Question 63 of 380

A $1m$ long wire is broken into two unequal parts $X$ and $Y$. The $X$ part of the wire is stretched 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$Y$.

A$1:4$B$1:2$C$4:1$D$2:1$

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Answer: (B) $1:2$

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Resistance of a wire is given by $R=\dfrac{\rho L}{A}$, where, $\rho$ is conductivity, $L$ is length of wire and $A$ is cross-sectional area of wire.
For a single wire, $A$ and $\rho$ are constant.
Thus, for parts $X$ and $Y$, resistance, $\dfrac{{R}_{X}}{{R}_{Y}}=\dfrac{{l}_{X}}{{l}_{Y}}$.
When wire is stretched to double of its length, then resistance becomes four times, thus, ${R}_{W}=4{R}_{X}=2{R}_{Y}$.
Hence, the ratio is $\dfrac{{R}_{X}}{{R}_{Y}}=\dfrac{1}{2}$.
So, $\dfrac{{l}_{X}}{{l}_{Y}}=\dfrac{1}{2}$