Question

A current I flows in the anticlockwise direction through a square loop of side a lying in the xoy plane with its center at the origin. The magnetic induction at the center of the square loop is

• A. $\frac{2 \sqrt{2} \mu_0 I}{\pi a }\hat{e}_x$
• B. $\frac{2 \sqrt{2} \mu_0 I}{\pi a }\hat{e}_z$
• C. $\frac{2 \sqrt{2} \mu_0 I}{\pi a^2 }\hat{e}_z$
• D. $\frac{2 \sqrt{2} \mu_0 I}{\pi a^2 }\hat{e}_x$

Answer 1

Answer: (B)

$\frac{2 \sqrt{2} \mu_0 I}{\pi a }\hat{e}_z$

Answer 2

Field due to one side of loop at $O$
$= \frac{\mu_0 I}{4 \pi ( \frac{a}{2})}$
Field at $O$ due to all four sides is along unit vector $\hat{e}_z$
$\therefore$ Total field
$= 4. \frac{\mu_{0}I}{4\pi\left(\frac{a}{2}\right)}\left(2 \sin 45^{\circ}\right) = \frac{2\sqrt{2} \mu_{0}I}{\pi a}$