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 x/15(∈ ₀A)/b. The value of x is ____________.

Solution

Physics sir · 3 steps

Pair each top stair with the bottom stair directly below it: three capacitors with gaps b, 3b and 5b.

1. Read the gaps. Measure every height from the top stair. The upper plate has stairs at 0, b and 2b down. The lower plate has stairs at 3b, 4b and 5b down, with its highest stair on the right.

   Column	Upper stair	Lower stair	Gap	Capacitance
   Left	0	5b	5b	(ε₀ A)/5b
   Middle	b	4b	3b	(ε₀ A)/3b
   Right	2b	3b	b	(ε₀ A)/b

2. Parallel combination. All the upper stairs are joined by wire, so they form one conductor at one potential. The same holds for the lower stairs. So all three capacitors share the same two terminals and are in parallel, and their capacitances add.

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

3. So. Compare this with x/15(ε₀ A)/b. That gives x = 23.

The idea to keep. When stepped plates are each one connected conductor, every facing pair of stairs is a separate capacitor across the same two terminals. Find each gap, then add the capacitances in parallel.

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