Three wires each of length L, cross-sectional area A and resistivity 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?
A. L_1 = 3L and A_1 = 2A
The resistance of a single wire of length L, cross-sectional area A, and resistivity is given by R = L A . Between points X and Y, there are two such wires connected in parallel. The equivalent resistance R_ XY is: 1 R_ XY = 1 R + 1 R = 2 R R_ XY = R 2 = L 2A Between points Y and Z, there is one such wire. Its resistance is R_ YZ = R = L A . The total resistance between X and Z is the series combination of R_ XY and R_ YZ : R_ XZ = R_ XY + R_ YZ = L 2A + L A = 3 L 2A The new wire has length L_1, area A_1, and the same resistivity . Its resistance is R_ new = L_1 A_1 . Since the new wire replaces the combination without changing the resistance, we have: R_ new = R_ XZ L_1 A_1 = 3 L 2A L_1 A_1 = 3L 2A Checking the given options: (a) L_1 = 3L, A_1 = 2A L_1 A_1 = 3L 2A (Matches) (b) L_1 = L, A_1 = A L_1 A_1 = L A (Incorrect) (c) L_1 = 2L, A_1 = 3A L_1 A_1 = 2L 3A (Incorrect) (d) L_1 = 2L, A_1 = 2A L_1 A_1 = L A (Incorrect) Answer: L_1 = 3L and A_1 = 2A
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