A metal rod of length l cuts across a uniform magnetic field B with a velocity v. If the resistance of the circuit of which the rod forms a part is r, then the force required to move the rod is

Options

A. `(B^(2)l^(2)v)/(r)`
B. `(Blv)/(r)`
C. `(B^(2)lv)/(r)`
D. `(Blv^(2))/(r)`

Answer

To find the force required to move a metal rod of length \( l \) across a uniform magnetic field \( B \) with a velocity \( v \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Induced EMF**:
The induced electromotive force (EMF) \( \mathcal{E} \) in the rod moving through the magnetic field can be calculated using the formula:
\[ \mathcal{E} = B \cdot l \cdot v \]
where \( B \) is the magnetic field strength, \( l \) is the length of the rod, and \( v \) is the velocity of the rod.

**Hint**: Remember that the induced EMF is generated due to the motion of the conductor in the magnetic field.

2. **Calculate the Current**:
The current \( I \) flowing through the circuit can be calculated using Ohm's law:
\[ I = \frac{\mathcal{E}}{r} = \frac{B \cdot l \cdot v}{r} \]
where \( r \) is the resistance of the circuit.

**Hint**: Use Ohm's law to relate EMF, current, and resistance.

3. **Determine the Magnetic Force**:
The magnetic force \( F_B \) acting on the rod due to the current in the magnetic field can be calculated using the formula:
\[ F_B = B \cdot I \cdot l \]
Substituting the expression for current \( I \):
\[ F_B = B \cdot \left(\frac{B \cdot l \cdot v}{r}\right) \cdot l = \frac{B^2 \cdot l^2 \cdot v}{r} \]

**Hint**: The magnetic force on a current-carrying conductor in a magnetic field is given by the product of the magnetic field, the current, and the length of the conductor.

4. **Force Required to Move the Rod**:
To keep the rod moving with a constant velocity \( v \), we need to apply an external force \( F \) that is equal in magnitude but opposite in direction to the magnetic force \( F_B \):
\[ F = F_B = \frac{B^2 \cdot l^2 \cdot v}{r} \]

**Hint**: The applied force must counteract the magnetic force to maintain constant velocity.

### Final Answer:
The force required to move the rod is:
\[ F = \frac{B^2 \cdot l^2 \cdot v}{r} \]