The ratio of lengths of two wires is 2:3 and the ratio of their resistivities is 3:2. If they have the same resistance then the ratio of their respective diameters is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 1 : 1 \)
The resistance of a wire of length \( l \), resistivity \( \rho \) and diameter \( d \) is
\( \displaystyle R = \frac{\rho\,l}{A} = \frac{\rho\,l}{\pi d^{2}/4} = \frac{4\rho\,l}{\pi d^{2}} \)
For equal resistances, \( d^{2} \propto \rho\,l \):
\( \displaystyle \frac{d_1^{2}}{d_2^{2}} = \frac{\rho_1\,l_1}{\rho_2\,l_2} = \frac{3}{2} \times \frac{2}{3} = 1 \)
\( \displaystyle \frac{d_1}{d_2} = 1 \)
So the ratio of diameters is \( 1 : 1 \).