First slide

Magnetic field due to current element

Question

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

Moderate

A

$\frac{μ_{0} i}{π R} (\tan \frac{π}{n})$
B

$\frac{μ_{0} i}{4 π R} (\tan \frac{π}{n})$
C

$\frac{μ_{0} i}{2 π R} (\tan \frac{π}{n})$
D

$\frac{μ_{0} i}{2 π R} (\cos \frac{π}{n})$

Solution

$B = \frac{μ_{0} I}{4 π r} [\sin \frac{π}{n} + \sin \frac{π}{n}]$

But $\cos π / n = \frac{r}{R}$ or $r = R \cos π / n$
$∴ B = \frac{μ_{0} I}{4 π R \cos π / n} [2 \sin π / n] ⇒ B = \frac{μ_{0} I}{2 π R} [\tan π / n]$

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