Question

A regular polygon of n sides is formed by bending a wire of total length $2\pi r$ which carries a current $i$. Find the magnetic field $B$ at the centre of the polygon.

Answer 1

$a=\frac{1}{2}\left(\frac{2\pi r}{n}\right)=\frac{\pi r}{n}$

$x=acot\frac{\pi }{n}=\left(\frac{\pi r}{n}\right)=\left(\frac{pir}{n}\right)cot\left(\frac{\pi }{n}\right)$

$B=n[\frac{{\mu }_o}{4\pi }\frac{i}{x}(sin\frac{\pi }{n}+sin\frac{\pi }{n})]$

$=n[\frac{{\mu }_o}{4\pi }\frac{i}{\left(\pi r/n\right)cot\pi /n}\left(2sin\pi /n\right)]$

$=\frac{{\mu }_{o}i{n}^{2}sin\left(\frac{\pi }{n}\right)tan\left(\frac{\pi }{n}\right)}{2{\pi }^{2}r}$