  The energy--momentum tensor and the tensor continuity equation serve as the
conservation laws of energy, linear momentum, and angular momentum for a
continuous flow. Previously, we derived equations of motion for macroscopic
electromagnetic fields in a homogeneous linear dielectric medium that is draped
with a gradient-index antireflection coating (J. Math Phys. 55, 042901 (2014)
). These results are consistent with the electromagnetic tensor continuity
equation in the limit that reflections and the accompanying surface forces are
negligible thereby satisfying the condition of an unimpeded flow in a
thermodynamically closed system. Here, we take the next step and derive
equations of motion for the macroscopic fields in the limiting case of a
piecewise-homogeneous simple linear dielectric medium. The presence of
radiation surface forces on the interface between two different homogeneous
linear materials means that the energy--momentum formalism must be modified to
treat separate homogeneous media in which the fields are connected by boundary
conditions at the interfaces. We demonstrate the explicit separation of the
total momentum into a field component and a material motion component, we
derive the radiation pressure that transfers momentum from the field to the
material, we derive the electromagnetic continuity equations for a piecewise
homogeneous dielectric, and we provide a lucid reinterpretation of the Jones
and Richards experiment.
