Classical Electrodynamics · Problem 11.28

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · moving-electric-dipole-potentials

Problem

Revisit Problems 6.21 and 6.22 from the viewpoint of Lorentz transformations. An electric dipole instantaneously at rest at the origin in the frame

K'

has potentials,

\Phi' = \mathbf{p} \cdot \mathbf{r}'/r'^3

, and

\mathbf{A}' = 0

(and thus only an electric field). The frame

K'

moves with uniform velocity

\mathbf{v} = \boldsymbol{\beta}c

in the frame

K

. (a) Show that in frame

K

to first order in

\beta

the potentials are

\Phi = \frac{\mathbf{p} \cdot \mathbf{R}}{R^3}, \quad \mathbf{A} = \boldsymbol{\beta}\frac{(\mathbf{p} \cdot \mathbf{R})}{R^3}

where

\mathbf{R} = \mathbf{x} - \mathbf{x}_0(t)

, with

\mathbf{v} = d\mathbf{x}_0/dt

at time

t

. (b) Show explicitly that the potentials in

K

satisfy the Lorentz condition. (c) Show that to first order in

\beta

the electric field

\mathbf{E}

K

is just the electric dipole field (centered at

\mathbf{x}_0

), or a dipole field plus time-dependent higher multipoles, if viewed from a fixed origin, and the magnetic field is

\mathbf{B} = \boldsymbol{\beta} \times \mathbf{E}

. Where is the effective magnetic dipole moment of Problem 6.21 or 11.27a?