Find the maximum induced emf.

Answer

To find the maximum induced electromotive force (emf) in a rotating coil placed in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have a coil placed in a magnetic field that is perpendicular to the plane of the coil. The coil is rotated with an angular velocity \( \omega \).

2. **Magnetic Flux Calculation**:
The magnetic flux \( \Phi \) through the coil can be expressed as:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
where \( B \) is the magnetic field strength, \( A \) is the area of the coil, and \( \theta \) is the angle between the magnetic field and the normal to the coil's surface. As the coil rotates, \( \theta \) changes with time, and we can express it as:
\[ \theta = \omega t \]
Therefore, the flux becomes:
\[ \Phi = B \cdot A \cdot \cos(\omega t) \]

3. **Induced EMF Calculation**:
According to Faraday's law of electromagnetic induction, the induced emf \( \mathcal{E} \) is given by the negative rate of change of magnetic flux:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Substituting the expression for flux:
\[ \mathcal{E} = -\frac{d}{dt}(B \cdot A \cdot \cos(\omega t)) \]
This simplifies to:
\[ \mathcal{E} = B \cdot A \cdot \omega \cdot \sin(\omega t) \]

4. **Finding Maximum Induced EMF**:
The maximum value of \( \sin(\omega t) \) is 1. Therefore, the maximum induced emf \( \mathcal{E}_{\text{max}} \) can be expressed as:
\[ \mathcal{E}_{\text{max}} = B \cdot A \cdot \omega \]

5. **Area of the Coil**:
If we assume the coil is circular with radius \( r \), the area \( A \) is given by:
\[ A = \pi r^2 \]
Substituting this into the equation for maximum induced emf:
\[ \mathcal{E}_{\text{max}} = B \cdot (\pi r^2) \cdot \omega \]

### Final Expression:
Thus, the maximum induced emf is:
\[ \mathcal{E}_{\text{max}} = B \cdot \pi r^2 \cdot \omega \]