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

• Correct answer
  A

  $\mathrm{n} \frac{\mu_{0} \mathrm{I}}{2 \pi \mathrm{R}} \tan \frac{\pi}{\mathrm{n}}$
• B

  $\frac{\mu_{0} I}{2 R}$
• C

  $\mathrm{n} \frac{\mu_{0} \mathrm{I}}{2 \pi \mathrm{R}} \tan \frac{2 \pi}{\mathrm{n}}$
• D

  $\frac{\mu_{0} \mathrm{I}}{\pi \mathrm{R}} \tan \frac{\pi}{\mathrm{n}}$

Solution

$\begin{aligned} \mathrm{B}_{\text {net }} &=\mathrm{n} \times \mathrm{B}_{1} \\ &=\mathrm{n} \cdot \frac{\mu_{0}}{4 \pi} \cdot \frac{\mathrm{I}}{\mathrm{R} \cos \frac{\pi}{\mathrm{n}}} 2 \sin \frac{\pi}{\mathrm{n}} \end{aligned}$

Hence, the correct answer is option: 'A'.