29, the less it affects the circuit being measured.

* Effects of Meter Resistance It is important to know the sensitivity of a meter, for in many cases the resistance of the meter can seriously affect your results. Take the following Example. EXAMPLE 26–17 Voltage reading versus true voltage. Suppose you are testing an electronic circuit which has two resistors, R1 and R2 , each 15 k, connected in series as shown in Fig. 26–32a. The battery maintains 8.0 V across them and has negligible internal resistance. A voltmeter whose sensitivity is 10,000 V is put on the 5.0-V scale. What voltage does the meter read when connected across R1 , Fig. 26–32b, and what error is caused by the finite resistance of the meter? APPROACH The meter acts as a resistor in parallel with R1 . We use parallel and series resistor analyses and Ohm's law to find currents and voltages. SOLUTION On the 5.0-V scale, the voltmeter has an internal resistance of (5.0 V)(10,000 V) = 50,000 . When connected across R1 , as in Fig. 26–32b, we have this 50 k in parallel with R1 = 15 k. The net resistance Req of these two is given by 1 1 1 13 = + = ; Req 50 k 15 k 150 k so Req = 11.5 k. This Req = 11.5 k is in series with R2 = 15 k, so the total resistance of the circuit is now 26.5 k (instead of the original 30 k). Hence the current from the battery is 8.0 V I = = 3.0 * 10 –4 A = 0.30 mA. 26.5 k Then the voltage drop across R1 , which is the same as that across the voltmeter, is A3.0 * 10 –4 ABA11.5 * 103 B = 3.5 V. [The voltage drop across R2 is A3.0 * 10 –4 AB A15 * 103 B = 4.5 V, for a total of 8.0 V.] If we assume the meter is precise, it will read 3.5 V. In the original circuit, without the meter, R1 = R2 so the voltage across R1 is half that of the battery, or 4.0 V. Thus the voltmeter, because of its internal resistance, gives a low reading. In this case it is off by 0.5 V, or more than 10%.

P H Y S I C S

A P P L I E D

Correcting for meter resistance a

V (a) V a

V (b) FIGURE 26–32 Example 26–17.

Example 26–17 illustrates how seriously a meter can affect a circuit and give a misleading reading. If the resistance of a voltmeter is much higher than the resistance of the circuit, however, it will have little effect and its readings can be trusted, at least to the manufactured precision of the meter, which for ordinary analog meters is typically 3% to 4% of full-scale deflection. An ammeter also can interfere with a circuit, but the effect is minimal if its resistance is much less than that of the circuit as a whole. For both voltmeters and ammeters, the more sensitive the galvanometer, the less effect it will have. A 50,000-V meter is far better than a 1000-V meter.

* Digital Meters Digital meters (see Fig. 26–27b) are used in the same way as analog meters: they are inserted directly into the circuit, in series, to measure current (Fig. 26–31b), and connected "outside," in parallel with the circuit, to measure voltage (Fig. 26–31c). The internal construction of digital meters, however, is different from that of analog meters in that digital meters do not use a galvanometer. The electronic circuitry and digital readout are more sensitive than a galvanometer, and have less effect on the circuit to be measured. When we measure dc voltages, a digital meter's resistance is very high, commonly on the order of 10 to 100 M A107 –108 B, and doesn't change significantly when different voltage scales are selected. A 100-M digital meter draws off very little current when connected across even a 1-M resistance. The precision of digital meters is exceptional, often one part in 104 ( = 0.01%) or better. This precision is not the same as accuracy, however. A precise meter of internal resistance 108 will not give accurate results if used to measure a voltage across a 108- resistor—in which case it is necessary to do a calculation like that in Example 26–17. Whenever we make a measurement on a circuit, to some degree we affect that circuit (Example 26–17). This is true for other types of measurement as well: when we make a measurement on a system, we affect that system in some way. On a temperature measurement, for example, the thermometer can exchange heat with the system, thus altering its temperature. It is important to be able to make needed corrections, as we saw in Example 26–17.

*SECTION 26–7

697

Summary A device that transforms another type of energy into electrical energy is called a source of emf. A battery behaves like a source of emf in series with an internal resistance. The emf is the potential difference determined by the chemical reactions in the battery and equals the terminal voltage when no current is drawn. When a current is drawn, the voltage at the battery's terminals is less than its emf by an amount equal to the potential decrease Ir across the internal resistance. When resistances are connected in series (end to end in a single linear path), the equivalent resistance is the sum of the individual resistances: Req = R1 + R2 + p .

(26–3)

In a series combination, Req is greater than any component resistance. When resistors are connected in parallel, the reciprocal of the equivalent resistance equals the sum of the reciprocals of the individual resistances: 1 1 1 = + + p. Req R1 R2

(26–4)

In a parallel connection, the net resistance is less than any of the individual resistances. Kirchhoff's rules are helpful in determining the currents and voltages in circuits. Kirchhoff's junction rule is based on conservation of electric charge and states that the sum of all currents entering any junction equals the sum of all currents leaving that junction. The second, or loop rule, is based on conservation of energy and states that the algebraic sum of the

changes in potential around any closed path of the circuit must be zero. When an RC circuit containing a resistor R in series with a capacitance C is connected to a dc source of emf, the voltage across the capacitor rises gradually in time characterized by an exponential of the form A1 - e –tRC B, where the time constant, (26–7)

t = RC,

is the time it takes for the voltage to reach 63 percent of its maximum value. The current through the resistor decreases as e –tRC. A capacitor discharging through a resistor is characterized by the same time constant: in a time t = RC, the voltage across the capacitor drops to 37 percent of its initial value. The charge on the capacitor, and voltage across it, decreases as e –tRC, as does the current. Electric shocks are caused by current passing through the body. To avoid shocks, the body must not become part of a complete circuit by allowing different parts of the body to touch objects at different potentials. Commonly, shocks are caused by one part of the body touching ground and another part touching a high electric potential. [*An ammeter measures current. An analog ammeter consists of a galvanometer and a parallel shunt resistor that carries most of the current. An analog voltmeter consists of a galvanometer and a series resistor. An ammeter is inserted into the circuit whose current is to be measured. A voltmeter is external, being connected in parallel to the element whose voltage is to be measured. Digital voltmeters have greater internal resistance and affect the circuit to be measured less than do analog meters.]

Questions 1. Explain why birds can sit on power lines safely, whereas leaning a metal ladder up against a power line to fetch a stuck kite is extremely dangerous. 2. Discuss the advantages and disadvantages of Christmas tree lights connected in parallel versus those connected in series. 3. If all you have is a 120-V line, would it be possible to light several 6-V lamps without burning them out? How? 4. Two lightbulbs of resistance R1 and R2 AR2 7 R1 B and a battery are all connected in series. Which bulb is brighter? What if they are connected in parallel? Explain.

9. When applying Kirchhoff's loop rule (such as in Fig. 26–33), does the sign (or direction) of a battery's emf depend on the direction of current through the battery? What about the terminal voltage?

r = 1.0 Ω = 18 V FIGURE 26–33 Question 9.

r = 2.0 Ω

R = 6.6 Ω

= 12 V

5. Household outlets are often double outlets. Are these connected in series or parallel? How do you know?

10. Compare and discuss the formulas for resistors and for capacitors when connected in series and in parallel.

6. With two identical lightbulbs and two identical batteries, how would you arrange the bulbs and batteries in a circuit to get the maximum possible total power to the lightbulbs? (Assume the batteries have negligible internal resistance.)

11. For what use are batteries connected in series? For what use are they connected in parallel? Does it matter if the batteries are nearly identical or not in either case?

7. If two identical resistors are connected in series to a battery, does the battery have to supply more power or less power than when only one of the resistors is connected? Explain. 8. You have a single 60-W bulb on in your room. How does the overall resistance of your room's electric circuit change when you turn on an additional 100-W bulb?

698

CHAPTER 26

DC Circuits

12. Can the terminal voltage of a battery ever exceed its emf? Explain. 13. Explain in detail how you could measure the internal resistance of a battery. 14. In an RC circuit, current flows from the battery until the capacitor is completely charged. Is the total energy supplied by the battery equal to the total energy stored by the capacitor? If not, where does the extra energy go?

15. Given the circuit shown in Fig. 26–34, use the words "increases," "decreases," or "stays the same" to complete the following statements: (a) If R7 increases, the potential difference between A and A and e. E _____. Assume no resistance in ~ (b) If R7 increases, the potential difference between A and A and e have resistance. E _____. Assume ~ (c) If R7 increases, the voltage drop across R4 __________. (d) If R2 decreases, the current through R1 __________. (e) If R2 decreases, the current through R6 __________. ( f ) If R2 decreases, the current R4 through R3 _____. R5 (g) If R5 increases, the voltage C D drop across R2 _____. R6 (h) If R5 increases, the voltage drop across R4 _____. (i) If R2 , R5 , and R7 increase, e (r = 0) _____.

R2 FIGURE 26–34 Question 15. R2 , R5 , and R7 are variable resistors (you can change their resistance), . given the symbol 16. Figure 26–35 is a diagram of microphone. The changing air causes one plate of the capacitor C to move back and forth. Explain how a current of the same frequency as the sound wave is produced.

R3 B

R1 A

+ –

a capacitor (or condenser) pressure in a sound wave

Sound pressure

17. Design a circuit in which two different switches of the type shown in Fig. 26–36 can be used to operate the same lightbulb from opposite sides of a room. Wire Wire

* 18. What is the main difference between an analog voltmeter and an analog ammeter? * 19. What would happen if you mistakenly used an ammeter where you needed to use a voltmeter? * 20. Explain why an ideal ammeter would have zero resistance and an ideal voltmeter infinite resistance. * 21. A voltmeter connected across a resistor always reads less than the actual voltage across the resistor when the meter is not present. Explain. * 22. A small battery-operated flashlight requires a single 1.5-V battery. The bulb is barely glowing, but when you take the battery out and check it with a voltmeter, it registers 1.5 V. How would you explain this? 23. Different lamps might have batteries connected in either of the two arrangements shown in Fig. 26–37. What would be the advantages of each scheme?

+ +

R V output FIGURE 26–35 Diagram of a capacitor microphone. Question 16.

Wire

FIGURE 26–36 Question 17.

Movable plate (diaphragm)

+ –

FIGURE 26–37 (a) Question 23.

(b)

Problems 26–1 Emf and Terminal Voltage 1. (I) Calculate the terminal voltage for a battery with an internal resistance of 0.900 and an emf of 6.00 V when the battery is connected in series with (a) an 81.0- resistor, and (b) an 810- resistor. 2. (I) Four 1.50-V cells are connected in series to a 12- lightbulb. If the resulting current is 0.45 A, what is the internal resistance of each cell, assuming they are identical and neglecting the resistance of the wires? 3. (II) A 1.5-V dry cell can be tested by connecting it to a lowresistance ammeter. It should be able to supply at least 25 A. What is the internal resistance of the cell in this case, assuming it is much greater than that of the ammeter? 4. (II) What is the internal resistance of a 12.0-V car battery whose terminal voltage drops to 8.4 V when the starter draws 95 A? What is the resistance of the starter?

26–2 Resistors in Series and Parallel In these Problems neglect the internal resistance of a battery unless the Problem refers to it.

5. (I) A 650- and a 2200- resistor are connected in series with a 12-V battery. What is the voltage across the 2200- resistor? 6. (I) Three 45- lightbulbs and three 65- lightbulbs are connected in series. (a) What is the total resistance of the circuit? (b) What is the total resistance if all six are wired in parallel? 7. (I) Suppose that you have a 680-, a 720-, and a 1.20-k resistor. What is (a) the maximum, and (b) the minimum resistance you can obtain by combining these? 8. (I) How many 10- resistors must be connected in series to give an equivalent resistance to five 100- resistors connected in parallel?

Problems

699

9. (II) Suppose that you have a 9.0-V battery and you wish to apply a voltage of only 4.0 V. Given an unlimited supply of 1.0- resistors, how could you connect them so as to make a "voltage divider" that produces a 4.0-V output for a 9.0-V input? 10. (II) Three 1.70-k resistors can be connected together in four different ways, making combinations of series and/or parallel circuits. What are these four ways, and what is the net resistance in each case? 11. (II) A battery with an emf of 12.0 V shows a terminal voltage of 11.8 V when operating in a circuit with two lightbulbs, each rated at 4.0 W (at 12.0 V), which are connected in parallel. What is the battery's internal resistance? 12. (II) Eight identical bulbs are connected in series across a 110-V line. (a) What is the voltage across each bulb? (b) If the current is 0.42 A, what is the resistance of each bulb, and what is the power dissipated in each? 13. (II) Eight bulbs are connected in parallel to a 110-V source by two long leads of total resistance 1.4 . If 240 mA flows through each bulb, what is the resistance of each, and what fraction of the total power is wasted in the leads? 14. (II) The performance of the starter circuit in an automobile can be significantly degraded by a small amount of corrosion on a battery terminal. Figure 26–38a depicts a properly functioning circuit with a battery (12.5-V emf, 0.02- internal resistance) attached via corrosion-free cables to a starter motor of = 12.5 V resistance RS = 0.15 . S Suppose that later, corror = 0.02 Ω sion between a battery terminal and a starter cable introduces an extra series resistance of just RS = 0.15 Ω into the RC = 0.10 (a) circuit as suggested in Fig. 26–38b. Let P0 be the power delivered to the RC = 0.10 Ω starter in the circuit free of corrosion, and let P be the power delivered to the S circuit with corrosion. r = 0.02 Ω Determine the ratio PP0.

FIGURE 26–38 Problem 14.

RS = 0.15 Ω (b)

15. (II) A close inspection of an electric circuit reveals that a 480- resistor was inadvertently soldered in the place where a 370- resistor is needed. How can this be fixed without removing anything from the existing circuit? 16. (II) Determine (a) the equivalent resistance of the circuit shown in Fig. 26–39, and (b) the voltage across each resistor. 820 Ω FIGURE 26–39 Problem 16.

700

CHAPTER 26

960 Ω

680 Ω 12.0 V

DC Circuits

17. (II) A 75-W, 110-V bulb is connected in parallel with a 25-W, 110-V bulb. What is the net resistance? 18. (II) (a) Determine the equivalent resistance of the "ladder" of equal 125- resistors shown in Fig. 26–40. In other words, what resistance would an ohmmeter read if connected between points A and B? (b) What is the current through each of the three resistors on the left if a 50.0-V battery is connected between points A and B?

A FIGURE 26–40 Problem 18.

R R

19. (II) What is the net resistance of the circuit connected to the battery in Fig. 26–41?

R B

R A

R C

R FIGURE 26–41 Problems 19 and 20.

20. (II) Calculate the current through each resistor in Fig. 26–41 if each resistance R = 1.20 k and V = 12.0 V. What is the potential difference between points A and B? 21. (II) The two terminals of a voltage source with emf e and internal resistance r are connected to the two sides of a load resistance R. For what value of R will the maximum power be delivered from the source to the load? 22. (II) Two resistors when connected in series to a 110-V line use one-fourth the power that is used when they are connected in parallel. If one resistor is 3.8 k, what is the resistance of the other? 23. (III) Three equal resistors (R) are connected to a battery as shown in Fig. 26–42. Qualitatively, what happens to (a) the voltage drop across each of these resistors, (b) the current flow through each, and (c) the terminal voltage of the battery, when the switch S is opened, after having been closed for a long time? (d) If the emf of the battery is 9.0 V, what is its terminal voltage when the switch is closed if the internal resistance r is 0.50 and R = 5.50 ? (e) What is the terminal R S voltage when the switch is open? r R R FIGURE 26–42 Problem 23. 24. (III) A 2.8-k and a 3.7-k resistor are connected in parallel; this combination is connected in series with a 1.8-k resistor. If each resistor is rated at 12 W (maximum without overheating), what is the maximum voltage that can be applied across the whole network?

25. (III) Consider the network of resistors shown in Fig. 26–43. Answer qualitatively: (a) What happens to the voltage across each resistor when the switch S is closed? (b) What happens to the current through each when the switch is closed? (c) What happens to the power output of the battery when the switch is closed? (d) Let R1 = R2 = R3 = R4 = 125 and V = 22.0 V. Determine the current through each resistor before and after closing the switch. Are your qualitative predictions confirmed? R1 S

V FIGURE 26–43 Problem 25.

FIGURE 26–47 Problem 29.

R = R0

R R

FIGURE 26–48 Problem 30.

31. (II) (a) What is the potential difference between points a and d in Fig. 26–49 (similar to Fig. 26–13, Example 26–9), and (b) what is the 34 Ω h terminal voltage of

Gas chamber

I2 FIGURE 26–49 Problem 31.

26–3 Kirchhoff's Rules

47 Ω

r= 2= 1 Ω 45 V

b 1= 75 V

18 Ω

r= 1Ω

32. (II) Calculate the currents in each resistor of Fig. 26–50.

27. (I) Calculate the current in the circuit of Fig. 26–45, and show that the sum of all the r = 2.0 Ω voltage changes around the circuit is zero. 9.0 V 9.5 Ω FIGURE 26–45 Problem 27.

12.0 Ω

28. (II) Determine the terminal voltage of each battery in Fig. 26–46. r2 = 2.0 Ω = 18 V

r1 = 1.0 Ω FIGURE 26–46 Problem 28.

1.5 V

30. (II) (a) A network of five equal resistors R is connected to a battery e as shown in Fig. 26–48. Determine the current I that flows out of the battery. (b) Use the value determined for I to find the single resistor Req that is equivalent to the five-resistor network.

each battery?

FIGURE 26–44 Problem 26.

26. (III) You are designing a wire resistance heater to heat an enclosed volume of gas. For the apparatus to function properly, this heater must transfer heat to the gas at a very constant rate. While in operation, the resistance of the heater will always be close to the value R = R0 , but may fluctuate slightly causing its resistance to vary a small amount ¢R A V R0 B. To maintain the heater at constant power, you design the circuit shown in Fig. 26–44, which includes two resistors, each of resistance r. Determine the value for r so that the heater power will remain constant even if its resistance R fluctuates by a small amount. [Hint: If ¢R V R0 , then ¢P L ¢R dP P .] dR

29. (II) For the circuit shown in Fig. 26–47, find the potential difference between points a R and b. Each resistor has R = 130 and each battery a 1.5 V is 1.5 V.

= 12 V

58 V

25 Ω 3.0 V

120 Ω

64 Ω

110 Ω

82 Ω FIGURE 26–50 Problem 32. 33. (II) Determine the magnitudes and directions of the currents through R1 and R2 in V1 = 9.0 V R1 = 22 Ω Fig. 26–51.

R2 = 18 Ω

R = 6.6 Ω FIGURE 26–51 Problem 33.

V3 = 6.0 V

Problems

701

34. (II) Determine the magnitudes and directions of the currents in each resistor shown in Fig. 26–52. The batteries have emfs of e1 = 9.0 V and e2 = 12.0 V and the resistors have values of R1 = 25 , R1 and R3 = 35 . R2 = 48 , (a) Ignore internal resistance of the 1 R2 batteries. (b) Assume each battery has internal resistance r = 1.0 .

FIGURE 26–56 Problem 41.

FIGURE 26–52 Problem 34.

35. (II) A voltage V is applied to n identical resistors connected in parallel. If the resistors are instead all connected in series with the applied voltage, show that the power transformed is decreased by a factor n2. 36. (III) (a) Determine the currents I1 , I2 , and I3 in Fig. 26–53. Assume the internal resistance of each battery is r = 1.0 . (b) What is the terminal voltage of the 6.0-V battery?

12.0 V

12 Ω

18 Ω

r I3

6.0 V

37. (III) What would the current I1 be in Fig. 26–53 if the 12- resistor is shorted out (resistance = 0)? Let r = 1.0 . 38. (III) Determine the current through each of the resistors in Fig. 26–54. 22 Ω

12 Ω

25 Ω

14 Ω

6.0 V

39. (III) If the 25- resistor in Fig. 26–54 is shorted out (resistance = 0), what then would be the current through the 15- resistor? 40. (III) Twelve resistors, each of resistance R, are connected as the edges of a cube as shown in Fig. 26–55. Determine the equivalent resistance (a) between points a and b, the ends of a side; (b) between points a and c, the ends of a face diagonal; (c) between y points a and d, the ends of the volume diagonal. a [Hint: Apply an emf and b determine currents; use symmetry at junctions.]

702

FIGURE 26–55 Problem 40.

d z

26–4 Emfs Combined, Battery Charging 42. (II) Suppose two batteries, with unequal emfs of 2.00 V and 3.00 V, are connected as shown in Fig. 26–57. If each internal resistance is r = 0.450 , and R = 4.00 Ω R = 4.00 , what is the voltage across the resistor R? = 2.00 V

FIGURE 26–57 Problem 42.

= 3.00 V r

43. (I) Estimate the range of resistance needed to make a variable timer for typical intermittent windshield wipers if the capacitor used is on the order of 1 mF. 44. (II) In Fig. 26–58 (same as Fig. 26–17a), the total resistance is 15.0 k, and the battery's emf is 24.0 V. If the time constant is measured to be 24.0 ms, calculate (a) the total capacitance of the R circuit and (b) the time it takes for the voltage across the resistor to reach – 16.0 V after the switch is closed. C + FIGURE 26–58 Problems 44 and 46.

15 Ω

FIGURE 26–54 Problems 38 and 39.

26–5 RC Circuits

12.0 V

FIGURE 26–53 Problems 36 and 37.

11 Ω

15 Ω

r 22 Ω

41. (III) Determine the net resistance in Fig. 26–56 (a) between points a and c, and (b) between points a and b. Assume R¿ Z R. [Hint: Apply an emf and determine currents; use symmetry at junctions.] R R R′ b

45. (II) Two 3.8-mF capacitors, two 2.2-k resistors, and a 12.0-V source are connected in series. Starting from the uncharged state, how long does it take for the current to drop from its initial value to 1.50 mA? 46. (II) How long does it take for the energy stored in a capacitor in a series RC circuit (Fig. 26–58) to reach 75% of its maximum value? Express answer in terms of the time constant t = RC. 47. (II) A parallel-plate capacitor is filled with a dielectric of dielectric constant K and high resistivity r (it conducts very slightly). This capacitor can be modeled as a pure capacitance C in parallel with a resistance R. Assume a battery places a charge ±Q and –Q on the capacitor's opposing plates and is then disconnected. Show that the capacitor discharges with a time constant t = Ke0 r (known as the dielectric relaxation time). Evaluate t if the dielectric is glass with r = 1.0 * 1012 m and K = 5.0. 48. (II) The RC circuit of Fig. 26–59 (same as Fig. 26–18a) has R = 8.7 k and C = 3.0 mF. The capacitor is at voltage V0 at t = 0, when the switch is closed. How long does it take the capacitor to discharge to 0.10% of its initial – R V0 C voltage? + FIGURE 26–59 Problem 48.

49. (II) Consider the circuit shown in Fig. 26–60, where all resistors have the same resistance R. At t = 0, with the capacitor C uncharged, the switch is closed. (a) At t = 0, the three currents can be determined by analyzing a simpler, but equivalent, circuit. Identify this simpler circuit and use it to find the values of I1 , I2 , and I3 at t = 0. (b) At t = q, the currents can be determined by analyzing a simpler, equivalent circuit. Identify this simpler circuit and implement it in finding the values of I1 , I2 , and I3 at t = q. (c) At t = q, what is the potential difference across the capacitor?

R I1 I2 FIGURE 26–60 Problem 49.

I3 C

R Q Q

* 56. (II) A galvanometer has an internal resistance of 32 and deflects full scale for a 55-mA current. Describe how to use this galvanometer to make (a) an ammeter to read currents up to 25 A, and (b) a voltmeter to give a full scale deflection of 250 V. * 57. (II) A particular digital meter is based on an electronic module that has an internal resistance of 100 M and a full-scale sensitivity of 400 mV. Two resistors connected as shown in Fig. 26–63 can be used to change the voltage range. Assume R1 = 10 M. Find the Meter module value of R2 that will r 100 MΩ result in a voltmeter with a full-scale range of 40 V. FIGURE 26–63 Problem 57.

R1 R2

* 58. (II) A milliammeter reads 25 mA full scale. It consists of a 0.20- resistor in parallel with a 33- galvanometer. How can you change this ammeter to a voltmeter giving a full 50. (III) Determine the time constant for charging the capacitor scale reading of 25 V without taking the ammeter apart? in the circuit of Fig. 26–61. [Hint: Use Kirchhoff's rules.] What will be the sensitivity (V) of your voltmeter? (b) What is the maximum R1 * 59. (II) A 45-V battery of negligible internal resistance is charge on the capacitor? connected to a 44-k and a 27-k resistor in series. What reading will a voltmeter, of internal resistance 95 k, give R2 C when used to measure the voltage across each resistor? FIGURE 26–61 What is the percent inaccuracy due to meter resistance for Problem 50. each case? * 60. (II) An ammeter whose internal resistance is 53 reads 51. (III) Two resistors and two uncharged capacitors are 5.25 mA when connected in a circuit containing a battery arranged as shown in Fig. 26–62. Then a potential difference and two resistors in series whose values are 650 and of 24 V is applied across the combination as shown. 480 . What is the actual current when the ammeter is (a) What is the potential at point a with switch S open? (Let absent? V = 0 at the negative terminal of the source.) (b) What is 61. (II) A battery with e = 12.0 V and internal resistance the potential at point b with the switch open? (c) When the * r = 1.0 is connected to two 7.5-k resistors in series. An switch is closed, what is the final potential of ammeter of internal resistance 0.50 measures the current, point b? (d) How much charge flows through the switch S and at the same time a voltmeter with internal resistance after it is closed? 15 k measures the voltage across one of the 7.5-k resistors in the circuit. What do the ammeter and voltmeter read? 8.8 Ω 0.48 mF * 62. (II) A 12.0-V battery (assume the internal resistance = 0) is connected to two resistors in series. A voltmeter whose + internal resistance is 18.0 k measures 5.5 V and 4.0 V, a b 24 V respectively, when connected across each of the resistors. S – What is the resistance of each resistor? 4.4 Ω 0.36 mF * 63. (III) Two 9.4-k resistors are placed in series and connected to a battery. A voltmeter of sensitivity 1000 V is on the FIGURE 26–62 3.0-V scale and reads 2.3 V when placed across either Problems 51 and 52. resistor. What is the emf of the battery? (Ignore its internal resistance.) 52. (III) Suppose the switch S in Fig. 26–62 is closed. What is 64. (III) When the resistor R in Fig. 26–64 is 35 , the the time constant (or time constants) for charging the * high-resistance voltmeter reads 9.7 V. When R is replaced by capacitors after the 24 V is applied? a 14.0- resistor, the voltmeter reading drops to 8.1 V. * 26–7 Ammeters and Voltmeters What are the emf and internal resistance of the battery? * 53. (I) An ammeter has a sensitivity of 35,000 V. What V current in the galvanometer produces full-scale deflection? * 54. (I) What is the resistance of a voltmeter on the 250-V scale if the meter sensitivity is 35,000 V? * 55. (II) A galvanometer has a sensitivity of 45 kV and R internal resistance 20.0 . How could you make this into FIGURE 26–64 (a) an ammeter that reads 2.0 A full scale, or (b) a voltmeter Problem 64. reading 1.00 V full scale?

Problems

703

General Problems 65. Suppose that you wish to apply a 0.25-V potential difference between two points on the human body. The resistance is about 1800 , and you only have a 1.5-V battery. How can you connect up one or more resistors to produce the desired voltage? 66. A three-way lightbulb can produce 50 W, 100 W, or 150 W, at 120 V. Such a bulb contains two filaments that can be connected to the 120 V individually or in parallel. (a) Describe how the connections to the two filaments are made to give each of the three wattages. (b) What must be the resistance of each filament? 67. Suppose you want to run some apparatus that is 65 m from an electric outlet. Each of the wires connecting your apparatus to the 120-V source has a resistance per unit length of 0.0065 m. If your apparatus draws 3.0 A, what will be the voltage drop across the connecting wires and what voltage will be applied to your apparatus? 68. For the circuit shown in Fig. 26–18a, show that the decrease in energy stored in the capacitor from t = 0 until one time constant has elapsed equals the energy dissipated as heat in the resistor. 69. A heart pacemaker is designed to operate at 72 beatsmin using a 6.5-mF capacitor in a simple RC circuit. What value of resistance should be used if the pacemaker is to fire (capacitor discharge) when the voltage reaches 75% of maximum? 70. Suppose that a person's body resistance is 950 . (a) What current passes through the body when the person accidentally is connected to 110 V? (b) If there is an alternative path to ground whose resistance is 35 , what current passes through the person? (c) If the voltage source can produce at most 1.5 A, how much current passes through the person in case (b)? 71. A Wheatstone bridge is a type of "bridge circuit" used to make measurements of resistance. The unknown resistance to be measured, Rx , is placed in the circuit with accurately known resistances R1 , R2 , and R3 (Fig. 26–65). One of these, R3 , is a variable resistor which is adjusted so that when the A shows zero switch is closed momentarily, the ammeter ~ current flow. (a) Determine Rx in terms of R1 , R2 , and B R3 . (b) If a Wheatstone R3 Rx bridge is "balanced" when I3 R1 = 630 , R2 = 972 , A and R3 = 78.6 , what A Ammeter C is the value of the unknown resistance? S I1

FIGURE 26–65 Problems 71 and 72. Wheatstone bridge.

CHAPTER 26

DC Circuits

FIGURE 26–66 Problem 76. 77. The current through the 4.0-k resistor in Fig. 26–67 is 3.10 mA. What is the terminal voltage Vba of 4.0 kΩ the "unknown" battery? Vba 5.0 kΩ (There are two answers. Why?) a b

D + –

8.0 kΩ

1.0 Ω FIGURE 26–67 Problem 77.

12.0 V

78. A battery produces 40.8 V when 7.40 A is drawn from it, and 47.3 V when 2.80 A is drawn. What are the emf and internal resistance of the battery? 79. In the circuit shown in Fig. 26–68, the 33- resistor dissipates 0.80 W. What is 68 Ω the battery voltage? 33 Ω

72. An unknown length of platinum wire 1.22 mm in diameter is placed as the unknown resistance in a Wheatstone bridge (see Problem 71, Fig. 26–65). Arms 1 and 2 have resistance of 38.0 and 29.2 , respectively. Balance is achieved when R3 is 3.48 . How long is the platinum wire? 73. The internal resistance of a 1.35-V mercury cell is 0.030 , whereas that of a 1.5-V dry cell is 0.35 . Explain why three mercury cells can more effectively power a 2.5-W hearing aid that requires 4.0 V than can three dry cells.

704

74. How many 12-W resistors, each of the same resistance, must be used to produce an equivalent 3.2-k, 3.5-W resistor? What is the resistance of each, and how must they be connected? Do not exceed P = 12 W in each resistor. 75. A solar cell, 3.0 cm square, has an output of 350 mA at 0.80 V when exposed to full sunlight. A solar panel that delivers close to 1.3 A of current at an emf of 120 V to an external load is needed. How many cells will you need to create the panel? How big a panel will you need, and how should you connect the cells to one another? How can you optimize the output of your solar panel? 76. A power supply has a fixed output voltage of 12.0 V, but you need VT = 3.0 V output for an experiment. (a) Using the voltage divider shown in Fig. 26–66, what should R2 be if R1 is 14.5 ? (b) What will the terminal voltage VT be if you R1 connect a load to the 3.0-V output, assuming the load 12.0 V has a resistance of 7.0 ? VT R2

FIGURE 26–68 Problem 79.

75 Ω

80. The current through the 20- resistor in Fig. 26–69 does not change whether the 20 Ω two switches S1 and S2 are both open or both closed. Use this clue to R determine the value of S2 10 S1 the unknown resisΩ tance R. 50 Ω FIGURE 26–69 6.0 V Problem 80.

* 81. (a) A voltmeter and an ammeter can be connected as shown in Fig. 26–70a to measure a resistance R. If V is the voltmeter reading, and I is the ammeter reading, the value of R will not quite be VI (as in Ohm's law) because some of the current actually goes through the voltmeter. Show that the actual value of R is given by 1 I 1 , = R V RV where RV is the voltmeter resistance. Note that R L VI if RV W R. (b) A voltmeter and an ammeter can also be connected as shown in Fig. 26–70b to measure a resistance R. Show in this case that V R = - RA , I where V and I are the voltmeter and ammeter readings and RA is the resistance of the ammeter. Note that R L VI if RA V R. A

(b)

(a)

FIGURE 26–70 Problem 81. 82. (a) What is the equivalent resistance of the circuit shown in Fig. 26–71? (b) What is the current in the 18- resistor? (c) What is the current in the 12- resistor? (d) What is the power dissipation in the 4.5- 6.0 V resistor? 12 Ω 18 Ω 25 Ω FIGURE 26–71 Problem 82.

4.5 Ω

83. A flashlight bulb rated at 2.0 W and 3.0 V is operated by a 9.0-V battery. To light the bulb at its rated voltage and power, a resistor R is connected in series as shown in Fig. 26–72. What value R should the resistor have? FIGURE 26–72 Problem 83.

9.0 V

84. Some light-dimmer switches use a variable resistor as shown in Fig. 26–73. The slide moves from position x = 0 to x = 1, and the resistance up to slide position x is proportional to x (the total resistance is Rvar = 150 at x = 1). What is the power expended in the lightbulb if (a) x = 1.00, (b) x = 0.65, (c) x = 0.35?

85. A potentiometer is a device to precisely measure potential differences or emf, using a "null" technique. In the simple potentiometer circuit shown in Fig. 26–74, R¿ represents the total resistance of the resistor from A to B (which could be a long uniform "slide" wire), whereas R represents the resistance of only the part from A to the movable contact at C. When the unknown emf to be measured, ex , is placed into the circuit as shown, the movable contact C is moved until the galvanometer G gives a null reading (i.e., zero) when the switch S is closed. The resistance between A and C for this situation we call Rx . Next, a standard emf, es , which is known precisely, is inserted into the circuit in place of ex and again the contact C is moved until zero current flows through the galvanometer when the switch S is closed. The resistance between A and C now is called Rs . (a) Show that the unknown emf is given by ex = ¢

where Rx , Rs , and es are all precisely known. The working battery is assumed to be fresh and to give a constant voltage. (b) A slide-wire potentiometer is balanced against a 1.0182-V standard cell when the slide wire is set at 33.6 cm out of a total length of 100.0 cm. For an unknown source, the setting is 45.8 cm. What is the emf of the unknown? (c) The galvanometer of a Working potentiometer has an battery internal resistance of 35 and can detect a RV current as small as 0.012 mA. What is the minimum uncertainty R' possible in measuring R an unknown voltage? A B (d) Explain the advanC tage of using this "null" method of S measuring emf. FIGURE 26–74 Potentiometer circuit. Problem 85.

FIGURE 26–73 Problem 84.

= 240 Ω

a Protector circuit

R C

Rbulb

+ x – or

86. Electronic devices often use an RC circuit to protect against power outages as shown in Fig. 26–75. (a) If the protector circuit is supposed to keep the supply voltage at least 75% of full voltage for as long as 0.20 s, how big a resistance R is needed? The capacitor is 8.5 mF. Assume the attached "electronics" draws negligible current. (b) Between which two terminals should the device be connected, a and b, b and c, or a and c?

Rvar = 150Ω 120 V

Rx ≤e Rs s

Power supply

Electronic device

Power outage protector

FIGURE 26–75 Problem 86.

General Problems

705

87. The circuit shown in Fig. 26–76 is a primitive 4-bit digitalto-analog converter (DAC). In this circuit, to represent each digit (2 n) of a binary number, a "1" has the nth switch closed whereas zero ("0") has the switch open. For example, 0010 is represented by closing switch n = 1, while all other switches are open. Show that the voltage V across the 1.0- resistor for the binary numbers 0001, 0010, 0100, and 1010 (which represent 16.0 kΩ n0 1, 2, 4, 10) follows the 8.0 kΩ n1 pattern that you expect 4.0 kΩ 16 V n 2 for a 4-bit 2.0 kΩ DAC. n3

* 91. Measurements made on circuits that contain large resistances can be confusing. Consider a circuit powered by a battery e = 15.000 V with a 10.00-M resistor in series with an unknown resistor R. As shown in Fig. 26–80, a particular voltmeter reads V1 = 366 mV when connected across the 10.00-M resistor, and this meter reads V2 = 7.317 V when connected across R. Determine the value of R. [Hint: Define RV as the voltmeter's internal resistance.]

10.00 MΩ

15.000 V

366 mV

1.0 Ω

FIGURE 26–76 Problem 87.

88. Determine the current in each resistor of the circuit shown in Fig. 26–77. 8.00 V + 10.00 V –

5.00 Ω 6.00 Ω

+ –

7.317 V

+ 4.00 V –

12.00 Ω

FIGURE 26–80 Problem 91.

FIGURE 26–77 Problem 88.

+ – 5.00 V

6.80 Ω

89. In the circuit shown in Fig. 26–78, switch S is closed at time t = 0. (a) After the capacitor is fully charged, what is the voltage across it? How much charge is on it? (b) Switch S is now opened. How long does it now take for the capacitor to discharge until S it has only 3.0% of its 10.0 Ω 1.0 Ω initial 12.0 V charge? C 2.2 mF 5.0 Ω 9.0 Ω FIGURE 26–78 Problem 89. 90. Figure 26–79 shows the circuit for a simple sawtooth oscillator. At time t = 0, its switch S is closed. The neon bulb has initially infinite resistance until the voltage across it reaches 90.0 V, and then it begins to conduct with very little resistance (essentially zero). It stops conducting (its resistance becomes essentially infinite) when the voltage drops down to 65.0 V. (a) At what time t1 does the neon bulb reach 90.0 V and start conducting? (b) At what time t2 does the bulb reach 90.0 V for a second time and again become conducting? (c) Sketch the sawtooth waveform between t = 0 and t = 0.70 s.

FIGURE 26–79 Problem 90.

10.00 MΩ

15.000 V

+ 100.0 V –

33.0 kΩ C = 4.00 mF

Neon Output bulb

* 92. A typical voltmeter has an internal resistance of 10 M and can only measure voltage differences of up to several hundred volts. Figure 26–81 shows the design of a probe to measure a very large voltage difference V using a voltmeter. If you want the voltmeter to read 50 V when V = 50 kV, what value R should be used in this probe?

Probe High voltage V

15 MΩ

FIGURE 26–81 Problem 92. * Numerical/Computer * 93. (II) An RC series circuit contains a resistor R = 15 k, a capacitor C = 0.30 mF, and a battery of emf e = 9.0 V. Starting at t = 0, when the battery is connected, determine the charge Q on the capacitor and the current I in the circuit from t = 0 to t = 10.0 ms (at 0.1-ms intervals). Make graphs showing how the charge Q and the current I change with time within this time interval. From the graphs find the time at which the charge attains 63% of its final value, Ce, and the current drops to 37% of its initial value, eR.

Answers to Exercises A: (a) 1.14 A; (b) 11.4 V; (c) PR = 13.1 W, Pr = 0.65 W. B: 6 and 25 .

D: 180 A; this high current through the batteries could cause them to become very hot: the power dissipated in the weak battery would be P = I 2r = (180 A)2(0.10 ) = 3200 W!

C: 41I3 - 45 + 21I2 - 80 = 0.

E: (a).

706

CHAPTER 26

DC Circuits

FPO I Magnets produce magnetic fields, but so do electric currents. An electric current flowing in this straight wire produces a magnetic field which causes the tiny pieces of iron (iron "filings") to align in the field. We shall see in this Chapter how magnetic field is defined, and that the magnetic field direction is along the iron filings. The magnetic field lines due to the electric current in this long wire are in the shape of circles around the wire. We also discuss how magnetic fields exert forces on electric currents and on charged particles, as well as useful applications of the interaction between magnetic fields and electric currents and moving electric charges.

P T E R

C H

Magnetism CHAPTER-OPENING QUESTION—Guess now! Which of the following can experience a force when placed in the magnetic field of a magnet? (a) (b) (c) (d)

An electric charge at rest. An electric charge moving. An electric current in a wire. Another magnet.

he history of magnetism begins thousands of years ago. In a region of Asia Minor known as Magnesia, rocks were found that could attract each other. These rocks were called "magnets" after their place of discovery. Not until the nineteenth century, however, was it seen that magnetism and electricity are closely related. A crucial discovery was that electric currents produce magnetic effects (we will say "magnetic fields") like magnets do. All kinds of practical devices depend on magnetism, from compasses to motors, loudspeakers, computer memory, and electric generators.

27–1

CONTENTS

27–1 Magnets and Magnetic Fields 27–2 Electric Currents Produce Magnetic Fields 27–3 Force on an Electric Current inBa Magnetic Field; Definition of B 27–4 Force on an Electric Charge Moving in a Magnetic Field 27–5 Torque on a Current Loop; Magnetic Dipole Moment *27–6 Applications: Motors, Loudspeakers, Galvanometers 27–7 Discovery and Properties of the Electron 27–8 The Hall Effect *27–9 Mass Spectrometer

Magnets and Magnetic Fields

We have all observed a magnet attract paper clips, nails, and other objects made of iron, Fig. 27–1. Any magnet, whether it is in the shape of a bar or a horseshoe, has two ends or faces, called poles, which is where the magnetic effect is strongest. If a bar magnet is suspended from a fine thread, it is found that one pole of the magnet will always point toward the north. It is not known for sure when this fact was discovered, but it is known that the Chinese were making use of it as an aid to navigation by the eleventh century and perhaps earlier. This is the principle of a compass.

FIGURE 27–1 A horseshoe magnet attracts iron tacks and paper clips.

707

Repulsive N

Attractive FIGURE 27–2 Like poles of a magnet repel; unlike poles attract. Red arrows indicate force direction.

FIGURE 27–3 If you split a magnet, you won't get isolated north and south poles; instead, two new magnets are produced, each with a north and a south pole. S

S S

N N

C A U T I O N

Magnets do not attract all metals

A compass needle is simply a bar magnet which is supported at its center of gravity so that it can rotate freely. The pole of a freely suspended magnet that points toward geographic north is called the north pole of the magnet. The other pole points toward the south and is called the south pole. It is a familiar observation that when two magnets are brought near one another, each exerts a force on the other. The force can be either attractive or repulsive and can be felt even when the magnets don't touch. If the north pole of one bar magnet is brought near the north pole of a second magnet, the force is repulsive. Similarly, if the south poles of two magnets are brought close, the force is repulsive. But when a north pole is brought near the south pole of another magnet, the force is attractive. These results are shown in Fig. 27–2, and are reminiscent of the forces between electric charges: like poles repel, and unlike poles attract. But do not confuse magnetic poles with electric charge. They are very different. One important difference is that a positive or negative electric charge can easily be isolated. But an isolated single magnetic pole has never been observed. If a bar magnet is cut in half, you do not obtain isolated north and south poles. Instead, two new magnets are produced, Fig. 27–3, each with north (N) and south (S) poles. If the cutting operation is repeated, more magnets are produced, each with a north and a south pole. Physicists have searched for isolated single magnetic poles (monopoles), but no magnetic monopole has ever been observed. Only iron and a few other materials, such as cobalt, nickel, gadolinium, and some of their oxides and alloys, show strong magnetic effects. They are said to be ferromagnetic (from the Latin word ferrum for iron). Other materials show some slight magnetic effect, but it is very weak and can be detected only with delicate instruments. We will look in more detail at ferromagnetism in Section 28–7. In Chapter 21, we used the concept of an electric field surrounding an electric charge. In a similar way, we can picture a magnetic field surrounding a magnet. The force one magnet exerts on another can then be described as the interaction between one magnet and the magnetic field of the other. Just as we drew electric field lines, we can also draw magnetic field lines. They can be drawn, as for electric field lines, so that (1) the direction of the magnetic field is tangent to a field line at any point, and (2) the number of lines per unit area is proportional to the strength of the magnetic field.

FIGURE 27–4 (a) Visualizing magnetic field lines around a bar magnet, using iron filings and compass needles. The red end of the bar magnet is its north pole. The N pole of a nearby compass needle points away from the north pole of the magnet. (b) Magnetic field lines for a bar magnet. (a)

C A U T I O N

Magnetic field lines form closed loops, unlike electric field lines

708

CHAPTER 27

Magnetism

(b)

The direction of the magnetic field at a given point can be defined as the direction that the north pole of a compass needle would point if placed at that point. (A more precise definition will be given in Section 27–3.) Figure 27–4a shows how thin iron filings (acting like tiny magnets) reveal the magnetic field lines by lining up like the compass needles. The magnetic field determined in this way for the field surrounding a bar magnet is shown in Fig. 27–4b. Notice that because of our definition, the lines always point out from the north pole and in toward the south pole of a magnet (the north pole of a magnetic compass needle is attracted to the south pole of the magnet). Magnetic field lines continue inside a magnet, as indicated in Fig. 27–4b. Indeed, given the lack of single magnetic poles, magnetic field lines always form closed loops, unlike electric field lines that begin on positive charges and end on negative charges.

North geographic pole ("true north")

Rotation axis Declination Magnetic pole

Compass N

The Earth's magnetic field is shown in Fig. 27–5. The pattern of field lines is as if there were an imaginary bar magnet inside the Earth. Since the north pole (N) of a compass needle points north, the Earth's magnetic pole which is in the geographic north is magnetically a south pole, as indicated in Fig. 27–5 by the S on the schematic bar magnet inside the Earth. Remember that the north pole of one magnet is attracted to the south pole of another magnet. Nonetheless, Earth's pole in the north is still often called the "north magnetic pole," or "geomagnetic north," simply because it is in the north. Similarly, the Earth's southern magnetic pole, which is near the geographic south pole, is magnetically a north pole (N). The Earth's magnetic poles do not coincide with the geographic poles, which are on the Earth's axis of rotation. The north magnetic pole, for example, is in the Canadian Arctic,† about 900 km from the geographic north pole, or "true north." This difference must be taken into account when you use a compass (Fig. 27–6). The angular difference between magnetic north and true (geographical) north is called the magnetic declination. In the U.S. it varies from 0° to about 20°, depending on location. Notice in Fig. 27–5 that the Earth's magnetic field at most locations is not tangent to the Earth's surface. The angle that the Earth's magnetic field makes with the horizontal at any point is referred to as the angle of dip.

Earth's Magnetic Field

Magnetic pole

South geographic pole

FIGURE 27–5 The Earth acts like a huge magnet; but its magnetic poles are not at the geographic poles, which are on the Earth's rotation axis.

EXERCISE A Does the Earth's magnetic field have a greater magnitude near the poles or near the equator? [Hint: Note the field lines in Fig. 27–5.]

P H Y S I C S

A P P L I E D

Use of a compass

FIGURE 27–6 Using a map and compass in the wilderness. First you align the compass case so the needle points away from true north (N) exactly the number of degrees of declination as stated on the map (15° for the place shown on this topographic map of a part of California). Then align the map with true north, as shown, not with the compass needle.

Uniform Magnetic Field The simplest magnetic field is one that is uniform—it doesn't change in magnitude or direction from one point to another. A perfectly uniform field over a large area is not easy to produce. But the field between two flat parallel pole pieces of a magnet is nearly uniform if the area of the pole faces is large compared to their separation, as shown in Fig. 27–7. At the edges, the field "fringes" out somewhat: the magnetic field lines are no longer quite parallel and uniform. The parallel evenly spaced field lines in the central region of the gap indicate that the field is uniform at points not too near the edges, much like the electric field between two parallel plates (Fig. 23–16).

FIGURE 27–7 Magnetic field between two wide poles of a magnet is nearly uniform, except near the edges.

†

Magnetic north is moving many kilometers a year at present. Magnetism in rocks suggests that the Earth's poles have not only moved significantly over geologic time, but have also reversed direction 400 times over the last 330 million years.

SECTION 27–1

Magnets and Magnetic Fields

709

FIGURE 27–8 (a) Deflection of compass needles near a currentcarrying wire, showing the presence and direction of the magnetic field. (b) Magnetic field lines around an electric current in a straight wire. See also the Chapter-Opening photo. (c) Right-hand rule for remembering the direction of the magnetic field: when the thumb points in the direction of the conventional current, the fingers wrapped around the wire point in the direction of the magnetic field.

(a)

27–2 I

FIGURE 27–9 Magnetic field lines due to a circular loop of wire. Right-hand-rule 1: magnetic field direction produced by electric current FIGURE 27–10 Right-hand rule for determining the direction of the magnetic field relative to the current.

(b)

(c)

Electric Currents Produce Magnetic Fields

During the eighteenth century, many scientists sought to find a connection between electricity and magnetism. A stationary electric charge and a magnet were shown to have no influence on each other. But in 1820, Hans Christian Oersted (1777–1851) found that when a compass needle is placed near a wire, the needle deflects as soon as the two ends of the wire are connected to the terminals of a battery and the wire carries an electric current. As we have seen, a compass needle is deflected by a magnetic field. So Oersted's experiment showed that an electric current produces a magnetic field. He had found a connection between electricity and magnetism. A compass needle placed near a straight section of current-carrying wire experiences a force, causing the needle to align tangent to a circle around the wire, Fig. 27–8a. Thus, the magnetic field lines produced by a current in a straight wire are in the form of circles with the wire at their center, Fig. 27–8b. The direction of these lines is indicated by the north pole of the compasses in Fig. 27–8a. There is a simple way to remember the direction of the magnetic field lines in this case. It is called a right-hand rule: grasp the wire with your right hand so that your thumb points in the direction of the conventional (positive) current; then your fingers will encircle the wire in the direction of the magnetic field, Fig. 27–8c. The magnetic field lines due to a circular loop of current-carrying wire can be determined in a similar way using a compass. The result is shown in Fig. 27–9. Again the right-hand rule can be used, as shown in Fig. 27–10. Unlike the uniform field shown in Fig. 27–7, the magnetic fields shown in Figs. 27–8 and 27–9 are not uniform—the fields are different in magnitude and direction at different points.

Magnetic field EXERCISE B A straight wire carries a current directly toward you. In what direction are the magnetic field lines surrounding the wire?

27–3

Force on an Electric Current inB a Magnetic Field; Definition of B

In Section 27–2 we saw that an electric current exerts a force on a magnet, such as a compass needle. By Newton's third law, we might expect the reverse to be true as well: we should expect that a magnet exerts a force on a current-carrying wire. Experiments indeed confirm this effect, and it too was first observed by Oersted.

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Force is up

I B

B S

Force is down

–

B –

22 V

(a)

(b)

(c)

Suppose a straight wire is placed in the magnetic field between the poles of a horseshoe magnet as shown in Fig. 27–11. When a current flows in the wire, experiment shows that a force is exerted on the wire. But this force is not toward one or the other pole of the magnet. Instead, the force is directed at right angles to the magnetic field direction, downward in Fig. 27–11a. If the current is reversed in direction, the force is in the opposite direction, upward as shown in Fig. 27–11b. Experiments show that the direction of the force is always perpendicular to the direction of the current and B also perpendicular to the direction of the magnetic field, B. The direction of the force is given by another right-hand rule, as illustrated in Fig. 27–11c. Orient your right hand until your outstretched fingers can point in the direction of the conventional current I, and when you bend your fingers they point B in the direction of the magnetic field lines, B. Then your outstretched thumb will B point in the direction of the force F on the wire. This right-hand rule describes the direction of the force. What about the magnitude of the force on the wire? It is found experimentally that the magnitude of the force is directly proportional to the current I in the wire, and to the length l of wire exposed to the magnetic field (assumed uniform). Furthermore, if the magnetic field B is made stronger, the force is found to be proportionally greater. The force also depends on the angle u between the current direction and the magnetic field (Fig. 27–12), being proportional to sin u. Thus, the force on a wire carrying a current I with length l in a uniform magnetic field B is given by F r IlB sin u. When the current is perpendicular to the field lines (u = 90°), the force is strongest. When the wire is parallel to the magnetic field lines (u = 0°), there is no force at all. Up to now we have not defined the magnetic field strength precisely. In fact, the magnetic field B can be conveniently defined in terms of the above proportion so that the proportionality constant is precisely 1. Thus we have

B Right-hand rule

FIGURE 27–11 (a) Force on a current-carrying wire placed in a B magnetic field B; (b) same, but current reversed; (c) right-hand rule for setup in (b).

Right-hand-rule 2: B force on current exerted by B

FIGURE 27–12 Current-carrying wire in a magnetic field. Force on the wire is directed into the page.

(27–1)

F = IlB sin u. B

If the direction of the current is perpendicular to the field B (u = 90°), then the force is C current ⊥ B D (27–2) B

Fmax = IlB.

If the current is parallel to the field (u = 0°), the force is zero. The magnitude of B can be defined using Eq. 27–2 as B = FmaxIl, where Fmax is the magnitude of the force B on a straight length l of wire carrying a current I when the wire is perpendicular to B. B The relation between the force F on a wire carrying current I, and the B magnetic field B that causes the force, can be written as a vector equation. To do B so, we recall that the direction of F is given by the right-hand rule (Fig. 27–11c), and the magnitude by Eq. 27–1. This is consistent with the definition of the vector cross product (see Section 11–2), so we can write B

F = I L * B;

(27–3)

here, L is a vector whose magnitude is the length of the wire and its direction is along the wire (assumed straight) in the direction of the conventional (positive) current. SECTION 27–3

Force on an Electric Current in a Magnetic Field; Definition of B

711

Equation 27–3 applies if the magnetic field is uniform and the wire is B straight. If B is not uniform, or if the wire does not everywhere make the same B angle u with B, then Eq. 27–3 can be written more generally as B

dF = I dL * B, B

(27–4) B

where dF is the infinitesimal force acting on a differential length dL of the wire. The total force on the wire is then found by integrating. Equation 27–4 can serve (just as well as Eq. 27–2 or 27–3) as a practical B B definition of B. An equivalent way to define B, in terms of the force on a moving electric charge, is discussed in the next Section. EXERCISE C A wire carrying current I is perpendicular to a magnetic field of strength B. Assuming a fixed length of wire, which of the following changes will result in decreasing the force on the wire by a factor of 2? (a) Decrease the angle from 90° to 45°; (b) decrease the angle from 90° to 30°; (c) decrease the current in the wire to I2; (d) decrease the magnetic field strength to B2; (e) none of these will do it.

EXAMPLE 27–1 Magnetic force on a current-carrying wire. A wire carrying a 30-A current has a length l = 12 cm between the pole faces of a magnet at an angle u = 60° (Fig. 27–12). The magnetic field is approximately uniform at 0.90 T. We ignore the field beyond the pole pieces. What is the magnitude of the force on the wire?

The SI unit for magnetic field B is the tesla (T). From Eqs. 27–1, 2, 3, or 4, we see that 1 T = 1 NA m. An older name for the tesla is the "weber per meter squared" A1 Wbm2 = 1 TB. Another unit sometimes used to specify magnetic field is a cgs unit, the gauss (G): 1 G = 10–4 T. A field given in gauss should always be changed to teslas before using with other SI units. To get a "feel" for these units, we note that the magnetic field of the Earth at its surface is about 12 G or 0.5 * 10–4 T. On the other hand, the field near a small magnet attached to your refrigerator may be 100 G (0.01 T) whereas strong electromagnets can produce fields on the order of 2 T and superconducting magnets can produce over 10 T.

FIGURE 27–12 (Repeated for Example 27–1.) Current-carrying wire in a magnetic field. Force on the wire is directed into the page.

APPROACH We use Eq. 27–1, F = IlB sin u. SOLUTION The force F on the 12-cm length of wire within the uniform field B is F = IlB sin u = (30 A)(0.12 m)(0.90 T)(0.866) = 2.8 N. EXERCISE D A straight power line carries 30 A and is perpendicular to the Earth's magnetic field of 0.50 * 10–4 T. What magnitude force is exerted on 100 m of this power line?

On a diagram, when we want to represent an electric current or a magnetic field that is pointing out of the page (toward us) or into the page, we use or * , respectively. The is meant to resemble the tip of an arrow pointing directly toward the reader, whereas the * or resembles the tail of an arrow moving away. (See Figs. 27–13 and 27–14.) EXAMPLE 27–2 Measuring a magnetic field. A rectangular loop of wire B hangs vertically as shown in Fig. 27–13. A magnetic field B is directed horizontally, perpendicular to the wire, and points out of the page at all points as represented B by the symbol . The magnetic field B is very nearly uniform along the horizontal portion of wire ab (length l = 10.0 cm) which is near the center of the gap of a large magnet producing the field. The top portion of the wire loop is free of the field. The loop hangs from a balance which measures a downward magnetic force (in addition to the gravitational force) of F = 3.48 * 10–2 N when the wire carries a current I = 0.245 A. What is the magnitude of the magnetic field B?

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APPROACH Three straight sections of the wire loop are in the magnetic field: a horizontal section and two vertical sections. We apply Eq. 27–1 to each section and use the right-hand rule. SOLUTION The magnetic force on the left vertical section of wire points to the left; the force on the vertical section on the right points to the right. These two forces are equal and in opposite directions and so add up to zero. Hence, the net magnetic force on the loop is that on the horizontal section ab, whose length is B l = 0.100 m. The angle u between B and the wire is u = 90°, so sin u = 1. Thus Eq. 27–1 gives

B (toward viewer)

F 3.48 * 10–2 N = = 1.42 T. Il (0.245 A)(0.100 m)

B =

10.0 cm

NOTE This technique can be a precise means of determining magnetic field strength. EXAMPLE 27–3 Magnetic force on a semicircular wire. A rigid wire, carrying a current I, consists of a semicircle of radius R and two straight portions as shown in Fig. 27–14. The wire lies in a plane perpendicular to a uniform B magnetic field B0 . Note choice of x and y axis. The straight portions each have length l within the field. Determine the net force on the wire due to the magnetic B field B0 .

b B

F FIGURE 27–13 Measuring a B magnetic field B. Example 27–2.

APPROACH The forces on the two straight sections are equal A = IlB0 B and in opposite directions, so they cancel. Hence the net force is that on the semicircular portion. y x B

B (away from B0 viewer)

df R FIGURE 27–14 Example 27–3.

f F = IlB0

F = IlB0 l

SOLUTION We divide the semicircle into short lengths dl = R df as indicated B B in Fig. 27–14, and use Eq. 27–4, dF = I dL * B, to find B

dF = IB0 R df, B

where dF is the force on the length dl = R df, and the angle between dL and B0 B is 90° (so sin u = 1 in the cross product). The x component of the force dF on B the segment dL shown, and the x component of dF for a symmetrically located dL on the other side of the semicircle, will cancel each other. Thus for the entire semicircle there will be no x component of force. Hence we need be concerned only with the y components, each equal to dF sin f, and the total force will have magnitude B

F =

dF sin f = IB0 R

sin f df = –IB0 R cos f 2

= 2IB0 R, 0

with direction vertically upward along the y axis in Fig. 27–14. SECTION 27–3

Force on an Electric Current in a Magnetic Field; Definition of B

713

27–4

Right-hand rule B

Force on an Electric Charge Moving in a Magnetic Field

We have seen that a current-carrying wire experiences a force when placed in a magnetic field. Since a current in a wire consists of moving electric charges, we might expect that freely moving charged particles (not in a wire) would also experience a force when passing through a magnetic field. Indeed, this is the case. From what we already know we can predict the force on a single moving electric charge. If N such particles of charge q pass by a given point in time t, they constitute a current I = Nqt. We let t be the time for a charge q to travel a B distance l in a magnetic field B; then L = vt where v is the velocity of the B B particle. Thus, the force on these N particles is, by Eq. 27–3, F = I L * B = B B (Nqt)(vt) * B = Nqv * B. The force on one of the N particles is then B

+qv B –qv B

F B

F = qv * B.

(27–5a)

This basic and important result can be considered as an alternative way of defining B the magnetic field B, in place of Eq. 27–4 or 27–3. The magnitude of the force in Eq. 27–5a is F = qvB sin u. (27–5b) This gives the magnitude of the force on a particle of charge q moving with velocity v B at a point where the magnetic field has magnitude B. The angle between v and B is u. B The force is greatest when the particle moves perpendicular to B (u = 90°): B

FIGURE 27–15 Force on charged particles due to a magnetic field is perpendicular to the magnetic field B direction. If v is horizontal, then F is vertical. Right-hand-rule 3: B force on moving charge exerted by B B

Fmax = qvB.

Cv ⊥ B D B

The force is zero if the particle moves parallel to the field lines (u = 0°). The B direction of the force is perpendicular to the magnetic field B and to the velocity v of the particle. It is given again by a right-hand rule (as for any cross product): you orient your right hand so that your outstretched fingers point along the direction of the particle's velocity (v), and when you bend your fingers they must point B along the direction of B. Then your thumb will point in the direction of the force. This is true only for positively charged particles, and will be "up" for the positive particle shown in Fig. 27–15. For negatively charged particles, the force is in exactly the opposite direction, "down" in Fig. 27–15. B

CONCEPTUAL EXAMPLE 27–4 Negative charge near a magnet. A negative charge –Q is placed at rest near a magnet.Will the charge begin to move? Will it feel a force? What if the charge were positive, ±Q? RESPONSE No to all questions. A charge at rest has velocity equal to zero. Magnetic fields exert a force only on moving electric charges (Eqs. 27–5). EXERCISE E Return to the Chapter-Opening Question, page 707, and answer it again now. Try to explain why you may have answered differently the first time.

FIGURE 27–16 Example 27–5. vB (up) B

Moving into

(a)

(b)

F (west)

page (north)

EXAMPLE 27–5 Magnetic force on a proton. A magnetic field exerts a force of 8.0 * 10–14 N toward the west on a proton moving vertically upward at a speed of 5.0 * 106 ms (Fig. 27–16a). When moving horizontally in a northerly direction, the force on the proton is zero (Fig. 27–16b). Determine the magnitude and direction of the magnetic field in this region. (The charge on a proton is q = ±e = 1.6 * 10–19 C. ) APPROACH Since the force on the proton is zero when moving north, the field must be in a north–south direction. In order to produce a force to the west when the proton B moves upward, the right-hand rule tells us that B must point toward the north. (Your thumb points west and the outstretched fingers of your right hand point upward only B when your bent fingers point north.) The magnitude of B is found using Eq. 27–5b. SOLUTION Equation 27–5b with u = 90° gives B =

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8.0 * 10–14 N F = = 0.10 T. qv A1.6 * 10–19 CBA5.0 * 106 msB

EXAMPLE 27–6 ESTIMATE Magnetic force on ions during a nerve pulse. Estimate the magnetic force due to the Earth's magnetic field on ions crossing a cell membrane during an action potential. Assume the speed of the ions is 10–2 ms (Section 25–10). APPROACH Using F = qvB, set the magnetic field of the Earth to be roughly B L 10 –4 T, and the charge q L e L 10–19 C. SOLUTION F L A10 –19 CBA10–2 msBA10–4 TB = 10 –25 N. NOTE This is an extremely small force. Yet it is thought migrating animals do somehow detect the Earth's magnetic field, and this is an area of active research. The path of a charged particle moving in a plane perpendicular to a uniform magnetic field is a circle as we shall now show. In Fig. 27–17 the magnetic field is directed into the paper, as represented by *'s. An electron at point P is moving to the right, and the force on it at this point is downward as shown (use the right-hand rule and reverse the direction for negative charge). The electron is thus deflected toward the page bottom. A moment later, say, when it reaches point Q, the force is still perpendicular to the velocity and is in the direction shown. Because the force is always perpendicular to v, the magnitude of v does not change—the electron moves at constant speed. We saw in Chapter 5 that if the force on a particle is always perpendicular to its velocity v, the particle moves in a circle and has a centripetal acceleration a = v2r (Eq. 5–1). Thus a charged particle moves in a circular path with constant centripetal acceleration in a uniform magnetic field (see Example 27–7). The electron moves clockwise in Fig. 27–17. A positive particle in this field would feel a force in the opposite direction and would thus move counterclockwise. B

–P

vB Q

–

EXAMPLE 27–7 Electron's path in a uniform magnetic field. An electron travels at 2.0 * 107 ms in a plane perpendicular to a uniform 0.010-T magnetic field. Describe its path quantitatively. APPROACH The electron moves at speed v in a curved path and so must have a centripetal acceleration a = v2r (Eq. 5–1). We find the radius of curvature using Newton's second law. The force is given by Eq. 27–5b with sin u = 1: F = qvB. SOLUTION We insert F and a into Newton's second law: ©F = ma mv2 . qvB = r We solve for r and find mv . r = qB B Since F is perpendicular to v, the magnitude of v doesn't change. From this B equation we see that if B = constant, then r = constant, and the curve must be a circle as we claimed above. To get r we put in the numbers: A9.1 * 10–31 kgBA2.0 * 107 msB r = = 1.1 * 10–2 m = 1.1 cm. A1.6 * 10–19 CB(0.010 T) B

Path of electron

B is into the page

FIGURE 27–17 Force exerted by a uniform magnetic field on a moving charged particle (in this case, an electron) produces a circular path.

FIGURE 27–18 The blue ring inside the glass tube is the glow of a beam of electrons that ionize the gas molecules. The red coils of currentcarrying wire produce a nearly uniform magnetic field, illustrating the circular path of charged particles in a uniform magnetic field.

NOTE See Fig. 27–18.

The time T required for a particle of charge q moving with constant speed v to B make one circular revolution in a uniform magnetic field B (⊥ v) is T = 2prv, where 2pr is the circumference of its circular path. From Example 27–7, r = mvqB, so 2pm . T = qB Since T is the period of rotation, the frequency of rotation is qB 1 . = f = (27–6) T 2pm This is often called the cyclotron frequency of a particle in a field because this is the frequency at which particles revolve in a cyclotron (see Problem 66). B

SECTION 27–4

715

CONCEPTUAL EXAMPLE 27–8 Stopping charged particles. Can a magnetic field be used to stop a single charged particle, as an electric field can? RESPONSE No, because the force is always perpendicular to the velocity of the particle and thus cannot change the magnitude of its velocity. It also means the magnetic force cannot do work on the particle and so cannot change the kinetic energy of the particle.

L SO VIN OBL E PR

by an electric field is parallel to the direction of the field (and unaffected by the velocity of the particle). 2. The right-hand rule, in its different forms, is intended to help you determine the directions of magnetic field, and the forces they exert, and/or the directions of electric current or charged particle velocity. The right-hand rules (Table 27–1) are designed to deal with the "perpendicular" nature of these quantities.

Magnetic Fields Magnetic fields are somewhat analogous to the electric fields of Chapter 21, but there are several important differences to recall: 1. The force experienced by a charged particle moving in a magnetic field is perpendicular to the direction of the magnetic field (and to the direction of the velocity of the particle), whereas the force exerted TABLE 27–1 Summary of Right-hand Rules ( = RHR) Physical Situation

Example

1. Magnetic field produced by current (RHR-1)

Result

Wrap fingers around wire with thumb pointing in direction of current I

Fingers point in direction of B

Fingers point straight along current I, then bend along B magnetic field B

Thumb points in direction B of the force F

Fingers point along particle's B velocity vB, then along B

Thumb points in direction B of the force F

Fig. 27–8c

B B

2. Force on electric current I due to magnetic field (RHR-2)

How to Orient Right Hand

Fig. 27–11c

3. Force on electric charge ± q due to magnetic field (RHR-3)

vB B

CONCEPTUAL EXAMPLE 27–9 A helical path. What is the path of a charged particle in a uniform magnetic field if its velocity is not perpendicular to the magnetic field?

vB)

Fig. 27–15

vB FIGURE 27–19 Example 27–9.

RESPONSE The velocity vector can be broken down into components parallel and perpendicular to the field. The velocity component parallel to the field lines experiences no force Au = 0 2 , so this component remains constant. The velocity component perpendicular to the field results in circular motion about the field lines. Putting these two motions together produces a helical (spiral) motion around the field lines as shown in Fig. 27–19. EXERCISE F What is the sign of the charge in Fig. 27–19? How would you modify the drawing if the sign were reversed?

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P H Y S I C S

* Aurora Borealis Charged ions approach the Earth from the Sun (the "solar wind") and enter the atmosphere mainly near the poles, sometimes causing a phenomenon called the aurora borealis or "northern lights" in northern latitudes. To see why, consider Example 27–9 and Fig. 27–20 (see also Fig. 27–19). In Fig. 27–20 we imagine a stream of charged particles approaching the Earth. The velocity component perpendicular to the field for each particle becomes a circular orbit around the field lines, whereas the velocity component parallel to the field carries the particle along the field lines toward the poles. As a particle approaches the N pole, the magnetic field is stronger and the radius of the helical path becomes smaller. A high concentration of charged particles ionizes the air, and as the electrons recombine with atoms, light is emitted (Chapter 37) which is the aurora. Auroras are especially spectacular during periods of high sunspot activity when the solar wind brings more charged particles toward Earth.

A P P L I E D

The aurora borealis Charged particle approaching Earth

(a)

Lorentz Equation B

If a particle of charge q moves with velocity v in the presence of both a magnetic B B field B and an electric field E, it will feel a force F = qAE + v * B B B

(27–7)

where we have made use of Eqs. 21–3 and 27–5a. Equation 27–7 is often called the Lorentz equation and is considered one of the basic equations in physics.

CONCEPTUAL EXAMPLE 27–10 Velocity selector, or filter: Crossed E and B fields. Some electronic devices and experiments need a beam of charged particles all moving at nearly the same velocity. This can be achieved using both a uniform electric field and a uniform magnetic field, arranged so they are at right angles to each other. As shown in Fig. 27–21a, particles of charge q pass through slit S1 B B and enter the region where B points into the page and E points down from the positive plate toward the negative plate. If the particles enter with different velocities, show how this device "selects" a particular velocity, and determine what this velocity is.

(b) FIGURE 27–20 (a) Diagram showing a negatively charged particle that approaches the Earth and is "captured" by the magnetic field of the Earth. Such particles follow the field lines toward the poles as shown. (b) Photo of aurora borealis (here, in Kansas, where it is a rare sight).

RESPONSE After passing through slit S1 , each particle is subject to two forces as FIGURE 27–21 A velocity selector: shown in Fig. 27–21b. If q is positive, the magnetic force is upwards and the if v = EB, the particles passing electric force downwards. (Vice versa if q is negative.) The exit slit, S2 , is assumed through S1 make it through S2 . to be directly in line with S1 and the particles' velocity v. Depending on the B B (into page) magnitude of v, some particles will be bent upwards and some downwards. The only ones to make it through the slit S2 will be those for which the net S2 S1 force is zero: ©F = qvB - qE = 0. Hence this device selects particles whose E velocity is q v B

E. v = B

(27–8)

(a) FB = qvB × B B

This result does not depend on the sign of the charge q.

EXERCISE G A particle in a velocity selector as diagrammed in Fig. 27–21 hits below the exit hole, S2 . This means that the particle (a) is going faster than the selected speed; (b) is going slower than the selected speed; (c) answer a is true if q 7 0, b is true if q 6 0; (d) answer a is true if q 6 0, b is true if q 7 0.

SECTION 27–4

vB FE = q E B

(b)

Force on an Electric Charge Moving in a Magnetic Field

717

27–5

Axis of rotation

I (a)

F1 B

Axis

I B

F2 (b) B

B = NIA ( to coil face)

θ Axis

When an electric current flows in a closed loop of wire placed in an external magnetic field, as shown in Fig. 27–22, the magnetic force on the current can produce a torque. This is the principle behind a number of important practical devices, including motors and analog voltmeters and ammeters, which we discuss in the next Section. The interaction between a current and a magnetic field is important in other areas as well, including atomic physics. Current flows through the rectangular loop in Fig. 27–22a, whose face we B B assume is parallel to B. B exerts no force and no torque on the horizontal segments of wire because they are parallel to the field and sin u = 0 in Eq. 27–1. But the magnetic field does exert a force on each of the vertical sections of wire B B as shown, F1 and F2 (see also top view, Fig. 27–22b). By right-hand-rule 2 (Fig. 27–11c or Table 27–1) the direction of the force on the upward current on the B left is in the opposite direction from the equal magnitude force F2 on the downward current on the right. These forces give rise to a net torque that acts to rotate the coil about its vertical axis. B Let us calculate the magnitude of this torque. From Eq. 27–2 Acurrent ⊥ B B, the force F = IaB, where a is the length of the vertical arm of the coil. The lever arm for each force is b2, where b is the width of the coil and the "axis" is at the B B midpoint. The torques produced by F1 and F2 act in the same direction, so the total torque is the sum of the two torques:

b b + IaB = IabB = IAB, 2 2

t = IaB

I (c)

Torque on a Current Loop; Magnetic Dipole Moment

where A = ab is the area of the coil. If the coil consists of N loops of wire, the current is then NI, so the torque becomes

FIGURE 27–22 Calculating the torque on a current loop in a B magnetic field B. (a) Loop face B parallel to B field lines; (b) top view; B (c) loop makes an angle to B, reducing the torque since the lever arm is reduced.

t = NIAB. If the coil makes an angle u with the magnetic field, as shown in Fig. 27–22c, the forces are unchanged, but each lever arm is reduced from 12 b to 12 b sin u. Note that B the angle u is taken to be the angle between B and the perpendicular to the face of the coil, Fig. 27–22c. So the torque becomes t = NIAB sin u.

(27–9)

This formula, derived here for a rectangular coil, is valid for any shape of flat coil. The quantity NIA is called the magnetic dipole moment of the coil and is considered a vector: B

= NIA, B

(27–10)

where the direction of A (and therefore of ) is perpendicular to the plane of the coil (the green arrow in Fig. 27–22c) consistent with the right-hand rule (cup your right hand so your fingers wrap around the loop in the direction of current flow, B then your thumb points in the direction of and A). With this definition of , we can rewrite Eq. 27–9 in vector form: B

T = NIA * B B

T = * B, B

(27–11)

which gives the correct magnitude and direction for the torque T. Equation 27–11 has the same form as Eq. 21–9b for an electric dipole B B (with electric dipole moment p) in an electric field E, which is T = p * E. B And just as an electric dipole has potential energy given by U = –p E when in an electric field, we expect a similar form for a magnetic dipole in a magnetic field. In order to rotate a current loop (Fig. 27–22) so as to increase u, we must do work against the torque due to the magnetic field. B

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Hence the potential energy depends on angle (see Eq. 10–22, the work-energy principle for rotational motion) as U =

t du

NIAB sin u du

= –B cos u + C.

If we choose U = 0 at u = p2, then the arbitrary constant C is zero and the potential energy is B U = –B cos u = – B, (27–12) as expected (compare Eq. 21–10). Bar magnets and compass needles, as well as current loops, can be considered as magnetic dipoles. Note the striking similarities of the fields produced by a bar magnet and a current loop, Figs. 27–4b and 27–9. B

EXAMPLE 27–11 Torque on a coil. A circular coil of wire has a diameter of 20.0 cm and contains 10 loops. The current in each loop is 3.00 A, and the coil is placed in a 2.00-T external magnetic field. Determine the maximum and minimum torque exerted on the coil by the field. APPROACH Equation 27–9 is valid for any shape of coil, including circular loops. Maximum and minimum torque are determined by the angle u the coil makes with the magnetic field. SOLUTION The area of one loop of the coil is A = pr2 = p(0.100 m)2 = 3.14 * 10 –2 m2. The maximum torque occurs when the coil's face is parallel to the magnetic field, so u = 90° in Fig. 27–22c, and sin u = 1 in Eq. 27–9: t = NIAB sin u = (10)(3.00 A)A3.14 * 10 –2 m2 B(2.00 T)(1) = 1.88 N m.

The minimum torque occurs if sin u = 0, for which u = 0°, and then t = 0 from Eq. 27–9. NOTE If the coil is free to turn, it will rotate toward the orientation with u = 0°. EXAMPLE 27–12 Magnetic moment of a hydrogen atom. Determine the magnetic dipole moment of the electron orbiting the proton of a hydrogen atom at a given instant, assuming (in the Bohr model) it is in its ground state with a circular orbit of radius 0.529 * 10–10 m. [This is a very rough picture of atomic structure, but nonetheless gives an accurate result.] APPROACH We start by setting the electrostatic force on the electron due to the proton equal to ma = mv2r since the electron's acceleration is centripetal. SOLUTION The electron is held in its orbit by the coulomb force, so Newton's second law, F = ma, gives mv2 e2 = ; r 4p 0 r2 so e2 v = C 4p 0 mr =

A8.99 * 109 N m2C 2 BA1.60 * 10 –19 CB A9.11 * 10 –31 kgBA0.529 * 10 –10 mB

= 2.19 * 106 ms.

Since current is the electric charge that passes a given point per unit time, the revolving electron is equivalent to a current e ev , I = = T 2pr where T = 2prv is the time required for one orbit. Since the area of the orbit is A = pr2, the magnetic dipole moment is ev = IA = Apr 2 B = 12 evr 2pr = 12 A1.60 * 10–19 CBA2.19 * 106 msBA0.529 * 10–10 mB = 9.27 * 10 –24 Am2, or 9.27 * 10 –24 JT.

SECTION 27–5

Torque on a Current Loop; Magnetic Dipole Moment

719

* 27–6

Applications: Motors, Loudspeakers, Galvanometers

* Electric Motors P H Y S I C S

A P P L I E D

DC motor Armature B

S B

F Brushes – + Voltage source FIGURE 27–23 Diagram of a simple dc motor.

An electric motor changes electric energy into (rotational) mechanical energy. A motor works on the principle that a torque is exerted on a coil of current-carrying wire suspended in the magnetic field of a magnet, described in Section 27–5. The coil is mounted on a large cylinder called the rotor or armature, Fig. 27–23, so that it can rotate continuously in one direction. Actually, there are several coils, although only one is indicated in the Figure. The armature is mounted on a shaft or axle. When the armature is in the position shown in Fig. 27–23, the magnetic field exerts forces on B the current in the loop as shown (perpendicular to B and to the current direction). However, when the coil, which is rotating clockwise in Fig. 27–23, passes beyond the vertical position, the forces would then act to return the coil back to vertical if the current remained the same. But if the current could somehow be reversed at that critical moment, the forces would reverse, and the coil would continue rotating in the same direction. Thus, alternation of the current is necessary if a motor is to turn continuously in one direction. This can be achieved in a dc motor with the use of commutators and brushes: as shown in Fig. 27–24, input current passes through stationary brushes that rub against the conducting commutators mounted on the motor shaft. At every half revolution, each commutator changes its connection over to the other brush. Thus the current in the coil reverses every half revolution as required for continuous rotation. Lead wires to armature coil Commutator

Brushes Battery (dc)

FIGURE 27–26 Loudspeaker.

Rigid metal Coil of wire frame (attached to speaker cone)

S N

Magnet I

* Loudspeakers

Lead-in wires

720

FIGURE 27–25 Motor with many windings.

Most motors contain several coils, called windings, each located in a different place on the armature, Fig. 27–25. Current flows through each coil only during a small part of a revolution, at the time when its orientation results in the maximum torque. In this way, a motor produces a much steadier torque than can be obtained from a single coil. An ac motor, with ac current as input, can work without commutators since the current itself alternates. Many motors use wire coils to produce the magnetic field (electromagnets) instead of a permanent magnet. Indeed the design of most motors is more complex than described here, but the general principles remain the same.

FIGURE 27–24 The commutator-brush arrangement in a dc motor ensures alternation of the current in the armature to keep rotation continuous. The commutators are attached to the motor shaft and turn with it, whereas the brushes remain stationary.

CHAPTER 27

Cone

Magnetism

A loudspeaker also works on the principle that a magnet exerts a force on a current-carrying wire. The electrical output of a stereo or TV set is connected to the wire leads of the speaker. The speaker leads are connected internally to a coil of wire, which is itself attached to the speaker cone, Fig. 27–26. The speaker cone is usually made of stiffened cardboard and is mounted so that it can move back and forth freely. A permanent magnet is mounted directly in line with the coil of wire. When the alternating current of an audio signal flows through the wire coil, which is free to move within the magnet, the coil experiences a force due to the magnetic field of the magnet. (The force is to the right at the instant shown in Fig. 27–26.)

As the current alternates at the frequency of the audio signal, the coil and attached speaker cone move back and forth at the same frequency, causing alternate compressions and rarefactions of the adjacent air, and sound waves are produced. A speaker thus changes electrical energy into sound energy, and the frequencies and intensities of the emitted sound waves can be an accurate reproduction of the electrical input.

* Galvanometer The basic component of analog meters (those with pointer and dial), including analog ammeters, voltmeters, and ohmmeters, is a galvanometer. We have already seen how these meters are designed (Section 26–7), and now we can examine how the crucial element, a galvanometer, works. As shown in Fig. 27–27, a galvanometer consists of a coil of wire (with attached pointer) suspended in the magnetic field of a permanent magnet. When current flows through the loop of wire, the magnetic field exerts a torque on the loop, as given by Eq. 27–9,

b a

S B

ts = kf, where k is the stiffness constant of the spring. The coil and attached pointer rotate to the angle where the torques balance. When the needle is in equilibrium at rest, the torques are equal: kf = NIAB sin u, or f =

NIAB sin u . k

The deflection of the pointer, f, is directly proportional to the current I flowing in B the coil, but also depends on the angle u the coil makes with B. For a useful meter we need f to depend only on the current I, independent of u. To solve this problem, magnets with curved pole pieces are used and the galvanometer coil is wrapped around a cylindrical iron core as shown in Fig. 27–28. The iron tends to concentrate B the magnetic field lines so that B always points parallel to the face of the coil at the wire outside the core. The force is then always perpendicular to the face of the coil, and the torque will not vary with angle. Thus f will be proportional to I as required.

27–7

Spring

t = NIAB sin u. This torque is opposed by a spring which exerts a torque ts approximately proportional to the angle f through which it is turned (Hooke's law). That is,

Pivot

FIGURE 27–27 Galvanometer. FIGURE 27–28 Galvanometer coil wrapped on an iron core. Pointer B

S B

F Iron core

Discovery and Properties of the Electron

The electron plays a basic role in our understanding of electricity and magnetism today. But its existence was not suggested until the 1890s. We discuss it here because magnetic fields were crucial for measuring its properties. Toward the end of the nineteenth century, studies were being done on the discharge of electricity through rarefied gases. One apparatus, diagrammed in Fig. 27–29, was a glass tube fitted with electrodes and evacuated so only a small amount of gas remained inside. When a very high voltage was applied to the electrodes, a dark space seemed to extend outward from the cathode (negative electrode) toward the opposite end of the tube; and that far end of the tube would glow. If one or more screens containing a small hole was inserted as shown, the glow was restricted to a tiny spot on the end of the tube. It seemed as though something being emitted by the cathode traveled to the opposite end of the tube. These "somethings" were named cathode rays. There was much discussion at the time about what these rays might be. Some scientists thought they might resemble light. But the observation that the bright spot at the end of the tube could be deflected to one side by an electric or magnetic field suggested that cathode rays could be charged particles; and the direction of the deflection was consistent with a negative charge. Furthermore, if the tube contained certain types of rarefied gas, the path of the cathode rays was made visible by a slight glow.

FIGURE 27–29 Discharge tube. In some models, one of the screens is the anode (positive plate). Screens Cathode

Glow Anode

–

High + voltage

SECTION 27–7

721

+ Anode

I +

–

FIGURE 27–30 Cathode rays deflected by electric and magnetic fields.

+ +

b –

High voltage

–

c Electric field plates

–

Coils to produce magnetic field

Estimates of the charge e of the (assumed) cathode-ray particles, as well as of their charge-to-mass ratio em, had been made by 1897. But in that year, J. J. Thomson (1856–1940) was able to measure em directly, using the apparatus shown in Fig. 27–30. Cathode rays are accelerated by a high voltage and then pass between a pair of parallel plates built into the tube. The voltage applied to the plates produces an electric field, and a pair of coils produces a magnetic field. When only the electric field is present, say with the upper plate positive, the cathode rays are deflected upward as in path a in Fig. 27–30. If only a magnetic field exists, say inward, the rays are deflected downward along path c. These observations are just what is expected for a negatively charged particle. The force on the rays due to the magnetic field is F = evB, where e is the charge and v is the velocity of the cathode rays. In the absence of an electric field, the rays are bent into a curved path, so we have, from F = ma, evB = m

v2 , r

and thus

722

CHAPTER 27

v . e = m Br The radius of curvature r can be measured and so can B. The velocity v can be found by applying an electric field in addition to the magnetic field. The electric field E is adjusted so that the cathode rays are undeflected and follow path b in Fig. 27–30. This is just like the velocity selector of Example 27–10 where the force due to the electric field, F = eE, is balanced by the force due to the magnetic field, F = evB. Thus eE = evB and v = EB. Combining this with the above equation we have E . e = (27–13) m B2r The quantities on the right side can all be measured so that although e and m could not be determined separately, the ratio em could be determined. The accepted value today is em = 1.76 * 1011 Ckg. Cathode rays soon came to be called electrons. It is worth noting that the "discovery" of the electron, like many others in science, is not quite so obvious as discovering gold or oil. Should the discovery of the electron be credited to the person who first saw a glow in the tube? Or to the person who first called them cathode rays? Perhaps neither one, for they had no conception of the electron as we know it today. In fact, the credit for the discovery is generally given to Thomson, but not because he was the first to see the glow in the tube. Rather it is because he believed that this phenomenon was due to tiny negatively charged particles and made careful measurements on them. Furthermore he argued that these particles were constituents of atoms, and not ions or atoms themselves as many thought, and he developed an electron theory of matter. His view is close to what we accept today, and this is why Thomson is credited with the "discovery." Note, however, that neither he nor anyone else ever actually saw an electron itself. We discuss this briefly, for it illustrates the fact that discovery in science is not always a clear-cut matter. In fact some philosophers of science think the word "discovery" is often not appropriate, such as in this case.

Thomson believed that an electron was not an atom, but rather a constituent, or part, of an atom. Convincing evidence for this came soon with the determination of the charge and the mass of the cathode rays. Thomson's student J. S. Townsend made the first direct (but rough) measurements of e in 1897. But it was the more refined oil-drop experiment of Robert A. Millikan (1868–1953) that yielded a precise value for the charge on the electron and showed that charge comes in discrete amounts. In this experiment, tiny droplets of mineral oil carrying an electric charge were allowed to fall under gravity between two parallel plates, Fig. 27–31. The electric field E between the plates was adjusted until the drop was suspended in midair. The downward pull of gravity, mg, was then just balanced by the upward force due to the electric field. Thus qE = mg, so the charge q = mgE. The mass of the droplet was determined by measuring its terminal velocity in the absence of the electric field. Sometimes the drop was charged negatively, and sometimes positively, suggesting that the drop had acquired or lost electrons (by friction, leaving the atomizer). Millikan's painstaking observations and analysis presented convincing evidence that any charge was an integral multiple of a smallest charge, e, that was ascribed to the electron, and that the value of e was 1.6 * 10–19 C. This value of e, combined with the measurement of em, gives the mass of the electron to be A1.6 * 10–19 CBA1.76 * 1011 CkgB = 9.1 * 10 –31 kg. This mass is less than a thousandth the mass of the smallest atom, and thus confirmed the idea that the electron is only a part of an atom. The accepted value today for the mass of the electron is me = 9.11 * 10 –31 kg.

Atomizer

Droplets

– – –

–

Telescope

FIGURE 27–31 Millikan's oil-drop experiment.

CRT, Revisited The cathode ray tube (CRT), which can serve as the picture tube of TV sets, oscilloscopes, and computer monitors, was discussed in Chapter 23. There, in Fig. 23–22, we saw a design using electric deflection plates to maneuver the electron beam. Many CRTs, however, make use of the magnetic field produced by coils to maneuver the electron beam. They operate much like the coils shown in Fig. 27–30.

27–8

The Hall Effect

When a current-carrying conductor is held fixed in a magnetic field, the field exerts a sideways force on the charges moving in the conductor. For example, if electrons move to the right in the rectangular conductor shown in Fig. 27–32a, the inward B B magnetic field will exert a downward force on the electrons FB = –evd * B, where vd is the drift velocity of the electrons (Section 25–8). Thus the electrons will tend to move nearer to face D than face C. There will thus be a potential difference between faces C and D of the conductor. This potential difference builds up until the electric B B field EH that it produces exerts a force, eEH , on the moving charges that is equal and opposite to the magnetic force. This effect is called the Hall effect after E. H. Hall, who discovered it in 1879. The difference of potential produced is called the Hall emf. B The electric field due to the separation of charge is called the Hall field, EH , and points downward in Fig. 27–32a, as shown. In equilibrium, the force due to this electric field is balanced by the magnetic force evd B, so eEH = evd B. Hence EH = vd B. The Hall emf is then (Eq. 23–4b, assuming the conductor is long and thin so EH is uniform) B

eH = EH d = vd Bd,

FIGURE 27–32 The Hall effect. (a) Negative charges moving to the right as the current. (b) Positive charges moving to the left as the current.

C +

EH –

–

B–

–

D (a)

(27–14)

where d is the width of the conductor. A current of negative charges moving to the right is equivalent to positive charges moving to the left, at least for most purposes. But the Hall effect can distinguish these two. As can be seen in Fig. 27–32b, positive particles moving to the left are deflected downward, so that the bottom surface is positive relative to the top surface. This is the reverse of part (a). Indeed, the direction of the emf in the Hall effect first revealed that it is negative particles that move in metal conductors.

–

C –

–

EH +

SECTION 27–8

+ d +

D (b)

The Hall Effect

723

The magnitude of the Hall emf is proportional to the strength of the magnetic field. The Hall effect can thus be used to measure magnetic field strengths. First the conductor, called a Hall probe, is calibrated with known magnetic fields. Then, for the same current, its emf output will be a measure of B. Hall probes can be made very small and are convenient and accurate to use. The Hall effect can also be used to measure the drift velocity of charge carriers when the external magnetic field B is known. Such a measurement also allows us to determine the density of charge carriers in the material. C +

+ B

– –

–

D (a) FIGURE 27–32a (Repeated here for Example 27–13.)

EXAMPLE 27–13 Drift velocity using the Hall effect. A long copper strip 1.8 cm wide and 1.0 mm thick is placed in a 1.2-T magnetic field as in Fig. 27–32a. When a steady current of 15 A passes through it, the Hall emf is measured to be 1.02 mV. Determine the drift velocity of the electrons and the density of free (conducting) electrons (number per unit volume) in the copper. APPROACH We use Eq. 27–14 to obtain the drift velocity, and Eq. 25–13 of Chapter 25 to find the density of conducting electrons. SOLUTION The drift velocity (Eq. 27–14) is vd =

eH 1.02 * 10 –6 V = = 4.7 * 10–5 ms. Bd (1.2 T)A1.8 * 10 –2 mB

The density of charge carriers n is obtained from Eq. 25–13, I = nevd A, A is the cross-sectional area through which the current I flows. Then n =

where

I 15 A = –19 –5 evd A A1.6 * 10 CBA4.7 * 10 msBA1.8 * 10 –2 mBA1.0 * 10–3 mB = 11 * 1028 m–3.

This value for the density of free electrons in copper, n = 11 * 1028 per m3, is the experimentally measured value. It represents more than one free electron per atom, which as we saw in Example 25–14 is 8.4 * 1028 m–3.

* 27–9 P H Y S I C S

A P P L I E D

The mass spectrometer FIGURE 27–33 Bainbridge-type mass spectrometer. The magnetic fields B and B¿ point out of the paper (indicated by the dots), for positive ions.

– + B

B and B E

Detector or film

Mass Spectrometer

A mass spectrometer is a device to measure masses of atoms. It is used today not only in physics but also in chemistry, geology, and medicine, often to identify atoms (and their concentration) in given samples. As shown in Fig. 27–33, ions are produced by heating, or by an electric current, in the source or sample S. The particles, of mass m and electric charge q, pass through slit S1 and enter crossed electric and magnetic fields. Ions follow a straight-line path in this "velocity selector" (as in Example 27–10) if the electric force qE is balanced by the magnetic force qvB: that is, if qE = qvB, or v = EB. Thus only those ions whose speed is v = EB will pass through undeflected and emerge through slit S2 . In the semicircular region, after S2 , there is only a magnetic field, B¿, so the ions follow a circular path. The radius of the circular path is found from their mark on film (or detectors) if B¿ is fixed; or else r is fixed by the position of a detector and B¿ is varied until detection occurs. Newton's second law, ©F = ma, applied to an ion moving in a circle under the influence only of the magnetic field B¿ gives qvB¿ = mv2r. Since v = EB, we have m =

qB¿r qBB¿r . = v E

All the quantities on the right side are known or can be measured, and thus m can be determined.

724

CHAPTER 27

Magnetism

Historically, the masses of many atoms were measured this way. When a pure substance was used, it was sometimes found that two or more closely spaced marks would appear on the film. For example, neon produced two marks whose radii corresponded to atoms of mass 20 and 22 atomic mass units (u). Impurities were ruled out and it was concluded that there must be two types of neon with different masses. These different forms were called isotopes. It was soon found that most elements are mixtures of isotopes, and the difference in mass is due to different numbers of neutrons (discussed in Chapter 41). EXAMPLE 27–14 Mass spectrometry. Carbon atoms of atomic mass 12.0 u are found to be mixed with another, unknown, element. In a mass spectrometer with fixed B¿, the carbon traverses a path of radius 22.4 cm and the unknown's path has a 26.2-cm radius. What is the unknown element? Assume the ions of both elements have the same charge. APPROACH The carbon and unknown atoms pass through the same electric and magnetic fields. Hence their masses are proportional to the radius of their respective paths (see equation on previous page). SOLUTION We write a ratio for the masses, using the equation at the bottom of the previous page: qBB¿rxE mx 26.2 cm = = = 1.17. mC qBB¿rCE 22.4 cm Thus mx = 1.17 * 12.0 u = 14.0 u. The other element is probably nitrogen (see the Periodic Table, inside the back cover). NOTE The unknown could also be an isotope such as carbon-14 A 146CB. See Appendix F. Further physical or chemical analysis would be needed.

Summary A magnet has two poles, north and south. The north pole is that end which points toward geographic north when the magnet is freely suspended. Like poles of two magnets repel each other, whereas unlike poles attract. We can picture that a magnetic field surrounds every magnet. The SI unit for magnetic field is the tesla (T). Electric currents produce magnetic fields. For example, the lines of magnetic field due to a current in a straight wire form circles around the wire, and the field exerts a force on magnets (or currents) near it. A magnetic field exerts a force on an electric current. The B force on an infinitesimal length of wire dL carrying a current I in B a magnetic field B is B

If the field B is uniform over a straight length L of wire, then the force is B B B (27–3) F = IL * B which has magnitude (27–1)

F = IlB sin u B

where u is the angle between magnetic field B and the wire. The direction of the force is perpendicular to the wire and to the magnetic field, and is given by the right-hand rule. This relation B serves as the definition of magnetic field B. B Similarly, a magnetic field B exerts a force on a charge q moving with velocity vB given by B

F = qvB * B. The magnitude of the force is F = qvB sin u, B where u is the angle between vB and B.

F = qE + qvB * B.

(27–7) B

The torque on a current loop in a magnetic field B is B

B TB = * B,

(27–11)

B where is the magnetic dipole moment of the loop:

(27–4)

dF = I dL * B. B

The path of a charged particle moving perpendicular to a uniform magnetic field is a circle. B B If both electric and magnetic fields (E and B) are present, B the force on a charge q moving with velocity v is

(27–5a) (27–5b)

B = NIA.

(27–10)

Here N is the number of coils carrying current I in the loop B and A is a vector perpendicular to the plane of the loop (use right-hand rule, fingers along current in loop) and has magnitude equal to the area of the loop. The measurement of the charge-to-mass ratio (em) of the electron was done using magnetic and electric fields. The charge e on the electron was first measured in the Millikan oil-drop experiment and then its mass was obtained from the measured value of the em ratio. [*In the Hall effect, moving charges in a conductor placed in a magnetic field are forced to one side, producing an emf between the two sides of the conductor.] [*A mass spectrometer uses magnetic and electric fields to measure the mass of ions.]

Summary

725

Questions 1. A compass needle is not always balanced parallel to the Earth's surface, but one end may dip downward. Explain. 2. Draw the magnetic field lines around a straight section of wire carrying a current horizontally to the left. 3. A horseshoe magnet is held vertically with the north pole on the left and south pole on the right. A wire passing between the poles, equidistant from them, carries a current directly away from you. In what direction is the force on the wire? B B B 4. In the relation F = I L * B, which pairs of the vectors B B B (F, L , B) are always at 90°? Which can be at other angles? 5. The magnetic field due to current in wires in your home can affect a compass. Discuss the effect in terms of currents, including if they are ac or dc. 6. If a negatively charged particle enters a region of uniform magnetic field which is perpendicular to the particle's velocity, will the kinetic energy of the particle increase, decrease, or stay the same? Explain your answer. (Neglect gravity and assume there is no electric field.) 7. In Fig. 27–34, charged particles move in the vicinity of a current-carrying wire. For each charged particle, the arrow indicates the direction of motion of the particle, and the ± or – indicates the sign of the charge. For each of the particles, indicate the direction of the magnetic force due to the magnetic + – field produced by the wire. I FIGURE 27–34 Question 7.

–

8. A positively charged particle in a nonuniform magnetic field follows the trajectory shown in Fig. 27–35. Indicate the direction of the magnetic field at points near the path, assuming the path is always in the plane of the page, and indicate the relative magnitudes of the field in each region. FIGURE 27–35 Question 8.

9. Note that the pattern of magnetic field lines surrounding a bar magnet is similar to that of the electric field around an electric dipole. From this fact, predict how the magnetic field will change with distance (a) when near one pole of a very long bar magnet, and (b) when far from a magnet as a whole. 10. Explain why a strong magnet held near a CRT television screen causes the picture to become distorted. Also, explain why the picture sometimes goes completely black where the field is the strongest. [But don't risk damage to your TV by trying this.] 11. Describe the trajectory of a negatively charged particle in the velocity selector of Fig. 27–21 if its speed exceeds EB. What is its trajectory if v 6 EB? Would it make any difference if the particle were positively charged? 12. Can you set a resting electron into motion with a steady magnetic field? With an electric field? Explain.

726

CHAPTER 27

Magnetism

13. A charged particle is moving in a circle under the influence of a uniform magnetic field. If an electric field that points in the same direction as the magnetic field is turned on, describe the path the charged particle will take. 14. The force on a particle in a magnetic field is the idea behind electromagnetic pumping. It is used to pump metallic fluids (such as sodium) and to pump blood in artificial heart machines. The basic design is shown in Fig. 27–36. An electric field is applied perpendicular to a blood vessel and to a magnetic field. Explain how ions are caused to move. Do positive and negative ions feel a force in the same direction? +

FIGURE 27–36 Electromagnetic pumping in a blood vessel. Question 14.

−

15. A beam of electrons is directed toward a horizontal wire carrying a current from left to right (Fig. 27–37). In what direction is the beam deflected?

I FIGURE 27–37 Question 15.

Electron direction

16. A charged particle moves in a straight line through a particular region of space. Could there be a nonzero magnetic field in this region? If so, give two possible situations. 17. If a moving charged particle is deflected sideways in some B region of space, can we conclude, for certain, that B Z 0 in that region? Explain. 18. How could you tell whether moving electrons in a certain region of space are being deflected by an electric field or by a magnetic field (or by both)? 19. How can you make a compass without using iron or other ferromagnetic material? 20. Describe how you could determine the dipole moment of a bar magnet or compass needle. 21. In what positions (if any) will a current loop placed in a uniform magnetic field be in (a) stable equilibrium, and (b) unstable equilibrium? * 22. A rectangular piece of semiconductor is inserted in a magnetic field and a battery is connected to its ends as shown in Fig. 27–38. When a sensitive voltmeter is connected between points a and b, B B it is found that point a is at a higher potential than b. What is a the sign of the charge carriers in b this semiconductor material? FIGURE 27–38 Question 22. * 23. Two ions have the same mass, but one is singly ionized and the other is doubly ionized. How will their positions on the film of the mass spectrometer of Fig. 27–33 differ?

Problems 27–3 Force on Electric Current in Magnetic Field 1. (I) (a) What is the force per meter of length on a straight wire carrying a 9.40-A current when perpendicular to a 0.90-T uniform magnetic field? (b) What if the angle between the wire and field is 35.0°? 2. (I) Calculate the magnitude of the magnetic force on a 240-m length of wire stretched between two towers and carrying a 150-A current. The Earth's magnetic field of 5.0 * 10–5 T makes an angle of 68° with the wire. 3. (I) A 1.6-m length of wire carrying 4.5 A of current toward the south is oriented horizontally. At that point on the Earth's surface, the dip angle of the Earth's magnetic field makes an angle of 41° to the wire. Estimate the magnitude of the magnetic force on the wire due to the Earth's magnetic field of 5.5 * 10 –5 T at this point. 4. (II) The magnetic force per meter on a wire is measured to be only 25 percent of its maximum possible value. Sketch the relationship of the wire and the field if the force had been a maximum, and sketch the relationship as it actually is, calculating the angle between the wire and the magnetic field. 5. (II) The force on a wire is a maximum of 7.50 * 10–2 N when placed between the pole faces of a magnet. The current flows horizontally to the right and the magnetic field is vertical. The wire is observed to "jump" toward the observer when the current is turned on. (a) What type of magnetic pole is the top pole face? (b) If the pole faces have a diameter of 10.0 cm, estimate the current in the wire if the field is 0.220 T. (c) If the wire is tipped so that it makes an angle of 10.0° with the horizontal, what force will it now feel? 6. (II) Suppose a straight 1.00-mm-diameter copper wire could just "float" horizontally in air because of the force due to B the Earth's magnetic field B, which is horizontal, perpendicular to the wire, and of magnitude 5.0 * 10–5 T. What current would the wire carry? Does the answer seem feasible? Explain briefly. 7. (II) A stiff wire 50.0 cm long is bent at a right angle in the middle. One section lies along the z axis and the other is along the line y = 2x in the xy plane. A current of 20.0 A flows in the wire—down the z axis and out the line in the xy plane. The wire passes through a uniform magnetic field B given by B = A0.318 iˆ B T. Determine the magnitude and direction of the total force on the wire. 8. (II) A long wire stretches along the x axis and carries a 3.0-A current to the right (± x). The wire is in a uniform B magnetic field B = A0.20 ˆi - 0.36 jˆ + 0.25 kˆ B T. Determine the components of the force on the wire per cm of length. 9. (II) A current-carrying circular loop of wire (radius r, current I) is partially immersed in a magnetic field of constant magnitude B0 directed out of the page as shown in Fig. 27–39. Determine the net force B0 B0 on the loop due to the θ0 field in terms of u0 . B0 B0 r (Note that u0 points to the dashed line, above which B = 0.) I

10. (II) A 2.0-m-long wire carries a current of 8.2 A and is B immersed within a uniform magnetic field B. When this wire B lies along the ±x axis, a magnetic force F = A –2.5 jˆ B N acts on the wire, and when it lies on the ±y axis, the force is B B F = A2.5 ˆi - 5.0 kˆ B N. Find B. 11. (III) A curved wire, connecting two points a and b, lies in a B plane perpendicular to a uniform magnetic field B and carries a current I. Show that the resultant magnetic force on the wire, no matter what its shape, is the same as that on a curved straight wire connecting the two b points carrying the same current I. t I igh See Fig. 27–40. stra a I FIGURE 27–40 Problem 11. 12. (III) A circular loop of wire, of radius r, carries current I. B It is placed in a magnetic B field whose straight lines B B B seem to diverge from a point B r I a distance d below the loop on its axis. (That is, the field makes an angle u with the loop at all points, Fig. 27–41, d where tan u = rd. ) Deterθ mine the force on the loop.

FIGURE 27–41 Problem 12.

27–4 Force on Charge Moving in Magnetic Field 13. (I) Determine the magnitude and direction of the force on an electron traveling 8.75 * 105 ms horizontally to the east in a vertically upward magnetic field of strength 0.45 T. 14. (I) An electron is projected vertically upward with a speed of 1.70 * 106 ms into a uniform magnetic field of 0.480 T that is directed horizontally away from the observer. Describe the electron's path in this field. 15. (I) Alpha particles of charge q = ±2e and mass m = 6.6 * 10–27 kg are emitted from a radioactive source at a speed of 1.6 * 107 ms. What magnetic field strength would be required to bend them into a circular path of radius r = 0.18 m? 16. (I) Find the direction of the force on a negative charge for each diagram shown in Fig. 27–42, where vB (green) is the B velocity of the charge and B (blue) is the direction of the magnetic field. ( means the vector points inward. means it points outward, toward you.)

vB vB FIGURE 27–42 Problem 16.

vB (b)

B (a)

y FIGURE 27–39 Problem 9.

B vB

(c)

(d)

Problems

727

(e)

(f)

17. (I) Determine the direction of B for each case in B Fig. 27–43, where F represents the maximum magnetic force on a positively charged particle moving with velocity vB. B

F (a)

(b)

F (c)

19. (II) A doubly charged helium atom whose mass is 6.6 * 10 –27 kg is accelerated by a voltage of 2700 V. (a) What will be its radius of curvature if it moves in a plane perpendicular to a uniform 0.340-T field? (b) What is its period of revolution?

20. (II) A proton (mass mp), a deuteron Am = 2mp , Q = eB, and an alpha particle Am = 4mp , Q = 2eB are accelerated by the same potential difference V and then enter a uniform B magnetic field B, where they move in circular paths B perpendicular to B. Determine the radius of the paths for the deuteron and alpha particle in terms of that for the proton. 21. (II) For a particle of mass m and charge q moving in a circular path in a magnetic field B, (a) show that its kinetic energy is proportional to r2, the square of the radius of curvature of its path, and (b) show that its angular momentum is L = qBr 2, about the center of the circle. 22. (II) An electron moves with velocity vB = A7.0 ˆi - 6.0 jˆ B * 104 ms B in a magnetic field B = A – 0.80 iˆ + 0.60 jˆ B T. Determine the magnitude and direction of the force on the electron.

23. (II) A 6.0-MeV (kinetic energy) proton enters a 0.20-T field, in a plane perpendicular to the field. What is the radius of its path? See Section 23–8. 24. (II) An electron experiences the greatest force as it travels 2.8 * 106 ms in a magnetic field when it is moving northward. The force is vertically upward and of magnitude 8.2 * 10 –13 N. What is the magnitude and direction of the magnetic field? 25. (II) A proton moves through a region of space where there B is a magnetic field B = A0.45 ˆi + 0.38 jˆ B T and an electric B field E = A3.0 ˆi - 4.2 jˆ B * 103 Vm. At a given instant, the proton's velocity is vB = A6.0 ˆi + 3.0 jˆ - 5.0 kˆ B * 103 ms. Determine the components of the total force on the proton. B 26. (II) An electron experiences a force F = A3.8 iˆ - 2.7 jˆ B * 10–13 N B when passing through a magnetic field B = (0.85 T) kˆ . Determine the components of the electron's velocity. 27. (II) A particle of charge q moves in a circular path of B radius r in a uniform magnetic field B. If the magnitude of the magnetic field is doubled, and the kinetic energy of the particle remains constant, what happens to the angular momentum of the particle?

CHAPTER 27

18. (II) What is the velocity of a beam of electrons that goes undeflected when passing through perpendicular electric and magnetic fields of magnitude 8.8 * 103 Vm and 7.5 * 10 –3 T, respectively? What is the radius of the electron orbit if the electric field is turned off?

728

F FIGURE 27–43 Problem 17.

28. (II) An electron enters a uniform magnetic field B = 0.28 T B at a 45° angle to B. Determine the radius r and pitch p (distance between loops) of the electron's helical path assuming its speed is 3.0 * 106 ms. See Fig. 27–44.

Magnetism

FIGURE 27–44 Problem 28.

29. (II) A particle with charge q and momentum p, initially moving along the x axis, enters a region where a uniform B magnetic field B = B0 kˆ extends over a width x = l as shown in Fig. 27–45. y The particle is deflected B0 a distance d in the ±y direction as it traverses the field. Determine (a) whether q is positive or negative, and d (b) the magnitude of its momentum p. q x B p

FIGURE 27–45 Problem 29.

30. (II) The path of protons emerging from an accelerator must be bent by 90° by a "bending magnet" so as not to strike a barrier in their path a distance d from their exit hole in the B accelerator. Show that the field B in the bending magnet, which we assume is uniform and can extend over an area 1 d * d, must have magnitude B A2mKe2d 2 B 2, where m is the mass of a proton and K is its kinetic energy. 31. (III) Suppose the Earth's magnetic field at the equator has magnitude 0.50 * 10 –4 T and a northerly direction at all points. Estimate the speed a singly ionized uranium ion (m = 238 u, q = e) would need to circle the Earth 5.0 km above the equator. Can you ignore gravity? [Ignore relativity.] 32. (III) A 3.40-g bullet moves with a speed of 155 ms perpendicular to the Earth's magnetic field of 5.00 * 10 –5 T. If the bullet possesses a net charge of 18.5 * 10–9 C, by what distance will it be deflected from its path due to the Earth's magnetic field after it has traveled 1.00 km? 33. (III) A proton moving with speed v = 1.3 * 105 ms in a field-free region abruptly enters an essentially uniform B magnetic field B = 0.850 T AB ⊥ vB B. If the proton enters the magnetic field region at a 45° angle as shown in Fig. 27–46, (a) at what angle θ does it leave, and (b) at what distance x does it exit the field?

FIGURE 27–46 Problem 33.

45°

34. (III) A particle with charge ± q and mass m travels in a B uniform magnetic field B = B0 kˆ . At time t = 0, the particle's speed is v0 and its velocity vector lies in the xy plane directed at an angle of 30° with respect to the y axis as shown in Fig. 27–47. At a later time t = ta , the particle will cross the x axis at x = a. In terms of q, m, v0 , and B0 , determine (a) a, and (b) ta .

* 27–6 Motors, Galvanometers * 41. (I) If the current to a motor drops by 12%, by what factor does the output torque change? * 42. (I) A galvanometer needle deflects full scale for a 63.0-mA current. What current will give full-scale deflection if the magnetic field weakens to 0.800 of its original value? * 43. (I) If the restoring spring of a galvanometer weakens by 15% over the years, what current will give full-scale deflection if it originally required 46 mA?

y B0

27–7 Discovery of Electron 30°

44. (I) What is the value of qm for a particle that moves in a circle of radius 8.0 mm in a 0.46-T magnetic field if a crossed 260-Vm electric field will make the path straight?

v0 FIGURE 27–47 Problem 34.

q, m

27–5 Torque on a Current Loop; Magnetic Moment 35. (I) How much work is required to rotate the current B loop (Fig. 27–22) in a uniform magnetic field B from B B (a) u = 0° A 7 B B to u = 180°, (b) u = 90° to u = –90°? 36. (I) A 13.0-cm-diameter circular loop of wire is placed with the plane of the loop parallel to the uniform magnetic field between the pole pieces of a large magnet. When 4.20 A flows in the coil, the torque on it is 0.185 mN. What is the magnetic field strength? 37. (II) A circular coil 18.0 cm in diameter and containing twelve loops lies flat on the ground. The Earth's magnetic field at this location has magnitude 5.50 * 10–5 T and points into the Earth at an angle of 66.0° below a line pointing due north. If a 7.10-A clockwise current passes through the coil, determine (a) the torque on the coil, and (b) which edge of the coil rises up, north, east, south, or west. 38. (II) Show that the magnetic dipole moment of an electron orbiting the proton nucleus of a hydrogen atom is related to the orbital momentum L of the electron by e = L. 2m 39. (II) A 15-loop circular coil 22 cm in diameter lies in the xy plane. The current in each loop of the coil is 7.6 A clockwise, B and an external magnetic field B = A0.55 ˆi + 0.60 jˆ - 0.65 kˆ B T passes through the coil. Determine (a) the magnetic B moment of the coil, ; (b) the torque on the coil due to the external magnetic field; (c) the potential energy U of the coil in the field (take the same zero for U as we did in our discussion of Fig. 27–22). 40. (III) Suppose a nonconducting rod of length d carries a uniformly distributed charge Q. It is rotated with angular velocity v about an axis perpendicular to the rod at one end, Fig. 27–48. Show that the magnetic dipole moment of this rod is 16 Qvd2. [Hint: Consider the motion of each infinitesimal length of the rod.] FIGURE 27–48 Problem 40.

Axis

45. (II) An oil drop whose mass is determined to be 3.3 * 10–15 kg is held at rest between two large plates separated by 1.0 cm as in Fig. 27–31. If the potential difference between the plates is 340 V, how many excess electrons does this drop have?

27–8 Hall Effect 46. (II) A Hall probe, consisting of a rectangular slab of current-carrying material, is calibrated by placing it in a known magnetic field of magnitude 0.10 T. When the field is oriented normal to the slab's rectangular face, a Hall emf of 12 mV is measured across the slab's width. The probe is then placed in a magnetic field of unknown magnitude B, and a Hall emf of 63 mV is measured. Determine B assuming that the angle u between the unknown field and the plane of the slab's rectangular face is (a) u = 90°, and (b) u = 60°. 47. (II) A Hall probe used to measure magnetic field strengths consists of a rectangular slab of material (free-electron density n) with width d and thickness t, carrying a current I along its length l. The slab is immersed in a magnetic field of magnitude B oriented perpendicular to its rectangular face (of area ld), so that a Hall emf eH is produced across its width d. The probe's magnetic sensitivity, defined as KH = eHIB, indicates the magnitude of the Hall emf achieved for a given applied magnetic field and current. A slab with a large KH is a good candidate for use as a Hall probe. (a) Show that KH = 1ent. Thus, a good Hall probe has small values for both n and t. (b) As possible candidates for the material used in a Hall probe, consider (i) a typical metal (n L 1 * 1029m3) and (ii) a (doped) semiconductor (n L 3 * 1022m3). Given that a semiconductor slab can be manufactured with a thickness of 0.15 mm, how thin (nm) should a metal slab be to yield a KH value equal to that of the semiconductor slab? Compare this metal slab thickness with the 0.3-nm size of a typical metal atom. (c) For the typical semiconductor slab described in part (b), what is the expected value for eH when I = 100 mA and B = 0.1 T? 48. (II) A rectangular sample of a metal is 3.0 cm wide and 680 mm thick. When it carries a 42-A current and is placed in a 0.80-T magnetic field it produces a 6.5-mV Hall emf. Determine: (a) the Hall field in the conductor; (b) the drift speed of the conduction electrons; (c) the density of free electrons in the metal.

Problems

729

49. (II) In a probe that uses the Hall effect to measure magnetic fields, a 12.0-A current passes through a 1.50-cm-wide 1.30-mm-thick strip of sodium metal. If the Hall emf is 1.86 mV, what is the magnitude of the magnetic field (take it perpendicular to the flat face of the strip)? Assume one free electron per atom of Na, and take its specific gravity to be 0.971. 50. (II) The Hall effect can be used to measure blood flow rate because the blood contains ions that constitute an electric current. (a) Does the sign of the ions influence the emf? (b) Determine the flow velocity in an artery 3.3 mm in diameter if the measured emf is 0.13 mV and B is 0.070 T. (In actual practice, an alternating magnetic field is used.) * 27–9 Mass Spectrometer * 51. (I) In a mass spectrometer, germanium atoms have radii of curvature equal to 21.0, 21.6, 21.9, 22.2, and 22.8 cm. The largest radius corresponds to an atomic mass of 76 u. What are the atomic masses of the other isotopes? * 52. (II) One form of mass spectrometer accelerates ions by a voltage V before they enter a magnetic field B. The ions are assumed to start from rest. Show that the mass of an ion is m = qB2R 22V, where R is the radius of the ions' path in the magnetic field and q is their charge.

* 53. (II) Suppose the electric field between the electric plates in the mass spectrometer of Fig. 27–33 is 2.48 * 104 Vm and the magnetic fields are B = B¿ = 0.58 T. The source contains carbon isotopes of mass numbers 12, 13, and 14 from a long dead piece of a tree. (To estimate atomic masses, multiply by 1.66 * 10–27 kg.) How far apart are the lines formed by the singly charged ions of each type on the photographic film? What if the ions were doubly charged? * 54. (II) A mass spectrometer is being used to monitor air pollutants. It is difficult, however, to separate molecules with nearly equal mass such as CO (28.0106 u) and N2 (28.0134 u). How large a radius of curvature must a spectrometer have if these two molecules are to be separated at the film or detectors by 0.65 mm? * 55. (II) An unknown particle moves in a straight line through crossed electric and magnetic fields with E = 1.5 kVm and B = 0.034 T. If the electric field is turned off, the particle moves in a circular path of radius r = 2.7 cm. What might the particle be?

General Problems 56. Protons move in a circle of radius 5.10 cm in a 0.625-T magnetic field. What value of electric field could make their paths straight? In what direction must the electric field point? 57. Protons with momentum 3.8 * 10–16 kgms are magnetically steered clockwise in a circular path 2.0 km in diameter at Fermi National Accelerator Laboratory in Illinois. Determine the magnitude and direction of the field in the magnets surrounding the beam pipe. 58. A proton and an electron have the same kinetic energy upon entering a region of constant magnetic field. What is the ratio of the radii of their circular paths? 59. Two stiff parallel wires a distance d apart in a horizontal plane act as rails to support a light metal rod of mass m B (perpendicular to each rail), Fig. 27–49. A magnetic field B, directed vertically upward (outward in diagram), acts throughout. At t = 0, a constant current I begins to flow through the system. Determine the speed of the rod, which starts from rest at t = 0, as a function of time (a) assuming no friction between the rod and the rails, and (b) if the coefficient of friction is mk . (c) In which direction does the rod move, east or west, if the current through it heads north?

B West I

North

B d East

I B

South

FIGURE 27–49 Looking down on a rod sliding on rails. Problems 59 and 60.

730

CHAPTER 27

Magnetism

60. Suppose the rod in Fig. 27–49 (Problem 59) has mass m = 0.40 kg and length 22 cm and the current through it is I = 36 A. If the coefficient of static friction is B ms = 0.50, determine the minimum magnetic field B (not necessarily vertical) that will just cause the rod to B slide. Give the magnitude of B and its direction relative to the vertical (outwards towards us). 61. Near the equator, the Earth's magnetic field points almost horizontally to the north and has magnitude B = 0.50 * 10–4 T. What should be the magnitude and direction for the velocity of an electron if its weight is to be exactly balanced by the magnetic force? 62. Calculate the magnetic force on an airplane which has acquired a net charge of 1850 mC and moves with a speed of 120 ms perpendicular to the Earth's magnetic field of 5.0 * 10 –5 T. 63. A motor run by a 9.0-V battery has a 20 turn square coil with sides of length 5.0 cm and total resistance 24 . When spinning, the magnetic field felt by the wire in the coil is 0.020 T. What is the maximum torque on the motor? 64. Estimate the approximate maximum deflection of the electron beam near the center of a CRT television screen due to the Earth's 5.0 * 10–5 T field. Assume the screen is 18 cm from the electron gun, where the electrons are accelerated (a) by 2.0 kV, or (b) by 28 kV. Note that in color TV sets, the beam must be directed accurately to within less than 1 mm in order to strike the correct phosphor. Because the Earth's field is significant here, mu-metal shields are used to reduce the Earth's field in the CRT. (See Section 23–9.) 65. The rectangular loop of wire shown in Fig. 27–22 has mass m and carries current I. Show that if the loop is oriented at an angle u V 1 (in radians), then when it is released it will execute simple harmonic motion about u = 0. Calculate the period of the motion.

66. The cyclotron (Fig. 27–50) is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point A with some initial velocity travel in circular orbits in the magnetic field B. The particles are accelerated to higher speeds each time they pass in the gap between the metal "dees," where there is an electric field E. (There is no electric field within the hollow metal dees.) The electric field changes direction each half-cycle, due to an ac voltage V = V0 sin 2pft, so that the particles are increased in speed at each passage through the gap. (a) Show that the frequency f of the voltage must be f = Bq2pm, where q is the charge on the particles and m their mass. (b) Show that the kinetic energy of the particles increases by 2qV0 each revolution, assuming that the gap is small. (c) If the radius of the cyclotron is 0.50 m and the magnetic field strength is 0.60 T, B B what will be the maximum kinetic energy of accelerated protons in E MeV? A B

FIGURE 27–50 A cyclotron. Problem 66.

∼

Gap "Dees"

67. Magnetic fields are very useful in particle accelerators for "beam steering"; that is, magnetic fields can be used to change the beam's direction without altering its speed (Fig. 27–51). Show how this could work with a beam of protons. What happens to protons that are not moving with the speed that the magnetic field is designed for? If the field extends over a region 5.0 cm wide and has a magnitude of 0.38 T, by approximately what angle will a Magnet beam of protons traveling at 0.85 * 107 ms be bent? B

FIGURE 27–51 Problem 67.

Evacuated tubes, inside of which the protons move with velocity indicated by the green arrows

68. A square loop of aluminum wire is 20.0 cm on a side. It is to carry 15.0 A and rotate in a uniform 1.35-T magnetic field as shown in Fig. 27–52. (a) Determine the minimum diameter of the wire so that it will not fracture from tension or shear. Assume a safety factor of 10. (See Table 12–2.) (b) What is B the resistance of a single loop B I of this wire? FIGURE 27–52 Problem 68.

69. A sort of "projectile launcher" is shown in Fig. 27–53. A large current moves in a closed loop composed of fixed rails, a power supply, and a very light, almost frictionless bar touching the rails. A 1.8 T magnetic field is perpendicular to the plane of the circuit. If the rails are a distance d = 24 cm apart, and the bar has a mass of 1.5 g, what constant current flow is needed to accelerate the bar from rest to 25 ms in a distance of 1.0 m? In what direction must the field point?

I I

FIGURE 27–53 Problem 69. 70. (a) What value of magnetic field would make a beam of electrons, traveling to the right at a speed of 4.8 * 106 ms, go undeflected through a region where there is a uniform electric field of 8400 Vm pointing vertically up? (b) What is the direction of the magnetic field if it is known to be perpendicular to the electric field? (c) What is the frequency of the circular orbit of the electrons if the electric field is turned off? 71. In a certain cathode ray tube, electrons are accelerated horizontally by 25 kV. They then pass through a uniform magnetic field B for a distance of 3.5 cm, which deflects them upward so they reach the top of the screen 22 cm away, 11 cm above the center. Estimate the value of B. 72. Zeeman effect. In the Bohr model of the hydrogen atom, the electron is held in its circular orbit of radius r about its proton nucleus by electrostatic attraction. If the atoms are B placed in a weak magnetic field B, the rotation frequency of B electrons rotating in a plane perpendicular to B is changed by an amount ¢f = &

eB 4pm

where e and m are the charge and mass of an electron. B (a) Derive this result, assuming the force due to B is much less than that due to electrostatic attraction of the nucleus. (b) What does the & sign indicate? 73. A proton follows a spiral path through a gas in a magnetic field of 0.018 T, perpendicular to the plane of the spiral, as shown in Fig. 27–54. In two successive loops, at points P and Q, the radii are 10.0 mm and 8.5 mm, respectively. Calculate the change in the kinetic energy of the proton as it travels from P to Q.

Q P

FIGURE 27–54 Problem 73.

General Problems

731

74. The net force on a current loop whose face is perpendicular to a uniform magnetic field is zero, since contributions to the net force from opposite sides of the loop cancel. However, if the field varies in magnitude from one side of the loop to the other, then there can be a net force on the loop. Consider a square loop with sides whose length is a, located with one side at x = b in the xy plane (Fig. 27–55). A magnetic field is directed along z, with a magnitude that varies with x according to B = B0 a 1 -

x b. b

If the current in the loop circulates counterclockwise (that is, the magnetic dipole moment of the loop is y along the z axis), find B B an expression for the net force on the loop.

a FIGURE 27–55 Problem 74.

75. The power cable for an electric trolley (Fig. 27–56) carries a horizontal current of 330 A toward the east. The Earth's magnetic field has a strength 5.0 * 10–5 T and makes an angle of dip of 22° at this location. Calculate the magnitude and direction of the magnetic force on a 5.0-m length of this cable. I = 330 A

76. A uniform conducting rod of length d and mass m sits atop a fulcrum, which is placed a distance d4 from the rod's left-hand end and is immersed in a uniform magnetic field of magnitude B directed into the page (Fig. 27–57). An object whose mass M is 8.0 times greater than the rod's mass is hung from the rod's left-hand end. What current (direction and magnitude) should flow through the rod in order for it to be "balanced" (i.e., be at rest horizontally) on the fulcrum? (Flexible connecting wires which d d/4 B exert negligible force B on the rod are not m shown.) FIGURE 27–57 Problem 76.

77. In a simple device for measuring the magnitude B of a magnetic field, a conducting rod (length d = 1.0 m, mass m = 150 g) hangs from a friction-free pivot and is oriented so that its axis of rotation is aligned with the direction of the magnetic field to be measured. Thin flexible wires (which exert negligible force on the rod) carry a current I = 12 A, which causes the rod to deflect an angle u with respect to the vertical, where it remains at rest (Fig. 27–58). (a) Is the current flowing upward (toward the pivot) or downward in Fig. 27–58? (b) If B B θ B B u = 13°, determine B. d m (c) What is the largest magnetic field magniI 12 A tude that can be measured using this device?

FIGURE 27–56 Problem 75.

FIGURE 27–58 Problem 77.

Voltage source

Answers to Exercises A: Near the poles, where the field lines are closer together.

E: (b), (c), (d).

B: Counterclockwise. C: (b), (c), (d).

F: Negative; the direction of the helical path would be reversed (still going to the right).

D: 0.15 N.

G: (d).

732

CHAPTER 27

Magnetism

A long coil of wire with many closely spaced loops is called a solenoid. When a long solenoid carries an electric current, a nearly uniform magnetic field is produced within the loops as suggested by the alignment of the iron filings in this photo. The magnitude of the field inside a solenoid is readily found using Ampère's law, one of the great general laws of electromagnetism, relating magnetic fields and electric currents. We examine these connections in detail in this Chapter, as well as other means for producing magnetic fields.

P T E R

C H

Sources of Magnetic Field CHAPTER-OPENING QUESTION—Guess now! Which of the following will produce a magnetic field? (a) An electric charge at rest. (b) A moving electric charge. (c) An electric current. (d) The voltage of a battery not connected to anything. (e) Any piece of iron. (f) A piece of any metal. n the previous Chapter, we discussed the effects (forces and torques) that a magnetic field has on electric currents and on moving electric charges. We also saw that magnetic fields are produced not only by magnets but also by electric currents (Oersted's great discovery). It is this aspect of magnetism, the production of magnetic fields, that we discuss in this Chapter. We will now see how magnetic field strengths are determined for some simple situations, and discuss some general relations between magnetic fields and their sources, electric current. Most elegant is Ampère's law. We also study the Biot-Savart Law, which can be very helpful for solving practical problems.

CONTENTS 28–1 Magnetic Field Due to a Straight Wire 28–2 Force between Two Parallel Wires 28–3 Definitions of the Ampere and the Coulomb 28–4 Ampère's Law 28–5 Magnetic Field of a Solenoid and a Toroid 28–6 Biot-Savart Law 28–7 Magnetic Materials— Ferromagnetism

*28–8 Electromagnets and

Solenoids—Applications

*28–9 Magnetic Fields in Magnetic Materials; Hysteresis

*28–10 Paramagnetism and Diamagnetism

733

28–1

Magnetic Field Due to a Straight Wire

We saw in Section 27–2 that the magnetic field due to the electric current in a long straight wire is such that the field lines are circles with the wire at the center (Fig. 28–1). You might expect that the field strength at a given point would be greater if the current flowing in the wire were greater; and that the field would be less at points farther from the wire. This is indeed the case. Careful experiments show that the magnetic field B due to a long straight wire at a point near it is directly proportional to the current I in the wire and inversely proportional to the distance r from the wire: I. B r r

FIGURE 28–1 Same as Fig. 27–8b. Magnetic field lines around a long straight wire carrying an electric current I.

This relation B r Ir is valid as long as r, the perpendicular distance to the wire, is much less than the distance to the ends of the wire (i.e., the wire is long). The proportionality constant is written† as m02p; thus, m0 I . B = [near a long straight wire] (28–1) 2p r The value of the constant m0 , which is called the permeability of free space, is m0 = 4p * 10–7 TmA.

FIGURE 28–2 Example 28–1.

EXAMPLE 28–1 Calculation of B near a wire. An electric wire in the wall of a building carries a dc current of 25 A vertically upward. What is the magnetic field due to this current at a point P, 10 cm due north of the wire (Fig. 28–2)?

I 10 cm

APPROACH We assume the wire is much longer than the 10-cm distance to the point P so we can apply Eq. 28–1. SOLUTION According to Eq. 28–1:

A4p * 10–7 TmAB(25 A) m0 I B = = = 5.0 * 10–5 T, 2pr (2p)(0.10 m)

or 0.50 G. By the right-hand rule (Table 27–1, page 716), the field due to the current points to the west (into the page in Fig. 28–2) at point P. NOTE The wire's field has about the same magnitude as Earth's magnetic field, so a compass at P would not point north but in a northwesterly direction. NOTE Most electrical wiring in buildings consists of cables with two wires in each cable. Since the two wires carry current in opposite directions, their magnetic fields cancel to a large extent, but may still affect sensitive electronic devices.

C A U T I O N

A compass, near a current, may not point north

EXERCISE A In Example 25–10 we saw that a typical lightning bolt produces a 100-A current for 0.2 s. Estimate the magnetic field 10 m from a lightning bolt. Would it have a significant effect on a compass?

EXAMPLE 28–2 Magnetic field midway between two currents. Two parallel straight wires 10.0 cm apart carry currents in opposite directions (Fig. 28–3). Current I1 = 5.0 A is out of the page, and I2 = 7.0 A is into the page. Determine the magnitude and direction of the magnetic field halfway between the two wires.

FIGURE 28–3 Example 28–2. Wire 1 carrying current I1 out towards us, and wire 2 carrying current I2 into the page, produce magnetic fields whose lines are circles around their respective wires.

B2 I2 (in)

I1 (out) 5.0 cm

APPROACH The magnitude of the field produced by each wire is calculated from Eq. 28–1. The direction of each wire's field is determined with the right-hand rule. The total field is the vector sum of the two fields at the midway point. SOLUTION The magnetic field lines due to current I1 form circles around the wire of I1 , and right-hand-rule-1 (Fig. 27–8c) tells us they point counterclockwise around the wire. The field lines due to I2 form circles around the wire of I2 and point clockwise, Fig. 28–3. At the midpoint, both fields point upward as shown, and so add together.

5.0 cm †

The constant is chosen in this complicated way so that Ampère's law (Section 28–4), which is considered more fundamental, will have a simple and elegant form.

734

CHAPTER 28

Sources of Magnetic Field

The midpoint is 0.050 m from each wire, and from Eq. 28–1 the magnitudes of B1 and B2 are A4p * 10 –7 T mAB(5.0 A) m0 I1 B1 = = = 2.0 * 10–5 T; 2pr 2p(0.050 m) B2

A4p * 10 –7 T mAB(7.0 A) m0 I2 = = = 2.8 * 10–5 T. 2pr 2p(0.050 m) 1

The total field is up with a magnitude of B = B1 + B2 = 4.8 * 10–5 T. EXERCISE B Suppose both I1 and I2 point into the page in Fig. 28–3. What then is the field midway between the two wires?

CONCEPTUAL EXAMPLE 28–3 Magnetic field due to four wires. Figure 28–4 shows four long parallel wires which carry equal currents into or out of the page as shown. In which configuration, (a) or (b), is the magnetic field greater at the center of the square?

2 1

2 3 4

(a)

(b)

FIGURE 28–4 Example 28–3.

RESPONSE It is greater in (a). The arrows illustrate the directions of the field produced by each wire; check it out, using the right-hand rule to confirm these results. The net field at the center is the superposition of the four fields, which will point to the left in (a) and is zero in (b).

28–2

2 3

Force between Two Parallel Wires

We have seen that a wire carrying a current produces a magnetic field (magnitude given by Eq. 28–1 for a long straight wire). Also, a current-carrying wire feels a force when placed in a magnetic field (Section 27–3, Eq. 27–1). Thus, we expect that two current-carrying wires will exert a force on each other. Consider two long parallel wires separated by a distance d, as in Fig. 28–5a. They carry currents I1 and I2 , respectively. Each current produces a magnetic field that is "felt" by the other, so each must exert a force on the other. For example, the magnetic field B1 produced by I1 in Fig. 28–5 is given by Eq. 28–1, which at the location of wire 2 is m0 I1 . B1 = 2p d See Fig. 28–5b, where the field due only to I1 is shown. According to Eq. 27–2, the force F2 exerted by B1 on a length l2 of wire 2, carrying current I2 , is F2 = I2 B1 l2 . Note that the force on I2 is due only to the field produced by I1 . Of course, I2 also produces a field, but it does not exert a force on itself. We substitute B1 into the formula for F2 and find that the force on a length l2 of wire 2 is m0 I1 I2 [parallel wires] (28–2) F2 = l . 2p d 2 If we use right-hand-rule-1 of Fig. 27–8c, we see that the lines of B1 are as shown in Fig. 28–5b. Then using right-hand-rule-2 of Fig. 27–11c, we see that the force exerted on I2 will be to the left in Fig. 28–5b. That is, I1 exerts an attractive force on I2 (Fig. 28–6a). This is true as long as the currents are in the same direction. If I2 is in the opposite direction, the right-hand rule indicates that the force is in the opposite direction. That is, I1 exerts a repulsive force on I2 (Fig. 28–6b). Reasoning similar to that above shows that the magnetic field produced by I2 exerts an equal but opposite force on I1 . We expect this to be true also from Newton's third law. Thus, as shown in Fig. 28–6, parallel currents in the same direction attract each other, whereas parallel currents in opposite directions repel. SECTION 28–2

Wire 1 Wire 2 (a)

(b)

FIGURE 28–5 (a) Two parallel conductors carrying currents I1 and I2 . B (b) Magnetic field B1 produced by I1 . (Field produced by I2 is not shown.) B B1 points into page at position of I2 . FIGURE 28–6 (a) Parallel currents in the same direction exert an attractive force on each other. (b) Antiparallel currents (in opposite directions) exert a repulsive force on each other.

F F

(a)

(b)

Force between Two Parallel Wires

735

EXAMPLE 28–4 Force between two current-carrying wires. The two wires of a 2.0-m-long appliance cord are 3.0 mm apart and carry a current of 8.0 A dc. Calculate the force one wire exerts on the other. APPROACH Each wire is in the magnetic field of the other when the current is on, so we can apply Eq. 28–2. SOLUTION Equation 28–2 gives F =

A4p * 10–7 TmAB(8.0 A)2(2.0 m) (2p)A3.0 * 10–3 mB

= 8.5 * 10 –3 N.

The currents are in opposite directions (one toward the appliance, the other away from it), so the force would be repulsive and tend to spread the wires apart. I1 = 80A d = 20 cm

FB mg B

I2 = ?

FIGURE 28–7 Example 28–5.

EXAMPLE 28–5 Suspending a wire with a current. A horizontal wire carries a current I1 = 80 A dc. A second parallel wire 20 cm below it (Fig. 28–7) must carry how much current I2 so that it doesn't fall due to gravity? The lower wire has a mass of 0.12 g per meter of length. APPROACH If wire 2 is not to fall under gravity, which acts downward, the magnetic force on it must be upward. This means that the current in the two wires must be in the same direction (Fig. 28–6). We can find the current I2 by equating the magnitudes of the magnetic force and the gravitational force on the wire. SOLUTION The force of gravity on wire 2 is downward. For each 1.0 m of wire length, the gravitational force has magnitude F = mg = A0.12 * 10–3 kgmB(1.0 m)A9.8 ms2 B = 1.18 * 10–3 N. The magnetic force on wire 2 must be upward, and Eq. 28–2 gives F =

m0 I1 I2 l 2p d

where d = 0.20 m and I1 = 80 A. We solve this for I2 and set the two force magnitudes equal (letting l = 1.0 m): I2 =

A1.18 * 10–3 NmB 2p(0.20 m) 2pd F a b = = 15 A. m0 I1 l (1.0 m) A4p * 10 –7 TmAB(80 A)

28–3

Definitions of the Ampere and the Coulomb

You may have wondered how the constant m0 in Eq. 28–1 could be exactly 4p * 10 –7 T mA. Here is how it happened. With an older definition of the ampere, m0 was measured experimentally to be very close to this value. Today, m0 is defined to be exactly 4p * 10 –7 TmA. This could not be done if the ampere were defined independently. The ampere, the unit of current, is now defined in terms of the magnetic field B it produces using the defined value of m0 . In particular, we use the force between two parallel current-carrying wires, Eq. 28–2, to define the ampere precisely. If I1 = I2 = 1 A exactly, and the two wires are exactly 1 m apart, then A4p * 10–7 TmAB (1 A)(1 A) m0 I1 I2 F = = = 2 * 10–7 Nm. l 2p d (2p) (1 m) Thus, one ampere is defined as that current flowing in each of two long parallel wires 1 m apart, which results in a force of exactly 2 * 10 –7 N per meter of length of each wire.

736

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Sources of Magnetic Field

This is the precise definition of the ampere. The coulomb is then defined as being exactly one ampere-second: 1 C = 1 As. The value of k or 0 in Coulomb's law (Section 21–5) is obtained from experiment. This may seem a rather roundabout way of defining quantities. The reason behind it is the desire for operational definitions of quantities—that is, definitions of quantities that can actually be measured given a definite set of operations to carry out. For example, the unit of charge, the coulomb, could be defined in terms of the force between two equal charges after defining a value for 0 or k in Eqs. 21–1 or 21–2. However, to carry out an actual experiment to measure the force between two charges is very difficult. For one thing, any desired amount of charge is not easily obtained precisely; and charge tends to leak from objects into the air. The amount of current in a wire, on the other hand, can be varied accurately and continuously (by putting a variable resistor in a circuit). Thus the force between two current-carrying conductors is far easier to measure precisely. This is why we first define the ampere, and then define the coulomb in terms of the ampere. At the National Institute of Standards and Technology in Maryland, precise measurement of current is made using circular coils of wire rather than straight lengths because it is more convenient and accurate. Electric and magnetic field strengths are also defined operationally: the electric field in terms of the measurable force on a charge, via Eq. 21–3; and the magnetic field in terms of the force per unit length on a current-carrying wire, via Eq. 27–2.

28–4

Ampère's Law

In Section 28–1 we saw that Eq. 28–1 gives the relation between the current in a long straight wire and the magnetic field it produces. This equation is valid only for a long straight wire. Is there a general relation between a current in a wire of any shape and the magnetic field around it? The answer is yes: the French scientist André Marie Ampère (1775–1836) proposed such a relation shortly after Oersted's discovery. Consider an arbitrary closed path around a current as shown in Fig. 28–8, and imagine this path as being made up of short segments each of length ¢l. First, we take the product of the length of each segment times the B component of B parallel to that segment (call this component B∑∑). If we now sum all these terms, according to Ampère, the result will be equal to m0 times the net current Iencl that passes through the surface enclosed by the path:

FIGURE 28–8 Arbitrary path enclosing a current, for Ampère's law. The path is broken down into segments of equal length ¢l.

a B∑∑ ¢l = m0 Iencl . The lengths ¢l are chosen so that B∑∑ is essentially constant along each length. The sum must be made over a closed path; and Iencl is the net current passing through the surface bounded by this closed path (orange in Fig. 28–8). In the limit ¢l S 0, this relation becomes

§ B # dL B

Closed path made up of segments of length l

Area enclosed by the path

= m0 Iencl ,

(28–3)

AMPÈRE'S LAW

where dL is an infinitesimal length vector and the vector dot product assures B that the parallel component of B is taken. Equation 28–3 is known as Ampère's law. The integrand in Eq. 28–3 is taken around a closed path, and Iencl is the current passing through the space enclosed by the chosen path or loop.

SECTION 28–4

Ampère's Law

737

To understand Ampère's law better, let us apply it to the simple case of a single long straight wire carrying a current I which we've already examined, and which served as an inspiration for Ampère himself. Suppose we want to B find the magnitude of B at some point A which is a distance r from the wire (Fig. 28–9). We know the magnetic field lines are circles with the wire at their center. So to apply Eq. 28–3 we choose as our path of integration a circle of radius r. The choice of path is ours, so we choose one that will be convenient: at B any point on this circular path, B will be tangent to the circle. Furthermore, since all points on the path are the same distance from the wire, by symmetry we expect B to have the same magnitude at each point. Thus for any B short segment of the circle (Fig. 28–9), B will be parallel to that segment, and (setting Iencl = I)

m0 I =

FIGURE 28–9 Circular path of radius r.

§ B # dL B

§ B dl

= B dl = B(2pr),

where dl = 2pr, the circumference of the circle. We solve for B and obtain B =

FIGURE 28–10 Magnetic field lines around two long parallel wires whose equal currents, I1 and I2 , are coming out of the paper toward the viewer. I1

m0 I . 2pr

This is just Eq. 28–1 for the field near a long straight wire as discussed earlier. Ampère's law thus works for this simple case. A great many experiments indicate that Ampère's law is valid in general. However, as with Gauss's law for the electric field, its practical value as a means to calculate the magnetic field is limited mainly to simple or symmetric situations. Its importance is that it relates the magnetic field to the current in a direct and mathematically elegant way. Ampère's law is thus considered one of the basic laws of electricity and magnetism. It is valid for any situation where the currents and fields are steady and not changing in time, and no magnetic materials are present. We now can see why the constant in Eq. 28–1 is written m02p. This is done so that only m0 appears in Eq. 28–3, rather than, say, 2pk if we had used k in Eq. 28–1. In this way, the more fundamental equation, Ampère's law, has the simpler form. B It should be noted that the B in Ampère's law is not necessarily due only to the current Iencl . Ampère's law, like Gauss's law for the electric field, is valid in B general. B is the field at each point in space along the chosen path due to all sources—including the current I enclosed by the path, but also due to any other sources. For example, the field surrounding two parallel current-carrying wires is the vector sum of the fields produced by each, and the field lines are shown in Fig. 28–10. If the path chosen for the integral (Eq. 28–3) is a circle centered on one of the wires with radius less than the distance between the wires (the dashed line in Fig. 28–10), only the current AI1 B in the encircled wire is included B on the right side of Eq. 28–3. B on the left side of the equation must be the B B total B at each point due to both wires. Note also that B # dL for the path shown in Fig. 28–10 is the same whether the second wire is present or not (in both cases, it equals m0 I1 according to Ampère's law). How can this be? It can be so because the fields due to the two wires partially cancel one another at some B points between them, such as point C in the diagram (B = 0 at a point midway between the wires if I1 = I2); at other points, such as D in Fig. 28–10, the fields B add together to produce a larger field. In the sum, B # dL , these effects just B balance so that B # dL = m0 I1 , whether the second wire is there or not. The B B integral B # dL will be the same in each case, even though B will not be the same at every point for each of the two cases. B

738

CHAPTER 28

Sources of Magnetic Field

EXAMPLE 28–6 Field inside and outside a wire. A long straight cylindrical wire conductor of radius R carries a current I of uniform current density in the conductor. Determine the magnetic field due to this current at (a) points outside the conductor (r 7 R), and (b) points inside the conductor (r 6 R). See Fig. 28–11. Assume that r, the radial distance from the axis, is much less than the length of the wire. (c) If R = 2.0 mm and I = 60 A, what is B at r = 1.0 mm, r = 2.0 mm, and r = 3.0 mm? APPROACH We can use symmetry: Because the wire is long, straight, and cylindrical, we expect from symmetry that the magnetic field must be the same at all points that are the same distance from the center of the conductor. There is no reason why any such point should have preference over others at the same distance from the wire (they are physically equivalent). So B must have the same B value at all points the same distance from the center. We also expect B to be tangent to circles around the wire (Fig. 28–1), so we choose a circular path of integration as we did in Fig. 28–9. SOLUTION (a) We apply Ampère's law, integrating around a circle (r 7 R) centered on the wire (Fig. 28–11a), and then Iencl = I:

§ B # dL B

= B(2pr) = m0 Iencl

or m0 I , [r 7 R] 2pr which is the same result as for a thin wire. (b) Inside the wire (r 6 R), we again choose a circular path concentric with the B cylinder; we expect B to be tangential to this path, and again, because of the symmetry, it will have the same magnitude at all points on the circle. The current enclosed in this case is less than I by a factor of the ratio of the areas: B =

r B

B r R

(a)

Iencl = I

r=R (b)

FIGURE 28–11 Magnetic field inside and outside a cylindrical conductor (Example 28–6).

pr 2 . pR2

So Ampère's law gives

§ B # dL B

= m0 Iencl

B(2pr) = m0 I a

pr 2 b pR 2

so B =

m0 Ir 2pR 2

[r 6 R]

The field is zero at the center of the conductor and increases linearly with r until r = R; beyond r = R, B decreases as 1r. This is shown in Fig. 28–11b. Note that these results are valid only for points close to the center of the conductor as compared to its length. For a current to flow, there must be connecting wires (to a battery, say), and the field due to these conducting wires, if not very far away, will destroy the assumed symmetry. (c) At r = 2.0 mm, the surface of the wire, r = R, so B =

C A U T I O N

Connecting wires can destroy assumed symmetry

A4p * 10–7 T mAB(60 A) m0 I = = 6.0 * 10–3 T. 2pR (2p)A2.0 * 10–3 mB

We saw in (b) that inside the wire B is linear in r. So at r = 1.0 mm, B will be half what it is at r = 2.0 mm or 3.0 * 10–3 T. Outside the wire, B falls off as 1r, so at r = 3.0 mm it will be two-thirds as great as at r = 2.0 mm, or B = 4.0 * 10–3 T. To check, we use our result in (a), B = m0 I2pr, which gives the same result. SECTION 28–4

Ampère's Law

739

P H Y S I C S

A P P L I E D

Coaxial cable (shielding)

Insulating sleeve

Cylindrical braid Solid wire

FIGURE 28–12 Coaxial cable. Example 28–7. FIGURE 28–13 Exercise C. C A B

CONCEPTUAL EXAMPLE 28–7 Coaxial cable. A coaxial cable is a single wire surrounded by a cylindrical metallic braid, as shown in Fig. 28–12. The two conductors are separated by an insulator. The central wire carries current to the other end of the cable, and the outer braid carries the return current and is usually considered ground. Describe the magnetic field (a) in the space between the conductors, and (b) outside the cable. RESPONSE (a) In the space between the conductors, we can apply Ampère's law for a circular path around the center wire, just as we did for the case shown in Figs. 28–9 and 28–11. The magnetic field lines will be concentric circles centered on the center of the wire, and the magnitude is given by Eq. 28–1. The current in the outer conductor has no bearing on this result. (Ampère's law uses only the current enclosed inside the path; as long as the currents outside the path don't affect the symmetry of the field, they do not contribute to the field along the path at all). (b) Outside the cable, we can draw a similar circular path, for we expect the field to have the same cylindrical symmetry. Now, however, there are two currents enclosed by the path, and they add up to zero. The field outside the cable is zero. The nice feature of coaxial cables is that they are self-shielding: no stray magnetic fields exist outside the cable. The outer cylindrical conductor also shields external electric fields from coming in (see also Example 21–14). This makes them ideal for carrying signals near sensitive equipment. Audiophiles use coaxial cables between stereo equipment components and even to the loudspeakers. EXERCISE C In Fig. 28–13, A and B are wires each carrying a 3.0-A current but in B opposite directions. On the circle C, which statement is true? (a) B = 0; (b) B # dL = 0; B # (c) B = 3m0 ; (d) B = –3m0 ; (e) B dL = 6m0 . B

FIGURE 28–14 Example 28–8. l

EXAMPLE 28–8 A nice use for Ampère's law. Use Ampère's law to show that in any region of space where there are no currents the magnetic field cannot be both unidirectional and nonuniform as shown in Fig. 28–14a.

(a)

a d (impossible)

APPROACH The wider spacing of lines near the top of Fig. 28–14a indicates the B field B has a smaller magnitude at the top than it does lower down. We apply Ampère's law to the rectangular path abcd shown dashed in Fig. 28–14a. SOLUTION Because no current is enclosed by the chosen path, Ampère's law gives

§ B # dL B

= 0. B

The integral along sections ab and cd is zero, since B ⊥ dL . Thus

§ B # dL B

= Bbcl - Bdal = ABbc - Bda Bl,

which is not zero since the field Bbc along the path bc is less than the field Bda along B path da. Hence we have a contradiction: B # dL cannot be both zero (since I = 0) and nonzero. Thus we have shown that a nonuniform unidirectional field is not consistent with Ampère's law. A nonuniform field whose direction also changes, as in Fig. 28–14b, is consistent with Ampère's law (convince yourself this is so), and possible. The fringing of a permanent magnet's field (Fig. 27–7) has this shape. B

(b)

(possible)

L SO VIN OBL E PR

Ampère's Law 1. Ampère's law, like Gauss's law, is always a valid statement. But as a calculation tool it is limited mainly to systems with a high degree of symmetry. The first step in applying Ampère's law is to identify useful symmetry. 2. Choose an integration path that reflects the symmetry (see the Examples). Search for paths where B has constant magnitude along the entire path or along segments of the path. Make sure your integration path passes through the point where you wish to evaluate the magnetic field.

740

CHAPTER 28

Sources of Magnetic Field

3. Use symmetry to determine the direction of B along B the integration path. With a smart choice of path, B will be either parallel or perpendicular to the path. 4. Determine the enclosed current, Iencl. Be careful with signs. Let the fingers of your right hand curl along the B direction of B so that your thumb shows the direction of positive current. If you have a solid conductor and your integration path does not enclose the full current, you can use the current density (current per unit area) multiplied by the enclosed area (as in Example 28–6).

28–5

Magnetic Field of a Solenoid and a Toroid

A long coil of wire consisting of many loops is called a solenoid. Each loop produces a magnetic field as was shown in Fig. 27–9. In Fig. 28–15a, we see the field due to a solenoid when the coils are far apart. Near each wire, the field lines are very nearly circles as for a straight wire (that is, at distances that are small compared to the curvature of the wire). Between any two wires, the fields due to each loop tend to cancel. Toward the center of the solenoid, the fields add up to give a field that can be fairly large and fairly uniform. For a long solenoid with closely packed coils, the field is nearly uniform and parallel to the solenoid axis within the entire cross section, as shown in Fig. 28–15b. The field outside the solenoid is very small compared to the field inside, except near the ends. Note that the same number of field lines that are concentrated inside the solenoid, spread out into the vast open space outside. We now use Ampère's law to determine the magnetic field inside a very long (ideally, infinitely long) closely packed solenoid. We choose the path abcd shown in Fig. 28–16, far from either end, for applying Ampère's law. We will consider this path as made up of four segments, the sides of the rectangle: ab, bc, cd, da. Then the left side of Eq. 28–3, Ampère's law, becomes

§ B # dL B

B # dL

B # dL B

B # dL

B # dL .

(a)

(b) FIGURE 28–15 Magnetic field due to a solenoid: (a) loosely spaced turns, (b) closely spaced turns.

The field outside the solenoid is so small as to be negligible compared to the field inside. B Thus the first term in this sum will be zero. Furthermore, B is perpendicular to the segments bc and da inside the solenoid, and is nearly zero between and outside the coils, Current out of page B

d a

Current into page

FIGURE 28–16 Cross-sectional view into a solenoid. The magnetic field inside is straight except at the ends. Red dashed lines indicate the path chosen for use in Ampère's law. and are electric current direction (in the wire loops) out of the page and into the page.

so these terms too are zero. Therefore we have reduced the integral to the segment cd B where B is the nearly uniform field inside the solenoid, and is parallel to dL , so B

§ B # dL B

B # dL B

= Bl,

where l is the length cd. Now we determine the current enclosed by this loop for the right side of Ampère's law, Eq. 28–3. If a current I flows in the wire of the solenoid, the total current enclosed by our path abcd is NI where N is the number of loops our path encircles (five in Fig. 28–16). Thus Ampère's law gives us Bl = m0 NI. If we let n = Nl be the number of loops per unit length, then B = m0 nI. [solenoid] (28–4) This is the magnitude of the magnetic field within a solenoid. Note that B depends only on the number of loops per unit length, n, and the current I. The field does not depend on position within the solenoid, so B is uniform. This is strictly true only for an infinite solenoid, but it is a good approximation for real ones for points not close to the ends. EXAMPLE 28–9 Field inside a solenoid. A thin 10-cm-long solenoid used for fast electromechanical switching has a total of 400 turns of wire and carries a current of 2.0 A. Calculate the field inside near the center. APPROACH We use Eq. 28–4, where the number of turns per unit length is n = 4000.10 m = 4.0 * 103 m–1. SOLUTION B = m0 nI = A4p * 10 –7 T # mABA4.0 * 103 m–1 B(2.0 A) 1.0 * 10–2 T.

SECTION 28–5

741

A close look at Fig. 28–15 shows that the field outside of a solenoid is much like that of a bar magnet (Fig. 27–4). Indeed, a solenoid acts like a magnet, with one end acting as a north pole and the other as south pole, depending on the direction of the current in the loops. Since magnetic field lines leave the north pole of a magnet, the north poles of the solenoids in Fig. 28–15 are on the right. Solenoids have many practical applications, and we discuss some of them later in the Chapter, in Section 28–8. I

Path 1

Path 2 (a)

EXAMPLE 28–10 Toroid. Use Ampère's law to determine the magnetic field (a) inside and (b) outside a toroid, which is like a solenoid bent into the shape of a circle as shown in Fig. 28–17a. APPROACH The magnetic field lines inside the toroid will be circles concentric with the toroid. (If you think of the toroid as a solenoid bent into a circle, B the field lines bend along with the solenoid.) The direction of B is clockwise. We choose as our path of integration one of these field lines of radius r inside the toroid as shown by the dashed line labeled "path 1" in Fig. 28–17a. We make this choice to use the symmetry of the situation, so B will be tangent to the path and will have the same magnitude at all points along the path (although it is not necessarily the same across the whole cross section of the toroid). This chosen path encloses all the coils; if there are N coils, each carrying current I, then Iencl = NI. SOLUTION (a) Ampère's law applied along this path gives

§ B # dL

= m0 Iencl

B(2pr) = m0 NI, where N is the total number of coils and I is the current in each of the coils. Thus m0 NI . B = 2pr

(b) FIGURE 28–17 (a) A toroid. (b) A section of the toroid showing direction of the current for three loops: means current toward you, means current away from you.

The magnetic field B is not uniform within the toroid: it is largest along the inner edge (where r is smallest) and smallest at the outer edge. However, if the toroid is large, but thin (so that the difference between the inner and outer radii is small compared to the average radius), the field will be essentially uniform within the toroid. In this case, the formula for B reduces to that for a straight solenoid B = m0 nI where n = N(2pr) is the number of coils per unit length. (b) Outside the toroid, we choose as our path of integration a circle concentric with the toroid, "path 2" in Fig. 28–17a. This path encloses N loops carrying current I in one direction and N loops carrying the same current in the opposite direction. (Figure 28–17b shows the directions of the current for the parts of the loop on the inside and outside of the toroid.) Thus the net current enclosed by path 2 is zero. For a very tightly packed toroid, all points on path 2 are equidistant from the toroid and equivalent, so we expect B to be the same at all points along the path. Hence, Ampère's law gives

§ B # dL B

= m0 Iencl

B(2pr) = 0 or B = 0. The same is true for a path taken at a radius smaller than that of the toroid. So there is no field exterior to a very tightly wound toroid. It is all inside the loops.

742

CHAPTER 28

Sources of Magnetic Field

28–6

Biot-Savart Law B

The usefulness of Ampère's law for determining the magnetic field B due to particular electric currents is restricted to situations where the symmetry of the B given currents allows us to evaluate B # dL readily. This does not, of course, invalidate Ampère's law nor does it reduce its fundamental importance. Recall the electric case, where Gauss's law is considered fundamental but is limited in its use B B for actually calculating E. We must often determine the electric field E by another method summing over contributions due to infinitesimal charge elements dq via Coulomb's law: dE = A14p 0 BAdqr2 B. A magnetic equivalent to this infinitesimal form of Coulomb's law would be helpful for currents that do not have great symmetry. Such a law was developed by Jean Baptiste Biot (1774–1862) and Felix Savart (1791–1841) shortly after Oersted's discovery in 1820 that a current produces a magnetic field. According to Biot and Savart, a current I flowing in any path can be considered as many tiny (infinitesimal) current elements, such as in the wire of Fig. 28–18. If dL represents any infinitesimal length along which the current is B flowing, then the magnetic field, dB, at any point P in space, due to this element of current, is given by B

m0 I dL * rˆ , dB = 4p r2 B

(28–5)

Biot-Savart law

where r is the displacement vector from the element dL to the point P, and rˆ = rr is the unit vector (magnitude = 1) in the direction of r (see Fig. 28–18). B

d B (out) P I

FIGURE 28–18 Biot-Savart law: theB field at P due to current element B B IdL is dB = Am0 I4pB AdL * rˆ r2 B.

dL I

Equation 28–5 is known as the Biot-Savart law. The magnitude of dB is dB =

m0 Idl sin u , 4pr2

(28–6)

where u is the angle between dL and r (Fig. 28–18). The total magnetic field at point P is then found by summing (integrating) over all current elements: B

B =

m0 I dB = 4p B

dL * rˆ . r2

Note that this is a vector sum. The Biot-Savart law is the magnetic equivalent of Coulomb's law in its infinitesimal form. It is even an inverse square law, like Coulomb's law. An important difference between the Biot-Savart law and Ampère's law B B (Eq. 28–3) is that in Ampère's law C B # dL = m0 Iencl D, B is not necessarily due only to the current enclosed by the path of integration. But in the B Biot-Savart law the field dB in Eq. 28–5 is due only, and entirely, to the current B element IdL . To find the total B at any point in space, it is necessary to include all currents. B

SECTION 28–6

Biot-Savart Law

743

d B (inward)

EXAMPLE 28–11 B due to current I in straight wire. For the field near a long straight wire carrying a current I, show that the Biot-Savart law gives the same result as Eq. 28–1, B = m0 I2pr. APPROACH We calculate the magnetic field in Fig. 28–19 at point P, which is a perpendicular distance R from an infinitely long wire. The current is moving upwards, and both dL and rˆ , which appear in the cross product of Eq. 28–5, are in the plane of B the page. Hence the direction of the field dB due to each element of current must be directed into the plane of the page as shown (right-hand rule for the cross product B B dL * rˆ ). Thus all the dB have the same direction at point P, and add up to give B the same direction consistent with our previous results (Figs. 28–1 and 28–11). B SOLUTION The magnitude of B will be

rB dL B

FIGURE 28–19 Determining B due to a long straight wire using the Biot-Savart law.

B =

m0 I 4p

±q

dy sin u r2

y = –q

where dy = dl and r2 = R 2 + y 2. Note that we are integrating over y (the length of the wire) so R is considered constant. Both y and u are variables, but they are not independent. In fact, y = –Rtan u. Note that we measure y as positive upward from point 0, so for the current element we are considering y 6 0. Then R du R du r2 du . = = 2 2 R sin u (Rr)

dy = ±R csc2 u du =

From Fig. 28–19 we can see that y = – q corresponds to u = 0 and that y = ± q corresponds to u = p radians. So our integral becomes B =

m0 I 1 4p R

sin u du = –

u=0

p m0 I m0 I . cos u 2 = 4pR 2pR 0

This is just Eq. 28–1 for the field near a long wire, where R has been used instead of r. B

EXAMPLE 28–12 Current loop. Determine B for points on the axis of a circular loop of wire of radius R carrying a current I, Fig. 28–20.

FIGURE 28–20 Determining B due to a current loop.

APPROACH For an element of current at the top of the loop, the magnetic field dB at point P on the axis is perpendicular to r as shown, and has magnitude (Eq. 28–5)

dL R

d B⊥

f B

P dB

m0 I dl 4pr2 B since dL is perpendicular to r so ∑dL * rˆ ∑ = dl. We can break dB down into components dB∑∑ and dB⊥ , which are parallel and perpendicular to the axis as shown. SOLUTION When we sum over all the elements of the loop, symmetry tells us that the perpendicular components will cancel on opposite sides, so B⊥ = 0. B Hence, the total B will point along the axis, and will have magnitude dB =

I (in)

B = B∑∑ =

dB cos f

dB r

dB AR

R 2

+ x B2

2 1

where x is the distance of P from the center of the ring, and r2 = R 2 + x2. Now we put in dB from the equation above and integrate around the current loop, 1 noting that all segments dL of current are the same distance, AR 2 + x 2 B 2, from point P: m0 I m0 IR2 R B = dl = 3 3 4p AR 2 + x 2 B 2 2AR2 + x2 B 2 B

since dl = 2pR, the circumference of the loop. NOTE At the very center of the loop (where x = 0 ) the field has its maximum value B =

744

CHAPTER 28

Sources of Magnetic Field

m0 I . 2R

[at center of current loop]

Recall from Section 27–5 that a current loop, such as that just discussed (Fig. 28–20), is considered a magnetic dipole. We saw there that a current loop has a magnetic dipole moment = NIA, where A is the area of the loop and N is the number of coils in the loop, each carrying current I. We also saw in Chapter 27 that a magnetic dipole placed in an external magnetic field experiences a torque and possesses potential energy, just like an electric dipole. In Example 28–12, we looked at another aspect of a magnetic dipole: the magnetic field produced by a magnetic dipole has magnitude, along the dipole axis, of B =

m0 IR 2

2AR 2 + x2 B 2 3

We can write this in terms of the magnetic dipole moment = IA = IpR2 (for a single loop N = 1): B =

m0 . 3 2 2p AR + x2 B 2

[magnetic dipole] (28–7a) I

(Be careful to distinguish for dipole moment from m0 , the magnetic permeability constant.) For distances far from the loop, x W R, this becomes B L

m0 . 2p x3

on axis, d (28–7b) magnetic dipole, x W R

The magnetic field on the axis of a magnetic dipole decreases with the cube of the distance, just as the electric field does for an electric dipole. B decreases as the cube of the distance also for points not on the axis, although the multiplying factor is not the same. The magnetic field due to a current loop can be determined at various points using the Biot-Savart law and the results are in accord with experiment. The field lines around a current loop are shown in Fig. 28–21.

FIGURE 28–21 Magnetic field due to a circular loop of wire. (Same as Fig. 27–9.)

EXAMPLE 28–13 B due to a wire segment. One quarter of a circular loop of wire carries a current I as shown in Fig. 28–22. The current I enters and leaves on straight segments of wire, as shown; the straight wires are along the radial direction from the center C of the circular portion. Find the magnetic field at point C. APPROACH The current in the straight sections produces no magnetic field at point C because dL and rˆ in the Biot-Savart law (Eq. 28–5) are parallel and therefore dL * rˆ = 0. Each piece dL of the curved section of wire produces a B field dB that points into the page at C (right-hand rule). B SOLUTION The magnitude of each dB due to each dl of the circular portion of wire is (Eq. 28–6)

FIGURE 28–22 Example 28–13. I I B

dB =

r=R C

m0 I dl 4pR 2

where r = R is the radius of the curved section, and sin u in Eq. 28–6 is sin 90° = 1. With r = R for all pieces dL , we integrate over a quarter of a circle. B

B =

m0 I 4pR 2

m0 I 1 m0 I . a 2pR b = 2 8R 4pR 4 SECTION 28–6

Biot-Savart Law

745

28–7 Magnetic Materials—Ferromagnetism

(a)

(b)

FIGURE 28–23 (a) An unmagnetized piece of iron is made up of domains that are randomly arranged. Each domain is like a tiny magnet; the arrows represent the magnetization direction, with the arrowhead being the N pole. (b) In a magnet, the domains are preferentially aligned in one direction (down in this case), and may be altered in size by the magnetization process.

FIGURE 28–24 Iron filings line up along magnetic field lines due to a permanent magnet.

C A U T I O N B

B lines form closed loops, B E start on and end on

746

CHAPTER 28

Magnetic fields can be produced (1) by magnetic materials (magnets) and (2) by electric currents. Common magnetic materials include ordinary magnets, iron cores in motors and electromagnets, recording tape, computer hard drives and magnetic stripes on credit cards. We saw in Section 27–1 that iron (and a few other materials) can be made into strong magnets. These materials are said to be ferromagnetic. We now look into the sources of ferromagnetism. A bar magnet, with its two opposite poles near either end, resembles an electric dipole (equal-magnitude positive and negative charges separated by a distance). Indeed, a bar magnet is sometimes referred to as a "magnetic dipole." There are opposite "poles" separated by a distance. And the magnetic field lines of a bar magnet form a pattern much like that for the electric field of an electric dipole: compare Fig. 21–34a with Fig. 27–4 (or 28–24). Microscopic examination reveals that a piece of iron is made up of tiny regions known as domains, less than 1 mm in length or width. Each domain behaves like a tiny magnet with a north and a south pole. In an unmagnetized piece of iron, these domains are arranged randomly, as shown in Fig. 28–23a. The magnetic effects of the domains cancel each other out, so this piece of iron is not a magnet. In a magnet, the domains are preferentially aligned in one direction as shown in Fig. 28–23b (downward in this case). A magnet can be made from an unmagnetized piece of iron by placing it in a strong magnetic field. (You can make a needle magnetic, for example, by stroking it with one pole of a strong magnet.) The magnetization direction of domains may actually rotate slightly to be more nearly parallel to the external field, and the borders of domains may move so domains with magnetic orientation parallel to the external field grow larger (compare Figs. 28–23a and b). We can now explain how a magnet can pick up unmagnetized pieces of iron like paper clips. The field of the magnet's south pole (say) causes a slight realignment of the domains in the unmagnetized object, which then becomes a temporary magnet with its north pole facing the south pole of the permanent magnet; thus, attraction results. Similarly, elongated iron filings in a magnetic field acquire aligned domains and align themselves to reveal the shape of the magnetic field, Fig. 28–24. An iron magnet can remain magnetized for a long time, and is referred to as a "permanent magnet." But if you drop a magnet on the floor or strike it with a hammer, you can jar the domains into randomness and the magnet loses some or all of its magnetism. Heating a permanent magnet can also cause loss of magnetism, for raising the temperature increases the random thermal motion of atoms, which tends to randomize the domains. Above a certain temperature known as the Curie temperature (1043 K for iron), a magnet cannot be made at all. Iron, nickel, cobalt, gadolinium, and certain alloys are ferromagnetic at room temperature; several other elements and alloys have low Curie temperature and thus are ferromagnetic only at low temperatures. Most other metals, such as aluminum and copper, do not show any noticeable magnetic effect (but see Section 28–10). The striking similarity between the fields produced by a bar magnet and by a loop of electric current (Figs. 27–4b and 28–21) offers a clue that perhaps magnetic fields produced by electric currents may have something to do with ferromagnetism. According to modern atomic theory, atoms can be visualized as having electrons that orbit around a central nucleus. The electrons are charged, and so constitute an electric current and therefore produce a magnetic field; but the fields due to orbiting electrons generally all add up to zero. Electrons themselves produce an additional magnetic field, as if they and their electric charge were spinning about their own axes. It is the magnetic field due to electron spin† that is believed to produce ferromagnetism in most ferromagnetic materials. It is believed today that all magnetic fields are caused by electric currents. This means that magnetic field lines always form closed loops, unlike electric field lines which begin on positive charges and end on negative charges. † The name "spin" comes from an early suggestion that this intrinsic magnetic moment arises from the electron "spinning" on its axis (as well as "orbiting" the nucleus) to produce the extra field. However this view of a spinning electron is oversimplified and not valid.

EXERCISE D Return to the Chapter-Opening Question, page 733, and answer it again now. Try to explain why you may have answered differently the first time.

* 28–8

Electromagnets and Solenoids—Applications

A long coil of wire consisting of many loops of wire, as discussed in Section 28–5, is called a solenoid. The magnetic field within a solenoid can be fairly large since it will be the sum of the fields due to the current in each loop (see Fig. 28–25). The solenoid acts like a magnet; one end can be considered the north pole and the other the south pole, depending on the direction of the current in the loops (use the right-hand rule). Since the magnetic field lines leave the north pole of a magnet, the north pole of the solenoid in Fig. 28–25 is on the right. If a piece of iron is placed inside a solenoid, the magnetic field is increased greatly because the domains of the iron are aligned by the magnetic field produced by the current. The resulting magnetic field is the sum of that due to the current and that due to the iron, and can be hundreds or thousands of times larger than that due to the current alone (see Section 28–9). This arrangement is called an electromagnet. The alloys of iron used in electromagnets acquire and lose their magnetism quite readily when the current is turned on or off, and so are referred to as "soft iron." (It is "soft" only in a magnetic sense.) Iron that holds its magnetism even when there is no externally applied field is called "hard iron." Hard iron is used in permanent magnets. Soft iron is usually used in electromagnets so that the field can be turned on and off readily. Whether iron is hard or soft depends on heat treatment, type of alloy, and other factors. Electromagnets have many practical applications, from use in motors and generators to producing large magnetic fields for research. Sometimes an iron core is not present—the magnetic field comes only from the current in the wire coils. When the current flows continuously in a normal electromagnet, a great deal of waste heat (I 2R power) can be produced. Cooling coils, which are tubes carrying water, are needed to absorb the heat in larger installations. For some applications, the current-carrying wires are made of superconducting material kept below the transition temperature (Section 25–9). Very high fields can be produced with superconducting wire without an iron core. No electric power is needed to maintain large current in the superconducting coils, which means large savings of electricity; nor must huge amounts of heat be dissipated. It is not a free ride, though, because energy is needed to keep the superconducting coils at the necessary low temperature. Another useful device consists of a solenoid into which a rod of iron is partially inserted. This combination is also referred to as a solenoid. One simple use is as a doorbell (Fig. 28–26). When the circuit is closed by pushing the button, the coil effectively becomes a magnet and exerts a force on the iron rod. The rod is pulled into the coil and strikes the bell. A large solenoid is used in the starters of cars; when you engage the starter, you are closing a circuit that not only turns the starter motor, but activates a solenoid that first moves the starter into direct contact with the gears on the engine's flywheel. Solenoids are used as switches in many devices. They have the advantage of moving mechanical parts quickly and accurately.

l FIGURE 28–25 Magnetic field of a solenoid. The north pole of this solenoid, thought of as a magnet, is on the right, and the south pole is on the left. P H Y S I C S

A P P L I E D

Electromagnets and solenoids

P H Y S I C S

A P P L I E D

Doorbell, car starter Iron rod Bell Spring Switch Voltage

I Voltage FIGURE 28–26 Solenoid used as a doorbell.

* Magnetic Circuit Breakers Modern circuit breakers that protect houses and buildings from overload and fire contain not only a "thermal" part (bimetallic strip as described in Section 25–6, Fig. 25–19) but also a magnetic sensor. If the current is above a certain level, the magnetic field it produces pulls an iron plate that breaks the same contact points as in Fig. 25–19b and c. In more sophisticated circuit breakers, including ground fault circuit interrupters (GFCIs—discussed in Section 29–8), a solenoid is used. The iron rod of Fig. 28–26, instead of striking a bell, strikes one side of a pair of points, opening them and opening the circuit. Magnetic circuit breakers react quickly (6 10 msec), and for buildings are designed to react to the high currents of shorts (but not shut off for the start-up surges of motors).

P H Y S I C S

A P P L I E D

Magnetic circuit breakers

*SECTION 28–8

747

* 28–9 I

Magnetic Fields in Magnetic Materials; Hysteresis

The field of a long solenoid is directly proportional to the current. Indeed, Eq. 28–4 tells us that the field B0 inside a solenoid is given by B0 = m0 nI. This is valid if there is only air inside the coil. If we put a piece of iron or other ferromagnetic material inside the solenoid, the field will be greatly increased, often by hundreds or thousands of times. This occurs because the domains in the iron become preferentially aligned by the external field. The resulting magnetic field is the sum of that due to the current and that due to the iron. It is sometimes convenient to write the total field in this case as a sum of two terms: B

B = B0 + BM . FIGURE 28–27 Iron-core toroid. FIGURE 28–28 Total magnetic field B in an iron-core toroid as a function of the external field B0 (B0 is caused by the current I in the coil). 1.20

B = mnI;

1.00

0.60 0.40 0.20 a

B0 (T) 0.40 0.80 1.20 10 −3 10 −3 10 −3

FIGURE 28–29 Hysteresis curve.

1.20 B (T) 1.00 c 0.80 0.60 0.40 0.20 a 0

B0 (T) 0.40 0.80 1.20 10 −3 10 −3 10 −3

f e

748

CHAPTER 28

(28–9)

m is called the magnetic permeability of the material (do not confuse it with for magnetic dipole moment). For ferromagnetic materials, m is much greater than m0 . For all other materials, its value is very close to m0 (Section 28–10). The value of m, however, is not constant for ferromagnetic materials; it depends on the value of the external field B0 , as the following experiment shows. Measurements on magnetic materials are generally done using a toroid, which is essentially a long solenoid bent into the shape of a circle (Fig. 28–27), so that B practically all the lines of B remain within the toroid. Suppose the toroid has an iron core that is initially unmagnetized and there is no current in the windings of the toroid. Then the current I is slowly increased, and B0 increases linearly with I. The total field B also increases, but follows the curved line shown in the graph of Fig. 28–28. (Note the different scales: B W B0 .) Initially, point a, the domains (Section 28–7) are randomly oriented. As B0 increases, the domains become more and more aligned until at point b, nearly all are aligned. The iron is said to be approaching saturation. Point b is typically 70% of full saturation. (If B0 is increased further, the curve continues to rise very slowly, and reaches 98% saturation only when B0 reaches a value about a thousandfold above that at point b; the last few domains are very difficult to align.) Next, suppose the external field B0 is reduced by decreasing the current in the toroid coils. As the current is reduced to zero, shown as point c in Fig. 28–29, the domains do not become completely random. Some permanent magnetism remains. If the current is then reversed in direction, enough domains can be turned around so B = 0 (point d). As the reverse current is increased further, the iron approaches saturation in the opposite direction (point e). Finally, if the current is again reduced to zero and then increased in the original direction, the total field follows the path efgb, again approaching saturation at point b. Notice that the field did not pass through the origin (point a) in this cycle. The fact that the curves do not retrace themselves on the same path is called hysteresis. The curve bcdefgb is called a hysteresis loop. In such a cycle, much energy is transformed to thermal energy (friction) due to realigning of the domains. It can be shown that the energy dissipated in this way is proportional to the area of the hysteresis loop. B

0.80

Here, B0 refers to the field due only to the current in the wire (the "external field"). It is the field that would be present in the absence of a ferromagnetic B material. Then BM represents the additional field due to the ferromagnetic material B B itself; often BM W B0 . The total field inside a solenoid in such a case can also be written by replacing the constant m0 in Eq. 28–4 by another constant, m, characteristic of the material inside the coil:

B (T) b

(28–8)

Sources of Magnetic Field

At points c and f, the iron core is magnetized even though there is no current in the coils. These points correspond to a permanent magnet. For a permanent magnet, it is desired that ac and af be as large as possible. Materials for which this is true are said to have high retentivity. Materials with a broad hysteresis curve as in Fig. 28–29 are said to be magnetically "hard" and make good permanent magnets. On the other hand, a hysteresis curve such as that in Fig. 28–30 occurs for "soft" iron, which is preferred for electromagnets and transformers (Section 29–6) since the field can be more readily switched off, and the field can be reversed with less loss of energy. A ferromagnetic material can be demagnetized—that is, made unmagnetized. This can be done by reversing the magnetizing current repeatedly while decreasing its magnitude. This results in the curve of Fig. 28–31. The heads of a tape recorder are demagnetized in this way. The alternating magnetic field acting at the heads due to a handheld demagnetizer is strong when the demagnetizer is placed near the heads and decreases as it is moved slowly away. Video and audio tapes themselves can be erased and ruined by a magnetic field, as can computer hard disks, other magnetic storage devices, and the magnetic stripes on credit cards.

* 28–10

Paramagnetism and Diamagnetism

FIGURE 28–30 Hysteresis curve for soft iron. FIGURE 28–31 Successive hysteresis loops during demagnetization. B

All materials are magnetic to at least a tiny extent. Nonferromagnetic materials fall into two principal classes: paramagnetic, in which the magnetic permeability m is very slightly greater than m0 ; and diamagnetic, in which m is very slightly less than m0 . The ratio of m to m0 for any material is called the relative permeability Km : Km =

m . m0

Another useful parameter is the magnetic susceptibility xm defined as xm = Km - 1. xm 7 0, whereas diamagnetic Paramagnetic substances have Km 7 1 and x substances have Km 6 1 and m 6 0. See Table 28–1, and note how small the effect is. TABLE 28–1 Paramagnetism and Diamagnetism: Magnetic Susceptibilities Xm

Paramagnetic substance Aluminum Calcium Magnesium Oxygen (STP) Platinum Tungsten

2.3 1.9 1.2 2.1 2.9 6.8

* * * * * *

10 –5 10 –5 10 –5 10 –6 10 –4 10 –5

Diamagnetic substance Copper Diamond Gold Lead Nitrogen (STP) Silicon

–9.8 –2.2 –3.6 –1.7 –5.0 –4.2

* * * * * *

10–6 10–5 10–5 10–5 10–9 10–6

The difference between paramagnetic and diamagnetic materials can be understood theoretically at the molecular level on the basis of whether or not the molecules have a permanent magnetic dipole moment. One type of paramagnetism occurs in materials whose molecules (or ions) have a permanent magnetic dipole moment.† In the absence of an external field, the molecules are randomly oriented and no magnetic effects are observed. However, when an external magnetic field is applied, say, by putting the material in a solenoid, the applied field exerts a torque on the magnetic dipoles (Section 27–5), tending to align them parallel to the field. The total magnetic field (external plus that due to aligned magnetic dipoles) will be slightly greater than B0 . The thermal motion of the molecules reduces the alignment, however. † Other types of paramagnetism also occur whose origin is different from that described here, such as in metals where free electrons can contribute.

*SECTION 28–10

Paramagnetism and Diamagnetism

749

A useful quantity is the magnetization vector, M, defined as the magnetic dipole moment per unit volume, , V B

M =

where is the magnetic dipole moment of the sample and V its volume. It is found experimentally that M is directly proportional to the external magnetic field (tending to align the dipoles) and inversely proportional to the kelvin temperature T (tending to randomize dipole directions). This is called Curie's law, after Pierre Curie (1859–1906), who first noted it: B

M = C

B, T

where C is a constant. If the ratio BT is very large (B very large or T very small) Curie's law is no longer accurate; as B is increased (or T decreased), the magnetization approaches some maximum value, Mmax . This makes sense, of course, since Mmax corresponds to complete alignment of all the permanent magnetic dipoles. However, even for very large magnetic fields, L 2.0 T, deviations from Curie's law are normally noted only at very low temperatures, on the order of a few kelvins. Ferromagnetic materials, as mentioned in Section 28–7, are no longer ferromagnetic above a characteristic temperature called the Curie temperature (1043 K for iron). Above this Curie temperature, they generally are paramagnetic. Diamagnetic materials (for which mm is slightly less than m0) are made up of molecules that have no permanent magnetic dipole moment. When an external magnetic field is applied, magnetic dipoles are induced, but the induced magnetic dipole moment is in the direction opposite to that of the field. Hence the total field will be slightly less than the external field. The effect of the external field—in the crude model of electrons orbiting nuclei—is to increase the "orbital" speed of electrons revolving in one direction, and to decrease the speed of electrons revolving in the other direction; the net result is a net dipole moment opposing the external field. Diamagnetism is present in all materials, but is weaker even than paramagnetism and so is overwhelmed by paramagnetic and ferromagnetic effects in materials that display these other forms of magnetism.

Summary The magnetic field B at a distance r from a long straight wire is directly proportional to the current I in the wire and inversely proportional to r: B =

m0 I . 2p r

(28–1)

The magnetic field lines are circles centered at the wire. The force that one long current-carrying wire exerts on a second parallel current-carrying wire 1 m away serves as the definition of the ampere unit, and ultimately of the coulomb as well. Ampère's law states that the line integral of the magnetic B field B around any closed loop is equal to m0 times the total net current Iencl enclosed by the loop:

§ B # dL B

= m0 Iencl .

(28–3)

The magnetic field inside a long tightly wound solenoid is B = m0 nI

(28–4)

where n is the number of coils per unit length and I is the current in each coil.

750

CHAPTER 28

Sources of Magnetic Field

The Biot-Savart law is useful for determining the magnetic field due to a known arrangement of currents. It states that B m0 I dL * rˆ B , (28–5) dB = 4p r2 where dB is the contribution to the total fieldBat some point P due to a current I along an infinitesimal length dL of its path, and rˆ is the unit vector along the direction of the displacement vector Br B B B from dL to P. The total field B will be the integral over all dB. Iron and a few other materials can be made into strong permanent magnets. They are said to be ferromagnetic. Ferromagnetic materials are made up of tiny domains—each a tiny magnet—which are preferentially aligned in a permanent magnet, but randomly aligned in a nonmagnetized sample. [*When a ferromagnetic material is placed in a magnetic field B0 due to a current, say inside a solenoid or toroid, the material becomes magnetized. When the current is turned off, however, the material remains magnetized, and when the current is increased in the opposite direction (and then again reversed), a graph of the total field B versus B0 is a hysteresis loop, and the fact that the curves do not retrace themselves is called hysteresis.] [*All materials exhibit some magnetic effects. Nonferromagnetic materials have much smaller paramagnetic or diamagnetic properties.]

Questions 1. The magnetic field due to current in wires in your home can affect a compass. Discuss the problem in terms of currents, depending on whether they are ac or dc, and their distance away. 2. Compare and contrast the magnetic field due to a long straight current and the electric field due to a long straight line of electric charge at rest (Section 21–7). 3. Two insulated long wires carrying equal currents I cross at right angles to each other. Describe the magnetic force one exerts on the other.

16. A type of magnetic switch similar to a solenoid is a relay (Fig. 28–32). A relay is an electromagnet (the iron rod inside the coil does not move) which, when activated, attracts a piece of iron on a pivot. Design a relay to close an electrical switch. A relay is used when you need to switch on a circuit carrying a very large current but you do not want that large current flowing through the main switch. For example, the starter switch of a car is connected to a relay so that the large current needed for the starter doesn't pass to the dashboard switch.

4. A horizontal wire carries a large current. A second wire carrying a current in the same direction is suspended below it. Can the current in the upper wire hold the lower wire in suspension against gravity? Under what conditions will the lower wire be in equilibrium? 5. A horizontal current-carrying wire, free to move in Earth's gravitational field, is suspended directly above a second, parallel, current-carrying wire. (a) In what direction is the current in the lower wire? (b) Can the upper wire be held in stable equilibrium due to the magnetic force of the lower wire? Explain. 6. (a) Write Ampère's law for a path that surrounds both conductors in Fig. 28–10. (b) Repeat, assuming the lower current, I2 , is in the opposite direction AI2 = –I1 B. 7. Suppose the cylindrical conductor of Fig. 28–11a has a concentric cylindrical hollow cavity inside it (so it looks like B a pipe). What can you say about B in the cavity?

8. Explain why a field such as that shown in Fig. 28–14b is consistent with Ampère's law. Could the lines curve upward instead of downward? 9. What would be the effect on B inside a long solenoid if (a) the diameter of all the loops was doubled, or (b) the spacing between loops was doubled, or (c) the solenoid's length was doubled along with a doubling in the total number of loops. 10. Use the Biot-Savart law to show that the field of the current loop in Fig. 28–21 is correct as shown for points off the axis. B

11. Do you think B will be the same for all points in the plane of the current loop of Fig. 28–21? Explain. 12. Why does twisting the lead-in wires to electrical devices reduce the magnetic effects of the leads? 13. Compare the Biot-Savart law with Coulomb's law. What are the similarities and differences? 14. How might you define or determine the magnetic pole strength (the magnetic equivalent of a single electric charge) for (a) a bar magnet, (b) a current loop? 15. How might you measure the magnetic dipole moment of the Earth?

FIGURE 28–32 Question 16. 17. A heavy magnet attracts, from rest, a heavy block of iron. Before striking the magnet the block has acquired considerable kinetic energy. (a) What is the source of this kinetic energy? (b) When the block strikes the magnet, some of the latter's domains may be jarred into randomness; describe the energy transformations. 18. Will a magnet attract any metallic object, such as those made of aluminum, or only those made of iron? (Try it and see.) Why is this so? 19. An unmagnetized nail will not attract an unmagnetized paper clip. However, if one end of the nail is in contact with a magnet, the other end will attract a paper clip. Explain. 20. Can an iron rod attract a magnet? Can a magnet attract an iron rod? What must you consider to answer these questions? 21. How do you suppose the first magnets found in Magnesia were formed? 22. Why will either pole of a magnet attract an unmagnetized piece of iron? 23. Suppose you have three iron rods, two of which are magnetized but the third is not. How would you determine which two are the magnets without using any additional objects? 24. Two iron bars attract each other no matter which ends are placed close together. Are both magnets? Explain. * 25. Describe the magnetization curve for (a) a paramagnetic substance and (b) a diamagnetic substance, and compare to that for a ferromagnetic substance (Fig. 28–29). * 26. Can all materials be considered (a) diamagnetic, (b) paramagnetic, (c) ferromagnetic? Explain.

Problems 28–1 and 28–2 Straight Wires, Magnetic Field, and Force 1. (I) Jumper cables used to start a stalled vehicle often carry a 65-A current. How strong is the magnetic field 3.5 cm from one cable? Compare to the Earth's magnetic field (5.0 * 10 - 5 T).

2. (I) If an electric wire is allowed to produce a magnetic field no larger than that of the Earth A0.50 * 10–4 TB at a distance of 15 cm from the wire, what is the maximum current the wire can carry?

Problems

751

3. (I) Determine the magnitude and direction of the force between two parallel wires 25 m long and 4.0 cm apart, each carrying 35 A in the same direction. 4. (I) A vertical straight wire carrying an upward 28-A current exerts an attractive force per unit length of 7.8 * 10–4 Nm on a second parallel wire 7.0 cm away. What current (magnitude and direction) flows in the second wire? 5. (I) In Fig. 28–33, a long straight wire carries current I out of the page toward the viewer. Indicate, with appropriate B arrows, the direction of B at each of the points C, D, and E in the plane of the page. C D I FIGURE 28–33 Problem 5.

13.0 cm 10. 0c m

6. (II) An experiment on the Earth's magnetic field is being carried out 1.00 m from an electric cable. What is the maximum allowable current in the cable if the experiment is to be accurate to &2.0%? 7. (II) Two long thin parallel wires 13.0 cm apart carry 35-A currents in the same direction. Determine the magnetic field vector at a point 10.0 cm from one wire and 6.0 cm from the 6.0 other (Fig. 28–34). cm P

FIGURE 28–34 Problem 7. 8. (II) A horizontal compass is placed 18 cm due south from a straight vertical wire carrying a 43-A current downward. In what direction does the compass needle point at this location? Assume the horizontal component of the Earth's field at this point is 0.45 * 10–4 T and the magnetic declination is 0°. 9. (II) A long horizontal wire carries 24.0 A of current due north. What is the net magnetic field 20.0 cm due west of the wire if the Earth's field there points downward, 44° below the horizontal, and has magnitude 5.0 * 10 –5 T? 10. (II) A straight stream of protons passes a given point in space at a rate of 2.5 * 109 protonss. What magnetic field do they produce 2.0 m from the beam? 11. (II) Determine the magnetic field midway between two long straight wires 2.0 cm apart in terms of the current I in one when the other carries 25 A. Assume these currents are (a) in the same direction, and (b) in opposite directions. 12. (II) Two straight parallel wires are separated by 6.0 cm. There is a 2.0-A current flowing in the first wire. If the magnetic field strength is found to be zero between the two wires at a distance of 2.2 cm from the first wire, what is the magnitude and direction of the current in the second wire?

752

CHAPTER 28

Sources of Magnetic Field

13. (II) Two long straight wires each carry a current I out of the page toward the viewer, Fig. 28–35. Indicate, with approB priate arrows, the direction of B at each of the points 1 to 6 in the plane of the I 4 page. State if the field is zero at any of the points. 1 2

3 FIGURE 28–35 Problem 13.

14. (II) A long pair of insulated wires serves to conduct 28.0 A of dc current to and from an instrument. If the wires are of negligible diameter but are 2.8 mm apart, what is the magnetic field 10.0 cm from their midpoint, in their plane (Fig. 28–36)? Compare to the magnetic field of the Earth.

10.0 cm

2.8 mm FIGURE 28–36 Problems 14 and 15. 15. (II) A third wire is placed in the plane of the two wires shown in Fig. 28–36 parallel and just to the right. If it carries 25.0 A upward, what force per meter of length does it exert on each of the other two wires? Assume it is 2.8 mm from the nearest wire, center to center. 16. (II) A power line carries a current of 95 A west along the tops of 8.5-m-high poles. (a) What is the magnitude and direction of the magnetic field produced by this wire at the ground directly below? How does this compare with the Earth's field of about 12 G? (b) Where would the line's field cancel the Earth's? 17. (II) A compass needle points 28° E of N outdoors. However, when it is placed 12.0 cm to the east of a vertical wire inside a building, it points 55° E of N. What is the magnitude and direction of the current in the wire? The Earth's field there is 0.50 * 10–4 T and is horizontal. 18. (II) A rectangular loop of wire is placed next to a straight wire, as shown in Fig. 28–37. There is a current of 3.5 A in both wires. Determine 3.5 A the magnitude and direction of the net 3.0 cm 3.5 A force on the loop. 5.0 cm FIGURE 28–37 Problem 18.

10.0 cm

19. (II) Let two long parallel wires, a distance d apart, carry equal currents I in the same direction. One wire is at x = 0, B the other at x = d, Fig. 28–38. Determine B along the x axis between the wires as a function of x.

a a a I

d I

FIGURE 28–38 Problems 19 and 20.

FIGURE 28–41 Problem 24.

28–4 and 28–5 Ampère's Law, Solenoids and Toroids

20. (II) Repeat Problem 19 if the wire at x = 0 carries twice the current (2I) as the other wire, and in the opposite direction. 21. (II) Two long wires are oriented so that they are perpendicular to each other. At their closest, they are 20.0 cm apart (Fig. 28–39). What is the magnitude of the magnetic field at a point midway between them if the top one carries a current of 20.0 A and the bottom one carries IT = 20.0 A 12.0 A? 10.0 cm

B=? 10.0 cm FIGURE 28–39 Problem 21.

IB = 12.0 A

Bottom wire

22. (II) Two long parallel wires 8.20 cm apart carry 16.5-A currents in the same direction. Determine the magnetic field vector at a point P, 12.0 cm from one wire and 13.0 cm from the other. See Fig. 28–40. [Hint: Use the law of cosines.] P

12.0 cm 13.0 cm

8.20 cm FIGURE 28–40 Problem 22. 23. (III) A very long flat conducting strip of width d and negligible thickness lies in a horizontal plane and carries a uniform current I across its cross section. (a) Show that at points a distance y directly above its center, the field is given by B =

24. (III) A triangular loop of side length a carries a current I (Fig. 28–41). If this loop is placed a distance d away from a very long straight wire carrying a current I¿, determine the force on the loop. I I

m0 I d , tan–1 pd 2y

assuming the strip is infinitely long. [Hint: Divide the strip into many thin "wires," and sum (integrate) over these.] (b) What value does B approach for y W d? Does this make sense? Explain.

25. (I) A 40.0-cm-long solenoid 1.35 cm in diameter is to produce a field of 0.385 mT at its center. How much current should the solenoid carry if it has 765 turns of wire? 26. (I) A 32-cm-long solenoid, 1.8 cm in diameter, is to produce a 0.30-T magnetic field at its center. If the maximum current is 4.5 A, how many turns must the solenoid have? 27. (I) A 2.5-mm-diameter copper wire carries a 33-A current (uniform across its cross section). Determine the magnetic field: (a) at the surface of the wire; (b) inside the wire, 0.50 mm below the surface; (c) outside the wire 2.5 mm from the surface. 28. (II) A toroid (Fig. 28–17) has a 50.0-cm inner diameter and a 54.0-cm outer diameter. It carries a 25.0 A current in its 687 coils. Determine the range of values for B inside the toroid. 29. (II) A 20.0-m-long copper wire, 2.00 mm in diameter including insulation, is tightly wrapped in a single layer with adjacent coils touching, to form a solenoid of diameter 2.50 cm (outer edge). What is (a) the length of the solenoid and (b) the field at the center when the current in the wire is 16.7 A? B 30. (II) (a) Use Eq. 28–1, and the vector nature of B, to show that the magnetic field lines around two long parallel wires carrying equal currents I1 = I2 are as shown in Fig. 28–10. (b) Draw the equipotential lines around two stationary positive electric charges. (c) Are these two diagrams similar? Identical? Why or why not? 31. (II) A coaxial cable consists of a solid inner conductor of radius R1 , surrounded by a concentric cylindrical tube of inner radius R2 and outer radius R3 (Fig. 28–42). The conductors carry equal and opposite currents I0 distributed uniformly across their cross sections. Determine the magnetic field at a distance R from the axis for: (a) R 6 R1 ; (b) R1 6 R 6 R2 ; (c) R2 6 R 6 R3 ; (d) R 7 R3 . I0 (e) Let I0 = 1.50 A, R1 = 1.00 cm, R2 = 2.00 cm, and R3 = 2.50 cm. R1 I0 Graph B from R = 0 to R = 3.00 cm.

FIGURE 28–42 Problems 31 and 32.

32. (III) Suppose the current in the coaxial cable of Problem 31, Fig. 28–42, is not uniformly distributed, but instead the current density j varies linearly with distance from the center: j1 = C1 R for the inner conductor and j2 = C2 R for the outer conductor. Each conductor still carries the same total current I0 , in opposite directions. Determine the magnetic field in terms of I0 in the same four regions of space as in Problem 31.

Problems

753

28–6 Biot-Savart Law 33. (I) The Earth's magnetic field is essentially that of a magnetic dipole. If the field near the North Pole is about 1.0 * 10 –4 T, what will it be (approximately) 13,000 km above the surface at the North Pole? 34. (II) A wire, in a plane, has the shape shown in Fig. 28–43, two arcs of a circle connected by radial lengths of wire. B Determine B at point C in terms I of R1 , R2 , u, and the current I. R2 I I θ

FIGURE 28–43 Problem 34.

35. (II) A circular conducting ring of radius R is connected to two exterior straight wires at two ends of a diameter (Fig. 28–44). The current I splits into unequal portions (as 0.65 I shown) while passing through B the ring. What is B at the center of the ring? I I R

0.35 I

FIGURE 28–44 Problem 35.

36. (II) A small loop of wire of radius 1.8 cm is placed at the center of a wire loop with radius 25.0 cm. The planes of the loops are perpendicular to each other, and a 7.0-A current flows in each. Estimate the torque the large loop exerts on the smaller one. What simplifying assumption did you make? 37. (II) A wire is formed into the shape of two half circles connected by equal-length straight sections as shown in Fig. 28–45. A current I flows in the circuit clockwise as shown. Determine (a) the magnitude and direction of the magnetic field at the center, C, and (b) the magnetic dipole R1 moment of the circuit. I C R2 FIGURE 28–45 Problem 37.

38. (II) A single point charge q is moving with velocity vB. Use B the Biot-Savart law to show that the magnetic field B it produces at a point P, whose position vector relative to the charge is Br (Fig. 28–46), is given by B

B =

m0 qvB * Br . 4p r3

(Assume v is much less than the speed of light.)

P B

FIGURE 28–46 Problem 38.

754

CHAPTER 28

Sources of Magnetic Field

39. (II) A nonconducting circular disk, of radius R, carries a uniformly distributed electric charge Q. The plate is set spinning with angular velocity v about an axis perpendicular to the plate through its center (Fig. 28–47). Determine (a) its magnetic dipole moment and (b) the magnetic field at points on its axis a distance x from its center; (c) does R Eq. 28–7b apply in this case for x x W R? v FIGURE 28–47 Q Problem 39. 40. (II) Consider a straight section of wire of length d, as in Fig. 28–48, which carries a current I. (a) Show that the magnetic field at a point P a distance R from the wire along its perpendicular bisector is d 2 I m0 I d . B = 2pR Ad2 + 4R 2 B 12 R P (b) Show that this is consistent with Example 28–11 for an infi- d nite wire. 2 FIGURE 28–48 Problem 40. 41. (II) A segment of wire of length d carries a current I as shown in Fig. 28–49. (a) Show that for points along the positive x axis (the axis of the wire), y such as point Q, the magnetic B P field B is zero. (b) Determine a formula for the field at points along the y axis, such as point P. I d

FIGURE 28–49 Problem 41. 42. (III) Use the result of Problem 41 to find the magnetic field at point P in Fig. 28–50 due to the current in the square loop. P d I d I I I FIGURE 28–50 d Problem 42. 43. (III) A wire is bent into the shape of a regular polygon with n sides whose vertices are a distance R from the center. (See Fig. 28–51, which shows the special case of n = 6.) If the wire carries a current I0 , I0 (a) determine the magnetic field at the center; (b) if n is allowed to become very large (n S q), R show that the formula in part (a) reduces to that for a circular loop (Example 28–12). FIGURE 28–51 Problem 43. 44. (III) Start with the result of Example 28–12 for the magnetic field along the axis of a single loop to obtain the field inside a very long solenoid with n turns per meter (Eq. 28–4) that stretches from ± q to – q.

45. (III) A single rectangular loop of wire, with sides a and b, carries a current I. An xy coordinate system has its origin at the lower left corner of the y rectangle with the x axis parallel to side b (Fig. 28–52) and the y axis parallel to side a. I Determine the magnetic field B I at all points (x, y) within the a loop. I I x FIGURE 28–52 0 b Problem 45. 46. (III) A square loop of wire, of side d, carries a current I. (a) Determine the magnetic field B at points on a line perpendicular to the plane of the square which passes through the center of the square (Fig. 28–53). Express B as a function of x, the distance along the line from the center of the square. (b) For I x W d, does the square appear to be a magnetic dipole? If so, d x what is its dipole moment? FIGURE 28–53 Problem 46.

* 28–9 Magnetic Materials; Hysteresis * 48. (I) The following are some values of B and B0 for a piece of annealed iron as it is being magnetized: B0(10 4 T) B(T)

0.0 0.0

B0(10 4 T) B(T)

1.9 2.5 1.01 1.18

0.13 0.25 0.50 0.63 0.78 1.0 1.3 0.0042 0.010 0.028 0.043 0.095 0.45 0.67 6.3 13.0 1.44 1.58

130 1300 10,000 1.72 2.26 3.15

Determine the magnetic permeability m for each value and plot a graph of m versus B0 . * 49. (I) A large thin toroid has 285 loops of wire per meter, and a 3.0-A current flows through the wire. If the relative permeability of the iron is mm0 = 2200, what is the total field B inside the toroid? * 50. (II) An iron-core solenoid is 38 cm long and 1.8 cm in diameter, and has 640 turns of wire. The magnetic field inside the solenoid is 2.2 T when 48 A flows in the wire. What is the permeability m at this high field strength?

28–7 Magnetic Materials—Ferromagnetism 47. (II) An iron atom has a magnetic dipole moment of about 1.8 * 10–23 A m2. (a) Determine the dipole moment of an iron bar 9.0 cm long, 1.2 cm wide, and 1.0 cm thick, if it is 100 percent saturated. (b) What torque would be exerted on this bar when placed in a 0.80-T field acting at right angles to the bar?

General Problems

FIGURE 28–54 Problems 51, 52, and 53.

3.5

51. Three long parallel wires are 3.5 cm from one another. (Looking along them, they are at three corners of an equilateral triangle.) The current in each wire is 8.00 A, but its direction in wire M is opposite to that in wires N and P (Fig. 28–54). Determine the magnetic force per unit length on each wire due to the other two. M

3.5 cm

52. In Fig. 28–54, determine the magnitude and direction of the magnetic field midway between points M and N.

55. Two long straight parallel wires are 15 cm apart. Wire A carries 2.0-A current. Wire B's current is 4.0 A in the same direction. (a) Determine the magnetic field due to wire A at the position of wire B. (b) Determine the magnetic field due to wire B at the position of wire A. (c) Are these two magnetic fields equal and opposite? Why or why not? (d) Determine the force per unit length on wire A due to wire B, and that on wire B due to wire A. Are these two forces equal and opposite? Why or why not? 56. A rectangular loop of wire carries a 2.0-A current and lies in a plane which also contains a very long straight wire carrying a 10.0-A current as shown in Fig. 28–55. Determine (a) the net force and (b) the net torque on the 5.0 cm loop due to the straight wire.

10.0 A

53. In Fig. 28–54 the top wire is 1.00-mm-diameter copper wire and is suspended in air due to the two magnetic forces from the bottom two wires. The current is 40.0 A in each of the two bottom wires. Calculate the required current flow in the suspended wire. 54. An electron enters a large solenoid at a 7.0° angle to the axis. If the field is a uniform 3.3 * 10–2 T, determine the radius and pitch (distance between loops) of the electron's helical path if its speed is 1.3 * 107 ms.

2.0 A

FIGURE 28–55 Problem 56.

2.0 A 26 cm

7.0 cm

General Problems

755

57. A very large flat conducting sheet of thickness t carries a B uniform current density j throughout (Fig. 28–56). Determine the magnetic field (magnitude and direction) at a distance y above the plane. P (Assume the plane is infinitely long and wide.) y FIGURE 28–56 B j Problem 57. 58. A long horizontal wire carries a current of 48 A. A second wire, made of 1.00-mm-diameter copper wire and parallel to the first, is kept in suspension magnetically 5.0 cm below (Fig. 28–57). (a) Determine the magnitude and direction of the current in the lower wire. (b) Is the lower wire in stable equilibrium? (c) Repeat parts (a) and (b) if the second wire is suspended 5.0 cm above I = 48 A the first due to the first's magnetic field. 5.0 cm FIGURE 28–57 I=? Problem 58. 59. A square loop of wire, of side d, carries a current I. Show that the magnetic field at the center of the square is 2 22 m0 I . B = pd B [Hint: Determine B for each segment of length d.] 60. In Problem 59, if you reshaped the square wire into a circle, would B increase or decrease at the center? Explain. 61. Helmholtz coils are two identical circular coils having the same radius R and the same number of turns N, separated by a distance equal to the radius R and carrying the same current I in the same direction. (See Fig. 28–58.) They are used in scientific instruments to generate nearly uniform magnetic fields.(They can be seen in the photo, Fig. 27–18.) (a) Determine the magnetic field B at points x along the line joining their centers. Let x = 0 at the center of one coil, and x = R at the center of the other. (b) Show that the field midway between the coils is particularly uniform by dB d2B = 0 at the midpoint between = 0 and showing that dx dx 2 the coils. (c) If R = 10.0 cm, N = 250 turns and I = 2.0 A, what is the field at the y midpoint between the coils, x = R2?

R FIGURE 28–58 Problem 61.

62. For two long parallel wires separated by a distance d, carrying currents I1 and I2 as in Fig. 28–10, show directly (Eq. 28–1) that Ampère's law is valid (but do not use Ampère's law) for a circular path of radius r (r 6 d) centered on I1:

§ B # dL B

756

CHAPTER 28

= m0 I1.

Sources of Magnetic Field

63. Near the Earth's poles the magnetic field is about 1 G A1 * 10–4 TB. Imagine a simple model in which the Earth's field is produced by a single current loop around the equator. Estimate roughly the current this loop would carry. 64. A 175-g model airplane charged to 18.0 mC and traveling at 2.8 ms passes within 8.6 cm of a wire, nearly parallel to its path, carrying a 25-A current. What acceleration (in g's) does this interaction give the airplane? 65. Suppose that an electromagnet uses a coil 2.0 m in diameter made from square copper wire 2.0 mm on a side; the power supply produces 35 V at a maximum power output of 1.0 kW. (a) How many turns are needed to run the power supply at maximum power? (b) What is the magnetic field strength at the center of the coil? (c) If you use a greater number of turns and this same power supply, will a greater magnetic field result? Explain. 66. Four long straight parallel wires located at the corners of a square of side d carry equal currents I0 perpendicular to the page as shown in Fig. 28–59. Determine the magnitude and B direction of B at the center C of the square.

FIGURE 28–59 Problem 66.

67. Determine the magnetic field at the point P due to a very long wire with a square bend as shown in Fig. 28–60. The point P is halfway between the two corners. [Hint: You can use the results of Problems 40 and 41.] a

I a FIGURE 28–60 Problem 67.

68. A thin 12-cm-long solenoid has a total of 420 turns of wire and carries a current of 2.0 A. Calculate the field inside the solenoid near the center. 69. A 550-turn solenoid is 15 cm long. The current into it is 33 A. A 3.0-cm-long straight wire cuts through the center of the solenoid, along a diameter. This wire carries a 22-A current downward (and is connected by other wires that don't concern us). What is the force on this wire assuming the solenoid's field points due east? 70. You have 1.0 kg of copper and want to make a practical solenoid that produces the greatest possible magnetic field for a given voltage. Should you make your copper wire long and thin, short and fat, or something else? Consider other variables, such as solenoid diameter, length, and so on.

71. A small solenoid (radius ra) is inside a larger solenoid (radius rb 7 ra). They are coaxial with na and nb turns per unit length, respectively. The solenoids carry the same current, but in opposite directions. Let r be the radial distance from the common axis of the solenoids. If the magnetic field inside the inner solenoid Ar 6 ra B is to be in the opposite direction as the field between the solenoids Ara 6 r 6 rb B, but have half the magnitude, determine the required ratio nbna . 72. Find B at the center of the 4.0-cm-radius semicircle in Fig. 28–61. The straight wires extend a great distance outward to the left and carry a current I = 6.0A.

FIGURE 28–61 Problem 72.

73. The design of a magneto-optical atom trap requires a magnetic field B that is directly proportional to position x along an axis. Such a field perturbs the absorption of laser light by atoms in the manner needed to spatially confine atoms in the trap. Let us demonstrate that "anti-Helmholtz" coils will provide the required field B = Cx, where C is a constant. AntiHelmholtz coils consist of two identical circular wire coils, each with radius R and N turns, carrying current I in opposite directions (Fig. 28–62). The coils share a common axis (defined as the x axis with x = 0 at the midpoint (0) between the coils). Assume that the centers of the coils are separated by a distance equal to the radius R of the coils. (a) Show that the magnetic field at position x along the x axis is given by B(x) =

74. You want to get an idea of the magnitude of magnetic fields produced by overhead power lines. You estimate that a transmission wire is about 12 m above the ground. The local power company tells you that the line operates at 15 kV and provide a maximum of 45 MW to the local area. Estimate the maximum magnetic field you might experience walking under such a power line, and compare to the Earth's field. [For an ac current, values are rms, and the magnetic field will be changing.] * Numerical/Computer * 75. (II) A circular current loop of radius 15 cm containing 250 turns carries a current of 2.0 A. Its center is at the origin and its axis lies along the x axis. Calculate the magnetic field B at a point x on the x axis for x = – 40 cm to ±40 cm in steps of 2 cm and make a graph of B as a function of x. * 76. (III) A set of Helmholtz coils (see Problem 61, Fig. 28–58) have a radius R = 10.0 cm and are separated by a distance R = 10.0 cm. Each coil has 250 loops carrying a current I = 2.0 A. (a) Determine the total magnetic field B along the x axis (the center line for the two coils) in steps of 0.2 cm from the center of one coil (x = 0) to the center of the other (x = R). (b) Graph B as a function of x. (c) By what % does B vary from x = 5.0 cm to x = 6.0 cm?

4 m0 NI 2x 2 – 32 2x 2 – 32 e c 4 + a1 b d - c4 + a1 + b d f. R R R

(b) For small excursions from the origin where x V R, show that the magnetic field is given by B L Cx, where the constant C = 48 m0 NI25 15 R2. (c) For optimal atom trapping, dBdx should be about 0.15 Tm. Assume an atom trap uses anti-Helmholtz coils with R = 4.0 cm and N = 150. What current should flow through the coils? [Coil separation equal to coil radius, as assumed in this problem, is not a strict I I requirement for antiHelmholtz coils.] R 0 FIGURE 28–62 Problem 75.

Answers to Exercises A: 2 * 10 –6 T; not at this distance, and then only briefly.

C: (b).

B: 0.8 * 10 –5 T, up.

D: (b), (c).

General Problems

757

One of the great laws of physics is Faraday's law of induction, which says that a changing magnetic flux produces an induced emf. This photo shows a bar magnet moving inside a coil of wire, and the galvanometer registers an induced current. This phenomenon of electromagnetic induction is the basis for many practical devices, including generators, alternators, transformers, tape recording, and computer memory.

P T E R

C H

CONTENTS 29–1 Induced EMF 29–2 Faraday's Law of Induction; Lenz's Law 29–3 EMF Induced in a Moving Conductor

*29–4 Electric Generators 29–5 Back EMF and Counter Torque; Eddy Currents 29–6 Transformers and Transmission of Power 29–7

A Changing Magnetic Flux Produces an Electric Field

*29–8 Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI

758

Electromagnetic Induction and Faraday's Law CHAPTER-OPENING QUESTION—Guess now! In the photograph above, the bar magnet is inserted into the coil of wire, and is left there for 1 minute; then it is removed from the coil. What would an observer watching the galvanometer see? (a) No change; without a battery there is no current to detect. (b) A small current flows while the magnet is inside the coil of wire. (c) A current spike as the magnet enters the coil, and then nothing. (d) A current spike as the magnet enters the coil, and then a steady small current. (e) A current spike as the magnet enters the coil, then nothing, and then a current spike in the opposite direction as the magnet leaves the coil. n Chapter 27, we discussed two ways in which electricity and magnetism are related: (1) an electric current produces a magnetic field; and (2) a magnetic field exerts a force on an electric current or moving electric charge. These discoveries were made in 1820–1821. Scientists then began to wonder: if electric currents produce a magnetic field, is it possible that a magnetic field can produce an electric current? Ten years later the American Joseph Henry (1797–1878) and the Englishman Michael Faraday (1791–1867) independently found that it was possible. Henry actually made the discovery first. But Faraday published his results earlier and investigated the subject in more detail. We now discuss this phenomenon and some of its world-changing applications including the electric generator.

29–1

Induced EMF

In his attempt to produce an electric current from a magnetic field, Faraday used an apparatus like that shown in Fig. 29–1. A coil of wire, X, was connected to a battery. The current that flowed through X produced a magnetic field that was intensified by the ring-shaped iron core around which the wire was wrapped. Faraday hoped that a strong steady current in X would produce a great enough magnetic field to produce a current in a second coil Y wrapped on the same iron ring. This second circuit, Y, contained a galvanometer to detect any current but contained no battery. Galvanometer + –

Iron

Switch

– 10

Battery

FIGURE 29–1 Faraday's experiment to induce an emf.

He met no success with constant currents. But the long-sought effect was finally observed when Faraday noticed the galvanometer in circuit Y deflect strongly at the moment he closed the switch in circuit X. And the galvanometer deflected strongly in the opposite direction when he opened the switch in X. A constant current in X produced a constant magnetic field which produced no current in Y. Only when the current in X was starting or stopping was a current produced in Y. Faraday concluded that although a constant magnetic field produces no current in a conductor, a changing magnetic field can produce an electric current. Such a current is called an induced current. When the magnetic field through coil Y changes, a current occurs in Y as if there were a source of emf in circuit Y. We therefore say that a changing magnetic field induces an emf. Faraday did further experiments on electromagnetic induction, as this phenomenon is called. For example, Fig. 29–2 shows that if a magnet is moved quickly into a coil of wire, a current is induced in the wire. If the magnet is quickly B removed, a current is induced in the opposite direction (B through the coil decreases). Furthermore, if the magnet is held steady and the coil of wire is moved toward or away from the magnet, again an emf is induced and a current flows. Motion or change is required to induce an emf. It doesn't matter whether the magnet or the coil moves. It is their relative motion that counts.

C A U T I O N B

Changing B, not B itself, induces current

C A U T I O N

Relative motion—magnet or coil moving induces current

FIGURE 29–2 (a) A current is induced when a magnet is moved toward a coil, momentarily increasing the magnetic B field through the coil. (b) The induced current is opposite when the magnet is moved away from the coil (B decreases). Note that the galvanometer zero is at the center of the scale and the needle deflects left or right, depending on the direction of the current. In (c), no current is induced if the magnet does not move relative to the coil. It is the relative motion that counts here: the magnet can be held steady and the coil moved, which also induces an emf. – 10

– 10

(a)

– 10

I Magnet moves up toward coil B (B in coil increasing)

(b)

I=0

Magnet moves down B (B in coil decreasing)

(c)

No movement B (B in coil constant)

S SECTION 29–1

759

EXERCISE A Return to the Chapter-Opening Question, page 758, and answer it again now. Try to explain why you may have answered differently the first time.

29–2 B

B||

u B⊥

Faraday investigated quantitatively what factors influence the magnitude of the emf induced. He found first of all that the more rapidly the magnetic field changes, the greater the induced emf. He also found that the induced emf depends on the area of the circuit loop. Thus we say that the emf is proportional to the rate of change of the magnetic flux, £ B , passing through the circuit or loop of area A. Magnetic flux for a uniform magnetic field is defined in the same way we did for electric flux in Chapter 22, namely as

u A = l2

Faraday's Law of Induction; Lenz's Law

FIGURE 29–3 Determining the flux through a flat loop of wire. This loop is square, of side l and area A = l2. FIGURE 29–4 Magnetic flux £ B is proportional to the number of B lines of B that pass through the loop.

B dA. B

£B =

(29–1b) B

=0 (a)

θ = 45° B = BA cos 45° (b)

θ = 0° B = BA (c)

FIGURE 29–5 Example 29–1.

Wire loop

As we saw in Chapter 27, the lines of B (like lines of E) can be drawn such that the number of lines per unit area is proportional to the field strength. Then the flux £ B can be thought of as being proportional to the total number of lines passing through the area enclosed by the loop. This is illustrated in Fig. 29–4, where the loop is viewed from the side (on edge). For u = 90°, no magnetic field lines pass through the loop and £ B = 0, whereas £ B is a maximum when u = 0°. The unit of magnetic flux is the tesla-meter 2; this is called a weber: 1 Wb = 1 T m2. CONCEPTUAL EXAMPLE 29–1 Determining flux. A square loop of wire B encloses area A 1 as shown in Fig. 29–5.A uniform magnetic field B perpendicular to the loop extends over the area A 2 . What is the magnetic flux through the loop A 1?

RESPONSE We assume that the magnetic field is zero outside the area A 2 . The total magnetic flux through area A 1 is the flux through area A 2 , which by Eq. 29–1a for a uniform field is BA 2 , plus the flux through the remaining area A = A 1 - A 2 B, which is zero because B = 0. So the total flux is £ B = BA 2 + 0AA 1 - A 2 B = BA 2 . It B is not equal to BA 1 because B is not uniform over A 1 .

Area A2

Area A1

[B uniform] (29–1a)

Here B⊥ is the component of the magnetic field B perpendicular to the face of the B B loop, and u is the angle between B and the vector A (representing the area) whose direction is perpendicular to the face of the loop. These quantities are shown in Fig. 29–3 for a square loop of side l whose area is A = l2. If the area is B of some other shape, or B is not uniform, the magnetic flux can be written†

θ = 90°

£ B = B⊥ A = BA cos u = B A.

With our definition of flux, Eqs. 29–1, we can now write down the results of Faraday's investigations: The emf induced in a circuit is equal to the rate of change of magnetic flux through the circuit: FARADAY'S LAW OF INDUCTION

e = –

d£ B . dt

(29–2a)

This fundamental result is known as Faraday's law of induction, and is one of the basic laws of electromagnetism. †

The integral is taken over an open surface—that is, one bounded by a closed curve such as a circle or square. In the present discussion, the area is that enclosed by the loop under discussion. The area is not an enclosed surface as we used in Gauss's law, Chapter 22.

760

CHAPTER 29

Electromagnetic Induction and Faraday's Law

If the circuit contains N loops that are closely wrapped so the same flux passes through each, the emfs induced in each loop add together, so

e = –N

d£ B . dt

[N loops] (29–2b)

FARADAY'S LAW OF INDUCTION

EXAMPLE 29–2 A loop of wire in a magnetic field. A square loop of wire of side l = 5.0 cm is in a uniform magnetic field B = 0.16 T. What B is the magnetic flux in the loop (a) when B is perpendicular to the face B B of the loop and (b) when B is at an angle of 30° to the area A of the loop? (c) What is the magnitude of the average current in the loop if it has a resistance of 0.012 and it is rotated from position (b) to position (a) in 0.14 s? B

APPROACH We use the definition £ B = B A to calculate the magnetic flux. Then we use Faraday's law of induction to find the induced emf in the coil, and from that the induced current (I = eR). 2 SOLUTION The area of the coil is A = l2 = A5.0 * 10 –2 mB = 2.5 * 10–3 m2, B and the direction of A is perpendicular to the face of the loop (Fig. 29–3). B B (a) B is perpendicular to the coil's face, and thus parallel to A (Fig. 29–3), so B

£ B = B A = BA cos 0° = (0.16 T)A2.5 * 10–3 m2 B(1) = 4.0 * 10–4 Wb. B

(b) The angle between B and A is 30°, so B

£ B = B A = BA cos u = (0.16 T)A2.5 * 10–3 m2 B cos 30° = 3.5 * 10–4 Wb. (c) The magnitude of the induced emf is

e =

A4.0 * 10–4 WbB - A3.5 * 10 –4 WbB ¢£ B = = 3.6 * 10–4 V. ¢t 0.14 s

The current is then I =

e 3.6 * 10–4 V = = 0.030 A = 30 mA. R 0.012

The minus signs in Eqs. 29–2a and b are there to remind us in which direction the induced emf acts. Experiments show that a current produced by an induced emf moves in a direction so that the magnetic field created by that current opposes the original change in flux. This is known as Lenz's law. Be aware that we are now discussing two distinct magnetic fields: (1) the changing magnetic field or flux that induces the current, and (2) the magnetic field produced by the induced current (all currents produce a field). The second field opposes the change in the first. Lenz's law can be said another way, valid even if no current can flow (as when a circuit is not complete):

C A U T I O N

Distinguish two different magnetic fields

An induced emf is always in a direction that opposes the original change in flux that caused it. SECTION 29–2

Faraday's Law of Induction; Lenz's Law

761

– 10

(a)

I Magnet moves up toward coil B (B in coil increasing)

(b)

Magnet moves down B (B in coil decreasing)

FIGURE 29–2 (repeated).

FIGURE 29–6 A current can be induced by changing the area of the coil, even though B doesn't change. Here the area is reduced by pulling on its sides: the flux through the coil is reduced as we go from (a) to (b). Here the brief induced current acts in the direction shown so as to try to maintain the original flux (£ = BA) by producing its own magnetic field into the page. That is, as the area A decreases, the current acts to increase B in the original (inward) direction.

Let us apply Lenz's law to the relative motion between a magnet and a coil, Fig. 29–2. The changing flux through the coil induces an emf in the coil, producing a current. This induced current produces its own magnetic field. In Fig. 29–2a the distance between the coil and the magnet decreases. The magnet's magnetic field (and number of field lines) through the coil increases, and therefore the flux increases. The magnetic field of the magnet points upward. To oppose the upward increase, the magnetic field inside the coil produced by the induced current needs to point downward. Thus, Lenz's law tells us that the current moves as shown (use the right-hand rule). In Fig. 29–2b, the flux decreases (because the magnet is moved away and B decreases), so the induced current in the coil produces an upward magnetic field through the coil that is "trying" to maintain the status quo. Thus the current in Fig. 29–2b is in the opposite direction from Fig. 29–2a. It is important to note that an emf is induced whenever there is a change in flux through the coil, and we now consider some more possibilities.

Flux through coil is decreased because A decreased

B (inward) (a)

(b)

Since magnetic flux £ B = B dA = B cos u dA, we see that an emf can be induced in three ways: (1) by a changing magnetic field B; (2) by changing the area A of the loop in the field; or (3) by changing the loop's orientation u with respect to the field. Figures 29–1 and 29–2 illustrated case 1. Examples of cases 2 and 3 are illustrated in Figs. 29–6 and 29–7, respectively. B

(inward) FIGURE 29–7 A current can be induced by rotating a coil in a magnetic field. The flux through the coil changes from (a) to (b) because u (in Eq. 29–1a, £ = BA cos u) went from 0° (cos u = 1) to 90° (cos u = 0).

Flux decreasing

(a) Maximum flux

FIGURE 29–8 Example 29–3: An induction stove.

(b) Zero flux

CONCEPTUAL EXAMPLE 29–3 Induction stove. In an induction stove (Fig. 29–8), an ac current exists in a coil that is the "burner" (a burner that never gets hot). Why will it heat a metal pan but not a glass container? RESPONSE The ac current sets up a changing magnetic field that passes through the pan bottom. This changing magnetic field induces a current in the pan bottom, and since the pan offers resistance, electric energy is transformed to thermal energy which heats the pot and its contents. A glass container offers such high resistance that little current is induced and little energy is transferred AP = V 2RB.

762

CHAPTER 29

Electromagnetic Induction and Faraday's Law

L SO VIN OBL E PR

Lenz's Law Lenz's law is used to determine the direction of the (conventional) electric current induced in a loop due to a change in magnetic flux inside the loop. To produce an induced current you need (a) a closed conducting loop, and (b) an external magnetic flux through the loop that is changing in time. 1. Determine whether the magnetic flux (£B = BA cos u) inside the loop is decreasing, increasing, or unchanged. 2. The magnetic field due to the induced current: (a) points in the same direction as the external

field if the flux is decreasing; (b) points in the opposite direction from the external field if the flux is increasing; or (c) is zero if the flux is not changing. 3. Once you know the direction of the induced magnetic field, use the right-hand rule to find the direction of the induced current. 4. Always keep in mind that there are two magnetic fields: (1) an external field whose flux must be changing if it is to induce an electric current, and (2) a magnetic field produced by the induced current.

FIGURE 29–9 Example 29–4.

(a) Pulling a round loop to the right out of a magnetic field which points out of the page

(b) Shrinking a loop in a magnetic field pointing into the page

S N

(d) N magnetic pole moving toward loop in the plane of the page

(e) Rotating the loop by pulling the left side toward us and pushing the right side in; the magnetic field points from right to left

CONCEPTUAL EXAMPLE 29–4 Practice with Lenz's law. In which direction is the current induced in the circular loop for each situation in Fig. 29–9? RESPONSE (a) Initially, the magnetic field pointing out of the page passes through the loop. If you pull the loop out of the field, magnetic flux through the loop decreases; so the induced current will be in a direction to maintain the decreasing flux through the loop: the current will be counterclockwise to produce a magnetic field outward (toward the reader). (b) The external field is into the page. The coil area gets smaller, so the flux will decrease; hence the induced current will be clockwise, producing its own field into the page to make up for the flux decrease. (c) Magnetic field lines point into the S pole of a magnet, so as the magnet moves toward us and the loop, the magnet's field points into the page and is getting stronger. The current in the loop will be induced in the counterclockwise B direction in order to produce a field B out of the page. (d) The field is in the plane of the loop, so no magnetic field lines pass through the loop and the flux through the loop is zero throughout the process; hence there is no change in external magnetic flux with time, and there will be no induced emf or current in the loop. (e) Initially there is no flux through the loop. When you start to rotate the loop, the external field through the loop begins increasing to the left. To counteract this change in flux, the loop will have current induced in a counterclockwise direction so as to produce its own field to the right. SECTION 29–2

C A U T I O N

Magnetic field created by induced current opposes change in external flux, not necessarily opposing the external field

Faraday's Law of Induction; Lenz's Law

763

B = 0.600 T B=0 B

Fext

5.00 cm FIGURE 29–10 Example 29–5. The square coil in a magnetic field B = 0.600 T is pulled abruptly to the right to a region where B = 0.

EXAMPLE 29–5 Pulling a coil from a magnetic field. A 100-loop square coil of wire, with side l = 5.00 cm and total resistance 100 , is positioned perpendicular to a uniform 0.600-T magnetic field, as shown in Fig. 29–10. It is B quickly pulled from the field at constant speed (moving perpendicular to B) to a region where B drops abruptly to zero. At t = 0, the right edge of the coil is at the edge of the field. It takes 0.100 s for the whole coil to reach the field-free region. Find (a) the rate of change in flux through the coil, and (b) the emf and current induced. (c) How much energy is dissipated in the coil? (d) What was the average force required (Fext)? APPROACH We start by finding how the magnetic flux, £ B = BA, changes during the time interval ¢ t = 0.100 s. Faraday's law then gives the induced emf and Ohm's law gives the current. 2 SOLUTION (a) The area of the coil is A = l2 = A5.00 * 10 –2 mB = 2.50 * 10–3 m2. The flux through one loop is initially £ B = BA = (0.600 T)A2.50 * 10 –3 m2 B = 1.50 * 10–3 Wb. After 0.100 s, the flux is zero. The rate of change in flux is constant (because the coil is square), equal to 0 - A1.50 * 10–3 WbB ¢£ B = = –1.50 * 10–2 Wbs. ¢t 0.100 s (b) The emf induced (Eq. 29–2) in the 100-loop coil during this 0.100-s interval is

e = –N

¢£ B = –(100)A –1.50 * 10 –2 WbsB = 1.50 V. ¢t

The current is found by applying Ohm's law to the 100- coil: FIGURE 29–11 Exercise B.

I =

e 1.50 V = = 1.50 * 10 –2 A = 15.0 mA. R 100 B

I increasing (a) I decreasing

By Lenz's law, the current must be clockwise to produce more B into the page and thus oppose the decreasing flux into the page. (c) The total energy dissipated in the coil is the product of the power A = I 2RB and the time: E = Pt = I 2Rt = A1.50 * 10–2 AB 2(100 )(0.100 s) = 2.25 * 10–3 J. (d) We can use the result of part (c) and apply the work-energy principle: the energy dissipated E is equal to the work W needed to pull the coil out of the field (Chapters 7 and 8). Because W = fd where d = 5.00 cm, then

(b)

f =

W 2.25 * 10 –3 J = = 0.0450 N. d 5.00 * 10–2 m B

I constant

(c)

I increasing

Alternate Solution (d) We can also calculate the force directly using F = I L * B, B Eq. 27–3, which here for constant B is F = IlB. The force the magnetic field exerts on the top and bottom sections of the square coil of Fig. 29–10 are in B opposite directions and cancel each other. The magnetic force FM exerted on the left vertical section of the square coil acts to the left as shown because the B current is up (clockwise). The right side of the loop is in the region where B = 0. Hence the external force, to the right, needed to just overcome the magnetic force to the left (on N = 100 loops) is Fext = NIlB = (100)(0.0150 A)(0.0500 m)(0.600 T) = 0.0450 N, which is the same answer, confirming our use of energy conservation above.

(d)

764

CHAPTER 29

EXERCISE B What is the direction of the induced current in the circular loop due to the current shown in each part of Fig. 29–11?

Electromagnetic Induction and Faraday's Law

29–3

EMF Induced in a Moving Conductor

Another way to induce an emf is shown in Fig. 29–12a, and this situation helps B illuminate the nature of the induced emf. Assume that a uniform magnetic field B is perpendicular to the area bounded by the U-shaped conductor and the movable rod resting on it. If the rod is made to move at a speed v, it travels a distance dx = v dt in a time dt. Therefore, the area of the loop increases by an amount dA = l dx = lv dt in a time dt. By Faraday's law there is an induced emf e whose magnitude is given by d£ B B dA Blv dt e = = = = Blv. (29–3) dt dt dt Equation 29–3 is valid as long as B, l, and v are mutually perpendicular. (If they are not, we use only the components of each that are mutually perpendicular.) An emf induced on a conductor moving in a magnetic field is sometimes called motional emf. We can also obtain Eq. 29–3 without using Faraday's law. We saw in Chapter 27 that a charged particle moving perpendicular to a magnetic field B with speed v B B experiences a force F = qv * B (Eq. 27–5a). When the rod of Fig. 29–12a moves to the right with speed v, the electrons in the rod also move with this speed. Therefore, B since v ⊥ B, each electron feels a force F = qvB, which acts up the page as shown in Fig. 29–12b. If the rod was not in contact with the U-shaped conductor, electrons would collect at the upper end of the rod, leaving the lower end positive (see signs in Fig. 29–12b). There must thus be an induced emf. If the rod is in contact with the U-shaped conductor (Fig. 29–12a), the electrons will flow into the U. There will then be a clockwise (conventional) current in the loop. To calculate the emf, we determine the work W needed to move a charge q from one end of the rod to the other against this potential difference: W = force * distance = (qvB)(l). The emf equals the work done per unit charge, so e = Wq = qvBlq = Blv, the same result† as from Faraday's law above, Eq. 29–3.

dA B

B (outward) (a)

v dt

– Force on electron

–

EXERCISE C In what direction will the electrons flow in Fig. 29–12 if the rod moves to the left, decreasing the area of the current loop?

EXAMPLE 29–6 ESTIMATE Does a moving airplane develop a large emf? An airplane travels 1000 kmh in a region where the Earth's magnetic field is about 5 * 10–5 T and is nearly vertical (Fig. 29–13). What is the potential difference induced between the wing tips that are 70 m apart? APPROACH We consider the wings to be a 70-m-long conductor moving through the Earth's magnetic field. We use Eq. 29–3 to get the emf. B SOLUTION Since v = 1000 kmh = 280 ms, and v ⊥ B, we have e = Blv = A5 * 10–5 TB(70 m)(280 ms) L 1 V. NOTE Not much to worry about.

(b) FIGURE 29–12 (a) A conducting rod is moved to the right on a U-shaped conductor in a uniform B magnetic field B that points out of the page. The induced current is clockwise. (b) Upward force on an electron in the metal rod (moving to B the right) due to B pointing out of the page; hence electrons can collect at top of rod, leaving + charge at bottom. FIGURE 29–13 Example 29–6.

EXAMPLE 29–7 Electromagnetic blood-flow measurement. The rate of blood flow in our body's vessels can be measured using the apparatus shown in Fig. 29–14, since blood contains charged ions. Suppose that the blood vessel is 2.0 mm in diameter, the magnetic field is 0.080 T, and the measured emf is 0.10 mV. What is the flow velocity v of the blood? B APPROACH The magnetic field B points horizontally from left to right (N pole toward S pole). The induced emf acts over the width l = 2.0 mm of the blood B vessel, perpendicular to B and v (Fig. 29–14), just as in Fig. 29–12. We can then use Eq. 29–3 to get v. (v in Fig. 29–14 corresponds to v in Fig. 29–12.) SOLUTION We solve for v in Eq. 29–3: A1.0 * 10–4 VB e v = = = 0.63 ms. Bl (0.080 T)A2.0 * 10 –3 mB NOTE In actual practice, an alternating current is used to produce an alternating magnetic field. The induced emf is then alternating.

P H Y S I C S

A P P L I E D

Blood-flow measurement

FIGURE 29–14 Measurement of blood velocity from the induced emf. Example 29–7.

† This force argument, which is basically the same as for the Hall effect (Section 27–8), explains this one way of inducing an emf. It does not explain the general case of electromagnetic induction.

Voltmeter

S vB

SECTION 29–3

765

EXAMPLE 29–8 Force on the rod. To make the rod of Fig. 29–12a move to the right at constant speed v, you need to apply an external force on the rod to the right. (a) Explain and determine the magnitude of the required force. (b) What external power is needed to move the rod? (Do not confuse this external force on the rod with the upward force on the electrons shown in Fig. 29–12b.)

dA B

B (outward)

v dt

(a)

APPROACH When the rod moves to the right, electrons flow upward in the rod according to the right-hand rule. So the conventional current is downward in the rod. We can see this also from Lenz's law: the outward magnetic flux through the loop is increasing, so the induced current must oppose the increase. Thus the current is clockwise so as to produce a magnetic field into the page B B (right-hand rule). The magnetic force on the moving rod is F = I L * B for a B constant B (Eq. 27–3). The right-hand rule tells us this magnetic force is to the left, and is thus a "drag force" opposing our effort to move the rod to the right. SOLUTION (a) The magnitude of the external force, to the right, needs to balance the magnetic force F = IlB, to the left. The current I = eR = BlvR (see Eq. 29–3), and the resistance R is that of the whole circuit: the rod and the U-shaped conductor. The force F required to move the rod is thus B

– Force on electron

–

+ (b) FIGURE 29–12 (repeated) (a) A conducting rod is moved to the right on a U-shaped conductor in a B uniform magnetic field B that points out of the page. The induced current is clockwise. (b) Upward force on an electron in the metal rod (moving to B the right) due to B pointing out of the page; hence electrons can collect at top of rod, leaving + charge at bottom.

F = IlB = ¢

Blv B2l2 ≤ lB = v. R R

If B, l, and R are constant, then a constant speed v is produced by a constant external force. (Constant R implies that the parallel rails have negligible resistance.) (b) The external power needed to move the rod for constant R is Pext = Fv =

B2l2v2 . R

The power dissipated in the resistance is P = I 2R. With I = eR = BlvR, PR = I 2R =

B2l2v2 , R

so the power input equals the power dissipated in the resistance at any moment.

29–4 FIGURE 29–15 An ac generator. Axle rotated mechanically Armature b I N

I I

766

b Brushes

CHAPTER 29

We discussed alternating currents (ac) briefly in Section 25–7. Now we examine how ac is generated: by an electric generator or dynamo, one of the most important practical results of Faraday's great discovery. A generator transforms mechanical energy into electric energy, just the opposite of what a motor does. A simplified diagram of an ac generator is shown in Fig. 29–15. A generator consists of many loops of wire (only one is shown) wound on an armature that can rotate in a magnetic field. The axle is turned by some mechanical means (falling water, steam turbine, car motor belt), and an emf is induced in the rotating coil. An electric current is thus the output of a generator. Suppose in Fig. 29–15 that the armature is rotating clockwise; B B then F = qv * B applied to charged particles in the wire (or Lenz's law) tells us that the (conventional) current in the wire labeled b on the armature is outward, toward us; therefore the current is outward from brush b. (Each brush is fixed and presses against a continuous slip ring that rotates with the armature.) After one-half revolution, wire b will be where wire a is now in the drawing, and the current then at brush b will be inward. Thus the current produced is alternating. B

Slip rings

Electric Generators

Electromagnetic Induction and Faraday's Law

Let us assume the loop is being made to rotate in a uniform magnetic field B with constant angular velocity v. From Faraday's law (Eq. 29–2a), the induced emf is

e = –

d£ B d B B d = – B dA = – [BA cos u] dt dt dt B

where A is the area of the loop and u is the angle between B and A. Since v = dudt, then u = u0 + vt. We arbitrarily take u0 = 0, so

e = –BA

d (cos vt) = BAv sin vt. dt

emf 0

Time

If the rotating coil contains N loops,

e = NBAv sin vt = e0 sin vt.

(29–4)

Thus the output emf is sinusoidal (Fig. 29–16) with amplitude e0 = NBAv. Such a rotating coil in a magnetic field is the basic operating principle of an ac generator. The frequency f ( = v2p) is 60 Hz for general use in the United States and Canada, whereas 50 Hz is used in many countries. Most of the power generated in the United States is done at steam plants, where the burning of fossil fuels (coal, oil, natural gas) boils water to produce high-pressure steam that turns a turbine connected to the generator axle. Falling water from the top of a dam (hydroelectric) is also common (Fig. 29–17). At nuclear power plants, the nuclear energy released is used to produce steam to turn turbines. Indeed, a heat engine (Chapter 20) connected to a generator is the principal means of generating electric power. The frequency of 60 Hz or 50 Hz is maintained very precisely by power companies, and in doing Problems, we will assume it is at least as precise as other numbers given.

FIGURE 29–16 An ac generator produces an alternating current. The output emf e = e0 sin vt, where e0 = NABv (Eq. 29–4). P H Y S I C S

A P P L I E D

Power plants FIGURE 29–17 Water-driven generators at the base of Bonneville Dam, Oregon.

EXAMPLE 29–9 An ac generator. The armature of a 60-Hz ac generator rotates in a 0.15-T magnetic field. If the area of the coil is 2.0 * 10–2 m2, how many loops must the coil contain if the peak output is to be e0 = 170 V? APPROACH From Eq. 29–4 we see that the maximum emf is e0 = NBAv. SOLUTION We solve Eq. 29– 4 for N with v = 2pf = (6.28)A60 s–1 B = 377 s–1: N =

e0 170 V = 150 turns. = BAv (0.15 T)A2.0 * 10 –2 m2 BA377 s–1 B

A dc generator is much like an ac generator, except the slip rings are replaced by split-ring commutators, Fig. 29–18a, just as in a dc motor (Section 27–6). The output of such a generator is as shown and can be smoothed out by placing a capacitor in parallel with the output (Section 26–5). More common is the use of many armature windings, as in Fig. 29–18b, which produces a smoother output.

(a)

(b)

A P P L I E D

FIGURE 29–18 (a) A dc generator with one set of commutators, and (b) a dc generator with many sets of commutators and windings.

P H Y S I C S

DC generator

SECTION 29–4

Electric Generators

767

FIGURE 29–19 (a) Simplified schematic diagram of an alternator. The input current to the rotor from the battery is connected through continuous slip rings. Sometimes the rotor electromagnet is replaced by a permanent magnet. (b) Actual shape of an alternator. The rotor is made to turn by a belt from the engine. The current in the wire coil of the rotor produces a magnetic field inside it on its axis that points horizontally from left to right, thus making north and south poles of the plates attached at either end. These end plates are made with triangular fingers that are bent over the coil—hence there are alternating N and S poles quite close to one another, with magnetic field lines between them as shown by the blue lines. As the rotor turns, these field lines pass through the fixed stator coils (shown on the right for clarity, but in operation the rotor rotates within the stator), inducing a current in them, which is the output. Loops of wire (in which current is induced)

Stator coil (emf induced in) Current to produce B field

Input current

North pole N

South pole S

S Slip rings S

(Rotating electromagnet)

Stator coil (emf induced in)

Output current (induced)

Slip rings S Coil (produces B ) Rotor

(a)

Rotates Stator assembly (b)

P H Y S I C S

A P P L I E D

Alternators

Automobiles used to use dc generators. Today they mainly use alternators, which avoid the problems of wear and electrical arcing (sparks) across the split-ring commutators of dc generators. Alternators differ from generators in that an electromagnet, called the rotor, is fed by current from the battery and is made to rotate by a belt from the engine. The magnetic field of the turning rotor passes through a surrounding set of stationary coils called the stator (Fig. 29–19), inducing an alternating current in the stator coils, which is the output. This ac output is changed to dc for charging the battery by the use of semiconductor diodes, which allow current flow in one direction only.

* 29–5

Back EMF and Counter Torque; Eddy Currents

* Back EMF, in a Motor A motor turns and produces mechanical energy when a current is made to flow in it. From our description in Section 27–6 of a simple dc motor, you might expect that the armature would accelerate indefinitely due to the torque on it. However, as the armature of the motor turns, the magnetic flux through the coil changes and an emf is generated. This induced emf acts to oppose the motion (Lenz's law) and is called the back emf or counter emf. The greater the speed of the motor, the greater the back emf. A motor normally turns and does work on something, but if there were no load, the motor's speed would increase until the back emf equaled the input voltage. When there is a mechanical load, the speed of the motor may be limited also by the load. The back emf will then be less than the external applied voltage. The greater the mechanical load, the slower the motor rotates and the lower is the back emf (e r v, Eq. 29–4).

768

CHAPTER 29

Electromagnetic Induction and Faraday's Law

EXAMPLE 29–10 Back emf in a motor. The armature windings of a dc motor have a resistance of 5.0 . The motor is connected to a 120-V line, and when the motor reaches full speed against its normal load, the back emf is 108 V. Calculate (a) the current into the motor when it is just starting up, and (b) the current when the motor reaches full speed. APPROACH As the motor is just starting up, it is turning very slowly, so there is no induced back emf. The only voltage is the 120-V line. The current is given by Ohm's law with R = 5.0 . At full speed, we must include as emfs both the 120-V applied emf and the opposing back emf. SOLUTION (a) At start up, the current is controlled by the 120 V applied to the coil's 5.0- resistance. By Ohm's law, V 120 V I = = = 24 A. R 5.0 (b) When the motor is at full speed, the back emf must be included in the equivalent circuit shown in Fig. 29–20. In this case, Ohm's law (or Kirchhoff's rule) gives 120 V - 108 V = I(5.0 ). Therefore 12 V I = = 2.4 A. 5.0 NOTE This result shows that the current can be very high when a motor first starts up. This is why the lights in your house may dim when the motor of the refrigerator (or other large motor) starts up. The large initial current causes the voltage to the lights and at the outlets to drop, since the house wiring has resistance and there is some voltage drop across it when large currents are drawn. CONCEPTUAL EXAMPLE 29–11 Motor overload. When using an appliance such as a blender, electric drill, or sewing machine, if the appliance is overloaded or jammed so that the motor slows appreciably or stops while the power is still connected, the device can burn out and be ruined. Explain why this happens.

Back emf induced in armature winding

Windings of motor 5.0 Ω

induced = 108

= 120 V FIGURE 29–20 Circuit of a motor showing induced back emf. Example 29–10.

P H Y S I C S

A P P L I E D

Burning out a motor

RESPONSE The motors are designed to run at a certain speed for a given applied voltage, and the designer must take the expected back emf into account. If the rotation speed is reduced, the back emf will not be as high as expected (e r v, Eq. 29–4), and the current will increase, and may become large enough that the windings of the motor heat up to the point of ruining the motor.

* Counter Torque In a generator, the situation is the reverse of that for a motor. As we saw, the mechanical turning of the armature induces an emf in the loops, which is the output. If the generator is not connected to an external circuit, the emf exists at the terminals but there is no current. In this case, it takes little effort to turn the armature. But if the generator is connected to a device that draws current, then a current flows in the coils of the armature. Because this current-carrying coil is in an external magnetic field, there will be a torque exerted on it (as in a motor), and this torque opposes the motion (use the right-hand rule for the force on a wire in Fig. 29–15). This is called a counter torque. The greater the electrical load—that is, the more current that is drawn—the greater will be the counter torque. Hence the external applied torque will have to be greater to keep the generator turning. This makes sense from the conservation of energy principle. More mechanical-energy input is needed to produce more electrical-energy output. EXERCISE D A bicycle headlight is powered by a generator that is turned by the bicycle wheel. (a) If you pedal faster, how does the power to the light change? (b) Does the generator resist being turned as the bicycle's speed increases, and if so how?

*SECTION 29–5

Back EMF and Counter Torque; Eddy Currents

769

* Eddy Currents

B (inward)

(a)

de (b)

FIGURE 29–21 Production of eddy currents in a rotating wheel. The grey lines in (b) indicate induced current.

FIGURE 29–22 Airport metal detector.

P H Y S I C S

A P P L I E D

Airport metal detector

FIGURE 29–23 Repairing a stepdown transformer on a utility pole.

Induced currents are not always confined to well-defined paths such as in wires. Consider, for example, the rotating metal wheel in Fig. 29–21a. An external magnetic field is applied to a limited area of the wheel as shown and points into the page. The section of wheel in the magnetic field has an emf induced in it because the conductor is moving, carrying electrons with it. The flow of induced (conventional) current in the wheel is upward in the region of the magnetic field (Fig. 29–21b), and the current follows a downward return path outside that region. Why? According to Lenz's law, the induced currents oppose the change that causes them. Consider the part of the wheel labeled c in Fig. 29–21b, where the magnetic field is B zero but is just about to enter a region where B points into the page. To oppose this inward increase in magnetic field, the induced current is counterclockwise to produce a field pointing out of the page (right-hand-rule 1). Similarly, region d is B about to move to e, where B is zero; hence the current is clockwise to produce an inward field opposed to this decreasing flux inward. These currents are referred to as eddy currents. They can be present in any conductor that is moving across a magnetic field or through which the magnetic flux is changing. B In Fig. 29–21b, the magnetic field exerts a force F on the induced currents it has created, and that force opposes the rotational motion. Eddy currents can be used in this way as a smooth braking device on, say, a rapid-transit car. In order to stop the car, an electromagnet can be turned on that applies its field either to the wheels or to the moving steel rail below. Eddy currents can also be used to dampen (reduce) the oscillation of a vibrating system. Eddy currents, however, can be a problem. For example, eddy currents induced in the armature of a motor or generator produce heat (P = Ie) and waste energy. To reduce the eddy currents, the armatures are laminated; that is, they are made of very thin sheets of iron that are well insulated from one another. The total path length of the eddy currents is confined to each slab, which increases the total resistance; hence the current is less and there is less wasted energy. Walk-through metal detectors at airports (Fig. 29–22) detect metal objects using electromagnetic induction and eddy currents. Several coils are situated in the walls of the walk-through at different heights. In a technique called "pulse induction," the coils are given repeated brief pulses of current (on the order of microseconds), hundreds or thousands of times a second. Each pulse in a coil produces a magnetic field for a very brief period of time. When a passenger passes through the walk-through, any metal object being carried will have eddy currents induced in it. The eddy currents persist briefly after each input pulse, and the small magnetic field produced by the persisting eddy current (before the next external pulse) can be detected, setting off an alert or alarm. Stores and libraries sometimes use similar systems to discourage theft.

29–6

Transformers and Transmission of Power

A transformer is a device for increasing or decreasing an ac voltage. Transformers are found everywhere: on utility poles (Fig. 29–23) to reduce the high voltage from the electric company to a usable voltage in houses (120 V or 240 V), in chargers for cell phones, laptops, and other electronic devices, in CRT monitors and in your car to give the needed high voltage (to the spark plugs), and in many other applications. A transformer consists of two coils of wire known as the primary and secondary coils. The two coils can be interwoven (with insulated wire); or they can be linked by an iron core which is laminated to minimize eddy-current losses (Section 29–5), as shown in Fig. 29–24. Transformers are designed so that (nearly) all the magnetic flux produced by the current in the primary coil also passes through the secondary coil, and we assume this is true in what follows. We also assume that energy losses (in resistance and hysteresis) can be ignored—a good approximation for real transformers, which are often better than 99% efficient.

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When an ac voltage is applied to the primary coil, the changing magnetic field it produces will induce an ac voltage of the same frequency in the secondary coil. However, the voltage will be different according to the number of loops in each coil. From Faraday's law, the voltage or emf induced in the secondary coil is d£ B , VS = NS dt where NS is the number of turns in the secondary coil, and d£ Bdt is the rate at which the magnetic flux changes. The input primary voltage, VP , is related to the rate at which the flux changes through it, d£ B , VP = NP dt where NP is the number of turns in the primary coil. This follows because the changing flux produces a back emf, NP d£ Bdt, in the primary that exactly balances the applied voltage VP if the resistance of the primary can be ignored (Kirchhoff's rules). We divide these two equations, assuming little or no flux is lost, to find VS NS . (29–5) = VP NP This transformer equation tells how the secondary (output) voltage is related to the primary (input) voltage; VS and VP in Eq. 29–5 can be the rms values (Section 25–7) for both, or peak values for both. DC voltages don't work in a transformer because there would be no changing magnetic flux. If the secondary coil contains more loops than the primary coil ANS 7 NP B, we have a step-up transformer. The secondary voltage is greater than the primary voltage. For example, if the secondary coil has twice as many turns as the primary coil, then the secondary voltage will be twice that of the primary voltage. If NS is less than NP , we have a step-down transformer. Although ac voltage can be increased (or decreased) with a transformer, we don't get something for nothing. Energy conservation tells us that the power output can be no greater than the power input. A well-designed transformer can be greater than 99% efficient, so little energy is lost to heat. The power output thus essentially equals the power input. Since power P = IV (Eq. 25–6), we have IP VP = IS VS , or IS NP . = (29–6) IP NS

Secondary coil

Primary coil

(input)

NP turns

(output)

NS turns

Laminated iron core FIGURE 29–24 Step-up transformer ANP = 4, NS = 12B.

EXAMPLE 29–12 Cell phone charger. The charger for a cell phone contains a transformer that reduces 120-V (or 240-V) ac to 5.0-V ac to charge the 3.7-V battery (Section 26–4). (It also contains diodes to change the 5.0-V ac to 5.0-V dc.) Suppose the secondary coil contains 30 turns and the charger supplies 700 mA. Calculate (a) the number of turns in the primary coil, (b) the current in the primary, and (c) the power transformed. APPROACH We assume the transformer is ideal, with no flux loss, so we can use Eq. 29–5 and then Eq. 29–6. SOLUTION (a) This is a step-down transformer, and from Eq. 29–5 we have (30)(120 V) VP NP = NS = = 720 turns. VS (5.0 V) (b) From Eq. 29–6 NS 30 = (0.70 A) ¢ ≤ = 29 mA. NP 720 (c) The power transformed is IP = IS

P = IS VS = (0.70 A)(5.0 V) = 3.5 W. NOTE The power in the primary coil, P = (0.029 A)(120 V) = 3.5 W, is the same as the power in the secondary coil. There is 100% efficiency in power transfer for our ideal transformer.

SECTION 29–6

771

Switch opened

Switch closed

12 V

VP 0

(a)

Time

10 kV 0 10 kV 20 kV

Time (b)

FIGURE 29–25 A dc voltage turned on and off as shown in (a) produces voltage pulses in the secondary (b). Voltage scales in (a) and (b) are not the same. P H Y S I C S

A P P L I E D

Transformers help power transmission

A transformer operates only on ac. A dc current in the primary coil does not produce a changing flux and therefore induces no emf in the secondary. However, if a dc voltage is applied to the primary through a switch, at the instant the switch is opened or closed there will be an induced voltage in the secondary. For example, if the dc is turned on and off as shown in Fig. 29–25a, the voltage induced in the secondary is as shown in Fig. 29–25b. Notice that the secondary voltage drops to zero when the dc voltage is steady. This is basically how, in the ignition system of an automobile, the high voltage is created to produce the spark across the gap of a spark plug that ignites the gas-air mixture. The transformer is referred to simply as an "ignition coil," and transforms the 12 V of the battery (when switched off in the primary) into a spike of as much as 30 kV in the secondary. Transformers play an important role in the transmission of electricity. Power plants are often situated some distance from metropolitan areas, so electricity must then be transmitted over long distances (Fig. 29–26). There is always some power loss in the transmission lines, and this loss can be minimized if the power is transmitted at high voltage, using transformers, as the following Example shows.

High voltage transmission line Step-down transformer (substation)

Power plant

Step-down transformer

Step-up transformer

12,000 V

Home

240,000 V

2400 V 240 V

FIGURE 29–26 The transmission of electric power from power plants to homes makes use of transformers at various stages.

EXAMPLE 29–13 Transmission lines. An average of 120 kW of electric power is sent to a small town from a power plant 10 km away. The transmission lines have a total resistance of 0.40 . Calculate the power loss if the power is transmitted at (a) 240 V and (b) 24,000 V. APPROACH We cannot use P = V2R because if R is the resistance of the transmission lines, we don't know the voltage drop along them; the given voltages are applied across the lines plus the load (the town). But we can determine the current I in the lines ( = PV), and then find the power loss from PL = I 2R, for both cases (a) and (b). SOLUTION (a) If 120 kW is sent at 240 V, the total current will be I =

P 1.2 * 105 W = = 500 A. V 2.4 * 102 V

The power loss in the lines, PL , is then PL = I 2R = (500 A)2(0.40 ) = 100 kW. Thus, over 80% of all the power would be wasted as heat in the power lines!

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(b) If 120 kW is sent at 24,000 V, the total current will be P 1.2 * 105 W I = = = 5.0 A. V 2.4 * 104 V The power loss in the lines is then PL = I 2R = (5.0 A)2(0.40 ) = 10 W, 1 which is less than 100 of 1%: a far better efficiency! NOTE We see that the higher voltage results in less current, and thus less power is wasted as heat in the transmission lines. It is for this reason that power is usually transmitted at very high voltages, as high as 700 kV. The great advantage of ac, and a major reason it is in nearly universal use, is that the voltage can easily be stepped up or down by a transformer. The output voltage of an electric generating plant is stepped up prior to transmission. Upon arrival in a city, it is stepped down in stages at electric substations prior to distribution. The voltage in lines along city streets is typically 2400 V or 7200 V (but sometimes less), and is stepped down to 240 V or 120 V for home use by transformers (Figs. 29–23 and 29–26). Fluorescent lights require a very high voltage initially to ionize the gas inside the bulb. The high voltage is obtained using a step-up transformer, called a ballast, and can be replaced independently of the bulb in many fluorescent light fixtures. When the ballast starts to fail, the tube is slow to light. Replacing the bulb will not solve the problem. In newer compact fluorescent bulbs designed to replace incandescent bulbs, the ballast (transformer) is part of the bulb, and is very small.

29–7

P H Y S I C S

A P P L I E D

Fluorescent lightbulb ballast

A Changing Magnetic Flux Produces an Electric Field

We have seen in earlier Chapters (especially Chapter 25, Section 25–8) that when an electric current flows in a wire, there is an electric field in the wire that does the work of moving the electrons in the wire. In this Chapter we have seen that a changing magnetic flux induces a current in the wire, which implies that there is an electric field in the wire induced by the changing magnetic flux. Thus we come to the important conclusion that a changing magnetic flux produces an electric field. This result applies not only to wires and other conductors, but is actually a general result that applies to any region in space. Indeed, an electric field will be produced at any point in space where there is a changing magnetic field.

Faraday's Law—General Form We can put these ideas into mathematical form by generalizing our relation between an electric field and the potential difference between two points a and b: bB Vab = a E dL (Eq. 23–4a) where dL is an element of displacement along the path of integration. The emf e induced in a circuit is equal to the work done per unit charge B by the electric field, which equals the integral of E dL around the closed path: B

e =

§ E dL. B

(29–7)

We combine this with Eq. 29–2a, to obtain a more elegant and general form of Faraday's law

§ E dL B

= –

d£ B dt

(29–8)

FARADAY'S LAW (general form)

which relates the changing magnetic flux to the electric field it produces. The integral on the left is taken around a path enclosing the area through which the magnetic flux £ B is changing. This more elegant statement of Faraday's law (Eq. 29–8) is valid not only in conductors, but in any region of space. To illustrate this, let us take an Example. SECTION 29–7

A Changing Magnetic Flux Produces an Electric Field

773

FIGURE 29–27 Example 29–14. B (a) Side view of nearly constant B. (b) Top view, for determining the B electric field E at point P. (c) Lines of B B E produced by increasing B (pointing outward). (d) Graph of E vs. r. S

APPROACH The changing magnetic flux through a circle of radius r, shown dashed in Fig. 29–27b, will produce an emf around this circle. Because all points on the dashed circle are equivalent physically, the electric field too will show this B B symmetry and will be in the plane perpendicular to B. Thus we can expect E to B be perpendicular to B and to be tangent to the circle of radius r. The direction of B B E will be as shown in Fig. 29–27b and c, since by Lenz's law the induced E needs to be capable of producing a current that generates a magnetic field opposing the B B original change in B. By symmetry, we also expect E to have the same magnitude at all points on the circle of radius r. SOLUTION We take the circle shown in Fig 29–27b as our path of integration in Eq. 29–8. We ignore the minus sign so we can concentrate on magnitude since we B already found the direction of E from Lenz's law, and obtain

B N (a)

EXAMPLE 29–14 E produced by changing B. A magnetic field B between the pole faces of an electromagnet is nearly uniform at any instant over a circular area of radius r0 as shown in Figs. 29–27a and b. The current in the windings of B the electromagnet is increasing in time so that B changes in time at a constant B B rate dBdt at each point. Beyond the circular region Ar 7 r0 B, we assume B = 0 B at all times. Determine the electric field E at any point P a distance r from the B center of the circular area due to the changing B.

r r0 B

E(2pr) = Apr 2 B

Cr 6 r0 D

dB , dt

since £ B = BA = BApr2 B at any instant. We solve for E and obtain

(b)

E = B

Cr 6 r0 D

r dB . 2 dt

This expression is valid up to the edge of the circle Ar r0 B, beyond which B B = 0. If we now consider a point where r 7 r0 , the flux through a circle of radius r is £ B = pr20B. Then Eq. 29–8 gives

E(2pr) = pr 20

Cr 7 r0 D

dB dt

or E =

r 20 dB . 2r dt

Cr 7 r0 D

Thus the magnitude of the induced electric field increases linearly from zero at the center of the magnet to E = (dBdt)Ar02B at the edge, and then decreases inversely with distance in the region beyond the edge of the magnetic field. The electric field lines are circles as shown in Fig. 29–27c. A graph of E vs. r is shown in Fig. 29–27d. EXERCISE E Consider the magnet shown in Fig. 29–27 with a radius r0 = 6.0 cm. If the magnetic field changes uniformly from 0.040 T to 0.090 T in 0.18 s, what is the magnitude of the resulting electric field at (a) r = 3.0 cm and (b) r = 9.0 cm?

* Forces Due to Changing B are Nonconservative r0 (d)

Example 29–14 illustrates an important difference between electric fields produced by changing magnetic fields and electric fields produced by electric charges at rest (electrostatic fields). Electric field lines produced in the electrostatic case (Chapters 21 to 24) start and stop on electric charges. But the electric field lines produced by a changing magnetic field are continuous; they form closed loops. This distinction goes even further and is an important one. In the electrostatic case, the potential difference between two points is given by (Eq. 23–4a) b

Vba = Vb - Va = –

E dL. B

If the integral is around a closed loop, so points a and b are the same, then Vba = 0.

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CHAPTER 29

Electromagnetic Induction and Faraday's Law

Hence the integral of E dL around a closed path is zero:

§ E dL B

= 0.

[electrostatic field]

This followed from the fact that the electrostatic force (Coulomb's law) is a conservative force, and so a potential energy function could be defined. Indeed, the B relation above, E dL = 0, tells us that the work done per unit charge around any closed path is zero (or the work done between any two points is independent of path—see Chapter 8), which is a property only of a conservative force. But in the nonelectrostatic case, when the electric field is produced by a changing magnetic field, the integral around a closed path is not zero, but is given by Eq. 29–8: B

§ E dL B

= –

Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI

* Microphone There are various types of microphones, and many operate on the principle of induction. In one form, a microphone is just the inverse of a loudspeaker (Section 27–6). A small coil connected to a membrane is suspended close to a small permanent magnet, as shown in Fig. 29–28. The coil moves in the magnetic field when sound waves strike the membrane and this motion induces an emf. The frequency of the induced emf will be just that of the impinging sound waves, and this emf is the "signal" that can be amplified and sent to loudspeakers, or sent to a recorder.

* Read/Write on Tape and Disks Recording and playback on tape or disks is done by magnetic heads. Recording tapes for use in audio and video tape recorders contain a thin layer of magnetic oxide on a thin plastic tape. During recording, the audio and/or video signal voltage is sent to the recording head, which acts as a tiny electromagnet (Fig. 29–29) that magnetizes the tiny section of tape passing over the narrow gap in the head at each instant. In playback, the changing magnetism of the moving tape at the gap causes corresponding changes in the magnetic field within the soft-iron head, which in turn induces an emf in the coil (Faraday's law). This induced emf is the output signal that can be amplified and sent to a loudspeaker (audio) or to the picture tube (video). In audio and video recorders, the signals may be analog—they vary continuously in amplitude over time. The variation in degree of magnetization of the tape at any point reflects the variation in amplitude and frequency of the audio or video signal. Digital information, such as used on computer hard drives or on magnetic computer tape and some types of digital tape recorders, is read and written using heads that are basically the same as just described (Fig. 29–29). The essential difference is in the signals, which are not analog, but are digital, and in particular binary, meaning that only two values are possible for each of the extremely high number of predetermined spaces on the tape or disk. The two possible values are usually referred to as 1 and 0. The signal voltage does not vary continuously but rather takes on only two values, ±5 V and 0 V, for example, corresponding to the 1 or 0. Thus, information is carried as a series of bits, each of which can have only one of two values, 1 or 0. *SECTION 29–8

Small coil of wire Magnet

d£ B . dt

We thus come to the conclusion that the forces due to changing magnetic fields are nonconservative. We are not able therefore to define a potential energy, or potential function, at a given point in space for the nonelectrostatic case. Although static electric fields are conservative fields, the electric field produced by a changing magnetic field is a nonconservative field.

* 29–8

Membrane

N S

To recorder or amplifier FIGURE 29–28 Diagram of a microphone that works by induction. FIGURE 29–29 (a) Read/Write (playback/recording) head for tape or disk. In writing or recording, the electric input signal to the head, which acts as an electromagnet, magnetizes the passing tape or disk. In reading or playback, the changing magnetic field of the passing tape or disk induces a changing magnetic field in the head, which in turn induces in the coil an emf that is the output signal. (b) Photo of a hard drive showing several platters and read/write heads that can quickly move from the edge of the disk to the center. Electric signal input (or output)

Coil

Read/Write head Gap

Moving magnetic tape or disk (a)

(b)

Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI

775

Coil

Suspension springs

* Credit Card Swipe Permanent magnet

When you swipe your credit card at a store or gas station, the magnetic stripe on the back of the card passes over a read head just as in a tape recorder or computer. The magnetic stripe contains personal information about your account and connects by telephone line for approval if your account is in order.

* Seismograph FIGURE 29–30 One type of seismograph, in which the coil is fixed to the case and moves with the Earth. The magnet, suspended by springs, has inertia and does not move instantaneously with the coil (and case), so there is relative motion between magnet and coil.

In geophysics, a seismograph measures the intensity of earthquake waves using a magnet and a coil of wire. Either the magnet or the coil is fixed to the case, and the other is inertial (suspended by a spring; Fig. 29–30). The relative motion of magnet and coil when the Earth shakes induces an emf output.

* Ground Fault Circuit Interrupter (GFCI) Fuses and circuit breakers (Sections 25–6 and 28–8) protect buildings from fire, and apparatus from damage, due to undesired high currents. But they do not turn off the current until it is very much greater than that which causes permanent damage to humans or death (L 100 mA). If fast enough, they may protect in case of a short. A ground fault circuit interrupter (GFCI) is meant to protect humans; GFCIs can react to currents as small as 5 mA.

Simple electronic circuit

Solenoid circuit breaker

Sensing coil

FIGURE 29–31 A ground fault circuit interrupter (GFCI). Hot

120 V

Neutral

Iron ring

FIGURE 29–32 (a) A GFCI wall outlet. GFCIs can be recognized because they have "test" and "reset" buttons. (b) Add-on GFCI that plugs into outlet.

(a)

(b)

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CHAPTER 29

I I

Electric circuit with one or more devices (possible sources of trouble)

Electromagnetic induction is the physical basis of a GFCI. As shown in Fig. 29–31, the two conductors of a power line leading to an electrical circuit or device (red) pass through a small iron ring. Around the ring are many loops of thin wire that serve as a sensing coil. Under normal conditions (no ground fault), the current moving in the hot wire is exactly balanced by the returning current in the neutral wire. If something goes wrong and the hot wire touches the ungrounded metal case of the device or appliance, some of the entering current can pass through a person who touches the case and then to ground (a ground fault). Then the return current in the neutral wire will be less than the entering current in the hot wire, so there is a net current passing through the GFCI's iron ring. Because the current is ac, it is changing and the current difference produces a changing magnetic field in the iron, thus inducing an emf in the sensing coil wrapped around the iron. For example, if a device draws 8.0 A, and there is a ground fault through a person of 100 mA ( = 0.1 A), then 7.9 A will appear in the neutral wire. The emf induced in the sensing coil by this 100-mA difference is amplified by a simple transistor circuit and sent to its own solenoid circuit breaker that opens the circuit at the switch S, thus protecting your life. If the case of the faulty device is grounded, the current difference is even higher when there is a fault, and the GFCI trips immediately. GFCIs can sense currents as low as 5 mA and react in 1 msec, saving lives. They can be small enough to fit as a wall outlet (Fig. 29–32a), or as a plug-in unit into which you plug a hair dryer or toaster (Fig. 29–32b). It is especially important to have GFCIs installed in kitchens, bathrooms, outdoors, and near swimming pools, where people are most in danger of touching ground. GFCIs always have a "test" button (to be sure it works) and a "reset" button (after it goes off).

Electromagnetic Induction and Faraday's Law

Summary The magnetic flux passing through a loop is equal to the product of the area of the loop times the perpendicular component of B the (uniform) magnetic field: £ B = B⊥ A = BA cos u. If B is not uniform, then

B dA. B

£B =

(29–1b)

If the magnetic flux through a coil of wire changes in time, an emf is induced in the coil. The magnitude of the induced emf equals the time rate of change of the magnetic flux through the loop times the number N of loops in the coil:

e = –N

d£ B . dt

(29–2b)

This is Faraday's law of induction. The induced emf can produce a current whose magnetic field opposes the original change in flux (Lenz's law). We can also see from Faraday's law that a straight wire of length l moving with speed v perpendicular to a magnetic field of strength B has an emf induced between its ends equal to:

e = Blv.

(29–3)

Faraday's law also tells us that a changing magnetic field produces an electric field. The mathematical relation is

§ E dL B

= –

d£ B dt

(29–8)

and is the general form of Faraday's law. The integral on the left is taken around the loop through which the magnetic flux £ B is changing.

An electric generator changes mechanical energy into electrical energy. Its operation is based on Faraday's law: a coil of wire is made to rotate uniformly by mechanical means in a magnetic field, and the changing flux through the coil induces a sinusoidal current, which is the output of the generator. [*A motor, which operates in the reverse of a generator, acts like a generator in that a back emf is induced in its rotating coil; since this counter emf opposes the input voltage, it can act to limit the current in a motor coil. Similarly, a generator acts somewhat like a motor in that a counter torque acts on its rotating coil.] A transformer, which is a device to change the magnitude of an ac voltage, consists of a primary coil and a secondary coil. The changing flux due to an ac voltage in the primary coil induces an ac voltage in the secondary coil. In a 100% efficient transformer, the ratio of output to input voltages AVSVP B equals the ratio of the number of turns NS in the secondary to the number NP in the primary: VS NS . = VP NP

(29–5)

The ratio of secondary to primary current is in the inverse ratio of turns: IS NP . = IP NS

(29–6)

[*Microphones, ground fault circuit interrupters, seismographs, and read/write heads for computer drives and tape recorders are applications of electromagnetic induction.]

Questions 1. What would be the advantage, in Faraday's experiments (Fig. 29–1), of using coils with many turns? 2. What is the difference between magnetic flux and magnetic field? 3. Suppose you are holding a circular ring of wire and suddenly thrust a magnet, south pole first, away from you toward the center of the circle. Is a current induced in the wire? Is a current induced when the magnet is held steady within the ring? Is a current induced when you withdraw the magnet? In each case, if your answer is yes, specify the direction. 4. Two loops of wire are moving in the vicinity of a very long straight wire carrying a steady current as shown in Fig. 29–33. I Find the direction of the induced vB current in each loop. vB

6. Suppose you are looking along a line through the centers of two circular (but separate) wire loops, one behind the other. A battery is suddenly connected to the front loop, establishing a clockwise current. (a) Will a current be induced in the second loop? (b) If so, when does this current start? (c) When does it stop? (d) In what direction is this current? (e) Is there a force between the two loops? (f ) If so, in what direction? 7. The battery mentioned in Question 6 is disconnected. Will a current be induced in the second loop? If so, when does it start and stop? In what direction is this current? 8. In what direction will the current flow in Fig. 29–12a if the rod moves to the left, which decreases the area of the loop to the left? 9. In Fig. 29–34, determine the direction of the induced current in resistor RA (a) when coil B is moved toward coil A, (b) when coil B is moved away from A, Coil B Coil A (c) when the resistance RB is increased.

FIGURE 29–33 Questions 4 and 5. 5. Is there a force between the two loops discussed in Question 4? If so, in what direction?

FIGURE 29–34 Question 9.

Questions

777

10. In situations where a small signal must travel over a distance, a shielded cable is used in which the signal wire is surrounded by an insulator and then enclosed by a cylindrical conductor carrying the return current (Fig. 28–12). Why is a "shield" necessary? 11. What is the advantage of placing the two insulated electric wires carrying ac close together or even twisted about each other? 12. Which object will fall faster in a nonuniform magnetic field, a conducting loop with radius l or a straight wire of length l2? 13. A region where no magnetic field is desired is surrounded by a sheet of low-resistivity metal. (a) Will this sheet shield the interior from a rapidly changing magnetic field outside? Explain. (b) Will it act as a shield to a static magnetic field? (c) What if the sheet is superconducting (resistivity = 0)? 14. A cell phone charger contains a transformer. Why can't you just buy one universal charger to charge your old cell phone, your new cell phone, your drill, and your toy electric train? 15. An enclosed transformer has four wire leads coming from it. How could you determine the ratio of turns on the two coils without taking the transformer apart? How would you know which wires paired with which? 16. The use of higher-voltage lines in homes—say, 600 V or 1200 V—would reduce energy waste. Why are they not used? 17. A transformer designed for a 120-V ac input will often "burn out" if connected to a 120-V dc source. Explain. [Hint: The resistance of the primary coil is usually very low.] * 18. Explain why, exactly, the lights may dim briefly when a refrigerator motor starts up. When an electric heater is turned on, the lights may stay dimmed as long as the heater is on. Explain the difference.

* 20. Will an eddy current brake (Fig. 29–21) work on a copper or aluminum wheel, or must the wheel be ferromagnetic? Explain. * 21. It has been proposed that eddy currents be used to help sort solid waste for recycling. The waste is first ground into tiny pieces and iron removed with a dc magnet. The waste then is allowed to slide down an incline over permanent magnets. How will this aid in the separation of nonferrous metals (Al, Cu, Pb, brass) from nonmetallic materials? * 22. The pivoted metal bar with slots in Fig. 29–35 falls much more quickly through a magnetic field than does a solid bar. Pivot Explain. FIGURE 29–35 Question 22. * 23. If an aluminum sheet is held between the poles of a large bar magnet, it requires some force to pull it out of the magnetic field even though the sheet is not ferromagnetic and does not touch the pole faces. Explain. * 24. A bar magnet falling inside a vertical metal tube reaches a terminal velocity even if the tube is evacuated so that there is no air resistance. Explain. * 25. A metal bar, pivoted at one end, oscillates freely in the absence of a magnetic field; but in a magnetic field, its oscillations are quickly damped out. Explain. (This magnetic damping is used in a number of practical devices.) * 26. Since a magnetic microphone is basically like a loudspeaker, could a loudspeaker (Section 27–6) actually serve as a microphone? That is, could you speak into a loudspeaker and obtain an output signal that could be amplified? Explain. Discuss, in light of your response, how a microphone and loudspeaker differ in construction.

* 19. Use Fig. 29–15 plus the right-hand rules to show why the counter torque in a generator opposes the motion.

Problems 29–1 and 29–2 Faraday's Law of Induction 1. (I) The magnetic flux through a coil of wire containing two loops changes at a constant rate from – 58 Wb to ± 38 Wb in 0.42 s. What is the emf induced in the coil? 2. (I) The north pole of the magnet in Fig. 29–36 is being inserted into the coil. In which direction is the induced current flowing through the resistor R?

S FIGURE 29–36 Problem 2.

3. (I) The rectangular loop shown in Fig. 29–37 is pushed into the magnetic field which points inward. In what direction is the induced current?

FIGURE 29–37 Problem 3.

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4. (I) A 22.0-cm-diameter loop of wire is initially oriented perpendicular to a 1.5-T magnetic field. The loop is rotated so that its plane is parallel to the field direction in 0.20 s. What is the average induced emf in the loop? 5. (II) A circular wire loop of radius r = 12 cm is immersed in a uniform magnetic field B = 0.500 T with its plane normal to the direction of the field. If the field magnitude then decreases at a constant rate of –0.010 Ts, at what rate should r increase so that the induced emf within the loop is zero? 6. (II) A 10.8-cm-diameter wire coil is initially oriented so that its plane is perpendicular to a magnetic field of 0.68 T pointing up. During the course of 0.16 s, the field is changed to one of 0.25 T pointing down. What is the average induced emf in the coil? 7. (II) A 16-cm-diameter circular loop of wire is placed in a 0.50-T magnetic field. (a) When the plane of the loop is perpendicular to the field lines, what is the magnetic flux through the loop? (b) The plane of the loop is rotated until it makes a 35° angle with the field lines. What is the angle u in Eq. 29–1a for this situation? (c) What is the magnetic flux through the loop at this angle?

Electromagnetic Induction and Faraday's Law

8. (II) (a) If the resistance of the resistor in Fig. 29–38 is slowly increased, what is the direction of the current induced in the small circular loop inside the larger loop? (b) What would it be if the small loop were placed outside the larger one, to the left? I

FIGURE 29–38 Problem 8. 9. (II) If the solenoid in Fig. 29–39 is being pulled away from the loop shown, in what direction is the induced current in the loop?

FIGURE 29–39 Problem 9. 10. (II) The magnetic field perpendicular to a circular wire loop 8.0 cm in diameter is changed from ± 0.52 T to –0.45 T in 180 ms, where ± means the field points away from an observer and – toward the observer. (a) Calculate the induced emf. (b) In what direction does the induced current flow? 11. (II) A circular loop in the plane of the paper lies in a 0.75-T magnetic field pointing into the paper. If the loop's diameter changes from 20.0 cm to 6.0 cm in 0.50 s, (a) what is the direction of the induced current, (b) what is the magnitude of the average induced emf, and (c) if the coil resistance is 2.5 , what is the average induced current? 12. (II) Part of a single rectangular loop of wire with dimensions shown in Fig. 29–40 is situated inside a region of uniform magnetic field of 0.650 T. The total resistance of the loop is 0.280 . Calculate the force required to pull the loop from the field (to the right) at a constant velocity of 3.40 ms. Neglect gravity.

0.350 m

0.750 m FIGURE 29–40 Problem 12. 13. (II) While demonstrating Faraday's law to her class, a physics professor inadvertently moves the gold ring on her finger from a location where a 0.80-T magnetic field points along her finger to a zero-field location in 45 ms. The 1.5-cm-diameter ring has a resistance and mass of 55 m and 15 g, respectively. (a) Estimate the thermal energy produced in the ring due to the flow of induced current. (b) Find the temperature rise of the ring, assuming all of the thermal energy produced goes into increasing the ring's temperature. The specific heat of gold is 129 Jkg C°.

14. (II) A 420-turn solenoid, 25 cm long, has a diameter of 2.5 cm. A 15-turn coil is wound tightly around the center of the solenoid. If the current in the solenoid increases uniformly from 0 to 5.0 A in 0.60 s, what will be the induced emf in the short coil during this time? 15. (II) A 22.0-cm-diameter coil consists of 28 turns of circular copper wire 2.6 mm in diameter. A uniform magnetic field, perpendicular to the plane of the coil, changes at a rate of 8.65 * 10–3 Ts. Determine (a) the current in the loop, and (b) the rate at which thermal energy is produced. 16. (II) A power line carrying a sinusoidally varying current with frequency f = 60 Hz and peak value I0 = 55 kA runs at a height of 7.0 m across a farmer's land (Fig. 29–41). The farmer constructs a vertically oriented 2.0-m-high 10-turn rectangular wire coil below the power line. The farmer hopes to use the induced voltage in this coil to power 120-Volt electrical equipment, which requires a sinusoidally varying voltage with frequency f = 60 Hz and peak value V0 = 170 V. What should the length l of the coil be? Would this be unethical?

I0 55 kA f 60 Hz

7.0 m

l Coil

2.0 m

FIGURE 29–41 Problem 16.

17. (II) The magnetic field perpendicular to a single 18.2-cmdiameter circular loop of copper wire decreases uniformly from 0.750 T to zero. If the wire is 2.35 mm in diameter, how much charge moves past a point in the coil during this operation? 18. (II) The magnetic flux through each loop of a 75-loop coil is given by A8.8t - 0.51t3 B * 10 –2 T m2, where the time t is in seconds. (a) Determine the emf e as a function of time. (b) What is e at t = 1.0 s and t = 4.0 s? 19. (II) A 25-cm-diameter circular loop of wire has a resistance of 150 . It is initially in a 0.40-T magnetic field, with its B plane perpendicular to B, but is removed from the field in 120 ms. Calculate the electric energy dissipated in this process. 20. (II) The area of an elastic circular loop decreases at a constant rate, dAdt = –3.50 * 10–2 m2s. The loop is in a magnetic field B = 0.28 T whose direction is perpendicular to the plane of the loop. At t = 0, the loop has area A = 0.285 m2. Determine the induced emf at t = 0, and at t = 2.00 s. 21. (II) Suppose the radius of the elastic loop in Problem 20 increases at a constant rate, drdt = 4.30 cms. Determine the emf induced in the loop at t = 0 and at t = 1.00 s.

Problems

779

22. (II) A single circular loop of wire is placed inside a long solenoid with its plane perpendicular to the axis of the solenoid. The area of the loop is A 1 and that of the solenoid, which has n turns per unit length, is A 2 . A current I = I0 cos vt flows in the solenoid turns. What is the induced emf in the small loop? 23. (II) We are looking down on an elastic conducting loop with resistance R = 2.0 , immersed in a magnetic field. The field's magnitude is uniform spatially, but varies with time t according to B(t) = at, where a = 0.60 Ts. The area A of the loop also increases at a constant rate, according to A(t) = A 0 + bt, where A 0 = 0.50 m2 and b = 0.70 m2s. Find the magnitude and direction (clockwise or counterclockwise, when viewed from above the page) of the induced current within the loop at time t = 2.0 s if the magnetic field (a) is parallel to the plane of the loop to the right; (b) is perpendicular to the plane of the loop, down. 24. (II) Inductive battery chargers, which allow transfer of electrical power without the need for exposed electrical contacts, are commonly used in appliances that need to be safely immersed in water, such as electric toothbrushes. Consider the following simple model for the power transfer in an inductive charger (Fig. 29–42). Within the charger's plastic base, a primary coil of diameter d with nP turns per unit length is connected to a home's ac wall outlet so that a current I = I0 sin(2pft) flows within it. When the toothbrush is seated on the base, an N-turn secondary coil inside the toothbrush has a diameter only slightly greater than d and is centered on the primary. Find an expression for the emf induced in the secondary coil. [This induced emf recharges the battery.] d

Primary coil (in charger base) Secondary coil (in toothbrush)

FIGURE 29–42 Problem 24.

I I0sin(2pft)

25. (III) (a) Determine the magnetic flux through a square loop of side a (Fig. 29–43) if one side is parallel to, and a distance b from, a straight wire that carries a current I. (b) If the loop is pulled away from the wire at speed v, what emf is induced in it? (c) Does the induced current flow clockwise or counterclockwise? (d) Determine the force F required to pull the loop away. I a r

FIGURE 29–43 Problems 25 and 26.

26. (III) Determine the emf induced in the square loop in Fig. 29–43 if the loop stays at rest and the current in the straight wire is given by I(t) = (15.0 A) sin(2500 t) where t is in seconds. The distance a is 12.0 cm, and b is 15.0 cm.

29–3 Motional EMF 27. (I) The moving rod in Fig. 29–12b is 13.2 cm long and generates an emf of 120 mV while moving in a 0.90-T magnetic field. What is its speed? 28. (I) The moving rod in Fig. 29–12b is 12.0 cm long and is pulled at a speed of 15.0 cms. If the magnetic field is 0.800 T, calculate the emf developed. 29. (II) In Fig. 29–12a, the rod moves to the right with a speed of 1.3 ms and has a resistance of 2.5 . The rail separation is l = 25.0 cm. The magnetic field is 0.35 T, and the resistance of the U-shaped conductor is 25.0 at a given instant. Calculate (a) the induced emf, (b) the current in the U-shaped conductor, and (c) the external force needed to keep the rod's velocity constant at that instant. 30. (II) If the U-shaped conductor in Fig. 29–12a has resistivity r, whereas that of the moving rod is negligible, derive a formula for the current I as a function of time. Assume the rod starts at the bottom of the U at t = 0, and moves with uniform speed v in the magnetic field B. The cross-sectional area of the rod and all parts of the U is A. 31. (II) Suppose that the U-shaped conductor and connecting rod in Fig. 29–12a are oriented vertically (but still in contact) so that the rod is falling due to the gravitational force. Find the terminal speed of the rod if it has mass m = 3.6 grams, length l = 18 cm, and resistance R = 0.0013 . It is falling in a uniform horizontal field B = 0.060 T. Neglect the resistance of the U-shaped conductor. 32. (II) When a car drives through the Earth's magnetic field, an emf is induced in its vertical 75.0-cm-long radio antenna. If the Earth's field A5.0 * 10–5 TB points north with a dip angle of 45°, what is the maximum emf induced in the antenna and which direction(s) will the car be moving to produce this maximum value? The car's speed is 30.0 ms on a horizontal road. 33. (II) A conducting rod rests on two long frictionless parallel B rails in a magnetic field B (⊥ to the rails and rod) as in Fig. 29–44. (a) If the rails are horizontal and the rod is given an initial push, will the rod travel at constant speed even though a magnetic field is present? (b) Suppose at t = 0, when the rod has speed v = v0 , the two rails are connected electrically by a wire from point a to point b. Assuming the rod has resistance R and the rails have negligible resistance, determine the speed of the rod as a function of time. Discuss your answer.

a dr

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Electromagnetic Induction and Faraday's Law

a B

B b

FIGURE 29–44 Problems 33 and 34.

34. (III) Suppose a conducting rod (mass m, resistance R) rests on two frictionless and resistanceless parallel rails a distance l B apart in a uniform magnetic field B (⊥ to the rails and to the rod) as in Fig. 29–44. At t = 0, the rod is at rest and a source of emf is connected to the points a and b. Determine the speed of the rod as a function of time if (a) the source puts out a constant current I, (b) the source puts out a constant emf e0. (c) Does the rod reach a terminal speed in either case? If so, what is it? 35. (III) A short section of wire, of length a, is moving with velocity vB, parallel to a very long wire carrying a current I as shown in Fig. 29–45. The near end of the wire section is a distance b from the long wire. Assuming the vertical wire is very long compared to a + b, determine the emf between the ends of the short section. Assume vB is (a) in the same direction as I, (b) in the opposite direction to I. a b I

vB FIGURE 29–45 Problem 35.

29–4 Generators 36. (I) The generator of a car idling at 875-rpm produces 12.4 V. What will the output be at a rotation speed of 1550 rpm assuming nothing else changes? 37. (I) A simple generator is used to generate a peak output voltage of 24.0 V. The square armature consists of windings that are 5.15 cm on a side and rotates in a field of 0.420 T at a rate of 60.0 revs. How many loops of wire should be wound on the square armature? 38. (II) A simple generator has a 480-loop square coil 22.0 cm on a side. How fast must it turn in a 0.550-T field to produce a 120-V peak output? 39. (II) Show that the rms output of an ac generator is Vrms = NABv22 where v = 2pf. 40. (II) A 250-loop circular armature coil with a diameter of 10.0 cm rotates at 120 revs in a uniform magnetic field of strength 0.45 T. What is the rms voltage output of the generator? What would you do to the rotation frequency in order to double the rms voltage output? * 29–5 Back EMF, Counter Torque; Eddy Current * 41. (I) The back emf in a motor is 72 V when operating at 1200 rpm. What would be the back emf at 2500 rpm if the magnetic field is unchanged? * 42. (I) A motor has an armature resistance of 3.05 . If it draws 7.20 A when running at full speed and connected to a 120-V line, how large is the back emf? * 43. (II) What will be the current in the motor of Example 29–10 if the load causes it to run at half speed? * 44. (II) The back emf in a motor is 85 V when the motor is operating at 1100 rpm. How would you change the motor's magnetic field if you wanted to reduce the back emf to 75 V when the motor was running at 2300 rpm?

* 45. (II) A dc generator is rated at 16 kW, 250 V, and 64 A when it rotates at 1000 rpm. The resistance of the armature windings is 0.40 . (a) Calculate the "no-load" voltage at 1000 rpm (when there is no circuit hooked up to the generator). (b) Calculate the full-load voltage (i.e. at 64 A) when the generator is run at 750 rpm. Assume that the magnitude of the magnetic field remains constant.

29–6 Transformers [Assume 100% efficiency, unless stated otherwise.]

46. (I) A transformer has 620 turns in the primary coil and 85 in the secondary coil. What kind of transformer is this, and by what factor does it change the voltage? By what factor does it change the current? 47. (I) Neon signs require 12 kV for their operation. To operate from a 240-V line, what must be the ratio of secondary to primary turns of the transformer? What would the voltage output be if the transformer were connected backward? 48. (II) A model-train transformer plugs into 120-V ac and draws 0.35 A while supplying 7.5 A to the train. (a) What voltage is present across the tracks? (b) Is the transformer step-up or step-down? 49. (II) The output voltage of a 75-W transformer is 12 V, and the input current is 22 A. (a) Is this a step-up or a step-down transformer? (b) By what factor is the voltage multiplied? 50. (II) If 65 MW of power at 45 kV (rms) arrives at a town from a generator via 3.0- transmission lines, calculate (a) the emf at the generator end of the lines, and (b) the fraction of the power generated that is wasted in the lines. 51. (II) Assume a voltage source supplies an ac voltage of amplitude V0 between its output terminals. If the output terminals are connected to an external circuit, and an ac current of amplitude I0 flows out of the terminals, then the equivalent resistance of the external circuit is Req = V0I0. (a) If a resistor R is connected directly to the output terminals, what is Req? (b) If a transformer with NP and NS turns in its primary and secondary, respectively, is placed between the source and the resistor as shown in Fig. 29–46, what is Req? [Transformers can be used in ac circuits to alter the apparent resistance of circuit elements, such as loud speakers, in order to maximize transfer of power.]

AC voltage V0 source

R (a)

Output terminal

AC voltage V0 source

(b) FIGURE 29–46 Problem 51.

Problems

781

52. (III) Design a dc transmission line that can transmit 225 MW of electricity 185 km with only a 2.0% loss. The wires are to be made of aluminum and the voltage is 660 kV. 53. (III) Suppose 85 kW is to be transmitted over two 0.100- lines. Estimate how much power is saved if the voltage is stepped up from 120 V to 1200 V and then down again, rather than simply transmitting at 120 V. Assume the transformers are each 99% efficient. B

29–7 Changing ≥B Produces E B

54. (II) In a circular region, there is a uniform magnetic field B pointing into the page (Fig. 29–47). An xy coordinate system has its origin at the circular region's center. A free positive point charge ± Q = 1.0 mC is initially at rest at a position x = ± 10 cm on the x axis. If the magnitude of the magnetic field is now decreased at a rate of – 0.10 Ts, what force (magnitude and direction) will act on ± Q?

55. (II) The betatron, a device used to accelerate electrons to high energy, consists of a circular vacuum tube placed in a magnetic field (Fig. 29–48), into which electrons are injected. The electromagnet produces a field that (1) keeps the electrons in their circular orbit inside the tube, and (2) increases the speed of the electrons when B changes. (a) Explain how the electrons are accelerated. (See Fig. 29–48.) (b) In what direction are the electrons moving in Fig. 29–48 (give directions as if looking down from above)? (c) Should B increase or decrease to accelerate the electrons? (d) The magnetic field is actually 60 Hz ac; show that the electrons can be accelerated only during 14 of a cycle 1 A 240 sB. (During this time they make N hundreds of thousands of revolutions B B and acquire very high energy.) S

y FIGURE 29–48 Problems 55 and 56. B

B Q 1.0 mC

0 x 10 cm

FIGURE 29–47 Problem 54.

Tube in which electrons orbit

56. (III) Show that the electrons in a betatron, Problem 55 and Fig. 29–48, are accelerated at constant radius if the magnetic field B0 at the position of the electron orbit in the tube is equal to half the average value of the magnetic field ABavg B over the area of the circular orbit at each moment: B0 = 12 Bavg . (This is the reason the pole faces have a rather odd shape, as indicated in Fig. 29–48.) 57. (III) Find a formula for the net electric field in the moving rod of Problem 34 as a function of time for each case, (a) and (b).

General Problems 58. Suppose you are looking at two current loops in the plane of the page as shown in Fig. 29–49. When the switch S is closed in the left-hand coil, (a) what is the direction of the induced current in the other loop? (b) What is the situation after a "long" time? (c) What is the direction of the induced current in the right-hand loop if that loop is quickly pulled horizontally to the right (S having been closed for a long time)? FIGURE 29–49 Problem 58.

S + −

59. A square loop 27.0 cm on a side has a resistance of 7.50 . It is initially in a 0.755-T magnetic field, with its plane B perpendicular to B, but is removed from the field in 40.0 ms. Calculate the electric energy dissipated in this process. 60. Power is generated at 24 kV at a generating plant located 85 km from a town that requires 65 MW of power at 12 kV. Two transmission lines from the plant to the town each have a resistance of 0.10 km. What should the output voltage of the transformer at the generating plant be for an overall transmission efficiency of 98.5%, assuming a perfect transformer? 61. A circular loop of area 12 m2 encloses a magnetic field perpendicular to the plane of the loop; its magnitude is B(t) = (10 Ts)t. The loop is connected to a 7.5- resistor and a 5.0-pF capacitor in series. When fully charged, how much charge is stored on the capacitor?

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62. The primary windings of a transformer which has an 85% efficiency are connected to 110-V ac. The secondary windings are connected across a 2.4-, 75-W lightbulb. (a) Calculate the current through the primary windings of the transformer. (b) Calculate the ratio of the number of primary windings of the transformer to the number of secondary windings of the transformer. 63. A pair of power transmission lines each have a 0.80- resistance and carry 740 A over 9.0 km. If the rms input voltage is 42 kV, calculate (a) the voltage at the other end, (b) the power input, (c) power loss in the lines, and (d) the power output. 64. Show that the power loss in transmission lines, PL , is given by PL = APT B 2RLV2, where PT is the power transmitted to the user, V is the delivered voltage, and RL is the resistance of the power lines. 65. A high-intensity desk lamp is rated at 35 W but requires only 12 V. It contains a transformer that converts 120-V household voltage. (a) Is the transformer step-up or stepdown? (b) What is the current in the secondary coil when the lamp is on? (c) What is the current in the primary coil? (d) What is the resistance of the bulb when on? 66. Two resistanceless rails rest 32 cm apart on a 6.0° ramp. They are joined at the bottom by a 0.60- resistor. At the top a copper bar of mass 0.040 kg (ignore its resistance) is laid across the rails. The whole apparatus is immersed in a vertical 0.55-T field. What is the terminal (steady) velocity of the bar as it slides frictionlessly down the rails?

Electromagnetic Induction and Faraday's Law

67. A coil with 150 turns, a radius of 5.0 cm, and a resistance of 12 surrounds a solenoid with 230 turnscm and a radius of 4.5 cm; see Fig. 29–50. The current in the solenoid changes at a constant rate from 0 to 2.0 A in 0.10 s. Calculate the magnitude and direction of the induced current in the 150-turn coil. FIGURE 29–50 Problem 67.

68. A search coil for measuring B (also called a flip coil) is a small coil with N turns, each of cross-sectional area A. It is connected to a so-called ballistic galvanometer, which is a device to measure the total charge Q that passes through it in a short time. The flip coil is placed in the magnetic field to be measured with its face perpendicular to the field. It is then quickly rotated 180° about a diameter. Show that the total charge Q that flows in the induced current during this short "flip" time is proportional to the magnetic field B. In particular, show that B is given by B =

69.

70.

* 71.

72.

QR 2NA

where R is the total resistance of the circuit, including that of the coil and that of the ballistic galvanometer which measures the charge Q. A ring with a radius of 3.0 cm and a resistance of 0.025 is rotated about an axis through its diameter by 90° in a magnetic field of 0.23 T perpendicular to that axis. What is the largest number of electrons that would flow past a fixed point in the ring as this process is accomplished? A flashlight can be made that is powered by the induced current from a magnet moving through a coil of wire. The coil and magnet are inside a plastic tube that can be shaken causing the magnet to move back and forth through the coil. Assume the magnet has a maximum field strength of 0.05 T. Make reasonable assumptions and specify the size of the coil and the number of turns necessary to light a standard 1-watt, 3-V flashlight bulb. A small electric car overcomes a 250-N friction force when traveling 35 kmh. The electric motor is powered by ten 12-V batteries connected in series and is coupled directly to the wheels whose diameters are 58 cm. The 270 armature coils are rectangular, 12 cm by 15 cm, and rotate in a 0.60-T magnetic field. (a) How much current does the motor draw to produce the required torque? (b) What is the back emf? (c) How much power is dissipated in the coils? (d) What percent of the input power is used to drive the car? What is the energy dissipated as a function of time in a circular loop of 18 turns of wire having a radius of 10.0 cm and a resistance of 2.0 if the plane of the loop is perpendicular to a magnetic field given by

73. A thin metal rod of length l rotates with angular velocity v about an axis through one end (Fig. 29–51). The rotation axis is perpendicular to the rod and is parallel to a B uniform magnetic field B. axis Determine the emf developed l between the ends of the rod. FIGURE 29–51 Problem 73.

* 74. The magnetic field of a "shunt-wound" dc motor is produced by field coils placed in parallel with the armature coils. Suppose that the field coils have a resistance of 36.0 and the armature coils 3.00 . The back emf at full speed is 105 V when the motor is connected to 115 V dc. (a) Draw the equivalent circuit for the situations when the motor is just starting and when it is running full speed. (b) What is the total current drawn by the motor at start up? (c) What is the total current drawn when the motor runs at full speed? 75. Apply Faraday's law, in the form of Eq. 29–8, to show that the static electric field between the plates of a parallel-plate capacitor cannot drop abruptly to zero at the edges, but must, in fact, fringe. Use the path shown dashed in Fig. 29–52.

FIGURE 29–52 Problem 75.

76. A circular metal disk of radius R rotates with angular velocity v about an axis through its center perpendicular to its face. The disk rotates in a uniform magnetic field B whose direction is parallel to the rotation axis. Determine the emf induced between the center and the edges. 77. What is the magnitude and direction of the electric field at each point in the rotating disk of Problem 76? 78. A circular-shaped circuit of radius r, containing a resistance R and capacitance C, is situated with its plane B perpendicular to a spatially uniform magnetic field B directed into the page (Fig. 29–53). Starting at time t = 0, the voltage difference Vba = Vb - Va across the capacitor plates is observed to increase with time t according to Vba = V0 A1 - e –tt B, where V0 and t are positive constants. Determine dBdt, the rate at which the B magnetic field magniB tude changes with time. Is B becoming larger or smaller as r Vb time increases?

C Va

B(t) = B0 e –tt with B0 = 0.50 T and t = 0.10 s?

FIGURE 29–53 Problem 78.

General Problems

783

79. In a certain region of space near Earth's surface, a uniform horizontal magnetic field of magnitude B exists above a level defined to be y = 0. Below y = 0, the field abruptly becomes zero (Fig. 29–54). A vertical square wire loop has resistivity r, mass density rm , diameter d, and side length l. It is initially at rest with its lower horizontal side at y = 0 and is then allowed to fall under gravity, with its plane perpendicular to the direction of the magnetic field. (a) While the loop is still partially immersed in the magnetic field (as it falls into the zero-field region), determine the magnetic "drag" force that acts on it at the moment when its speed is v. (b) Assume that the loop achieves a terminal velocity vT before its upper horizontal side exits the field. Determine a formula for vT. (c) If the loop is made of copper and B = 0.80 T, find vT. B

l d l

* Numerical/Computer * 80. (III) In an experiment, a coil was mounted on a low-friction cart that moved through the magnetic field B of a permanent magnet. The speed of the cart v and the induced voltage V were simultaneously measured, as the cart moved through the magnetic field, using a computer-interfaced motion sensor and a voltmeter. The Table below shows the collected data: Speed, v (ms) 0.367 0.379 0.465 Induced voltage, V (V) 0.128 0.135 0.164

0.623 0.221

(a) Make a graph of the induced voltage, V, vs. the speed, v. Determine a best-fit linear equation for the data. Theoretically, the relationship between V and v is given by V = BNlv where N is the number of turns of the coil, B is the magnetic field, and l is the average of the inside and outside widths of the coil. In the experiment, B = 0.126 T, N = 50, and l = 0.0561 m. (b) Find the % error between the slope of the experimental graph and the theoretical value for the slope. (c) For each of the measured speeds v, determine the theoretical value of V and find the % error of each.

y=0 FIGURE 29–54 Problem 79.

gB vB

Answers to Exercises A: (e).

D: (a) increases; (b) yes; increases (counter torque).

B: (a) Counterclockwise; (b) clockwise; (c) zero; (d) counterclockwise.

E: (a) 4.2 * 10 –3 Vm; (b) 5.6 * 10 –3 Vm.

C: Electrons flow clockwise (conventional current counterclockwise).

784

CHAPTER 29

0.630 0.222

Electromagnetic Induction and Faraday's Law

A sp rk plug in c r receives high volt ge, which produces high enough electric field in the ir cross its g p to pull electrons off the toms in the ir-g soline mixture nd form sp rk. The high volt ge is produced, from the b sic 12V of the c r b ttery, by n induction coil which is b sic lly tr nsformer or mutu l induct nce. Any coil of wire h s self-induct nce, nd ch nging current in it c uses n emf to be induced. Such inductors re useful in m ny circuits.

T £

Inductance, Electromagnetic ° Oscillations, and A ircuits CHAPTER-OPENING QUESTION—Guess now! Consider a ir uit with only a apa itor C and a oil of many loops of wire ( alled an indu tor, L ) as shown. If the apa itor is in itia lly harged (Q =

the same as the measured speed of light in vacuum. Light had been shown some 60 years previously to behave like a wave (we'll discuss this in hapter 34). But nobody knew what kind of wave it was. What is it that is oscillating in a light wave? Maxwell, on the basis of the calculated speed of EM waves, argued that light must be an electromagnetic wave. This idea soon came to be generally accepted by scientists, but not fully until after EM waves were experi mentally detected. EM waves were first generated and detected experimentally by Heinrich Hertz (1857-1894) in 1887, eight years after Maxwell's death. Hertz used a spark-gap apparatus in which charge was made to rush back and forth for a short time, generating waves whose frequency was about 109Hz. He detected them some distance away using a loop of wire in which an emf was produced when a changing magnetic field passed through. These waves were later shown to travel at the speed of light, 3.00 X 10 m/s, and to exhibit all the characteristics of light such as reflection, refraction, and interference. The only difference was that they were not visible. Hertz's experiment was a strong confirmation of Maxwell's theory. The wavelengths of visible light were measured in the first decade of the nineteenth century, long before anyone imagined that light was an electromagnetic wave. The wavelengths were found to lie between 4.0 X 10-7 m and 7.5 X 10-7 m, or 400 nm to 750 nm (l nm = 109m). The frequencies of visible light can be found using Eq. 15-1 or 31-8, which we rewrite here: c (31-14) A /, where / and A are the frequency and wavelength, respectively, of the wave. Here, c is the speed of light, 3.00 X 10 m/s; it gets the special symbol c because of its universality for all EM waves in free space. Equation 31 -14 tells us that the frequencies of visible light are between 4.0 X 1014Hz and 7.5 X 1014Hz. (Recall that 1 Hz = 1 cycle per second = 1 s-1.) But visible light is only one kind of EM wave. As we have seen, Hertz produced EM waves of much lower frequency, about 109Hz. These are now called r dio w ves, because frequencies in this range are used to transmit radio and TV signals. Electromagnetic waves, or EM radiation as we sometimes call it, have been produced or detected over a wide range of frequencies. They are usually categorized as shown in Fig. 31-12, which is known as the electrom gnetic spectrum. FIGURE 31 - 12

Electrom gnetic spectrum. W velength (m) 3 x 104m 3m

3 x IO-4m Infr red

R dio w ves 60 Hz ( c current)

r dio

3 x 10-8 m

3 x 10-12m

Ultr violet

G mm r ys X-r ys

Microw ves (e.g., r d r) fm Cellul r S tellite TV I TV phones TV ch2-6 |ch7&upj-j—1 ^-------- 1

Frequency (Hz)

/ = 4x 1014Hz

SECTION 31 - 6

Visible light

7 .5 x 1014H z

Light as a Electromag etic Wave a d the Electromag etic Spectrum

823

Radio waves and microwaves can be produced in the laboratory using electronic equipment (Fig. 31 -7). Higher-frequency waves are very difficult to produce electronically. These and other types of EM waves are produced in natural processes, as emission from atoms, molecules, and nuclei (more on this later). EM waves can be produced by the acceleration of electrons or other charged particles, such as electrons in the antenna of Fig. 31-7. Another example is X-rays, which are produced ( hapter 35) when fast-moving electrons are rapidly decelerated upon striking a metal target. Even the visible light emitted by an ordinary incandescent light is due to electrons undergoing acceleration within the hot filament. We will meet various types of EM waves later. However, it is worth mentioning here that infrared (IR) radiation (EM waves whose frequency is just less than that of visible light) is mainly responsible for the heating effect of the Sun. The Sun emits not only visible light but substantial amounts of IR and UV (ultraviolet) as well. The molecules of our skin tend to "resonate" at infrared frequencies, so it is these that are preferentially absorbed and thus warm us up. We humans experience EM waves differently, depending on their wavelengths: Our eyes detect wavelengths between about 4 X 10" 7 m and 7.5 X 10 7 m (visible light), whereas our skin detects longer wavelengths (IR). Many EM wavelengths we don't detect directly at all. I EXERCISE B Return to the Ch pter-Opening Question, p ge 812, nd nswer it g in now.

| Try to expl in why you m y h ve nswered differently the first time. /j\ CAUTI ON S o u n d a n d E M

w a ves

a re d iffe r e n t

Light and other electromagnetic waves travel at a speed of 3 X 10 m/s. ompare this to sound, which travels (see hapter 16) at a speed of about 300 m/s in air, a million times slower; or to typical freeway speeds of a car, 30 m/s (100 km/h, or 60 mi/h), 10 million times slower than light. EM waves differ from sound waves in another big way: sound waves travel in a medium such as air, and involve motion of air molecules; EM waves do not involve any material— only fields, and they can travel in empty space. U 5 E H I E H 8 Wavelengths of EM waves. alculate the wavelength (a) of a 60-Hz EM wave, (b) of a 93.3-MHz FM radio wave, and (c) of a beam of visible red light from a laser at frequency 4.74 X 1014Hz. APPR ACH All of these waves are electromagnetic waves, so their speed is c = 3.00 X 10 m/s. We solve for Ain Eq. 31-14: A = c /f. , s c 3.00 X 10 m/s * — — = 5.0 X 106m, S LUTI N (fl) A = T / 60s 1 or 5000 km. 60 Hz is the frequency of ac current in the United States, and, as we see here, one wavelength stretches all the way across the continental USA. .

3.00 x I t f . / ,

w 93.3 X 10 s The length of an FM antenna is about half this (|A), or l|m . _

= 3.00 X 10*m/s = 6.33 x l 0 - 7m ( = 633nm). 4.74 X 10 s v J

EXERCISE C Wh t re the frequencies of X-r y of w velength 5.5 X 10-11 m?

(a )

n 80-m-w velength r dio w ve, nd

(b )

EXAMPLE 31-4 ESTIMATE 1 Cell phone antenna. The antenna of a cell phone is often \ wavelength long. A particular cell phone has an 8.5-cm-long straight rod for its antenna. Estimate the operating frequency of this phone. APPR ACH The basic equation relating wave speed, wavelength, and frequency is c = A/; the wavelength Aequals four times the antenna's length. S LUTI N The antenna is ^A long, so A = 4(8.5 cm) = 34 cm = 0.34 m. Then f = c /A = (3.0 X 10 m/s)/(0.34m) = . X 108Hz = 880 MHz. N TE Radio antennas are not always straight conductors. The conductor may be a round loop to save space. See Fig. 31-21b. EXERCISE D How long should

824

CHAPTER 31

\ A ntenn be for n ircr ft r dio oper ting t 165 MHz?

Maxwell's Equatio s a d Electromag etic Waves

Electromagnetic waves can travel along transmission lines as well as in empty space. When a source of emf is connected to a transmission line — be it two parallel wires or a coaxial cable (Fig. 31 - 13) — the electric field within the wire is not set up immediately at all points along the wires. This is based on the same argument we used in Section 31 - 4 with reference to Fig. 31 - 7. Indeed, it can be shown that if the wires are separated by empty space or air, the electrical signal travels along the wires at the speed c = 3.0 X 10 m /s. For example, when you flip a light switch, the light actually goes on a tiny fraction of a second later. If the wires are in a medium whose electric permittivity is e and magnetic permeability is fi (Sections 24 - 5 and 28 - 9, respectively), the speed is not given by Eq. 31-12, but by

FIGURE 31 - 13

oaxial cable,

\ 1

v =

v e /

EXAMPLE 31 -5 ESTIMATE I Phone call time lag. You make a telephone call from New York to a friend in London. Estimate how long it will take the electrical signal generated by your voice to reach London, assuming the signal is (a) carried on a telephone cable under the Atlantic Ocean, and (b) sent via satellite 36.000 km above the ocean. Would this cause a noticeable delay in either case? APPR ACH The signal is carried on a telephone wire or in the air via satellite. In either case it is an electromagnetic wave. Electronics as well as the wire or cable slow things down, but as a rough estimate we take the speed to be c = 3.0 X 10 m/s. S LUTI N The distance from New York to London is about 5000 km. (a) The time delay via the cable is t = d / c « (5 X 106m)/(3.0 X 10 m/s) = 0.017 s. ( b ) Via satellite the time would be longer because communications satellites, which are usually geosynchronous (Example 6- 6), move at a height of 36.000 km. The signal would have to go up to the satellite and back down, or about 72,000 km. The actual distance the signal would travel would be a little more than this as the signal would go up and down on a diagonal. Thus t = d /c » 7.2 X 107 m/(3 X 10s m /s) = 0.24 s. N TE When the signal travels via the underwater cable, there is only a hint of a delay and conversations are fairly normal. When the signal is sent via satellite, the delay is noticeable. The length of time between the end of when you speak and your friend receives it and replies, and then you hear the reply, is about a half second beyond the normal time in a conversation. This is enough to be noticeable, and you have to adjust for it so you don't start talking again while your friend's reply is on the way back to you. EXERCISE E If you are on the phone via satellite to someone only 100 km away, would you hear the same effect?

If your voice traveled as a sound wave, how long would it take to go from

INew York to London? EXERCISE F

3 1 -7 M e suring the Speed o f Light Galileo attempted to measure the speed of light by trying to measure the time required for light to travel a known distance between two hilltops. He stationed an assistant on one hilltop and himself on another, and ordered the assistant to lift the cover from a lamp the instant he saw a flash from Galileo's lamp. Galileo measured the time between the flash of his lamp and when he received the light from his assistant's lamp. The time was so short that Galileo concluded it merely represented human reaction time, and that the speed of light must be extremely high. The first successful determination that the speed of light is finite was made by the Danish astronomer Ole Roemer (1644-1710). Roemer had noted that the carefully measured orbital period of Io, a moon of Jupiter with an average period of 42.5 h, varied slightly, depending on the relative position of Earth and Jupiter. He attributed this variation in the apparent period to the change in distance between the Earth and Jupiter during one of Io's periods, and the time it took light to travel the extra distance. Roemer concluded that the speed of light— though great — is finite.

SECTI N 31- 7

825

Since then a number of techniques have been used to measure the speed of light. Among the most important were those carried out by the American Albert A. Michelson (1852-1931). Michelson used the rotating mirror apparatus diagrammed in Fig. 31-14 for a series of high-precision experiments carried out from 1880 to the 1920s. Light from a source would hit one face of a rotating eight-sided mirror. (Mt. B ldy) The reflected light traveled to a stationary mirror a large distance away and back Light again as shown. If the rotating mirror was turning at just the right rate, the source (Mt. Wilson) returning beam of light would reflect from one of the eight mirrors into a small telescope through which the observer looked. If the speed of rotation was only -35 km slightly different, the beam would be deflected to one side and would not be seen FIGURE 31 - 14 Michelson 's speedby the observer. From the required speed of the rotating mirror and the known of-light pp r tus (not to sc le). distance to the stationary mirror, the speed of light could be calculated. In the 1920s, Michelson set up the rotating mirror on the top of Mt. Wilson in southern alifornia and the stationary mirror on Mt. Baldy (Mt. San Antonio) 35 km away. He later measured the speed of light in vacuum using a long evacuated tube. Today the speed of light, c, in vacuum is taken as

Eight-sided & Observer rot ting mirror

c = 2.99792458

10 m/s,

and is defined to be this value. This means that the standard for length, the meter, is no longer defined separately. Instead, as we noted in Section 1 -4, the meter is now formally defined as the distance light travels in vacuum in 1/299,792,458 of a second. We usually round off c to c = 3.00

10 m/s

when extremely precise results are not required. In air, the speed is only slightly less.

31—8 Energy in EMW ves; the PoyntingVector Electromagnetic waves carry energy from one region of space to another. This energy is associated with the moving electric and magnetic fields. In Section 24 -4, we saw that the energy density uE (J/m 3) stored in an electric field E is u e = \ eoE2 (Eq. 24 - 6). The energy density stored in a magnetic field B, as we discussed in Section 30-3, is given by uB = \ B2/ /jlq (Eq. 30-7). Thus, the total energy stored per unit volume in a region of space where there is an electromagnetic wave is u = uE + uB = ^€0£ 2 + Z

(31-15)

Z ^ 0

In this equation, E and B represent the electric and magnetic field strengths of the wave at any instant in a small region of space. We can write Eq. 31-15 in terms of the E field alone, using Eqs. 31-11 (B = E /c) and 31-12 (c = l/V ^o^o) to obtain 1 t-2 , 1 e0fi0E 2 u = —e0E + ----------(31-16 ) 2 2 vo Note here that the energy density associated with the B field equals that due to the E field, and each contributes half to the total energy. We can also write the energy density in terms of the B field only: d2

u = 60E 2 = €oc2B2 = — >

(31-16b)

or in one term containing both E and B, u = e0E 2 =

e0EB

= J — EB.

(31 - 16c)

Equations 31-16 give the energy density in any region of space at any instant. Now let us determine the energy the wave transports per unit time per unit area. This is given by a vector S, which is called the Poynting vector.1 The units of S are W /m2. The direction of S is the direction in which the energy is transported, which is the direction in which the wave is moving. fAfter J. H. Poynting (1852-1914).

826

CHAPTER 31

Let us imagine the wave is passing through an area A perpendicular to the x axis as shown in Fig. 31-15. In a short time dt, the wave moves to the right a distance dx = c dt where c is the wave speed. The energy that passes through A in the time dt is the energy that occupies the volume dV = A dx = Ac dt. The energy density u is u = e0E 2 where E is the electric field in this volume at the given instant. So the total energy dU contained in this volume dV is the energy density u times the volume: dU = u dV = (e0E 2)(Ac dt). Therefore the energy crossing the area A per time dt is = 1 dU Adt

e0cE2.

(31-17)

FIGURE 31 - 15 Electrom gnetic w ve c rrying energy through re A.

Since E = cB and c = l / \ / e 0fjL0, this can also be written: S = encE2 =

cB2 Mo

EB Mo

The direction of S is along v, perpendicular to E and B, so the Poynting vector S can be written

S = -(Bxi).

(31-18)

Equation 31-17 or 31-18 gives the energy transported per unit area per unit time at any instant. We often want to know the average over an extended period of time since the frequencies are usually so high we don't detect the rapid time variation. If E and B are sinusoidal, then E2 = E \/2, just as for electric currents and voltages (Section 25 -7), where E0 is the maximum value of E. Thus we can write for the magnitude of the Poynting vector, on the average, 1 _ ^ B2 = E0B0 2 Mo

(31-19 )

2//,0

where B0is the maximum value of B. This time averaged value of S is the intensity, defined as the average power transferred across unit area (Section 15-3). We can also write for the average value of S: S =

(31 - 19b)

where Erms and BTms are the rms values (£rms = \ / e 2 , Brms = \Z l¥ ). EXAMPLE 31 -6 _ E and B from the Sun. Radiation from the Sun reaches the Earth (above the atmosphere) at a rate of about 1350J/s - m 2(= 1350W /m2). Assume that this is a single EM wave, and calculate the maximum values of E and B. APPR ACH We solve Eq. 31-19a (5 = §e 0c£o) for ^o in terms of S using S = 1350 J/s - m 2. 2(1350 J/s - m 2)

S LUTI N

(8.85 = 1.01

10"12 2/N - m 2)(3.00

10 m/s)

103V/m.

From Eq. 31-11, B = E /c , so _ E0 1.01 X 103V/m Bn = — = ---------- , „ , = 3.37 c 3.00 X 10 m/s

10 6T.

N TE Although B has a small numerical value compared to E (because of the way the different units for E and B are defined), B contributes the same energy to the wave as E does, as we saw earlier (Eqs. 31-15 and 16). SECTION 31 - 8

E ergy i EM Waves; the Poy ti g Vector

827

31—9 R di tion Pressure If electromagnetic waves carry energy, then we might expect them to also carry linear momentum. When an electromagnetic wave encounters the surface of an object, a force will be exerted on the surface as a result of the momentum transfer (F = dp/dt), just as when a moving object strikes a surface. The force per unit area exerted by the waves is called r di tion pressure, and its existence was predicted by Maxwell. He showed that if a beam of EM radiation (light, for example) is completely absorbed by an object, then the momentum transferred is Ap

radiation fully absorbed

= —

(31-20 )

where A t/ is the energy absorbed by the object in a time At, and c is the speed of light.1" If instead, the radiation is fully reflected (suppose the object is a mirror), then the momentum transferred is twice as great, just as when a ball bounces elastically off a surface: radiation 2 AU fully (31-20b) Ap = — —

reflected If a surface absorbs some of the energy, and reflects some of it, then Ap = a AU/c, where a is a factor between 1 and 2. Using Newton's second law we can calculate the force and the pressure exerted by radiation on the object. The force F is given by dp F = - j -dt The average rate that energy is delivered to the object is related to the Poynting vector by dT

where A is the cross-sectional area of the object which intercepts the radiation. The radiation pressure P (assuming full absorption) is given by (see Eq. 31-20a) -

Adt

Ac dt

[M l,

I absorbed I

If the light is fully reflected, the pressure is twice as great (Eq. 31-20b):

f - T

R0. (c) Plot B0(R) in tesla for the range 0 < R < 10 cm.

31 - 5 EM Waves . (I) If the electric field in an EM wave has a peak magnitude of 0.57 X 10_4V/m, what is the peak magnitude of the magnetic field strength?

9. (I) If the magnetic field in a traveling EM wave has a peak magnitude of 12.5 nT, what is the peak magnitude of the electric field? 10. (I) In an EM wave traveling west, the B field oscillates vertically and has a frequency of 80.0 kHz and an rms strength of 7.75 X 10_9T. Determine the frequency and rms strength of the electric field. What is its direction? 11. (II) The electric field of a plane EM wave is given by Ex = E0cos(kz + (tit), Ey = Ez = 0. Determine (a) the direc tion of propagation and (b) the magnitude and direction of B. 12. (Ill) onsider two possible candidates E (x,t) as solutions of the wave equation for an EM wave's electric field. Let A and a be constants. Show that (a) E (x ,t) = Ae~a^x~vt^2 satisfies the wave equation, and that (b) E (x ,t) = A e ~ ^ ~ vt^ does not satisfy the wave equation.

31 - 6 Electromagnetic Spectrum 13. (I) What is the frequency of a microwave whose wavelength is 1.50 cm? 14. (I) (a) What is the wavelength of a 25.75 X 109Hz radar signal? (b) What is the frequency of an X-ray with wave length 0.12 nm? 15. (I) How long does it take light to reach us from the Sun, 1.50 X 10 km away? 16. (I) An EM wave has frequency 8.56 X 1014 Hz. What is its wavelength, and how would we classify it? 17. (I) Electromagnetic waves and sound waves can have the same frequency, (a) What is the wavelength of a 1.00-kHz electromagnetic wave? (b) What is the wavelength of a 1.00-kHz sound wave? (The speed of sound in air is 341 m/s.) (c) an you hear a 1.00-kHz electromagnetic wave? 18. (II) Pulsed lasers used for science and medicine produce very brief bursts of electromagnetic energy. If the laser light wavelength is 1062 nm (Neodymium -YAG laser), and the pulse lasts for 38 picoseconds, how many wavelengths are found within the laser pulse? How brief would the pulse need to be to fit only one wavelength? 19. (II) How long would it take a message sent as radio waves from Earth to reach Mars (a) when nearest Earth, (b) when farthest from Earth? 20. (II) An electromagnetic wave has an electric field given by E = i(225 V/m) sin[(0.077 m-1)z —(2.3 X 107 rad/s)?]. (a) What are the wavelength and frequency of the wave? (b) Write down an expression for the magnetic field. Problems

833

31 - 7 Speed of Light 21. (II) What is the minimum angular speed at which Michelson's eight-sided mirror would have had to rotate to reflect light into an observer's eye by succeeding mirror faces (1/8 of a revolution, Fig. 31 -14)?

31 - 8 EM Wave Energy; Poynting Vector 22. (I) The E field in an EM wave has a peak of 26.5 mV/m. What is the average rate at which this wave carries energy across unit area per unit time? 23. (II) The magnetic field in a traveling EM wave has an rms strength of 22.5 nT. How long does it take to deliver 335 J of energy to 1.00 cm2 of a wall that it hits perpendicularly? 24. (II) How much energy is transported across a 1.00 cm2 area per hour by an EM wave whose E field has an rms strength of 32.8 mV/m? 25. (II) A spherically spreading EM wave comes from a 1500-W source. A t a distance of 5.0 m, what is the intensity, and what is the rms value of the electric field? 26. (II) If the amplitude of the B field of an EM wave is 2.5 X 10- 7 T, (a) what is the amplitude of the E field? (b) What is the average power per unit area of the EM wave? 27. (II) What is the energy contained in a 1.00-m3 volume near the E arth's surface due to radiant energy from the Sun? See Example 31 - 6. 28. (II) A 15.8-mW laser puts out a narrow beam 2.00 mm in diameter. What are the rms values of E and B in the beam? 29. (II) Estimate the average power output of the Sun, given that about 1350 W /m 2 reaches the upper atmosphere of the Earth. 30. (II) A high-energy pulsed laser emits a 1.0-ns-long pulse of average power 1.8 X 1011 W. The beam is 2.2 X 10_3m in radius. Determine (a) the energy delivered in each pulse, and (b) the rms value of the electric field. 31. (II) How practical is solar power for various devices? Assume that on a sunny day, sunlight has an intensity of 1000 W /m 2 at the surface of Earth and that, when illumi nated by that sunlight, a solar-cell panel can convert 10% of the sunlight's energy into electric power. For each device given below, calculate the area A of solar panel needed to power it. (a) A calculator consumes 50 mW. Find A in cm2. Is A small enough so that the solar panel can be mounted directly on the calculator that it is powering? (b) A hair dryer consumes 1500 W. Find A in m2. Assuming no other electronic devices are operating within a house at the same time, is A small enough so that the hair dryer can be powered by a solar panel mounted on the house's roof? (c) A car requires 20 hp for highway driving at constant velocity (this car would perform poorly in situations requiring acceleration). Find A in m2. Is A small enough so that this solar panel can be mounted directly on the car and power it in "real time"? (27) 32. (Ill) (a ) Show that the Poynting vector S points radially inward toward the center of a circular parallel-plate capacitor when it is being charged as in Example 31 -1. (b) Integrate S over the cylindrical boundary of the capacitor gap to show that the rate at which energy enters the capacitor is equal to the rate at which electrostatic energy is being stored in the electric field of the capacitor (Section 24 - 4). Ignore fringing of E.

834

HAPTER 31

33. (Ill) The Arecibo radio telescope in Puerto Rico can detect a radio wave with an intensity as low as 1 X 10_23W /m 2. As a "best-case" scenario for communication with extrater restrials, consider the following: suppose an advanced civilization located at point A, a distance x away from Earth, is somehow able to harness the entire power output of a Sun-like star, converting that power completely into a radio-wave signal which is transmitted uniformly in all directions from A. (a) In order for Arecibo to detect this radio signal, what is the maximum value for x in light-years ( l l y ~ 1 0 16m)? (b) How does this maximum value compare with the 100,000-ly size of our Milky Way galaxy? The intensity of sunlight at Earth's orbital distance from the Sun is 1350 W /m 2.

31 - 9 Radiation Pressure 34. (II) Estimate the radiation pressure due to a 75-W bulb at a distance of 8.0 cm from the center of the bulb. Estimate the force exerted on your fingertip if you place it at this point. 35. (II) Laser light can be focused (at best) to a spot with a radius r equal to its wavelength A. Suppose that a 1.0-W beam of green laser light (A = 5 X 10-7 m) is used to form such a spot and that a cylindrical particle of about that size (let the radius and height equal r) is illuminated by the laser as shown in Fig. 31 -23. Estimate the acceleration of the particle, if its density equals that of water and it absorbs the radiation. [This order-of-magnitude calculation convinced researchers of the feasibility of "optical tweezers," p. 829.]

FIGURE 31-23 Problem 35.

36. (II) The powerful laser used in a laser light show provides a 3-mm diameter beam of green light with a power of 3 W. When a space-walking astronaut is outside the Space Shuttle, her colleague inside the Shuttle playfully aims such a laser beam at the astronaut's space suit. The masses of the suited astronaut and the Space Shuttle are 120 kg and 103,000 kg, respectively, (a) Assuming the suit is perfectly reflecting, determine the "radiation - pressure" force exerted on the astronaut by the laser beam, (b) Assuming the astro naut is separated from the Shuttle's center of mass by 20 m, model the Shuttle as a sphere in order to estimate the grav itation force it exerts on the astronaut, (c) Which of the two forces is larger, and by what factor? 37. What size should the solar panel on a satellite orbiting Jupiter be if it is to collect the same amount of radiation from the Sun as a 1.0-m2 solar panel on a satellite orbiting Earth?

Maxwell's Equations and Electromagnetic Waves

31 - 10 Radio, TV, Wireless 38. (I) What is the range of wavelengths for (a) FM radio ( MHz to 108 MHz) and (b) AM radio (535 kHz to 1700 kHz)? 39. (I) Estimate the wavelength for 1.9-GHz cell phone reception. 40. (I) The variable capacitor in the tuner of an AM radio has a capacitance of 2200 pF when the radio is tuned to a station at 550 kHz. What must the capacitance be for a station near the other end of the dial, 1610 kHz?

41. (II) A certain FM radio tuning circuit has a fixed capacitor = 620 pF. Tuning is done by a variable inductance. What range of values must the inductance have to tune stations from MHz to 108 MHz? 42. (II) A satellite beams microwave radiation with a power of 12 kW toward the Earth's surface, 550 km away. When the beam strikes Earth, its circular diameter is about 1500 m. Find the rms electric field strength of the beam at the surface of the Earth.

Gener l Problems___________ 43. A 1.60-m-long FM antenna is oriented parallel to the elec tric field of an EM wave. How large must the electric field be to produce a 1.00-mV (rms) voltage between the ends of the antenna? What is the rate of energy transport per square meter? 44. Who will hear the voice of a singer first: a person in the balcony 50.0 m away from the stage (see Fig. 31 - 24), or a person 1500 km away at home whose ear is next to the radio listening to a live broadcast? Roughly how much sooner? Assume the microphone is a few centimeters from the singer and the temperature is 20° . h

50.0 m

48.

49.

50.

51.

FIGURE 31-24 Problem 44. 45. Light is emitted from an ordinary lightbulb filament in wave-train bursts about 10_ 8s in duration. What is the length in space of such wave trains? 46. Radio -controlled clocks throughout the United States receive a radio signal from a transmitter in Fort ollins, olorado, that accurately (within a microsecond) marks the beginning of each minute. A slight delay, however, is introduced because this signal must travel from the trans mitter to the clocks. Assuming Fort ollins is no more than 3000 km from any point in the U.S., what is the longest travel-time delay? 47. A radio voice signal from the Apollo crew on the Moon (Fig. 31 - 25) was beamed to a listening crowd from a radio speaker. If you were standing 25 m from the loudspeaker, what was the total time lag between when you heard the sound and when the sound entered a microphone on the Moon and traveled to Earth?

52.

53.

54.

55.

osmic microwave background radiation fills all space with an average energy density of 4 X 10_14J/m 3. (a) Find the rms value of the electric field associated with this radiation. (b) How far from a 7.5-kW radio transmitter emitting uniformly in all directions would you find a comparable value? What are E0 and B0 2.00 m from a 75-W light source? Assume the bulb emits radiation of a single frequency uniformly in all directions. Estimate the rms electric field in the sunlight that hits Mars, knowing that the Earth receives about 1350 W /m 2 and that Mars is 1.52 times farther from the Sun (on average) than is the Earth. At a given instant in time, a traveling EM wave is noted to have its maximum magnetic field pointing west and its maximum electric field pointing south. In which direction is the wave traveling? If the rate of energy flow is 560 W /m 2, what are the maximum values for the two fields? How large an emf (rms) will be generated in an antenna that consists of a circular coil 2.2 cm in diameter having 320 turns of wire, when an EM wave of frequency 810 kHz transporting energy at an average rate of 1.0 X 10-4 W /m 2 passes through it? [Hint, you can use Eq. 29 - 4 for a generator, since it could be applied to an observer moving with the coil so that the magnetic field is oscillating with the frequency / = a)/2it.\ The average intensity of a particular TV station's signal is 1.0 X 10-13 W /m 2 when it arrives at a 33-cm -diameter satel lite TV antenna, (a) alculate the total energy received by the antenna during 6.0 hours of viewing this station's programs. (b) What are the amplitudes of the E and B fields of the EM wave? A radio station is allowed to broadcast at an average power not to exceed 25 kW. If an electric field amplitude of 0.020 V /m is considered to be acceptable for receiving the radio transmission, estimate how many kilometers away you might be able to hear this station. A point source emits light energy uniformly in all directions at an average rate P0 with a single frequency / . Show that the peak electric field in the wave is given by E. =

V qcP q lir r 2

FIGURE 31 -25

Problem 47.

56. Suppose a 35-kW radio station emits EM waves uniformly in all directions, (a) How much energy per second crosses a 1 .0-m2 area 1.0 km from the transmitting antenna? (ib) What is the rms magnitude of the E field at this point, assuming the station is operating at full power? What is the rms voltage induced in a 1 .0-m-long vertical car antenna (c) 1.0 km away, (d) 50 km away?

General Problems

835

57. What is the maximum power level of a radio station so as to avoid electrical breakdown of air at a distance of 0.50 m from the transmitting antenna? Assume the antenna is a point source. Air breaks down in an electric field of about 3 X 106V/m. 58. In free space ("vacuum"), where the net charge and current flow is zero, the speed of an EM wave is given by v = l/V e o ^ o - If, instead, an EM wave travels in a noncon ducting ("dielectric") material with dielectric constant K, then v = l / y l e ^ . For frequencies corresponding to the visible spectrum (near 5 X 1014 Hz), the dielectric constant of water is 1.77. Predict the speed of light in water and compare this value (as a percentage) with the speed of light in a vacuum. 59. The metal walls of a microwave oven form a cavity of dimen sions 37 cm X 37 cm X 20 cm. When 2.45-GHz microwaves are continuously introduced into this cavity, reflection of inci dent waves from the walls set up standing waves with nodes at the walls. Along the 37-cm dimension of the oven, how many nodes exist (excluding the nodes at the wall) and what is the distance between adjacent nodes? [Because no heating occurs at these nodes, most microwaves rotate food while operating.] 60. Imagine that a steady current I flows in a straight cylindrical wire of radius R 0 and resistivity p. (a) If the current is then changed at a rate d i/d t, show that a displacement current / D exists in the wire of magnitude e0p (d l/d t). (b) If the current in a copper wire is changed at the rate of 1.0 A /m s, deter mine the magnitude of / D. (c) Determine the magnitude of the magnetic field BD (T) created by /d at the surface of a copper wire with R 0 = 1.0 mm. ompare (as a ratio) BD with the field created at the surface of the wire by a steady current of 1.0 A.

61. The electric field of an EM wave pulse traveling along the x axis in free space is given by Ey = E0 exp [ - a 2* 2 - (32t2 + 2a(3xt], where E q, a, and /3 are positive constants, (a) Is the pulse moving in the Hjic - or —x direction? (b) Express /3 in terms of a and c (speed of light in free space), (c) Determine the expression for the magnetic field of this EM wave. 62. Suppose that a right-moving EM wave overlaps with a leftmoving EM wave so that, in a certain region of space, the total electric field in the y direction and magnetic field in the z direc tion are given by Ey = Eq sin(kx — cot) + Eq sin(kx + cot) and Bz = Bq sin(/:x - cot) — Bq sin(/:x + cot), (a) Find the mathematical expression that represents the standing electric and magnetic waves in the y and z directions, respectively. (b) Determine the Poynting vector and find the x locations at which it is zero at all times. 63. The electric and magnetic fields of a certain EM wave in free space are given by E = E 0 sin (A:* - cot)j + E qcos (kx —cot) k and B = i?0c o s ^ x ~~ t)j - B0sin(kx — cot)k. (a) Show that E and B are perpendicular to each other at all times, (b) For this wave, E and B are in a plane parallel to the yz plane. Show that the wave moves in a direction perpendicular to both E and B. (c) A t any arbitrary choice of position x and time t, show that the magnitudes of E and B always equal E q and Bq, respectively. (d) A t x = 0, draw the orientation of E and B in the yz plane at t = 0. Then qualitatively describe the motion of these vectors in the yz plane as time increases. [Note: The EM wave in this Problem is "circularly polarized."]

Answers to Exercises A: (c).

(b). : (a) 3.8

836

106Hz; (b) 5.5

HAPTER 31

101 Hz.

45 cm.

E: Yes; the signal still travels 72,000 km. F: Over 4 hours.

Maxwell's Equations and Electromagnetic Waves

Reflection from still w ter, s from gl ss mirror, c n be n lyzed using the r y model of light. Is this picture right side up? How c n you tell? Wh t re the clues? Notice the people nd position of the Sun. R y di gr ms, which we will le rn to dr w in this Ch pter, c n provide the nswer. See Ex mple 32 -3. In this first Ch pter on light nd optics, we use the r y model of light to underst nd the form tion of im ges by mirrors, both pl ne nd curved (spheric l). We lso begin our study of refr ction— how light r ys bend when they go from one medium to nother — which prep res us for our study in the next Ch pter of lenses, which re the cruci l p rt of so m ny optic l instruments.

Light: Reflection and Refraction CHAPTER-OPENING QUESTION — Guess now! A 2.0-m-tall person is standing 2.0 m from a flat vertical mirror staring at her image. What minimum height must the mirror have if the person is to see her entire body, from the top of her head to her feet? ( ) 0.50 m.

(b) 1.0m. (c) 1.5 m. (d) 2.0 m. (e) 2.5 m. he sense of sight is extremely important to us, for it provides us with a large part of our information about the world. How do we see? What is the something called lig t that enters our eyes and causes the sensation of sight? How does light behave so that we can see everything that we do? We saw in hapter 31 that light can be considered a form of electromagnetic radiation. We now examine the subject of light in detail in the next four hapters. We see an object in one of two ways: (1) the object may be a source of light, such as a lightbulb, a flame, or a star, in which case we see the light emitted directly from the source; or, more commonly, (2) we see an object by light reflected from it. In the latter case, the light may have originated from the Sun, artificial lights, or a campfire. An understanding of how objects emit light was not achieved until the 1920s, and will be discussed in hapter 37. How light is reflected from objects was understood earlier, and will be discussed here, in Section 32 -2.

CONTENTS 32 - 1 The R y Model of Light 3 2 - 2 Reflection; Im ge Form tion by Pl ne Mirror 32 - 3 Form tion of Im ges by Spheric l Mirrors 3 2 - 4 Index of Refr ction 32 - 5 Refr ction: Snell's L w 32 - 6 Visible Spectrum nd Dispersion 3 2 - 7 Tot l Intern l Reflection; Fiber Optics *32 - 8 Refr ction t Surf ce

Spheric l

837

3 2 —1 The R y Model o f Light

This bundle

FIGURE 32 - 1 Light r ys come from e ch single point on n object. A sm ll bundle of r ys le ving one point is shown entering person's eye.

A great deal of evidence suggests that lig t travels in straig t lines under a wide variety of circumstances. For example, a source of light like the Sun casts distinct shadows, and the light from a laser pointer appears to be a straight line. In fact, we infer the positions of objects in our environment by assuming that light moves from the object to our eyes in straight-line paths. Our orientation to the physical world is based on this assumption. This reasonable assumption is the basis of the r y model of light. This model assumes that light travels in straight-line paths called light r ys. Actually, a ray is an idealization; it is meant to represent an extremely narrow beam of light. When we see an object, according to the ray model, light reaches our eyes from each point on the object. Although light rays leave each point in many different directions, normally only a small bundle of these rays can enter an observer's eye, as shown in Fig. 32-1. If the person's head moves to one side, a different bundle of rays will enter the eye from each point. We saw in hapter 31 that light can be considered as an electromagnetic wave. Although the ray model of light does not deal with this aspect of light (we discuss the wave nature of light in hapters 34 and 35), the ray model has been very successful in describing many aspects of light such as reflection, refraction, and the formation of images by mirrors and lenses.1 Because these explanations involve straight-line rays at various angles, this subject is referred to as geometric optics.

3 2 —2 Reflection; Im ge Form tion by Pl ne Mirror When light strikes the surface of an object, some of the light is reflected. The rest can be absorbed by the object (and transformed to thermal energy) or, if the object is transparent like glass or water, part can be transmitted through. For a very smooth shiny object such as a silvered mirror, over 95% of the light may be reflected. When a narrow beam of light strikes a flat surface (Fig. 32 -2), we define the ngle of incidence, 6 to be the angle an incident ray makes with the normal (perpendicular) to the surface, and the ngle of reflection, 0r , to be the angle the reflected ray makes with the normal. It is found that the incident and reflected rays lie in t e same plane wit t e normal to t e surface, and that the ngle of reflection equ ls the ngle of incidence, 0r = 0\. This is the l w of reflection, and it is depicted in Fig. 32-2. It was known to the ancient Greeks, and you can confirm it yourself by shining a narrow flashlight beam or a laser pointer at a mirror in a darkened room.

L w of reflection: ( ) Shows 3 -D view of n incident r y being reflected t the top of fl t surf ce; (b) shows side or "end -on " view, which we will usu lly use bec use of its cl rity.

Norm l to surf ce I I I

FIGURE 32 - 2

A n alp n f

Incident light r y

Norm l to surf ce

I A n rrl*=» n f

Reflected light r y

fIn ignoring the w ve properties of light we must be c reful th t when the light r ys p ss by objects or through pertures, these must be l rge comp red to the w velength of the light (so the w ve phenomen of interference nd diffr ction, s discussed in Ch pter 15, c n be ignored), nd we ignore wh t h ppens to the light t the edges of objects until we get to Ch pters 34 nd 35.

838

CHAPTER 32

Light: Reflectio a d Refractio

When light is incident upon a rough surface, even microscopically rough such as this page, it is reflected in many directions, as shown in Fig. 32 - 3. This is called diffuse reflection. The law of reflection still holds, however, at each small section of the surface. Because of diffuse reflection in all directions, an ordinary object can be seen at many different angles by the light reflected from it. When you move your head to the side, different reflected rays reach your eye from each point on the object (such as this page), Fig. 32 - 4a. Let us compare diffuse reflection to reflection from a mirror, which is known as specul r reflection. ("Speculum" is Latin for mirror.) When a narrow beam of light shines on a mirror, the light will not reach your eye unless your eye is positioned at just the right place where the law of reflection is satisfied, as shown in Fig. 32 - 4b. This is what gives rise to the special image-forming properties of mirrors. Eye t both

The eye here does not see reflected light

Diffuse reflection rough surf ce.

FIGURE 3 2 - 3

from

_ , The eye here does see

FIG URE 3 2 - 4 A n rrow be m of light shines on ( ) white p per, nd (b) mirror. In p rt ( ), you c n see with your eye the white light reflected t v rious positions bec use of diffuse reflection. But in p rt (b), you see the reflected light only when your eye is pl ced correctly (0r = 6[); mirror reflection is lso known s specul r reflection. (G lileo, using simil r rguments, showed th t the M oon must h ve rough surf ce r ther th n highly polished surf ce like mirror, s some people thought.)

EXAMPLE 32-1 Reflection from flat mirrors. Two flat mirrors are perpen dicular to each other. An incoming beam of light makes an angle of 15° with the first mirror as shown in Fig. 32 - 5a. What angle will the outgoing beam make with the second mirror? APPR ACH We sketch the path of the beam as it reflects off the two mirrors, and draw the two normals to the mirrors for the two reflections. We use geometry and the law of reflection to find the various angles. FIG URE 3 2 - 5

Ex mple 32 - 1.

S LUTI N In Fig. 32-5b,

+ 15° = 90°, so Qx = 75°; by the law of reflection

02 = 0i = 75° too. The two normals to the two mirrors are perpendicular to each as for any triangle. Thus other, so 02 + 03 + 90° = 180° 03 = 180° - 90° - 75° = 15°. By the law of reflection, 04 = 03 = 15°, so 05 = 75° is the angle the reflected ray makes with the second mirror surface.

N TE The outgoing ray is parallel to the incoming ray. Red reflectors on bicycles and cars use this principle. When you look straight into a mirror, you see what appears to be yourself as well as various objects around and behind you, Fig. 32 - 6. Your face and the other objects look as if they are in front of you, beyond the mirror. But what you see in the mirror is an im ge of the objects, including yourself, that are in front of the mirror. SECTION 3 2 - 2

FIGURE 3 2 - 6 When you look in mirror, you see n im ge of yourself nd objects round you. You don 't see yourself s others see you, bec use left nd right ppe r reversed in the im ge.

Reflectio ; Image Formatio

by a Pla e Mirror

839

A "plane" mirror is one with a smooth flat reflecting surface. Figure 32 - 7 shows how an image is formed by a plane mirror according to the ray model. We are viewing the mirror, on edge, in the diagram of Fig. 32 - 7, and the rays are shown reflecting from the front surface. (Good mirrors are generally made by putting a highly reflective metallic coating on one surface of a very flat piece of glass.) Rays from two different points on an object (the bottle on the left in Fig. 32 - 7) are shown: two rays are shown leaving from a point on the top of the bottle, and two more from a point on the bottom. Rays leave each point on the object going in many directions, but only those that enclose the bundle of rays that enter the eye from each of the two points are shown. Each set of diverging rays that reflect from the mirror and enter the eye appear to come from a single point (called the image point) behind the mirror, as shown by the dashed lines. That is, our eyes and brain interpret any rays that enter an eye as having traveled straightline paths. The point from which each bundle of rays seems to come is one point on the image. For each point on the object, there is a corresponding image point. Pl ne mirror

FIGURE 3 2 - 7 Form tion of im ge by pl ne mirror.

virtu l

Let us concentrate on the two rays that leave point A on the object in Fig. 32 -7, and strike the mirror at points B and B'. We use geometry for the rays at B. The angles ADB and DB are right angles; and because of the law of reflection, dx = 0r at point B. Therefore, angles ABD and BD are also equal. The two triangles ABD and BD are thus congruent, and the length AD = D. That is, the image appears as far behind the mirror as the object is in front. The im ge dist nce, dx(perpendicular distance from mirror to image, Fig. 32-7), equals the object dist nce, d0 (perpendicular distance from object to mirror). From the geom etry, we can also see that the height of the image is the same as that of the object. The light rays do not actually pass through the image location itself in Fig. 32 -7. (Note where the red lines are dashed to show they are our projections, not rays.) The image would not appear on paper or film placed at the location of the image. Therefore, it is called a virtu l im ge. This is to distinguish it from a re l im ge in which the light does pass through the image and which therefore could appear on film or in an electronic sensor, and even on a white sheet of paper or screen placed at the position of the image. Our eyes can see both real and virtual images, as long as the diverging rays enter our pupils. We will see that curved mirrors and lenses can form real images, as well as virtual. A movie projector lens, for example, produces a real image that is visible on the screen. © - PHYSI CS APPLI ED H ow tall a mirror do you need to see a reflection o f your entire self?

840

CHAPTER 32

EXAMPLE 32-2 How tall must a full-length mirror be? A woman 1.60 m tall stands in front of a vertical plane mirror. What is the minimum height of the mirror, and how close must its lower edge be to the floor, if she is to be able to see her whole body? Assume her eyes are 10 cm below the top of her head. APPR ACH For her to see her whole body, light rays from the top of her head and from the bottom of her foot must reflect from the mirror and enter her eye: see Fig. 32 -8. We don't show two rays diverging from each point as we did in Fig. 32 -7, where we wanted to find where the image is. Now that we know the image is the same distance behind a plane mirror as the object is in front, we only need to show one ray leaving point G (top of head) and one ray leaving point A (her toe), and then use simple geometry.

FIGURE 32 - 8 Seeing oneself in mirror. Ex mple 32 - 2.

S LUTI N First consider the ray that leaves her foot at A, reflects at B, and enters the eye at E. The mirror needs to extend no lower than B. The angle of reflection equals the angle of incidence, so the height BD is half of the height AE. Because AE = 1.60 m - 0.10 m = 1.50 m, then BD = 0.75 m. Similarly, if the woman is to see the top of her head, the top edge of the mirror only needs to reach point F, which is 5 cm below the top of her head (half of GE = 10 cm). Thus, DF = 1.55 m, and the mirror needs to have a vertical height of only (1.55 m - 0.75 m) = 0.80 m. The mirror's bottom edge must be 0.75 m above the floor. N TE We see that a mirror, if positioned well, need be only half as tall as a person for that person to see all of himself or herself. I EXERCISE A Does the result of Ex mple 32 - 2 depend on your dist nce from the mirror? (Try it.) EXERCISE B Return to the Ch pter-Opening Question, p ge 837, nd nswer it g in now. Try to expl in why you m y h ve nswered differently the first time. EXERCISE C Suppose you re st nding bout 3 m in front of mirror in h ir s lon. You c n see yourself from your he d to your w ist, where the end of the mirror cuts off the rest of your im ge. If you w lk closer to the mirror (a) you will not be ble to see ny more of your im ge; (b) you will be ble to see more of your im ge, below your w ist; (c) you will see less of your im ge, with the cutoff rising to be bove your w ist.

C NCEPTUAL EXAMPLE 32-3~l Is the photo upside down? lose examination of the photograph on the first page of this hapter reveals that in the top portion, the image of the Sun is seen clearly, whereas in the lower portion, the image of the Sun is partially blocked by the tree branches. Show why the reflection is not the same as the real scene by drawing a sketch of this situation, showing the Sun, the camera, the branch, and two rays going from the Sun to the camera (one direct and one reflected). Is the photograph right side up? RESP NSE We need to draw two diagrams, one assuming the photo on p. 837 is right side up, and another assuming it is upside down. Figure 32-9 is drawn assuming the photo is upside down. In this case, the Sun blocked by the tree would be the direct view, and the full view of the Sun the reflection: the ray which reflects off the water and into the camera travels at an angle below the branch, whereas the ray that travels directly to the camera passes through the branches. This works. Try to draw a diagram assuming the photo is right side up (thus assuming that the image of the Sun in the reflection is higher above the horizon than it is as viewed directly). It won't work. The photo on p. 837 is upside down. Also, what about the people in the photo? Try to draw a diagram showing why they don't appear in the reflection. [Hint: Assume they are not sitting on the edge of poolside, but back from the edge a bit.] Then try to draw a diagram of the reverse (i.e., assume the photo is right side up so the people are visible only in the reflection). Reflected images are not perfect replicas when different planes (distances) are involved.

FIGURE 3 2 - 9

Ex mple 32 - 3.

SECTION 3 2 - 2

841

FIGURE 32-10 Mirrors with convex and concave spherical surfaces. Note that 0r = 6[ for each ray.

(b)

3 2 —3

Formation of Images by Spherical Mirrors

R eflecting surfaces do n o t have to b e flat. T he m ost com m on curved m irrors are sp erical, w hich m eans they form a section of a sphere. A spherical m irro r is called co nvex if th e reflection takes place on th e o u te r surface of th e spherical shape so th a t th e c en ter of th e m irro r surface bulges o u t tow ard th e view er (Fig. 3 2 - 10a). A m irror is called concav e if the reflecting surface is on th e inner surface of the sphere so th a t th e m irror surface sinks away from the view er (like a "cave"), Fig. 32 - 10b. oncave m irrors are used as shaving o r cosm etic m irrors (Fig. 32 - 1 1 a) because they magnify, and convex m irrors are som etim es used on cars and trucks (rearview m irrors) and in shops (to w atch for thieves), because they ta k e in a w ide field of view (Fig. 3 2 - l l b ) .

FIGURE 32-11 (a) A concave cosmetic mirror gives a magnified image, (b) A convex mirror in a store reduces image size and so includes a wide field of view.

(a)

Focal

(b)

oint and Focal Length

To see how spherical m irrors form images, we first consider an object th at is very far from a concave m irror. For a distant object, as shown in Fig. 32 - 12, the rays from each point on th e object th at strike the m irror will be nearly parallel. For an object infinitely fa r aw ay (th e Sun and stars approach this), t e rays w o u ld be precisely parallel.

essentially parallel.

FIGURE 32-12 If the object's distance is large compared to the size of the mirror (or lens), the rays are nearly parallel.They are parallel for an object at infinity (oo).

842

HAPTER 32

Light: Reflection and Refraction

Now consider such parallel rays falling on a concave mirror as in Fig. 32-13. The law of reflection holds for each of these rays at the point each strikes the mirror. As can be seen, they are not all brought to a single point. In order to form a sharp image, the rays must come to a point. Thus a spherical mirror will not make as sharp an image as a plane mirror will. However, as we show below, if the mirror is small compared to its radius of curvature, so that a reflected ray makes only a small angle with the incident ray (20 in Fig. 32 -14), then the rays will cross each other at very nearly a single point, or focus. In the case shown in Fig. 32 -14, the incoming rays are parallel to the princip l xis, which is defined as the straight line perpendicular to the curved surface at its center (line A in Fig. 32-14). The point F, where incident parallel rays come to a focus after reflection, is called the foc l point of the mirror. The distance between F and the center of the mirror, length FA, is called the foc l length, / , of the mirror. The focal point is also the image point for an object infinitely far away along the principal axis. The image of the Sun, for example, would be at F.

FIGURE 32 - 13 P r llel r ys striking conc ve spheric l mirror do not intersect (or focus) t precisely single point. (This "defect" is referred to s "spheric l berr tion.")

R ys p r llel to the princip l xis of conc ve spheric l mirror come to focus t F, the foc l point, s long s the mirror is sm ll in width s comp red to its r dius of curv ture, r, so th t the r ys re "p r xi l"— th t is, m ke only sm ll ngles with the horizont l xis. FIGURE 32 - 14

Now we will show, for a mirror whose reflecting surface is small compared to its radius of curvature, that the rays very nearly meet at a common point, F, and we will also calculate the focal length / . In this approximation, we consider only rays that make a small angle with the principal axis; such rays are called p r xi l r ys, and their angles are exaggerated in Fig. 32 -14 to make the labels clear. First we consider a ray that strikes the mirror at B in Fig. 32-14. The point is the center of curvature of the mirror (the center of the sphere of which the mirror is a part). So the dashed line B is equal to r, the radius of curvature, and B is normal to the mirror's surface at B. The incoming ray that hits the mirror at B makes an angle 6 with this normal, and hence the reflected ray, BF, also makes an angle 0 with the normal (law of reflection). Note that angle B F is also 0 as shown. The triangle BF is isosceles because two of its angles are equal. Thus we have length F = BF. We assume the mirror surface is small compared to the mirror's radius of curvature, so the angles are small, and the length FB is nearly equal to length FA. In this approximation, FA = F . But FA = / , the focal length, and A = 2 X FA = r. Thus the focal length is half the radius of curvature: [spherical mirror] (32 -1) We assumed only that the angle 0 was small, so this result applies for all other incident paraxial rays. Thus all paraxial rays pass through the same point F, the focal point. Since it is only approximately true that the rays come to a perfect focus at F, the more curved the mirror, the worse the approximation (Fig. 32 -13) and the more blurred the image. This "defect" of spherical mirrors is called spheric l berr tion; we will discuss it more with regard to lenses in hapter 33. A p r bolic mirror, on the other hand, will reflect the rays to a perfect focus. However, because parabolic shapes are much harder to make and thus much more expensive, spherical mirrors are used for most purposes. (Many astronomical telescopes use parabolic mirrors.) We consider here only spherical mirrors and we will assume that they are small compared to their radius of curvature so that the image is sharp and Eq. 32 -1 holds. SECTION 3 2 - 3

Formatio of Images by Spherical Mirrors

843

Image Formation —Ray Diagrams We saw that for an object at infinity, the image is located at the focal point of a concave spherical mirror, where / = r/2. But where does the image lie for an object not at infinity? First consider the object shown as an arrow in Fig. 32 -15a, which is placed between F and at point O (O for object). Let us determine where the image will be for a given point O' at the top of the object. ( ) R y 1 goes out from O ' p r llel to the xis nd reflects through F.

FIGURE 32 - 15 R ys le ve point O' on the object ( n rrow). Shown re the three most useful r ys for determining where the im ge I' is formed. [Note th t our mirror is not sm ll comp red to / , so our di gr m will not give the precise position of the im ge.]

(b) R y 2 goes through F nd then reflects b ck p r llel to the xis.

(c)

R y 3 is chosen perpendicul r to mirror, nd so must reflect b ck on itself nd go through C (center of curv ture). Diverging r ys he ding tow rd eye

RAY

DI AGRAM

F in d in g t

e im a g e p o s itio n

f o r a c u r v e d m ir r o r

PROBLEM

SOLVI NG

I m a g e p o i n t is w

ere

r e fle c te d r a y s in te r s e c t

844

CHAPTER 32

To do this we can draw several rays and make sure these reflect from the mirror such that the angle of reflection equals the angle of incidence. Many rays could be drawn leaving any point on an object, but determining the image position is simpli fied if we deal with three particularly simple rays. These are the rays labeled 1, 2, and 3 in Fig. 32-15 and we draw them leaving object point O' as follows: Ray 1 is drawn parallel to the axis; therefore after reflection it must pass along a line through F (Fig. 32 -15a). Ray 2 leaves O' and is made to pass through F (Fig. 32-15b); therefore it must reflect so it is parallel to the axis. Ray 3 passes through , the center of curvature (Fig. 32-15c); it is along a radius of the spherical surface and is perpendicular to the mirror, so it is reflected back on itself. All three rays leave a single point O' on the object. After reflection from a (small) mirror, the point at which these rays cross is the image point I'. All other rays from the same object point will also pass through this image point. To find the image point for any object point, only these three types of rays need to be drawn. Only two of these rays are needed, but the third serves as a check. We have shown the image point in Fig. 32-15 only for a single point on the object. Other points on the object are imaged nearby, so a complete image of the object is formed, as shown by the dashed arrow in Fig. 32-15c. Because the light actually passes through the image itself, this is a re l im ge that will appear on a piece of paper or film placed there. This can be compared to the virtual image formed by a plane mirror (the light does not actually pass through that image, Fig. 32-7). The image in Fig. 32-15 can be seen by the eye when the eye is placed to the left of the image, so that some of the rays diverging from each point on the image (as point I') can enter the eye as shown in Fig. 32-15c. (See also Figs. 32-1 and 32-7.)

Mirror Equation and Magnification Image points can be determined, roughly, by drawing the three rays as just described, Fig. 32 -15. But it is difficult to draw small angles for the "paraxial" rays as we assumed. For more accurate results, we now derive an equation that gives the image distance if the object distance and radius of curvature of the mirror are known. To do this, we refer to Fig. 32 -16. The object dist nce, dQ, is the distance of the object (point O) from the center of the mirror. The im ge dist nce, d \, is the distance of the image (point I) from the center of the mirror. The height of the object O O ' is called Qand the height of the image, I'l, is x.

FIG URE 3 2 - 1 6 Di gr m for deriving the mirror equ tion. For the deriv tion, we ssume the mirror size is sm ll comp red to its r dius of curv ture.

Two rays leaving O ' are shown: O 'FB I' (same as ray 2 in Fig. 32 - 15) and O 'A I', which is a fourth type of ray that reflects at the center of the mirror and can also be used to find an image point. The ray O 'A I' obeys the law of reflection, so the two right triangles O 'A O and I'A I are similar. Therefore, we have o =

[

For the other ray shown, O 'FB I', the triangles O 'FO and AFB are also similar because the angles are equal and we use the approximation AB = x (mirror small compared to its radius). Furthermore FA = / , the focal length of the mirror, so ho

_ OF _ dQ FA /

The left sides of the two preceding expressions are the same, so we can equate the right sides: do

~di

f f

We now divide both sides by dQand rearrange to obtain — H— ~ = dQ

(32 - 2)

Mirror equation

This is the equation we were seeking. It is called the mirror equ tion and relates the object and image distances to the focal length / (where / = r / 2). The l ter l m gnific tion, ra, of a mirror is defined as the height of the image divided by the height of the object. From our first set of similar triangles above, or the first equation on this page, we can write: \ m

7hT 0 =

di (3 2 " 3)

The minus sign in Eq. 32 - 4 is inserted as a convention. Indeed, we must be careful about the signs of all quantities in Eqs. 32 - 2 and 32 -3. Sign conventions are chosen so as to give the correct locations and orientations of images, as predicted by ray diagrams. SECTION 3 2 - 3

Formatio of Images by Spherical Mirrors

845

FIGURE 32 - 16

(Repe ted from

previous p ge.)

j PROBLEM Sign conventions

SOLVI NG

The sign conventions we use are: the image height { is positive if the image is upright, and negative if inverted, relative to the object (assuming Qis taken as positive); d± or dQis positive if image or object is in front of the mirror (as in Fig. 32-16); if either image or object is behind the mirror, the corresponding distance is negative (an example can be seen in Fig. 32-18, Example 32 -6). Thus the magnification (Eq. 32 - 3) is positive for an upright image and negative for an inverted image (upside down). We summarize sign conventions more fully in the Problem Solving Strategy following our discussion of convex mirrors later in this Section.

oncave Mirror Examples b =»:fJ iMJI = l» g .■ Image in a concave mirror. A 1.50-cm-high diamond ring is placed 20.0 cm from a concave mirror with radius of curvature 30.0 cm. Determine (a) the position of the image, and (b) its size. APPR ACH We determine the focal length from the radius of curvature (Eq. 32 -1), / = r /2 = 15.0 cm. The ray diagram is basically like that shown in Fig. 32-16 (repeated here on this page), since the object is between F and . The position and size of the image are found from Eqs. 32 - 2 and 32-3. S LUTI N Referring to Fig. 32-16, we have A = r = 30.0 cm, FA = / = 15.0 cm, and OA = dQ = 20.0 cm. (a) From Eq. 32 -2, 1_ _

1 _ 1

di ~ f

A CAUTI ON Remember to take t e reciprocal

d 1

15.0 cm

20.0 cm

= 0.0167 cm

So di = 1/(0.0167 cm x) = 60.0 cm. Because is positive, the image is 60.0 cm in front of the mirror, on the same side as the object. (b) From Eq. 32 -3, the magnification is m =

—

60.0 cm 20.0 cm

= - 3.00.

The image is 3.0 times larger than the object, and its height is i = m Q = ( - 3.00)(1.5 cm) = - 4.5 cm. The minus sign reminds us that the image is inverted, as in Fig. 32-16. N TE When an object is further from a concave mirror than the focal point, we can see from Fig. 32-15 or 32-16 that the image is always inverted and real.

846

CHAPTER 32

Light: Reflectio a d Refractio

C NCEPTUAL EXAMPLE 32 -5 I Reversible rays. If the object in Example 32 -4 is placed instead where the image is (see Fig. 32-16), where will the new image be? RESP NSE The mirror equation is symmetric in d0 and d{. Thus the new image will be where the old object was. Indeed, in Fig. 32-16 we need only reverse the direction of the rays to get our new situation. ■ bject closer to concave mirror. A 1.00-cm-high object is placed 10.0 cm from a concave mirror whose radius of curvature is 30.0 cm. (a) Draw a ray diagram to locate (approximately) the position of the image. (b) Determine the position of the image and the magnification analytically. APPR ACH We draw the ray diagram using the rays as in Fig. 32-15, page 844. An analytic solution uses Eqs. 32 - 1,32 - 2, and 32-3. S LUTI N (a) Since / = r/2 = 15.0 cm, the object is between the mirror and the focal point. We draw the three rays as described earlier (Fig. 32-15); they are shown leaving the tip of the object in Fig. 32-17. Ray 1 leaves the tip of our object heading toward the mirror parallel to the axis, and reflects through F. Ray 2 cannot head toward F because it would not strike the mirror; so ray 2 must point as if it started at F (dashed line) and heads to the mirror, and then is reflected parallel to the principal axis. Ray 3 is perpendicular to the mirror, as before. The rays reflected from the mirror diverge and so never meet at a point. They appear, however, to be coming from a point behind the mirror. This point locates the image of the tip of the arrow. The image is thus behind the mirror and is virtual. (Why?) (b) We use Eq. 32 - 2 to find di when dQ = 10.0 cm:

J_ _ 1 _ J_ _ __ 1________ 1__ _

30.0 cm

15.0 cm

10.0 cm

1 30.0 cm

Therefore, d{ = - 30.0 cm. The minus sign means the image is behind the mirror, which our diagram also told us. The magnification is m = - d J d Q = - ( - 30.0cm )/(10.0cm ) = +3.00. So the image is 3.00 times larger than the object. The plus sign indicates that the image is upright (same as object), which is consistent with the ray diagram, Fig. 32-17. N TE The image distance cannot be obtained accurately by measuring on Fig. 32-17, because our diagram violates the paraxial ray assumption (we draw rays at steeper angles to make them clearly visible). N TE When the object is located inside the focal point of a concave mirror (d0 < f )> the image is always upright and vertical. And if the object O in Fig. 32 -17 is you, you see yourself clearly, because the reflected rays at point O are diverging. Your image is upright and enlarged.

---

> ---1

PHYSI CS APPLI ED Seeing yourself uprig t and magnified in a concave mirror

[ 1 O

i i i

Object pl ced within the foc l point F. The im ge is be ind the mirror nd is virtual, Ex mple 32 - 6. [Note th t the vertic l sc le (height of object = 1.0 cm) is different from the horizont l (O A = 10.0 cm) for e se of dr wing, nd reduces the precision of the dr wing.] FIGURE 32 - 17

SECTION 3 2 - 3

Formatio of Images by Spherical Mirrors

847

P HYS I CS APPLI ED M a g n ify in g m ir r o r (s

a v in g /c o s m e tic )

It is useful to compare Figs. 32-15 and 32-17. We can see that if the object is within the focal point (dQ< /) , as in Fig. 32-17, the image is virtual, upright, and magnified. This is how a shaving or cosmetic mirror is used—you must place your head closer to the mirror than the focal point if you are to see yourself right-side up (see the photograph of Fig. 32-11 a). If the object is beyond the focal point, as in Fig. 32-15, the image is real and inverted (upside down— and hard to use!). Whether the magnification has magnitude greater or less than 1.0 in the latter case depends on the position of the object relative to the center of curvature, point . Practice making ray diagrams with various object distances. The mirror equation also holds for a plane mirror: the focal length is / = r/2 = oo, and Eq. 32 - 2 gives d{ = ~ d Q.

Seeing the Image For a person's eye to see a sharp image, the eye must be at a place where it intercepts diverging rays from points on the image, as is the case for the eye's position in Figs. 32-15 and 32-16. Our eyes are made to see normal objects, which always means the rays are diverging toward the eye as shown in Fig. 32 -1. (Or, for very distant objects like stars, the rays become essentially parallel, as in Fig. 32-12.) If you placed your eye between points O and I in Fig. 32-16, for example, converging rays from the object OO' would enter your eye and the lens of your eye could not bring them to a focus; you would see a blurry image. [We will discuss the eye more in hapter 33.]

FIGURE 32 - 18 You c n see cle r inverted im ge of your f ce when you re beyond C (dQ > 2 /), bec use the r ys th t rrive t your eye re diverging. St nd rd r ys 2 nd 3 re shown le ving point O on your nose. R y 2 ( nd other ne rby r ys) enters your eye. Notice th t r ys re diverging s they move to the left of im ge point I.

Convex mirror: ( ) the foc l point is t F, behind the mirror; (b) the im ge I of the object t O is virtu l, upright, nd sm ller th n the object. [Not to sc le for Ex mple 32 - 7.] FIGURE 32 - 19

If you are the object OO' in Fig. 32 -16, situated between F and , and are trying to see yourself in the mirror, you would see a blur; but the person whose eye is shown in Fig. 32 -16 can see you clearly. You can see yourself clearly, but upside down, if you are to the left of in Fig. 32 -16, where dQ> 2/. Why? Because then the rays reflected from the image will be diverging at your position as demonstrated in Fig. 32 -18, and your eye can then focus them. You can also see yourself clearly, and right-side up, if you are closer to the mirror than its focal point (dQ< /) , as we saw in Example 32 -6, Fig. 32-17.

onvex Mirrors The analysis used for concave mirrors can be applied to convex mirrors. Even the mirror equation (Eq. 32 -2) holds for a convex mirror, although the quantities involved must be carefully defined. Figure 32 -19a shows parallel rays falling on a convex mirror. Again spherical aberration is significant (Fig. 32-13), unless we assume the mirror's size is very small compared to its radius of curvature. The reflected rays diverge, but seem to come from point F behind the mirror. This is the foc l point, and its distance from the center of the mirror (point A) is the foc l length, / . It is easy to show that again / = r/2. We see that an object at infinity produces a virtual image in a convex mirror. Indeed, no matter where the object is

848

CHAPTER 32

Light: Reflectio a d Refractio

placed on the reflecting side of a convex mirror, the image will be virtual and upright, as indicated in Fig. 32-19b. To find the image we draw rays 1 and 3 according to the rules used before on the concave mirror, as shown in Fig. 32-19b. Note that although rays 1 and 3 don't actually pass through points F and , the line along which each is drawn does (shown dashed). The mirror equation, Eq. 32-2, holds for convex mirrors but the focal length / must be considered negative, as must the radius of curvature. The proof is left as a Problem. It is also left as a Problem to show that Eq. 32-3 for the magnification is also valid. S O L V /

9.0

B l

c > ______________________________________________

Spherical Mirrors 1. Always dr w r y di gr m even though you are going to make an analytic calculation—the diagram serves as a check, even if not precise. From one point on the object, draw at least two, preferably three, of the easy-to-draw rays using the rules described in Fig. 32-15. The image point is where the reflected rays intersect or appear to intersect. 2. Apply the mirror equ tion, Eq. 32 -2, and the m gnific tion equ tion, Eq. 32-3. It is crucially important to follow the sign conventions—see the next point.

3. Sign Conventions (a) When the object, image, or focal point is on the reflecting side of the mirror (on the left in our drawings), the corresponding distance is positive. If any of these points is behind the mirror (on the right) the corresponding distance is negatived (b) The image height i is positive if the image is upright, and negative if inverted, relative to the object ( Qis always taken as positive). 4. Check that the analytical solution is consistent with the ray diagram.

fObject dist nces re positive for m teri l objects, but c n be neg tive in systems with more th n one mirror or lens—see Section 33-3.

EXAMPLE 32-7 Convex rearview mirror. An external rearview car mirror is convex with a radius of curvature of 16.0 m (Fig. 32-20). Determine the location of the image and its magnification for an object 10.0 m from the mirror.

@ PHYSI CS

a p p l i e d

Convex rearview mirror

APPR ACH We follow the steps of the Problem Solving Strategy explicitly. S LUTI N 1. Dr w r y di gr m. The ray diagram will be like Fig. 32-19b, but the large object distance (dQ = 10.0 m) makes a precise drawing difficult. We have a convex mirror, so r is negative by convention. 2. Mirror nd m gnific tion equ tions. The center of curvature of a convex mirror is behind the mirror, as is its focal point, so we set r = - 16.0 m so that the focal length is / = r/ 2 = - 8.0 m. The object is in front of the mirror, dQ = 10.0 m. Solving the mirror equation, Eq. 32-2, for 1/d {gives 1

- .0 m

1 10.0 m

- 10.0 - .0 80.0 m

18 80.0 m

FIGURE 32 - 20

Example 32-7.

Thus di = — 80.0m/18 = - 4.4 m. Equation 32 -3 gives the magnification d{

( - 4 .4 m)

= + 0 '4 4 ' m = ~T0 = ~ 3. Sign conventions. The image distance is negative, - 4.4 m, so the image is be ind the mirror. The magnification is m = +0.44, so the image is uprig t (same orientation as object) and less than half as tall as the object. 4. Check. Our results are consistent with Fig. 32-19b. onvex rearview mirrors on vehicles sometimes come with a warning that objects are closer than they appear in the mirror. The fact that d{ may be smaller than dQ(as in Example 32 -7) seems to contradict this observation. The real reason the object seems farther away is that its image in the convex mirror is smaller than it would be in a plane mirror, and we judge distance of ordinary objects such as other cars mostly by their size. SECTION 3 2 - 3

Formatio of Images by Spherical Mirrors

849

TABLE 32-1 Indices of Refraction1

3 2 —4 Index o f Refr ction We saw in

hapter 31 that the speed of light in vacuum is

M teri l

c = 2.99792458 X 10 m/s, V cuum Air ( t STP)

1.0000 1.0003

W ter Ethyl lcohol Gl ss Fused qu rtz Crown gl ss Light flint

1.33

Lucite or Plexigl s Sodium chloride Di mond

1.51

1.36 1.46 1.52 1.58

which is usually rounded off to 3.00 X 10 m/s when extremely precise results are not required. In air, the speed is only slightly less. In other transparent materials such as glass and water, the speed is always less than that in vacuum. For example, in water light travels at about fc. The ratio of the speed of light in vacuum to the speed v in a given material is called the index of refr ction, n, of that material:

1.53

fA = 589i

(32 -4)

n = —•

2.42

The index of refraction is never less than 1, and values for various materials are given in Table 32-1. For example, since n = 2.42 for diamond, the speed of light in diamond is V

(3.00 X 10 m/s) = 1.24 X 10 m/s. 2.42

As we shall see later, n varies somewhat with the wavelength of the light—except in vacuum— so a particular wavelength is specified in Table 32 -1, that of yellow light with wavelength A = 589 nm. That light travels more slowly in matter than in vacuum can be explained at the atomic level as being due to the absorption and reemission of light by atoms and molecules of the material.

3 2 -5 Refr ction: Snell's L w

CAUTI ON

A n g le s d i a n d d2 a re m e a s u r e d fr o m t

e p e r p e n d ic u la r , n o t f r o m

s u r fa c e

When light passes from one transparent medium into another with a different index of refraction, part of the incident light is reflected at the boundary. The remainder passes into the new medium. If a ray of light is incident at an angle to the surface (other than perpendicular), the ray changes direction as it enters the new medium. This change in direction, or bending, is called refr ction. Figure 32-21 a shows a ray passing from air into water. Angle 61is the angle the incident ray makes with the normal (perpendicular) to the surface and is called the ngle of incidence. Angle 02 is the ngle of refr ction, the angle the refracted ray makes with the normal to the surface. Notice that the ray bends toward the normal when entering the water. This is always the case when the ray enters a medium where the speed of light is less (and the index of refraction greater, Eq. 32-4). If light travels from one medium into a second where its speed is greater, the ray bends away from the normal; this is shown in Fig. 32-21b for a ray traveling from water to air. Norm l

Normal

J Retimed

Refracted

Refr ction. ( ) Light refr cted when p ssing from ir ( n i ) into w ter (n 2 ): n 2 > n \ . (b) Light refr cted when p ssing from > n2. w ter («i) into ir ( n 2 ) : FIGURE 32 - 21

Refracted

ray

H2 >H| (a) Ray hends toward 1

850

CHAPTER 32

Light: Reflectio a d Refractio

; ,1> n l

(b ) Ray bonds away from

FIGURE 32 - 22 R y di gr m showing why person 's legs look shorter when st nding in w ist-deep w ter: the p th of light tr veling from the b ther's foot to the observer's eye bends t the w ter's surf ce, nd our br in interprets the light s h ving tr veled in str ight line, from higher up (d shed line). FIGURE 32 - 23 A str w in w ter looks bent even when it isn 't.

Refraction is responsible for a number of common optical illusions. For example, a person standing in waist-deep water appears to have shortened legs. As shown in Fig. 32-22, the rays leaving the person's foot are bent at the surface. The observer's brain assumes the rays to have traveled a straight-line path (dashed red line), and so the feet appear to be higher than they really are. Similarly, when you put a straw in water, it appears to be bent (Fig. 32-23).

Snell's Law The angle of refraction depends on the speed of light in the two media and on the incident angle. An analytical relation between Qx and 02 in Fig- 32-21 was arrived at experimentally about 1621 by Willebrord Snell (1591-1626). It is known as Snell's l w and is written: riismOx = n2sin02.

(32 -5)

SNELL'S LAW (LAW OF REFRACTION)

6i is the angle of incidence and d2 is the angle of refraction; nx and n2 are the respective indices of refraction in the materials. See Fig. 32-21. The incident and refracted rays lie in the same plane, which also includes the perpendicular to the surface. Snell's law is the l w of refr ction. (Snell's law was derived in Section 15-10 where Eq. 15-19 is just a combination of Eqs. 32 -5 and 32-4. We also derive it in hapter 34 using the wave theory of light.) It is clear from Snell's law that if n2 > n1, then 02 < #i •That is, if light enters a medium where n is greater (and its speed is less), then the ray is bent toward the normal. And if n2 < nx, then 02 > 0 i, so the ray bends away from the normal. This is what we saw in Fig. 32-21. EXERCISE D Light p sses from medium with n = 1.3 into medium with n = 1.5. Is the light bent tow rd or w y from the perpendicul r to the interf ce?

Refraction through flat glass. Light traveling in air strikes a flat piece of uniformly thick glass at an incident angle of 60°, as shown in Fig. 32-24. If the index of refraction of the glass is 1.50, (a) what is the angle of refraction 0Ain the glass; (b) what is the angle 0Bat which the ray emerges from the glass?

FIGURE 32 - 24 Light p ssing through piece of gl ss (Ex mple 32 - 8).

APPR ACH We apply Snell's law at the first surface, where the light enters the glass, and again at the second surface where it leaves the glass and enters the air. S LUTI N (a) The incident ray is in air, so nx = 1.00 and n2 = 1.50. Applying Snell's law where the light enters the glass (0! = 60°) gives sin0A = ^

sin 60° = 0.5774,

so 0A = 35.3°. (b) Since the faces of the glass are parallel, the incident angle at the second surface is just 0A(simple geometry), so sin 0A = 0.5774. At this second interface, rii = 1.50 and n2 = 1.00. Thus the ray re-enters the air at an angle 0B( =d2) given by 1.50 sin0B = T o o sin0A = ° ' 866 '

and 0B = 60°. The direction of a light ray is thus unchanged by passing through a flat piece of glass of uniform thickness. N TE This result is valid for any angle of incidence. The ray is displaced slightly to one side, however. You can observe this by looking through a piece of glass (near its edge) at some object and then moving your head to the side slightly so that you see the object directly. It "jumps."

f object

/ "Im ge" (where object ppe rs to be) when viewed through the

SECTION 3 2 - 5

851

FIGURE 32 - 25

Ex mple 32 - 9.

EXAMPLE 32 -9 Apparent depth of a pool. A swimmer has dropped her goggles to the bottom of a pool at the shallow end, marked as 1.0 m deep. But the goggles don't look that deep. Why? How deep do the goggles appear to be when you look straight down into the water? APPR ACH We draw a ray diagram showing two rays going upward from a point on the goggles at a small angle, and being refracted at the water's (flat) surface, Fig. 32-25. The two rays traveling upward from the goggles are refracted away from the normal as they exit the water, and so appear to be diverging from a point above the goggles (dashed lines), which is why the water seems less deep than it actually is. S LUTI N To calculate the apparent depth d' (Fig. 32-25), given a real depth d = 1.0 m, we use Snell's law with ^ = 1.33 for water and n2 = 1.0 for air: sin 02 = Wi sin

•

We are considering only small angles, so sin 0 « tan 0 « 0, with 0 in radians. So Snell's law becomes 02 ~

Wi 0 i .

From Fig. 32-25, we see that X

02 ~ tan 02 = —

and

6X « tan d1 = —■ d

Putting these into Snell's law, 02 ~ n161, we get

or 1.0 m .. d d ~ — = ~ r ^ = 0.75 m. nx 1.33

The pool seems only three-fourths as deep as it actually is.

3 2 - 6 Visible Spectrum nd Dispersion

FIGURE 32 - 26 The spectrum of visible light, showing the r nge of w velengths for the v rious colors s seen in ir. M ny colors, such s brown, do not ppe r in the spectrum; they re m de from mixture of w velengths.

An obvious property of visible light is its color. olor is related to the wavelengths or frequencies of the light. (How this was discovered will be discussed in hapter 34.) Visible light—that to which our eyes are sensitive— has wavelengths in air in the range of about 400 nm to 750 nm.f This is known as the visible spectrum, and within it lie the different colors from violet to red, as shown in Fig. 32-26. Light with wavelength shorter than 400 nm (= violet) is called ultr violet (UV), and light with wavelength greater than 750 nm (= red) is called infr red (IR).* Although human eyes are not sensitive to UV or IR, some types of photographic film and digital cameras do respond to them. A prism can separate white light into a rainbow of colors, as shown in Fig. 32-27. This happens because the index of refraction of a material depends on the wavelength, as shown for several materials in Fig. 32-28. White light is a 4

fi(K) nm

7(XJ nm

J------------ 1------------ 1------------ 1------------ 1------------ 1------------ 1------------ 1

(A)

unit is used when referring to light: 1 A = 1 fSometimes the ngstrom light f lls in the w velength r nge of 4000 to 7500

*The complete electrom gnetic spectrum is illustr ted in Fig. 31-12.

852

CHAPTER 32

Light: Reflectio a d Refractio

10 10m. Then visible

Violet Blue FIGURE 32 - 27 White light p ssing through prism is broken down into its constituent colors.

W velength (nm) Green Yellow Or nge

Red

Index of refr ction s function of w velength for v rious tr nsp rent solids. FIGURE 32 - 28

mixture of all visible wavelengths, and when incident on a prism, as in Fig. 32-29, the different wavelengths are bent to varying degrees. Because the index of refraction is greater for the shorter wavelengths, violet light is bent the most and red the least, as indicated. This spreading of white light into the full spectrum is called dispersion. Rainbows are a spectacular example of dispersion— by drops of water. You can see rainbows when you look at falling water droplets with the Sun behind you. Figure 32 -30 shows how red and violet rays are bent by spherical water droplets and are reflected off the back surface of the droplet. Red is bent the least and so reaches the observer's eyes from droplets higher in the sky, as shown in the diagram. Thus the top of the rainbow is red. FIGURE 32 - 30

( ) R y di gr m expl ining how

White light prism into the visible

FIGURE 3 2 - 29

dispersed by spectrum.

® lP H Y S I C S

APPLIED

R a in b o w s

r inbow (b) is formed.

These two r ys re seen by observer (not to sc le)

(b)

(a)

The visible spectrum, Fig. 32-26, does not show all the colors seen in nature. For example, there is no brown in Fig. 32-26. Many of the colors we see are a mixture of wavelengths. For practical purposes, most natural colors can be reproduced using three primary colors. They are red, green, and blue for direct source viewing such as TV and computer monitors. For inks used in printing, the primary colors are cyan (the color of the margin notes in this book), yellow, and magenta (the color we use for light rays in diagrams). For any wave, its velocity v is related to its wavelength A and frequency / by v = /A (Eq. 15-1 or 31-14). When a wave travels from one material into another, the frequency of the wave does not change across the boundary since a point (an atom) at the boundary oscillates at that frequency. Thus if light goes from air into a material with index of refraction n, the wavelength becomes (recall Eq. 32^): f

n= 7 = ~ f = A

(32"6>

where Ais the wavelength in vacuum or air and \ n is the wavelength in the material with index of refraction n.

SECTION 3 2 - 6

853

C NCEPTUAL EXAMPLE 32-10 I bserved color of light under water. We said that color depends on wavelength. For example, for an object emitting 650 nm light in air, we see red. But this is true only in air. If we observe this same object when under water, it still looks red. But the wavelength in water A„ is (Eq. 32-6) 650nm/1.33 = 489 nm. Light with wavelength 489 nm would appear blue in air. an you explain why the light appears red rather than blue when observed under water? RESP NSE It must be that it is not the wavelength that the eye responds to, but rather the frequency. For example, the frequency of 650 nm red light in air is / = c/A = (3.0 X 108m/s)/(650 X 10_9m) = 4.6 X 1014Hz, and does not change when the light travels from one medium to another. Only Achanges. N TE If we classified colors by frequency, the color assignments would be valid for any material. We typically specify colors by wavelengt in air (even if less general) not just because we usually see objects in air, but because wavelength is what is commonly measured (it is easier to measure than frequency).

3 2 - 7 Tot l Intern l Reflection; Fiber Optics When light passes from one material into a second material where the index of refraction is less (say, from water into air), the light bends away from the normal, as for rays I and J in Fig. 32-31. At a particular incident angle, the angle of refraction will be 90°, and the refracted ray would skim the surface (ray K) in this

FIGURE 32 - 31 Since n 2 < « i , light r ys re tot lly intern lly reflected if the incident ngle > 6q , s for r y L. If di < 6C, s for r ys I nd J, only p rt of the light is reflected, nd the rest is refr cted.

case. The incident angle at which this occurs is called the critic l ngle, 0 . From Snell's law, 0 is given by n2 n2 sm 6C = — sin 90° = — nx ni

CAUTI ON T o ta l in te r n a l r e fle c tio n (o c c u r s o n ly i f r e fr a c tiv e

i n d e x is s m a l l e r b e y o n d b o u n d a r y )

(32 -7)

For any incident angle less than 0 , there will be a refracted ray, although part of the light will also be reflected at the boundary. However, for incident angles greater than 0 , Snell's law would tell us that sin0 2 is greater than 1.00. Yet the sine of an angle can never be greater than 1.00. In this case there is no refracted ray at all, and all o f t e lig t is reflected, as for ray L in Fig. 32-31. This effect is called tot l intern l reflection. Total internal reflection can occur only when light strikes a boundary where the medium beyond has a lower index of refraction. EXERCISE E Fill sink with w ter. Pl ce w terproof w tch just below the surf ce with the w tch 's fl t cryst l p r llel to the w ter surf ce. From bove you c n still see the w tch re ding. As you move your he d to one side f r enough wh t do you see, nd why?

854

CHAPTER 32

Light: Reflectio a d Refractio

C NCEPTUAL EXAMPLE 32-11 I View up from under water. Describe what a person would see who looked up at the world from beneath the perfectly smooth surface of a lake or swimming pool. RESP NSE For an air - water interface, the critical angle is given by 1.00

sin 0C = — — = 0.750. c 1.33 Therefore, 0 = 49°. Thus the person would see the outside world compressed into a circle whose edge makes a 49° angle with the vertical. Beyond this angle, the person would see reflections from the sides and bottom of the lake or pool (Fig. 32-32).

FIGURE 32 - 32 ( ) Light r ys, nd (b) view looking upw rd from bene th the w ter (the surf ce of the w ter must be very smooth). Ex mple 32 - 11.

Diamonds achieve their brilliance from a combination of dispersion and total internal reflection. Because diamonds have a very high index of refraction of about 2.4, the critical angle for total internal reflection is only 25°. The light dispersed into a spectrum inside the diamond therefore strikes many of the internal surfaces of the diamond before it strikes one at less than 25° and emerges. After many such reflections, the light has traveled far enough that the colors have become sufficiently separated to be seen individually and brilliantly by the eye after leaving the diamond. Many optical instruments, such as binoculars, use total internal reflection within a prism to reflect light. The advantage is that very nearly 100% of the light is reflected, whereas even the best mirrors reflect somewhat less than 100%. Thus the image is brighter, especially after several reflections. For glass with n = 1.50,0 = 41.8°. Therefore, 45° prisms will reflect all the light internally, if oriented as shown in the binoculars of Fig. 32-33.

FIGURE 32 - 33 Tot l intern l reflection of light by prisms in binocul rs.

EXERCISE F If 45.0° pl stic lenses re used in binocul rs, wh t minimum index of refr ction must the pl stic h ve?

Fiber Optics Total internal reflection is the principle behind fiber optics. Glass and plastic fibers as thin as a few micrometers in diameter are common. A bundle of such tiny fibers is called a light pipe or cable, and light1" can be transmitted along it with almost no loss because of total internal reflection. Figure 32 -34 shows how light traveling down a thin fiber makes only glancing collisions with the walls so that total internal reflection occurs. Even if the light pipe is bent into a complicated shape, the critical angle still won't be exceeded, so light is transmitted practically undiminished to the other end. Very small losses do occur, mainly by reflection at the ends and absorption within the fiber. Important applications of fiber-optic cables are in communications and medicine. They are used in place of wire to carry telephone calls, video signals, and computer data. The signal is a modulated light beam (a light beam whose intensity can be varied) and data is transmitted at a much higher rate and with less loss and less interference than an electrical signal in a copper wire. Fibers have been developed that can support over one hundred separate wavelengths, each modulated to carry up to 10 gigabits (lO10bits) of information per second. That amounts to a terabit (lO12bits) per second for the full one hundred wavelengths.

FIGURE 32 - 34 Light reflected tot lly t the interior surf ce of gl ss or tr nsp rent pl stic fiber.

PHYSI CS

APPLIED

F ib e r o p tic s in c o m m u n ic a tio n s

fFiber optic devices use not only visible light but lso infr red light, ultr violet light, nd microw ves.

SECTION 3 2 - 7

Total I ter al Reflectio ; Fiber Optics

855

PHYSI CS

APPLIED

The sophisticated use of fiber optics to transmit a clear picture is particularly useful in medicine, Fig. 32-35. For example, a patient's lungs can be examined by inserting a light pipe known as a bronchoscope through the mouth and down the bronchial tube. Light is sent down an outer set of fibers to illuminate the lungs. The reflected light returns up a central core set of fibers. Light directly in front of each fiber travels up that fiber. At the opposite end, a viewer sees a series of bright and dark spots, much like a TV screen— that is, a picture of what lies at the opposite end. Lenses are used at each end. The image may be viewed directly or on a monitor screen or film. The fibers must be optically insulated from one another, usually by a thin coating of material with index of refraction less than that of the fiber. The more fibers there are, and the smaller they are, the more detailed the picture. Such instruments, including bronchoscopes, colonoscopes (for viewing the colon), and endoscopes (stomach or other organs), are extremely useful for examining hard-to-reach places.

3 2 —8 Refr ction t

FIGURE 32 - 35 ( ) How fiber-optic im ge is m de, (b) Ex mple of fiber-optic device inserted through the mouth to view the g stro intestin l tr ct, with im ge on screen.

Spheric l Surf ce

We now examine the refraction of rays at the spherical surface of a transparent material. Such a surface could be one face of a lens or the cornea of the eye. To be general, let us consider an object which is located in a medium whose index of refraction is n1, and rays from each point on the object can enter a medium whose index of refraction is n2. The radius of curvature of the spherical boundary is R, and its center of curvature is at point , Fig. 32-36. We now show that all rays leaving a point O on the object will be focused at a single point I, the image point, if we consider only paraxial rays: rays that make a small angle with the axis.

FIGURE 32 - 36 R ys from point O on n object will be focused t single im ge point I by spheric l bound ry between two tr nsp rent m teri ls (n2 > ni), s long s the r ys m ke sm ll ngles with the xis.

To do so, we consider a single ray that leaves point O as shown in Fig. 32-37. From Snell's law, Eq. 32 -5, we have sin Qx = n2sin 02• We are assuming that angles d1,d2, a, /3, and 7 are small, so sin 0 ~ 0 (in radians), and Snell's law becomes, approximately, All ^1 = ^2 ^2 •

Also, /3 + 0 = 180° and 02 + 7 + (f> = 180°, so /3 =

7 + 02 .

Similarly for triangle OP , = a + j3.

FIGURE 32 - 37 Di gr m for showing th t ll p r xi l r ys from O focus t the s me point I (n2 > ni).

856

CHAPTER 32

Light: Reflectio a d Refractio

These three relations can be combined to yield

+ n2y = (n2 — «i)/3. Since we are considering only the case of small angles, we can write, approximately, _ "

Ta '

where d0 and d\ are the object and image distances and is the height as shown in Fig. 32 -37. We substitute these into the previous equation, divide through by , and obtain r + n i = n2 — rix (32 - 8) dr

For a given object distance dQ, this equation tells us the image distance, does not depend on the angle of a ray. Hence all paraxial rays meet at the same point I. This is true only for rays that make small angles with the axis and with each other, and is equivalent to assuming that the width of the refracting spherical surface is small compared to its radius of curvature. If this assumption is not true, the rays will not converge to a point; there will be spherical aberration, just as for a mirror (see Fig. 32 - 13), and the image will be blurry. (Spherical aberration will be discussed further in Section 33 -10.) We derived Eq. 32 - 8 using Fig. 32 - 37 for which the spherical surface is convex (as viewed by the incoming ray). It is also valid for a concave surface — as can be seen using Fig. 32 - 38 — if we use the following conventions: 1. If the surface is convex (so the center of curvature is on the side of the surface opposite to that from which the light comes), R is positive; if the surface is concave ( on the same side from which the light comes) R is negative. 2. The image distance, d{, follows the same convention: positive if on the opposite side from where the light comes, negative if on the same side. 3. The object distance is positive if on the same side from which the light comes (this is the normal case, although when several surfaces bend the light it may not be so), otherwise it is negative. For the case shown in Fig. 32 - 38 with a concave surface, both R and dxare negative when used in Eq. 32 - 8. Note, in this case, that the image is virtual.

FIG URE 3 2 - 3 8 R ys from O refr cted by conc ve surf ce form virtu l im ge (n2 > ni). Per our conventions, R < 0, < 0, dQ > 0.

EXAMPLE 32-12

Apparent depth II. A person looks vertically down into a 1.0-m-deep pool. How deep does the water appear to be?

APPR ACH Example 32 - 9 solved this problem using Snell's law. Here we use Eq. 32 - 8.

FIGURE 3 2 - 3 9

Ex mple 32 - 12.

S LUTI N A ray diagram is shown in Fig. 32 -39. Point O represents a point on the pool's bottom. The rays diverge and appear to come from point I, the image. We have d0 = 1.0 m and, for a flat surface, R = oo. Then Eq. 32 - 8 becomes 1.33 1.0 m

1.00 _ (1.00 - 1.33) di oo

Hence d\ = —(1.0m )/(1.33) = - 0.75 m. So the pool appears to be only threefourths as deep as it actually is, the same result we found in Example 32 -9. The minus sign tells us the image point I is on the same side of the surface as O, and the image is virtual. A t angles other than vertical, this conclusion must be modified. *SECTION 3 2 - 8

Refractio at a Spherical Surface

857

EXAMPLE 3 2 - 1 3 A spherical "lens." A point source of light is placed at a distance of 25.0 cm from the center of a glass sphere (n = 1.5) of radius 10.0 cm, Fig. 32 - 40. Find the image of the source.

APPR ACH As shown in Fig. 32 - 40, there are two refractions, and we treat them successively, one at a time. The light rays from the source first refract from the convex glass surface facing the source. We analyze this first refraction, treating it as in Fig. 32 - 36, ignoring the back side of the sphere. S LUTI N Using Eq. 32 - 8 (assuming paraxial rays) with n x = 1.0, n2 = 1.5, R = 10.0 cm, and dQ = 25.0 cm - 10.0 cm = 15.0 cm, we solve for the image distance as form ed at surface 1 ,d xX\ 1

d\\

1 f ( n 2 - n x)

Wl\

n 2\

d j

1.0 \ 15.0 c m /

J _ | ( 1.5 - 1.0 1.5 '^ 10.0 cm

90.0 cm

Thus, the image of the first refraction is located 90.0 cm to t e left of the front surface. This image (1 ^ now serves as the object for the refraction occurring at the back surface (surface 2) of the sphere. This surface is concave so R = - 1 0 .0 cm, and we consider a ray close to the axis. Then the object distance is d 02 = 90.0 cm + 2(10.0 cm) = 110.0 cm, and Eq. 32 - 8 yields, with n x = 1.5, n 2 = 1 .0,

J _ / L 0________________ 1.5 1.5 d[2

1.0 V - 10.0 cm

4.0 110.0 cm

110.0 cm

so dl2 = 28 cm. Thus, the final image is located a distance 28 cm from the back side of the sphere.

FIG U R E 3 2 - 4 0

Example 32-13.

Source

R = 10.0 cm

Summ ry Light appears to travel along straight-line paths, called r ys, at a speed v that depends on the index of refr ction, n, of the material; that is (32-4) where c is the speed of light in vacuum. When light reflects from a flat surface, t e angle o f reflection equals t e angle o f incidence. This l w of reflection explains why mirrors can form im ges. In a pl ne mirror, the image is virtual, upright, the same size as the object, and is as far behind the mirror as the object is in front. A spheric l mirror can be concave or convex. A conc ve spher ical mirror focuses parallel rays of light (light from a very distant object) to a point called the foc l point. The distance of this point from the mirror is the foc l length / of the mirror and

f =

(32-1)

where r is the radius of curvature of the mirror. Parallel rays falling on a convex mirror reflect from the mirror as if they diverged from a common point behind the mirror. The distance of this point from the mirror is the focal length and is considered negative for a convex mirror. For a given object, the position and size of the image formed by a mirror can be found by ray tracing. Algebraically,

858

HAPTER 32

Light: Reflection and Refraction

the relation between image and object distances, d[ and dQ, and the focal length / , is given by the mirror equ tion: JL i - I dQ + di f

(32-2)

The ratio of image height to object height, which equals the magnification m of a mirror, is i m = 7TQ =

di - ~ dQ T'

If the rays that converge to form an image actually pass through the image, so the image would appear on film or a screen placed there, the image is said to be a re l im ge. If the rays do not actually pass through the image, the image is a virtu l im ge. When light passes from one transparent medium into another, the rays bend or refract. The l w of refr ction (Snell's l w) states that ni sin0! = n2sin02> (32-5) where nx and are the index of refraction and angle with the normal to the surface for the incident ray, and n2 and 02 are for the refracted ray. When light of wavelength A enters a medium with index of refraction n, the wavelength is reduced to K = ~

The frequency does not change.

0 2 -6 )

The frequency or wavelength of light determines its color. The visible spectrum in air extends from about 400 nm (violet) to about 750 nm (red). Glass prisms break white light down into its constituent colors because the index of refraction varies with wavelength, a phenomenon known as dispersion.

When light rays reach the boundary of a material where the index of refraction decreases, the rays will be tot lly intern lly reflected if the incident angle, Qi, is such that Snell's law would predict sin0 2 > 1. This occurs if d\ exceeds the critical angle 6C given by sin 0

= —*

(32-7)

Questions 1. What would be the appearance of the Moon if it had (a) a rough surface; (b) a polished mirrorlike surface? 2. Archimedes is said to have burned the whole Roman fleet in the harbor of Syracuse by focusing the rays of the Sun with a huge spherical mirror. Is this1'reasonable? 3. What is the focal length of a plane mirror? What is the magnification of a plane mirror? 4. An object is placed along the principal axis of a spherical mirror. The magnification of the object is - 3.0. Is the image real or virtual, inverted or upright? Is the mirror concave or convex? On which side of the mirror is the image located? 5. Using the rules for the three rays discussed with reference to Fig. 32-15, draw ray 2 for Fig. 32-19b. 6. Does the mirror equation, Eq. 32 -2, hold for a plane mirror? Explain. 7. If a concave mirror produces a real image, is the image necessarily inverted? . How might you determine the speed of light in a solid, rectangular, transparent object? 9. When you look at the Moon's reflection from a ripply sea, it appears elongated (Fig. 32-41). Explain.

14. When a wide beam of parallel light enters water at an angle, the beam broadens. Explain. 15. You look into an aquarium and view a fish inside. One ray of light from the fish as it emerges from the tank is shown in Fig. 32-42. The apparent position of the fish is also shown. In the drawing, indicate the approximate position of the actual fish. Briefly justify your answer.

FIGURE 32-42 Question 15.

16. How can you "see" a round drop of water on a table even though the water is transparent and colorless? 17. A ray of light is refracted through three different materials (Fig. 32-43). Rank the materials according to their index of refraction, least to greatest. FIGURE 32-43 Question 17. 18.

an a light ray traveling in air be totally reflected when it strikes a smooth water surface if the incident angle is chosen correctly? Explain. 19. When you look up at an object in air from beneath the surface in a swimming pool, does the object appear to be the same size as when you see it directly in air? Explain. 20. What type of mirror is shown in Fig. 32-44?

FIGURE 32-41 Question 9. 10. How can a spherical mirror have a negative object distance? 11. What is the angle of refraction when a light ray is incident perpendicular to the boundary between two transparent materials? 12. When you look down into a swimming pool or a lake, are you likely to overestimate or underestimate its depth? Explain. How does the apparent depth vary with the viewing angle? (Use ray diagrams.) 13. Draw a ray diagram to show why a stick looks bent when part of it is under water (Fig. 32-23). Students t MIT did

fe sibility study. See www.mit.edu.

FIGURE 32Question 20. 21. Light rays from stars (including our Sun) always bend toward the vertical direction as they pass through the Earth's atmosphere, (a) Why does this make sense? (b) What can you conclude about the apparent positions of stars as viewed from Earth?

Questions

859

Problems 32 - 2 Reflection; Plane Mirrors

32 - 3 Spherical Mirrors

1. (I) When you look at yourself in a 60-cm-tall plane mirror, you see the same amount of your body whether you are close to the mirror or far away. (Try it and see.) Use ray diagrams to show why this should be true.

9. (I) A solar cooker, really a concave mirror pointed at the Sun, focuses the Sun's rays 18.8 cm in front of the mirror. What is the radius of the spherical surface from which the mirror was made? 10. (I) How far from a concave mirror (radius 24.0 cm) must an object be placed if its image is to be at infinity? 11. (I) When walking toward a concave mirror you notice that the image flips at a distance of 0.50 m. What is the radius of curvature of the mirror? 12. (II) A small candle is 35 cm from a concave mirror having a radius of curvature of 24 cm. (a) What is the focal length of the mirror? (b) Where will the image of the candle be located? (c) Will the image be upright or inverted? 13. (II) You look at yourself in a shiny 9.2 -cm -diameter hristmas tree ball. If your face is 25.0 cm away from the ball's front surface, where is your image? Is it real or virtual? Is it upright or inverted? 14. (II) A mirror at an amusement park shows an upright image of any person who stands 1.7 m in front of it. If the image is three times the person's height, what is the radius of curvature of the mirror? (See Fig. 32 - 44.) 15. (II) A dentist wants a small mirror that, when 2.00 cm from a tooth, will produce a 4.0 X upright image. What kind of mirror must be used and what must its radius of curvature be? 16. (II) Some rearview mirrors produce images of cars to your rear that are smaller than they would be if the mirror were flat. Are the mirrors concave or convex? What is a mirror's radius of curvature if cars 18.0 m away appear 0.33 their normal size? 17. (II) You are standing 3.0 m from a convex security mirror in a store. You estimate the height of your image to be half of your actual height. Estimate the radius of curvature of the mirror. 18. (II) An object 3.0 mm high is placed 18 cm from a convex mirror of radius of curvature 18 cm. (a) Show by ray tracing that the image is virtual, and estimate the image distance. (b) Show that the (negative) image distance can be computed from Eq. 32 - 2 using a focal length of - 9 .0 cm. (c) ompute the image size, using Eq. 32 - 3. 19. (II) The image of a distant tree is virtual and very small when viewed in a curved mirror. The image appears to be 16.0cm behind the mirror. What kind of mirror is it, and what is its radius of curvature? 20. (II) Use two techniques, (a) a ray diagram, and (b) the mirror equation, to show that the magnitude of the magnifi cation of a concave mirror is less than 1 if the object is beyond the center of curvature (d0 > r), and is greater than 1 if the object is within (d0 < r). 21. (II) Show, using a ray diagram, that the magnification m of a convex mirror is m = ~ d[/d0 , just as for a concave mirror. [Hint: onsider a ray from the top of the object that reflects at the center of the mirror.] 22. (II) Use ray diagrams to show that the mirror equation, Eq. 32 - 2, is valid for a convex mirror as long as / is considered negative. 23. (II) The magnification of a convex mirror is +0.55 X for objects 3.2 m from the mirror. What is the focal length of this mirror?

2. (I) Suppose that you want to take a photograph of yourself as you look at your image in a mirror 2.8 m away. For what distance should the camera lens be focused? 3. (II) Two plane mirrors meet at a 135° angle, Fig. 32 -45. If light rays strike one mirror at 38° as shown, at what angle (f> do they leave the second mirror?

FIGURE 32-45 Problem 3. 4. (II) A person whose eyes are 1.64 m above the floor stands 2.30 m in front of a vertical plane mirror whose bottom edge is 38 cm above the floor, Fig. 32 -46. What is the horizontal distance x to the base of the wall supporting the mirror of the nearest point on the floor - 2JO m that can be seen reflected in fr the mirror? y £" 1.64 m J

FIGURE 32-46 Problem 4.

MK cm

'■*-£—

5. (II) Show that if two plane mirrors meet at an angle 4>, a single ray reflected successively from both mirrors is deflected through an angle of 2 independent of the inci dent angle. Assume < 90° and that only two reflections, one from each mirror, take place. 6. (II) Suppose you are cm from a plane mirror. What area of the mirror is used to reflect the rays entering one eye from a point on the tip of your nose if your pupil diameter is 4.5 mm? 7. (II) Stand up two plane mirrors so they form a 90.0° angle as in Fig. 32 -47. When you look into this double mirror, you see yourself as others see you, instead of reversed as in a single mirror. Make a ray diagram to show how this occurs.

FIGURE 32-47 Problems 7 and . . (Ill) Suppose a third mirror is placed beneath the two shown in Fig. 32-47, so that all three are perpendicular to each other. (a) Show that for such a "corner reflector," any incident ray will return in its original direction after three reflections. (b) What happens if it makes only two reflections?

860

HAPTER 32

Light: Reflection and Refraction

24. (II) (a) Where should an object be placed in front of a concave mirror so that it produces an image at the same location as the object? (b) Is the image real or virtual? (c) Is the image inverted or upright? (d) What is the magnifica tion of the image? 25. (II) A 4.5-cm tall object is placed 26 cm in front of a spherical mirror. It is desired to produce a virtual image that is upright and 3.5 cm tall, (a) What type of mirror should be used? (b) Where is the image located? (c) What is the focal length of the mirror? (d) What is the radius of curvature of the mirror? 26. (II) A shaving or makeup mirror is designed to magnify your face by a factor of 1.35 when your face is placed 20.0cm in front of it. (a) What type of mirror is it? (b) Describe the type of image that it makes of your face. (c) alculate the required radius of curvature for the mirror. 27. (II) A concave mirror has focal length / . When an object is placed a distance dQ > f from this mirror, a real image with magnification m is formed, (a) Show that m = / / ( / - dQ). (ib) Sketch m vs. dQ over the range f < d0 < +oo where / = 0.45 m. (c) For what value of dQ will the real image have the same (lateral) size as the object? (d) To obtain a real image that is much larger than the object, in what general region should the object be placed relative to the mirror? 28. (II) Let the focal length of a convex mirror be written as / = - | / | . Show that the magnification m of an object a distance dQfrom this mirror is given by m = \f\/(d0 + |/|). Based on this relation, explain why your nose looks bigger than the rest of your face when looking into a convex mirror. 29. (II) A spherical mirror of focal length / produces an image of an object with magnification m. (a) Show that the object is a distance d0 = f ( 1 — —) from the reflecting side of V m) the mirror. (b) Use the relation in part (a) to show that, no matter where an object is placed in front of a convex mirror, its image will have a magnification in the range 0 < m < + 1 . 30. (Ill) An object is placed a distance r in front of a wall, where r exactly equals the radius of curvature of a certain concave mirror. A t what distance from the wall should this mirror be placed so that a real image of the object is formed on the wall? What is the magnification of the image? 31. (Ill) A short thin object (like a short length of wire) of length i is placed along the axis of a spherical mirror (perpendicular to the glass surface). Show that its image has length I' = m 2l so the longitudinal magnification is equal to —m 2 where m is the normal "lateral" magnification, Eq. 32 - 3. Why the minus sign? [Hint: Find the image posi tions for both ends of the wire, and assume £ is very small.]

32 - 4 Index of Refraction 32. (I) The speed of light in ice is 2.29 X 10 m /s. What is the index of refraction of ice? 33. (I) What is the speed of light in (a) ethyl alcohol, (b) lucite, (c) crown glass? 34. (I) Our nearest star (other than the Sun) is 4.2 light years away. That is, it takes 4.2 years for the light to reach Earth. How far away is it in meters? 35. (I) How long does it take light to reach us from the Sun, 1.50 X 10 km away?

36. (II) The speed of light in a certain substance is % of its value in water. What is the index of refraction of that substance? 37. (II) Light is emitted from an ordinary lightbulb filament in wave-train bursts of about 10 s in duration. What is the length in space of such wave trains?

32 - 5 Snell's Law 38. (I) A diver shines a flashlight upward from beneath the water at a 38.5° angle to the vertical. A t what angle does the light leave the water? 39. (I) A flashlight beam strikes the surface of a pane of glass (n = 1.56) at a 63° angle to the normal. What is the angle of refraction? 40. (I) Rays of the Sun are seen to make a 33.0° angle to the vertical beneath the water. A t what angle above the horizon is the Sun? 41. (I) A light beam coming from an underwater spotlight exits the water at an angle of 56.0°. A t what angle of incidence did it hit the air - w ater interface from below the surface? 42. (II) A beam of light in air strikes a slab of glass (n = 1.56) and is partially reflected and partially refracted. Determine the angle of incidence if the angle of reflection is twice the angle of refraction. 43. (II) A light beam strikes a 2.0-cm-thick piece of plastic with a refractive index of 1.62 at a 45° angle. The plastic is on top of a 3.0-cmthick piece of glass for which n = 1.47. What is the distance D in Fig. 32 -48?

FIGURE 32-48 Problem 43. 44. (II) An aquarium filled with water has flat glass sides whose index of refraction is 1.56. A beam of light from outside the aquarium strikes the glass at a 43.5° angle to the perpen dicular (Fig. 32 -49). What is the angle of this light ray when it enters (a) the glass, and then (b) the water? (c) What would Glass be the refracted angle if the ray Air W ter entered the water directly?

FIGURE 32-49

4 3 .5 ^ ^

Problem 44. 45. (II) In searching the bottom of a pool at night, a watchman shines a narrow beam of light from his flashlight, 1.3 m above the water level, onto the surface of the water at a point 2.5 m from his foot at the edge of the pool (Fig. 32-50). Where does the spot of light hit the bottom of the pool, measured from the bottom of the wall beneath his foot, if the pool is 2.1 m deep?

FIGURE 32-50 Problem 45.

Problems

861

46. (II) The block of glass (n = 1.5) shown in cross section in Fig. 32-51 is surrounded by air. A ray of light enters the block at its left-hand face with incident angle 6\ and reemerges into the air from the right-hand face directed parallel to the block's base. Determine 6 i.

FIGURE 32-51 Problem 46. 47. (II) A laser beam of diameter d\ = 3.0 mm in air has an incident angle 6\ = 25° at a flat air-glass surface. If the index of refraction of the glass is n = 1.5, determine the diameter d2 of the beam after it enters the glass. 48. (II) Light is incident on an equilateral glass prism at a 45.0° angle to one face, Fig. 32-52. alculate the angle at which light emerges from the opposite face. Assume that n = 1.54.

FIGURE 32-52 Problems 48 and 65. 49. (II) A triangular prism made of crown glass (n = 1.52) with base angles of 30.0° is surrounded by air. If parallel rays are incident normally on its base as shown in Fig. 32-53, what is the angle between the two emerging rays?

54. (II) A parallel beam of light containing two wavelengths, Xl = 465 nm and A2 = 652 nm, enters the silicate flint glass of an equilateral prism as shown in Fig. 32-54. At what angle does each beam leave the prism (give angle with normal to the face)? See Fig. 32-28. FIGURE 32-54 Problem 54. 55. (I ll) A ray of light with wavelength Ais incident from air at precisely 60° ( = 0) on a spherical water drop of radius r and index of refraction n (which depends on A). When the ray reemerges into the air from the far side of the drop, it has been deflected an angle cf>from its original direction as shown in Fig. 32-55. By how ^ much does the value \ _ ^no\ of for violet light (n = 1.341) differ from the value for red light (n = 1.330)?

FIGURE 32-55 Problem 55. 56. (Ill) For visible light, the index of refraction n of glass is roughly 1.5, although this value varies by about 1% across the visible range. onsider a ray of white light incident from air at angle Qi onto a flat piece of glass, (a) Show that, upon entering the glass, the visible colors contained in this inci dent ray will be dispersed over a range A02 of refracted angles given approximately by A 02 ~

sin di

\ / n 2 - sin26i n FIGURE 32-53 Problem 49. 50. (II) Show in general that for a light beam incident on a uniform layer of transparent material, as in Fig. 32-24, the direction of the emerging beam is parallel to the incident beam, independent of the incident angle 0. Assume the air on the two sides of the transparent material is the same. 51. (Ill) A light ray is incident on a flat piece of glass with index of refraction n as in Fig. 32-24. Show that if the incident angle 0 is small, the emerging ray is displaced a distance d = td(n — 1 )/n, where t is the thickness of the glass, 0 is in radians, and d is the perpendicular distance between the incident ray and the (dashed) line of the emerging ray (Fig. 32-24).

32 - 6 Visible Spectrum; Dispersion 52. (I) By what percent is the speed of blue light (450 nm) less than the speed of red light (680 nm), in silicate flint glass (see Fig. 32-28)? 53. (I) A light beam strikes a piece of glass at a 60.00° incident angle. The beam contains two wavelengths, 450.0 nm and 700.0 nm, for which the index of refraction of the glass is 1.4831 and 1.4754, respectively. What is the angle between the two refracted beams?

862

HAPTER 32

Light: Reflection and Refraction

[Hint: For x in radians, (d/dx)(siri -1*) = i/V i - *2.] (b) If Qi = 0°, what is A02 in degrees? (c) If di = 90°, what is A02 in degrees?

32 - 7 Total Internal Reflection 57. (I) What is the critical angle for the interface between water and diamond? To be internally reflected, the light must start in which material? 58. (I) The critical angle for a certain liquid - air surface is 49.6°. What is the index of refraction of the liquid? 59. (II) A beam of light is emitted in a pool of water from a depth of 72.0 cm. Where must it strike the air - water inter face, relative to the spot directly above it, in order that the light does n o t exit the water? 60. (II) A ray of light, after entering a light fiber, reflects at an angle of 14.5° with the long axis of the fiber, as in Fig. 32-56. alculate the distance along the axis of the fiber that the light ray travels between successive reflections off the sides of the fiber. Assume that the fiber has an index of refraction of 1.55 and is 1.40 X 10_4m in diameter.

FIGURE 32 -56

Problem 60.

61. (II) A beam of light is emitted 8.0 cm beneath the surface of a liquid and strikes the surface 7.6 cm from the point directly above the source. If total internal reflection occurs, what can you say about the index of refraction of the liquid? 62. (II) Figure 32 - 57 shows a liquid -detecting prism device that might be used inside a washing machine or other liquid - containing appliance. If no liquid covers the prism's hypotenuse, total internal reflection of the beam from the light source produces a large signal in the light sensor. If liquid covers the hypotenuse, some light escapes from the prism into the liquid and the light sensor's signal decreases. Thus a large signal from the light sensor indi cates the absence of liquid in the reservoir. If this device is designed to detect the presence of water, determine the allowable range for the prism's index of refraction n. Will the device work properly if the prism is constructed from (inexpensive) lucite? For lucite, n = 1.5.

64. (II) (a) What is the minimum index of refraction for a glass or plastic prism to be used in binoculars (Fig. 32 - 33) so that total internal reflection occurs at 45°? (b) Will binoculars work if their prisms (assume n = 1.58) are immersed in water? (c) What minimum n is needed if the prisms are immersed in water? 65. (Ill) Suppose a ray strikes the left face of the prism in Fig. 32 - 52 at 45.0° as shown, but is totally internally reflected at the opposite side. If the apex angle (at the top) is 6 = 60.0°, what can you say about the index of refraction of the prism? 66. (Ill) A beam of light enters the end of an optic fiber as

shown in Fig. 32 -59. (a) Show that we can guarantee total internal reflection at the side surface of the material (at point A), if the index of refraction is greater than about 1.42. In other words, regardless of the angle a, the light beam reflects back into the material at point A, assuming air outside.

FIGURE 32-57 Problem 62.

FIGURE 32-59 Problem 66.

material

*3 2 - 8 Refraction at Spherical Surface *67. (II) A 13.0-cm-thick plane piece of glass (n = 1.58) lies on the surface of a 12.0-cm-deep pool of water. How far below the top of the glass does the bottom of the pool seem, as viewed from directly above?

63. (II) Two rays A and B travel down a cylindrical optical fiber of diameter d = 75.0 /z,m, length £ = 1.0 km, and index of refraction ni = 1.465. Ray A travels a straight path down the fiber's axis, whereas ray B propagates down the fiber by repeated reflections at the critical angle each time it impinges on the fiber's boundary. Determine the extra time Af it takes for ray B to travel down the entire fiber in comparison with ray A (Fig. 32 - 58), assuming (a) the fiber is surrounded by air, (b) the fiber is surrounded by a cylin drical glass "cladding" with index of refraction n2 = 1.460.

* 6 . (II) A fish is swimming in water inside a thin spherical glass bowl of uniform thickness. Assuming the radius of curvature of the bowl is 28.0 cm, locate the image of the fish if the fish is located: (a) at the center of the bowl; (b) 20.0 cm from the side of the bowl between the observer and the center of the bowl. Assume the fish is small.

"69. (Ill) In Section 32-8, we derived Eq. 32 -8 for a convex spherical surface with n2 > nt. Using the same conventions and using diagrams similar to Fig. 32-37, show that Eq. 32-8 is valid also for (a) a convex spherical surface with n2 < nu (b) a concave spherical surface with n2 > nt, and (c) a concave spherical surface with n2 < n x.

75.0 /xm j - g = 1.465 FIGURE 32 - 58

Problem 63.

*70. (Ill) A coin lies at the bottom of a 0.75-m-deep pool. If a viewer sees it at a 45° angle, where is the image of the coin, relative to the coin? [Hint. The image is found by tracing back to the intersection of two rays.]

Problems

863

| Gener l Problems 71. Two identical concave mirrors are set facing each other 1.0 m apart. A small lightbulb is placed halfway between the mirrors. A small piece of paper placed just to the left of the bulb prevents light from the bulb from directly shining on the left mirror, but light reflected from the right mirror still reaches the left mirror. A good image of the bulb appears on the left side of the piece of paper. What is the focal length of the mirrors? 72. A slab of thickness D, whose two faces are parallel, has index of refraction n. A ray of light incident from air onto one face of the slab at incident angle Qi splits into two rays A and B. Ray A reflects directly back into the air, while B travels a total distance ft within the slab before reemerging from the slab's face a distance d from its point of entry (Fig. 32 -60). (a) Derive expressions for I and d in terms of D, n, and 6 i. (b) For normal incidence (i.e., Qi = 0°) show that your expressions yield the expect values for £ and

76. The critical angle of a certain piece of plastic in air is 0 = 39.3°. What is the critical angle of the same plastic if it is immersed in water? 77. Each student in a physics lab is assigned to find the location where a bright object may be placed in order that a concave mirror, with radius of curvature r = 46 cm, will produce an image three times the size of the object. Two students complete the assignment at different times using identical equipment, but when they compare notes later, they discover that their answers for the object distance are not the same. Explain why they do not necessarily need to repeat the lab, and justify your response with a calculation. 78. A kaleidoscope makes symmetric patterns with two plane mirrors having a 60° angle between them as shown in Fig. 32 -63. Draw the location of the images (some of them images of images) of the object placed between the mirrors.

FIGURE 32-63 Problem 78.

FIGURE 32-60 Problem 72.

73. Two plane mirrors are facing each other 2.2 m apart as in Fig. 32 -61. You stand 1.5 m away from one of these mirrors and look into it. You will see multiple images of yourself. (a) How far away from you are the first three images of yourself in the mirror in front of you? (b) Are these first three images facing toward you or away from you?

79. When light passes through a prism, the angle that the refracted ray makes relative to the incident ray is called the deviation angle 5, Fig. 32 -64. Show that this angle is a minimum when the ray passes through the prism symmetri cally, perpendicular to the bisector of the apex angle (f>, and show that the minimum deviation angle, + Sm) sin 0/2

[Hint: For 0 in radians, (d/d0)(sin ~ 16) =

1/V l - 02.]

FIGURE 32-61 Problem 73.

FIGURE 32-64 Problems 79 and 80. 74. We wish to determine the depth of a swimming pool filled with water by measuring the width (x = 5.50 m) and then noting that the bottom edge of the pool is just visible at an angle of 13.0° above the horizontal as shown in Fig. 32 -62. alculate the depth of the pool.

75. A 1.80-m-tall person stands 3.80 m from a convex mirror and notices that he looks precisely half as tall as he does in a plane mirror placed at the same distance. What is the radius of curvature of the convex mirror? (Assume that sin 0 ~ 0.) [Hint: The viewing angle is half.]

864

HAPTER 32

Light: Reflection and Refraction

80. If the apex angle of a prism is = 72° (see Fig. 32 -64), what is the minimum incident angle for a ray if it is to emerge from the opposite side (i.e., not be totally internally reflected), given n = 1.58? 81. Ferm t's principle states that "light travels between two points along the path that requires the least time, as compared to other nearby paths." From Fermat's principle derive (a) the law of reflection (6[ = 0r) and (b) the law of refraction (Snell's law). [Hint: hoose two appropriate points so that a ray between them can undergo reflection or refraction. Draw a rough path for a ray between these points, and write down an expression of the time required for light to travel the arbitrary path chosen. Then take the derivative to find the minimum.] *82. Suppose Fig. 32 - 36 shows a cylindrical rod whose end has a radius of curvature R = 2.0 cm, and the rod is immersed in water with index of refraction of 1.33. The rod has index of refraction 1.53. Find the location and height of the image of an object 2.0 mm high located 23 cm away from the rod.

83. An optical fiber is a long transparent cylinder of diameter d and index of refraction n. If this fiber is bent sharply, some light hitting the side of the cylinder may escape rather than reflect back into the fiber (Fig. 32-65). What is the smallest radius r at a short bent section for which total internal reflection will be assured for light initially travelling parallel to the axis of the fiber?

FIGURE 32-65 Problem 83.

h -d -H

84. An object is placed 15 cm from a certain mirror. The image is half the height of the object, inverted, and real. How far is the image from the mirror, and what is the radius of curvature of the mirror? 85. The end faces of a cylindrical glass rod (n = 1.51) are perpendicular to the sides. Show that a light ray entering an end face at any angle will be totally internally reflected inside the rod when it strikes the sides. Assume the rod is in air. What if it were in water? 6. The paint used on highway signs often contains small trans

parent spheres which provide nighttime illumination of the sign's lettering by retro -reflecting vehicle headlight beams. onsider a light ray from air incident on one such sphere of radius r and index of refraction n. Let 6 be its incident angle, and let the ray follow the path shown in Fig. 32-66, so that the ray exits the sphere in the direction exactly antiparallel to its incoming direction. onsid\ ering only rays for which sin 0 can be approximated as 0, determine the required value for n.

HNumerical/ omputer H87. (II) The index of refraction, n, of crown flint glass at different wavelengths (A) of light are given in the Table below. A(nm) n

546.1 1.51978

1060 1.50586

365.0 1.54251

312.5 1.5600

Make a graph of n versus A. The variation in index of refraction with wavelength is given by the auchy equation n = A + B / A2. Make another graph of n versus 1 /A 2 and determine the constants A and B for the glass by fitting the data with a straight line. H . (Ill) onsider a ray of sunlight incident from air on a spher ical raindrop of radius r and index of refraction n. Defining 0 to be its incident angle, the ray then follows the path shown in Fig. 32-67, exiting the drop at a "scattering angle" compared with its original incoming direction, (a) Show that (f) = 180° + 20 - 4 siiT1(sin 0/n). (b) The parallel rays of sunlight illuminate a raindrop with rays of all possible incident angles from 0° to 90°. Plot 4> vs. 0 in the range 0° < 0 < 90°, in 0.5° steps, assuming n = 1.33 as is appro priate for water at visible-light wavelengths. (c) From your plot, you should find that a fairly large fraction of the incident angles have nearly the same scattering angle. Approximately what fraction of the possible incident angles is within roughly 1° of (f> = 139°? [This subset of incident rays is what creates the rainbow. Wavelength dependent variations in n cause the rainbow to form at slightly different for the various visible colors.]

FIGURE 32-67 Problem

FIGURE 32-66 Problem 6.

Answers to Exercises A: No. B:

(b).

: (a).

Toward.

E: The face becomes shiny: total internal reflection. F: 1.414.

General Problems

865

■

■' ma t I A ^ l i u ^ n lit f i t

f y irtp

Of the m ny optic l devices we discuss this Ch pter, the m gnifying gl ss is the simplest. Here it is m gnifying p rt of p ge 886 of this Ch pter, which describes how the m gnifying gl ss works ccording to the r y model. In this Ch pter we ex mine thin lenses in det il, seeing how to determine im ge position s function of object position nd the foc l length of the lens, b sed on the r y model of light. We then ex mine optic l devices including film nd digit l c mer s, the hum n eye, eyegl sses, telescopes, nd microscopes.

t> T £ k

n „ r *

? 3 - ? l u . ihL

Ml Itk.

HJin* k-n» pn*Jui\ri * tm iu l

SL!w

lo Incunw II If ihr eye n ivbrteil.

mparison t>I [art ( j ) mitu* r^*vnip*ni. 'Viewed at the near point wifftv wi^n >ou *4 ibe rtf M ih lc n d tr r i u u f t g Lbc unaided eye, with the object a\ the nea L h e e y e 0 and dx < 0), and for an inverted image the magnification is negative ( { < 0 and d{ > 0). From sign convention 1, it follows that the power (Eq. 33 -1) of a converging lens, in diopters, is positive, whereas the power of a diverging lens is negative. A converging lens is sometimes referred to as a positive lens, and a diverging lens as a neg tive lens. S O L V ;

' V.

? 8

Thin Lenses 1. Draw a r y di gr m, as precise as possible, but even a rough one can serve as confirmation of analytic results. hoose one point on the object and draw at least two, or preferably three, of the easy-to-draw rays described in Figs. 33 -6 and 33 -8. The image point is where the rays intersect.

2. For analytic solutions, solve for unknowns in the thin lens equ tion (Eq. 33 -2) and the m gnific tion equ tion (Eq. 33 -3). The thin lens equation involves reciprocals— don't forget to take the reciprocal. 3. Follow the sign conventions listed just above. 4. heck that your analytic answers are consistent with your ray diagram.

SECTION 33 -2

The Thi Le s Equatio ; Mag ificatio

871

EXAMPLE 33-2 Image formed by converging lens. What is (a) the position, and (b) the size, of the image of a 7.6-cm-high leaf placed 1.00 m from a +50.0-mm-focal-length camera lens? APPR ACH We follow the steps of the Problem Solving Strategy explicitly. S LUTI N 1. R y di gr m. Figure 33-11 is an approximate ray diagram, showing only rays 1 and 3 for a single point on the leaf. We see that the image ought to be a little behind the focal point F, to the right of the lens. 2. Thin lens nd m gnific tion equ tions, (a) We find the image position analytically using the thin lens equation, Eq. 33 - 2. The camera lens is converging, with / = +5.00 cm, and dQ = 100 cm, and so the thin lens equation gives 1

5.00 cm

100 cm

20.0 - 1.0 100 cm

19.0 100 cm

Then, taking the reciprocal, 100 cm

1 9 0

= 5 2 6 cm '

or 52.6 mm behind the lens. (b) The magnification is di 5.26 cm m = - — = ~ —- ----- = -0.0526, 100 cm dQ so i = m Q = ( - 0.0526) (7.6 cm) = - 0.40 cm. The image is 4.0 mm high. 3. Sign conventions. The image distance di came out positive, so the image is behind the lens. The image height is { = - 0.40 cm; the minus sign means the image is inverted. 4. Consistency. The analytic results of steps (2) and (3) are consistent with the ray diagram, Fig. 33-11: the image is behind the lens and inverted. N TE Part (a) tells us that the image is 2.6 mm farther from the lens than the image for an object at infinity, which would equal the focal length, 50.0 mm. Indeed, when focusing a camera lens, the closer the object is to the camera, the farther the lens must be from the sensor or film. EXERCISE A If the le f (object) of Ex mple 33 - 2 is moved f rther from the lens, does the im ge move closer to or f rther from the lens? (Don 't c lcul te!)

872

CHAPTER 33

Le ses a d Optical I strume ts

EXAMPLE 33 - 3 bject close to converging lens. An object is placed 10 cm from a 15-cm-focal-length converging lens. Determine the image position and size (a) analytically, and (b) using a ray diagram.

APPR ACH We first use Eqs. 33-2 and 33-3 to obtain an analytic solution, and then confirm with a ray diagram using the special rays 1,2, and 3 for a single object point. S LUTI N (a) Given / = 15 cm and dQ = 10 cm, then ^

15 cm

10 cm

30 cm

and di = - 3 0 cm. (Remember to take the reciprocal!) Because di is negative, the image must be virtual and on the same side of the lens as the object. The magnification m

A CAUTI ON D on 't forget to take t e reciprocal

- 3 0 cm = 3.0. 10 cm

The image is three times as large as the object and is upright. This lens is being used as a simple magnifying glass, which we discuss in more detail in Section 33-7. (b) The ray diagram is shown in Fig. 33-12 and confirms the result in part (a). We choose point O' on the top of the object and draw ray 1, which is easy. But ray 2 may take some thought: if we draw it heading toward F', it is going the wrong way—so we have to draw it as if coming from F' (and so dashed), striking the lens, and then going out parallel to the lens axis. We project it back parallel, with a dashed line, as we must do also for ray 1, in order to find where they cross. Ray 3 is drawn through the lens center, and it crosses the other two rays at the image point, I'. N TE From Fig. 33 -12 we can see that, whenever an object is placed between a converging lens and its focal point, the image is virtual.

FIGURE 33 - 12 An object pl ced within the foc l point of converging lens produces virtu l im ge. Ex mple 33 - 3.

EXAMPLE 33 -4 Diverging lens. Where must a small insect be placed if a 25-cm-focal-length diverging lens is to form a virtual image 20 cm from the lens, on the same side as the object?

FIGURE 33 - 13

Exercise C.

APPR ACH The ray diagram is basically that of Fig. 33 -10 because our lens here is diverging and our image is in front of the lens within the focal distance. (It would be a valuable exercise to draw the ray diagram to scale, precisely, now.) The insect's distance, dQ, can be calculated using the thin lens equation. S LUTI N The lens is diverging, so / is negative: / = - 2 5 cm. The image distance must be negative too because the image is in front of the lens (sign conventions), so dx = - 2 0 cm. Equation 33 -2 gives JL _ i _ 1 dQ f d{

25 cm

20 cm

~4 + 5 100 cm

1 100 cm

So the object must be 100 cm in front of the lens. EXERCISE B Return to the Ch pter-Opening Question, p ge 866, nd nswer it g in now. Try to expl in why you m y h ve nswered differently the first time. EXERCISE C Figure 33 - 13 shows converging lens held bove three equ l-sized letters A. In ( ) the lens is 5 cm from the p per, nd in (b) the lens is 15 cm from the p per. Estim te the foc l length of the lens. Wh t is the im ge position for e ch c se?

SECTION 3 3 - 2

The Thi Le s Equatio ; Mag ificatio

873

3 3 —3 Combin tions o f Lenses Optical instruments typically use lenses in combination. When light passes through more than one lens, we find the image formed by the first lens as if it were alone. This image becomes the object for the second lens, and we find the image then formed by this second lens, which is the final image if there are only two lenses. The total magnification will be the product of the separate magnifications of each lens, as we shall see. Even if the second lens intercepts the light from the first lens before it forms an image, this technique still works. A two-lens system. Two converging lenses, A and B, with focal lengths f A = 20.0 cm and f B = 25.0 cm, are placed 80.0 cm apart, as shown in Fig. 33-14a. An object is placed 60.0 cm in front of the first lens as shown in Fig. 33-14b. Determine (a) the position, and (b) the magnification, of the final image formed by the combination of the two lenses. APPR ACH Starting at the tip of our object O, we draw rays 1, 2, and 3 for the first lens, A, and also a ray 4 which, after passing through lens A, acts as "ray 3" (through the center) for the second lens, B. Ray 2 for lens A exits parallel, and so is ray 1 for lens B. To determine the position of the image IA formed by lens A, we use Eq. 33 - 2 with f A = 20.0 cm and doA = 60.0 cm. The distance of IA (lens A's image) from lens B is the object distance doB for lens B. The final image is found using the thin lens equation, this time with all distances relative to lens B. For (b) the magnifications are found from Eq. 33 - 3 for each lens in turn. S LUTI N (a) The object is a distance doA = +60.0 cm from the first lens, A, and this lens forms an image whose position can be calculated using the thin lens equation: 1

d\a A

CAUTI ON Object distance for second lens is not equal to t e image distance for first lens

1 /

doA

2 0 .0

3 - 1

60.0 cm

30.0 cm

So the first image IA is at diA = 30.0 cm behind the first lens. This image becomes the object for the second lens, B. It is a distance doB = 80.0 cm - 30.0 cm = 50.0 cm in front of lens B, as shown in Fig. 33-14b. The image formed by lens B, again using the thin lens equation, is at a distance diB from the lens B: 1

d{B

1 /

doB

25.0 cm

50.0 cm

2 - 1

50.0 cm

Hence dlB = 50.0 cm behind lens B. This is the final image— see Fig. 33-14b.

Lens A

FIGURE 3 3 - 1 4 Two lenses, A nd B, used in combin tion, Ex mple 33 - 5. The sm ll numbers refer to the e sily dr wn r ys.

( )

874

CHAPTER 33

80.0 cm A

(b)

Le ses a d Optical I strume ts

Lens B

(ib) Lens A has a magnification (Eq. 33 - 3) m A

dlA doA

30.0 cm = -0.500. 60.0 cm

Thus, the first image is inverted and is half as high as the object (again Eq. 33 - 3): \A

rnA oA

0.500/ioA.

Lens B takes this image as object and changes its height by a factor r B = loB

50.0 cm = - 1 .0 0 0 . 50.0 cm

The second lens reinverts the image (the minus sign) but doesn't change its size. The final image height is (remember oB is the same as iA) iB =

m B 0B =

m Q iA =

m Qm A oA =

(m tot l)/ioA.

The total magnification is the product of m A and raB, which here equals wtotai = mAmB = ( —1.000)( —0.500) = +0.500, or half the original height, and the final image is upright. EXAMPLE 33 -6 Measuring / for a diverging lens. To measure the focal length of a diverging lens, a converging lens is placed in contact with it, as shown in Fig. 33 - 15. The Sun's rays are focused by this combination at a point 28.5 cm, behind the lenses as shown. If the converging lens has a focal length f c of 16.0 cm, what is the focal length / Dof the diverging lens? Assume both lenses are thin and the space between them is negligible. APPR ACH The image distance for the first lens equals its focal length (16.0 cm) since the object distance is infinity (o o ). The position of this image, even though it is never actually formed, acts as the object for the second (diverging) lens. We apply the thin lens equation to the diverging lens to find its focal length, given that the final image is at dx = 28.5 cm. S LUTI N If the diverging lens was absent, the converging lens would form the image at its focal point— that is, at a distance f c = 16.0 cm behind it (dashed lines in Fig. 33 - 15). When the diverging lens is placed next to the converging lens, we treat the image formed by the first lens as the object for the second lens. Since this object lies to the right of the diverging lens, this is a situation where dQis negative (see the sign conventions, page 871). Thus, for the diverging lens, the object is virtual and d0 = - 16.0 cm. The diverging lens forms the image of this virtual object at a distance d[ = 28.5 cm away (given). Thus, 1

= - 0.0274 cm" 16.0 cm 28.5 cm fo dQ d[ We take the reciprocal to find / D = - 1/(0.0274 cm-1) = - 36.5 cm. N TE If this technique is to work, the converging lens must be "stronger" than the diverging lens—that is, it must have a focal length whose magnitude is less than that of the diverging lens. (Rays from the Sun are focused 28.5 cm behind the combination, so the focal length of the total combination is / T = 28.5 cm.) Im ge point

FIGURE 33 - 15 Determining the foc l length of diverging lens. Ex mple 33 - 6.

SECTION 3 3 - 3

Combi atio s of Le ses

875

* 3 3 -4 Lensm ker's Equ tion In this Section, we will show that parallel rays are brought to a focus at a single point for a thin lens. At the same time, we will also derive an equation that relates the focal length of a lens to the radii of curvature of its two surfaces and its index of refraction, which is known as the lensmaker's equation. In Fig. 33-16, a ray parallel to the axis of a lens is refracted at the front surface of the lens at point A 1 and is refracted at the back surface at point A 2. This ray then passes through point F, which we call the focal point for this ray. Point A xis a height h x above the axis, and point A 2 is height h 2 above the axis. i and 2 are the centers of curvature of the two lens surfaces; so the length i A x = R i, the radius of curvature of the front surface, and 2A 2 = R2 is the radius of the second surface. We consider a double convex lens and by convention choose the radii of both lens surfaces as positive. The thickness of the lens has been grossly exagger ated so that the various angles would be clear. But we will assume that the lens is actually very thin and that angles between the rays and the axis are small. In this approximation, the sines and tangents of all the angles will be equal to the angles themselves in radians. For example, sin 61 « tan dx ~ Qx (radians).

FIGURE 3 3 - 1 6 Di gr m of r y p ssing through lens for deriv tion of the lensm ker's equ tion.

To this approximation, then, Snell's law tells us that Qx = nd2 04 = nd3 where n is the index of refraction of the glass, and we assume that the lens is surrounded by air (n = 1). Notice also in Fig. 33-16 that

0i «

sin0]

*2 h2

/ '

The last follows because the distance from F to the lens (assumed very thin) is /. From the diagram, the angle 7 is 7 = 0 ! - 02.

A careful examination of Fig. 33-16 shows also that a =

876

CHAPTER 33

d3 -

This can be seen by drawing a horizontal line to the left from point A 2, which divides the angle 03 into two parts. The upper part equals 7 and the lower part equals a. (The opposite angles between an oblique line and two parallel lines are equal.) Thus, 03 = 7 + a. Furthermore, by drawing a horizontal line to the right from point A 2, we divide 04 into two parts. The upper part is a and the lower is /3.

Thus 04 = a + (3. We now combine all these equations: 04

a = 03 - 7 = — - (0i - 02) = - + - - 0i n K J n n or /?2 ______ ^ __

nR2

^1 | hi nf

nRx

Because the lens is thin, x « 2and all 's can be canceled from all the numerators. We then multiply through by n and rearrange to find that

■ f - stop = —

where / is the focal length of the lens and D is the diameter of the lens opening (Fig. 33-18). For example, when a 50-mm-focal-length lens has an opening D = 25 mm, we say it is set at //2 . When this lens is set at / / , the opening is only 6 \ mm (50/6^ = ). For faster shutter speeds, or low light conditions, a wider lens opening must be used to get a proper exposure, which corresponds to a smaller / - stop number. The smaller the / - stop number, the larger the opening and the more light passes through the lens to the sensor or film. The smallest / - number of a lens (largest opening) is referred to as the speed of the lens. The best lenses may have a speed of //2.0, or even faster. The advantage of a fast lens is that it allows pictures to be taken under poor lighting conditions. Good quality lenses consist of several elements to reduce the defects present in simple thin lenses (Section 33-10). Standard / - stops are 1.0, 1.4, 2.0, 2.8, 4.0, 5.6, , 11, 16, 22, and 32 (Fig. 33-21). Each of these stops corresponds to a diameter reduction by a factor of about V2 ~ 1.4. Because the amount of light reaching the film is proportional to the area of the opening, and therefore proportional to the diameter squared, each standard / - stop corresponds to a factor of 2 in light intensity reaching the film.

Focusing Focusing is the operation of placing the lens at the correct position relative to the sensor or film for the sharpest image. The image distance is smallest for objects at infinity (the symbol oo is used for infinity) and is equal to the focal length. For closer objects, the image distance is greater than the focal length, as can be seen from the lens equation, 1 / / = l / d Q+ l / d l (Eq. 33 -2). To focus on nearby objects, the lens must therefore be moved away from the sensor or film, and this is usually done on a manual camera by turning a ring on the lens. different films h ve different sensitivities to light, referred to s the "film speed" nd specified s n "ISO (or ASA) number."A "f ster" film is more sensitive nd needs less light to produce good im ge. F ster films re gr inier so offer less sh rpness (resolution) when enl rged. Digit l c mer s m y h ve "g in" or "ISO " djustment for sensitivity. Adjusting CCD to be "f ster" for low light conditions results in "noise," the digit l equiv lent of gr ininess.

SECTION 3 3 - 5

879

FIGURE 3 3 - 2 2 Photos t ken with c mer lens ( ) focused on ne rby object with dist nt object blurry, nd (b) focused on more dist nt object with ne rby object blurry.

(a)

(b)

If the lens is focused on a nearby object, a sharp image of it will be formed, but the image of distant objects may be blurry (Fig. 33 -22). The rays from a point on the distant object will be out of focus— they will form a circle on the sensor or film as shown (exaggerated) in Fig. 33-23. The distant object will thus produce an image consisting of overlapping circles and will be blurred. These circles are called circles of confusion. To include near and distant objects in the same photo, you (or the camera) can try setting the lens focus at an intermediate position. For a given distance setting, there is a range of distances over which the circles of confusion will be small enough that the images will be reasonably sharp. This is called the depth of field. The depth of field varies with the lens opening. If the lens opening is smaller, only rays through the central part of the lens are accepted, and these form smaller circles of confusion for a given object distance. Hence, at smaller lens openings, a greater range of object distances will fit within the circle of confusion criterion, so the depth of field is greater.f For a sensor or film width of 36 mm (including 35-mm film cameras), the depth of field is usually based on a maximum circle of confusion diameter of 0.003 mm. FIGURE 3 3 - 2 3 When the lens is positioned to focus on ne rby object, points on dist nt object produce circles nd re therefore blurred. (The effect is shown gre tly ex gger ted.)

R ys from ne iby object (in locus} +lCirclc of confusion" for distant objcct (greatly exuggonikxt)

EXAMPLE 33 -8 Camera focus. How far must a 50.0-mm-focal-length camera lens be moved from its infinity setting to sharply focus an object 3.00 m away? APPR ACH For an object at infinity, the image is at the focal point, by definition. For an object distance of 3.00 m, we use the thin lens equation, Eq. 33 -2, to find the image distance (distance of lens to film or sensor). S LUTI N When focused at infinity, the lens is 50.0 mm from the film. When focused at dQ = 3.00 m, the image distance is given by the lens equation, J_ _ 1 _ J _ _ _ _ 1 _________ 1 3000 - 50 2950 _ _ d[ f dQ 50.0 mm 3000 mm (3000)(50.0) mm 150,000 mm We solve for dxand find dx = 50.8 mm, so the lens needs to move 0.8 mm away from the film or digital sensor. fSm ller lens openings, however, result in reduced resolutions due to diffr ction (discussed in Ch pter 35). Best resolution is typic lly found round //8 .

880

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Le ses a d Optical I strume ts

C NCEPTUAL EXAMPLE 55 - 9 I Shutter speed. To improve the depth of field, you " stop down" your amera lens by two /-stops from //4 to / / 8 .What should you do to the shutter speed to maintain the same exposure? RESP NSE The amount of light admitted by the lens is proportional to the area of the lens opening. Redu ing the lens opening by two /-stops redu es the diameter by a fa tor of 2, and the area by a fa tor of 4. To maintain the same exposure, the shutter must be open four times as long. If the shutter speed had been ^ s , you would have to in rease the exposure time to ^5 s.

* Pictur Sharpn ss The sharpness of a pi ture depends not only on a urate fo using and short exposure times, but also on the graininess of the film , or the number of pixels for a digital amera. Fine-grained films are " slower," meaning they require longer exposures for a given light level. Digital ameras have averaging (or " ompression" ) programs, su h as JPEG, whi h redu e memory size by averaging over pixels where little ontrast is dete ted. Hen e it is unusual to use all pixels available. They also average over pixels in low light onditions, resulting in a less sharp photo. The quality of the lens strongly affe ts the image quality, and we dis uss lens reso lution and diffra tion effe ts in Se tions 33-10 and 35-4. The sharpness, or resolution, of a lens is often given as so many lines per millimeter, measured by photographing a standard set of parallel lines on fine-grain film or high quality sensor, or as so many dots per in h (dpi). The minimum spa ing of distinguishable lines or dots gives the resolution; 50 lines/mm is reasonable, 100 lines/mm is quite good ( = 100 dots/mm « 2500 dpi on the sensor). EXAMPLE 33 - 10 Pixels and resolution. A 6 -MP (6 -megapixel) digital amera offers a maximum resolution of 2000 X 3000 pixels on a 16-mm X 24-mm CCD sensor. How sharp should the lens be to make use of this resolution? APPR ACH We find the number of pixels per m illim eter and require the lens to be at least that good. S LUTI N We an either take the image height (2000 pixels in 16 mm) or the width (3000 pixels in 24 mm): 3000 pixels — — --------- = 125 pixels/mm. 24 mm We would want the lens to be able to resolve at least 125 lines or dots per mm as well, whi h would be a very good lens. If the lens is not this good, fewer pixels and less memory ould be used. N TE In reasing lens resolution is a tougher problem today than is squeezing more pixels on a CCD or CMOS. The sensor for high MP ameras must also be physi ally larger for greater light sensitivity (low light onditions). EXAMPLE 33-11 Blown-up photograph. An enlarged photograph looks sharp at normal viewing distan es if the dots or lines are resolved to about 10 dots/mm. Would an 8 X 10-in h enlargement of a photo taken by the amera in Example 33-10 seem sharp? To what maximum size ould you enlarge this 2000 X 3000-pixel image? APPR ACH We assume the image is 2000 X 3000 pixels on a 16 X 24-mm CCD as in Example 33-10, or 125 pixels/mm. We make an enlarged photo 8 X 10 in. = 20 m X 25 m. S LUTI N The short side of the CCD is 16 mm = 1.6 m long, and that side of the photograph is 8 in hes or 20 m. Thus the size is in reased by a fa tor of 20 m/1.6 m = 12.5X (or 25 m/2.4 m « 10X). To fill the 8 X 10-in. paper, we assume the enlargement is 12.5 X . The pixels are thus enlarged 12.5 X ; so the pixel ount of 125/mm on the CCD be omes 10 per mm on the print. Hen e an 8 X 10-in h print is just about the maximum possible for a sharp photograph with 6 megapixels. If you feel 7 dots per mm is good enough, you an enlarge to maybe 11 X 14 in hes. SECTI N 33 - 5 —

Cameras: Film and Digital

881

T l photos and Wid -angl s

Prism

Mirror FIGURE 33-24 Single-lens reflex

(SLR) c mer , showing how the im ge is viewed through the lens with the help of mov ble mirror nd prism.

Camera lenses are ategorized into normal, telephoto, and wide angle, a ording to fo al length and film size. A norm l lens overs the sensor or film w ith a field of view that orresponds approximately to that o f normal vision. A normal lens fo r 35-mm film has a fo al length in the vi inity o f 50 mm. The best digital ameras aim for a sensor o f the same size1" (24 mm X 36mm). ( If the sensor is smaller, digital ameras sometimes spe ify fo al lengths to orrespond w ith lassi 35-mm ameras.) Telephoto lenses a t like teles opes to magnify images. They have longer fo al lengths than a normal lens: as we saw in Se tion 33 -2 (Eq. 33 -3), the height of the image fo r a given obje t distan e is proportional to the image distan e, and the image distan e w ill be greater fo r a lens w ith longer fo al length. For distant obje ts, the image height is very nearly proportional to the fo al length. Thus a 200-mm telephoto lens fo r use w ith a 35-mm amera gives a 4X m agnifi ation over the normal 50-mm lens. A wide- ngle lens has a shorter fo al length than normal: a wider fie ld o f view is in luded, and obje ts appear smaller. A zoom lens is one whose fo al length an be hanged so that you seem to zoom up to, or away from , the subje t as you hange the fo al length. D igita l ameras may have an optic l zoom meaning the lens an hange fo al length and m aintain resolution. But an " ele troni " or digit l zoom just enlarges the dots (pixels) w ith loss of sharpness. D iffe re n t types of viewing systems are used in ameras. In some ameras, you view through a small window just above the lens as in Fig. 33 -18. In a single-lens reflex amera (SLR ), you a tually view through the lens w ith the use o f prisms and m irrors (Fig. 33 -24). A m irro r hangs at a 45° angle behind the lens and flips up out o f the way just before the shutter opens. SLRs have the advantage that you an see almost exa tly what you w ill get. D ig ita l ameras use an LCD display, and it too an show what you w ill get on the photo if it is arefully designed.

33-6 The Hum n Eye; Corrective Lenses 0

PHYSICS

APPLIED

T e eye FIGURE 33-25 Di gr m of

hum n eye. Cili ry muscles Aqueous humor

Retin Fove

Pupil Corne

Optic nerve Cili ry muscles

The human eye resembles a amera in its basi stru ture (Fig. 33 -25), but is far more sophisti ated. The in te rio r o f the eye is fille d w ith a transparent gel-like substan e alled the vitreou s u m or w ith index o f refra tion n = 1.337. L ight enters this en losed volume through the ornea and lens. Between the ornea and lens is a watery flu id , the aqueous hum or (aqua is " w ater" in Latin ) w ith n = 1.336. A diaphragm, alled the iris (the olored part o f your eye), adjusts autom ati ally to ontrol the amount o f ligh t entering the eye, sim ilar to a amera. The hole in the iris through whi h lig h t passes (the pupil) is bla k be ause no lig h t is refle ted from it ( it's a hole), and very little ligh t is refle ted ba k out from the in te rio r o f the eye. The retin , whi h plays the role of the film or sensor in a amera, is on the urved rear surfa e o f the eye. The retina onsists o f a omplex array o f nerves and re eptors known as rods and cones whi h a t to hange lig h t energy into ele tri al signals that travel along the nerves. The re onstru tion o f the image from all these tin y re eptors is done m ainly in the brain, although some analysis may also be done in the omplex inter onne ted nerve netw ork at the retina itself. A t the enter o f the retina is a small area alled the fove , about 0.25 mm in diameter, where the ones are very losely pa ked and the sharpest image and best olor dis rimination are found. U nlike a amera, the eye ontains no shutter. The equivalent operation is arried out by the nervous system, whi h analyzes the signals to form images at the rate o f about 30 per se ond. This an be ompared to m otion pi ture or television ameras, whi h operate by taking a series o f s till pi tures at a rate o f 24 (movies) or 30 (U.S. television) per se ond; their rapid proje tion on the s reen gives the appearan e o f motion.

fA "35-mm c mer "uses film th t is physic lly 35mm wide; th t 35mm is not to be confused with foc l length. 35-mmfilmh s sprocket holes, so only 24 mmof its height is used for the photo; the width is usu lly 36 mm for stills. Thus one fr me is 24 mm X 36 mm. Movie fr mes re 18mm X 24 mm.

882

CHAPTER 33

Le ses a d Optical I strume ts

The lens of the eye (n = 1.386 to 1.406) does little of the bending of the light rays. Most of the refra tion is done at the front surfa e of the corne ( n = 1.376) at its interfa e with air ( n = 1.0). The lens a ts as a fine adjustment for fo using at different distan es. This is a omplished by the iliary mus les (Fig. 33 -25), whi h hange the urvature of the lens so that its fo al length is hanged. To fo us on a distant obje t, the iliary mus les of the eye are relaxed and the lens is thin, as shown in Fig. 33-26a, and parallel rays fo us at the fo al point (on the retina). To fo us on a nearby obje t, the mus les ontra t, ausing the enter of the lens to thi ken, Fig. 33-26b, thus shortening the fo al length so that images of nearby obje ts an be fo used on the retina, behind the new fo al point. This fo using adjustment is alled

Foc l point of lens ndcorne

Object t oo

Foc l point of lens ndcorne

ccommod tion. The losest distan e at whi h the eye an fo us learly is alled the ne r point of the eye. For young adults it is typi ally 25 m, although younger hildren an often fo us on obje ts as lose as 10 m. As people grow older, the ability to a ommodate is redu ed and the near point in reases. A given person's f r point is the farthest distan e at whi h an obje t an be seen learly. For some purposes it is useful to speak of a norm l eye (a sort of average over the population), defined as an eye having a near point of 25 m and a far point of infinity. To he k your own near point, pla e this book lose to your eye and slowly move it away until the type is sharp. The " normal" eye is sort of an ideal. Many people have eyes that do not a ommodate w ithin the " normal" range of 25 m to infinity, or have some other defe t. Two ommon defe ts are nearsightedness and farsightedness. Both an be orre ted to a large extent with lenses— either eyeglasses or onta t lenses. In ne rsightedness, or m yopia, the eye an fo us only on nearby obje ts. The far point is not infin ity but some shorter distan e, so distant obje ts are not seen learly. It is usually aused by an eyeball that is too long, although sometimes it is the urvature of the ornea that is too great. In either ase, images of distant obje ts are fo used in front of the retina. A diverging lens, be ause it auses parallel rays to diverge, allows the rays to be fo used at the retina (Fig. 33-27a) and thus orre ts this defe t. In f rsightedness, or yperopia, the eye annot fo us on nearby obje ts. Although distant obje ts are usually seen learly, the near point is somewhat greater than the " normal" 25 m, whi h makes reading diffi ult. This defe t is aused by an eyeball that is too short or (less often) by a ornea that is not suffi iently urved. It is orre ted by a onverging lens, Fig. 33-27b. Similar to hyperopia is presbyopia, whi h refers to the lessening ability of the eye to a om modate as a person ages, and the near point moves out. Converging lenses also ompensate for this.

Object

FIGURE 33-26 Accommod tion by norm l eye: ( ) lens rel xed, focused t infinity; (b) lens thickened, focused on ne rby object.

PHYSICS

APPLIED

Corrective lenses

Object at°°

____

FIGURE 33-27 Correcting eye defects with lenses: ( ) ne rsighted eye, which c nnot focus cle rly on dist nt objects, c n be corrected by use of diverging lens; (b) f rsighted eye, which c nnot focus cle rly on ne rby objects, c n be corrected by use of converging lens.

Ne rsighted eye

SECTION 3 3 - 6

The Huma Eye; Corrective Le ses

883

Im ge (line)

Object (point) FIGURE 33 -28 A cylindric l lens forms line im ge of point object bec use it is converging in one pl ne only.

Astigm tism is usually aused by an out-of-round ornea or lens so that point obje ts are fo used as short lines, whi h blurs the image. It is as if the ornea were spheri al with a ylindri al se tion superimposed. As shown in Fig. 33-28, a ylindri al lens fo uses a point into a line parallel to its axis. An astigmati eye may fo us rays in one plane, su h as the verti al plane, at a shorter distan e than it does for rays in a horizontal plane. Astigmatism is orre ted with the use of a ompensating ylindri al lens. Lenses for eyes that are nearsighted or farsighted as well as astigmati are ground with superimposed spheri al and ylindri al surfa es, so that the radius of urvature of the orre ting lens is different in different planes. EXAMPLE 33 -12 Farsighted eye. Sue is farsighted with a near point of 100 m. Reading glasses must have what lens power so that she an read a newspaper at a distan e of 25 m? Assume the lens is very lose to the eye. APPR ACH When the obje t is pla ed 25 m from the lens, we want the image to be 100 m away on the same side of the lens (so the eye an fo us it), and so the image is virtual, as shown in Fig. 33-29, and d\ = -100 m w ill be negative. We use the thin lens equation (Eq. 33 -2) to determine the needed fo al length. Optometrists' pres riptions spe ify the power (P = 1/ / , Eq. 33 -1) given in diopters ( l D = 1 m-1). S LUTI N Given that dQ = 25 m and dx = -100 m, the thin lens equation gives

= JL

25 m

-100 m

1 -

dQ + d{

4- 1 100 m

1 33 m

So / = 33 m = 0.33 m. The power P of the lens is P = 1 / f = +3.0 D. The plus sign indi ates that it is a onverging lens. N TE We hose the image position to be where the eye an a tually fo us. The lens needs to put the image there, given the desired pla ement of the obje t (newspaper).

FIGURE 33 -29 Lens of re ding gl sses (Ex mple 33-12).

FIGURE 33 -30

Ex mple 33-13. 2cm-|

Object t oo

(F rpoint) (a)

EXAMPLE 33 -13 Nearsighted eye. A nearsighted eye has near and far points of 12 m and 17 m, respe tively, (a) What lens power is needed for this person to see distant obje ts learly, and ( b ) what then w ill be the near point? Assume that the lens is 2.0 m from the eye (typi al for eyeglasses). APPR ACH For a distant obje t (d Q = oo), the lens must put the image at the far point of the eye as shown in Fig. 33-30a, 17 m in front of the eye. We an use the thin lens equation to find the fo al length of the lens, and from this its lens power. The new near point (as shown in Fig. 33-30b) an be al ulated for the lens by again using the thin lens equation. S LUTI N (a) For an obje t at infinity (d Q = oo), the image must be in front of the lens 17 m from the eye or (17 m — 2 m) = 15 m from the lens; hen e di = - 1 5 m. We use the thin lens equation to solve for the fo al length of the needed lens:

_ JL

1 _ JL di

-1 5 m

15 m

So / = - 1 5 m = -0.15 m or P = 1 / f = - 6 .7 D. The minus sign indi ates that it must be a diverging lens for the myopi eye.

884

CHAPTER 33

Le ses a d Optical I strume ts

(b) The near point when glasses are worn is where an obje t is pla ed (d Q) so that the lens forms an image at the " near point of the naked eye," namely 12 m from the eye. That image point is (12 m — 2 m) = 10 m in front of the lens, so di = - 0.10 m and the thin lens equation gives

1 1 + 0.15 m 0.10 m

-2 + 3 0.30 m

1 0.30 m

So dQ = 30 m, whi h means the near point when the person is wearing glasses is 30 m in front of the lens, or 32 m from the eye. Suppose onta t lenses are used to orre t the eye in Example 33-13. Sin e onta ts are pla ed dire tly on the ornea, we would not subtra t out the 2.0 m for the image distan es. That is, for distant obje ts rfi = / = - 1 7 m , so P = 1 / f = - 5 .9 D. The new near point would be 41 m. Thus we see that a onta t lens and an eyeglass lens w ill require slightly different powers, or fo al lengths, for the same eye be ause of their different pla ements relative to the eye. We also see that glasses in this ase give a better near point than onta ts.

@ PHYSICS

APPLIED

Contact lenses

EXERCISE E Wh t power cont ct lens is needed for n eye to see dist nt objects if its f r point is 25 cm? When your eyes are under water, distant underwater obje ts look blurry be ause at the water- ornea interfa e, the differen e in indi es of refra tion is very small: n = 1.33 for water, 1.376 for the ornea. Hen e light rays are bent very little and are fo used far behind the retina, Fig. 33 -3la. If you wear goggles or a fa e mask, you restore an air- ornea interfa e (n = 1.0 and 1.376, respe tively) and the rays an be fo used, Fig. 33-31b.

PHYSICS

APPLIED

Underwater vision

FIGURE 33-31 ( ) Under w ter, we see blurry im ge bec use light r ys re bent much less th n in ir. (b) If we we r goggles, we g in h ve n ir-corne interf ce nd c n see cle rly.

FIGURE 33-32 When the s me object is viewed t shorter dist nce, the im ge on the retin is gre ter, so the object ppe rs l rger nd more det il c n be seen.The ngle 6 th t the object subtends in ( ) is gre ter th n in (b). Note:This is not norm l r y di gr mbec use we re showing only one r y frome ch point.

33-7 M gnifying Gl ss Mu h of the remainder of this Chapter w ill deal w ith opti al devi es that are used to produ e magnified images of obje ts. We first dis uss the simple m gnifier, or m gnifying gl ss, whi h is simply a onverging lens (see ChapterOpening Photo). How large an obje t appears, and how mu h detail we an see on it, depends on the size of the image it makes on the retina. This, in turn, depends on the angle subtended by the obje t at the eye. For example, a penny held 30 m from the eye looks twi e as ta ll as one held 60 m away be ause the angle it subtends is twi e as great (Fig. 33 -32). When we want to examine detail on an obje t, we bring it up lose to our eyes so that it subtends a greater angle. However, our eyes an a ommodate only up to a point (the near point), and we w ill assume a standard distan e of N = 25 m as the near point in what follows.

IniiWf

(b)

SECTION 3 3 - 7

Mag ifyi g Glass

885

Im ge

-N-

(= 25 cmfor norm l eye) (b) Le f viewed ( ) through m gnifying gl ss, nd (b) with the un ided eye. The eye is focused t its ne r point in both c ses. FIGURE 33 -33

A magnifying glass allows us to pla e the obje t loser to our eye so that it subtends a greater angle. As shown in Fig. 33-33a, the obje t is pla ed at the fo al point or just within it. Then the onverging lens produ es a virtual image, whi h must be at least 25 m from the eye if the eye is to fo us on it. If the eye is relaxed, the image w ill be at infinity, and in this ase the obje t is exa tly at the fo al point. (You make this slight adjustment yourself when you " fo us" on the obje t by moving the magnifying glass.) A omparison of part (a) of Fig. 33-33 with part (b), in whi h the same obje t is viewed at the near point with the unaided eye, reveals that the angle the obje t subtends at the eye is mu h larger when the magnifier is used. The ngul r m gnific tion or m gnifying power, M, of the lens is defined as the ratio of the angle subtended by an obje t when using the lens, to the angle subtended using the unaided eye, with the obje t at the near point N of the eye (N = 25 m for a normal eye):

M = j,

(33 -5)

where 0 and 0' are shown in Fig. 33-33. We an write M in terms of the fo al length by noting that 0 = /N (Fig. 33-33b) and O' = /d Q (Fig. 33-33a), where is the height of the obje t and we assume the angles are small so 0 and 0 ' equal their sines and tangents. If the eye is relaxed (for least eye strain), the image w ill be at infin ity and the obje t w ill be pre isely at the fo al point; see Fig. 33-34. = /N as before (Fig. 33-33b). Thus Then dQ = f and 0' = /f , whereas

M = — 0

lL /N

\ eye fo used at oo; \_N = 25 m for normal eye

= K. f

(33-6 )

We see that the shorter the fo al length of the lens, the greater the magnifi ation .1 The magnifi ation of a given lens an be in reased a b it by moving the lens and adjusting your eye so it fo uses on the image at the eye's near point. In this ase, d{ = —N (see Fig. 33-33a) if your eye is very near the magnifier. fSimple single-lens m gnifiers re limited to bout 2 or 3X bec use of blurring due to spheric l berr tion (Section 33-10).

FIGURE 33 - 34 With the eye rel xed, the object is pl ced t the foc l point, nd the im ge is t infinity. Comp re to Fig. 33-33 where the im ge is t the eye's ne r point.

Im ge h

r F

i /

j 1 f

886

CHAPTER 33

Le ses a d Optical I strume ts

Then the obje t distan e d Qis given by

J- = i _ 1 = i do~ f

J-

~ f + N'

We see from this equation that dQ = f N / ( f + N ) < f , as shown in Fig. 33-33a, sin e N / ( f + N ) must be less than 1. W ith O' = /d Q the magnifi ation is = e

= N /N

1_ N

[

eye fo used at near point, N; 1 N = 25 m for normal eye J ^

We see that the magnifi ation is slightly greater when the eye is fo used at its near point, as ompared to when it is relaxed.

EXAMPLE 33 - 14 ESTIMATE I A jeweler's "loupe." An 8 - m -fo al-length onverging lens is used as a "jeweler's loupe," whi h is a magnifying glass. Esti mate (a) the magnifi ation when the eye is relaxed, and ( b ) the magnifi ation if the eye is fo used at its near point N = 25 m. APPR ACH The magnifi ation when the eye is relaxed is given by Eq. 33- 6 a. When the eye is fo used at its near point, we use Eq. 33-6b and we assume the lens is near the eye. S LUTI N (a) W ith the relaxed eye fo used at infinity, N 25 m M = — = —----- ~ 3 X .

8 m

FIGURE 33 -35 ( ) Objective lens (mounted now in n ivory fr me) fromthe telescope with which G lileo m de his world-sh king discoveries, including the moons of Jupiter, (b) L ter telescopes m de by G lileo.

(ft) The magnifi ation when the eye is fo used at its near point (N = 25 m), and the lens is near the eye, is M = l + ^ = l + ^ - « 4 x . J o

33-8 Telescopes A teles ope is used to magnify obje ts that are very far away. In most ases, the obje t an be onsidered to be at infinity. Galileo, although he did not invent it,f developed the teles ope into a usable and im portant instrument. He was the first to examine the heavens w ith the teles ope (Fig. 33 -35), and he made world-shaking dis overies, in luding the moons of Jupiter, the phases of Venus, sunspots, the stru ture of the M oon's surfa e, and that the M ilky Way is made up of a huge number of individual stars. G lileo built his first telescope in 1609 fter h ving he rd of such n instrument existing in Holl nd. The first telescopes m gnified only three to four times, but G lileo soon m de 30-power instrument. The first Dutch telescope seems to d te from bout 1604, but there is reference suggesting it m y h ve been copied from n It li n telescope built s e rly s 1590. Kepler (see Ch pter 6 ) g ve r y description (in 1611) of the Kepleri n telescope, which is n med for him bec use he first described it, lthough he did not build it.

(b) SECTION 3 3 - 8

Telescopes

887

FIGURE 33 -36 Astronomic l telescope (refr cting). P r llel light from one point on dist nt object (dQ= oo) is brought to focus by the objective lens in its foc l pl ne. This im ge (IJ is m gnified by the eyepiece to form the fin l im ge I2. Only two of the r ys shown entering the objective re st nd rd r ys (2 nd 3) s described in Fig. 33-6.

P r llel r ys from object t

Several types of stronomic l telescope exist. The ommon refr cting type, sometimes alled Kepleri n, ontains two onverging lenses lo ated at opposite ends of a long tube, as illustrated in Fig. 33-36. The lens losest to the obje t is alled the objective lens (fo al length / Q) and forms a real image I4 of the distant obje t in the plane of its fo al point F0 (or near it if the obje t is not at infinity). The se ond lens, alled the eyepiece (fo al length / e), a ts as a magnifier. That is, the eyepie e magnifies the image I x formed by the obje tive lens to produ e a se ond, greatly magnified image, I 2, whi h is virtual and inverted. If the viewing eye is relaxed, the eyepie e is adjusted so the image I 2 is at infinity. Then the real image 1 is at the fo al point Fg of the eyepie e, and the distan e between the lenses is f + / efor an obje t at infinity. To find the total angular magnifi ation of this teles ope, we note that the angle an obje t subtends as viewed by the unaided eye is just the angle subtended at the teles ope obje tive. From Fig. 33-36 we an see that 0 ~ / f 0 , where is the height of the image and we assume 0 is small so that tan 0 « 0. Note, too, that the thi kest of the three rays drawn in Fig. 33-36 is parallel to the axis before it strikes the eyepie e and therefore is refra ted through the eyepie e fo al point Fe on the far side. Thus, 0' « / f e and the tot l m gnifying power (that is, angular magnifi ation, whi h is what is always quoted) of this teles ope is

FIGURE 33 -37 This l rge refr cting telescope w s built in 1897 nd is housed t Yerkes Observ tory inWisconsin. The objective lens is 102 cm (40 inches) in di meter, nd the telescope tube is bout 19m long. Ex mple 33-15.

( /f e )

M = — = j-— \ =

-~ T >

[teles ope] (33 -7)

Jo )

where we have inserted a minus sign to indi ate that the image is inverted. To a hieve a large magnifi ation, the obje tive lens should have a long fo al length and the eyepie e a short fo al length. EXAMPLE 33 -15 Telescope magnification. The largest opti al refra ting teles ope in the world is lo ated at the Yerkes Observatory in Wis onsin, Fig. 33-37. It is referred to as a " 40-in h" teles ope, meaning that the diameter of the obje tive is 40 in., or 102 m. The obje tive lens has a fo al length of 19 m, and the eyepie e has a fo al length of 10 m. (a) Cal ulate the total magnifying power of this teles ope. ( b ) Estimate the length of the teles ope. APPR ACH Equation 33 -7 gives the magnifi ation. The length of the teles ope is the distan e between the two lenses. S LUTI N (a) From Eq. 33 -7 we find fo 19 m = -190X 0.10 m fc

888

CHAPTER 33

(b ) For a relaxed eye, the image 1 is at the fo al point of both the eyepie e and the obje tive lenses. The distan e between the two lenses is thus f Q + / e ~ 19 m, whi h is essentially the length of the teles ope.

EXERCISE F A 40 X telescope h s 1.2-cm foc l length eyepiece. Wh t is the foc l length of the objective lens?

For an astronomi al teles ope to produ e bright images of faint stars, the obje tive lens must be large to allow in as mu h light as possible. Indeed, the diameter of the obje tive lens (and hen e its " light-gathering power" ) is an important parameter for an astronomi al teles ope, whi h is why the largest ones are spe ified by giving the obje tive diameter (su h as the 10-meter Ke k teles ope in Hawaii). The onstru tion and grinding of large lenses is very diffi ult. Therefore, the largest teles opes are reflecting telescopes whi h use a urved m irror as the obje tive, Fig. 33-38. A m irror has only one surfa e to be ground and an be supported along its entire surfa e1 (a large lens, supported at its edges, would sag under its own weight). Often, the eyepie e lens or m irror (see Fig. 33-38) is removed so that the real image formed by the obje tive m irror an be re orded dire tly on film or on an ele troni sensor (CCD or CMOS, Se tion 33-5).

Conc ve mirror (objective)

Second ry mirror

(b) (c) (d) FIGURE 33 -38 A conc ve mirror c n be used s the objective of n stronomic l telescope. Arr ngement ( ) is c lled the Newtoni n focus, nd (b) the C ssegr ini n focus. Other rr ngements re lso possible, (c) The 200-inch (mirror di meter) H le telescope on P lom r Mount in in C liforni , (d) The 10-meter Keck telescope on M un Ke , H w ii. The Keck combines thirty-six 1.8-meter six-sided mirrors into the equiv lent of very l rge single reflector, 10m in di meter. A terrestri l telescope, for viewing obje ts on Earth, must provide an upright image— seeing normal obje ts upside down would be diffi ult (mu h less important for viewing stars). Two designs are shown in Fig. 33-39. The G lile n type, whi h Galileo used for his great astronomi al dis overies, has a diverging lens as eyepie e whi h inter epts the onverging rays from the obje tive lens before they rea h a fo us, and a ts to form a virtual upright image, Fig. 33-39a. This design is still used in opera glasses. The tube is reasonably short, but the field of view is small. The se ond type, shown in Fig. 33-39b, is often alled a spygl ss and makes use of a third onvex lens that a ts to make the image upright as shown. A spyglass must be quite long. The most pra ti al design today is the prism binocul r whi h was shown in Fig. 32-33. The obje tive and eyepie e are onverging lenses. The prisms refle t the rays by total internal refle tion and shorten the physi al size of the devi e, and they also a t to produ e an upright image. One prism reinverts the image in the verti al plane, the other in the horizontal plane. fAnother dv nt ge of mirrors is th t they exhibit no chrom tic berr tion bec use the light doesn't p ss through them; nd they c n be ground in p r bolic sh pe to correct for spheric l berr tion (Section 33-10). The reflecting telescope w s first proposed by Newton.

FIGURE 33 -39 Terrestri l telescopes th t produce n upright im ge: ( ) G lile n; (b) spygl ss, or erector type.

PHYSICS

APPLIED

Microscopes

The ompound microscope, like the teles ope, has both obje tive and eyepie e (or o ular) lenses, Fig. 33-40. The design is different from that for a teles ope be ause a mi ros ope is used to view obje ts that are very lose, so the obje t distan e is very small. The obje t is pla ed just beyond the obje tive's fo al point as shown in Fig. 33-40a. The image formed by the obje tive lens is real, quite far from the obje tive lens, and mu h enlarged. The eyepie e is positioned so that this image is near the eyepie e fo al point Fe. The image I x is magnified by the eyepie e into a very large virtual image, I 2, whi h is seen by the eye and is inverted. Modern mi ros opes use a third " tube" lens behind the obje tive, but we w ill analyze the simpler arrangement shown in Fig. 33-40a.

FIGURE 33-40 Compound microscope: ( ) r y di gr m, (b) photogr ph (illumin tion comes from the lower right, then up through the slide holding the object). The overall magnifi ation of a mi ros ope is the produ t of the magnifi ations produ ed by the two lenses. The image ^ formed by the obje tive lens is a fa tor m Q greater than the obje t itself. From Fig. 33-40a and Eq. 33-3 for the lateral magnifi ation of a simple lens, we have i Q

di dQ

e - U dQ

m0 = — = — = — j— >

(33 -8)

where d Q and d{ are the obje t and image distan es for the obje tive lens, £ is the distan e between the lenses (equal to the length of the barrel), and we ignored the minus sign in Eq. 33-3 whi h only tells us that the image is inverted. We set di = £ - / e, whi h is true only if the eye is relaxed, so that the image 1 is at the eyepie e fo al point Fe. The eyepie e a ts like a simple magnifier. If we assume that the eye is relaxed, the eyepie e angular magnifi ation M eis (from Eq. 33- 6 a)

Me = — >

(33 -9)

where the near point N = 25 m for the normal eye. Sin e the eyepie e enlarges the image formed by the obje tive, the overall angular magnifi ation M is the produ t of the lateral magnifi ation of the obje tive lens, m 0, times the angular magnifi ation, M e, of the eyepie e lens (Eqs. 33 -8 and 33 -9): M = Mc m

J ^

[mi ros ope] (33-10 )

« ~

[fo and / e «

f] (33-10b)

J e Jo

The approximation, Eq. 33 - 10b, is a urate when / eand f Qare small ompared to £, so £ - / e « £, and the obje t is near F0 so d Q « f (Fig. 33-40a). This is a good

890

CHAPTER 33

Le ses a d Optical I strume ts

approximation for large magnifi ations, whi h are obtained when f Q and / e are very small (they are in the denominator of Eq. 33-10b). To make lenses of very short fo al length, ompound lenses involving several elements must be used to avoid serious aberrations, as dis ussed in the next Se tion.

EXAMPLE 33-16 Microscope. A ompound mi ros ope onsists of a 10 X eyepie e and a 50 X obje tive 17.0 m apart. Determine (a) the overall magnifi ation, ( b ) the fo al length of ea h lens, and ( ) the position of the obje t when the final image is in fo us with the eye relaxed. Assume a normal eye, so N = 25 m. APPR ACH The overall magnifi ation is the produ t of the eyepie e magni fi ation and the obje tive magnifi ation. The fo al length of the eyepie e is found from Eq. 33-6a or 33 -9 for the magnifi ation of a simple magnifier. For the obje tive lens, it is easier to next find d Q (part ) using Eq. 33 -8 before we find f Q. S LUTI N (a) The overall magnifi ation is (10X)(50X) = 500X. (b) The eyepie e fo al length is (Eq. 33 -9) / e = N /M e = 25 m/10 = 2.5 m. Next we solve Eq. 33 -8 for d Q, and find £ - /e (17.0 m - 2.5 m) dQ = --------- = ----------- — ----------- = 0.29 m. mQ 50

Then, from the thin lens equation for the obje tive with dx = £ — / e = 14.5 m (see Fig. 33-40a), —

0.29 m

14.5 m

3 52

•

cm - 1 -

m ,

so f Q = l/(3.52 m _1) = 0.28 m. ( ) We just al ulated d = 0.29 m, whi h is very lose to f Q.

*33—10 Aberr tions of Lenses nd Mirrors Earlier in this Chapter we developed a theory of image formation by a thin lens. We found, for example, that all rays from ea h point on an obje t are brought to a single point as the image point. This result, and others, were based on approxima tions for a thin lens, mainly that all rays make small angles with the axis and that we an use sin ~ 0. Be ause of these approximations, we expe t deviations from the simple theory, whi h are referred to as lens berr tions. There are several types of aberration; we w ill briefly dis uss ea h of them separately, but all may be present at one time. Consider an obje t at any point (even at infinity) on the axis of a lens with spheri al surfa es. Rays from this point that pass through the outer regions of the lens are brought to a fo us at a different point from those that pass through the enter of the lens. This is alled spheric l berr tion, and is shown exaggerated in Fig. 33-41. Consequently, the image seen on a s reen or film w ill not be a point but a tiny ir ular pat h of light. If the sensor or film is pla ed at the point C, as indi ated, the ir le w ill have its smallest diameter, whi h is referred to as the circle of le st confusion. Spheri al aberration is present whenever spheri al surfa es are used. It an be redu ed by using nonspheri al (= aspheri al) lens surfa es, but grinding su h lenses is d iffi u lt and expensive. Spheri al aberration an be redu ed by the use of several lenses in ombination, and by using prim arily the entral part of lenses.

*SECTION 33 - 10

PHYSICS

APPLIED

Lens aberrations

FIGURE 33-41 Spheric l berr tion (ex gger ted). Circle of le st confusion is t C. C

Aberratio s of Le ses a d Mirrors

891

Axis r r

P -

(a)

(b) Distortion: lenses m y im ge squ re grid of perpendicul r lines to produce ( ) b rrel distortion or (b) pincushion distortion. These distortions c n be seen in the photogr ph of Fig. 33-2d. FIGURE 33 - 42

White

Chrom tic berr tion. Different colors re focused t different points. FIGURE 33 - 43

FIGURE 33 - 44

Achrom tic

doublet.

0 P H Y S I C S

APPLIED

Human eye

For obje t points o ff the lens axis, additional aberrations o ur. Rays passing through the different parts of the lens ause spreading of the image that is non ir ular. There are two effe ts: com (be ause the image of a point is omet-shaped rather than a ir le) and off- xis stigm tism.1 Furthermore, the image points for obje ts off the axis but at the same distan e from the lens do not fall on a flat plane but on a urved surfa e— that is, the fo al plane is not flat. (We expe t this be ause the points on a flat plane, su h as the film in a amera, are not equidistant from the lens.) This aberration is known as curv ture of field and is a problem in ameras and other devi es where the film is pla ed in a flat plane. In the eye, however, the retina is urved, whi h ompensates for this effe t. Another aberration, distortion, is a result of variation of magnifi ation at different distan es from the lens axis. Thus a straight-line obje t some distan e from the axis may form a urved image. A square grid of lines may be distorted to produ e " barrel distortion," or " pin ushion distortion," Fig. 33-42. The former is ommon in extreme wide-angle lenses. A ll the above aberrations o ur for mono hromati light and hen e are referred to as monoc romatic aberrations. Normal light is not mono hromati , and there w ill also be chrom tic berr tion. This aberration arises be ause of dispersion— the variation of index of refra tion of transparent materials with wave length (Se tion 32 -6). For example, blue light is bent more than red light by glass. So if white light is in ident on a lens, the different olors are fo used at different points, Fig. 33-43, and have slightly different magnifi ations resulting in olored fringes in the image. Chromati aberration an be eliminated for any two olors (and redu ed greatly for all others) by the use of two lenses made of different materials with different indi es of refra tion and dispersion. Normally one lens is onverging and the other diverging, and they are often emented together (Fig. 33-44). Su h a lens ombination is alled an chrom tic doublet (or " olor- orre ted" lens). To redu e aberrations, high -quality lenses are compound lenses onsisting of many simple lenses, referred to as elements. A typi al high -quality amera lens may ontain six to eight (or more) elements. For simpli ity we w ill usually indi ate lenses in diagrams as if they were simple lenses. The human eye is also subje t to aberrations, but they are minimal. Spheri al aberration, for example, is minimized be ause ( 1 ) the ornea is less urved at the edges than at the enter, and (2 ) the lens is less dense at the edges than at the enter. Both effe ts ause rays at the outer edges to be bent less strongly, and thus help to redu e spheri al aberration. Chromati aberration is partially ompensated for be ause the lens absorbs the shorter wavelengths appre iably and the retina is less sensitive to the blue and violet wavelengths. This is just the region of the spe trum where dispersion— and thus hromati aberration — is greatest (Fig. 32-28). Spheri al m irrors (Se tion 32 -3) also suffer aberrations in luding spheri al aberration (see Fig. 32-13). M irrors an be ground in a paraboli shape to orre t for aberrations, but they are mu h harder to make and therefore very expensive. Spheri al mirrors do not, however, exhibit hromati aberration be ause the light does not pass through them (no refra tion, no dispersion). fAlthough the effect is the s me s for stigm tism in the eye (Section 33-6), the c use is different. Off-£ stigm tism is no problem in the eye bec use objects re cle rly seen only t the fove , on the lens xis.

Summary A lens uses refr ction to produce re l or virtu l im ge. P r llel r ys of light re focused to point, c lled the foc l point, by converging lens. The dist nce of the foc l point from the lens is c lled the foc l length / of the lens. After p r llel r ys p ss through diverging lens, they ppe r to diverge from point, its foc l point; nd the corre sponding foc l length is considered neg tive.

892

CHAPTER 33

Le ses a d Optical I strume ts

The power P of lens, which is P = J

(33-1)

is given in diopters, which re units of inverse meters (m-1). For given object, the position nd size of the im ge formed by lens c n be found pproxim tely by r y tr cing.

Algebrai ally, the relation between image and obje t distan es, d[ and d Q, and the fo al length / , is given by the thin lens equ tion:

= 1

dQ + d{

(33 - 2)

obje ts. Converging lenses are used to orre t fo r defe ts in whi h the eye annot fo us on lose obje ts. A simple m gnifier is a onverging lens that forms a virtual image o f an obje t pla ed at (or w ithin) the fo al point. The ngul r m gnific tion, when viewed by a relaxed normal eye, is

The ratio o f image height to obje t height, whi h equals the

l ter l m gnific tion ra, is A, n

(33 -6 ) "

dc,

(33 - 3)

When using the various equations of geometri opti s, it is important to remember the sign conventions for all quantities involved: arefully review them (page 871) when doing Problems. When two (or more) thin lenses are used in ombination to produ e an image, the thin lens equation an be used fo r ea h lens in sequen e. The image produ ed by the firs t lens a ts as the obje t fo r the se ond lens. [*The lensm ker's equ tion relates the radii of urvature of the lens surfa es and the len's index o f refra tion to the fo al length o f the lens.] A c mer lens forms an image on film , or on an ele troni sensor (CCD or CMOS) in a digital amera, by allowing light in through a shutter. The image is fo used by moving the lens rela tive to the film , and the / - stop (or lens opening) must be adjusted fo r the brightness o f the s ene and the hosen shutter speed. The /- sto p is defined as the ratio o f the fo al length to the diameter of the lens opening. The human eye also adjusts fo r the available lig h t— by opening and losing the iris. It fo uses not by moving the lens, but by adjusting the shape o f the lens to vary its fo al length. The image is formed on the retina, whi h ontains an array of re eptors known as rods and ones. Diverging eyeglass or onta t lenses are used to orre t the defe t of a nearsighted eye, whi h annot fo us w ell on distant

- T

where / is the fo al length of the lens and N is the near point of the eye (25 m fo r a " norm al" eye). A n stronomic l telescope onsists of an objective lens or m irror, and an eyepiece that magnifies the real image formed by the obje tive. The m gnific tion is equal to the ratio o f the obje tive and eyepie e fo al lengths, and the image is inverted:

(33 - 7) [*A ompound microscope also uses obje tive and eye pie e lenses, and the final image is inverted. The total magnifi a tion is the produ t of the magnifi ations o f the two lenses and is approximately M

« —

(33 -10b)

/e /o

where £ is the distan e between the lenses, N is the near point of the eye, and f Q and / e are the fo al lengths of obje tive and eyepie e, respe tively.] [*Mi ros opes, teles opes, and other opti al instruments are lim ited in the form ation o f sharp images by lens berr tions. These in lude spheric l berr tion, in whi h rays passing through the edge of a lens are not fo used at the same point as those that pass near the enter; and chrom tic berr tion, in whi h different olors are fo used at different points. Compound lenses, onsisting o f several elements, an largely orre t fo r aberrations.]

Questions 1. Where must the film be pla ed if a amera lens is to make a sharp image o f an obje t far away?

2. A photographer moves loser to his subje t and then refo uses. Does the amera lens move farther away from or loser to the sensor? Explain.

3. Can a diverging lens form a real image under any ir um stan es? Explain.

9. A lens is made of a material w ith an index of refra tion n = 1.30. In air, it is a onverging lens. W ill it still be a onverging lens if pla ed in water? Explain, using a ray diagram.

10. Explain how you ould have a virtual obje t. 11. A dog w ith its ta il in the air stands fa ing a onverging lens. If the nose and the ta il are ea h fo used on a s reen in turn, whi h w ill have the greater magnifi ation?

4. Use ray diagrams to show that a real image formed by a

12. A at w ith its ta il in the air stands fa ing a onverging lens.

thin lens is always inverted, whereas a virtual image is always upright if the obje t is real.

Under what ir umstan es (if any) would the image o f the nose be virtual and the image of the ta il be real? Where would the image o f the rest of the at be?

5. Light rays are said to be " reversible." Is this onsistent w ith the thin lens equation? Explain. 6 . Can real images be proje ted on a s reen? Can virtual

images? Can either be photographed? Dis uss arefully.

13. Why, in Example 33 -6, must the

onverging lens have a shorter fo al length than the diverging lens if the latter's fo al length is to be determined by ombining them?

Does the real image formed hange (a) in position, ( oo so Ay be omes the infinitesim al width dy. In this ase, the diagrams of Fig. 35-8 be ome smooth urves, one of whi h is shown in Fig. 35-9. For any angle 0, the wave amplitude on the s reen is E e, equal to the hord in Fig. 35-9. The length of the ar is E 0, as before. If r is the radius of urvature of the ar , then

Ee . P = rsin — • 2

—

Using radian measure for j8/2, we also have

FIGURE 35 -9 Determining mplitude Ee s function of 6 for single-slit diffr ction.

Eo _ 2

P r2

We ombine these to obtain sin B

Ee =

(35-5)

The angle /3 is the phase differen e between the waves from the top and bottom edges of the slit. The path differen e for these two rays is D sin 0 (see Fig. 35 -7 as well as Eq. 35 -4), so

/3 = - p D sin .

(35 -6)

Intensity is proportional to the square of the wave amplitude, so the intensity Ie at any angle 0 is, from Eq. 35 -5, ( sin @/

2\2

* = \fijr )

(35_7)

where I (o Eq) is the intensity at 9 = 0 (the entral maximum). We an ombine Eqs. 35 -7 and 35 -6 (although it is often simpler to leave them as separate equations) to obtain sin

7rD sin 0 ttD sin 6

(35-8)

A A ording to Eq. 35-8, minima (le = 0) o ur where sin(7rDsin0/A) = 0, whi h means ttD sin 0/A must be 77, 2 ir, 377, and so on, or D sin0 = raA,

m = 1,2,3, •••

[minima]

whi h is what we have obtained previously, Eq. 35-2. Noti e that m annot be zero: when /3/2 = 7rZ)sin0/A = 0, the denominator as well as the numerator in Eqs. 35 -7 or 35 -8 vanishes. We an evaluate the intensity in this ase by taking the lim it as 0 — » 0 (or /3 — > 0); for very small angles, sin/3/2 « (3/2, so (sin/3/2)/(jS/2) —> 1 and Ie = 70, the maximum at the enter of the pattern. The intensity Ie as a fun tion of 0, as given by Eq. 35 -8, orresponds to the diagram of Fig. 35-4. ■ ESTIMATE I Intensity at secondary maxima. Estimate the intensities of the first two se ondary maxima to either side of the entral maximum. APPR ACH The se ondary maxima o ur lose to halfway between the minima, at about ttD sin 0 /3 - = -----------

. u « (m + i)TT.

m = 1,2,3,•••

The a tual maxima are not quite at these points— their positions an be determined by differentiating Eq. 35-7 (see Problem 14)— but we are only seeking an estimate. S LUTI N Using these values for f3 in Eq. 35-7 or 35-8, with sin(m + 5)77 = 1, gives Ie = ?(ra r W2) 77r — +

926

CHAPTER 35

Diffractio a d Polarizatio

m = 1' 2' 3> '"

For m = 1 and 2, we get

222

° - 0457 °

= 617 = ° 0167°-

[m = 2]

The first maximum to the side of the entral peak has only 1/22, or 4.5%, the intensity of the entral peak, and su eeding ones are smaller still, just as we an see in Fig. 35 -4 and the photo of Fig. 35-2 . D iffra tion by a ir ular opening produ es a similar pattern (though ir ular rather than re tangular) and is of great pra ti al importan e, sin e lenses are essentially ir ular apertures through whi h light passes. We w ill dis uss this in Se tion 35 -4 and see how diffra tion lim its the resolution (or sharpness) of images.

*35-3 Diffr ction in the Double-Slit Experiment When we analyzed Young's double-slit experiment in Se tion 34 -4, we assumed that the entral portion of the s reen was uniform ly illuminated. This is equivalent to assuming the slits are infinitesim ally narrow, so that the entral diffra tion peak is spread out over the whole s reen. This an never be the ase for real slits; diffra tion redu es the intensity of the bright interferen e fringes to the side of enter so they are not all of the same height as they were shown in Fig. 34-14. (They were shown more orre tly in Fig. 34-9b.) To al ulate the intensity in a double-slit interferen e pattern, in luding diffra tion, let us assume the slits have equal widths D and their enters are separated by a distan e d. Sin e the distan e to the s reen is large ompared to the slit separation d, the wave amplitude due to ea h slit is essentially the same at ea h point on the s reen. Then the total wave amplitude at any angle w ill no longer be

g Eeo = 2E0

c o s

—>

as was given by Eq. 34-5b. Rather, it must be modified, be ause of diffra tion, by Eq. 35 -5, so that

eo =

( sin/3/2^ 8 \ ~ p / ~ J os 2

Thus the intensity w ill be given by r ( sin /3 /2 \2/' = J o l - ^ - 1 I o s -]

" H m

(35 -9)

) V

where / 0 = 4£ q, and from Eqs. 35 -6 and 34 -4 we have (3

7T ^ .

— = — D sin0 2 A

and

77 , .

a sm . — = 2 A

*SECTION 3 5 - 3

Diffractio i the Double-Slit Experime t

927

10°

V |V

5° 0 5° ( ) Diffr ction f ctor, (sin2/3/2)/(/3/2)2vs. 6

10°

V |V

10°

V |V

5° 0 5° (b) Interference f ctor, cos2 ^ vs. 6

10°

FIGURE 35 - 10

( ) Diffr ction f ctor, (b) interference f ctor, nd (c) the result nt intensity Ie, plotted s function of 6 for d = 6D = 60A. (c) Intensity, I q v s .

Equation 35 -9 for the intensity in a double-slit pattern, as we just saw, is sin j8/2

0/2 Photogr phs of double-slit interference p ttern using l ser be m showing effects of diffr ction. In both c ses d = 0.50 mm, where s D = 0.040 mm in ( ) nd 0.080 mm in (b). FIGURE 35 -11

8 CO S —

(35 - 9)

The first term in parentheses is sometimes alled the " diffra tion fa tor" and the se ond one the " interferen e fa tor." These two fa tors are plotted in Fig. 35-10a and b for the ase when d = 6D , and D = 10A. (Figure 35-10b is essentially the same as Fig. 34-14.) Figure 35-10 shows the produ t of these two urves (times 70) whi h is the a tual intensity as a fun tion of 0 (or as a fun tion of position on the s reen for not too large) as given by Eq. 35-9. As indi ated by the dashed lines in Fig. 35-10 , the diffra tion fa tor a ts as a sort of envelope that lim its the interferen e peaks.

Diffraction plus interference. Show why the entral diffra tion peak in Fig. 35-10 ontains 11 interferen e fringes. APPR ACH The firs firstt minimum in the diffra tion pattern o urs where • „ sm e = -ASin e d = 6D , d sin 0 =

6D I

I = 6A.

S LUTI N From Eq. 34 -2a, interferen e peaks (maxima) o ur fo r d sin 0 = raA where ra an be 0,1,••• or any integer. Thus the diffra tion minimum (d sin = 6A) oin ides w ith tn = 6 in the interferen e pattern, so the m = 6 peak won't appear. Hen e the entral diffra tion peak en loses the entral interferen e peak (ra = 0) and five peaks ( ra = 1 to 5) on ea h side fo r a total o f 11. Sin e the sixth order doesn't appear, it is said to be a " missing order."

(b)

928

HAPTER 35

Noti e from Example 35 -4 that the number o f interferen e fringes in the entral diffra tion peak depends only on the ratio d /D . It does not depend on wavelength A. The a tual spa ing (in angle, or in position on the s reen) does depend on A. For the ase illustrated, D = 10A, and so the first diffra tion minimum o urs at sin0 = X /D = 0.10 or about 6°. The de rease in intensity o f the interferen e fringes away from the enter, as graphed in Fig. 35 -10, is shown in Fig. 35-11.

Int rf r nc vs. Diffraction The patterns due to interferen e and diffra tion arise from the same phenomenon— the superposition of oherent waves of different phase. The distin tion between them is thus not so mu h physi al as for onvenien e of des ription, as in this Se tion where we analyzed the two -slit pattern in terms of interferen e and diffra tion separately. In general, we use the word " diffra tion" when referring to an analysis by superposition of many infinitesimal and usually ontiguous sour es, su h as when we subdivide a sour e into infinitesimal parts. We use the term " interferen e" when we superpose the wave from a finite (and usually small) number of oherent sour es.

35-4 Limits o f Resolution; Circul r Apertures The ability of a lens to produ e distin t images of two point obje ts very lose together is alled the resolution of the lens. The loser the two images an be and still be seen as distin t (rather than overlapping blobs), the higher the resolution. The resolution of a amera lens, for example, is often spe ified as so many dots or lines per m illim eter, as mentioned in Se tion 33-5. Two prin ipal fa tors lim it the resolution of a lens. The first is lens aberrations. As we saw in Chapter 33, be ause of spheri al and other aberrations, a point obje t is not a point on the image but a tiny blob. Careful design of ompound lenses an redu e aberrations signifi antly, but they annot be eliminated entirely. The se ond fa tor that lim its resolution is diffraction , whi h annot be orre ted for be ause it is a natural result of the wave nature of light. We dis uss it now. In Se tion 35 -1, we saw that be ause light travels as a wave, light from a point sour e passing through a slit is spread out into a diffra tion pattern (Figs. 35 -2 and 35 -4). A lens, be ause it has edges, a ts like a round slit. When a lens forms the image of a point obje t, the image is a tually a tiny diffra tion pattern. Thus an image w ould be blurred even if aberrations w ere absent.

In the analysis that follows, we assume that the lens is free of aberrations, so we an on entrate on diffra tion effe ts and how mu h they lim it the resolution of a lens. In Fig. 35 -4 we saw that the diffra tion pattern produ ed by light passing through a re tangular slit has a entral maximum in whi h most of the light falls. This entral peak falls to a minimum on either side of its enter at an angle given by sin0 = X /D (this is Eq. 35 -1), where D is the slit width and A the wavelength of light used. 0 is the angular half-width of the entral maximum, and for small angles an be written

0 « sin

There are also low -intensity fringes beyond. For a lens, or any ir ular hole, the image of a point obje t w ill onsist of a circular entral peak ( alled the diffraction spot or A iry disk) surrounded by faint ir ular fringes, as shown in Fig. 35-12a. The entral maximum has an angular half width given by

I (b) FIGURE 35-12 Photogr phs of im ges (gre tly m gnified) formed by lens, showing the diffr ction p ttern of n im ge for: ( ) single point object; (b) two point objects whose im ges re b rely resolved.

= L22A D

where D is the diameter of the ir ular opening. This is a theoreti al result for a perfe t ir le or lens. For real lenses or ir les, the fa tor is on the order of 1 to 2. This formula differs from that for a slit (Eq. 35 -1) by the fa tor 1.22. This fa tor appears be ause the width of a ir ular hole is not uniform (like a re tangular slit) but varies from its diameter D to zero. A mathemati al analysis shows that the " average" width is D/1.22. Hen e we get the equation above rather than Eq. 35-1. The intensity of light in the diffra tion pattern from a point sour e of light passing through a ir ular opening is shown in Fig. 35-13. The image for a non -point sour e is a superposition of su h patterns. For most purposes we need onsider only the entral spot, sin e the on entri rings are so mu h dimmer. If two point obje ts are very lose, the diffra tion patterns of their images w ill overlap as shown in Fig. 3 5 - 12b. As the obje ts are moved loser, a separation is

FIGURE 35-13 Intensity of light cross the diffr ction p ttern of circul r hole. Intensity

SE TION 35 -4

929

rea hed where you an't tell if there are two overlapping images or a single image. The separation at whi h this happens may be judged differently by different observers. However, a generally a epted riterion is that proposed by Lord Rayleigh (1842-1919). This Rayleigh riterion states that tw o images are just resolvable w en t e center o f t e diffraction disk o f one image is directly over t e first minimum in t e diffraction pattern o f t e ot er. This is shown in Fig. 35-14. Sin e the first minimum is at an angle 0 = 1.22A/ D from the entral maximum, Fig. 35-14 shows that two obje ts an be onsidered ju st resolvable if they are

separated by at least an angle 0 given by Q =

criterion. Two im ges re just resolv ble when the center of the diffr ction pe k of one is directly over the first minimum in the diffr ction p ttern of the other. The two point objects O nd O' subtend n ngle 6 t the lens; only one r y (it p sses through the center of the lens) is dr wn for e ch object, to indic te the center of the diffr ction p ttern of its im ge. FIGURE 35 -15 Hubble Sp ce Telescope, with E rth in the b ckground. The fl t or nge p nels re sol r cells th t collect energy from the Sun.

1.22A

[0 in radians] (35 -10)

In this equation, D is the diameter of the lens, and applies also to a m irror diameter. This is the lim it on resolution set by the wave nature of light due to diffra tion. A smaller angle means better resolution: you an make out loser obje ts. We see from Eq. 35-10 that using a shorter wavelength A an redu e 0 and thus in rease resolution. EXERCISE B Green light (550 nm) p sses through 25-mm-di meter c mer lens. Wh t is the ngul r h lf-width of the resulting diffr ction p ttern? (a) 2.7 X IO5 - degrees, (b) 1.5 X 1(T3degrees, (c) 3.2°, (d) 27°, (e) 1.5 X 103degrees.

I u lJI Hubble Space Telescope. The Hubble Spa e Teles ope (HST) is a refle ting teles ope that was pla ed in orbit above the Earth's atmosphere, so its resolution would not be lim ited by turbulen e in the atmosphere (Fig. 35-15). Its obje tive diameter is 2.4 m. For visible light, say A = 550 nm, estimate the improvement in resolution the Hubble offers over Earth-bound teles opes, whi h are lim ited in resolution by movement of the Earth's atmosphere to about half an ar se ond. (Ea h degree is divided into 60 minutes ea h ontaining 60 se onds, so 1° = 3600 ar se onds.) APPR ACH Angular resolution for the Hubble is given (in radians) by Eq. 35-10. The resolution for Earth teles opes is given, and we first onvert it to radians so we an ompare. S LUTI N Earth-bound teles opes are lim ited to an angular resolution of

1 \ ° ( ir rad 3600/ V 360 ■ = 2-4 x 1 0 - ra d. The Hubble, on the other hand, is lim ited by diffra tion (Eq. 35-10) whi h for A = 550 nm is 1.22(550 X IO-9 m) 1.22A 2.8 X 10-7 rad, 0 = D 2.4 m thus giving almost ten times better resolution (2.4 X IO-6 rad/2.8 X 10" 7rad « 9x). EXAMPLE 35 - 6 ESTIMATE Eye resolution. You are in an airplane at an altitude of 10,000 m. If you look down at the ground, estimate the minimum separation s between obje ts that you ould distinguish. Could you ount ars in a parking lot? Consider only diffra tion, and assume your pupil is about 3.0 mm in diameter and A = 550 nm. 0

PHYSICS

APPLIED

How well t e eye can see

APPR ACH We use the Rayleigh riterion, Eq. 35-10, to estimate 0. The separation s of obje ts is s = id, where i = 104m and 0 is in radians. S LUTI N In Eq. 35-10, we set D = 3.0 mm for the opening of the eye: s = id = i

1.22A

(lO4m)(1.22)(550 X l(T 9m) 3.0 X 10-3m

= 2.2 m.

Yes, you ould just resolve a ar (roughly 2 m wide by 3 or 4 m long) and ount them.

930

CHAPTER 35

Diffractio a d Polarizatio

35— 5 Resolution o f Telescopes nd Microscopes; the A Limit You might think that a mi ros ope or teles ope ould be designed to produ e any desired magnifi ation, depending on the hoi e of fo al lengths and quality of the lenses. But this is not possible, be ause of diffra tion. An in rease in magnifi ation above a ertain point merely results in magnifi ation of the diffra tion patterns. This an be highly misleading sin e we might think we are seeing details of an obje t when we are really seeing details of the diffra tion pattern. To examine this problem, we apply the Rayleigh riterion: two obje ts (or two nearby points on one obje t) are just resolvable if they are separated by an angle (Fig. 35-14) given by Eq. 35-10:

1.22A e = ^ r This formula is valid for either a mi ros ope or a teles ope, where D is the diameter of the obje tive lens or mirror. For a teles ope, the resolution is spe ified by stating as given by this equation.1

FIGURE 35 -16 The 300-meter r diotelescope inArecibo, Puerto Rico, uses r dio w ves (Fig. 31-12) inste d of visible light.

Telescope resolution (radio wave vs. visible light). What is the theoreti al minimum angular separation of two stars that an just be resolved by (a) the 200-in h teles ope on Palomar Mountain (Fig. 33-38 ); and (b) the Are ibo radioteles ope (Fig. 35-16), whose diameter is 300 m and whose radius of urvature is also 300 m. Assume A = 550 nm for the visible -light teles ope in part (a), and A = 4 m (the shortest wavelength at whi h the radioteles ope has been operated) in part ( b ). APPR ACH We apply the Rayleigh riterion (Eq. 35-10) for ea h teles ope. S LUTI N (a) Sin e D = 200 in. = 5.1 m, we have from Eq. 35-10 that n

1.22A D

. _ 7 (1.22)(5.50 X 10-7m) = — — = 1.3 x 10 rad, (5.1 m)

= ——

or 0.75 X 10-5 deg. (Note that this is equivalent to resolving two points less than 1 m apart from a distan e of 100 km!) (b) For radio waves with A = 0.04 m emitted by stars, the resolution is

(1.22) (0.04 m) x ; = 1.6 X 10 rad. (300 m)

The resolution is less be ause the wavelength is so mu h larger, but the larger obje tive olle ts more radiation and thus dete ts fainter obje ts. N TE In both ases, we determined the lim it set by diffra tion. The resolution for a visible -light Earth-bound teles ope is not this good be ause of aberrations and, more importantly, turbulen e in the atmosphere. In fa t, large-diameter obje tives are not justified by in reased resolution, but by their greater light-gathering ability — they allow more light in, so fainter obje ts an be seen. Radioteles opes are not hindered by atmospheri turbulen e, and the resolution found in ( b ) is a good estimate.

fE rth-bound telescopes with l rge-di meter objectives re usu lly limited not by diffr ction but by other effects such s turbulence in the tmosphere. The resolution of high-qu lity microscope, on the other h nd, norm lly is limited by diffr ction; microscope objectives re complex compound lenses cont ining m ny elements of sm ll di meter (since / is sm ll), thus reducing berr tions.

SECTION 35 - 5

Resolutio of Telescopes a d Microscopes; the A Limit

931

For a mi ros ope, it is more onvenient to spe ify the a tual distan e, s, between two points that are just barely resolvable: see Fig. 35-14. Sin e obje ts are normally pla ed near the fo al point of the mi ros ope obje tive, the angle = s / f , so s = fd . If we ombine this with subtended by two obje ts is Eq. 35-10, we obtain the resolving power (RP) of a mi ros ope 1.22A/ RP = 5 fe = — (35- 11)

where / is the obje tive lens' fo al length (not frequen y). This distan e s is alled the resolving power of the lens be ause it is the minimum separation of two obje t points that an just be resolved— assuming the highest quality lens sin e this lim it is imposed by the wave nature of light. A smaller RP means better resolution, better detail.

EXERCISE C Wh t is the resolving power of microscope with 5-mm-di meter objective | which h s / = 9mm? (a) 550nm, (b ) 750nm, (c) 1200nm, (d) 0.05nm, (e) 0.005nm. D iffra tion sets an ultimate lim it on the detail that an be seen on any obje t. In Eq. 35-11 for resolving power of a mi ros ope, the fo al length of the lens annot pra ti ally be made less than (approximately) the radius of the lens, and even that is very diffi ult (see the lensmaker's equation, Eq. 33 -4). In this best ase, Eq. 35-11 gives, with / « D /2 .

(35-12) Thus we an say, to within a fa tor of 2 or so, that

it is not possible to resolve det il of objects sm ller th n the w velength of the r di tion being used.

A limits resolution

This is an important and useful rule of thumb. Compound lenses in mi ros opes are now designed so well that the a tual lim it on resolution is often set by diffra tion — that is, by the wavelength of the light used. To obtain greater detail, one must use radiation of shorter wavelength. The use of U V radiation an in rease the resolution by a fa tor of perhaps 2. Far more important, however, was the dis overy in the early twentieth entury that ele trons have wave properties (Chapter 37) and that their wavelengths an be very small. The wave nature of ele trons is utilized in the ele tron mi ros ope (Se tion 37-8), whi h an magnify 100 to 1000 times more than a visiblelight mi ros ope be ause of the mu h shorter wavelengths. X-rays, too, have very short wavelengths and are often used to study obje ts in great detail (Se tion 35-10). *

35—6 Resolution of the Hum n Eye nd Useful M gnific tion The resolution of the human eye is limited by several fa tors, all of roughly the same order of magnitude. The resolution is best at the fovea, where the - (= 3000nm ). The diameter of the pupil one spa ing is smallest, about 3 /um varies from about 0.1 m to about 0.8 m. So for A = 550 nm (where the eye's sensitivity is greatest), the diffra tion lim it is about « 1.22k/D ~ 8 X 10" 5rad to 6 X 10" 4rad. The eye is about 2 m long, giving a resolving power (Eq. 35-11) of i ~ ( 2 x 10" 2m)(8 X 10_5rad) « 2 /im at best, to about 10/xm at worst (pupil small). Spheri al and hromati aberration also lim it the resolution to about 10 /Am. The net result is that the eye an just resolve obje ts whose angular separation is around r -. best eye 5 X 10-4 rad. [resolution]

932

HAPTER 35

This orresponds to obje ts separated by 1 m at a distan e of about 20 m. The typi al near point of a human eye is about 25 m. A t this distan e, the eye an just resolve obje ts that are (25 m)(5 X 10_4rad) » 10_4m = ^m m apart. Sin e the best light mi ros opes an resolve obje ts no smaller than about 200 nm at best (Eq. 35-12 for violet light, A = 400 n m ), the useful magnifi ation

[= (resolution by naked eye)/(resolution by mi ros ope)] is lim ited to about 10"4m maximum useful 500 X mi ros ope m agnifi ation] 200 X 1(T9m In pra ti e, magnifi ations of about 1000 X are often used to minimize eyestrain. Any greater magnifi ation would simply make visible the diffra tion pattern produ ed by the mi ros ope obje tive lens. Now you have the answers to the Chapter-Opening Question: ( ), by the equation above, and ( ) by the A rule.

[

35—7 Diffr ction Gr ting A large number of equally spa ed parallel slits is alled a diffraction grating, although the term " interferen e grating" might be as appropriate. Gratings an be made by pre ision ma hining of very fine parallel lines on a glass plate. The untou hed spa es between the lines serve as the slits. Photographi transparen ies of an original grating serve as inexpensive gratings. Gratings ontaining 10,000 lines per entimeter are ommon, and are very useful for pre ise measurements of wavelengths. A diffra tion grating ontaining slits is alled a transmission grating. Another type of diffra tion grating is the reflection grating, made by ruling fine lines on a metalli or glass surfa e from whi h light is refle ted and analyzed. The analysis is basi ally the same as for a transmission grating, whi h we now dis uss. The analysis of a diffra tion grating is mu h like that of Young's double-slit experiment. We assume parallel rays of light are in ident on the grating as shown in Fig. 35-17. We also assume that the slits are narrow enough so that diffra tion by ea h of them spreads light over a very wide angle on a distant s reen beyond the grating, and interferen e an o ur with light from all the other slits. Light rays = 0°) interfere onstru tively to that pass through ea h slit without deviation produ e a bright line at the enter of the s reen. Constru tive interferen e also o urs at an angle su h that rays from adja ent slits travel an extra distan e of M = raA, where ra is an integer. If d is the distan e between slits, then we see from Fig. 35-17 that M = d sin0, and

diffra tion grating, 1 naxima J v [ prin ipal maxima 7 is the riterion to have a brightness maximum. This is the same equation as for the double-slit situation, and again ra is alled the order of the pattern. There is an im portant differen e between a double -slit and a m ultiple -slit pattern. The bright maxima are mu h s arper and narrow er for a grating. Why? Suppose that the angle is in reased just slightly beyond that required fo r a maximum. In the ase of only two slits, the two waves w ill be only slightly out of phase, so nearly fu ll onstru tive interferen e o urs. This means the maxima are wide (see Fig. 34 -9). For a grating, the waves from two adja ent slits w ill also not be signifi antly out of phase. But waves from one slit and those from a se ond one a few hundred slits away may be exa tly out of phase; all or nearly all the light an an el in pairs in this way. For example, suppose the angle 0 is very slightly different from its first-order maximum, so that the extra path length for a pair of adja ent slits is not exa tly A but rather 1.0010A. The wave through one slit and another one 500 slits below w ill have a path differen e of 1A + (500)(0.0010A) = 1.5000A, or 1 \ wavelengths, so the two w ill an el. A pair of slits, one below ea h of these, w ill also an el. That is, the light from slit 1 an els with that from slit 501; light from slit 2 an els with that from slit 502, and so on. Thus even for a tiny anglef orresponding to an extra path length of o tA , there is mu h destru tive interferen e, and so the maxima are very narrow. The more lines there are in a grating, the sharper w ill be the peaks (see Fig. 35 -18). Be ause a grating produ es mu h sharper lines than two slits alone an (and mu h brighter lines be ause there are many more slits), a grating is a far more pre ise devi e for measuring wavelengths. ra = 0,1,2,

depending on the tot l number of slits, there m y or m y not be complete c ncell tion for such n ngle, so there will be very tiny pe ks between the m in m xim (see Fig. 35-18b), but they re usu lly much too sm ll to be seen.

Diffraction grating is analyzed using interferenceformulas, not diffraction formulas FIGURE 35 -18 Intensity s function of viewing ngle 0 (or position on the screen) for ( ) two slits, (b) six slits. For diffr ction gr ting, the number of slits is very l rge («104) nd the pe ks re n rrower still.

AAA m= 1

ra = 0

ra = 1

(a)

.a a |a ^a a |a ^^a |a a ,_ r =1

r =0

r = 1

(b)

SE TION 35 -7

933

Suppose the light striking a diffra tion grating is not mono hromati , but onsists of two or more distin t wavelengths. Then for all orders other than m = 0, ea h wave length w ill produ e a maximum at a different angle (Fig. 35-19a), just as for a double slit. If white light strikes a grating, the entral (ra = 0) maximum w ill be a sharp white peak. But for all other orders, there w ill be a distin t spe trum of olors spread out over a ertain angular width, Fig. 35-19b. Be ause a diffra tion grating spreads out light into its omponent wavelengths, the resulting pattern is alled a spe trum.

m =2 FIGURE 35 -19 Spectr produced by gr ting: ( ) two w velengths, 400nm nd 700nm; (b) white light. The second order will norm lly be dimmer th n the first order. (Higher orders re not shown.) If gr ting sp cing is sm ll enough, the second nd higher orders will be missing.

m = 2m = l m= 1

700 nm

400 700 nm nm

400 nm

m =0

Both A ()

m= 1 m= 1m= 2

400 nm

m =2

700 400 nm nm

700 nm

m =2

m=1

m =0

m= 1

m =2

R inbow (f inter)

R inbow

White

R inbow

R inbow (f inter)

■

(b)

■

EXAMPLE 35 - 8 Diffraction grating: lines. Determine the angular positions of the first- and se ond-order maxima for light of wavelength 400 nm and 700 nm in ident on a grating ontaining 10,000 lines/ m. APPR ACH First we find the distan e d between grating lines: if the grating has N lines in lm , then the distan e between lines must be d = 1 /N meters. Then we use Eq. 35-13, sin 0 = m X /d, to find the angles for the two wavelengths for ra = 1 and 2 . S LUTI N The grating ontains 1.00 X 104 lines/ m = 1.00 X 106 lines/m, whi h means the distan e between lines is d = (l/l.O O X 106)m = 1.00 X 10" 6m = 1.00 fim. In first order (ra = 1), the angles are sin 04OO =

(1)(4.00 X 10-7 m)

____

= — — — ---- — 7------ = 0.400

1.00 X 10 m

(1)(7.00 X 10- 7 m) sin 07oo = ---- — r ----- - = 0.700 1.00 X 10 m so 04OO= 23.6°

and

07OO= 44.4°. In se ond order,

2A (2)(4.00 X 10~7 m) sin 0 4OO = — = — ------------ 7----- - = 0.800 400 d 1.00 X 10 m (2) (7.00 X 10-7 m) sin 07OO = — ------------ t----- - = 1.40 700 1.00 X 10 m so 04OO= 53.1°. But the se ond order does not exist for A = 700 nm be ause sin0 annot ex eed 1. No higher orders w ill appear. I frfJ iMJI * V g l Spectra overlap. White light ontaining wavelengths from 400 nm to 750 nm strikes a grating ontaining 4000 lines/ m. Show that the blue at A = 450 nm of the third -order spe trum overlaps the red at 700 nm of the se ond order. APPR ACH We use sin 0 = m X /d to al ulate the angular positions of the ra = 3 blue maximum and the ra = 2 red one.

934

CHAPTER 35

Diffractio a d Polarizatio

S LUTI N The grating spa ing is d = (1/4000) m = 2.50 X 10 6m. The blue of the third order o urs at an angle 0 given by mX

sin0 = —

(3)(4.50 X 10-7 m) ^ = 0.540. (2.50 X 10 m)

Red in se ond order o urs at sin0 =

(2)(7.00 X 10-7 m) ^ = 0.560, (2.50 X 10 m)

whi h is a greater angle; so the se ond order overlaps into the beginning of the third -order spe trum. C NCEPTUAL EXAMPLE 55 - 10 I Compact disk. When you look at the surfa e of a musi CD (Fig. 35-20), you see the olors of a rainbow, (a) Estimate the distan e between the urved lines (they are read by a laser). (b ) Estimate the distan e between lines, noting that a CD ontains at most 80 min of musi , that it rotates at speeds from 200 to 500 rev/m in, and that \ of its 6 m radius ontains the lines.

FIGURE 35-20 A ompa t disk, Example 35-10.

RESP NSE (a) The CD a ts like a refle tion diffra tion grating. To satisfy Eq. 35-13, we might estimate the line spa ing as one or a few (2 or 3) wavelengths (A « 550 nm) or 0.5 to 1.5 /xm. (b) Average rotation speed of 350 rev/m in times 80 min gives 28,000 total rotations or 28,000 lines, whi h are spread over (|)(6 m) = 4 m. So we have a sort of refle tion diffra tion grating with about (28,000 lines)/(4 m) = 7000 lines/ m. The distan e d between lines is roughly 1 m/7000 lines « 1.4 X 10_6m = 1.4 /xm. Our results in (a) and ( b ) agree.

35-8 The Spectrometer nd Spectroscopy A spectrometer or spectroscope, Fig. 35-21, is a devi e to measure wavelengths a urately using a diffra tion grating (or a prism) to separate different wavelengths of light. Light from a sour e passes through a narrow slit S in the " ollimator." The slit is at the fo al point of the lens L, so parallel light falls on the grating. The movable teles ope an bring the rays to a fo us. Nothing w ill be seen in the viewing teles ope unless it is positioned at an angle 0 that orresponds to a diffra tion peak (first order is usually used) of a wavelength emitted by the sour e. The angle 0 an be measured to very high a ura y, so the wavelength of a line an be determined to high a ura y using Eq. 35-13: • 0, „ A = —d sin

FIGURE 35 -21

Spe trometer or

spe tros ope,

where m is an integer representing the order, and d is the distan e between grating lines. The line you see in a spe trometer orresponding to ea h wavelength is a tu ally an image of the slit S. A narrower slit results in dimmer light but we an measure the angular positions more pre isely. If the light ontains a ontinuous range of wavelengths, then a ontinuous spe trum is seen in the spe tros ope. The spe trometer in Fig. 35-21 uses a transmission grating. Others may use a refle tion grating, or sometimes a prism. A prism works be ause of dispersion (Se tion 32 -6), bending light of different wavelengths into different angles. (A prism is not a linear devi e and must be alibrated.) An important use of a spe trometer is for the identifi ation of atoms or mole ules. When a gas is heated or an ele tri urrent is passed through it, the gas emits a hara teristi line spectrum. That is, only ertain dis rete wavelengths of light are emitted, and these are different for different elements and ompounds.* Figure 35-22 shows the line spe tra for a number of elements in the gas state. Line spe tra o ur only for gases at high temperatures and low pressure and density. The light from heated solids, su h as a lightbulb filament, and even from a dense gaseous obje t su h as the Sun, produ es a continuous spectrum in luding a wide range of wavelengths. tWhy toms nd molecules emit line spectr w s gre t mystery for m ny ye rs nd pl yed role in the development of modern qu ntum theory, s we sh ll see in Ch pter 37.

SECTION 3 5 - 8

centr l

The Spectrometer a d Spectroscopy

935

Atomic hydrogen

Mercury

Sodium

Sol r bsorption spectrum FIGURE 35 -22 Line spectr for the g ses indic ted, nd the spectrum from the Sun showing bsorption lines.

PHYSICS

APPLIED

C emical and bioc emical analysis by spectroscopy

Figure 35-22 also shows the Sun's " ontinuous spe trum," whi h ontains a number of dark lines (only the most prominent are shown), alled absorption lines. Atoms and mole ules an absorb light at the same wavelengths at whi h they emit light. The Sun's absorption lines are due to absorption by atoms and mole ules in the ooler outer atmosphere of the Sun, as well as by atoms and mole ules in the Earth's atmosphere. A areful analysis of all these thousands of lines reveals that at least two -thirds of all elements are present in the Sun's atmosphere. The presen e of elements in the atmosphere of other planets, in interstellar spa e, and in stars, is also determined by spe tros opy. Spe tros opy is useful for determining the presen e of ertain types of mole ules in laboratory spe imens where hemi al analysis would be diffi ult. For example, biologi al D N A and different types of protein absorb light in parti ular regions of the spe trum (su h as in the U V). The material to be examined, whi h is often in solution, is pla ed in a mono hromati light beam whose wavelength is sele ted by the pla ement angle of a diffra tion grating or prism. The amount of absorption, as ompared to a standard solution without the spe imen, an reveal not only the presen e of a parti ular type of mole ule, but also its on entration. Light emission and absorption also o ur outside the visible part of the spe trum, su h as in the U V and IR regions. Glass absorbs light in these regions, so refle tion gratings and mirrors (in pla e of lenses) are used. Spe ial types of film or sensors are used for dete tion. EXAMPLE 35-11 Hydrogen spectrum. Light emitted by hot hydrogen gas is observed with a spe tros ope using a diffra tion grating having 1.00 X 104lines/ m. The spe tral lines nearest to the enter (0°) are a violet line at 24.2°, a blue line at 25.7°, a blue-green line at 29.1°, and a red line at 41.0° from the enter. What are the wavelengths of these spe tral lines of hydrogen? APPR ACH The wavelengths an be determined from the angles by using A = (d /m ) sin 0 where d is the spa ing between slits, and m is the order of the spe trum (Eq. 35-13). S LUTI N Sin e these are the losest lines to 0 = 0°, this is the first-order spe trum (m = 1). The slit spa ing is d = l/(l.0 0 X 104 m x) =1.00 X 10- 6 m. The wavelength of the violet line is d \ . . ( 1.00 X 1(T6m sin0 = m

A= I—

sin 24.2° = 4.10 X 10" 7m = 410 nm.

The other wavelengths are: blue: A = (l.OO X 10-6 m) sin 25.7° = 434 nm,

936

HAPTER 35

blue-green:

A = (l.OO X 10_6m) sin 29.1° = 486 nm,

red:

A = (l.OO X 10-6m) sin 41.0° = 656 nm.

N TE In an unknown mixture of gases, these four spe tral lines need to be seen to identify that the mixture ontains hydrogen.

35—9 Pe k Widths nd Resolving Power for

Diffr ction Gr ting

We now look at the pattern of maxima produ ed by a m ultiple -slit grating using phasor diagrams. We an determine a formula for the width of ea h peak, and we w ill see why there are tiny maxima between the prin ipal maxima, as indi ated in Fig. 35-18b. First of all, it should be noted that the two -slit and six-slit patterns shown in Fig. 35-18 were drawn assuming very narrow slits so that diffra tion does not lim it the height of the peaks. For real diffra tion gratings, this is not normally the ase: the slit width D is often not mu h smaller than the slit separation d , and diffra tion thus lim its the intensity of the peaks so the entral peak (m = 0) is brighter than the side peaks. We won't worry about this effe t on intensity ex ept to note that if a diffra tion minimum oin ides with a parti ular order of the inter feren e pattern, that order w ill not appear. (For example, if d = 2D , all the even orders, m = 2, 4, •••, w ill be missing. Can you see why? H int: See Example 35 -4.) Figures 35-23 and 35-24 show phasor diagrams for a two-slit and a six-slit grating, respe tively. Ea h short arrow represents the amplitude of a wave from a single slit, and their ve tor sum (as phasors) represents the total amplitude for a given viewing angle 0. Part (a) of ea h Figure shows the phasor diagram at 0 = 0°, at the enter of the pattern, whi h is the entral maximum (ra = 0). Part (b) of ea h Figure shows the ondition for the adja ent minimum: where the arrows first lose on themselves (add to zero) so the amplitude Ed is zero. For two slits, this o urs when the two separate ampli tudes are 180° out of phase. For six slits, it o urs when ea h amplitude makes a 60° angle with its neighbor. For two slits, the minimum o urs when the phase between slits is = t t (in radians); for six slits it o urs when the phase is 2tt/6 ; and in the general ase of N slits, the minimum o urs for a phase differen e between adja ent slits of 8 = ~ (35-14)

8 2 /2

What does this orrespond to in 0? First note that

d sin

8is related to 0 by (Eq. 34-4)

cs 27T = ——d sin 0. or (35-15) A A Let us all A0Othe angular position of the minimum next to the peak at 0 = 0. Then d sin A0r _8_

2 tt

2tt

We insert Eq. 35-14 for sinA0n =

E =0 180° (b) Minimum: 8= 180° Ph sor di gr mfor two slits ( ) t the centr l m ximum, (b) t the ne rest minimum. FIGURE 35 -23

FIGURE 35 -24 Ph sor di gr mfor six slits ( ) t the centr l m ximum, (b) t the ne rest minimum.

( ) Centr l m ximum: 6=0, 8=0

8and find A

(35-16a)

Nd Sin e A0Ois usually small (N is usually very large for a grating), sin A0O » A0O, so

in the small angle lim it we an write (35-16b) Ae° = w It is lear from either of the last two relations that the larger N is, the narrower w ill be the entral peak. (For N = 2, sin A0O= A/2 d, whi h is what we obtained earlier for the double slit, Eq. 34-2b with ra = 0 .) Either of Eqs. 35-16 shows why the peaks be ome narrower for larger N. The origin of the small se ondary maxima between the prin ipal peaks (see Fig. 35-18b) an be dedu ed from the diagram of Fig. 35-25. This is just a ontinuation of Fig. 35-24b (where = 60°); but now the phase has been in reased to almost 90°, where Ee is a relative maximum. Note that Ee is mu h less than E (Fig. 35-24a), so the intensity in this se ondary maximum is mu h smaller than in a prin ipal peak. As (and 0) is in reased further, E e again de reases to zero (a " double ir le" ), then rea hes another tiny maximum, and so on. Eventually the diagram unfolds again and = 360°, all the amplitudes again lie in a straight line (as in Fig. 35-24a) when orresponding to the next prin ipal maximum (ra = 1 in Eq. 35-13).

( ) Centr l m ximum: 6= 0, 8= 0

£ = 60° (b) Minimum: 8 = 60°, Ee = 0

FIGURE 35 -25 Ph sor di gr mfor the second ry pe k.

♦SE TION 35 -9

937

Equation 35-16 b gives the half-width of the entral (ra = 0) peak. To determine the half-width of higher order peaks, A m for order ra, we differentiate Eq. 35-15 so as to relate the hange A in S, to the hange AO in the angle : dd . n ird .. A « — AO = ----- os 0 AO.

If A 0m represents the half-width of a peak of order ra (ra = 1,2, •••)— that is, the angle between the peak maximum and the minimum to either side— then A = 2 tt /N as given by Eq. 35-14. We insert this into the above relation and find

A0„, =

. . . A

0- '

(35-17)

N d os m

where 0m is the angular position of the rath peak as given by Eq. 35-13. This derivation is valid, of ourse, only for small A ( = 2 t t/ N ) whi h is indeed the ase for real gratings sin e N is on the order of 104or more. An important property of any diffra tion grating used in a spe trometer is its ability to resolve two very losely spa ed wavelengths (wavelength differen e = AA). The resolving power R of a grating is defined as

R = i r

W ith a little work, using Eq. 35-17, we an show that AA = X /N m where N is the total number of grating lines and ra is the order. Then we have

R = XT = N m .

(35-19)

AA The larger the value of R, the loser two wavelengths an be resolvable. If R is given, the minimum separation AA between two wavelengths near A, is (by Eq. 35-18) aa = A . EXAMPLE 35 -12 Resolving two close lines. Yellow sodium light, whi h onsists of two wavelengths, Ax = 589.00 nm and X = 589.59 nm, falls on a 7500-line/ m diffra tion grating. Determine (a) the maximum order ra that w ill be present for sodium light, (b ) the width of grating ne essary to resolve the two sodium lines.

FIGURE 35 -26 X-r y tube. Electrons emitted by he ted fil ment in v cuum tube re cceler ted by high volt ge. When they strike the surf ce of the node, the "t rget,"X-r ys re emitted.

APPR ACH We first find d = 1 m/7500 = 1.33 X 10" 6m, and then use Eq. 35-13 to find ra. For (b) we use Eqs. 35-18 and 35-19. S LUTI N (a) The maximum value of ra at A = 589 nm, using Eq. 35-13 with sin0 < 1, is d 1.33 x 1fT ° m d . ra = 2.26, ~ x S1 ~ x = 5.89 X 10_/ m so ra = 2 is the maximum order present (b) The resolving power needed is A _ 589 nm = 1000. R = 0.59 nm AA From Eq. 35-19, the total number N of lines needed for the ra = 2 order is N = R /ra = 1000/2 = 500, so the grating need only be 500/7500 m-1 = 0.0667 m wide. A typi al grating is a few entimeters wide, and so w ill easily resolve the two lines.

35-10 X-R ys nd X-R y Diffr ction In 1895, W. C. Roentgen (1845-1923) dis overed that when ele trons were a elerated by a high voltage in a va uum tube and allowed to strike a glass or metal surfa e inside the tube, fluores ent minerals some distan e away would glow, and photographi film would be ome exposed. Roentgen attributed these effe ts to a new type of radiation (different from athode rays). They were given the name X -rays after the algebrai symbol jc, meaning an unknown quantity. He soon found that X -rays penetrated through some materials better than through others, and w ithin a few weeks he presented the first X -ray photograph (of his w ife's hand). The produ tion of X -rays today is usually done in a tube (Fig. 35 -26) similar to Roentgen's, using voltages of typi ally 30 kV to 150 kV.

Investigations into the nature of X-rays indi ated they were not harged parti les (su h as ele trons) sin e they ould not be defle ted by ele tri or magneti fields. It was suggested that they might be a form of invisible light. However, they showed no diffra tion or interferen e effe ts using ordinary gratings. Indeed, if their wavelengths were mu h smaller than the typi al grating spa ing of 10-6 m (= 103nm), no effe ts would be expe ted. Around 1912, Max von Laue (1879-1960) suggested that if the atoms in a rystal were arranged in a regular array (see Fig. 17-2a), su h a rystal might serve as a diffra tion grating for very short wavelengths on the order of the spa ing between atoms, estimated to be about IO-10 m (= 10-1 nm). Experiments soon showed that X-rays s attered from a rystal did indeed show the peaks and valleys of a diffra tion pattern (Fig. 35-27). Thus it was shown, in a single blow, that X-rays have a wave nature and that atoms are arranged in a regular way in rystals. Today, X-rays are re ognized as ele tromagneti radiation with wavelengths in the range of about 10-2 nm to 10 nm, the range readily produ ed in an X -ray tube. We saw in Se tion 35-5 that light of shorter wavelength provides greater resolu tion when we are examining an obje t mi ros opi ally. Sin e X-rays have mu h shorter wavelengths than visible light, they should in prin iple offer mu h greater resolution. However, there seems to be no effe tive material to use as lenses for the very short wavelengths of X-rays. Instead, the lever but ompli ated te hnique of X - ray diffraction (or crystallography) has proved very effe tive for examining the mi ros opi world of atoms and mole ules. In a simple rystal su h as NaCl, the atoms are arranged in an orderly ubi al fashion, Fig. 35-28, with atoms spa ed a distan e d apart. Suppose that a beam of X-rays is in ident on the rystal at an angle cf) to the surfa e, and that the two rays shown are refle ted from two subsequent planes of atoms as shown. The two rays w ill onstru tively interfere if the extra distan e ray I travels is a whole number of wavelengths farther than the distan e ray II travels. This extra distan e is 2d sin . Therefore, onstru tive interferen e w ill o ur when mX = 2d sin cf),

m = 1,2, 3, •■•,

FIGURE 35 -27 This X-r y diffr ction p ttern is one of the first observed by M x von L ue in 1912 when he imed be m of X-r ys t zinc sulfide cryst l. The diffr ction p ttern w s detected directly on photogr phic pl te. FIGURE 35 -28

X-r y diffr ction

by cryst l.

(35-20)

where m an be any integer. (Noti e that to the direction in which / max is obtained, has intensity 1 + p cos 2(f> i + P /max where p = P / 100 is the "fractional polarization."

| Gener l Problems__________ 63. When violet light of wavelength 415 nm falls on a single slit, it creates a central diffraction peak that is 8.20 cm wide on a screen that is 2.85 m away. How wide is the slit? 64. A series of polarizers are each placed at a 10° interval from the previous polarizer. Unpolarized light is incident on this series of polarizers. How many polarizers does the light have to go through before it is \ of its original intensity? 65. The wings of a certain beetle have a series of parallel lines across them. When normally incident 480-nm light is reflected from the wing, the wing appears bright when viewed at an angle of 56°. How far apart are the lines? 66. A teacher stands well back from an outside doorway 0.88 m wide, and blows a whistle of frequency 850 Hz. Ignoring reflections, estimate at what angle(s) it is not possible to hear the whistle clearly on the playground outside the doorway. Assume 340 m /s for the speed of sound. 67. Light is incident on a diffraction grating with 7600 lines/cm and the pattern is viewed on a screen located 2.5 m from the grating. The incident light beam consists of two wavelengths, Ai = 4.4 X 10_7m and A2 = 6.8 X 10_7m. alculate the linear distance between the first-order bright fringes of these two wavelengths on the screen. 68. How many lines per centimeter must a grating have if there is to be no second -order spectrum for any visible wavelength?

69. When yellow sodium light, A = 589 nm, falls on a diffrac tion grating, its first-order peak on a screen 66.0 cm away falls 3.32 cm from the central peak. Another source produces a line 3.71 cm from the central peak. What is its wavelength? How many lines/cm are on the grating? 70. Two of the lines of the atomic hydrogen spectrum have wave lengths of 656 nm and 410 nm. If these fall at normal incidence on a grating with 8100 lines/cm, what will be the angular separation of the two wavelengths in the first-order spectrum? 71. (a) How far away can a human eye distinguish two car headlights 2.0 m apart? onsider only diffraction effects and assume an eye diameter of 6.0 mm and a wavelength of 560 nm. (b) What is the minimum angular separation an eye could resolve when viewing two stars, considering only diffraction effects? In reality, it is about V of arc. Why is it not equal to your answer in (b)l 72. A laser beam passes through a slit of width 1.0 cm and is pointed at the Moon, which is approximately 380,000 km from the Earth. Assume the laser emits waves of wave length 633 nm (the red light of a H e - N e laser). Estimate the width of the beam when it reaches the Moon. 73. A H e - N e gas laser which produces monochromatic light of wavelength A = 6.328 X 10_7m is used to calibrate a reflection grating in a spectroscope. The first-order diffraction line is found at an angle of 21.5° to the incident beam. How many lines per meter are there on the grating?

General Problems

949

74. The entrance to a boy's bedroom consists of two door ways, each 1.0 m wide, which are separated by a distance of 3.0 m. The boy's m other yells at him through the two doors as shown in Fig. 35 - 48, telling him to clean up his room. H er voice has a frequency of 400 Hz. Later, when the mother discovers the room is still a mess, the boy says he never heard her telling him to clean his room. The velocity of sound is 340 m/s. (a) Find all of the angles 0 (Fig. 35-48) at which no sound will be heard within the bedroom when the mother yells. Assume sound is completely absorbed Boy ■ when it strikes a bedroom wall. (b) If the boy was at the / / position shown when his / mother yelled, does he have a / / good explanation for not / 8.0 m having heard her? Explain. / / / / Ti.o,m

/ = 400 Hz 3.0 m

|W Tl.Om

FIGURE 35 - 48

Problem 74.

I Mom

-5.0 m-

75. A t what angle above the horizon is the Sun when light reflecting off a smooth lake is polarized most strongly? 76. Unpolarized light falls on two polarizer sheets whose axes are at right angles, (a) What fraction of the incident light intensity is transmitted? (b) What fraction is transmitted if a third polarizer is placed between the first two so that its axis makes a 66° angle with the axis of the first polarizer? (c) What if the third polarizer is in front of the other two? 77 A t what angle should the axes of two Polaroids be placed so as to reduce the intensity of the incident unpolarized light by an additional factor (after the first Polaroid cuts it in half) of (a) 4, (b) 10, (c) 100? 78. Four polarizers are placed in succession with their axes vertical, at 30.0° to the vertical, at 60.0° to the vertical, and at 90.0° to the vertical, (a) alculate what fraction of the incident unpolarized light is transmitted by the four polarizers, (b) an the transmitted light be decreased by removing one of the polarizers? If so, which one? (c) an the transmitted light intensity be extinguished by removing polarizers? If so, which one(s)? 79. Spy planes fly at extremely high altitudes (25 km) to avoid interception. Their cameras are reportedly able to discern features as small as 5 cm. What must be the minimum aperture of the camera lens to afford this resolution? (Use A = 580 nm.) 80. Two polarizers are oriented at 48° to each other and plane polarized light is incident on them. If only 25% of the light gets through both of them, what was the initial polarization direction of the incident light?__________________________

81. X-rays of wavelength 0.0973 nm are directed at an unknown crystal. The second diffraction maximum is recorded when the X-rays are directed at an angle of 23.4° relative to the crystal surface. What is the spacing between crystal planes? 82. X -rays of wavelength 0.10 nm fall on a microcrystalline powder sample. The sample is located 12 cm from the photographic film. The crystal structure of the sample has an atomic spacing of 0.22 nm. alculate the radii of the diffraction rings corresponding to first- and second -order scattering. Note in Fig. 35-28 that the X -ray beam is deflected through an angle 2. 83. The Hubble Space Telescope with an objective diameter of 2.4 m, is viewing the Moon. Estimate the minimum distance between two objects on the Moon that the Hubble can distinguish. onsider diffraction of light with wavelength 550 nm. Assume the Hubble is near the Earth. 84. The Earth and Moon are separated by about 400 X 106m. When Mars is 8 X 1010m from Earth, could a person standing on Mars resolve the Earth and its Moon as two separate objects without a telescope? Assume a pupil diam eter of 5 mm and A = 550 nm. 85. A slit of width D = 22 fxm is cut through a thin aluminum plate. Light with wavelength A = 650 nm passes through this slit and forms a single-slit diffraction pattern on a screen a distance £ = 2.0 m away. Defining x to be the distance between the first minima (ra = +1 and ra = - 1 ) in this diffraction pattern, find the change Ax in this distance when the temperature T of the metal plate is changed by amount AT = 55 °. [Hint: Since A « D, the first minima occur at a small angle.]

*Num rical/Comput r *86. (II) A student shined a laser light onto a single slit of width 0.04000 mm. He placed a screen at a distance of 1.490 m from the slit to observe the diffraction pattern of the laser light. The accompanying Table shows the distances of the dark fringes from the center of the central bright fringe for different orders. Order number,ra: 1 2 3 4 5 6 7 8 Distance (m) 0.0225 0.0445 0.0655 0.0870 0.1105 0.1320 0.1540 0.1775 Determine the angle of diffraction, 0, and sin 0 for each order. Make a graph of sin 0 vs. order number, ra, and find the wave length, A, of the laser from the best-fit straight line. *87. (Ill) Describe how to rotate the plane of polarization of a plane-polarized beam of light by 90° and produce only a 10% loss in intensity, using polarizers. Let N be the number of polar izers and 0 be the (same) angle between successive polarizers. 88. (Ill) The "full-width at half-maximum" (FWHM) of the central peak for single-slit diffraction is defined as the angle A0 between the two points on either side of center where the intensity is \ I 0. (a) Determine A0 in terms of (A/D ). Use graphs or a spreadsheet to solve sin a = a / v 2 . (b) Determine A0 (in degrees) for D = A and for D = 100A.

Answers to Exercises

D: 0.28 nm.

A: Narrower. B: (ib).

E: Zero for both (a) and (b), because the two successive polarizers at 90° cancel all light. The 45° Polaroid must be inserted between the other two if any transmission is to occur.

: (c).

950

CHAPTER 35

Diffraction and Polarization

A n e rly science f nt sy book (1940), c lled M r Tompkins in W onderland by physicist G eorge G mow, im gined world in which the speed of light w s only 10 m /s (20 m i/h ). Mr Tompkins h d studied rel tivity nd when he beg n "speeding " on bicycle, he "expected th t he would be im m edi tely shortened, nd w s very h ppy bout it s his incre sing figure h d l tely c used him som e nxiety. To his gre t surprise, however, nothing h ppened to him or to his cycle. On the other h nd, the picture round him com pletely ch nged. The streets grew shorter, the windows o f the shops beg n to look like n rrow slits, nd the policem n on the corner bec m e the thinnest m n he h d ever seen. 'By Jove! ' excl im ed Mr Tompkins excitedly, 'I see the trick now. This is where the word relativity com es in.' " Rel tivity does indeed predict th t objects m oving rel tive to us t high speed, close to the speed of light c, re shortened in length. We d on 't notice it s Mr Tompkins did, bec use c = 3 X 108 m /s is incredibly f st. We will study length contr ction, time dil tion, simult neity non greem ent, nd how energy nd m ss re equiv lent

(£ = me2)

T he Speci l T h eory o f R e l tiv ity HAPTER -OPENING QUESTION — Guess now! A rocket is headed away from Earth at a speed of 0.80c. The rocket fires a missile at a speed of 0.70c (the missile is aimed away from Earth and leaves the rocket at 0.70c relative to the rocket). How fast is the missile moving relative to Earth? (a) 1.50c; (b) a little less than 1.50c; (c) a little over c; (d) a little under c; (e) 0.75c. hysics at the end of the nineteenth century looked back on a period of great progress. The theories developed over the preceding three centuries had been very successful in explaining a wide range of natural phenomena. Newtonian mechanics beautifully explained the motion of objects on Earth and in the heavens. Furthermore, it formed the basis for successful treatments of fluids, wave motion, and sound. Kinetic theory explained the behavior of gases and other materials. Maxwell's theory of electromagnetism not only brought together and explained electric and magnetic phenomena, but it predicted the existence of electromagnetic waves that would behave in every way just like light—so light came to be thought of as an electromagnetic wave. Indeed, it seemed that the natural world, as seen through the eyes of physicists, was very well explained. A few puzzles remained, but it was felt that these would soon be explained using already known principles.

ONTENTS 36 - 1 *36 - 2 36 - 3 36 - 4 36 - 5 36 - 6 36 - 7 36 - 8 36 - 9 36 - 10 36-11 36 - 12 36 - 13

G lile n-Newtoni n Rel tivity The Michelson-Morley Experiment Postul tes of the Speci l Theory of Rel tivity Simult neity Time Dil tion nd the Twin P r dox Length Contr ction Four-Dimension l Sp ce-Time G lile n nd Lorentz Tr nsform tions Rel tivistic Momentum The Ultim te Speed E = m e2; M ss nd Energy Doppler Shift for Light The Imp ct of Speci l Rel tivity

951

It did not turn out so simply. Instead, these puzzles were to be solved only by the introduction, in the early part of the twentieth century, of two revolutionary new theories that changed our whole conception of nature: the t eory o f relativity and quantum t eory. Physics as it was known at the end of the nineteenth century (what we've covered up to now in this book) is referred to as cl ssic l physics. The new physics that grew out of the great revolution at the turn of the twentieth century is now called modern physics. In this hapter, we present the special theory of relativity, which was first proposed by Albert Einstein (1879-1955; Fig. 36 -1) in 1905. In hapter 37, we introduce the equally momentous quantum theory.

3 6 -1 G lile n-Newtoni n Rel tivity FIGURE 36 - 1 A lbert Einstein (1879 - 1955), one o f the gre t minds o f the twentieth century, w s the cre tor o f the speci l nd gener l theories of rel tivity.

FIGURE 3 6 - 2 A coin is dropped by person in m oving c r. The upper views show the m om ent of the coin 's rele se, the lower views re short time l ter, ( ) In the reference fr me of the c r, the coin f lls str ight down ( nd the tree m oves to the left), (b) In reference fr me fixed on the E rth, the coin h s n initi l velocity ( = to c r 's) nd follow s curved (p r bolic) p th.

Einstein's special theory of relativity deals with how we observe events, particularly how objects and events are observed from different frames of reference. This subject had, of course, already been explored by Galileo and Newton. The special theory of relativity deals with events that are observed and measured from so-called inerti l reference fr mes (Sections 4 - 2 and 11-8), which are reference frames in which Newton's first law is valid: if an object experiences no net force, the object either remains at rest or continues in motion with constant speed in a straight line. It is usually easiest to analyze events when they are observed and measured by observers at rest in an inertial frame. The Earth, though not quite an inertial frame (it rotates), is close enough that for most purposes we can consider it an inertial frame. Rotating or otherwise accelerating frames of reference are noninertial frames,f and won't concern us in this hapter (they are dealt with in Einstein's general theory of relativity). A reference frame that moves with constant velocity with respect to an inertial frame is itself also an inertial frame, since Newton's laws hold in it as well. When we say that we observe or make measurements from a certain reference frame, it means that we are at rest in that reference frame. Both Galileo and Newton were aware of what we now call the rel tivity principle applied to mechanics: that t e basic laws o f p ysics are t e same in all inertial reference frames. You may have recognized its validity in everyday life. For example, objects move in the same way in a smoothly moving (constant-velocity) train or airplane as they do on Earth. (This assumes no vibrations or rocking which would make the reference frame noninertial.) When you walk, drink a cup of soup, play pool, or drop a pencil on the floor while traveling in a train, airplane, or ship moving at constant velocity, the objects move just as they do when you are at rest on Earth. Suppose you are in a car traveling rapidly at constant velocity. If you drop a coin from above your head inside the car, how will it fall? It falls straight downward with respect to the car, and hits the floor directly below the point of release, Fig. 36-2a. tOn rot ting pl tform (s y merry-go-round), for ex mple, n object t rest st rts moving outw rd even though no object exerts force on it. This is therefore not n inerti l fr me. See Section 11-8.

(a)

Reference frame = car

952

CHAPTER 36

The Special Theory of Relativity

(b)

Reference frame = Barth

This is just how objects fall on the Earth —straight down—and thus our experiment in the moving car is in accord with the relativity principle. (If you drop the coin out the car's window, this won't happen because the moving air drags the coin backward relative to the car.) Note in this example, however, that to an observer on the Earth, the coin follows a curved path, Fig. 36-2b. The actual path followed by the coin is different as viewed from different frames of reference. This does not violate the relativity principle because this principle states that the laws of physics are the same in all inertial frames. The same law of gravity, and the same laws of motion, apply in both reference frames. The acceleration of the coin is the same in both reference frames. The difference in Figs. 36 -2a and b is that in the Earth's frame of reference, the coin has an initial velocity (equal to that of the car). The laws of physics therefore predict it will follow a parabolic path like any projectile ( hapter 3). In the car's reference frame, there is no initial velocity, and the laws of physics predict that the coin will fall straight down. The laws are the same in both reference frames, although the specific paths are different. Galilean -Newtonian relativity involves certain unprovable assumptions that make sense from everyday experience. It is assumed that the lengths of objects are the same in one reference frame as in another, and that time passes at the same rate in different reference frames. In classical mechanics, then, space and time intervals are considered to be bsolute: their measurement does not change from one reference frame to another. The mass of an object, as well as all forces, are assumed to be unchanged by a change in inertial reference frame. The position of an object, however, is different when specified in different reference frames, and so is velocity. For example, a person may walk inside a bus toward the front with a speed of 2 m/s. But if the bus moves 10 m/s with respect to the Earth, the person is then moving with a speed of 12 m /s with respect to the Earth. The acceleration of an object, however, is the same in any inertial reference frame according to classical mechanics. This is because the change in velocity, and the time interval, will be the same. For example, the person in the bus may accelerate from 0 to 2 m/s in 1.0 seconds, so a = 2 m /s2 in the reference frame of the bus. With respect to the Earth, the acceleration is (12 m/s - 10 m /s)/(1.0 s) = 2 m/s2, which is the same. Since neither F, m, nor a changes from one inertial frame to another, then Newton's second law, F = ma, does not change. Thus Newton's second law satisfies the relativity principle. It is easily shown that the other laws of mechanics also satisfy the relativity principle. That the laws of mechanics are the same in all inertial reference frames implies that no one inertial frame is special in any sense. We express this important conclusion by saying that ll inerti l reference fr mes re equiv lent for the description of mechanical phenomena. No one inertial reference frame is any better than another. A reference frame fixed to a car or an aircraft traveling at constant velocity is as good as one fixed on the Earth. When you travel smoothly at constant velocity in a car or airplane, it is just as valid to say you are at rest and the Earth is moving as it is to say the reverse.* There is no experiment you can do to tell which frame is "really" at rest and which is moving. Thus, there is no way to single out one particular reference frame as being at absolute rest. A complication arose, however, in the last half of the nineteenth century. Maxwell's comprehensive and successful theory of electromagnetism ( hapter 31) predicted that light is an electromagnetic wave. Maxwell's equations gave the velocity of light c as 3.00 X 108m/s; and this is just what is measured. The question then arose: in what reference frame does light have precisely the value predicted by Maxwell's theory? It was assumed that light would have a different speed in different frames of reference. For example, if observers were traveling on a rocket ship at a speed of 1.0 X 108m /s away from a source of light, we might expect them to measure the speed of the light reaching them to be (3.0 X 108m/s) - (1.0 X 108m/s) = 2.0 X 108m/s. But Maxwell's equations have no provision for relative velocity. They predicted the speed of light to be c = 3.0 X 108m/s, which seemed to imply that there must be some preferred reference frame where c would have this value. fWe re ignoring the rot tion nd curv ture of the E rth.

AUT I ON ___________

L a w s are t e same, but pat s m ay be different in different reference fram es

/j\

AUT I ON _____________

Position and velocity are different in different reference frames, but lengt is t e sam e (classical)

SECTI N 36-1

953

We discussed in hapters 15 and 16 that waves can travel on water and along ropes or strings, and sound waves travel in air and other materials. Nineteenth -century physicists viewed the material world in terms of the laws of mechanics, so it was natural for them to assume that light too must travel in some medium. They called this transparent medium the ether and assumed it permeated all spaced It was therefore assumed that the velocity of light given by Maxwell's equations must be with respect to the ether. At first it appeared that Maxwell's equations did not satisfy the relativity principle. They were simplest in the frame where c = 3.00 X 108m/s; that is, in a reference frame at rest in the ether. In any other reference frame, extra terms would have to be added to take into account the relative velocity. Thus, although most of the laws of physics obeyed the relativity principle, the laws of electricity and magnetism apparently did not. Einstein's second postulate (Section 36 -3) resolved this problem: Maxwell's equations do satisfy relativity. Scientists soon set out to determine the speed of the Earth relative to this absolute frame, whatever it might be. A number of clever experiments were designed. The most direct were performed by A. A. Michelson and E. W. Morley in the 1880s. They measured the difference in the speed of light in different directions using Michelson's interferometer (Section 34-6). They expected to find a difference depending on the orientation of their apparatus with respect to the ether. For just as a boat has different speeds relative to the land when it moves upstream, downstream, or across the stream, so too light would be expected to have different speeds depending on the velocity of the ether past the Earth. Strange as it may seem, they detected no difference at all. This was a great puzzle. A number of explanations were put forth over a period of years, but they led to contradictions or were otherwise not generally accepted. This null result was one of the great puzzles at the end of the nineteenth century. Then in 1905, Albert Einstein proposed a radical new theory that reconciled these many problems in a simple way. But at the same time, as we shall see, it completely changed our ideas of space and time.

*3 6 —2 Th Mich lson-Morl y Exp rim nt The Michelson-Morley experiment was designed to measure the speed of the et er—the medium in which light was assumed to travel—with respect to the Earth. The experimenters thus hoped to find an absolute reference frame, one that could be considered to be at rest. One of the possibilities nineteenth-century scientists considered was that the ether is fixed relative to the Sun, for even Newton had taken the Sun as the center of the universe. If this were the case (there was no guarantee, of course), the Earth's speed of about 3 X 104m/s in its orbit around the Sun could produce a change of 1 part in 104 in the speed of light (3.0 X 108m/s). Direct measurement of the speed of light to this precision was not possible. But A. A. Michelson, later with the help of E. W. Morley, was able to use his interferometer (Section 34 -6) to measure the difference in the speed of light in different directions to this precision. This famous experiment is based on the principle shown in Fig. 36-3. Part (a) is a diagram of the Michelson interferometer, and it is assumed that the "ether wind" is moving with speed v to the right. (Alternatively, the Earth is assumed to move to the left with respect to the ether at speed v.) The light from a source is split into two beams by a half-silvered mirror Ms . One beam travels to mirror Mxand the other to mirror M2. The beams are reflected by Mxand M2 and are joined again after passing through Ms . The now superposed beams interfere with each other and the resultant is viewed by the observer's eye as an interference pattern (discussed in Section 34-6). Whether constructive or destructive interference occurs at the center of the interference pattern depends on the relative phases of the two beams after they have traveled their separate paths. Let us consider an analogy of a boat traveling up and

954

CHAPTER 36

trThe medium for light w ves could not be ir, since light tr vels from the Sun to E rth through ne rly empty sp ce. Therefore, nother medium w s postul ted, the ether. The ether w s not only tr nsp rent but, bec use of difficulty in detecting it, w s ssumed to h ve zero density.

M, Jither Wind ' 2/c 2

This expression can be considerably simplified if we assume that v/c « case we can use the binomial expansion (Appendix A), so 1 y2 1 H— ^ c2

1 - v2/c 2

j and

1. In this

1 . 1 v2 — - « 1 + — V l - ^ /c 2

Then

fc +

Now we assume v = 3.0 X 104m/s, the speed of the Earth in its orbit around the Sun. In Michelson and Morley's experiments, the arms ^ and t 2 were about 11 m long. The time difference would then be about (22m)(3.0 X 104m/s)2 (3.0 X 108m/s)3

956

CHAPTER 36

For visible light of wavelength A = 5.5 X 10_7m, say, the frequency would be / = c/A = (3.0 X 108m/s)/(5.5 X 10_7m) = 5.5 X 1014Hz, which means that wave crests pass by a point every l/(5.5 X 1014Hz) = 1.8 X 10-15s. Thus, with a time difference of 7.3 X 10-16s, Michelson and Morley should have noted a movement in the interference pattern of (7.3 X 10-16s)/(l.8 X 10-15s) = 0.4 fringe. They could easily have detected this, since their apparatus was capable of observing a fringe shift as small as 0.01 fringe. But they found no significant fringe s ift w ateverl They set their apparatus at various orientations. They made observations day and night so that they would be at various orientations with respect to the Sun (due to the Earth's rotation).

They tried at different seasons of the year (the Earth at different locations due to its orbit around the Sun). Never did they observe a significant fringe shift. This null result was one of the great puzzles of physics at the end of the nineteenth century. To explain it was a difficult challenge. One possibility to explain the null result was put forth independently by G. E Fitzgerald and H. A. Lorentz (in the 1890s) in which they proposed that any length (including the arm of an interferometer) contracts by a factor y / l — v2/c 2 in the direction of motion through the ether. According to Lorentz, this could be due to the ether affecting the forces between the molecules of a substance, which were assumed to be electrical in nature. This theory was eventually replaced by the far more comprehen sive theory proposed by Albert Einstein in 1905—the special theory of relativity.

3 6 -3 Postul tes of the Speci l Theory o f Rel tivity The problems that existed at the start of the twentieth century with regard to electromagnetic theory and Newtonian mechanics were beautifully resolved by Einstein's introduction of the theory of relativity in 1905. Unaware of the Michelson-Morley null result, Einstein was motivated by certain questions regarding electromagnetic theory and light waves. For example, he asked himself: "What would I see if I rode a light beam?" The answer was that instead of a traveling electromagnetic wave, he would see alternating electric and magnetic fields at rest whose magnitude changed in space, but did not change in time. Such fields, he realized, had never been detected and indeed were not consistent with Maxwell's electromagnetic theory. He argued, therefore, that it was unreasonable to think that the speed of light relative to any observer could be reduced to zero, or in fact reduced at all. This idea became the second postulate of his theory of relativity. In his famous 1905 paper, Einstein proposed doing away completely with the idea of the ether and the accompanying assumption of a preferred or absolute reference frame at rest. This proposal was embodied in two postulates. The first postulate was an extension of the Galilean-Newtonian relativity principle to include not only the laws of mechanics but also those of the rest of physics, including electricity and magnetism: First postulate (t e relativity principle): The l ws of physics h ve the s me form in ll inerti l reference fr mes. The first postulate can also be stated as: T ere is no experiment you can do in an inertial reference frame to tell if you are at rest or moving uniformly at constant velocity. The second postulate is consistent with the first: Secondpostulate (constancy of t e speed of lig t): Light prop g tes through empty sp ce with definite speed c independent of the speed of the source or observer. These two postulates form the foundation of Einstein's speci l theory of rel tivity. It is called "special" to distinguish it from his later "general theory of relativity," which deals with noninertial (accelerating) reference frames ( hapter 44). The special theory, which is what we discuss here, deals only with inertial frames. The second postulate may seem hard to accept, for it seems to violate common sense. First of all, we have to think of light traveling through empty space. Giving up the ether is not too hard, however, since it had never been detected. But the second postulate also tells us that the speed of light in vacuum is always the same, 3.00 X 108m/s, no matter what the speed of the observer or the source. Thus, a person traveling toward or away from a source of light will measure the same speed for that light as someone at rest with respect to the source. This conflicts with our everyday experience: we would expect to have to add in the velocity of the observer. On the other hand, perhaps we can't expect our everyday experience to be helpful when dealing with the high velocity of light. Furthermore, the null result of the Michelson-Morley experiment is fully consistent with the second postulated trThe Michelson-Morley experiment c n lso be considered s evidence for the first postul te, since it w s intended to me sure the motion of the E rth rel tive to n bsolute reference fr me. Its f ilure to do so implies the bsence of ny such preferred fr me.

SECTION 3 6 - 3

957

Einstein's proposal has a certain beauty. By doing away with the idea of an absolute reference frame, it was possible to reconcile classical mechanics with Maxwell's electromagnetic theory. The speed of light predicted by Maxwell's equations is the speed of light in vacuum in any reference frame. Einstein's theory required us to give up common sense notions of space and time, and in the following Sections we will examine some strange but interesting consequences of special relativity. Our arguments for the most part will be simple ones. We will use a technique that Einstein himself did: we will imagine very simple experimental situations in which little mathematics is needed. In this way, we can see many of the consequences of relativity theory without getting involved in detailed calculations. Einstein called these "thought" experiments.

3 6 —4 Simult neity An important consequence of the theory of relativity is that we can no longer regard time as an absolute quantity. No one doubts that time flows onward and never turns back. But the time interval between two events, and even whether or not two events are simultaneous, depends on the observer's reference frame. By an event, which we use a lot here, we mean something that happens at a particular place and at a particular time. Two events are said to occur simultaneously if they occur at exactly the same time. But how do we know if two events occur precisely at the same time? If they occur at the same point in space— such as two apples falling on your head at the same time —it is easy. But if the two events occur at widely separated places, it is more difficult to know whether the events are simultaneous since we have to take into account the time it takes for the light from them to reach us. Because light travels at finite speed, a person who sees two events must calculate back to find out when they actually occurred. For example, if two events are observed to occur at the same time, but one actually took place farther from the observer than the other, then the more distant one must have occurred earlier, and the two events were not simultaneous. We now imagine a simple thought experiment. Assume an observer, called O, is located exactly halfway between points A and B where two events occur, Fig. 36 -4. Suppose the two events are lightning that strikes the points A and B, as shown. For brief events like lightning, only short pulses of light (blue in Fig. 36-4) will travel outward from A and B and reach O. Observer O "sees" the events when the pulses of light reach point O. If the two pulses reach O at the same time, then the two events had to be simultaneous. This is because the two light pulses travel at the same speed (postulate 2), and since the distance OA equals OB, the time for the light to travel from A to O and B to O must be the same. Observer O can then definitely state that the two events occurred simultaneously. On the other hand, if O sees the light from one event before that from the other, then the former event occurred first.

FIGURE 3 6 - 4 A m om ent fter lightning strikes t points A nd B, the pulses of light (shown s blue w ves) re tr veling tow rd the observer O, but O "sees " the lightning only when the light re ches O. Light coming from the two events t A nd B

958

CHAPTER 36

The Special Theory of Relativity

The question we really want to examine is this: if two events are simultaneous to an observer in one reference frame, are they also simultaneous to another observer moving with respect to the first? Let us call the observers and 0 2and assume they are fixed in reference frames 1 and 2 that move with speed v relative to one another. These two reference frames can be thought of as two rockets or two trains (Fig. 36-5). 0 2 says that Ox is moving to the right with speed v, as in Fig. 36-5a; and says 0 2 is moving to the left with speed v, as in Fig. 36-5b. Both viewpoints are legitimate according to the relativity principle. [There is no third point of view which will tell us which one is "really" moving.] Now suppose that observers and 0 2 observe and measure two lightning strikes. The lightning bolts mark both trains where they strike: at A 1 and Bxon Oi's train, and at A 2 and B2 on 0 2's train, Fig. 36-6a. For simplicity, we assume that is exactly halfway between A xand Bx, and that 0 2is halfway between A2 and B2. Let us first put ourselves in 0 2's reference frame, so we observe ^ moving to the right with speed v. Let us also assume that the two events occur simultaneously in 0 2's frame, and just at the instant when Ox and 0 2 are opposite each other, Fig. 36-6a. A short time later, Fig. 36-6b, the light from A 2 and B2 reaches 0 2 at the same time (we assumed this). Since 0 2 knows (or measures) the distances 0 2A 2 and 0 2B2 as equal, 0 2 knows the two events are simultaneous in the 0 2 reference frame.

o 2. (a)

•0 1

(b) FIGURE 3 6 - 5 Observers O i nd 0 2 , on two different tr ins (two different reference fr mes), re m oving with rel tive speed v. 0 2 s ys th t Ox is moving to the right ( ); O i s ys th t 0 2 is moving to the left (b). Both viewpoints re legitim te: it ll depends on your reference fr me.

FIGURE 3 6 - 6 Thought experiment on simult neity. In both ( ) nd (b) we re in the reference fr me of observer 0 2 , who sees the reference fr me of O i moving to the right. In ( ), one lightning bolt strikes the two reference fr mes t A i nd A 2 , nd second lightning bolt strikes t nd B 2 . (b) A moment l ter, the light (shown in blue) from the two events re ches 0 2 t the s me time. So ccording to observer 0 2 , the two bolts of lightning struck simult neously. But in O i 's reference fr me, the light from B i h s lre dy re ched O i , where s the light from A i h s not yet re ched O x. So in O i 's reference fr me, the event t B i must h ve preceded the event t A x . Simult neity in time is not bsolute.

But what does observer Ox observe and measure? From our ( 0 2) reference frame, we can predict what Oi will observe. We see that Ox moves to the right during the time the light is traveling to from A 1 and Bx. As shown in Fig. 36 -6b, we can see from our 0 2 reference frame that the light from has already passed Ol5 whereas the light from A 1 has not yet reached ^. That is, Ox observes the light coming from before observing the light coming from A x. Given (1) that light travels at the same speed c in any direction and in any reference frame, and (2) that the distance ^Ax equals ^Bx, then observer Oi can only conclude that the event at Bxoccurred before the event at A i. The two events are not simultaneous for , even though they are for 0 2. We thus find that two events which take place at different locations and are simultaneous to one observer, are actually not simultaneous to a second observer who moves relative to the first. It may be tempting to ask: "Which observer is right, Ox or 0 2?" The answer, according to relativity, is that they are bot right. There is no "best" reference frame we can choose to determine which observer is right. Both frames are equally good. We can only conclude that simultaneity is not an absolute concept, but is relative. We are not aware of this lack of agreement on simultaneity in everyday life because the effect is noticeable only when the relative speed of the two reference frames is very large (near c), or the distances involved are very large. EXERCISE A Ex mine the experiment of Fig. 36 - 6 from O^s reference fr me. In this c se, O x will be t rest nd will see event B t occur before A x. Will O x recognize th t 0 2, who is moving with speed v to the left, will see the two events s simult neous? [Hint: Dr w di gr m equiv lent to Fig. 36 - 6.]

SECTION 3 6 - 4

Simulta eity

959

Mirror

source

(a )

FIGURE 3 6 - 7 Time dil tion c n be shown by thought experiment: the time it t kes for light to tr vel cross sp ceship nd b ck is longer for the observer on E rth (b) th n for the observer on the sp ceship ( ).

Recsivtfr

(k)

Clock timer

Earth

3 6 —5 Time Dil tion nd the Twin P r dox The fact that two events simultaneous to one observer may not be simultaneous to a second observer suggests that time itself is not absolute. ould it be that time passes differently in one reference frame than in another? This is, indeed, just what Einstein's theory of relativity predicts, as the following thought experiment shows. Figure 36 -7 shows a spaceship traveling past Earth at high speed. The point of view of an observer on the spaceship is shown in part (a), and that of an observer on Earth in part (b). Both observers have accurate clocks. The person on the spaceship (Fig. 36-7a) flashes a light and measures the time it takes the light to travel directly across the spaceship and return after reflecting from a mirror (the rays are drawn at a slight angle for clarity). In the reference frame of the spaceship, the light travels a distance 2D at speed c; so the time required to go across and back, which we call At0, is At0 = 2D/c. The observer on Earth, Fig. 36-7b, observes the same process. But to this observer, the spaceship is moving. So the light travels the diagonal path shown going across the spaceship, reflecting off the mirror, and returning to the sender. Although the light travels at the same speed to this observer (the second postulate), it travels a greater distance. Hence the time required, as measured by the observer on Earth, will be greater than that measured by the observer on the spaceship. Let us determine the time interval At measured by the observer on Earth between sending and receiving the light. In time At, the spaceship travels a distance 21 = v At where v is the speed of the spaceship (Fig. 36 - 7b). The light travels a total distance on its diagonal path (Pythagorean theorem) of 2 \/D 2 + I 2, where I = v At/2. Therefore 2 V - D2 + f2 2 V ® 2 + v2(At)2/4 At At = = We square both sides, AD2 + v2, (At)2 and solve for At, to find 2D At = c \ / l - v2/c 2 We combine this equation for At with the formula above, At0 = 2D/c, and find | TIM E D IL A T IO N \

At =

AtQ

V l -

960

CHAPTER 36

(36 - la)

f 2/c 2

Since \ / l —v2/c2 is always less than 1, we see that At > At0. That is, the time interval between the two events (the sending of the light, and its reception on the

spaceship) is greater for the observer on Earth than for the observer on the spaceship. This is a general result of the theory of relativity, and is known as time dil tion. Stated simply, the time dilation effect says that clocks moving rel tive to n observer re me sured to run more slowly ( s comp red to clocks t rest rel tive to th t observer). However, we should not think that the clocks are somehow at fault. Time is actually measured to pass more slowly in any moving reference frame as compared to your own. This remarkable result is an inevitable outcome of the two postulates of the theory of relativity. The factor l / \ / l — v2/c 2 occurs so often in relativity that we often give it the shorthand symbol 7 (the Greek letter "gamma"), and write Eq. 36 - la as At = 7 At0

(36-lb )

where 7 =

. 1 (36 -2) \ / \ - v2/c 2 Note that 7 is never less than one, and has no units. At normal speeds, 7 = 1 to a few decimal places; in general, 7 > 1. The concept of time dilation may be hard to accept, for it contradicts our experience. We can see from Eqs. 36-1 that the time dilation effect is indeed negligible unless v is reasonably close to c. If v is much less than c, then the term v2/c 2 is much smaller than the 1 in the denominator of Eq. 36 - la, and then At « At0 (see Example 36-2). The speeds we experience in everyday life are much smaller than c, so it is little wonder we don't ordinarily notice time dilation. Experiments have tested the time dilation effect, and have confirmed Einstein's predictions. In 1971, for example, extremely precise atomic clocks were flown around the Earth in jet planes. The speed of the planes (lO3 km/h) was much less than c, so the clocks had to be accurate to nanoseconds (lO-9 s) in order to detect any time dilation. They were this accurate, and they confirmed Eqs. 36-1 to within experimental error. Time dilation had been confirmed decades earlier, however, by observations on "elementary particles" which have very small masses (typically IO-30 to 10"27kg) and so require little energy to be accelerated to speeds close to the speed of light, c. Many of these elementary particles are not stable and decay after a time into lighter particles. One example is the muon, whose mean lifetime is 2.2 juls when at rest. areful experiments showed that when a muon is traveling at high speeds, its lifetime is measured to be longer than when it is at rest, just as predicted by the time dilation formula. EXAMPLE 36-1 Lifetime of a moving muon, (a) What will be the mean lifetime of a muon as measured in the laboratory if it is traveling at v = 0.60c = 1.80 X 108m /s with respect to the laboratory? Its mean lifetime at rest is 2.20 /jls = 2.20 X 10_6s. (b) How far does a muon travel in the laboratory, on average, before decaying? APPR ACH If an observer were to move along with the muon (the muon would be at rest to this observer), the muon would have a mean life of 2.20 X 10-6 s. To an observer in the lab, the muon lives longer because of time dilation. We find the mean lifetime using Eq. 3 6 - la and the average distance using d = v At. S LUTI N (a) From Eq. 3 6 - la with v = 0.60c, we have

Ar =

Af" \ / l — v2/c 2

\ / l — 0.36c2/c 2

y/0.64

(fe) Relativity predicts that a muon with speed 1.80 X 108m/s would travel an average distance d = v At = (1.80 X 108m/s)(2.8 X 10"6s) = 500 m, and this is the distance that is measured experimentally in the laboratory. N TE At a speed of 1.8 X 108m/s, classical physics would tell us that with a mean life of 2.2/as, an average muon would travel d = vt = (1.8 X 108m/s)(2.2 X 10-6 s) = 400 m. This is shorter than the distance measured. EXERCISE B W h t is the m uon 's me n lifetim e (Ex m ple 3 6 - 1 ) if it is tr veling 0.90c? (a) 0.42 / a s ; (b) 2.3 ju ls; (c) 5.0 / a s ; (d) 5.3 / a s ; (e) 12.0 / a s .

SECTION 36 —5

961

AUT I ON

P roper tim e At0 is fo r 2 events at t e sam e p o in t in space.

We need to clarify how to use Eqs. 36 -1, and the meaning of At and At0. The equation is true only when At0represents the time interval between the two events in a reference frame where the two events occur at t e same point in space (as in Fig. 36 - 7a where the two events are the light flash being sent and being received). This time interval, At0, is called the proper time. Then At in Eqs. 36 -1 represents the time interval between the two events as measured in a reference frame moving with speed v with respect to the first. In Example 36-1 above, At0(and not At) was set equal to 2.2 X 10-6 s because it is only in the rest frame of the muon that the two events ("birth" and "decay") occur at the same point in space. The proper time At0 is the shortest time between the events any observer can measure. In any other moving reference frame, the time At is greater. ____________________ EXAMPLE 36 - 2 Time dilation at 100 km/h. Let us check time dilation for

everyday speeds. A car traveling 100 km/h covers a certain distance in 10.00 s according to the driver's watch. What does an observer at rest on Earth measure for the time interval? APPR ACH The car's speed relative to Earth is 100 km/h = (l.OO X 105m)/(3600 s)

= 27.8 m/s. The driver is at rest in the reference frame of the car, so we set At0 = 10.00 s in the time dilation formula. S LUTI N We use Eq. 36 - la: At =

At0 ,______ 1 - — c2

BLEM S LVING Use o f t e binom ial expansion

10.00 s ,____________________ 27.8 m/s \2 1 3.00 X 108m/s /

10.00 s ,_______________ V 1 " (8-59 x 10_15)

If you put these numbers into a calculator, you will obtain At = 10.00 s, since the denominator differs from 1 by such a tiny amount. Indeed, the time measured by an observer on Earth would show no difference from that measured by the driver, even with the best instruments. A computer that could calculate to a large number of decimal places would reveal a difference between At and At0. We can estimate the difference using the binomial expansion (Appendix A), (1 + x)n « 1 + nx.

[for x «

In our time dilation formula, we have the factor 7 = (l ,2\-; 1 v2 A t = y A t o = Atnl 1 Atr 1 + 27 10.00 s 1 + -

27.8 m/s 3.00 X 108m/s

v 2/ c 2) ~ 2.

Thus

10.00 s + 4 X 10"

So the difference between At and At0is predicted to be 4 X 10 14s, an extremely small amount. EXERCISE C A cert in tomic clock keeps perfect time on E rth. If the clock is t ken on sp ceship tr veling t speed v = 0.60c, does this clock now run slow ccording to the people (a) on the sp ceship, (b) on E rth?

EXAMPLE 36 - 3 Reading a magazine on a spaceship. A passenger on a high-speed spaceship traveling between Earth and Jupiter at a steady speed of 0.75c reads a magazine which takes 10.0 min according to her watch, (a) How long does this take as measured by Earth-based clocks? (b) How much farther is the spaceship from Earth at the end of reading the article than it was at the beginning? APPR ACH (a) The time interval in one reference frame is related to the time

interval in the other by Eq. 36 - la or b. (b) At constant speed, distance is speed X time. Since there are two times (a At and a At0) we will get two distances, one for each reference frame. [This surprising result is explored in the next Section (36-6).]

962

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The Special Theory of Relativity

S LUTI N (a) The given 10.0-min time interval is the proper time— starting and finishing the magazine happen at the same place on the spaceship. Earth clocks measure 10.00 min . At = —, = —, = = 15.1 mm. 1 _1? V I - (0.75)2

(b) In the Earth frame, the rocket travels a distance D = v At = (0.75c)(15.1 min) = (0.75)(3.0 X 108m/s)(15.1min X 60s/min) = 2.04 X 10n m. In the spaceship's frame, the Earth is moving away from the spaceship at 0.75c, but the time is only 10.0 min, so the distance is measured to be D1 = v A t 0 = (2.25 X 108m/s)(600s) = 1.35 X 10n m. Values for 7 = l / \ / l - v2/c 2 at a few speeds v are given in Table 36-1.

TABLE 36-1 Values of 7

0.01c

1.000 1.000

0.10c

1.005

Sp ce Tr vel? Time dilation has aroused interesting speculation about space travel. According to classical (Newtonian) physics, to reach a star 100 light-years away would not be possible for ordinary mortals (1 light-year is the distance light can travel in 1 year = 3.0 X 108m/s X 3.16 X 107s = 9.5 X 1015m). Even if a spaceship could travel at close to the speed of light, it would take over 100 years to reach such a star. But time dilation tells us that the time involved could be less. In a spaceship traveling at v = 0.999c, the time for such a trip would be only about At0 = At y j l - v2/c 2 = (100 y r)v "l " (0.999)2 = 4.5 yr. Thus time dilation allows such a trip, but the enormous practical problems of achieving such speeds may not be possible to overcome, certainly not in the near future. In this example, 100 years would pass on Earth, whereas only 4.5 years would pass for the astronaut on the trip. Is it just the clocks that would slow down for the astronaut? No. All processes, including aging and other life processes, run more slowly for the astro naut according to the Earth observer. But to the astronaut, time would pass in a normal way. The astronaut would experience 4.5 years of normal sleeping, eating, reading, and so on. And people on Earth would experience 100 years of ordinary activity.

0.50c

1.15

0.90c

2.3

0.99c

7.1

Twin P r dox Not long after Einstein proposed the special theory of relativity, an apparent paradox was pointed out. According to this twin paradox, suppose one of a pair of 20-year-old twins takes off in a spaceship traveling at very high speed to a distant star and back again, while the other twin remains on Earth. According to the Earth twin, the astronaut twin will age less. Whereas 20 years might pass for the Earth twin, perhaps only 1 year (depending on the spacecraft's speed) would pass for the traveler. Thus, when the traveler returns, the earthbound twin could expect to be 40 years old whereas the traveling twin would be only 21. This is the viewpoint of the twin on the Earth. But what about the traveling twin? If all inertial reference frames are equally good, won't the traveling twin make all the claims the Earth twin does, only in reverse? an't the astronaut twin claim that since the Earth is moving away at high speed, time passes more slowly on Earth and the twin on Earth will age less? This is the opposite of what the Earth twin predicts. They cannot both be right, for after all the spacecraft returns to Earth and a direct comparison of ages and clocks can be made. There is, however, no contradiction here. One of the viewpoints is indeed incorrect. The consequences of the special theory of relativity—in this case, time dilation—can be applied only by observers in an inertial reference frame. The Earth is such a frame (or nearly so), whereas the spacecraft is not. The spacecraft accelerates at the start and end of its trip and when it turns around at the far point of its journey. During the accel eration, the twin on the spacecraft is not in an inertial frame. In between, the astronaut twin may be in an inertial frame (and is justified in saying the Earth twin's clocks run slow), but it is not always the same frame. So she cannot use special relativity to predict their relative ages when she returns to Earth. The Earth twin stays in the same inertial frame, and we can thus trust her predictions based on special relativity. Thus, there is no paradox. The prediction of the Earth twin that the traveling twin ages less is the proper one.

SECTI N 36-5

963

PHYSICS

APPLIED

Global positioning system (GPS)

Airplanes, cars, boats, and hikers use global positioning system (GPS) receivers to tell them quite accurately where they are, at a given moment. The 24 global positioning system satellites send out precise time signals using atomic clocks. Your receiver compares the times received from at least four satellites, all of whose times are carefully synchronized to within 1 part in 1013. By comparing the time differences with the known satellite positions and the fixed speed of light, the receiver can determine how far it is from each satellite and thus where it is on the Earth. It can do this to a typical accuracy of 15 m, if it has been constructed to make corrections such as the one below due to special relativity. C NCEPTUAL EXAMPLE 56 - 4 I A relativity correction to CPS. GPS satellites move at about 4km /s = 4000 m/s. Show that a good GPS receiver needs to correct for time dilation if it is to produce results consistent with atomic clocks accurate to 1 part in 1013. RESP NSE Let us calculate the magnitude of the time dilation effect by inserting v = 4000 m /s into Eq. 36 - la:

i--------------------------4 X 103m /s\ 2 3 X 108m/s /

—

= —

V l - u x IO"10

Af0.

We use the binomial expansion: (1 + x )n « 1 + nx for x « which here is (1 - x)~^ « 1 + \x . That is At

(1 + | ( 1 . 8 X 1 0 " 10)) A t0 =

1 (see Appendix A)

(1 + 9 X 1 0 " n ) A t0 .

The time "error" divided by the time interval is (At - Atn) --------------- -- =

A f0

1 + 9 X 1 0 -11 -

1 =

9 X 1 0 -11 «

1 0 " 10.

Time dilation, if not accounted for, would introduce an error of about 1 part in 1010, which is 1000 times greater than the precision of the atomic clocks. Not correcting for time dilation means a receiver could give much poorer position accuracy. N TE GPS devices must make other corrections as well, including effects associated with general relativity.

3 6 - 6 Length Contr ction Time intervals are not the only things different in different reference frames. Space intervals— lengths and distances—are different as well, according to the special theory of relativity, and we illustrate this with a thought experiment. Observers on Earth watch a spacecraft traveling at speed v from Earth to, say, Neptune, Fig. 36 -8a. The distance between the planets, as measured by the Earth observers, is £0. The time required for the trip, measured from Earth, is £ At = [Earth observer] In Fig. 36 -8b we see the point of view of observers on the spacecraft. In this frame of reference, the spaceship is at rest; Earth and Neptune movef with speed v. The time between departure of Earth and arrival of Neptune (observed from the spacecraft) is the "proper time," since the two events occur at the same point in space (i.e., on the spacecraft). Therefore the time interval is less for the spacecraft

964

CHAPTER 36

fWe ssume v is much gre ter th n the rel tive speed of Neptune nd E rth, so the l tter c n be ignored.

Earth

Nepiune

(a) Eiirlh al nxsl

Eiidh (b) Spacecraft j ( rest

FIGURE 3 6 - 8 ( ) A sp ceship tr veling t very high speed from E rth to the pl net Neptune, s seen from E rth 's fr me o f reference. (b) According to n observer on the sp ceship, E rth nd N eptune re m oving t the very high speed v : E rth le ves the sp ceship, nd time At0 l ter N eptune rrives t the sp ceship.

N eptune

observers than for the Earth observers. That is, because of time dilation (Eq. 36 - la), the time for the trip as viewed by the spacecraft is At0 = At V l " v2/c 2 = At/y. [spacecraft observer] Because the spacecraft observers measure the same speed but less time between these two events, they also measure the distance as less. If we let £ be the distance between the planets as viewed by the spacecraft observers, then I = v At0, which we can rewrite as i = v At0 = v At y / l - v2/c 2 = £0\ / l - v2/c 2. Thus we have the important result that (36-3 )

i = f0\ / i - ^ / c 2

L E N G T H C O N T R A C T IO N

or, using 7 (Eq. 36 -2), I =

(36-3b)

This is a general result of the special theory of relativity and applies to lengths of objects as well as to distance between objects. The result can be stated most simply in words as: the length of n object moving rel tive to n observer is me sured to be shorter long its direction of motion th n when it is t rest. This is called length contr ction. The length £0 in Eqs. 36 - 3 is called the proper length. It is the length of the object (or distance between two points whose positions are measured at the same time) as determined by observers at rest with respect to the object. Equations 36 -3 give the length i that will be measured by observers when the object travels past them at speed v. It is important to note that length contraction occurs only along t e direction o f motion. For example, the moving spaceship in Fig. 36 - 8a is shortened in length, but its height is the same as when it is at rest. Length contraction, like time dilation, is not noticeable in everyday life because the factor y / l —v2/c 2in Eq. 36-3a differs from 1.00 significantly only when v is very large. tTiIj T T T S M Painting's contraction. A rectangular painting measures 1.00 m tall and 1.50 m wide. It is hung on the side wall of a spaceship which is moving past the Earth at a speed of 0.90c. See Fig. 36 -9a. (a) What are the dimensions of the picture according to the captain of the spaceship? (b) What are the dimensions as seen by an observer on the Earth? APPR ACH We apply the length contraction formula, Eq. 36-3a, to the dimension parallel to the motion; v is the speed of the painting relative to the observer. S LUTI N (a) The painting is at rest (v = 0) on the spaceship so it (as well as everything else in the spaceship) looks perfectly normal to everyone on the spaceship. The captain sees a 1.00-m by 1.50-m painting. (b) Only the dimension in the direction of motion is shortened, so the height is unchanged at 1.00 m, Fig. 36-9b. The length, however, is contracted to

/j\

AUT I ON

P roper lengt is m easured in reference fram e w ere t e tw o position s are at rest

FIGURE 3 6 - 9

Ex m ple 3 6 - 5 .

t = A

1 - —2

MX) m

1.50 ni

l rIX) m

1 lb )

= (1.50 m ) \ / l " (0.90)2 = 0.65 m. So the picture has dimensions 1.00 m

0.65 m.

SECTI N 36-6

965

EXAMPLE 36 - 6 A fantasy supertrain. A very fast train with a proper length of 500 m is passing through a 200-m-long tunnel. Let us imagine the train's speed to be so great that the train fits completely within the tunnel as seen by an observer at rest on the Earth. That is, the engine is just about to emerge from one end of the tunnel at the time the last car disappears into the other end. What is the train's speed? APPR ACH Since the train just fits inside the tunnel, its length measured by the

person on the ground is 200 m. The length contraction formula, Eq. 36 - 3a or b, can thus be used to solve for v. S LUTI N Substituting I = 200 m and £0 = 500 m into Eq. 36 - 3a gives

L if 200 m = 500 m -w 1 _ ^2 ' dividing both sides by 500 m and squaring, we get (0.40)2 = 1 - ^

^ = \ A - (0.40)2 v = 0.92c. N TE No real train could go this fast. But it is fun to think about. N TE An observer on the train would not see the two ends of the train inside the tunnel at the same time. Recall that observers moving relative to each other do not agree about simultaneity.

EXERCISE D Wh t is the length o f the tunnel Ex m ple 36 - 6?

s m e sured by observers on the tr in in

C NCEPTUAL EXAMPLE 56-7~| Resolving the train and tunnel length.

Observers at rest on the Earth see a very fast 200-m-long train pass through a 200-m-long tunnel (as in Example 36-6) so that the train momentarily disappears from view inside the tunnel. Observers on the train measure the train's length to be 500 m and the tunnel's length to be only 80 m (Exercise D, using Eq. 36-3a). learly a 500-m-long train cannot fit inside an 80-m-long tunnel. How is this apparent inconsistency explained? RESP NSE Events simultaneous in one reference frame may not be simultaneous in another. Let the engine emerging from one end of the tunnel be "event A," and the last car disappearing into the other end of the tunnel "event B." To observers in the Earth frame, events A and B are simultaneous. To observers on the train, however, the events are not simultaneous. In the train's frame, event A occurs before event B. As the engine emerges from the tunnel, observers on the train observe the last car as still 500 m - 80 m = 420 m from the entrance to the tunnel.

966

CHAPTER 36

(a)

A9 ± i

FIGURE 3 6 - 1 0 According to n ccur te clock on f st - moving tr in, person ( ) begins dinner t 7:00 nd (b) finishes t 7:15. A t the beginning of the m e l, two observers on E rth set their w tches to correspond with the clock on the tr in. These observers m e sure the e ting time s 20 minutes.

(b)

3 6 - 7 Four-Dimension l Sp ce-Time Let us imagine a person is on a train moving at a very high speed, say 0.65c, Fig. 36-10. This person begins a meal at 7:00 and finishes at 7:15, according to a clock on the train. The two events, beginning and ending the meal, take place at the same point on the train. So the proper time between these two events is 15 min. To observers on Earth, the meal will take longer— 20 min according to Eqs. 36-1. Let us assume that the meal was served on a 20-cm-diameter plate. To observers on the Earth, the plate is only 15 cm wide (length contraction). Thus, to observers on the Earth, the meal looks smaller but lasts longer. In a sense the two effects, time dilation and length contraction, balance each other. When viewed from the Earth, what an object seems to lose in size it gains in length of time it lasts. Space, or length, is exchanged for time. onsiderations like this led to the idea of four-dimension l sp ce-time: space takes up three dimensions and time is a fourth dimension. Space and time are intimately connected. Just as when we squeeze a balloon we make one dimension larger and another smaller, so when we examine objects and events from different reference frames, a certain amount of space is exchanged for time, or vice versa. Although the idea of four dimensions may seem strange, it refers to the idea that any object or event is specified by four quantities— three to describe where in space, and one to describe when in time. The really unusual aspect of four-dimensional space -time is that space and time can intermix: a little of one can be exchanged for a little of the other when the reference frame is changed. It is difficult for most of us to understand the idea of four-dimensional space-time. Somehow we feel, just as physicists did before the advent of relativity, that space and time are completely separate entities. Yet we have found in our thought experiments that they are not completely separate. And think about Galileo and Newton. Before Galileo, the vertical direction, that in which objects fall, was considered to be distinctly different from the two horizontal dimensions. Galileo showed that the vertical dimension differs only in that it happens to be the direction in which gravity acts. Otherwise, all three dimensions are equivalent, a viewpoint we all accept today. Now we are asked to accept one more dimension, time, which we had previously thought of as being somehow different. This is not to say that there is no distinction between space and time. What relativity has shown is that space and time determinations are not independent of one another. In Galilean -Newtonian relativity, the time interval between two events, At, and the distance between two events or points, Ax, are invariant quantities no matter what inertial reference frame they are viewed from. Neither of these quantities is invariant according to Einstein's relativity. But there is an invariant quantity in four-dimensional space -time, called the sp ce-time interv l, which is (As)2 = (c At)2 - (Ax)2. We leave it as a Problem (97) to show that this quantity is indeed invariant under a Lorentz transformation (Section 36 -8).

♦SECTI N 36-7

967

3 6 —8 G lile n nd LorentzTr nsform tions We now examine in detail the mathematics of relating quantities in one inertial reference frame to the equivalent quantities in another. In particular, we will see how positions and velocities transform (that is, change) from one frame to the other. We begin with the classical or Galilean viewpoint. onsider two inertial reference frames S and S' which are each characterized by a set of coordinate axes, Fig. 36-11. The axes x and y (z is not shown) refer to S and x' and y' to S'. The x' and x axes overlap one another, and we assume that frame S' moves to the right in the x direction at constant speed v with respect to S. For simplicity let us assume the origins 0 and O' of the two reference frames are superimposed at time t = 0.

y S FIGURE 36 - 11 Inertial reference frame S' moves to the right at constant speed v with respect to frame S.

-------------- v t -------------- -

—x - —

•aT r)

*------------------ x ---------

Now consider an event that occurs at some point P (Fig. 36-11) represented by the coordinates *', y', z' in reference frame S' at the time t' . What will be the coordinates of P in S? Since S and S' initially overlap precisely, after a time t', S' will have moved a distance vt'. Therefore, at time t', x = x' + v t'. The y and z coordinates, on the other hand, are not altered by motion along the x axis; thus y = y' and z = z '. Finally, since time is assumed to be absolute in Galilean -Newtonian physics, clocks in the two frames will agree with each other; so t = t'. We summarize these in the following G lile n tr nsform tion equ tions: x = x' + vt'

y = y' z = z' t = t'.

[Galilean] (36 -4)

These equations give the coordinates of an event in the S frame when those in the S' frame are known. If those in the S frame are known, then the S' coordinates are obtained from x' = x — vt,

y' = y,

z' = z,

t' = t.

[Galilean]

These four equations are the "inverse" transformation and are very easily obtained from Eqs. 36 -4. Notice that the effect is merely to exchange primed and unprimed quantities and replace v by —v. This makes sense because from the S' frame, S moves to the left (negative x direction) with speed v. Now suppose the point P in Fig. 36-11 represents a particle that is moving. Let the components of its velocity vector in S' be u'x , u'y , u'z . (We use u to distinguish it from the relative velocity of the two frames, v.) Now u'x = dx'/ d t', u'y = dy' I dt' and u'z = d z'/dt'. The velocity of P as seen from S will have components ux ,u y , and uz . We can show how these are related to the velocity components in S' by differentiating Eqs. 36-4. For ux we get dx d(x' + vt') u* = dF = dF'------------ = u* + v since v is assumed constant. For the other components, u'y = uy and u'z = uz , so

968

CHAPTER 36

The Special Theory of Relativity

we have = u'r + V Uy = u'y u7 = u'7.

[Galilean] (36 -5)

These are known as the G lile n velocity tr nsform tion equ tions. We see that the y and z components of velocity are unchanged, but the x components differ by v : ux = u'x + v. This is just what we have used before (see hapter 3, Section 3 - 9) when dealing with relative velocity. The Galilean transformations, Eqs. 36 - 4 and 36 -5, are valid only when the velocities involved are much less than c. We can see, for example, that the first of Eqs. 36 -5 will not work for the speed of light: light traveling in S' with speed u'x = c would have speed c + v in S, whereas the theory of relativity insists it must be c in S. learly, then, a new set of transformation equations is needed to deal with relativistic velocities. We derive the required equation, looking again at Fig. 36-11. We will try the simple assumption that the transformation is linear and of the form = y(x' + vt'),

y = y

z = z

(i) That is, we modify the first of Eqs. 36 -4 by multiplying by a constant 7 which is yet to be determined1(7 = 1 non-relativistically). But we assume the y and z equations are unchanged since there is no length contraction in these directions. We will not assume a form for t, but will derive it. The inverse equations must have the same form with v replaced by —v. (The principle of relativity demands it, since S' moving to the right with respect to S is equivalent to S moving to the left with respect to S'.) Therefore x' = y(x - vt).

(ii)

Now if a light pulse leaves the common origin of S and S' at time t = t' = 0, after a time t it will have traveled a distance x = ct or x' = ct' along the x axis. Therefore, from Eqs. (i) and (ii) above, ct = 7 (ct' + vt') = 7 (c + v ) t',

(iii)

ct' = 7 (ct - vt) = 7 (c - v)t.

(iv)

We substitute t' from Eq. (iv) into Eq. (iii) and find ct = 7(c + v)J(c — v)(t/c) = 72(c2 - v2)t/c. We cancel out the t on each side and solve for 7 to find 1 7 = \ / l - v2/c 2 The constant 7 here has the same value as the 7 we used before, Eq. 36-2. Now that we have found 7, we need only find the relation between t and t'. To do so, we combine x' = 7(x - vt) with x = J(x' + vt'): x' = J(x — vt) = 7(7[x' + vt'] — vt). We solve for t and find t = l ( t' + v x '/c 2). In summary, X = J (x ' + vt')

y = y' z = z'

t = y\t' +

(36 -6)

L O R E N T Z T R A N SF O R M A T IO N S

vx'

These are called the Lorentz tr nsform tion equ tions. They were first proposed, in a slightly different form, by Lorentz in 1904 to explain the null result of the Michelson-Morley experiment and to make Maxwell's equations take the same form in all inertial reference frames. A year later Einstein derived them independently based on his theory of relativity. Notice that not only is the x equation modified as compared to the Galilean transformation, but so is the t equation; indeed, we see directly in this last equation how the space and time coordinates mix. f7 here is not ssumed to be given by Eq. 36-2.

SECTION 3 6 - 8

Galilea a d Lore tz Tra sformatio s

969

D riving L ngth Contraction We now derive the length contraction formula, Eq. 36-3, from the Lorentz transformation equations. We consider two reference frames S and S' as in Fig. 36-11. Let an object of length £0be at rest on the x axis in S. The coordinates of its two end points are x x and x2, so that x2 — x 1 = £0• At any instant in S', the end points will be at Xi and x 2 as given by the Lorentz transformation equations. The length measured in S' is I = x 2 — x [ . An observer in S' measures this length by measuring x2 and x\ at the same time (in the S' reference frame), so t2 = t[. Then, from the first of Eqs. 36-6, lQ = X2 -

X l

1 0 / A *2 + Vt'2 - xi - Vti).

y l - vl /c l

Since t2 = t \ , we have 1 f° = y /,l - v2/c 2/22

I " X0 = v Tl1 _ v 2l /c2 '

1 = W i - «2/c 2, which is Eq. 36-3. D riving Tim Dilation We now derive the time-dilation formula, Eq. 36 - la, using the Lorentz transformation equations. The time At0between two events that occur at the same place (x 2 = x^) in S' is measured to be At0 = t2 — t[. Since x2 = x [ , then from the last of Eqs. 36 -6, the time At between the events as measured in S is 1

I ,

VX2

VX1

1 (* - kT/rac2; assume me2 » 3kT. Evaluate Af / f 0 for a gas of hydrogen atoms at 550 K. [This "Doppler - broadening" effect is commonly used to measure gas temperature, such as in astronomy.]

| Gener l Problems 69. An atomic clock is taken to the North Pole, while another stays at the Equator. How far will they be out of synchro nization after 2.0 years has elapsed? {Hint: Use the binomial expansion, Appendix A.] 70. A spaceship in distress sends out two escape pods in opposite directions. One travels at a speed Vi = —0.60c in one direction, and the other travels at a speed v2 = +0.50c in the other direction, as observed from the spaceship. What speed does the first escape pod measure for the second escape pod? 71. An airplane travels 1300 km /h around the Earth in a circle of radius essentially equal to that of the Earth, returning to the same place. Using special relativity, estimate the differ ence in time to make the trip as seen by Earth and airplane observers. [Hint: Use the binomial expansion, Appendix A.] 72. The nearest star to Earth is Proxima entauri, 4.3 lightyears away, (a) A t what constant velocity must a spacecraft travel from Earth if it is to reach the star in 4.6 years, as measured by travelers on the spacecraft? (b) How long does the trip take according to Earth observers? 73. A quasar emits familiar hydrogen lines whose wave-lengths are 2.5 times longer than what we measure in the laboratory, (a) What is the speed of this quasar? (b) What result would you obtain if you used the "classical" Doppler shift discussed in hapter 16? 74. A healthy astronaut's heart rate is 60beats/min. Flight doctors on Earth can monitor an astronaut's vital signs remotely while in flight. How fast would an astronaut have to be flying away from Earth in order for the doctor to measure her having a heart rate of 30 beats/min? 75. A spacecraft (reference frame S') moves past Earth (refer ence frame S) at velocity v, which points along the x and x ' axes. The spacecraft emits a light beam (speed c) along its y' axis as shown in Fig. 36 - 17. (a) What angle 0 does this light beam make with yf the x axis in the E arth's S' reference frame? (b) Use velocity transformations to show that the light moves c = cj with speed c also in the E arth's reference frame, (c) ompare these relativistic results to what you would have obtained classically (Galilean transformations). Earth

FIGURE 36 -17

Problem 75.

v S

76. Rocket A passes Earth at a speed of 0.65c. A t the same time, rocket B passes Earth moving 0.85c relative to Earth in the same direction. How fast is B moving relative to A when it passes A? 77. (a) What is the speed v of an electron whose kinetic energy is 14,000 times its rest energy? You can state the answer as the difference c - v. Such speeds are reached in the Stan ford Linear Accelerator, SLA . (b) If the electrons travel in the lab through a tube 3.0 km long (as at SLA ), how long is this tube in the electrons' reference frame? [Hint: Use the binomial expansion.]

78. As a rough rule, anything traveling faster than about 0.1c is called relativistic— that is, special relativity is a significant effect. Determine the speed of an electron in a hydrogen atom (radius 0.53 X 10-10 m) and state whether or not it is relativistic. (Treat the electron as though it were in a circular orbit around the proton.) 79. What minimum amount of electromagnetic energy is needed to produce an electron and a positron together? A positron is a particle with the same mass as an electron, but has the opposite charge. (Note that electric charge is conserved in this process. See Section 37 - 5.) 80. How many grams of matter would have to be totally destroyed to run a 75-W lightbulb for 1.0 year? 81. If E is the total energy of a particle with zero potential energy, show that dE /dp = v, where p and v are the momentum and velocity of the particle, respectively. 82. A free neutron can decay into a proton, an electron, and a neutrino. Assume the neutrino's mass is zero; the other masses can be found in the Table inside the front cover. Determine the total kinetic energy shared among the three particles when a neutron decays at rest. 83. The Sun radiates energy at a rate of about 4 X 1026W. (a) A t what rate is the Sun's mass decreasing? (b) How long does it take for the Sun to lose a mass equal to that of Earth? (c) Estimate how long the Sun could last if it radi ated constantly at this rate. 84. An unknown particle is measured to have a negative charge and a speed of 2.24 X 108m/s. Its momentum is determined to be 3.07 X 10_22kg - m /s. Identify the particle by finding its mass. 85. How much energy would be required to break a helium nucleus into its constituents, two protons and two neutrons? The rest masses of a proton (including an electron), a neutron, and neutral helium are, respectively, 1.00783 u, 1.00867 u, and 4.00260 u. (This energy difference is called the total binding energy of the |H e nucleus.) 86. Show analytically that a particle with momentum p and energy E has a speed given by v =

PC E

pc V»

+ PL

87. Two protons, each having a speed of 0.985c in the labora tory, are moving toward each other. Determine (a) the momentum of each proton in the laboratory, (b) the total momentum of the two protons in the laboratory, and (c) the momentum of one proton as seen by the other proton. 88. When two moles of hydrogen molecules (H 2) and one mole of oxygen molecules ( 0 2) react to form two moles of water (H 20 ) , the energy released is 484 kJ. How much does the mass decrease in this reaction? What % of the total original mass of the system does this mass change represent? 89. The fictional starship Enterprise obtains its power by combining matter and antimatter, achieving complete conversion of mass into energy. If the mass of the Enterprise is approximately 6 X 109kg, how much mass must be converted into kinetic energy to accelerate it from rest to one - tenth the speed of light?

General Problems

985

90. A spaceship and its occupants have a total mass of 180.000 kg. The occupants would like to travel to a star that is 35 light-years away at a speed of 0.70c. To accelerate, the engine of the spaceship changes mass directly to energy. How much mass will be converted to energy to accelerate the spaceship to this speed? Assume the acceleration is rapid, so the speed for the entire trip can be taken to be 0.70c, and ignore decrease in total mass for the calculation. How long will the trip take according to the astronauts on board? 91. In a nuclear reaction two identical particles are created, traveling in opposite directions. If the speed of each particle is 0.85c, relative to the laboratory frame of reference, what is one particle's speed relative to the other particle? 92. A 32,000-kg spaceship is to travel to the vicinity of a star 6.6 light-years from Earth. Passengers on the ship want the (one-way) trip to take no more than 1.0 year. How much work must be done on the spaceship to bring it to the speed necessary for this trip? 93. Suppose a 14,500-kg spaceship left Earth at a speed of 0.98c. What is the spaceship's kinetic energy? ompare with the total U.S. annual energy consumption (about IO20J). 94. A pi meson of mass ra^ decays at rest into a muon (mass m^) and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is K,jl = ~ f f ij2c2/ ( 2 m j . 95. Astronomers measure the distance to a particular star to be 6.0 light-years (1 ly = distance light travels in 1 year). A spaceship travels from Earth to the vicinity of this star at steady speed, arriving in 2.50 years as measured by clocks on the spaceship, (a) How long does the trip take as measured by clocks in E arth's reference frame (assumed inertial)? (b) What distance does the spaceship travel as measured in its own reference frame? 96. A 1.88-kg mass oscillates on the end of a spring whose spring stiffness constant is k = 84.2 N/m. If this system is in a spaceship moving past Earth at 0.900c, what is its period of oscillation according to (a) observers on the ship, and (ib) observers on Earth? 97. Show that the space - time interval, (c A t)2 - (Ax)2, is invariant, meaning that all observers in all inertial reference frames calculate the same number for this quantity for any pair of events.

98. A slab of glass with index of refraction n moves to the right with speed v. A flash of light is emitted at point A (Fig. 36 - 18) and passes through the glass arriving at point B a distance £ away. The glass has thickness d in the refer ence frame where it is at rest, and the speed of light in the glass is c/n. How long does it take the light to go from point A to point B according to an observer at rest with respect to points A and B? heck your answer for the cases v = c, v = 0, and n = 1. glass

FIGURE 36 -18

Problem 98.

*Num rical/Comput r * 99. (II) For a 1.0-kg mass, make a plot of the kinetic energy as a function of speed for speeds from 0 to 0.9c, using both the classical formula (K = \ m v2) and the correct relativistic formula (K = (7 - l)m c2). *100. (Ill) A particle of mass m is projected horizontally at a relativistic speed v0 in the +x direction. There is a constant downward force F acting on the particle. Using the definition of relativistic momentum p = 7 m \ and Newton's second law F = d\>/dt, (a) show that the x and y components of the velocity of the particle at time t are given by % (0

P o c /(m 2c2 + Po + F 2t2)2

vy(t) = ~ F ct/(m 2c2 + Po + F 2t2)i where p 0 is the initial momentum of the particle, (b) Assume the particle is an electron (ra = 9.11 X 10 31 kg), with v0 = 0.50 c and F = 1.00 X 10-15N. alculate the values of vx and vy of the electron as a function of time t from t = 0 to t = 5.00 /as in intervals of 0.05 /jls. Graph the values to show how the velocity components change with time during this interval, (c) Is the path parabolic, as it would be in classical mechanics (Sections 3 -7 and 3 -8)? Explain.

Answers to Exercises A: Yes.

E: 0.030c, same as classical, to an accuracy of better than 0.1%.

B: (c).

F: (d).

: (a) No; (b) yes.

G: (d).

D: 80 m.

986

CHAPTER 36

H: No.

The Special Theory of Relativity

Electron microscopes produce im ges using electrons which h ve w ve properties just s light does. Since the w velength of electrons c n be much sm ller th n th t of visible light, much gre ter resolution nd m gnific tion c n be obt ined. A sc nning electron microscope (SEM) c n produce im ges with three-dimension l qu lity, s for these Giardia cells inside hum n sm ll intestine. M gnific tion here is bout 2000X. Giardia is on the minds of b ckp ckers bec use it h s become too common in untre ted w ter, even in the high mount ins, nd c uses n unple s nt intestin l infection not e sy to get rid of.

T £

r ly Q u

n tu m T h eo ry

n d M o d e ls o f th e A to m HAPTER -OPENING QUESTION — Guess now! It has been found experimentally that (a) light behaves as a wave. (b) light behaves as a particle. (c) electrons behave as particles. (d) electrons behave as waves. (e) all of the above are true. (f> none of the above are true. he second aspect of the revolution that shook the world of physics in the early part of the twentieth century was the quantum theory (the other was Einstein's theory of relativity). Unlike the special theory of relativity, the revolution of quantum theory required almost three decades to unfold, and many scientists contributed to its development. It began in 1900 with Planck's quantum hypothesis, and culminated in the mid-1920s with the theory of quantum mechanics of Schrodinger and Heisenberg which has been so effective in explaining the structure of matter.

3 7 —1 Pl nck's Qu ntum Hypothesis; Bl ckbody R di tion

ONTENTS 37 - 1 37 - 2 37 - 3 37 - 4 37 - 5 37 - 6 37 - 7 *37 - 8 37 - 9 37 - 10

Pl nck's Qu ntum Hypothesis; Bl ckbody R di tion Photon Theory of Light nd the Photoelectric Effect Energy, M ss, nd Momentum of Photon Compton Effect Photon Inter ctions; P ir Production W ve-P rticle Du lity; the Principle of Complement rity W ve N ture of M tter Electron Microscopes E rly Models of the Atom Atomic Spectr : Key to the Structure of the Atom

37 -11 The Bohr Model 37 - 12 de Broglie's Hypothesis Applied to Atoms

Blackbody Radiation One of the observations that was unexplained at the end of the nineteenth century was the spectrum of light emitted by hot objects. We saw in Section 19-10 that all objects emit radiation whose total intensity is proportional to the fourth power of the Kelvin (absolute) temperature (T4). At normal temperatures ( ~ 300K), we are not aware of this electromagnetic radiation because of its low intensity.

987

Intensity

Frequency (Hz)

Visible

W velength (nm)

FIGURE 37 - 1 M e sured spectr of w velengths nd frequencies em itted by bl ckbody t three different temper tures.

At higher temperatures, there is sufficient infrared radiation that we can feel heat if we are close to the object. At still higher temperatures (on the order of 1000 K), objects actually glow, such as a red -hot electric stove burner or the heating element in a toaster. At temperatures above 2000 K, objects glow with a yellow or whitish color, such as white-hot iron and the filament of a lightbulb. The light emitted is of a continuous range of wavelengths or frequencies, and the spectrum is a plot of intensity vs. wavelength or frequency. As the temperature increases, the electromagnetic radiation emitted by objects not only increases in total intensity but reaches a peak at higher and higher frequencies. The spectrum of light emitted by a hot dense object is shown in Fig. 37-1 for an idealized bl ckbody. A blackbody is a body that would absorb all the radiation falling on it (and so would appear black under reflection when illuminated by other sources). The radiation such an idealized blackbody would emit when hot and luminous, called bl ckbody r di tion (though not necessarily black in color), approximates that from many real objects. The 6000-K curve in Fig. 37-1, corresponding to the temperature of the surface of the Sun, peaks in the visible part of the spectrum. For lower temperatures, the total radiation drops considerably and the peak occurs at longer wavelengths (or lower frequencies). (This is why objects glow with a red color at around 1000 K.) It is found experimentally that the wavelength at the peak of the spectrum, AP, is related to the Kelvin temperature T by APT = 2.90 X 10"3m -K.

(37 -1)

This is known as Wien's l w. EXAMPLE 37-1 The Sun's surface temperature. Estimate the temperature of the surface of our Sun, given that the Sun emits light whose peak intensity occurs in the visible spectrum at around 500 nm. APPR ACH We assume the Sun acts as a blackbody, and use AP = 500 nm in

Wien's law (Eq. 37-1). S LUTI N Wien's law gives 2.90 X 10"3m •K

2.90 X 10"3m •K 500 X 10_9m

6000 K.

EXAMPLE 37 - 2 Star color. Suppose a star has a surface temperature of 32,500 K. What color would this star appear? APPR ACH We assume the star emits radiation as a blackbody, and solve for AP

in Wien's law, Eq. 37-1. S LUTI N From Wien's law we have FIGURE 37 - 2 Comp rison o f the W ien nd the R yleigh - Je ns theories to th t o f Pl nck, which closely follow s experiment. The d shed lines show l ck o f greem ent o f older theories.

2.90 X 10"3m •K 2.90 X 10"3m •K = 89.2 nm. T 3.25 X 104K The peak is in the UV range of the spectrum, and will be way to the left in Fig. 37-1. In the visible region, the curve will be descending, so the shortest visible wavelengths will be strongest. Hence the star will appear bluish (or blue-white). N TE This example helps us to understand why stars have different colors (reddish for the coolest stars, orangish, yellow, white, bluish for "hotter" stars.) Ap —

Planck's Quantum Hypoth sis

Frequency (Hz)

988

CHAPTER 37

A major problem facing scientists in the 1890s was to explain blackbody radiation. Maxwell's electromagnetic theory had predicted that oscillating electric charges produce electromagnetic waves, and the radiation emitted by a hot object could be due to the oscillations of electric charges in the molecules of the material. Although this would explain where the radiation came from, it did not correctly predict the observed spectrum of emitted light. Two important theoretical curves based on classical ideas were those proposed by W. Wien (in 1896) and by Lord Rayleigh (in 1900). The latter was modified later by J. Jeans and since then has been known as the Rayleigh-Jeans theory. As experimental data came in, it became clear that neither Wien's nor the Rayleigh-Jeans formulations were in accord with experiment (see Fig. 37-2).

In the year 1900 Max Planck (1858-1947) proposed an empirical formula that nicely fit the data (now often called Planck's radiation formula): I U T) =

e c/XkT _ 1

/(A, T) is the radiation intensity as a function of wavelength Aat the temperature T; k is Boltzman's constant, c is the speed of light, and is a new constant, now called Pl nck's const nt. The value of was estimated by Planck by fitting his formula for the blackbody radiation curve to experiment. The value accepted today is = 6.626 X IO-34J •s. To provide a theoretical basis for his formula, Planck made a new and radical assumption: that the energy of the oscillations of atoms within molecules cannot have just any value; instead each has energy which is a multiple of a minimum value related to the frequency of oscillation by E = f. Planck's assumption suggests that the energy of any molecular vibration could be only a whole number multiple of the minimum energy f: E = n f, n = 1 ,2 ,3 ,- , (37 -2) where n is called a qu ntum number ("quantum" means "discrete amount" as opposed to "continuous"). This idea is often called Pl nck's qu ntum hypothesis, although little attention was brought to this point at the time. In fact, it appears that Planck considered it more as a mathematical device to get the "right answer" rather than as an important discovery in its own right. Planck himself continued to seek a classical explanation for the introduction of . The recognition that this was an important and radical innovation did not come until later, after about 1905 when others, particularly Einstein, entered the field. The quantum hypothesis, Eq. 37-2, states that the energy of an oscillator can be E = f, or 2 f, or 3 f, and so on, but there cannot be vibrations with energies between these values. That is, energy would not be a continuous quantity as had been believed for centuries; rather it is qu ntized—it exists only in discrete amounts. The smallest amount of energy possible ( f) is called the qu ntum of energy. Recall from hapter 14 that the energy of an oscillation is proportional to the amplitude squared. Another way of expressing the quantum hypothesis is that not just any amplitude of vibration is possible. The possible values for the amplitude are related to the frequency /. A simple analogy may help. ompare a ramp, on which a box can be placed at any height, to a flight of stairs on which the box can have only certain discrete amounts of potential energy, as shown in Fig. 37-3.

(b) FIGURE 3 7 - 3 R mp versus st ir n logy, ( ) On r mp, box c n h ve continuous v lues of potenti l energy, (b) But on st irs, the box c n h ve only discrete (qu ntized) v lues o f energy.

3 7 - 2 Photon Theory of Light nd the Photoelectric Effect In 1905, the same year that he introduced the special theory of relativity, Einstein made a bold extension of the quantum idea by proposing a new theory of light. Planck's work had suggested that the vibrational energy of molecules in a radiating object is quantized with energy E = n f, where n is an integer and / is the frequency of molecular vibration. Einstein argued that when light is emitted by a molecular oscillator, the molecule's vibrational energy of n f must decrease by an amount f (or by 2 f, etc.) to another integer times f, such as (n — 1) f. Then to conserve energy, the light ought to be emitted in packets, or quanta, each with an energy E = f, (37 -3) where / is here the frequency of the emitted light. Again is Planck's constant. Since all light ultimately comes from a radiating source, this suggests that perhaps lig t is transmitted as tiny particles, or photons, as they are now called, as well as via waves predicted by Maxwell's electromagnetic theory. The photon theory of light was a radical departure from classical ideas. Einstein proposed a test of the quantum theory of light: quantitative measurements on the photoelectric effect.

SECTI N 37-2

989

Lighl

Photocell

V V

I------------- ^

------------ 1

FIGURE 37-4 The photoele tri effe t.

When light shines on a metal surface, electrons are found to be emitted from the surface. This effect is called the photoelectric effect and it occurs in many materials, but is most easily observed with metals. It can be observed using the apparatus shown in Fig. 37-4. A metal plate P and a smaller electrode are placed inside an evacuated glass tube, called a photocell. The two electrodes are connected to an ammeter and a source of emf, as shown. When the photocell is in the dark, the ammeter reads zero. But when light of sufficiently high frequency illuminates the plate, the ammeter indicates a current flowing in the circuit. We explain completion of the circuit by imagining that electrons, ejected from the plate by the impinging radiation, flow across the tube from the plate to the "collector" as indicated in Fig. 37-4. That electrons should be emitted when light shines on a metal is consistent with the electromagnetic (EM) wave theory of light: the electric field of an EM wave could exert a force on electrons in the metal and eject some of them. Einstein pointed out, however, that the wave theory and the photon theory of light give very different predictions on the details of the photoelectric effect. For example, one thing that can be measured with the apparatus of Fig. 37 - 4 is the maximum kinetic energy ( ^ max) of the emitted electrons. This can be done by using a variable voltage source and reversing the terminals so that electrode is negative and P is positive. The electrons emitted from P will be repelled by the negative electrode, but if this reverse voltage is small enough, the fastest electrons will still reach and there will be a current in the circuit. If the reversed voltage is increased, a point is reached where the current reaches zero — no electrons have sufficient kinetic energy to reach . This is called the stopping potential, or stopping voltage, V0, and from its measurement, Kmax can be determined using conservation of energy (loss of kinetic energy = gain in potential energy): -^max —

•

Now let us examine the details of the photoelectric effect from the point of view of the wave theory versus Einstein's particle theory. First the wave theory, assuming monochromatic light. The two important prop erties of a light wave are its intensity and its frequency (or wavelength). When these two quantities are varied, the wave theory makes the following predictions: W ave t eory predictions

1. If the light intensity is increased, the number of electrons ejected and their maximum kinetic energy should be increased because the higher intensity means a greater electric field amplitude, and the greater electric field should eject electrons with higher speed. 2. The frequency of the light should not affect the kinetic energy of the ejected electrons. Only the intensity should affect Kmax. The photon theory makes completely different predictions. First we note that in a monochromatic beam, all photons have the same energy ( = f ). Increasing the intensity of the light beam means increasing the number of photons in the beam, but does not affect the energy of each photon as long as the frequency is not changed. According to Einstein's theory, an electron is ejected from the metal by a collision with a single photon. In the process, all the photon energy is transferred to the electron and the photon ceases to exist. Since electrons are held in the metal by attractive forces, some minimum energy W0 is required just to get an electron out through the surface. W0 is called the work function, and is a few electron volts (l eV = 1.6 X IO-19j) for most metals. If the frequency f of the incoming light is so low that f is less than W0, then the photons will not have enough energy to eject any electrons at all. If f > W0, then electrons will be ejected and energy will be conserved in the process. That is, the input energy (of the photon), f, will equal the outgoing kinetic energy K of the electron plus the energy required to get it out of the metal, W: f = K + W.

990

CHAPTER 37

(37-4 )

The least tightly held electrons will be emitted with the most kinetic energy (X™ ). in which case W in this equation becomes the work function W0,

and K becomes Kmax: f = K max + W0.

[least bound electrons] (37-4b)

Many electrons will require more energy than the bare minimum (W0) to get out of the metal, and thus the kinetic energy of such electrons will be less than the maximum. From these considerations, the photon theory makes the following predictions: 1. An increase in intensity of the light beam means more photons are incident, so more electrons will be ejected; but since the energy of each photon is not changed, the maximum kinetic energy of electrons is not changed by an increase in intensity. 2. If the frequency of the light is increased, the maximum kinetic energy of the electrons increases linearly, according to Eq. 37-4b. That is, Km„ =

P oton t eory predictions

f - W0.

This relationship is plotted in Fig. 37 -5. 3. If the frequency / is less than the "cutoff" frequency / 0, where f0 = W0, no electrons will be ejected, no matter how great the intensity of the light. These predictions of the photon theory are clearly very different from the predictions of the wave theory. In 1913-1914, careful experiments were carried out by R. A. Millikan. The results were fully in agreement with Einstein's photon theory. One other aspect of the photoelectric effect also confirmed the photon theory. If extremely low light intensity is used, the wave theory predicts a time delay before electron emission so that an electron can absorb enough energy to exceed the work function. The photon theory predicts no such delay—it only takes one photon (if its frequency is high enough) to eject an electron— and experiments showed no delay. This too confirmed Einstein's photon theory. Photon energy.

Frequency of lig h t/

FIGURE 37 - 5 Photoelectric effect: the m ximum kinetic energy of ejected electrons incre ses line rly with the frequency of incident light. N o electrons re em itted if / < / 0.

alculate the energy of a photon of blue

light, A = 450 nm in air (or vacuum). APPR ACH The photon has energy E = S LUTI N Since / = c/A, we have

f (Eq. 37 -3) where / = c/A.

e (6.63 X 10"34J-s)(3.0 X 108m/s) f = — = ------------t -------- — = 4.4 X 10"19J, J A (4.5 X 10 m)

E =

or (4.4 X 10-19j)/(l.60 X 10_19J/eV) = 2.8 eV. Section 23 -8, 1 eV = 1.60 X 10"19J.)

(See definition of eV in

ESTIMATE"! Photons from a lightbulb. Estimate how many visible light photons a 100-W lightbulb emits per second. Assume the bulb has a typical efficiency of about 3% (that is, 97% of the energy goes to heat). APPR ACH Let's assume an average wavelength in the middle of the visible

spectrum, A « 500 nm. The energy of each photon is E = f = e/A. Only 3% of the 100-W power is emitted as light, or 3 W = 3 J/s. The number of photons emitted per second equals the light output of 3 J/s divided by the energy of each photon. S LUTI N The energy emitted in one second ( = 3 J) is E = N f where N is the number of photons emitted per second and / = c/A. Hence a

_____

______

(3J)(500 X 10- rn J v_________________________ i _____________

_____________ v

i r \18

(6.63 X 10"34J-s)(3.0 X 10s m/s)

per second, or almost 1019photons emitted per second, an enormous number. EXERCISE A Comp re light be m th t cont ins infr red light of single w velength, 1000 nm, with be m of monochrom tic U V t 100 nm, both of the s me intensity. A re there m ore 100 - nm photons or m ore 1000 - nm photons?

SECTI N 37-2

991

EXAMPLE 37 - 5 Photoelectron speed and energy. What is the kinetic energy and the speed of an electron ejected from a sodium surface whose work function is W0 = 2.28 eV when illuminated by light of wavelength (a) 410 nm, (b) 550 nm? APPR ACH We first find the energy of the photons (E =

f = c /\). If the energy is greater than WQ, then electrons will be ejected with varying amounts of kinetic energy, with a maximum of Kmax = f — W0. S LUTI N (a) For A = 410 nm, e f = — = 4.85 x 10 19J A

3.03 eV.

The maximum kinetic energy an electron can have is given by Eq. 37-4b, Km„ = 3.03 eV - 2.28 eV = 0.75 eV, or (0.75 eV)(l.60 x IO-19J/eV) = 1.2 x 10"19J. Since K = \trnf1 where m = 9.11 X H r 31 kg, '

= 5.1

105m/s.

Most ejected electrons will have less kinetic energy and less speed than these maximum values. (ft) For A = 550 nm, f = c /\ = 3.61 X 10-19J = 2.26 eV. Since this photon energy is less than the work function, no electrons are ejected. N TE In (a) we used the nonrelativistic equation for kinetic energy. If v had turned out to be more than about 0.1c, our calculation would have been inaccurate by at least a percent or so, and we would probably prefer to redo it using the relativistic form (Eq. 36-10). EXERCISE B D eterm ine the low est frequency nd the longest w velength needed to emit electrons from sodium. FIGURE 3 7 - 6 Optic l sound tr ck on m ovie film. In the projector, light from sm ll source (different from th t for the picture) p sses through the sound tr ck on the m oving film. The light nd d rk re s on the sound tr ck v ry the intensity o f the tr nsmitted light which re ches the photocell, w hose output current is then replic of the origin l sound. This output is mplified nd sent to the loudspe kers. High - qu lity projectors c n show m ovies cont ining sever l p r llel sound tr cks to go to different spe kers round the the ter. Picture

992

CHAPTER 37

It is easy to show that the energy of a photon in electron volts, when given the wavelength Ain nm, is 1.240 X 103eV -nm ri E (eV) = -------- —---- ---------- [photon energy in eV] A (nm)

Tri

Applications of th Photo l ctric Eff ct The photoelectric effect, besides playing an important historical role in confirming the photon theory of light, also has many practical applications. Burglar alarms and automatic door openers often make use of the photocell circuit of Fig. 37 -4. When a person interrupts the beam of light, the sudden drop in current in the circuit activates a switch—often a solenoid — which operates a bell or opens the door. UV or IR light is sometimes used in burglar alarms because of its invisibility. Many smoke detectors use the photoelectric effect to detect tiny amounts of smoke that interrupt the flow of light and so alter the electric current. Photographic light meters use this circuit as well. Photocells are used in many other devices, such as absorption spectrophotometers, to measure light intensity. One type of film sound track is a variably shaded narrow section at the side of the film. Light passing through the film is thus "modulated," and the output electrical signal of the photocell detector follows the frequencies on the sound track. See Fig. 37-6. For many applications today, the vacuum-tube photocell of Fig. 37 - 4 has been replaced by a semiconductor device known as a photodiode (Section 40 -9). In these semiconductors, the absorption of a photon liberates a bound electron, which changes the conductivity of the material, so the current through a photodiode is altered.

Early Qua tum Theory a d Models of the Atom

3 7 -3 Energy, M ss, nd Momentum o f Photon We have just seen (Eq. 37 -3) that the total energy of a single photon is given by E = f. Because a photon always travels at the speed of light, it is truly a relativistic particle. Thus we must use relativistic formulas for dealing with its energy and momentum. The momentum of any particle of mass m is given by p = m v / \ / l — v2/c 2. Since v = c for a photon, the denominator is zero. To avoid having an infinite momentum, we conclude that the photon's mass must be zero: m = 0. This makes sense too because a photon can never be at rest (it always moves at the speed of light). A photon's kinetic energy is its total energy: K = E =

[photon]

The momentum of a photon can be obtained from the relativistic formula (Eq. 36-13) E 2 = p 2c2 + m2cA where we set m = 0, so E 2 = p2c2 or r[photon] 1 i

E• p = — Since E =

/f\

AUT I ON __________

M om entum o f p oton is n ot m v

f for a photon, its momentum is related to its wavelength by E f p = - = i = -•

(37 -5)

EXAMPLE 37 - 6 Photon momentum and force. Suppose the 1019 photons emitted per second from the 100-W lightbulb in Example 37 -4 were all focused onto a piece of black paper and absorbed, (a) alculate the momentum of one photon and (b) estimate the force all these photons could exert on the paper. APPR ACH Each photon's momentum is obtained from Eq. 37 -5, p = /k.

Next, each absorbed photon's momentum changes from p = /k to zero. We use Newton's second law, F = A p/ At, to get the force. Let A = 500 nm. S LUTI N (a) Each photon has a momentum p = — = y A

6.63 X 10-34J - s „ „ „ ~7l ---- q---- = 1.3 X 10 27kg - m/s. 500 X 10 m 6

(b) Using Newton's second law for N = 1019 photons (Example 37 -4) whose momentum changes from /k to 0, we obtain F =

= NH^ ~ ° = N j f» (lO19s-1)(l0 -27 kg •m/s) « 10"8N.

This is a tiny force, but we can see that a very strong light source could exert a measurable force, and near the Sun or a star the force due to photons in electromagnetic radiation could be considerable. See Section 31-9. EXAMPLE 37 - 7 Photosynthesis. In p otosynt esis, pigments such as chlorophyll in plants capture the energy of sunlight to change 0 2 to useful carbohydrate. About nine photons are needed to transform one molecule of 0 2to carbohydrate and 0 2. Assuming light of wavelength A = 670 nm (chlorophyll absorbs most strongly in the range 650 nm to 700 nm), how efficient is the photosynthetic process? The reverse chemical reaction releases an energy of 4.9 eV/molecule of 0 2. APPR ACH The efficiency is the minimum energy required (4.9 eV) divided by

the actual energy absorbed, nine times the energy ( f) of one photon. S LUTI N The energy of nine photons, each of energy f = c/k is (9)(6.63 X 10"34J-s)(3.0 X 108m/s)/(6.7 X 10"7m) = 2.7 X 10"18J or 17eV. Thus the process is (4.9 eV/17 eV) = 29% efficient. SECTION 3 7 - 3

E ergy, Mass, a d Mome tum of a Photo

993

BEFORE COLLISION y±

AFTER COLLISION Sc ttered photon (A")j

Incident photon (A) Electron^ t rest initi lly FIGURE 37 - 7 The Compton effect. A single photon of w velength A strikes n electron in some m teri l, knocking it out of its tom. The sc ttered photon h s less energy (som e energy is given to the electron) nd hence h s longer w velength A'.

FIGURE 37 - 8 Plots of intensity of r di tion sc ttered from t rget such s gr phite (c rbon), for three different ngles. The v lues for A' m tch Eq. 37 - 6. For ( ) (f> = 0°, A' = A0. In (b) nd (c) pe k is found not only t A' due to photons sc ttered from free electrons (or very ne rly free), but lso pe k t lmost precisely A0 . The l tter is due to sc ttering from electrons very tightly bound to their toms so the m ss in Eq. 3 7 - 6 becomes very l rge (m ss of the tom) nd AA becomes very sm ll.

3 7 - 4 Compton Effect Besides the photoelectric effect, a number of other experiments were carried out in the early twentieth century which also supported the photon theory. One of these was the Compton effect (1923) named after its discoverer, A. H. ompton (1892-1962). ompton scattered short-wavelength light (actually X-rays) from various materials. He found that the scattered light had a slightly longer wavelength than did the incident light, and therefore a slightly lower frequency indicating a loss of energy. He explained this result on the basis of the photon theory as incident photons colliding with electrons of the material, Fig. 37 -7. Using Eq. 37 -5 for momentum of a photon, ompton applied the laws of conservation of momentum and energy to the collision of Fig. 37 -7 and derived the following equation for the wavelength of the scattered photons: A' = A H-------- (1 - cos d>), m Pc v '

(37-6 )

where rae is the mass of the electron. For = 0, the wavelength is unchanged (there is no collision for this case of the photon passing straight through). At any other angle, A' is longer than A. The difference in wavelength, AA = A'

A = ----- (1 m ec

cos (f>),

(37-6b)

is called the Compton shift. The quantity /m ec, which has the dimensions of length, is called the Compton w velength Ac of a free electron, Ar = ----- = 2.43 X 10 3nm = 2.43 pm. m Pc

[electron]

Equations 37 - 6 predict that A' depends on the angle = 0, and 1 - cos (f> = 1. So A' = A

mec

____ 0.140 nm

6.63 X 10-34J •s (9.11 X 10-31 kg)(3.00 X 108m/s)

= 0.140 nm + 2.4 X 10-12m = 0.142 nm; that is, the wavelength is longer by one an electron).

994

CHAPTER 37

ompton wavelength ( = 0.0024 nm for

= 0.140 nm + 2(0.0024 nm) = 0.145 nm.

m ec

N TE The maximum shift in wavelength occurs for backward scattering, and it is twice the ompton wavelength.

The ompton effect has been used to diagnose bone disease such as osteoporosis. Gamma rays, which are photons of even shorter wavelength than X-rays, coming from a radioactive source are scattered off bone material. The total intensity of the scattered radiation is proportional to the density of electrons, which is in turn proportional to the bone density. hanges in the density of bone can indicate the onset of osteoporosis.

PHYSICS

APPLIED

Measuring bone density

* D rivation of Compton Shift If the incoming photon in Fig. 37 -7 has wavelength A, then its total energy and momentum are EZ7 = i rf = —c

anda

p = -•

After the collision of Fig. 37-7, the photon scattered at the angle We apply conservation of momentum to the x and y components of momentum: A

—

OS (f> +

m P v cos 6 - \ /i _ v2/ c2

m ev sin 6 0 = — sin (f) A' V l - «2/c2 We can combine these three equations to eliminate v and 6, and we obtain, as ompton did, an equation for the wavelength of the scattered photon in terms of its scattering angle 0: A' = A H--------(1 - cos 22.8 nm, the photon has too little energy to cause ionization.

1008

HAPTER 37

In this last Example, we saw that E1for the He+ ion is four times more negative than that for hydrogen. Indeed, the energy-level diagram for He+ looks just like that for hydrogen, Fig. 37-26, except that the numerical values for each energy level are four times larger. Note, however, that we are talking here about the He+ ion. Normal (neutral) helium has two electrons and its energy level diagram is entirely different.

ON EPTUAL EXAMPLE 37-17 I Hydrogen at 20° . Estimate the average kinetic energy of whole hydrogen atoms (not just the electrons) at room temperature, and use the result to explain why nearly all H atoms are in the ground state at room temperature, and hence emit no light. RESP NSE According to kinetic theory ( hapter 18), the average kinetic energy of atoms or molecules in a gas is given by Eq. 18-4: K = \k T , where k = 1.38 X 10-23J/K is Boltzmann's constant, and T is the kelvin (absolute) temperature. Room temperature is about T = 300 K, so K = |(1.38

10"23J/K)(300K) = 6.2

10"21J,

or, in electron volts:

*The average kinetic energy of an atom as a whole is thus very small compared to the energy between the ground state and the next higher energy state (13.6 eV - 3.4 eV = 10.2 eV). Any atoms in excited states quickly fall to the ground state and emit light. Once in the ground state, collisions with other atoms can transfer energy of only 0.04 eV on the average. A small fraction of atoms can have much more energy (see Section 18 -2 on the distribution of molecular speeds), but even a kinetic energy that is 10 times the average is not nearly enough to excite atoms into states above the ground state. Thus, at room temperature, nearly all atoms are in the ground state. Atoms can be excited to upper states by very high temperatures, or by passing a current of high energy electrons through the gas, as in a discharge tube (Fig. 37-19).

orrespondence Principle We should note that Bohr made some radical assumptions that were at variance with classical ideas. He assumed that electrons in fixed orbits do not radiate light even though they are accelerating (moving in a circle), and he assumed that angular momentum is quantized. Furthermore, he was not able to say how an electron moved when it made a transition from one energy level to another. On the other hand, there is no real reason to expect that in the tiny world of the atom electrons would behave as ordinary-sized objects do. Nonetheless, he felt that where quantum theory overlaps with the macroscopic world, it should predict classical results. This is the correspondence principle, already mentioned in regard to relativity (Section 36-13). This principle does work for Bohr's theory of the hydrogen atom. The orbit sizes and energies are quite different for n = 1 and n = 2, say. But orbits with n = 100,000,000 and 100,000,001 would be very close in radius and energy (see Fig. 37-26). Indeed, jumps between such large orbits (which would approach macroscopic sizes), would be imperceptible. Such orbits would thus appear to be continuously spaced, which is what we expect in the everyday world. Finally, it must be emphasized that the well-defined orbits of the Bohr model do not actually exist. The Bohr model is only a model, not reality. The idea of electron orbits was rejected a few years later, and today electrons are thought of ( hapter 39) as forming "probability clouds."

3 7 -1 2 de B roglie's H y p o th esis A p p lied to A tom s Bohr's theory was largely of an ad oc nature. Assumptions were made so that theory would agree with experiment. But Bohr could give no reason why the orbits were quantized, nor why there should be a stable ground state. Finally, ten years later, a reason was proposed by Louis de Broglie. We saw in Section 37 - 7 that in 1923, de Broglie proposed that material particles, such as electrons, have a wave nature; and that this hypothesis was confirmed by experiment several years later.

SECTI N 37 -1]

1009

One of de Broglie's original arguments in favor of the wave nature of electrons was that it provided an explanation for Bohr's theory of the hydrogen atom. According to de Broglie, a particle of mass m moving with a nonrelativistic speed v would have a wavelength (Eq. 37 -7) of

FIG U R E 3 7 - 2 7 A n ordin ry st nding w ve comp red to circul r st nding w ve. FIG U R E 3 7 - 2 8 W hen w ve does not close ( nd hence interferes destructively with itself), it r pidly dies out.

A - -* -• mv Each electron orbit in an atom, he proposed, is actually a standing wave. As we saw in hapter 15, when a violin or guitar string is plucked, a vast number of wavelengths are excited. But only certain ones— those that have nodes at the ends— are sustained. These are the resonant modes of the string. Waves with other wavelengths interfere with themselves upon reflection and their amplitudes quickly drop to zero. With electrons moving in circles, according to Bohr's theory, de Broglie argued that the electron wave was a circular standing wave that closes on itself, Fig. 31-21. If the wavelength of a wave does not close on itself, as in Fig. 37-28, destructive interference takes place as the wave travels around the loop, and the wave quickly dies out. Thus, the only waves that persist are those for which the circumference of the circular orbit contains a whole number of wavelengths, Fig. 37-29. The circumference of a Bohr orbit of radius rn is 2irrn, so to have constructive interference, we need 2irrn = nk,

FIG U R E 3 7 - 2 9 St nding circul r w ves for two, three, nd five w velengths on the circumference; the number of w velengths, is lso the qu ntum number.

n = 1,2,3,

When we substitute A = /m v , we get 2irrn = n /m v, or n mVrn 2 ir' This is just the quantum condition proposed by Bohr on an ad oc basis, Eq. 37-10. It is from this equation that the discrete orbits and energy levels were derived. Thus we have a first explanation for the quantized orbits and energy states in the Bohr model: they are due to the wave nature of the electron, and only resonant "standing" waves can persist.1This implies that the wave-particle duality is at the root of atomic structure. In viewing the circular electron waves of Fig. 37-29, the electron is not to be thought of as following the oscillating wave pattern. In the Bohr model of hydrogen, the electron moves in a circle. The circular wave, on the other hand, represents the amplitude of the electron "matter wave," and in Fig. 37-29 the wave amplitude is shown superimposed on the circular path of the particle orbit for convenience. Bohr's theory worked well for hydrogen and for one-electron ions. But it did not prove successful for multi-electron atoms. Bohr's theory could not predict line spectra even for the next simplest atom, helium. It could not explain why some emission lines are brighter than others, nor why some lines are split into two or more closely spaced lines ("fine structure"). A new theory was needed and was indeed developed in the 1920s. This new and radical theory is called quantum mec anics. It finally solved the problem of atomic structure, but it gives us a very different view of the atom: the idea of electrons in well-defined orbits was replaced with the idea of electron "clouds." This new theory of quantum mechanics has given us a wholly different view of the basic mechanisms underlying physical processes. fWe note, however, th t Eq. 37 - 10 is no longer considered v lid, s discussed in Ch pter 39.

Summary Qu ntum theory h s its origins in Pl nck's qu ntum hypothesis th t m olecul r oscill tions re qu ntized: their energy c n only be integer ( ) m ultiples of where is Pl nck 's const nt nd / is the n tur l frequency of oscill tion:

n E=nf.

(3 7 - 2 )

Einstein proposed th t for som e experiments, light could be pictured s being em itted nd bsorbed s qu nt (p rticles), which w e now c ll photons, e ch with energy

This hypothesis expl ined the spectrum of r di tion em itted by bl ckbody t high temper ture.

1010

HAPTER 37

E= f E_ f_= c

(3 7 - 3 )

nd m om entum

Early Quantum Theory and Models of the Atom

(37-5)

He proposed the photoelectric effect as a test for the photon theory of light. In the photoelectric effect, the photon theory says that each incident photon can strike an electron in a mate rial and eject it if the photon has sufficient energy. The maximum energy of ejected electrons is then linearly related to the frequency of the incident light. The photon theory is also supported by the Compton effect and the observation of electron -positron p ir production. The w ve-p rticle du lity refers to the idea that light and matter (such as electrons) have both wave and particle proper ties. The wavelength of an object is given by

A = —>

(37-7)

P where p is the momentum of the object (p = mv for a particle of mass m and speed v). The principle of complement rity states that we must be aware of both the particle and wave properties of light and of matter for a complete understanding of them. Early models of the atom include the plum-pudding model, and Rutherford's planetary (or nuclear) model of an atom which consists of a tiny but massive positively charged nucleus surrounded (at a relatively great distance) by electrons. To explain the line spectr emitted by atoms, as well as the stability of atoms, Bohr's theory postulated that: (1) electrons bound in an atom can only occupy orbits for which the angular

momentum is quantized, which results in discrete values for the radius and energy; (2) an electron in such a st tion ry st te emits no radiation; (3) if an electron jumps to a lower state, it emits a photon whose energy equals the difference in energy between the two states; (4) the angular momentum L of atomic electrons is quantized by the rule

where n is an integer called the qu ntum number. The n = 1 state is the ground st te, which in hydrogen has an energy Ei = —13.6 eV. Higher values of n correspond to excited st tes, and their energies are Z2 En = —(13.6 eV) —r nl

(37-14b)

Atoms are excited to these higher states by collisions with other atoms or electrons, or by absorption of a photon of just the right frequency. De Broglie's hypothesis that electrons (and other matter) have a wavelength A = /m v gave an explanation for Bohr's quantized orbits by bringing in the wave-particle duality: the orbits correspond to circular standing waves in which the circumference of the orbit equals a whole number of wavelengths.

Q uestions 1. What can be said about the relative temperatures of whitish-yellow, reddish, and bluish stars? Explain. 2. If energy is radiated by all objects, why can we not see most of them in the dark?

3. Does a lightbulb at a temperature of 2500 K produce as white a light as the Sun at 6000 K? Explain. 4. Darkrooms for developing black-and-white film were some times lit by a red bulb. Why red? Would such a bulb work in a darkroom for developing color photographs?

5. If the threshold wavelength in the photoelectric effect increases when the emitting metal is changed to a different metal, what can you say about the work functions of the two metals? 6. Explain why the existence of a cutoff frequency in the photoelectric effect more strongly favors a particle theory rather than a wave theory of light.

7. UV light causes sunburn, whereas visible light does not. Suggest a reason. 8. The work functions for sodium and cesium are 2.28 eV and 2.14 eV, respectively. For incident photons of a given frequency, which metal will give a higher maximum kinetic energy for the electrons? 9. (a) Does a beam of infrared photons always have less energy than a beam of ultraviolet photons? Explain. (b) Does a single photon of infrared light always have less energy than a single photon of ultraviolet light? 10. Light of 450-nm wavelength strikes a metal surface, and a stream of electrons emerges from the metal. If light of the same intensity but of wavelength 400 nm strikes the surface, are more electrons emitted? Does the energy of the emitted electrons change? Explain.

11. Explain how the photoelectric circuit of Fig. 37 -4 could be used in (a) a burglar alarm, (b) a smoke detector, (c) a photographic light meter. 12. Kan X-ray photon is scattered by an electron, does the photon's wavelength change? If so, does it increase or decrease? 13. In both the photoelectric effect and in the ompton effect, a photon collides with an electron causing the electron to fly off. What then, is the difference between the two processes?

14.

onsider a point source of light. How would the intensity of light vary with distance from the source according to (a) wave theory, (b) particle (photon) theory? Would this help to distinguish the two theories? 15. If an electron and a proton travel at the same speed, which has the shorter de Broglie wavelength? Explain.

16. Why do we say that light has wave properties? Why do we say that light has particle properties?

17. Why do we say that electrons have wave properties? Why do we say that electrons have particle properties?

18. What are the differences between a photon and an electron? Be specific: make a list.

19. In Rutherford's planetary model of the atom, what keeps the electrons from flying off into space?

20. How can you tell if there is oxygen near the surface of the Sun?

21. When a wide spectrum of light passes through hydrogen gas at room temperature, absorption lines are observed that correspond only to the Lyman series. Why don't we observe the other series? 22. Explain how the closely spaced energy levels for hydrogen near the top of Fig. 37-26 correspond to the closely spaced spectral lines at the top of Fig. 37-21.

Questions

1011

23. Is it possible for the de Broglie wavelength of a "particle" to

27. Use conservation of momentum to explain why photons

be greater than the dimensions of the particle? To be smaller? Is there any direct connection? 24. In a helium atom, which contains two electrons, do you think that on average the electrons are closer to the nucleus or farther away than in a hydrogen atom? Why? 25. How can the spectrum of hydrogen contain so many lines when hydrogen contains only one electron? 26. The Lyman series is brighter than the Balmer series because this series of transitions ends up in the most common state for hydrogen, the ground state. Why then was the Balmer series discovered first?

emitted by hydrogen atoms have slightly less energy than that predicted by Eq. 37 - 9. 28. Suppose we obtain an emission spectrum for hydrogen at very high temperature (when some of the atoms are in excited states), and an absorption spectrum at room temper ature, when all atoms are in the ground state. Will the two spectra contain identical lines?

P ro b lem s 37 -1 Planck's Quantum Hypothesis 1. (I) Estimate the peak wavelength for radiation from (a) ice at 273K, (ib) a floodlamp at 3500 K, (c) helium at 4.2 K, (d) for the universe at T = 2.725 K, assuming blackbody emission. In what region of the EM spectrum is each? 2. (I) How hot is metal being welded if it radiates most strongly at 460 nm? 3. (I) An H 1 molecule vibrates with a natural frequency of 8.1 X 1013 Hz. What is the difference in energy (in joules and electron volts) between successive values of the oscilla tion energy? 4. (II) Estimate the peak wavelength of light issuing from the pupil of the human eye (which approximates a blackbody) assuming normal body temperature. 5. (Ill) Planck's radiation law is given by:

11. (I) What minimum frequency of light is needed to eject electrons from a metal whose work function is 4.8 X 1 0 19J?

12. (I) What is the longest wavelength of light that will emit electrons from a metal whose work function is 3.70 eV?

13. (II) What wavelength photon would have the same energy 14.

15.

16.

^ where /(A, T) is the rate energy is radiated per unit surface area per unit wavelength interval at wavelength A and Kelvin temperature T. (a) Show that Wien's displacement law follows from this relationship, (b) Determine the value of from the experimental value of APT given in the text. [You may want to use graphing techniques.] (c) Derive the r 4 dependence of the rate at which energy is radiated (as in the Stefan-Boltzmann law, Eq. 19-17), by integrating Planck's formula over all wavelengths; that is, show that j

/(A, T) dX oc T \

17.

18.

19.

20.

37 - 2 and 37 -3 Photons and the Photoelectric Effect 6. (I) What is the energy of photons (in joules) emitted by a 104.1-MHz FM radio station? 7. (I) What is the energy range (in joules and eV) of photons in the visible spectrum, of wavelength 410 nm to 750 nm? 8. (I) A typical gamma ray emitted from a nucleus during radioactive decay may have an energy of 380 keV. What is its wavelength? Would we expect significant diffraction of this type of light when it passes through an everyday opening, such as a door? 9. (I) About 0.1 eV is required to break a "hydrogen bond" in a protein molecule. alculate the minimum frequency and maximum wavelength of a photon that can accomplish this. 10. (I) alculate the momentum of a photon of yellow light of wavelength 6.20 X 10 7m.

1012

HAPTER 37

21.

22.

23.

as a 145-gram baseball moving 30.0 m/s? (II) The human eye can respond to as little as 10-18 J of light energy. For a wavelength at the peak of visual sensitivity, 550 nm, how many photons lead to an observable flash? (II) The work functions for sodium, cesium, copper, and iron are 2.3, 2.1, 4.7, and 4.5 eV, respectively. Which of these metals will not emit electrons when visible light shines on it? (II) In a photoelectric -effect experiment it is observed that no current flows unless the wavelength is less than 520 nm. (a) What is the work function of this material? (b) What is the stopping voltage required if light of wavelength 470 nm is used? (II) What is the maximum kinetic energy of electrons ejected from barium (W0 = 2.48 eV) when illuminated by white light, A = 410 to 750 nm? (II) Barium has a work function of 2.48 eV. What is the maximum kinetic energy of electrons if the metal is illumi nated by UV light of wavelength 365 nm? What is their speed? (II) When UV light of wavelength 285 nm falls on a metal surface, the maximum kinetic energy of emitted electrons is 1.70 eV. What is the work function of the metal? (II) The threshold wavelength for emission of electrons from a given surface is 320 nm. What will be the maximum kinetic energy of ejected electrons when the wavelength is changed to (a) 280 nm, (b) 360 nm? (II) When 230-nm light falls on a metal, the current through a photoelectric circuit (Fig. 37 -4) is brought to zero at a stop ping voltage of 1.84 V. What is the work function of the metal? (II) A certain type of film is sensitive only to light whose wavelength is less than 630 nm. What is the energy (eV and kcal/mol) needed for the chemical reaction to occur which causes the film to change? (II) The range of visible light wavelengths extends from about 410 nm to 750 nm. (a) Estimate the minimum energy (eV) necessary to initiate the chemical process on the retina that is responsible for vision, (b) Speculate as to why, at the other end of the visible range, there is a threshold photon energy beyond which the eye registers no sensation of sight. Determine this threshold photon energy (eV).

Early Quantum Theory and Models of the Atom

24. (II) In a photoelectric experiment using a clean sodium

surface, the maximum energy of the emitted electrons was measured for a number of different incident frequencies, with the following results. Frequency (x 1014 Hz)

Energy (eV)

11.8 10.6 9.9 9.1 8.2 6.9

2.60 2.11 1.81 1.47 1.10 0.57

Plot the graph of these results and find: (a) Planck's constant; (b) the cutoff frequency of sodium; (c) the work function. 25. (II) A photomultiplier tube (a very sensitive light sensor), is

based on the photoelectric effect: incident photons strike a metal surface and the resulting ejected electrons are collected. By counting the number of collected electrons, the number of incident photons (i.e., the incident light intensity) can be determined. (a) If a photomultiplier tube is to respond properly for incident wavelengths throughout the visible range (410 nm to 750 nm), what is the maximum value for the work function W0 (eV) of its metal surface? (b) If W0 for its metal surface is above a certain threshold value, the photomultiplier will only function for incident ultraviolet wavelengths and be unresponsive to visible light. Determine this threshold value (eV). 26. (Ill) A group of atoms is confined to a very small (point-like) volume in a laser-based tom tr p. The incident laser light causes each atom to emit 1.0 X 106 photons of wavelength 780 nm every second. A sensor of area 1.0 cm2 measures the light intensity emanating from the trap to be 1.6 nW when placed 25 cm away from the trapped atoms. Assuming each atom emits photons with equal probability in all directions, determine the number of trapped atoms. 27. (Ill) Assume light of wavelength A is incident on a metal surface, whose work function is known precisely (i.e., its uncertainty is better than 0.1% and can be ignored). Show that if the stopping voltage can be determined to an accuracy of A ^ , the fractional uncertainty (magnitude) in wavelength is

Determine this fractional uncertainty if A = 550 nm.

= 0.01 V and

37 - 4 Compton Effect 28. (I) A high-frequency photon is scattered off of an electron

and experiences a change of wavelength of 1.5 X 10-4 nm. At what angle must a detector be placed to detect the scattered photon (relative to the direction of the incoming photon)? 29. (II) Determine the ompton wavelength for (a) an elec tron, (b) a proton, (c) Show that if a photon has wavelength equal to the ompton wavelength of a particle, the photon's energy is equal to the rest energy of the particle. 30. (II) X-rays of wavelength A = 0.120 nm are scattered from carbon. What is the expected ompton wavelength shift for photons detected at angles (relative to the incident beam) of exactly {a) 60°, (b) 90°, (c) 180°?

31. (II) In the

ompton effect, determine the ratio (AA/A) of the maximum change AA in a photon's wavelength to the photon's initial wavelength A, if the photon is (a) a visiblelight photon with A = 550 nm, (b) an X-ray photon with A = 0.10 nm. 32. (II) A 1.0-MeV gamma-ray photon undergoes a sequence of ompton-scattering events. If the photon is scattered at an angle of 0.50° in each event, estimate the number of events required to convert the photon into a visible-light photon with wavelength 555 nm. You can use an expansion for small 0; see Appendix A. [Gamma rays created near the center of the Sun are transformed to visible wavelengths as they travel to the Sun's surface through a sequence of small-angle ompton scattering events.] 33. (Ill) In the ompton effect, a 0.160-nm photon strikes a free electron in a head-on collision and knocks it into the forward direction. The rebounding photon recoils directly backward. Use conservation of (relativistic) energy and momentum to determine (a) the kinetic energy of the elec tron, and (b) the wavelength of the recoiling photon. Use Eq. 37 -5, but not Eq. 37-6. 34. (Ill) In the ompton effect (see Fig. 37-7), use the rela tivistic equations for conservation of energy and of linear momentum to show that the ompton shift in wavelength is given by Eq. 37-6. 37 - 5 Pair Production

35. (I) How much total kinetic energy will an electron -positron 36. 37.

38.

39.

pair have if produced by a 2.67-MeV photon? (II) What is the longest wavelength photon that could produce a proton - antiproton pair? (Each has a mass of 1.67 X 10"27kg.) (II) What is the minimum photon energy needed to produce a /jl+pair? The mass of each /jl (muon) is 207 times the mass of an electron. What is the wavelength of such a photon? (II) An electron and a positron, each moving at 2.0 X 105m/s, collide head on, disappear, and produce two photons moving in opposite directions, each with the same energy and momentum. Determine the energy and momentum of each photon. (II) A gamma-ray photon produces an electron and a positron, each with a kinetic energy of 375 keV. Determine the energy and wavelength of the photon.

37 - 7 Wave Nature of Matter alculate the wavelength of a 0.23-kg ball traveling at 0.10 m/s. 41. (I) What is the wavelength of a neutron (ra = 1.67 X 10_27kg) traveling at 8.5 X 104m/s? 42. (I) Through how many volts of potential difference must an electron be accelerated to achieve a wavelength of 0.21 nm? 43. (II) What is the theoretical limit of resolution for an elec tron microscope whose electrons are accelerated through 85 kV? (Relativistic formulas should be used.) 44. (II) The speed of an electron in a particle accelerator is 0.98c. Find its de Broglie wavelength. (Use relativistic momentum.) 45. (II) alculate the ratio of the kinetic energy of an electron to that of a proton if their wavelengths are equal. Assume that the speeds are nonrelativistic.

40. (I)

Problem s

1013

46. (II) Neutrons can be used in diffraction experiments to

47.

48. 49.

50.

51.

probe the lattice structure of crystalline solids. Since the neutron's wavelength needs to be on the order of the spacing between atoms in the lattice, about 0.3 nm, what should the speed of the neutrons be? (II) An electron has a de Broglie wavelength A = 6.0 X IO-10 m. (a) What is its momentum? (b) What is its speed? (c) What voltage was needed to accelerate it to this speed? (II) What is the wavelength of an electron of energy (a) 20 eV, (b) 200 eV, (c) 2.0 keV? (II) Show that if an electron and a proton have the same nonrelativistic kinetic energy, the proton has the shorter wavelength. (II) alculate the de Broglie wavelength of an electron in a TV picture tube if it is accelerated by 33,000 V. Is it rela tivistic? How does its wavelength compare to the size of the "neck" of the tube, typically 5 cm? Do we have to worry about diffraction problems blurring our picture on the screen? (II) After passing through two slits separated by a distance of 3.0 /mi, a beam of electrons creates an interference pattern with its second -order maximum at an angle of 55°. Find the speed of the electrons in this beam.

*37 - 8 Electron Microscope *52. (II) What voltage is needed to produce electron wavelengths of 0.28 nm? (Assume that the electrons are nonrelativistic.) *53. (II) Electrons are accelerated by 3450 V in an electron microscope. Estimate the maximum possible resolution of the microscope.

59. (II) What is the longest wavelength light capable of ionizing a hydrogen atom in the ground state?

60. (II) In the Sun, an ionized helium (He+) atom makes a tran sition from the n = 5 state to the n = 2 state, emitting a photon. an that photon be absorbed by hydrogen atoms present in the Sun? If so, between what energy states will the hydrogen atom transition occur? 61. (II) What wavelength photon would be required to ionize a hydrogen atom in the ground state and give the ejected electron a kinetic energy of 20.0 eV? 62. (II) For what maximum kinetic energy is a collision between an electron and a hydrogen atom in its ground state definitely elastic?

63. (II)

onstruct the energy-level diagram for the H e+ ion (like Fig. 37 -26).

64. (II) 65.

66.

67.

68.

37 - 10 and 37 -11 Bohr Model 54. (I) For the three hydrogen transitions indicated below, with n being the initial state and n' being the final state, is the transition an absorption or an emission? Which is higher, the initial state energy or the final state energy of the atom? Finally, which of these transitions involves the largest energy photon? (a) n = 1, n' = 3; (b) n = 6, n' = 2; (c) n = 4, n' = 5. 55. (I) How much energy is needed to ionize a hydrogen atom in the n = 3 state? 56. (I) (a) Determine the wavelength of the second Balmer line (n = 4 to n = 2 transition) using Fig. 37 -26. Determine likewise (b) the wavelength of the third Lyman line and (c) the wavelength of the first Balmer line. 57. (I) alculate the ionization energy of doubly ionized lithium, Li2+, which has Z = 3. 58. (I) Evaluate the Rydberg constant R using the Bohr model (compare Eqs. 37 - 8 and 37 -15) and show that its value is R = 1.0974 X 107m _1.

70.

onstruct the energy-level diagram (like Fig. 37-26) for doubly ionized lithium, Li2+. (II) Determine the electrostatic potential energy and the kinetic energy of an electron in the ground state of the hydrogen atom. (II) An excited hydrogen atom could, in principle, have a diameter of 0.10 mm. What would be the value of n for a Bohr orbit of this size? What would its energy be? (II) Is the use of nonrelativistic formulas justified in the Bohr atom? To check, calculate the electron's velocity, v, in terms of c, for the ground state of hydrogen, and then calcu late \ / l - v2lc 2. (II) A hydrogen atom has an angular momentum of 5.273 X 10-34 kg •m2/s. According to the Bohr model, what is the energy (eV) associated with this state? (II) Assume hydrogen atoms in a gas are initially in their ground state. If free electrons with kinetic energy 12.75 eV collide with these atoms, what photon wavelengths will be emitted by the gas? (II) Suppose an electron was bound to a proton, as in the hydrogen atom, but by the gravitational force rather than by the electric force. What would be the radius, and energy, of the first Bohr orbit?

71. (II) Correspondence principle: Show that for large values of n, the difference in radius Ar between two adjacent orbits (with quantum numbers n and n - 1) is given by Ar = rn - rn- i « — > n so Ar/rn —>• 0 as n —» oo in accordance with the correspon dence principle. [Note that we can check the correspondence principle by either considering large values of n (n —» oo) or by letting —» 0. Are these equivalent?]

|G n ral Probl ms__________ 72. If a 75-W lightbulb emits 3.0% of the input energy as visible

74. A beam of 125-eV electrons is scattered from a crystal, as in

light (average wavelength 550 nm) uniformly in all direc tions, estimate how many photons per second of visible light will strike the pupil (4.0 mm diameter) of the eye of an observer 250 m away. 73. A t low temperatures, nearly all the atoms in hydrogen gas will be in the ground state. What minimum frequency photon is needed if the photoelectric effect is to be observed?

X -ray diffraction, and a first-order peak is observed at 6 = 38°. What is the spacing between planes in the diffracting crystal? (See Section 35 -10.) 75. A microwave oven produces electromagnetic radiation at A = 12.2 cm and produces a power of 860 W. alculate the number of microwave photons produced by the microwave oven each second.

1014

HAPTER 37

Early Quantum Theory and Models of the Atom

76. Sunlight reaching the Earth has an intensity of about 1350 W /m2. Estimate how many photons per square meter per second this represents. Take the average wavelength to be 550 nm. 77. A beam of red laser light (A = 633 nm) hits a black wall and is fully absorbed. If this light exerts a total force F = 6.5 nN on the wall, how many photons per second are hitting the wall? 78. The Big Bang theory states that the beginning of the universe was accompanied by a huge burst of photons. Those photons are still present today and make up the so-called cosmic microwave background radiation. The universe radiates like a blackbody with a temperature of about 2.7 K. alculate the peak wavelength of this radiation. 79. An electron and a positron collide head on, annihilate, and create two 0.755-MeV photons traveling in opposite directions. What were the initial kinetic energies of electron and positron? 80. By what potential difference must (a) a proton (m = 1.67X 10-27 kg), and (b) an electron (m = 9.11 X 10-31 kg), be accelerated to have a wavelength A = 6.0 X 10-12 m? 81. In some of Rutherford's experiments (Fig. 37-17) the a particles (mass = 6.64 X 10 27kg) had a kinetic energy of 4.8 MeV. How close could they get to the center of a silver nucleus (charge = +47e)? Ignore the recoil motion of the nucleus. 82. Show that the magnitude of the electrostatic potential energy of an electron in any Bohr orbit of a hydrogen atom is twice the magnitude of its kinetic energy in that orbit. 83. alculate the ratio of the gravitational force to the electric force for the electron in a hydrogen atom. an the gravitational force be safely ignored? 84. Electrons accelerated by a potential difference of 12.3 V pass through a gas of hydrogen atoms at room temperature. What wavelengths of light will be emitted? 85. In a particular photoelectric experiment, a stopping potential of 2.70 V is measured when ultraviolet light of wavelength 380 nm is incident on the metal. If blue light of wavelength 440 nm is used, what is the new stopping potential? 86. In an X-ray tube (see Fig. 35-26 and discussion in Section 35-10), the high voltage between filament and target is V. After being accelerated through this voltage, an electron strikes the target where it is decelerated (by positively charged nuclei) and in the process one or more X-ray photons are emitted, (a) Show that the photon of shortest wavelength will have

(b) What is the shortest wavelength of X-ray emitted when accelerated electrons strike the face of a 33-kV television picture tube? 87. The intensity of the Sun's light in the vicinity of Earth is about 1350 W /m2. Imagine a spacecraft with a mirrored square sail of dimension 1.0 km. Estimate how much thrust (in newtons) this craft will experience due to collisions with the Sun's photons. [Hint: Assume the photons bounce perpendicularly off the sail with no change in the magnitude of their momentum.] 88. Photons of energy 9.0 eV are incident on a metal. It is found that current flows from the metal until a stopping potential of 4.0 V is applied. If the wavelength of the incident photons is doubled, what is the maximum kinetic energy of the ejected electrons? What would happen if the wavelength of the incident photons was tripled?

89. Light of wavelength 360 nm strikes a metal whose work function is 2.4 eV. What is the shortest de Broglie wavelength for the electrons that are produced as photoelectrons?

90. Visible light incident on a diffraction grating with slit spacing of 0.012 mm has the first maximum at an angle of 3.5° from the central peak. If electrons could be diffracted by the same grating, what electron velocity would produce the same diffraction pattern as the visible light?

91. (a) Suppose an unknown element has an absorption spectrum with lines corresponding to 2.5, 4.7, and 5.1 eV above its ground state, and an ionization energy of 11.5 eV. Draw an energy level diagram for this element. (b) If a 5.1-eV photon is absorbed by an atom of this substance, in which state was the atom before absorbing the photon? What will be the energies of the photons that can subsequently be emitted by this atom?

92. Light of wavelength 424 nm falls on a metal which has a work function of 2.28 eV. (a) How applied to bring the current to maximum speed of the emitted the de Broglie wavelength of these

much voltage should be zero? (b) What is the electrons? (c) What is electrons?

93. Apply Bohr's assumptions to the Earth - Moon system to calculate the allowed energies and radii of motion. Given the known distance between Earth and the Moon, is the quantization of the energy and radius apparent?

94. Show that the wavelength of a particle of mass ra with kinetic energy K is given by the relativistic formula A = c /y /K 2 + 2mc2K .

95. A small flashlight is rated at 3.0 W. As the light leaves the flashlight in one direction, a reaction force is exerted on the flashlight in the opposite direction. Estimate the size of this reaction force. 96. At the atomic-scale, the electron volt and nanometer are well-suited units for energy and distance, respectively. (a) Show that the energy E in eV of a photon, whose wavelength Ais in nm, is given by ^ _ 1240 eV •nm A (nm) (b) How much energy (eV) does a 650-nm photon have?

*97. Three fundamental constants of nature —the gravitational constant G, Planck's constant , and the speed of light c — have the dimensions of [L3/M T 2], [ML2/T], and [L/T], respectively, (a) Find the mathematical combination of these fundamental constants that has the dimension of time. This combination is called the "Planck time" tP and is thought to be the earliest time, after the creation of the universe, at which the currently known laws of physics can be applied. (b) Determine the numerical value of tP. (c) Find the mathematical combination of these fundamental constants that has the dimension of length. This combination is called the "Planck length" AP and is thought to be the smallest length over which the currently known laws of physics can be applied, (d) Determine the numerical value of AP.

98. Imagine a free particle of mass ra bouncing back and forth between two perfectly reflecting walls, separated by distance L Imagine that the two oppositely directed matter waves associated with this particle interfere to create a standing wave with a node at each of the walls. Show that the ground state (first harmonic) and first excited state (second harmonic) have (non-relativistic) kinetic energies A2/8ra£2 and /z2/2mf2, respectively. General Problem s

1015

99. (a) A rubidium atom (ra = 85 u) is at rest with one elec tron in an excited energy level. When the electron jumps to the ground state, the atom emits a photon of wavelength A = 780 nm. Determine the resulting (nonrelativistic) recoil speed v of the atom. (b) The recoil speed sets the lower limit on the temperature to which an ideal gas of rubidium atoms can be cooled in a laser-based tom tr p. Using the kinetic theory of gases ( hapter 18), estimate this "lowest achievable" temperature. 100. A rubidium atom (atomic mass 85) is initially at room temper ature and has a velocity v = 290 m/s due to its thermal motion. onsider the absorption of photons by this atom from a laser beam of wavelength A = 780 nm. Assume the rubidium atom's initial velocity v is directed into the laser beam (the photons are moving right and the atom is moving left) and that the atom absorbs a new photon every 25 ns. How long will it take for this process to completely stop ("cool") the rubidium atom? [Note: a more detailed analysis predicts that the atom can be slowed to about 1 cm/s by this light absorption process, but it cannot be completely stopped.]

*Numerical/ omputer *101. (Ill) (a) Graph Planck's radiation formula (top of page 989) as a function of wavelength from A = 20 nm to 2000 nm in 20 nm steps for two lightbulb filaments, one at 2700 K and the other at 3300 K. Plot both curves on the same set of axes. (b) Approximately how much more intense is the visible light from the hotter bulb? Use numerical integration. *102. (Ill) Estimate what % of emitted sunlight energy is in the visible range. Use Planck's radiation formula (top of page 989) and numerical integration. *103. (Ill) Potassium has one of the lowest work functions of all metals and so is useful in photoelectric devices using visible light. Light from a source is incident on a potassium surface. Data for the stopping voltage V0 as a function of wave length Ais shown below, (a) Explain why a graph of V0 vs. 1/A is expected to yield a straight line. What are the theoretical expectations for the slope and the ^-intercept of this line? (b ) Using the data below, graph V q vs. 1/A and show that a straight-line plot does indeed result. Determine the slope a and ^-intercept b of this line. Using your values for a and b, determine (c) potassium's work function (eV) and (d) Planck's constant (J - s). A (/mi)

0^00

0430

0460

0490

0520~

V0 (V)

0.803

0.578

0.402

0.229

0.083

Answers to Exercises A: More 1000-nm photons (lower frequency). B: 5.50 X 1014Hz, 545 nm. : Only A.

1016

HAPTER 37

Decrease. E: (e). F: (c). D:

Early Q uantum Theory and M odels of th e Atom

P rticip nts in the 1927 Solv y Conference. Albert Einstein, se ted t front center, h d difficulty ccepting th t n ture could beh ve ccording to the rules of qu ntum mech nics. We present brief summ ry of qu ntum mech nics in this Ch pter st rting with the w ve function nd the uncert inty principle. We ex mine the Schrodinger equ tion nd its solutions for som e simple c ses: free p rticles, the squ re well, nd tunneling through b rrier.

T £ A. Picc rd, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder, E. Schrodinger, E. Versch ffelt, W.P uli, W.Heisenberg, R. H. Fowler, L. Brillouin, P.Debye, M.Knudsen, W.L.Br gg, H.A.Kr mers, P.A.M.Dir c, A.H. Compton, L.deBroglie, M.Bom, N.Bohr, I.Lngmuir, M.Pl nck, MmeCurie, H.A.Lorentz, A.Einstein, P.L ngevin, Ch.E.Guye, C.T.R.Wilson, O.W.Rich rdson

Quantum Mechanics HAPTER -OPENING QUESTION — Guess now! The uncertainty principle states that (a) no measurement can be perfect because it is technologically impossible to make perfect measuring instruments. (b) it is impossible to measure exactly where a particle is, unless it is at rest. (c) it is impossible to simultaneously know both the position and the momentum of a particle with complete certainty. (d) a particle cannot actually have a completely certain value of momentum.

ohr's model of the atom gave us a first (though rough) picture of what an atom is like. It proposed explanations for why there is emission and absorption of light by atoms at only certain wavelengths. The wavelengths of the line spectra and the ionization energy for hydrogen (and one-electron ions) are in excellent agreement with experiment. But the Bohr theory had important limitations. It was not able to predict line spectra for more complex atoms— not even for the neutral helium atom, which has only two electrons. Nor could it explain why emission lines, when viewed with great precision, consist of two or more very closely spaced lines (referred to as fine structure). The Bohr model also did not explain why some spectral lines were brighter than others. And it could not explain the bonding of atoms in molecules or in solids and liquids. From a theoretical point of view, too, the Bohr theory was not satisfactory: it was a strange mixture of classical and quantum ideas. Moreover, the wave -particle duality was not really resolved.

ONTENTS 3 8 -1

Qu ntum Mech nics— A New Theory

3 8 -2

The W ve Function nd Its Interpret tion; the Double Slit Experiment

3 8 -3

The Heisenberg Uncert inty Principle

3 8 -4

Philosophic Implic tions; Prob bility Versus Determinism

3 8 -5

The Schrodinger Equ tion in One Dimension — TimeIndependent Form

* 38 - 6

Time - Dependent Schrodinger Equ tion

3 8 -7

Free P rticles; Pl ne W ves nd W ve P ckets

3 8 -8

P rticle in n Infinitely Deep Squ re Well Potenti l ( Rigid Box)

3 8 -9

Finite Potenti l Well

38 - 10 Tunneling through

B rrier

1017

Erwin Schrodinger with Lise M eitner (see Ch pter 42).

FIG U R E 3 8 - 1

FIG U R E 3 8 - 2 Werner H eisenberg (center) on L ke Com o with Enrico Fermi (left) nd Wolfg ng P uli (right).

We mention these limitations of the Bohr theory not to disparage it — for it was a landmark in the history of science. Rather, we mention them to show why, in the early 1920s, it became increasingly evident that a new, more comprehensive theory was needed. It was not long in coming. Less than two years after de Broglie gave us his matter-wave hypothesis, Erwin Schrodinger (1887-1961; Fig. 38 -1) and Werner Heisenberg (1901-1976; Fig. 38 -2) independently developed a new comprehensive theory.

3 8 - 1 Qu ntum Mech nics—A New Theory The new theory, called quantum mechanics, has been extremely successful. It unifies the wave-particle duality into a single consistent theory and has successfully dealt with the spectra emitted by complex atoms, even the fine details. It explains the relative brightness of spectral lines and how atoms form molecules. It is also a much more general theory that covers all quantum phenomena from blackbody radiation to atoms and molecules. It has explained a wide range of natural phenomena and from its predictions many new practical devices have become possible. Indeed, it has been so successful that it is accepted today by nearly all physicists as the fundamental theory underlying physical processes. Quantum mechanics deals mainly with the microscopic world of atoms and light. But this new theory, when it is applied to macroscopic phenomena, must be able to produce the old classical laws. This, the correspondence principle (already mentioned in Section 37-11), is satisfied fully by quantum mechanics. This doesn't mean we should throw away classical theories such as Newton's laws. In the everyday world, the latter are far easier to apply and they give sufficiently accurate descriptions. But when we deal with high speeds, close to the speed of light, we must use the theory of relativity; and when we deal with the tiny world of the atom, we use quantum mechanics. Although we won't go into the detailed mathematics of quantum mechanics, we will discuss the main ideas and how they involve the wave and particle properties of matter to explain atomic structure and other applications.

3 8 —2 The Wave Function and Its Interpretation; the Double-Slit Experiment

1018

CHAPTER 38

The important properties of any wave are its wavelength, frequency, and amplitude. For an electromagnetic wave, the frequency (or wavelength) determines whether the light is in the visible spectrum or not, and if so, what color it is. We also have seen (Eq. 37-3) that the frequency is a measure of the energy of the corresponding photons (E = f). The amplitude or displacement of an electromagnetic wave at any point is the strength of the electric (or magnetic) field at that point, and is related to intensity of the wave (the brightness of the light).

For material particles such as electrons, quantum mechanics relates the wavelength to momentum according to de Broglie's formula, A = /p, Eq. 37-7. But what corresponds to the amplitude or displacement of a matter wave? The amplitude of an electromagnetic wave is represented by the electric and magnetic fields, E and B. In quantum mechanics, this role is played by the wave function, which is given the symbol M* (the Greek capital letter psi, pronounced "sigh"). Thus 'P represents the wave displacement, as a function of time and position, of a new kind of field which we might call a "matter" field or a matter wave. To understand how to interpret the wave function ''P', we make an analogy with light using the wave -particle duality. We saw in hapter 15 that the intensity I of any wave is proportional to the square of the amplitude. This holds true for light waves as well, as we saw in hapter 31. That is, I

E 2,

where E is the electric field strength. From the particle point of view, the intensity of a light beam (of given frequency) is proportional to the number of photons, N, that pass through a given area per unit time. The more photons there are, the greater the intensity. Thus /

This proportion can be turned around so that we have N

E 2.

That is, the number of photons (striking a page of this book, say) is proportional to the square of the electric field strength. If the light beam is very weak, only a few photons will be involved. Indeed, it is possible to "build up" a photograph in a camera using very weak light so the effect of individual photons can be seen. If we are dealing with only one photon, the relationship above (N oc E 2) can be interpreted in a slightly different way. At any point the square of the electric field strength, E 2, is a measure of the probability that a photon will be at that location. At points where E 2 is large, there is a high probability the photon will be there; where E 2 is small, the probability is low. We can interpret matter waves in the same way, as was first suggested by Max Born (1882-1970) in 1927. The wave function 'V P may vary in magnitude from point to point in space and time. If V describes a collection of many electrons, then l^ l2 at any point will be proportional to the number of electrons expected to be found at that point.1When dealing with small numbers of electrons we can't make very exact predictions, so |^ |2 takes on the character of a probability. If 'V P, which depends on time and position, represents a single electron (say, in an atom), then |^ |2 is interpreted as follows: |^ |2 at a certain point in space and time represents t e probability o f finding t e electron at t e given position and time. Thus |^ |2 is often referred to as the probability density or probability distribution.

Double-Slit Interference Experiment for Electrons To understand this better, we take as a thought experiment the familiar double-slit experiment, and consider it both for light and for electrons. onsider two slits whose size and separation are on the order of the wavelength of whatever we direct at them, either light or electrons, Fig. 38-3. We know very well what would happen in this case for light, since this is just Young's double-slit experiment (Section 34-3): an interference pattern would be seen on the screen behind. If light were replaced by electrons with wavelength comparable to the slit size, they too would produce an interference pattern (recall Fig. 37-11). In the case of light, the pattern would be visible to the eye or could be recorded on film. For electrons, a fluorescent screen could be used (it glows where an electron strikes).

FIGURE 3 8 - 3 P r llel be m, of light or electrons, f lls on two slits w hose sizes re comp r ble to the w velength. A n interference p ttern is observed. Light or electrons

fThe w ve function is gener lly complex qu ntity (th t is, it involves ) nd hence is not directly observ ble. On the other h nd, |2, the bsolute v lue of M* squ red, is lw ys re l qu ntity nd it is to |^ |2 th t we c n give physic l interpret tion.

SE TION 3 8 - 2

1019

FIG U R E 3 8 - 4 Young 's double - slit experim ent done with electrons — note th t the p ttern is not evident with only few electrons (top p hoto), but with m ore nd m ore electrons (second nd third photos), the f mili r double - slit interference p ttern (Ch pter 34) is seen.

If we reduced the flow of electrons (or photons) so they passed through the slits one at a time, we would see a flash each time one struck the screen. At first, the flashes would seem random. Indeed, there is no way to predict just where any one electron would hit the screen. If we let the experiment run for a long time, and kept track of where each electron hit the screen, we would soon see a pattern emerging—the interference pattern predicted by the wave theory; see Fig. 38 -4. Thus, although we could not predict where a given electron would strike the screen, we could predict probabilities. (The same can be said for photons.) The probability, as mentioned before, is proportional to l^ l2. Where |Mf|2 is zero, we would get a minimum in the interference pattern. And where |^ |2 is a maximum, we would get a peak in the interference pattern. The interference pattern would thus occur even when electrons (or photons) passed through the slits one at a time. So the interference pattern could not arise from the interaction of one electron with another. It is as if an electron passed through both slits at the same time, interfering with itself. This is possible because an electron is not precisely a particle. It is as much a wave as it is a particle, and a wave could travel through both slits at once. But what would happen if we covered one of the slits so we knew that the electron passed through the other slit, and a little later we covered the second slit so the electron had to have passed through the first slit? The result would be that no interference pattern would be seen. We would see, instead, two bright areas (or diffraction patterns) on the screen behind the slits. This confirms our idea that if both slits are open, the screen shows an interference pattern as if each electron passed through both slits, like a wave. Yet each electron would make a tiny spot on the screen as if it were a particle. The main point of this discussion is this: if we treat electrons (and other particles) as if they were waves, then 'V P represents the wave amplitude. If we treat them as particles, then we must treat them on a probabilistic basis. The square of the wave function, |'V P|2, gives the probability of finding a given electron at a given point. We cannot predict— or even follow— the path of a single electron precisely through space and time.

3 8 —3 The Heisenberg Uncertainty Principle Whenever a measurement is made, some uncertainty is always involved. For example, you cannot make an absolutely exact measurement of the length of a table. Even with a measuring stick that has markings 1 mm apart, there will be an inaccuracy of perhaps \ mm or so. More precise instruments will produce more precise measurements. But there is always some uncertainty involved in a measurement, no matter how good the measuring device. We expect that by using more precise instruments, the uncertainty in a measurement can be made indefinitely small. But according to quantum mechanics, there is actually a limit to the precision of certain measurements. This limit is not a restriction on how well instruments can be made; rather, it is inherent in nature. It is the result of two factors: the wave -particle duality, and the unavoidable interaction between the thing observed and the observing instrument. Let us look at this in more detail. To make a measurement on an object without disturbing it, at least a little, is not possible. onsider trying to locate a Ping-Pong ball in a completely dark room. You grope about trying to find its position; and just when you touch it with your finger, you bump it and it bounces away. Whenever we measure the position of an object, whether it is a ball or an electron, we always touch it with something else that gives us the information about its position. To locate a lost Ping-Pong ball in a dark room, you could probe about with your hand or a stick; or you could shine a light and detect the light reflecting off the ball. When you search with your hand or a stick, you find the ball's position when you touch it, but at the same time you unavoidably bump it and give it some momentum.

1020

HAPTER 38

Quantum Mechanics

Thus you won't know its future position. The same would be true if you observe the Ping-Pong ball using light. In order to "see" the ball, at least one photon must scatter from it, and the reflected photon must enter your eye or some other detector. When a photon strikes an ordinary-sized object, it does not appreciably alter the motion or position of the object. But when a photon strikes a very tiny object like an electron, it can transfer momentum to the object and thus greatly change the object's motion and position in an unpredictable way. The mere act of measuring the position of an object at one time makes our knowledge of its future position imprecise. Now let us see where the wave -particle duality comes in. Imagine a thought experiment in which we are trying to measure the position of an object, say an electron, with photons, Fig. 38 -5. (The arguments would be similar if we were using, instead, an electron microscope.) As we saw in hapter 35, objects can be seen to a precision at best of about the wavelength of the radiation used due to diffraction. If we want a precise position measurement, we must use a short wave length. But a short wavelength corresponds to high frequency and large momentum (p = / A); and the more momentum the photons have, the more momentum they can give the object when they strike it. If we use photons of longer wavelength, and correspondingly smaller momentum, the object's motion when struck by the photons will not be affected as much. But the longer wavelength means lower resolution, so the object's position will be less accurately known. Thus the act of observing produces an uncertainty in both the position and the momentum of the electron. This is the essence of the uncertainty principle first enunciated by Heisenberg in 1927. Quantitatively, we can make an approximate calculation of the magnitude of this effect. If we use light of wavelength A, the position can be measured at best to a precision of about A. That is, the uncertainty in the position measurement, Ax, is approximately Ax

i Electron

Light source

(a)

(b)

Thought experim ent for observing n electron with powerful light microscope. A t le st one photon must sc tter from the electron (tr nsferring som e momentum to it) nd enter the microscope. FIG U R E 3 8 - 5

Suppose that the object can be detected by a single photon. The photon has a momentum px = /X (Eq. 37-5). When the photon strikes our object, it will give some or all of this momentum to the object, Fig. 38-5. Therefore, the final x momentum of our object will be uncertain in the amount " i since we can't tell beforehand how much momentum will be transferred. The product of these uncertainties is (Ax)(Ap x) «

The uncertainties could be worse than this, depending on the apparatus and the number of photons needed for detection. A more careful mathematical calculation shows the product of the uncertainties as, at best, about

(38-1)

UNCERTAINTY PRINCIPLE (Ax and Ap)

where Ap x is the uncertainty of the momentum in the x direction.* This is a mathematical statement of the Heisenberg uncert inty principle, or, as it is sometimes called, the indetermin ncy principle. It tells us that we cannot measure both the position and momentum of an object precisely at the same time. The more accurately we try to measure the position so that Ax is small, the greater will be the uncertainty in momentum, Ap x . If we try to measure the momentum very accurately, then the uncertainty in the position becomes large. fNote, however, th t qu ntum mech nics does llow simult neous precise me surements of th t is, > 0.

(Ay)(Apx)

SE TION 3 8 - 3

px y: nd

The Heisenberg Uncertainty Principle

1021

/l\

AUT I ON _____________

Usntrceurm taein tieesn oietd uye, toin n td fic n c butinerentinnature(wave-particle)

The uncertainty principle does not forbid individual precise measurements, however. For example, in principle we could measure the position of an object exactly. But then its momentum would be completely unknown. Thus, although we might know the position of the object exactly at one instant, we could have no idea at all where it would be a moment later. The uncertainties expressed here are inherent in nature, and reflect the best precision theoretically attainable even with the best instruments. | EXERCISE A Return to the Ch pter -Opening Question on p. 1017, nd nswer it g in now.

Another useful form of the uncertainty principle relates energy and time, and we examine this as follows. The object to be detected has an uncertainty in position Ax « A. The photon that detects it travels with speed c, and it takes a time At « Ax /c « A/c to pass through the distance of uncertainty. Hence, the measured time when our object is at a given position is uncertain by about A At « - • c Since the photon can transfer some or all of its energy ( = f = e/A) to our object, the uncertainty in energy of our object as a result is a 77

c A' The product of these two uncertainties is (AE )(A t) ~

A more careful calculation gives UNCERTAINTY PRINCIPLE (AE and At)

(A E )(A t) a ~

(38 -2)

This form of the uncertainty principle tells us that the energy of an object can be uncertain (or can be interpreted as briefly nonconserved) by an amount AE for a time At « /(2ir AE). The quantity ( /2ir) appears so often in quantum mechanics that for convenience it is given the symbol ("h-bar"). That is, 277

= 6.626 X 10~34J - s _ 2tt

10_34j.s

By using this notation, Eqs. 38 -1 and 38 - 2 for the uncertainty principle can be written (Ax)(Apx) ^

and

(AE)(At) ^

We have been discussing the position and velocity of an electron as if it were a particle. But it isn't simply a particle. Indeed, we have the uncertainty principle because an electron— and matter in general—has wave as well as particle properties. What the uncertainty principle really tells us is that if we insist on thinking of the electron as a particle, then there are certain limitations on this simplified view—namely, that the position and velocity cannot both be known precisely at the same time; and even that the electron does not ave a precise position and momentum at the same time (because it is not simply a particle). Similarly, the energy can be uncertain in the amount AE for a time At « /A E . Because Planck's constant, , is so small, the uncertainties expressed in the uncertainty principle are usually negligible on the macroscopic level. But at the level of atomic sizes, the uncertainties are significant. Because we consider ordinary objects to be made up of atoms containing nuclei and electrons, the uncertainty principle is relevant to our understanding of all of nature. The uncertainty principle expresses, perhaps most clearly, the probabilistic nature of quantum mechanics. It thus is often used as a basis for philosophic discussion.

1022

HAPTER 38

Quantum Mechanics

EXAMPLE 38-1 Position uncertainty of an electron. An electron moves in 1.10 X 106m/s which has been a straight line with a constant speed v measured to a precision of 0.10%. What is the maximum precision with which its position could be simultaneously measured?

APPR ACH The momentum is p = mv, and the uncertainty in p is Ap = 0.0010/7. The uncertainty principle (Eq. 38-1) gives us the lowest Ax using the equals sign. S LUTI N The momentum of the electron is p = mv = (9.11 x IO"31kg)(l.l0 x 106m/s) = 1.00 X IO"24kg - m/s. The uncertainty in the momentum is 0.10% of this, or Ap = 1.0 X 10 27 kg - m/s. From the uncertainty principle, the best simultaneous position measurement will have an uncertainty of 1.055 X 10-34J - s „„ Ax ~ - — = — ---- -------------- - = 1.1 X 10 m, Ap 1.0 X 10 7kg -m/s or 110 nm. N TE This is about 1000 times the diameter of an atom. EXAMPLE 38-2 Position uncertainty of a baseball. What is the uncertainty in position, imposed by the uncertainty principle, on a 150-g baseball thrown at (93 + 2) mi/h = (42 + 1) m/s? APPR ACH The uncertainty in the speed is Av = 1 m/s. We multiply Av by m to get Ap and then use the uncertainty principle, solving for Ax. S LUTI N The uncertainty in the momentum is Ap = m Av = (0.150kg)(l m/s) = 0.15 kg - m/s. Hence the uncertainty in a position measurement could be as small as Ax _ A _ 1.055 X 10~34J •s Ap 0.15 kg - m/s N TE This distance is far smaller than any we could imagine observing or measuring. It is trillions of trillions of times smaller than an atom. Indeed, the uncertainty principle sets no relevant limit on measurement for macroscopic objects. ESTIMATE! j/tff lifetime calculated. The J/if/ meson, discovered in 1974, was measured to have an average mass of 3100 MeV/c2 (note the use of energy units since E = me2) and an intrinsic width of 63keV/c2. By this we mean that the masses of different J/*f/ mesons were actually measured to be slightly different from one another. This mass "width" is related to the very short lifetime of the J/if/ before it decays into other particles. From the uncertainty principle, if the particle exists for only a time At, its mass (or rest energy) will be uncertain by AE « /A t. Estimate the J/i/f lifetime. APPR ACH We use the energy-time version of the uncertainty principle, Eq. 38-2. S LUTI N The uncertainty of 63keV/c2 in the J / ^ 's mass is an uncertainty in its rest energy, which in joules is AE = (63

Then we expect its lifetime

103eV)(l.60

10"19J/eV) = 1.01

10"14J.

(= At using Eq. 3 8 - 2 ) to be

1.06 X 10-34J - s 1, 1 X 10_20s. AE 1.01X 10"14J Lifetimes this short are difficult to measure directly, and the assignment of very short lifetimes depends on this use of the uncertainty principle. (See hapter 43.) .

The uncertainty principle applies also for angular variables: ( A L Z) ( A < « S

where L is the component of angular momentum along a given axis (z) and (f>is the angular position in a plane perpendicular to that axis. SE TION 3 8 - 3

1023

3 8 —4 P h ilo so p h ic Im plications; P ro b ab ility Versus D eterm inism The classical Newtonian view of the world is a deterministic one (see Section 6 -5). One of its basic ideas is that once the position and velocity of an object are known at a particular time, its future position can be predicted if the forces on it are known. For example, if a stone is thrown a number of times with the same initial velocity and angle, and the forces on it remain the same, the path of the projectile will always be the same. If the forces are known (gravity and air resistance, if any), the stone's path can be precisely predicted. This mechanistic view implies that the future unfolding of the universe, assumed to be made up of particulate objects, is completely determined. This classical deterministic view of the physical world has been radically altered by quantum mechanics. As we saw in the analysis of the double-slit experiment (Section 38-2), electrons all prepared in the same way will not all end up in the same place. According to quantum mechanics, certain probabilities exist that an electron will arrive at different points. This is very different from the classical view, in which the path of a particle is precisely predictable from the initial position and velocity and the forces exerted on it. According to quantum mechanics, the position and velocity of an object cannot even be known accurately at the same time. This is expressed in the uncertainty principle, and arises because basic entities, such as electrons, are not considered simply as particles: they have wave properties as well. Quantum mechanics allows us to calculate only the probability* that, say, an electron (when thought of as a particle) will be observed at various places. Quantum mechanics says there is some inherent unpredictability in nature. Since matter is considered to be made up of atoms, even ordinary-sized objects are expected to be governed by probability, rather than by strict determinism. For example, quantum mechanics predicts a finite (but negligibly small) probability that when you throw a stone, its path will suddenly curve upward instead of following the downward-curved parabola of normal projectile motion. Quantum mechanics predicts with extremely high probability that ordinary objects will behave just as the classical laws of physics predict. But these predictions are considered probabilities, not certainties. The reason that macroscopic objects behave in accordance with classical laws with such high probability is due to the large number of molecules involved: when large numbers of objects are present in a statistical situation, deviations from the average (or most probable) approach zero. It is the average configuration of vast numbers of molecules that follows the so-called fixed laws of classical physics with such high probability, and gives rise to an apparent "determinism." Deviations from classical laws are observed when small numbers of molecules are dealt with. We can say, then, that although there are no precise deterministic laws in quantum mechanics, there are statistical laws based on probability. It is important to note that there is a difference between the probability imposed by quantum mechanics and that used in the nineteenth century to understand thermodynamics and the behavior of gases in terms of molecules ( hapters 18 and 20). In thermodynamics, probability is used because there are far too many particles to keep track of. But the molecules are still assumed to move and interact in a deterministic way following Newton's laws. Probability in quantum mechanics is quite different; it is seen as in erent in nature, and not as a limitation on our abilities to calculate or to measure. The view presented here is the generally accepted one and is called the Copenh gen interpret tion of quantum mechanics in honor of Niels Bohr's home, since it was largely developed there through discussions between Bohr and other prominent physicists. Because electrons are not simply particles, they cannot be thought of as following particular paths in space and time. This suggests that a description of matter in space and time may not be completely correct. This deep and far-reaching

1024 CHAPTER 38

tNote th t these prob bilities c n be c lcul ted precisely, just like ex ct predictions of prob bilities t rolling dice or pl ying c rds, but they re unlike predictions of prob bilities t sporting events or for n tur l or m n-m de dis sters, which re only estim tes.

conclusion has been a lively topic of discussion among philosophers. Perhaps the most important and influential philosopher of quantum mechanics was Bohr. He argued that a space-time description of actual atoms and electrons is not possible. Yet a description of experiments on atoms or electrons must be given in terms of space and time and other concepts familiar to ordinary experience, such as waves and particles. We must not let our descriptions of experiments lead us into believing that atoms or electrons themselves actually move in space and time as classical particles.

3 8 - 5 The Schrodinger Equation in One D im ension—Time-Independent Form In order to describe physical systems quantitatively using quantum mechanics, we must have a means of determining the wave function 'V P mathematically. The basic equation (in the nonrelativistic realm) for determining is the Sc rodinger equation. We cannot, however, derive the Schrodinger equation from some higher principles, just as Newton's second law, for example, cannot be derived. The relation F = ma was invented by Newton to describe how the motion of an object is related to the net applied force. As we saw early in this book, Newton's second law works exceptionally well. In the realm of classical physics it is the starting point for analytically solving a wide range of problems, and the solutions it yields are fully consistent with experiment. The validity of any fundamental equation resides in its agreement with experiment. The Schrodinger equation forms part of a new theory, and it too had to be invented — and then checked against experiment, a test that it passed splendidly. The Schrodinger equation can be written in two forms: the time-dependent version and the time-independent version. We will mainly be interested in steadystate situations— that is, when there is no time dependence— and so we mainly deal with the time-independent version. (We briefly discuss the time-dependent version in the optional Section 38 -6.) The time-independent version involves a wave function with only spatial dependence which we represent by lowercase psi, ijj(x), for the simple one-dimensional case we deal with here. In three dimensions, we write if/(x, y, z) or if/(r, 6, ). In classical mechanics, we solved problems using two approaches: via Newton's laws with the concept of force, and by using the energy concept with the conservation laws. The Schrodinger equation is based on the energy approach. Even though the Schrodinger equation cannot be derived, we can suggest what form it might take by using conservation of energy and considering a very simple case: that of a free particle on which no forces act, so that its potential energy U is constant. We assume that our particle moves along the x axis, and since no force acts on it, its momentum remains constant and its wavelength (A = /p) is fixed. To describe a wave for a free particle such as an electron, we expect that its wave function will satisfy a differential equation that is akin to (but not identical to) the classical wave equation. Let us see what we can infer about this equation. onsider a simple traveling wave of a single wavelength A whose wave displacement, as we saw in hapter 15 for mechanical waves and in hapter 31 for electromagnetic waves, is given by A sin(A:* - cot), or more generally as a superposition of sine and cosine: A sin (kx - cot) + B cos(kx - (ot). We are only interested in the spatial dependence, so we consider the wave at a specific moment, say t = 0. Thus we write as the wave function for our free particle i[/(x) = A s in k x + B co skx,

(38 -3a)

where A and B are constants1and k = 2 / \ (Eq. 15-11). For a particle of mass m and velocity v, the de Broglie wavelength is A = /p, where p = mv is the particle's momentum. Hence 277

2 d

k = T -~T = T fIn qu ntum mech nics, const nts c n be complex (i.e., with

re l nd/or im gin ry p rt).

SECTI N 38 - 5

1025

One requirement for our wave equation, then, is that it have the wave function as given by Eq. 38 -3 as a solution for a free particle. A second requirement is that it be consistent with the conservation of energy, which we can express as P2 ^ - + U = E, 2m where E is the total energy, U is the potential energy, and (since we are considering the nonrelativistic realm) the kinetic energy K of our particle of mass m is K = \m v 2 = p 2/2m. Since p = k (Eq. 38 -3b), we can write the conservation of energy condition as 2k2 — V U = E. (38 -4) 2m Thus we are seeking a differential equation that satisfies conservation of energy (Eq. 38 -4) when if/(x) is its solution. Now, note that if we take two derivatives of our expression for if/(x), Eq. 38 -3a, we get a factor - k2 multiplied by if/(x): dif/(x) d —------= — (A sin k x + B co skx) = k iA c o s k x — B sin kx ) dx dx d2if/(x) d i2-----=

" B sin k x) = - k (A sin k x + B co skx) = - k ijj(x ).

an this last term be related to the k 2term in Eq. 38 -4? Indeed, if we multiply this last relation by — 2/2m, we obtain V d2H x ) 2m dx

Vic2 , . . 2m

The right side is just the first term on the left in Eq. 38 - 4 multiplied by ifj(x). If we multiply Eq. 38 -4 through by if/(x), and make this substitution, we obtain 2 d2ifj(x)

SCHRODINGER EQUATION (time-independentform)

(38- 5)

This is, in fact, the one-dimensional time-independent Schrodinger equation, where for generality we have written U = U(x). It is the basis for solving problems in nonrelativistic quantum mechanics. For a particle moving in three dimensions there would be additional derivatives with respect to y and z (see hapter 39). Note that we have by no means derived the Schrodinger equation. Although we have made a good argument in its favor, other arguments could also be made which might or might not lead to the same equation. The Schrodinger equation as written (Eq. 38 -5) is useful and valid only because it has given results in accord with experiment for a wide range of situations. There are some requirements we impose on any wave function that is a solution of the Schrodinger equation in order that it be physically meaningful. First, we insist that it be a continuous function; after all, if \if/\2 represents the probability of finding a particle at a certain point, we expect the probability to be continuous from point to point and not to take discontinuous jumps. Second, we want the wave function to be normalized. By this we mean that for a single particle, the probability of finding the particle at one point or another (i.e., the probabilities summed over all space) must be exactly 1 (or 100%). For a single particle, |i/f|2 represents the probability of finding the particle in unit volume. Then |

47re0 r where we distinguish U ' from the U in Fig. 40-10. This formula works for our two ions if we set £7' = 0 at r = oo, whi h in the plot of Fig. 40-10 orresponds to U = +1.53 eV (Fig. 40-10 is drawn for U = 0 for the free atoms). The point rx in Fig. 40-10 orresponds to U ' = -1.53 eV relative to the two free ions. We solve the U ' equation above for r, setting q = + e and q = —e : 1 q iq2 (9.0 X 109 N •m2/C 2)( —1.60 X 10" 19C)( + 1.60 X 10" 19C)

(-1.53 eV)(l.60 X 10-19 J/eV) = 0.94 nm, whi h is just the measured value. (b) A t r = 0.24 nm, the potential energy of the two ions (relative to r = oo for the two ions) is 477€n U '

£7' =

1 4776,

(9.0 X 109 N • m2/C 2)( -1.60 X 10" 19 C)(+1.60 X 10" 19C)

6.0 eV. (0.24 X 10" 9m )(l.60 X 10" 19J/eV ) Thus, we estimate that 6.0 eV of energy is given up when Na+ and C r ions form a NaCl bond. Put another way, it takes 6.0 eV to break the NaCl bond and form the Na+ and Cl- ions. To get the binding energy— the energy to separate the NaCl into Na and Cl atom s — we need to subtra t out the 1.53 eV (the " bump" in Fig. 40-10) needed to ionize them: binding energy = 6.0 eV - 1.53 eV = 4.5 eV. The measured value (shown on Fig. 40-10) is 4.2 eV. The differen e an be attributed to the energy asso iated with the repulsion of the ele tron shells at this distan e. PHYSICS

APPLIED

ATP and energy in t e cell 1076

CHAPTER 40

Sometimes the potential energy of a bond looks like that of Fig. 40-11. In this ase, the energy of the bonded mole ule, at a separation rQ, is greater than when there is no bond (r = oo). That is, an energy input is required to make the bond (hen e the binding energy is negative), and there is energy release when the bond is broken.

Su h a bond is stable only be ause there is the barrier of the a tivation energy. This type of bond is important in living ells, for it is in su h bonds that energy an be stored effi iently in ertain mole ules, parti ularly ATP (adenosine triphosphate). The bond that onne ts the last phosphate group (designated ® in Fig. 40-11) to the rest of the mole ule (ADP, meaning adenosine diphosphate, sin e it ontains only two phosphates) has potential energy of the shape shown in Fig. 40-11. Energy is stored in this bond. When the bond is broken (ATP — ADP + ® ), energy is released and this energy an be used to make other hemi al rea tions " go." In living ells, many hemi al rea tions have a tivation energies that are often on the order of several eV. Su h energy barriers are not easy to over ome in the ell. This is where enzymes ome in. They a t as catalysts, whi h means they a t to lower the a tivation energy so that rea tions an o ur that otherwise would not. Enzymes a t via the ele trostati for e to distort the bonding ele tron louds, so that the initial bonds are easily broken.

FIGURE 40-11 Potenti l-energy di gr mfor the form tion of ATP fromADP nd phosph te (0 ).

EXAMPLE 40 -2 Bond length. Suppose a diatomi mole ule has a potential energy given by U = - (1/4776 o)(e /r ) + B /r 6 where B = 1.0 X 10-78 J -m 6 and r is the distan e between the enters of the two atoms. Determine the expe ted equilibrium separation of the two atoms (bond length of the mole ule).

APPR ACH The for e between the two atoms is given by F = - d U / d r (analo gous to Eq. 8 -7). The lassi al equilibrium separation is found by setting F = 0.

S LUTI N

F = - d U / d r = (1 / 47re 0)( — e2/ r 2) - B ( - / r 7) = —(l/4 ir e )(e / r 2) + 6 B /r = 0.

Then

5 6B(47re0)/e 2= 2.6 X 10-50m5.

r =

So r = 1 2 X 10" 10m = 0.12 nm. N TE This an be shown to be a stable equilibrium point by he king the sign of d U /d r 2, or by evaluating F at positions slightly larger and smaller than the equilibrium position.

40— 3 We k (v n der W ls) Bonds On e a bond between two atoms or ions is made, energy must normally be supplied to break the bond and separate the atoms. As mentioned in Se tion 40 -1, this energy is alled the bond energy or binding energy. The binding energy for ovalent and ioni bonds is typi ally 2 to 5 eV. These bonds, whi h hold atoms together to form mole ules, are often alled strong bonds to distinguish them from so- alled " weak bonds." The term weak bond, as we use it here, refers to atta hments between mole ules due to simple ele trostati attra tion — su h as between polar mole ules (and not wit in a polar mole ule, whi h is a strong bond). The strength of the atta hment is mu h less than for the strong bonds. Binding energies are typi ally in the range 0.04 to 0.3 eV — hen e their name " weak bonds." Weak bonds are generally the result of attra tion between dipoles (Se tions 21-11, 23-6). For example, Fig. 40-12 shows two mole ules, whi h have permanent dipole moments, attra ting one another. Besides su h dipole-dipole bonds, there an also be dipole-induced dipole bonds, in whi h a polar mole ule with a permanent dipole moment an indu e a dipole moment in an otherwise ele tri ally balan ed (nonpolar) mole ule, just as a single harge an indu e a separation of harge in a nearby obje t (see Fig. 21-7). There an even be an attra tion between two nonpolar mole ules, be ause their ele trons are moving about: at any instant there may be a transient separation of harge, reating a brief dipole moment and weak attra tion. A ll these weak bonds are referred to as van der Waals bonds, and the for es involved van der Waals forces. The potential energy has the general shape shown in Fig. 40-8, with the attra tive van der Waals potential energy varying as 1 /r6. When one of the atoms in a dipole -dipole bond is hydrogen, as in Fig. 40-12, it is alled a hydrogen bond. A hydrogen bond is generally the strongest of the weak bonds, be ause the hydrogen atom is the smallest atom and an be approa hed more losely. Hydrogen bonds also have a partial " ovalent" hara ter: that is, ele trons between the two dipoles may be shared to a small extent, making a stronger, more lasting bond.

FIGURE 40 -12 TheC+ — CT nd H+— N- dipoles ttr ct e ch other. (These dipoles m y be p rt of, for ex mple, the nucleotide b ses cytosine ndgu nine inDNA molecules. See Fig. 40-13.) The + nd —ch rges typic lly h ve m gnitudes of fr ction of e.

SE TION 40 -3

1077

FIGURE 40 -13 ( ) Section of DNA double helix. The red dots represent hydrogen bonds between the two str nds, (b) "Close-up"view: cytosine (C) nd gu nine (G) molecules on sep r te str nds of DNA double helix re held together by the hydrogen bonds (red dots) involving n H+ on one molecule ttr cted to n N_ or C+— O of molecule on the dj cent ch in. See lso Section 21-12 nd Figs. 21-47 nd 21-48. (a)

PHYSICS

APPLIED

DNA

Weak bonds are important in liquids and solids when strong bonds are absent (see Se tion 40-5). They are also very important for understanding the a tivities of ells, su h as the double helix shape of DNA (Fig. 40-13), and DNA repli ation. The average kineti energy of mole ules in a living ell at normal temperatures (T ~ 300 K) is around \ k T « 0.04 eV, about the magnitude of weak bonds.This means that a weak bond an readily be broken just by a mole ular ollision. Hen e weak bonds are not very permanent— they are, instead, brief atta hments. This helps them play parti ular roles in the ell. On the other hand, strong bonds— those that hold mole ules together— are almost never broken simply by mole ular ollision. Thus they are relatively perma nent. They an be broken by hemi al a tion (the making of even stronger bonds), and this usually happens in the ell with the aid of an enzyme, whi h is a protein mole ule.

Nucleotide energy. Cal ulate the potential energy between a C = O dipole of the nu leotide base ytosine and the nearby H — N dipole of guanine, assuming that the two dipoles are lined up as shown in Fig. 40-12. Dipole moment (= q i) measurements (see Table 23 -2 and Fig. 40-12) give 3.0 X IO - 30C -m = 3.0 X 10-20C = 0.19 . Qh = V

(40 -8a)

£ = 0 , 1 , 2 ,£ = 1,2,3,

(40 -8b)

where we have used Eqs. 40 -3 and 40 -7. Note that fo r £ —> £ - 1 transitions, annot be zero be ause there is then no state w ith £ = - 1 . Equations 40 -8 predi t an absorption spe trum like that shown s hemati ally in Fig. 40 -21, w ith transitions £ —> £ - 1 on the le ft and £ — >• £ + 1 on the right. Figure 40 -22 shows the mole ular absorption spe trum of HC1, whi h follows this pattern very well.

fEqs. 40-8 re for bsorption; for emission of photon, the tr nsitionwould be v —>v —1, £ —» £ + 1. A£

= -1

A£

= -1 = -1

FIGURE 40 -21 Expe ted spe trum for transitions between ombined rotational and vibrational states.

FIGURE 40 - 22 Absorption spe trum for HC1 mole ules. Lines on the left orrespond to transitions where £ —> £ — 1; those on the right are for £ —> £ + 1. Ea h line has a double peak be ause hlorine has two isotopes of different mass and different moment of inertia.

1084

HAPTER 40

Energy (eV)

A£ A£ A£ A£ :+ l = + 1 = + 1 = + 1

(Ea h line in the spe trum of Fig. 40-22 is split into two be ause Cl onsists of two isotopes of different mass; hen e there are two kinds of HC1 mole ule with different moments of inertia I.)

estim ate! Hie Hd molecule. Estimate the moment of inertia of the HC1 mole ule using the absorption spe trum shown in Fig. 40-22. For the purposes of a rough estimate you an ignore the differen e between the two isotopes. APPR ACH The lo ations of the peaks in Fig. 40-22 should

orrespond to Eqs. 40-8. We don't know what value of I ea h peak orresponds to in Fig. 40-22, but we an estimate the energy differen e between peaks to be about AE ' = 0.0025 eV. S LUTI N From Eqs. 40 -8, the energy differen e between two peaks is given by A fi = K -

A E ' = AEi 0? We expe t that some (at least) of the ele trons w ill in rease in energy due to thermal motion. Classi ally, the distribution of o upied states would be given by the Boltzmann fa tor, e~E/kT (see Eqs. 39-16). But for our ele tron gas, a quantum -me hani al system obeying the ex lusion prin iple, the probability of a given state of energy E being o upied is given by the Ferm i - D irac probability function (or Fermi factor): m FIGURE 40 -27 The Fermi-Dir c prob bility function for two temper tures, T = OK (bl ckline) nd T = 1200 K (blue curve). For f ( E ) = 1, st te with energy E is cert inly occupied. For f ( E ) = 0.5, which occurs t E = EF, the st te with EF h s 50%ch nce of being occupied.

1088

HAPTER 40

< *-« >

where EF is the Fermi energy. This fun tion is plotted in Fig. 40-27 for two temperatures, T = OK and T = 1200K (just below the melting point of opper). A t T = 0 (or as T approa hes zero) the fa tor e^E~E^/kT in Eq. 40-14 is zero if E < EF and is oo if E > EF. Thus f(E ) =

1 0

E 92) do not o ur naturally on Earth and are only produ ed artifi ially (that is, in the laboratory), as are many nu lides with Z < 92. The approximate size of nu lei was determined originally by Rutherford from the s attering of harged parti les by thin metal foils (Fig. 37-17). We annot speak about a definite size for nu lei be ause of the wave-parti le duality: their spatial extent must remain somewhat fuzzy. Nonetheless a rough " size" an be measured by s attering high-speed ele trons off nu lei. It is found that nu lei have a roughly spheri al shape with a radius that in reases with A a ording to the approximate formula r ra (1.2 X 10" 15m)(A3).

(41-1)

Sin e the volume of a sphere is V = f n r 3, we see that the volume of a nu leus is approximately proportional to the number of nu leons, V oc A. This is what we would expe t if nu leons were like impenetrable billiard balls: if you double the number of balls, you double the total volume. Hen e, all nu lei have nearly the same density, and it is enormous (see Example 41-1). The metri abbreviation for 10-15m is the ferm i (after Enri o Fermi) or the femtometer, fm (Table 1 - 4 or inside the front over). Thus 1.2 X 10" 15 m = 1.2 fm 1 or 1.2 fermis. Be ause nu lear radii vary as A 3, the largest nu lei, su h as uranium with A = 238, have a radius only about ^23 8 « 6 times that of the smallest, hydrogen (A = 1).

EXAMPLE 41-1

ESTIMATE"! Nuclear and atomic densities. Compare the

density of nu lear matter to the density of normal solids.

APPR ACH The density of normal liquids and solids is on the order of 103 to 104 kg/m 3 (see Table 13-1), and be ause the atoms are lose pa ked, atoms have about this density too. We therefore ompare the density (mass per volume) of a nu leus to that of its atom as a whole. S LUTI N The mass of a proton is greater than the mass of an ele tron by a fa tor 1.7 X 10-27kg 9.1 X IO-31 kg ~ 2 X 10 ' Thus, over 99.9% of the mass of an atom is in the nu leus, and for our estimate we an say the mass of the atom equals the mass of the nu leus, w nu l/m atom = 1 . Atoms have a radius of about IO-10 m (Chapter 37) and nu lei on the order of IO-15 m (Eq. 41-1). Thus the ratio of nu lear density to atomi density is about Pnu l _ (mnu l/Ku l) _ / ^nu l \ j ^atom ^ ( 1 ) ^ ^ = 1015 Patom (w atom/Ktom) \ matom/ f ^ n iid (10 15)3 The nu leus is 1015 times more dense than ordinary matter.

/ f \ C A U T I O N ______

Masses arefor neutral atom (nucleusplus electrons)

1106

HAPTER 41

The masses of nu lei an be determined from the radius of urvature of fastmoving nu lei (as ions) in a known magneti field using a mass spe trometer, as dis ussed in Se tion 27-9. Indeed the existen e of different isotopes of the same element (different number of neutrons) was dis overed using this devi e. Nu lear masses an be spe ified in unified atomi mass units (u). On this s ale, a neutral 12eC atom is given the pre ise value 12.000000 u. A neutron then has a measured mass of 1.008665 u, a proton 1.007276 u, and a neutral hydrogen atom }H (proton plus ele tron) 1.007825 u. The masses of many nu lides are given in Appendix F. It should be noted that the masses in this Table, as is ustomary, are for the neutral atom (in luding ele trons), and not for a bare nu leus.

Nuclear Physics and Radioactivity

Masses are often spe ified using the ele tron -volt energy unit. This an be done be ause mass and energy are related, and the pre ise relationship is given by Einstein's equation E = m c 2 (Chapter 36). Sin e the mass of a proton is 1.67262 X IO-27 kg, or 1.007276 u, then J 1.67262 X 10" 27 kg \ 1.0000 u = (1.0000 u) ----- ----------------- - = 1.66054 X 10 v \ 1.007276 u )

kg; 5'

this is equivalent to an energy (see Table inside front over) in MeV (= 106 eV) of E = me

2 _

(1.66054 X 10" 27kg)(2.9979 X 108 m /s )2

= 931.5 MeV.

(1.6022 X 10-19 J/eV)

Thus lu

= 1.6605 X 10" 27kg = 931.5M eV / 2.

The masses of some of the basi parti les are given in Table 41-1.

TABLE 41-1 Masses in Kilograms, Unified Atomic Mass Units, and MeV/c2 Object Electron Proton iH tom Neutron

9.1094 X 1.67262 X 1.67353 X 1.67493 X

10" 3 1 IO" 2 7 10" 2 7 10" 2 7

M ss u

MeV/c2

0.00054858 1.007276 1.007825 1.008665

0.51100 938.27 938.78 939.57

Just as an ele tron has intrinsi spin and orbital angular momentum quantum numbers, so too do nu lei and their onstituents, the proton and neutron. Both the proton and the neutron are spin -| parti les, just like the ele tron. A nu leus, made up of protons and neutrons, has a nuclear spin quantum number I that is the ve tor sum of the spins of all its nu leons (plus any orbital angular momentum), and an be either integer or half integer, depending on whether it is made up of an even or an odd number of nu leons. [O rbital angular momentum is integer and doesn't affe t half integer or whole for /.] The total nuclear angular momentum of a nu leus is given, as might be expe ted (see Se tion 39-2 and Eq. 39-15), by V /( / + 1) . Nuclear magnetic mom ents are measured in terms of the nuclear magneton eh

M'N

(41-2)

2m T

whi h is defined by analogy with the Bohr magneton for ele trons (|xB = e /2 m e, Se tion 39-7). Sin e fxN ontains the proton mass, rap, instead of the ele tron mass, it is about 2000 times smaller. The ele tron spin magneti moment is about 2 Bohr magnetons. The proton's magneti moment |xp has been measured to be |xp = 2.7928 |xN. There is no satisfa tory explanation for this large fa tor. The neutron has a magneti moment |xn = -1.9135 |xN, whi h suggests that, although the neutron arries no net harge, it may have internal stru ture (quarks, as we dis uss later). The minus sign for |xn indi ates that its magneti moment is opposite to its spin. Im portant appli ations based on nu lear spin are nu lear magneti resonan e (NM R) and magneti resonan e imaging (M R I). They are dis ussed in the next Chapter (Se tion 42-10).

SE TION 41 -1

Structure and Properties of the Nucleus

1107

41—2 Binding Energy nd Nucle r Forces Binding En rgi s The total mass of a stable nu leus is always less than the sum of the masses of its separate protons and neutrons, as the following Example shows.

jHe mass compared to its constituents. Compare the mass of a ^He atom to the total mass of its onstituent parti les. P R

B L

EM S O L V I N G Keep track of electron masses

APPR ACH The jH e nu leus ontains 2 protons and 2 neutrons. Tables normally give the masses of neutral atoms— that is, nu leus plus its Z ele trons— sin e this is how masses are measured. We must therefore be sure to balan e out the ele trons when we ompare masses. Thus we use the mass of }H rather than that of a proton alone. We look up the mass of the ^He atom in Appendix F (it in ludes the mass of 2 ele trons), as well as the mass for the 2 neutrons and 2 hydrogen atoms (= 2 protons + 2 ele trons). S LUTI N The mass of a neutral jH e atom, from Appendix F, is 4.002603 u. The mass of two neutrons and two H atoms (2 protons in luding the 2 ele trons) is 2 mn = 2(1.008665 u) = 2.017330 u 2m (jH ) = 2(1.007825 u) = 2.015650 u sum = 4.032980 u. Thus the mass of ^He is measured to be less than the masses of its onstituents by an amount 4.032980 u - 4.002603 u = 0.030377 u. Where has this lost mass of 0.030377 u disappeared to? It must be E = m e1. If the four nu leons suddenly ame together to form a ^He nu leus, the mass " loss" would appear as energy of another kind (su h as y radiation, or kineti energy). The mass (or energy) differen e in the ase of ^He, given in energy units, is (0.030377 u )(931.5 M eV /u) = 28.30 MeV. This differen e is referred to as the total binding energy of the nu leus. The total binding energy represents the amount of energy that must be put into a nu leus in order to break it apart into its onstituents. If the mass of, say, a ^He nu leus were exa tly equal to the mass of two neutrons plus two protons, the nu leus ould fall apart without any input of energy. To be stable, the mass of a nu leus must be less than that of its onstituent nu leons, so that energy input is needed to break it apart. Note that the binding energy is not something a nu leus has— it is energy it " la ks" relative to the total mass of its separate onstituents. We saw in Chapter 37 that the binding energy of the one ele tron in the hydrogen atom is 13.6 eV; so the mass of a }H atom is less than that of a single proton plus a single ele tron by 13.6 eV/ 2. Compared to the total mass of the hydrogen atom (939 M eV / 2), this is in redibly small, 1 part in 108. The binding energies of nu lei are on the order of MeV, so the eV binding energies of ele trons an be ignored. Note that nu lear binding energies, ompared to nu lear masses, are on the order of (28 MeV/4000 M eV) » 1 X 10 2, where we used helium's binding energy (see above) and mass « 4 X 940 MeV « 4000 MeV.

EXERCISEA Determine how much less the m ss of the ^Li nucleus is comp red to th t of I its constituents. The binding energy per nucleon is defined as the total binding energy of a nu leus divided by A , the total number of nu leons. We al ulated above that the binding energy of ^He is 28.3 MeV, so its binding energy per nu leon is 28.3 M eV/4 = 7.1 MeV. Figure 41-1 shows the binding energy per nu leon as a fun tion of A for stable nu lei. The urve rises as A in reases and rea hes a plateau at about 8.7 MeV per nu leon above A « 40. Beyond A « 80, the urve de reases slowly, indi ating that larger nu lei are held together a little less tightly than those in the middle of the Periodi Table. We w ill see later that these hara teristi s allow the release of nu lear energy in the pro esses of fission and fusion.

108

HAPTER 41

Nuclear Physics and Radioactivity

FIGURE 41-1 Binding energy per nu leon for the more stable nu lides as a fun tion of mass number A.

EXAMPLE 41 - 3 Binding energy for iron. Cal ulate the total binding energy and the binding energy per nu leon for ^Fe, the most ommon stable isotope of iron. APPR ACH We subtra t the mass o f an

atom from the total mass o f 26 hydrogen atoms and 30 neutrons, all found in Appendix F. Then we onvert mass units to energy units; fin ally we divide by A = 56, the total number o f nu leons. S LUTI N ^Fe has 26 protons and 30 neutrons whose separate masses are 26m (}H ) = (26) (1.007825 u) =

26.20345 u (in ludes 26 ele trons)

30mn = (30) (1.008665 u) =

30.25995 u

sum =

56.46340 u.

t Subtra t mass o f 56 gjFe: = -55.93494 u (Appendix F)

Am =

0.52846 u.

The total binding energy is thus (0.52846 u) (931.5 M eV /u) = 492.26 M eV and the binding energy per nu leon is _ 8 .7 9 M eV. 56 nu leons N TE The binding energy per nu leon graph (Fig. 41-1) peaks about here, fo r iron, so the iron nu leus (and its neighbors) is the most stable of nu lei. I EXERCISE B Determine the binding energy per nu leon for 1gO.

EXAMPLE 41 - 4

Binding energy of last neutron. W hat is the binding energy

APPR ACH I f ^C lost one neutron, it would be ^C . We subtra t the mass o f ^C from the masses o f and a free neutron. S LUTI N Obtaining the masses from Appendix F, we have Mass " C = 12.000000 u Mass Jn = 1.008665 u Total = 13.008665 u. Subtra t mass of 1lC -13.003355 u 6' : 0.005310 u Am = whi h in energy is (931.5 M eV /u) (0.005310 u) = 4.95 M eV. That is, it would require 4.95 M eV input of energy to remove one neutron from 1gC.

SE TION 41 - 2

1109

Nud ar Forc s

--120 ^

--1 0 0

St ble nuclei.;ij:

We an analyze nu lei not only from the point of view of energy, but also from the point of view of the for es that hold them together. We would not expe t a olle tion of protons and neutrons to ome together spontaneously, sin e protons are all positively harged and thus exert repulsive ele tri for es on ea h other. Sin e stable nu lei do stay together, it is lear that another for e must be a ting. Be ause this new for e is stronger than the ele tri for e, it is alled the strong nuclear force. The strong nu lear for e is an attra tive for e that a ts between all nu leons— protons and neutrons alike. Thus protons attra t ea h other via the strong nu lear for e at the same time they repel ea h other via the ele tri for e. Neutrons, sin e they are ele tri ally neutral, only attra t other neutrons or protons via the strong nu lear for e. The strong nu lear for e turns out to be far more ompli ated than the gravi tational and ele tromagneti for es. One important aspe t of the strong nu lear for e is that it is a short-range for e: it a ts only over a very short distan e. It is very strong between two nu leons if they are less than about 10 " 15 m apart, but it is essentially zero if they are separated by a distan e greater than this. Compare this to ele tri and gravitational for es, whi h de rease as / r 2 but ontinue a ting over any distan es and are therefore alled long-range for es. The strong nu lear for e has some strange features. For example, if a nu lide ontains too many or too few neutrons relative to the number of protons, the binding of the nu leons is redu ed; nu lides that are too unbalan ed in this regard are unstable. As shown in Fig. 41-2, stable nu lei tend to have the same number of protons as neutrons (N = Z ) up to about A = 30. Beyond this, stable nu lei ontain more neutrons than protons. This makes sense sin e, as Z in reases, the ele tri al repulsion in reases, so a greater number of neutrons— whi h exert only the attra tive strong nu lear for e— are required to maintain stability. For very large Z, no number of neutrons an over ome the greatly in reased ele tri repulsion. Indeed, there are no ompletely stable nu lides above Z = 82. What we mean by a stable nucleus is one that stays together indefinitely. What then is an unstable nucleus? It is one that omes apart; and this results in radioa tive de ay. Before we dis uss the important subje t of radioa tivity (next Se tion), we note that there is a se ond type of nu lear for e that is mu h weaker than the strong nu lear for e. It is alled the weak nuclear force, and we are aware of its existen e only be ause it shows itself in ertain types of radioa tive de ay. These two nu lear for es, the strong and the weak, together with the gravitational and ele tromagneti for es, omprise the four basi types of for e in nature.

Protonnumber (Z) FIGURE 41-2 Number of neutrons versus number of protons for st ble nuclides,which rerepresentedby dots. The str ight line represents N = Z.

41—3 R dio ctivity FIGURE 41-3 M rie nd Pierre Curie in their l bor tory ( bout 1906) where r diumw s discovered.

1110

HAPTER 41

Nu lear physi s had its beginnings in 1896. In that year, Henri Be querel (1852-1908) made an important dis overy: in his studies of phosphores en e, he found that a ertain mineral (whi h happened to ontain uranium) would darken a photographi plate even when the plate was wrapped to ex lude light. It was lear that the mineral emitted some new kind of radiation that, unlike X-rays, o urred without any external stimulus. This new phenomenon eventually ame to be alled radioactivity. Soon after Be quereFs dis overy, Marie Curie (1867-1934) and her husband, Pierre Curie (1859-1906), isolated two previously unknown elements that were very highly radioa tive (Fig. 41-3). These were named polonium and radium. Other radioa tive elements were soon dis overed as well. The radioa tivity was found in every ase to be unaffe ted by the strongest physi al and hemi al treatments, in luding strong heating or ooling or the a tion of strong hemi al reagents. It was suspe ted that the sour e of radioa tivity must be deep within the atom, emanating from the nu leus. It be ame apparent that radioa tivity is the result of the disintegration or decay of an unstable nu leus. Certain isotopes are not stable, and they de ay with the emission of some type of radiation or " rays."

Nuclear Physics and Radioactivity

Many unstable isotopes o ur in nature, and su h radioa tivity is alled " natural radioa tivity." Other unstable isotopes an be produ ed in the laboratory by nu lear rea tions (Se tion 42-1); these are said to be produ ed " artifi ially" and to have " artifi ial radioa tivity." Radioa tive isotopes are sometimes referred to as radioisotopes or radionuclides. Rutherford and others began studying the nature of the rays emitted in radioa tivity about 1898. They lassified the rays into three distin t types a ording to their penetrating power. One type of radiation ould barely penetrate a pie e of paper. The se ond type ould pass through as mu h as 3 mm of aluminum. The third was extremely penetrating: it ould pass through several entimeters of lead and still be dete ted on the other side. They named these three types of radiation alpha (a), beta (/3), and gamma (7), respe tively, after the first three letters of the Greek alphabet. Ea h type of ray was found to have a different harge and hen e is bent differently in a magneti field, Fig. 41^1; a rays are positively harged, /3 rays are negatively harged, and 7 rays are neutral. It was soon found that all three types of radiation onsisted of fam iliar kinds of parti les. Gamma rays are very high-energy photons whose energy is even higher than that of X-rays. Beta rays are ele trons, identi al to those that orbit the nu leus, but they are reated w ithin the nu leus itself. Alpha rays (or a parti les) are simply the nu lei of helium atoms, ^He; that is, an a ray onsists of two protons and two neutrons bound together. We now dis uss ea h of these three types of radioa tivity, or de ay, in more detail.

s mple (r dium) Alph nd bet r ys re bent in opposite directions by m gnetic field, where s g mm r ys re not bent t ll. FIGURE 41 - 4

4 1 -4 Alph Dec y Experiments show that when nu lei de ay, the number of nu leons ( = mass number A ) is onserved, as well as ele tri harge (= Z e). When a nu leus emits an a parti le (^He), the remaining nu leus w ill be different from the original: it has lost two protons and two neutrons. Radium 226 (2gfRa), for example, is an a emitter. It de ays to a nu leus with Z = 88 - 2 = 86 and A = 226 — 4 = 222. The nu leus with Z = 86 is radon (Rn)— see Appendix F or the Periodi Table. Thus the radium de ays to radon with the emission of an a parti le. This is written 2|R a - * 2iR n + 4He. See Fig. 41-5. When a de ay o urs, a different element is formed. The daughter nu leus (2ggRn in this ase) is different from the parent nu leus (2|sRa in this ase). This hanging of one element into another is alled transmutation. Alpha de ay an be written in general as Z -2N ' + ^He

FIGURE 41 -5 R dio ctive dec y of r dium to r don with emission of n lph p rticle.

/1 3 8 ® \

/l3 6 ® \

V 88® /

\8 6

226 88 Ra Ka

222 Rn 86^

w 2He

[a: de ay]

where N is the parent, N ' the daughter, and Z and A are the atomi number and atomi mass number, respe tively, of the parent.

^Dy dec ys by a emission to wh t element? (a) Pb, (b ) Gd, (c) Sm, I (d)Er,(e)Yb.

I EXERCISE C

Alpha de ay o urs be ause the strong nu lear for e is unable to hold very large nu lei together. The nu lear for e is a short-range for e: it a ts only between neighboring nu leons. But the ele tri for e a ts all the way a ross a large nu leus. For very large nu lei, the large Z means the repulsive ele tri for e be omes so large (Coulomb's law) that the strong nu lear for e is unable to hold the nu leus together.

SE TION 41 - 4

Alpha Decay

1111

We an express the instability of the parent nu leus in terms of energy (or mass): the mass of the parent nu leus is greater than the mass of the daughter nu leus plus the mass of the a parti le. The mass differen e appears as kineti energy, whi h is arried away by the a parti le and the re oiling daughter nu leus. The total energy released is alled the disintegration energy, Q, or the 0 -value of the de ay. From onservation of energy,

MPc2 =

M d c2

+ m ac2+ Q ,

where Q equals the kineti energy of the daughter and a parti le, and MP, M D, and m a are the masses of the parent, daughter, and a parti le, respe tively. Thus

Q = Mj>c - (AfD + m a)c 2.

(41 -3)

I f the parent had less mass than the daughter plus the a parti le (so Q < 0), the de ay ould not o ur spontaneously, for the onservation of energy law would be violated.

EXAMPLE 41-5 Uranium decay energy release. Cal ulate the disintegration energy when (mass = 232.037156 u) de ays to 29(jTh (228.028741 u) with the emission of an a parti le. (As always, masses given are for neutral atoms.)

APPR ACH We use onservation of energy as expressed in Eq. 41-3. g2U is the parent, 29§Th is the daughter. S LUTI N Sin e the mass of the ^He is 4.002603 u (Appendix F), the total mass in the final state is 228.028741 u + 4.002603 u = 232.031344 u.

The mass lost when the HU de ays is 232.037156 u - 232.031344 u = 0.005812 u. Sin e 1 u = 931.5 MeV, the energy Q released is Q = (0.005812 u) (931.5 M eV /u) = 5.4 MeV

and this energy appears as kineti nu leus.

energy of the a parti le and the daughter

Kinetic energy of the a in 2||U decay. For the l U de ay of Example 41-5, how mu h of the 5.4-MeV disintegration energy w ill be arried off by the a parti le? a m Ja

APPR ACH In any rea tion, momentum must be onserved as well as energy. S LUTI N Before disintegration, the nu leus an be assumed to be at rest, so

D ughter nucleus mT>%

FIGURE 41 -6 Momentum conserv tion in Ex mple 41-6.

the total momentum was zero. A fter disintegration, the total ve tor momentum must still be zero so the magnitude of the a parti le's momentum must equal the magnitude of the daughter's momentum (Fig. 41-6): m a va = m DvD.

Thus va = m T>vT)/ m a and the a 's kineti energy is ^

Ka =

2 m av a

D\2 =

( m DV

! 2^ 0^ W D — =

—

* D

228.028741 u K » = 51Kjy, 4.002603 u The total disintegration Hen e K« = § 2

energy is

Q = K a + K B = 57K B + K B = 58Kjy.

= 5.3 MeV.

The lighter a particle carries off (57/58) or 98% of the total kinetic energy.

1112

HAPTER 41

Nuclear Physics and Radioactivity

a-D cay Th ory—Tunn ling I f the mass of the daughter nu leus plus the mass of the a parti le is less than the mass of the parent nu leus (so the parent is energeti ally allowed to de ay), why are there any parent nu lei at all? That is, why haven't radioa tive nu lei all de ayed long ago, right after they were formed (in supernovae)? We an under stand de ay using a model of a nu leus that has an alpha parti le trapped inside it. The potential energy " seen" by the a parti le would have a shape something like that shown in Fig. 41-7. The potential energy well (approximately square) between r = 0 and r = R represents the short-range attra tive nu lear for e.

FIGURE 41-7 Potenti l energy for lph p rticle nd (d ughter) nucleus, showing the Coulomb b rrier through which the a p rticle must tunnel to esc pe. The Q-v lue of the re ction is lso shown. This plot ssumes spheric l symmetry, so the centr l well h s di meter 2R0. Beyond the nu lear radius, R 0, the Coulomb repulsion dominates (sin e the / r dependen e of the nu lear for e drops to zero) and we see the hara teristi Coulomb potential. The a parti le, trapped w ithin the nu leus, an be thought of as moving ba k and forth between the potential walls. Sin e the potential energy just beyond r = R is greater than the energy of the a parti le (dashed line), the a parti le ould not es ape the nu leus if it were governed by lassi al physi s. But a ording to quantum me hani s, there is a nonzero probability that the a parti le an tunnel through the Coulomb barrier, from point A to point B in Fig. 41-7, as we dis ussed in Se tion 38-10. The height and width of the barrier affe t the rate at whi h nu lei de ay (Se tion 41 -8). Be ause of this barrier, the lifetim es of a-unstable nu lei an be quite long, from a fra tion of a mi rose ond to over IO10 years. Note in Fig. 41-7 that the Q -value represents the total kineti energy when the a parti le is far from the nu leus. A simple way to look at tunneling is via the un ertainty prin iple whi h tells us that energy onservation an be violated by an amount A E for a length of time A t given by

(A £ )(A 0

Thus quantum me hani s allows onservation of energy to be violated for brief periods that may be long enough for an a parti le to " tunnel" through the barrier. A E would represent the energy differen e between the average barrier height and the parti le's energy, and A t the time to pass through the barrier. The higher and wider the barrier, the less time the a parti le has to es ape and the less likely it is to do so. It is therefore the height and width of this barrier that ontrols the rate of de ay and half-life of an isotope.

SE TION 41 - 4

Alpha Decay

1113

Why a Particl s? Why, you may wonder, do nu lei emit this ombination of four nu leons alled an a parti le? Why not just four separate nu leons, or even one? The answer is that the a parti le is very strongly bound, so that its mass is signifi antly less than that of four separate nu leons. As we saw in Example 41-2, two protons and two neutrons separately have a total mass of about 4.032980 u (ele trons in luded). For the de ay of dis ussed in Example 41-5, the total mass of the daughter 29(jTh plus four separate nu leons is 232.061721 u, whi h is greater than the mass of the 292U parent (232.037156). Su h a de ay ould not o ur be ause it would violate —> 29§Th + 2n + 2p. the onservation of energy. Indeed, we have never seen Similarly, it is almost always true that the emission of a single nu leon is energeti ally not possible.

Smok D t ctors—An Application @

P HY S I CS APPLIED

Smoke detector

One widespread appli ation of nu lear physi s is present in nearly every home in the form of an ordinary smoke dete tor. The most ommon type of dete tor ontains about 0.2 mg of the radioa tive ameri ium isotope, ^ A m , in the form of A m 0 2. The radiation ontinually ionizes the nitrogen and oxygen mole ules in the air spa e between two oppositely harged plates. The resulting ondu tivity allows a small steady ele tri urrent. If smoke enters, the radiation is absorbed by the smoke parti les rather than by the air mole ules, thus redu ing the urrent. The urrent drop is dete ted by the devi e's ele troni s and sets o ff the alarm. The radiation dose that es apes from an inta t ameri ium smoke dete tor is mu h less than the natural radioa tive ba kground, and so an be onsidered relatively harmless. There is no question that smoke dete tors save lives and redu e property damage.

4 1 -5 Bet Dec y P~ D cay Transmutation of elements also o urs when a nu leus de ays by (3 de ay— that is, with the emission of an ele tron or (3~ parti le. The nu leus l\C, for example, emits an ele tron when it de ays: 1JC — > 14N + e_ + neutrino,

/ j \ C A U T I O N _________

fi-decay e~ comes from nucleus (it is not an orbital electron)

where e" is the symbol for the ele tron. The parti le known as the neutrino, whose harge q = 0 and whose mass is very small or zero, was not initia lly dete ted and was only later hypothesized to exist, as we shall dis uss later in this Se tion. No nu leons are lost when an ele tron is emitted, and the total number of nu leons, A , is the same in the daughter nu leus as in the parent. But be ause an ele tron has been emitted from the nu leus itself, the harge on the daughter nu leus is + l e greater than that on the parent. The parent nu leus in the de ay written above had Z = + 6 , so from harge onservation the nu leus remaining behind must have a harge of + le . So the daughter nu leus has Z = 7, whi h is nitrogen. It must be arefully noted that the ele tron emitted in /3 de ay is not an orbital ele tron. Instead, the ele tron is reated wit in t e nucleus itself. What happens is that one of the neutrons hanges to a proton and in the pro ess (to onserve harge) emits an ele tron. Indeed, free neutrons a tually do de ay in this fashion: n — > p + e" + neutrino. To remind us of their origin in the nu leus, the ele trons emitted in j8 de ay are often referred to as "(3 parti les." They are, nonetheless, indistinguishable from orbital ele trons.

1114

HAPTER 41

Nuclear Physics and Radioactivity

EXAMPLE 41 - 7 Energy release in when 1gC de ays to 14N by /3 emission?

decay. How mu h energy is released

APPR ACH We find the mass differen e before and after de ay, Am. The energy released is E = (A m )c 2. The masses given in Appendix F are those of the neutral atom, and we have to keep tra k of the ele trons involved. Assume the parent nu leus has six orbiting ele trons so it is neutral; its mass is 14.003242 u. The daughter in this de ay, 14N, is not neutral sin e it has the same six orbital ele trons ir ling it but the nu leus has a harge of + le . However, the mass of this daughter with its six ele trons, plus the mass of the emitted ele tron (whi h makes a total of seven ele trons), is just the mass of a neutral nitrogen atom. S LUTI N The total mass in the final state is

/? \

caution

____________

Be careful wit atomic and electron masses in f3 decay

(mass of 14N nu leus + 6 ele trons) + (mass of 1 ele tron), and this is equal to mass of neutral 14N (in ludes 7 ele trons), whi h from Appendix F is a mass of 14.003074 u. So the mass differen e is 14.003242 u - 14.003074 u = 0.000168 u, whi h is equivalent to an energy hange A m e = (0.000168 u) (931.5 M eV /u) = 0.156 MeV orl56 ke V . N TE The neutrino doesn't ontribute to either the mass or harge balan e sin e it has q = 0 and m ~ 0 .

A ording to Example 41-7, we would expe t the emitted ele tron to have a kineti energy of 156 keV. (The daughter nu leus, be ause its mass is very mu h larger than that of the ele tron, re oils with very low velo ity and hen e gets very little of the kineti energy— see Example 41 -6.) Indeed, very areful measurements indi ate that a few emitted j8 parti les do have kineti energy lose to this al ulated value. But the vast m ajority of emitted ele trons have somewhat less energy. In fa t, the energy of the emitted ele tron an be anywhere from zero up to the maximum value as al ulated above. This range of ele tron kineti energy was found for any jS de ay. It was as if the law of onservation of energy was being violated, and indeed Bohr a tually onsidered this possibility. Careful experiments indi ated that linear momentum and angular momentum also did not seem to be onserved. Physi ists were troubled at the prospe t of giving up these laws, whi h had worked so well in all previous situations. In 1930, Wolfgang Pauli proposed an alternate solution: perhaps a new parti le that was very d iffi u lt to dete t was emitted during /3 de ay in addition to the ele tron. This hypothesized parti le ould be arrying o ff the energy, momentum, and angular momentum required to maintain the onservation laws. This new parti le was named the neutrino— meaning " little neutral one" — by the great Italian physi ist Enri o Fermi (1901-1954; Fig. 41 -8), who in 1934 worked out a detailed theory of /3 de ay. (It was Fermi who, in this theory, postu lated the existen e of the fourth for e in nature whi h we all the w eak nuclear force.) The ele tron neutrino has zero harge, spin of \ , and was long thought to have zero mass, although today we are quite sure that it has a very tiny mass (< 0.14 eV / 2). If its mass were zero, it would be mu h like a photon in that it is neutral and would travel at the speed of light. But the neutrino is very diffi ult to dete t. In 1956, omplex experiments produ ed further eviden e for the existen e of the neutrino; but by then, most physi ists had already a epted its existen e. The symbol for the neutrino is the Greek letter nu (v). The orre t way of writing the de ay of 1JC is then

FIGURE 41-8 Enrico Fermi, s portr yed on US post ge st mp. Fermi contributed signific ntly to both theoretic l nd experiment l physics, fe t lmost unique in modern times.

^C -> 14N + e" + v. The bar (- ) over the neutrino symbol is to indi ate that it is an " antineutrino." (Why this is alled an antineutrino rather than simply a neutrino need not on ern us now; it is dis ussed in Chapter 43.)

SE TION 41- 5

Beta Decay

1115

P + Decay

- 120 I

--1 0 0

St ble j : nuclei.ii;:

Many isotopes de ay by ele tron emission. They are always isotopes that have too many neutrons ompared to the number of protons. That is, they are isotopes that lie above the stable isotopes plotted in Fig. 41-2. But what about unstable isotopes that have too few neutrons ompared to their number of protons— those that fall below the stable isotopes of Fig. 41-2? These, it turns out, de ay by emitting a positron instead of an ele tron. A positron (sometimes alled an e+ or /3+ parti le) has the same mass as the ele tron, but it has a positive harge of + le . Be ause it is so like an ele tron, ex ept for its harge, the positron is alled the antiparticle1^to the ele tron. An example of a (3+ de ay is that of JoNe: JgNe - ► 19F + e+ + v,

Protonnumber (Z) FIGURE 41-2 (Repe ted.) Number of neutrons versus number of protons for st ble nuclides, which re represented by dots. The str ight line represents N = Z.

where e+ stands for a positron. Note that the v emitted here is a neutrino, whereas that emitted in (3 de ay is alled an antineutrino. Thus an antiele tron (= positron) is emitted with a neutrino, whereas an antineutrino is emitted with an ele tron; this gives a ertain balan e as dis ussed in Chapter 43. We an write /3" and /3+ de ay, in general, as follows: zN - > z+iN ' + e" + v zN - > z - iN ' + e+ + v,

[/3" de ay] [p + de ay]

where N is the parent nu leus and N ' is the daughter.

El ctron Captur Besides /3" and /3+ emission, there is a third related pro ess. This is electron capture (abbreviated EC in Appendix F) and o urs when a nu leus absorbs one of its orbiting ele trons. An example is jBe, whi h as a result be omes ^Li. The pro ess is written jBe + e" — » ^Li + v, or, in general, + e" — >• z - iN ' + v.

[ele tron apture]

Usually it is an ele tron in the innermost (K ) shell that is aptured, in whi h ase the pro ess is alled K -capture. The ele tron disappears in the pro ess, and a proton in the nu leus be omes a neutron; a neutrino is emitted as a result. This pro ess is inferred experimentally by dete tion of emitted X-rays (due to other ele trons jumping down to fill the empty state) of just the proper energy. In jS de ay, it is the weak nu lear for e that plays the ru ial role. The neutrino is unique in that it intera ts with matter only via the weak for e, whi h is why it is so hard to dete t.

FIGURE 41-9 Energy-level di gr m showing how 12B c n dec y to the ground st te of l\C by j8 dec y (tot l energy rele sed = 13.4 MeV), or c n inste d j8 dec y to n excited st te of X\C (indic ted by *), which subsequently dec ys to its ground st te by emitting 4.4-MeV 7 r y.

1116

HAPTER 41

4 1 -6 G mm Dec y Gamma rays are photons having very high energy. They have their origin in the de ay of a nu leus, mu h like emission of photons by ex ited atoms. Like an atom, a nu leus itself an be in an ex ited state. When it jumps down to a lower energy state, or to the ground state, it emits a photon whi h we all a 7 ray. The possible energy levels of a nu leus are mu h farther apart than those of an atom: on the order of keV or MeV, as ompared to a few eV for ele trons in an atom. Hen e, the emitted photons have energies that an range from a few keV to several MeV. For a given de ay, the 7 ray always has the same energy. Sin e a 7 ray arries no harge, there is no hange in the element as a result of a 7 de ay. How does a nu leus get into an ex ited state? It may o ur be ause of a violent ollision with another parti le. More ommonly, the nu leus remaining after a previous radioa tive de ay may be in an ex ited state. A typi al example is shown in the energy-level diagram of Fig. 41-9. ^B an de ay by /3 de ay dire tly tDiscussed in Ch pter 43. Briefly, n ntip rticle h s the s me m ss s its corresponding p rticle, but opposite ch rge. A p rticle nd its ntip rticle c n quickly nnihil te e ch other, rele sing energy (7 r ys).

to the ground state of ^C; or it an go by /3 de ay to an ex ited state of 1gC, whi h then de ays by emission of a 4.4-MeV 7 ray to the ground state. We an write 7 de ay as an*

_1-7,

[7 de ay]

where the asterisk means " ex ited state" of that nu leus. What, you may wonder, is the differen e between a 7 ray and an X-ray? They both are ele tromagneti radiation (photons) and, though 7 rays usually have higher energy than X-rays, their range of energies overlap to some extent. The differen e is not intrinsi . We use the term X -ray if the photon is produ ed by an ele tron-atom intera tion, and 7 ray if the photon is produ ed in a nu lear pro ess.

Isom rs; Int rnal Conv rsion In some ases, a nu leus may remain in an ex ited state for some time before it emits a 7 ray. The nu leus is then said to be in a met st ble st te and is alled an isomer. An ex ited nu leus an sometimes return to the ground state by another pro ess known as intern l conversion with no 7 ray emitted. In this pro ess, the ex ited nu leus intera ts with one of the orbital ele trons and eje ts this ele tron from the atom with the same kineti energy (minus the binding energy of the ele tron) that an emitted 7 ray would have had.

4 1 -7 Conserv tion o f Nudeon Number nd Other Conserv tion L ws In all three types of radioa tive de ay, the lassi al onservation laws hold. Energy, linear momentum, angular momentum, and ele tri harge are all onserved. These quantities are the same before the de ay as after. But a new onservation law is also revealed, the l w of conserv tion of nucleon number. A ording to this law, the total number of nu leons {A ) remains onstant in any pro ess, although one type an hange into the other type (protons into neutrons or vi e versa). This law holds in all three types of de ay. Table 41-2 gives a summary of a, /3, and 7 de ay. [In Chapter 43 we w ill generalize this and all it onservation of baryon number.]

TABLE 41-2 The Three Types of Radioactive Decay a decay: zN

Z -2N' + ^He

fi decay: zN z+iN' + e" + v |N -► |_ iN' + e+ + v + e" -► z -iN' + v [EC]f 7 decay: |N * - ►

+ y

f Electron c pture. *Indic tes the excited st te of nucleus.

41—8 H lf-Life nd R te o f Dec y A ma ros opi sample of any radioa tive isotope onsists of a vast number of radioa tive nu lei. These nu lei do not all de ay at one time. Rather, they de ay one by one over a period of time. This is a random pro ess: we an not predi t exa tly when a given nu leus w ill de ay. But we an determine, on a probabilisti basis, approximately how many nu lei in a sample w ill de ay over a given time period, by assuming that ea h nu leus has the same probability of de aying in ea h se ond that it exists. The number of de ays A N that o ur in a very short time interval A t is then proportional to A t and to the total number N of radioa tive nu lei present: AN =

-X N At

(41 - 4 )

where the minus sign means N is de reasing. We rewrite this to get the rate of de ay: AN — =

-X N .

(41 - 4b)

In these equations, A is a onstant of proportionality alled the dec y const nt, whi h is different for different isotopes. The greater A is, the greater the rate of de ay and the more " radioa tive" that isotope is said to be. The number of de ays that o ur in the short time interval A t is designated A N be ause ea h de ay that

SE TION 41 - 8

Half-Life and Rate of Decay

1117

o FIGURE 41-10 R dio ctive nuclei dec y one by one. Hence, the number of p rent nuclei in s mple is continu lly decre sing. When 1gCnucleus emits n electron, the nucleus becomes 14N nucleus.

° o ° o ° o o o o

(a)

/ 0 o o o o o (b)

o o o o ° o o ° °

Legend Q tom (p rent) O

o o (C)

N tom (d ughter)

o urs orresponds to a de rease by one in the number N of nu lei present. That is, radioa tive de ay is a " one-shot" pro ess, Fig. 41-10. On e a parti ular parent nu leus de ays into its daughter, it annot do it again. If we take the lim it A t — ► 0 in Eq. 41-4, A N w ill be small ompared to N, and we an write the equation in infinitesim al form as dN =

(41 - 5)

- X N dt.

We an determine Af as a fun tion of t by rearranging this equation to dN N ~

and then integrating from t = 0 to t = t\ •N

= - I A dt,

where N is the number of parent nu lei present at t = 0 and N is the number remaining at time t. The integration gives iIn —N

= -A t

or (41 - 6) Equation 41-6 is alled the radioactive decay law. It tells us that the number of radioa tive nu lei in a given sample de reases exponentially in time. This is shown in Fig. 41-1 la for the ase of 1JC whose de ay onstant is 3.83 X 10 -12 -1 The rate of de ay in a pure sample, or number of de ays per se ond, is dN dt

also alled the activity of the sample.We use absolute value signs to make a tivity a posi tive number (d N /d t is negative be ause the number of parent nu lei N is de reasing). The symbol R is also used for a tivity, R = \dN /dt\.

FIGURE 41-11 ( ) The number N of p rent nuclei in given s mple of decre ses exponenti lly. We set N0 = 1.00 X 102 2 here, s we do in the text shortly, (b) The number of dec ys per second lso decre ses exponenti lly. The h lf-life (Eq. 41-8) of 1gCis bout 5730yr, which me ns th t the number of p rent nuclei, N, nd the r te of dec y, \dN/dt\, decre se by h lf every 5730 yr. 1.0 X 1022)

5730

11,460 Time, t (yr)

17,190

(a)

1118

HAPTER 41

Nuclear Physics and Radioactivity

(b)

From Eqs. 41-5 and 41-6, dN dt

= \N = \N0e M.

(41-7a)

A t t = 0, the a tivity is dN dt

= AA/q.

(41 - 7b)

Hen e, at any other time t the a tivity is dN dt

dN *TAr, dt o

(41 - 7c)

so the a tivity de reases exponentially in time at the same rate as for N (Fig. 4 1 - llb ). Equation 41-7 is sometimes referred to as the r dio ctive dec y l w (so is Eq. 41-6), and an be written using R to represent a tivity, R = \d N /d t\, as

(41 - 7d)

R = R e ~ xt.

The rate of de ay of any isotope is often spe ified by giving its half-life rather than the de ay onstant A. The h lf-life of an isotope is defined as the time it takes for half the original amount of isotope in a given sample to de ay. For example, the half-life of 1JC is about 5730 years. If at some time a pie e of petrified wood ontains, say, 1.00 X 1022 1JC nu lei, then 5730 yr later it w ill ontain only 0.50 X 1022 nu lei. A fter another 5730 yr it w ill ontain 0.25 X 1022 nu lei, and so on. This is hara teristi of the exponential fun tion, and is shown in Fig. 4 1 - lla . Sin e the rate of de ay \dN /dt\ is proportional to N , it too de reases by a fa tor of 2 every half-life, Fig. 4 1 - llb .

The h lf-life of 22N is 2.6 ye rs. How much will be left of 1.0-^tg s mple fter 5.2yr? (a) None, (b) I fig. (c) I fig. (,d) \fig. (e) 0.693 fig.

EXERCISE D

of 22N

I EXERCISE E Return to the Ch pter-Opening Question, p ge 1104, nd nswer it g in I now. Try to expl in why you m y h ve nswered differently the first time. The half-lives of known radioa tive isotopes vary from as short as 10_22s to about 1028s (about 1021yr). The half-lives of many isotopes are given in Appendix F. It should be lear that the half-life (whi h we designate 71) bears an inverse relationship to the de ay onstant. The longer the half-life of an isotope, the more slowly it de ays, and hen e A is smaller. The pre ise relation is obtained from Eq. 41-6 by setting N = N0/2 at t = Tv. y

= N o e -x i

ekTi = 2 .

We take natural logs of both sides (" In" and "e " are inverse operations, meaning ln(e*) = x) and find ln(eA7l) = In 2,

71 = — 2

= 9^3.

A71 = In 2

(41 - 8)

We an then write Eq. 41-6 as

0 693t/Tl.

N = N e~°'

Sometimes the me n life r of an isotope is quoted, whi h is defined as r = 1/A (see also Problem 80), so then Eq. 41-6 an be written

N = N e~,/T

just as for R C and LR ir uits (Chapters 26 and 30, where r is alled the time onstant). Sin e ^ (41 - 9 ) A 0.693 the mean life and half-life differ by a fa tor 0.693; onfusing them an ause serious error. The radioactive decay law , Eq. 41-7d, an be written simply as

R = Roe~t/r.

(41-9b)

SE TION 41-8

1119

EXAMPLE 41 - 8 Sample activity. The isotope l\C has a half-life of 5730 yr. If a sample ontains 1.00 X 1022 arbon-14 nu lei, what is the a tivity of the sample?

APPR ACH We first use the half-life to find the de ay onstant (Eq. 41-8), and use that to find the a tivity, Eq. 41-7b or 41-5. The number of se onds in a year is (60s/m in)(60m in/h)(24h/d)(365^d/yr) = 3.156 X 107 s. S LUTI N The de ay onstant A from Eq. 41-8 is 0.693 0.693 ' ------------------------------------------------------= 3.83 X 10 71 (5730 yr)(3.156 X 107 s/yr) From Eq. 41-7b, the a tivity or rate of de ay is dN dt

= XN

s .

(3.83 X 10-12 s_1)(l.00 X 1022) = 3.83 X 1010 de ays/s.

Noti e that the graph of Fig. 41-1 lb starts at this value, orresponding to the original value of N = 1.0 X 1022 nu lei in Fig. 41-1 la. N TE The unit " de ays/s" is often written simply as s-1 sin e " de ays" is not a unit but refers only to the number. This simple unit of a tivity is alled the be querel: 1 Bq = 1 de ay/s, as dis ussed in Chapter 42.

| EXERCISE F Determine the dec y const nt for r dium (71 = 1600 yr). C NCEPTUAL EXAMPLE 41 -9 I Safety: Activity versus half-life. One might think that a short half-life material is safer than a long half-life material be ause it w ill not last as long. Is that true? RESP NSE No. A shorter half-life means the a tivity is higher and thus more " radioa tive" and an ause more biologi al damage. On the other hand, a longer half-life for the same sample size N means a lower a tivity but we have to worry about it for longer and find safe storage until it rea hes a safe (low) level of a tivity. [2 2 H I3 S H H E 1 A sample of radioactive ]|N. A laboratory has 1.49 /xg of pure 13N, whi h has a half-life of 10.0min (600 s). (a) How many nu lei are present initially?

(b) What is the a tivity initially? ( ) What is the a tivity after 1.00 h? (d) A fter approximately how long w ill the a tivity drop to less than one per se ond (= Is -1)?

APPR ACH We use the definition of the mole and Avogadro's number (Se tions 17-7 and 17-9) to find the number of nu lei. For ( b ) we get A from the given half-life and use Eq. 41-7b for the a tivity. For ( ) and ( d ) we use Eq. 41-7 , and/or make a Table of the times. S LUTI N (a) The atomi mass is 13.0, so 13.0 g w ill ontain 6.02 X 1023 nu lei (Avogadro's number). We have only 1.49 X 10_6 g, so the number of nu lei N that we have in itia lly is given by the ratio N 1.49 X 10_6g

0 X 1023 " 6.02 Solving, we find N 0= 6.90 X 1016 nu lei.

13.0 g

(b) From Eq. 41-8,

A = 0.693/71 = (0.693)/(600s) = 1.155 X 10"3 s_1. Then, at t = 0 (Eq. 41-7b), ^

dt o

= AJV0 = (1.155 x 10-3 s_1)(6.90 x 1016) = 7.97 x 1013 de ays/s.

( ) A fte r 1.00 h = 3600 s, the magnitude of the a tivity w ill be (Eq. 41-7 ) dN dt

g-A* = (7>9? x 1013s- l) e-(l.l55xl0-3s-1)(3600s) = L25 X 1012 S" 1. I (d) We want to determine the time t when \dN /dt\ = 1.00 s-1. From Eq. 41-7 , we have dt

\dN/dt\ \dN/dt\0

1120

HAPTER 41

1.00 s - 1

= 1.25 X 10'

We take the natural log (In) of both sides (In In(1.25 X 1(T14)

ekt= Xt) and divide by Ato find

= 2.77 X 104s = 7.70 h.

EasyAlternate Solution to (c) 1.00 h = 60.0 minutes is 6 half-lives, so the activity will decrease to (§)(§)(§)(§)(§)(§) = (i )6 = ® of its original value, or (7.97 X 1013)/(64) = 1.25 X 1012 per second. EXERCISE G Technicium 43TC h s h lf- life of 4.2 X 10° yr. of 28.79 yr. Which st tements re true? ( ) The dec y const nt of Sr is gre ter th n the dec y const nt of Tc. The ctivity o f 100 g o f Sr is less th n the ctivity of 100 g of Tc. (c) The long h lf - life of Tc me ns th t it dec ys by lph dec y. A Tc tom h s higher prob bility o f dec ying in 1 yr th n Sr tom. (e) 28.79 g of Sr h s the s me ctivity s 4.2 X 10 6 g of Tc.

h lf - life

(b) (d)

41—9 Dec y Series It is often the case that one radioactive isotope decays to another isotope that is also radioactive. Sometimes this daughter decays to yet a third isotope which also is radioactive. Such successive decays are said to form a dec y series. An important example is illustrated in Fig. 41-12. As can be seen, decays by a emission to 29oTh, which in turn decays by /3 decay to 294Pa. The series continues as shown, with several possible branches near the bottom, ending at the stable lead isotope, 2g2Pb. The two last decays can be 2°fPb + e-

7% n or

206i 82r

a. Other radioactive series also exist. 84r u

—r

(71 = 4.2 min) (71 = 138 days) 238,

FIGURE 41-12 D ec y series beginning with 292U . N uclei in the series re specified by dot nd Z v lues. representing H lf - lives re given in seconds (s), minutes (min), hours (h), d ys (d), or ye rs (yr). N ote th t horizont l rrow represents j8 dec y does not ch nge), where s di gon l line represents dec y ch nges by 4, Z ch nges by 2). For the four nuclides shown th t c n dec y by both nd /3 dec y, the m ore prom inent dec y (in these four c ses, > 99.9%) is shown s solid rrow nd the less com mon dec y (< 0 .1%) s d shed rrow.

(A (.A

SE TION 4 1 - 9

1121

Because of such decay series, certain radioactive elements are found in nature that otherwise would not be. When the solar system (including Earth) was formed about 5 billion years ago, it is believed that nearly all nuclides were present, having been formed (by fusion and neutron capture, Sections 42^4 and 44-2) in a nearby supernova explosion (Section 44-2). Many isotopes with short half-lives decayed quickly and no longer are detected in nature today. But long-lived isotopes, such as 29fU with a half-life of 4.5 X 109yr, still do exist in nature today. Indeed, about half of the original still remains. We might expect, however, that radium (2|fRa), with a half-life of 1600 yr, would long since have disappeared from the Earth. Indeed, the original 2i|R a nuclei must by now have all decayed. However, because 2ifU decays (in several steps) to 2i|Ra, the supply of 2||R a is continually replen ished, which is why it is still found on Earth today. The same can be said for many other radioactive nuclides. ON EPTUAL EXAMPLE 41 - 11 I Decay chain. In the decay chain of Fig. 41-12, if we start looking below 2^U , we see four successive nuclides with half-lives of 250,000 yr, 75,000 yr, 1600 yr, and a little under 4 days. Each decay in the chain has an alpha particle of a characteristic energy, and so we can monitor the radioactive decay rate of each nuclide. Given a sample that was pure a million years ago, which alpha decay would you expect to have the highest activity in the sample? RESPONSE The first instinct is to say that the process with the shortest half-life would show the highest activity. Surprisingly, perhaps, the activities of the four nuclides in this sample are all the same. The reason is that in each case the decay of the parent acts as a bottleneck to the decay of the daughter. ompared to the 1600-yr half-life of 2|§Ra, for example, its daughter 2|^Rn decays almost immediately, but it cannot decay until it is made. (This is like an automobile assembly line: if worker A takes 20 minutes to do a task and then worker B takes only 1 minute to do the next task, worker B still does only one car every 20 minutes.)

4 1 -1 0 R dio ctive D ting S APPLI ED Carbon - 14 dating

1122

HAPTER 41

Radioactive decay has many interesting applications. One is the technique of radioactive dating by which the age of ancient materials can be determined. The age of any object made from once-living matter, such as wood, can be determined using the natural radioactivity of 1J . All living plants absorb carbon dioxide ( 0 2) from the air and use it to synthesize organic molecules. The vast majority of these carbon atoms are l\C, but a small fraction, about 1.3 X 10"12, is the radioactive isotope ^ . The ratio of 1J to l\C in the atmosphere has remained roughly constant over many thousands of years, in spite of the fact that ^ decays with a half-life of about 5730 yr. This is because energetic nuclei in the cosmic radi ation, which impinges on the Earth from outer space, strike nuclei of atoms in the atmosphere and break those nuclei into pieces, releasing free neutrons. Those neutrons can collide with nitrogen nuclei in the atmosphere to produce the nuclear transformation n + ^ + p. That is, a neutron strikes and is absorbed by a nucleus, and a proton is knocked out in the process. The remaining nucleus is 1g . This continual production of JJ in the atmosphere roughly balances the loss of C by radioactive decay. As long as a plant or tree is alive, it continually uses the carbon from carbon dioxide in the air to build new tissue and to replace old. Animals eat plants, so they too are continually receiving a fresh supply of carbon for their tissues. Organisms cannot distinguish ^ from ^ , and since the ratio of ^ to ^ in the atmosphere remains nearly constant, the ratio of the two isotopes within the living organism remains nearly constant as well. When an organism dies, carbon dioxide is no longer absorbed and utilized. Because the ^ decays radioactively, the ratio of XJ to X\C in a dead organism decreases over time. Since the half-life of ^ is about 5730 yr, the u6C /l\C ratio decreases by half every 5730 yr. If, for example, fOrg nisms oper te lmost exclusively vi chemic l re ctions—which involve only the outer orbit l electrons of the tom; extr neutrons in the nucleus h ve essenti lly no effect.

the l\C /l\C ratio of an ancient wooden tool is half of what it is in living trees, then the object must have been made from a tree that was felled about 5730 years ago. Actually, corrections must be made for the fact that the ratio in the atmosphere has not remained precisely constant over time. The determination of what this ratio has been over the centuries has required techniques such as comparing the expected ratio to the actual ratio for objects whose age is known, such as very old trees whose annual rings can be counted reasonably accurately. ■ m u j i s i B n An ancient animal. The mass of carbon in an animal bone fragment found in an archeological site is 200 g. If the bone registers an activity of 16 decays/s, what is its age? APPROA H First we determine how many 1J atoms there were in our 200-g sample when the animal was alive, given the known fraction of ^ , 1.3 X 1 0 12. Then we use Eq. 41-7b to find the activity back then, and Eq. 41-7c to find out how long ago that was by solving for the time t. SOLUTION The 200 g of carbon is nearly all ^ ; 12.0 g of 6.02 X 1023 atoms, so 200 g contains

PHYSICS

APPLIED

Archeological dating

contains

6.02 X 1023atoms/mol (200 g) = 1.00 X 1025 atoms. 12 g/mol When the animal was alive, the ratio of The number of 1g nuclei at that time was

in the bone was 1.3 X 10-12.

N0 = (l.OO X 1025 atoms)(l.3 X 10 12) = 1.3 X 1013 atoms. :ample L (Example 41-8), the magni From Eq. 41-7b, with A = 3.83 X 10 12s 1 for tude of the activity when the animal was still alive (t 0) was dN dt

= AA/q = (3.83 X 10_12s_1)(l.3 X 1013) = 50 s_1.

From Eq. 41-7c, dN dt

dN dt

where \dN/dt\ is given as 16 s 1. Then =

\dN/dt\0 \dN/dt\

50 s- 1 16 s_1

We take the natural log (In) of both sides (and divide by A) to get 1 r \dN/dt\0l 50 s- i' In A" 16 s- l A |_ \dN/dt\

= 2.98 X 1011 s = 9400 yr, which is the time elapsed since the death of the animal.

Geological Time Scale Dating arbon dating is useful only for determining the age of objects less than about 60,000 years old. The amount of ^ remaining in objects older than that is usually too small to measure accurately, although new techniques are allowing detection of even smaller amounts of 1J , pushing the time frame further back. On the other hand, radioactive isotopes with longer half-lives can be used in certain circum stances to obtain the age of older objects. For example, the decay of 29fU, because of its long half-life of 4.5 X 109years, is useful in determining the ages of rocks on a geologic time scale. When molten material on Earth long ago solidified into rock as the temperature dropped, different compounds solidified according to the melting points, and thus different compounds separated to some extent. Uranium SE TION 41 - 10

PHYSICS

APPLIED

Geological dating

Radioactive Dating

1123

PHYSICS

APPLIED

OldestEart roacnkds earliestlife

Thin window

Wire electrode = +) Met l tube (c thode = —) Insul tor

103V

-VWV--- r - ^ 1— To counter FIGURE 4 1 1 3 Di gr m o f Geiger counter.

FIGURE 4 1 1 4 Scintill tion counter with photom ultiplier tube. Incoming

Scintill tion

(to counter)

1124

HAPTER 41

present in a material became fixed in position and the daughter nuclei that result from the decay of uranium were also fixed in that position. Thus, by measuring the amount of remaining in the material relative to the amount of daughter nuclei, the time when the rock solidified can be determined. Radioactive dating methods using and other isotopes have shown the age of the oldest Earth rocks to be about 4 X 109yr. The age of rocks in which the oldest fossilized organisms are embedded indicates that life appeared more than 3§ billion years ago. The earliest fossilized remains of mammals are found in rocks 200 million years old, and humanlike creatures seem to have appeared about 2 million years ago. Radioactive dating has been indispensable for the reconstruc tion of Earth's history.

41—11 Detection o f R di tion Individual particles such as electrons, protons, a particles, neutrons, and 7 rays are not detected directly by our senses. onsequently, a variety of instruments have been developed to detect them.

ounters One of the most common is the Geiger counter. As shown in Fig. 41-13, it consists of a cylindrical metal tube filled with a certain type of gas. A long wire runs down the center and is kept at a high positive voltage (« 103V) with respect to the outer cylinder. The voltage is just slightly less than that required to ionize the gas atoms. When a charged particle enters through the thin "window" at one end of the tube, it ionizes a few atoms of the gas. The freed electrons are attracted toward the positive wire, and as they are accelerated they strike and ionize additional atoms. An "avalanche" of electrons is quickly produced, and when it reaches the wire anode, it produces a voltage pulse. The pulse, after being amplified, can be sent to an electronic counter, which counts how many particles have been detected. Or the pulses can be sent to a loudspeaker and each detection of a particle is heard as a "click." Only a fraction of the radiation emitted by a sample is detected by any detector. A scintillation counter makes use of a solid, liquid, or gas known as a scintillator or phosphor. The atoms of a scintillator are easily excited when struck by an incoming particle and emit visible light when they return to their ground states. Typical scintillators are crystals of Nal and certain plastics. One face of a solid scintillator is cemented to a photomultiplier tube, and the whole is wrapped with opaque material to keep it light-tight (in the dark) or is placed within a light-tight container. The photomultiplier (PM) tube converts the energy of the scintillator-emitted photon(s) into an electric signal. A PM tube is a vacuum tube containing several electrodes (typically to 14), called dynodes, which are maintained at successively higher voltages as shown in Fig. 41-14. At its top surface is a photoelectric surface, called the p otocat ode, whose work function (Section 37-2) is low enough that an electron is easily released when struck by a photon from the scintillator. Such an electron is accelerated toward the positive voltage of the first dynode. When it strikes the first dynode, the electron has acquired sufficient kinetic energy so that it can eject two to five more electrons. These, in turn, are accelerated toward the higher voltage second dynode, and a multiplication process begins. The number of electrons striking the last dynode may be 106 or more. Thus the passage of a particle through the scintillator results in an electric signal at the output of the PM tube that can be sent to an electronic counter just as for a Geiger tube. Solid scintillators are much more dense than the gas of a Geiger counter, and so are much more efficient detectors— especially for 7 rays, which interact less with matter than do a or /3 particles. Scintillators that can measure the total energy deposited are much used today and are called calorimeters.

Nuclear Physics and Radioactivity

In tracer work (Section 42-8), liquid scintillators are often used. Radioactive samples taken at different times or from different parts of an organism are placed directly in small bottles containing the liquid scintillator. This is particularly conve nient for detection of /3 rays from 3H and l\C, which have very low energies and have difficulty passing through the outer covering of a crystal scintillator or Geiger tube. A PM tube is still used to produce the electric signal from the liquid scintillator. A semiconductor detector consists of a reverse-biased pn junction diode (Sections 40-8 and 40-9). A particle passing through the junction can excite electrons into the conduction band, leaving holes in the valence band. The freed charges produce a short electrical pulse that can be counted just as for Geiger and scintillation counters. Hospital workers and others who work around radiation may carry film badges which detect the accumulation of radiation exposure. The film inside is periodically replaced and developed, the darkness being related to total exposure (see Section 42-6).

Visualization The devices discussed so far are used for counting the number of particles (or decays of a radioactive isotope). There are also devices that allow the track of charged particles to be seen. Very important are semiconductor detectors. Silicon wafer semiconductors have their surface etched into separate tiny pixels, each providing particle position information. They are much used in elementary particle physics ( hapter 43) to track the positions of particles produced and to determine their point of origin and/or their momentum (with the help of a magnetic field). The pixel arrangement can be D or MOS (Section 33-5), the latter able to incorporate electronics inside, allowing fast readout. One of the oldest tracking devices is the photographic emulsion, which can be small and portable, used now particularly for cosmic-ray studies from balloons. A charged particle passing through an emulsion ionizes the atoms along its path. These points undergo a chemical change, and when the emulsion is developed (like film) the particle's path is revealed. In a cloud chamber, used in the early days of nuclear physics, a gas is cooled to a temperature slightly below its usual condensation point ("supercooled"), and gas molecules condense on any ionized molecules present. Tiny droplets form around ions produced when a charged particle passes through (Fig. 41-15). Light scattering from these droplets reveals the track of the particle. The bubble chamber, invented in 1952 by D. A. Glaser (1926- ), makes use of a superheated liquid kept close to its normal boiling point. Bubbles characteristic of boiling form around ions produced by the passage of a charged particle, revealing paths of particles that recently passed through. Because a bubble chamber uses a liquid, often liquid hydrogen, many more interactions can occur than in a cloud chamber. A magnetic field is usually applied across the chamber so the momentum of the moving particles can be determined from the radius of curvature of their paths. A wire drift chamber consists of a set of closely spaced fine wires immersed in a gas (Fig. 41-16). Many wires are grounded, and the others between are kept at very high voltage. A charged particle passing through produces ions in the gas. Freed electrons drift toward the nearest high voltage wire, creating an "avalanche" of many more ions, and producing an electric pulse or signal at that wire. The positions of the particles are determined electronically by the position of the wire and by the time it takes the pulses to reach "readout" electronics at the ends of the wires. The paths of the particles are reconstructed electronically by computers which can "draw" a picture of the tracks, as shown in the photo at the start of hapter 43. An external magnetic field curves the paths, allowing the momentum of the particles to be measured. In many detectors, the energy of the particles can be measured by the strength of the electronic signal; such detectors are referred to as calorimeters, as already mentioned. SE TION 41 - 11

P th of p rticle

FIGURE 4 1 1 5 In cloud ch mber or bubble ch mber, droplets or bubbles re formed round ions produced by the p ss ge of ch rged p rticle. FIGURE 4 1 1 6 Wire-drift ch mber inside the Collider D etector t Fermil b (CDF). The photo t the st rt of Ch pter 43 (p ge 1164) w s done with this detector.

Detection of Radiation

1125

Summ ry Nuclear physics is the study of atomic nuclei. Nuclei contain protons and neutrons, which are collectively known as nucleons. The total number of nucleons, A , is the nucleus's atomic mass number. The number of protons, Z, is the atomic number. The number of neutrons equals A - Z. Isotopes are nuclei with the same Z, but with different numbers of neutrons. For an element X, an isotope of given Z and A is represented by

The nuclear radius is approximately proportional to A indicating that all nuclei have about the same density. Nuclear masses are specified in unified atomic mass units (u), where the mass of (including its 6 electrons) is defined as exactly 12.000000 u. In terms of the energy equivalent (because E = me2), lu

= 931.5 M eV /c 2 = 1.66 X 10" 27kg.

The mass of a stable nucleus is less than the sum of the masses of its constituent nucleons. The difference in mass (times c2) is the total binding energy. It represents the energy needed to break the nucleus into its constituent nucleons. The binding energy per nucleon averages about MeV per nucleon, and is lowest for low mass and high mass nuclei. Unstable nuclei undergo radioactive decay; they change into other nuclei with the emission of an a, (3, or 7 particle. An a particle is a 2He nucleus; a /3 particle is an electron or positron; and a 7 ray is a high-energy photon. In /3 decay, a neutrino is also emitted. The transformation of the parent into the daughter nucleus is called transmutation of the elements. Radioactive decay occurs spontaneously only when the mass of the products is less than the mass of the parent nucleus. The loss in mass appears as kinetic energy of the products. Alpha decay occurs via the purely quantum mechanical process of tunneling through a barrier.

Nuclei are held together by the strong nuclear force. The weak nuclear force makes itself apparent in (3 decay. These two forces, plus the gravitational and electromagnetic forces, are the four known types of force. Electric charge, linear and angular momentum, mass energy, and nucleon number are conserved in all decays. Radioactive decay is a statistical process. For a given type of radioactive nucleus, the number of nuclei that decay (AN) in a time AMs proportional to the number N of parent nuclei present: A N = - \ N At; (41 - 4a) the minus sign means N decreases in time. The proportionality constant A is called the decay constant and is characteristic of the given nucleus. The number N of nuclei remaining after a time t decreases exponentially N = N0e~M,

(41 - 6)

as does the activity, R = \dN/dt\: dN dt

dN dt

(41 -7c)

The half-life, 7i, is the time required for half the nuclei of a radioactive sample to decay. It is related to the decay constant by 71 =

0.693

(41 - 8)

Radioactive decay can be used to determine the age of certain objects, such as once-living biological material (^ ) or geological formations ( ^ U ) . Particle detectors include Geiger counters, scintillators with attached photomultiplier tubes, and semiconductor detectors. Detectors that can image particle tracks include semiconductors, photographic emulsions, bubble chambers, and wire drift chambers.

Questions 1. What do different isotopes of a given element have in common? How are they different? 2. What are the elements represented by the X in the following: (a) *lX ; (b) ^X ; (c) JX; (d) gX ; (e) 2$X ? 3. How many protons and how many neutrons do each of the isotopes in Question 2 have? 4. Identify the element that has nucleons and 50 neutrons. 5. Why are the atomic masses of many elements (see the Peri odic Table) not close to whole numbers? 6. How do we know there is such a thing as the strong nuclear force? 7. What are the similarities and the differences between the strong nuclear force and the electric force? . What is the experimental evidence in favor of radioactivity being a nuclear process? 9. The isotope 29 U is unusual in that it can decay by 7, (3 , and IQ+emission. What is the resulting nuclide for each case? 10. A 2q|U nucleus decays via a decay to a nucleus containing how many neutrons?

1126

HAPTER 41

Nuclear Physics and Radioactivity

11. Describe, in as many ways as you can, the difference between a, (3, and 7 rays. 12. What element is formed by the radioactive decay of (a) i}Na (fir); ( 6) i?Na (/3+); (e) ^P o (a)1 13. What element is formed by the decay of (a) 15P (fi ); ( b ) H Sf/r);

14. Fill in the missing particle or nucleus: (a) 2o a —» ? + e _ + v (ib) § u* -► ? + 7 (c) g r 46y + ? (d) 2%\Pu ? + a (e) 2jgNp 2$P u + ? 15. Immediately after a 298U nucleus decays to 29oTh + 2He, the daughter thorium nucleus may still have 92 electrons circling it. Since thorium normally holds only 90 electrons, what do you suppose happens to the two extra ones? 16. When a nucleus undergoes either (3~ or j + decay, what happens to the energy levels of the atomic electrons? What is likely to happen to these electrons following the decay?

17. The alpha particles from a given alpha-emitting nuclide are generally monoenergetic; that is, they all have the same kinetic energy. But the beta particles from a beta-emitting nuclide have a spectrum of energies. Explain the difference between these two cases. 18. Do isotopes that undergo electron capture generally lie above or below the stable nuclides in Fig. 41-2? 19. an hydrogen or deuterium emit an a particle? Explain. 20. Why are many artificially produced radioactive isotopes rare in nature? 21. An isotope has a half-life of one month. After two months, will a given sample of this isotope have completely decayed? If not, how much remains? 22. Why are none of the elements with Z > 92 stable?

23. 24. 25.

26. 27. 28.

A proton strikes a 3Li nucleus. As a result, an a particle and another particle are released. What is the other particle? an ^ dating be used to measure the age of stone walls and tablets of ancient civilizations? Explain. In both internal conversion and fi decay, an electron is emitted. How could you determine which decay process occurred? Describe how the potential energy curve for an a particle in an a-emitting nucleus differs from that for a stable nucleus. Explain the absence of fi+ emitters in the radioactive decay series of Fig. 41-12. As 2ggRn decays into ^ P b , how many alpha and beta parti cles are emitted? Does it matter which path in the decay series is chosen? Why or why not?

Problems_________________ 41 - 1 N uclear P ro p e rtie s 1. (I) A pi meson has a mass of 139 MeV/c2. What is this in atomic mass units? 2. (I) What is the approximate radius of an alpha particle (^He)? 3. (I) By what % is the radius of 21$\J greater than the radius of 2jgU? 4. (II) (a) What is the approximate radius of a 14§ d nucleus? (b) Approximately what is the value of A for a nucleus whose radius is 3.7 X 10-15 m? 5. (II) What is the mass of a bare a particle (without electrons) in MeV/c2? 6. (II) Suppose two alpha particles were held together so they were just touching. Estimate the electrostatic repulsive force each would exert on the other. What would be the accelera tion of an alpha particle subjected to this force? 7. (II) (a) Show that the density of nuclear matter is essen tially the same for all nuclei. (b) What would be the radius of the Earth if it had its actual mass but had the density of nuclei? (c) What would be the radius of a 21$\J nucleus if it had the density of the Earth? . (II) What stable nucleus has approximately half the radius of a uranium nucleus? [Hint Find A and use Appendix F to get Z.] 9. (II) If an alpha particle were released from rest near the surface of a looFm nucleus, what would its kinetic energy be when far away? 10. (II) (a) What is the fraction of the hydrogen atom's mass that is in the nucleus? (b) What is the fraction of the hydrogen atom's volume that is occupied by the nucleus? 11. (II) Approximately how many nucleons are there in a 1.0-kg object? Does it matter what the object is made of? Why or why not? 12. (II) How much kinetic energy must an a particle have to just "touch" the surface of a 29§U nucleus? 41 - 2 B inding Energy 13. (I) Estimate the total binding energy for 29 U, using Fig. 41-1. 14. (II) Use Appendix F to calculate the binding energy of 2H (deuterium). 15. (II) Determine the binding energy of the last neutron in a 15P nucleus.

16. (II) alculate the total binding energy, and the binding energy per nucleon, for (a) ^Li, (b) ^ A u . Use Appendix F. 17. (II) ompare the average binding energy of a nucleon in §N a to that in 24Na. 18. (Ill) How much energy is required to remove (a) a proton, (b) a neutron, from 1yN? Explain the difference in your answers. 19. (Ill) (a) Show that the nucleus ®Be (mass = 8.005305 u) is unstable and will decay into two a particles. (b) Is stable against decay into three a particles? Show why or why not. 41 - 3 to 41 - 7 R adioactive D ecay 20. (I) How much energy is released when tritium, fH, decays by fi~ emission? 21. (I) What is the maximum kinetic energy of an electron emitted in the fi decay of a free neutron? 22. (I) Show that the decay 1J —►^B + p is not possible because energy would not be conserved. 23. (I) The ^Li nucleus has an excited state 0.48 MeV above the ground state. What wavelength gamma photon is emitted when the nucleus decays from the excited state to the ground state? 24. (II) Give the result of a calculation that shows whether or not the following decays are possible: ->■ (a) + n; (b) 14N 13N + n; 19K + n. (c) gK 25. (II) 2iNa is radioactive, (a) Is it a fi~ or fi+ emitter? (b) Write down the decay reaction, and estimate the maximum kinetic energy of the emitted fi. 26.(11) When ?§Ne (mass = 22.9945 u) decays to §Na (mass = 22.9898 u), what is the maximum kinetic energy of the emitted electron? What is its minimum energy? What is the energy of the neutrino in each case? Ignore recoil of the daughter nucleus. 27. (II) A 298U nucleus emits an a particle with kinetic energy = 4.20 MeV. (a) What is the daughter nucleus, and (b) what is the approximate atomic mass (in u) of the daughter atom? Ignore recoil of the daughter nucleus. 28. (II) What is the maximum kinetic energy of the emitted fi particle during the decay of 27 0?

Problems

1127

29. (II) A nucleus of mass 256 u, initially at rest, emits an a particle with a kinetic energy of 5.0 MeV. What is the kinetic energy of the recoiling daughter nucleus? 30. (II) The isotope 2 4Po can decay by either a or j8" emission. What is the energy release in each case? The mass of 2^P o is 218.008965 u. 31. (II) The nuclide 32P decays by emitting an electron whose maximum kinetic energy can be 1.71 MeV. (a) What is the daughter nucleus? (b) alculate the daughter's atomic mass (in u). 32. (II) A photon with a wavelength of 1.00 X 10-13m is ejected from an atom. alculate its energy and explain why it is a 7 ray from the nucleus or a photon from the atom. 33. (II) How much energy is released in electron capture by beryllium: 4Be + e _ —> 3Li + v l 34. (II) How much recoil energy does a i§K nucleus get when it emits a 1.46-MeV gamma ray? 35. (II) Determine the maximum kinetic energy of j + particles released when X\C decays to ^B . What is the maximum energy the neutrino can have? What is the minimum energy of each? 36. (Ill) The a particle emitted when decays has 4.20 MeV of kinetic energy. alculate the recoil kinetic energy of the daughter nucleus and the Q -value of the decay. 37. (Ill) What is the energy of the a particle emitted in the decay ^ P o —» ^ P b + a? Take into account the recoil of the daughter nucleus. 38. (Ill) Show that when a nucleus decays by /3+ decay, the total energy released is equal to (MP - Md — 2m e)c2, where MP and MD are the masses of the parent and daughter atoms (neutral), and rae is the mass of an electron or positron. 41 - to 41 - 1 0 H alf - L ife, D ecay R a te s, D ecay S e rie s, D atin g 39. (I) (a) What is the decay constant of whose half-life is 4.5 X 109yr? (b) The decay constant of a given nucleus is 3.2 X 10" 5 s-1. What is its half-life? 40. (I) A radioactive material produces 1280 decays per minute at one time, and 3.6 h later produces 320 decays per minute. What is its half-life? 41. (I) What fraction of a sample of ^G e, whose half-life is about 9 months, will remain after 2.0 yr? 42. (I) What is the activity of a sample of ^ that contains 8.1 X IO20nuclei? 43. (I) What fraction of a sample is left after exactly 6 half-lives? 44. (II) A sample of 27 0 and a sample of 153I both have N0 atoms at t = 0. How long will it take until both have the same activity? (Use Appendix F for half-life data.) 45. (II) How many nuclei of 21$\J remain in a rock if the activity registers 340 decays per second? 46. (II) In a series of decays, the nuclide becomes ^ P b . How many a and j _ particles are emitted in this series? 47. (II) The iodine isotope 133I is used in hospitals for diagnosis of thyroid function. If 782 fig are ingested by a patient, determine the activity (a) immediately, (b) 1.00 h later when the thyroid is being tested, and (c) 4.0 months later. Use Appendix F.

1128

HAPTER 41

Nuclear Physics and Radioactivity

48. (II) 155 s has a half-life of 30.8 s. (a) If we have 7.8 fig initially, how many s nuclei are present? (b) How many are present 2.6 min later? (c) What is the activity at this time? (d) After how much time will the activity drop to less than about 1 per second? 49. (II) alculate the mass of a sample of pure 19K with an initial decay rate of 2.0 X 105 s-1. The half-life of 4 1033 yr.

How long would you expect to wait for one proton in your body to decay (approximate your body as all water)? 67. Show, using the decays given in Section 41 -5, that the neutrino has either spin \ or §• 6 . The original experiments which established that an atom

has a heavy, positive nucleus were done by shooting alpha particles through gold foil. The alpha particles used had a kinetic energy of 7.7 MeV. What is the closest they could get to a gold nucleus? How does this compare with the size of the nucleus? 69. How long must you wait (in half-lives) for a radioactive sample to drop to 1 .00% of its original activity? 70. If the potassium isotope i§K gives 45 decays/s in a liter of milk, estimate how much jcjK and regular 19K are in a liter of milk. Use Appendix F. 71. (a) In a decay of, say, a 2ggRa nucleus, show that the nucleus carries away a fraction 1/(1 + °f the total energy available, where A D is the mass number of the daughter nucleus. [Hint: Use conservation of momentum as well as conservation of energy.] (b) Approximately what percentage of the energy available is thus carried off by the a particle when 2ggRa decays? 72. Strontium -90 is produced as a nuclear fission product of uranium in both reactors and atomic bombs. Look at its location in the Periodic Table to see what other elements it might be similar to chemically, and tell why you think it might be dangerous to ingest. It has too many neutrons, and it decays with a half-life of about 29 yr. How long will we have to wait for the amount of ^S r on the E arth's surface to reach 1 % of its current level, assuming no new material is scattered about? Write down the decay reaction, including the daughter nucleus. The daughter is radioactive: write down its decay. 73. Using the uncertainty principle and the radius of a nucleus, estimate the minimum possible kinetic energy of a nucleon in, say, iron. Ignore relativistic corrections. [Hint: A particle can have a momentum at least as large as its momentum uncertainty.]

74. (a)

alculate the kinetic energy of the a particle emitted when 292U decays. (b) Use Eq. 41-1 to estimate the radius of an a particle and a 29oTh nucleus. Use this to estimate (c) the maximum height of the oulomb barrier, and (d) its width AB in Fig. 41-7.

75. The nuclide ^ O s decays with (3~ energy of 0.14 MeV accompanied by J rays of energy 0.042 MeV and 0.129 MeV. (a) What is the daughter nucleus? (b) Draw an energy-level diagram showing the ground states of the parent and daughter and excited states of the daughter, (c) To which of the daughter states does (3~ decay of ^ O s occur?

76. Determine the activities of (a) 1.0 g of ^ I (71 = 8.02 days) and (b) 1.0 g of 29328U (Ti = 4.47 X 109yr).

77. Use Fig. 41-1 to estimate the total binding energy for copper and then estimate the energy, in joules, needed to break a 3.0-g copper penny into its constituent nucleons.

78. Instead of giving atomic masses for nuclides as in Appendix F, some Tables give the mass excess, A, defined as A = M — A , where A is the atomic mass number and M is the mass in u. Determine the mass excess, in u and in M eV/c2, for: (a) 2He; (b) ; (c) gSr; (d) 292U. (e) From a glance at Appendix F, can you make a generalization about the sign of A as a function of Z or A l

79. When water is placed near an intense neutron source, the neutrons can be slowed down by collisions with the water molecules and eventually captured by a hydrogen nucleus to form the stable isotope called deuterium, 2H, giving off a gamma ray. What is the energy of the gamma ray?

80. (a) Show that the mean life of a radioactive nuclide, defined as OO t N (t) dt )________ [ N (t) dt is r = 1/A. (b) What fraction of the original number of nuclei remains after one mean life?

81. (a) A 72-gram sample of natural carbon contains the usual fraction of . Estimate how long it will take before there is only one nucleus left. (b) How does the answer in (a) change if the sample is 270 grams? What does this tell you about the limits of carbon dating?

82. If the mass of the proton were just a little closer to the mass of the neutron, the following reaction would be possible even at low collision energies: e _ + p —> n + v. (fl) Why would this situation be catastrophic? (b) By what percentage would the proton's mass have to be increased to make this reaction possible?

83. What is the ratio of the kinetic energies for an alpha particle and a beta particle if both make tracks with the same radius of curvature in a magnetic field, oriented perpendicular to the paths of the particles?

General Problems

1129

84. A 1.00-g sample of natural samarium emits a particles at a rate of 120 s_1 due to the presence of 16?Sm. The natural abundance of ^ S m is 15%. alculate the half-life for this decay process. 85. Almost all of naturally occurring uranium is 21$\J with a half-life of 4.468 X 109yr. Most of the rest of natural uranium is with a half-life of 7.04 X 10 yr. Today a sample contains 0.720% ^ U . (a) What was this percentage 1.0 billion years ago? (b) What percentage of will remain 100 million years from now? 6. A typical banana contains 400 mg of potassium, of which a small fraction is the radioactive isotope }§K (see Appendix F). Estimate the activity of an average banana due to i§K. 87. Some radioactive isotopes have half-lives that are larger than the age of the universe (like gadolinium or samarium). The only way to determine these half-lives is to monitor the decay rate of a sample that contains these isotopes. For example, suppose we find an asteroid that currently contains about 15,000 kg of ^ G d (gadolinium) and we detect an activity of 1 decay/s. What is the half-life of gadolinium (in years)?

18Numerical/ om puter "89. (I) A laboratory has a 1.80-^tg sample of radioactive 13N whose decay constant A = 1.16 X 10- 3 s-1. alculate the initial number of nuclei, N q, present in the sample. Use the radioactive decay law, N = N0e~kt, to determine the number of nuclei N present at time t for t = 0 to 30 minutes (1800 s) in steps of 0.5 min (30 s). Make a graph of N versus t and from the graph determine the half-life of the sample. "90. (II) onstruct a spreadsheet (or other numerical tool) that will reproduce Fig. 41-1, the graph of binding energy per nucleon (in MeV) vs. the mass number A . Using Appendix F, calculate the binding energy per nucleon for the most stable isotope of each possible mass number A > 2. [The first few values will be for jH, ^He (it is more stable than 3H), ^He, §Li, and 3IJ (since it is more stable than jBe).] To reduce the amount of data, for A > 20 plot only points for even values of A , and plot to a maximum of A = 142.

. Decay series, such as that shown in Fig. 41-12, can be classi fied into four families, depending on whether the mass numbers have the form An, An + 1, An + 2, or An + 3, where n is an integer. Justify this statement and show that for a nuclide in any family, all its daughters will be in the same family.

A n sw e rs to E x ercises A: 0.042130 u.

E: (a).

B: 7.98 MeV/ nucleon.

F: 1.37 X l O ^ V 1.

: (b).

G: (a).

D: (c).

1130

HAPTER 41

Nuclear Physics and Radioactivity

Di gr m of ITER (Intern tion l Thermonucle r Experim ent l Re ctor), which will hopefully begin oper tion round 2016. Inside its c vity, over 12 m in di meter, pl sm of electrons nd light nuclei will be he ted to high temper tures th t riv l the Sun. Confining pl sm by m gnetic fields h s proved difficult, nd intense rese rch is needed if the fusion of light nuclei is to fulfill its promise s source o f bund nt nd rel tively cle n power. This Ch pter covers the b sic physics topics of nucle r re ctions, nucle r fission, nucle r fusion, nd how we obt in nucle r energy. We lso ex mine the he lth spects of r di tion dosimetry, ther py, nd im g in g b y CAT, P E T , S P E T , n d M R I.

Nuclear Energy; Effects and Uses of Radiation HAPTER -OPENING QUESTIONS — Guess now! 1. The Sun is powered by (a) nuclear alpha decay. (b) nuclear beta decay. (c) nuclear gamma decay. (d) nuclear fission. (e) nuclear fusion. 2. Which radiation induces the most biological damage for a given amount of energy deposited in tissue? (a) Alpha particles. (b) Gamma radiation. (c) Beta radiation. (d) They all do the same damage for the same deposited energy. (e) It depends on the type of tissue.

e continue our study of nuclear physics in this hapter. We begin with a discussion of nuclear reactions, and then we examine the important large energy-releasing processes of fission and fusion. This hapter also deals with the effects of nuclear radiation passing through matter, particularly biological matter, and how radiation is used medically for therapy, diagnosis, and imaging techniques.

ONTENTS 4 2 -1

Nucle r Re ctions nd the Tr nsmut tion of Elements

4 2 -2

Cross Section

4 2 -3

Nucle r Fission; Nucle r Re ctors

4 2 -4

Nucle r Fusion

4 2 -5

P ss ge of R di tion Through M tter; R di tion D m ge

4 2 -6

Me surement of R di tion — Dosimetry

* 4 2 -7

R di tion Ther py

* 4 2 -8

Tr cers in Rese rch nd Medicine

* 4 2 -9

Im ging by Tomogr phy: CAT Sc ns nd Emission Tomogr phy

* 42 - 10 Nucle r M gnetic Reson nce (NMR); M gnetic Reson nce Im ging (MRI)

1131

4 2 —1 Nucle r Re ctions nd the Tr nsmut tion o f Elements When a nucleus undergoes a or /3 decay, the daughter nucleus is a different element from the parent. The transformation of one element into another, called transmutation, also occurs by means of nuclear reactions. A nuclear reaction is said to occur when a given nucleus is struck by another nucleus, or by a simpler particle such as a 7 ray or neutron, and an interaction takes place. Ernest Rutherford was the first to report seeing a nuclear reaction. In 1919 he observed that some of the a particles passing through nitrogen gas were absorbed and protons emitted. He concluded that nitrogen nuclei had been transformed into oxygen nuclei via the reaction 2He + 14N - > 17sO + }H, where 2He is an a particle, and }H is a proton. Since then, a great many nuclear reactions have been observed. Indeed, many of the radioactive isotopes used in the laboratory are made by means of nuclear reactions. Nuclear reactions can be made to occur in the laboratory, but they also occur regularly in nature. In hapter 41 we saw an example: 1J is continually being made in the atmosphere via the reaction n + 14N —> ^ + p. Nuclear reactions are sometimes written in a shortened form: for example, the reaction n + 14N —» U6C + p can be written (n, p) " . The symbols outside the parentheses on the left and right represent the initial and final nuclei, respectively. The symbols inside the parentheses represent the bombarding particle (first) and the emitted small particle (second). In any nuclear reaction, both electric charge and nucleon number are conserved. These conservation laws are often useful, as the following Example shows. ON EPTUAL EXAMPLE 42 - 1 I Deuterium reaction. A neutron is observed to strike an 1|0 nucleus, and a deuteron is given off. (A deuteron, or deuterium, is the isotope of hydrogen containing one proton and one neutron, 2H; it is sometimes given the symbol d or D.) What is the nucleus that results? RESPONSE We have the reaction n + J|0 —>> ? + ?H. The total number of nucleons initially is 1 + 16 = 17, and the total charge is 0 + = . The same totals apply after the reaction. Hence the product nucleus must have Z = 7 and A = 15. From the Periodic Table, we find that it is nitrogen that has Z = 7, so the nucleus produced is ^N. EXERCISE A D eterm ine the resulting nucleus in the re ction n + ^ B

? + 7.

Energy and momentum are also conserved in nuclear reactions, and can be used to determine whether or not a given reaction can occur. For example, if the total mass of the final products is less than the total mass of the initial particles, this decrease in mass (recall AE = Am c2) is converted to kinetic energy (K) of the outgoing particles. But if the total mass of the products is greater than the total mass of the initial reactants, the reaction requires energy. The reaction will then not occur unless the bombarding particle has sufficient kinetic energy. onsider a nuclear reaction of the general form a + X —» Y + b,

(4 2 - 1 )

where a is a projectile particle (or small nucleus) that strikes nucleus X, producing

1132

HAPTER 42

Nuclear Energy; Effects and Uses of Radiation

nucleus Y and particle b (typically, p, n, a, 7). We define the reaction energy, or Q-yalue, in terms of the masses involved, as Q = (Ma + Mx - Mb - M y ) c2.

(42 -2a)

For a J ray, M = 0. Because energy is conserved, Q has to be equal to the change in kinetic energy (final minus initial): Q = K + KY - K a - Kx .

(42-2b)

If X is a target nucleus at rest (or nearly so) struck by incoming particle a, then Kx = 0. For Q > 0, the reaction is said to be exot ermic or exoergic; energy is released in the reaction, so the total kinetic energy is greater after the reaction than before. If Q is negative (Q < 0), the reaction is said to be endot ermic or endoergic: the final total kinetic energy is less than the initial kinetic energy, and an energy input is required to make the reaction happen. The energy input comes from the kinetic energy of the initial colliding particles (a and X). EXAMPLE 4 2 - 2

A slow - neutron reaction. The nuclear reaction

n + X§B

]U + ^He

is observed to occur even when very slow-moving neutrons (mass Mn = 1.0087 u) strike a boron atom at rest. For a particular reaction in which Kn « 0, the outgoing helium (MHe = 4.0026 u) is observed to have a speed of 9.30 X 106m/s. Determine (a) the kinetic energy of the lithium (MLi = 7.0160 u), and (b) the (2-value of the reaction. APPROA H Since the neutron and boron are both essentially at rest, the total momentum before the reaction is zero; momentum is conserved and so must be zero afterward as well. Thus, VU = ^He^HeWe solve this for vLi and substitute it into the equation for kinetic energy. In (b) we use Eq. 42-2b. SOLUTION (a) We can use classical kinetic energy with little error, rather than relativistic formulas, because vHe = 9.30 X 106m/s is not close to the speed of light c, and vL{will be even less since MLi > MHe. Thus we can write: KLi ~ 2 M u vu - 2 M^ ^Hc vHe We put in numbers, changing the mass in u to kg and recall that 1.60 x 10"13J = 1 MeV: K\; —

(4.0026 u) 2(1.66 x 10~27 kg/u)2(9.30 x 106m /s )2 2(7.0160 u )(l .66

= 1.64

10-27 kg/u)

10-13J = 1.02 MeV.

(b) We are given the data Ka = Kx = 0 in Eq. 42-2b, so Q = K i + K He, where ^H e

\ ^ H e vHe

= | (4.0026 u )(l .66 = 2.87

10"27kg/u)(9.30

106m /s )2

10-13J = 1.80 MeV.

Hence, Q = 1.02 MeV + 1.80 MeV = 2.82 MeV. SE TION 42 - 1

Nuclear Reactions and the Transmutation of Elements

1133

EXAMPLE 42 - 3

Will the reaction "go"?

p + ^ occur when ^

an the reaction

-► 13N + n

is bombarded by 2.0 -MeV protons?

APPROA H The reaction will "go" if the reaction is exothermic (Q > 0) and even if Q < 0 if the input momentum and kinetic energy are sufficient. First we calculate Q from the difference between final and initial masses using Eq. 42 -2a, and looking up the masses in Appendix F. SOLUTION The total masses before and after the reaction are: Before

M(1l ) = 13.003355 M0H) = 1.007825 14.011180

After

M(^N) = 13.005739 M(n) = 1.008665 14.014404

(We must use the mass of the }H atom rather than that of the bare proton because the masses of l\C and 13N include the electrons, and we must include an equal number of electron masses on each side of the equation since none are created or destroyed.) The products have an excess mass of (14.014404 - 14.011180)u = 0.003224 u X 931.5 M eV /u = 3.00 MeV. Thus Q = - 3.00 MeV, and the reaction is endothermic. This reaction requires energy, and the 2.0-MeV protons do not have enough to make it go. NOTE The incoming proton in this Example would have to have somewhat more than 3.00 MeV of kinetic energy to make this reaction go; 3.00 MeV would be enough to conserve energy, but a proton of this energy would produce the 13N and n with no kinetic energy and hence no momentum. Since an incident 3.0-MeV proton has momentum, conservation of momentum would be violated. A calculation using conservation of energy and of momentum, as we did in Examples 41 -6 and 42-2, shows that the minimum proton energy, called the threshold energy, is 3.23 MeV in this case ( = Problem 16).

Neutron Physics Neutron c ptured by 2^ U .

2f2U

dec ys by j8 dec y to neptunium -239.

2||N p itself dec ys by j8 dec y to produce plutonium - 239.

FIGURE 42 - 1 N eptunium nd plutonium re produced in this series o f re ctions, fter bom b rdm ent o f by neutrons.

21$\J

The artificial transmutation of elements took a great leap forward in the 1930s when Enrico Fermi realized that neutrons would be the most effective projectiles for causing nuclear reactions and in particular for producing new elements. Because neutrons have no net electric charge, they are not repelled by positively charged nuclei as are protons or alpha particles. Hence the probability of a neutron reaching the nucleus and causing a reaction is much greater than for charged projectiles,f particularly at low energies. Between 1934 and 1936, Fermi and his co -workers in Rome produced many previously unknown isotopes by bombarding different elements with neutrons. Fermi realized that if the heaviest known element, uranium, is bombarded with neutrons, it might be possible to produce new elements with atomic numbers greater than that of uranium. After several years of hard work, it was suspected that two new elements had been produced, neptunium (Z = 93) and plutonium (Z = 94). The full confirmation that such "transuranic" elements could be produced came several years later at the University of alifornia, Berkeley. The reactions are shown in Fig. 42-1. It was soon shown that what Fermi had actually observed when he bombarded uranium was an even stranger process—one that was destined to play an extraor dinary role in the world at large. We discuss it in Section 42-3. trTh t is, positively ch rged p rticles. Electrons r rely c use nucle r re ctions bec use they do not inter ct vi the strong nucle r force.

1134

HAPTER 42

Nuclear Energy; Effects and Uses of Radiation

4 2 —2 Cross Section Some reactions have a higher probability of occurring than others. The reaction probability is specified by a quantity called the cross section. Although the size of a nucleus, like that of an atom, is not a clearly defined quantity since the edges are not distinct like those of a tennis ball or baseball, we can nonetheless define a cross section for nuclei undergoing collisions by using an analogy. Suppose that projectile particles strike a stationary target of total area A and thickness £, as shown in Fig. 42-2. Assume also that the target is made up of identical objects (such as marbles or nuclei), each of which has a cross-sectional area cr, and we assume the incoming projectiles are small by comparison. We assume that the target objects are fairly far apart and the thickness £ is so small that we don't have to worry about overlapping. This is often a reasonable assumption because nuclei have diameters on the order of 10-14 m but are at least 10-10 m (atomic size) apart even in solids. If there are n nuclei per unit volume, the total cross-sectional area of all these tiny targets is A ' = nAlcr since nA l = (n)(volume) is the total number of targets and cr is the crosssectional area of each. If A ' « A, most of the incident projectile particles will pass through the target without colliding. If R0 is the rate at which the projectile particles strike the target (number/second), the rate at which collisions occur, R, is R — Rn

— Rn

^ 0 °0 0

Projectiles 0

---

0 0 ---0 u 0

h£H -FIGURE 4 2 2 Projectile particles strike a target of area A and thickness £made up of n nuclei per unit volume.

nA£a

so R = R0n£(i. Thus, by measuring the collision rate, R, we can determine cr: cr =

R Rn n£

(4 2 - 3 )

If nuclei were simple billiard balls, and R the number of particles that are deflected per second, cr would represent the real cross-sectional area of each ball. But nuclei are complicated objects that cannot be considered to have distinct boundaries. Furthermore, collisions can be either elastic or inelastic, and reactions can occur in which the nature of the particles can change. By measuring R for each possible process, we can determine an effective cross section, cr, for each process. None of these cross sections is necessarily related to a geometric cross-sectional area. Rather, cr is an "effective" target area. It is a measure o f t e probability o f a collision or o f a particular reaction occurring per target nucleus, independent of the dimensions of the entire target. The concept of cross section is useful because cr depends only on the properties of the interacting particles, whereas R depends on the thickness and area of the physical (macroscopic) target, on the number of particles in the incident beam, and so on. When a given pair of particles interact, we define their elastic cross section crel using Eq. 42-3, where R for a given experimental setup is the rate of elastic collisions (or elastic scattering), by which we mean collisions for which the final particles are the same as the initial particles (a = b, X = Y in Eq. 42-1) and 13N + n or p + ^ —> ^B + jHe, and so on. The sum of all the separate reaction cross sections (for a given pair of initial particles) is called the total reaction cross section, crR. The total cross section, crT, is °T

°el +

°m el +

and is a measure of all possible interactions or collisions starting with given initial particles. Said another way, crx is a measure of how many of the incident particles interact in some way and hence are eliminated from the incident beam.

SE TION 4 2 - 2

1135

0.001

0.01 0.1 1.0 10 100 Energy (eY)

FIGURE 4 2 3 The neutron cross section for c dmium - 114 s function o f incom ing neutron kinetic energy. It is extr ordin rily l rge for ^ 1 eV. N o te th t both sc les re log rithmic.

We can also define differential cross sections, which represent the probability of the deflected (or emitted) particles leaving at particular angles. It is said that when one of the first nuclear cross sections was measured, a physicist, surprised that it was as large as it was (~ 10_2 m2), remarked, "it's as big as a barn." Ever since then nuclear cross sections have been measured in "barns," where 1 barn (bn) = 10"2 m2. The value of cr for a given reaction depends on, among other things, the incident kinetic energy. Typical nuclear cross sections are on the order of barns, but they can vary from millibarns to kilobarns or more. Figure 42-3 shows the cross section for neutron capture in cadmium (n + 'JJ d - > ^ d + y) as a function of neutron kinetic energy. Neutron cross sections for most materials are greater at low energies, as in Fig. 42-3. To produce nuclear reactions at a high rate it is therefore desirable that the bombarding neutrons have low energy. Neutrons that have been slowed down and have reached equilibrium with matter at room temperature (§kT ~ 0.04 eV at T = 300 K) are called thermal neutrons. EXAMPLE 4 2 - 4

Using cross section. The reaction

p + —> 27 0 + n has a cross section of 0.65 bn for a particular incident proton energy. Suppose the iron target has an area of 1.5 cm2, and is 2.0/um thick. The density of iron is 7.8 X 103kg/m3. If the protons are incident at a rate of 2.0 X 1013particles/s, calculate the rate at which neutrons are produced.

FIGURE 4 2 4 Fission o f U nucleus fter c pture o f neutron, ccording to the liquid - drop m odel.

APPROA H We use Eq. 42-3 in the form: R = R0nicr. SOLUTION We are given R0 = 2.0 X 1013particles/s, i = 2.0 X 10_6m, and c = 0.65 bn. Recalling from hapter 17 that one mole (mass = 56 g for iron) contains 6.02 X 1023 atoms, then the number of iron atoms per unit volume is x (7.8 X 103kg/m3) n = (6.02 X 1023 atoms/mole) 7----------- ------------ r = 8.4 X 102 atoms/m3. v ' (56 X 10" 3kg/mole) Then we find that the rate at which neutrons are produced is R = R0nicr = (2.0 X 1013particles/s)(8.4 = 2.2 X 10 particles/s.

102 atoms/m 3)(2.0

10" 6m)(0.65

10" 2 m2)

4 2 —3 Nucle r Fission; Nucle r Re ctors In 1938, the German scientists Otto Hahn and Fritz Strassmann made an amazing discovery. Following up on Fermi's work, they found that uranium bombarded by neutrons sometimes produced smaller nuclei that were roughly half the size of the original uranium nucleus. Lise Meitner and Otto Frisch quickly realized what had happened: the uranium nucleus, after absorbing a neutron, actually had split into two roughly equal pieces. This was startling, for until then the known nuclear reactions involved knocking out only a tiny fragment (for example, n, p, or a) from a nucleus.

Nuclear Fission and

(d)

1136

HAPTER 42

hain Reactions

This new phenomenon was named nuclear fission because of its resemblance to biological fission (cell division). It occurs much more readily for than for the more common 29 U. The process can be visualized by imagining the uranium nucleus to be like a liquid drop. According to this liquid-drop model, the neutron absorbed by the 292U nucleus gives the nucleus extra internal energy (like heating a drop of water). This intermediate state, or compound nucleus, is (because of the absorbed neutron). The extra energy of this nucleus—it is in an excited state—appears as increased motion of the individual nucleons, which causes the nucleus to take on abnormal elongated shapes, Fig. 42-4. When the nucleus elongates (in this model) into the shape shown in Fig. 42-4c, the attraction of the two ends via the short-range nuclear force is greatly weakened by the increased separation distance, and the electric repulsive force becomes dominant, and the nucleus splits in two (Fig. 42-4d). The two resulting nuclei, X 1 and X2, are called fission fragments, and in the process a number

of neutrons (typically two or three) are also given off. The reaction can be written n + 2gU

Xj + X 2 + neutrons.

(4 2 - 4 )

The compound nucleus, 2i2U, exists for less than IO-12 s, so the process occurs very quickly. The two fission fragments, X 1 and X2, more often split the original uranium mass as about 40%-60% rather than precisely half and half. A typical fission reaction is n + 23i U -> ^ B a + gKr + 3n,

(4 2 - 5 )

although many others also occur. ON EPTUAL EXAMPLE 4 2 - 5 I ounting nucleons. Identify the element X in the fission reaction n + 292U —> ^X + ^Sr + 2n. RESPONSE The number of nucleons is conserved (Section 41-7). The uranium nucleus with 235 nucleons plus the incoming neutron make 235 + 1 = 236 nucleons. So there must be 236 nucleons after the reaction. The Sr has 93 nucleons, and the two neutrons make 95 nucleons, so X has A = 236 - 95 = 141. Electric charge is also conserved: before the reaction, the total charge is 92e. After the reaction the total charge is (Z + 38) e and must equal 92e. Thus Z = 92 - 38 = 54. The element with Z = 54 is xenon (see Appendix F or the Periodic Table inside the back cover), so the isotope is 'gXe. Figure 42-5 shows the distribution of 292U fission fragments according to mass. Only rarely (about 1 in 104) does a fission result in equal mass fragments (arrow in Fig. 42-5). A tremendous amount of energy is released in a fission reaction because the mass of 292U is considerably greater than the total mass of the fission fragments plus released neutrons. This can be seen from the binding-energy-per-nucleon curve of Fig. 41-1; the binding energy per nucleon for uranium is about 7.6 MeV/nucleon, but for fission fragments that have intermediate mass (in the center portion of the graph, A ~ 100), the average binding energy per nucleon is about 8.5 MeV/nucleon. Since the fission fragments are more tightly bound, the sum of their masses is less than the mass of the uranium. The difference in mass, or energy, between the original uranium nucleus and the fission fragments is about 8.5 —7.6 = 0.9 MeV per nucleon. Since there are 236 nucleons involved in each fission, the total energy released per fission is (0.9 MeV/nucleon)(236 nucleons) « 200 MeV. (4 2 - 6 ) This is an enormous amount of energy for one single nuclear event. At a practical level, the energy from one fission is tiny. But if many such fissions could occur in a short time, an enormous amount of energy at the macroscopic level would be available. A number of physicists, including Fermi, recognized that the neutrons released in each fission (Eqs. 42-4 and 42-5) could be used to create a chain reaction. That is, one neutron initially causes one fission of a uranium nucleus; the two or three neutrons released can go on to cause additional fissions, so the process multiplies as shown schematically in Fig. 42-6.

FIGURE 4 2 - 6

70 80 90 100 110 120 130 140 150 160 170

M ss number,

FIGURE 4 2 - 5 M ss distribution of fission fr gments from 292U + n. The sm ll rrow indic tes equ l m ss fr gments X (236 117, ssuming 2 neutrons re liber ted). N ote th t the vertic l sc le is log rithmic.

— 2)=

Ch in re ction.

SE TION 4 2 - 3

1137

FIGURE 4 2 7 This is the only photograph of the first nuclear reactor, built by Fermi under the grandstand of Stagg Field at the University of hicago. It is shown here under construction as a layer of graphite (used as moderator) was being placed over a layer of natural uranium. On December 2,1942, Fermi slowly withdrew the cadmium control rods and the reactor went critical. This first self-sustaining chain reaction was announced to Washington, via telephone, by Arthur ompton who witnessed the event and reported: "The Italian navigator has just landed in the new world."

If a self-sustaining chain reaction was actually possible in practice, the enormous energy available in fission could be released on a larger scale. Fermi and his co-workers (at the University of hicago) showed it was possible by constructing the first nuclear reactor in 1942 (Fig. 42-7).

N udear Reactors

FIGURE 4 2 8 If the amount of uranium exceeds the critical mass, as in (b), a sustained chain reaction is possible. If the mass is less than critical, as in (a), too many neutrons escape before additional fissions occur, and the chain reaction is not sustained.

()

1138

HAPTER 42

Several problems have to be overcome to make any nuclear reactor function. First, the probability that a ^ U nucleus will absorb a neutron is large only for slow neutrons, but the neutrons emitted during a fission (which are needed to sustain a chain reaction) are moving very fast. A substance known as a moderator must be used to slow down the neutrons. The most effective moderator will consist of atoms whose mass is as close as possible to that of the neutrons. (To see why this is true, recall from hapter 9 that a billiard ball striking an equal mass ball at rest can itself be stopped in one collision; but a billiard ball striking a heavy object bounces off with nearly unchanged speed.) The best moderator would thus contain }H atoms. Unfortunately, }H tends to absorb neutrons. But the isotope of hydrogen called deuterium, 2H, does not absorb many neutrons and is thus an almost ideal moderator. Either {H or 2H can be used in the form of water. In the latter case, it is heavy water, in which the hydrogen atoms have been replaced by deuterium. Another common moderator is grap ite, which consists of 126C atoms. A second problem is that the neutrons produced in one fission may be absorbed and produce other nuclear reactions with other nuclei in the reactor, rather than produce further fissions. In a "light-water" reactor, the }H nuclei absorb neutrons, as does 2^U to form 2^U in the reaction n + 2^U —» 2^U + 7. Naturally occurring uranium 1 contains 99.3% 2^U and only 0.7% fissionable ^ U . To increase the probability of fission of ^ U nuclei, natural uranium can be enriched to increase the percentage of 2^U by using processes such as diffusion or centrifugation. Enrichment is not usually necessary for reactors using heavy water as moderator since heavy water doesn't absorb neutrons. The third problem is that some neutrons will escape through the surface of the reactor core before they can cause further fissions (Fig. 42-8). Thus the mass of fuel must be sufficiently large for a self-sustaining chain reaction to take place. The minimum mass of uranium needed is called the critical mass. The value of the crit ical mass depends on the moderator, the fuel (299Pu may be used instead of ^ U ) , and how much the fuel is enriched, if at all. Typical values are on the order of a few kilograms (that is, neither grams nor thousands of kilograms). To have a self-sustaining chain reaction, on average at least one neutron produced in each fission must go on to produce another fission. The average number of neutrons per fission that do go on to produce further fissions is called the multiplication factor, / . For a self-sustaining chain reaction, we must have / > 1. t292U will fission, but only with f st neutrons (^fU is more st ble th n ^ U ) . The prob bility of bsorbing f st neutron nd producing fission is too low to produce self-sust ining ch in re ction.

If / < 1, the reactor is "subcritical." If / > 1, it is "supercritical" (and could become dangerously explosive). Reactors are equipped with movable control rods (good neutron absorbers like cadmium or boron), whose function is to absorb neutrons and maintain the reactor at just barely "critical," / = 1. The release of neutrons and subsequent fissions occur so quickly that manipulation of the control rods to maintain / = 1 would not be possible if it weren't for the small percentage (« 1%) of so-called delayed neutrons. They come from the decay of neutron-rich fission fragments (or their daughters) having lifetimes on the order of seconds—sufficient to allow enough reaction time to operate the control rods and maintain / = 1 . Nuclear reactors have been built for use in research and to produce electric power. Fission produces many neutrons and a "research reactor" is basically an intense source of neutrons. These neutrons can be used as projectiles in nuclear reactions to produce nuclides not found in nature, including isotopes used as tracers and for therapy. A "power reactor" is used to produce electric power. The energy released in the fission process appears as heat, which is used to boil water and produce steam to drive a turbine connected to an electric generator (Fig. 42-9). The core of a nuclear reactor consists of the fuel and a moderator (water in most U.S. commercial reactors). The fuel is usually uranium enriched so that it contains 2 to 4 percent ^ U . Water at high pressure or other liquid (such as liquid sodium) is allowed to flow through the core. The thermal energy it absorbs is used to produce steam in the heat exchanger, so the fissionable fuel acts as the heat input for a heat engine ( hapter 20). Prim ry system

Core (fuel nd moder tor)

Hot w ter (or liquid sodium)

...

Second ry system

He t exch nger

H Electric gener tor

FIGURE 4 2 9 A nucle r re ctor. The he t gener ted by the fission process in the fuel rods is c rried off by hot w ter or liquid sodium nd is used to boil w ter to ste m in the he t exch nger. The ste m drives turbine to gener te electricity nd is then cooled in the condenser.

FIGURE 4 2 1 0 D ev st tion round Chernobyl in Russi , fter the nucle r power pl nt m eltdown in

Shielding

1986. Cooling w ter

There are problems associated with nuclear power plants. Besides the usual thermal pollution associated with any heat engine (Section 20-11), there is the serious problem of disposal of the radioactive fission fragments produced in the reactor, plus radioactive nuclides produced by neutrons interacting with the structural parts of the reactor. Fission fragments, like their uranium or plutonium parents, have about 50% more neutrons than protons. Nuclei with atomic number in the typical range for fission fragments (Z « 30 to 60) are stable only if they have more nearly equal numbers of protons and neutrons (see Fig. 41-2). Hence the highly neutron-rich fission fragments are very unstable and decay radioactively. The accidental release of highly radioactive fission fragments into the atmosphere poses a serious threat to human health (Section 42-5), as does possible leakage of the radioactive wastes when they are disposed of. The accidents at Three Mile Island, Pennsylvania (1979), and at hernobyl, Russia (1986), have illustrated some of the dangers and have shown that nuclear plants must be constructed, maintained, and operated with great care and precision (Fig. 42-10). SE TION 4 2 - 3

Nuclear Fission; Nuclear Reactors

1139

Finally, the lifetime of nuclear power plants is limited to 30-some years, due to buildup of radioactivity and the fact that the structural materials themselves are weakened by the intense conditions inside. "Decommissioning" of a power plant could take a number of forms, but the cost of any method of decommissioning a large plant is very great. So-called breeder reactors were proposed as a solution to the problem of limited supplies of fissionable uranium, ^ U . A breeder reactor is one in which some of the neutrons produced in the fission of are absorbed by 292U, and 2g9Pu is produced via the set of reactions shown in Fig. 42-1. 2g9Pu is fissionable with slow neutrons, so after separation it can be used as a fuel in a nuclear reactor. Thus a breeder reactor "breeds" new fuel (299Pu) from otherwise useless ^fU. Since natural uranium is 99.3% 292U, this means that the supply of fissionable fuel could be increased by more than a factor of 100. But breeder reactors have the same problems as other reactors, plus other serious problems. Not only is plutonium considered to be a serious health hazard in itself (radioactive with a half-life of 24,000 years), but plutonium produced in a reactor can readily be used in a bomb, increasing the danger of nuclear proliferation and theft of fuel by terrorists to produce a bomb. Nuclear power presents risks. Other large-scale energy-conversion methods, such as conventional oil and coal-burning steam plants, also present health and environmental hazards; some of them were discussed in Section 20-11, and include air pollution, oil spills, and the release of 0 2 gas which can trap heat as in a greenhouse to raise the Earth's temperature. The solution to the world's needs for energy is not only technological, but also economic and political. A major factor surely is to "conserve"—to minimize our use of energy. "Reduce, reuse, recycle." EXAMPLE 4 2 - 6

Uranium fuel am ount. Estimate the minimum amount of

292U that needs to undergo fission in order to run a 1000-MW power reactor per

year of continuous operation. Assume an efficiency ( hapter 20) of about 33%. APPROA H At 33% efficiency, we need 3 X 1000 MW = 3000 X 106J/s input. Each fission releases about 200 MeV (Eq. 42-6), so we divide the energy for a year by 200 MeV to get the number of fissions needed per year. Then we multiply by the mass of one uranium atom. SOLUTION For 1000 MW output, the total power generation needs to be 3000 MW, of which 2000 MW is dumped as "waste" heat. Thus the total energy release in 1 yr (3 X 107 s) from fission needs to be about (3 X 109J / s)(3 X 107 s)

« 1017J.

If each fission releases 200 MeV of energy, the number of fissions required for a year is 7-------

----- -—-—J --------—-— r r 3 x 1027fissions. 10"19J/eV)

10 eV/fission)(l.6

The mass of a single uranium atom is about (235 u )(l .66 4 X 10 25kg, so the total uranium mass needed is (4

PHYSICS

APPLIED

Energy in coal vs. uranium

1140

HAPTER 42

10_25kg/fission)(3

10- 27kg/u) ra

1027 fissions) ra 1000 kg,

or about a ton of 292U. NOTE Since 292U makes up only 0.7% of natural uranium, the yearly requirement for uranium is on the order of a hundred tons. This is orders of magnitude less than coal, both in mass and volume. oal releases 2.8 X 107 J/kg, whereas 292U can release 1017 J/10 3kg = 1014J/kg. For natural uranium, the figure is 100 times less, 1012J/kg.

Nuclear Energy; Effects and Uses of Radiation

EXERCISE B

A nuclear-powered submarine needs 6000-kW input power. How many

292U fissions is this per second?

Atom Bomb The first use of fission, however, was not to produce electric power. Instead, it was first used as a fission bomb (called the "atomic bomb"). In early 1940, with Europe already at war, Germany's leader, Adolf Hitler, banned the sale of uranium from the zech mines he had recently taken over. Research into the fission process suddenly was enshrouded in secrecy. Physicists in the United States were alarmed. A group of them approached Einstein —a man whose name was a household word—to send a letter to President Franklin Roosevelt about the possibilities of using nuclear fission for a bomb far more powerful than any previously known, and inform him that Germany might already have begun development of such a bomb. Roosevelt responded by authorizing the program known as the Manhattan Project, to see if a bomb could be built. Work began in earnest after Fermi's demonstration in 1942 that a sustained chain reaction was possible. A new secret laboratory was developed on an isolated mesa in New Mexico known as Los Alamos. Under the direction of J. Robert Oppenheimer (1904-1967; Fig. 42-11), it became the home of famous scientists from all over Europe and the United States. To build a bomb that was subcritical during transport but that could be made supercritical (to produce a chain reaction) at just the right moment, two pieces of uranium were used, each less than the critical mass but together greater than the critical mass. The two masses, kept separate until the moment of detonation, were then forced together quickly by a kind of gun, and a chain reaction of explosive proportions occurred. An alternate bomb detonated conventional explosives (TNT) surrounding a plutonium sphere to compress it by implosion to double its density, making it more than critical and causing a nuclear explosion. The first fission bomb was tested in the New Mexico desert in July 1945. It was successful. In early August, a fission bomb using uranium was dropped on Hiroshima and a second, using plutonium, was dropped on Nagasaki (Fig. 42-12), both in Japan. World War II ended shortly thereafter. Besides its great destructive power, a fission bomb produces many highly radioactive fission fragments, as does a nuclear reactor. When a fission bomb explodes, these radioactive isotopes are released into the atmosphere and are known as radioactive fallout. Testing of nuclear bombs in the atmosphere after World War II was a cause of concern, for the movement of air masses spread the fallout all over the globe. Radioactive fallout eventually settles to the Earth, particularly in rainfall, and is absorbed by plants and grasses and enters the food chain. This is a far more serious problem than the same radioactivity on the exterior of our bodies, since a and /3 particles are largely absorbed by clothing and the outer (dead) layer of skin. But inside our bodies via food, the isotopes are in direct contact with living cells. One particularly dangerous radioactive isotope is ^Sr, which is chemically much like calcium and becomes concentrated in bone, where it causes bone cancer and destruction of bone marrow. The 1963 treaty signed by over 100 nations that bans nuclear weapons testing in the atmosphere was motivated because of the hazards of fallout.

FIGURE 4 2 1 1 J. Robert Oppenheimer, on the left, with General Leslie Groves, who was the administrative head of Los Alamos during World War II. The photograph was taken at the Trinity site in the New Mexico desert, where the first atomic bomb was exploded.

FIGURE 4 2 1 2 Photo taken a month after the bomb was dropped on Nagasaki. The shacks were constructed afterwards from debris in the ruins.

4 2 —4 Nucle r Fusion The mass of every stable nucleus is less than the sum of the masses of its constituent protons and neutrons. For example, the mass of the helium isotope 2He is less than the mass of two protons plus the mass of two neutrons, as we saw in Example 41-2. Thus, if two protons and two neutrons were to come together to form a helium nucleus, there would be a loss of mass. This mass loss is manifested in the release of a large amount of energy. SE TION 4 2 - 4

Nuclear Fusion

1141

FIGURE 4 2 1 3 Aver ge binding energy per nucleon s function of m ss number for st ble nuclei. S me s Fig. 41 - 1.

Number of nucleons,

(m ss number)

Nuclear Fusion; Stars The process of building up nuclei by bringing together individual protons and neutrons, or building larger nuclei by combining small nuclei, is called nuclear fusion. A glance at Fig. 42-13 (same as Fig. 41-1) shows why small nuclei can combine to form larger ones with the release of energy: it is because the binding energy per nucleon is smaller for light nuclei than it is for those of increasing mass (up to about A « 60). It is believed that many of the elements in the universe were orig inally formed through the process of fusion (see hapter 44), and that fusion is today continually taking place within the stars, including our Sun, producing the prodigious amounts of radiant energy they emit. EXAMPLE 4 2 - 7 Fusion energy release. One of the simplest fusion reactions involves the production of deuterium, 2H, from a neutron and a proton: }H + n —> iH + If. How much energy is released in this reaction? APPROA H The energy released equals the difference in mass (times c2) between the initial and final masses. SOLUTION From Appendix F, the initial mass is 1.007825 u + 1.008665 u = 2.016490 u, and after the reaction the mass is that of the ?H, namely 2.014082 u (the 7 is mass less). The mass difference is 2.016490 u - 2.014082 u = 0.002408 u, so the energy released is (Am)c2 = (0.002408 u) (931.5 MeV/u) = 2.24 MeV, and it is carried off by the ?H nucleus and the 7 ray. The energy output of our Sun is believed to be due principally to the following sequence of fusion reactions: }H + }H —> ?H + e+ + v

(0.44 MeV)

(42 -7a)

}H + ?H —> |H e + J

(5.48 MeV)

(42-7b)

(12.86 MeV)

(42-7c)

iHe + iHe - > ^He + {H + jH

where the energy released (Q-value) for each reaction is given in parentheses. The net effect of this sequence, which is called the proton-proton cycle, is for four protons to

1142

HAPTER 42

Nuclear Energy; Effects and Uses of Radiation

combine to form one ^He nucleus plus two positrons, two neutrinos, and two gamma rays: 4 }H —>• ^He + 2e+ + 2v + 27. (42 -8) Note that it takes two of each of the first two reactions (Eqs. 42-7a and b) to produce the two |H e for the third reaction. So the total energy release for the net reaction, Eq. 42-8, is (2 X 0.44 MeV + 2 X 5.48 MeV + 12.86 MeV) = 24.7 MeV. In addi tion, each of the two e+ (Eq. 42-7a) quickly annihilates with an electron to produce 2m e c 2 = 1.02 MeV; so the total energy released is (24.7 MeV + 2 X 1.02 MeV) = 26.7 MeV. The first reaction, the formation of deuterium from two protons (Eq. 42-7a), has a very low probability, and the infrequency of that reaction serves to limit the rate at which the Sun produces energy. EX ER C ISE C Return to the first Ch pter -Opening Question, p ge 1131, nd nswer it g in now. Try to expl in why you m y h ve nswered it differently the first time. EX ER C ISE D If the Sun is gener ting const nt mount of energy vi fusion, the m ss of the Sun must be (a) incre sing, (b) decre sing, (c) const nt, ( d) irregul r.

EXAMPLE 4 2 - 8 ESTIMATE"! Estimating fusion energy. Estimate the energy released if the following reaction occurred: ?H + ?H —» ^He. APPROA H We use Fig. 42-13 for a quick estimate. SOLUTION We see in Fig. 42-13 that each 2H has a binding energy of about l^MeV/nucleon, which for 2 nuclei of mass 2 is 4 X ( lj) « 5 MeV. The ^He has a binding energy per nucleon of about 7 MeV for a total of 4 X 7 MeV = 28 MeV. Hence the energy release is about 28 MeV - 5 MeV = 23 MeV. In stars hotter than the Sun, it is more likely that the energy output comes principally from the carbon (or NO) cycle, which comprises the following sequence of reactions: 126C + }H 13N 136C + }H -> + }H

+ 7 + e+ + v + 7 + 7 + e+ + v

+ }H

n6C + ^He.

It is easy to see (see Problem 47) that no carbon is consumed in this cycle (see first and last equations) and that the net effect is the same as the proton - proton cycle, Eq. 42-8 (plus one extra 7). The theory of the proton - proton cycle and of the carbon cycle as the source of energy for the Sun and stars was first worked out by Hans Bethe (1906-2005) in 1939. ON EPTUAL EXAMPLE 4 2 - 9 I Stellar fusion. What is the heaviest element likely to be produced in fusion processes in stars? RESPONSE Fusion is possible if the final products have more binding energy (less mass) than the reactants, for then there is a net release of energy. Since the binding energy curve in Fig. 42-13 (or Fig. 41-1) peaks near A ~ 56 to 58 which corresponds to iron or nickel, it would not be energetically favorable to produce elements heavier than that. Nevertheless, in the center of massive stars or in supernova explosions, there is enough initial kinetic energy available to drive endothermic reactions that produce heavier elements, as well. SE TION 4 2 - 4

Nuclear Fusion

1143

Possible Fusion Reactors 0

PHYSICS

APPLIED

Fusion energy reactors

The possibility of utilizing the energy released in fusion to make a power reactor is very attractive. The fusion reactions most likely to succeed in a reactor involve the isotopes of hydrogen, ?H (deuterium) and ?H (tritium), and are as follows, with the energy released given in parentheses: ?H + ?H -► l a + }H

(4.00 MeV)

(42-9a)

iHe + n

(3.23 MeV)

(42-9b)

?H + lH - > 2He + n.

(17.57 MeV)

(42-9c)

?H + ?H

omparing these energy yields with that for the fission of ^ U , we can see that the energy released in fusion reactions can be greater for a given mass of fuel than in fission. Furthermore, as fuel, a fusion reactor could use deuterium, which is very plentiful in the water of the oceans (the natural abundance of ?H is 0.0115% on average, or about 1g of deuterium per 80 L of water). The simple proton -proton reaction of Eq. 42-7a, which could use a much more plentiful source of fuel, }H, has such a small probability of occurring that it cannot be considered a possibility on Earth. Although a useful fusion reactor has not yet been achieved, considerable progress has been made in overcoming the inherent difficulties. The problems are associated with the fact that all nuclei have a positive charge and repel each other. However, if they can be brought close enough together so that the short-range attractive strong nuclear force can come into play, it can pull the nuclei together and fusion will occur. For the nuclei to get close enough together, they must have large kinetic energy to overcome the electric repulsion. High kinetic energies are easily attainable with particle accelerators ( hapter 43), but the number of parti cles involved is too small. To produce realistic amounts of energy, we must deal with matter in bulk, for which high kinetic energy means higher temperatures. Indeed, very high temperatures are required for fusion to occur, and fusion devices are often referred to as thermonuclear devices. The interiors of the Sun and other stars are very hot, many millions of degrees, so the nuclei are moving fast enough for fusion to take place, and the energy released keeps the temperature high so that further fusion reactions can occur. The Sun and the stars represent huge self-sustaining thermonuclear reactors that stay together because of their great gravitational mass; but on Earth, containment of the fast-moving nuclei at the high temperatures and densities required has proven difficult. It was realized after World War II that the temperature produced within a fission (or "atomic") bomb was close to 10 K. This suggested that a fission bomb could be used to ignite a fusion bomb (popularly known as a thermonuclear or hydrogen bomb) to release the vast energy of fusion. The uncontrollable release of fusion energy in an H-bomb (in 1952) was relatively easy to obtain. But to realize usable energy from fusion at a slow and controlled rate has turned out to be a serious challenge. ESTIMATE I Temperature needed for d - t fusion. Estimate the temperature required for deuterium -tritium fusion (d - t) to occur. APPROA H We assume the nuclei approach head-on, each with kinetic energy K, and that the nuclear force comes into play when the distance between their centers equals the sum of their nuclear radii. The electrostatic potential energy ( hapter 23) of the two particles at this distance equals the minimum total kinetic energy of the two particles when far apart. The average kinetic energy is related to Kelvin temperature by Eq. 18-4. SOLUTION The radii of the two nuclei (A d = 2 and A t = 3) are given by Eq. 41-1: rd « 1.5 fm, rt « 1.7 fm, so rd + rt 3.2 X 10"15m. We equate the kinetic energy of the two initial particles to the potential energy when at this distance:

2K ~ 4lreo (/d + t) 0N - m2\ (1.6 X 10"19 )2 9.0 X 109— 5- t ---------- „ .. ---------— r « 0.45 MeV. 2 J (3.2 x 10 m )(l .6 x 10 J/eV)

1144

HAPTER 42

Thus, K « 0.22 MeV, and if we ask that the average kinetic energy be this high,

then from Eq. 18-4, §kT = K, we have a temperature of T =

2(0.22 M eV)(l .6

3k ~

3(1.38

10"13J/MeV)

10" 23J/K)

2 X 10 K.

NOTE More careful calculations show that the temperature required for fusion is actually about an order of magnitude less than this rough estimate, partly because it is not necessary that the average kinetic energy be 0.22 MeV—a small percentage with this much energy (particles in the high-energy tail of the Maxwell distribution, Fig. 18-3) would be sufficient. Reasonable estimates for a usable fusion reactor are in the range T ^ 1 to 4 X 10 K. It is not only a high temperature that is required for a fusion reactor. There must also be a high density of nuclei to ensure a sufficiently high collision rate. A real difficulty with controlled fusion is to contain nuclei long enough and at a high enough density for sufficient reactions to occur that a usable amount of energy is obtained. At the temperatures needed for fusion, the atoms are ionized, and the resulting collection of nuclei and electrons is referred to as a plasma. Ordinary materials vaporize at a few thousand degrees at most, and hence cannot be used to contain a high-temperature plasma. Two major containment techniques are magnetic confinement and inertial confinement. In magnetic confinement, magnetic fields are used to try to contain the hot plasma. A simple approach is the "magnetic bottle" shown in Fig. 42-14. The paths of the charged particles in the plasma are bent by the magnetic field; where magnetic field lines are close together, the force on the particles reflects them back toward the center. Unfortunately, magnetic bottles develop "leaks" and the charged particles slip out before sufficient fusion takes place. The most promising design today is the tokamak, first developed in Russia. A tokamak (Fig. 42-15) is toroid-shaped and involves complicated magnetic fields: current-carrying conductors produce a magnetic field directed along the axis of the toroid ("toroidal" field); an additional field is produced by currents within the plasma itself ("poloidal" field). The combination produces a helical field as shown in Fig. 42-15, confining the plasma, at least briefly, so it doesn't touch the vacuum chamber's metal walls.

Current-c rrving wires

M gnetic field lines FIGURE 4 2 - 1 4 used to confine

Pl sm

"M gnetic b ottle '

Toroid l v cuum ch mber

FIGURE 4 2 - 1 5 Tok m k configur tion, showing the tot l B field due to extern l current plus current in the pl sm itself.

In 1957, J. D. Lawson showed that the product of ion density n and confine ment time r must exceed a minimum value of approximately n r > 3 X IO20s/m3. This Lawson criterion must be reached to produce ignition, meaning fusion that continues after all external heating is turned off. Practically, it is expected to be achieved with n ~ 1 to 3 X IO20m-3 and r ~ 1 -3 s. To reach break-even, the point at which the energy output due to fusion is equal to the energy input to heat the plasma, requires an m about an order of magnitude less. The break-even point was very closely approached in the 1990s at the Tokamak Fusion Test Reactor (TFTR) at Princeton, and the very high temperature needed for ignition (4 X 10 K) was exceeded—although not both of these at the same time. SE TION 4 2 - 4

Nuclear Fusion

1145

Tokamak fusion research continues throughout the world, from the Princeton Plasma Physics Laboratory (PPPL) in the U.S. to the KSTAR tokamak in South Korea. This research will help us in developing the huge multinational test device (European Union, India, Japan, South Korea, Russia, hina, and the U.S.), called ITER (International Thermonuclear Experimental Reactor). It is hoped that ITER will be finished and running by 2016, in southeast France, with an expected power output of about 500 MW, 10 times the input energy. ITER (see hapter-Opening Photograph on page 1131) is planned to be the final research step before building a commercial reactor. The second method for containing the fuel for fusion is inertial confinement: a small pellet or capsule of deuterium and tritium is struck simultaneously from several directions by very intense laser beams. The intense influx of energy heats and ionizes the pellet into a plasma, compressing it and heating it to temperatures at which fusion can occur. The confinement time is on the order of 10-11 to 10-9 s, during which time the ions do not move appreciably because of their own inertia, and fusion can take place.

4 2 —5 P ss ge o f R di tion Through M tter; R di tion D m ge When we speak of radiation, we include a, /3, 7, and X-rays, as well as protons, neutrons, and other particles such as pions (see hapter 43). Because charged particles can ionize the atoms or molecules of any material they pass through, they are referred to as ionizing radiation. And because radiation produces ionization, it can cause considerable damage to materials, particularly to biological tissue. harged particles, such as a and f3 rays and protons, cause ionization because of electric forces. That is, when they pass through a material, they can attract or repel electrons strongly enough to remove them from the atoms of the material. Since the a and /3 rays emitted by radioactive substances have energies on the order of 1 MeV (104 to 107 eV), whereas ionization of atoms and molecules requires on the order of 10 eV, it is clear that a single a or /3 particle can cause thousands of ionizations. Neutral particles also give rise to ionization when they pass through materials. For example, X-ray and 7-ray photons can ionize atoms by knocking out electrons by means of the photoelectric and ompton effects ( hapter 37). Furthermore, if a 7 ray has sufficient energy (greater than 1.02 MeV), it can undergo pair production: an electron and a positron are produced (Section 37-5). The charged particles produced in all of these processes can themselves go on to produce further ionization. Neutrons, on the other hand, interact with matter mainly by collisions with nuclei, with which they interact strongly. Often the nucleus is broken apart by such a collision, altering the molecule of which it was a part. The fragments produced can in turn cause ionization. Radiation passing through matter can do considerable damage. Metals and other structural materials become brittle and their strength can be weakened if the radiation is very intense, as in nuclear reactor power plants and for space vehicles that must pass through areas of intense cosmic radiation.

*Biological Damage PHYSICS

APPLIED

Biological radiation damage

1146

HAPTER 42

The radiation damage produced in biological organisms is due primarily to ionization produced in cells. Several related processes can occur. Ions or radicals are produced that are highly reactive and take part in chemical reactions that inter fere with the normal operation of the cell. All forms of radiation can ionize atoms by knocking out electrons. If these are bonding electrons, the molecule may break apart, or its structure may be altered so it does not perform its normal function or may perform a harmful function. In the case of proteins, the loss of one molecule is not serious if there are other copies of it in the cell and additional copies can be

Nuclear Energy; Effects and Uses of Radiation

made from the gene that codes for it. However, large doses of radiation may damage so many molecules that new copies cannot be made quickly enough, and the cell dies. Damage to the DNA is more serious, since a cell may have only one copy. Each alteration in the DNA can affect a gene and alter the molecule it codes for (Section 40-3), so that needed proteins or other materials may not be made at all. Again the cell may die. The death of a single cell is not normally a problem, since the body can replace it with a new one. (There are exceptions, such as neurons, which are mostly not replaceable, so their loss is serious.) But if many cells die, the organism may not be able to recover. On the other hand, a cell may survive but be defective. It may go on dividing and produce many more defective cells, to the detriment of the whole organism. Thus radiation can cause cancer—the rapid uncontrolled production of cells. The possible damage done by the medical use of X-rays and other radiation must be balanced against the medical benefits and prolongation of life as a result of their use.

4 2 - 6 Me surement o f R di tion— Dosimetry Although the passage of ionizing radiation through the human body can cause considerable damage, radiation can also be used to treat certain diseases, particularly cancer, often by using very narrow beams directed at a cancerous tumor in order to destroy it (Section 42-7). It is therefore important to be able to quantify the amount, or dose, of radiation. This is the subject of dosimetry. The strength of a source can be specified at a given time by stating the source activity: how many nuclear decays (or disintegrations) occur per second. The traditional unit is the curie ( i), defined as 1 i = 3.70

PHYSICS

APPLIED

Dosimetry

IO10decays per second.

(This number comes from the original definition as the activity of exactly one gram of radium.) Although the curie is still in common use, the SI unit for source activity is the becquerel (Bq), defined as 1 Bq = 1 decay/s. ommercial suppliers of radionuclides (radioactive nuclides) specify the activity at a given time. Since the activity decreases over time, more so for short-lived isotopes, it is important to take this into account. The magnitude of the source activity \dN/dt\ is related to the number of radioactive nuclei present, N, and to the half-life, 71, by (see Section 41-8): dN dt

= AiV =

EXAMPLE 42 - 11 Radioactivity taken up by cells. In a certain experiment, 0.016 /x i of 15P is injected into a medium containing a culture of bacteria. After 1.0 h the cells are washed and a 70% efficient detector (counts 70% of emitted (3 rays) records 720 counts per minute from the cells. What percentage of the orig inal 15P was taken up by the cells? APPROA H The half-life of 15P is about 14 days (Appendix F), so we can ignore any loss of activity over 1 hour. From the given activity, we find how many /3 rays are emitted. We can compare 70% of this to the (720/min)/(60s/min) = 12 per second detected. SOLUTION The total number of decays per second originally was (0.016 X 10_6)(3.7 X 1010) = 590. The counter could be expected to count 70% of this, or 410 per second. Since it counted 720/60 = 12 per second, then 12/410 = 0.029 or 2.9% was incorporated into the cells. SE TION 4 2 - 6

Measurement of Radiation -Dosimetry

1147

Another type of measurement is the exposure or absorbed dose—that is, the effect the radiation has on the absorbing material. The earliest unit of dosage was the roentgen (R), defined in terms of the amount of ionization produced by the radiation (1 R = 1.6 X 1012 ion pairs per gram of dry air at standard conditions). Today, 1 R is defined as the amount of X or 7 radiation that deposits 0.878 X 10-2 J of energy per kilogram of air. The roentgen was largely superseded by another unit of absorbed dose applicable to any type of radiation, the rad: 1 rad is t at amount o f radiation w ic deposits energy per unit mass o f1.00 X 10 2J/kg in any absorbing material. (This is quite close to the roentgen for X- and 7 rays.) The proper SI unit for absorbed dose is the gray (Gy): 1 Gy = lJ /k g = 100 rad.

TABLE 42 - 1 Quality Factor (QF) of Different Kinds of Radiation Type

X - nd 7 r ys F st protons

1 «1 1

Slow neutrons

«3

F st neutrons

U p to 10

/3 (electrons)

U p to 20

p rticles nd he vy ions

(42-10)

The absorbed dose depends not only on the strength of a given radiation beam (number of particles per second) and the energy per particle, but also on the type of material that is absorbing the radiation. Bone, for example, absorbs more of the radiation normally used than does flesh, so the same beam passing through a human body deposits a greater dose (in rads or grays) in bone than in flesh. The gray and the rad are physical units of dose—the energy deposited per unit mass of material. They are, however, not the most meaningful units for measuring the biological damage produced by radiation because equal doses of different types of radiation cause differing amounts of damage. For example, 1 rad of a radiation does 10 to 20 times the amount of damage as 1 rad of (3 or 7 rays. This difference arises largely because a rays (and other heavy particles such as protons and neutrons) move much more slowly than f3 and 7 rays of equal energy due to their greater mass. Hence, ionizing collisions occur closer together, so more irreparable damage can be done. The relative biological effectiveness (RBE) or quality factor (QF) of a given type of radiation is defined as the number of rads of X or 7 radiation that produces the same biological damage as 1 rad of the given radiation. Table 42-1 gives the QF for several types of radiation. The numbers are approximate since they depend somewhat on the energy of the particles and on the type of damage that is used as the criterion. The effective dose can be given as the product of the dose in rads and the QF, and this unit is known as the rem (which stands for rad equivalent man): effective dose (in rem) = dose (in rad)

QF.

(4 2 - lla)

This unit is being replaced by the SI unit for "effective dose," the sievert (Sv): effective dose (Sv) = dose (Gy)

QF.

(42 - llb)

By these definitions, 1 rem (or 1 Sv) of any type of radiation does approximately the same amount of biological damage. For example, 50 rem of fast neutrons does the same damage as 50 rem of 7 rays. But note that 50 rem of fast neutrons is only 5 rads, whereas 50 rem of 7 rays is 50 rads. EXERCISE E Return to the second Ch pter - Opening Q uestion, p ge 1131, nd nswer it g in now. Try to expl in why you m ny h ve nswered it differently the first time.

IHuman Exposure to Radiation 0

PHYSICS

APPLIED

Radon

PHYSICS

APPLIED

Human radiation exposure

1148

HAPTER 42

We are constantly exposed to low-level radiation from natural sources: cosmic rays, natural radioactivity in rocks and soil, and naturally occurring radioactive isotopes in our food, such as 4!]K. Radon, ^ R n , is of considerable concern today. It is the product of radium decay and is an intermediate in the decay series from uranium (see Fig. 41-12). Most intermediates remain in the rocks where formed, but radon is a gas that can escape from rock (and from building material like concrete) to enter the air we breathe, and attack the interior of the lung. The natural radioactive background averages about 0.30 rem (300 mrem) per year per person in the U.S., although there are large variations. From medical X-rays and scans, the average person receives about 50 to 60 mrem per year, giving an average total dose of about 360 mrem (3.6 mSv) per person. Government regulators suggest an upper limit of allowed radiation for an individual in the

general populace at about 100 mrem (1 mSv) per year in addition to natural back ground. It is believed that even low doses of radiation increase the chances of cancer or genetic defects; there is no safe level or threshold of radiation exposure. The upper limit for people who work around radiation—in hospitals, in power plants, in research—has been set higher, a maximum of 5 rem (50 mSv) whole-body dose in any one year, and significantly less averaged over more years (below 2rem /yr averaged over 5 years). To monitor exposure, those people who work around radiation generally carry some type of dosimeter, one common type being a radiation film badge which is a piece of film wrapped in light-tight material. The passage of ionizing radiation through the film changes it so that the film is darkened upon development, and thus indicates the received dose. Newer types include the t ermoluminescent dosimeter (TLD). Dosimeters and badges do not protect the worker, but high levels detected suggest reassignment or modified work practices to reduce radiation exposure to acceptable levels. Large doses of radiation can cause unpleasant symptoms such as nausea, fatigue, and loss of body hair. Such effects are sometimes referred to as radiation sickness. Large doses can be fatal, although the time span of the dose is important. A short dose of 1000 rem (10 Sv) is nearly always fatal. A 400-rem (4-Sv) dose in a short period of time is fatal in 50% of the cases. However, the body possesses remarkable repair processes, so that a 400-rem dose spread over several weeks is usually not fatal. It will, nonetheless, cause considerable damage to the body. The effects of low doses over a long time are difficult to determine and are not well known as yet.

PHYSICS

APPLIED

PHYSICS

APPLIED

R adiabtio Film adnw georkerexposure

Radiationsickness

U 5 E E H E H 3 Whole - body dose. What whole-body dose is received by a 70-kg laboratory worker exposed to a 40-m i 27 0 source, assuming the person's body has cross-sectional area 1.5 m2 and is normally about 4.0 m from the source for 4.0 h per day? 27 0 emits 7 rays of energy 1.33 MeV and 1.17 MeV in quick succession. Approximately 50% of the 7 rays interact in the body and deposit all their energy. (The rest pass through.) APPROA H Of the given energy emitted, only a fraction passes through the worker, equal to er area divided by the total area over a full sphere of radius 4.0 m (Fig. 42-16). SOLUTION The total 7-ray energy per decay is (1.33 + 1.17) MeV = 2.50 MeV, so the total energy emitted by the source per second is (0.040 i)(3.7

1010decays/ i*s)(2.50 MeV) = 3.7

109MeV/s.

The proportion of this energy intercepted by the body is its 1.5-m2 area divided by the area of a sphere of radius 4.0 m (Fig. 42-16): 1.5 m2 1.5 m2 ^Airr — r = 47t(4.0 , m) = 7 -5 x 10 •

FIGURE 4 2 1 6 R di tion spre ds out in ll directions. A person 4.0 m w y intercepts only fr ction: her cross - section l re divided by the re of sphere of r dius 4.0 m. Ex m ple 42 - 12.

So the rate energy is deposited in the body (remembering that only 50% of the 7 rays interact in the body) is E = g)(7.5

10"3)(3.7

109M eV /s)(l .6

10"13J/MeV) = 2.2

10" 6J/s.

Since lG y = lJ /k g , the whole-body dose rate for this 70-kg person is (2.2 X 10-6 J/s)/(70kg) = 3.1 X 10_ Gy/s. In 4.0 h, this amounts to a dose of (4.0 h )(3600 s/h)(3.1

10" Gy/s) = 4.5

10"4Gy.

Since QF « 1 for gammas, the effective dose is 450 /iSv (Eqs. 42 - llb and 42-10) or: (100 rad/Gy)(4.5 X 10_4Gy)(l rem/rad) = 45 mrem = 0.45 mSv. NOTE This 45-mrem effective dose is almost 50% of the normal allowed dose for a whole year (100 mrem/yr), or 1 % of the maximum one-year allowance for radi ation workers. This worker should not receive such a large dose every day and should seek ways to reduce it (shield the source, vary the work, work farther from the source, work less time this close to source, etc.). SE TION 4 2 - 6

Measurement of Radiation -Dosimetry

1149

P HY S I CS

APPLIED

Radonexposure

EXAMPLE 42 - 13 Radon exposure. In the U.S., yearly deaths from radon exposure (the second leading cause of lung cancer) are estimated to exceed the yearly deaths from drunk driving. The Environmental Protection Agency recommends taking action to reduce the radon concentration in living areas if it exceeds 4p i/L of air. In some areas 50% of houses exceed this level from naturally occurring radon in the soil. Estimate the mass of radon that emits 4.0 p i of ^ R n radiation.

APPROA H We can use the definition of the curie to determine how many decays per second correspond to 4 p i, then Eq. 41-7b to determine how many nuclei of radon it takes to have this activity \dN/dt\. SOLUTION We saw at the start of Section 42-6 that 1 i = 3.70 X IO10 decays/s. Thus dN dt

= 4.0 p i = (4.0

10-

i)(3.70 x 1010decays/s/ i)

= 0.148 s-1. From Eqs. 41-7a and 41dN dt

0.693 = AN = —= ~ N . 712

Appendix F tells us 71 = 3.8232 days, so N =

dN dt 0.693 (°-148s ^

= 7.05

(3.8232 days)(8.64 x 104s/day) 0693

104atoms of radon-222.

The molar mass (222 u) and Avogadro's number are used to find the mass:

(7.05

6.02

104atoms)(222 g/mol) X

1023atoms/mol

= 2.6

IO" 17 g

or 26 attograms in 1L of air. This 2.6 X 10 17g/L = 2.6 X 10 14g/m 3 of radon is about 150 atoms decaying per second in every meter of air at the limit of 4 p i/L. NOTE Each radon atom emits 4 a particles and 4 /3 particles before the decay sequence reaches a stable element, each one capable of causing many harmful ionizations.

*4 2 - 7 R di tion Ther py

PHYSICS

1150

APPLIED

Radiationt erapy

HAPTER 42

The medical application of radioactivity and radiation to human beings involves two basic aspects: ( 1 ) radiation therapy—the treatment of disease (mainly cancer)—which we discuss in this Section; and (2) the diagnosis of disease, which we discuss in the following Sections of this hapter. Radiation can cause cancer. It can also be used to treat it. Rapidly growing cancer cells are especially susceptible to destruction by radiation. Nonetheless, large doses are needed to kill the cancer cells, and some of the surrounding normal cells are inevitably killed as well. It is for this reason that cancer patients receiving radiation therapy often suffer side effects characteristic of radiation sickness. To

Nuclear Energy; Effects and Uses of Radiation

minimize the destruction of normal cells, a narrow beam of 7 or X-rays is often used when a cancerous tumor is well localized. The beam is directed at the tumor, and the source (or body) is rotated so that the beam passes through various parts of the body to keep the dose at any one place as low as possible—except at the tumor and its immediate surroundings, where the beam passes at all times (Fig. 42-17). The radiation may be from a radioactive source such as 27 0, or it may be from an X-ray machine that produces photons in the range 200 keV to 5 MeV. Protons, neutrons, electrons, and pions, which are produced in particle accelerators (Section 43-1), are also being used in cancer therapy. Protons used to kill tumors have a special property that makes them particularly useful. As shown in Fig. 42-18, when protons enter tissue, most of their energy is deposited at the end of their path. The protons' initial kinetic energy can be chosen so that most of the energy is deposited at the depth of the tumor itself, to destroy it. The incoming protons deposit only a small amount of energy in the tissue in front of the tumor, and none at all behind the tumor, thus having less negative effect on healthy tissue than X- or 7 rays. Because tumors have physical size, even several centimeters in diameter, a range of proton energies is often used. Heavier ions, such as a particles or carbon ions, are similarly useful. This proton therapy technique is more than a half century old, but the necessity of having a large accelerator has meant that few hospitals have used the technique until now. Many such "proton centers" are now being built.

FIGURE 4 2 - 1 7 R di tion source rot tes so th t the be m lw ys p sses through the dise sed tissue, but m inimizes the dose in the rest of the body.

PHYSICS

APPLIED

Proton t erapy

FIGURE 4 2 - 1 8 Energy deposited in tissue s function o f depth for 170 - M eV protons (red curve) nd 190 - M eV protons (green). The pe k of e ch curve is often c lled the Br gg pe k.

Depth in tissue (cm)

Another form of treatment is to insert a tiny radioactive source directly inside a tumor, which will eventually kill the majority of the cells. A similar technique is used to treat cancer of the thyroid with the radioactive isotope ^ I . The thyroid gland concentrates iodine present in the bloodstream, particularly in any area where abnormal growth is taking place. Its intense radioactivity can destroy the defective cells. Another application of radiation is for sterilizing bandages, surgical equipment, and even packaged foods, since bacteria and viruses can be killed or deactivated by large doses of radiation.

* 4 2 -8 Tr cers in Rese rch nd Medicine Radioactive isotopes are commonly used in biological and medical research as A given compound is artificially synthesized using a radioactive isotope such as 1g or ^H. Such "tagged" molecules can then be traced as they move through an organism or as they undergo chemical reactions. The presence of these tagged molecules (or parts of them, if they undergo chemical change) can be detected by a Geiger or scintillation counter, which detects emitted radiation (see Section 41-11). How food molecules are digested, and to what parts of the body they are diverted, can be traced in this way.

tracers.

*SE TION 4 2 - 8

( PHYSICS APPLIED Tracers in medicine and biology

Tracers in Research and Medicine

1151

( )

(b) FIGURE 4 2 1 9 ( ) Autor diogr ph o f le f exposed for 30 s to 14C 0 2 . The photosynthetic (green) tissue h s becom e r dio ctive; the nonphotosynthetic tissue o f the veins is free of 14C nd therefore does not bl cken the X - r y sheet. This technique is useful in following p tterns of nutrient tr nsport in pl nts, (b) Autor diogr ph of chromosom l D N A . The d shed rr ys of film gr ins show the Y - sh ped growing point of replic ting D N A .

Radioactive tracers have been used to determine how amino acids and other essential compounds are synthesized by organisms. The permeability of cell walls to various molecules and ions can be determined using radioactive isotopes: the tagged molecule or ion is injected into the extracellular fluid, and the radioactivity present inside and outside the cells is measured as a function of time. In a technique known as autoradiography, the position of the radioactive isotopes is detected on film. For example, the distribution of carbohydrates produced in the leaves of plants from absorbed 0 2 can be observed by keeping the plant in an atmosphere where the carbon atom in the 0 2 is x\ . After a time, a leaf is placed firmly on a photographic plate and the emitted radiation darkens the film most strongly where the isotope is most strongly concentrated (Fig. 42-19a). Autoradiography using labeled nucleotides (components of DNA) has revealed much about the details of DNA replication (Fig. 42-19b). For medical diagnosis, the radionuclide commonly used today is 94™Tc, a long-lived excited state of technetium -99 (the "m" in the symbol stands for "metastable" state). It is formed when 92Mo decays. The great usefulness of 943*Tc derives from its convenient half-life of 6 h (short, but not too short) and the fact that it can combine with a large variety of compounds. The compound to be labeled with the radionuclide is so chosen because it concentrates in the organ or region of the anatomy to be studied. Detectors outside the body then record, or image, the distribution of the radioactively labeled compound. The detection could be done by a single detector (Fig. 42-20a) which is moved across the body, measuring the intensity of radioactivity at a large number of points. The image represents the relative intensity of radioactivity at each point. The relative radioactivity is a diagnostic tool. For example, high or low radioactivity may represent overactivity or underactivity of an organ or part of an organ, or in another case may represent a lesion or tumor. More complex gamma cameras make use of many detectors which simultaneously record the radioac tivity at many points. The measured intensities can be displayed on a TV or computer monitor. The image is sometimes called a scintigram (after scintillator), Fig. 42-20b. Gamma cameras are relatively inexpensive, but their resolution is limited (by non -perfect collimation*). But they allow "dynamic" studies (that is, images that change in time, like a movie) to be performed. fTo "collim te" me ns to "m ke p r llel," usu lly by blocking non-p r llel r ys with inside le d, s in Fig. 42-20 .

n rrow tube

-Photomultiplier tube

Scintill tor cryst l Le d collim tor - Collim ting hole

P tient

( )

(b)

FIGURE 4 2 2 0 ( ) Collim ted g mm -r y detector for sc nning (moving) over p tient. The collim tor selects 7 r ys th t com e in (ne rly) str ight line from the p tient. Without the collim tor, 7 r ys from ll p rts of the body could strike the scintill tor, producing poor im ge. Detectors tod y usu lly h ve m ny collim tor tubes nd re c lled (b) G mm c mer im ge (scintigr m) of leg with f tigue fr cture detecting 7s from " 43TC. The im ge is gr y sc le, but is colored here for visibility.

gammacameras,

1152

HAPTER 42

*42—9 Im ging by Tomogr phy: CAT Sc ns nd Emission Tomogr phy *Normal X-ray Image For a conventional medical or dental X-ray photograph, the X-rays emerging from the tube (Section 35-10) pass through the body and are detected on photographic film or a fluorescent screen, Fig. 42-21. The rays travel in very nearly straight lines through the body with minimal deviation since at X-ray wavelengths there is little diffraction or refraction. There is absorption (and scattering), however; and the difference in absorption by different structures in the body is what gives rise to the image produced by the transmitted rays. The less the absorption, the greater the transmission and the darker the film. The image is, in a sense, a "shadow" of what the rays have passed through. The X-ray image is not produced by focusing rays with lenses as for the instruments discussed in hapter 33.

@ PHYSICS APPLIED NormalX-ray image isasort ofs adow (no lenses involved)

FIGURE 4 2 - 2 1 Convention l X - r y im ging, which is essenti lly sh dowing.

Tomography Images ( T) In conventional X-ray images, the entire thickness of the body is projected onto the film; structures overlap and in many cases are difficult to distinguish. In the 1970s, a revolutionary new X-ray technique was developed called computed tomography ( T), which produces an image of a slice through the body. (The word tomography comes from the Greek: tomos = slice, grap = picture.) Structures and lesions previously impossible to visualize can now be seen with remarkable clarity. The principle behind T is shown in Fig. 42-22: a thin collimated beam of X-rays (to "collimate" means to "make parallel") passes through the body to a detector that measures the transmitted intensity. Measurements are made at a large number of points as the source and detector are moved past the body together. The apparatus is then rotated slightly about the body axis and again scanned; this is repeated at (perhaps) 1° intervals for 180°. The intensity of the transmitted beam for the many points of each scan, and for each angle, are sent to a computer that reconstructs the image of the slice. Note that the imaged slice is perpendicular to the long axis of the body. For this reason, T is sometimes called com puterized axial tomography ( AT), although the abbreviation AT, as in AT scan, can also be read as computer- assisted tomography. The use of a single detector as in Fig. 42-22 would require a few minutes for the many scans needed to form a complete image. Much faster scanners use Video monitor

Collim tor

source

W p H Y S l C S— A P P L I E D Computed tomograp y images

FIGURE 4 2 - 2 2 Tomogr phic im ging: the X - r y source nd detector m ove together cross the body, the tr nsmitted intensity being me sured t l rge number of points. Then the source - detector ssembly is rot ted slightly (s y, 1°) nd nother sc n is m de. This process is repe ted for perh ps 180°. The com puter reconstructs the im ge of the slice nd it is presented on TV or computer monitor.

omography: AT Scans and Emission Tomography

1153

Electron source

M gnetic deflection coil

Electron be m X -r ys

( )

(b)

FIGURE 4 2 2 3 ( ) F n - be m sc nner. R ys tr nsmitted through the entire body re m e sured sim ult neously t e ch ngle. The source nd detector rot te to t ke m e surem ents t different ngles. In nother type of f n - be m sc nner, there re detectors round the entire 360° of the circle which rem in fixed s the source moves, (b) In still nother type, be m o f electrons from source is directed by m gnetic fields t tungsten t rgets surrounding the p tient.

a fan beam, Fig. 42-23a, in which beams passing through the entire cross section of the body are detected simultaneously by many detectors. The source and detectors are then rotated about the patient, and an image requires only a few seconds. Even faster, and therefore useful for heart scans, are fixed source machines wherein an electron beam is directed (by magnetic fields) to tungsten targets surrounding the patient, creating the X-rays. See Fig. 42-23b. FIGURE 4 2 2 4 Two CT im ges, with different resolutions, e ch showing cross section o f br in. Photo ( ) is of low resolution. Photo (b), o f higher resolution, shows br in tumor, nd uses f lse color to highlight it.

( )

*Image Formation But how is the image formed? We can think of the slice to be imaged as being divided into many tiny picture elements (or pixels), which could be squares. (See Fig. 35-42.) For T, the width of each pixel is chosen according to the width of the detectors and/or the width of the X-ray beams, and this determines the resolution of the image, which might be 1 mm. An X-ray detector measures the intensity of the transmitted beam. When we subtract this value from the intensity of the beam at the source, we obtain the total absorption (called a "projection") along that beam line. omplicated mathematical techniques are used to analyze all the absorption projections for the huge number of beam scans measured (see the next Subsec tion), obtaining the absorption at each pixel and assigning each a "grayness value" according to how much radiation was absorbed. The image is made up of tiny spots (pixels) of varying shades of gray. Often the amount of absorption is color-coded. The colors in the resulting false-color image have nothing to do, however, with the actual color of the object. The real medical images are monochromatic (various shades of gray). Only visible light has color, X-rays and y rays don't. Figure 42-24 illustrates what actual T images look like. It is generally agreed that T scanning has revolutionized some areas of medicine by providing much less invasive, and/or more accurate, diagnosis. omputed tomography can also be applied to ultrasound imaging (Section 16-9) and to emissions from radioisotopes and nuclear magnetic resonance, which we discuss in Section 42-10.

*Tomographic Image Reconstruction

(b)

1154

CHAPTER 42

How can the "grayness" of each pixel be determined even though all we can measure is the total absorption along each beam line in the slice? It can be done only by using the many beam scans made at a great many different angles. Suppose the image is to be an array of 100 X 100 elements for a total of 104 pixels. If we have 100 detectors and measure the absorption projections at 100 different angles, then we get 104 pieces of information. From this information, an image can be reconstructed, but not precisely. If more angles are measured, the reconstruction of the image can be done more accurately.

To suggest how mathematical reconstruction is done, we consider a very simple case using the "iterative" technique ("to iterate" is from the Latin "to repeat"). Suppose our sample slice is divided into the simple 2 X 2 pixels as shown in Fig. 42-25. The number inside each pixel represents the amount of absorption by the material in that area (say, in tenths of a percent): that is, 4 represents twice as much absorption as 2. But we cannot directly measure these values—they are the unknowns we want to solve for. All we can measure are the projections—the total absorption along each beam line—and these are shown in the diagram as the sum of the absorptions for the pixels along each line at four different angles. These projections (given at the tip of each arrow) are what we can measure, and we now want to work back from them to see how close we can get to the true absorption value for each pixel. We start our analysis with each pixel being assigned a zero value, Fig. 42-26a. In the iterative technique, we use the projections to estimate the absorption value in each square, and repeat for each angle. The angle 1 projections are 7 and 13. We divide each of these equally between their two squares: each square in the left column gets 3 \ (half of 7), and each square in the right column gets 6 \ (half of 13); see Fig. 42-26b. Next we use the projections at angle 2. We

FIGURE 4 2 2 5 A simple 2 X 2 im ge showing true bsorption v lues nd m e sured projections.

FIGURE 4 2 2 6 Reconstructing the im ge using projections in n iter tive procedure. 7

| Angle 1 |

— ►14

(a)

Angle 2 (me sured)

5\ (c)

(b)

(d) 11

calculate the difference between the measured projections at angle 2 (6 and 14) and the projections based on the previous estimate (top row: 3 \ + 6 \ = 10; same for bottom row). Then we distribute this difference equally to the squares in that row. For the top row, we have 3b

6 -10

and for the bottom row, „i 14 - 10 3 \ + ---- ------

6 -10

and

14 - 10

These values are inserted as shown in Fig. 42-26c. Next, the projection at angle 3 gives 11 - 10 11 - 10 (upper left) l \ + = 2 and (lower right) = 9: + and that for angle 4 gives 9 — 10 (lower left) 5 \ + = 5

and

(upper right) A\ + = 4. ° ' z 2 The result, shown in Fig. 42-26d, corresponds exactly to the true values. (In real situations, the true values are not known, which is why these computer techniques are required.) To obtain these numbers exactly, we used six pieces of information (two each at angles 1 and 2, one each at angles 3 and 4). For the much larger number of pixels used for actual images, exact values are generally not attained. Many iterations may be needed, and the calculation is considered sufficiently precise when the difference between calculated and measured projections is sufficiently small. The above example illustrates the "convergence" of the process: the first iteration (b to c in Fig. 42-26) changed the values by 2, the last iteration (c to d) by only \ • 2

♦SE TION 4 2 - 9

1155

E m is s io n T o m o g ra p h y It is possible to image the emissions of a radioactive tracer (see Section 42-8) in a single plane or slice through a body using computed tomography techniques. A basic gamma detector (Fig. 42-20a) can be moved around the patient to measure the radioactive intensity from the tracer at many points and angles; the data are processed in much the same way as for X-ray T scans. This technique is referred to as single photon emission tomography (SPET), or SPE T (single photon emission computed tomography); see Fig. 42-27. Another important technique is positron emission tomography (PET), which makes use of positron emitters such as ^ , 13N, ^O, and 1 F. These isotopes are incorporated into molecules that, when inhaled or injected, accumulate in the FIGURE 4 2 2 7 SPECT sc n of organ or region of the body to be studied. When such a nuclide undergoes br in (f lse color) with M esul m 's jS+ decay, the emitted positron travels at most a few millimeters before it collides syndrome, l beled with " ^Tc. with a normal electron. In this collision, the positron and electron are annihilated, PHYSICS APPLIED producing two 7 rays (e+ + e" —» 27). The two 7 rays fly off in opposite directions (180° + 0.25°) since they must have almost exactly equal and opposite momenta to conserve momentum (the momenta of the initial e+ and e" are essentially zero compared to the momenta of the 7 rays). Because the photons travel FIGURE 4 2 2 8 Positron em ission along the same line in opposite directions, their detection in coincidence by tomogr phy (PET) system showing ring o f detectors to detect the two rings of detectors surrounding the patient (Fig. 42-28) readily establishes the nnihil tion 7 r ys (e+ + e " —►27) line along which the emission took place. If the difference in time of em itted t 180° to e ch other. arrival of the two photons could be determined accurately, the actual position of the emitting nuclide along that line could be calculated. Present-day electronics can measure times to at best + 300 ps, so at the 7 ray's speed (c = 3 X 10 m/s), the actual position could be determined to an accuracy on the order of about d = vt ~ (3 X 108m/s)(300 X 10"12s) ~ 10 cm, which f? A is not very useful. Although there may be future potential for time-of-flight measurements to determine position, today computed tomography techniques are used instead, similar to those for X-ray T, which can reconstruct PET images with a resolution on the order of 3-5 mm. One big advantage of PET is that no collimators are needed (as for detection of a single photon —see Fig. 42-20a). Thus, fewer photons are "wasted" and lower doses can be administered to the patient with PET. Both PET and SPET systems can give images related to biochemistry, metabolism, and function. This is to be compared to X-ray T scans, whose images reflect shape and structure—that is, the anatomy of the imaged region.

Emissio(S nto og,P raE pTy) PEm CT

42-10 Nude r M gnetic Reson nce (NMR); M gnetic Reson nce Im ging (MRI) FIGURE 4 2 2 9 Schem tic picture o f proton in m gnetic field B (pointing upw rd) with the two possible st tes o f proton spin, up nd down.

6 Up

D own

1156

HAPTER 42

Nuclear magnetic resonance (NMR) is a phenomenon which soon after its discovery in 1946 became a powerful research tool in a variety of fields from physics to chem istry and biochemistry. It is also an important medical imaging technique. We first briefly discuss the phenomenon, and then look at its applications.

Nuclear Magnetic Resonance (NMR) We saw in hapter 39 that the energy levels in atoms are split when they are placed in a magnetic field B (the Zeeman effect) according to the angular momentum or spin of the state. The splitting is proportional to B and to the magnetic moment, |x. Nuclei too have magnetic moments (Section 41-1), and we examine only the simplest, the hydrogen (|H) nucleus which consists of a single proton. Its spin angular momentum (and its magnetic moment), like that of the elec tron, can take on only two values when placed in a magnetic field: spin up (parallel to the field) and spin down (antiparallel to the field) as suggested in Fig. 42-29.

Nuclear Energy; Effects and Uses of Radiation

/ Eq-

(P =

Spin down

f = AE = 2|XpB

FIGURE 4 2 3 0 Energy in the bsence of m gnetic field splits into two levels in the presence of m gnetic field.

Spin up

When a magnetic field is present, an energy state splits into two levels as shown in Fig. 42-30 with the spin-up state (parallel to field) having the lower energy. The spin-down state acquires an additional energy fJLpJBx and the spin-up state has its energy changed by - |x pBx (Eq. 27-12 and Section 39-7), where BT is the total magnetic field at the nucleus. The difference in energy between the two states (Fig. 42-30) is thus AE = 2 |xpi?x , where is the magnetic moment of the proton. In a standard nuclear magnetic resonance (NMR) setup, the sample to be examined is placed in a static magnetic field. A radiofrequency (RF) pulse of elec tromagnetic radiation (that is, photons) is applied to the sample. If the frequency,/, of this pulse corresponds precisely to the energy difference between the two energy levels (Fig. 42-30), so that f = AE = 2\LpBTf

(42-12)

then the photons of the RF beam will be absorbed, exciting many of the nuclei from the lower state to the upper state. This is a resonance phenomenon since there is significant absorption only if /is very near / = 2jjlpBT/ . Hence the name "nuclear magnetic resonance." For free {H nuclei, the frequency is 42.58 MHz for a field B = 1.0 T (Example 42-14). If the H atoms are bound in a molecule, the total magnetic field BT at the H nuclei will be the sum of the external applied field (Bext) plus the local magnetic field (B{oc) due to electrons and nuclei of neighboring atoms. Since / is proportional to BT, the value of / for a given external field will be slightly different for bound H atoms than for free atoms: f = 2/ip (Bext + Bloc). This change in frequency, which can be measured, is called the "chemical shift." A great deal has been learned about the structure of molecules and bonds using this NMR technique. EXAMPLE 42 - 1 4 NMR for free protons. for free protons in a 1.000-T magnetic field.

alculate the resonant frequency

APPROA H We use Eq. 42-12, where the magnetic moment of the proton (Section 41-1) is ( pfi, \ ( p = 2.7928|jln = 2.7928 - — = 2.7928 x2mpJ \ 4irmp SOLUTION We solve for / i n Eq. 42-12 and find AE

_ 2(ipB

* ~ ( eB \ f (1-6022 X 10~19 )(1.000 T) = (2.7928) ------- = 2.7928 ' 2nm pJ ' L 2ir(l.6726 X IO-27 kg) = 42.58 MHz. *SE TION 4 2 - 1 0

Nuclear Magnetic Resonance (NMR); Magnetic Resonance Imaging (MRI)

1157

v£/ P HYS I S —A P P L I E D — NMR imaging (MRI)

P tient

B low

FIGURE 4 2 3 2 A st tic field th t is stronger t the bottom th n t the top. The frequency o f bsorbed or em itted r di tion is proportion l to B in NM R. FIGURE 4 2 3 3 F lse - color N M R im ge (M RI) o f vertic l section through the he d showing structures in the norm l br in.

1158

HAPTER 42

*M a g n e tic R e s o n a n c e Im a g in g (M RI) For producing medically useful NMR images—now commonly called MRI, or magnetic resonance imaging—the element most used is hydrogen since it is the commonest element in the human body and gives the strongest NMR signals. The experimental apparatus is shown in Fig. 42-31. The large coils set up the static magnetic field, and the RF coils produce the RF pulse of electromagnetic waves (photons) that cause the nuclei to jump from the lower state to the upper one (Fig. 42-30). These same coils (or another coil) can detect the absorption of energy or the emitted radiation (also of frequency / = AE/ ) when the nuclei jump back down to the lower state.

FIGURE 4 2 3 1

Typic l M RI im ging setup: ( ) di gr m; (b) photogr ph.

The formation of a two-dimensional or three-dimensional image can be done using techniques similar to those for computed tomography (Section 42-9). The simplest thing to measure for creating an image is the intensity of absorbed and/or reemitted radiation from many different points of the body, and this would be a measure of the density of H atoms at each point. But how do we determine from what part of the body a given photon comes? One technique is to give the static magnetic field a gradient; that is, instead of applying a uniform magnetic field, Bt , the field is made to vary with position across the width of the sample (or patient). Since the frequency absorbed by the H nuclei is proportional to BT (Eq. 42-12), only one plane within the body will have the proper value of BT to absorb photons of a particular frequency /. By varying /, absorption by different planes can be measured. Alternately, if the field gradient is applied after the RF pulse, the frequency of the emitted photons will be a measure of where they were emitted. See Fig. 42-32. If a magnetic field gradient in one direction is applied during excitation (absorption of photons) and photons of a single frequency are trans mitted, only H nuclei in one thin slice will be excited. By applying a gradient during reemission in a direction perpendicular to the first, the frequency / of the reemitted radiation will represent depth in that slice. Other ways of varying the magnetic field throughout the volume of the body can be used in order to correlate NMR frequency with position. A reconstructed image based on the density of H atoms (that is, the intensity of absorbed or emitted radiation) is not very interesting. More useful are images based on the rate at which the nuclei decay back to the ground state, and such images can produce resolution of 1 mm or better. This NMR technique (sometimes called spin-echo) produces images of great diagnostic value, both in the delineation of structure (anatomy) and in the study of metabolic processes. An NMR image is shown in Fig. 42-33, color enhanced—no medical imaging uses visible light, so the colors shown here are added. The original images, those looked at by your doctor, are various shades of gray, representing intensity (or counts).

Nuclear Energy; Effects and Uses of Radiation

N M R imaging is considered to be noninvasive. We can calculate the energy of the photons involved: as determ ined in E xam ple 42- 14, in a 1.0 - T m agnetic field, / = 42.58 M Hz for }H. This corresponds to an energy of f = (6.6 X IO 34 J •s )(43 X 106Hz) « 3 X IO 26 J or about IO 7 eV. Since molec ular bonds are on the order of 1 eV, it is clear that the R F photons can cause little cellular disruption. This should be compared to X - or 7 rays whose energies are 104 to 106eV and thus can cause significant damage. The static magnetic fields, though often large ( ~ 0.1 to I T ) , are believed to be harmless (except for people wearing heart pacemakers). Table 42 2 lists the major techniques we have discussed for imaging the interior of the human body, along with the optimum resolution attainable today. Resolution is just one factor that must be considered, as the different imaging techniques provide different types of information, useful for different types of diagnosis.

TABLE 4 2 - 2 M e d ica l Im a g in g T e c h n iq u e s Technique

Resolution

onventional X -ray T scan, X -ray Nuclear medicine (tracers) SPET (single photon emission) PET (positron emission) NMR (MRI) Ultrasound (Section 16-9)

2 mm \ mm 1 cm 1 cm

3 - 5 mm \ - l mm 2 mm

Summ ry A nuclear reaction occurs when two nuclei collide and two or more other nuclei (or particles) are produced. In this process, as in radioactivity, transmutation (change) of elements occurs. The reaction energy or g - value of a reaction a + X —» Y + b is Q = (Ma + Mx - M = K

- M y ) c2

+ KY - Ka - Kx .

(42-2a) (42-2b)

The effective cross section a for a reaction is a measure of the reaction probability per target nucleus. In fission, a heavy nucleus such as uranium splits into two intermediate -sized nuclei after being struck by a neutron. 2;j2U is fissionable by slow neutrons, whereas some fissionable nuclei require fast neutrons. Much energy is released in fission (~ 200 MeV per fission) because the binding energy per nucleon is lower for heavy nuclei than it is for intermediate sized nuclei, so the mass of a heavy nucleus is greater than the total mass of its fission products. The fission process releases neutrons, so that a chain reaction is possible. The critical mass is the minimum mass of fuel needed to sustain a chain reaction. In a nuclear reactor or nuclear bomb, a moderator is used to slow down the released neutrons. The fusion process, in which small nuclei combine to form larger ones, also releases energy. The energy from our Sun origi nates in the fusion reactions known as the proton - proton cycle in which four protons fuse to form a 2He nucleus producing 25 MeV of energy. A useful fusion reactor for power gener ation has not yet proved possible because of the difficulty in containing the fuel (e.g., deuterium) long enough at the extremely

high temperature required ( ~ 10 K). Nonetheless, great progress has been made in confining the collection of charged ions known as a plasma. The two main methods are magnetic confinement, using a magnetic field in a device such as the toroidal-shaped tokamak, and inertial confinement in which intense laser beams compress a fuel pellet of deuterium and tritium. Radiation can cause damage to materials, including biolog ical tissue. Quantifying amounts of radiation is the subject of dosimetry. The curie ( i) and the becquerel (Bq) are units that measure the source activity or rate of decay of a sample: 1 i = 3.70 X IO10 decays per second, whereas lB q = 1 decay/s. The absorbed dose, often specified in rads, measures the amount of energy deposited per unit mass of absorbing material: lr a d is the amount of radiation that deposits energy at the rate of 10-2 J/kg of material. The SI unit of absorbed dose is the gray: 1 Gy = U /k g = 100 rad. The effective dose is often specified by the rem = rad X QF, where QF is the "quality factor" of a given type of radiation; lre m of any type of radiation does approximately the same amount of biological damage. The average dose received per person per year in the United States is about 360mrem. The SI unit for effective dose is the sievert: 1 Sv = 102 rem. [*Nuclear radiation is used in medicine for cancer therapy, and for imaging of biological structure and processes. Tomographic imaging of the human body, which can provide 3-dimensional detail, includes several types: T scans, PET, SPET (= SPE T), and MRI; the latter makes use of nuclear magnetic resonance (NMR).]

Questions (NOTE: Masses are found in Appendix F.) 1. Fill in the missing particles or nuclei: (a) n + ^ B a - * ? + 7; 0b) n + ^ B a - » ^ s + ?; (c) d + 2H —>• 2He + ?; (d) a + ^ A u - » ? + d where d stands for deuterium. 2. The isotope 32P is produced by the reaction: n + ? —» 32P + p. What must be the target nucleus?

3. When 22Na is bombarded by deuterons (2H), an a particle is emitted. What is the resulting nuclide? 4. Why are neutrons such good projectiles for producing nuclear reactions? 5. A proton strikes a 2oNe nucleus, and an a particle is observed to emerge. What is the residual nucleus? Write down the reaction equation. 6. Are fission fragments (3+ or (3~ emitters? Explain. 7. The energy from nuclear fission appears in the form of thermal energy—but the thermal energy of what?

Questions

1159

. 292U releases an average of 2.5 neutrons per fission compared to 2.9 for ^ P u . Which of these two nuclei do you think would have the smaller critical mass? Explain. 9. If 292U released only 1.5 neutrons per fission on the average, would a chain reaction be possible? If so, how would the chain reaction be different than if 3 neutrons were released per fission? 10. Why can't uranium be enriched by chemical means? 11. How can a neutron, with practically no kinetic energy, excite a nucleus to the extent shown in Fig. 42-4? 12. Why would a porous block of uranium be more likely to explode if kept under water rather than in air? 13. A reactor that uses highly enriched uranium can use ordi nary water (instead of heavy water) as a moderator and still have a self-sustaining chain reaction. Explain. 14. Why must the fission process release neutrons if it is to be useful? 15. Why are neutrons released in a fission reaction? 16. What is the reason for the "secondary system" in a nuclear reactor, Fig. 42-9? That is, why is the water heated by the fuel in a nuclear reactor not used directly to drive the turbines? 17. What is the basic difference between fission and fusion?

18. Discuss the relative merits and disadvantages, including pollution and safety, of power generation by fossil fuels, nuclear fission, and nuclear fusion. 19. A higher temperature is required for deuterium - deuterium ignition than for deuterium - tritium. Explain. 20. Light energy emitted by the Sun and stars comes from the fusion process. What conditions in the interior of stars make this possible? 21. How do stars, and our Sun, maintain confinement of the plasma for fusion? 22. Why is the recommended maximum radiation dose higher for women beyond the child-bearing age than for younger women? 23. People who work around metals that emit alpha particles are trained that there is little danger from proximity or touching the material, but they must take extreme precautions against ingesting it. Why? (Eating and drinking while working are forbidden.) 24. What is the difference between absorbed dose and effective dose? What are the SI units for each? 25. Radiation is sometimes used to sterilize medical supplies and even food. Explain how it works. *26. How might radioactive tracers be used to find a leak in a pipe?

Problems (NOTE: Masses are found in Appendix F.) 42 - 1 N u c le a r R e a c tio n s , T ra n s m u ta tio n 1. (I) Natural aluminum is all 13Al. If it absorbs a neutron, what does it become? Does it decay by j + or j3- ? What will be the product nucleus? 2. (I) Determine whether the reaction 2H + 2H —> |H e + n requires a threshold energy. 3. (I) Is the reaction n + neutrons? Explain.

—> 2^ \J + J possible with slow

4. (II) Does the reaction p + ^Li —> ^He + a require energy, or does it release energy? How much energy? 5. (II) alculate the energy released (or energy input required) for the reaction a + 9Be —» 1g + n. 6. (II) (a)

an the reaction n + 2^Mg —» 23Na + d occur if the bombarding particles have 16.00 MeV of kinetic energy? (d stands for deuterium, 2H.) (b) If so, how much energy is released? If not, what kinetic energy is needed?

7. (II) (a) an the reaction p + ^Li —» 2He + a occur if the incident proton has kinetic energy = 3500 keV? (b) If so, what is the total kinetic energy of the products? If not, what kinetic energy is needed? . (II) In the reaction a + 14N —> 1gO + p, the incident a particles have 9.68 MeV of kinetic energy. The mass of 17sO is 16.999132 u. (a) an this reaction occur? (b) If so, what is the total kinetic energy of the products? If not, what kinetic energy is needed? alculate the g - value for the "capture" reaction CL + 110 2oNe + y.

9. (II)

10. (II) alculate the total kinetic energy of the products of the reaction d + l\C —►14N + n if the incoming deuteron has kinetic energy K = 44.4 MeV.

1160

HAPTER 42

11. (II) Radioactive is produced in the atmosphere when a neutron is absorbed by 14N. Write the reaction and find its Q-value. 12. (II) An example of a stripping nuclear reaction is d + 3Li —> X + p. (a) What is X, the resulting nucleus? (b) Why is it called a "stripping" reaction? (c) What is the Q -value of this reaction? Is the reaction endothermic or exothermic? 13. (II) An example of a pick-up nuclear reaction is fHe + x\ —» X + a. (a) Why is it called a "pick -up" reaction? (b) What is the resulting nucleus? (c) What is the (2-value of this reaction? Is the reaction endothermic or exothermic? 14. (II) (a) omplete the following nuclear reaction, p + ? —> fgS + J. (b) What is the Q-value? 15. (II) The reaction p + 1gO —►18F + n requires an input of energy equal to 2.438 MeV. What is the mass of 1$F? 16. (Ill) Use conservation of energy and momentum to show that a bombarding proton must have an energy of 3.23 MeV in order to make the reaction ^ ( p , n )13N occur. (See Example 42-3.) 17. (Ill) How much kinetic energy (if any) would the proton require for the reaction 1^ (p, n )14N to proceed? 4 2 -2 ross S e c tio n 18. (I) The cross section for the reaction n + 1§B —►^Li + ^He is about 40 bn for an incident neutron of low energy (kinetic energy « 0). The boron is contained in a gas with n = 1.7 X 1021 nuclei/m 3 and the target has thickness £ = 12.0 cm. What fraction of incident neutrons will be scattered? 19. (I) What is the effective cross section for the collision of two hard spheres of radius Ri and R 2?

Nuclear Energy; Effects and Uses of Radiation

(II) When the target is thick, the rate at which projectile particles collide with nuclei in the rear of the target is less than in the front of the target, since some scattering (i.e., collisions) takes place in the front layers. Let R 0 be the rate at which incident particles strike the front of the target, and R x be the rate at a distance x into the target (Rx = R0 at x = 0). Then show that the rate at which particles are scattered (and therefore lost from the incident beam) in a thickness dx is —dRx = R x no-dx, where the minus sign means that R x is decreasing and n is the number of nuclei per unit volume. Then show that R x = R 0e~nax, where a is the total cross section. If the thickness of the target is £, what does R x = R 0e ~n£

simple

olliding Beams High-energy physics experiments were once done by aiming a beam of particles from an accelerator at a stationary target. But to obtain the maximum possible collision energy from a given accelerator, two beams of particles are now accelerated to very high energy and are steered so that they collide head-on. One way to accomplish such colliding beams with a single accelerator is through the use of storage rings, in which oppositely circulating beams can be repeatedly brought into collision with one another at particular points. For example, in the experiments that provided strong evidence for the top quark ( hapter-Opening Photo and Section 43 -9), the Fermilab Tevatron accelerated protons and antiprotons each to 900 GeV, so that the combined energy of head-on collisions was 1.8 TeV. The largest collider is the Large Hadron ollider (LH ) at ERN, with a circumference of 26.7 km (Fig. 43 -5), scheduled to begin operating as this book is about to be published. The two colliding beams each carry 7-TeV protons for a total interaction energy of 14 TeV.

FIGURE 4 3 - 5 The l rge circle represents the position o f the tunnel, bout 100 m below the ground t CER N (ne r G enev ) on the French -Swiss border, which houses the LHC. The sm ller circle shows the position of the Super Proton Synchrotron th t will be used for cceler ting protons prior to injection into the LHC.

SE TION 43 - 1

High-Energy Particles and Accelerators

1169

FIGURE 4 3 6 ATLAS, one of sever l l rge com plex detectors t the LHC, is shown here s it w s being built. It is hoped the LH C will provide evidence for the H iggs boson (to help underst nd the St nd rd M odel), nd perh ps to find supersymmetric p rticles which re c ndid tes for the unknown d rk m tter th t m kes up l rge p rt of the m ss - en ergy o f the universe. We will touch on these topics l ter in this Ch pter.

Figure 43 -6 shows part of one of the detectors (ATLAS) as it was being constructed at the LH . The detectors within ATLAS include silicon semicon ductor detectors with huge numbers of pixels used to track particle paths, to find their point of interaction, and to measure their radius of curvature in a magnetic field and thus determine their momentum (Section 27-4). Their energy is determined in "calorimeters" utilizing plastic, liquid, or dense metal compound crystal scintillators (Section 41-11). In the planning stage is the International Linear ollider (IL ) which would have colliding beams of e_ and e+ at around 0.3 to ITeV, with semiconductor detectors using MOS (Section 33-5) with embedded transistors to allow fast readout. Vi Protons a t relativistic speeds. Determine the energy required to accelerate a proton in a high-energy accelerator (a) from rest to v = 0.900 c, and (b) from v = 0.900 c to v = 0.999 c. (c) What is the kinetic energy achieved by the proton in each case? APPROA H We use the work-energy principle, which is still valid relativistically as mentioned in Section 36-11: W = A K SOLUTION The kinetic energy of a proton of mass m is given by Eq. 36-10, K = (J - 1)mc2, where the relativistic factor 7 is 1

V l - v2/c 2 The work-energy theorem becomes W = b K = (AT)mc2 since m and c are constant. (a) For v = 0, 7 = 1; and for v = 0.900c 2.29. \ / l - (0.900)2 For a proton, me2 = 938 MeV, so the work (or energy) needed to accelerate it from rest to v = 0.900c is W = AK = (A7) me2 = (2.29 - 1.00)(938 MeV) = 1.21 GeV. (b) For v = 0.999 c, 1

(0.999)2

22.4.

So the work needed to accelerate a proton from 0.900 c to 0.999 c is W = AK = (A7) me2 = (22.4 - 2.29)(938 MeV) = 18.9 GeV, which is 15 times as much. (c) The kinetic energy reached by the proton in (a) is just equal to the work done on it, K = 1.21 GeV. The final kinetic energy of the proton in (b), moving at v = 0.999 c, is K = (7 - l)m c 2 = (21.4)(938 MeV) = 20.1 GeV, which makes sense since, starting from rest, we did work W = 1.21 GeV + 18.9 GeV = 20.1 GeV on it.

1170

HAPTER 43

Elementary Particles

EXAMPLE 43 - 4 Speed of a 1.0-TeV proton. What is the speed of a 1.0-TeV proton produced at Fermilab?

APPROA H The kinetic energy K = 1.0 TeV = 1.0 X 1012eV is much greater than the mass of the proton, 0.938 X 109eV, so relativistic calculations must be used. In particular, we use Eq. 36-10: K = (7 - l)m c

2 _

SOLUTION ompared to K = 1.0 be neglected, so we write

me2 \ / \ - v2/c 2 X

1012eV, the rest energy (« 10 3TeV) can

K = Then 1 - *

or v _ c V

/racM 2 _ I _ (938 X 106e V \2 \K J V \ 1.0 X 10 eV

v = 0.9999996 c. The proton is traveling at a speed extremely close to c, the speed of light.

4 3 - 2 Beginnings of Element ry P rticle Physics—P rticle Exch nge The accepted model for elementary particles today views quarks and leptons as the basic constituents of ordinary matter. To understand our present-day view of elementary particles, it is necessary to understand the ideas leading up to its formulation. Elementary particle physics might be said to have begun in 1935 when the Japanese physicist Hideki Yukawa (1907-1981) predicted the existence of a new particle that would in some way mediate the strong nuclear force. To understand Yukawa's idea, we first consider the electromagnetic force. When we first discussed electricity, we saw that the electric force acts over a distance, without contact. To better perceive how a force can act over a distance, we used the idea of a field. The force that one charged particle exerts on a second can be said to be due to the electric field set up by the first. Similarly, the magnetic field can be said to carry the magnetic force. Later ( hapter 31), we saw that electromagnetic (EM) fields can travel through space as waves. Finally, in hapter 37, we saw that electromag netic radiation (light) can be considered as either a wave or as a collection of particles called p otons. Because of this wave -particle duality, it is possible to imagine that the electromagnetic force between charged particles is due to (1) the EM field set up by one charged particle and felt by the other, or (2) an exchange of photons (7 particles) between them. It is (2) that we want to concentrate on here, and a crude analogy for how an exchange of particles could give rise to a force is suggested in Fig. 43 -7. In part (a), two children start throwing heavy pillows at each other; each throw and each catch results in the child being pushed backward by the impulse. This is the equivalent of a repulsive force. On the other hand, if the two children exchange pillows by grabbing them out of the other person's hand, they will be pulled toward each other, as when an attractive force acts. SE TION 4 3 - 2

FIGURE 4 3 7 Forces equiv lent to p rticle exch nge, ( ) Repulsive force (children on roller sk tes throwing pillows t e ch other). (b) Attr ctive force (children gr bbing pillows from e ch other 's h nds).

Jy S k -

- AA

( ) Repulsive force (children throwing pillows)

fc j (b) Attr ctive force (children gr bbing pillows from e ch other's h nds)

Beginnings of Elementary Particle Physics—Particle Exchange

1171

FIGURE 4 3 8 Feynm n di gr m showing photon cting s the c rrier of the electrom gnetic force betw een two electrons. This is sort of n vs. gr ph, with incre sing upw rd. St rting t the bottom , two electrons ppro ch e ch other (the dist nce betw een them decre ses in tim e). A s they get close, m omentum nd energy get tr nsferred from one to the other, c rried by photon (or, perh ps, by m ore th n one), nd the two electrons bounce p rt.

FIGURE 4 3 9 E rly m odel showing m eson exch nge when proton nd neutron inter ct vi the strong nucle r force. (Tod y, s we sh ll see shortly, w e view the strong force s c rried by gluons betw een qu rks.) p

For the electromagnetic force, it is photons that are exchanged between two charged particles that give rise to the force between them. A simple diagram describing this photon exchange is shown in Fig. 43-8. Such a diagram, called a Feynman diagram after its inventor, the American physicist Richard Feynman (1918-1988), is based on the theory of quantum electrodynamics (QED). Figure 43 -8 represents the simplest case in QED, in which a single photon is exchanged. One of the charged particles emits the photon and recoils somewhat as a result; and the second particle absorbs the photon. In any collision or interaction, energy and momentum are transferred from one charged particle to the other, carried by the photon. The photon is absorbed by the second particle after it is emitted by the first and is not observable. Hence it is referred to as a virtual photon, in contrast to one that is free and can be detected by instruments. The photon is said to mediate, or carry, the electromagnetic force. By analogy with photon exchange that mediates the electromagnetic force, Yukawa argued that there ought to be a particle that mediates the strong nuclear force—the force that holds nucleons together in the nucleus. Yukawa called this predicted particle a meson (meaning "medium mass"). Figure 43 -9 is a Feynman diagram showing the original model of meson exchange: a meson carrying the strong force between a neutron and a proton. A rough estimate of the mass of the meson can be made as follows. Suppose the proton on the left in Fig. 43 -9 is at rest. For it to emit a meson would require energy (to make the meson's mass) which, coming from nowhere, would violate conservation of energy. But the uncertainty principle allows nonconservation of energy by an amount AE if it occurs only for a time At given by (A E)(At) « /277. We set AE equal to the energy needed to create the mass m of the meson: AE = me2. onservation of energy is violated only as long as the meson exists, which is the time At required for the meson to pass from one nucleon to the other, where it is absorbed and disappears. If we assume the meson travels at relativistic speed, close to the speed of light c, then A^ need be at most about At = d/c, where d is the maximum distance that can separate the inter acting nucleons. Thus we can write AEAt « A 2ir

Meson

d\ me21 — n

me*

(4 3 - 3 )

The range of the strong nuclear force (the maximum distance away it can be felt) is small—not much more than the size of a nucleon or small nucleus (see Eq. 41 - 1) — so let us take d tt 1.5 x 10" 15 m. Then from Eq. 43 -3, me

(6.6

2ird ~

10-34 J •s)(3.0

(6.28)(l.5

10 m/s)

10- 15m)

2.1

10"n J = 130 MeV.

The mass of the predicted meson, roughly 130 MeV/c2, is about 250 times the electron mass of 0.51 MeV/c2. EXERCISE C Wh t effect does n incre se in the m ss of the virtu l exch nge p rticle h ve on the r nge o f the force it m edi tes? D ecre ses it; ( ) incre ses it; (c) h s no decre ses the r nge for ch rged p rticles nd incre ses the r nge ppreci ble effect; for neutr l p rticles.

(d)

(a)

Note that since the electromagnetic force has infinite range, Eq. 43 -3 with d = oo tells us that the exchanged particle for the electromagnetic force, the photon, will have zero mass, which it does.

1172

HAPTER 43

Elementary Particles

The particle predicted by Yukawa was discovered in cosmic rays by . F. Powell and G. Occhialini in 1947, and is called the "77" or pi meson, or simply the pion. It comes in three charge states: +e, —e, or 0, where e = 1.6 X 10_19 . The 7r+ and ~ have mass of 139.6 MeV/c 2 and the 77° a mass of 135.0 MeV/c2, all close to Yukawa's prediction. All three interact strongly with matter. Reactions observed in the laboratory, using a particle accelerator, include p + p p + p

P +

p + n +

77°,

(4 3 - 4 )

The incident proton from the accelerator must have sufficient energy to produce the additional mass of the free pion. Yukawa's theory of pion exchange as carrier of the strong force has been superseded by quantum c romodynamics in which protons, neutrons, and other strongly interacting particles are made up of basic entities called quarks, and the basic carriers of the strong force are gluons, as we shall discuss shortly. But the basic idea of the earlier theory, that forces can be understood as the exchange of particles, remains valid. There are four known types of force— or interactions—in nature. The electro magnetic force is carried by the photon, the strong force by gluons. What about the other two: the weak force and gravity? These too are believed to be mediated by particles. The particles that transmit the weak force are referred to as the W+, W ", and Z°, and were detected in 1983 (Fig. 43-10). The quantum (or carrier) of the gravitational force is called the graviton, and if it exists it has not yet been observed.

FIGURE 4 3 1 0 ( ) Computer reconstruction of Z - p rticle dec y into n electron nd positron (Z c + e ) whose tr cks re shown in white, which took pl ce in the U A 1 detector t CERN. (b) P hoto of the U A 1 detector t CER N s it w s being built.

A comparison of the four forces is given in Table 43-1, where they are listed according to their (approximate) relative strengths. Notice that although gravity may be the most obvious force in daily life (because of the huge mass of the Earth), on a nuclear scale it is by far the weakest of the four forces, and its effect at the particle level can nearly always be ignored. TABLE 43 - 1 The Four Forces in Nature Type

Rel tive Strength ( pprox., for 2 protons in nucleus)

Field P rticle

Gluons

Electrom gnetic

IO"2

Photon

We k

10"6 10 - 38

Strong

Gr vit tion l

SE TION 4 3 - 2

nd Z °

Gr viton (?)

Beginnings of Elementary Particle Physics—Particle Exchange

1173

4 3 - 3 P rticles nd Antip rticles

norm l electrons +m 2

f>2me2

Photon

-mc* neg tive se

( ) FIGURE 4 3 1 1 ( ) Possible energy st tes for n electron. N ote the v st se of fully occupied electron st tes t (b) A n electron in the neg tive se is hit by photon 2 nd knocks it up to norm l positive energy st te. The positive "h o le " left behind cts like positive electron — it is positron.

E me2)

The positron, as we discussed in Sections 37-5 (pair production) and 41-5 (/3+ decay), is basically a positive electron. That is, many of its properties are the same as for the electron, such as mass, but it has the opposite electric charge (+e). Other quantum numbers that we will discuss shortly are also reversed. The positron is said to be the antiparticle to the electron. The original idea for antiparticles came from a relativistic wave equation developed in 1928 by the Englishman P. A. M. Dirac (1902-1984). Recall that the non-relativistic Schrodinger equation took conservation of energy as a starting point. The Dirac equation too was based in part on conservation of energy. In hapter 36, we saw that the total energy E of a particle with mass m and momentum p and zero potential energy is given by Eq. 36-13, E 2 = p2c2 + ra2c4. Thus E = ± v V c 2 + m2c4. Dirac applied his new equation and found that it included solutions with both + and - signs. He could not ignore the solution with the negative sign, which we might have thought unphysical. If those negative energy states are real, then we would expect normal free electrons to drop down into those states, emitting photons— never experimentally seen. To deal with this difficulty, Dirac postulated that all those negative energy states are normally occupied. That is, what we thought was the vacuum is instead a vast sea of electrons in negative energy states (Fig. 43 - lla). These electrons are not normally observable. But if a photon strikes one of these negative energy electrons, that electron can be knocked up to a normal (E > me2) energy state as shown in Fig. 43 - llb. (Note in Fig. 43-11 that there are no energy states between E = - mc 2 and E = +mc2 because p 2 cannot be negative in the equation E ± V p 2c2 + m 2c 4 . ) The photon that knocks an e" from the negative sea up to a normal state (Fig. 43 - llb) must have an energy greater than 2me2. What is left behind is a hole (as in semiconductors, Sections 40-7 and 40-8) with positive charge. We call that "hole" a positron, and it can move around as a free particle with positive energy. Thus Fig. 43 - llb represents (Section 37-5) pair production: 7 —> e"e+. The positron was first detected as a curved path in a cloud chamber in a magnetic field by arl Anderson in 1932. It was predicted that other particles also would have antiparticles. It was decades before another type was found. Finally, in 1955 the antiparticle to the proton, the antiproton (p), which carries a negative charge (Fig. 43-12), was discovered at the University of alifornia, Berkeley,

FIGURE 4 3 - 1 2 Liquid - hydrogen bubble ch mber photogr ph of n ntiproton (p) colliding with proton t rest, producing X i - n ti - X i p ir (p + p —» H - + H+) th t subsequently dec y into other p rticles. The dr wing indic tes the ssignment of p rticles to e ch tr ck, which is b sed on how or if th t p rticle dec ys, nd on m ss v lues estim ted from m e surem ent of m om entum (curv ture of tr ck in m gnetic field) nd energy (thickness of tr ck, for ex m ple). Neutr l p rticle p ths re shown by d shed lines since neutr l p rticles produce no bubbles nd hence no tr cks.

1174

HAPTER 43

Elementary Particles

by Emilio Segre (1905-1989, Fig. 43-13) and Owen hamberlain (1920-2006). A bar, such as over the p, is used in general to indicate the antiparticle (p). Soon after, the antineutron (n) was found. All particles have antiparticles. But a few, like the photon and the 77°, do not have distinct antiparticles—we say that they are their own antiparticles. Antiparticles are produced in nuclear reactions when there is sufficient energy available to produce the required mass, and they do not live very long in the pres ence of matter. For example, a positron is stable when by itself; but if it encounters an electron, the two annihilate each other. The energy of their vanished mass, plus any kinetic energy they possessed, is converted into the energy of 7 rays or of other particles. Annihilation also occurs for all other particle -antiparticle pairs. The vast sea of electrons with negative energy in Fig. 43-11 is the vacuum (or vacuum state). According to quantum mechanics, the vacuum is not empty, but contains electrons and other particles as well. The uncertainty principle allows a particle to jump briefly up to a normal energy, thus creating a particle-antiparticle pair. It is possible that they could be the source of the recently discovered dark energy that fills the universe ( hapter 44). We still have a lot to learn. Antimatter is a term referring to material that would be made up of "antiatoms" in which antiprotons and antineutrons would form the nucleus around which positrons (antielectrons) would move. The term is also used for antiparticles in general. If there were pockets of antimatter in the universe, a huge explosion would occur if it should encounter normal matter. It is believed that antimatter was prevalent in the very early universe (Section 44-7).

4 3 - 4 P rticle Inter ctions nd Conserv tion L ws One of the important uses of high-energy accelerators is to study the interactions of elementary particles with each other. As a means of ordering this subnuclear world, the conservation laws are indispensable. The laws of conservation of energy, of momentum, of angular momentum, and of electric charge are found to hold precisely in all particle interactions. A study of particle interactions has revealed a number of new conservation laws which (just like the old ones) are ordering principles: they help to explain why some reactions occur and others do not. For example, the following reaction has never been observed: p + n\»p + p + p even though charge, energy, and so on, are conserved (\> means the reaction does not occur). To understand why such a reaction does not occur, physicists hypothesized a new conservation law, the conservation of baryon number. (Baryon number is a generalization of nucleon number, which we saw earlier is conserved in nuclear reactions and decays.) All nucleons are defined to have baryon number B = +1, and all antinucleons (antiprotons, antineutrons) have B = - 1. All other types of particles, such as photons, mesons, and electrons and other leptons, have B = 0. The reaction shown at the start of this paragraph does not conserve baryon number since the left side has B = ( + 1) + ( + 1) = +2, and the right has B = ( + 1) + ( + 1) + ( - 1 ) = +1. On the other hand, the following reaction does conserve B and does occur if the incoming proton has sufficient energy: p + p - > p + p + p + p, B = +1 + 1 = +1 + 1 - 1 + 1. As indicated, B = +2 on both sides of this equation. From these and other reactions, the conservation of baryon number has been established as a basic principle of physics. Also useful are conservation laws for the three lepton numbers, associated with weak interactions including decays. In ordinary j decay, an electron or positron is emitted along with a neutrino or antineutrino. In another type of decay, a particle known as a "/u" or mu meson, or muon, can be emitted instead of an electron. The muon (discov ered in 1937) seems to be much like an electron, except its mass is 207 times larger (106 MeV/c2). The neutrino (ve) that accompanies an emitted electron is found to be different from the neutrino (v^) that accompanies an emitted muon.

buk-

0& 2* & Up £U■vt+1-

* / t^

FIGURE 4 3 1 3 Em ilio Segre: he worked with Fermi in the 1930s, l ter discovered the first "m n - m de " elem ent, technetium , nd other elem ents, nd then the ntiproton. The inscription below the photo is from book by Segre given to this b ook 's uthor.

/ j\ C A U T I

edoiffe ttyaplesofneutrinos aTren tidreenntic SE TION 4 3 - 4

1175

Each of these neutrinos has an antiparticle: ve and v . In ordinary /3 decay we have, for example, n —» p + e_ + ve but not n \> p + e_ + v^. To explain why these do not occur, the concept of electron lepton number, Le, was invented. If the electron (e") and the electron neutrino (ve) are assigned Le = + 1 , and e+ and ve are assigned Le = - 1 , whereas all other particles have Le = 0, then all observed decays conserve Le. For example, in n —> p + e" + ve, initially Le = 0, and afterward Le = 0 + ( + 1 ) + ( - 1 ) = 0. Decays that do not conserve Le, even though they would obey the other conservation laws, are not observed to occur. In a decay involving muons, such as 7T+ -> fl+ + Vp, a second quantum number, muon lepton number ( L ll) , is conserved. The /jl and are = —1 , whereas all other assigned L l l = + 1 , and their antiparticles fi+and have = 0. particles have too is conserved in interactions and decays. Similar assign ments can be made for the tau lepton number, L T, associated with the r lepton (discovered in 1976 with mass more than 3000 times the electron mass) and its neutrino, vT. Keep in mind that antiparticles have not only opposite electric charge from their particles, but also opposite B, Le, L^, and Lr . For example, a neutron has B = +1, an antineutron has B = - 1 (and all the L 's are zero). ON EPTUAL EXAMPLE 4 5 - 5 I Lepton number in muon decay. Which of the following decay schemes is possible for muon decay: (a) / jl~ —> e" + ve; (b) /jl~ e _ + v&+ ; (c) /jl~ —» e _ + vQl All of these particles have L T = 0. = +1 and Le = 0. This is the initial state for all decays RESPONSE A fi has = +1, Le = 0. In (a), the final state given, and the final state must also have = 0 + 0 = 0, and Le = +1 - 1 = 0; LMwould not be conserved and has indeed this decay is not observed to occur. The final state of (b) has = 0 + 0 + 1 = +1 and Le = +1 - 1 + 0 = 0, so both L M and Le are conserved. This is in fact the most common decay mode of the Lastly, (c) does not occur because Le (= +2 in the final state) is not conserved, nor is L^. EXAMPLE 4 3 - 6 Energy and momentum are conserved. In addition to the "number" conservation laws which help explain the decay schemes of particles, we can also apply the laws of conservation of energy and momentum. The decay of a 2 + particle at rest with a mass of 1189 MeV/c 2 (Table 43-2 in Section 43-6) commonly yields a proton (mass = 938 MeV/c2) and a neutral pion, 7r° (mass = 135 MeV/c2): 2 + -► p + 77°. What are the kinetic energies of the decay products, assuming the 2 + parent particle was at rest? APPROA H We find the energy release from the change in mass (E = me2) as we did for nuclear processes (Eq. 41 -3 or 42 -2a), and apply conservation of energy and momentum. SOLUTION The energy released, or Q-value, is the change in mass times c2: Q = [™?+ ~ (w*p + rn^)]c2 = [1189 - (938 + 135)] MeV = 116MeV. This energy Q becomes the kinetic energy of the resulting decay particles, p and 77°: Q = Kp + K v 0 with each particle's kinetic energy related to its momentum by (Eqs. 36-11 and 13): Kp = Ep - mvc2 = V G v T " ~+Tm^c2f mpc2, and similarly for the pion. From momentum conservation, the proton and pion have the same magnitude of momentum since the original 2 + was at rest: pv = /V = p • Then, Q = K v + K,« gives 116 MeV = \ \ / ( p c f + (938 MeV )2 - 938 MeV] + W ( p c f + (135 MeV )2 - 135 MeV], We solve this for pc, which gives pc = 189 MeV. Substituting into the expression above for the kinetic energy, first for the proton, then for the pion, we obtain Kp = 19 MeV and Kvo = 97 MeV.

1176

HAPTER 43 Elementary Particles

43— 5 Neutrinos—Recent Results We first met neutrinos with regard to (3~ decay in Section 41-5. The study of neutrinos is a "hot" subject today. Experiments are being carried out in deep underground laboratories, sometimes in deep mine shafts. The thick layer of earth above is meant to filter out all other "background" particles, leaving mainly the very weakly interacting neutrinos to arrive at the detectors. Some very important results have come to the fore in recent years. First there was the solar neutrino problem. The energy output of the Sun is believed to be due to the nuclear fusion reactions discussed in hapter 42, Eqs. 42 -7 and 42-8. The neutrinos emitted in these reactions are all ve (accompanied by e+). But the rate at which ve arrive at Earth is measured to be much less than expected based on the power output of the Sun. It was then proposed that perhaps, on the long trip between Sun and Earth, ve might turn into or vT. Subsequent experiments confirmed this hypothesis.Thus the three neutrinos, ve,v jJi,v T, can change into one another in certain circumstances, a phenomenon called neutrino flavor oscillation (each of the three neutrino types being called, whimsically, a different "flavor"). This result suggests that the lepton numbers Le, , and L Tare not perfectly conserved. But the sum, Le + + L T, is believed to be always conserved. The second exceptional result has long been speculated on: are neutrinos massless as originally thought, or do they have a nonzero mass? Rough upper limits on the masses have been made. Today astrophysical experiments show that the sum of all three neutrino masses is less than about 0.14 eV/c2. But can the masses be zero? Not if there are the flavor oscillations discussed above. It seems likely that at least one neutrino type has a mass of at least 0.04 eV/c2. As a result of neutrino oscillations, the three types of neutrino may not be exactly what we thought they were (e, /jl, t ) . If not, the three basic neutrinos, called 1, 2, and 3, are linear combinations of and vT. Another outstanding question is whether or not neutrinos are in the category called Majorana particles/ meaning they would be their own antiparticles. If so, a lot of other questions (and answers) would appear.

*Neutrino Mass Estimate from a Supernova The supernova of 1987 offered an opportunity to estimate electron neutrino mass. If neutrinos do have mass, then v < c and neutrinos of different energy would take different times to travel the 170,000 light-years from the supernova to Earth. To get an idea of how such a measurement could be done, suppose two neutrinos from "SN1987a" were emitted at the same time and detected on Earth (via the reaction v&+ p —> n + e+) 10 seconds apart, with measured kinetic energies of about 20 MeV and 10 MeV. Since we expect the neutrino mass to be surely less than 100 eV (from other laboratory measurements), and since our neutrinos have kinetic energy of 20 MeV and 10 MeV, we can make the approximation mvc2 « E, so that E (the total energy) is essentially equal to the kinetic energy. We use Eq. 36-11, which tells us

We solve this for v, the velocity of a neutrino with energy E\

where we have used the binomial expansion (1 —x ¥ = 1 —\ x + •■*, and we ignore higher-order terms since ra2c4 « E 2. The time t for a neutrino to travel a distance d ( = 170,000 ly) is

where again we used the binomial expansion

[(1 - x) 1 = 1 + x +

trThe brilli nt young physicist Ettore M jor n (1906-1938) dis ppe red from circumst nces in 1938 t the ge of 31.

ship under mysterious

SE TION 4 3 - 5

1177

The difference in arrival times for our two neutrinos of energies E x = 20 MeV and E2 = 10 MeV is 11 —

d mA vc ~ c 2

1 E\

We solve this for m vc2 and set t2 — tx = 10 s:

m„c

E \E \ E\ - E \a

-[ 2(3.0 [(1(1.7 /7 = 22

10 m /s)(10s)

105 ly)(l.0

(400 MeV 2)(l00 MeV2) "Ii

1016m/ly) (400 MeV 2 - 100 MeV2)

]

K r 6MeV = 22 eV.

We thus estimate the mass of the neutrino to be 22 eV /c2, but there would of course be experimental uncertainties, not to mention the unwarranted assumption that the two neutrinos were emitted at the same time. Theoretical models of supernova explosions suggest that the neutrinos are emitted in a burst that lasts from a second or two up to perhaps 10 s. If we assume the neutrinos are not emitted simultaneously but rather at any time over a 10-s interval, what then could we say about the neutrino mass based on the data given above? The 10-s difference in their arrival times could be due to a 10-s difference in their emission time. In this case our data would be consistent with zero mass, and it puts an approximate upper limit on the neutrino mass of 22 eV /c2. The actual detection of these neutrinos was brilliant— it was a rare event that allowed us to detect something other than EM radiation from beyond the solar system, and was an exceptional confirmation of theory. In the experiments, the most sensitive detector consisted of several thousand tons of water in an under ground chamber. It detected 11 events in 12 seconds, probably via the reaction ve + p —> n + e+. There was not a clear correlation between energy and time of arrival. Nonetheless, a careful analysis of that experiment set a rough upper limit on the electron anti-neutrino mass of about 4 eV /c2. The more recent results mentioned above are much more definitive— they provide evidence that mass is much smaller, and that it is not zero.

4 3 - 6 P rticle C l ssific tion In the decades following the discovery of the meson in the late 1940s, hundreds of other subnuclear particles were discovered. One way of arranging the particles in categories is according to their interactions, since not all particles interact by means of all four of the forces known in nature (though all interact via gravity). Table 43 - 2 (next page) lists some of the more common particles classified in this way along with many of their properties. At the top of Table 43 - 2 are the so-called "fundamental" particles which we believe have no internal structure. Below them are some of the "composite" particles which are made up of quarks, according to the Standard Model. The fundamental particles include the gauge bosons (so -named after the theory that describes them, "gauge theory"), which include the gluons, the photon, and the W and Z particles; these are the particles that mediate the strong, electromagnetic, and weak interactions, respectively. Also fundamental are the leptons, which are particles that do not interact via the strong force but do interact via the weak nuclear force. Leptons that carry electric charge also interact via the electromagnetic force. The leptons include the electron, the muon, and the tau, and three types of neutrino: the electron neutrino (ve), the muon neutrino (i'J , and the tau neutrino (vT). Each has an antiparticle.

1178

HAPTER 43

Elementary Particles

TABLE 43 - 2 Particles (selected)^ C tegory

Forces involved

P rticle n me

Symbol

Antip rticle

Spin

M ss (M eV /c2)

Le Lp Lt S

[ ntip rticles h ve opposite sign]

Me n life (s)

Princip l Dec y Modes

Fund ment l G uge bosons str (torce em c rriers) w, em w Leptons

w, em*

Gluons Photon W Z

Self Self

W+ Z°

W "

Electron Neutrino (e) Muon Neutrino (/x) T u Neutrino (r)

Self

1 1 1 1

0 0 0 0

St ble St ble

80.40 X 103 91.19 X 103

0 0 0 0

0.511 0 ( K° + n, the initia l state has S = 0 and the final state has S ' = + l + 0 = + l , so strangeness would not be onserved; and this rea tion is not observed. To explain the de ay of strange parti les, it is assumed that strangeness is onserved in the strong intera tion but is not conserved in t e w eak interaction. Thus, strange parti les were forbidden by strangeness onservation to de ay to nonstrange parti les of lower mass via the strong intera tion, but ould de ay by means of the weak intera tion at the observed longer lifetimes of 10-10 to 10-8 s. The onservation of strangeness was the first example of a partially conserved quantity. In this ase, the quantity strangeness is onserved by strong intera tions but not by weak.

SE TION 43 -8

c a u t i o n

____________

Partially conserved quantities

Strange Particles? harm? Towards a New Model

1181

C NCEPTUAL EXAMPLE 45-8~| Guess the missing particle. Using the conser vation laws for parti le intera tions, determine the possibilities for the missing parti le in the rea tion 7T~ + p

K° + ?

in addition to K° + A0mentioned above.

RESP NSE We write equations for the onserved numbers in this rea tion, with B, L e, S, and Q as unknowns whose determination w ill reveal what the possible

parti le might be: Baryon number: 0 Lepton number: 0 Charge: -1 Strangeness: 0 The unknown produ t parti le B =

+ 1 = 0 + B + 0 = 0 + Le + 1 = 0 + Q + 0 = 1 + S. would have to have these hara teristi s:

Le = 0

Q = 0

S =

- 1.

In addition to A0, a neutral sigma parti le, 2°, is also onsistent with these numbers. In the next Se tion we w ill dis uss another partially onserved quantity whi h was given the name charm. The dis overy in 1974 of a parti le with harm helped solidify a new theory involving quarks, whi h we now dis uss.

4 3 -9 Qu rks A ll parti les, ex ept the gauge bosons (Se tion 43 -6), are either leptons or hadrons. One differen e between these two groups is that the hadrons intera t via the strong intera tion, whereas the leptons do not. There is another major differen e. The six leptons (e" , /jl~, t ~, v e, , vT) are onsidered to be truly fundamental parti les be ause they do not show any internal stru ture, and have no measurable size. (Attempts to determine the size of leptons have put an upper lim it of about 10-18 m.) On the other hand, there are hundreds of hadrons, and experiments indi ate they do have an internal stru ture. In 1963, M. Gell-Mann and G. Zweig proposed that none of the hadrons, not even the proton and neutron, are tru ly fundamental, but instead are made up of ombinations of three more fundamental pointlike entities alled (somewhat whimsi ally) quarks.1 Today, the quark theory is well-a epted, and quarks are onsidered truly fundamental parti les, like leptons. The three quarks originally proposed were labeled u, d, s, and have the names up, down, and strange. The theory today has six quarks, just as there are six leptons— based on a presumed symmetry in nature. The other three quarks are alled charmed, bottom, and top. The names apply also to new properties of ea h (quantum numbers c ,b ,t) that distinguish the new quarks from the old quarks (see Table 43 -3), and whi h (like strangeness) are onserved in strong, but not weak, intera tions. tGell-M nn chose the word from phr se in J mes Joyce's Finnegans Wake.

TABLE 43-3 Properties of Quarks

(Antiquarks have opposite sign Q, B, S, c, t, b)

Qu rks N me

Symbol

Up Down

1182

5 95

Str nge

Ch rmed Bottom

Top

HAPTER 43

M ss (MeV/c2)

1250 4200 173,000

Elementary Particles

Ch rge

B ryon Number

+ le ~ \e + ie +ie

l 3 1 3 1 3 1 3 1 3 1 3

Str ngeness Ch rm

Bottomness

Topness

0 0

0 0 0

0 0

+1 0

0 -1

0 0

-1

Mesons

B ryons

Anti p rticle

P rticle D+ D°

D" D°

J/iA (3097) Y (9460) B" B°

D5 Self Self B+ B°

Ac 2C ++

A-

2+ 2°

2 r 2" 2°

Spin 0 0 0

1869.4

1 1 0 0

3096.9 9460 5279 5279

2286 2454 2453 2454

2 1 2 1 2 1 2 1 2

1864.5 1968

Str ngeness S

Ch rm c

Bottomness: b

Me n life 00

0 0 0 0 0 0 0

0 0 +1 0 0 0 0

+1 +1

0 0 0 0 0 -1 -1

10.6 X 10"13 4.1 X 10"13

0 0 0 0

+1 +1 +1

0 0 0 0

-1

2.0 X 10-13 « 10"21 « 10"21 « 10"21 1.2 X 10"12

+1 +1 +1 +1

5620

II b-

C tegory

B ryon Number M ss B (MeV/c2)

II 4

TABLE 43-4 Partial List of Heavy Hadrons, with Charm and Bottomness (Le =

+1 0 0 0 0 +1

A ll quarks have spin \ and an ele tri harge of either + \ e or —\e (that is, a fra tion of the previously thought smallest harge e). Antiquarks have opposite sign of ele tri harge Q, baryon number B, strangeness S, harm , bottomness b, and topness t. Other properties of quarks are shown in Table 43-3. A ll hadrons are onsidered to be made up of ombinations of quarks (plus the gluons that hold them together), and their properties are des ribed by looking at their quark ontent. Mesons onsist of a quark -antiquark pair. For example, a 7r+ meson is a ud ombination: note that for the ud pair (Table 43-3), Q = \ e + \ e = + le , B = \ — \ = 0, 5 = 0 + 0 = 0, as they must for a 7r+; and a K + = us, with Q = +1, B = 0, S = +1. Baryons, on the other hand, onsist of three quarks. For example, a neutron is n = ddu, whereas an antiproton is p = u u d . See Fig. 43-15. Strange parti les all ontain an s or s quark, whereas harmed parti les ontain a or quark. A few of these hadrons are listed in Table 43-4. A fter the quark theory was proposed, physi ists began looking for these fra tionally harged parti les, but dire t dete tion has not been su essful. Current models suggest that quarks may be so tightly bound together that they may not ever exist singly in the free state. But observations of very high energy ele trons s attered off protons suggest that protons are indeed made up of onstituents. Today, the truly fundamental particles are onsidered to be the six quarks, the six leptons, and the gauge bosons that arry the fundamental for es. See Table 43 -5, where the quarks and leptons are arranged in three " families" or " generations." Ordinary matter— atoms made of protons, neutrons, and ele trons— is ontained in the " first generation." The others are thought to have existed in the very early universe, but are seen by us today only at powerful a elerators or in osmi rays. A ll of the hundreds of hadrons an be a ounted for by ombi nations of the six quarks and six antiquarks.

5.0 X 10"13 « io - 20 « io - 20 1.6 X 10"12 1.5 X 10"12

Princip l Dec y Modes K + others, e + others K + others, juor e + others K + others H drons, e+e_, /jl+/jT H drons, e+e", t+t" D° + others D° + others H drons (e.g., A + others) A+7T+ A a, then C < 2irr.)

FIGURE 44 —18 Example of a two-dimensional surfa e with negative urvature.

Another way to test the urvature of spa e is to measure the radius r and ir umferen e C of a large ir le. On a plane surfa e, C = 2irr. But on a two dimensional spheri al surfa e, C is less than 2irr, as an be seen in Fig. 44-17. The proportionality between C and r is less than 2ir. Su h a surfa e is said to have po sitive curvature. On the saddlelike surfa e o f Fig. 44-18, the ir umferen e o f a ir le is greater than 2irr, and the sum of the angles of a triangle is less than 180°. Su h a surfa e is said to have a negative curvature.

Curvatur of th Univ rs What about our universe? On a large s ale (not just near a large mass), what is the overall urvature of the universe? Does it have positive urvature, negative urva ture, or is it fla t (zero urvature)? We per eive our world as Eu lidean (flat), but we an not ex lude the possibility that spa e ould have a urvature so slight that we don't normally noti e it. This is a ru ial question in osmology, and it an be answered only by pre ise experimentation. If the universe had a positive urvature, the universe would be closed , or finite in volume. This would n ot mean that the stars and galaxies extended out to a ertain boundary, beyond whi h there is empty spa e. There is no boundary or edge in su h a universe. The universe is all there is. I f a parti le were to move in a straight line in a parti ular dire tion, it would eventually return to the starting point— perhaps eons of time later. On the other hand, if the urvature of spa e was zero or negative, the universe would be open. It ould just go on forever. A n open universe ould be infinite', but a ording to re ent resear h, even that may not ne essarily be so. Today the eviden e is very strong that the universe on a large s ale is very lose to being flat. Indeed, it is so lose to being fla t that we an't te ll if it m ight have very slightly positive or very slightly negative urvature.

Black Hol s Rubber-sheet analogy for spa e-time urved by matter. FIGURE 44 - 19

Weight

1208

HAPTER 44

A ording to Einstein's theory, spa e-time is urved near massive obje ts. We m ight think of spa e as being like a thin rubber sheet: if a heavy weight is hung from it, it urves as shown in Fig. 44-19. The weight orresponds to a huge mass that auses spa e (spa e itself!) to urve. Thus, in Einstein's theory1 we do not speak of the " for e" of gravity a ting on obje ts. Instead we say that obje ts and light rays move as they do be ause spa e-time is urved. A n obje t starting at rest or moving slowly near the great mass of Fig. 44-19 would follow a geodesi (the equivalent of a straight line in plane geometry) toward that great mass. Alex nder Pope (1688-1744) wrote n epit ph for Newton: "N ture, nd N ture's l ws l y hid in night: God s id, Let Newton be! nd ll w s light." Sir John Squire (1884-1958), perh ps uncomfort ble with Einstein's profound thoughts, dded: "It did not l st: the Devil howling 'Ho! Let Einstein be/ ' restored the st tus quo."

The extreme urvature of spa e-time shown in Fig. 44-19 ould be produ ed by a black hole. A bla k hole, as we mentioned in Se tion 44-2, is so dense that even light annot es ape from it. To be ome a bla k hole, an obje t of mass M must undergo gravitational collapse, ontra ting by gravitational self-attra tion to within a radius alled the Schwarzschild radius: _ 2G M where G is the gravitational onstant and c the speed of light. If an obje t ollapses to within this radius, it is predi ted by general relativity to rapidly (« IO-5 s) ollapse to a point at r = 0, forming an infinitely dense singularity. This predi tion is un ertain, however, be ause in this realm we need to ombine quantum me hani s with gravity, a unifi ation of theories not yet a hieved (Se tion 43-12). | EXERCISE C Wh t is the Schw rzschild r dius for n object with 2 sol r m sses?

The S hwarzs hild radius also represents the event horizon of a bla k hole. By an ever rea h us, and thus inform us of events that happen beyond that surfa e. As a star ollapses toward a bla k hole, the light it emits is pulled harder and harder by gravity, but we an still see it. On e the matter passes within the event horizon, the emitted light annot es ape but is pulled ba k in by gravity. A ll we an know about a bla k hole is its mass, its angular momentum (there ould be rotating bla k holes), and its ele tri harge. No other information, no details of its stru ture or the kind of matter it was formed of, an be known be ause no information an es ape. How might we observe bla k holes? We annot see them be ause no light an es ape from them. They would be bla k obje ts against a bla k sky. But they do exert a gravitational for e on nearby obje ts. The bla k hole believed to be at the enter of our Galaxy (M ~ 2 X 106MSun) was dis overed by examining the motion of matter in its vi inity. Another te hnique is to examine stars whi h appear to move as if they were one member of a binary system (two stars rotating about their ommon enter of mass), but without a visible ompanion. If the unseen star is a bla k hole, it might be expe ted to pull o ff gaseous material from its visible ompanion (as in Fig. 44-10). As this matter approa hed the bla k hole, it would be highly a eler ated and should emit X-rays of a hara teristi type before plunging inside the event horizon. Su h X-rays, plus a suffi iently high mass estimate from the rotational motion, an provide eviden e for a bla k hole. One of the many andidates for a bla k hole is in the binary-star system Cygnus X - l. It is widely believed that the enter of most galaxies is o upied by a bla k hole with a mass 106 to 109times the mass of a typi al star like our Sun. event horizon we mean the surfa e beyond whi h no emitted signals

EXERCISE D A bl ck hole h s r dius R. Its m ss is proportion l to (a) R, (b) R2, (c) R3. Justify your nswer.

4 4 —5 The Exp nding Universe: Redshift nd Hubble's L w We dis ussed in Se tion 44 -2 how individual stars evolve from their birth to their death as white dwarfs, neutron stars, and bla k holes. But what about the universe as a whole: is it stati , or does it hange? One of the most important s ientifi dis overies of the twentieth entury was that distant galaxies are ra ing away from us, and that the farther they are from us, the faster they are moving away. How astronomers arrived at this astonishing idea, and what it means for the past history of the universe as well as its future, w ill o upy us for the remainder of the book.

SE TION 44 -5

The Expanding Universe: Redshift and Hubble's Law

1209

Low redshift g l xy spectrum z = 0.004

FIGURE 44-20 A tom s nd molecules emit nd bsorb light of p rticul r frequencies depending on the sp cing of their energy levels, s we s w in Ch pters 37 to 40. ( ) The spectrum of light received from rel tively slow-moving g l xy. (b) Spectrum of g l xy moving w y from us t much higher speed. Note how the pe ks (or lines) in the spectrum h ve moved to longer w velengths. The redshift is

500 ( )

600 W velength (nm)

700

Higher redshift g l xy spectrum z = 0.104

Z = (^-obs — ^rest)/^-rest •

500 (b)

600 W velength (nm)

700

That the universe is expanding was first put forth by Edwin Hubble in 1929. This idea was based on distan e measurements of galaxies (Se tion 44 -3), and determination of their velo ities by the Doppler shift of spe tral lines in the light re eived from them (Fig. 44-20). In Chapter 16 we saw how the frequen y of sound is higher and the wavelength shorter if the sour e and observer move toward ea h other. If the sour e moves away from the observer, the frequen y is lower and the wavelength longer. The Doppler effect o urs also for light, and we saw in Se tion 36-12 (Eq. 36-15) that a ording to spe ial relativity, the Doppler shift is given by (44-3)

where Arest is the emitted wavelength as seen in a referen e frame at rest with respe t to the sour e, and Aobs is the wavelength observed in a frame moving with velo ity v away from the sour e along the line of sight. (For relative motion tow ard ea h other, v < 0 in this formula.) When a distant sour e emits light of a parti ular wavelength, and the sour e is moving away from us, the wavelength appears longer to us: the olor of the light (if it is visible) is shifted toward the red end of the visible spe trum, an effe t known as a redshift. ( If the sour e moves toward us, the olor shifts toward the blue or shorter wavelength.) In the spe tra of stars in other galaxies, lines are observed that orrespond to lines in the known spe tra of parti ular atoms (see Se tion 37-10 and Figs. 35-22 and 37-20). What Hubble found was that the lines seen in the spe tra from distant galaxies were generally reds ifted, and that the amount of shift seemed to be approx imately proportional to the distan e of the galaxy from us. That is, the velo ity v of a galaxy moving away from us is proportional to its distan e d from us: H U BBLE 'S LAW

v = Hd.

(44-4)

This is Hubble 's law, one of the most fundamental astronomi al ideas. The onstant H is alled the Hubble parameter. The value of H until re ently was un ertain by over 20%, and thought to be between 50 and 80km /s/M p . But re ent measurements now put its value more pre isely at H = 71 km /s/M p (that is, 71 km /s per megaparse of distan e). The urrent un ertainty is about 5%, or + 4 km /s/M p . If we use light-years for distan e, then H = 22 km /s per m illion light-years of distan e: H = 22 km /s/M ly

1210

HAPTER 44

with an estimated uncertainty of +1 km/s/Mly.

R dshift Origins Galaxies very near us seem to be moving randomly relative to us: some move towards us (blueshifted), others away from us (redshifted); their speeds are on the order of 0.001 . But for more distant galaxies, the velo ity of re ession is mu h greater than the velo ity of lo al random motion, and so is dominant and Hubble's law (Eq. 44 -4) holds very well. More distant galaxies have higher re ession velo ity and a larger redshift, and we all their redshift a cosmologic l redshift. We interpret this redshift today as due to the expansion o f space itself. We an think of the originally emitted wavelength Arest as being stret hed out (be oming longer) along with the expanding spa e around it, as suggested in Fig. 44-21. Although Hubble thought of the redshift as a Doppler shift, now we understand it in this sense of expanding spa e. Contrast the osmologi al redshift, due to the expansion of spa e itself, with an ordinary D oppler reds ift whi h is due to the relative motion of emitter and observer in a spa e that an be onsidered fixed over the time interval of observation. There is a third way to produ e a redshift, whi h we mention for ompleteness: a gr vit tion l redshift. Light leaving a massive star is gaining in gravitational potential energy (just like a stone thrown upward from Earth). So the kineti energy of ea h photon, f, must be getting smaller (to onserve energy). A smaller frequen y /means a larger (longer) wavelength A { = c / f ) , whi h is a redshift. The amount of a redshift is spe ified by the redshift p r meter, z, defined as ^•obs — ^-rest

z = ----- ---------- = - —

' (44 - 5 ) ^rest ^rest where Arest is a wavelength as seen by an observer at rest relative to the sour e, and Aobs is the wavelength measured by a moving observer. Equation 44-5a an also be written as ^obs i (44 - 5b) z = --------- 1 Arest

z + 1 =

(44 - 5c)

For low speeds not lose to the speed of light (v ^ 0.1 ), the Doppler formula (Eq. 44-3) an be used to show (Problem 29) that z is proportional to the speed of the sour e toward or away from us:

FIGURE 44-21 Simplified model of 2-dimension l universe, im gined s b lloon. As you blow up the b lloon ( = exp nding universe), the w velength of w ve on its surf ce gets longer.

FIGURE 44 -22 Hubble Ultr Deep Field photogr ph showing wh t m y be mong the most dist nt g l xies from us (sm ll red dots, indic ted by green squ res), with z ~ 5 or 6, existing when the universe w s only bout 800 million ye rs old. The two dist nt g l xies in this photo re shown enl rged below.

^•obs — ^rest V r a\ « — [v « ] (44 - 6) ^rest But redshifts are not always small, in whi h ase the approximation of Eq. 44 -6 is not valid. Modern teles opes regularly observe galaxies with z ~ 5 (Fig. 44-22); for large z galaxies, not even Eq. 44-3 applies be ause the redshift is due to the expansion of spa e ( osmologi al redshift), not the Doppler effe t. z =

Scal Factor The expansion of spa e an be des ribed as a simple s aling of the typi al distan e between two points or obje ts in the universe. If two distant galaxies are a distan e d0 apart at some initial time, then a time t later they w ill be separated by a greater distan e d (t). The sc le f ctor is the same as for light, expressed in Eq. 44-5a. That is, d { t ) — d0

= z

or m d0

= 1 + z.

Thus, for example, if a galaxy has z = 3, then the s ale fa tor is now (1 + 3) = 4 times larger than when the light was emitted from that galaxy. That is, the average distan e between galaxies has be ome 4 times larger. Thus the fa tor by whi h the wavelength has in reased sin e it was emitted tells us by what fa tor the universe (or the typi al distan e between obje ts) has in reased in size.

SE TION 44 -5

The Expanding Universe: Redshift and Hubble's Law

1211

FIGURE 44 -23 Exp nsion of the universe looks the s me from ny point in the universe. If you re on E rth s shown in p rt ( ) or you re inste d t point A (which is t rest in the reference fr me shown in (b)), ll other g l xies ppe r to be r cing w y from you.

Expansion, and th Cosmological Principl What does it mean that distant galaxies are all moving away from us, and with ever greater speed the farther they are from us? It seems to suggest some kind of explo sive expansion that started at some very distant time in the past. And at first sight we seem to be in the middle of it all. But we aren't. The expansion appears the same from any other point in the universe. To understand why, see Fig. 44-23. In Fig. 44-23a we have the view from Earth (or from our Galaxy). The velo ities of surrounding galaxies are indi ated by arrows, pointing away from us, and the arrows are longer for galaxies more distant from us. Now, what if we were on the galaxy labeled A in Fig. 44-23a? From Earth, galaxy A appears to be moving to the right at a velo ity, all it vA, represented by the arrow pointing to the right. If we were on galaxy A , Earth would appear to be moving to the left at velo ity — vA. To determine the velo ities of other galaxies relative to A , we ve torially add the velo ity ve tor, - v A, to all the velo ity arrows shown in Fig. 44-23a. This yields Fig. 44-23b, where we see learly that the universe is expanding away from galaxy A as well; and the velo ities of galaxies re eding from A are proportional to their urrent distan e from A. The universe looks pretty mu h the same from different points. Thus the expansion of the universe an be stated as follows: all galaxies are ra ing away from eac ot er at an average rate of about 22 km /s per m illion light-years of distan e between them. The ramifi ations of this idea are profound, and we dis uss them in a moment. A basi assumption in osmology has been that on a large s ale, the universe would look the same to observers at different pla es at the same time. In other words, the universe is both isotropic (looks the same in all dire tions) and omogeneous (would look the same if we were lo ated elsewhere, say in another galaxy). This assumption is alled the cosmological principle. On a lo al s ale, say in our solar system or within our Galaxy, it learly does not apply (the sky looks different in different dire tions). But it has long been thought to be valid if we look on a large enough s ale, so that the average population density of galaxies and lusters of galaxies ought to be the same in different areas of the sky. This seems to be valid on distan es greater than about 200 Mp (700 M ly). The expan sion of the universe (Fig. 44-23) is onsistent with the osmologi al prin iple; and the near uniform ity of the osmi mi rowave ba kground radiation (dis ussed in Se tion 44 -6) supports it. Another way to state the osmologi al prin iple is that our place in t e universe is not special.

The expansion of the universe, as des ribed by Hubble's law, strongly suggests that galaxies must have been loser together in the past than they are now. This is, in fa t, the basis of the Big Bang theory of the origin of the universe, whi h pi tures the universe as a relentless expansion starting from a very hot and ompressed beginning. We dis uss the Big Bang in detail shortly, but first let us see what an be said about the age of the universe.

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Astrophysics and osmology

One way to estimate the age of the universe uses the Hubble parameter. W ith H « 22 km /s per 106 light-years, the time required for the galaxies to arrive at their present separations would be approximately (starting with v = d / t and using Hubble's law, Eq. 44 -4), f =

d = A . = JL v

(I0 6ly)(0.95 x 10 n km /ly) ^ (22km/s)(3.16 X 107s/yr)

or 14 billion years. The age of the universe

al ulated in this way is alled the

c aracteristic expansion time or " Hubble age." It is a very rough estimate and assumes

the rate of expansion of the universe was onstant (whi h today we are quite sure is not true). Today's best measurements give the age of the universe as 13.7 X 109yr, in remarkable agreement w ith the rough Hubble age estimate.

* St ady-Stat Mod l Before dis ussing the Big Bang in detail, we mention one alternative to the Big Bang— the steady-state model— whi h assumed that the universe is infinitely old and on average looks the same now as it always has. (This assumed uniform ity in time as well as spa e was alled the perfect cosm ological principle.) A ording to the steady-state model, no large-s ale hanges have taken pla e in the universe as a whole, parti ularly no Big Bang. To maintain this view in the fa e of the re ession of galaxies away from ea h other, matter must be reated ontinuously to maintain the assumption of uniform ity. The rate of mass reation required is very small— about one nu leon per ubi meter every 109years. The steady-state model provided the Big Bang model with healthy ompetition in the mid-twentieth entury. But the dis overy of the osmi mi rowave ba kground radiation (next Se tion), as well as other observations of the universe, has made the Big Bang model universally a epted.

4 4 —( The Big B ng nd the Cosmic Microw ve B ckground The expansion of the universe suggests that typi al obje ts in the universe were on e mu h loser together than they are now. This is the basis for the idea that the universe began about 14 billion years ago as an expansion from a state of very high density and temperature known affe tionately as the Big Bang. The birth of the universe was not an explosion, be ause an explosion blows pie es out into the surrounding spa e. Instead, the Big Bang was the start of an expansion of spa e itself. The observable universe was very small at the start and has been expanding ever sin e. The in itia l tiny universe of extremely dense matter is not to be thought of as a on entrated mass in the midst of a mu h larger spa e around it. The in itia l tiny but dense universe was the entire universe. There wouldn't have been anything else. When we say that the universe was on e smaller than it is now, we mean that the average separation between obje ts (su h as galaxies) was less. It is thought the universe was infinite in extent then, and it still is (only bigger). The observable universe, however, is finite. A major pie e of eviden e supporting the Big Bang is the cosmic microwave background radiation (or CMB) whose dis overy ame about as follows. In 1964, Arno Penzias and Robert Wilson pointed their radioteles ope (a large antenna devi e for dete ting radio waves) into the night sky (Fig. 44-24). W ith it they dete ted widespread emission, and be ame onvin ed that it was oming from outside our Galaxy. They made pre ise measurements at a wave length A = 7.35 m, in the mi rowave region of the ele tromagneti spe trum (Fig. 31 -12). The intensity of this radiation was found in itia lly not to vary by day or night or time of year, nor to depend on dire tion. It ame from all dire tions in the universe with equal intensity, to a pre ision of better than 1%. It ould only be on luded that this radiation ame from the universe as a whole.

SE TION 44 - 6

FIGURE 44 - 24 Arno Penzi s (left) nd Robert Wilson, nd behind them their "horn ntenn ."

The Big Bang and the osmic Microwave Background

1213

W velength (cm)

FIGURE 44 -25 Spectrum of cosmic microw ve b ckground r di tion, showing bl ckbody curve nd experiment l me surements including t the frequency detected by Penzi s nd Wilson. (Th nks to G. F. Smoot nd D. Scott. The vertic l b rs represent the most recent experiment l uncert inty in me surement.)

1.0

0.1

| J

100

1000

Frequency (GHz) FIGURE 44 - 26 COBE scientists John M ther (left, chief scientist nd responsible for me suring the bl ckbody form of the spectrum) nd George Smoot (chief investig tor for nisotropy experiment) shown here during celebr tions for their Dec. 2006 Nobel Prize, given for their discovery of the spectrum nd nisotropy of the CMB using the COBE instrument.

They measured this osmi mi rowave ba kground radiation at A = 7.35 m, and its intensity orresponds to bla kbody radiation (see Se tion 37 -1) at a temperature of about 3 K. When radiation at other wavelengths was measured by the COBE satellite (COsmi Ba kground Explorer), the intensities were found to fall on a nearly perfe t bla kbody urve as shown in Fig. 44-25, orresponding to a temperature of 2.725 K (+ 0.002 K ). The remarkable uniform ity of the osmi mi rowave ba kground radiation was in a ordan e w ith the osmologi al prin iple. But theorists fe lt that there needed to be some small inhomogeneities, or " anisotropies," in the CMB that would have provided " seeds" around whi h galaxy form ation ould have started. Small areas of slightly higher density, whi h ould have ontra ted under gravity to form stars and galaxies, were indeed found. These tiny inhomo geneities in density and temperature were dete ted first by the COBE satellite experiment in 1992, led by John M ather and George Smoot (Fig. 44 -26). This dis overy of the anisotropy of the CMB ranks with the dis overy of the CMB itself in the history of osmology. It was the ulmination of de ades of resear h by pioneers su h as Paul Ri hards and David Wilkinson. Subsequent experiments with greater detail ulminated in 2003 with the WMAP (Wilkinson Mi rowave Anisotropy Probe) results. See Fig. 44-27 whi h presents the latest (2006) results. The CMB provides strong eviden e in support of the Big Bang, and gives us information about onditions in the very early universe. In fa t, in the late 1940s, George Gamow and his ollaborators al ulated that a Big Bang origin of the universe should have generated just su h a mi rowave ba kground radiation. To understand why, let us look at what a Big Bang might have been like. (Today we usually use the term " Big Bang" to refer to the process , starting from the birth of the universe through the subsequent expansion.) The temperature must have been extremely high at the start, so high that there ould not have been any atoms in the very early stages of the universe. Instead, the universe

FIGURE 44 -27 Me surements of the cosmic microw ve b ckground r di tion over the entire sky, color-coded to represent differences in temper ture from the ver ge 2.725 K: the color sc le r nges from +200 /x,K (red) to - 2 0 0 fiK (d rk blue), representing slightly hotter nd colder spots ( ssoci ted with v ri tions in density). Results re from the WMAP s tellite in 2006: the ngul r resolution is 0.2°. The white lines re dded to show the me sured pol riz tion direction of the e rliest light, which gives further clues to the e rly universe.

1214

HAPTER 44

Astrophysics and osmology

would have onsisted solely of radiation (photons) and a plasma of harged ele trons and other elementary parti les. The universe would have been opaque— the photons in a sense " trapped," traveling very short distan es before being s attered again, prim arily by ele trons. Indeed, the details of the mi rowave ba kground radiation is strong eviden e that matter and radiation were on e in equilibrium at a very high temperature. As the universe expanded, the energy spread out over an in reasingly larger volume and the temperature dropped. Only when the temperature had fallen to about 3000 K, some 380,000years later, ould nu lei and ele trons ombine together as atoms. W ith the disappearan e of free ele trons, as they ombined with nu lei to form atoms, the radiation would have been freed— decoupled from matter, we say. The universe be ame transparent be ause photons were now free to travel nearly unimpeded straight through the universe. It is this radiation, from 380,000 years after the birth of the universe, that we now see as the CMB. As the universe expanded, so too the wavelengths of the radiation lengthened, thus redshifting to longer wavelengths that orrespond to lower temperature (re all Wien's law, APT = onstant, Se tion 37 -1), until they would have rea hed the 2.7-K ba kground radiation we observe today.

Looking Back toward th Big Bang—Lookback Tim Figure 44-28 shows our Earth point of view, looking out in all dire tions ba k toward the Big Bang and the brief (380,000-year-long) period when radiation was trapped in the early plasma (yellow band). The time it takes light to rea h us from an event is alled its lookback time. The " lose-up" insert in Fig. 44-28 shows a photon s attering repeatedly inside that early plasma and then exiting the plasma in a straight line. No matter what dire tion we look, our view of the very early universe is blo ked by this wall of plasma. It is like trying to look into a very thi k fog or into the surfa e of the Sun— we an see only as far as its surfa e, alled the surface of last scattering, but not into it. Wavelengths from there are redshifted by z ~ 1100. Time A t ' in Fig. 44-28 is the lookba k time (not real time that goes forward). Re all that when we view an obje t far away, we are seeing it as it was then, when the light was emitted, not as it would appear today. FIGURE 44 - 28 When we look out from the E rth, we look b ck in time. Any other observer in the universe would see more or less the s me thing. The f rther n object is from us, the longer go the light we see h d to h ve left it. We c nnot see quite s f r s the Big B ng; we c n see only s f r s the "surf ce of l st sc ttering," which r di ted the CMB. The insert on the lower right shows the e rliest 380,000 ye rs of the universe when it w s op que: photon is shown sc ttering m ny times nd then ( t decoupling, 380,000 yr fter the birth of the universe) becoming free to tr vel in str ight line. If this photon w sn 't he ding our w y when "liber ted," m ny others were. G l xies re not shown, but would be concentr ted close to E rth in this di gr m bec use they were cre ted rel tively recently. Note: This di gr m is not norm l m p. M ps show section of the world s might be seen ll at a given time. This di gr m shows sp ce (like m p), but e ch point is not t the s me time. The light coming from point dist nce r from E rth took time At' = r/c to re ch E rth, nd thus shows n event th t took pl ce long go, time A£' = r/c in the p st, which we c ll its "lookb ck time." The universe beg n Atb = 13.7 Gyr go.

Th Obs rvabl Univ rs Figure 44-28 is a b it dangerous: it is not a pi ture of the universe at a given instant, but is intended to suggest how we look out in all dire tions from our observation point (the Earth, or near it). Be areful not to think that the birth of the universe took pla e in a ir le or a sphere surrounding us as if Fig. 44-28 were a photo taken at a given moment. What Fig. 44-28 does show is what we an see, the observable universe. Better yet, it shows the m ost we ould see.

SECTI N 4 4 - 6

1215

FIGURE 44 -29 Two observers, on widely sep r ted g l xies, h ve different horizons, different observ ble universes.

We would undoubtedly be arrogant to think that we ould see the entire universe. Indeed, theories assume that we annot see everything, that the entire universe is greater than the observ ble universe, whi h is a sphere of radius r0 = ct0 entered on the observer, with t0 being the age of the universe. We an never see further ba k than the time it takes light to rea h us. Consider, for example, an observer in another galaxy, very far from us, lo ated to the left of our observation point in Fig. 44-28. That observer would not yet have seen light oming from the far right of the large ir le in Fig. 44-28 that we see— it w ill take some time for that light to rea h her. But she w ill have already, some time ago, seen the light oming from the left that we are seeing now. In fa t, her observable universe, superimposed on ours, is suggested by Fig. 44-29. The edge of our observable universe is alled the horizon. We ould, in prin iple, see as far as the horizon, but not beyond it. An observer in another galaxy, far from us, w ill have a different horizon.

4 4 —7 The St nd rd Cosmologic l Model: E rly History of the Universe In the last de ade or two, a onvin ing theory of the origin and evolution of the universe has been developed, now alled the St nd rdCosmologic l Model, or (sometimes) the concordance model. Part of this theory is based on re ent theoreti al and experimental advan es in elementary parti le physi s, and part from observations of the universe in luding COBE and WMAP. Indeed, osmology and elementary parti le physi s have ross-fertilized to a surprising extent. Let us go ba k to the earliest of times— as lose as possible to the Big Bang— and follow a Standard Model s enario of events as the universe expanded and ooled after the Big Bang. In itia lly we talk of extremely small time intervals as well as extremely high temperatures, far beyond anything in the universe today. Figure 44-30 is a ompressed graphi al representation of the events, and it may be helpful to onsult it as we go along. FIGURE 44 -30 Compressed gr phic l represent tion of the development of the universe fter the Big B ng, ccording to modern cosmology. [The time sc le is mostly log rithmic (e ch f ctor of 10 in time gets equ l tre tment), except t the st rt (there c n be no t = 0 on log sc le), nd just fter t = 10 -35 s (to s ve sp ce). The vertic l height is rough indic tion of the size of the universe, m inly to suggest exp nsion of the universe

Decoupling

3000k | ^ ' ^ —

St rs nd I g l xies

R di tion er Infl tion

Universe Universe op que

Pl nck Birth of universe

H dron er

1032K

Dark^T

GUT 10 43s (Pl nck time)

INucleosynthesis Is 10s 102s 103s Time

energy | 380,000 yr

1010 yr [Now]

Th History

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CHAPTER 44

We begin at a time only a minus ule fra tion of a se ond after the b irth of the universe, 10-43 s. This time is sometimes referred to as the Pl nck time, whi h is a value determined by the fundamental onstants. It is related to the Planck lengt AP whi h we obtained in Chapter 1 (Example 1 -10) by dimensional analysis: AP = \ j G j ? ~ 4 X 10" 35m. The Plan k time is the time it takes light to travel the Plan k length: tF = AP/ c « (4 X 10" 35m )/(3 X 108m /s) « 10_43s. This is an unimag inably short time, and predi tions an be only spe ulative. Earlier, we an say nothing be ause we do not have a theory of quantum gravity whi h would be needed for the in redibly high densities and temperatures during this " Plan k era." It is thought that,

perhaps as early as IO-43 s, the four for es in nature were unified— there was only one for e (Chapter 43, Rg. 43-19). The temperature would have been about 1032K, orresponding to randomly moving parti les with an average kineti energy K of 1019GeV (see Eq. 18-4): K

(1.4 X IO-23J /K )(l0 32K) « k T « ------------------ 7^7 -------- - « 10 eV = 10 GeV. 1.6 X 10 J/eV

(Note that the fa tor § in Eq. 18-4 is usually ignored in su h order of magnitude al ulations.) A t t = 10_43s, a kind of " phase transition" is believed to have o urred during whi h the gravitational for e, in effe t, " ondensed out" as a separate for e. This, and subsequent phase transitions, are analogous to the phase transitions water undergoes as it ools from a gas, ondenses into a liquid, and with further ooling freezes into i e.* The symmetry of the four for es was broken, but the strong, weak, and ele tromagneti for es were still unified, and the universe entered the grand unified era (G UT — see Chapter 43). There was no distin tion between quarks and leptons; baryon and lepton numbers were not onserved. Very shortly thereafter, as the universe expanded onsiderably and the temperature had dropped to about 1027K, there was another phase transition and the strong for e ondensed out at about 10" 35s after the Big Bang. Now the universe was filled with a " soup" of leptons and quarks. The quarks were initially free, but soon began to " ondense" into more normal parti les: nu leons and the other hadrons and their antiparti les. W ith this confinement of quarks, the universe entered the hadron era. About this time, when the universe was only 10"35s old, a strange thing may have happened, a ording to theorists. A brilliant idea, proposed around 1980, suggests that the universe underwent an in redible exponential expansion, in reasing in size by a fa tor of IO40or maybe mu h more, in a tiny fra tion of a se ond, perhaps 10-35 s. The usefulness of this inflationary scenario is that it solved major problems with earlier Big Bang models, su h as explaining why the universe is flat, as well as the thermal equilibrium to provide the nearly uniform CMB, as dis ussed below. A fter the very brief inflationary period, the universe would have settled ba k into its more regular expansion. The universe was now a " soup" of leptons and hadrons. We an think of this " soup" as a plasma of parti les and antiparti les, as well as photons— all in roughly equal numbers— olliding with one another frequently and ex hanging energy. By the time the universe was only about a mi rose ond (l0 _6s) old, it had ooled to about 1013K, orresponding to an average kineti energy of 1 GeV, and the vast m ajority of hadrons disappeared. To see why, let us fo us on the most fam iliar hadrons: nu leons and their antiparti les. When the average kineti energy of parti les was somewhat higher than 1 GeV, protons, neutrons, and their antiparti les were ontinually being reated out of the energies of ollisions involving photons and other parti les, su h as photons — > p + p — > n + n. But just as qui kly, parti les and antiparti les would annihilate: for example p + p — » photons or leptons. So the pro esses of reation and annihilation of nu leons were in equilibrium. The numbers of nu leons and antinu leons were high— roughly as many as there were ele trons, positrons, or photons. But as the universe expanded and ooled, and the average kineti energy of parti les dropped below about 1 GeV, whi h is the minimum energy needed in a typi al ollision to reate nu leons and antinu leons (about 940 MeV ea h), the pro ess of nu leon reation ould not ontinue. The pro ess of annihilation ould ontinue, however, with antinu leons annihilating nu leons, until there were almost no nu leons left. But not quite zero. Somehow we need to explain our present world of matter (nu leons and ele trons) with very little antimatter in sight. fIt m y be interesting to note th t our story of origins here be rs some resembl nce to ncient ccounts th t mention the "void," "formless w stel nd" (or "d rkness over the deep"), " byss," "divide the w ters" (possibly ph se tr nsition?), not to mention the sudden ppe r nce of light.

SE TION 44 -7

1217

To explain our world of matter, we might suppose that earlier in the universe, perhaps around IO-35 s after the Big Bang, a slight ex ess of quarks over antiquarks was formed.1 This would have resulted in a slight ex ess of nu leons over antinu leons. And it is these " leftover" nu leons that we are made of today. The ex ess of nu leons over anti nu leons was probably about one part in 109. During the hadron era,there should have been about as many nu leons as photons. A fter it ended, the " leftover" nu leons thus numbered only about one nu leon per 109photons, and this ratio has persisted to this day. Protons, neutrons, and all other heavier parti les were thus tremendously redu ed in number by about IO-6 s after the Big Bang. The lightest hadrons, the pions, soon disappeared, about IO-4 s after the Big Bang; be ause they are the lightest mass hadrons (140 MeV), they were the last hadrons to be able to be reated as the temperature (and average kineti energy) dropped. Lighter parti les, in luding ele trons and neutrinos, were the dominant form of matter, and the universe entered the lepton era. By the time the first fu ll se ond had passed ( learly the most eventful se ond in history!), the universe had ooled to about 10 billion degrees, 1010K. The average kineti energy was about 1MeV. This was still suffi ient energy to reate ele trons and positrons and balan e their annihilation rea tions, sin e their masses orrespond to about 0.5 M eV So there were about as many e+ and e" as there were photons. But within a few more se onds, the temperature had dropped suffi iently so that e+ and e" ould no longer be formed. Annihilation (e+ + e_ —> photons) ontinued. And, like nu leons before them, ele trons and positrons all but disappeared from the universe— ex ept for a slight ex ess of ele trons over positrons (later to join with nu lei to form atoms). Thus, about t = 10 s after the Big Bang, the universe entered the radiation era (Fig. 44-30). Its major onstituents were photons and neutrinos. But the neutrinos, partaking only in the weak for e, rarely intera ted. So the universe, until then experien ing signifi ant amounts of energy in matter and in radiation, now be ame radiation-dominated: mu h more energy was ontained in radiation than in matter, a situation that would last more than 50,000 years. FIGURE 44 -30 (Repe ted.) Compressed gr phic l represent tion of the development of the universe fter the Big B ng, ccording to modern cosmology. Decoupling 3000K ^ St rs nd g l xies R di tion er Infl tion Universe op que

Pl nck Birth of

Universe I tr nsp rent11

H dron er

1032K GUT 10-43 s (Pl nck time)

D rk " energy

jNucleosynthesis 10- 6 s 10- 4 s

Is 10sl02sl03s

380,000 yr

Time

1010 yr [Now]

Meanwhile, during the next few minutes, ru ial events were taking pla e. Beginning about 2 or 3 minutes after the Big Bang, nu lear fusion began to o ur. The temperature had dropped to about 109K, orresponding to an average kineti energy K « 100 keV, where nu leons ould strike ea h other and be able to fuse (Se tion 42 -4), but now ool enough so newly formed nu lei would not be imme diately broken apart by subsequent ollisions. Deuterium, helium, and very tiny amounts of lithium nu lei were made. But the universe was ooling too qui kly, and larger nu lei were not made. A fte r only a few minutes, probably not even a quarter of an hour after the Big Bang, the temperature dropped far enough that nu leosynthesis stopped, not to start again for millions of years (in stars). fWhy this could h ve h ppened is

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CHAPTER 44

Astrophysics and Cosmology

question for which we re seeking n nswer tod y.

Thus, after the first quarter hour or so of the universe, matter onsisted mainly of bare nu lei of hydrogen (about 75%) and helium (about 25%)t and ele trons. But radiation (photons) ontinued to dominate. Our story is almost omplete. The next important event is thought to have o urred 380,000 years later. The universe had expanded to about ^ ° f its present s ale, and the temperature had ooled to about 3000 K. The average kineti energy of nu lei, ele trons, and photons was less than an ele tron volt. Sin e ionization ener gies of atoms are on the order of eV, then as the temperature dropped below this point, ele trons ould orbit the bare nu lei and remain there (without being eje ted by ollisions), thus forming atoms. This period is often alled the recombination epo h (a misnomer sin e ele trons had never before been ombined with nu lei to form atoms). W ith the disappearan e of free ele trons and the birth of atoms, the photons— whi h had been ontinually s attering from the free ele trons— now be ame free to spread throughout the universe. As mentioned in the previous Se tion, we say that the photons be ame decoupled from matter. Thus decoupling o urred at recombination. The total energy ontained in radiation had been de reasing (lengthening in wavelength as the universe expanded); and at about t = 56,000 yr (even before de oupling) the total energy ontained in matter be ame dominant over radiation. The universe was said to have be ome matter-dominated (marked on Fig. 44-30). As the universe ontinued to expand, the ele tromagneti radiation ooled further, to 2.7 K today, forming the osmi mi rowave ba kground radiation we dete t from everywhere in the universe. A fter the birth of atoms, then stars and galaxies ould begin to form: by self gravitation around mass on entrations (inhomogeneities). Stars began to form about 200 m illion years after the Big Bang, galaxies after almost 109 years. The universe ontinued to evolve until today, some 14 billion years after it started. * * * This s enario, like other s ientifi models, annot be said to be " proven." Yet this model is remarkably effe tive in explaining the evolution of the universe we live in, and makes predi tions whi h an be tested against the next generation of observations. A major event, and something only dis overed very re ently, is that when the universe was more than half as old as it is now (about 5 Gyr ago), its expansion began to a elerate. This was a big surprise be ause it was assumed the expansion of the universe would slow down due to gravitational attra tion of all obje ts towards ea h other. Another major re ent dis overy is that ordinary matter makes up very little of the total mass-energy of the universe (« 5%). Instead, as we dis uss in Se tion 44-9, the major ontributors to the energy density of the universe are dark matter and dark energy. On the right in Fig. 44-30 is a narrow verti al strip that represents the most re ent 5 billion years of the universe, during whi h dark energy seems to have dominated.

4 4 —8 Infl tion: Expl ining Fl tness, Uniformity, nd Structure The idea that the universe underwent a period of exponential inflation early in its life, expanding by a fa tor of 1040or more (previous Se tion), was first put forth by Alan Guth in 1981. Many more sophisti ated models have sin e been proposed. The energy required for this wild expansion may have been released when the ele troweak for e separated from the strong for e (end of GUT era, Fig. 43-19). So far, the eviden e for inflation is indire t; yet it is a feature of most viable osmologi al models be ause it is able to provide natural explanations for several remarkable features of our universe. trThis St nd rd Model prediction of 25% primordi l production of helium grees with wh t we observe tod y—the universe does cont in bout 25% He— nd it is strong evidence in support of the St nd rd Big B ng Model. Furthermore, the theory s ys th t 25% He bund nce is fully consistent with there being three neutrino types, which is the number we observe. And it sets n upper limit of four to the m ximum number of possible neutrino types. This is striking ex mple of the exciting interf ce between p rticle physics nd cosmology.

SE TION 44 - 8

Inflation: Explaining Flatness, Uniformity, and Structure

1219

FIGURE 44-31 ( ) Simple 2-D model of the entire universe; the observ ble universe is suggested by the sm ll circle centered on us (blue dot), (b) Edge of universe is essenti lly fl t fter the 1040- fold exp nsion during infl tion. ( ) Before infl tion

(b) After infl tion

Flatn ss First of all, our best measurements suggest that the universe is flat, that it has zero urvature. As s ientists, we would like some reason for this remarkable result. To see how inflation explains flatness, let us onsider a simple 2-dimensional model of the universe (as we did earlier in Figs. 44-16 and 44-21). A ir le on the surfa e of this 2-dimensional universe (a sphere, Fig. 44-31) represents the observable universe as seen by an observer at the ir le's enter. A possible hypothesis is that inflation o urred over a time interval that very roughly doubled the age of the universe, from let us say, t = 1 X IO-35 s to t = 2 X IO-35 s. The size of the observ able universe (r = ct) would have in reased by a fa tor of two during inflation, while the radius of urvature of the entire universe in reased by an enormous fa tor of IO40or more. Thus the edge of our 2-D sphere representing the entire universe would have seemed flat to a high degree of pre ision, Fig. 44-31b. Even if the time of inflation was a fa tor of 10 or 100 (instead of 2), the expansion fa tor of IO40 or more would have blotted out any possibility of observing anything but a flat universe.

CMB Uniformity Inflation also explains why the CMB is so uniform. W ithout inflation, the tiny universe at 10-35 s was too large for all parts of it to have been in onta t so as to rea h the same temperature (inform ation annot travel faster than ). Imagine a universe about 1 m in diameter at t = 10-36 s, as per original Big Bang theory. In that 10_36s, light ould have traveled d = ct = (3 X 108m /s)(l0 _36s) = 10-27m, way too small for opposite sides of a 1- m-wide universe to have been in ommuni ation. But if the universe had been IO40 times smaller (= 10_42m), as proposed by the inflation model, there ould have been onta t and thermal equilibrium to produ e the observed nearly uniform CMB. Inflation, by making the early universe very small, assures that all parts ould have been in thermal equilibrium; and after inflation the universe ould be large enough to give us today's observable universe.

Galaxy S ds, Fluctuations, Magn tic Monopol s

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CHAPTER 44

Inflation also gives us a lue as to how the present stru ture of the universe (galaxies and lusters of galaxies) ame about. We saw earlier that, a ording to the un ertainty prin iple, energy may not be onserved by an amount A E for a time A t ~ /A E . For es, whether ele tromagneti or other types, an undergo su h tiny quantum fluctuations a ording to quantum theory, but they are so tiny they are not dete table unless magnified in some way. That is what inflation might have done: it ould have magnified those flu tuations perhaps IO40 times in size, whi h would give us the density irregularities seen in the osmi mi rowave ba kground (WMAP, Fig. 44-27). That would be very ni e, be ause the density variations we see in the CMB are what we believe were the seeds that later oales ed under gravity into galaxies and galaxy lusters, in luding their substru tures (stars and planets), and our models fit the data extremely well. Sometimes it is said that the quantum flu tuations o urred in the vacuum state or va uum energy. This ould be possible be ause the va uum is no longer onsidered to be empty, as we dis ussed in Se tion 43-3 relative to positrons and a negative energy sea of ele trons. Indeed, the va uum is thought to be filled with fields and parti les o upying all the possible negative energy states. Also, the virtual ex hange parti les that arry the for es, as dis ussed in Chapter 43, ould leave their brief virtual states and a tually be ome real as a result of the IO40 magnifi ation of spa e (a ording to inflation) and the very short time over whi h it o urred (A t = /A E ).

Inflation helps us too with the puzzle of why m gnetic monopoles have never been observed, yet isolated magneti poles may well have been opiously produ ed at the start After inflation, they would have been so far apart that we have never stumbled on one. Some theorists have proposed that inflation may not have o urred in the entire universe. Perhaps only some regions of that tiny early universe be ame unstable (maybe it was a quantum flu tuation) and inflated into osmi "bubbles." If so, we would be living in one of the bubbles. The universe outside the bubble would be hopelessly unobserv ble to us. Inflation may solve outstanding problems, but it needs to be onfirmed and we may need new physi s just to understand how inflation o urred.

4 4 —9 D rk M tter nd D rk Energy A ording to the Standard Big Bang Model, the universe is evolving and hanging. Individual stars are being reated, evolving, and then dying to be ome white dwarfs, neutron stars, bla k holes. A t the same time, the universe as a whole is expanding. One important question is whether the universe w ill ontinue to expand forever. U ntil the late 1990s, the universe was thought to be dominated by matter whi h intera ts by gravity, and this question was onne ted to the urvature of spa e-time (Se tion 44 -4). If the universe had negative urvature, the expansion of the universe would never stop, although the rate of expansion would de rease due to the gravitational attra tion of its parts. Su h a universe would be open and infinite. If the universe is flat (no urvature), it would still be open and infinite but its expansion would slowly approa h a zero rate. Finally, if the universe had positive urvature, it would be closed and finite; the effe t of gravity would be strong enough that the expansion would eventually stop and the universe would begin to ontra t, ollapsing ba k onto itself in a big crunch.

ICritic l Density

EXERCISE E Return to the Ch pter-Opening Question, p ge 1193, nd nswer it g in. Try to expl in why you m y h ve nswered differently the first time.

A ording to the above s enario (whi h does not in lude inflation or the re ently dis overed a eleration of the universe), the fate of the universe would depend on the average mass-energy density in the universe. For an average mass density greater than a riti al value known as the critic l density, estimated to be about p

« 10-26 kg/m 3

(i.e., a few nu leons/m3 on average throughout the universe), gravity would prevent expansion from ontinuing forever. Eventually (if p > pc) gravity would pull the universe ba k into a big run h and spa e-time would have a positive urvature. If instead the a tual density was equal to the riti al density, p = p , the universe would be flat and open. If the a tual density was less than the riti al density, p < p , the universe would have negative urvature. See Fig. 44-32. Today we believe the universe is very lose to flat. But re ent eviden e suggests the universe is expanding at an accelerating rate, as dis ussed below.

FIGURE 44 -32 Three future possibilities for the universe, depending on the density p of ordin ry m tter, plus fourth possibility th t includes d rk energy. Note th t ll curves h ve been chosen to h ve the s me slope (= H , the Hubble p r meter) right now. Looking b ck in time, the Big B ng occurs where e ch curve touches the horizont l (time) xis.

depends on model)

SE TION 44 - 9

Dark Matter and Dark Energy

1221

EXAMPLE 44 - 7 ESTIMATE~| Critical density of the universe. Use energy onservation and es ape velo ity (Se tion 8 -7) to estimate the riti al density of the universe.

APPR ACH A t the riti al density, p , any given galaxy of mass m w ill just be able to " es ape" away from our Galaxy. As we saw in Se tion 8 -7, es ape an just o ur if the total energy E of the galaxy satisfies E = K + U = \m v 2 - G ^ ~ = 0. R Here R is the distan e of that galaxy m from us.We approximate the total mass M that pulls inward on m as the total mass within a sphere of radius R around us (Appendix D). If we assume the density of galaxies is roughly onstant, then M = f TrpcR 2.' S LUTI N Substituting this M into the equation above, and setting v = H R

(Hubble's law, Eq. 44 -4), we obtain GM

= w

or R

We solve for p : _ 3 P

^ 3[(22 k m /s /M ly )(l M ly/1019km )]2

8ttG ~

8(3.14)(6.67 X ltT 11N • m2/kg 2)

3 ~

8/m '

Dark Matt r W MAP and other experiments have onvin ed s ientists that the universe is flat and p = p . But this p annot be only normal baryoni matter (atoms are 99.9% baryons— protons and neutrons— by weight). These re ent experiments put the amount of normal baryoni matter in the universe at only about 5% of the riti al density. What is the other 95%? There is strong eviden e for a signifi ant amount of nonluminous matter in the universe referred to as dark matter, whi h a ts normally under gravity, but does not absorb or radiate light. For example, observa tions of the rotation of galaxies suggest that they rotate as if they had onsiderably more mass than we an see. Re all from Chapter 6, Eq. 6 -5, that for a satellite of mass m revolving around Earth (mass M ) _ mM v2 m — = G — zr r and hen e v = V G M /r . If we apply this equation to stars in a galaxy, we see that their speed depends on gala ti mass. Observations show that stars farther from the gala ti enter revolve mu h faster than expe ted if there is only the pull of visible matter, suggesting a great deal of invisible matter. Similarly, observations of the motion of galaxies within lusters also suggest that they have onsiderably more mass than an be seen. W ithout dark matter, galaxies and stars probably would not have formed and would not exist; it would seem to hold the universe together. But what might this nonluminous matter in the universe be? We don't know yet. But we hope to find out soon. It annot be made of ordinary (baryoni ) matter, so it must onsist of some other sort of elementary parti le, perhaps reated at a very early time. Perhaps it is a supersymmetri parti le (Se tion 43-12), maybe the lightest one. We are anxiously awaiting details both from parti le a elerators and the osmos. Dark matter makes up about 20% of the mass-energy of the universe, a ording to the latest observations and models. Thus the total mass-energy is 20% dark matter plus 5% baryons for a total of about 25%, whi h does not bring p up to p . What is the other 75%? We are not sure about that either, but we have given it a name: " dark energy."

1222

HAPTER 44

Astrophysics and osmology

D ark Energy — osm ic Acceleration In 1998, just before the turn of the millennium, osmologists re eived a huge surprise. Gravity was assumed to be the predominant for e on a large s ale in the universe, and it was thought that the expansion of the universe ought to be slowing down in time be ause gravity a ts as an attra tive for e between obje ts. But measurements of type la supernovae (SNIa, our best standard andles— see Se tion 44 -3) unexpe tedly showed that very distant (high z) SNIa's were dimmer than expe ted. That is, given their great distan e d as determined from their low brightness, their speed v as determined from the measured z was less than expe ted a ording to Hubble's law. This result suggests that nearer galaxies are moving away from us relatively faster than those very distant ones, meaning the expansion of the universe in more re ent epo hs has sped up. This acceleration in the expan sion of the universe (in pla e of the expe ted de eleration due to gravitational attra tion between masses) seems to have begun roughly 5 billion years ago (5 Gyr, whi h would be 8 to 9 Gyr after the Big Bang). What ould be ausing the universe to a elerate in its expansion, against the attra tive for e of gravity? Does our understanding of gravity need to be revised? We don't know the answers to these questions. Many s ientists say dark energy is the biggest mystery fa ing physi al s ien e today. There are several spe ulations. But somehow it seems to have a long-range repulsive effe t on spa e, like a negative gravity, ausing obje ts to speed away from ea h other ever faster. Whatever it is, it has been given the name dark energy. One idea is a sort of quantum field given the name quintessence. Another possibility suggests an energy latent in spa e itself (vacuum energy) and relates to an aspe t of General Relativity known as the cosmological constant (symbol A). When Einstein developed his equations, he found that they offered no solutions for a stati universe. In those days (1917) it was thought the universe was stati — un hanging and everlasting. Einstein added an arbitrary onstant to his equations to provide solutions for a stati universe. A de ade later, when Hubble showed us an expanding universe, Einstein dis arded his osmologi al onstant as no longer needed (A = 0). But today, measurements are onsistent with dark energy being due to a nonzero osmologi al onstant, although further measurements are needed to see subtle differen es among theories. There is in reasing eviden e that the effe ts of some form of dark energy are very real. Observations of the CMB, supemovae, and large-s ale stru ture (Se tion 44-10) agree well with theories and omputer models when they input dark energy as providing 75% of the mass-energy in the universe, and when the total mass-energy density equals the riti al density p . Today's best estimate of how the mass-energy in the universe is distributed is FIGURE 4 4 - 3 3 Portions of tot l m ss -energy in the universe. approximately (Fig. 44-33): Norm l m tter =5%

75% dark energy

St rs nd g l xies

25% matter, subje t to the known gravitational for e. O f this 25%, about 20% is dark matter 5% is baryons (what atoms are made of); of this 5% only ^ is readily visible matter— stars and galaxies (that is, 0.5% of the total); the other jq of ordinary matter, whi h is not visible, is mainly gaseous plasma. It is remarkable that only 0.5% of all the mass-energy in the universe is visible as stars and galaxies. The idea that the universe is dominated by ompletely unknown forms of energy seems bizarre. Nonetheless, the ability of our present model to pre isely explain observations of the CMB anisotropy, osmi expansion, and large-s ale stru ture (next Se tion) presents a ompelling ase.

SE TION 44 - 9

Dark Matter and Dark Energy

1223

FIGURE 44 -34 Distribution of some 50,000 g l xies in 2.5° slice through lmost h lf of the sky bove the equ tor, s me sured by the Slo n Digit l Sky Survey (SDSS). E ch dot represents g l xy. The dist nce from us is obt ined from the redshift nd Hubble 's l w, nd is given in units of 109 light-ye rs (Gly). A t gre ter dist nces, fewer g l xies re bright enough to be detected, thus resulting in n pp rent thinning out of g l xies. The point 0 represents us, our observ tion point. Note the "w lls" nd "voids " of g l xies.

Gly

0 (Our G l xy)

4 4—10 L rge-Sc le Structure o f the Universe The beautiful W MAP pi tures of the sky (Fig. 44-27 and Chapter-Opening Photo) show small but signifi ant inhomogeneities in the temperature of the CMB. These anisotropies refle t ompressions and expansions in the prim ordial plasma just before de oupling, from whi h stars, galaxies, and lusters of galaxies formed. Analysis of the irregularities in the CMB using mammoth omputer simulations predi t a large-s ale distribution of galaxies very similar to what is seen today (Fig. 44-34). These simulations are very su essful if they ontain dark energy and dark_matter; and the dark matter needs to be cold (slow speed— think of Eq. 18-4, \m v 2 = \k T where T is temperature), rather than " hot" dark matter su h as neutrinos whi h move at or very near the speed of light. Indeed, the modern cosmological model is alled the A C D M model, where lambda (A ) stands fo r the osmologi al onstant, and CDM is cold dark matter. Cosmologists have gained substantial onfiden e in this osmologi al model from su h a pre ise fit between observations and theory. They an also extra t very pre ise values for osmologi al parameters whi h previously were only known with low a ura y. The CMB is su h an important osmologi al observable that every effort is being made to extra t all of the information it ontains. A new generation of ground, balloon, and satellite experiments w ill observe the CMB with greater resolution and sensitivity. They may dete t intera tion of gravity waves (produ ed in the inflationary epo h) with the CMB and thereby provide dire t eviden e for osmi inflation, and also provide information about elementary parti le physi s at energies far beyond the rea h of man-made a elerators.

44—11 Fin lly . . .

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CHAPTER 44

When we look up into the night sky, we see stars; and with the best teles opes, we see galaxies and the exoti obje ts we dis ussed earlier, in luding rare supernovae. But even with our best instruments we do not see the pro esses going on inside stars and supernovae that we hypothesized (and believe). We are dependent on brilliant theorists who ome up with viable theories and verifiable models. We depend on ompli ated omputer models whose parameters are varied until the outputs ompare favorably with our observations and analyses of W MAP and other experiments. And we now have a surprisingly pre ise idea about some aspe ts of our universe: it is flat, it is about 14 billion years old, it ontains only 5% " normal" baryoni matter (for atoms), and so on. The questions raised by osmology are diffi ult and profound, and may seem removed from everyday " reality." We an always say, " the Sun is shining, it's going to burn on for an unimaginably long time, all is well." Nonetheless, the questions of

cosmology are deep ones that fascinate the human intellect. One aspect that is especially intriguing is this: calculations on the formation and evolution of the universe have been performed that deliberately varied the values— just slightly— of certain fundamental physical constants. The result? A universe in which life as we know it could not exist. [For example, if the difference in mass between proton and neutron were zero, or small (less than the mass of the electron, 0 .5 11 M eV /c2), there would be no atoms: electrons would be captured by protons never to be freed again.] Such results have contributed to a philosophical idea called the A n thropi prin iple, which says that if the universe were even a little different than it is, we could not be here. We physicists are trying to find out if there are some undis covered fundamental laws that determined those conditions that allowed us to exist. A poet might say that the universe is exquisitely tuned, almost as if to accommodate us.

Summ ry The night sky ontains myriads of stars in luding those in the M ilky Way, whi h is a " side view" of our G l xy looking along the plane of the disk. Our Galaxy in ludes over 1011 stars. Beyond our Galaxy are billions of other galaxies. Astronomi al distan es are measured in light-ye rs ( l ly « 1013km). The nearest star is about 4 ly away and the nearest large galaxy is 2 m illion ly away. Our Gala ti disk has a diameter of about 100,000 ly. Distan es are often spe ified in p rsecs, where 1 parse = 3.26 ly. Stars are believed to begin life as ollapsing masses of gas (protostars), largely hydrogen. As they ontra t, they heat up (potential energy is transformed to kineti energy). When the temperature rea hes about 10 m illion degrees, nu lear fusion begins and forms heavier elements (nucleosynthesis), mainly helium at first. The energy released during these rea tions heats the gas so its outward pressure balan es the inward gravitational for e, and the young star stabilizes as a m in-sequence star. The tremendous luminosity of stars omes from the energy released during these thermonu lear rea tions. A fter billions of years, as helium is olle ted in the ore and hydrogen is used up, the ore ontra ts and heats further. The envelope expands and ools, and the star be omes a red gi nt (larger diameter, redder olor). The next stage of stellar evolution depends on the mass of the star, whi h may have lost mu h of its original mass as its outer envelope es aped into spa e. Stars of residual mass less than about 1.4 solar masses ool further and be ome white dw rfs, eventually fading and going out altogether. Heavier stars ontra t further due to their greater gravity: the density approa hes nu lear density, the huge pressure for es ele trons to ombine with protons to form neutrons, and the star be omes essentially a huge nu leus of neutrons. This is a neutron st r, and the energy released from its final ore ollapse is believed to produ e supernov explosions. If the star is very massive, it may ontra t even further and form a bl ck hole, whi h is so dense that no matter or light an es ape from it. In the gener l theory of rel tivity, the equiv lence principle states that an observer annot distinguish a eleration from a gravitational field. Said another way, gravitational and inertial masses are the same. The theory predi ts gravitational bending of light rays to a degree onsistent with experiment. Gravity is treated as a urvature in spa e and time, the urvature being greater near massive obje ts. The universe as a whole may be urved. W ith suffi ient mass, the urvature of the universe would be positive, and the universe is closed and finite', otherwise, it would be open and infinite. Today we believe the universe is fl t. Distant galaxies display a redshift in their spe tral lines, originally interpreted as a Doppler shift. The universe seems to be

exp nding, its galaxies ra ing away from ea h other at speeds (v) proportional to the distan e (d) between them: (44-4)

v = H d,

whi h is known as Hubble's l w (H is the Hubble p r meter). This expansion of the universe suggests an explosive origin, the Big B ng, whi h o urred about 13.7 billion years ago. It is not like an ordinary explosion, but rather an expansion of spa e itself. The cosmologic l principle assumes that the universe, on a large s ale, is homogeneous and isotropi . Im portant eviden e for the Big Bang model of the universe was the dis overy of the cosmic microw ve b ckground radia tion (CMB), whi h onforms to a bla kbody radiation urve at a temperature of 2.725 K. The St nd rd Model of the Big Bang provides a possible s enario as to how the universe developed as it expanded and ooled after the Big Bang. Starting at 10-43 se onds after the Big Bang, a ording to this model, there were a series of ph se tr nsitions during whi h previously unified for es of nature " ondensed out" one by one. The infl tion ry scen rio assumes that during one of these phase transitions, the universe underwent a brief but rapid exponential expansion. U ntil about 10 35s, there was no distin tion between quarks and leptons. Shortly thereafter, quarks were confined into hadrons (the h dron er ). About 10 4s after the Big Bang, the majority of hadrons disappeared, having ombined with anti-hadrons, produ ing photons, leptons, and energy, leaving mainly photons and leptons to freely move, thus introdu ing the lepton er . By the time the universe was about 10 s old, the ele trons too had mostly disappeared, having ombined with their antiparti les; the universe was r di tion-domin ted. A ouple of minutes later, nu leosynthesis began, but lasted only a few minutes. It then took almost four hundred thousand years before the universe was ool enough for ele trons to ombine with nu lei to form atoms (recombin tion). Photons, up to then ontinually being s attered off of free ele trons, ould now move freely— they were decoupled from matter and the universe be ame trans parent. The ba kground radiation had expanded and ooled so mu h that its total energy be ame less than the energy in matter, and m tter domin ted in reasingly over radiation. Then stars and galaxies formed, produ ing a universe not mu h different than it is today— some 14 billion years later. Re ent observations indi ate that the universe is flat, that it ontains an as-yet unknown type of d rk m tter, and that it is dominated by a mysterious d rk energy whi h exerts a sort of negative gravity ausing the expansion of the universe to a elerate. The total ontributions of baryoni (normal) matter, dark matter, and dark energy sum up to the critic l density.

Summary

1225

Questions 1. The M ilky Way was on e thought to be " murky" or " m ilky" but is now onsidered to be made up of point sour es. Explain.

14. Compare an explosion on Earth to the Big Bang. Consider

2. A star is in equilibrium when it radiates at its surfa e all the energy generated in its ore. What happens when it begins to generate more energy than it radiates? Less energy? Explain.

3. Des ribe a red giant star. List some of its properties. 4. Sele t a point on the H - R diagram. Mark several dire tions away from this point. Now des ribe the hanges that would take pla e in a star moving in ea h of these dire tions. 5. Does the H -R diagram reveal anything about the ore of a star? 6. Why do some stars end up as white dwarfs, and others as neutron stars or bla k holes?

15. 16. 17. 18.

7. Can we tell, by looking at the population on the H - R diagram, that hotter main-sequen e stars have shorter lives? Explain. 8. If you were measuring star parallaxes from the Moon instead of Earth, what orre tions would you have to make? What hanges would o ur if you were measuring parallaxes from Mars? 9. Cep eid variable stars hange in luminosity with a typi al period of several days. The period has been found to have a definite relationship with the intrinsi luminosity of the star. How ould these stars be used to measure the distan e to galaxies?

10. What is a geodesi ? What is its role in General Relativity? 11. If it were dis overed that the redshift of spe tral lines of galaxies was due to something other than expansion, how might our view of the universe hange? Would there be onfli ting eviden e? Dis uss. 12. A ll galaxies appear to be moving away from us. Are we therefore at the enter of the universe? Explain. 13. If you were lo ated in a galaxy near the boundary of our observable universe, would galaxies in the dire tion of the M ilky Way appear to be approa hing you or re eding from you? Explain.

19. 20. 21. 22. 23.

24.

su h questions as: Would the debris spread at a higher speed for more distant parti les, as in the Big Bang? Would the debris ome to rest? What type of universe would this orrespond to, open or losed? If nothing, not even light, es apes from a bla k hole, then how an we tell if one is there? What mass w ill give a S hwarzs hild radius equal to that of the hydrogen atom in its ground state? The Earth's age is often given as about 4 billion years. Find that time on Fig. 44-30. People have lived on Earth on the order of a m illion years. Where is that on Fig. 44-30? Explain what the 2.7-K osmi mi rowave ba kground radi ation is. Where does it ome from? Why is its temperature now so low? Why were atoms, as opposed to bare nu lei, unable to exist until hundreds of thousands of years after the Big Bang? (a) Why are type la supernovae so useful for determining the distan es of galaxies? (b) How are their distan es a tually measured? Explain why the CMB radiation should not be that of a perfe t bla kbody. (The deviations from a bla kbody spe trum are slightly less than one part in 104.) Under what ir umstan es would the universe eventually ollapse in on itself? When stable nu lei first formed, about 3 minutes after the Big Bang, there were about 7 times more protons than neutrons. Explain how this leads to a ratio of the mass of hydrogen to the mass of helium of 3:1. This is about the a tual ratio observed in the universe. (a) Why did astronomers expe t that the expansion rate of the universe would be de reasing (de elerating) with time? (b) How, in prin iple, ould astronomers hope to determine whether the universe used to expand faster than it does now?

| Problems_________________ 44 - 1 to 4 4 - 3 Stars, Galaxies, Stellar Evolution, Distances 1. (I) The parallax angle of a star is 0.00029°. How far away is the star? 2. (I) A star exhibits a parallax of 0.27 se onds of ar . How far away is it?

3. (I) What is the parallax angle for a star that is 65 ly away? How many parse s is this? 4. (I) A star is 56 p away. What is its parallax angle? State (a) in se onds of ar , and ( b ) in degrees.

8. (II) We saw earlier (Chapter 19) that the rate energy rea hes the Earth from the Sun (the " solar onstant" ) is about 1.3 X 103W /m 2. What is (a) the apparent brightness b of the Sun, and ( b ) the intrinsi luminosity L of the Sun?

9. (II) When our Sun be omes a red giant, what w ill be its average density if it expands out to the orbit of Mer ury (6 X 1010 m, from the Sun)?

10. (II) Estimate the angular width that our Galaxy would subtend if observed from the nearest galaxy to us (Table 44 -1). Compare to the angular width of the Moon from Earth.

5. (I) If one star is twi e as far away from us as a se ond star, w ill the parallax angle of the farther star be greater or less than that of the nearer star? By what fa tor?

11. (II) Cal ulate the Q-values for the He burning rea tions of

6. (II) A star is 85 p away. How long does it take for its light to rea h us?

be about the size of the Moon. What angular width would it subtend from the present distan e to Earth?

7. (II) What is the relative brightness of the Sun as seen from Jupiter, as ompared to its brightness from Earth? (Jupiter is 5.2 times farther from the Sun than the Earth is.)

13. (II) Cal ulate the density of a white dwarf whose mass is

1226

HAPTER 44

Astrophysics and osmology

Eq. 44-2. (The mass of the very unstable ®Be is 8.005305 u.)

12. (II) When our Sun be omes a white dwarf, it is expe ted to

equal to the Sun's and whose radius is equal to the Earth's. How many times larger than Earth's density is this?

14. (II) A neutron star whose mass is 1.5 solar masses has a radius

26. (II) If an absorption line of al ium is normally found at a

of about 11 km. Cal ulate its average density and ompare to that for a white dwarf (Problem 13) and to that of nu lear matter.

wavelength of 393.4 nm in a laboratory gas, and you measure it to be at 423.4 nm in the spe trum of a galaxy, what is the approximate distan e to the galaxy? (II) What is the speed of a galaxy with z = 0.060? (II) What would be the redshift parameter z for a galaxy traveling away from us at v = 0.075 ? (II) Starting from Eq. 44-3, show that the Doppler shift in wavelength is AA/Arest « v /c (Eq. 44 -6) for v « . [Hint: Use the binomial expansion.] (II) Radioteles opes are designed to observe 21- m waves emitted by atomi hydrogen gas. A signal from a distant radio -emitting galaxy is found to have a wavelength that is 0.10 m longer than the normal 21- m wavelength. Estimate the distan e to this galaxy.

15. ( Ill) Stars lo ated in a ertain luster are thought to be about the same distan e from us. Two su h stars have spe tra that peak at = 470 nm and A2 = 720 nm, and the ratio of their apparent brightness is b i/b 2 = 0.091. Estimate their relative sizes (give ratio of their diameters) using Wien'slaw and the Stefan-Boltzmann equation, Eq. 19-17.

16. ( Ill) Suppose two stars of the same apparent brightness

are also believed to be the same size. The spe trum of one star peaks at 750 nm whereas that of the other peaks at 450 nm. Use Wien's law and the Stefan-Boltzmann equation (Eq. 19-17) to estimate their relative distan es from us.

44 - 4 General Relativity, Gravity and Curved Space

17. (I) Show that the S hwarzs hild radius for a star with mass equal to that of Earth is 8.9 mm. 18. (II) What is the S hwarzs hild radius for a typi al galaxy (like ours)? 19. (II) What is the maximum sum-of-the-angles for a triangle on a sphere? 20. (II) Cal ulate the es ape velo ity, using Newtonian me hani s, from an obje t that has ollapsed to its S hwarzs hild radius. 21. (II) What is the apparent defle tion of a light beam in an elevator (Fig. 44-13) whi h is 2.4 m wide if the elevator is a elerating downward at 9.8 m/s2?

44 - 5 Redshift, Hubble's Law

22. (I) The redshift of a galaxy indi ates a velo ity of 1850 km/s. How far away is it?

23. (I) If a galaxy is traveling away from us at 1.5% of the speed of light, roughly how far away is it?

24. (II) A galaxy is moving away from Earth. The "blue" hydrogen line at 434 nm emitted from the galaxy is measured on Earth to be 455 nm. (a) How fast is the galaxy moving? (ib) How far is it from Earth? 25. (II) Estimate the wavelength shift for the 656-nm line in the Balmer series of hydrogen emitted from a galaxy whose distan e from us is (a) 7.0 X 106ly, (b) 7.0 X 107ly.

27. 28. 29. 30.

4 4 - 6 to 4 4 - 8 The Big Bang, CMB, Universe Expansion

31. (I) Cal ulate the wavelength at the peak of the bla kbody radiation distribution at 2.7 K using Wien's law.

32. (II) Cal ulate the peak wavelength of the CMB at 1.0s after the birth of the universe. In what part of the EM spe trum is this radiation? 33. (II) The riti al density for losure of the universe is p « 10_26kg/m 3. State p in terms of the average number of nu leons per ubi meter. 34. (II) The s ale fa tor of the universe (average distan e between galaxies) at any given time is believed to have been inversely proportional to the absolute temperature. Estimate the size of the universe, ompared to today, at (a) t = 106yr, (b) t = 1 s, ( ) t = 10-6 s, and (d) t = 10-35 s. 35. (II) A t approximately what time had the universe ooled below the threshold temperature for produ ing (a) kaons (M « 500 M eV / 2), (b) Y (M ~ 9500 M eV / 2), and ( c) muons (M « 100 M eV / 2)?

4 4 - 9 Dark Matter, Dark Energy

36. (II) Only about 5% of the energy in the universe is omposed of baryoni matter, (a) Estimate the average density of baryoni matter in the observable universe with a radius of 14 billion light-years that ontains 1011 galaxies, ea h with about 1011stars like our Sun. (b) Estimate the density of dark matter in the universe.

| Gener l Problems__________ 37. The evolution of stars, as dis ussed in Se tion 44-2, an lead to a white dwarf, a neutron star, or even a bla k hole, depending on the mass, (a) Referring to Se tions 44-2 and 44-4, give the radius of (i) a white dwarf of 1 solar mass, (ii) a neutron star of 1.5 solar masses, and (iii) a bla k hole of 3 solar masses, (ib) Express these three radii as ratios ( ^ : : 7^).

41. Suppose that three main-sequen e stars ould undergo the three hanges represented by the three arrows, A , B, and C, in the H - R diagram of Fig. 44-35. For ea h ase, des ribe the hanges in temperature, intrinsi luminosity, and size.

38. Use onservation of angular momentum to estimate the angular velo ity of a neutron star whi h has ollapsed to a diameter of 16 km, from a star whose radius was equal to that of our Sun (7 X 108m). Assume its mass is 1.5 times that of the Sun, and that it rotated (like our Sun) about on e a month.

39. By what fa tor does the rotational kineti energy hange when the star in Problem 38 ollapses to a neutron star?

40. Assume that the nearest stars to us have an intrinsi luminosity about the same as the Sun's. Their apparent brightness, however, is about 1011times fainter than the Sun. From this, estimate the distan e to the nearest stars. (Newton did this al ulation, although he made a numeri al error of a fa tor of 100.)

FIGURE 44 -35

Problem 41.

1227

42. A ertain pulsar, believed to be a neutron star o f mass 1.5 times that of the Sun, w ith diameter 16 km, is observed to have a rotation speed o f 1.0 rev/s. I f it loses rotational kineti energy at the rate of 1 part in 109 per day, whi h is all transformed into radiation, what is the power output of the star? 43. The nearest large galaxy to our Galaxy is about 2 X 106ly away. I f both galaxies have a mass o f 3 X 1041kg, w ith what gravitational for e does ea h galaxy attra t the other? 44. Estimate what neutrino mass (in eV / 2) would provide the riti al density to lose the universe. Assume the neutrino density is, like photons, about 109 times that of nu leons, and that nu leons make up only (a) 2% o f the mass needed, or (b) 5% o f the mass needed. 45. Two stars, whose spe tra peak at 660 nm and 480 nm, respe tively, both lie on the main sequen e. Use W ien's law, the Stefan-Boltzmann equation, and the H - R diagram (Fig. 44 -6) to estimate the ratio o f their diameters. 46. (a) In order to measure distan es w ith parallax at 100 parse s, what minimum angular resolution (in degrees) is needed? (b) W hat diameter m irror or lens would be needed? 47. What is the temperature that orresponds to 1.96-TeV o lli sions at the Fermilab ollider? To what era in osmologi al history does this orrespond? [Hint: See Fig. 44 -30.] 48. Astronomers have measured the rotation o f gas around a possible supermassive bla k hole o f about 2 b illion solar masses at the enter of a galaxy. I f the radius from the gala ti enter to the gas louds is 68 light-years, estimate the measured value o f z. 49. In the later stages of stellar evolution, a star (if massive enough) w ill begin fusing arbon nu lei to form , fo r example, magnesium: 126c +

24-

gM g + 7.

(a) How mu h energy is released in this rea tion (see Appendix F)? ( b ) How mu h kineti energy must ea h arbon nu leus have (assume equal) in a head-on ollision if they are just to " tou h" (use Eq. 41 -1) so that the strong for e an ome into play? ( ) W hat temperature does this kineti energy orrespond to? 50. Consider the rea tion

+ ijjo - > g s i + ^He, and answer the same questions as in Problem 49.

51. Cal ulate the S hwarzs hild radius using a semi- lassi al (Newtonian) gravitational theory, by al ulating the minimum radius R fo r a sphere o f mass M su h that a photon an es ape from the surfa e. (General R elativity gives R = 2G M /c 1.) 52. How large would the Sun be if its density equaled the riti al density o f the universe, p ~ 10_26kg/m 3? Express your answer in light-years and ompare w ith the Earth -Sun distan e and the diameter o f our Galaxy.

53. The Large Hadron C ollider in Geneva, Switzerland, an ollide two beams o f protons at an energy o f 14 TeV. E sti mate the tim e after the Big Bang probed by this energy.

54.

(a) Use spe ial relativity and Newton's law o f gravitation to show that a photon of mass m = E / c 2 just grazing the Sun w ill be defle ted by an angle A 6 given by Ad =

2G M c R

where G is the gravitational onstant, R and M are the radius and mass of the Sun, and c is the speed o f light. ( b ) Put in values and show A0 = 0.87". (General R elativity predi ts an angle twi e as large, 1.74".)

55. Astronomers use an pp rent m gnitude (m) s ale to des ribe apparent brightness. It uses a logarithmi s ale, where a higher number orresponds to a less bright star. (For example, the Sun has magnitude - 2 7 , whereas most stars have positive magnitudes.) On this s ale, a hange in apparent magnitude by +5 orresponds to a de rease in apparent brightness by a fa tor of 100. If Venus has an apparent magnitude of - 4 .4 , whereas Sirius has an apparent magnitude of — 1.4, whi h is brighter? What is the ratio of the apparent brightness of these two obje ts?

56. Estimate the radius of a white dwarf whose mass is equal to that o f the Sun by the follow ing method, assuming there are N nu leons and \ N ele trons (why §?): (a) Use Ferm i-Dira

statisti s (Se tion 40 -6) to show that the total energy o f all the ele trons is , 1 /3 JV\3 3 fl Ee = - \ - N 5 V2 7 8rae V 2 V J

[Hint: See Eqs. 40 -12 and 40 -13; we assume ele trons fill energy levels from 0 up to the Fermi energy.] (b) The

nu leons ontribute to the total energy m ainly via the grav itational for e (note that the Fermi energy fo r nu leons is negligible ompared to that fo r ele trons— why?). Use a gravitational form of Gauss's law to show that the total gravitational potential energy o f a uniform sphere of radius R is 3 GM2 5 R by onsidering the potential energy of a spheri al shell of radius r due only to the mass inside it (why?) and integrate from r = 0 to r = R. (See also Appendix D.) ( ) W rite the total energy as a sum o f these two terms, and set d E /d R = 0 to find the equilibrium radius, and evaluate it fo r a mass equal to the Sun's (2.0 X IO30kg).

57. Determine the radius of a neutron star using the same argu ment as in Problem 56 but fo r N neutrons only. Show that the radius of a neutron star, of 1.5 solar masses, is about 11 km.

58. Use

dim ensional analysis w ith the fundamental onstants c, G , and to estimate the value of the so- alled Planck time.

It is thought that physi s as we know it about the universe before this time.

an say nothing

Answers to Exercises A:

urselves, 2 years ago.

C: 6 km.

B: 600 ly (estimating L from Fig. 44 -6 as L « 8 X 1026W; note that on a log s ale, 6000 K is loser to 7000 K than it is to 5000 K ).

12 28

HAPTER 44

Astrophysics and

osmology

D : (a); not the usual R 3, but R: see form ula fo r the S hwarzs hild radius, g.

?, N D

th e m tic l F o r m u l s

A—1 Qu dr tic Formul If then

ax2 + bx + c = 0 - b ± \ / b 2 - 4ac 2a

A—2 Binomi l Exp nsion { 1 ± x )n = 1 ± n x + ^ (x + y r = xJ 1 + l Y

x, ± < ! L ^ l ^ l x3

= xJ 1 + n l + n ( n - l ) y 2 Xj \ X

A—3 Other Exp nsions jl.

e* = 1 + * + -

+ -

2 3 4 ln (l + *) = x ~ Y + Y ~ T + '"

sine = e -

|j - + |j - -

- . - . - S - S - . t n0 = e + J + — d5 + ••• In gener l:

f ( x ) = /( 0 ) +

: +

|0| < | f - + •••

A—4 Exponents ( n)( m) =

'I+m

= a-"

(a")(f ") = (a*)" ( ")m =

®"

A—5 Are s nd Volumes Object

Surf ce re

Volume

Circle, r dius r Sphere, r dius r Right circul r cylinder, r dius r, height Right circul r cone, r dius r, height

Trr1 4ttt2 2ttr2 + 27rr ttt2 + irr\/r2 + 2

— Iirr3 7rr2 \irr2

A-1

A—6 Pl ne Geometry

FIGURE A - 1 If line ax is parallel to line a2 , then 61 = 02.

FIGURE A - 2 If« !_ L « 2 and bx _Lb2 , then d1 = 02.

3. The sum of the angles in any plane triangle is 180°.

4. Pyt agorean t eorem : In any right triangle (one angle = 90°) of sides a, b, and : a2 + b2 = c2

where c is the length of the hypotenuse (opposite the 90° angle).

FIGURE A - 3

5. Similar triangles: Two triangles are said to be similar if all three of their angles are equal (in Fig. A - 4 , 61 = i,d2 = 4>2, and 03 = (f>3). Similar triangles an have different sizes and different orientations. (a) Two triangles are similar if any two of their angles are equal. (This follows be ause the third angles must also be equal sin e the sum of the angles of a triangle is 180°.) (b) The ratios of orresponding sides of two similar triangles are equal (Fig. A - 4 ): al

FIGURE A - 4

6. Congruent triangles: Two triangles are ongruent if one an be pla ed pre isely

on top of the other. That is, they are similar triangles and they have the same size. Two triangles are ongruent if any of the following holds: (a) The three orresponding sides are equal. (b) Two sides and the en losed angle are equal (" side-angle-side" ). ( ) Two angles and the en losed side are equal (" angle-side-angle" ).

A—7 Log rithms Logarithms are defined in the following way: if

y = Ax,

then

x = log A y.

That is, the logarithm of a number y to the base A is that number whi h, as the exponent of A , gives ba k the number y. For common logarithms, the base is 10, so if

y = 10*,

then

x = logy.

The subs ript 10 on log10 is usually omitted when dealing with ommon logs. Another important base is the exponential base e = 2.718 ..., a natural number. Su h logarithms are alled natural logarithms and are written In. Thus, if

y = ex,

then

x = In y.

For any number y, the two types of logarithm are related by ln y = 2.3026 logy. Some simple rules for logarithms are as follows: log (ab) = log a + log b,

(i)

whi h is true be ause if a = 10" and b = 10m, then ab = 10" +m. From the APPENDIX A

definition of logarithm, log a = n, log b = ra, and log (ab) = n + m; hen e, \og(ab) = n + m = log a + \ogb. In a similar way, we an show that lo g ^ j = log a

log b

(ii)

and log a" = ralog 0 .

(iii)

These three rules apply to any kind of logarithm. If you do not have a al ulator that al ulates logs, you an easily use a log table, su h as the small one shown here (Table A - 1 ): the number N whose log we want is given to two digits. The first digit is in the verti al olumn to the left, the se ond digit is in the horizontal row a ross the top. For example, Table A - 1 tells us that log 1.0 = 0.000, log 1.1 = 0.041, and log 4.1 = 0.613. Table A - 1 does not in lude the de imal point. The Table gives logs for numbers between 1.0 and 9.9. For larger or smaller numbers, we use rule (i) above, log (ab) = log a + log b. For example, log(380) = log(3.8 X 102) = log(3.8) + log(l02). From the Table, log 3.8 = 0.580; and from rule (iii) above log(l02) = 21og(10) = 2, sin e log (10) = 1. [This follows from the definition of the logarithm: if 10 = 101, then 1 = log(10).] Thus, log (380) = log(3.8) + log(l02) = 0.580 + 2 = 2.580. Similarly, log (0.081) = log(8.1) + log (l0 -2) = 0.908 - 2 = -1.092. The reverse pro ess of finding the number N whose log is, say, 2.670, is alled " taking the antilogarithm." To do so, we separate our number 2.670 into two parts, making the separation at the de imal point: log N = 2.670 = 2 + 0.670 = log 102 + 0.670. We now look at Table A - 1 to see what number has its log equal to 0.670; none does, so we must interpolate: we see that log 4.6 = 0.663 and log 4.7 = 0.672. So the number we want is between 4.6 and 4.7, and loser to the latter by \ • Approxi mately we an say that log 4.68 = 0.670. Thus log N = 2 + 0.670 = log(l02) + log(4.68) = log(4.68 X 102), so N = 4.68 X 102 = 468.

If the given logarithm is negative, say, -2.180, we pro eed as follows: log TV = -2.180 = - 3 + 0.820 = loglO -3 + log 6.6 = log 6.6 X 10-3, so N = 6.6 X 10 3. Noti e that we added to our given logarithm the next largest

integer (3 in this ase) so that we have an integer, plus a de imal number between 0 and 1.0 whose antilogarithm an be looked up in the Table.

TABLE A-1 Short Table of Common Logarithms N

0.0

000 301 477 602

2 3 4 5 6 7 8 9

699 778 845 903 954

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

041

079 342 505

114

146 380 531

176 398 544

204

362

415 556

255 447 580

279 462

681 763 833

591 690 771 839

892 944 991

898 949 996

322 491 613 708 785 851 908 959

623 716 792

519 633 724 799

643 732 806

653 740 813

663 748 820

230 431 568 672 756 826

857 914 964

863 919 968

869 924 973

875 929 978

881 935 982

886 940 987

SE TION A-7

Logarithms

A -3

A—8 Vectors Vector addition is overed in Se tions 3 -2 to 3 -5. Vector multiplication is overed in Se tions 3 - 3 ,7 - 2 , and 11-2.

A—9 Trigonometric Functions nd Identities The trigonometri fun tions are defined as follows (see Fig. A - 5 , o = side opposite, a = side adja ent, = hypotenuse. Values are given in Table A - 2 ): 1 sin0 1 se 0 = COS0 1 ot 0 = tan0

• sin 0^ = —°

CSC0 =

FIGURE A - 5

os 0 = —

sin 0 o tan 0 = — = -----a os 0

o a a o

and re all that a2 + o2 = FIGURE A - 6 First Qu dr nt (0° to 90°)

x>0

Second Qu dr nt (90° to 180°) xO cos0 fl2]

[x2 < a 2

\a - x J

! - " " ( f ) - - " *" ( ! )

B—5 A Few Definite Integr ls I,o JV

roo 77 x2e~" ' dx = ' 16fl3 Jo roo x ^ -^ d x = 2a2 Jo

n\ an+1 pTT V 4a 1

[ x e -^ d x = J Jo 2la

f1°°j:2V _'" x2ne~ a^2dx dx = 1 ' 3 ' 5 ' " ( 2 "---- 51 J z Jo 2 " fl V fl

B—6 Integr tion by P rts Sometimes

difficult integr l c n be simplified by c refully choosing the functions u nd v in the identity:

j u d v = uv — Jv du.

[Integr tion by p rts]

This identity follows from the property of deriv tives d . , dv du -(u v ) = u - + v or s differenti ls: d(uv) = u dv + vdu. For ex mple Jxe~ x dx c n be integr ted by choosing u = x nd dv = e xdx in the "integr tion by p rts" equ tion bove: | xe~x dx = (x ) ( - e ~x) + j e~x dx

= - xe ~x —e~x = —(x + l)e~x.

SE TION B-6 A-7

More on Dimensional Analysis An important use of dimensional analysis (Se tion 1 -7) is to obtain the form of an equation: how one quantity depends on others. To take a on rete example, let us try to find an expression for the period T of a simple pendulum. First, we try to figure out what T ould depend on, and make a list of these variables. It might depend on its length £, on the mass m of the bob, on the angle of swing 0, and on the a eleration due to gravity, g. It might also depend on air resistan e (we would use the vis osity of air), the gravitational pull of the Moon, and so on; but everyday experien e suggests that the Earth's gravity is the major for e involved, so we ignore the other possible for es. So let us assume that T is a fun tion of £, m, 0, and g, and that ea h of these fa tors is present to some power: T = C£wm x6yg z . C is a dimensionless onstant, and w , x, y, and z are exponents we want to solve

for. We now write down the dimensional equation (Se tion 1 -7) for this relationship: [T] = [L] w[M ] x [L /T 2]z . Be ause 0 has no dimensions (a radian is a length divided by a length — see Eq. 1 0 - la ), it does not appear. We simplify and obtain [T] = [L] w+z[M ] x [T]~2z

To have dimensional onsisten y, we must have 1 = —2z 0 = w + z 0 = x.

We solve these equations and find that z = desired equation must be t

= cVt/gf(e)

and x = 0. Thus our (C - i)

where /(0 ) is some fun tion of 0 that we annot determine using this te hnique. Nor an we determine in this way the dimensionless onstant C. (To obtain C and / , we would have to do an analysis su h as that in Chapter 14 using Newton's laws, whi h reveals that C = 2ir and / « 1 for small 0). But look what we ave found, using only dimensional onsisten y. We obtained the form of the expression that relates the period of a simple pendulum to the major variables of the situation, £ and g (see Eq. 14 -12 ), and saw that it does not depend on the mass m. How did we do it? And how useful is this te hnique? Basi ally, we had to use our intuition as to whi h variables were important and whi h were not. This is not always easy, and often requires a lot of insight. As to usefulness, the final result in our example ould have been obtained from Newton's laws, as in Chapter 14. But in many physi al situations, su h a derivation from other laws annot be done. In those situations, dimensional analysis an be a powerful tool. In the end, any expression derived by the use of dimensional analysis (or by any other means, for that matter) must be he ked against experiment. For example, in our derivation of Eq. C - l, we an ompare the periods of two pendulums of different lengths, lx and l 2 >whose amplitudes (0) are the same. For, using Eq. C - l, we would have Ti

o /ijg m

t2 cV U gf(e) Vi 2 Be ause C and /(0 ) are the same for both pendula, they an el out, so we an experimentally determine if the ratio of the periods varies as the ratio of the square roots of the lengths. This omparison to experiment he ks our derivation, at least in part; C and /(0 ) ould be determined by further experiments. A -8

APPENDIX

Jewel Stoliker

Hendrickson Walking Beam Suspension Diagram

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