Question

Identical metal plates are located in air at equal distance $d$ from one another. The area of each plate is equal to $A$. If the capacitance of the system between $P$ and $Q$ if the plates are interconnected as shown in the figure $x{\epsilon }_0\frac{A}{d}$. Find $x$

Answer 1

Above is the equivalent circuit including parallel plate capacitors $C1C2andC3$

The Capacitance of all the three parallel plate capacitors is equal and is given as $C1=C2=C3=C=\frac{{\epsilon }_{0}A}{d}$

As $C2andC3$ are connected in series the resultant capacitance is equal to $\frac{C}{2}$

The resultant capacitance is in parallel with $C1$ thus the resultance capacitance of the given circuit $C_{eq}=\frac{C}{2}+C=\frac{3C}{2}$

$\therefore C_{eq}=\frac{3{\epsilon }_{0}A}{2d}$

comparing with given $C_{eq}=\frac{x{\epsilon }_{0}A}{d}$ we get $x=\frac{3}{2}$

Thus $x=\frac{3}{2}$