Classical Electrodynamics · Problem 16.8

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-damping-classical-models-charged-particles

Problem

(a) Show that for relativistic motion in one dimension the equation of motion of Problem 16.7 can be written in the form,

\dot{p} - \frac{2e^2}{3mc^3}\left(\ddot{p} - \frac{p\dot{p}^2}{p^2 + m^2c^2}\right) = \sqrt{1 + \frac{p^2}{m^2c^2}}\,f(\tau)

where

p

is the momentum in the direction of motion, a dot means differentiation with respect to proper time, and

f(\tau)

is the ordinary Newtonian force as a function of proper time. (b) Show that the substitution of

p = mc\sinh y

reduces the relativistic equation to the Abraham--Lorentz form (16.9) in

y

and

\tau

. Write down the general solution for

p(\tau)

, with the initial condition that

p(\tau) = p_0 \quad \text{at} \quad \tau = 0