On the Geselowitz formula in biomagnetics

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Abstract
A detailed analysis of the Geselowitz formula for the magnetic induction and for the electric potential fields, due to a localized dipole current density, is provided. It is shown that the volume integral, which describes the contribution of the conductive tissue to the magnetic field, exhibits a hyper-singular behaviour at the point where the dipole source is located. This singularity is handled both via local regularization of the volume integral as well as through calculation of the total flux it generates. The analysis reveals that the contribution of the primary dipole to the volume integral is equal to the one third of the magnetic field generated by the primary dipole while the rest is due to the distributed conductive tissue surrounding the singularity. Furthermore, multipole expansion is introduced, which expresses the magnetic field in terms of polyadic moments of the electric potential over the surface of the conductor.