First slide

Magnetic field due to a current carrying circular ring

Question

What will be the magnitude of magnetic induction at the centre oi a square loop of side l, if a current i is flowing in the loop ?

Difficult

A

$\frac{2 \sqrt{2} μ_{0} i}{πl}$
B

$\frac{μ_{0} i}{πl}$
C

$\frac{2 \sqrt{2} μ_{0} i}{l}$
D

$\frac{2 μ_{0} i}{πl}$

Solution

The magnetic induction due to the loop at O is
B= 4x B_AB
Here B_AB= $\frac{μ_{0} i}{4 π (l / 2)} [\sin 45^{\circ} + \sin 45^{\circ}]$
$\begin{aligned} = \frac{2 μ_{0} i}{4 πl} \times [\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}] = \frac{2 \sqrt{2} μ_{0} i}{4 πl} \\ ∴ B = 4 \times B_{AB} = \frac{4 \times 2 \sqrt{2} μ_{0} i}{4 πl} = \frac{2 \sqrt{2} μ_{0} i}{πl} \end{aligned}$

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