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A parallel plate capacitor is made up of a stair-like structure with a plate area A of each stair, the steps being of size b: the upper plate is a staircase descending in steps of b and the lower plate a staircase rising in steps of b, so the three facing pairs of stairs, each of area A, are separated by 5b, 3b and b respectively and act in parallel. The capacitance of the arrangement is x/(15)(ε₀ A)/b. The value of x is ______.

Asked in JEE Main 27th June 2nd Shift 2022 · Capacitance of a conductor and a capacitor

Figure: Capacitance of a conductor and a capacitor
Answer: 23

Step-by-step solution

Each facing pair of stairs is a parallel plate capacitor of area A and its own separation, and all three share the same pair of terminals, so they are in parallel and simply add.

C=(ε₀ A)/(5b)+(ε₀ A)/(3b)+(ε₀ A)/b.

=(ε₀ A)/b(1/5+1/3+1)=(ε₀ A)/b·(3+5+15)/(15)=(23)/(15)(ε₀ A)/b.

x=23.

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