Classical Electrodynamics · Problem 16.11

Radiating Systems, Multipole Fields and Radiation — John David Jackson

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

Problem

A nonrelativistic particle of charge ee and mass mm is accelerated in one-dimensional motion across a gap of width dd by a constant electric field. The mathematical idealization is that the particle has applied to it a constant force mαm\alpha while its coordinate lies in the interval (0,d)(0, d). Without radiation damping the particle, having initial velocity v0v_0, is accelerated uniformly for a time T=(−v0/α)+(v02/α2)+(2d/α)T = (-v_0/\alpha) + \sqrt{(v_0^2/\alpha^2) + (2d/\alpha)}, emerging at x=dx = d with a final velocity v1=v02+2αdv_1 = \sqrt{v_0^2 + 2\alpha d}. With radiation damping the motion is altered so that the particle takes a time T′T' to cross the gap and emerges with a velocity v1′v_1'. (a) Solve the integro-differential equation of motion, including damping, assuming TT and T′T' large compared to τ\tau. Sketch a velocity-versus-time diagram for the motion with and without damping. (b) Show that to lowest order in τ\tau, T′=T−τ(1−v0v1)T' = T - \tau\left(1 - \frac{v_0}{v_1}\right) v1′=v1−α2τv1Tv_1' = v_1 - \frac{\alpha^2\tau}{v_1}T (c) Verify that the sum of the energy radiated and the change in the particle's kinetic energy is equal to the work done by the applied field.
Intuitive explanation

The key to this problem is recognizing that radiation damping introduces a velocity-dependent retarding force that slightly reduces both the final velocity and transit time. Since the damping time τ\tau is much smaller than the crossing time TT, you can treat the radiation reaction as a small perturbation—the particle still accelerates nearly uniformly, but loses energy continuously to electromagnetic radiation. The integro-differential equation couples the particle's acceleration to its past motion through the radiation reaction term, which you can solve perturbatively by expanding around the undamped solution and keeping only first-order corrections in τ\tau.

Intuitive explanation

The key to this problem is recognizing that radiation damping introduces a velocity-dependent retarding force that slightly reduces both the final velocity and transit time. Since the damping time τ\tau is much smaller than the crossing time TT, you can treat the radiation reaction as a small perturbation—the particle still accelerates nearly uniformly, but loses energy continuously to electromagnetic radiation. The integro-differential equation couples the particle's acceleration to its past motion through the radiation reaction term, which you can solve perturbatively by expanding around the undamped solution and keeping only first-order corrections in τ\tau.

Related theory

Browse the problem library