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Solved Problem-4

The region between two conducting plates at x = 0 and x = d is filled with a perfect dielectric of nonuniform permittivity,
Description: 169570.png
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
We know that Description: 169562.png and Description: 169554.png
Using the identity for a scalar S and a vector Description: 169545.png,
Description: 169538.png
For a charge-free region, Description: 169521.png, so that,
Description: 169509.png
Considering the variation of ε and V only with respect to x, the equation becomes,
or, Description: 169501.png
Integrating, we get,
or, Description: 219574.png
Integrating again, we get,
Description: 169485.png
Applying the boundary conditions,
(1) At x = 0, V = 0,
Description: 169478.png

(2) At x = dV V0,
Description: 169468.png

(a) So, the potential is given as,
Description: 169461.png
(b) The field intensity is,
Description: 169452.png

(c) Flux density, Description: 169445.png
So, the surface charge densities are,
Description: 169436.png

(d) Charge on either plate per unit area is,
Description: 169429.png 
Thus, capacitance per unit area is given as,
Description: 169418.png

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