Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 36D (Electrodynamics)

A dielectric is a medium whose constituent charges become polarized by an applied electric field; it may also acquire a magnetization. Spatial variation of $P$ leaves uncompensated dipole charge, giving ${\rho }_{\mathrm{bd}}=-\nabla \cdot P$. Time-dependent polarization transports charge and aligned microscopic current loops contribute magnetization current, giving the bound charge and bound current

Split the microscopic sources into free and bound parts and defineSubstitution of $\rho ={\rho }_{\mathrm{f}}-\nabla \cdot P$ into Gauss's law gives $\nabla \cdot D={\rho }_{\mathrm{f}}$. Similarly, using $J=J_{\mathrm{f}}+{\partial }_{t}P+\nabla \times M$ in the Ampère-Maxwell equation gives $\nabla \times H=J_{\mathrm{f}}+{\partial }_{t}D$. The homogeneous Maxwell equations are unchanged. Thus

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 $b_3$.