Light and electromagnetic waves
We saw in the last chapter that among their solutions, Maxwell’s equations have waves of electricity and magnetism. These waves correspond to the phenomena of radio, light, x-rays, and so on, depending on the wavelength. We have already studied light in great detail in Vol. I. In this chapter we want to tie together the two subjects—we want to show that Maxwell’s equations can indeed form the base for our earlier treatment of the phenomena of light.  When we studied light, we began by writing down equations for the electric and magnetic fields produced by a charge which moves in any arbitrary way. Those equations were \begin{equation} \label{Eq:II:21:1} \FLPE=\frac{q}{4\pi\epsO}\biggl[ \frac{\FLPe_{r'}}{r'^2}+\frac{r'}{c}\,\ddt{}{t}\biggl( \frac{\FLPe_{r'}}{r'^2}\biggr)+\frac{1}{c^2}\,\frac{d^2}{dt^2}\,\FLPe_{r'} \biggr] \end{equation} and \begin{equation*} c\FLPB=\FLPe_{r'}\times\FLPE. \end{equation*} [See Eqs. (28.3) and (28.4), Vol. I. As explained below, the signs here are the negatives of the old ones.]  If a charge moves in an arbitrary way, the electric field we would find now at some point depends only on the position and motion of the charge not now, but at an earlier time—at an instant which is earlier by the time it would take light, going at the speed $c$, to travel the distance $r'$ from the charge to the field point. In other words, if we want the electric field at point $(1)$ at the time $t$, we must calculate the location $(2')$ of the charge and its motion at the time $(t-r'/c)$, where $r'$ is the distance to the point $(1)$ from the position of the charge $(2')$ at the time $(t-r'/c)$. The prime is to remind you that $r'$ is the so-called “retarded distance” from the point $(2')$ to the point $(1)$, and not the actual distance between point $(2)$, the position of the charge at the time $t$, and the field point $(1)$ (see Fig. 21-1). Note that we are using a different convention now for the direction of the unit vector $\FLPe_r$. In Chapters 28 and 34 of Vol. I it was convenient to take $\FLPr$ (and hence $\FLPe_r$) pointing toward the source. Now we are following the definition we took for Coulomb’s law, in which $\FLPr$ is directed from the charge, at $(2)$, toward the field point at $(1)$. The only difference, of course, is that our new $\FLPr$ (and $\FLPe_r$) are the negatives of the old ones.  We have also seen that if the velocity $v$ of a charge is always much less than $c$, and if we consider only points at large distances from the charge, so that only the last term of Eq. (21.1) is important, the fields can also be written as  \begin{equation*} \label{Eq:II:21:1s} \FLPE=\frac{q}{4\pi\epsO c^2r'} \begin{bmatrix} \text{acceleration of the charge at $(t-r'/c)$}\\[-.5ex] \text{projected at right angles to $r'$}\phantom{(t-r'/c)} \end{bmatrix} \tag{$21.1'$} \end{equation*} and \begin{equation*} c\FLPB=\FLPe_{r'}\times\FLPE. \end{equation*} 
 \begin{equation*} \label{Eq:II:21:1s} \FLPE=\frac{q}{4\pi\epsO c^2r'} \begin{bmatrix} \text{acceleration of the charge}\\[-.5ex] \text{at $(t-r'/c)$}\\[-.5ex] \text{projected at right angles to $r'$} \end{bmatrix} \tag{$21.1'$} \end{equation*} and \begin{equation*} c\FLPB=\FLPe_{r'}\times\FLPE. \end{equation*} 
 Let’s look at what the complete equation, Eq. (21.1), says in a little more detail. The vector $\FLPe_{r'}$ is the unit vector to point $(1)$ from the retarded position $(2')$. The first term, then, is what we would expect for the Coulomb field of the charge at its retarded position—we may call this “the retarded Coulomb field.”  The electric field depends inversely on the square of the distance and is directed away from the retarded position of the charge (that is, in the direction of $\FLPe_{r'}$).  But that is only the first term. The other terms tell us that the laws of electricity do not say that all the fields are the same as the static ones, but just retarded (which is what people sometimes like to say). To the “retarded Coulomb field” we must add the other two terms. The second term says that there is a “correction” to the retarded Coulomb field which is the rate of change of the retarded Coulomb field multiplied by $r'/c$, the retardation delay. In a way of speaking, this term tends to compensate for the retardation in the first term. The first two terms correspond to computing the “retarded Coulomb field” and then extrapolating it toward the future by the amount $r'/c$, that is, right up to the time $t$!  The extrapolation is linear, as if we were to assume that the “retarded Coulomb field” would continue to change at the rate computed for the charge at the point $(2')$. If the field is changing slowly, the effect of the retardation is almost completely removed by the correction term, and the two terms together give us an electric field that is the “instantaneous Coulomb field”—that is, the Coulomb field of the charge at the point $(2)$—to a very good approximation.  Finally, there is a third term in Eq. (21.1) which is the second derivative of the unit vector $\FLPe_{r'}$. For our study of the phenomena of light, we made use of the fact that far away from the charge the first two terms went inversely as the square of the distance and, for large distances, became very weak in comparison to the last term, which decreases as $1/r$. So we concentrated entirely on the last term, and we showed that it is (again, for large distances) proportional to the component of the acceleration of the charge at right angles to the line of sight. (Also, for most of our work in Vol. I, we took the case in which the charges were moving nonrelativistically. We considered the relativistic effects in only one chapter, Chapter 34.)  Now we should try to connect the two things together. We have the Maxwell equations, and we have Eq. (21.1) for the field of a point charge. We should certainly ask whether they are equivalent. If we can deduce Eq. (21.1) from Maxwell’s equations, we will really understand the connection between light and electromagnetism. To make this connection is the main purpose of this chapter.  It turns out that we won’t quite make it—that the mathematical details get too complicated for us to carry through in all their gory details. But we will come close enough so that you should easily see how the connection could be made. The missing pieces will only be in the mathematical details. Some of you may find the mathematics in this chapter rather complicated, and you may not wish to follow the argument very closely. We think it is important, however, to make the connection between what you have learned earlier and what you are learning now, or at least to indicate how such a connection can be made. You will notice, if you look over the earlier chapters, that whenever we have taken a statement as a starting point for a discussion, we have carefully explained whether it is a new “assumption” that is a “basic law,” or whether it can ultimately be deduced from some other laws. We owe it to you in the spirit of these lectures to make the connection between light and Maxwell’s equations. If it gets difficult in places, well, that’s life—there is no other way. 