A metallic wire is folded to form a square loop of side a.It carries current i and is kept perpendicular to the region of uniform magnetic field B. If the shape of the loop is changed from square to an equilateral triangle without changing the length of the wire and current, the amount of work done in doing so is

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$B i a^{2} (1 - \frac{4 \sqrt{3}}{9})$

$B i a^{2} (1 - \frac{\sqrt{3}}{9})$

$\frac{2}{3} B i a^{2}$

Zero

answer is A.

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Detailed Solution

Emf induced in coil, $ε = - \frac{∆ ϕ}{∆ t} = - \frac{(ϕ_{2} - ϕ_{1})}{t_{2} - t_{1}}$

$ϕ_{2} = B A_{2} = \frac{4 a}{2 \times 3} \times \frac{4 a}{3} \times \sin 60^{0} = \frac{4 \sqrt{3}}{9} B a^{2}$

$ϕ_{1} = B a^{2}$

Work done, $W = ε q = ε I ∆ t = B i a^{2} (1 - \frac{4 \sqrt{3}}{9}) i$

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