Question

A current $i$ flows through a thin wire shaped as a regular polygon of $n$ sides which can be inscribed in a circle of radius $R$. The magnetic field induction at the centre of the polygon due to one side of the polygon is:

• A. $\frac{{\mu }_{o}i}{2\pi r}.tan(\frac{\pi }{n})$
• B. $\frac{{\mu }_{o}i}{4\pi r}.tan(\frac{\pi }{n})$
• C. $\frac{{\mu }_{o}i}{\pi r}.tan(\frac{\pi }{n})$
• D. None of these

Answer 1

The correct option is A $\frac{{\mu }_{o}i}{2\pi r}.tan(\frac{\pi }{n})$

$B_{oneside}=\frac{{\mu }_{o}i}{4\pi r}(sin\alpha +sin\beta )-E{q}^n\left(1\right)$
Here, $\alpha =\beta =\frac{\pi }{n}$
$cos\frac{\pi }{n}=\frac{r}{R}$
Hence, $r=R\times cos\frac{\pi }{n}$
Substituting these values in (1),
we get
$B=\frac{{\mu }_{o}i}{\pi r}.tan(\frac{\pi }{n})$