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. $\frac{\mu_{0} n i}{2 \pi a} \tan \frac{\pi}{n}$
• C. $\frac{2}{\pi} \frac{n i}{a} \mu_{0} \tan \frac{\pi}{n}$
• D. $\frac{n i}{2 a} \mu_{0} \tan \frac{\pi}{n}$

Answer 1

Answer: (B)

Magnetic field at the centre due to one side
$B_{1}=\frac{\mu_{0}}{4 \pi} \cdot \frac{2 i \sin \theta}{r}$
where $r=a \cos \theta$
So $B_{1}=\frac{\mu_{0}}{4 \pi} \cdot \frac{2 i \sin \theta}{a \cos \theta}=\frac{\mu_{0} i}{2 \pi a} \tan \theta$
Hence net magnetic field
$B_{\text {net }}=n \times \frac{\mu_{0} i}{2 \pi a} \tan \frac{\pi}{n}$