Question

An infinitely long wire carries a current $I$ flowing towards positive $z$ - axis. Consider a circle in $x-y$ plane with centre at $(2\text{m},0,0)$ and radius $1\text{m}$ as shown in figure.

The maximum value of path integral $\oint \overrightarrow{B}.\overrightarrow{dl}$ of the magnetic field $\overrightarrow{B}$ due to current carrying wire between two points $\text{P}$ & $\text{Q}$ lying along the circular path is

• A. $\frac{{\mu }_{0}I}{12}$
• B. $\frac{{\mu }_{0}I}{8}$
• C. $\frac{{\mu }_{0}I}{6}$
• D. Zero

Answer 1

The correct option is C $\frac{{\mu }_{0}I}{6}$

Two point $\text{P}$ and $\text{Q}$ will be equidistant from the current carrying wire placed at origin due to the symmetry.

Also, the distance of any point on the arc $\text{PQ}$ from wire position $\text{O}$ will be equal.

The magnetic field due to wire will be along clockwise direction on the arc $\text{PQ}$.

Magnetic field due to wire,

$B=\frac{{\mu }_{0}I}{2\pi d}=\frac{{\mu }_{0}I}{2\pi \left(OP\right)}$

Now $\oint \overrightarrow{B}.d\overrightarrow{l}=\oint Bdl\cos 0^{\circ }$

or, $\oint \overrightarrow{B}.d\overrightarrow{l}=B\oint dl$

Here $\oint dl$ is the length of arc $\text{PQ}$

$\text{PQ}=\left(\text{OP}\right)\left(2\theta \right)$

Thus, $\oint \overrightarrow{B}.d\overrightarrow{l}=\frac{{\mu }_{0}I}{2\pi \left(OP\right)}\times \left(OP\right)\left(2\theta \right).........\left(1\right)$

For the maximum value of $\oint \overrightarrow{B}.d\overrightarrow{l}$, according to equation, the value of $2\theta$ should be maximum. It can happen only if $\text{P}$ and $\text{Q}$ are at position such that $\text{OP}$ & $\text{OQ}$ are tangents to the circle.


$\sin \theta =\frac{PC}{OC}$

$\Rightarrow \sin \theta =\frac{1}{2}$

$\therefore \theta ={30}^{\circ }$

Hence, $2\theta ={60}^{\circ }\Rightarrow 2\theta =\frac{\pi }{3}$

The maximum value of $\oint \overrightarrow{B}.\overrightarrow{dl}$ is given by,

$(\oint \overrightarrow{B}.\overrightarrow{dl}{)}_{max}=\frac{{\mu }_{0}I}{2\pi }\times \frac{\pi }{3}=\frac{{\mu }_{0}I}{6}$