A coil is rotated in a uniform magnetic field about an axis perpendicular to the field. The emf induced in the coil would be maximum when the plane of coil is :

Options

A. parallel to the field
B. perpendicular to the field
C. at `45^(@)` to the field
D. none of these

Answer

To determine when the induced electromotive force (emf) in a coil is maximum while it is rotated 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 uniform magnetic field. The axis of rotation is perpendicular to the direction of the magnetic field.

2. **Defining Magnetic Flux**:
- The magnetic flux (Φ_B) through the coil is given by the formula:
\[ \Phi_B = B \cdot A \cdot \cos(\theta) \]
where:
- \(B\) is the magnetic field strength,
- \(A\) is the area of the coil,
- \(\theta\) is the angle between the magnetic field and the normal (perpendicular) to the coil's surface.

3. **Induced EMF**:
- According to Faraday's law of electromagnetic induction, the induced emf (ε) in the coil is given by:
\[ \epsilon = -\frac{d\Phi_B}{dt} \]
- Substituting the expression for magnetic flux:
\[ \epsilon = -\frac{d}{dt}(B \cdot A \cdot \cos(\theta)) \]

4. **Differentiating the Flux**:
- Since \(B\) and \(A\) are constant, we can simplify:
\[ \epsilon = -B \cdot A \cdot \frac{d}{dt}(\cos(\theta)) \]
- The derivative of \(\cos(\theta)\) is:
\[ \frac{d}{dt}(\cos(\theta)) = -\sin(\theta) \cdot \frac{d\theta}{dt} \]
- Therefore, we can write:
\[ \epsilon = B \cdot A \cdot \sin(\theta) \cdot \frac{d\theta}{dt} \]

5. **Maximizing the Induced EMF**:
- The induced emf will be maximum when \(\sin(\theta)\) is maximum. The maximum value of \(\sin(\theta)\) is 1, which occurs when:
\[ \theta = 90^\circ \]
- This means that the area vector of the coil is perpendicular to the magnetic field.

6. **Conclusion**:
- Therefore, the induced emf in the coil is maximum when the plane of the coil is **parallel to the magnetic field**.

### Final Answer:
The emf induced in the coil would be maximum when the plane of the coil is parallel to the magnetic field.

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