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Electromagnetic Fields and Waves
By Vladimir Rojansky Dover Publications, Inc.
Copyright © 1979 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-14773-4
CHAPTER 1
Scalar Fields
The theory of electromagnetic fields discussed in this book was developed in the 1860's by James Clerk Maxwell (1832-1879). It rests on four equations, called Maxwell's field equations and shown in the chart in the preface. These equations presumably describe all electromagnetic phenomena, except for quantum modifications in the atomic domain. They are built upon and include the law of interaction of static electric charges announced in 1785 by Charles Augustin de Coulomb (1736-1806), the law of interaction of steady currents formulated in 1822 by André Marie Ampère (1775-1836), and the law of induction of electric fields by time-varying magnetic fields discovered in 1831 by Michael Faraday (1791-1867). They also include the current law and the voltage law for electric circuits stated in 1848 by Gustav Robert Kirchhoff (1824-1887).
The roots of Maxwell's theory lie deep in Faraday's graphic concepts, which guided Maxwell in his work and which he eventually expressed in mathematical form. Some of these concepts, such as "lines of force," are still as useful today as they were to Faraday and Maxwell—for picturing electric and magnetic fields, for solving simple problems, and for making the implications of some of the mathematical formulas more vivid.
The reader has several tasks before him: to understand the physical content of Maxwell's equations, to learn the mathematical shorthand in which they are usually written, and to master the language of the theory—a language so blended of physics and mathematics that sometimes it is hard to tell which is which. We will lead up to Maxwell's equations by reviewing the more familiar laws of electricity and magnetism (such as the circuit laws illustrated for simple cases in §1), restating each in field-theoretic language, and rewriting each in field-theoretic symbols. As we do this, the reader will have to work in detail through quite a few unsophisticated preliminaries.
The portions of Maxwell's theory discussed in this book involve scalar fields and vector fields. Examples of scalar fields (conductivity and charge density) are given in §§2 and 3, without waiting for the mathematical cautions hinted at in Chapter 2. Scalar fields in general are taken up in §4, and vector fields in Chapter 6.
1. CIRCUIT QUANTITIES AND CIRCUIT EQUATIONS
Most of the topics presented in this section are presumably already familiar to the reader. We will nevertheless touch upon them, partly to have on hand certain equations for later reference, and partly because the concept of "field quantities" can be made clearer by contrasting it with the concept of "circuit quantities."
Units. The reader should gradually familiarize himself with Appendix B, which describes the MKS-Giorgi units of measurement used in this book, and with the conversion table on the inside front cover. For expository reasons, these units appear in the text in a different order from that in the appendix. For example, the definition of the ampere refers to forces between current-carrying wires. Therefore, although we will begin at once to call our unit of current the "ampere," the reader will have to wait until §89 before we make sure that this unit is in fact the ampere defined in the appendix. The MKS-Giorgi system is especially convenient for our purposes, but older systems —particularly the cgs Gaussian system, described in other books—have advantages in discussions of other aspects of electricity and magnetism.
Static charges. Conductors and insulators consist of positive electric charges (atomic nuclei) and negative charges (electrons), all in constant motion. Any extra charges placed on or in a body—say the extra electrons on the negative plate of a capacitor—are also in constant motion. The frequencies of the oscillation of all these charges are, however, very high—of the order of optical frequencies—so that in experiments not involving such frequencies only the time-averages of the fields of these charges come into play. It is, therefore, useful to introduce the term static charges, which stands for actual charges imagined to be held fixed in their average positions. The electric fields produced by static charges are called electrostatic.
Conduction currents. In metals, some of the electrons, called conduction electrons, are bound to the atomic nuclei only loosely and can be easily moved from place to place. A copper wire, for example, has one such electron per copper atom. If a piece of wire is connected across the terminals of a dry cell, an extra electric field is set up inside the wire in addition to the ever-present atomic electric fields, and the conduction electrons drift along the wire, amounting to a current. (We will not discuss "semiconductors," such as germanium, or "semimetals," such as arsenic, in which the situation is more complicated.)
The drift speed of the conduction electrons is much lower than the speed of electric signals along a wire (Exercise 2). The reason for this is the stop-and-go zigzag drift of the conduction electrons, caused by continual collisions with atoms. At each collision, some of the kinetic energy of a conduction electron is transferred to an atom and converted into vibrational energy of the atomic lattice, and thus into heat.
Despite their continual motion, the electrons that are tightly bound to atomic nuclei do not contribute to the conduction current flowing, say, through an ammeter—they move around the nuclei so fast and so close to the nuclei that from the large-scale viewpoint they can be regarded as standing still. Their motion is ignored in this book, which deals with large-scale (macroscopic) and not with small-scale (microscopic) phenomena. The motions of the tightly bound electrons cause magnetic effects, of course, effects that do not necessarily average out if neighboring atoms interact with one another sufficiently strongly. These effects must be considered in detail if one wants to account on the microscopic basis for, say, the magnetic permeability of a medium, which is itself a macroscopic quantity that describes an average effect.
We follow a standard convention and say that electric current is flowing in a branch ab of a circuit from a to b if in effect a positive electric charge is moving in this branch from a to b. For example, if the branch ab of a circuit is a copper wire, the statement that the current in this branch flows from a to b means in fact that conduction electrons move in this branch from b to a.
Circuit quantities. When we refer to such points as a and b in Fig. 1, we let the subscripts ab and (ab) mean, respectively, "from a to b" and "between a and b." The simpler circuit quantities are the resistance, say R(ab), between the terminals a and b of a conductor (resistor); the potential drop, say Vab, from a to b; the current, say iab, flowing from a to b; the capacitance between the terminals of a capacitor (condenser), and so on. In many conductors the current iab is directly proportional to Vab; these conductors are said to satisfy a law formulated in 1826 by Georg Simon Ohm (1787-1854). In this book we consider only these "ohmic" conductors and take it for granted that in the equation
iab = Vab/R(ab) (1)
the resistance R(ab) is a constant; this equation then states Ohm's law. An example of a "nonohmic" conductor is thyrite, in which iab is roughly proportional to (Vab), so that the resistance of a thyrite rod drops rapidly when the potential difference between its ends is increased.
Another way of writing (1) is
iab = G(ab)Vabi (2)
here G(ab) is the conductance of the conductor between the terminals a and b, defined as
G(ab) = 1/R(ab). (3)
We express resistance in ohms, and hence our unit for measuring conductance is the "reciprocal ohm," called the mho.
Equation (2) is usually written i = GV. We used double subscripts above to stress the fact that such circuit quantities as current, conductance, and potential drop each pertain not to a single point but to a pair of terminals (which are points in principle, but perhaps binding posts in practice). For example, let a and b be the terminals of a thick wire and let P be a point inside the wire. Then the question "What is the magnitude of the current flowing from a to b?" has a clear-cut meaning; the answer might be "five amperes." But the question "What is the magnitude of the current flowing at the point P?" is meaningless, because current is not the kind of quantity that can be evaluated at an individual point. (A proper question to ask is "What is the current density at P?")
Electromotive forces. The current i in Fig. 1 is caused by the chemical cell that has a positive terminal, marked plus, and a negative terminal. A simple cell is shown in Fig. 1(b), where the dashed line indicates a semi-permeable partition separating the CuSO4 and the ZnSO4 solutions, and where chemical action of the copper and the zinc ions keeps the copper plate charged positively relative to the zinc plate. Outside the cell, the conventional current flows "from plus to minus," but inside the cell it flows "from minus to plus." This means that the cell has a rather special property: it forces charges to move in an "unnatural" direction, and hence it does work on them. This property is described by saying that the cell is a source of an electromotive force (emf). The magnitude of this emf, say E, is the work that the cell does per coulomb of positive charge that it moves from its negative to its positive plate. We express E in joules per coulomb or simply volts. The resistance R in the figure includes the internal resistance of the cell.
Given the values of E and R in the circuit of Fig. 1, the current i can be found by using Kirchhoff's loop law. In the case of constant currents in circuits consisting of resistances and sources of emf's, this law can be stated as follows: If we go around any closed loop of a circuit, the sum of the potential drops that we encounter along the resistances will be equal to the sum of the emfs included in the loop. Suppose, for example, that in Fig. 1 we go from a to b and then, through the cell, back to a. On the way from a to b we move "with the current," and hence encounter the potential drop Vab; as we cross the cell "from minus to plus" we encounter the emf E. Kirchhoff's law states in this case that Vab = E. By Ohm's law, the potential drop Vab is the "iR drop" along the resistor. Accordingly, iR = E and i = E/R.
Junction law. If a circuit comprises more than one loop, Kirchhoff's junction law comes into play: The sum of currents flowing toward a junction of wires is equal to the sum of the currents flowing away from this junction. Figure 2 shows three equivalent examples of two 1-ampere currents flowing toward a junction and one 2-ampere current leaving it.
Branch currents. The circuit shown in Fig. 3(a) has three branches: that containing the cell and its internal resistance r, that containing R1, and that containing R2. The corresponding currents are labeled i, i1, and i2. The directions along the branches, marked by the arrowheads, are chosen arbitrarily, with the understanding that if i1, say, actually flows in the direction opposite to that of the arrowhead, then the value of the number i1will prove to be negative. [In Fig. 3(a) the three branch currents obviously do flow as indicated by the arrowheads, and hence each of the three numbers i, i1, and i2 is positive.] At the starred junction we have
i = i1 + i2. (4)
At the other junction we have the equation i1 + i2 = i, which adds nothing new.
The circuit has three loops (or meshes): that containing the cell and R1, and marked α; that containing R1 and R2, and marked β; and that containing the cell and R2—the "big" loop. If we start at the star and go counterclockwise around the first loop, we get
i1R1 + ir = E. (5)
If we start at the star and go counterclockwise around the second loop, we get
i2R2 - i1R1 = 0. (6)
The big loop contributes nothing new. We now have three equations—(4), (5), and (6)—ready to be solved for the three unknown currents. The solution for i proves to be
i = R1 + R2/R1R2 + r(R1 + R2) E. (7)
Loop currents. To prepare for another method of computing the currents in Fig. 3, we imagine that the loops α and β carry the "loop currents" (or "mesh currents") iα and iβ, pictured in Fig. 3(b), and we count up the potential drops and the emf's along these loops.
Starting at the star, we go around loop α. As we go along R1, the net current flowing "with us" is iα —iβ, so the potential drop we encounter along R1 is (iα—iβ)R1. The potential drop along r is iαr and consequently
(iα - iβ)R1 + iαr = E, (8)
which can be written in the form (9). Similarly, loop β yields the equation iβR2 + (iβ —iα)R1 = 0, recorded below as (10). Thus the two loop currents satisfy the two equations
(R1 + r)iα - R1iβ = E, (9)
-R1iα + (R1 + R2)iβ = 0, (10)
which are, of course, consistent with the three equations (4), (5), and (6) of the branch-current method (Exercise 5).
The method of loop currents has two advantages: Kirchhoff's junction equations need not be written explicitly, and the coefficients in such sets of equations as (9) and (10) are easy to check. For example, the respective coefficients of iα in (9) and of iβ in (10) are the total resistances of the corresponding loops; the coefficient of -iβ in (9) and of -iα in (10) is the resistance shared by the two loops.
(Continues...)
Excerpted from Electromagnetic Fields and Waves by Vladimir Rojansky. Copyright © 1979 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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