Question

If the value charge density of a dielectric sphere with a cavity (a shown in the figure) is $\rho$. Find the electrostatic self energy.

Answer 1

Self energy is equal to interaction energy of charges constituting the thick spherical shell.
Consider core of outer radius $r$ having charge $4q$, layer of infinitesimal thickness $dr$ with charge $dq$.
Since core can be assumed to be a point charge at the centre with
$q=\rho .\frac{4}{3}\pi (r^3-R^3)$
$\therefore$ Electrostatic interaction energy of point charge and the layer is given by
$dU=\frac{1}{4\pi {\epsilon }_0}\frac{q.dq}{r}=\frac{1}{4\pi {\epsilon }_0}\frac{p\frac{4}{3}\pi (r^3-R^3).\rho 4\pi r^{2}dr}{r}$
$U=\stackrel{2R}{\underset{R}{\int }}\frac{{\rho }^2}{3{\epsilon }_0}(r^4-R^{3}r).dr$
$U=\frac{{\rho }^{2}4\pi }{3{\epsilon }_0}{[\frac{r^5}{5}-\frac{R^3{r}^2}{2}]}_R^{2R}=\frac{94}{15}\frac{{\rho }^2\pi R^5}{{\epsilon }_0}$