Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 36C (Electrodynamics)

For a linear dielectric, $D=\epsilon E$ (equivalently $D={\epsilon }_{0}E+P$). With no free charge, Gauss law gives $\nabla \cdot D=0$. Across an interface without free surface charge, $D_n$ is continuous; electrostatic Faraday law makes $E_t$ continuous.

Imposing those two boundary conditions on the ansatz gives, with $\kappa =(\epsilon -{\epsilon }_0)/(\epsilon +2{\epsilon }_0)$,Thus the internal field is uniform and the external correction has dipolar angular form.

The far field of a dipole is $[3(p\cdot \hat{x})\hat{x}-p]/(4\pi {\epsilon }_0{r}^3)$. Comparison yields the induced electric dipole moment