Question

Seven identical large conducting plates are placed parallel to each other. The separation between any two consecutive plates is a. The area of each plate is $S$. The effective capacitance of the system between A and B is $\frac{2x}{11}\frac{{ϵ}_{0}S}{a}$. Then the value of $x$ is :

Answer 1

There are $(7-1)=6$ capacitors and the capacitance of each capacitor is $C=\frac{S{ϵ}_0}{a}.$
Above 2 capacitors are in parallel and below 3 capacitors are in parallel and these two groups and middle one capacitor are in series as shown in the figure below.
The total effective capacitance of the system is
$\frac{1}{C_{eff}}=\frac{1}{(C+C)}+\frac{1}{C}+\frac{1}{(C+C+C)}=\frac{11}{6C}$

or $C_{eff}=\frac{6C}{11}=\frac{6}{11}.\frac{S{ϵ}_0}{a}.$

thus, $\frac{2x}{11}=\frac{6}{11}\Rightarrow x=3$