A Physically Consistent Formulation of Macroscopic Electrodynamics in Matter

Abstract: Classical electrodynamics in material media, while essential, faces century-old inconsistencies regarding energy, momentum, and force. This work re-examines the theory from first principles to establish a rigorous description. Applying the force-energy consistency requirement, derived from the unambiguous Maxwell-Lorentz theory for free charges, reveals profound physical inconsistencies in conventional energy balances (using $\mathbf{D}, \mathbf{H}$) and major historical energy-momentum tensors (Minkowski, Abraham, Einstein-Laub). Their critical failure lies in accounting for energy dissipation, especially in stationary matter. This analysis justifies a unique formulation where the field's energy/momentum uses the vacuum tensor $T_{EM}(\mathbf{E},\mathbf{B})$, and interaction is solely the total Lorentz force $f_{Lorentz}$ acting on the total current $J_{total}$ (incorporating material polarization $\mathbf{P}$ and magnetization $\mathbf{M}$). This framework correctly treats $\mathbf{j}_{total} \cdot \mathbf{E}$ as the universal energy exchange gateway, handling storage and dissipation consistently, and clarifies $\mathbf{D}, \mathbf{H}$ as mathematical aids, not fundamental energy carriers. Addressing force density controversies, we demonstrate via spatial averaging analysis that microscopic force distributions are inherently indeterminable in any macroscopic theory. This justifies focusing on consistent energy/momentum accounting and provides a unified, physically sound, relativistically consistent foundation for electrodynamics in matter.