Classical Electrodynamics · Problem 14.3

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · heaviside-feynman-electric-field

Problem

The Heaviside–Feynman expression for the electric field of a particle of charge ee in arbitrary motion, an alternative to the Liénard–Wiechert expression (14.14), is E=e[nR2]ret+e[Rc]retddt[nR2]ret+ec2dt2[n]ret\mathbf{E} = e\left[\frac{\mathbf{n}}{R^2}\right]_\text{ret} + e\left[\frac{R}{c}\right]_\text{ret} \frac{d}{dt}\left[\frac{\mathbf{n}}{R^2}\right]_\text{ret} + e\frac{c^2}{dt^2}[\mathbf{n}]_\text{ret} where the time derivatives are with respect to the time at the observation point. The magnetic field is given by (14.13). Using the fact that the retarded time is t′=t−R(t′)/ct' = t - R(t')/c and that, as a result, dtdt′=1−β(t′)⋅n(t′)\frac{dt}{dt'} = 1 - \boldsymbol{\beta}(t') \cdot \mathbf{n}(t') show that the form above yields (14.14) when the time differentiations are performed.

Related theory

Browse the problem library