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In "Classical Electrodynamics" (Jackson) a theorem is proved on the averageof an electrostatic or magnetostatic field over a spherical volume. The proofof the theorem is based on an expansion in spherical harmonics and it is usefulfor deriving the singular behaviour of dipole fields. In this paper we give asimple proof of this theorem, the key element being to use reciprocity. Themethod also highlights the relation between the dipole singularity and thedepolarizing field of a sphere, which is thus also found very easily. Inaddition this link enables us to extend the theorem to a more generalellipsoidal volume. Unfortunately one can only prove the existence of thedepolarizing tensor in this way and one still needs another method for actuallycalculating the tensor. Finally the theory is also extended to an anisotropicbackground medium.

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