question_answer

A current I flows through a thin wire shaped as regular polygon of n sides which can be inscribed in a circle of radius R. The magnetic field induction at the center of polygon due to one side of the polygon is  

A) \[\frac{{{\mu }_{0}}I}{\pi R}\left( \tan \frac{\pi }{n} \right)\]

B) \[\frac{{{\mu }_{0}}I}{4\pi R}\left( \tan \frac{\pi }{n} \right)\]

C) \[\frac{{{\mu }_{0}}I}{2\pi R}\left( \tan \frac{\pi }{n} \right)\]

D) \[\frac{{{\mu }_{0}}I}{2\pi R}\left( \cos \frac{\pi }{n} \right)\]

Correct Answer: C

Solution :

[c] \[B=\frac{{{\mu }_{0}}I}{4\pi r}\left[ 2\,\sin \frac{\pi }{n} \right]\] \[\cos \frac{\pi }{n}=\frac{r}{R}\] But or \[r=R\cos \frac{\pi }{n}\] \[\therefore \,\,\,B=\frac{{{\mu }_{0}}I}{4\pi \,R\,\cos \frac{\pi }{n}}\left[ 2\,\sin \frac{\pi }{n} \right]=\frac{{{\mu }_{0}}I}{2\pi R}\left[ \tan \,\frac{\pi }{n} \right]\]