At the centre of a fixed large circular coil of radius $\mathrm{R}$, a much smaller circular coil of radius…

At the centre of a fixed large circular coil of radius $\mathrm{R}$, 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.
  1. $\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}} \omega \mathrm{r}^2 \sin \omega \mathrm{t}$
  2. $\frac{\mu_0 I}{4 R} \omega \pi r^2 \sin \omega t$
  3. $\frac{\mu_0 I}{2 R} \omega \pi r^2 \sin \omega t$
  4. $\frac{\mu_0 \mathrm{I}}{4 \mathrm{R}} \omega \mathrm{r}^2 \sin \omega \mathrm{t}$

Solution

According to Faraday's law of electromagnetic induction, $ \begin{array}{r} e=-\frac{d \phi}{d t} \text { and } \phi=B A \cos \omega t=B \pi r^2 \cos \omega t \\ \Rightarrow \quad e=-\frac{d}{d t}\left(\pi r^2 B \cos \omega t\right)=\pi r^2 B \sin \omega t(\omega) \\ \therefore e=\frac{\mu_0 I}{2 R} \pi \omega r^2 \sin \omega t\left(\because B=\frac{\mu_0 I}{2 R}\right) \end{array} $

Asked in: JEE Main 2018 (15 Apr Shift 2 Online)

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