Two forms of ohmic heating of astrophysical secondaries have received particular attention: unipolar-generator heating with currents running between the primary and secondary, and magnetic induction heating due to the primary's time-varying field. Neither appears to cause significant dissipation in the contemporary solar system. But these discussions have overlooked heating derived from the spatial variation of the primary's field across the interior of the secondary. This leads to Lorentz-force-driven currents around paths entirely internal to the secondary, with resulting ohmic heating. We examine three ways to drive such currents, by the cross product of (1) the secondary's azimuthal orbital velocity with the nonaxially symmetric field of the primary, (2) the radial velocity (due to nonzero eccentricity) of the secondary with the primary's field, or (3) the out-of-plane velocity (due to nonzero inclination) with the primary's field. The first of these operates even for a spin-locked secondary whose orbit has zero eccentricity, in strong contrast to tidal dissipation. We show that Jupiter's moon Io today could dissipate about 600 GW (more than likely current radiogenic heating) in the outer 100 m of its metallic core by this mechanism. Had Io ever been at 3 Jovian radii instead of its current 5.9, it could have been dissipating 15,000 GW. Ohmic dissipation provides a mechanism that could operate in any solar system to drive inward migration of secondaries that then necessarily comes to a halt upon reaching a sufficiently close distance to the primary.