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 long horizontal metallic rod falling under gravity, we can follow these steps:

### Step 1: Understand the Setup
We have a long metallic rod oriented in the east-west direction and it is falling vertically under the influence of gravity. The falling motion of the rod will induce an electromotive force (EMF) due to its motion through the Earth's magnetic field.

**Hint:** Identify the orientation of the rod and the direction of the gravitational force acting on it.

### Step 2: Identify the Relevant Formula
The induced EMF (ε) in a conductor moving through a magnetic field can be given by the formula:
\[ \text{EMF} = B \cdot L \cdot v \]
where:
- \( B \) is the magnetic field strength,
- \( L \) is the length of the rod,
- \( v \) is the velocity of the rod.

**Hint:** Recall that EMF is related to the magnetic field, the length of the conductor, and its velocity.

### Step 3: Determine the Velocity of the Rod
As the rod falls under gravity, its velocity \( v \) increases over time. The velocity can be expressed using the equations of motion:
\[ v = u + gt \]
Since the initial velocity \( u = 0 \) (the rod starts from rest), we have:
\[ v = gt \]

**Hint:** Use the equations of motion to express the velocity of the falling rod.

### Step 4: Substitute the Velocity into the EMF Formula
Substituting \( v = gt \) into the EMF formula gives us:
\[ \text{EMF} = B \cdot L \cdot (gt) \]
This indicates that the induced EMF is directly proportional to time \( t \).

**Hint:** Substitute the expression for velocity into the EMF formula to see how it changes over time.

### Step 5: Conclusion about the Potential Difference
Since the EMF is proportional to time, it means that the potential difference between the two ends of the rod increases as the rod continues to fall. Therefore, the potential difference will increase with time.

**Hint:** Analyze the relationship between EMF and time to conclude about the behavior of the potential difference.

### Final Answer
The potential difference between the two ends of the rod will **increase with time**.