The ratio of lengths of two wires is 2 : 3 and the ratio of their res…

The correct answer is Option (4) → 1 : 1

Given:

Length ratio: $\frac{L_1}{L_2} = \frac{2}{3}$

Resistivity ratio: $\frac{\rho_1}{\rho_2} = \frac{3}{2}$

Same resistance: $R_1 = R_2$

Resistance of a wire: $R = \rho \frac{L}{A} = \rho \frac{4 L}{\pi d^2}$ (since $A = \pi d^2 /4$)

Equating resistances:

$\rho_1 \frac{L_1}{d_1^2} = \rho_2 \frac{L_2}{d_2^2}$

Substitute ratios:

$\frac{3}{2} \cdot \frac{2}{3} \cdot \frac{1}{d_1^2} = \frac{1}{d_2^2} \Rightarrow \frac{1}{d_1^2} = \frac{1}{d_2^2}$

Wait, calculate carefully:

$\frac{\rho_1 L_1}{d_1^2} = \frac{\rho_2 L_2}{d_2^2}$

Substitute values: $\rho_1 L_1 / \rho_2 L_2 = d_1^2 / d_2^2$

$\frac{3/2 \cdot 2/3}{1} = 1 \Rightarrow \frac{d_1^2}{d_2^2} = 1 \Rightarrow d_1 : d_2 = 1 : 1$

Answer: Ratio of diameters $d_1 : d_2 = 1 : 1$