Classical Electrodynamics · Problem 14.11

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-external-electric-magnetic-fields

Problem

A particle of charge

ze

and mass

m

moves in external electric and magnetic fields

\mathbf{E}

and

\mathbf{B}

. (a) Show that the classical relativistic result for the instantaneous energy radiated per unit time can be written

P = \frac{2}{3}\frac{z^4 e^4}{m^2 c^3}\gamma^2[(\mathbf{E} + \boldsymbol{\beta} \times \mathbf{B})^2 - (\boldsymbol{\beta} \cdot \mathbf{E})^2]

where

\mathbf{E}

and

\mathbf{B}

are evaluated at the position of the particle and

\gamma

is the particle's instantaneous Lorentz factor. (b) Show that the expression in part a can be put into the manifestly Lorentz-invariant form,

P = \frac{2z^4 r_0^2}{3m^2 c} \cdot F^{\mu\nu} p_\nu p^\lambda F_{\lambda\mu}

where

r_0 = e^2/mc^2

is the classical charged particle radius.

Intuitive explanation

The key to part (a) is recognizing that the radiated power depends on the particle's acceleration in its own rest frame, but we must express this in the lab frame using the Lorentz force. The electric force $z\mathbf{E}$ and magnetic force $z\boldsymbol{\beta} \times \mathbf{B}$ combine to give the total acceleration; the magnetic field doesn't radiate energy directly (it does no work), so you need to isolate the component of the effective force perpendicular to the velocity. Part (b) then asks you to rewrite this result using the electromagnetic field tensor $F^{\mu\nu}$ and four-momentum $p^\mu$ , which automatically encodes the Lorentz transformation properties—this guarantees the formula holds in any reference frame, not just the lab frame.

Intuitive explanation

Related theory

Energy In Electric Magnetic Fields

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