Introduction to Electrodynamics (PDF) by David J. Griffiths (2013)

INTRODUCTION TO ELECTRODYNAMICS This page intentionally left blank INTRODUCTION TO ELECTRODYNAMICS Fourth Edition David J. Grifﬁths Reed College Executive Editor: Jim Smith Senior Project Editor: Martha Steele Development Manager: Laura Kenney Managing Editor: Corinne Benson Production Project Manager: Dorothy Cox Production Management and Composition: Integra Cover Designer: Derek Bacchus Manufacturing Buyer: Dorothy Cox Marketing Manager: Will Moore Credits and acknowledgments for materials borrowed from other sources and reproduced, with permission, in this textbook appear on the appropriate page within the text. Copyright c 2013, 1999, 1989 Pearson Education, Inc. All rights reserved. Manufactured in the United States of America. This publication is protected by Copyright, and permis- sion should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, 1900 E. Lake Ave., Glenview, IL 60025. For information regarding permissions, call (847) 486-2635. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publication Data Griffiths, David J. (David Jeffery), 1942- Introduction to electrodynamics/ David J. Griffiths, Reed College. – Fourth edition. pages cm Includes index. ISBN-13: 978-0-321-85656-2 (alk. paper) ISBN-10: 0-321-85656-2 (alk. paper) 1. Electrodynamics–Textbooks. I. Title. QC680.G74 2013 537.6–dc23 2012029768 ISBN 10: 0-321-85656-2 ISBN 13: 978-0-321-85656-2 www.pearsonhighered.com 1 2 3 4 5 6 7 8 9 10—CRW—16 15 14 13 12 Contents Preface xii Advertisement xiv 1 Vector Analysis 1 1.1 Vector Algebra 1 1.1.1 Vector Operations 1 1.1.2 Vector Algebra: Component Form 4 1.1.3 Triple Products 7 1.1.4 Position, Displacement, and Separation Vectors 8 1.1.5 How Vectors Transform 10 1.2 Differential Calculus 13 1.2.1 “Ordinary” Derivatives 13 1.2.2 Gradient 13 1.2.3 The Del Operator 16 1.2.4 The Divergence 17 1.2.5 The Curl 18 1.2.6 Product Rules 20 1.2.7 Second Derivatives 22 1.3 Integral Calculus 24 1.3.1 Line, Surface, and Volume Integrals 24 1.3.2 The Fundamental Theorem of Calculus 29 1.3.3 The Fundamental Theorem for Gradients 29 1.3.4 The Fundamental Theorem for Divergences 31 1.3.5 The Fundamental Theorem for Curls 34 1.3.6 Integration by Parts 36 1.4 Curvilinear Coordinates 38 1.4.1 Spherical Coordinates 38 1.4.2 Cylindrical Coordinates 43 1.5 The Dirac Delta Function 45 1.5.1 The Divergence of r̂/r 2 45 1.5.2 The One-Dimensional Dirac Delta Function 46 1.5.3 The Three-Dimensional Delta Function 50 v vi Contents 1.6 The Theory of Vector Fields 52 1.6.1 The Helmholtz Theorem 52 1.6.2 Potentials 53 2 Electrostatics 59 2.1 The Electric Field 59 2.1.1 Introduction 59 2.1.2 Coulomb’s Law 60 2.1.3 The Electric Field 61 2.1.4 Continuous Charge Distributions 63 2.2 Divergence and Curl of Electrostatic Fields 66 2.2.1 Field Lines, Flux, and Gauss’s Law 66 2.2.2 The Divergence of E 71 2.2.3 Applications of Gauss’s Law 71 2.2.4 The Curl of E 77 2.3 Electric Potential 78 2.3.1 Introduction to Potential 78 2.3.2 Comments on Potential 80 2.3.3 Poisson’s Equation and Laplace’s Equation 83 2.3.4 The Potential of a Localized Charge Distribution 84 2.3.5 Boundary Conditions 88 2.4 Work and Energy in Electrostatics 91 2.4.1 The Work It Takes to Move a Charge 91 2.4.2 The Energy of a Point Charge Distribution 92 2.4.3 The Energy of a Continuous Charge Distribution 94 2.4.4 Comments on Electrostatic Energy 96 2.5 Conductors 97 2.5.1 Basic Properties 97 2.5.2 Induced Charges 99 2.5.3 Surface Charge and the Force on a Conductor 103 2.5.4 Capacitors 105 3 Potentials 113 3.1 Laplace’s Equation 113 3.1.1 Introduction 113 3.1.2 Laplace’s Equation in One Dimension 114 3.1.3 Laplace’s Equation in Two Dimensions 115 3.1.4 Laplace’s Equation in Three Dimensions 117 3.1.5 Boundary Conditions and Uniqueness Theorems 119 3.1.6 Conductors and the Second Uniqueness Theorem 121 Contents vii 3.2 The Method of Images 124 3.2.1 The Classic Image Problem 124 3.2.2 Induced Surface Charge 125 3.2.3 Force and Energy 126 3.2.4 Other Image Problems 127 3.3 Separation of Variables 130 3.3.1 Cartesian Coordinates 131 3.3.2 Spherical Coordinates 141 3.4 Multipole Expansion 151 3.4.1 Approximate Potentials at Large Distances 151 3.4.2 The Monopole and Dipole Terms 154 3.4.3 Origin of Coordinates in Multipole Expansions 157 3.4.4 The Electric Field of a Dipole 158 4 Electric Fields in Matter 167 4.1 Polarization 167 4.1.1 Dielectrics 167 4.1.2 Induced Dipoles 167 4.1.3 Alignment of Polar Molecules 170 4.1.4 Polarization 172 4.2 The Field of a Polarized Object 173 4.2.1 Bound Charges 173 4.2.2 Physical Interpretation of Bound Charges 176 4.2.3 The Field Inside a Dielectric 179 4.3 The Electric Displacement 181 4.3.1 Gauss’s Law in the Presence of Dielectrics 181 4.3.2 A Deceptive Parallel 184 4.3.3 Boundary Conditions 185 4.4 Linear Dielectrics 185 4.4.1 Susceptibility, Permittivity, Dielectric Constant 185 4.4.2 Boundary Value Problems with Linear Dielectrics 192 4.4.3 Energy in Dielectric Systems 197 4.4.4 Forces on Dielectrics 202 5 Magnetostatics 210 5.1 The Lorentz Force Law 210 5.1.1 Magnetic Fields 210 5.1.2 Magnetic Forces 212 5.1.3 Currents 216 5.2 The Biot-Savart Law 223 5.2.1 Steady Currents 223 5.2.2 The Magnetic Field of a Steady Current 224 viii Contents 5.3 The Divergence and Curl of B 229 5.3.1 Straight-Line Currents 229 5.3.2 The Divergence and Curl of B 231 5.3.3 Ampère’s Law 233 5.3.4 Comparison of Magnetostatics and Electrostatics 241 5.4 Magnetic Vector Potential 243 5.4.1 The Vector Potential 243 5.4.2 Boundary Conditions 249 5.4.3 Multipole Expansion of the Vector Potential 252 6 Magnetic Fields in Matter 266 6.1 Magnetization 266 6.1.1 Diamagnets, Paramagnets, Ferromagnets 266 6.1.2 Torques and Forces on Magnetic Dipoles 266 6.1.3 Effect of a Magnetic Field on Atomic Orbits 271 6.1.4 Magnetization 273 6.2 The Field of a Magnetized Object 274 6.2.1 Bound Currents 274 6.2.2 Physical Interpretation of Bound Currents 277 6.2.3 The Magnetic Field Inside Matter 279 6.3 The Auxiliary Field H 279 6.3.1 Ampère’s Law in Magnetized Materials 279 6.3.2 A Deceptive Parallel 283 6.3.3 Boundary Conditions 284 6.4 Linear and Nonlinear Media 284 6.4.1 Magnetic Susceptibility and Permeability 284 6.4.2 Ferromagnetism 288 7 Electrodynamics 296 7.1 Electromotive Force 296 7.1.1 Ohm’s Law 296 7.1.2 Electromotive Force 303 7.1.3 Motional emf 305 7.2 Electromagnetic Induction 312 7.2.1 Faraday’s Law 312 7.2.2 The Induced Electric Field 317 7.2.3 Inductance 321 7.2.4 Energy in Magnetic Fields 328 7.3 Maxwell’s Equations 332 7.3.1 Electrodynamics Before Maxwell 332 7.3.2 How Maxwell Fixed Ampère’s Law 334 7.3.3 Maxwell’s Equations 337 Contents ix 7.3.4 Magnetic Charge 338 7.3.5 Maxwell’s Equations in Matter 340 7.3.6 Boundary Conditions 342 8 Conservation Laws 356 8.1 Charge and Energy 356 8.1.1 The Continuity Equation 356 8.1.2 Poynting’s Theorem 357 8.2 Momentum 360 8.2.1 Newton’s Third Law in Electrodynamics 360 8.2.2 Maxwell’s Stress Tensor 362 8.2.3 Conservation of Momentum 366 8.2.4 Angular Momentum 370 8.3 Magnetic Forces Do No Work 373 9 Electromagnetic Waves 382 9.1 Waves in One Dimension 382 9.1.1 The Wave Equation 382 9.1.2 Sinusoidal Waves 385 9.1.3 Boundary Conditions: Reﬂection and Transmission 388 9.1.4 Polarization 391 9.2 Electromagnetic Waves in Vacuum 393 9.2.1 The Wave Equation for E and B 393 9.2.2 Monochromatic Plane Waves 394 9.2.3 Energy and Momentum in Electromagnetic Waves 398 9.3 Electromagnetic Waves in Matter 401 9.3.1 Propagation in Linear Media 401 9.3.2 Reﬂection and Transmission at Normal Incidence 403 9.3.3 Reﬂection and Transmission at Oblique Incidence 405 9.4 Absorption and Dispersion 412 9.4.1 Electromagnetic Waves in Conductors 412 9.4.2 Reﬂection at a Conducting Surface 416 9.4.3 The Frequency Dependence of Permittivity 417 9.5 Guided Waves 425 9.5.1 Wave Guides 425 9.5.2 TE Waves in a Rectangular Wave Guide 428 9.5.3 The Coaxial Transmission Line 431 10 Potentials and Fields 436 10.1 The Potential Formulation 436 10.1.1 Scalar and Vector Potentials 436 10.1.2 Gauge Transformations 439 x Contents 10.1.3 Coulomb Gauge and Lorenz Gauge 440 10.1.4 Lorentz Force Law in Potential Form 442 10.2 Continuous Distributions 444 10.2.1 Retarded Potentials 444 10.2.2 Jeﬁmenko’s Equations 449 10.3 Point Charges 451 10.3.1 Liénard-Wiechert Potentials 451 10.3.2 The Fields of a Moving Point Charge 456 11 Radiation 466 11.1 Dipole Radiation 466 11.1.1 What is Radiation? 466 11.1.2 Electric Dipole Radiation 467 11.1.3 Magnetic Dipole Radiation 473 11.1.4 Radiation from an Arbitrary Source 477 11.2 Point Charges 482 11.2.1 Power Radiated by a Point Charge 482 11.2.2 Radiation Reaction 488 11.2.3 The Mechanism Responsible for the Radiation Reaction 492 12 Electrodynamics and Relativity 502 12.1 The Special Theory of Relativity 502 12.1.1 Einstein’s Postulates 502 12.1.2 The Geometry of Relativity 508 12.1.3 The Lorentz Transformations 519 12.1.4 The Structure of Spacetime 525 12.2 Relativistic Mechanics 532 12.2.1 Proper Time and Proper Velocity 532 12.2.2 Relativistic Energy and Momentum 535 12.2.3 Relativistic Kinematics 537 12.2.4 Relativistic Dynamics 542 12.3 Relativistic Electrodynamics 550 12.3.1 Magnetism as a Relativistic Phenomenon 550 12.3.2 How the Fields Transform 553 12.3.3 The Field Tensor 562 12.3.4 Electrodynamics in Tensor Notation 565 12.3.5 Relativistic Potentials 569 A Vector Calculus in Curvilinear Coordinates 575 A.1 Introduction 575 A.2 Notation 575 Contents xi A.3 Gradient 576 A.4 Divergence 577 A.5 Curl 579 A.6 Laplacian 581 B The Helmholtz Theorem 582 C Units 585 Index 589 Preface This is a textbook on electricity and magnetism, designed for an undergradu- ate course at the junior or senior level. It can be covered comfortably in two semesters, maybe even with room to spare for special topics (AC circuits, nu- merical methods, plasma physics, transmission lines, antenna theory, etc.) A one-semester course could reasonably stop after Chapter 7. Unlike quantum me- chanics or thermal physics (for example), there is a fairly general consensus with respect to the teaching of electrodynamics; the subjects to be included, and even their order of presentation, are not particularly controversial, and textbooks differ mainly in style and tone. My approach is perhaps less formal than most; I think this makes difﬁcult ideas more interesting and accessible. For this new edition I have made a large number of small changes, in the in- terests of clarity and grace. In a few places I have corrected serious errors. I have added some problems and examples (and removed a few that were not effective). And I have included more references to the accessible literature (particularly the American Journal of Physics). I realize, of course, that most readers will not have the time or inclination to consult these resources, but I think it is worthwhile anyway, if only to emphasize that electrodynamics, notwithstanding its venerable age, is very much alive, and intriguing new discoveries are being made all the time. I hope that occasionally a problem will pique your curiosity, and you will be inspired to look up the reference—some of them are real gems. I have maintained three items of unorthodox notation: The Cartesian unit vectors are written x̂, ŷ, and ẑ (and, in general, all unit vectors inherit the letter of the corresponding coordinate). The distance from the z axis in cylindrical coordinates is designated by s, to avoid confusion with r (the distance from the origin, and the radial coordi- nate in spherical coordinates). The script letter r denotes the vector from a source point r to the ﬁeld point r (see Figure). Some authors prefer the more explicit (r − r ). But this makes many equations distractingly cumbersome, especially when the unit vector r̂ is involved. I realize that unwary readers are tempted to interpret r as r—it certainly makes the integrals easier! Please take note: r ≡ (r − r ), which is not the same as r. I think it’s good notation, but it does have to be handled with care.1 1 In MS Word, r is “Kaufmann font,” but this is very difﬁcult to install in TeX. TeX users can download xii a pretty good facsimile from my web site. Preface xiii z Source point dτ⬘ r Field point r r⬘ y x As in previous editions, I distinguish two kinds of problems. Some have a speciﬁc pedagogical purpose, and should be worked immediately after reading the section to which they pertain; these I have placed at the pertinent point within the chapter. (In a few cases the solution to a problem is used later in the text; these are indicated by a bullet ( ) in the left margin.) Longer problems, or those of a more general nature, will be found at the end of each chapter. When I teach the subject, I assign some of these, and work a few of them in class. Unusually challenging problems are ﬂagged by an exclamation point (!) in the margin. Many readers have asked that the answers to problems be provided at the back of the book; unfortunately, just as many are strenuously opposed. I have compromised, supplying answers when this seems particularly appropriate. A complete solution manual is available (to instructors) from the publisher; go to the Pearson web site to order a copy. I have beneﬁtted from the comments of many colleagues. I cannot list them all here, but I would like to thank the following people for especially useful con- tributions to this edition: Burton Brody (Bard), Catherine Crouch (Swarthmore), Joel Franklin (Reed), Ted Jacobson (Maryland), Don Koks (Adelaide), Charles Lane (Berry), Kirk McDonald2 (Princeton), Jim McTavish (Liverpool), Rich Saenz (Cal Poly), Darrel Schroeter (Reed), Herschel Snodgrass (Lewis and Clark), and Larry Tankersley (Naval Academy). Practically everything I know about electrodynamics—certainly about teaching electrodynamics—I owe to Edward Purcell. David J. Grifﬁths 2 Kirk’sweb site, http://www.hep.princeton.edu/∼mcdonald/examples/, is a fantastic resource, with clever explanations, nifty problems, and useful references. Advertisement WHAT IS ELECTRODYNAMICS, AND HOW DOES IT FIT INTO THE GENERAL SCHEME OF PHYSICS? Four Realms of Mechanics In the diagram below, I have sketched out the four great realms of mechanics: Classical Mechanics Quantum Mechanics (Newton) (Bohr, Heisenberg, Schrödinger, et al.) Special Relativity Quantum Field Theory (Einstein) (Dirac, Pauli, Feynman, Schwinger, et al.) Newtonian mechanics is adequate for most purposes in “everyday life,” but for objects moving at high speeds (near the speed of light) it is incorrect, and must be replaced by special relativity (introduced by Einstein in 1905); for objects that are extremely small (near the size of atoms) it fails for different reasons, and is superseded by quantum mechanics (developed by Bohr, Schrödinger, Heisenberg, and many others, in the 1920’s, mostly). For objects that are both very fast and very small (as is common in modern particle physics), a mechanics that com- bines relativity and quantum principles is in order; this relativistic quantum me- chanics is known as quantum ﬁeld theory—it was worked out in the thirties and forties, but even today it cannot claim to be a completely satisfactory system. In this book, save for the last chapter, we shall work exclusively in the domain of classical mechanics, although electrodynamics extends with unique simplic- ity to the other three realms. (In fact, the theory is in most respects automat- ically consistent with special relativity, for which it was, historically, the main stimulus.) Four Kinds of Forces Mechanics tells us how a system will behave when subjected to a given force. There are just four basic forces known (presently) to physics: I list them in the order of decreasing strength: xiv Advertisement xv 1. Strong 2. Electromagnetic 3. Weak 4. Gravitational The brevity of this list may surprise you. Where is friction? Where is the “normal” force that keeps you from falling through the ﬂoor? Where are the chemical forces that bind molecules together? Where is the force of impact between two colliding billiard balls? The answer is that all these forces are electromagnetic. Indeed, it is scarcely an exaggeration to say that we live in an electromagnetic world— virtually every force we experience in everyday life, with the exception of gravity, is electromagnetic in origin. The strong forces, which hold protons and neutrons together in the atomic nu- cleus, have extremely short range, so we do not “feel” them, in spite of the fact that they are a hundred times more powerful than electrical forces. The weak forces, which account for certain kinds of radioactive decay, are also of short range, and they are far weaker than electromagnetic forces. As for gravity, it is so pitifully feeble (compared to all of the others) that it is only by virtue of huge mass con- centrations (like the earth and the sun) that we ever notice it at all. The electrical repulsion between two electrons is 1042 times as large as their gravitational at- traction, and if atoms were held together by gravitational (instead of electrical) forces, a single hydrogen atom would be much larger than the known universe. Not only are electromagnetic forces overwhelmingly dominant in everyday life, they are also, at present, the only ones that are completely understood. There is, of course, a classical theory of gravity (Newton’s law of universal gravitation) and a relativistic one (Einstein’s general relativity), but no entirely satisfactory quantum mechanical theory of gravity has been constructed (though many people are working on it). At the present time there is a very successful (if cumbersome) theory for the weak interactions, and a strikingly attractive candidate (called chro- modynamics) for the strong interactions. All these theories draw their inspiration from electrodynamics; none can claim conclusive experimental veriﬁcation at this stage. So electrodynamics, a beautifully complete and successful theory, has be- come a kind of paradigm for physicists: an ideal model that other theories emulate. The laws of classical electrodynamics were discovered in bits and pieces by Franklin, Coulomb, Ampère, Faraday, and others, but the person who completed the job, and packaged it all in the compact and consistent form it has today, was James Clerk Maxwell. The theory is now about 150 years old. The Uniﬁcation of Physical Theories In the beginning, electricity and magnetism were entirely separate subjects. The one dealt with glass rods and cat’s fur, pith balls, batteries, currents, electrolysis, and lightning; the other with bar magnets, iron ﬁlings, compass needles, and the North Pole. But in 1820 Oersted noticed that an electric current could deﬂect xvi Advertisement a magnetic compass needle. Soon afterward, Ampère correctly postulated that all magnetic phenomena are due to electric charges in motion. Then, in 1831, Faraday discovered that a moving magnet generates an electric current. By the time Maxwell and Lorentz put the ﬁnishing touches on the theory, electricity and magnetism were inextricably intertwined. They could no longer be regarded as separate subjects, but rather as two aspects of a single subject: electromagnetism. Faraday speculated that light, too, is electrical in nature. Maxwell’s theory pro- vided spectacular justiﬁcation for this hypothesis, and soon optics—the study of lenses, mirrors, prisms, interference, and diffraction—was incorporated into electromagnetism. Hertz, who presented the decisive experimental conﬁrmation for Maxwell’s theory in 1888, put it this way: “The connection between light and electricity is now established... In every ﬂame, in every luminous parti- cle, we see an electrical process... Thus, the domain of electricity extends over the whole of nature. It even affects ourselves intimately: we perceive that we possess... an electrical organ—the eye.” By 1900, then, three great branches of physics–electricity, magnetism, and optics–had merged into a single uniﬁed the- ory. (And it was soon apparent that visible light represents only a tiny “window” in the vast spectrum of electromagnetic radiation, from radio through microwaves, infrared and ultraviolet, to x-rays and gamma rays.) Einstein dreamed of a further uniﬁcation, which would combine gravity and electrodynamics, in much the same way as electricity and magnetism had been combined a century earlier. His uniﬁed ﬁeld theory was not particularly success- ful, but in recent years the same impulse has spawned a hierarchy of increasingly ambitious (and speculative) uniﬁcation schemes, beginning in the 1960s with the electroweak theory of Glashow, Weinberg, and Salam (which joins the weak and electromagnetic forces), and culminating in the 1980s with the superstring the- ory (which, according to its proponents, incorporates all four forces in a single “theory of everything”). At each step in this hierarchy, the mathematical difﬁcul- ties mount, and the gap between inspired conjecture and experimental test widens; nevertheless, it is clear that the uniﬁcation of forces initiated by electrodynamics has become a major theme in the progress of physics. The Field Formulation of Electrodynamics The fundamental problem a theory of electromagnetism hopes to solve is this: I hold up a bunch of electric charges here (and maybe shake them around); what happens to some other charge, over there? The classical solution takes the form of a ﬁeld theory: We say that the space around an electric charge is permeated by electric and magnetic ﬁelds (the electromagnetic “odor,” as it were, of the charge). A second charge, in the presence of these ﬁelds, experiences a force; the ﬁelds, then, transmit the inﬂuence from one charge to the other—they “mediate” the interaction. When a charge undergoes acceleration, a portion of the ﬁeld “detaches” itself, in a sense, and travels off at the speed of light, carrying with it energy, momen- tum, and angular momentum. We call this electromagnetic radiation. Its exis- Advertisement xvii tence invites (if not compels) us to regard the ﬁelds as independent dynamical entities in their own right, every bit as “real” as atoms or baseballs. Our interest accordingly shifts from the study of forces between charges to the theory of the ﬁelds themselves. But it takes a charge to produce an electromagnetic ﬁeld, and it takes another charge to detect one, so we had best begin by reviewing the essential properties of electric charge. Electric Charge 1. Charge comes in two varieties, which we call “plus” and “minus,” because their effects tend to cancel (if you have +q and −q at the same point, electrically it is the same as having no charge there at all). This may seem too obvious to warrant comment, but I encourage you to contemplate other possibilities: what if there were 8 or 10 different species of charge? (In chromodynamics there are, in fact, three quantities analogous to electric charge, each of which may be positive or negative.) Or what if the two kinds did not tend to cancel? The extraordinary fact is that plus and minus charges occur in exactly equal amounts, to fantastic precision, in bulk matter, so that their effects are almost completely neutralized. Were it not for this, we would be subjected to enormous forces: a potato would explode violently if the cancellation were imperfect by as little as one part in 1010. 2. Charge is conserved: it cannot be created or destroyed—what there is now has always been. (A plus charge can “annihilate” an equal minus charge, but a plus charge cannot simply disappear by itself—something must pick up that electric charge.) So the total charge of the universe is ﬁxed for all time. This is called global conservation of charge. Actually, I can say something much stronger: Global conservation would allow for a charge to disappear in New York and instantly reappear in San Francisco (that wouldn’t affect the total), and yet we know this doesn’t happen. If the charge was in New York and it went to San Fran- cisco, then it must have passed along some continuous path from one to the other. This is called local conservation of charge. Later on we’ll see how to formulate a precise mathematical law expressing local conservation of charge—it’s called the continuity equation. 3. Charge is quantized. Although nothing in classical electrodynamics requires that it be so, the fact is that electric charge comes only in discrete lumps—integer multiples of the basic unit of charge. If we call the charge on the proton +e, then the electron carries charge −e; the neutron charge zero; the pi mesons +e, 0, and −e; the carbon nucleus +6e; and so on (never 7.392e, or even 1/2e).3 This fundamental unit of charge is extremely small, so for practical purposes it is usually appropriate to ignore quantization altogether. Water, too, “really” con- sists of discrete lumps (molecules); yet, if we are dealing with reasonably large 3 Actually, protons and neutrons are composed of three quarks, which carry fractional charges (± 23 e and ± 13 e). However, free quarks do not appear to exist in nature, and in any event, this does not alter the fact that charge is quantized; it merely reduces the size of the basic unit. xviii Advertisement quantities of it we can treat it as a continuous ﬂuid. This is in fact much closer to Maxwell’s own view; he knew nothing of electrons and protons—he must have pictured charge as a kind of “jelly” that could be divided up into portions of any size and smeared out at will. Units The subject of electrodynamics is plagued by competing systems of units, which sometimes render it difﬁcult for physicists to communicate with one another. The problem is far worse than in mechanics, where Neanderthals still speak of pounds and feet; in mechanics, at least all equations look the same, regardless of the units used to measure quantities. Newton’s second law remains F = ma, whether it is feet-pounds-seconds, kilograms-meters-seconds, or whatever. But this is not so in electromagnetism, where Coulomb’s law may appear variously as q1 q2 1 q1 q2 1 q1 q2 F= r̂ (Gaussian), or F = r̂ (SI), or F = r̂ (HL). r 2 4π 0 r 2 4π r2 Of the systems in common use, the two most popular are Gaussian (cgs) and SI (mks). Elementary particle theorists favor yet a third system: Heaviside-Lorentz. Although Gaussian units offer distinct theoretical advantages, most undergradu- ate instructors seem to prefer SI, I suppose because they incorporate the familiar household units (volts, amperes, and watts). In this book, therefore, I have used SI units. Appendix C provides a “dictionary” for converting the main results into Gaussian units. CHAPTER 1 Vector Analysis 1.1 VECTOR ALGEBRA 1.1.1 Vector Operations If you walk 4 miles due north and then 3 miles due east (Fig. 1.1), you will have gone a total of 7 miles, but you’re not 7 miles from where you set out—you’re only 5. We need an arithmetic to describe quantities like this, which evidently do not add in the ordinary way. The reason they don’t, of course, is that displace- ments (straight line segments going from one point to another) have direction as well as magnitude (length), and it is essential to take both into account when you combine them. Such objects are called vectors: velocity, acceleration, force and momentum are other examples. By contrast, quantities that have magnitude but no direction are called scalars: examples include mass, charge, density, and temperature. I shall use boldface (A, B, and so on) for vectors and ordinary type for scalars. The magnitude of a vector A is written |A| or, more simply, A. In diagrams, vec- tors are denoted by arrows: the length of the arrow is proportional to the magni- tude of the vector, and the arrowhead indicates its direction. Minus A (−A) is a vector with the same magnitude as A but of opposite direction (Fig. 1.2). Note that vectors have magnitude and direction but not location: a displacement of 4 miles due north from Washington is represented by the same vector as a displacement 4 miles north from Baltimore (neglecting, of course, the curvature of the earth). On a diagram, therefore, you can slide the arrow around at will, as long as you don’t change its length or direction. We deﬁne four vector operations: addition and three kinds of multiplication. 3 mi 4 mi 5 mi A −A FIGURE 1.1 FIGURE 1.2 1 2 Chapter 1 Vector Analysis B −B A (A+B) (B+A) A (A−B) A B FIGURE 1.3 FIGURE 1.4 (i) Addition of two vectors. Place the tail of B at the head of A; the sum, A + B, is the vector from the tail of A to the head of B (Fig. 1.3). (This rule generalizes the obvious procedure for combining two displacements.) Addition is commutative: A + B = B + A; 3 miles east followed by 4 miles north gets you to the same place as 4 miles north followed by 3 miles east. Addition is also associative: (A + B) + C = A + (B + C). To subtract a vector, add its opposite (Fig. 1.4): A − B = A + (−B). (ii) Multiplication by a scalar. Multiplication of a vector by a positive scalar a multiplies the magnitude but leaves the direction unchanged (Fig. 1.5). (If a is negative, the direction is reversed.) Scalar multiplication is distributive: a(A + B) = aA + aB. (iii) Dot product of two vectors. The dot product of two vectors is deﬁned by A · B ≡ AB cos θ, (1.1) where θ is the angle they form when placed tail-to-tail (Fig. 1.6). Note that A · B is itself a scalar (hence the alternative name scalar product). The dot product is commutative, A · B = B · A, and distributive, A · (B + C) = A · B + A · C. (1.2) Geometrically, A · B is the product of A times the projection of B along A (or the product of B times the projection of A along B). If the two vectors are parallel, then A · B = AB. In particular, for any vector A, A · A = A2. (1.3) If A and B are perpendicular, then A · B = 0. 1.1 Vector Algebra 3 2A A A θ B FIGURE 1.5 FIGURE 1.6 Example 1.1. Let C = A − B (Fig. 1.7), and calculate the dot product of C with itself. Solution C · C = (A − B) · (A − B) = A · A − A · B − B · A + B · B, or C 2 = A2 + B 2 − 2AB cos θ. This is the law of cosines. (iv) Cross product of two vectors. The cross product of two vectors is de- ﬁned by A × B ≡ AB sin θ n̂, (1.4) where n̂ is a unit vector (vector of magnitude 1) pointing perpendicular to the plane of A and B. (I shall use a hat ( ˆ ) to denote unit vectors.) Of course, there are two directions perpendicular to any plane: “in” and “out.” The ambiguity is resolved by the right-hand rule: let your ﬁngers point in the direction of the ﬁrst vector and curl around (via the smaller angle) toward the second; then your thumb indicates the direction of n̂. (In Fig. 1.8, A × B points into the page; B × A points out of the page.) Note that A × B is itself a vector (hence the alternative name vector product). The cross product is distributive, A × (B + C) = (A × B) + (A × C), (1.5) but not commutative. In fact, (B × A) = −(A × B). (1.6) 4 Chapter 1 Vector Analysis A C A θ θ B B FIGURE 1.7 FIGURE 1.8 Geometrically, |A × B| is the area of the parallelogram generated by A and B (Fig. 1.8). If two vectors are parallel, their cross product is zero. In particular, A×A=0 for any vector A. (Here 0 is the zero vector, with magnitude 0.) Problem 1.1 Using the deﬁnitions in Eqs. 1.1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive, a) when the three vectors are coplanar; ! b) in the general case. Problem 1.2 Is the cross product associative? ? (A × B) × C = A × (B × C). If so, prove it; if not, provide a counterexample (the simpler the better). 1.1.2 Vector Algebra: Component Form In the previous section, I deﬁned the four vector operations (addition, scalar mul- tiplication, dot product, and cross product) in “abstract” form—that is, without reference to any particular coordinate system. In practice, it is often easier to set up Cartesian coordinates x, y, z and work with vector components. Let x̂, ŷ, and ẑ be unit vectors parallel to the x, y, and z axes, respectively (Fig. 1.9(a)). An arbitrary vector A can be expanded in terms of these basis vectors (Fig. 1.9(b)): z z A z Azz x Ax x y y y x (a) x Ayy (b) FIGURE 1.9 1.1 Vector Algebra 5 A = A x x̂ + A y ŷ + A z ẑ. The numbers A x , A y , and A z , are the “components” of A; geometrically, they are the projections of A along the three coordinate axes (A x = A · x̂, A y = A · ŷ, A z = A · ẑ). We can now reformulate each of the four vector operations as a rule for manipulating components: A + B = (A x x̂ + A y ŷ + A z ẑ) + (Bx x̂ + B y ŷ + Bz ẑ) = (A x + Bx )x̂ + (A y + B y )ŷ + (A z + Bz )ẑ. (1.7) Rule (i): To add vectors, add like components. aA = (a A x )x̂ + (a A y )ŷ + (a A z )ẑ. (1.8) Rule (ii): To multiply by a scalar, multiply each component. Because x̂, ŷ, and ẑ are mutually perpendicular unit vectors, x̂ · x̂ = ŷ · ŷ = ẑ · ẑ = 1; x̂ · ŷ = x̂ · ẑ = ŷ · ẑ = 0. (1.9) Accordingly, A · B = (A x x̂ + A y ŷ + A z ẑ) · (Bx x̂ + B y ŷ + Bz ẑ) = A x B x + A y B y + A z Bz. (1.10) Rule (iii): To calculate the dot product, multiply like components, and add. In particular, A · A = A2x + A2y + A2z , so A= A2x + A2y + A2z. (1.11) (This is, if you like, the three-dimensional generalization of the Pythagorean theorem.) Similarly,1 x̂ × x̂ = ŷ × ŷ = ẑ × ẑ = 0, x̂ × ŷ = −ŷ × x̂ = ẑ, ŷ × ẑ = −ẑ × ŷ = x̂, ẑ × x̂ = −x̂ × ẑ = ŷ. (1.12) 1 These signs pertain to a right-handed coordinate system (x-axis out of the page, y-axis to the right, z-axis up, or any rotated version thereof). In a left-handed system (z-axis down), the signs would be reversed: x̂ × ŷ = −ẑ, and so on. We shall use right-handed systems exclusively. 6 Chapter 1 Vector Analysis Therefore, A × B = (A x x̂ + A y ŷ + A z ẑ) × (Bx x̂ + B y ŷ + Bz ẑ) (1.13) = (A y Bz − A z B y )x̂ + (A z Bx − A x Bz )ŷ + (A x B y − A y Bx )ẑ. This cumbersome expression can be written more neatly as a determinant: x̂ ŷ ẑ A × B = A x A y A z . (1.14) B x B y Bz Rule (iv): To calculate the cross product, form the determinant whose ﬁrst row is x̂, ŷ, ẑ, whose second row is A (in component form), and whose third row is B. Example 1.2. Find the angle between the face diagonals of a cube. Solution We might as well use a cube of side 1, and place it as shown in Fig. 1.10, with one corner at the origin. The face diagonals A and B are A = 1 x̂ + 0 ŷ + 1 ẑ; B = 0 x̂ + 1 ŷ + 1 ẑ. z (0, 0, 1) B θ A (0, 1, 0) y x (1, 0, 0) FIGURE 1.10 So, in component form, A · B = 1 · 0 + 0 · 1 + 1 · 1 = 1. On the other hand, in “abstract” form, √ √ A · B = AB cos θ = 2 2 cos θ = 2 cos θ. Therefore, cos θ = 1/2, or θ = 60◦. Of course, you can get the answer more easily by drawing in a diagonal across the top of the cube, completing the equilateral triangle. But in cases where the geometry is not so simple, this device of comparing the abstract and component forms of the dot product can be a very efﬁcient means of ﬁnding angles. 1.1 Vector Algebra 7 Problem 1.3 Find the angle between the body diagonals of a cube. Problem 1.4 Use the cross product to ﬁnd the components of the unit vector n̂ perpendicular to the shaded plane in Fig. 1.11. 1.1.3 Triple Products Since the cross product of two vectors is itself a vector, it can be dotted or crossed with a third vector to form a triple product. (i) Scalar triple product: A · (B × C). Geometrically, |A · (B × C)| is the volume of the parallelepiped generated by A, B, and C, since |B × C| is the area of the base, and |A cos θ | is the altitude (Fig. 1.12). Evidently, A · (B × C) = B · (C × A) = C · (A × B), (1.15) for they all correspond to the same ﬁgure. Note that “alphabetical” order is preserved—in view of Eq. 1.6, the “nonalphabetical” triple products, A · (C × B) = B · (A × C) = C · (B × A), have the opposite sign. In component form, Ax Ay Az A · (B × C) = Bx By Bz . (1.16) Cx Cy Cz Note that the dot and cross can be interchanged: A · (B × C) = (A × B) · C (this follows immediately from Eq. 1.15); however, the placement of the parenthe- ses is critical: (A · B) × C is a meaningless expression—you can’t make a cross product from a scalar and a vector. z 3 n A n 2 y θ 1 C x B FIGURE 1.11 FIGURE 1.12 8 Chapter 1 Vector Analysis (ii) Vector triple product: A × (B × C). The vector triple product can be simpliﬁed by the so-called BAC-CAB rule: A × (B × C) = B(A · C) − C(A · B). (1.17) Notice that (A × B) × C = −C × (A × B) = −A(B · C) + B(A · C) is an entirely different vector (cross-products are not associative). All higher vec- tor products can be similarly reduced, often by repeated application of Eq. 1.17, so it is never necessary for an expression to contain more than one cross product in any term. For instance, (A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C); A × [B × (C × D)] = B[A · (C × D)] − (A · B)(C × D). (1.18) Problem 1.5 Prove the BAC-CAB rule by writing out both sides in component form. Problem 1.6 Prove that [A × (B × C)] + [B × (C × A)] + [C × (A × B)] = 0. Under what conditions does A × (B × C) = (A × B) × C? 1.1.4 Position, Displacement, and Separation Vectors The location of a point in three dimensions can be described by listing its Cartesian coordinates (x, y, z). The vector to that point from the origin (O) is called the position vector (Fig. 1.13): r ≡ x x̂ + y ŷ + z ẑ. (1.19) z Source point r r (x, y, z) r r⬘ z y r Field point O O x x y FIGURE 1.13 FIGURE 1.14 1.1 Vector Algebra 9 I will reserve the letter r for this purpose, throughout the book. Its magnitude, r = x 2 + y2 + z2, (1.20) is the distance from the origin, and r x x̂ + y ŷ + z ẑ r̂ = = (1.21) r x 2 + y2 + z2 is a unit vector pointing radially outward. The inﬁnitesimal displacement vector, from (x, y, z) to (x + d x, y + dy, z + dz), is dl = d x x̂ + dy ŷ + dz ẑ. (1.22) (We could call this dr, since that’s what it is, but it is useful to have a special notation for inﬁnitesimal displacements.) In electrodynamics, one frequently encounters problems involving two points—typically, a source point, r , where an electric charge is located, and a ﬁeld point, r, at which you are calculating the electric or magnetic ﬁeld (Fig. 1.14). It pays to adopt right from the start some short-hand notation for the separation vector from the source point to the ﬁeld point. I shall use for this purpose the script letter r: r ≡ r − r. (1.23) Its magnitude is r = |r − r |, (1.24) and a unit vector in the direction from r to r is r r − r r̂ = =. (1.25) r |r − r | In Cartesian coordinates, r = (x − x )x̂ + (y − y )ŷ + (z − z )ẑ, (1.26) r = (x − x )2 + (y − y )2 + (z − z )2 , (1.27) (x − x )x̂ + (y − y )ŷ + (z − z )ẑ r̂ = (1.28) (x − x )2 + (y − y )2 + (z − z )2 (from which you can appreciate the economy of the script-r notation). Problem 1.7 Find the separation vector r from the source point (2,8,7) to the ﬁeld point (4,6,8). Determine its magnitude (r), and construct the unit vector r̂. 10 Chapter 1 Vector Analysis 1.1.5 How Vectors Transform2 The deﬁnition of a vector as “a quantity with a magnitude and direction” is not altogether satisfactory: What precisely does “direction” mean? This may seem a pedantic question, but we shall soon encounter a species of derivative that looks rather like a vector, and we’ll want to know for sure whether it is one. You might be inclined to say that a vector is anything that has three components that combine properly under addition. Well, how about this: We have a barrel of fruit that contains N x pears, N y apples, and Nz bananas. Is N = N x x̂ + N y ŷ + Nz ẑ a vector? It has three components, and when you add another barrel with Mx pears, M y apples, and Mz bananas the result is (N x + Mx ) pears, (N y + M y ) apples, (Nz + Mz ) bananas. So it does add like a vector. Yet it’s obviously not a vector, in the physicist’s sense of the word, because it doesn’t really have a direction. What exactly is wrong with it? The answer is that N does not transform properly when you change coordi- nates. The coordinate frame we use to describe positions in space is of course entirely arbitrary, but there is a speciﬁc geometrical transformation law for con- verting vector components from one frame to another. Suppose, for instance, the x, y, z system is rotated by angle φ, relative to x, y, z, about the common x = x axes. From Fig. 1.15, A y = A cos θ, A z = A sin θ, while A y = A cos θ = A cos(θ − φ) = A(cos θ cos φ + sin θ sin φ) = cos φ A y + sin φ A z , A z = A sin θ = A sin(θ − φ) = A(sin θ cos φ − cos θ sin φ) = − sin φ A y + cos φ A z. z z A y θ θ φ y FIGURE 1.15 2 This section can be skipped without loss of continuity. 1.1 Vector Algebra 11 We might express this conclusion in matrix notation: Ay cos φ sin φ Ay =. (1.29) Az − sin φ cos φ Az More generally, for rotation about an arbitrary axis in three dimensions, the transformation law takes the form ⎛ ⎞ ⎛ ⎞⎛ ⎞ Ax Rx x Rx y Rx z Ax ⎝ A y ⎠ = ⎝ R yx R yy R yz ⎠ ⎝ A y ⎠ , (1.30) Az Rzx Rzy Rzz Az or, more compactly, 3 Ai = Ri j A j , (1.31) j=1 where the index 1 stands for x, 2 for y, and 3 for z. The elements of the ma- trix R can be ascertained, for a given rotation, by the same sort of trigonometric arguments as we used for a rotation about the x axis. Now: Do the components of N transform in this way? Of course not—it doesn’t matter what coordinates you use to represent positions in space; there are still just as many apples in the barrel. You can’t convert a pear into a banana by choosing a different set of axes, but you can turn A x into A y. Formally, then, a vector is any set of three components that transforms in the same manner as a displace- ment when you change coordinates. As always, displacement is the model for the behavior of all vectors.3 By the way, a (second-rank) tensor is a quantity with nine components, Tx x , Tx y , Tx z , Tyx ,... , Tzz , which transform with two factors of R: T x x = Rx x (Rx x Tx x + Rx y Tx y + Rx z Tx z ) + Rx y (Rx x Tyx + Rx y Tyy + Rx z Tyz ) + Rx z (Rx x Tzx + Rx y Tzy + Rx z Tzz ),... or, more compactly, 3 3 T ij = Rik R jl Tkl. (1.32) k=1 l=1 3 If you’re a mathematician you might want to contemplate generalized vector spaces in which the “axes” have nothing to do with direction and the basis vectors are no longer x̂, ŷ, and ẑ (indeed, there may be more than three dimensions). This is the subject of linear algebra. But for our purposes all vectors live in ordinary 3-space (or, in Chapter 12, in 4-dimensional space-time.) 12 Chapter 1 Vector Analysis In general, an nth-rank tensor has n indices and 3n components, and transforms with n factors of R. In this hierarchy, a vector is a tensor of rank 1, and a scalar is a tensor of rank zero.4 Problem 1.8 (a) Prove that the two-dimensional rotation matrix (Eq. 1.29) preserves dot prod- ucts. (That is, show that A y B y + A z B z = A y B y + A z Bz.) (b) What constraints must the elements (Ri j ) of the three-dimensional rotation ma- trix (Eq. 1.30) satisfy, in order to preserve the length of A (for all vectors A)? Problem 1.9 Find the transformation matrix R that describes a rotation by 120◦ about an axis from the origin through the point (1, 1, 1). The rotation is clockwise as you look down the axis toward the origin. Problem 1.10 (a) How do the components of a vector5 transform under a translation of coordi- nates (x = x, y = y − a, z = z, Fig. 1.16a)? (b) How do the components of a vector transform under an inversion of coordinates (x = −x, y = −y, z = −z, Fig. 1.16b)? (c) How do the components of a cross product (Eq. 1.13) transform under inver- sion? [The cross-product of two vectors is properly called a pseudovector be- cause of this “anomalous” behavior.] Is the cross product of two pseudovectors a vector, or a pseudovector? Name two pseudovector quantities in classical me- chanics. (d) How does the scalar triple product of three vectors transform under inversions? (Such an object is called a pseudoscalar.) z z z x a y } y y y x x x (a) z (b) FIGURE 1.16 4A scalar does not change when you change coordinates. In particular, the components of a vector are not scalars, but the magnitude is. 5 Beware: The vector r (Eq. 1.19) goes from a speciﬁc point in space (the origin, O) to the point P = (x, y, z). Under translations the new origin (Ō) is at a different location, and the arrow from Ō to P is a completely different vector. The original vector r still goes from O to P, regardless of the coordinates used to label these points. 1.2 Differential Calculus 13 1.2 DIFFERENTIAL CALCULUS 1.2.1 “Ordinary” Derivatives Suppose we have a function of one variable: f (x). Question: What does the derivative, d f /d x, do for us? Answer: It tells us how rapidly the function f (x) varies when we change the argument x by a tiny amount, d x: df df = d x. (1.33) dx In words: If we increment x by an inﬁnitesimal amount d x, then f changes by an amount d f ; the derivative is the proportionality factor. For example, in Fig. 1.17(a), the function varies slowly with x, and the derivative is correspond- ingly small. In Fig. 1.17(b), f increases rapidly with x, and the derivative is large, as you move away from x = 0. Geometrical Interpretation: The derivative d f /d x is the slope of the graph of f versus x. 1.2.2 Gradient Suppose, now, that we have a function of three variables—say, the temperature T (x, y, z) in this room. (Start out in one corner, and set up a system of axes; then for each point (x, y, z) in the room, T gives the temperature at that spot.) We want to generalize the notion of “derivative” to functions like T , which depend not on one but on three variables. A derivative is supposed to tell us how fast the function varies, if we move a little distance. But this time the situation is more complicated, because it depends on what direction we move: If we go straight up, then the temperature will prob- ably increase fairly rapidly, but if we move horizontally, it may not change much at all. In fact, the question “How fast does T vary?” has an inﬁnite number of answers, one for each direction we might choose to explore. Fortunately, the problem is not as bad as it looks. A theorem on partial deriva- tives states that ∂T ∂T ∂T dT = dx + dy + dz. (1.34) ∂x ∂y ∂z f f x x (a) (b) FIGURE 1.17 14 Chapter 1 Vector Analysis This tells us how T changes when we alter all three variables by the inﬁnites- imal amounts d x, dy, dz. Notice that we do not require an inﬁnite number of derivatives—three will sufﬁce: the partial derivatives along each of the three co- ordinate directions. Equation 1.34 is reminiscent of a dot product: ∂T ∂T ∂T dT = x̂ + ŷ + ẑ · (d x x̂ + dy ŷ + dz ẑ) ∂x ∂y ∂z = (∇T ) · (dl), (1.35) where ∂T ∂T ∂T ∇T ≡ x̂ + ŷ + ẑ (1.36) ∂x ∂y ∂z is the gradient of T. Note that ∇T is a vector quantity, with three components; it is the generalized derivative we have been looking for. Equation 1.35 is the three-dimensional version of Eq. 1.33. Geometrical Interpretation of the Gradient: Like any vector, the gradient has magnitude and direction. To determine its geometrical meaning, let’s rewrite the dot product (Eq. 1.35) using Eq. 1.1: dT = ∇T · dl = |∇T ||dl| cos θ, (1.37) where θ is the angle between ∇T and dl. Now, if we ﬁx the magnitude |dl| and search around in various directions (that is, vary θ ), the maximum change in T evidentally occurs when θ = 0 (for then cos θ = 1). That is, for a ﬁxed distance |dl|, dT is greatest when I move in the same direction as ∇T. Thus: The gradient ∇T points in the direction of maximum increase of the function T. Moreover: The magnitude |∇T | gives the slope (rate of increase) along this maximal direction. Imagine you are standing on a hillside. Look all around you, and ﬁnd the di- rection of steepest ascent. That is the direction of the gradient. Now measure the slope in that direction (rise over run). That is the magnitude of the gradient. (Here the function we’re talking about is the height of the hill, and the coordinates it depends on are positions—latitude and longitude, say. This function depends on only two variables, not three, but the geometrical meaning of the gradient is easier to grasp in two dimensions.) Notice from Eq. 1.37 that the direction of maximum descent is opposite to the direction of maximum ascent, while at right angles (θ = 90◦ ) the slope is zero (the gradient is perpendicular to the contour lines). You can conceive of surfaces that do not have these properties, but they always have “kinks” in them, and correspond to nondifferentiable functions. What would it mean for the gradient to vanish? If ∇T = 0 at (x, y, z), then dT = 0 for small displacements about the point (x, y, z). This is, then, a stationary point of the function T (x, y, z). It could be a maximum (a summit), 1.2 Differential Calculus 15 a minimum (a valley), a saddle point (a pass), or a “shoulder.” This is analogous to the situation for functions of one variable, where a vanishing derivative signals a maximum, a minimum, or an inﬂection. In particular, if you want to locate the extrema of a function of three variables, set its gradient equal to zero. Example 1.3. Find the gradient of r = x 2 + y 2 + z 2 (the magnitude of the position vector). Solution ∂r ∂r ∂r ∇r = x̂ + ŷ + ẑ ∂x ∂y ∂z 1 2x 1 2y 1 2z = x̂ + ŷ + ẑ 2 x 2 + y2 + z2 2 x 2 + y2 + z2 2 x 2 + y2 + z2 x x̂ + y ŷ + z ẑ r = = = r̂. x +y +z 2 2 2 r Does this make sense? Well, it says that the distance from the origin increases most rapidly in the radial direction, and that its rate of increase in that direction is 1... just what you’d expect. Problem 1.11 Find the gradients of the following functions: (a) f (x, y, z) = x 2 + y 3 + z 4. (b) f (x, y, z) = x 2 y 3 z 4. (c) f (x, y, z) = e x sin(y) ln(z). Problem 1.12 The height of a certain hill (in feet) is given by h(x, y) = 10(2x y − 3x 2 − 4y 2 − 18x + 28y + 12), where y is the distance (in miles) north, x the distance east of South Hadley. (a) Where is the top of the hill located? (b) How high is the hill? (c) How steep is the slope (in feet per mile) at a point 1 mile north and one mile east of South Hadley? In what direction is the slope steepest, at that point? Problem 1.13 Let r be the separation vector from a ﬁxed point (x , y , z ) to the point (x, y, z), and let r be its length. Show that (a) ∇(r2 ) = 2r. (b) ∇(1/r) = −r̂/r2. (c) What is the general formula for ∇(rn )? 16 Chapter 1 Vector Analysis ! Problem 1.14 Suppose that f is a function of two variables (y and z) only. Show that the gradient ∇ f = (∂ f /∂ y)ŷ + (∂ f /∂z)ẑ transforms as a vector un- der rotations, Eq. 1.29. [Hint: (∂ f /∂ y) = (∂ f /∂ y)(∂ y/∂ y) + (∂ f /∂z)(∂z/∂ y), and the analogous formula for ∂ f /∂z. We know that y = y cos φ + z sin φ and z = −y sin φ + z cos φ; “solve” these equations for y and z (as functions of y and z), and compute the needed derivatives ∂ y/∂ y, ∂z/∂ y, etc.] 1.2.3 The Del Operator The gradient has the formal appearance of a vector, ∇, “multiplying” a scalar T : ∂ ∂ ∂ ∇T = x̂ + ŷ + ẑ T. (1.38) ∂x ∂y ∂z (For once, I write the unit vectors to the left, just so no one will think this means ∂ x̂/∂ x, and so on—which would be zero, since x̂ is constant.) The term in paren- theses is called del: ∂ ∂ ∂ ∇ = x̂ + ŷ + ẑ. (1.39) ∂x ∂y ∂z Of course, del is not a vector, in the usual sense. Indeed, it doesn’t mean much until we provide it with a function to act upon. Furthermore, it does not “multiply” T ; rather, it is an instruction to differentiate what follows. To be precise, then, we say that ∇ is a vector operator that acts upon T , not a vector that multiplies T. With this qualiﬁcation, though, ∇ mimics the behavior of an ordinary vector in virtually every way; almost anything that can be done with other vectors can also be done with ∇, if we merely translate “multiply” by “act upon.” So by all means take the vector appearance of ∇ seriously: it is a marvelous piece of notational simpliﬁcation, as you will appreciate if you ever consult Maxwell’s original work on electromagnetism, written without the beneﬁt of ∇. Now, an ordinary vector A can multiply in three ways: 1. By a scalar a : Aa; 2. By a vector B, via the dot product: A · B; 3. By a vector B via the cross product: A × B. Correspondingly, there are three ways the operator ∇ can act: 1. On a scalar function T : ∇T (the gradient); 2. On a vector function v, via the dot product: ∇ · v (the divergence); 3. On a vector function v, via the cross product: ∇ × v (the curl). 1.2 Differential Calculus 17 We have already discussed the gradient. In the following sections we examine the other two vector derivatives: divergence and curl. 1.2.4 The Divergence From the deﬁnition of ∇ we construct the divergence: ∂ ∂ ∂ ∇ · v = x̂ + ŷ + ẑ · (vx x̂ + v y ŷ + vz ẑ) ∂x ∂y ∂z ∂vx ∂v y ∂vz = + +. (1.40) ∂x ∂y ∂z Observe that the divergence of a vector function6 v is itself a scalar ∇ · v. Geometrical Interpretation: The name divergence is well chosen, for ∇ · v is a measure of how much the vector v spreads out (diverges) from the point in question. For example, the vector function in Fig. 1.18a has a large (positive) divergence (if the arrows pointed in, it would be a negative divergence), the func- tion in Fig. 1.18b has zero divergence, and the function in Fig. 1.18c again has a positive divergence. (Please understand that v here is a function—there’s a differ- ent vector associated with every point in space. In the diagrams, of course, I can only draw the arrows at a few representative locations.) Imagine standing at the edge of a pond. Sprinkle some sawdust or pine needles on the surface. If the material spreads out, then you dropped it at a point of positive divergence; if it collects together, you dropped it at a point of negative divergence. (The vector function v in this model is the velocity of the water at the surface— this is a two-dimensional example, but it helps give one a “feel” for what the divergence means. A point of positive divergence is a source, or “faucet”; a point of negative divergence is a sink, or “drain.”) (a) (b) (c) FIGURE 1.18 6 A vector function v(x, y, z) = v (x, y, z) x̂ + v (x, y, z) ŷ + v (x, y, z) ẑ is really three functions— x y z one for each component. There’s no such thing as the divergence of a scalar. 18 Chapter 1 Vector Analysis Example 1.4. Suppose the functions in Fig. 1.18 are va = r = x x̂ + y ŷ + z ẑ, vb = ẑ, and vc = z ẑ. Calculate their divergences. Solution ∂ ∂ ∂ ∇ · va = (x) + (y) + (z) = 1 + 1 + 1 = 3. ∂x ∂y ∂z As anticipated, this function has a positive divergence. ∂ ∂ ∂ ∇ · vb = (0) + (0) + (1) = 0 + 0 + 0 = 0, ∂x ∂y ∂z as expected. ∂ ∂ ∂ ∇ · vc = (0) + (0) + (z) = 0 + 0 + 1 = 1. ∂x ∂y ∂z Problem 1.15 Calculate the divergence of the following vector functions: (a) va = x 2 x̂ + 3x z 2 ŷ − 2x z ẑ. (b) vb = x y x̂ + 2yz ŷ + 3zx ẑ. (c) vc = y 2 x̂ + (2x y + z 2 ) ŷ + 2yz ẑ. Problem 1.16 Sketch the vector function r̂ v= , r2 and compute its divergence. The answer may surprise you... can you explain it? ! Problem 1.17 In two dimensions, show that the divergence transforms as a scalar under rotations. [Hint: Use Eq. 1.29 to determine v y and v z , and the method of Prob. 1.14 to calculate the derivatives. Your aim is to show that ∂v y /∂ y + ∂v z /∂z = ∂v y /∂ y + ∂vz /∂z.] 1.2.5 The Curl From the deﬁnition of ∇ we construct the curl: x̂ ŷ ẑ ∇ × v = ∂/∂ x ∂/∂ y ∂/∂z vx vy vz ∂vz ∂v y ∂vx ∂vz ∂v y ∂vx = x̂ − + ŷ − + ẑ −. (1.41) ∂y ∂z ∂z ∂x ∂x ∂y 1.2 Differential Calculus 19 z z y y x (a) x (b) FIGURE 1.19 Notice that the curl of a vector function7 v is, like any cross product, a vector. Geometrical Interpretation: The name curl is also well chosen, for ∇ × v is a measure of how much the vector v swirls around the point in question. Thus the three functions in Fig. 1.18 all have zero curl (as you can easily check for yourself), whereas the functions in Fig. 1.19 have a substantial curl, pointing in the z direction, as the natural right-hand rule would suggest. Imagine (again) you are standing at the edge of a pond. Float a small paddlewheel (a cork with toothpicks pointing out radially would do); if it starts to rotate, then you placed it at a point of nonzero curl. A whirlpool would be a region of large curl. Example 1.5. Suppose the function sketched in Fig. 1.19a is va = −y x̂ + x ŷ, and that in Fig. 1.19b is vb = x ŷ. Calculate their curls. Solution x̂ ŷ ẑ ∇ × va = ∂/∂ x ∂/∂ y ∂/∂z = 2ẑ, −y x 0 and x̂ ŷ ẑ ∇ × vb = ∂/∂ x ∂/∂ y ∂/∂z = ẑ. 0 x 0 As expected, these curls point in the +z direction. (Incidentally, they both have zero divergence, as you might guess from the pictures: nothing is “spreading out”... it just “swirls around.”) 7 There’s no such thing as the curl of a scalar. 20 Chapter 1 Vector Analysis Problem 1.18 Calculate the curls of the vector functions in Prob. 1.15. Problem 1.19 Draw a circle in the x y plane. At a few representative points draw the vector v tangent to the circle, pointing in the clockwise direction. By comparing adjacent vectors, determine the sign of ∂vx /∂ y and ∂v y /∂ x. According to Eq. 1.41, then, what is the direction of ∇ × v? Explain how this example illustrates the geo- metrical interpretation of the curl. Problem 1.20 Construct a vector function that has zero divergence and zero curl everywhere. (A constant will do the job, of course, but make it something a little more interesting than that!) 1.2.6 Product Rules The calculation of ordinary derivatives is facilitated by a number of rules, such as the sum rule: d df dg ( f + g) = + , dx dx dx the rule for multiplying by a constant: d df (k f ) = k , dx dx the product rule: d dg df ( f g) = f +g , dx dx dx and the quotient rule: g d f − f dg d f = dx 2 dx. dx g g Similar relations hold for the vector derivatives. Thus, ∇( f + g) = ∇ f + ∇g, ∇ · (A + B) = (∇ · A) + (∇ · B), ∇ × (A + B) = (∇ × A) + (∇ × B), and ∇(k f ) = k∇ f, ∇ · (kA) = k(∇ · A), ∇ × (kA) = k(∇ × A), as you can check for yourself. The product rules are not quite so simple. There are two ways to construct a scalar as the product of two functions: fg (product of two scalar functions), A·B (dot product of two vector functions), 1.2 Differential Calculus 21 and two ways to make a vector: fA (scalar times vector), A×B (cross product of two vectors). Accordingly, there are six product rules, two for gradients: (i) ∇( f g) = f ∇g + g∇ f, (ii) ∇(A · B) = A × (∇ × B) + B × (∇ × A) + (A · ∇)B + (B · ∇)A, two for divergences: (iii) ∇ · ( f A) = f (∇ · A) + A · (∇ f ), (iv) ∇ · (A × B) = B · (∇ × A) − A · (∇ × B), and two for curls: (v) ∇ × ( f A) = f (∇ × A) − A × (∇ f ), (vi) ∇ × (A × B) = (B · ∇)A − (A · ∇)B + A(∇ · B) − B(∇ · A). You will be using these product rules so frequently that I have put them inside the front cover for easy reference. The proofs come straight from the product rule for ordinary derivatives. For instance, ∂ ∂ ∂ ∇ · ( f A) = ( f Ax ) + ( f A y ) + ( f Az ) ∂x ∂y ∂z ∂f ∂ Ax ∂f ∂ Ay ∂f ∂ Az = Ax + f + Ay + f + Az + f ∂x ∂x ∂y ∂y ∂z ∂z = (∇ f ) · A + f (∇ · A). It is also possible to formulate three quotient rules: f g∇ f − f ∇g ∇ = , g g2 A g(∇ · A) − A · (∇g) ∇· = , g g2 A g(∇ × A) + A × (∇g) ∇× =. g g2 However, since these can be obtained quickly from the corresponding product rules, there is no point in listing them separately. 22 Chapter 1 Vector Analysis Problem 1.21 Prove product rules (i), (iv), and (v). Problem 1.22 (a) If A and B are two vector functions, what does the expression (A · ∇)B mean? (That is, what are its x, y, and z components, in terms of the Cartesian compo- nents of A, B, and ∇?) (b) Compute (r̂ · ∇)r̂, where r̂ is the unit vector deﬁned in Eq. 1.21. (c) For the functions in Prob. 1.15, evaluate (va · ∇)vb. Problem 1.23 (For masochists only.) Prove product rules (ii) and (vi). Refer to Prob. 1.22 for the deﬁnition of (A · ∇)B. Problem 1.24 Derive the three quotient rules. Problem 1.25 (a) Check product rule (iv) (by calculating each term separately) for the functions A = x x̂ + 2y ŷ + 3z ẑ; B = 3y x̂ − 2x ŷ. (b) Do the same for product rule (ii). (c) Do the same for rule (vi). 1.2.7 Second Derivatives The gradient, the divergence, and the curl are the only ﬁrst derivatives we can make with ∇; by applying ∇ twice, we can construct ﬁve species of second deriva- tives. The gradient ∇T is a vector, so we can take the divergence and curl of it: (1) Divergence of gradient: ∇ · (∇T ). (2) Curl of gradient: ∇ × (∇T ). The divergence ∇ · v is a scalar—all we can do is take its gradient: (3) Gradient of divergence: ∇(∇ · v). The curl ∇ × v is a vector, so we can take its divergence and curl: (4) Divergence of curl: ∇ · (∇ × v). (5) Curl of curl: ∇ × (∇ × v). This exhausts the possibilities, and in fact not all of them give anything new. Let’s consider them one at a time: ∂ ∂ ∂ ∂T ∂T ∂T (1) ∇ · (∇T ) = x̂ + ŷ + ẑ · x̂ + ŷ + ẑ ∂x ∂y ∂z ∂x ∂y ∂z ∂2T ∂2T ∂2T = + +. (1.42) ∂x2 ∂ y2 ∂z 2 1.2 Differential Calculus 23 This object, which we write as ∇ 2 T for short, is called the Laplacian of T ; we shall be studying it in great detail later on. Notice that the Laplacian of a scalar T is a scalar. Occasionally, we shall speak of the Laplacian of a vector, ∇ 2 v. By this we mean a vector quantity whose x-component is the Laplacian of vx , and so on:8 ∇ 2 v ≡ (∇ 2 vx )x̂ + (∇ 2 v y )ŷ + (∇ 2 vz )ẑ. (1.43) This is nothing more than a convenient extension of the meaning of ∇ 2. (2) The curl of a gradient is always zero: ∇ × (∇T ) = 0. (1.44) This is an important fact, which we shall use repeatedly; you can easily prove it from the deﬁnition of ∇, Eq. 1.39. Beware: You might think Eq. 1.44 is “obvi- ously” true—isn’t it just (∇ × ∇)T , and isn’t the cross product of any vector (in this case, ∇) with itself always zero? This reasoning is suggestive, but not quite conclusive, since ∇ is an operator and does not “multiply” in the usual way. The proof of Eq. 1.44, in fact, hinges on the equality of cross derivatives: ∂ ∂T ∂ ∂T =. (1.45) ∂x ∂y ∂y ∂x If you think I’m being fussy, test your intuition on this one: (∇T ) × (∇S). Is that always zero? (It would be, of course, if you replaced the ∇’s by an ordinary vector.) (3) ∇(∇ · v) seldom occurs in physical applications, and it has not been given any special name of its own—it’s just the gradient of the divergence. Notice that ∇(∇ · v) is not the same as the Laplacian of a vector: ∇ 2 v = (∇ · ∇)v = ∇(∇ · v). (4) The divergence of a curl, like the curl of a gradient, is always zero: ∇ · (∇ × v) = 0. (1.46) You can prove this for yourself. (Again, there is a fraudulent short-cut proof, using the vector identity A · (B × C) = (A × B) · C.) (5) As you can check from the deﬁnition of ∇: ∇ × (∇ × v) = ∇(∇ · v) − ∇ 2 v. (1.47) So curl-of-curl gives nothing new; the ﬁrst term is just number (3), and the sec- ond is the Laplacian (of a vector). (In fact, Eq. 1.47 is often used to deﬁne the 8 Incurvilinear coordinates, where the unit vectors themselves depend on position, they too must be differentiated (see Sect. 1.4.1). 24 Chapter 1 Vector Analysis Laplacian of a vector, in preference to Eq. 1.43, which makes explicit reference to Cartesian coordinates.) Really, then, there are just two kinds of second derivatives: the Laplacian (which is of fundamental importance) and the gradient-of-divergence (which we seldom encounter). We could go through a similar ritual to work out third derivatives, but fortunately second derivatives sufﬁce for practically all physical applications. A ﬁnal word on vector differential calculus: It all ﬂows from the operator ∇, and from taking seriously its vectorial character. Even if you remembered only the deﬁnition of ∇, you could easily reconstruct all the rest. Problem 1.26 Calculate the Laplacian of the following functions: (a) Ta = x 2 + 2x y + 3z + 4. (b) Tb = sin x sin y sin z. (c) Tc = e−5x sin 4y cos 3z. (d) v = x 2 x̂ + 3x z 2 ŷ − 2x z ẑ. Problem 1.27 Prove that the divergence of a curl is always zero. Check it for func- tion va in Prob. 1.15. Problem 1.28 Prove that the curl of a gradient is always zero. Check it for function (b) in Prob. 1.11. 1.3 INTEGRAL CALCULUS 1.3.1 Line, Surface, and Volume Integrals In electrodynamics, we encounter several different kinds of integrals, among which the most important are line (or path) integrals, surface integrals (or ﬂux), and volume integrals. (a) Line Integrals. A line integral is an expression of the form b v · dl, (1.48) a where v is a vector function, dl is the inﬁnitesimal displacement vector (Eq. 1.22), and the integral is to be carried out along a prescribed path P from point a to point b (Fig. 1.20). If the path in question forms a closed loop (that is, if b = a), I shall put a circle on the integral sign: v · dl. (1.49) At each point on the path, we take the dot product of v (evaluated at that point) with the displacement dl to the next point on the path. To a physicist,the most familiar example of a line integral is the work done by a force F: W = F · dl. Ordinarily, the value of a line integral depends critically on the path taken from a to b, but there is an important special class of vector functions for which the line 1.3 Integral Calculus 25 z y dl b 2 b (2) (ii) a 1 y a (i) (1) x 1 2 x FIGURE 1.20 FIGURE 1.21 integral is independent of path and is determined entirely by the end points. It will be our business in due course to characterize this special class of vectors. (A force that has this property is called conservative.) Example 1.6. Calculate the line integral of the function v = y 2 x̂ + 2x(y + 1) ŷ from the point a = (1,1, 0) to the point b = (2, 2, 0), along the paths (1) and (2) in Fig. 1.21. What is v · dl for the loop that goes from a to b along (1) and returns to a along (2)? Solution As always, dl = d x x̂ + dy ŷ + dz ẑ. Path (1) consists of two parts. Along the “horizontal” segment, dy = dz = 0, so 2 (i) dl = d x x̂, y = 1, v · dl = y 2 d x = d x, so v · dl = 1 d x = 1. On the “vertical” stretch, d x = dz = 0, so (ii) dl = dy ŷ, x = 2, v · dl = 2x(y + 1) dy = 4(y + 1) dy, so 2 v · dl = 4 (y + 1) dy = 10. 1 By path (1), then, b v · dl = 1 + 10 = 11. a Meanwhile, on path (2) x = y, d x = dy, and dz = 0, so dl = d x x̂ + d x ŷ, v · dl = x 2 d x + 2x(x + 1) d x = (3x 2 + 2x) d x, and b 2 2 v · dl = (3x 2 + 2x) d x = (x 3 + x 2 )1 = 10. a 1 (The strategy here is to get everything in terms of one variable; I could just as well have eliminated x in favor of y.) 26 Chapter 1 Vector Analysis For the loop that goes out (1) and back (2), then, v · dl = 11 − 10 = 1. (b) Surface Integrals. A surface integral is an expression of the form v · da, (1.50) S where v is again some vector function, and the integral is over a speciﬁed surface S. Here da is an inﬁnitesimal patch of area, with direction perpendicular to the surface (Fig. 1.22). There are, of course, two directions perpendicular to any surface, so the sign of a surface integral is intrinsically ambiguous. If the surface is closed (forming a “balloon”), in which case I shall a

Introduction to Electrodynamics (PDF) by David J. Griffiths (2013)

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