A conducing rod of length `l` is falling with a velocity `v` perpendicular to a unifrorm horizontal magnetic field `B` . The potential difference between its two ends will be

Options

A. 2 B/v
B. B/v
C. `(1)/(2)B//v`
D. `B^(2)l^(2)v^(2)`

Explanation

To find the potential difference between the two ends of a conducting rod of length \( l \) falling with a velocity \( v \) perpendicular to a uniform horizontal magnetic field \( B \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a conducting rod of length \( l \) falling vertically with a velocity \( v \).
- The rod is in a uniform magnetic field \( B \) that is horizontal.

2. **Identify the Forces Acting on the Electrons**:
- As the rod falls through the magnetic field, the electrons in the rod will experience a magnetic force due to their motion in the magnetic field. This force acts perpendicular to both the direction of the velocity of the rod and the direction of the magnetic field.

3. **Use the Lorentz Force Law**:
- The magnetic force \( F_B \) on an electron is given by:
\[ F_B = q(\mathbf{v} \times \mathbf{B}) \]
- Since the velocity \( \mathbf{v} \) and the magnetic field \( \mathbf{B} \) are perpendicular, we can simplify this to:
\[ F_B = qvB \]
- Here, \( q \) is the charge of the electron.

4. **Establish Charge Separation**:
- The magnetic force causes a separation of charges within the rod, leading to one end of the rod becoming negatively charged (excess electrons) and the other end becoming positively charged (deficiency of electrons).

5. **Electric Field Induced**:
- This charge separation creates an electric field \( E \) within the rod. The electric field can be expressed as:
\[ E = \frac{F_E}{q} = \frac{F_B}{q} = vB \]
- Here, \( F_E \) is the electric force acting on the charge due to the electric field.

6. **Calculate the Potential Difference**:
- The potential difference \( V \) between the two ends of the rod can be calculated using the relationship between electric field and potential difference:
\[ V = E \cdot l \]
- Substituting the expression for \( E \):
\[ V = (vB) \cdot l \]
- Thus, the potential difference between the two ends of the rod is:
\[ V = Blv \]

### Final Answer:
The potential difference between the two ends of the conducting rod is given by:
\[ V = Blv \]