Outside the core, put $p=p_0$. The angular-gradient contribution is
$$
F_{\mathrm{el}}
=\int_{r_0}^L2\pi r\,dr\,
\frac\kappa2\frac{p_0^2}{r^2}
=\boxed{
\pi\kappa p_0^2\log\frac L{r_0}
}.
$$
The omitted core contribution is finite and independent of $L$.


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 2 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-344/2/e/solution