A long horizontal metallic rod with length along the east-west direction is falling under gravity. The potential difference between its two ends will

Options

A. Be zero
B. Be constant
C. Increase with time
D. Decrease with time

Explanation

To solve the problem of finding the potential difference between the two ends of a falling metallic rod oriented in the east-west direction, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a long horizontal metallic rod that is falling under the influence of gravity. The rod is aligned in the east-west direction.
- As the rod falls, it cuts through the Earth's magnetic field, which is assumed to be uniform.

2. **Identifying the Relevant Physics**:
- The phenomenon we are dealing with here is called electromagnetic induction. Specifically, we are interested in the induced electromotive force (emf) due to the motion of the rod in a magnetic field.
- According to Faraday's law of electromagnetic induction, an emf is induced in a conductor moving through a magnetic field.

3. **Using the Formula for Induced EMF**:
- The induced emf (ε) in the rod can be calculated using the formula:
\[ \epsilon = B \cdot L \cdot V \]
where:
- \( B \) is the magnetic field strength (due to the Earth),
- \( L \) is the length of the rod,
- \( V \) is the velocity of the rod as it falls.

4. **Determining the Direction of Motion**:
- As the rod falls downwards, it moves perpendicular to the magnetic field lines (which are generally directed from north to south).
- The direction of the induced emf can be determined using the right-hand rule, which tells us that the induced current will flow from one end of the rod to the other due to the motion through the magnetic field.

5. **Conclusion on Potential Difference**:
- The potential difference between the two ends of the rod (let's call them A and B) will be equal to the induced emf:
\[ V_{AB} = \epsilon = B \cdot L \cdot V \]
- As the rod continues to fall, its velocity \( V \) will increase, leading to an increase in the induced emf and thus an increase in the potential difference between the ends of the rod over time.

### Final Answer:
The potential difference between the two ends of the falling metallic rod will increase with time as the rod falls under gravity.

---