Spherical decomposition of electromagnetic fields generated by quasi-static currents

The spherical decomposition of electromagnetic fields generated by an arbitrary distribution of quasi-static currents may be obtained from the electromagnetic Green’s functions under spherical coordinates. It is shown that the magnetic field on an observation sphere may be represented in terms of external and internal toroidal currents as well as the radial currents across the sphere. The last component corresponds to the poloidal mode currents which generate no magnetic field inside any source-free domain. The poloidal mode of the electric field does not vanish under the same conditions but still generates negligible induction inside a conductive spherical Earth, and has negligible tangential components at the surface of the Earth. These observations justify the standard assumption used in electromagnetic induction studies, that external sources may be taken to be purely toroidal.