A rectangular loop of wire is placed perpendicualr to a uniform magnetic field and then spun around one of its sides at frequency f. The induced emf is a maximum when the

Options

A. Flux is zero
B. Flux is maximum
C. Flux is half its maximum value
D. Derivative of the flux with respect to time is zero

Answer

To solve the problem, we need to analyze the situation of a rectangular loop of wire spinning in a uniform magnetic field and determine when the induced electromotive force (emf) is at its maximum.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a rectangular loop of wire placed perpendicular to a uniform magnetic field.
- The loop is spun around one of its sides at a frequency \( f \).

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

3. **Initial Condition**:
- Initially, when the loop is perpendicular to the magnetic field, \( \theta = 0 \) degrees. Thus, the magnetic flux is maximum:
\[ \Phi_{\text{max}} = B \cdot A \cdot \cos(0) = B \cdot A \]

4. **Rotation of the Loop**:
- As the loop spins, the angle \( \theta \) changes with time. If the loop spins with an angular velocity \( \omega \), then:
\[ \theta = \omega t \]
- The magnetic flux at time \( t \) can then be expressed as:
\[ \Phi(t) = B \cdot A \cdot \cos(\omega t) \]

5. **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 magnetic flux:
\[ \mathcal{E} = -\frac{d}{dt}(B \cdot A \cdot \cos(\omega t)) = B \cdot A \cdot \omega \cdot \sin(\omega t) \]

6. **Finding Maximum Induced EMF**:
- The induced emf is maximum when \( \sin(\omega t) = 1 \). This occurs at:
\[ \omega t = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
- At this point, \( \cos(\omega t) = 0 \), which means the magnetic flux \( \Phi \) is zero.

7. **Conclusion**:
- Therefore, the induced emf is maximum when the magnetic flux through the loop is zero.

### Final Answer:
The induced emf is maximum when the magnetic flux is zero.

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