A spherical capacitor with the radii of the plates $a$ and $b,$ where $a<b,$ is filled with an isotropic heterogeneous dielectric whose permittivity depends on the distance $r$ from the center of the system as $\epsilon =\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$r$}\right.,$ where $\alpha$is a constant. Find the capacitance of the capacitor.

In accordance with the definition of the capacitance of a capacitor $(C=\raisebox{1ex}{$q$}\!\left/ \!\raisebox{-1ex}{$U$}\right.),$ the problem is reduced to determination of the potential difference $U$ for a given charge $q:$

$U={\int }_a^{b}Edr ........\left(1\right)$

where we assume that the charge of the inner plate is $q>0.$ Let us find $E$ with the help of the Gauss theorem for vector $D:$

$E=\frac{q}{4\mathrm{\pi }\epsilon {\epsilon }_0}\frac{1}{r^2}=\frac{1}{4\pi {\epsilon }_0}\frac{q}{\epsilon r^2}=\frac{1}{4\pi {\epsilon }_0}=\frac{q}{\alpha r}.$ 

Substituting the latter expression into (1) and integrating, we find

$U=\frac{q}{4\pi {\epsilon }_0\alpha }\ln \frac{b}{a}, C=\frac{4\pi {\epsilon }_0\alpha }{\ln \left({\displaystyle \raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}\right)}$