At the center of a fixed large circular coil of radius R, a much smaller circular coil of radius r is placed. The two coils are concentric and are in the same plane. The larger coil carries a current I. The smaller coil is set to rotate with a constant angular velocity $ω$ about an axis along their common diameter. Calculate the emf induced in the smaller coil after a time t of its start of rotation

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$\frac{μ_{0} I}{4 R} {ωπr}^{2} sin ωt$

$\frac{μ_{0} I}{4 R} {ωr}^{2} sin ωt$

$\frac{μ_{0} I}{2 R} {ωr}^{2} sin ωt$

$\frac{μ_{0} I}{2 R} {ωπr}^{2} sin ωt$

answer is D.

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Detailed Solution

Emf induced in smaller coil is given by
$ε = - \frac{dϕ}{dt} = - B \times A \frac{d}{dt} (\cos θ)$
$= \frac{μ_{0} I}{2 R} ({πr}^{2}) (\sin θ) \times \frac{dθ}{dt} = \frac{μ_{0} I}{2 R} {ωπr}^{2} \sin ωt$
(B is assumed to be constant in small region)