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Q.

Two large parallel metal plates are oriented horizontally and separated by a distance $3 d$ . A grounded conducting wire joins them, and initially each plate carries no charge. Now a third identical plate carrying charge $Q$ is inserted between the two plates, parallel to them and located a distance $d$ from the upper plate, as in Figure.
What potential difference appears between the middle plate and each of the other plates? Each plate has area $A$ .

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a

$\frac{Q d}{3 A}$

b

$\frac{Q d}{3}$

c

$\frac{2 Q d}{3 ε_{0} A}$

d

$\frac{2 d}{ε_{0} A}$

answer is C.

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Detailed Solution

Imagine the centre plate is split along its midplane and pulled apart. We have two capacitors total in parallel supporting the same $△ V$ and carrying total charge $Q$ . The upper has a capacitance $C_{1} = \frac{ε_{0} A}{d}$ and the lower has a capacitance $C_{2} = \frac{ε_{0} A}{2 d}$ .

Charge flows from ground onto each of the outside plates of capacitors, so that

$Q_{1} + Q_{2} = Q$ ……(1)

and $△ V_{1} = △ V_{2} = △ V$

Question Image

$\Rightarrow \frac{Q_{1}}{C_{1}} = \frac{Q_{2}}{C_{2}} \Rightarrow \frac{Q_{1} d}{ε_{0} A} = \frac{Q_{2} 2 d}{ε_{0} A} \Rightarrow Q_{1} = 2 Q_{2}$

From (1), we get

$2 Q_{2} + Q_{2} = Q \Rightarrow Q_{2} = \frac{Q}{3}$

$△ V = \frac{Q_{1}}{C_{1}} = \frac{2 Q d}{3 ε_{0} A}$

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