5. Inhomogeneous dielectric. Consider two concentric spherical conducting surfaces of radii a and b (a < b). The volume between these surfaces is filled with an inhomoge- neous material with a dielectric constant (r) that depends on the radial position r via €(r) k(r) = = €0 = 1 1 - ar where a is a constant and k ≥ 1 throughout the material. The electric displacement vector D is related everywhere to the electric field E via D = €E (i.e., this is a linear dielectric.) A charge Q is placed on the inner conducting surface while the outer surface is grounded. Find... (a) ...D(r) in the region a < r < b. (b) ...the potential difference between the two conducting spheres. (c) the capacitance of the device. (d) ...the polarization density P(r) in the region a<r<b. (e) the bound charge volume density p₁ in the region a < r < b. (f) ...the bound charge surface densities σ at the surfaces r = a and r = b.

Fig: 1