Invertibility of current density from near-field electromagnetic data

The problem of determining a current density confined to a volume from measurements of the magnetic and electric fields it produces exterior to that volume is known to have nonunique solutions. Despite the nonuniqueness of the inversion we show that one may nevertheless uniquely determine certain moments of the vector spherical harmonic expansion of the current. It is demonstrated that the determination of these moments allows for the unique inversion of a current density confined to a spherical shell. Although unique the inversion may be ill conditioned and require a regularization of the inversion as demonstrated in an example numerical inversion