Volume conductor

A volume conductor is a conducting three-dimensional medium through which electric currents flow. In the quasistatic bioelectric model, the total current density is $\mathbf{J}={\mathbf{J}}^p-\sigma \nabla u$, comprising an impressed source current and its Ohmic return current. Electric charge conservation gives $\nabla \cdot (\sigma \nabla u)=\nabla \cdot {\mathbf{J}}^p$. An insulating boundary has zero normal total current.

For a homogeneous volume conductor with an insulating boundary $S$, the Geselowitz formula expresses the exterior magnetic field as the primary Biot-Savart law field minus a surface contribution from the induced electric potential. For a current dipole, ${\mathbf{B}}^p={\mu }_0\mathbf{Q}\times (\mathbf{r}-{\mathbf{r}}_0)/(4\pi |\mathbf{r}-{\mathbf{r}}_0{|}^3)$. On a spherical boundary, ${\mathbf{n}}^{\prime }$ is parallel to ${\mathbf{r}}^{\prime }$, so the surface term has zero radial component at the observation point.

A spherical volume conductor model approximates a conducting body by a sphere, usually with uniform or radially stratified conductivity. It provides explicit forward models for electroencephalography and magnetoencephalography. Its symmetry makes a radial current dipole magnetically silent outside the sphere; departures from spherical symmetry need not retain that property.

For the exterior of a spherical volume conductor model, put $P=|\mathbf{r}-{\mathbf{r}}_0|$, $r=|\mathbf{r}|$ and $F=r{P}^2+P\mathbf{r}\cdot (\mathbf{r}-{\mathbf{r}}_0)$. The displayed Sarvas formula follows by taking the radial component of the Geselowitz formula and integrating the magnetic scalar potential from infinity. It is independent of a spherically symmetric conductivity profile. The sign shown uses $\mathbf{B}=+{\mu }_0\nabla U/(4\pi )$; conventions defining a magnetic potential with a minus sign reverse $U$. Sarvas's original paper derives the spherical-conductor field and its consequences for the inverse problem.

For $r>b$ and $P=|\mathbf{r}-{\mathbf{r}}_0|$, the displayed kernel satisfies $\varphi ={\int }_r^{\infty }ds/[s|s\hat{\mathbf{r}}-{\mathbf{r}}_0|]$. The limit at $b=0$ is $1/r$. Differentiating along $\mathbf{Q}\times {\mathbf{r}}_0$, which preserves $b$, gives $(\mathbf{Q}\times {\mathbf{r}}_0)\cdot {\nabla }_{{\mathbf{r}}_0}\varphi =(\mathbf{Q}\times {\mathbf{r}}_0)\cdot \mathbf{r}/F$. Thus it generates the normalized magnetic scalar potential in the Sarvas formula.