Classical Electrodynamics · Problem 12.19

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · conservation-laws-angular-momentum-fields

Problem

Source-free electromagnetic fields exist in a localized region of space. Consider the various conservation laws that are contained in the integral of

\partial_\alpha M^{\alpha\beta\gamma} = 0

over all space, where

M^{\alpha\beta\gamma}

is defined by (12.117). (a) Show that when

\beta

and

\gamma

are both space indices conservation of the total field angular momentum follows. (b) Show that when

\beta = 0

the conservation law is

\frac{d\mathbf{X}}{dt} = \frac{c^2 \mathbf{P}_{em}}{E_{em}}

where

\mathbf{X}

is the coordinate of the center of mass of the electromagnetic fields, defined by

\mathbf{X}\int u\,d^3x = \int \mathbf{x}u\,d^3x

where

u

is the electromagnetic energy density and

E_{em}

and

\mathbf{P}_{em}

are the total energy and momentum of the fields.