Question

A wire bent in the shape of a regular $n$ polygonal loop carries a steady current $I$. Let $l$ be the perpendicular distance of a given segment and $R$ be the distance of a vertex both from the centre of the loop. The magnitude of the magnetic field at the centre of the loop is given by:

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

Answer 1

The correct option is A $\frac{n{\mu }_{0}I}{2\pi l}\sin \left(\pi /n\right)$

For n-sided regular polygon, magnetic field at centre due to one side is given by

$B=\frac{{\mu }_{0}i}{4\pi l}\times 2\sin \theta$
Where $2\theta =\frac{2\pi }{n}$

$B_0=\frac{{\mu }_{0}I}{2\pi l}sin\left(\frac{\pi }{n}\right)$

Hence due n sides it will be $B=n{B}_0$