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. $\frac{{\mu }_{0}I}{2R}\omega \pi r^2\sin \omega t$
• B. $\frac{{\mu }_{0}I}{4R}\omega \pi r^2\sin \omega t$
• C. $\frac{{\mu }_{0}I}{4R}\omega r^2\sin \omega t$
• D. $\frac{{\mu }_{0}I}{2R}\omega r^2\sin \omega t$

Answer 1

Answer: (A)

We know that electric flux $\varphi =\vec{B}\cdot \vec{A}$
$\Rightarrow \varphi =BA\cos \omega t$
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,
$\varphi =\frac{{\mu }_0}{2}\frac{I}{R}\pi r^2\cos \omega t$
Now induced emf $\epsilon =\frac{-d\varphi }{dt}=\frac{-d}{dt}\left(\frac{{\mu }_0}{2}\frac{I}{R}\pi r^2\cos \omega t\right)$
$\Rightarrow \epsilon =\frac{{\mu }_0}{2}\frac{I}{R}\pi r^2\omega \sin \omega t$