Question

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 induced at the centre of the polygon, due to its one side, is:

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

Answer 1

Answer: (C)

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

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