Question

When a rectangular coil is rotated about an axis in the plane of coil and perpendicular to magnetic field with steady speed then
A. Periodical emf is induced
B. No emf is induced
C. Unidirectional emf is induced
D. Multi-directional emf is induced

Answer 1

We can use the formula for the magnetic flux linked with a coil and the expression for the Lenz’s law which gives the emf induced in the coil. Using these two formulae, derive the relation for the emf induced in the rectangular coil and check whether this induced emf is periodic, unidirectional or multidirectional.

Formulae used:
The magnetic flux $\phi$ linked with a coil is given by
$\phi = BA\cos \theta$ …… (1)
Here, $B$ is the magnetic field, $A$ is the area of the coil and $\theta$ is the angle between the direction of the magnetic field and area vector.
The expression for Lenz’s law is given by
$e = - \dfrac{{d\phi }}{{dt}}$ ..…. (2)
Here, $e$ is the emf induced in the coil and $d\phi$ is the change in the magnetic flux linked with the coil in time $dt$.

Complete step by step answer:
We have given that a rectangular coil is rotated about an axis in the plane of the coil and perpendicular to the magnetic field with steady speed. Let $\theta$ be the angle between the normal to the axis of coil and direction of magnetic field and $N$ be the number of turns in the coil. Thus, according to equation (1), the magnetic flux linked with N turns of the coil is given by
$\phi = NBA\cos \theta$
Let now calculate the emf induced in the coil.
Substitute $NBA\cos \theta$ for $\phi$ in equation (2).
$e = - \dfrac{{d\left( {NBA\cos \theta } \right)}}{{dt}}$
$\Rightarrow e = - NBA\dfrac{{d\left( {\cos \theta } \right)}}{{dt}}$
$\therefore e = - NBA\left( { - \sin \theta } \right)$
This is the expression for the emf induced in the rotating coil.From the above expression for induced emf, we can see that all the quantities in this expression remain the same. Hence, the induced emf changes only due to change in the term $\sin \theta$.Due to this term, the induced emf in the coil is period.Therefore, the emf induced in the coil changes periodically.

Hence, the correct option is A.

Note: The students may think that we have given that the direction of the axis of the rectangular coil is perpendicular to the magnetic field then also we have considered the angle between the area vector and the magnetic field. But the students should keep in mind that when the rectangular coil rotates this angle changes and the value of the induced emf is the same after a periodic rotation.