Classical Electrodynamics · Problem 14.5

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-head-on-collision-nonrelativistic

Problem

A nonrelativistic particle of charge

ze

, mass

m

, and kinetic energy

E

makes a head-on collision with a fixed central force field of finite range. The interaction is repulsive and described by a potential

V(r)

, which becomes greater than

E

at close distances. (a) Show that the total energy radiated is given by

\Delta W = \frac{4}{3}\frac{z^2 e^2}{m^2 c^3}\sqrt{\frac{m}{2}} \int_{r_\text{min}}^\infty \left|\frac{dV}{dr}\right|^2 \frac{dr}{\sqrt{V(r_\text{min}) - V(r)}}

where

r_\text{min}

is the closest distance of approach in the collision. (b) If the interaction is a Coulomb potential

V(r) = zZe^2/r

, show that the total energy radiated is

\Delta W = \frac{8}{45}\frac{z m v_0^5}{Z c^3}

where

v_0

is the velocity of the charge at infinity.