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