On the definition of macroscopic electric and magnetic fields

Abstract

Macroscopic electric and magnetic fields and potential functions are usually defined as weighted averages of their microscopic counterparts. It is shown that this procedure is satisfactory for the field and the scalar potential due to electric point charges. When elementary electric dipoles are considered, the procedure can still be applied to the scalar potential but not to the field itself because the weighted average of the latter is not convergent. It is shown that the macroscopic field can then be defined as minus the gradient of the average potential. The usual macroscopic field equations are shown to hold, as well as the usual relation between the macroscopic field and the local field. Similar considerations can be made about the fields due to elementary magnetic dipoles. While the averaged magnetic field is divergent, the averaged scalar and vector potentials exist. Minus the gradient of the averaged scalar potential yields the macroscopic magnetic field. The curl of the averaged vector potential yields the macroscopic magnetic induction.