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, $\frac{R_{x}}{R_{y}}=\frac{l_{x}}{D_{x}^{2}} \times \frac{D_{y}^{2}}{l_{y}}$
$=\frac{\left(1_{y} / 2\right)}{\left(D_{y} / 2\right)^{2}} \times \frac{D_{y}^{2}}{1_{y}}$
$=\frac{4}{2}=\frac{2}{1}=2: 1$