Question

Two thin long parallel wires separated by a distance $b$ are carrying a current $i$ ampere each. The magnitude of the force per unit length exerted by one wire on the other, is

• A. $\frac{{\mu }_0{i}^2}{b^2}$
• B. $\frac{{\mu }_{0}i}{2\pi b^2}$
• C. $\frac{{\mu }_{0}i}{2\pi b}$
• D. $\frac{{\mu }_0{i}^2}{2\pi b}$

Answer 1

Answer: (D)

Let two long parallel thin wires $X$ and $Y$ carry current $i$ and separated by a distance $b$ apart.
The magnitude of magnetic field $\vec{B}$ at any point on $Y$ due to current $i_1$ in $X$ is given by

$B=\frac{{\mu }_0}{2\pi }\frac{i_1}{b}$
The magnitude of force acting on length $l$ of $Y$ is
$F=i_{2}Bl=i_2\left(\frac{{\mu }_0}{2\pi }\frac{i_1}{b}\right)l$
Force per unit length is
$\frac{F}{l}=\frac{{\mu }_0}{2\pi }\frac{i_1{i}_2}{b}$
Given, $i_1=i_2=i$, therefore,
$\frac{F}{l}=\frac{{\mu }_0}{2\pi }\frac{i^2}{b}$