34 Relativistie Effeets in Radiation 34-1 Moving sources In the present chapter we shall describe a number of miscellaneous effects in 34-1 Moving sources connection with radiation,and then we shall be finished with the classical theory of light propagation.In our analysis of light,we have gone rather far and into 34-2 Finding the“apparent'motion considerable detail.The only phenomena of any consequence associated with 34-3 Synchrotron radiation electromagnetic radiation that we have not discussed is what happens if radiowaves are contained in a box with reflecting walls,the size of the box being comparable 34-4 Cosmic synchrotron radiation to a wavelength,or are transmitted down a long tube.The phenomena of so-called 34-5 Bremsstrahlung cavity resonators and waveguides we shall discuss later;we shall first use another physical example-sound-and then we shall return to this subject.Except for 34-6 The Doppler effect this,the present chapter is our last consideration of the classical theory of light. 34-7 The w,k four-vector We can summarize all the effects that we shall now discuss by remarking that 34-8 Aberration they have to do with the effects of moving sources.We no longer assume that the source is localized,with all its motion being at a relatively low speed near a fixed 34-9 The momentum of light point. We recall that the fundamental laws of electrodynamics say that,at large distances from a moving charge,the electric field is given by the formula g deR' E=-4c2d2 (34.1) The second derivative of the unit vector er which points in the apparent direction of the charge,is the determining feature of the electric field.This unit vector does not point toward the present position of the charge,of course,but rather in the direction that the charge would seem to be,if the information travels only at the finite speed c from the charge to the observer. Associated with the electric field is a magnetic field,always at right angles to the electric field and at right angles to the apparent direction of the source. given by the formula B -eR X E/c. (34.2) Until now we have considered only the case in which motions are nonrela- A tivistic in speed,so that there is no appreciable motion in the direction of the source Ro to be considered.Now we shall be more general and study the case where the mo- tion is at an arbitrary velocity,and see what different effects may be expected in those circumstances.We shall let the motion be at an arbitrary speed,but of course we shall still assume that the detector is very far from the source. Fig.34-1.The path of a moving We already know from our discussion in Chapter 28 that the only things charge.The true position at the time that count in d2eR/dt2 are the changes in the direction of eR'.Let the coor- T is at T,but the retarded position is at A. dinates of the charge be (x,y,z),with z measured along the direction of observa- tion (Fig.34-1).At a given moment in time,say the moment 7,the three compo- nents of the position are x(r),y(r),and z(r).The distance R is very nearly equal to R(r)=Ro z(r).Now the direction of the vector er'depends mainly on x and y,but hardly at all upon z:the transverse components of the unit vector are x/R and y/R,and when we differentiate these components we get things like R2 in the denominator: d(x/R)dx/dt dz x d R dt R2 34-1
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
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
and it is at right angles both to the field and to the velocity. As usual, the force is equal to the rate of change of momentum with time. If the field is directed upward out of the paper, the momentum of the particle and the force on it are as shown in Fig. 34-4. Since the force is at right angles to the velocity, the kinetic energy, and therefore the speed, remains constant. All the magnetic field does is to ge the direction of motion. In a short time At, the momentum vector changes at right angles to itself by an amount Ap= FAL, and therefore p turns through an angle 40= Ap/p= qUB At/p, since F= quB. But in this same time the particle has gone a distance As v AL. Evidently, the two lines AB and CD will intersect at a point O such that OA =OC = R, where As =R A0. Combining this with the find r△0/△ P in a circular (or helical)path in a uniform magnetic field qub/p Since this same argument can be applied during the next instant, the next, and so on,we conclude that the particle must be moving in a circle of radius R, with angu- The result that the momentum of the particle is equal to a charge times the radius times the magnetic field is a very important law that is used a great deal It is important for practical purposes because if we have elementary particles which all have the same charge and we observe them in a magnetic field, we can measure the radii of curvature of their orbits and, knowing the magnetic field, thus deter- mine the momenta of the particles. If we multiply both sides of Eq (34.7)by c, and express q in terms of the electronic charge, we can measure the momentum in units of the electron volt. In those units our formula is 3×105(q/q)BR where B, R, and the speed of light are all expressed in the mks system, the latter numerically. The mks unit of magnetic field is called a weber per square meter. There is an older unit which is still in common use, called a gauss. One weber/m is equal to 10* gauss. To give an idea of how big magnetic fields are, the strongest magnetic field that one can usually make in iron is about 1.5x 10* gauss; beyond that, the advantage of using iron disappears. Today, electromagnets conducting wire are able to produce steady fields of over 10 gauss strength--that is, 10 mks units. The field of the earth is a few tenths of a gauss at the equator. Returning to Eq. (34.9), we could imagine the synchrotron running at a billion electron volts, so pc would be 10 for a billion electron volts. (We shall come back to the energy in just a moment. Then, if we had a b corresponding to, say, 10,000 gauss,which is a good substantial field, one mks unit, then we see that R would have to be 3.3 meters. The actual radius of the Caltech synchrotron is 3.7 meters, the field is a little bigger, and the energy is 1.5 billion, but it is the same idea. S now we have a feeling for why the synchrotron has the size it has We have calculated the momentum, but we know that the total energy, including the rest energy, is given by W= vpac2 +m c4, and for an electron the rest energy corresponding to mc2 is 0.511 X 106 ev, so when pe is 10 ev we can neglect mc2, and so for all practical purposes w=pc when the speeds are relativistic. It is practically the same to say the energy of an electre billion electron volts as to say the momentum times c is a billion electron volts. If w 10%ev, it is easy to show that the speed differs from the speed of light by but one We turn now to the radiation emitted by such a particle. a particle moving on a circle of radius 3. 3 meters, or 20 meters circumference, goes around once in roughly the time it takes light to go 20 meters. So the wavelength that should be emitted by such a particle would be 20 meters-in the shortwave radio region But because of the piling up effect that we have been discussing(Fig. 34-3), and because the distance by which we must extend the radius to reach the speed c is
only one part in eight million of the radius, the cusps of the hypocycloid are enormously sharp compared with the distance between them. The acceleration which involves a second derivative with respect to time, gets twice the"compression factor" 8x 10 because the time scale is reduced by eight million twice in the neighborhood of the cusp. Thus we might expect the effective wavelength to be much shorter, to the extent of 64 times 102 smaller than 20 meters, and that corresponds to the x-ray region. (Actually, the cusp itself is not the entire determining factor; one must also include a certain region about the cusp thi changes the factor to the 3/2 power instead of the square, but still leaves us above the optical region. )Thus, even though a slowly moving electron would have radiated 20-meter radiowaves, the relativistic effect cuts down the wavelength so 2- much that we can see it! Clearly, the light should be polarized, with the electric Pulse from electron field perpendicular to the uniform magnetic field sug. To further appreciate what we would observe, suppose that we were to take th light (to simpli because these pulses are so fa rt in time shall just take one pulse)and direct it onto a diffraction grating, which is a lot of scattering wires. After this pulse comes away from the grating, what do we see (We should see red light, blue light, and so on, if we see any light at all. )what do we see? The pulse strikes the grating head-on, and all the oscillators in the grating, together, are violently moved up and then back down aga5.But the They then produce effects in various directions, as shown in Fig point P is closer to one end of the grating than to the other, so at this point the ig. 34-5. The light which strikes a electric field arrives first from wire A, next from B, and so on; finally, the pulse grating as a single, sharp pulse is scat from the last wire arrives. In short, the sum of the reflections from all the succes tered in various directions as different wires is as shown in Fig. 34-6(a); it is an electric field which is a series of pulses and it is very like a sine wave whose wavelength is the distance between the pulses just as it would be for monochromatic light striking he ting! So, we get colored light all right. But, by the same argument, will we not get light from any kind of a pulse"?No. Suppose that the curve were much smoother; then we would add all the scattered waves together, separated by a small time between them(Fig 34-66). Then we see that the field would not shake at all, it would be a very smooth because each pulse does not vary much in the time interval between pulse The electromagnetic radiation emitted by relativistic charged particles cir- ulating in a magnetic field is called synchrotron radiation. It is so named for obvi- Fig. 34-6. The total electric field due ous reasons, but it is not limited specifically to synchrotrons, or even to earthbound to a series of (a) sharp pulses and (b) laboratories. It is exciting and interesting that it also occurs in nature