Magnetostatic field of electric currents

The relationship between the magnetic scalar potential and an electric-current sheet is examined. It is shown that any magnetostatic field in a source-free region of space can always be produced by an electric current flowing on any smooth, closed surface bounding the region. This has an interesting application in the perturbation of a magnetostatic field by a perfectly conducting body. It is shown that the problem of finding the current distribution which will produce a prescribed magnetostatic field depends upon the solution of the exterior Neumann problem. The converse problem of deducing the field produced by a prescribed current distribution can be treated in the same way, without recourse to the Biot and Savart law or the construction of the magnetic vector potential. A number of illustrative examples are given.