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The first thing which will become apparent to a reader perusing this text is that it takes a very different approach to the theory of electrodynamics from that in other texts. The general approach taken by electrodynamics texts has hardly changed at all in the last 50 (and not very much in the last 100) years. During this time the way in which calculations are actually done has changed drastically, due to the introduction of computer techniques. Retention of the traditional pedagogical approach has been justified during the first two or three decades of the computer age by the arguments that computer techniques are useful only to advanced students who have access to expensive special equipment, and that they are too hard for beginning students to understand. The first argument has, needless to say, become ridiculous. The second argument is unfortunately true of most books on computer methods, which assume a thorough understanding of the continuum theory of electrodynamics and regard discretizations as approximations to this continuum theory.
The main point of the present text is that this order (continuum, then discrete) is both pedagogically and logically inferior to the reverse order. We introduce a very simple discrete version of the equations of electrodynamics (Maxwell's equations), from which the continuum equations follow in a natural way. As I hope the reader will see, this has four important advantages over the conventional order: It drastically simplifies the minimum mathematical preparation necessary for understanding electrodynamics, it allows a more logical approach to the subject, it develops intuition into electrodynamic phenomena much more quickly through experience with simulations, and it lays the groundwork both for the use of modern calculational tools in electrodynamics and for modern discrete theories in other fields of physics and engineering.
I would like to thank Jane Boyd for typing the initial draft of Part C and helping in the revision of Part A, and Joseph E. Cates for writing the Apple Pascal versions of the simulation programs and revising the Apple Basic versions. I also acknowledge the hospitality of Los Alamos National Laboratory, where parts of the manuscript were written.
Pieter B. Visscher
The subject of electrodynamics appeals to many students because it is a concise, elegant, rigorous, highly symmetrical theory which explains a very wide variety of real physical phenomena. It is also feared by many students because it requires a high level of continuum mathematics (particularly partial differential equations) which very few physics students ever learn on a rigorous level. They must therefore learn this mathematics (usually in the E&M course itself) on a "cookbook" level. The luster of the concise, elegant, rigorous, symmetrical theory mentioned above is considerably tarnished by the fact that one's understanding of it rests on a quagmire of cookbook rules for manipulating vector fields, which sometimes work and sometimes don't.
The purpose of the present book is to separate the concise, elegant, rigorous, highly symmetrical theory from the cookbook continuum mathematics. This is achieved by introducing the electric and magnetic fields, and the Maxwell equations which describe their evolution, in the context of a discrete lattice system. The Maxwell equations are again elegant and symmetrical, but now can be treated in a completely rigorous and geometrically transparent way without difficult mathematics. We can easily prove rigorously the law of conservation of energy, Coulomb's law, the macroscopic forms of Gauss', Faraday's, and Ampere's laws, the formulas for magnetization current and polarization charge, etc. Many of these derivations remain buried in vector calculus to most students using the conventional continuum approach, and the results must be treated as still more "cookbook rules" for computation. We can also do numerical computations to illustrate various physical phenomena, all without continuum mathematics. Of course one must eventually bite the bullet and consider the continuum limit, but this can now be done after the basic equations and the physical phenomena they describe are understood, in such a way that one's understanding of the elegant theory and the physical phenomenology does not depend on imperfectly understood continuum concepts. I do not want to leave the impression that this book is intended as a remedial text for students who are incapable of learning electrodynamics in the conventional way. I believe that even a student who can prove all the existence and uniqueness theorems of partial differential equations will learn Maxwell's equations more easily if they are first presented in a simple, intuitively graspable, but still rigorous way.
Many people who have learned electrodynamics in the traditional way react initially to a discrete approach with the objection that the continuum equations are more fundamental, and one should start out with the fundamental equations and worry about discrete approximations to them later. But if one thinks about how the derivatives in continuum equations are defined (as limits of discrete differences), it is at least as reasonable to regard the discrete equations as fundamental, with the continuum equations being derived from them by letting the grid spacing approach zero. Our prejudice in favor of continuum equations may be as much a consequence of our having been taught them first as of any real fundamentality. The question is academic in any event; from the point of view of pedagogy the choice is between beginning with the abstract (differential equations) and proceeding to the concrete (simulations using discrete equations describing direct cause-and-effect relationships between neighboring electric and magnetic fields) or beginning with the concrete and proceeding to the abstract. The latter is clearly preferable.
Another reason for our prejudice against discrete equations is that most discretizations of continuum equations lose some or all of the elegance of the continuum equations, in this case their high degree of symmetry and exact satisfaction of conservation laws and integral relations like Gauss' and Ampere's laws. The discretizations of Maxwell's equations used in this text have been carefully chosen to retain the elegant features of the continuum equations. The discrete Coulomb's law in spherical coordinates is given by exactly the same simple inverse-square formula as the continuum one. Stokes' theorem and the divergence theorem are exactly true, and can be proved with elementary algebra, unlike their continuum counterparts.
A by-product of this discrete approach, but a key reason for introducing it at this particular time, is that it lends itself perfectly to the use of computer simulation techniques, especially on microcomputers.
Another by-product of the discrete approach is that it allows a more logical arrangement of the topics in electrodynamics. This arrangement is discussed in detail in Sec. 2.1.1; basically the idea is that the laws of electrostatics, including Coulomb's law, are simple logical consequences of Maxwell's equations, and not the other way around. Traditional texts are forced to treat them in an illogical order because the mathematics required to introduce Maxwell's equations and show they imply Coulomb's law is too heavy to drop on a student at the beginning of the course. Using a discrete approach, this is no longer true, and it is possible (and therefore desireable) to use the logical order. This has the additional advantage that the phenomenon of "action at a distance", which seems so mysterious to many students learning Coulomb's law, is simply not present. All interactions are explicitly local: a field variable at one grid point influences only those at the neighboring points.
This book is designed for a comfortably-paced one-year course, or for a more compressed one-semester course for students with more mathematical sophistication. Ideally a student at this level would have had some experience with the discrete simulation of simple mechanical or population-diffusion systems, and so would not find the discrete Maxwell equations very strange (they are very similar to these). One could then skip or review briefly the first eight chapters (Part A, on non-electromagnetic dynamical systems). The course could start almost immediately with Maxwell's equations (Chapter 2.1). In the more likely event that the average student's background is devoid of any experience with dynamical systems or simulation, Part A would have to be treated more completely. Sections which could be skipped without serious loss of continuity are marked with an asterisk in the table of contents, although it should be pointed out that some of these (especially the earlier ones) may be useful background for later chapters for students who are unfamiliar with the topics they treat. In particular, I have included a chapter on symmetry which may seem overly explicit to many instructors. However, I have found that this is one of the hardest concepts for students (even very good ones) to grasp. It is often described in terms which have a perfectly specific meaning to the instructor, who has seen them applied to many examples, but which the student has heard only as vague generalities.