Principles of Electrodynamics

by Melvin Schwartz

Overview

Unlike most textbooks on electromagnetic theory, which treat electricity, magnetism, Coulomb's law and Faraday's law as almost independent subjects within the framework of the theory, this well-written text takes a relativistic point of view in which electric and magnetic fields are really different aspects of the same physical quantity.
Suitable for advanced undergraduates and graduate students, this volume offers a superb exposition of the essential unity of electromagnetism in its natural , relativistic framework while demonstrating the powerful constraint of relativistic invariance. It will be seen that all electromagnetism follows from electrostatics and from the requirement for the simplest laws allowable under the relativistic constraint. By means of these insights, the author hopes to encourage students to think about theories as yet undeveloped and to see this model as useful in other areas of physics.
After an introductory chapter establishing the mathematical background of the subject and a survey of some new mathematical ideas, the author reviews the principles of electrostatics. He then introduces Einstein's special theory of relativity and applies it throughout the rest of the book. Topics treated range from Gauss's theorem, Coulomb's law, the Faraday effect and Fresnel's equations to multiple expansion of the radiation field , interference and diffraction, waveguides and cavities and electric and magnetic susceptibility.
Carefully selected problems at the end of each chapter invite readers to test their grasp of the material. Professor Schwartz received his Ph.D. from Columbia University and has taught physics there and at Stanford University. He is perhaps best known for his experimental research in the field of high-energy physics and was a co-discoverer of the muon-type neutrino in 1962. He shared the 1988 Nobel Prize in Physics with Leon M. Lederman and Jack Steinberger.

Product Details

ISBN-13:	9780486654935
Publisher:	Dover Publications
Publication date:	10/01/1987
Series:	Dover Books on Physics
Edition description:	Revised ed.
Pages:	368
Product dimensions:	5.50(w) x 8.50(h) x (d)

Read an Excerpt

Principles of Electrodynamics

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By Melvin Schwartz

Dover Publications, Inc.

Copyright © 1972 Melvin Schwartz
All rights reserved.
ISBN: 978-0-486-65493-5

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CHAPTER 1

Mathematical Review and Survey of Some New Mathematical Ideas

It would be delightful if we could start right out doing physics without the need for a mathematical introduction. Unfortunately though, this would make much of our work immeasurably more laborious. Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates. Indeed, the beauty and elegance of the physical laws themselves are only apparent when expressed in the appropriate mathematical framework.

We shall try to cover a fair bit of the mathematics we will need in this introductory chapter. Several subjects are, however, best treated within the context of our physical development and will be covered later. It is assumed that the reader has a working familiarity with elementary calculus, three-dimensional vectors, and the complex number system. All other subjects will be developed as we go along.

1-1 VECTORS IN THREE DIMENSIONS; A REVIEW OF ELEMENTARY NOTIONS

We begin by reviewing what we have already learned about three-dimensional vectors. As we remember from our elementary physics, there are a large number of quantities that need three components for their specification. Position is, of course, the simplest of these quantities. Others include velocity and acceleration. Even though we rarely defined what was meant by a vector in mathematically rigorous terms, we were able to develop a certain fluency in dealing with them. For example, we learned to add two vectors by adding their components. That is, if r1 = (x1, y1, z1) and r2 = (x2, y2, z2) are two vectors, then

r1 + r2 = (x1 + x2, y1 + y2, z1 + z2)

If a is a number, then

ar1 = (ax1, ay1, az1)

We also found it convenient to represent a vector by means of an arrow whose magnitude was equal to the vector magnitude and whose direction was the vector direction. Doing this permitted us to add two vectors by placing the "tail" of one at the "head" of the other as in Fig. 1-1. We also learned how to obtain a so-called scalar quantity by carrying out a type of multiplication with two vectors. If r1 = (x1, y1, z1) and r2 = (x2, y2, z2) are two vectors, then r1 • r2 is defined by the equation

r1 • r2 = x1x2 + y1y2 + z1z2

It was also shown that r1 • r2 could be obtained by evaluating |r1| |r2| cos θ12, where |r1| and |r2| are, respectively, the magnitudes of r1 and r2 and θ12 is the angle between them. Another so-called vector was obtained by taking the cross product of r1 and r2. That is,

r1 x r2 = (y1z2 - y2z1, z1x2 - z2x1, x1y2 - y1x2)

We shall have much more to say about the true nature of this beast very shortly. At the moment we just recall that it appears in some respects to be a vector whose magnitude is equal to |r1| |r2| sin θ12 and whose direction, at right angles to both r1 and r2, is given by a so-called right-hand rrule I going from r1 to r2. If we look from the head toward the tail of r1 × r2, we would see the shortest rotation from r1 to r2 to be in the counterclockwise direction.

Unfortunately, we shall have to relearn much of the above within a more abstract framework if we are to make any progress beyond this point. We shall have to go back to our basic notions and see if we can define what we mean by vector in a more suitable, less intuitive manner. Only by doing so will we be prepared to say clearly which combinations of three numbers are vectors and which are not. We will also be able to define scalar in a reasonable way and will then see our way clear to an understanding of higher-rank tensors.

1-2 THE TRANSFORMATION PROPERTIES OF VECTORS UNDER SPATIAL ROTATION

To open the way for a more rigorous definition of vector, we proceed a bit further with our old intuitive notions. Let us consider a so-called position vector, that is, a vector from the origin of our coordinate system to the point (x, y, z). If we draw a unit vector along each of the three axes as shown in Fig. 1-2 and call them [??], [??], and [??], respectively, we can write r = x[??] + y[??] + z[??]. Now, we ask, what if we were to rotate our coordinate system to a new set of axes x', y', and z' with a new set of unit vectors [??]', [??]', and [??]'? How would r be expressed now? We answer this question very simply by expressing [??], [??], and [??] in terms of the new unit vectors [??]', [??]', and [??]'. (This is possible because any three-dimensional vector whatsoever can be expressed either in terms of [??], [??], and [??] or in terms of [??]', [??]', and [??]'.) We write

(1-2-1) [MATHEMATICAL EXPRESSION OMITTED]

We note the obvious fact that

[MATHEMATICAL EXPRESSION OMITTED]

This, of course, permits us immediately to express the unit vectors [??]', [??]', and [??]' in terms of [??], [??], and [??], viz.,

(1-2-2) [MATHEMATICAL EXPRESSION OMITTED]

We realize that not all the nine quantities aij can be chosen independently. After all, only three angles are necessary to specify the rotation of one coordinate system into another. We expect then to have six equations linking the coefficients. We obtain these equations by requiring that [??]', [??]', and [??]' form an orthogonal set of unit vectors.

(1-2-3) [MATHEMATICAL EXPRESSION OMITTED]

Now to return to our original vector r. We can write r in terms of its components in either of two ways:

r = x[??] + y[??] + z[??] or r = x'[??]' + y'[??]' + z'[??]'

Making use of Eqs. (1-2-1), we find immediately that

(1-2-4) x' = a11x + a12y + a13z y' = a21x + a22y + a23z z' = a31x + a32y + a33z

We have traditionally used a right-handed coordinate system to specify the components of a vector. That is to say we have chosen [??], [??], and [??] so that if we curl the fingers of our right hand from [??] to [??], our thumb will point along [??]. Expressing this in language somewhat more abstract and less anthropomorphic, we can say that [??] × [??] = [??] in such a system. Obviously there is nothing in nature that requires us to limit ourselves to right-handed coordinate systems, and we might ask if there is anything special about the set of numbers aij if the primed system should happen to be a left-handed system. For a left-handed system we can write

(1-2-5) ([??]' × [??]') [??]' = -1

Expressing [??]', [??]', and [??]' in terms of [??],[??], and [??], we can rewrite this equation as follows:

[MATHEMATICAL EXPRESSION OMITTED]

Carrying out the indicated multiplications, we find

(1-2-6) [MATHEMATICAL EXPRESSION OMITTED]

The expression on the left of Eq. (1-2-6) is called the determinant of the matrix of numbers aij or aij for short. It is often written in the notation

[MATHEMATICAL EXPRESSION OMITTED]

We see then that any transformation that takes us from a right-handed coordinate system to a left-handed coordinate system is characterized by having its determinant equal to -1. Indeed, as we can easily see, the determinant is equal to -1 whenever we change the handedness of our system and +1 if we keep it unchanged. By allowing transformations with either sign of determinant, we allow ourselves to deal with both rotations and reflections or with any combination of these transformations.

We have begun to think of our transformation as having an "identity" all its own. It is characterized by a set of nine numbers, which we have called a matrix. Furthermore we have seen in Eq. (1-2-4) that we can obtain the triplet (x', y', z') by "multiplying" the triplet (x, y, z) by this matrix, with the operation of multiplication being defined as

(1-2-7) [MATHEMATICAL EXPRESSION OMITTED]

We can represent the above operation symbolically by writing

(1-2-8) r' = ar

(In the future, a boldface sans serif symbol, such as a, will mean that the symbol is a matrix and not a number.)

Suppose now that we wish to undertake two successive transformations, the first characterized by a and the second by another matrix b. If we begin with the triplet r, then the first transformation leads to the triplet r' and the second to the triplet r". That is,

r' = ar

r" = br'

Alternatively, we might have gone directly from the unprimed to the double-primed coordinate system by means of a transformation c.

r" = cr

Writing out these transformations in detail will show that we could determine all the elements of c directly from a and b by means of the simple set of equations

c11 = b11a11 + b12a21 + b13a31

c12 = b11a12 + b12a22 + b13a32

or, in general,

cij = bi1a1j + bi2a2j + bi3a3j

We abbreviate this in the customary way by writing

(1-2-9) [MATHEMATICAL EXPRESSION OMITTED]

Thus the element cij can be obtained by taking the "scalar product," so to speak, of the ith row in b with the jth column in a.

The operation which we have defined above in Eq. (1-2-9) is called the product of two matrices a and b and can be represented by the expression c = ba. Matrix multiplication, unlike the multiplication of two numbers, is not in general commutative, as the reader can very easily convince himself. That is to say the product ab is not in general equal to the product ba. Multiplication is, however, associative. This means that we can in general write, for three transformations a, b, and c,

(1-2-10) a(bc) = (ab)c

To complete our picture we should point out that one of the possible transformations is the identity transformation which leaves the coordinate system unchanged. We write this matrix as 1 with the observation that

(1-2-11) [MATHEMATICAL EXPRESSION OMITTED]

Returning back to Eqs. (1-2-1) and (1-2-2), we see that for every transformation a there is also an inverse transformation a-1 such that

(1-2-12) aa-1 = a-1 a = 1

The inverse transformation is just given by the transposed matrix. That is to say

(1-2-13) aij-1 = aji

(For those whose mathematical sophistication is just a bit above average, we might point out that the set of all transformations defined above constitute what is known in the trade as a group. The detailed properties of groups play an important role in the development of much of quantum mechanics and should be studied at the earliest possible moment by those who intend to extend their horizons in physics beyond the classical domain.)

We can now think in terms of the complete set of all transformations from one orthogonal coordinate system to another, including within our set both rotations (det a = +1) and reflections (det a = -1). The definition of scalar, vector, and various other entities is now best done in terms of this set of transformations.

Let us begin with what is intuitively the simplest of these entities, the scalar. Imagine that we are given a set of explicit instructions for determining some number. We follow these instructions scrupulously, coming up with a value for the number. We can now rotate our coordinate system or change its handedness (by means of the transformation a). If the same set of rules for determining the number leads to the same result in the new system, regardless of the choice of rotation or reflection, then the number is a scalar.

Obviously there are innumerable trivial examples of scalars that we can readily cite. The number of cents in the dollar or the number of fingers on your hand have nothing to do with the coordinate system and hence are ipso facto scalars. Much less trivial, though, are numbers that are derived by means of rules which concern coordinates themselves. Let us take a simple example.

Suppose the rule tells us to take the x coordinate of a point, square it, add to that the square of the y coordinate of the same point, and add to the sum the square of the z coordinate of the point. We would have then a number equal to x2 + y2 + z2. If we transform to a new system and follow the same prescription in the new system, we come up with x'2 + y'2 + z'2. Unless we knew the Pythagorean theorem we would have no a priori expectation that the same rule applied in these different systems would give us the same result. Indeed it does because we have just determined the square of the distance from our point to the origin, and that quantity does not depend on the rotational orientation or the handedness of our system. Clearly then the number x2 + y2 + z2 is a scalar.

Let us try a more difficult example now. Consider two points whose coordinates in one system are (x1, y1, z1) and (x2, y2, z2). We can form the expression x1x2 + y1y2 + z1z2 and evaluate it in this coordinate system. We can now transform coordinates and evaluate the same expression in the new system, obtaining x'1x'2 + y'1y'2 + z'1z'2. Again we have no a priori expectation that the two numbers will come out to be the same. Making use of Eqs. (1-2-4) and (1-2-3), the reader can easily convince himself that this is, however, the case — the numbers are the same and so the expression x1x1 + y1y2 + z1z2 is a scalar. (The result is not entirely unanticipated for we remember that this expression is the scalar product of r1 and r2 and can also be written as |r1| |r2| cos θ. The latter formula does not depend on the coordinate system.)

There is a great temptation now to let every "constant" of nature, like charge and mass, be labeled a scalar. In fact we must be exceedingly careful since an attribute like charge is defined operationally in terms of forces by external fields, and we must investigate the behavior of the entire system under both rotation and reflection before we can conclude that the attribute is a scalar. We shall have more to say about this very shortly.

(Continues...)

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Excerpted from Principles of Electrodynamics by Melvin Schwartz. Copyright © 1972 Melvin Schwartz. Excerpted by permission of Dover Publications, Inc..
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Table of Contents

Contents

Preface, vii,
1 MATHEMATICAL REVIEW AND SURVEY OF SOME NEW MATHEMATICAL IDEAS, 1,
2 PRINCIPLES OF ELECTROSTATICS, 28,
3 ELECTROMAGNETISM AND ITS RELATION TO RELATIVITY, 105,
4 TIME-INDEPENDENT CURRENT DISTRIBUTIONS; MAGNETOSTATICS, 139,
5 THE VARIATION OF THE ELECTROMAGNETIC FIELD WITH TIME: FARADAY S LAW, DISPLACEMENT CURRENTS, THE RETARDED POTENTIAL, 186,
6 LET THERE BE LIGHT!, 212,
7 THE INTERACTION OF RADIATION WITH MATTER, 234,
8 MULTIPOLE EXPANSION OF THE RADIATION FIELD; SOME FURTHER CONSIDERATIONS ON THE INTERACTION OF RADIATION WITH MATTER; INTERFERENCE AND DIFFRACTION, 273,
9 WAVEGUIDES AND CAVITIES, 304,
10 ELECTRIC AND MAGNETIC SUSCEPTIBILITY, 321,
Tables, 339,
Index, 341,