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

Answer 2

An alternative way to see the shape of these two graphs is to think about the motional emf generated in the two sides of the coil directly, instead of differentiating the flux expression.

Consider the coil as a rectangular loop with two sides of length \( l \), each at perpendicular distance \( b/2 \) from the rotation axis, spinning about the vertical diameter with angular speed \( \omega \) inside the horizontal field \( B \).

Each of these two sides moves with a speed \( v = \omega (b/2) \) that is always directed horizontally, perpendicular to the rotation axis. As the coil turns, the angle between this velocity and the field \( B \) keeps changing, so the motional emf induced in each side, \( \varepsilon_\text{side} = Blv\sin\theta \) with \( \theta \) the angle between \( v \) and \( B \), also keeps changing with time.

At \( t = 0 \), the plane of the coil is parallel to \( B \), which means the velocity of the sides is perpendicular to \( B \) at that instant, so \( \theta = 90^\circ \) and the motional emf, and hence the induced emf of the full loop, is at its maximum magnitude right at \( t = 0 \).

As the coil rotates further, \( \theta \) decreases from \( 90^\circ \), so \( \sin\theta \) falls, and the total emf around the loop traces out a cosine-shaped curve in \( \omega t \), starting at its peak value at \( \omega t = 0 \) and crossing zero a quarter cycle later, when the sides move parallel to \( B \), that is, when the plane of the coil is perpendicular to \( B \).

The flux, being the quantity whose rate of change produces this emf, must be zero exactly where the emf is largest and largest exactly where the emf is zero, which fixes the flux curve as a sine function of \( \omega t \), starting from zero, a quarter cycle behind the cosine-shaped emf.

This motional-emf picture gives the same pair of curves: \( \phi \) against \( \omega t \) is a sine wave beginning at the origin, and \( e \) against \( \omega t \) is a cosine wave beginning at its maximum value, with the emf curve leading the flux curve by a quarter cycle.