Question

A current $i$ is flowing in a hexagonal coil of side $a$ as shown in figure. The magnetic induction at the centre of the coil will be :

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

Answer 1

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

From figure,
$OA=r\frac{\sqrt{3}}{2}$
$r^2-r^2(\frac{3}{4})=\frac{a^2}{4}$
$r=a\Rightarrow OA=a\frac{\sqrt{3}}{2}$
$\Rightarrow$ magnetic field due to a straight line wire at distance $x=\frac{{\mu }_{0}I}{4\pi x}\left(\sin {\theta }_2-\sin {\theta }_1\right)$
$=\frac{{2\mu }_{0}I}{4\pi \left(a\sqrt{3}\right)}\left(\sin \left({30}^o\right)-\sin (-{30}^o)\right)$
Now due to 6 wires of hexagon $\Rightarrow 12\frac{{\mu }_{0}I}{4\sqrt{3}\pi a}(\frac{1}{2}-\left(\frac{-1}{2}\right))$
$=\frac{\sqrt{3}{\mu }_{0}I}{\pi a}$