of-sight coordinate is ct, as shown at the right. (The figure shows the result for a rather complicated motion in a plane, but of course the motion may not be in one plane--it may be even more complicated than motion in a plane. )The point is that the horizontal (i.e, line-of-sight) distance now is no longer the old z, but is z +ct, and therefore is ct. Thus we have found a picture of the curve, x' (and y)against t! All we have to do to find the field is to look at the acceleration of this curve i e, to differentiate it twice. So the final answer is: in order to find the electric field for a moving charge, take the motion of the charge and translate it back at the speed c to"open it out"; then the curve, so drawn, is a curve of the xand y' positions of the function of t. The acceleration of this curve gives the electric field as a function of t. Or, if we wish, we can now imagine that this whole igid"curve moves forward at the speed c through the plane of sight, so that the point of intersection with the plane of sight has the coordinates xand y'. The acceleration of this point makes the electric field. This solution is just as exact as the formula we started with-it is simply a geometrical representation particle moving at constant speed v 0.94c, a circle. If the motion is relatively slow, for instance if we have an oscillator just going up and down slowly, then when we shoot that motion away at the speed of light, et, of course, a simple cosine curve, and tha king at for a long time: it gives the fiel A more interesting example is an electron moving rapidly, very nearly at the speed of light, in a circle. If we look in the plane of the circle, the retarded x(o shown in Fig. 34-3. What is thi from the center of the circle to the charge and if we extend this radial line a little bit past the charge, just a shade if it is going fast, then we come to a point on the line that goes at the speed of light. Therefore, when we translate the motion back at the speed of light, that corresponds to having a wheel with a charge on it rolling backward (without slipping) at the speed c; thus we find a curve which is very close to a cycloid-it is called a hypocycloid. If the charge is going very nearly at the speed of light, the"cusps"are very sharp indeed; if it went at exactly the speed of light, they would be actual cusps, infinitely sharp. "Infinitely sharp"is inter esting; it means that near a cusp the second derivative is enormous. Once in each cycle we get a sharp pulse of electric field. This is not at all what we would get from a nonrelativistic motion, where each time the charge goes around there is an oscillation which is of about the same"strength"all the time. Instead, there are very sharp pulses of electric field spaced at time intervals 1/To apart, where To is the period of revolution. These strong electric fields are emitted in a narrow cone in the direction of motion of the charge. When the charge is moving away from P, there is very little curvature and there is very little radiated field in the direction of p 34-3 Synchrotron radiation We have very fast electrons moving in circular paths in the synchrotron; they are travelling at very nearly the speed c, and it is possible to see the above radiation as actual light! Let us discuss this in more detail In the synchrotron we have electrons which go around in circles in a uniform magnetic field. First, let us see why they go in circles. From Eq(12. 10), we know that the force on a particle in a magnetic field is given by F=q×B