Classical Electrodynamics · Problem 14.6

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-nonzero-angular-momentum-collision

Problem

(a) Generalize the circumstances of the collision of Problem 14.5 to nonzero angular momentum (impact parameter) and show that the total energy radiated is given by ΔW=4z2e23m2c3(m2)1/2∫rmin∞(dVdr)2(E−V(r)−L22mr2)−1/2dr\Delta W = \frac{4z^2 e^2}{3m^2 c^3}\left(\frac{m}{2}\right)^{1/2} \int_{r_\text{min}}^\infty \left(\frac{dV}{dr}\right)^2 \left(E - V(r) - \frac{L^2}{2mr^2}\right)^{-1/2} dr where rminr_\text{min} is the closest distance of approach (root of E−V−L2/2mr2E - V - L^2/2mr^2), L=mbv0L = mbv_0, where bb is the impact parameter, and v0v_0 is the incident speed (E=mv02/2E = mv_0^2/2). (b) Specialize to a repulsive Coulomb potential V(r)=zZe2/rV(r) = zZe^2/r. Show that ΔW\Delta W can be written in terms of impact parameter as ΔW=2zmv05Zc3[−t−4+t−5(1+t23)tan⁡−1t]\Delta W = \frac{2zmv_0^5}{Zc^3}\left[-t^{-4} + t^{-5}\left(1 + \frac{t^2}{3}\right)\tan^{-1} t\right] where t=bmv02/zZe2t = bmv_0^2/zZe^2 is the ratio of twice the impact parameter to the distance of closest approach in a head-on collision. Show that in the limit of tt going to zero the result of Problem 14.5b is recovered, while for t≫1t \gg 1 one obtains the approximate result of Problem 14.7a. (c) Using the relation between the scattering angle θ\theta and tt (=cot⁡θ/2= \cot \theta/2), show that ΔW\Delta W can be expressed as ΔW=2zmv05Zc3tan⁡3θ2[16(π−θ)(1+3tan⁡2θ2)−tan⁡θ2]\Delta W = \frac{2zmv_0^5}{Zc^3}\tan^3\frac{\theta}{2}\left[\frac{1}{6}(\pi - \theta)\left(1 + 3\tan^2\frac{\theta}{2}\right) - \tan\frac{\theta}{2}\right] (d) What changes occur for an attractive Coulomb potential?

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