A conducting rod of mass m and length l is placed over a smooth horizontal surface. A uniform magnetic field B is acting perpendicular to the rod. Charge q is suddenly passed through the rod and it acquires an initial velocity v on the surface, then q is equal to

Options

A. `(2mv)/(BI)`
B. `(BI)/(2mv)`
C. `(mv)/(BI)`
D. `(BlV)/(2m)`

Answer

To solve the problem, we need to find the charge \( q \) that is passed through the conducting rod, given that it acquires an initial velocity \( v \) in a magnetic field \( B \). Here’s the step-by-step solution:

### Step 1: Understand the scenario
We have a conducting rod of mass \( m \) and length \( l \) placed on a smooth horizontal surface. A uniform magnetic field \( B \) is acting perpendicular to the rod. When a charge \( q \) is passed through the rod, it acquires an initial velocity \( v \).

### Step 2: Identify the forces acting on the rod
The force that causes the rod to move is the magnetic force. According to the Lorentz force law, the magnetic force \( F \) on a current-carrying conductor in a magnetic field is given by:
\[ F = I \cdot B \cdot l \]
where \( I \) is the current flowing through the rod.

### Step 3: Relate current to charge
The current \( I \) can be expressed in terms of charge \( q \) and the time \( \Delta t \) during which the charge flows:
\[ I = \frac{q}{\Delta t} \]
Substituting this into the force equation gives:
\[ F = \left(\frac{q}{\Delta t}\right) \cdot B \cdot l \]

### Step 4: Use the impulse-momentum theorem
The impulse imparted to the rod is equal to the change in momentum. Since the rod starts from rest and acquires a velocity \( v \), the change in momentum is:
\[ \text{Impulse} = \Delta p = m \cdot v \]

### Step 5: Relate impulse to force and time
Impulse can also be expressed as the product of force and time:
\[ \text{Impulse} = F \cdot \Delta t \]
Setting the two expressions for impulse equal gives:
\[ m \cdot v = F \cdot \Delta t \]

### Step 6: Substitute the expression for force
Substituting the expression for \( F \) from Step 3 into the impulse equation:
\[ m \cdot v = \left(\frac{q}{\Delta t} \cdot B \cdot l\right) \cdot \Delta t \]
This simplifies to:
\[ m \cdot v = q \cdot B \cdot l \]

### Step 7: Solve for charge \( q \)
Rearranging the equation to solve for \( q \) gives:
\[ q = \frac{m \cdot v}{B \cdot l} \]

### Final Answer
Thus, the charge \( q \) that is passed through the rod is:
\[ q = \frac{m \cdot v}{B \cdot l} \]