Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 1 / b / Solution

Along the circle with increasing $\varphi$, the line element is $d\mathbf{r}=a\sin \alpha {\mathbf{e}}_{\varphi }d\varphi$. Thus the two circulations areThe half-angle identities simplify them toHere $0<\alpha <\pi$ so both potentials are defined on the whole curve.