Electromagnetic fields in matter revisited

TL;DR

This paper revisits the electromagnetic force density and energy-momentum tensor in matter, deriving macroscopic quantities from microscopic Maxwell equations and clarifying the roles of free currents and polarization.

Contribution

It provides a microscopic derivation of macroscopic electromagnetic quantities in matter, confirming Minkowski's momentum density and clarifying the energy-momentum tensor structure.

Findings

The macroscopic force density depends linearly on average fields and derivatives.

The average current density includes free and dipolar contributions.

The field's energy-momentum tensor aligns with Minkowski's expression.

Abstract

The force density on matter and the kinetic energy-momentum tensor of the electromagnetic field in matter are obtained starting from Maxwell equations and Lorentz force at microscopic level and averaging over a small region of space-time. The macroscopic force density is taken to depend linearly on the average fields and their first derivatives and is shown to be determined by two phenomenological fields which are subsequently identified with the free current density and the polarization density tensor. It is shown that as expected, the average current density is the sum of the free current density and a dipolar contribution and that the average field satisfy the Maxwell equations. The macroscopic energy-momentum tensor of the field is shown to be equal to the standard empty-space energy-momentum tensor built with the macroscopic fields plus a dipolar correction. The density of momentum of the field is confirmed to be given by Minkowski's expression. The energy-momentum tensor of macroscopic matter is equal to the average of the microscopic energy-momentum tensor of matter plus the difference between the average tensor of microscopic fields and the macroscopic tensor of fields.