Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 76 / 1

Write $b=|{\mathbf{r}}_0|$, $r=|\mathbf{r}|$, $\mathbf{P}=\mathbf{r}-{\mathbf{r}}_0$ and $P=|\mathbf{P}|$. Initially assume $0<b<a$ and a nonzero current dipole. The possible singularities of the continued formulas occur when $r=0$, $P=0$ orBy the Cauchy-Schwarz inequality, $D\ge 0$. Away from the two endpoints it vanishes exactly when $\mathbf{P}$ is antiparallel to $\mathbf{r}$, namely on the open segment $\mathbf{r}=s{\mathbf{r}}_0$, $0<s<1$. Thus both formulas are regular in the physical exterior $r>a$; the singularities requested here belong to their analytic expressions continued into the sphere.

To determine the order on the segment, let $\mathbf{e}={\mathbf{r}}_0/b$ and approach an interior point as $\mathbf{r}=z\mathbf{e}+\rho {\mathbf{e}}_{\rho }$, with $0<z<b$ and ${\mathbf{e}}_{\rho }\perp \mathbf{e}$ a unit transverse direction. Expansion givesConsequently the leading singular terms in the electric potential and magnetic scalar potential areThus a dipole with a nonzero transverse moment has first-order line singularities on the open segment: their generic growth is $1/\rho$, the inverse distance to that segment. Angular coefficients can vanish along particular approach directions without removing the line singularity.

At the source point put $\mathbf{r}={\mathbf{r}}_0+P\mathbf{n}$ with a fixed direction $\mathbf{n}\ne -\mathbf{e}$. The explicit dipole term givesThe source therefore has a second-order electric dipole singularity and, for a nonradial moment, a first-order magnetic singularity. Approaches tangent to the singular segment need not have these fixed-angle scalings.

At the origin put $\mathbf{r}=r\mathbf{n}$, with $\mathbf{n}\ne \mathbf{e}$. The leading terms areThe electric expression has a first-order point singularity there. The magnetic expression has a generally direction-dependent finite radial limit, so the origin is a nonremovable endpoint singularity of radial order zero, rather than an isolated $1/r$ pole. It can still be unbounded along approaches whose angle to the segment also tends to zero. Stating the approach matters for this nonisolated singularity.

For a radial current dipole, $\mathbf{Q}=q\mathbf{e}$, the transverse line terms cancel and the electric line singularity is removable. Its only electric singularities are the first-order term $-q/(4\pi \sigma br)$ at the origin and the second-order dipole at ${\mathbf{r}}_0$. The magnetic potential is identically zero because $\mathbf{Q}\times {\mathbf{r}}_0=0$. Finally, if ${\mathbf{r}}_0=0$, the two terms in the given exterior electric expression combine toThere is then just a second-order electric singularity at the origin. A zero dipole moment makes both potentials zero.