A plane circular coil is rotated about its vertical diameter with a constant angular speed \( \omega \) in a uniform horizontal magnetic field. Initially the plane of the coil is parallel to the magnetic field. Draw plots showing the variation of the following physical quantities as a function of \( \omega t \), where \( t \) represents time elapsed:
Magnetic flux \( \phi \) linked with the coil, and
emf induced in the coil.
Show Hint
In rotating coil problems:
Flux → sine or cosine depending on initial angle
emf is derivative of flux → phase difference \( 90^\circ \)
If flux starts from zero, emf starts from maximum.
Concept:
Magnetic flux through a rotating coil:
\[
\phi = BA \cos \theta
\]
Where:
\( \theta = \omega t \)
Coil rotates with angular speed \( \omega \)
Induced emf:
\[
e = -\frac{d\phi}{dt}
\]
Step 1: Initial condition.
Given:
Plane of coil initially parallel to magnetic field.
So, angle between area vector and field = \( 90^\circ \).
Hence:
\[
\phi = 0 \text{ at } t = 0
\]
Step 2: Expression for magnetic flux.
As the coil rotates:
\[
\theta = \omega t + \frac{\pi}{2}
\]
So:
\[
\phi = BA \cos\left(\omega t + \frac{\pi}{2}\right)
= BA \sin(\omega t)
\]
Graph:
Magnetic flux varies sinusoidally with time, starting from zero.
So, \( \phi \) vs \( \omega t \) is a sine curve starting from origin.
Step 3: Induced emf.
\[
e = -\frac{d\phi}{dt}
= -BA\omega \cos(\omega t)
\]
Graph:
Cosine curve
Maximum at \( t = 0 \)
Phase difference of \( 90^\circ \) with flux
Step 4: Final Graph Description.
[(a)] \( \phi \) vs \( \omega t \): sine wave starting from zero.
[(b)] \( e \) vs \( \omega t \): cosine wave starting from maximum value.
