First slide

Magnetic field due to a straight current carrying wire

Question

A current f flows in 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

Difficult

A

$\tan \frac{π}{n} tesla$
B

$\frac{μ_{0}}{π} \frac{I}{R} [\tan \frac{π}{n}] tesla$
C

$\frac{μ_{0} I}{2 πR} [\tan \frac{π}{n}] tesla$
D

$\frac{μ_{0} I}{2 πR} [\cos \frac{π}{n}] tesla$

Solution

$\begin{array}{l} B = \frac{μ_{0} I}{4 πr} [2 \sin \frac{π}{n}] \\ But \cos \frac{π}{n} = \frac{r}{R} \end{array}$
$or r = R \cos \frac{π}{n}$
$\begin{array}{l} ∴ B = \frac{μ_{0} I}{4 πR \cos \frac{π}{n}} [2 \sin \frac{π}{n}] \\ ∴ B = \frac{μ_{0} I}{2 πR} [\tan \frac{π}{n}] \end{array}$

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