Answer:
Option D
Explanation:
B at centre of a circle = $\frac{\mu_{0}I}{2R}$
B at centre of a square
$=4\times \frac{\mu I}{4\pi.\frac{l}{2}}[\sin 45 ^{\circ} + \sin 45 ^{\circ}]$
$=4\sqrt{2}\frac{\mu I}{2\pi l}$
Now, $R=\frac{L}{2\pi} $ and $l= \frac{L}{4}$
(As L=$2\pi R= 4l$ )
Where , L= length of wire
$B_{A}=\frac{\mu_{0}I}{2.\frac{L}{2\pi}}=\frac{\pi\mu_{0}I}{L}=\pi [\frac{\mu_{0}I}{L}]$
$B_{B}=4\sqrt{2}\frac{\mu_{0}I}{2\pi(\frac{L}{4})}$
$=\frac{8\sqrt{2}\mu_{0}I}{\pi L}=\frac{8\sqrt{2}}{\pi}[\frac{\mu_{0}I}{L}]$
$\frac{B_{A}}{B_{B}}=\pi^{2}:8\sqrt{2}$