Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 9.9

Radiation of Electromagnetic Waves

Physics — Landau & Lifshitz · larmor-formula-relativistic

Problem

The relativistic generalization of the Larmor formula is

P = \frac{\mu_0 q^2 \gamma^6}{6\pi c}\left(\dot{v}^2 - \frac{v^2 \dot{v}^2}{c^2} + \frac{(\mathbf{v} \cdot \dot{\mathbf{v}})^2}{c^2}\right).

For a particle moving in a circle of radius

R

at speed

v

(so

\dot{v} = v^2/R

and

\mathbf{v}\cdot\dot{\mathbf{v}}=0

), show that

P = \frac{\mu_0 q^2 c \gamma^4}{6\pi R^2} \beta^4,

where

\beta = v/c

. Compare this result with the nonrelativistic formula.