Four identical thin, square metal sheets, and , each of side are kept parallel to each other with equal distance between them, as shown in the figure. Let where is the permittivity of free space. [2024]

Question

Four identical thin, square metal sheets, $S_{1}, S_{2}, S_{3}$ and $S_{4}$ , each of side $a$ are kept parallel to each other with equal distance $d (d ≪ a)$ between them, as shown in the figure. Let $C = \frac{ε_{0} a^{2}}{d},$ where $ε_{0}$ is the permittivity of free space.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option. [2024]

	List-I		List-II
(P)	The capacitance between $S_{1}$ and $S_{4}$ , with $S_{2}$ and $S_{3}$ not connected, is	(1)	$3 C_{0}$
(Q)	The capacitance between $S_{1}$ and $S_{4}$ , with $S_{2}$ shorted to $S_{3}$ , is	(2)	$\frac{C_{0}}{2}$
(R)	The capacitance between $S_{1}$ and $S_{3}$ , with $S_{2}$ shorted to $S_{4}$ , is	(3)	$\frac{C_{0}}{3}$
(S)	The capacitance between $S_{1}$ and $S_{2}$ , with $S_{3}$ shorted to $S_{1}$ and $S_{2}$ shorted to $S_{4}$ , is	(4)	$\frac{2 C_{0}}{3}$
		(5)	$2 C_{0}$

Smart Achievers · Accepted Answer

(3)

For P, the capacitance between $S_{1}$ and $S_{4}$ , with $S_{2}$ and $S_{3}$ not connected, all are in series.

$\frac{1}{C_{eq}} = \frac{1}{C_{1}} + \frac{1}{C_{2}} + \frac{1}{C_{3}} + \dots + \frac{1}{C_{n}}$

$∴$ $\frac{1}{C_{eq}} = \frac{1}{C_{0}} + \frac{1}{C_{0}} + \frac{1}{C_{0}} = \frac{3}{C_{0}}$

$∴$ $C_{eq} = \frac{C_{0}}{3}$

$S_{4}$

$C_{eq} = \frac{C_{0}}{2}$

$For R,$

$C_{eq} = \frac{2 C}{3}$

$For S,$

$C_{eq} = 3 C_{0}$

Solution

1 P → 3; Q → 2; R → 4; S → 5

2 P → 2; Q → 3; R → 2; S → 1

3 P → 3; Q → 2; R → 4; S → 1

4 P → 3; Q → 2; R → 2; S → 5

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