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Q.

A star shaped loop (with $l$ = length of each section) carries current $i$ . Magnetic field at the centroid of the loop is

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a

$\frac{3 μ_{0} i}{π l}$

b

$\frac{3 μ_{0} i}{2 π l}$

c

$(3 - \sqrt{3}) \frac{μ_{0} i}{π l}$

d

$(3 + \sqrt{3}) \frac{μ_{0} i}{2 π l}$

answer is C.

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Detailed Solution

$r = \frac{\sqrt{3} l}{2} B = 6 [\frac{μ_{0}}{4 π} \frac{i}{r} (\sin 60^{°} + \sin 60^{°})] - 6 [\frac{μ_{0}}{4 π} . \frac{i}{r} (\sin 30^{°} + \sin 30^{°})] = 6 \times \frac{μ_{0}}{4 π} . \frac{i}{r} (\sqrt{3} - 1) = 6 \times \frac{μ_{0}}{4 π} . \frac{i}{(\frac{\sqrt{3} l}{2})} (\sqrt{3} - 1) = (3 - \sqrt{3}) \frac{μ_{0} i}{π l}$

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