Under the influence of a uniform magnetic field a charged particle moves with constant speed V in a circle of radius R. The time period of rotation of the particle.

Options

A. Depends on R and not on V
B. Is independent of both V and R
C. Depends on both V and R
D. Depends on V and not on R

Answer

To find the time period of rotation of a charged particle moving in a uniform magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion of the Charged Particle:**
A charged particle moving in a magnetic field experiences a magnetic force that acts as the centripetal force, causing it to move in a circular path.

2. **Identify the Parameters:**
- Let \( V \) be the constant speed of the particle.
- Let \( R \) be the radius of the circular path.
- Let \( m \) be the mass of the particle.
- Let \( Q \) be the charge of the particle.

3. **Calculate the Distance Travelled in One Complete Revolution:**
The distance travelled in one complete revolution (circumference of the circle) is given by:
\[ \text{Distance} = 2\pi R \]

4. **Relate Time Period to Distance and Speed:**
The time period \( T \) (the time taken for one complete revolution) can be expressed as:
\[ T = \frac{\text{Distance}}{\text{Speed}} = \frac{2\pi R}{V} \]

5. **Use the Relationship Between Radius, Mass, Charge, and Speed:**
In a magnetic field, the radius \( R \) of the circular path is related to the speed \( V \), mass \( m \), and charge \( Q \) of the particle by the equation:
\[ R = \frac{mv}{QB} \]
where \( B \) is the magnetic field strength.

6. **Substitute the Expression for R into the Time Period Equation:**
Rearranging the equation for \( R \) gives:
\[ R = \frac{mv}{QB} \implies v = \frac{QRB}{m} \]
Substitute this expression for \( R \) back into the time period equation:
\[ T = \frac{2\pi R}{V} = \frac{2\pi \left(\frac{mv}{QB}\right)}{V} \]

7. **Simplify the Expression:**
This simplifies to:
\[ T = \frac{2\pi m}{QB} \]

8. **Conclusion:**
The time period \( T \) is given by:
\[ T = \frac{2\pi m}{QB} \]
This shows that the time period is independent of both the radius \( R \) and the speed \( V \) of the particle.

### Final Answer:
The time period of rotation of the charged particle is:
\[ T = \frac{2\pi m}{QB} \]