EM3. The volume between two concentric conducting spherical surfaces of radius a and b (where a < b) is filled with material with an inhomogeneous dielectric constant
$$\epsilon(r) = \frac{\epsilon_0}{1 + Kr}$$
where $\epsilon_0$ and $K$ are constants and $r$ is the radial coordinate, so that $D(r) = \epsilon(r)E(r)$. A charge $Q$ is placed on the inner surface, while the outer surface is grounded. Find:
(a) The displacement in the region $a < r < b$.
(b) The capacitance of the device.
(c) The polarization charge density in the region $a < r < b$.
(d) The surface polarization charge density at $r = a$ and $r = b$.
Gauss' law and spherical symmetry give
$$D = \frac{Q}{4\pi r^2}e_r, (a < r < b).$$
The electric field intensity is
$$E = \frac{Q}{4\pi\epsilon_0 r^2}(1 + Kr)e_r, (a < r < b).$$
