Question

A capacitor is made of a flat plate of area $A$ and a second plate having a stair-like structure as shown in figure. The width of each stair is $a$ and the height is $b$. Find the capacitance of the assembly.

Answer 1

In the given question, three capacitors are arranged in parallel

All having same surface area $a=\frac{A}{3}$

The capacitance of a capacitor is given by:

$C=\frac{{ϵ}_{o}A}{d}$

so,

$C_1=\frac{{ϵ}_{o}A}{3d}$

$C_2=\frac{{ϵ}_{o}A}{3(b+d)}$

$C_3=\frac{{ϵ}_{o}A}{3(2b+d)}$

The equivalent capacitance of the combination will be:

$C_{eq}=C_1+C_2+C_3$

$=\frac{{ϵ}_{o}A}{3}(\frac{1}{d}+\frac{1}{b+d}+\frac{1}{2b+d})$

$=\frac{{ϵ}_{o}A}{3d}\frac{(3d^2+6bd+2b^2)}{(b+d)(2b+d)}$