A capacitor is made of a flat plate of area ' $A$ ' and the second plate has staircase like structure. Width…

A capacitor is made of a flat plate of area ' $A$ ' and the second plate has staircase like structure. Width of each stair is ' $a$ ' and its height is ' $b$ '. The capacity of the capacitor is
  1. $\frac{\epsilon_0 \mathrm{~A}}{4 \mathrm{~d}}\left[\frac{\mathrm{b}+2 \mathrm{~d}}{\mathrm{~b}}\right]$
  2. $\frac{\epsilon_0 A}{4 d}\left[\frac{b+2 d}{d+b}\right]$
  3. $\in_0 A\left[\frac{2 d+b}{d-b}\right]$
  4. $\frac{\epsilon_0 A}{2 d}\left[\frac{2 d+b}{d+b}\right]$

Solution

Above capacitor can be considered as parallel combination of two capacitors of different width $\mathrm{d}$ and $\mathrm{b}+\mathrm{a}$ with each have cross-section $\frac{\mathrm{A}}{2}$. $C=C_1+C_2=\frac{\varepsilon_0\left(\frac{A}{2}\right)}{d}+\frac{\varepsilon_0\left(\frac{A}{2}\right)}{d+b}=\frac{\varepsilon_0 A}{2 d}\left\{\frac{2 d+b}{(d+b)}\right\}$

Asked in: MHT CET 2022 (05 Aug Shift 2)

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