Classical Electrodynamics · Problem 11.27

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · lorentz-transformation-dipole-moments

Problem

(a) A charge density

\rho'

of zero total charge, but with a dipole moment

\mathbf{p}

, exists in reference frame

K'

. There is no current density in

K'

. The frame

K'

moves with a velocity

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

in the frame

K

. Find the charge and current densities

\rho

and

\mathbf{J}

in the frame

K

and show that there is a magnetic dipole moment,

\mathbf{m} = (\mathbf{p} \times \boldsymbol{\beta})/2

, correct to first order in

\beta

. What is the electric dipole moment in

K

to the same order in

\beta

? (b) Instead of the charge density, but no current density, in

K'

, consider no charge density, but a current density

\mathbf{J}'

that has a magnetic dipole moment

\mathbf{m}

. Find the charge and current densities in

K

and show that to first order in

\beta

there is an electric dipole moment

\mathbf{p} = \boldsymbol{\beta} \times \mathbf{m}

in addition to the magnetic dipole moment.