Classical Electrodynamics · Problem 15.10

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · bremsstrahlung-hyperbolic-orbit-coulomb

Problem

A particle of charge zeze, mass mm, and velocity vv is deflected in a hyperbolic path by a fixed repulsive Coulomb potential, V(r)=Zze2/rV(r) = Zze^2/r. Assume the nonrelativistic dipole approximation (but no further approximations). (a) Show that the energy radiated per unit frequency interval by the particle when initially incident at impact parameter bb is ddωI(ω,b)=83π(zeaν)2c3e−(πω/ω0){[Kiω/ω0′(ωϵω0)]2+ϵ2−1ϵ2[Kiω/ω0(ωϵω0)]2}\frac{d}{d\omega}I(\omega,b) = \frac{8}{3\pi}\frac{(zea\nu)^2}{c^3}e^{-(\pi\omega/\omega_0)}\left\{\left[K'_{i\omega/\omega_0}\left(\frac{\omega\epsilon}{\omega_0}\right)\right]^2 + \frac{\epsilon^2-1}{\epsilon^2}\left[K_{i\omega/\omega_0}\left(\frac{\omega\epsilon}{\omega_0}\right)\right]^2\right\} (b) Show that the radiation cross section is ddωχ(ω)=163(zeav)2c3e−(πω/ω0)ωω0 Kiω/ω0(ωω0)[−Kiω/ω0′(ωω0)]\frac{d}{d\omega}\chi(\omega) = \frac{16}{3}\frac{(zeav)^2}{c^3}e^{-(\pi\omega/\omega_0)}\frac{\omega}{\omega_0}\, K_{i\omega/\omega_0}\left(\frac{\omega}{\omega_0}\right)\left[-K'_{i\omega/\omega_0}\left(\frac{\omega}{\omega_0}\right)\right] (c) Prove that the radiation cross section reduces to that obtained in the text for classical bremsstrahlung for ω≪ω0\omega \ll \omega_0. What is the limiting form for ω≫ω0\omega \gg \omega_0? (d) What modifications occur for an attractive Coulomb interaction? The hyperbolic path may be described by x=a(ϵ+cosh⁡ξ),y=−bsinh⁡ξ,ω0t=(ξ+ϵsinh⁡ξ)x = a(\epsilon + \cosh\xi), \qquad y = -b\sinh\xi, \qquad \omega_0 t = (\xi + \epsilon\sinh\xi) where a=Zze2/mv2a = Zze^2/mv^2, ϵ=1+(b/a)2\epsilon = \sqrt{1+(b/a)^2}, ω0=v/a\omega_0 = v/a.

Browse the problem library