Question

The length of a given cylindrical wire is increased to double of its original length. The percentage increase in the resistance of the wire will be ___$\%$.

Answer 1

$V ^{\prime}= V$
$\ell^{\prime} A =\ell A$
$2 \ell A ^{\prime}=\ell A$
$A ^{\prime}=\frac{ A }{2}$
$R =\rho \frac{\ell}{ A } \ldots$ (i)
$\ell^{\prime}=2 \ell$
$A ^{\prime}=\frac{ A }{2}$
$R ^{\prime}=\frac{\rho \ell^{\prime}}{ A ^{\prime}}=\frac{\rho 2 \ell}{\frac{ A }{2}}$
$R ^{\prime}=\frac{4 \rho \ell}{ A }$
$R^{\prime}=4 R$ from equation (i)
$\%$ increase in resistance
$=\frac{ R ^{\prime}- R }{ R } \times 100=\frac{4 R - R }{ R } \times 100 $
$=300 \%$