A conducting rod is moved with a constant velocity v in a magnetic field. A potential difference appears across the two ends

Options

A. if vec v|| vec l
B. if vec v ||vec B
C. if vec l || vec B
D. none of these.

Answer

To solve the problem of a conducting rod moving in a magnetic field and generating a potential difference across its ends, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
- Let the length of the conducting rod be \( L \).
- The rod is moving with a constant velocity \( v \).
- The magnetic field \( B \) is present in the region where the rod is moving.

2. **Understanding the EMF Generation**:
- When a conducting rod moves through a magnetic field, an electromotive force (EMF) is induced across its ends.
- The EMF (\( \mathcal{E} \)) can be calculated using the formula:
\[ \mathcal{E} = v \cdot B \cdot L \cdot \sin(\theta) \]
where \( \theta \) is the angle between the velocity vector \( v \) and the magnetic field vector \( B \).

3. **Conditions for Non-Zero EMF**:
- For the EMF to be non-zero, the vectors \( v \), \( B \), and \( L \) must not be parallel. This means:
- \( v \) must be perpendicular to \( B \), or
- \( v \) must be perpendicular to \( L \), or
- \( B \) must be perpendicular to \( L \).

4. **Analyzing the Given Options**:
- If \( v \) is parallel to \( L \), then \( \sin(0) = 0 \) and EMF will be zero.
- If \( v \) is parallel to \( B \), then again \( \sin(0) = 0 \) and EMF will be zero.
- Therefore, for EMF to exist, none of these conditions can hold true.

5. **Conclusion**:
- Since the EMF exists between the two ends of the rod, the only valid conclusion is that the conditions for EMF generation are not satisfied by the given options. Hence, the answer is that none of the options provided are correct.

### Final Answer:
The answer is: **None of these**.