2 interferenee 29-1 Electromagnetic waves In this chapter we shall discuss the subject of the preceding chapt 29-1 Electromagnetic waves mathematically. We have qualitatively demonstrated that there are maxima and minima in the radiation field from two sources, and our problem now is to describe 29-2 Energy of radiation the field in mathematical detail, not just qualitatively 29-3 Sinusoidal waves We have already physically analyzed the meaning of formula(28.6)quite 29-4 Two dipole radiators satisfactorily, but there are a few points to be made about it mathematically. In the first place, if a charge is accelerating up and down along a line, in a motion of 29-5 The mathematics of interference very small amplitude, the field at some angle 0 from the axis of the motion is in a direction at right angles to the line of sight and in the plane containing both the acceleration and the line of sight( Fig. 29-1). If the distance is called r, then at time t the electric field has the magnitude E(n where a(r -r/c) is the acceleration at the time(t-r/c), called the retarded acceleration Now it would be interesting to draw a picture of the field under different conditions. The thing that is interesting, of course, is the factor a(t-r/c), and to understand it we can take the simplest case, 0=900, and plot the field graphically. What we had been thinking of before is that we stand in one position and ask how Fig. 29-1. The electric field e due the field looks like at different positions in space at a given instant. So what we acceleration/ ge whose retarded the field there changes with tiRne. But instead of that, we are now going to see what to a positive che want is a"snapshot"picture which tells us what the field is in different places Of course it depends upon the acceleration of the charge. Suppose that the charge at first had some particular motion: it was initially standing still, and it suddenly accelerated in some manner, as shown in Fig. 29-2, and then stopped. Then little bit later, we measure the field at a different place. Then we may assert that he field will appear as shown in Fig. 29-3. At each point the field is determined by the acceleration of the charge at an earlier time, the amount earlier being the delay r/c. The field at farther and farther points is determined by the acceleration at earlier and earlier times. So the curve in Fig. 29-3 is really, in a sense, a"reversed lot of the acceleration as a function of time; the distance is related to time by a Fig. 29-2. The acceleration of a constant scale factor c, which we often take as unity. This is easily seen by consider is a function of time the mathematical behavior of a(t-r/c). Evidently, if we add a little time Ar, we get the same value for a(t-r/c)as we would have if we had subtracted a little distance:△r=-c△ Stated another way: if we add a little time At, we can restore a(t-r/)to its field moves as a wave outward from the source. That is the reason why we sometimes say light is propagated as waves. It is equivalent to saying that the field is delayed or to saying that the electric field is moving outward as time goes on An interesting special case is that where the charge q is moving up and down in an oscillatory manner. The case which we studied experimentally in the last chapter was one in which the displacement x at any time t was equal to a certain Fig. 29-3. The electric field as a constant xo, the magnitude of the oscillation times cos wr. Then the acceleration is function of position at a later time. (The o cOS wf= ao cos (292) I/r variation is ignored. 291