A parallel plate capacitor is made up of stair like structure with a plate area $A$ of each stair and that is connected with a wire of length $b$, as shown in the figure. The capacitance of the arrangement is $\dfrac{x}{15}\dfrac{\varepsilon_0 A}{b}$. The value of $x$ is ______ .
Numerical value type. Enter your answer.
Answer: 23
Each step of the upper plate faces a step of the lower plate. Counting risers of length $b$:
Right-hand steps: separation $b$.
Middle steps: separation $b + b + b = 3b$.
Left-hand steps: separation $5b$.
The three capacitors share the same two conductors, so they are in parallel:
$$C = \frac{\varepsilon_0A}{b} + \frac{\varepsilon_0A}{3b} + \frac{\varepsilon_0A}{5b} = \frac{15 + 5 + 3}{15}\frac{\varepsilon_0A}{b} = \frac{23}{15}\frac{\varepsilon_0A}{b}$$
$x = 23$.
Solution by Sreeraj P, M.Sc Physics