# 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

(a) The potential between the plates

(b) The field intensity

(c) Surface charge density,

(d) Capacitance per unit area for the plates

*x*= d is maintained at a voltage*V*_{0}and at*x*= 0 is grounded, find(a) The potential between the plates

(b) The field intensity

(c) Surface charge density,

*ρ*_{s}at*x*= 0,*d*(d) Capacitance per unit area for the plates

Solution

We know that and
Using the identity for a scalar
Here,

For a charge-free region, , so that,
Considering the variation of

or,
Integrating, we get,

or,
Integrating again, we get,
Applying the boundary conditions,

(1) At

(2) At

(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,

*S*and a vector ,*ε*and*V*only with respect to*x*, the equation becomes,or,

or,

(1) At

*x*= 0,*V*= 0,(2) At

*x*=*d*,*V*=*V*_{0},(a) So, the potential is given as,

(c) Flux density,

So, the surface charge densities are,

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