Question

At the centre 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 $\omega$ about an axis along their common diameter. Calculate the emf induced in the smaller coil after a time t of its start of rotation.

• A. $\frac{\mu_0 I}{2 R} \omega \, \pi \, r^2 \, \sin \omega \, t $
• B. $\frac{\mu_0 I}{4 R} \omega \, \pi \, r^2 \, \sin \omega \, t $
• C. $\frac{\mu_0 I}{4 R} \omega \, \, r^2 \, \sin \omega \, t $
• D. $\frac{\mu_0 I}{2 R} \omega \, \, r^2 \, \sin \omega \, t $

Answer 1

Answer: (A)

We know that electric flux $\phi=\vec{B} \cdot \vec{A}$
$\Rightarrow \phi=B A \cos \omega t$
Now $B=\frac{\mu_{0}}{2} \frac{I}{R}$ is magnetic field due to circular coil of radius $R$ and $A=\pi r^{2}$ is area of circular coil of radius $r .$ Therefore,
$\phi=\frac{\mu_{0}}{2} \frac{I}{R} \pi r^{2} \cos \omega t$
Now induced emf $\varepsilon=\frac{-d \phi}{d t}=\frac{-d}{d t}\left(\frac{\mu_{0}}{2} \frac{I}{R} \pi r^{2} \cos \omega t\right)$
$\Rightarrow \varepsilon=\frac{\mu_{0}}{2} \frac{I}{R} \pi r^{2} \omega \sin \omega t$