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A square loop of edge length $l$ carrying a current I is having magnetic field at its center $B_{A} .$ An equilateral triangle of same edge length and carrying same current has magnetic field $B_{B}$ at its centre. $B_{A} : B_{B} = 4 \sqrt{2} : k,$ where k is ___________

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answer is 9.

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$B_{A} = \frac{μ_{0} I}{4 π \frac{l}{2}} [\cos 45^{0} + \cos 45^{0}] \times 4 = \frac{2 \sqrt{2} μ_{0} I}{π l}$
$B_{B} = \frac{μ_{0} I}{4 π \frac{l}{2 \sqrt{3}}} [\cos 30^{0} + \cos 30^{0}] \times 3 = \frac{9 μ_{0} I}{2 π l}$
$B_{A} : B_{B} = 2 \sqrt{2} : \frac{9}{2} = 4 \sqrt{2} : 9 = 4 \sqrt{2} : k$
$k = 9$

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