Magnetic field arising from current dipoles randomly distributed in a homogeneous spherical volume conductor

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Abstract
Certain types of magnetic noise arising from physical as well as biological sources can be explained by the model of current dipoles randomly distributed in a volume conductor. The volume conductor investigated in the present study is the homogeneous sphere, which is the model commonly used in analyses of the magnetic field of the human brain (magnetoencephalogram). Uniform distributions of random dipoles in three different source spaces are considered: a spherical surface, a spherical shell, and a sphere. The main emphasis is put on the radial component of the magnetic field. It is shown that the expectation value for the product of magnetic fields simultaneously measured at two locations, the covariance, depends on only two parameters of the measurement locations: the angular distance and the product of the radii. The formulas derived for the covariance can be expressed in terms of elliptic integrals of the first and the second kind, so that a very efficient numerical calculation is possible. For the special case of the variance (two identical measurement locations) these formulas reduce to expressions composed of elementary functions. Numerical examples show that the noise produced by random dipoles in a sphere is similar to the noise generated by random dipoles on a spherical surface having a slightly smaller radius.

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