Classical Electrodynamics · Problem 15.2

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · bremsstrahlung-hard-sphere-collision

Problem

A nonrelativistic particle of charge ee and mass mm collides with a fixed, smooth, hard sphere of radius RR. Assuming that the collision is elastic, show that in the dipole approximation (neglecting retardation effects) the classical differential cross section for the emission of photons per unit solid angle per unit energy interval is d2σdΩ d(ℏω)=R212πe2ℏc(vc)21ℏω(2+3sin⁡2θ)\frac{d^2\sigma}{d\Omega\, d(\hbar\omega)} = \frac{R^2}{12\pi}\frac{e^2}{\hbar c}\left(\frac{v}{c}\right)^2\frac{1}{\hbar\omega}(2 + 3\sin^2\theta) where θ\theta is measured relative to the incident direction. Sketch the angular distribution. Integrate over angles to get the total bremsstrahlung cross section. Qualitatively, what factor (or factors) govern the upper limit to the frequency spectrum?

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