Question

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.

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.

Answer 1

Step 1: Initial condition.
The plane of the coil is initially parallel to the magnetic field. Hence, angle between area vector and field: \( \theta = 90^\circ \) \[ \phi = 0 \text{ at } t = 0 \]
Step 2: Magnetic flux as coil rotates.
For a coil rotating with angular speed \( \omega \): \[ \theta = \omega t + \frac{\pi}{2} \] \[ \phi = BA \cos\left(\omega t + \frac{\pi}{2}\right) = BA \sin(\omega t) \] Graph: Magnetic flux \( \phi \) varies sinusoidally with time, starting from zero. Hence, \( \phi \) vs \( \omega t \) is a sine curve starting at the origin.
Step 3: Induced emf. \[ e = -\frac{d\phi}{dt} = -BA\omega \cos(\omega t) \] Graph: The induced emf \( e \) follows a cosine curve: - Maximum at \( t = 0 \) - Phase difference of \( 90^\circ \) with flux
Step 4: Graph Summary.
(a) \( \phi \) vs \( \omega t \): sine wave starting from zero
(b) \( e \) vs \( \omega t \): cosine wave starting from maximum value