A rod of length l is rotated with angular speed `omega` about an axis passing through the centre of rod and perpendicular to its length. A uniform magnetic field B is applied parallel to the axis of rotation, Potential difference developed between the ends of the rod is

Options

A. `Bl(omega^(2))/(2)`
B. `Bl(omega^(2))`
C. `Bl(omega^(2))/(8)`
D. Zero

Explanation

To find the potential difference developed between the ends of a rotating rod in a uniform magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a rod of length \( L \) that is rotating about its center with an angular speed \( \omega \).
- A uniform magnetic field \( B \) is applied parallel to the axis of rotation.

2. **Identify the Points**:
- Let’s denote the ends of the rod as point A and point B. The center of the rod is point O.

3. **Determine the Length of Each Segment**:
- Since the rod is rotating about its center, the distance from the center to each end (A and B) is \( \frac{L}{2} \).

4. **Calculate the Linear Velocity**:
- The linear velocity \( v \) of any point on the rod can be calculated using the formula:
\[ v = r \cdot \omega \]
- For the ends of the rod (points A and B), \( r = \frac{L}{2} \), so:
\[ v_A = v_B = \frac{L}{2} \cdot \omega \]

5. **Use the Formula for Potential Difference**:
- The potential difference \( V_{AB} \) between the ends of the rod in a magnetic field can be calculated using the formula:
\[ V_{AB} = B \cdot L' \cdot \omega \]
- Here, \( L' \) is the effective length of the rod contributing to the potential difference, which is \( \frac{L}{2} \) for each half of the rod.

6. **Calculate the Potential Difference**:
- Since both ends of the rod are at the same distance from the center, we can write:
\[ V_{AB} = B \cdot \left(\frac{L}{2}\right) \cdot \omega \]
- Therefore, the total potential difference developed across the ends of the rod is:
\[ V_{AB} = \frac{B \cdot L \cdot \omega}{2} \]

### Final Answer:
The potential difference developed between the ends of the rod is:
\[ V_{AB} = \frac{B \cdot L \cdot \omega}{2} \]