Complete electromagnetic multipole expansion including toroidal moments

The electromagnetic multipole expansion presented in this paper is complete on two accounts: i) It is valid for all points in space, and ii) it recognizes the existence of toroidal moments. The electromagnetic field due to alternating poloidal currents in a toroidal solenoid is evaluated exactly via the solution of the inhomogeneous vector Helmholtz equations, using the outgoing wave Green function technique and the Debye potentials for sources and fields. The physical meaning of the toroidal moments can be appreciated when they are compared with the familiar electric and magnetic moments; the analysis of the long-wavelength limit of the exact results also explains the previous neglect of the toroidal moments. The magnetostatic limit and the point source limit are also physically and didactically interesting.