Question

A wire of radius r has resistance R. If it is stretched to a radius $\frac{r}{2}$, its resistance will be

• A. $16\text{R}$
• B. $2\text{R}$
• C. $4\text{R}$
• D. $0$

Answer 1

The correct option is A $16\text{R}$

Let length & cross sectional area of wire be $\ell$ & a respectively. $\rho$ be the specific resistance, then
$R=\rho \frac{\ell }{a}$
If radius becomes half, area becomes $\frac{1}{4}th$ or cross sectional area after the stretch $=\frac{a}{4}$. Let its length increases to ${\ell }^{\prime }$. Since volume remains the same in the process,
$\ell a={\ell }^{\prime }\times \frac{a}{4}\Rightarrow {\ell }^{\prime }=4\ell$
Let R' be the resistance of stretched wire,
$R^{\prime }=\rho \frac{4\ell }{a/4}=16\times \rho \frac{\ell }{a}=16R$