A rod of length l rotates with a uniform angular velocity omega about its perpendicular bisector. A uniform magnetic field B exists parallel to the axis of rotation. The potential difference between the two ends of the lrod is

Options

A. zero
B. `(1)/(2)Blomega^2`
C. `Blomega^2`
D. `2Blomega^2`

Answer

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

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a rod of length \( L \) rotating about its perpendicular bisector with an angular velocity \( \omega \).
- A uniform magnetic field \( B \) exists parallel to the axis of rotation.

2. **Identify Points on the Rod**:
- Let the center of the rod be point \( O \).
- The two ends of the rod are points \( P \) and \( Q \).
- The distance from the center \( O \) to either end (i.e., \( OP \) or \( OQ \)) is \( \frac{L}{2} \).

3. **Calculate the EMF Developed**:
- The potential difference (EMF) developed between the ends of the rod can be calculated using the formula:
\[ E = B \cdot v \cdot L \]
- Here, \( v \) is the linear velocity of the ends of the rod, which can be expressed as:
\[ v = \frac{L}{2} \cdot \omega \]
- Substituting \( v \) into the EMF formula gives:
\[ E = B \cdot \left(\frac{L}{2} \cdot \omega\right) \cdot L \]
- Simplifying this, we get:
\[ E = \frac{B \cdot \omega \cdot L^2}{2} \]

4. **Potential Difference Between Ends**:
- The potential difference between the ends of the rod \( V_P - V_Q \) is equal to the EMF calculated:
\[ V_P - V_Q = \frac{B \cdot \omega \cdot L^2}{2} \]

5. **Final Result**:
- Therefore, the potential difference between the two ends of the rod is:
\[ V_P - V_Q = \frac{B \cdot \omega \cdot L^2}{2} \]