Smoke rings re-visited: an analytic solution for the transient electric field produced by a vertical magnetic dipole on or in a homogeneous conductive half-space

Abstract

For a vertical magnetic dipole (VMD) on or under the surface of a conductive half-space, explicit analytic formulae, not Hankel transform expressions, have been derived for the transient electric field induced below the ground, and the corresponding Hertz potential. These formulae permit rapid calculation of surface and underground dB/dt and B-field components, as well as of the induced current distribution. The solution allows the current produced by the surface reflection to be isolated, revealing how the interaction with the ground surface is responsible for the downward diffusion of the smoke ring. The mathematical form of the solution has also permitted confirmation of a prediction that the current distribution from a buried dipole in the limit of late time is a scaled version of the current distribution produced by a VMD on the ground surface.