= Sarvas formula
{c}
{title2=$\mathbf B=\frac{\mu_0}{4\pi}\nabla\frac{(\mathbf Q\times\mathbf r_0)\cdot\mathbf r}{F}$}
For the exterior of a <spherical volume conductor model>, put $P=|\mathbf r-\mathbf r_0|$, $r=|\mathbf r|$ and $F=rP^2+P\mathbf r\cdot(\mathbf r-\mathbf r_0)$. 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 $U$. https://pubmed.ncbi.nlm.nih.gov/3823129/[Sarvas's original paper] derives the spherical-conductor field and its consequences for the inverse problem.
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