Question

In Figure, a circular loop of wire 10 cmin diameter (seen edge-on) is placed with its normal $\overrightarrow{\mathbf{N}}$at an angle$\mathit{\theta }\mathbf{=}{\mathbf{30}}^{\mathbf{°}}$with the direction of a uniform magnetic field $\overrightarrow{\mathbf{B}}$of magnitude 0.50 T. The loop is then rotated such thatrotates in a cone about the field direction at the rate 100 rev/min; angleremains unchanged during the process. What is the emf induced in the loop?

Short answer

The magnetic flux through the loop is zero, $\epsilon =0$.

Step-by-step solution

Given

1. Fig.30-33.
2. The diameter of the circular loop is, d = 10cm
3. The angle$\theta =30°$
4. The uniform magnetic field is$\overrightarrow{B}=0.50\mathrm{T}$
5. The normal$\overrightarrow{N}$rotates in the cone about the field direction at the rate 100 rev/min.
6. Angle remains unchanged during the process.

Determining the concept

Using the equation for the magnetic flux and the given information and applying Faraday’s law, find the emf induced in the loop.

Faraday's law of electromagnetic induction states, Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced in it.

Formulae are as follow:

The magnetic flux through the loop is,

Where,${\Phi }_B$is magnetic flux, B is magnetic field, A is area.

Determining the magnetic flux through the loop

The magnetic flux through the loop is,

${\Phi }_B=BA\cos \theta$

Since, the angle remains unchanged during the rotation, the magnetic field is uniform and the area is also constant.

Hence, the magnetic flux${\Phi }_B$through the loop is also unchanged during the rotation. That is,

${\Phi }_B=cons\tan t$

According to Faraday’s law, the emf is induced only if there is any change in the magnetic flux.

Therefore, the emf induced in the loop is zero. That is,

$$\begin{gathered} \epsilon =-\frac{d{\Phi }_B}{dt}...............................................(30-4) \\ \mathrm{But}{\Phi }_B=cons\tan t \\ \epsilon =0 \end{gathered}$$

Hence,the magnetic flux through the loop is zero, $\epsilon =0$.

Therefore, by using Faraday’s law and equation, the magnetic flux through the loop can be determined.