A 1 m long wire is broken into two unequal parts X and Y. The X part of the wire is streched into another wire W. Length of W is twice the length of X and the resistance of W is twice that of Y. Find the ratio of length of X and Y.

Solution

Physics sir · 4 steps

Stretching keeps the volume fixed, so doubling the length makes the resistance four times, not two.

1. Same wire, same A. X and Y come from one wire, so they share the same resistivity ρ and cross-section A. So R_X = (ρ l_X)/A and R_Y = (ρ l_Y)/A.
2. Stretch X into W. Stretching does not change the volume lA. When the length doubles to 2l_X, the area halves to A/2. Length goes up by 2 and area goes down by 2, so the resistance goes up by 4.

   R_W = (ρ (2l_X))/(A/2) = (4ρ l_X)/A

3. Use the condition. R_W = 2R_Y gives (4ρ l_X)/A = (2ρ l_Y)/A. The ρ and A cancel, leaving 4l_X = 2l_Y.
4. So. l_X/l_Y = 2/4 = 1/2, so the ratio is 1 : 2, option (b).

The idea to keep. When a wire is stretched n times its length, the volume stays the same, so the area becomes 1/n times and the resistance becomes n² times.

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