Question

A polygon shaped wire is inscribed in a circle of radius $R$. The magnetic induction at the centre of polygon, when current flows through the wire is

• A. $\frac{{\mu }_{0}nI}{2\pi R}\tan \left(\frac{2\pi }{n}\right)$
• B. $\frac{{\mu }_{0}nI}{2\pi R}\tan \left(\frac{4\pi }{n}\right)$
• C. $\frac{{\mu }_{0}nI}{2\pi R}\tan \left(\frac{\pi }{n}\right)$
• D. $\frac{{\mu }_{0}nI}{2\pi R}\tan \left(\frac{\pi }{n^2}\right)$

Answer 1

Answer: (C)

Angle subtended at centre by any of side $=\frac{2\pi }{n}$
$\Rightarrow 2\theta =\frac{2\pi }{n};\theta =\frac{\pi }{n}$
Field due to one side,
$B_1=\frac{{\mu }_{0}I}{4\pi r}(\sin \theta +\sin \theta )$
But, $r=R\cos \theta =R\cos \frac{\pi }{n}$ and $\sin \theta =\sin \frac{\pi }{n}$
$\therefore B_1=\frac{{\mu }_{0}I}{4\pi R\cos \frac{\pi }{n}}\times 2\sin \frac{\pi }{n}=\frac{{\mu }_{0}I}{2\pi R}\tan \frac{\pi }{n}$
and so field on $n$ sides at centre will add up to form net field
$B_{\text{centre }}=\frac{{\mu }_{0}nI}{2\pi R}\tan \frac{\pi }{n}.$