Modern Electrodynamics
An engaging writing style and a strong focus on the physics make this comprehensive, graduate-level textbook unique among existing classical electromagnetism textbooks.
Charged particles in vacuum and the electrodynamics of continuous media are given equal attention in discussions of electrostatics, magnetostatics, quasistatics, conservation laws, wave propagation, radiation, scattering, special relativity, and field theory. Extensive use of qualitative arguments similar to those used by working physicists makesModern Electrodynamicsa must-have for every student of this subject.
In 24 chapters, the textbook covers many more topics than can be presented in a typical two-semester course, making it easy for instructors to tailor courses to their specific needs. Close to 120 worked examples and 60 applications boxes help the reader build physical intuition and develop technical skill. Nearly 600 end-of-chapter homework problems encourage students to engage actively with the material. A solutions manual is available for instructors at www.cambridge.org/Zangwill.
Andrew Zangwillis a Professor of Physics at the Georgia Institute of Technology, with research interests in theoretical condensed matter physics. He is the author of the popular textbookPhysics at Surfaces(Cambridge University Press, 1988) and has taught classical electromagnetism at the graduate and undergraduate levels for 20 years.
CUUK1954-FM CUUK1954/Zangwill 978 0 521 89697 9 August 10, 2012 13:53
Modern
Electrodynamics
ANDREW ZANGWILL
School of Physics
Georgia Institute of Technology
CUUK1954-FM CUUK1954/Zangwill 978 0 521 89697 9 August 10, 2012 13:53
cambridge university press
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Published in the United States of America by Cambridge University Press, New York www.cambridge.org
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There are more things in heaven & earth connected with electromagnetism than are yet dream’t of in philosophy.
Joseph Henry, letter to Lewis C. Beck (1827)
Reprinted fromThe Papers of Joseph Henry, Volume I, edited by Nathan Reingold (Washington, DC: Smithsonian Institution Press). Used by permission of the Smithsonian Institution. Copyright 1972.
The search for reason ends at the shore of the known; on the immense expanse beyond it only the ineffable can glide.
Abraham Joshua Heschel,Man is Not Alone(1951)
Why repeat all this? Because there are new generations born every day. Because there are great ideas developed in the history of man, and these ideas do not last unless they are passed purposely and clearly from generation to generation.
Richard Feynman,The Meaning of It All(1963)
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Contents
List of Applications pagexiv
Preface xix
1 Mathematical Preliminaries 1
1.1 Introduction 1
1.2 Vectors 1
1.3 Derivatives 7
1.4 Integrals 9
1.5 Generalized Functions 11
1.6 Fourier Analysis 15
1.7 Orthogonal Transformations 18
1.8 Cartesian Tensors 20
1.9 The Helmholtz Theorem 22
1.10 Lagrange Multipliers 24
Sources, References, and Additional Reading 24
Problems 25
2 The Maxwell Equations 29
2.1 Introduction 29
2.2 The Maxwell Equations in Vacuum 33
2.3 Microscopic vs. Macroscopic 38
2.4 The Maxwell Equations in Matter 43
2.5 Quantum Limits and New Physics 46
2.6 SI Units 50
2.7 A Heuristic Derivation 51
Sources, References, and Additional Reading 53
Problems 55
3 Electrostatics 58
3.1 Introduction 58
3.2 Coulomb’s Law 59
3.3 The Scalar Potential 60
3.4 Gauss’ Law and Solid Angle 68
3.5 Electrostatic Potential Energy 74
3.6 Electrostatic Total Energy 76
3.7 The Electric Stress Tensor 81
Sources, References, and Additional Reading 84
Problems 85
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viii CONTENTS
4 Electric Multipoles 90
4.1 Introduction 90
4.2 The Electric Dipole 92
4.3 Electric Dipole Layers 98
4.4 The Electric Quadrupole 102
4.5 Spherical Mathematics 106
4.6 Spherical and Azimuthal Multipoles 109
4.7 Primitive and Traceless Multipole Moments 116
Sources, References, and Additional Reading 119
Problems 121
5 Conducting Matter 126
5.1 Introduction 126
5.2 Electrostatic Induction 126
5.3 Screening and Shielding 133
5.4 Capacitance 134
5.5 The Energy of a System of Conductors 142
5.6 Forces on Conductors 143
5.7 Real Conductors 149
Sources, References, and Additional Reading 151
Problems 152
6 Dielectric Matter 158
6.1 Introduction 158
6.2 Polarization 158
6.3 The Field Produced by Polarized Matter 162
6.4 The Total Electric Field 165
6.5 Simple Dielectric Matter 167
6.6 The Physics of the Dielectric Constant 175
6.7 The Energy of Dielectric Matter 178
6.8 Forces on Dielectric Matter 184
Sources, References, and Additional Reading 191
Problems 193
7 Laplace’s Equation 197
7.1 Introduction 197
7.2 Potential Theory 198
7.3 Uniqueness 199
7.4 Separation of Variables 201
7.5 Cartesian Symmetry 203
7.6 Azimuthal Symmetry 209
7.7 Spherical Symmetry 212
7.8 Cylindrical Symmetry 215
7.9 Polar Coordinates 218
7.10 The Complex Potential 221
7.11 A Variational Principle 226
Sources, References, and Additional Reading 228
Problems 229
CONTENTS ix
8 Poisson’s Equation 236
8.1 Introduction 236
8.2 The Key Idea: Superposition 236
8.3 The Method of Images 237
8.4 The Green Function Method 250
8.5 The Dirichlet Green Function 252
8.6 The Complex Logarithm Potential 260
8.7 The Poisson-Boltzmann Equation 262
Sources, References, and Additional Reading 264
Problems 265
9 Steady Current 272
9.1 Introduction 272
9.2 Current in Vacuum 273
9.3 Current in Matter 275
9.4 Potential Theory for Ohmic Matter 276
9.5 Electrical Resistance 277
9.6 Joule Heating 280
9.7 Electromotive Force 282
9.8 Current Sources 287
9.9 Diffusion Current: Fick’s Law 291
Sources, References, and Additional Reading 293
Problems 294
10 Magnetostatics 301
10.1 Introduction 301
10.2 The Law of Biot and Savart 304
10.3 Amp`ere’s Law 307
10.4 The Magnetic Scalar Potential 312
10.5 The Vector Potential 320
10.6 The Topology of Magnetic Field Lines 325
Sources, References, and Additional Reading 329
Problems 330
11 Magnetic Multipoles 336
11.1 Introduction 336
11.2 The Magnetic Dipole 337
11.3 Magnetic Dipole Layers 345
11.4 Exterior Multipoles 346
11.5 Interior Multipoles 353
11.6 Axially Symmetric Magnetic Fields 357
Sources, References, and Additional Reading 359
Problems 361
12 Magnetic Force and Energy 365
12.1 Introduction 365
12.2 Charged Particle Motion 366
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x CONTENTS
12.3 The Force between Steady Currents 368
12.4 The Magnetic Dipole 372
12.5 The Magnetic Stress Tensor 381
12.6 Magnetostatic Total Energy 384
12.7 Magnetostatic Potential Energy 389
12.8 Inductance 394
Sources, References, and Additional Reading 399
Problems 401
13 Magnetic Matter 407
13.1 Introduction 407
13.2 Magnetization 407
13.3 The Field Produced by Magnetized Matter 412
13.4 Fictitious Magnetic Charge 415
13.5 The Total Magnetic Field 419
13.6 Simple Magnetic Matter 421
13.7 The Energy of Magnetic Matter 433
13.8 Forces on Magnetic Matter 435
13.9 Permanent Magnetic Matter 443
Sources, References, and Additional Reading 447
Problems 448
14 Dynamic and Quasistatic Fields 455
14.1 Introduction 455
14.2 The Amp`ere-Maxwell Law 456
14.3 Faraday’s Law 460
14.4 Electromagnetic Induction 462
14.5 Slowly Time-Varying Charge in Vacuum 467
14.6 Slowly Time-Varying Current in Vacuum 470
14.7 Quasistatic Fields in Matter 472
14.8 Poor Conductors: Quasi-Electrostatics 473
14.9 Good Conductors: Quasi-Magnetostatics 475
14.10 The Skin Effect 477
14.11 Magnetic Diffusion 481
14.12 Eddy-Current Phenomena 483
14.13 AC Circuit Theory 486
Sources, References, and Additional Reading 493
Problems 494
15 General Electromagnetic Fields 501
15.1 Introduction 501
15.2 Symmetry 501
15.3 Electromagnetic Potentials 503
15.4 Conservation of Energy 507
15.5 Conservation of Linear Momentum 511
15.6 Conservation of Angular Momentum 516
15.7 The Center of Energy 519
15.8 Conservation Laws in Matter 522
CONTENTS xi
15.9 The Force on Isolated Matter 526
Sources, References, and Additional Reading 529
Problems 531
16 Waves in Vacuum 536
16.1 Introduction 536
16.2 The Wave Equation 537
16.3 Plane Waves 539
16.4 Polarization 545
16.5 Wave Packets 552
16.6 The Helmholtz Equation 557
16.7 Beam-Like Waves 558
16.8 Spherical Waves 565
16.9 Hertz Vectors 569
16.10 Forces on Particles in Free Fields 571
Sources, References, and Additional Reading 575
Problems 577
17 Waves in Simple Matter 584
17.1 Introduction 584
17.2 Plane Waves 584
17.3 Reflection and Refraction 588
17.4 Radiation Pressure 599
17.5 Layered Matter 602
17.6 Simple Conducting Matter 607
17.7 Anisotropic Matter 613
Sources, References, and Additional Reading 616
Problems 617
18 Waves in Dispersive Matter 624
18.1 Introduction 624
18.2 Frequency Dispersion 624
18.3 Energy in Dispersive Matter 627
18.4 Transverse and Longitudinal Waves 629
18.5 Classical Models for Frequency Dispersion 630
18.6 Wave Packets in Dispersive Matter 641
18.7 The Consequences of Causality 649
18.8 Spatial Dispersion 656
Sources, References, and Additional Reading 657
Problems 659
19 Guided and Confined Waves 666
19.1 Introduction 666
19.2 Transmission Lines 667
19.3 Planar Conductors 672
19.4 Conducting Tubes 675
19.5 Dielectric Waveguides 687
19.6 Conducting Cavities 693
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xii CONTENTS
19.7 Dielectric Resonators 704
Sources, References, and Additional Reading 706
Problems 707
20 Retardation and Radiation 714
20.1 Introduction 714
20.2 Inhomogeneous Wave Equations 715
20.3 Retardation 719
20.4 The Time-Dependent Electric Dipole 727
20.5 Radiation 730
20.6 Thin-Wire Antennas 737
20.7 Cartesian Multipole Radiation 743
20.8 Spherical Multipole Radiation 755
20.9 Radiation in Matter 762
Sources, References, and Additional Reading 765
Problems 767
21 Scattering and Diffraction 775
21.1 Introduction 775
21.2 The Scattering Cross Section 776
21.3 Thomson Scattering 777
21.4 Rayleigh Scattering 782
21.5 Two Exactly Solvable Problems 783
21.6 Two Approximation Schemes 790
21.7 The Total Cross Section 793
21.8 Diffraction by a Planar Aperture 797
21.9 Generalized Optical Principles 807
Sources, References, and Additional Reading 812
Problems 814
22 Special Relativity 822
22.1 Introduction 822
22.2 Galileo’s Relativity 823
22.3 Einstein’s Relativity 825
22.4 The Lorentz Transformation 826
22.5 Four-Vectors 834
22.6 Electromagnetic Quantities 839
22.7 Covariant Electrodynamics 848
22.8 Matter in Uniform Motion 858
Sources, References, and Additional Reading 863
Problems 865
23 Fields from Moving Charges 870
23.1 Introduction 870
23.2 The Li´enard-Wiechert Problem 870
23.3 Radiation in the Time Domain 880
23.4 Radiation in the Frequency Domain 886
23.5 Synchrotron Radiation 891
CONTENTS xiii
23.6 Radiation Reaction 899
23.7 Cherenkov Radiation 906
Sources, References, and Additional Reading 910
Problems 912
24 Lagrangian and Hamiltonian Methods 916
24.1 Introduction 916
24.2 Hamilton’s Principle 916
24.3 Lagrangian Description 918
24.4 Invariance and Conservation Laws 927
24.5 Hamiltonian Description 931
Sources, References, and Additional Reading 940
Problems 941
Appendix A List of Important Symbols 945
Appendix B Gaussian Units 949
Appendix C Special Functions 953
Appendix D Managing Minus Signs in Special Relativity 959
Index 964
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Table of Applications
Number Name Section
1.1 Two Identities for∇ ×L 1.2.7
1.2 Inversion and Reflection 1.8.1
2.1 Moving Point Charges 2.1.3
3.1 Field Lines for a Point Charge in a Uniform Field 3.4.4
3.2 The Ionization Potential of a Metal Cluster 3.6.3
4.1 Monolayer Electric Dipole Drops 4.3.2
4.2 Nuclear Quadrupole Moments 4.4.2
4.3 The Potential Produced byσ(θ)=σ0cosθ 4.6.3
4.4 The Liquid Drop Model of Nuclear Fission 4.6.4
4.5 The Dielectric PolarizationP(r) 4.7.1
5.1 σ(rS) for a Conducting Disk 5.2.2
5.2 Coulomb Blockade 5.5
6.1 A Uniformly Polarized Sphere 6.3
6.2 Refraction of Field Lines at a Dielectric Interface 6.5.4
6.3 A Classical Model for Quark Confinement 6.7.2
6.4 The Electric Force on an Embedded Volume 6.8.5
7.1 A Conducting Duct 7.5
7.2 Going Off the Axis 7.6.1
7.3 The Unisphere 7.7
7.4 An Electrostatic Lens 7.8.2
8.1 The Electrostatics of an Ion Channel 8.3.4
8.2 A Point Charge Outside a Grounded Tube 8.5.5
8.3 A Wire Array Above a Grounded Plane 8.6
9.1 Contact Resistance 9.5
9.2 The Electric Field Outside a Current-Carrying Wire 9.7.4
9.3 The Four-Point Resistance Probe 9.8
9.4 The Resting Potential of a Cell 9.9
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xvi TABLE OF APPLICATIONS
Number Name Section
10.1 Irrotational Current Sources 10.2
10.2 Magnetic Resonance Imaging 10.4.2
10.3 Chaotic Lines ofB 10.6.1
11.1 The Point Magnetic Monopole 11.2.3
11.2 Parity Violation and the Anapole Moment 11.4.1
11.3 The Helmholtz Anti-Coil 11.4.6
11.4 Strong Focusing by Quadrupole Magnetic Fields 11.5.2
11.5 The Principle of the Electron Microscope 11.6
12.1 The Magnetic Mirror 12.4.1
12.2 Magnetic Trapping 12.4.2
12.3 Magnetic Bacteria in a Rotating Field 12.4.4
12.4 The Magnetic Virial Theorem 12.5.1
12.5 The Solar Corona 12.6.1
13.1 A Uniformly Magnetized Sphere 13.4.1
13.2 Magnetic Shielding 13.6.4
13.3 The Magnetic Force on an Embedded Volume 13.8.5
13.4 Refrigerator Magnets 13.9.1
14.1 The Dielectric Constant of a Plasma 14.2.2
14.2 Faraday’s Disk Generator 4.4.1
14.3 The Quadrupole Mass Spectrometer 14.5.1
14.4 A Solenoid with a Time-Harmonic Current 14.6.1
14.5 The Power Dissipated in a Metal by a Passing Charge 14.8
14.6 Shielding of an AC Magnetic Field by a Cylindrical Shell 14.10.1
14.7 Thomson’s Jumping Ring 14.13.3
15.1 Uniqueness of Solutions and Boundary Conditions 15.4.1
15.2 PEMfor a Capacitor in a Uniform Magnetic Field 15.5.3
15.3 The Hidden Momentum of a Static System 15.7
16.1 The Importance of Being Angular 16.8.2
16.2 Whittaker’s Theorem Revisited 16.9.2
17.1 Alfv`en Waves 17.2.1
17.2 Bragg Mirrors 17.5.2
17.3 Plane Wave Propagation in a Uniaxial Crystal 17.7.1
18.1 Charge Relaxation in an Ohmic Medium 18.5.3
18.2 Reflection of Radio Waves by the Ionosphere 18.5.5
18.3 Lorentz Model Velocities 18.6.1
18.4 Whistlers 18.6.2
18.5 Sum Rules 18.7.4
19.1 Slow Waves for Charged Particle Acceleration 19.4.8
19.2 Quantum Billiards in a Resonant Cavity 19.6.4
19.3 Whispering Gallery Modes 19.7
20.1 The Fields of a Point Charge in Uniform Motion I 20.2
20.2. Schott’s Formulae 20.3.4
20.3 The Classical Zeeman Effect 20.7.1
21.1 The Polarization of Cosmic Microwave Radiation 21.3
21.2 Approximate Mie Scattering 21.7.1
21.3 Sub-wavelength Apertures and Near-Field Optics 21.9.1
TABLE OF APPLICATIONS xvii
Number Name Section
22.1 The Global Positioning System 22.3.1
22.2 Heavy Ion Collisions and the Quark Gluon Plasma 22.4.2
22.3 Charged Particle Motion in a Plane Wave 22.5.2
22.4 Reflection from a Moving Mirror 22.6.5
22.5 The Electric Field Outside a Pulsar 22.8.2
23.1 The Fields of a Point Charge in Uniform Motion II 23.2.4
23.2 A Collinear Acceleration Burst 23.4.4
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Preface
A textbook, as opposed to a treatise, should include everything a student must know, not everything the author does know.
Kenneth Johnson, quoted by Francis Low (1997)
In hisLectures on Physics, Richard Feynman asserts that “ten thousand years from now, there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics”. Whether this prediction is borne out or not, it is impossible to deny the significance of Maxwell’s achievement to the history, practice, and future of physics. That is why electrodynamics has a permanent place in the physics curriculum, along with classical mechanics, quantum mechanics, and statistical mechanics. Of these four, students often find electrodynamics the most challenging. One reason is surely the mathematical demands of vector calculus and partial differential equations. Another stumbling block is the non-algorithmic nature of electromagnetic problem-solving. There are many entry points to a typical electromagnetism problem, but it is rarely obvious which lead to a quick solution and which lead to frustrating complications. Finally, Freeman Dyson points to the “two-level” structure of the theory.1A first layer of linear equations relates the electric and magnetic fields to their sources and to each other. A second layer of equations for force, energy, and stress are quadratic in the fields. Our senses and measurements probe the second-layer quantities, which are determined only indirectly by the fundamental first-layer quantities.
Modern Electrodynamicsis a resource for graduate-level readers interested to deepen their under- standing of electromagnetism without minimizing the role of the mathematics. The book’s size was dictated by two considerations: first, my aim that it serve both as a classroom text and as a reference volume; second, my struggle to apply the epigraph at the top of the page.2Physicists are a prickly and opinionated bunch, so it is not surprising that there is very little agreement about “everything a student must know” about electromagnetism at the graduate level. Beyond a very basic core (the main content of undergraduate texts), the topics which appear in graduate-level textbooks and (electronically) pub- lished lecture notes depend strongly on the research background of the writer and whether he or she is a theorist or an experimenter. Some instructors view the subject as a convenient setting to illustrate the methods of mathematical physics and/or computational physics. Others see it as an opportunity to introduce topics (optics, plasma physics, astrophysics, biophysics, etc.) into a curriculum which might otherwise not include them. Still others teach electromagnetism for the main purpose of introducing the methods of relativistic field theory to their students.
Given the many uses of this foundational course,Modern Electrodynamicspurposely contains much more material than can be comfortably covered in a two-semester course. Presentations with quite
1 F.J. Dyson, “Why is Maxwell’s theory so hard to understand?”, inJames Clerk Maxwell Commemorative Booklet (The James Clerk Maxwell Foundation, Edinburgh, 1999). Available at www.clerkmaxwellfoundation.org/
DysonFreemanArticle.pdf.
2 From the preface to F.E. Low,Classical Field Theory(Wiley, New York, 1997), p. xi.
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xx PREFACE
different emphases can be constructed by making different choices from among the many topics offered for discussion. All instructors will omit various sections and probably entire chapters. Consistent with this point of view, I do not offer a single, idiosyncratic “vision” of electromagnetism. Rather, I have aimed to present what seemed (to me) to be the pedagogically soundest approach for students coming to this material after a first serious exposure at the junior/senior undergraduate level. In many cases, the same issue is examined from more than one point of view. The mathematics of the subject is given its proper due, but the qualitative and physical arguments I provide may ultimately remain with the reader longer.
The organization of this book reflects my personal experience as an instructor. After experimenting with relativity-first, Lagrangian-first, and radiation-first approaches, I concluded that the majority of students grasped the subject matter best when I used a traditional arrangement of topics. The text is purposely repetitive. This is done both to reinforce key ideas and to help readers who do not read the text in chapter order. My background as a condensed matter physicist appears in various places, including an emphasis on the practical (rather than the formal) aspects of microscopic averaging, a discussion of the limitations of the Lorentz model of dielectric and magnetic matter, and the presence of an entire chapter devoted to the experimentally important subject of quasistatics.
Every chapter of Modern Electrodynamics contains worked examples chosen either to develop problem-solving skills or to reveal subtleties of the subject which do not appear when one’s exposure is limited to a few standard examples. Every chapter also contains several “applications” drawn from all the major subfields of physics. By and large, these are topics I was unwilling to relegate to the end-of-chapter homework for fear many readers would never see them. About half the chapters include a boxed excursion into a issue (often historical) where words serve better than equations, and every chapter ends with an annotated list of Sources, References, & Additional Readingto acknowledge my debt to others and to stimulate inquisitive readers. Finally, every chapter contains a large number of homework problems. These range from undergraduate-type drill problems to more challenging problems drawn directly from the research literature. Like most textbook authors, I emphasize that active engagement with the homework problems is an important part of the learning process. This is particularly important for electromagnetism where the struggle with difficult problems has somehow (wrongly) been elevated to a rite of passage. My desired outcome is a reader who, after completion of a course based on this book, can comfortably read and understand (if not necessarily reproduce in detail) a non-trivial electromagnetic argument or calculation which appears in the course of his or her research or reading.
The modernity of the text indicated by its title isnotassociated with the use of particularly “modern”
mathematical methods. Rather, it derives from the inclusion of topics which have attracted new or renewed attention in recent decades. Examples include the electrostatics of ion channels, the modern theory of electric polarization, magnetic resonance imaging, the quantum Hall effect, optical tweezers, negative refraction, the time-domain approach to radiation, the polarization anisotropy of the cosmic microwave background, near-field optics, and relativistic heavy ion collisions. To keep the text finite, some familiar special topics from other texts have been omitted or barely touched upon. Examples include collisions and energy exchange between charged particles, the method of virtual quanta, transition radiation, energy loss in matter, and classical models of the electron. On the other hand, Modern Electrodynamicsincludes an overview of Dirac’s Hamiltonian approach to electrodynamics and an update on certain “perpetual” problems like the correctness of the Lorentz-Dirac equation of motion for classical point particles and the Abraham-Minkowski controversy over the electromagnetic energy-momentum tensor in matter. All of these are illustrative of the self-refreshing nature of a subject which is re-invented by every new generation to meet its needs.
Finally, two choices I have made may give pause to some readers. One is my use of SI units throughout. The other is my use of the imaginary numberito impose the metric in special relativity.
The technical rationale for using SI units is given in Section 2.6. An equally good reason is simply that
PREFACE xxi
this system has become the worldwide standard and nearly all undergraduate textbooks use it without apology. However, because the physics literature is replete with books and research papers which use Gaussian units, Appendix B discusses this system and provides an algorithm to painlessly convert from SI to Gaussian and vice versa.
My reason for using the “old-fashioned” Minkowski metric is purely pedagogical and cannot be stated more clearly than Nobel prize winner Gerard ’t Hooft did in the preface to hisIntroduction to General Relativity(2001), namely: “In special relativity, thei has considerable practical advantage:
Lorentz transformations are orthogonal, and all inner products only come with+signs. No confusion over signs remains”. Although he switches to a metric tensor to discuss general relativity (as he must),
’t Hooft further champions his use ofiin special relativity with the remark, “I see no reason to shield students against the phenomenon of changes in convention and notation. Such transitions are necessary whenever one switches from one research field to another. They better get used to it”. That being said, Appendix D outlines the use of the metric tensorgμνin special relativity.
It would have been impossible for me to write this book without help. At the top of the list, I gratefully acknowledge the contributions of Wayne Saslow (Texas A&M University) and Glenn Smith (Georgia Tech). These colleagues read and commented on many chapters and were always willing to talk electromagnetism with me. My colleague Brian Kennedy (Georgia Tech) repeatedly put aside his research work and helped on the many occasions when I managed to confuse myself. Special thanks go to Michael Cohen (University of Pennsylvania), who tried to teach me this subject when I was a graduate student and who generously shared many of his insights and thoughtful homework problems.
I regret that Mike never found the time to write his own book on this subject.
I am happy to acknowledge David Vanderbilt (Rutgers University) and Stephen Barnett (University of Strathclyde) who provided essential help for my discussions of dielectric polarization and radiation pressure, respectively. Olivier Darrigol (Universit´e Paris Diderot - Paris VII) answered my questions about the history of Lagrangians in electrodynamics and Andrew Scherbakov (Georgia Tech) gave assistance with the homework problems. I also thank the brave souls who provided feedback after testing some of this material in their classrooms: Roman Grigoriev (Georgia Tech), Peter McIntyre (Texas A&M University), Brian Tonner (University of Central Florida), Jiang Xiao (Fudan University), and Kapil Krishan (Jawaharlal Nehru University). Finally, I am delighted to thank my editor Simon Capelin and his first-rate staff at Cambridge University Press. Simon never failed to show enthusiasm for this project, now over more years than either of us would care to remember.
I dedicate this work to my wife Sonia and daughter Hannah. They suffered, but not in silence.
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1 Mathematical Preliminaries
The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious.
Eugene Wigner (1960)
1.1 Introduction
This chapter presents a collection of mathematical notation, definitions, identities, theorems, and transformations which play an important role in the study of electromagnetism. A brief discussion accompanies some of the less familiar topics and only a few proofs are given in detail. For more details and complete proofs, the reader should consult the books and papers listed in Sources, References, and Additional Reading at the end of the chapter. Appendix C at the end of the book summarizes the properties of Legendre polynomials, spherical harmonics, and Bessel functions.
1.2 Vectors
A vector is a geometrical object characterized by a magnitude and direction.1Although not necessary, it is convenient to discuss an arbitrary vector using its components defined with respect to a given coordinate system. An example is the right-handed coordinate system with orthogonal unit basis vectors (ˆe1,eˆ2,eˆ3) shown in Figure 1.1, where
ˆ
e1·eˆ1=1 eˆ2·ˆe2=1 eˆ3·eˆ3=1 (1.1) ˆ
e1·eˆ2=0 eˆ2·ˆe3=0 eˆ3·eˆ1=0 (1.2) ˆ
e1×eˆ2=eˆ3 eˆ2×eˆ3=eˆ1 eˆ3×eˆ1=eˆ2. (1.3) We express an arbitrary vectorVin this basis using componentsVk=ˆek·V,
V=V1ˆe1+V2eˆ2+V3eˆ3. (1.4) A vector can be decomposed in any coordinate system we please, so
3 k=1
Vkeˆk= 3 k=1
Vkeˆk. (1.5)
1 A more precise definition of a vector is given in Section 1.8.
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2 MATHEMATICAL PRELIMINARIES: DEFINITIONS, IDENTITIES, AND THEOREMS
Figure 1.1: An orthonormal set of unit vectors ˆe1, ˆe2, ˆe3.Vis an arbitrary vector.
z z
y
x
(a) Cylindrical
r z
y
x
(b) Spherical Figure 1.2: Two curvilinear coordinate systems.
1.2.1 Cartesian Coordinates
Our notation for Cartesian components and unit vectors is
V=Vxxˆ+Vyyˆ+Vzz.ˆ (1.6)
In particular,rkalways denotes the Cartesian components of the position vector,
r=xxˆ+yyˆ+zˆz. (1.7)
It is not obvious geometrically (see Example 1.7 in Section 1.8), but the gradient operator is a vector with the Cartesian representation
∇ =xˆ ∂
∂x+yˆ ∂
∂y +ˆz∂
∂z. (1.8)
The divergence, curl, and Laplacian operations are, respectively,
∇ ·V= ∂Vx
∂x +∂Vy
∂y +∂Vz
∂z (1.9)
∇ ×V= ∂Vz
∂y −∂Vy
∂z
ˆ x+
∂Vx
∂z −∂Vz
∂x
ˆ y+
∂Vy
∂x −∂Vx
∂y
ˆ
z (1.10)
∇2A= ∂2A
∂x2 +∂2A
∂y2 +∂2A
∂z2. (1.11)
1.2.2 Cylindrical Coordinates
Figure 1.2(a) defines cylindrical coordinates (ρ, φ, z). Our notation for the components and unit vectors in this system is
V=Vρρˆ+Vφφˆ+Vzz.ˆ (1.12)
The transformation to Cartesian coordinates is
x=ρcosφ y=ρsinφ z=z. (1.13)
1.2 Vectors 3
The volume element in cylindrical coordinates isd3r=ρdρdφdz. The unit vectors (ρˆ,φˆ, ˆz) form a right-handed orthogonal triad. ˆzis the same as in Cartesian coordinates. Otherwise,
ρˆ=xˆcosφ+yˆsinφ xˆ=ρˆcosφ−φˆsinφ (1.14) φˆ= −xˆsinφ+yˆcosφ yˆ=ρˆsinφ+φˆcosφ. (1.15) The gradient operator in cylindrical coordinates is
∇ =ρˆ ∂
∂ρ +φˆ ρ
∂
∂φ +zˆ∂
∂z. (1.16)
The divergence, curl, and Laplacian operations are, respectively,
∇ ·V= 1 ρ
∂(ρVρ)
∂ρ + 1 ρ
∂Vφ
∂φ +∂Vz
∂z (1.17)
∇ ×V= 1
ρ
∂Vz
∂φ −∂Vφ
∂z
ρˆ+ ∂Vρ
∂z −∂Vz
∂ρ
φˆ+ 1 ρ
∂(ρVφ)
∂ρ −∂Vρ
∂φ
ˆ
z (1.18)
∇2A= 1 ρ
∂
∂ρ
ρ∂A
∂ρ
+ 1 ρ2
∂2A
∂φ2 +∂2A
∂z2. (1.19)
1.2.3 Spherical Coordinates
Figure 1.2(b) defines spherical coordinates (r, θ, φ). Our notation for the components and unit vectors in this system is
V=Vrrˆ+Vθθˆ+Vφφ.ˆ (1.20)
The transformation to Cartesian coordinates is
x=rsinθcosφ y=rsinθsinφ z=rcosθ. (1.21) The volume element in spherical coordinates isd3r=r2sinθdrdθdφ.The unit vectors are related by
ˆ
r=xˆsinθcosφ+yˆsinθsinφ+zˆcosθ xˆ=rˆsinθcosφ+θˆcosθcosφ−φˆsinφ (1.22) θˆ=xˆcosθcosφ+yˆcosθsinφ−zˆsinθ yˆ=rˆsinθsinφ+θˆcosθsinφ+φˆcosφ (1.23) φˆ= −ˆxsinφ+yˆcosφ ˆz=rˆcosθ−θˆsinθ. (1.24) The gradient operator in spherical coordinates is
∇ =rˆ ∂
∂r +θˆ r
∂
∂θ + φˆ rsinθ
∂
∂φ. (1.25)
The divergence, curl, and Laplacian operations are, respectively,
∇ ·V= 1 r2
∂(r2Vr)
∂r + 1 rsinθ
∂(sinθ Vθ)
∂θ + 1
rsinθ
∂Vφ
∂φ (1.26)
∇ ×V= 1 rsinθ
∂(sinθ Vφ)
∂θ −∂Vθ
∂φ
ˆ r
(1.27) +1
r 1
sinθ
∂Vr
∂φ −∂(rVφ)
∂r
θˆ+1 r
∂(rVθ)
∂r −∂Vr
∂θ
φˆ
∇2A= 1 r2
∂
∂r
r2∂A
∂r
+ 1 r2sinθ
∂
∂θ
sinθ∂A
∂θ
+ 1 r2sin2θ
∂2A
∂φ2. (1.28)
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4 MATHEMATICAL PRELIMINARIES: DEFINITIONS, IDENTITIES, AND THEOREMS
1.2.4 The Einstein Summation Convention
Einstein (1916) introduced the following convention. An index which appears exactly twice in a mathematical expression is implicitly summed over all possible values for that index. The range of thisdummy indexmust be clear from context and the index cannot be used elsewhere in the same expression for another purpose. In this book, the range for a roman index likeiis from 1 to 3, indicating a sum over the Cartesian indicesx,y, andz. Thus,Vin (1.6) and its dot product with another vector Fare written
V= 3 k=1
Vkˆek≡Vkeˆk V·F= 3 k=1
VkFk≡VkFk. (1.29) In a Cartesian basis, the gradient of a scalarϕand the divergence of a vectorDcan be variously written
∇ϕ=ˆek∇kϕ=eˆk∂kϕ=eˆk
∂ϕ
∂rk
(1.30)
∇ ·D= ∇kDk=∂kDk= ∂Dk
∂rk
. (1.31)
If anN×NmatrixCis the product of anN×MmatrixAand anM×NmatrixB, Cik =
M j=1
AijBj k=AijBj k. (1.32)
1.2.5 The Kronecker and Levi-Civit`a Symbols
The Kronecker delta symbolδijand Levi-Civit`a permutation symbolij khave roman indicesi,j, and kwhich take on the Cartesian coordinate valuesx,y, andz. They are defined by
δij =
1 i=j,
0 i=j, (1.33)
and
ij k=
⎧⎨
⎩
1 ij k = xyz yzx zxy,
−1 ij k = xzy yxz zyx, 0 otherwise.
(1.34) Some useful Kronecker delta and Levi-Civit`a symbol identities are
ˆ
ei·ˆej =δij δkk=3 (1.35)
∂krj =δj k Vkδkj =Vj (1.36)
[V×F]i=ij kVjFk [∇ ×A]i =ij k∂jAk (1.37)
δijij k=0 ij kij k=6. (1.38)
A particularly useful identity involves a single sum over the repeated indexi:
ij kist=δj sδkt−δj tδks. (1.39) A generalization of (1.39) when there are no repeated indices to sum over is the determinant
kimpq =
δkm δim δ m δkp δip δ p
δkq δiq δ q
. (1.40)
1.2 Vectors 5
Finally, letCbe a 3×3 matrix with matrix elementsC11, C12, etc. The determinant ofCcan be written using either an expansion by columns,
detC=ij kCi1Cj2Ck3, (1.41)
or an expansion by rows,
detC=ij kC1iC2jC3k. (1.42)
A closely related identity we will use in Section 1.8.1 is
mndetC=ij kCiCmjCnk. (1.43)
1.2.6 Vector Identities in Cartesian Components
The Kronecker and Levi-Civit`a symbols simplify the proof of vector identities. An example is
a×(b×c)=b(a·c)−c(a·b). (1.44)
Using the left side of (1.37), theith component ofa×(b×c) is
[a×(b×c)]i =ij kaj(b×c)k=ij kajkmblcm. (1.45) The definition (1.34) tells us thatij k=kij. Therefore, the identity (1.39) gives
[a×(b×c)]i=kijkmajbcm=(δiδj m−δimδj )ajbcm=ajbicj−ajbjci. (1.46) The final result,bi(a·c)−ci(a·b), is indeed the ith component of the right side of (1.44). The same method of proof applies to gradient-, divergence-, and curl-type vector identities because the components of the∇operator transform like the components of a vector [see above (1.8)]. The next three examples illustrate this point.
Example 1.1 Prove that∇ ·(∇ ×g)=0.
Solution: Begin with ∇ ·(∇ ×g)=∂iij k∂jgk=12∂i∂jgkij k+12∂i∂jgkij k. Exchanging the dummy indicesiandjin the last term gives
∇ · ∇ ×g= 12∂i∂jgkij k+12∂j∂igkj ik= 12{ij k+j ik}∂i∂jgk=0.
The final zero comes fromij k= −j ik, which is a consequence of (1.34).
Example 1.2 Prove that∇ ×(A×B)=A∇ ·B−(A· ∇)B+(B· ∇)A−B∇ ·A.
Solution: Focus on theith Cartesian component and use the left side of (1.37) to write [∇ ×(A×B)]i =ij k∂j(A×B)k=ij kkst∂j(AsBt).
The cyclic properties of the Levi-Civit`a symbol and the identity (1.39) give
[∇ ×(A×B)]i=kijkst∂j(AsBt)=(δisδj t−δitδj s)(As∂jBt+Bt∂jAs).
Therefore,
[∇ ×(A×B)]i=Ai∂jBj−Aj∂jBi+Bj∂jAi−Bi∂jAj. This proves the identity because the choice ofiis arbitrary.
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6 MATHEMATICAL PRELIMINARIES: DEFINITIONS, IDENTITIES, AND THEOREMS
Example 1.3 Prove the “double-curl identity”∇ ×(∇ ×A)= ∇(∇ ·A)− ∇2A.
Solution: Consider theith Cartesian component. The identity on the left side of (1.37) and the invariance of the Levi-Civit`a symbol with respect to cyclic permutations of its indices give
[∇ ×(∇ ×A)]i=ij k∂j(∇ ×A)k=ij k∂jkpq∂pAq=kijkpq∂j∂pAq. Now apply the identity (1.39) to get
[∇ ×(∇ ×A)]i=(δipδj q−δiqδjp)∂j∂pAq=∂i∂jAj−∂j∂jAi= ∇i(∇ ·A)− ∇2Ai. The double-curl identity follows because
∇2A= ∇2(Axxˆ+Ayyˆ+Azz)ˆ =x∇ˆ 2Ax+y∇ˆ 2Ay+z∇ˆ 2Az.
1.2.7 Vector Identities in Curvilinear Components
Care is needed to interpret the vector identities in Examples 1.2 and 1.3 when the vectors in question are decomposed into spherical or cylindrical components such asA=Arrˆ+Aθθˆ+Aφφˆ. This can be seen from Example 1.3 where the final step is no longer valid because ˆr,θˆ, andφˆare not constant vectors. In other words,
∇2A = ∇ · ∇(Arrˆ+Aθθˆ+Aφφ)ˆ = ˆr∇2Ar+θˆ∇2Aθ +φˆ∇2Aφ. (1.47) One way to proceed is to work out the components of∇(Arˆr),∇(Aθθˆ), and∇(Aφφˆ). Alternatively, we may simply define the meaning of the operation∇2A whenA is expressed using curvilinear components. For example,
[∇2A]φ≡∂φ(∇ ·A)−[∇ ×(∇ ×A)]φ, (1.48) and similarly for (∇2A)rand (∇2A)θ.
Exactly the same issue arises when we examine the last step in Example 1.2, namely
[∇ ×(A×B)]i =Ai∇ ·B−(A· ∇)Bi+(B· ∇)Ai−Bi∇ ·A. (1.49) By construction, this equation makes sense whenistands forx,y, orz. It doesnotmake sense ifi stands for, say,r,θ, orφ. On the other hand, the full vector version of the identity is correct as long as we retain ther,θ, andφvariations of ˆr,θˆ, andφˆ. For example,
(A· ∇)B=
Ar
∂
∂r +Aθ
r
∂
∂θ + Aφ
rsinθ
∂
∂φ
(Brˆr+Bθθˆ+Bφφ).ˆ (1.50)
Application 1.1 Two Identities for∇ ×L
The−h=1 version of the quantum mechanical angular momentum operator,L= −ir× ∇, plays a useful role in the analysis of classical spherical systems. In this application, we prove two operator identities which will appear later in the text:
(A) ∇ ×L= −ir∇2+i∇(1+r· ∇) (B) ∇ ×L=(ˆr×L)
1 r
∂
∂rr
+rˆi rL2.
1.3 Derivatives 7
Proof of Identity (A):
We use (1.37), (1.39), and the cyclic property of the Levi-Civit`a symbol to evaluate thekth component of∇ ×Lacting on a scalar functionφ:
[∇ ×L]kφ= −i[∇ ×(r× ∇)]kφ= −imkmst∂rs∂tφ= −i[∂rk∂φ−∂r∂kφ]. (1.51) Because∂r=3 and∂rk=δk,
∇ ×Lφ=[−ir∇2+i2∇ +i(r· ∇)∇]φ. (1.52)
However,
∂k[r∂φ]=∂kφ+r∂∂kφ, (1.53) which is thekth component of∇(r· ∇)φ= ∇φ+(r· ∇)∇φ. Substituting the latter into (1.53) gives Identity (A).
Proof of Identity (B):
We decompose the gradient operator into its radial and angular pieces:
∇ =r(ˆˆr· ∇)−rˆ×(ˆr× ∇)=rˆ∂
∂r − i
rˆr×L. (1.54)
Equation (1.54) and the Levi-Civit`a formalism produce the intermediate result
∇ ×L=(ˆr×L)∂
∂r − i
r(ˆr×L)×L=(ˆr×L)∂
∂r − i r
rˆkLLk−rLˆ 2
. (1.55)
However, the angular momentum operator obeys commutation relations which can be summarized as L×L=iL. Therefore,
ˆ
r×(L×L)=irˆ×L ⇒ rˆkLLk−rˆkLkL=iˆr×L. (1.56) On the other hand,rkLk=0 becauseLis perpendicular to bothrand∇. Therefore, ˆrkLLk=iˆr×L, which we can substitute into (1.55). The result is identity (B) because, for any scalar functionφ,
1 r
∂
∂r(rφ)
= ∂φ
∂r +φ
r. (1.57)
1.3 Derivatives
1.3.1 Functions of r and | r |
The position vector isr=rrˆwithr=
x2+y2+z2. Iff(r) is a scalar function andf(r)=df/dr,
∇r=ˆr ∇ ×r=0 (1.58)
∇f =frˆ ∇2f =(r2f)
r2 (1.59)
∇ ·(fr)= (r3f)
r2 ∇ ×(fr)=0. (1.60)
Similarly, ifg(r) is a vector function andcis a constant vector,
∇ ·g=g·rˆ ∇ ×g=rˆ×g (1.61)
(g· ∇)r=g (r· ∇)g=rg (1.62)
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8 MATHEMATICAL PRELIMINARIES: DEFINITIONS, IDENTITIES, AND THEOREMS
∇(r·g)=g+(r·g)r
r ∇ ·(g×r)=0 (1.63)
∇ ×(g×r)=2g+rg−(r·g)r
r ∇(c·r)=c. (1.64)
1.3.2 Functions of r − r
LetR=r−r=(x−x)ˆx+(y−y)ˆy+(z−z)ˆz. Then,
∇f(R)=f(R) ˆR ∇ ·g(R)=g(R)·Rˆ ∇ ×g(R)=Rˆ×g(R). (1.65) Moreover, because
∇ =xˆ ∂
∂x +yˆ ∂
∂y +zˆ∂
∂z and ∇=xˆ ∂
∂x +yˆ ∂
∂y+zˆ ∂
∂z, (1.66)
it it straightforward to confirm that
∇f(R)= −∇f(R). (1.67)
1.3.3 The Convective Derivative
Letφ(r, t) be a scalar function of space and time. An observer who repeatedly samples the value of φ at a fixed point in space,r, records the time rate of change ofφ as the partial derivative∂φ/∂t. However, the same observer who repeatedly samplesφalong a trajectory in spacer(t) that moves with velocityυ(t)=r(t) records the time rate of change of˙ φas theconvective derivative,
dφ dt = ∂φ
∂t +dx dt
∂φ
∂x +dy dt
∂φ
∂y +dz dt
∂φ
∂z = ∂φ
∂t +(υ· ∇)φ. (1.68) For a vector functiong(r, t), the corresponding convective derivative is
dg dt = ∂g
∂t +(υ· ∇)g. (1.69)
1.3.4 Taylor’s Theorem
Taylor’s theorem in one dimension is f(x)=f(a)+(x−a)df
dx
x=a+ 1
2!(x−a)2 d2f dx2
x=a+ · · ·. (1.70)
An alternative form follows from (1.70) ifx→x+anda→x:
f(x+)=f(x)+df dx + 1
2!2d2f
dx2 + · · ·. (1.71) Equivalently,
f(x+)=
1+ d dx + 1
2!
d
dx 2
+ · · ·
f(x)=exp
d dx
f(x). (1.72) This generalizes for a function of three variables to
f(x+x, y+y, z+z)=exp
x
∂
∂x
exp
y
∂
∂y
exp
z
∂
∂z
f(x, y, z), (1.73) or
f(r+)=exp (· ∇)f(r)=
1+· ∇ + 1
2!(· ∇)2+ · · ·
f(r). (1.74)
1.4 Integrals 9
1.4 Integrals
1.4.1 Jacobian Determinant
The determinant of the Jacobian matrixJrelates volume elements when changing variables in an integral. For example, supposexandyareN-dimensional space vectors in two different coordinates systems, e.g., Cartesian and spherical. The volume elementsdNxanddNyare related by
dNx= |J(x,y)|dNy=
∂x1
∂y1
∂x1
∂y2
· · · ∂x1
∂yN
· · · ·
· · · ·
∂xN
∂y1
∂xN
∂y2 · · · ∂xN
∂yN
dNy. (1.75)
1.4.2 The Divergence Theorem
LetF(r) be a vector function defined in a volumeV enclosed by a surfaceSwith an outward normal n. Ifˆ dS=dSn, the divergence theorem isˆ
V
d3r∇ ·F=
S
dS·F. (1.76)
Special choices for the vector functionF(r) produce various integral identities based on (1.76). For example, ifcis an arbitrary constant vector, the reader can confirm that the choicesF(r)=cψ(r) and F(r)=A(r)×csubstituted into (1.76) respectively yield
V
d3r∇ψ=
S
dSψ (1.77)
V
d3r∇ ×A=
S
dS×A. (1.78)
1.4.3 Green’s Identities
The choiceF(r)=φ(r)∇ψ(r) in (1.76) leads toGreen’s first identity,
V
d3r[φ∇2ψ+ ∇φ· ∇ψ]=
S
dS·φ∇ψ. (1.79)
Writing (1.79) with the roles ofφandψexchanged and subtracting that equation from (1.79) itself givesGreen’s second identity,
V
d3r[φ∇2ψ−ψ∇2φ]=
S
dS·[φ∇ψ−ψ∇φ]. (1.80) The choiceF=P× ∇ ×Qin (1.76) and the identity∇ ·(A×B)=B· ∇ ×A−A· ∇ ×Bproduces a vector analog of Green’s first identity:
V
d3r[∇ ×P· ∇ ×Q−P· ∇ × ∇ ×Q]=
S
dS·(P× ∇ ×Q). (1.81)
