So, when we are far enough away the only terms we have to worry about are the variations of x and y. Thus we take out the factor Ro and get Ro Ey (343) where Ro is the distance, more or less, to g: let us take it as the distance oP to the origin of the coordinates (x, y, z). Thus the electric field is a constant multiplied by a very simple thing, the second derivatives of the x- and y-coordinates. (We could put it more mathematically by calling x and y the transverse components of the position vector r of the charge, but this would not add to the clarity. Of course, we realize that the coordinates must be measured at the retarded time. Here we find that z(o) does affect the retardation. What time is the retarded time? If the time of observation is called t(the time at P)then the time t to which this corresponds at d is not the time t, but is delayed by the total distance that the light has to go, divided by the speed of light. In the first approximation, this delay is Ro/c, a constant(an uninteresting feature), but in the next approximation we must include the effects of the position in the z-direction at the time T, because if g is a little farther back, there is a little more retardation. This is an effect that e have neglected before, and it is the only change needed in order to make our results valid for all speeds. What we must now do is to choose a certain value of t and calculate the value of T from it, and thus find out where x and y are at that T. These are then the retarded x and y, which we call xand y, whose second derivatives determine the field. Thus T is determined by R01z(7) T and x(1)=x(),y()=y(T) Now these are complicated equations, but it is easy enough to make a geometrical picture to describe their solution. This picture will give us a good qualitative feeling for how things work, but it still takes a lot of detailed mathematics to deduce the precise results of a complicated problem. 4y一 TO OBSERVER Fig. 34-2. A geometrical solution of Eq (34.5)to find x'(n). 342 Finding the“ apparent” motion The above equation has an interesting simplification. If we disregard the un interesting constant delay Ro/c, which just means that we must change the origin of t by a constant, then it says that ct= CT +z(r), x=x(o), y= y(r) Now we need to find xand y as functions of t, not T, and we can do this in the following way: Eq. (34.5)says that we should take the actual motion and add a constant(the speed of light) times T. What that turns out to mean is shown in Fig 34-2. We take the actual motion of the charge(shown at left)and imagine that as it is going around it is being swept away from the point p at the speed c (there are no contractions from relativity or anything like that; this is just a mathe- matical addition of the cr). In this way we get a new motion, in which the line 342