Effective area of coil in a magnetic field changes with time, the flux at any time is:

Options

A. `phi_(B) = B A cot omega t`
B. `phi_(B) = B A cos omega t`
C. `phi_(B) = B A tan omega t`
D. `phi_(B) = B A sec omega t`

Explanation

To solve the problem of finding the magnetic flux through a coil when its effective area changes with time, we can follow these steps:

### Step-by-Step Solution:

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

2. **Identifying Variables**:
In this case, we have a coil whose effective area \( A \) is changing with time, and the coil is rotating. The angle \( \theta \) can be expressed as:
\[ \theta = \omega t \]
where \( \omega \) is the angular velocity and \( t \) is time.

3. **Substituting the Angle**:
Substitute \( \theta \) in the flux equation:
\[ \Phi = B \cdot A \cdot \cos(\omega t) \]

4. **Final Expression for Flux**:
Thus, the expression for the magnetic flux at any time \( t \) becomes:
\[ \Phi(t) = B \cdot A \cdot \cos(\omega t) \]

### Final Answer:
The magnetic flux at any time \( t \) is given by:
\[ \Phi(t) = B \cdot A \cdot \cos(\omega t) \]