Introduction to Electrodynamics · Problem 8.23

Electromagnetic Waves — David J. Griffiths

Physics · electromagnetic-energy-momentum-media

Problem

(a) Carry through the argument in Sect. 8.1.2, starting with Eq. 8.6, but using

\mathbf{J}_f

in place of

\mathbf{J}

. Show that the Poynting vector becomes

\mathbf{S} = \mathbf{E} \times \mathbf{H}, \tag{8.46}

and the rate of change of the energy density in the fields is

\frac{\partial u}{\partial t} = \mathbf{E} \cdot \frac{\partial \mathbf{D}}{\partial t} + \mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t}.

For linear media, show that

u = \frac{1}{2}(\mathbf{E} \cdot \mathbf{D} + \mathbf{B} \cdot \mathbf{H}). \tag{8.47}

(b) In the same spirit, reproduce the argument in Sect. 8.2.2, starting with Eq. 8.15, with

\rho_f

and

\mathbf{J}_f

in place of

\rho

and

\mathbf{J}

. Don't bother to construct the Maxwell stress tensor, but do show that the momentum density is

\mathbf{g} = \mathbf{D} \times \mathbf{B}. \tag{8.48}