Electromagnetic theory is basic to classical and relativistic physics at every level. This text covers material for a year's course in advanced theoretical electromagnetism, at all times keeping mathematics ancillary to the physics. Prior knowledge of differential equations and vector analysis will prepare the reader for inquiry into:
• meaning and definition of vectors D and H inside matter
• interpretations of the conservation laws for energy, momentum and angular momentum
• localization and flow of electromagnetic energy
• concepts of invariance, form invariance and covariance in special relativity
Summation problems form a kind of pattern in treating all the topics, which include general boundary value, dielectrics, time-varying fields, time-harmonic currents and propagation in imperfect conductors. Problems (no solutions) and references (annotated) are found at the end of each chapter; a general bibliography is also appended. Students have discovered this book in its earlier hardcover edition to be a stimulating exploration of theory, self-contained and suitable for unsupervised study; physicists and professors will find more than one fresh insight into their field.
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THE CLASSICAL ELECTROMAGNETIC FIELD
By Leonard Eyges
Dover Publications, Inc.Copyright © 1972 Leonard Eyges
All rights reserved.
CONCEPTS OF A FIELD THEORY
This is a book on the electromagnetic field, one of the many examples of fields that are important in physics and natural science. In the words of Morse and Feshbach (B), "Practically all of modern physics deals with fields: potential fields, probability fields, electromagnetic fields, tensor fields, and spinor fields." Since the concept of field is applied so widely, it is perhaps natural that it has taken on somewhat different contexts with, of course, an underlying common denominator. In this first brief chapter, we shall try to analyze the concept and its variants and so highlight the essential aspects of the electromagnetic field.
One mathematical common denominator is easy to isolate. Mathematically, a field is a function, or a set of functions considered as an entity, of the coordinates of a point in space (and possibly of time). For example, if the temperature is defined at every point in some volume, we say there is a scalar temperature field throughout that volume. If, in a moving fluid, the three vector components of velocity are known as a function of position in the fluid, they constitute as a whole a vector velocity field. In the theory of elasticity, the relative vector displacements of points of an elastic solid from their unstrained positions are described in terms of a double vectorial or, as it is more usually called, a tensor field. In modern physics, the Schrödinger probability amplitude ψ or the generalized Dirac spinor amplitudes are examples of fields.
It is worth noting that in this above list, there are really two kinds of fields. The first kind, exemplified by the temperature or velocity fields, is an idealization that is really defined only in a certain approximation of coarseness or fineness. For example, the velocity field of hydrodynamics is meaningful only in an average or continuum approximation in which the atomic and grainy structure of the fluid is not considered. This point is discussed in some detail in Morse and Feshbach (B) and we shall not elaborate on it here. By contrast, the Schrödinger or Dirac fields are not approximations to an underlying discontinuous physical model but must be assumed to exist no matter how finely space is divided.
To describe the evolution of the electromagnetic field concept, we recall some history that begins with Newton (1642-1726). One of the great laws of physics is Newton's universal law of gravitation. This law embodies the concept of action at a distance, according to which gravitational masses exert the forces they do on each other by virtue of their positions in space, the intervening space playing no active role. This is meant to contrast with forces which work via contiguous action whereby two masses at a distance exert forces which are transmitted by the intervening medium. For example, if several billiard balls are in contact in a row on a table and the first one is struck, the last one will move. Thus, the one billiard ball exerts a force on the other distant one, but by a mechanism which involves successive actions of the intermediate balls, the one moving the next, moving the next, etc. The concept of contiguous action is then quite different from that of action at a distance where no intermediate mechanism or medium is considered.
The work of Newton is relevant in a second way. His laws of the motion of point particles and rigid bodies paved the way for the development of the continuum mechanics of fluids and later of elastic bodies. Some tentative beginnings on the subject of fluid flow were made by Newton himself, but the real groundwork was later laid by John Bernoulli (1667-1748) and Euler (1707 – 1783). They bypassed the problem of the actual microscopic structure of fluids by adopting a continuum model and then applied Newton's laws of point mechanics to small elements of the continuum. The same idea was later applied to elastic solids, and the vibrations of these solids was discussed by applying Newton's laws to a small element of the solid, assuming that the forces acting on it were the stresses due to the rest of the solid, plus any external forces. Hydrodynamics was therefore formulated in terms of the velocity and acceleration of the moving fluid at every point, i.e., in terms of velocity and acceleration vector fields. The theory of elastic solids was similarly formulated in terms of stress and strain tensorial fields.
So much for mechanics; we turn now to electromagnetism. A basic law of electrostatics is Coulomb's law for the force between two charged particles. Except for the fact that the electric force can be either attractive or repulsive, whereas the gravitational force is always attractive, this law is obviously similar to Newton's law of gravitation. It was then considered from the time of its discovery as an example of action at a distance, in which two charges act on each other in a way that has nothing to do with the intervening medium. But this view began to be questioned, at least in the mind of Faraday (1791-1867), by his work on dielectric polarization. This phenomenon led him to attribute more and more importance to the intervening medium. We cannot go into the details of Faraday's results and reasoning but, for illustration, shall concentrate on one of his findings. This has to do with the effect of insulators or dielectrics on the capacitance of condensers. Consider a parallel-plate capacitor with air between its plates; it has a certain capacitance. If the air is replaced by a dielectric medium, the capacitance will be increased. Faraday viewed the phenomenon of enhanced capacitance as somehow due to the fact that the electric force generated by the charges on the plates was weakened by the dielectric medium. But if changing the medium that intervened between the charges changed the force, then somehow the forces must depend on, or be transmitted by, the medium. As a corollary of this view, Faraday considered that the essential feature of the interaction between charged particles was the lines of force that carried the "stresses" of the medium from one charge to another. These lines of force that extend from charge to charge through the medium were considered primary; the charges merely happen to be the places where the lines of force start and stop. Although these views were mainly derived from experiments on dielectric polarization with strongly dielectric substances, they seemed to be equally valid for those whose dielectric constant was close to unity. By extension, then, Faraday considered them valid for that insulator whose dielectric constant is exactly unity, i.e., free space. In short, in Faraday's view, free space was a substance, qualitatively like all other insulating substances, that contained charges which could be separated and displaced, as charges were separated and displaced in the material insulators he used in his capacitor experiments.
The next great name in the history of the electromagnetic field is that of Maxwell (1831-1879). He took up Faraday's ideas on the nature of the force between charges and the importance of the intervening medium. He succeeded in showing that the forces that charge complexes exert on each other and their energies could be expressed not only in terms of the magnitudes of the charges and their positions but in terms of a stress energy tensor that was defined throughout the medium (even if that medium was free space) and that had as components certain functions of the field strengths. For example, the force that one point charge exerts on another could be calculated by either using Coulomb's law, or surrounding the charge by an imaginary surface and integrating over that surface the total electrical stress as given by the stress energy tensor. This concept of stresses in the medium gave no new result, but it did at least show that the Faraday-Maxwell conception of the "state of the medium" was consistent with the results, if not the concept, of action at a distance.
The next step in the development of the field concept was also due to Maxwell. This was, of course, the discovery of the equations that bear his name and, as a corollary, the discovery of electromagnetic waves and their identity with light waves. Two basic guides in this work were Faraday's theory of charge polarization or displacement, and the theory, well developed by Maxwell's time, of the vibrations of an elastic solid. By generalizing Faraday's idea of displacement to the time-varying case and by introducing the so-called displacement current D, Maxwell found that electromagnetic phenomena could be described in terms of four field vectors, E, B, D, H. But like the velocity fields of hydrodynamics or the stress or strain fields of elasticity, these fields were not considered to exist by themselves but were somehow considered to be vibrations or displacements of an underlying luminiferous ether whose properties were those of a somewhat special kind of elastic solid. Electromagnetic waves were then, so to speak, secondary: the ether could exist without electromagnetic waves but electromagnetic waves could not exist without the ether.
Maxwell's theory was a great triumph. But as the years after its discovery passed, it was accompanied by an increasing perplexity as to the nature of the hypothetical ether that underlay it. The history of the researches and speculations on the nature of the ether is beyond us here. It is well described in Whittaker (B) and some detail is given in Chapter 12 of this book. Suffice it to say here that a famous experiment by Michelson and Morley showed that the ether did not exist. Nonetheless, electromagnetic waves continued doggedly to be generated and propagated. The Michelson-Morley experiment thus served, so to speak, to emancipate the electromagnetic field. After it, the fields could not be considered as "merely" vibrations of an underlying medium. From the time of the disproof of the existence of the ether, the electromagnetic field had to be looked on as an entity in its own right, as real as matter and everywhere on a par with it.
In fact, the essence of the electromagnetic field theory is that the field does have properties that we usually associate with matter. It can possess energy, momentum, and angular momentum. The field is thus a dynamical concept and is not merely a mathematical function of the space coordinates and time. We shall highlight this essential aspect of the electromagnetic field by a simple example. In it, we presuppose a small amount of elementary knowledge on the reader's part; namely, that accelerated charged particles radiate and that this radiation is propagated with a finite velocity c.
Suppose two charged particles q1 and q2 are separated by a distance d. Imagine that we suddenly move particle 1, say, and then quickly bring it to rest again. Having been accelerated, the particle will emit a pulse of radiation which travels at velocity c and hence would make itself felt on particle 2 at a time t = d/c. If we look at this system of two charged particles at some time after the first particle is brought to rest but at a time which is less than d/c, we would see simply two charged particles, each at rest; they would constitute an isolated system with kinetic energy zero. Suddenly, however, at a time t = d/c, we would find that the second charged particle began to move. Superficially, the energy (and momentum) of this isolated system would appear to change even though there were no external forces acting on it. How can we reconcile this with the conservation laws of energy and momentum? The field theory gives one possible answer to this question. According to it, we have simply overlooked the fact that there are forms of energy other than kinetic; there is in fact another physical entity that we have not mentioned, the electromagnetic field, and this entity "contains" energy and momentum. The energy and momentum that begin to be transmitted to particle 2 at time t = d/c were, in fact, contained in the electromagnetic field for earlier times.
We shall spell out these conservation laws involving the electromagnetic field in Chapter 11. For the moment, however, we make another point: the conceptual difficulties with energy and momentum conservation come about primarily because of the finite velocity c of electromagnetic propagation. For if c were infinite, we could not set up, even in thought, the above experiment and the concomitant difficulties would not arise. For static fields, then, in which propagation velocities are not involved, we shall see that the field concept does not so inevitably impose itself. Many of the results of electrostatics and magnetostatics can be formulated in terms either of action at a distance or of a field.
In summary, the basic idea of electromagnetism as a field theory is that charges and currents produce at each point of space a field that has a reality of its own, that can contain and propagate energy, momentum, and angular momentum, and that acts on other charges. The field is produced by, and acts on, charges. Correspondingly, there are two sets of equations. These are, first, Maxwell's equations which describe the field produced by a given set of charges and currents. Second, there is the Lorentz force equation which shows how a given field acts on charges.
We should not leave the impression that classical field theory itself is without its difficulties. It has these too. There is a conceptual difficulty in describing the self-action of the field on the charge that produces it, which is not soluble in classical field theory. This is outlined in Section 14.9, and an extensive discussion is given in Rohrlich (B). For practical computational purposes, these difficulties are resolved in part by quantum electrodynamics, which is the extension of the classical field theory of this book to incorporate quantum concepts; there remain however difficulties in principle connected with the occurrence of infinite quantities. There have been attempts over the years to bypass the difficulties with field theory, by reviving a sophisticated version of the action-at- a-distance theory. Examples are the theories of Wheeler and Feynman (R) and a recent paper by Kennedy (R).CHAPTER 2
THE ELECTROSTATIC FIELD
It has been known for a long time that there are materials which, having been rubbed, will attract or repel other small pieces of matter. In particular, the Greek philosophers knew that amber possessed this property; it is from the Greek word for amber, ηλεκτρov, that the word "electricity" is derived. There appears to have been nothing extensive made of this knowledge, however, and the beginning of electrostatics as a science must be dated, along with many other beginnings, from the Renaissance. The detailed history of this is complicated and fascinating, but we can mention only some highlights. Gilbert (1540 – 1605) found that materials could be roughly divided into those that could be electrified by friction (electrics) and those that could not (nonelectrics). Today we call such materials insulators and conductors, respectively. DuFay (1698-1739) discovered that the "electrical fluid" was of two types, "vitreous" and "resinous"; this was the origin of the concept of positive and negative electrical charges. Moreover, it was found that vitreous and resinous electricity could not be produced from nothing; a change in the amount of one was accompanied by a like change in the amount of the other. This observation was the primitive basis of the idea of the conservation of charge; in effect, it justified our treating amounts of positive and negative electricity according to the algebraic laws for combining positive and negative quantities.
All the early experiments were done with frictional electricity, i.e., with electrical forces generated by rubbing bodies together. It is amusing to note that even today we still do not have a clear understanding of the detailed mechanism of this phenomenon, as Harper (R) discusses.
Excerpted from THE CLASSICAL ELECTROMAGNETIC FIELD by Leonard Eyges. Copyright © 1972 Leonard Eyges. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
1 - CONCEPTS OF A FIELD THEORY,
2 - THE ELECTROSTATIC FIELD,
3 - THE SUMMATION PROBLEM FOR CHARGES,
4 - BOUNDARY VALUE PROBLEMS WITH CONDUCTORS,
5 - GENERAL BOUNDARY VALUE PROBLEMS,
6 - DIELECTRICS,
7 - THE MAGNETOSTATIC FIELD,
8 - THE SUMMATION PROBLEM FOR STATIONARY CURRENTS,
9 - MAGNETIC MATERIALS AND BOUNDARY VALUE PROBLEMS,
10 - FORCE AND ENERGY IN STATIC FIELDS,
11 - TIME-VARYING FIELDS,
12 - ELECTRODYNAMICS AND RELATIVITY,
13 - THE SUMMATION PROBLEM FOR TIME-HARMONIC CURRENTS,
14 - POINT CURRENTS,
15 - PERFECT CONDUCTORS AND TIME HARMONIC FIELDS,
16 - TIME HARMONIC FIELDS IN MATTER,
17 - PROPAGATION IN IMPERFECT CONDUCTORS AND DIELECTRICS,