A capacitor is made of two square plates each of side 'a' making a very small angle α between them, as shown in figure. The capacitance will be close to :

Solution

Physics sir · 4 steps

The gap changes along the plate, so slice it into strips of constant gap and add them in parallel.

1. Gap at a point. Measure x from the edge where the plates are closest. Because α is very small, tan α ≈ α, so the gap there is d + α x.
2. One thin strip. A strip of width dx runs across the full side a, so its area is a dx. Its separation is almost constant, so dC = (ε₀ a dx)/(d + α x). All the strips share the same two plates, so they are in parallel and their capacitances add.

   C = ∫₀ᵃ (ε₀ a dx)/(d+α x) = (ε₀ a)/α ln(1+(α a)/d)

3. Expand the log. Since (α a)/d is small, use ln(1+y) ≈ y - y²/2. This gives C ≈ (ε₀ a)/α((α a)/d - (α² a²)/2d²).
4. So. C ≈ (ε₀ a²)/d(1 - (α a)/2d), which is option (c).

The idea to keep. When the plate separation varies, slice the capacitor into strips of constant gap. Strips on the same plates are in parallel, so integrate dC, then expand to first order.

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