On the magnetic field generated outside an inhomogeneous volume conductor by internal current sources

The magnetocardiogram results from the detection of magnetic fields generated outside the body by electric current sources in the heart. Let the source current dipole moment per unit volume be J
ⁱ
, the conductivity
g
, the electric potential
V
, and the electric field intensity
E
. Then it may be shown that the magnetic field
H
is given by either
H = (1/4\pi) \int J^{i} \times \nabla(1/r) dv + \sum_{i} \int (g' - g")(E \times dS_{i}/r)
, or
H = (1/4\pi) \int J^{i} \times \nabla(1/r) dv + \sum_{i} \int (g' - g")V\nabla(1/r) \times dS_{i}
, where the surface integral is over all surfaces S
i
separating regions of different conductivity, i.e.,
g
' and
g
", and
r
is the distance from the point of measurement to the element of volume or surface. The magnetic dipole moment
m
is given by
m = \frac{1}{2} \int r_{1} \times J^{i}dv - \frac{1}{2} \sum_{i} \int (g'-g")Vr_{1} \times dS_{i}
, where r
1
is a radius vector from an arbitrary origin.