A regular polygon of 6 sides is formed by bending a wire of length $4\pi$ metre. If an electric current of $4\pi\sqrt3\ \text{A}$ flows through the sides of the polygon, the magnetic field at the centre of the polygon would be $x\times10^{-7}\ \text{T}$. The value of $x$ is ______.
Numerical value type. Enter your answer.
Answer: 72
Side of the hexagon: $a = \dfrac{4\pi}{6} = \dfrac{2\pi}{3}\ \text{m}$. Distance of each side from the centre: $d = \dfrac{\sqrt3}{2}a = \dfrac{\pi}{\sqrt3}\ \text{m}$.
Each side subtends $30^\circ$ on either side of the perpendicular, so
$$B = 6\times\frac{\mu_0I}{4\pi d}(\sin30^\circ + \sin30^\circ) = \frac{6\times10^{-7}I}{d}$$
$$B = \frac{6\times10^{-7}\times4\pi\sqrt3}{\pi/\sqrt3} = 72\times10^{-7}\ \text{T} \Rightarrow x = 72$$
Solution by Sreeraj P, M.Sc Physics