Question

12.12 Headlight effect* A light beam is emitted at angle θ 0 with respect to the x, axis in (a) Find the angle 0 the beam makes with respect to the x axis in S (b) A source that radiates light uniformly in all directions in its rest frame radiates strongly in the forward direction in a frame in which it is moving with speed v close to c. This is called the head- light effect; is very pronounced in synchrotron light sources in which electrons moving at relativistic speeds emit light in a nar- row cone in the forward direction. Using the result of part (a) find the speed of a source for which half the radiation is emitted in a cone subtending 10-3 rad. (The sketch is considerably exag gerated, because 10-3 rad is only about 0.06 degree.) Rest frame 10-3 rad

Answer 1

Answer 2

The following figure shows the two frame S and S', angle O, that makes with respect to x' axis in S' frame. S S' 101 6 In the S frame, x=r cos e y=rsin e r = ct In the S' frame, x' =r' cos

y' = r' sin , r' = ct' B From Lorentz transformation velocity equation, the two frames are related as:

u +v uv 1+ cos 0 + v vcos, 1+ cos = 1 cos 6, +v v cos, 1+ с V cos 0 + 1+ ß cos cos +B cos = 1+ B cos (b) The forward pointing come into which half the source half of ß radiation has a half angle is given by,

022 2 1-B 1+ ß 1 1 = 11- B 2 Vi+ß 1000 10-6 - 1-B 1+B 4 B= 0.999995 V = 0.999995 с v = 0.999995c