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
$F=i_{2} B l=i_{2}\left(\frac{\mu_{0}}{2 \pi} \frac{i_{1}}{b}\right) l$
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,
$\frac{F}{l}=\frac{\mu_{0}}{2 \pi} \frac{i^{2}}{b}$