The gap between the plates of a parallel plate capacitor of area A and distance between plates d , is filled…

The gap between the plates of a parallel plate capacitor of area A and distance between plates d, is filled with a dielectric whose relative permittivity varies linearly from ϵ1 at one plate to ϵ2 at the other. The capacitance of the capacitor is
  1. ϵ0∈2-∈1Adln∈2/∈1
  2. ϵ0∈2+∈1A2d
  3. ϵ0∈1+∈2Ad
  4. ϵ0Adln∈2/∈1

Solution

1dC=dxϵ0ϵA

=dxϵ0ϵ1+ϵ2-ϵ1dxA

1C=∫1dC=1∈0A∫0xdx∈1+∈2-∈1d

=ln∈2/∈1d∈0A∈2-∈1

C=∈0A∈2-∈1ln∈2/∈1·d

Asked in: JEE Main 2014 (19 Apr Online)

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