When a coil is rotated in a magnetic field, with steady speed , then

Options

A. no e.m.f is induced
B. a periodic e.m.f is induced
C. unidirectional e.m.f is induced
D. multidirectional e.m.f is induced

Answer

To solve the question regarding the behavior of a coil rotating in a magnetic field at a steady speed, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a coil that is rotating in a uniform magnetic field. The coil can be circular or rectangular, and it is rotated with a steady angular velocity (ω).

2. **Magnetic Flux Calculation**:
- The magnetic flux (Φ) through the coil is given by the formula:
\[ \Phi = B \cdot S \cdot \cos(\theta) \]
where:
- \( B \) is the magnetic field strength,
- \( S \) is the area of the coil,
- \( \theta \) is the angle between the magnetic field and the normal to the surface of the coil.

3. **Relating Angle to Time**:
- As the coil rotates, the angle \( \theta \) changes with time. We can express this relationship as:
\[ \theta(t) = \omega t \]
- Substituting this into the magnetic flux equation gives:
\[ \Phi(t) = B \cdot S \cdot \cos(\omega t) \]

4. **Finding Induced EMF**:
- According to Faraday's law of electromagnetic induction, the induced EMF (ε) is given by the negative rate of change of magnetic flux:
\[ \epsilon = -\frac{d\Phi}{dt} \]
- Differentiating the flux with respect to time:
\[ \epsilon = -\frac{d}{dt}(B \cdot S \cdot \cos(\omega t)) \]
- Using the chain rule:
\[ \epsilon = -B \cdot S \cdot (-\sin(\omega t)) \cdot \omega \]
- This simplifies to:
\[ \epsilon = B \cdot S \cdot \omega \cdot \sin(\omega t) \]

5. **Conclusion**:
- The induced EMF is a sinusoidal function of time, which means it varies periodically as the coil rotates in the magnetic field. Therefore, the correct conclusion is that the induced EMF is periodic.

### Final Answer:
The induced EMF when a coil is rotated in a magnetic field with steady speed is **periodic**.

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