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

Answer 1

The correct option is C $\frac{\pi a^2{\mu }_{0}I}{2b}\omega \sin \left(\omega t\right)$

Radius of smaller loop is $a$

Area of smaller loop $A=\pi a^2$

Radius of bigger loop is $b$

Magnetic field intensity due to bigger loop at the centre $B=\frac{{\mu }_{o}I}{2b}$

Net flux through the smaller loop $\varphi =NBAcoswt$ where $N=1$

$⟹$ $\varphi =BAcoswt$

$\frac{d\varphi }{dt}=-BAw$ $sinwt$

Emf induced in smaller loop $E=-\frac{d\varphi }{dt}=BAw$ $sinwt$

$⟹$ $E=\frac{{\mu }_{o}I}{2b}\times \pi a^{2}w$ $sinwt$ $=\frac{\pi a^2{\mu }_{o}I}{2b}w$ $sinwt$