 Sarvas formula

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.