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Question

The gap between the plates of a parallel plate capacitor of area A and distance between platesd, is filled with a dielectric whose permittivity varies linearly from 1 at one plate to 2 at the other. The capacitance of capacitor is:

A
0(1+2)A/d
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B
0(1+2)A/2d
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C
0A/[d/n(2/1)]
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D
0(12)A/[d/n(2/1)]
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Solution

The correct option is D 0(12)A/[d/n(2/1)]
Asthepermittivityofdielectricvarieslinearlyfromε1atoneplatetoε2attheother,itisgovernedbyequation,k=(ε2ε1d)x+ε1considerasmallelementofthicknessdxatadistancexfromplate,Then,dV=EokdxV0dV=d0σε01(ε2ε1d)x+ε1dxV=dσε0(ε2ε1)ln(ε2ε1)Q=CVC=QV=σAdσε0(ε2ε1)ln(ε2ε1)=ε0(ε2ε1)Adln(ε2ε1)Hence,theoptionDisthecorrectanswer.

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