Question

A current is flowing in a hexagonal coil of side a shown in figure. Find the magnetic field induction at the center of the coil

• A. $\frac{\sqrt{2}{\mu }_{0}I}{\pi a}$
• B. $\frac{\sqrt{3}{\mu }_{0}I}{\pi a}$
• C. $\frac{2\pi a}{\sqrt{3}I}$
• D. $\frac{{\mu }_{0}I}{\pi a}$

Answer 1

Answer: (B)

Here $\angle EOD=60^{\circ }$
$\angle EON=30^{\circ }=\angle NOD$
$OE=OD=ED=a$
$ON=OE\cos 30^{\circ }=a\times \frac{\sqrt{3}}{2}$

Total magnetic field induction at $O$ due to current through all the six sides of hexagonal is
$B=6\times \frac{{\mu }_0}{4\pi }\times \frac{1}{a\frac{\sqrt{3}}{2}}\left(\sin 30^{\circ }+\sin 30^{\circ }\right)$
$B=\frac{\sqrt{3}{\mu }_{0}l}{\pi a}$