Question

A wire $X$ is half the diameter and half the length of a wire $Y$ of similar material. The ratio of resistance of $X$ to that of $Y$ is

• A. 8 : 1
• B. 4 : 1
• C. 2 : 1
• D. 1 : 1

Answer 1

Answer: (C)

Resistance of a wire is $R=\frac{\rho l}{\pi r^2}=\frac{\rho l}{\pi (d/2{)}^2}$
where $\rho =$ resistivity, $l=$ length and $d=$ diameter of wire.
As both have same material so resistivity is same for both.
Thus, $R\propto \frac{l}{d^2}$
Here, $l_x=l_y/2$ and $D_x=D_y/2$
So, $\frac{R_x}{R_y}=\frac{l_x}{D_x^2}\times \frac{D_y^2}{l_y}$
$=\frac{\left(1_y/2\right)}{{\left(D_y/2\right)}^2}\times \frac{D_y^2}{1_y}$
$=\frac{4}{2}=\frac{2}{1}=2:1$