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Question

A parallel plate capacitor, with plate area A and distance of separation d, is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as follows:
ϵ(x)=ϵ0+kx, for (0<xd2)
ϵ(x)=ϵ0+k(dx), for (d2xd)

A
kA2 ln(2ϵ0+kd2ϵ0)
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B
kA2 ln(2ϵ02ϵ0kd)
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C
(ϵ0 kd2)2k a
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D
0
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Solution

The correct option is A kA2 ln(2ϵ0+kd2ϵ0)

The net capacity will be the effective capacity of series combination of two capacitors formed by the two halves of the dielectric.
i.e. 1C=1C1+1C2

Taking an element of width dx at a distance
x from left plate (x<d2)
dC1=(ϵ0+kx)Adx

Capacitance of the first half of the capacitor is,
1C1=d201dc=1Ad20dxϵ0+kx

1C1=1kAln⎜ ⎜ ⎜ϵ0+kd2ϵ0⎟ ⎟ ⎟

Consider another element of width dx, at a distance
x from the center (x>d2)
dC2=A(ϵ0+k(dx))dx

Capacitance of the second half of the capacitor is,
1C1=dd21dC2=1Add2dxϵ0+kdkx

1C2=1kAln⎜ ⎜ ⎜ϵ0+kd2ϵ0⎟ ⎟ ⎟

As, 1C1=1C2

Ceq=C12=C22=kA2 ln(2ϵ0+kd2ϵ0)

Hence, (B) is the correct answer.

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