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We have two wires A and B of the same mass and the same material. The diameter of wire A is half that of wire B. If the resistance of wire A is 24 a9, then the resistance of wire B will be

1. 12 a9
2. 3.0 a9
3. 1.5 a9
4. None of the above

Correct answer: 1.5 a9

Solution

For wires of the same material and mass, volume is the same, so \(LA\) is constant. Since diameter of A is half of B, its area is one-fourth, so its length is four times larger. Thus \(R \propto \frac{L}{A} \propto \frac{1/A}{A} = \frac{1}{A^2}\), giving \(R_A/R_B = (A_B/A_A)^2 = 4^2 = 16\). Hence \(R_B = 24/16 = 1.5\,\Omega\).

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