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    Principles of Electrodynamics
    Principles of Electrodynamics
    Principles of Electrodynamics
    Ebook552 pages4 hours

    Principles of Electrodynamics

    By Melvin Schwartz

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    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.

    LanguageEnglish
    PublisherDover Publications
    Release dateApr 24, 2012
    ISBN9780486134673
    Principles of Electrodynamics

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      Principles of Electrodynamics - Melvin Schwartz

      Index

      Preface

      Electromagnetic theory is beautiful! When looked at from the relativistic point of view where electric and magnetic fields are really different aspects of the same physical quantity, it exhibits an aesthetically pleasing structure which has served as a model for much of modern theoretical physics. Unfortunately this beauty has been all but buried as most textbooks have treated electricity, magnetism, Coulomb’s law, and Faraday’s law as almost completely independent subjects with the ground work always supplied by means of empirical or historical example. Occasionally a chapter is devoted to the relativistic coalescence of the various aspects of electromagnetism but use is rarely made of the requirement of Lorentz invariance in deriving the fundamental laws.

      Our point of view here is quite different. Basically we have two purposes in mind—one is to exhibit the essential unity of electromagnetism in its natural, relativistic framework and the other is to show how powerful the constraint of relativistic invariance is. To these ends we shall show that all electromagnetism follows from electrostatics and the requirement that our laws be the simplest ones allowable under the relativistic constraint. The hope is that the student will make use of these new insights in thinking about theories that are as yet undeveloped and that the model we set here will be generally useful in other areas of physics.

      A word about units. Unfortunately one of the results of the completely disconnected way in which electricity and magnetism have been taught in the past has been the growing acceptance of the mks over the cgs system of units. We have no special preference for centimeters over meters or of grams over kilograms. We do, however, require a system wherein the electric field E and the magnetic field B are in the same units. Using the mks system, as it is presently constituted, for electromagnetic theory is akin to using a meterstick to measure along an East-West line and a yardstick to measure along a North-South line. To measure E and B in different units is completely antithetical to the entire notion of relativistic invariance. Accordingly we will make use of the cgs (gaussian) system of units exclusively. Conversion to practical units where necessary can be carried out with no difficulty.

      The author would like to express his most profound appreciation to Miss Margaret Hazzard for her patient and careful typing of the text.

      MELVIN SCHWARTZ

      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.

      1-1VECTORS 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

      If a is a number, then

      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

      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,

      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 rule in 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.

      Fig. 1-1 The addition of two vectors can be accomplished by placing the tail of one at the head of the other.

      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.

      1-2THE 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 . Now, we ask, what if we were to rotate our coordinate system to a new set of axes x′ y′, and z? How would r .) We write

      Fig. 1-2 The vector r are unit vectors along the x, y, z axes.

      We note the obvious fact that

      , viz.,

      We realize that not all the nine quantities aij form an orthogonal set of unit vectors.

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

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

      , we can rewrite this equation as follows:

      Carrying out the indicated multiplications, we find

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

      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

      We can represent the above operation symbolically by writing

      (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,

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

      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

      or, in general,

      We abbreviate this in the customary way by writing

      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,

      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

      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

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

      (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 + z1z. 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 x1x2 + 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.

      We go on now to the definition of another important entity, the pseudoscalar. The pseudoscalar differs from the scalar in only one important respect. The sign of the number we obtain by following our prescription in a left-handed coordinate system is opposite to that we obtain in a right-handed system. For pure rotations, scalars and pseudoscalars behave identically.

      To find an example of a pseudoscalar is not difficult at all. Let us take three points in space which in one coordinate system have the components (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3). We can construct a determinant D out of these nine numbers:

      (It is quite clear that D is equal to r1 · (r2 × r3) and has magnitude equal to the volume of the parallelopiped determined by r1, r2, and r3.) If we change coordinates by means of a pure rotation with det a = +1, we find that D′ as evaluated in the new system for the same three points is unchanged. On the other hand, if we change from a right-handed to a left-handed system, D′ changes sign. (Take the simplest such transformation corresponding to x′ = –x, y′ = –y, and z′ = –z and substitute above. Then rotate to any other left-handed system.)

      So far we have been dealing with prescriptions whereby we come up with single numbers. However, we have already discovered some entities which require three components for their specification, like the coordinates of a point. This brings us then to another class of mathematical objects which we shall call the polar vector.

      Imagine that we have a prescription for calculating a triplet and obtain (υ1, υ2, υ3) as the result of following this prescription. Consider next that we transform to a new coordinate system by means of the transformation a . These triplets are the components of a polar vector v if and only if, for any choice of a, we have

      Obviously the three coordinates of a point (x, y, z) constitute a polar vector. So do the three components of velocity and acceleration. Using the notation we have developed earlier, we can write

      An important characteristic of a polar vector is the fact that v changes sign under the pure inversion represented by

      Under this transformation, of course,

      On the other hand, we can imagine a triplet of numbers which behaves exactly like a polar vector under rotation but does not change sign under inversion. Such an entity comes under the classification of axial vector. To construct such a triplet we need only take the vector product of two polar vectors v and w:

      Under an inversion,

      We shall shortly discuss more complex mathematical entities with more than three components. For the moment though, let us pause and see if we can understand the physical importance of what we have done.

      One of our underlying physical principles is that there is no preferred direction or handedness¹ to the universe. This means that the basic laws of nature cannot depend on the coordinate system we choose to use for their formulation. Now basic physical laws are written down as equations. If the two sides of an equation do not have the same transformation properties then the form of the equation will depend on the coordinate system we chose, thereby violating our principle. For example, suppose our equation sets a scalar quantity equal to a pseudoscalar quantity. If we reverse the handedness of our system, one side of the equation would change sign and the other would not, leading to an obvious change in the appearance of the physical law. To avoid these problems we shall agree never to write down a basic equation in classical physics where the two sides do not behave identically under transformation. That means that we will always equate scalars with scalars, polar vectors with polar vectors, axial vectors with axial vectors, and so forth. In this way, if an equation is true in one coordinate system then the identical equation will be true in any system related to it by rotations or inversions.

      We should emphasize that in all the above we are only talking about those equations which describe the fundamental physical laws. In applying the physical laws to any specific situation, we will usually find that there is a preferred coordinate system to use and hence what we have said above would not necessarily hold true. For example, if we were studying the trajectory followed by a baseball near the earth’s surface we would naturally choose one of our coordinate axes in the upward direction. On the other hand, when we write down a general set of laws governing the behavior of magnetic and electric fields (Maxwell’s equations), we will certainly insist that no preference be given to any coordinate system or to a particular handedness of our system.

      The type of reasoning we have just described plays a particularly important role in electromagnetism, and we would like to take the liberty of drawing on some illustrations here even though we have not developed the subject yet. (Hopefully the student taking this course has already studied some elementary physics before.) The first discovery we will make in electrostatics is Coulomb’s law, where we will find that the force by one charge on another is proportional to the product of the charges and inversely proportional to the distance between them. Looking ahead to Eq. (2-1-1), we shall write

      F12 = force by charge 1 on charge 2

      Now, at the moment we do not know whether charge is a scalar or a pseudoscalar. Regardless of what it is though, the product of two charges q1q2 is a scalar. Hence the right side of the equation is a polar vector. This in turn means that force is a polar vector. Since force is equal to mass times acceleration, we next conclude that mass is a scalar (under these three-dimensional transformations).

      What about charge itself though? Whenever we see an elementary force in electromagnetism it is always proportional to the product of charges. Hence, there is no way of determining whether charge is a scalar or pseudo-scalar quantity. Since it makes no difference, we will assume it to be a scalar. This implies immediately that electric field E is a polar vector since it is just equal to force per unit charge.

      How about magnetic field? As we shall learn, a magnetic field B exerts a force on a charge q given by

      where c is the velocity of light. This tells us that B is an axial vector.

      In Chap. 3 we will discuss an experiment to search for magnetic monopoles. These, if they exist, are elementary magnetic charges which are acted upon by a magnetic field in the same way as ordinary electric charges are acted upon by an electric field. We would have then, for a magnetic charge qm, a force given by

      Since B is an axial vector, we conclude that qm is a pseudoscalar quantity. Obviously, if we had begun by choosing electric charge to be a pseudoscalar, then we would now come out with magnetic charge as a scalar.

      We can apply the principles we have discussed above in another manner to help us in determining the physical laws themselves. Suppose we have determined that a part of our physical law states that a particular component of a given polar (or axial) vector v is equal to the same component of another polar (or axial) vector w. We can choose our x axis along the direction in which the components are known to be equal and then summarize our knowledge by the statement that υx = wx in this coordinate system. If we further know that there is no preferred system for the laws we are uncovering, then we can deduce that υy = wy and υz = wz also. In other words, v = w in all systems. We shall make extensive use of this procedure when we introduce magnetism in Chap. 3. The only difference will be that we will be working in the four-dimensional world of special relativity. It is at that point that the full beauty of the notions we have developed here will become apparent.

      1-3DIFFERENTIATION OF VECTORS WITH RESPECT TO TIME AND POSITION; THE DEL OPERATOR (∇) AS A VECTOR

      Since we know how to add and

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