Question

A loop carrying current $I$ has the shape of a regular polygon of $n$ sides. If $R$ is the distance from the center to any vertex, then the magnitude of the magnetic induction vector $\overrightarrow{B}$ at the center of the loop is:

• A. $n\frac{{\mu }_{0}I}{2\pi R}\tan \frac{\pi }{n}$
• B. $n\frac{{\mu }_{0}I}{2\pi R}\tan \frac{2\pi }{n}$
• C. $\frac{{\mu }_{0}I}{2R}$
• D. $\frac{{\mu }_{0}I}{\pi R}\tan \frac{\pi }{n}$

Answer 1

The correct option is A $n\frac{{\mu }_{0}I}{2\pi R}\tan \frac{\pi }{n}$

B due to one side is $B=\frac{{\mu }_{o}i(sin\alpha +sin\beta )}{4\pi d}$

$\alpha =\beta =\frac{\pi }{n}$

$d=Rcos\frac{\pi }{n}$

due to n side total magnetic field is nB

So $B=n\times \frac{{\mu }_{o}I}{2\pi R}tan\frac{\pi }{n}$