Question

If a copper wire is stretched to increase its length by 0.1% then percentage increase in its resistance will be

• A. $0.2$
• B. $2$
• C. $1$
• D. $0.1$

Answer 1

The correct option is A $0.2$

The resistance of the wire is given by

$R=\frac{\rho L}{A}$

when wire is stretched to increase its length by 0.1% the new length is

$L^{\prime }=L+\frac{0.1}{100}L=1.001L$

The volume of wire remains same hence

$V=AL=A^{\prime }{L}^{\prime }$

where, A' is the new area of cross-section of wire.

$A^{\prime }=\frac{L}{L^{\prime }}\Rightarrow A^{\prime }=\frac{A}{1.001}$

Now, the changed resistance is

$R^{\prime }=\frac{\rho L^{\prime }}{A^{\prime }}$

$R^{\prime }=\frac{\rho \left(1.001\right)L}{\frac{A}{1.001}}$

$R^{\prime }=\frac{\rho L}{A}\left(1.001\right)\left(1.001\right)$

$R^{\prime }=R\left(1.002\right)$

$\frac{R^{\prime }}{R}=\frac{1.002}{1}$

$\frac{R^{\prime }-R}{R}=\frac{1.002-1}{1}$

$\frac{R^{\prime }-R}{R}=0.002$

Percent increase in resistance is

$\frac{R^{\prime }-R}{R}\times 100=0.002\times 100=0.2$%