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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$\frac{Q d}{3}$

$\frac{2 Q d}{3 ε_{0} A}$

$\frac{2 d}{ε_{0} A}$

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

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 $C_{2} = \frac{ε_{0} A}{2 d}$ .