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(∈1−∈2)A/[d/n(∈2/∈1)]
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Solution
The correct option is D∈0(∈1−∈2)A/[d/n(∈2/∈1)] Asthepermittivityofdielectricvarieslinearlyfromε1atoneplatetoε2attheother,itisgovernedbyequation,k=(ε2−ε1d)x+ε1considerasmallelementofthicknessdxatadistancexfromplate,Then,dV=Eokdx∫V0dV=∫d0σε01(ε2−ε1d)x+ε1dxV=dσε0(ε2−ε1)ln(ε2ε1)Q=CV⇒C=QV=σAdσε0(ε2−ε1)ln(ε2ε1)=ε0(ε2−ε1)Adln(ε2ε1)Hence,theoptionDisthecorrectanswer.