Eyges - The Classical Electromagnetic Field.

APPENDIX A CONVERSION TABLE FOR GAUSSIAN MKS UNITS

The following table gives the relations between the units of common quantities in the rationalized MKS system and in the Gaussian (CGS) system used in this book.

Read equals, with the numerical factor of column three, between the second and fourth columns. For example, one newton equals 105 dynes.

The first point to grasp about -functions is that they are not functions. They can be considered on occasion as a notational shorthand, or as the limit of a sequence of functions, or as a mnemonic for certain integration formulas, but they are not functions in the sense that sin x or e x are functions.

Their first guise in electromagnetic theory is perhaps this: We can write a general expression for the potential of any charge distribution by using the potential for a charge element and the principle of superposition. If the charge distribution is partially continuous with density p(r), and partially discrete with charges q at r i , then the expression for the potential at r consists of an integral and a sum:

(B.1)

Now the integral and the sum are more closely akin than they may look: the same physics, Coulombs law and superposition, is in both of them. It is then convenient to avoid the redundancy of writing both of them by generalizing the distribution p ( r ) so that it includes sharply localized integrals, i.e., point charges as a special case. We can then write the expression (B.1) in terms of the integral only, and this makes a very convenient shorthand.

To elaborate on this, consider one-dimensional charge distributions that are functions, say, of the x -coordinate. Suppose that we have such a distribution that is partially continuous and partially discrete. We can think of the discrete part, the one-dimensional point charges, as consisting of thin charge slabs at x 1, x 2,... , x n. The continuous part is described by a density distribution which is some function of x. How can we describe the point charges as well in terms of a density distribution ? Clearly the density distribution that describes a unit point source at, say, the origin must be a highly singular function. It must be nonvanishing essentially only at the origin, and moreover its integral must be unity. As a function of x, it is not an ordinary function, but if we call such a kind of function (x) it must have the properties

(B.2)

Moreover, since it is such a sharp function, if it appears in an integrand with any other function f ( x ) which is regular at the origin, then the contribution to the integral will come only from x = 0. Then we can evaluate f ( x ) in the integrand at the origin and get the basic defining equation of the -function,

(B.3)

With this definition, a source of strength q at x = a is represented by q (x a ), where the generalized defining equation is

(B.4)

The extension to three dimensions is then straightforward. If there is a point charge q at r = r1, i.e., at x = x 1, y = y , z = z , we can represent it by

p ( x, y, z ) = q(x x 1)(y y )(z z1)

or, as we shall write more succinctly,

p(r) = q(r r1).

A sum of charges with charge q i at r i is then represented by

(B.5)

Given this last expression, we can write the general expression for the potential of a charge distribution of any kind as

(B.6)

for it is easy to verify that this formula, combined with the representation of point charges given by Eq. (B.5), just leads back to Eq. (B.1).

From the above point of view, the -function is defined by its integral properties, and the only real requirement on it is that the representation of a point charge by a -function must, when substituted in (B.6), lead back to a known correct formula. We see immediately, then, how to represent -functions in coordinates other than Cartesian. For example, in spherical coordinates r, , , ( r r 1) is represented by

The factor 1 /r 2 sin is, of course, inserted here because in spherical coordinates the volume element is r sin d d dr.

As we have presented it up to now, the -function is a notational shorthand, and the integral properties of the function essentially define it But as happens so often in mathematics, once the concept has been defined, it invites generalization. For example, in the expression (B.6) we may want to integrate by parts, or change variables, or expand in some series. The question then arises as to how to treat the -function part of the distribution function in these processes. Can we generalize the concept of the -function to make it close enough to the ordinary concept of a function so that at least some of the processes of analysis are defined for it? The answer is a limited yes. Although we cannot represent the -function as an ordinary function, we can represent it as the limit of a sequence of ordinary functions.

To elaborate on this, we shall go from the specific to the general. Consider for example the function

This is a function which has the value / at the origin, which integrates to unity independently of the value of , and, which as a function of x , becomes sharper as increases. In the limit when tends to infinity, it is then infinite at the origin, of zero width and with unit integral. In this limit, we can consider it to be a representation of the -function, and so can write

This specific example is only one of an infinite number of similar representations. Start with any function g(x) that has an absolute maximum at the origin, that drops off at least as 1/| x | for x large, and whose integral from to is unity,