A coil of cross-sectional area A having n turns is placed in uniform magnetic field B. When it is rotated with an angular velocity `omega`, the maximum e.m.f. induced in the coil will be :

Options

A. `NABomega`
B. `(NAB)/(omega)`
C. `(NAomega)/(B)`
D. `(omegaB)/(NA)`

Answer

To find the maximum electromotive force (e.m.f.) induced in a coil placed in a uniform magnetic field when it is rotated, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Parameters**:
- Let the cross-sectional area of the coil be \( A \).
- The number of turns in the coil is \( n \).
- The uniform magnetic field strength is \( B \).
- The angular velocity of rotation is \( \omega \).

2. **Determine the Magnetic Flux**:
- The magnetic flux \( \Phi \) through the coil at any instant is given by:
\[ \Phi = n \cdot B \cdot A \cdot \cos(\theta) \]
- Here, \( \theta \) is the angle between the magnetic field and the normal to the surface of the coil. As the coil rotates, \( \theta \) changes with time and can be expressed as:
\[ \theta = \omega t \]
- Therefore, the magnetic flux can be rewritten as:
\[ \Phi = n \cdot B \cdot A \cdot \cos(\omega t) \]

3. **Apply Faraday's Law of Electromagnetic Induction**:
- According to Faraday's law, the induced e.m.f. \( \mathcal{E} \) is given by the negative rate of change of magnetic flux:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]

4. **Differentiate the Magnetic Flux**:
- To find the induced e.m.f., we differentiate the expression for magnetic flux:
\[ \mathcal{E} = -\frac{d}{dt}(n \cdot B \cdot A \cdot \cos(\omega t)) \]
- Using the chain rule, we have:
\[ \mathcal{E} = -n \cdot B \cdot A \cdot \frac{d}{dt}(\cos(\omega t)) = -n \cdot B \cdot A \cdot (-\omega \sin(\omega t)) \]
- This simplifies to:
\[ \mathcal{E} = n \cdot B \cdot A \cdot \omega \cdot \sin(\omega t) \]

5. **Determine the Maximum Induced e.m.f.**:
- The maximum value of \( \sin(\omega t) \) is 1. Therefore, the maximum induced e.m.f. \( \mathcal{E}_{\text{max}} \) is:
\[ \mathcal{E}_{\text{max}} = n \cdot B \cdot A \cdot \omega \]

### Final Answer:
The maximum e.m.f. induced in the coil is:
\[ \mathcal{E}_{\text{max}} = n \cdot B \cdot A \cdot \omega \]