Question

A wire $P$ has a resistance of $ 20\,\Omega $ . Another wire $Q$ of same material but length twice that of $P$ has resistance of $8\,\Omega $ . If $r$ is the radius of cross-section of $P$, the radius of cross-section of $Q$ is

• A. $ r $
• B. $ \frac{r}{\sqrt{2}} $
• C. $ \sqrt{5}\,r $
• D. $ 2\,r $

Answer 1

Answer: (C)

Resistance $R=\frac{\rho l}{A}$
For wire P
$20=\frac{\rho l}{\pi r^{2}}$
Simiiarly, for wire $Q$
$8=\frac{\rho(2 l)}{\pi\left(r'\right)^{2}}$
Dividing Eq. (i) by Eq. (ii), we have
$\frac{20}{8}=\frac{\rho l}{\pi r^{2}} \times \frac{\pi\left(r'\right)^{2}}{\rho(2 l)}$
$\Rightarrow 5=\left(\frac{r^{\prime}}{r}\right)^{2}$
$\Rightarrow r'=\sqrt{5} r$