Question

A regular polygon of n sides is formed by bending a wire of total length $2\pi r$ which carries a current $i$. By letting $n\to \infty$, deduce the expression for the magnetic field at the centre of a circular coil.

Answer 1

The above magnetic field can be written as,
$B=\frac{{\mu }_{o}i}{2r}\frac{{(\frac{sin\pi /n}{\pi /n})}^2}{cos\left(\pi /n\right)}$

As, $n\to \infty ,\frac{\pi }{n}\to 0$

Hence, ${lim}_{\pi /n\to 0}(\frac{sin\pi /n}{\pi /n})\to 1$

and ${lim}_{\pi /n\to 0}\left(cos\pi /n\right)\to 1$

$\Rightarrow B=\frac{{\mu }_{o}i}{2r}$