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
$B=6 \times \frac{\mu_{0}}{4 \pi} \times \frac{1}{a \frac{\sqrt{3}}{2}}\left(\sin 30^{\circ}+\sin 30^{\circ}\right)$
$B =\frac{\sqrt{3} \mu_{0} l }{\pi a }$