Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 36B / b

The electric polarization $\mathbf{P}$ is electric dipole moment per unit volume. It producesThe electric displacement field $\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}$ is useful becauseso it separates free charge from the bound charge represented by $\mathbf{P}$. In a linear isotropic dielectric, $\mathbf{D}=ϵ\mathbf{E}$.

Spherical symmetry and Gauss's law for $\mathbf{D}$ give, on both sides of the interface,ThusWith $\mathbf{n}={\mathbf{e}}_r$ pointing from medium one to medium two, the net bound sheet charge at $r=R$ is the polarization jumpIf one assigns surface charge separately to the inner body and outer medium, their contributions are ${\mathbf{P}}_1\cdot \mathbf{n}$ and $-{\mathbf{P}}_2\cdot \mathbf{n}$; their sum is the displayed physical interface charge.