Two wires $A$ and $B$ are made up of the same material and have the same mass. Wire A has radius of $2.0 \mathrm{~mm}$ and wire $B$ has radius of $4.0 \mathrm{~mm}$. The resistance of wire B is $2 \Omega$. The resistance of wire $\mathrm{A}$ is_______. $\Omega$.
JEE MAIN 2024, Diffcult
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$\because \mathrm{R}=\frac{\rho \ell}{A}=\frac{\rho \mathrm{V}}{\mathrm{A}^2}$
$\therefore \frac{\mathrm{R}_{\mathrm{A}}}{\mathrm{R}_{\mathrm{B}}}=\frac{\mathrm{A}_{\mathrm{B}}^2}{\mathrm{~A}_{\mathrm{A}}^2}=\frac{\mathrm{r}_{\mathrm{B}}^4}{\mathrm{r}_{\mathrm{A}}^4}$
$\Rightarrow \frac{\mathrm{R}_{\mathrm{A}}}{2}=\left[\frac{4 \times 10^{-3}}{2 \times 10^{-3}}\right]^4$
$\Rightarrow \mathrm{R}_{\mathrm{A}}=32 \Omega .$
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