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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<x≤d2) ϵ(x)=ϵ0+k(d−x), for (d2≤x≤d)

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=1A∫d20dxϵ0+kx 1C1=1kAln⎛⎜ ⎜ ⎜⎝ϵ0+kd2ϵ0⎞⎟ ⎟ ⎟⎠ Consider another element of width dx, at a distance x from the center (x>d2) dC2=A(ϵ0+k(d−x))dx Capacitance of the second half of the capacitor is, 1C1=∫dd21dC2=1A∫dd2dxϵ0+kd−kx 1C2=1kAln⎛⎜ ⎜ ⎜⎝ϵ0+kd2ϵ0⎞⎟ ⎟ ⎟⎠ As, 1C1=1C2 Ceq=C12=C22=kA2 ln(2ϵ0+kd2ϵ0) Hence, (B) is the correct answer.

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