Question

A regular polygon of $n$ sides is formed by bending a wire of total length $2\pi r$ which carries a current $i$. If number of sides of a polygon so formed has infinite sides, then the magnetic field at the centre of a circular current is $\frac{{\mu }_{0}i}{mR}$. The value of $m$ is

Answer 1

We know that, for a wire bent into a shape to form a polygon of $n$ sides we have,

$B=\frac{{\mu }_{0}i}{2R}\frac{\tan \left(\pi /n\right)}{\pi /n}\frac{\sin \left(\pi /n\right)}{\pi /n}$

Now, if there is a $n$ sided polygon i.e. it is nothing but a ring which is formed.
So, we can write,

$B=\frac{{\mu }_{0}i}{2R}{\text{lim}}_{n\to \infty }\frac{\tan \left(\pi /n\right)}{\pi /n}\frac{\sin \left(\pi /n\right)}{\pi /n}$

But, ${\text{lim}}_{n\to \infty }\frac{\tan \left(\pi /n\right)}{\pi /n}\frac{\sin \left(\pi /n\right)}{\pi /n}=1$

i.e. $B=\frac{{\mu }_{0}i}{2R}$