Classical Electrodynamics · Problem 14.23

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-N-charges-circular-orbit

Problem

Instead of a single charge ee moving with constant velocity ω0R\omega_0 R in a circular path of radius RR, as in Problem 14.15, NN charges qjq_j move with fixed relative positions θj\theta_j around the same circle. (a) Show that the power radiated into the mmth multiple of ω0\omega_0 is dPm(N)dΩ=dPm(1)dΩ Fm(N)\frac{dP_m(N)}{d\Omega} = \frac{dP_m(1)}{d\Omega}\, F_m(N) where dPm(1)/dΩdP_m(1)/d\Omega is the result of part a in Problem 14.15 with e→1e\to 1, and Fm(N)=∣∑j=1Nqjeimθj∣2F_m(N) = \left|\sum_{j=1}^N q_j e^{im\theta_j}\right|^2 (b) Show that, if the charges are all equal in magnitude and uniformly spaced around the circle, energy is radiated only into multiples of Nω0N\omega_0, but with an intensity N2N^2 times that for a single charge. Give a qualitative explanation of these facts. (c) For the situation of part b, without detailed calculations show that for nonrelativistic motion the dependence on NN of the total power radiated is dominantly as β2N\beta^{2N}, so that in the limit N→∞N\to\infty no radiation is emitted. (d) By arguments like those of part c show that for NN relativistic particles of equal charge and symmetrically arrayed, the radiated power varies with NN mainly as e−2N/3γ3e^{-2N/3\gamma^3} for N≫γ3N\gg\gamma^3, so that again in the limit N→∞N\to\infty no radiation is emitted. (e) What relevance have the results of parts c and d to the radiation properties of a steady current in a loop?

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