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A material ' $B$ ' has twice the specific resistance of ' $A$ '. A circular wire made of ' $B$ ' has twice the diameter of a wire made of ' $A$ '. Then for the two wires to have the same resistance, the ratio $\ell_A / \ell_B$ of their respective lengths must be
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$\mathrm{R}_1=\frac{\rho_{\mathrm{A}} \ell_{\mathrm{A}}}{\pi \mathrm{R}_{\mathrm{A}}^2} \quad \mathrm{R}_2=\frac{\rho_{\mathrm{B}} \ell_{\mathrm{B}}}{\pi \mathrm{R}_{\mathrm{B}}^2}$
$\frac{\ell_{\mathrm{A}}}{\ell_{\mathrm{B}}}=\frac{\rho_{\mathrm{B}} \mathrm{R}_{\mathrm{A}}^2}{\rho_{\mathrm{A}} \mathrm{R}_{\mathrm{B}}^2}=\frac{2 \rho_{\mathrm{A}} \mathrm{R}_{\mathrm{A}}^2}{\rho_{\mathrm{A}} \cdot 4 \mathrm{R}_{\mathrm{A}}^2} \Rightarrow \frac{\ell_{\mathrm{B}}}{\ell_{\mathrm{A}}}=2$
$\frac{\ell_{\mathrm{A}}}{\ell_{\mathrm{B}}}=\frac{\rho_{\mathrm{B}} \mathrm{R}_{\mathrm{A}}^2}{\rho_{\mathrm{A}} \mathrm{R}_{\mathrm{B}}^2}=\frac{2 \rho_{\mathrm{A}} \mathrm{R}_{\mathrm{A}}^2}{\rho_{\mathrm{A}} \cdot 4 \mathrm{R}_{\mathrm{A}}^2} \Rightarrow \frac{\ell_{\mathrm{B}}}{\ell_{\mathrm{A}}}=2$
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