Three wires each of length L, cross-sectional area A and resistivity \rho are connected as shown in the figure. These are to be replaced by another wire of same resistivity such that the resistance between points X and Z does not change. If L_1 is the length and A_1 is the cross-sectional area of the new wire, then which one among the following is correct?

  1. A. L_1 = 3L \text{ and } A_1 = 2A ✓
  2. B. L_1 = L \text{ and } A_1 = A
  3. C. L_1 = 2L \text{ and } A_1 = 3A
  4. D. L_1 = 2L \text{ and } A_1 = 2A

Correct Answer: A. L_1 = 3L \text{ and } A_1 = 2A

Explanation

Two wires in parallel between X and Y have resistance R/2. The single wire between Y and Z has resistance R. Total resistance is 1.5R = \frac{3}{2} \rho \frac{L}{A}. Substituting L_1 = 3L and A_1 = 2A yields the exact same resistance.

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