Classical Electrodynamics · Problem 16.9

Radiating Systems, Multipole Fields and Radiation — John David Jackson

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

Problem

(a) Show that the radiation reaction force in the Lorentz--Dirac equation of Problem 16.7 can be expressed alternatively as

F^\text{rad}_\mu = \frac{2e^2}{3mc^3}\left[\left(g_{\mu\nu} - \frac{p_\mu p_\nu}{m^2c^2}\right)\frac{d^2p^\nu}{d\tau^2}\right]

(b) The relativistic generalization of (16.10) can be obtained by replacing

d^2p^\nu/d\tau^2

g^{\nu\lambda}dF^\text{ext}_\lambda/d\tau

in the expression for

F^\text{rad}_\mu

. Show that the spatial part of the generalization of (16.10) becomes

\frac{d\mathbf{p}}{dt} = \mathbf{F} + \tau\left[\gamma\frac{d\mathbf{F}}{dt} - \frac{\gamma^3}{c^2}\frac{d\mathbf{v}}{dt} \times (\mathbf{v} \times \mathbf{F})\right]

where

\mathbf{F}

is the spatial part of

F^\mu_\text{ext}/\gamma

. For a charged particle in external electric and magnetic fields

\mathbf{F}

is the Lorentz force. Reference: G. W. Ford and R. F. O'Connell, Phys. Lett. A 174, 182 (1993).