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Q.
A wíre 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 :-
$R _{0}=1 \Omega \,\,\,\, R _{1}=?$
$\ell_{0}=1 m \,\,\,\, \ell_{1}=1.25 m$
$A _{0}= A$
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 $( R )=\frac{\rho \ell}{ A }$
$\frac{ R _{0}}{ R _{1}}=\frac{\ell_{0} / A _{0}}{\rho \ell_{1} / A _{1}}$
$\frac{1}{ R _{1}}=\frac{\ell_{0}}{ A _{0}}\left(\frac{\ell_{0} A _{0}}{\ell_{1} \times \ell_{1}}\right) \quad R _{1}=\frac{\ell_{1}^{2}}{\ell_{0}^{2}}=1.5625 \Omega$
So $\%$ change in resistance
$=\frac{ R _{1}- R _{0}}{ R _{0}} \times 100 \%$
$=\frac{1.5625-1}{1} \times 100 \%$
$=56.25 \%$