N identical point particles, each with charge q, move in a circle of radius a. The angular position of the j^th particle is φ_j(t) = ω₀(t − t₀) + θ_j.
(a) Prove that the power spectrum satisfies N identical point particles, each with charge q, move in a circle of radius a. The angular position of the j^th particle is φj (t) = ω0(t − t0) + θj .
(a) Prove that the power spectrum satisfies

(b) Specialize to the case where the charges are equally spaced around the circle at all times. Use part (a) to prove that this configuration radiates only at the frequencies kNω0 where k is a positive integer.
(c) Show that dPm/dwhen the charges are distributed randomly around the ring.

Transcribed Image Text:

dPm ΦΩ IN N Σ exp(-imt;) j=1 dPm dΩ 1