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Four identical conducting plates are placed parallel to each other at equal separations as shown below:
Plates A and D are fixed while B and C are connected with the help of a conducting spring of spring constant k. Initially, the spring is in relaxed position and B and C are at rest, now the spring is compressed slightly by displacing B and C and then plates B and C are released to perform SHM. Now the capacitance of the system across PQ [ Mass of each plate is m]

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remain constant

Increase continuously

deceases continuously

is varying sinusoidal with frequency $\frac{1}{2 π} \sqrt{\frac{2 k}{m}}$

answer is D.

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Detailed Solution

Let us consider separation between any two plates to be x, and cross- section area of plate be S. As plates B and C are connected by a conducting spring, they won’t take part in the equivalent capacitance.

Capacitance of the system is

$C_{e q} = \frac{C_{1} C_{2}}{C_{1} + C_{2}}$

where

$C_{1} = C_{2} = \frac{{Aε}_{0}}{x}$

$\Rightarrow C_{e q} = \frac{A ε_{0}}{2 x}$

x varies with a frequency of

$\frac{1}{2 π} \sqrt{\frac{2 k}{m}}$

Therefore, frequency of capacitance is also

$\frac{1}{2 π} \sqrt{\frac{2 k}{m}}$

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