Classical Electrodynamics · Problem 15.8

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · bremsstrahlung-identical-charged-particles

Problem

Two identical point particles of charge qq and mass mm interact by means of a short-range repulsive interaction that is equivalent to a hard sphere of radius RR in their relative separation. Neglecting the electromagnetic interaction between the two particles, determine the radiation cross section in the center-of-mass system for a collision between these identical particles to the lowest nonvanishing approximation. Show that the differential cross section for emission of photons per unit solid angle per unit energy interval is d2σd(ℏω) dΩ=(q2ℏc)⋅β2R248π1ℏω[β2(25+3sin⁡2θcos⁡2θ)+(ωRc)2(1−38sin⁡4θ)]\frac{d^2\sigma}{d(\hbar\omega)\, d\Omega} = \left(\frac{q^2}{\hbar c}\right)\cdot\frac{\beta^2 R^2}{48\pi}\frac{1}{\hbar\omega}\left[\beta^2\left(\frac{2}{5} + 3\sin^2\theta\cos^2\theta\right) + \left(\frac{\omega R}{c}\right)^2\left(1 - \frac{3}{8}\sin^4\theta\right)\right] where θ\theta is measured relative to the incident direction and cβc\beta is the relative speed. By integration over the angles of emission, show that the total cross section for radiation per unit photon energy ℏω\hbar\omega is dσd(ℏω)=(q2ℏc)⋅β2R215⋅1ℏω[β2+(ωRc)2]\frac{d\sigma}{d(\hbar\omega)} = \left(\frac{q^2}{\hbar c}\right)\cdot\frac{\beta^2 R^2}{15}\cdot\frac{1}{\hbar\omega}\left[\beta^2 + \left(\frac{\omega R}{c}\right)^2\right] Compare these results with that of Problem 15.2 as to frequency dependence, relative magnitude, etc.
Intuitive explanation

When two charged particles collide via hard-sphere repulsion, they undergo acceleration during the brief contact, radiating electromagnetic energy. The key is treating this as a classical scattering problem where the acceleration profile during collision determines the radiated power spectrum—you'll use the dipole approximation and angular distribution of bremsstrahlung radiation, then integrate over all possible impact parameters to build the cross section. The result separates naturally into two regimes: low-frequency radiation (dominated by β2\beta^2 terms reflecting the collision dynamics) and high-frequency radiation (suppressed by the finite collision timescale R/cR/c).

Intuitive explanation

When two charged particles collide via hard-sphere repulsion, they undergo acceleration during the brief contact, radiating electromagnetic energy. The key is treating this as a classical scattering problem where the acceleration profile during collision determines the radiated power spectrum—you'll use the dipole approximation and angular distribution of bremsstrahlung radiation, then integrate over all possible impact parameters to build the cross section. The result separates naturally into two regimes: low-frequency radiation (dominated by β2\beta^2 terms reflecting the collision dynamics) and high-frequency radiation (suppressed by the finite collision timescale R/cR/c).

Related theory

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