A $1\,m$ long wire is broken into two unequal parts $X$ and $Y$ The $X$ part of the wire is streched into another wire $W$. Length of $W$ is twice the length of $X$ and the resistance of $W$ is twice that of $Y$. Find the ratio of length of $X$ and $Y$.
JEE MAIN 2022, Medium
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$\frac{ R _{ X }}{ R _{ Y }}=\frac{\ell_{ X }}{\ell_{ Y }}$
When wire is stretched to double of its length, then resistance becomes 4 times
$R _{ W }=4 R _{ X }=2 R _{ Y }$
$\frac{ R _{ X }}{ R _{ Y }}=\frac{1}{2}$
So. $\frac{\ell_{ X }}{\ell_{ y }}=\frac{1}{2}$
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