A current I flows along a thin wire shaped as a regular polygon with n sides which can be inscribed into a circle of radius R. Find the magnetic induction at the centre of the polygon. Also, obtained expression at  $n\to \infty$.

In a regular polygon of n-sides, each of its side subtends an angle $\frac{2\pi }{n}$ at the centre of the polygon. The angle $\theta =\frac{\pi }{n}$ . If  $B_1$ is the field produced by each side of the polygon, then total magnetic induction 

$B=n{B}_1=n\frac{{\mu }_{0}I}{4\pi }\frac{[\sin \frac{\pi }{n}+\sin \frac{\pi }{n}]}{r}$

$where r=R\cos \frac{\pi }{n}$

$Thus, B=n\frac{{\mu }_0}{4\pi }I\times \frac{2\sin \pi /n}{R\cos \pi /n}=\frac{{\mu }_{0}nI}{2\pi R}\tan \frac{\pi }{n}$

$For n\to \infty , we can write$

$B=\frac{{\mu }_{0}I}{2R}\underset{n\to \infty }{\lim }\left[\frac{\tan \pi /n}{\pi /n}\right]=\frac{{\mu }_{0}I}{2R}$

Polygon of infinite sides is a circle, and so its magnetic induction at the centre is equal to that due to a circular loop.