Question

Two wires have their lengths in the ratio $1:3$ diameters in the ratio $2:3$ and their materials have specific resistances in the ratio $2:1$. Then the ratio of their resistances is :

• A. $1:3$
• B. $3:1$
• C. $2:3$
• D. $3:2$

Answer 1

Answer: (D)

$3:2$

The resistance $R$ is given by:

$R=\frac{\rho l}{A}=\frac{\rho l}{\pi r^2}$

So, $\frac{R_1}{R_2}=\frac{\left(\frac{{\rho }_1{l}_1}{\pi r_1^2}\right)}{\left(\frac{{\rho }_2{l}_2}{\pi r_2^2}\right)}$
$\frac{R_1}{R_2}=\frac{{\rho }_1{l}_1{r}_2^2}{{\rho }_2{l}_2{r}_1^2}$
But $\frac{{\rho }_1}{{\rho }_2}=\frac{2}{1},\frac{l_1}{l_2}=\frac{1}{3},\frac{r_1}{r_2}=\frac{2}{3}$
$\Rightarrow \frac{R_1}{R_2}=\frac{2}{1}\times \frac{1}{3}\times {\left(\frac{3}{2}\right)}^2=\frac{3}{2}$