Question

What is the phase difference between the flux linked with a coil rotating in a uniform magnetic field and the induced e.m.f. produced in it?

• A. \( \pi \)
• B. \( \frac{\pi}{2} \)
• C. \( \frac{\pi}{4} \)
• D. \( 0 \)

Hint:

The phase difference between the flux and the induced e.m.f. in a rotating coil is \( \frac{\pi}{2} \) because the rate of change of flux is proportional to the sine of the angle, while flux itself varies with the cosine of the angle.

Answer 1

The Correct Option is B

Step 1: Understanding the relation between magnetic flux and induced e.m.f.
The induced electromotive force (e.m.f.) in a coil rotating in a magnetic field is governed by Faraday's Law of Induction, which states that the induced e.m.f. is equal to the rate of change of the magnetic flux linked with the coil:
\[ \mathcal{E} = - \frac{d\Phi_B}{dt}, \]
where \( \Phi_B \) is the magnetic flux linked with the coil, which depends on the angle between the magnetic field and the normal to the plane of the coil.

Step 2: Magnetic flux linked with the rotating coil.
For a coil rotating in a magnetic field, the magnetic flux \( \Phi_B \) is given by:
\[ \Phi_B = B A \cos(\theta), \]
where:
- \( B \) is the magnetic field strength,
- \( A \) is the area of the coil,
- \( \theta \) is the angle between the magnetic field and the normal to the coil (which changes as the coil rotates).

Step 3: Induced e.m.f. from the flux.
The induced e.m.f. is given by:
\[ \mathcal{E} = - \frac{d\Phi_B}{dt} = - \frac{d}{dt} \left( B A \cos(\theta) \right) = B A \sin(\theta) \frac{d\theta}{dt}. \]

Step 4: Phase difference between flux and induced e.m.f.
Since the magnetic flux depends on \( \cos(\theta) \) and the induced e.m.f. depends on \( \sin(\theta) \), the phase difference between the flux and the e.m.f. is:
\[ \Delta \phi = \frac{\pi}{2}. \]
This is because the derivative of \( \cos(\theta) \) is \( -\sin(\theta) \), indicating that the e.m.f. leads the flux by \( \frac{\pi}{2} \) radians.
Final Answer:
Thus, the phase difference between the flux and the induced e.m.f. is:
\[ \boxed{\frac{\pi}{2}}. \]