A parallel-plate capacitor is constructed using a dielectric whose constant varies with position. The plates have area A. The bottom plate is at y = 0 and the top plate is at y = y0. The dielectric constant is given as a function of y according to κ = 1 + (3/y0)y.

(a) What is the capacitance?

(b) Find σb/σf on the surfaces of the dielectric.

(c) Use Gauss's law to find the induced volume charge density ρ(y) within this dielectric.

(d) Integrate the expression for the volume charge density found in (c) over the dielectric, and show that the total induced bound charge, including that on the surfaces, is zero.