Question

The three loops of wire shown in the figure are all subject to the same uniform magnetic field B that does not vary with time. Loop 1 oscillates back and forth as the bob in a pendulum, loop 2 rotates about a vertical axis, and loop 3 oscillates up and down at the end of a spring. Which loop, or loops, will have an induced emf?

 

Answer 1

Concepts and reason

The concepts required to solve this problem are the induced emf and the magnetic flux.

Initially, write the expression of the induced emf. Then, write the expression of the magnetic flux. Finally, refer the diagrams and find which of the loop will have an induced emf.

Fundamentals

The expression of the induced emf is,

$\varepsilon = - \frac{{d\phi }}{{dt}}$

Here, $\varepsilon$ is the induced emf and $\phi$ is the magnetic flux.

The expression of the magnetic flux is,

$\phi = B \cdot S$

Here, B is the magnetic field and S is the area.

The expression of the induced emf is,

$\varepsilon = - \frac{{d\phi }}{{dt}}$

Substitute $B \cdot S$ for $\phi$ .

$\varepsilon = - \frac{{d\left( {B \cdot S} \right)}}{{dt}}$

Simplify the above result.

$\varepsilon = - \frac{{d\left( {BS\cos \theta } \right)}}{{dt}}$

Refer the figures given in the question.

For the loop 1:

The magnetic field remains the same throughout the to and fro motion of the loop and the loop always face the direction of the magnetic field. There is no change in the magnetic field and the area. So, this will leads to no change in the magnetic flux. Thus, no emf will be induced as the emf is the rate of change of magnetic flux.

For the loop 2:

The angle between the magnetic field and the area of the loop is changes while loop 2 is rotating. This will lead to the change in the magnetic flux and the emf is induced in this loop.

For the loop 3:

The magnetic field remains the same throughout the loop, when it oscillates in the up and down at the end of a spring. The loop always faces the direction of the magnetic field. There is no change in the magnetic field and the area. So, this will leads to no change in the magnetic flux. Thus, no emf will be induced as the emf is the rate of change of magnetic flux.

Ans:

The emf induced will be induced in the loop 2.