An emf induced in a coil rotating in a uniform magnetic field is given by

Options

A. `e=e_(0)sin omegat`
B. `e_(0)=esinomegat`
C. `e=sinomegat`
D. `e=e_(0)sin omega`

Answer

To solve the problem of finding the induced electromotive force (emf) in a coil rotating in a uniform magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a coil rotating in a magnetic field. The coil rotates about its diameter, and the magnetic field is uniform and perpendicular to the plane of rotation.

2. **Defining Variables**:
- Let \( n \) be the number of turns in the coil.
- Let \( A \) be the area of the coil.
- Let \( B \) be the magnetic field strength.
- Let \( \omega \) be the angular velocity of the coil.
- The angle between the area vector \( A \) and the magnetic field \( B \) changes as the coil rotates.

3. **Calculating Magnetic Flux**:
- The magnetic flux \( \Phi \) through the coil is given by:
\[ \Phi = n \cdot A \cdot B \cdot \cos(\theta) \]
- Here, \( \theta \) is the angle between the area vector and the magnetic field vector. As the coil rotates, \( \theta \) changes with time as \( \theta = \omega t \).

4. **Substituting for Angle**:
- Substitute \( \theta \) into the flux equation:
\[ \Phi = n \cdot A \cdot B \cdot \cos(\omega t) \]

5. **Finding the Induced EMF**:
- According to Faraday's law of electromagnetic induction, the induced emf \( E \) is given by the negative rate of change of magnetic flux:
\[ E = -\frac{d\Phi}{dt} \]

6. **Differentiating the Flux**:
- Differentiate the expression for magnetic flux:
\[ \frac{d\Phi}{dt} = n \cdot A \cdot B \cdot \frac{d}{dt}(\cos(\omega t)) = n \cdot A \cdot B \cdot (-\omega \sin(\omega t)) \]
- Thus, the induced emf becomes:
\[ E = n \cdot A \cdot B \cdot \omega \sin(\omega t) \]

7. **Identifying Maximum EMF**:
- The maximum induced emf \( E_0 \) occurs when \( \sin(\omega t) = 1 \):
\[ E_0 = n \cdot A \cdot B \cdot \omega \]
- Therefore, we can express the induced emf as:
\[ E = E_0 \cdot \sin(\omega t) \]

8. **Conclusion**:
- The induced emf in the coil rotating in a uniform magnetic field is given by:
\[ E = E_0 \cdot \sin(\omega t) \]
- This matches with option A from the given choices.

### Final Answer:
The correct option is **A: \( E = E_0 \cdot \sin(\omega t) \)**.