# A planar loop of wire rotates in a uniform magnetic field.

Question:

A planar loop of wire rotates in a uniform magnetic field. Initially, at $t=0$, the plane of the loop is perpendicular to the magnetic field. If it rotates with a period of $10 \mathrm{~s}$ about an axis in its plane then the magnitude of induced emf will be maximum and minimum, respectively at:

1. (1) $2.5 \mathrm{~s}$ and $7.5 \mathrm{~s}$

2. (2) $2.5 \mathrm{~s}$ and $5.0 \mathrm{~s}$

3. (3) $5.0 \mathrm{~s}$ and $7.5 \mathrm{~s}$

4. (4) $5.0 \mathrm{~s}$ and $10.0 \mathrm{~s}$

Correct Option: , 4

Solution:

(4) We have given, time period, $T=10 \mathrm{~s}$

$\therefore \quad$ Angular velocity, $\omega=\frac{2 \pi}{10}=\frac{\pi}{5}$

Magnetic flux, $\phi(t)=B A \cos \omega t$

Emf induced, $E=\frac{-d \phi}{d t}=B A \omega \sin \omega t=B A \omega \sin (\omega t)$

Induced emf, $|\varepsilon|$ is maximum when $\omega t=\frac{\pi}{2}$

$\Rightarrow t=\frac{\pi}{\frac{2}{\frac{\pi}{5}}}=2.5 \mathrm{~s}$

For induced emf to be minimum i.e zero

$\omega t=\pi \quad \Rightarrow \quad t=\frac{\pi}{\frac{\pi}{5}}=5 \mathrm{~s}$

$\therefore$ Induced emf is zero at $\mathrm{t}=5 \mathrm{~s}$