A Circular coil made of a wire of length L, is placed in the OXY plane. The magnetic field in the region is $B_{0} \hat{k}$ . The coil is changed into a square, keeping the length L unchanged in time T. The average emf induced in the coil is

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$\frac{B_{0} L^{2}}{2 T} (\frac{1}{π} - \frac{1}{4})$

$\frac{B_{0} L^{2}}{4 T} (1 - \frac{1}{π})$

$\frac{B_{0} L^{2}}{4 T} (\frac{1}{π} - \frac{1}{4})$

$\frac{B_{0} L^{2}}{4 T} (\frac{1}{π} - \frac{1}{2})$

answer is C.

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Detailed Solution

Average $emf = \frac{Δϕ}{Δt} ϕ_{1} = π {(\frac{L}{2 π})}^{2} B_{0}, ϕ_{2} = {(\frac{L}{4})}^{2} B_{0}$
$E_{2 v} = |\frac{ϕ_{2} - ϕ_{1}}{T}| = |\frac{B_{0} L^{2}}{4 T} (\frac{1}{4} - \frac{1}{π})| = \frac{B_{0} L^{2}}{4 T} (\frac{1}{π} - \frac{1}{4})$