Four identical rectangular plates with length, l = 2 cm and breadth, b = 3/2 cm are arranged as shown in figure. The equivalent capacitance between A and C is (xε ₀)/d. The value of x is ______________. (Round off to the Nearest Integer)

Solution

Physics sir · 4 steps

The wire joins B and D, so the two capacitors on either side of C sit in parallel.

1. Three capacitors. Each neighbouring pair of plates forms a capacitor of area A = l × b = 2 × 3/2 = 3 cm² and gap d. So A–B, B–C and C–D each have C₀ = (ε₀ A)/d = 3ε₀/d, with the area kept in cm² as the answer format expects.
2. Same node. B and D are joined by a wire, so they are at the same potential. Both B–C and C–D then connect the same node (B, D) to C, so they are in parallel and give 2C₀.
3. Series with A–B. From A, charge reaches the (B, D) node through C₀, then goes on to C through 2C₀. These two are in series.

   C_(AC) = (C₀ × 2C₀)/(C₀ + 2C₀) = 2/3C₀ = 2/3 × 3ε₀/d = 2ε₀/d

4. So. Comparing with xε₀/d gives x = 2.

The common slip: 1. Ignoring the wire between B and D and putting all three capacitors in series gives C₀/3 = ε₀/d, so x = 1.

The idea to keep. In plate problems, first mark which plates are wired together, since they share one potential. Two capacitors joining the same pair of nodes are in parallel, even on opposite sides of a plate.

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