Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 36D / c / Solution

Since there are no currents, $\nabla \times B=0$ in each constant-permeability region, while $\nabla \cdot B=0$ gives ${\nabla }^2\psi =0$. The displayed dipole forms are therefore harmonic. Regularity at the origin and the field at infinity giveFor $\psi =(ar+b/r^2)\cos \theta$,The magnetic spherical-shell matching conditions, continuity of $B_r$ and $H_{\theta }$, give the remaining four equationsThe two $R_1$ equations yieldTogether with $b_1=0$ and $a_3=B_0$, these are the requested expressions. Substituting them into the two $R_2$ equations gives a two-by-two linear system for $a_1$ and $b_3$.