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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)

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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.

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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