# Solved Problem-4

Problem-4
The region between two conducting plates at x = 0 and x = d is filled with a perfect dielectric of nonuniform permittivity,

If the plate at x = d is maintained at a voltage V0 and at x = 0 is grounded, find
(a) The potential between the plates
(b) The field intensity
(c) Surface charge density, ρs at x = 0, d
(d) Capacitance per unit area for the plates

Solution
We know that  and

Using the identity for a scalar S and a vector ,

Here,

For a charge-free region, , so that,

Considering the variation of ε and V only with respect to x, the equation becomes,
or,

Integrating, we get,
or,

Integrating again, we get,

Applying the boundary conditions,
(1) At x = 0, V = 0,

(2) At x = dV V0,

(a) So, the potential is given as,

(b) The field intensity is,

(c) Flux density,
So, the surface charge densities are,

(d) Charge on either plate per unit area is,

Thus, capacitance per unit area is given as,