Question

In the following figure a wire bent in the form of a regular polygon of $n$ sides is inscribed in a circle of radius $a$. Net magnetic field at centre will be

• A. $\frac{{\mu }_{0}i}{2\pi a}\tan \frac{\pi }{n}$
• B. $\frac{{\mu }_{0}ni}{2\pi a}\tan \frac{\pi }{n}$
• C. $\frac{2}{\pi }\frac{ni}{a}{\mu }_0\tan \frac{\pi }{n}$
• D. $\frac{ni}{2a}{\mu }_0\tan \frac{\pi }{n}$

Answer 1

Answer: (B)

Magnetic field at the centre due to one side
$B_1=\frac{{\mu }_0}{4\pi }\cdot \frac{2i\sin \theta }{r}$
where $r=a\cos \theta$
So $B_1=\frac{{\mu }_0}{4\pi }\cdot \frac{2i\sin \theta }{a\cos \theta }=\frac{{\mu }_{0}i}{2\pi a}\tan \theta$
Hence net magnetic field
$B_{\text{net }}=n\times \frac{{\mu }_{0}i}{2\pi a}\tan \frac{\pi }{n}$