A particle of charge e and mass m moves relativistically in a helical path in a uniform

Question:

A particle of charge e and mass m moves relativistically in a helical path in a uniform magnetic field B. The pitch angle of the helix is ? (? = 0 corresponds to circular motion).

(a) By arguments similar to those of Section 14.4, show that an observer far from the helix would detect radiation with a fundamental frequency w₀ = w_B/cos²? and that the spectrum would extend up to frequencies in the order of w_c = 3/2 ?³w_B cos ? where w_B = eB/?mc. (Take care in determining the radius of curvature ? of the helical path.)

(b) From part a and the results of Section 14.6, show that the power received by the observer per unit solid angle and per unit circular frequency interval is

where w_B and a)c are defined above, ? = (w/2w_c) (l + ?²?²)^3/2, and ? is the angle of observation measured relative to the particle's velocity vector, as in Figure.