A wire of $1 \,\Omega$ has a length of $1\, m$. It is stetched till its length increases by $25\, \%$. The percentage change in resistance to the neartest integer is .....$\%$
JEE MAIN 2021, Diffcult
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$R _{0}=1\, \Omega$

$\ell_{0}=1\, m$

$A _{0}= A$

$R _{1}=?$

$\ell_{1}=1.25\, m$

As volume of wire remains constant so

$A _{0} \ell_{0}= A _{1} \ell_{1} \Rightarrow A _{1}=\frac{\ell_{0} A _{0}}{\ell_{1}}$

Now

Resistance $( R )=\frac{\rho \ell}{ A }$

$\frac{ R _{0}}{ R _{1}}=\frac{\ell_{0} / A _{0}}{\rho \ell_{1} / A _{1}}$

$\frac{1}{ R _{1}}=\frac{\ell_{0}}{ A _{0}}\left(\frac{\ell_{0} A _{0}}{\ell_{1} \times \ell_{1}}\right) R _{1}=\frac{\ell_{1}^{2}}{\ell_{0}^{2}}=1.5625\, \Omega$

So \% change in resistance

$=\frac{ R _{1}- R _{0}}{ R _{0}} \times 100\, \%$

$=\frac{1.5625-1}{1} \times 100 \,\%$

$=56.25\, \%$

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