Question

If a copper wire is stretched to make it $0.1\%$ longer. The percentage change in its resistance is

• A. $0.2\%$ increase
• B. $0.2\%$ decrease
• C. $0.1\%$ increase
• D. $0.1\%$ decrease

Answer 1

The correct option is A $0.2\%$ increase

For a given wire, the resistance to the electric current is given by,$$R=\frac{\rho L}{A}$$Since, $L\times A=\text{volume}=V=\text{constant}$

Therefore,
$R=\frac{\rho L}{\left(V/L\right)}=\frac{\rho L^2}{V}$

$\Rightarrow R\propto L^2$

$\Rightarrow \frac{\Delta R}{R}=2\frac{\Delta L}{L}$

$\therefore \frac{\Delta R}{R}=2\left(0.1\%\right)=0.2\%\text{increase}$

Hence, option (a) is the correct answer.

Why this question? Effect of percentage change in length of wire$$\frac{R_2}{R_1}=\frac{l^2{[1+\frac{X}{100}]}^2}{l^2}$$where, $l-\text{original length and X-\% increment}$ $\text{if X is quite small (say}<5\%)\text{then \% change in R is}$ $$\frac{R_2-R_1}{R_1}\times 100=\left[\frac{{(1+\frac{X}{100})}^2-1}{1}\right]\times 100\cong 2X\%$$