Über die Autorin bzw. den Autor

Anupam Garg is professor of physics and astronomy at Northwestern University.

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"This text provides a fresh, modern look at electrodynamics. It is comprehensive, chock full of interesting insights and anecdotes, and written with a clear enthusiasm."--Kenneth A. Intriligator, University of California, San Diego

"Garg displays considerable wisdom and courage in writing a long-overdue, modern treatment of electromagnetism. I wish I had this book when I was a student. It contains delightful morsels of deep insight (the introduction taught me that fields are as real as a rhinoceros, or as I might extend it, 'quantum fields are as real as quantum rhinos') and interesting topics that are rarely, if ever, treated in other texts."--A. Zee, author ofQuantum Field Theory in a Nutshell

"Garg demonstrates that while the mathematical beauty of his subject is deserving of the attention it gets, the physical implications are even more seductive. This book is a treasure trove of thoughtful and incisive nuggets. I expect to see it on the shelves of many students and professors the world over."--R. Shankar, Yale University

"An outstanding text. It offers delightful insights into the physics of electromagnetism rather than treating this subject as an excuse for laborious exercises in mathematical methods. It has the best treatment of electromagnetism in material media that I know. Remarkably lucid and enjoyable, and a worthy competitor to Jackson's classic text."--Jainendra Jain, Pennsylvania State University

"Classical Electromagnetism in a Nutshell is an interesting and elegant book, and an excellent text for a graduate-level course on the subject. Garg has a lively, modern writing style that will engage today's graduate students."--John D. Stack, University of Illinois, Urbana-Champaign

"Garg's textbook is truly excellent. It goes the extra mile to provide physical insight in ways that will enhance students' understanding, and includes rarely seen topics as well. I want to compliment the author on the obvious care and expertise with which he assembled this text. If I were to teach a yearlong graduate-level electromagnetism course, I would use this book."--John W. Belcher, Massachusetts Institute of Technology

Aus dem Klappentext

"This text provides a fresh, modern look at electrodynamics. It is comprehensive, chock full of interesting insights and anecdotes, and written with a clear enthusiasm."--Kenneth A. Intriligator, University of California, San Diego

"Garg displays considerable wisdom and courage in writing a long-overdue, modern treatment of electromagnetism. I wish I had this book when I was a student. It contains delightful morsels of deep insight (the introduction taught me that fields are as real as a rhinoceros, or as I might extend it, 'quantum fields are as real as quantum rhinos') and interesting topics that are rarely, if ever, treated in other texts."--A. Zee, author ofQuantum Field Theory in a Nutshell

"Garg demonstrates that while the mathematical beauty of his subject is deserving of the attention it gets, the physical implications are even more seductive. This book is a treasure trove of thoughtful and incisive nuggets. I expect to see it on the shelves of many students and professors the world over."--R. Shankar, Yale University

"An outstanding text. It offers delightful insights into the physics of electromagnetism rather than treating this subject as an excuse for laborious exercises in mathematical methods. It has the best treatment of electromagnetism in material media that I know. Remarkably lucid and enjoyable, and a worthy competitor to Jackson's classic text."--Jainendra Jain, Pennsylvania State University

"Classical Electromagnetism in a Nutshell is an interesting and elegant book, and an excellent text for a graduate-level course on the subject. Garg has a lively, modern writing style that will engage today's graduate students."--John D. Stack, University of Illinois, Urbana-Champaign

"Garg's textbook is truly excellent. It goes the extra mile to provide physical insight in ways that will enhance students' understanding, and includes rarely seen topics as well. I want to compliment the author on the obvious care and expertise with which he assembled this text. If I were to teach a yearlong graduate-level electromagnetism course, I would use this book."--John W. Belcher, Massachusetts Institute of Technology

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Classical Electromagnetism in a Nutshell

PRINCETON UNIVERSITY PRESS

Contents

Preface........................................................................................xvList of symbols................................................................................xxiSuggestions for using this book................................................................xxxi1 Introduction.................................................................................12 Review of mathematical concepts..............................................................183 Electrostatics in vacuum.....................................................................554 Magnetostatics in vacuum.....................................................................825 Induced electromagnetic fields...............................................................1146 Symmetries and conservation laws.............................................................1327 Electromagnetic waves........................................................................1528 Interference phenomena.......................................................................1789 The electromagnetic field of moving charges..................................................20010 Radiation from localized sources............................................................21711 Motion of charges and moments in external fields............................................24512 Action formulation of electromagnetism......................................................27313 Electromagnetic fields in material media....................................................28514 Electrostatics around conductors............................................................30215 Electrostatics of dielectrics...............................................................34416 Magnetostatics in matter....................................................................37017 Ohm's law, emf, and electrical circuits.....................................................40418 Frequency-dependent response of materials...................................................42719 Quasistatic phenomena in conductors.........................................................44320 Electromagnetic waves in insulators.........................................................47021 Electromagnetic waves in and near conductors................................................48722 Scattering of electromagnetic radiation.....................................................50523 Formalism of special relativity.............................................................52424 Special relativity and electromagnetism.....................................................55325 Radiation from relativistic sources.........................................................581Appendix A: Spherical harmonics................................................................605Appendix B: Bessel functions...................................................................617Appendix C: Time averages of bilinear quantities in electrodynamics............................625Appendix D: Caustics...........................................................................627Appendix E: Airy functions.....................................................................633Appendix F: Power spectrum of a random function................................................637Appendix G: Motion in the earth's magnetic field—the Stormer problem.....................643Appendix H: Alternative proof of Maxwell's receding image construction.........................651Bibliography...................................................................................655Index..........................................................................................659

Chapter One

1 The field concept

The central concept in the modern theory of electromagnetism is that of the electromagnetic field. The forces that electrical charges, currents, and magnets exert on each other were believed by early thinkers to be of the action-at-a-distance type, i.e., the forces acted instantaneously over arbitrarily large distances. Experiments have shown, however, that this is not true. A radio signal, for example, can be sent by moving electrons back and forth in a metallic antenna. This motion will cause electrons in a distant piece of metal to move back and forth in response-this is how the signal is picked up in a radio or cell phone receiver. We know that the electrons in the receiver cannot respond in a time less than that required by light to travel the distance between transmitter and receiver. Indeed, radio waves, or electromagnetic waves more generally, are a form of light.

Facts such as these have led us to abandon the notion of action at a distance. Instead, our present understanding is that electrical charges and currents produce physical entities called fields, which permeate the space around them and which in turn act on other charges and currents. When a charge moves, the fields that it creates change, but this change is not instantaneous at every point in space. For a complete description, one must introduce two vector fields, E(r, t), and B(r, t), which we will call the electric and magnetic fields, respectively. In other words, at every time t, and at every point in space r, we picture the existence of two vectors, E and B. This picture is highly abstract, and early physicists had great trouble in coming to grips with it. Because the fields did not describe particulate matter and could exist in vacuum, they seemed very intangible, and early physicists were reluctant to endow them with physical reality. The modern view is quite different. Not only do these fields allow us to describe the interaction of charges and currents with each other in the mathematically simplest and cleanest way, we now believe them to be absolutely real physical entities, as real as a rhinoceros. Light is believed to be nothing but a jumble of wiggling E and B vectors everywhere, which implies that these fields can exist independently of charges and currents. Secondly, these fields carry such concrete physical properties as energy, momentum, and angular momentum. When one gets to a quantum mechanical description, these three attributes become properties of a particle called the photon, a quantum of light. At sufficiently high energies, two of these particles can spontaneously change into an electron and a positron, in a process called pair production. Thus, there is no longer any reason for regarding the E and B fields as adjuncts, or aids to understanding, or to picture the interactions of charges through lines of force or flux. Indeed, it is the latter concepts that are now regarded as secondary, and the fields as primary.

The impossibility of action at a distance is codified into the modern theory of relativity. The principle of relativity as enunciated by Galileo states that the laws of physics are identical in all inertial reference frames. One goes from Galilean relativity to the modern theory by recognizing that there is a maximum speed at which physical influences or signals may propagate, and since this is a law of physics, the maximum speed must then be the same in all inertial frames. This speed immediately acquires the status of a fundamental constant of nature and is none other than the speed of light in vacuum. Needless to say, this law, and the many dramatic conclusions that follow from considering it in conjunction with the principle of relativity, are amply verified by experiment.

The application of the principle of relativity also leads us to discover that E and B are two aspects of the same thing. A static set of charges creates a time-independent electric field, and a steady current creates a time-independent magnetic field. Since a current can be regarded as a charge distribution in motion, it follows that E and B will, in general, transform into one another when we change reference frames. In fact, the relativistic invariance of the laws of electrodynamics is best expressed in terms of a single tensor field, generally denoted F. The fields E and B are obtained as different components of F. At low speeds, however, these two different components have so many dissimilar aspects that greater physical understanding is obtained by thinking of them as separate vector fields. This is what we shall do in this book.

2 The equations of electrodynamics

The full range of electromagnetic phenomena is very wide and can be very complicated. It is somewhat remarkable that it can be captured in a small number of equations of relatively simple form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

These laws are confirmed by extensive experience and the demands of consistency with general principles of symmetry and relativistic invariance, although their full content can be appreciated only after detailed study. We have written them in the two most widespread systems of units in use today and given them the names commonly used in the Western literature. The first four equations are also collectively known as the Maxwell equations, after James Clerk Maxwell, who discovered the last term on the right-hand side of the Ampere-Maxwell law in 1865 and thereby synthesized the, till then, separate subjects of electricity and magnetism into one.

We assume that readers have at least some familiarity with these laws and are aware of some of their more basic consequences. A brief survey is still useful, however. We begin by discussing the symbols. The parameter c is the speed of light, and [member of]₀ and µ₀ are constant scale factors or conversion factors used in the SI system. The quantity ρ is a scalar field ρ(r, t), denoting the charge distribution or density. Likewise, j is a vector field j(r, t), denoting the current distribution. This means that the total charge inside any closed region of space is the integral of ρ(r, t) over that space, and the current flowing across any surface is the integral of the normal component of j(r, t) over the surface. This may seem a roundabout way of specifying the position and velocity of all the charges, which we know, after all, to be made of discrete objects such as electrons and protons. But, it is in these terms that the equations for E and B are simplest. Further, in most macroscopic situations, one does not know where each charge is and how fast it is moving, so that, at least in such situations, this description is the more natural one anyway.

The four Maxwell equations allow one to find E and B if ρ and j are known. For this reason, the terms involving ρ and j are sometimes known as source terms, and the E and B fields are said to be "due to" the charges and currents. However, we began by talking of the forces exerted by charges on one another, and of this there is no mention in the Maxwell equations. This deficiency is filled by the last law in our table-the Lorentz force law-which gives the rule for how the fields acts on charges. According to this law, the force on a particle with charge q at a point r and moving with a velocity v depends only on the instantaneous value of the fields at the point r, which makes it a local law. Along with Newton's second law,

dp/dt = F, (2.2)

equating force to the rate of change of momentum, it allows us to calculate, in principle, the complete motion of the charges.

Let us now discuss some of the more salient features of the equations written above. First, the Maxwell equations are linear in E and B, and in ρ and j. This leads immediately to the superposition principle. If one set of charges and currents produces fields E1 and B1, and another set produces fields E2 and B2, then if both sets of charges and currents are simultaneously present, the fields produced will be given by E₁ +E₂, and B₁ + B₂. This fact enables one to simplify the calculation of the fields in many circumstances. In principle, one need only know the fields produced by a single moving charge, and the fields due to any distribution may be obtained by addition. In practice, the problem of addition is often not easy, and one is better off trying to solve the differential equations directly. A large part of electromagnetic theory is devoted to developing the classical mathematical machinery for this purpose. This includes the theorems named after Gauss, Stokes, and Green, and Fourier analysis and expansions in complete sets of orthogonal functions. With modern-day computers, direct numerical solution is the method of choice in many cases, but a sound grasp of the analytic techniques and concepts is essential if one is to make efficient use of computational resources.

The second point is that the equations respect the symmetries of nature. We discuss these in considerably greater detail in chapter 6, and here we only list the symmetries. The first of these is invariance with respect to space and time translations, i.e., the equivalence of two frames with different origins or zeros of time. As in mechanics, this symmetry is connected with the conservation of momentum and energy. The fact that it holds for Maxwell's equations automatically leads us to assign energy and momentum to the electromagnetic field itself. The second symmetry is rotational invariance, or the isotropy of space. That this holds can be seen directly from the vector nature of E and B, and the properties of the divergence and curl. It is connected with the conservation of angular momentum. The third symmetry is spatial inversion, or parity, which in conjunction with rotations is the same as mirror symmetry. We shall find that under inversion, E -> -E, in the same way that a "normal" vector like the velocity v behaves, but B -> B. One therefore says that E is a polar vector, or just a vector, while B is a pseudovector or axial vector. The fourth symmetry is time reversal, or what might be better called motion reversal. This is the symmetry that says that if one could make a motion picture of the world and run it backward, one would not be able to tell that it was running backward. The fifth symmetry is the already mentioned equivalence of reference frames, also known as relativistic invariance or Lorentz invariance. This symmetry is extremely special and, in contrast to the first three, is the essential way in which electromagnetism differs from Newtonian or pre-Einsteinian classical mechanics. We shall devote chapter 23 to its study. Historically, electromagnetism laid the seed for modern (Einsteinian) relativity. The problem was that the Maxwell equations are not Galilean invariant. This fact is mostly clearly seen by noting that light propagation, which is a consequence of the Maxwell equations, is described by a wave equation of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Here, f stands for any Cartesian component of E or B. As is well known, classical wave phenomena are not Galilean invariant. Sound, e.g., requires a material medium for its propagation, and the frame in which this medium is at rest is clearly special. The lack of Galilean invariance of Maxwell's equations was well known to physicists around the year 1900, but experimental support for the most commonly proposed cure, namely, that there was a special frame for light as well, and a special medium (the ether) filling empty space, through which light traveled, failed to materialize. Finally, in 1905, Einstein saw that Galilean invariance itself had to be given up. Although rooted in electromagnetism, this proposal has far-reaching consequences for all branches of physics. In mechanics, we mention the nonabsolute nature of time, the equivalence of mass and energy, and the impossibility of the existence of rigid bodies and elementary particles with finite dimensions. Today, relativity is not regarded as a theory of a particular phenomenon but as a framework into which all of physics must fit. Much of particle physics in the twentieth century can be seen as an outcome of this idea in conjunction with quantum mechanics.

Another feature of the Maxwell equations that may be described as a symmetry is that they imply charge conservation. If we add the time derivative of the first equation, Gauss's law, to the divergence of the second, the Ampere-Maxwell law, we obtain the continuity equation for charge,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

If we integrate this equation over any closed region of space, and any finite interval of time, the left-hand side gives the net increase in charge inside the region, while, by Gauss's theorem, the right-hand side gives the inflow of charge through the surface bounding the region. Thus, eq. (2.4) states that charge is locally conserved. This conservation law is intimately connected with a symmetry known as gauge invariance. We shall say more about this in chapter 12.

The last symmetry to be discussed is a certain duality between E and B. Let us consider the second and third Maxwell equations and temporarily ignore the current source term. The equations would then transform into one another under the replacements E -> B, B -> -E. The same is true of the remaining pair of equations if the charge source term is ignored. This makes it natural to ask whether we should not modify the equations for ∇ · B and ∇ × E to include magnetic charge and current densities ρ_m and j_m, in other words, to write (in the Gaussian system),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

∇ · B = 4πρ_m. (2.6)

All the existing experimental evidence to date, however, indicates that free magnetic charges or monopoles do not exist.

In the same connection, we should note that there is another source of magnetic field besides currents caused by charges in motion. All the charged elementary particles, the electron (and the other leptons, the muon and the taon) and the quarks, possess an intrinsic or spin magnetic moment. This moment cannot be understood as arising from a classical spinning charged object, however. The question then arises whether we should not add a source term to the equation for ∇ · B to take account of this magnetic moment. If we are interested only in describing the field classically, however, we can do equally well by thinking of these moments as idealized current loops of zero spatial extent and including this current in the source term proportional to j in the Ampere-Maxwell law. The integral of the divergence of this current over any finite volume is always zero, so the equation of continuity is unaffected, and we need never think of the charge distribution carried by these loops separately. In fact, the alternative of putting all or some of the source terms into the equation for ∇ · B is not an option, for it leads to unacceptable properties for the vector potential. We discuss this point further in section 26.

3 A lightspeed survey of electromagnetic phenomena

Having surveyed the essential properties of the equations of electrodynamics, let us now mention some of the most prominent phenomena implied by them. First, let us consider a set of static charges. This is the subject of electrostatics. Then j = 0, and ρ(r) is time independent. The simplest solution is then to take B = 0, and the E-field, which is also time independent, is given by Gauss's law. In particular, we can find E(r) for a point charge, and then, in combination with the Lorentz force law, we obtain Coulomb's force law-namely, that the force between two charges is proportional to the product of the charges, to the inverse square of their separation, and acts along the line joining the charges. We study electrostatics further in chapter 3.

(Continues...)

Excerpted from Classical Electromagnetism in a Nutshellby Anupam Garg Copyright © 2012 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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