On the exterior magnetic field and silent sources

Two main results are included in this paper. The first one deals with the leading asymp-totic term of the magnetic field outside any conductive medium. In accord with physical reality, it is proved mathematically that the leading approximation is a quadrupole term which means that the conductive brain tissue weakens the intensity of the magnetic field outside the head. The second one concerns the orientation of the silent sources when the geometry of the brain model is not a sphere but an ellipsoid which provides the best possible mathematical approximation of the human brain. It is shown that what char-acterizes a dipole source as “silent ” is not the collinearity of the dipole moment with its position vector, but the fact that the dipole moment lives in the Gaussian image space at the point where the position vector meets the surface of the ellipsoid. The appropriate representation for the spheroidal case is also included. 1. The magnetic field The mathematical theory of magnetoencephalography (MEG) is governed by the equa-tions of quasistatic theory of electromagnetism [11, 14, 15, 19, 20]. If we denote by V− the region occupied by the conductive brain tissue, with conductivity σ> 0 and magnetic permeability µ0> 0, then, as Geselowitz has shown [3, 9, 10], the magnetic field in the exterior of V − region, V+, due to the internal electric dipole current Jp(r)=Qδ(r − r0), r0 ∈V−, (1.1) assumes the representation B(r) = µ0 4π Q × r − r0∣∣r − r0∣∣3 − µ0