A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in figure. If the area of each stair is A/3 and the height is d, the capacitance of the arrangement is :

Solution

Physics sir · 4 steps

Each stair faces its own strip of the bottom plate: three capacitors in parallel, at gaps d, 2d and 3d.

1. Split the stairs. Each stair of area A/3 sits directly above its own strip of the flat plate, so it forms a separate parallel-plate capacitor. Measured from the bottom plate, the three gaps are d, 2d and 3d.
2. Parallel connection. All three stairs are part of one conductor, and so are the three strips below them. The three capacitors therefore share the same potential difference. That makes them parallel, so their capacitances add.

   C = (ε₀ A)/3d + (ε₀ A)/3(2d) + (ε₀ A)/3(3d) = (ε₀ A)/3d(1 + 1/2 + 1/3)

3. Add the fractions. 1 + 1/2 + 1/3 = (6 + 3 + 2)/6 = 11/6. So C = (ε₀ A)/3d × 11/6.
4. So. C = (11ε₀ A)/18d, which is option (d).

The idea to keep. When one plate is stepped, cut it into strips. Each strip faces the other plate across its own gap, and all strips share the same voltage, so the strips are in parallel and their capacitances add.

Answer from the NTA final answer key. Solution by PrepWiser.