Question

A very small circular loop of radius a is initially (at t = 0) coplanar and concentric with a much larger fixed circular loop of radius $b$. A constant current $l$ flows in the larger loop. The smaller loop is rotated with a constant angular speed $\omega$ about the common diameter. The emf induced in the smaller loop as a function of time $t$ is

• A. $\frac{\pi a^2{\mu }_{0}l}{2b}\omega \cos \left(\omega t\right)$
• B. $\frac{\pi a^2{\mu }_{0}l}{2b}\omega \sin \left({\omega }^2{t}^2\right)$
• C. $\frac{\pi a^2{\mu }_{0}l}{2b}\omega \sin \omega t$
• D. $\frac{\pi a^2{\mu }_{0}l}{2b}\omega {\sin }^2\omega t$

Answer 1

The correct option is C $\frac{\pi a^2{\mu }_{0}l}{2b}\omega \sin \omega t$

The angle which the area vector of the smaller loop forms with that of the larger loop is $\theta =\omega t$
Magnetic field due to current in the larger loop is $\frac{{\mu }_{0}I}{2b}$

Hence the flux that passes thought the smaller loop at time $t=\varphi =\frac{{\mu }_{0}I}{2b}.\pi a^{2}cos\omega t$

Hence the emf induced in the loop$=-\frac{d\varphi }{dt}$

$=\frac{\pi a^2{\mu }_{0}I}{2b}\omega sin\omega t$