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Q.
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
Moving Charges and Magnetism
Solution:
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
$B_{\text {centre }}=\frac{\mu_{0} n I}{2 \pi R} \tan \frac{\pi}{n} .$
