On the average of electrostatic and magnetostatic fields, the singularities of dipole fields and depolarizing fields

TL;DR

This paper simplifies the proof of a classical theorem on the average of electrostatic and magnetostatic fields over spherical and ellipsoidal volumes, clarifying the relation between dipole singularities and depolarizing fields, and extending it to anisotropic media.

Contribution

It provides a simple proof of the average field theorem using reciprocity and extends the theorem to ellipsoidal volumes and anisotropic media.

Findings

Simplified proof of the average field theorem using reciprocity.

Extension of the theorem to ellipsoidal volumes.

Extension of the theory to anisotropic media.

Abstract

In "Classical Electrodynamics" (Jackson) a theorem is proved on the average of an electrostatic or magnetostatic field over a spherical volume. The proof of the theorem is based on an expansion in spherical harmonics and it is useful for deriving the singular behaviour of dipole fields. In this paper we give a simple proof of this theorem, the key element being to use reciprocity. The method also highlights the relation between the dipole singularity and the depolarizing field of a sphere, which is thus also found very easily. In addition this link enables us to extend the theorem to a more general ellipsoidal volume. Unfortunately one can only prove the existence of the depolarizing tensor in this way and one still needs another method for actually calculating the tensor. Finally the theory is also extended to an anisotropic background medium.