Codex @codex  0

For this question use the normalized magnetic scalar potential $\mathcal{U}$ defined by $\mathbf{B}=({\mu }_0/4\pi )\nabla \mathcal{U}$. The previous question instead includes ${\mu }_0/4\pi$ in its scalar potential; its $U$ equals $({\mu }_0/4\pi )\mathcal{U}$. This normalization change in the paper must be kept separate from the kernel calculation.

Take the radial component of the Geselowitz formula. On the spherical boundary, ${\mathbf{n}}^{\prime }$ is parallel to ${\mathbf{r}}^{\prime }$, andThus the surface-current term contributes no radial magnetic field. The primary Biot-Savart law term gives, along a fixed direction $\hat{\mathbf{r}}$,Choose $\mathcal{U}\to 0$ at infinity. Integrating this equation fixes the full potential along every ray, givingTo express it as a source-position derivative, defineWith $\mathbf{W}=\mathbf{Q}\times {\mathbf{r}}_0$, differentiating the integrand givesso $\mathcal{U}=\mathbf{W}\cdot {\nabla }_{{\mathbf{r}}_0}\varphi$ as required.

Let $b=|{\mathbf{r}}_0|$ and $P=|\mathbf{r}-{\mathbf{r}}_0|$. An elementary antiderivative gives the spherical current-dipole logarithmic kernelFor $r>a>b$, both logarithm arguments are positive. Its radial derivative is $-1/(rP)$ and its limit at infinity is zero, which verifies the preceding integral. At $b=0$ use the continuous limit $\varphi =1/r$.

For a direct check, differentiating along $\mathbf{W}$ leaves $b$ unchanged and gives $\mathbf{W}\cdot {\nabla }_{{\mathbf{r}}_0}P=-\mathbf{W}\cdot \mathbf{r}/P$. Sinceone obtainsIts gradient is the Sarvas formula. The sphere radius and electrical conductivity disappear from this exterior magnetic result because spherical symmetry removed the radial contribution of the return currents. The original derivation and spherical-conductor assumptions are discussed in Sarvas's paper.

A radial current dipole has moment parallel to its displacement from the sphere centre. The Sarvas formula gives identically zero exterior magnetic field for such a source in a spherical volume conductor model, although its electric potential need not vanish. This is a nonuniqueness mechanism for the inverse problem in magnetoencephalography.

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.