A charge q enters into a magnetic field (B) perpendicularly with velocity V. The time rate of work done by the magnetic force on the charge is

Options

A. qB
B. `qB//V`
C. `qB//V^(2)`
D. None of these

Answer

To solve the problem, we need to analyze the situation where a charge \( q \) enters a magnetic field \( B \) perpendicularly with a velocity \( V \). We want to determine the time rate of work done by the magnetic force on the charge.

### Step-by-Step Solution:

1. **Understanding the Motion of the Charge:**
When a charged particle enters a magnetic field perpendicularly, it experiences a magnetic force that acts as a centripetal force, causing it to move in a circular path.

2. **Magnetic Force Calculation:**
The magnetic force \( F \) acting on the charge is given by:
\[ F = qV B \]
where \( q \) is the charge, \( V \) is the velocity, and \( B \) is the magnetic field strength.

3. **Centripetal Force:**
For circular motion, the centripetal force \( F_c \) required to keep the charge moving in a circle is given by:
\[ F_c = \frac{mV^2}{R} \]
where \( m \) is the mass of the charge and \( R \) is the radius of the circular path.

4. **Equating Forces:**
Since the magnetic force provides the necessary centripetal force, we can equate the two:
\[ qVB = \frac{mV^2}{R} \]

5. **Finding the Radius:**
Rearranging the equation gives us the radius \( R \) of the circular path:
\[ R = \frac{mV}{qB} \]

6. **Work Done by the Magnetic Force:**
The work done \( W \) by a force is defined as the product of the force and the displacement in the direction of the force:
\[ W = F \cdot d \]
However, in this case, the magnetic force is always perpendicular to the velocity of the charge. Since the displacement in the direction of the force is zero (the force does not do work on the charge), the work done is:
\[ W = 0 \]

7. **Rate of Work Done:**
The rate of work done (power) is given by:
\[ P = \frac{W}{t} \]
Since \( W = 0 \), we have:
\[ P = \frac{0}{t} = 0 \]

### Conclusion:
The time rate of work done by the magnetic force on the charge is zero.