New Perspective on the Reciprocity Theorem of Classical Electrodynamics

TL;DR

This paper offers a straightforward physical proof of the reciprocity theorem in classical electrodynamics applicable to general media, including linearly polarizable and magnetizable substances, extending to spatially dispersive media.

Contribution

It provides a simple, general proof of the reciprocity theorem for complex media, including conditions for spatial dispersion and tensor symmetry constraints.

Findings

Proof applies to media with spatially varying susceptibilities

Reciprocity holds if susceptibility tensors are symmetric or transpose-related

Extension to spatial dispersion media established

Abstract

We provide a simple physical proof of the reciprocity theorem of classical electrodynamics in the general case of material media that contain linearly polarizable as well as linearly magnetizable substances. The excitation source is taken to be a point-dipole, either electric or magnetic, and the monitored field at the observation point can be electric or magnetic, regardless of the nature of the source dipole. The electric and magnetic susceptibility tensors of the material system may vary from point to point in space, but they cannot be functions of time. In the case of spatially non-dispersive media, the only other constraint on the local susceptibility tensors is that they be symmetric at each and every point. The proof is readily extended to media that exhibit spatial dispersion: For reciprocity to hold, the electric susceptibility tensor Chi_E_mn that relates the complex-valued magnitude of the electric dipole at location r_m to the strength of the electric field at r_n must be the transpose of Chi_E_nm. Similarly, the necessary and sufficient condition for the magnetic susceptibility tensor is Chi_M_mn = Chi^T_M_nm.