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A $1 \mathrm{~m}$ long wire is broken into two unequal parts $\mathrm{X}$ and $\mathrm{Y}$ . The $\mathrm{X}$ part of the wire is streched into another wire $\mathrm{W}$ . Length of $\mathrm{W}$ is twice the length of $\mathrm{ X}$ and the resistance of $\mathrm{W}$ is twice that of $\mathrm{Y}$ . Find the ratio of length of $\mathrm{X}$ and $\mathrm{Y}$ .
Option: 1
$1:4$

Option: 2
$1:2$

Option: 3
$4:1$

Option: 4
$2:1$

Answers (1)

best_answer

$\mathrm{R_{w}=2R_{Y}\left ( Given \right )}$

$\mathrm{\frac{R_{x}}{R_{Y}}=\frac{\ell}{1-\ell}}\rightarrow (1)$

$\mathrm{\frac{R_{W}}{R_{X}}=\frac{\left (2\ell \right )^{2}}{\left (\ell \right )^{2}}=4}\rightarrow (2)$

From the above equations

$\mathrm{R_W=4 R_X =2 R_Y }$

$\mathrm{\frac{R_X}{R_Y}=\frac{1}{2} =\frac{\ell}{1-\ell} }$

$\mathrm{1-\ell =2 \ell }$

$\mathrm{3 \ell=1 }$

$\mathrm{\ell=\frac{1}{3} }$

$\mathrm{\frac{\ell_x}{\ell_Y}=\frac{\ell}{1-\ell} =\frac{1 / 3}{2 / 3}=\frac{1}{2}}$

Hence 2 is correct option

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