Question

Two wires made of same material have lengths in the ratio $1:2$ and their volumes in the same ratio. The ratio of their resistances is:

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

Answer 1

Answer: (D)

$1:4$

The resistance R is given by,
$R=\rho \left(\frac{l}{A}\right)$.
Now volume of the wire is $V=A\times l$ or $A=\frac{V}{l}$.
Using this in first equation:$R=\rho \left(\frac{l^2}{V}\right)$. So $R\propto l^2$ for the wires of same volume and same material.
$\Rightarrow \frac{R_1}{R_2}=\frac{l_1^2}{l_2^2}$
But $\frac{l_1}{l_2}=\frac{1}{2}$
$\Rightarrow \frac{R_1}{R_2}={\left(\frac{1}{2}\right)}^2=\frac{1}{4}$