Question

Determine the magnitude of magnetic field at the center $\left(P\right)$ of a regular hexagon shaped current loop of side $L$.

• A. $\frac{\sqrt{3}{\mu }_{0}i}{\pi L}$
• B. $\frac{3\sqrt{3}{\mu }_{0}i}{\pi L}$
• C. $\frac{3{\mu }_{0}i}{\pi L}$
• D. $\frac{{\mu }_{0}i}{3\pi L}$

Answer 1

The correct option is A $\frac{\sqrt{3}{\mu }_{0}i}{\pi L}$

The magnetic field due to a finite current carrying conductor at a point is given by,


$B=\frac{{\mu }_{0}i}{4\pi d}(\sin \alpha +\sin \beta )$

For one side $\left(AB\right)$ of the given loop,
$\alpha =\beta ={30}^{\circ }$ because $△ABP$ is equilateral triangle as polygon is regular hexagon.

Also, $d=AP\cos {30}^{\circ }=\frac{\sqrt{3}L}{2}$


So,

$B=\frac{{\mu }_{0}i}{4\pi \left(\sqrt{3}L/2\right)}(\sin {30}^{\circ }+\sin {30}^{\circ })=\frac{{\mu }_{0}i}{2\sqrt{3}\pi L}$

Now, due to all the sides, net magnetic field,

$B_{\text{net}}=\frac{{\mu }_{0}i}{2\sqrt{3}\pi L}\times 6=\frac{\sqrt{3}{\mu }_{0}i}{\pi L}$

Hence, option $\left(A\right)$ is the correct answer.

Why this Question? To understand the concept of magnetic field at a point due to number of straight conductors.