Classical Electrodynamics · Problem 11.30

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · electrodynamics-moving-medium

Problem

An isotropic linear material medium, characterized by the constitutive relations (in its rest frame

K'

\mathbf{D}' = \epsilon \mathbf{E}'

and

\mu\mathbf{H}' = \mathbf{B}'

, is in uniform translation with velocity

\mathbf{v}

in the inertial frame

K

. By exploiting the fact that

F_{\mu} = (\mathbf{E}, \mathbf{B})

and

G_{\mu} = (\mathbf{D}, \mathbf{H})

transform as second rank 4-tensors under Lorentz transformations, show that the macroscopic fields

\mathbf{D}

and

\mathbf{H}

are given in terms of

\mathbf{E}

and

\mathbf{B}

\mathbf{D} = \epsilon\mathbf{E} + \gamma^2\left(\epsilon - \frac{1}{\mu}\right)[\beta^2\mathbf{E}_{\perp} + \boldsymbol{\beta} \times \mathbf{B}]

\mathbf{H} = \frac{1}{\mu}\mathbf{B} + \gamma^2\left(\epsilon - \frac{1}{\mu}\right)[-\beta^2\mathbf{B}_{\perp} + \boldsymbol{\beta} \times \mathbf{E}]

where

\mathbf{E}_{\perp}

and

\mathbf{B}_{\perp}

are components perpendicular to

\mathbf{v}