Question

If the area of each plate is $A$ (figure) and the successive separations are $d,2d$ and $3d$, the equivalent capacitance across $A$ and $B$ is $\frac{{\epsilon }_{0}A}{xd}$. Find $x$ :

Answer 1

The equivalent circuit is shown in figure.
Here, $C_1=\frac{A{ϵ}_0}{d}$ and $C_2=\frac{A{ϵ}_0}{3d}=\frac{C_1}{3}$
As they are in series so equivalent capacitance is $C_{AB}=\frac{C_1{C}_2}{C_1+C_2}=\frac{C_1\left(C_1/3\right)}{C_1+\left(C_1/3\right)}=\frac{C_1}{4}=\frac{A{ϵ}_0}{4d}$