A charged particle of charge q and mass m enters perpendiculalry in a magnetic field B. Kinetic energy of particle is E, then frequency of rotation is

Options

A. `(qB)/(mpi)`
B. `qB/(2pim)`
C. `(qBE)/(2pim)`
D. `(qB)/(2piE)`

Answer

To solve the problem, we need to find the frequency of rotation of a charged particle with charge \( q \) and mass \( m \) that enters a magnetic field \( B \) perpendicularly, given that its kinetic energy is \( E \).

### Step-by-step Solution:

1. **Understanding the Kinetic Energy**:
The kinetic energy \( E \) of the charged particle is given by the formula:
\[ E = \frac{1}{2} mv^2 \]
where \( v \) is the velocity of the particle.

2. **Relating Kinetic Energy to Velocity**:
From the kinetic energy equation, we can express the velocity \( v \) in terms of kinetic energy \( E \):
\[ v = \sqrt{\frac{2E}{m}} \]

3. **Magnetic Force and Circular Motion**:
When the charged particle moves in a magnetic field perpendicularly, it experiences a magnetic force that acts as the centripetal force required for circular motion. The magnetic force is given by:
\[ F = qvB \]
The centripetal force needed for circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( r \) is the radius of the circular path.

4. **Setting the Forces Equal**:
Since the magnetic force provides the centripetal force, we can set these two forces equal:
\[ qvB = \frac{mv^2}{r} \]

5. **Solving for the Radius**:
Rearranging the equation gives us:
\[ r = \frac{mv}{qB} \]

6. **Finding Angular Frequency**:
The angular frequency \( \omega \) of the particle in circular motion is related to the velocity and radius by:
\[ \omega = \frac{v}{r} \]
Substituting \( r \) from the previous step:
\[ \omega = \frac{v}{\frac{mv}{qB}} = \frac{qB}{m} \]

7. **Finding Frequency**:
The frequency \( f \) is related to angular frequency \( \omega \) by:
\[ f = \frac{\omega}{2\pi} \]
Substituting \( \omega \):
\[ f = \frac{qB}{2\pi m} \]

### Final Answer:
Thus, the frequency of rotation of the charged particle is:
\[ f = \frac{qB}{2\pi m} \]