Question

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

Answer 1

$2\theta =\frac{2\pi }{n}\Rightarrow \theta =\frac{\pi }{n}$

$\tan \theta =\frac{\ell }{2x}\Rightarrow x=\frac{\ell }{2\tan \theta }$

$\frac{\ell }{2}=\frac{\pi r}{n}$

$B_{AB}=\frac{{\mu }_{0}i}{4\pi \left(X\right)}(\sin \theta +\sin \theta )=\frac{{\mu }_{0}i2\tan \theta \times 2\sin \theta }{4\pi \ell }$

$=\frac{{\mu }_{0}i2\tan \left(\pi /n\right)2\sin \left(\pi /n\right)n}{4\pi 2\pi r}=\frac{{\mu }_{0}in\tan \left(\pi /n\right)\sin \left(\pi /n\right)}{2{\pi }^{2}r}$

For n sides , $B_{net}=\frac{{\mu }_{0}in\tan \left(\pi /n\right)\sin \left(\pi /n\right)}{2{\pi }^{2}r}$