Question

The figure shows four wire loops, with edge lengths of either I or 2I. All four loops will move through a region of uniform magnetic field g at the same constant velocity. In which loop the emf induced in maximum:

A. I
B. II
C. III
D. IV

Answer 1

In order to solve this question, the concept of motional emf must be understood. Motional emf is defined as the emf induced in the metal conductor when it is in movement by cutting across a magnetic field in the direction normal to that of the magnetic field.

Complete step-by-step answer:
The Faraday’s law of electromagnetism states that:
Whenever there is change in the magnetic flux linked with the closed loop conductor, there is an emf induced at its end.
$Emf, E = - \dfrac{{d\phi }}{{dt}}$
where $\dfrac{{d\phi }}{{dt}}$= rate of change of magnetic flux in unit area.
Consider the case of 4 wire loops of length $l$ ($2l$ in the rectangles) moving in the magnetic field of strength B, moving at the same velocity.

When the loops move over the magnetic field, the magnetic flux linked with the coil changes with their movement. This motion that causes an emf to be induced in them, is called motional emf.
Flux linked with the loops, $\phi = B \times A$
The emf, $E = - \dfrac{{d\phi }}{{dt}}$
Substituting, we get –
$E = - \dfrac{d}{{dt}}\left( {B \times A} \right)$
Since, the value of B is constant here –
$E = - B\dfrac{{dA}}{{dt}} \\\ \to E \propto - \dfrac{{dA}}{{dt}} \\\$
Here, it is proved that the emf induced is equal to the rate of change of the area over time.
In other words, the loop that sweeps more area in the given time will have higher induced emf.
In the figure, among the 4, the loop that has maximum emf is IV, since it lies on the magnetic field, longer and hence, the area swept by it will be more compared to the other figures.

Hence, the correct option is Option D.

Note: The practical condition where you can see the motional emf is in trains. When trains move, an emf is induced at the ends of the axle since it cuts the Earth’s magnetic field while the train is moving. This emf also, produces a current. The reason why the emf induced is in negative sign is because it is induced in the opposite direction.