Classical Electrodynamics · Problem 6.9

Plane Electromagnetic Waves and Wave Propagation — John David Jackson

Physics — E&M · conservation-laws-macroscopic-medium-minkowski

Problem

Discuss the conservation of energy and linear momentum for a macroscopic system of sources and electromagnetic fields in a uniform, isotropic medium described by a permittivity ϵ\epsilon and a permeability μ\mu. Show that in a straightforward calculation the energy density, Poynting vector, field-momentum density, and Maxwell stress tensor are given by the Minkowski expressions, u=12(ϵE2+μH2)S=E×Hg=μϵE×HTij=[ϵEiEj+μHiHj−12δij(ϵE2+μH2)]\begin{aligned} u &= \frac{1}{2}(\epsilon E^2 + \mu H^2) \\ \mathbf{S} &= \mathbf{E} \times \mathbf{H} \\ \mathbf{g} &= \mu\epsilon\mathbf{E} \times \mathbf{H} \\ T_{ij} &= [\epsilon E_i E_j + \mu H_i H_j - \tfrac{1}{2}\delta_{ij}(\epsilon E^2 + \mu H^2)] \end{aligned} What modifications arise if ϵ\epsilon and μ\mu are functions of position?

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