A material '$B$' has twice the specific resistance of '$A$'. A circular wire made of '$B$' has twice the diameter ofa wire made of '$A$'. then for the two wires to have the same resistance, the ratio $\frac{{{l_B}}}{{{l_A}}}$ of their respective lengths must be
AIEEE 2006, Medium
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$\rho_{B}=2 \rho_{A}$
$d_{B}=2 d_{A}$
$R_{B}=R_{A} \Rightarrow \frac{\rho_{B} \ell_{B}}{A_{B}}=\frac{\rho_{A} \ell_{A}}{A_{A}}$
$\therefore \frac{\ell_{B}}{\ell_{A}}=\frac{\rho_{A}}{\rho_{B}} \times \frac{d_{B}^{2}}{d_{A}^{2}}=\frac{\rho_{A}}{2 \rho_{A}} \times \frac{4 d_{A}^{2}}{d_{A}^{2}}=2$
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