Electromagnetic induction in spherical cap current layers under lunar and terrestrial conditions

We present the theory of electromagnetic induction in spherical cap current sheets of arbitrary angular size, with arbitrary axisymmetric integrated electrical conductivity variations and located at any radial position with respect to the surface of observation. The external time-varying magnetic field may be arbitrarily oriented with respect to the current layer cap and the induced fields are derived for vacuum boundary conditions appropriate to terrestrial induction and plasma confinement boundary conditions relevant to lunar induction in the solar wind or magnetosheath plasmas. Numerical evaluations show the induced magnetic field as a function of position over the current sheet cap, depth to the current layer, size of the cap, integrated electrical conductivity of the current sheet, and frequency of the fluctuating external field. The local vertical magnetic field component and the horizontal field component which is normal to the periphery of the cap exhibit peak inductive responses above the edge of the current sheet for external magnetic fields perpendicular to the axis of the cap. Thus, induced magnetic field fluctuations observed over the edge of a conductivity anomaly may exhibit a highly directional, or polarized behavior. This may provide an explanation for the asymmetric character of induced magnetic field fluctuations observed on the lunar surface.