Radiation
That, then, is a summary of the world picture. Now let us use it to                         discuss the phenomena called radiation. To discuss these phenomena, we                         must select from Eq. (28.3) only that piece which varies                         inversely as the distance and not as the square of the distance. It                         turns out that when we finally do find that piece, it is so simple in                         its form that it is legitimate to study optics and electrodynamics in                         an elementary way by taking it as “the law” of the electric field                         produced by a moving charge far away. We shall take it temporarily as                         a given law which we will learn about in detail next year. Of the terms appearing in (28.3), the first one evidently                         goes inversely as the square of the distance, and the second is only a                         correction for delay, so it is easy to show that both of them vary                         inversely as the square of the distance. All of the effects we are                         interested in come from the third term, which is not very complicated,                         after all. What this term says is: look at the charge and note the                         direction of the unit vector (we can project the end of it onto the                         surface of a unit sphere). As the charge moves around, the unit vector                         wiggles, and the acceleration of that unit vector is what we are                             looking for. That is all. Thus                         \begin{equation}                         \label{Eq:I:28:5}                         \FLPE=\frac{-q}{4\pi\epsO c^2}\,                         \frac{d^2\FLPe_{r'}}{dt^2}                         \end{equation}                         is a statement of the laws of radiation, because that is the only                         important term when we get far enough away that the fields are varying                         inversely as the distance. (The parts that go as the square have                         fallen off so much that we are not interested in them.) Now we can go a little bit further in studying (28.5) to                         see what it means. Suppose a charge is moving in any manner                         whatsoever, and we are observing it from a distance. We imagine for a                         moment that in a sense it is “lit up” (although it is light that we                         are trying to explain); we imagine it as a little white dot. Then we                         would see this white dot running around. But we don’t see                         exactly how it is running around right now, because of                         the delay that we have been talking about. What counts is how it was                         moving earlier. The unit vector $\FLPe_{r'}$ is pointed toward                         the apparent position of the charge. Of course, the end                         of $\FLPe_{r'}$ goes on a slight curve, so that its acceleration has two                         components. One is the transverse piece, because the end of it goes up                         and down, and the other is a radial piece because it stays on a                         sphere. It is easy to demonstrate that the latter is much smaller and                         varies as the inverse square of $r$ when $r$ is very great. This is                         easy to see, for when we imagine that we move a given source farther                         and farther away, then the wigglings of $\FLPe_{r'}$ look smaller and                         smaller, inversely as the distance, but the radial component of                         acceleration is varying much more rapidly than inversely as the                         distance. So for practical purposes all we have to do is project the                         motion on a plane at unit distance. Therefore we find the following                         rule: Imagine that we look at the moving charge and that everything we                         see is delayed—like a painter trying to paint a scene on a screen at                         a unit distance. A real painter, of course, does not take into account                         the fact that light is going at a certain speed, but paints the world                         as he sees it. We want to see what his picture would look like. So we                         see a dot, representing the charge, moving about in the picture. The                         acceleration of that dot is proportional to the electric field. That                         is all—all we need. Thus Eq. (28.5) is the complete and correct formula for                         radiation; even relativity effects are all contained in it. However,                         we often want to apply it to a still simpler circumstance in which the                         charges are moving only a small distance at a relatively slow                         rate. Since they are moving slowly, they do not move an appreciable                         distance from where they start, so that the delay time is practically                         constant. Then the law is still simpler, because the delay time is                         fixed. Thus we imagine that the charge is executing a very tiny motion                         at an effectively constant distance. The delay at the distance $r$                         is $r/c$. Then our rule becomes the following: If the charged object is                         moving in a very small motion and it is laterally displaced by the                         distance $x(t)$, then the angle that the unit vector $\FLPe_{r'}$ is                         displaced is $x/r$, and since $r$ is practically constant, the                         $x$-component of $d^2\FLPe_{r'}/dt^2$ is simply the acceleration                         of $x$ itself at an earlier time divided by $r$, and so finally we get                         the law we want, which is                         \begin{equation}                         \label{Eq:I:28:6}                         E_x(t)=\frac{-q}{4\pi\epsO c^2r}\,a_x                         \Bigl(t-\frac{r}{c}\Bigr).                         \end{equation}                          Only the component $a_x$, perpendicular to the line of sight, is                         important. Let us see why that is. Evidently, if the charge is moving                         in and out straight at us, the unit vector in that direction does not                         wiggle at all, and it has no acceleration. So it is only the sidewise                         motion which is important, only the acceleration that we see projected                         on the screen.