Spherical current-dipole logarithmic kernel ($\varphi =\frac{1}{b}\log \frac{r+P+b}{r+P-b},b=|{\mathbf{r}}_0|$)

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.