Under the influence of a unifrom magnetic field a charged particle is moving on a circle of radius `R` with Constnant speed `v`. The time period of the motion

Options

A. depends on v and not on R
B. Depends on both R and v
C. is independent of both R and v
D. depends on R and not on v

Answer

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

### Step-by-Step Solution:

1. **Understanding the Forces**:
When a charged particle moves in a magnetic field, it experiences a magnetic force that acts as the centripetal force required for circular motion. The magnetic force \( F \) on a charged particle with charge \( Q \) moving with velocity \( v \) in a magnetic field \( B \) is given by:
\[ F = QvB \sin(\theta) \]
Since the particle moves perpendicular to the magnetic field, \( \theta = 90^\circ \) and \( \sin(90^\circ) = 1 \). Therefore, the magnetic force simplifies to:
\[ F = QvB \]

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

3. **Equating Forces**:
Since the magnetic force provides the necessary centripetal force, we can set the two forces equal to each other:
\[ QvB = \frac{mv^2}{R} \]

4. **Solving for Radius \( R \)**:
Rearranging the equation to solve for \( R \):
\[ R = \frac{mv}{QB} \]

5. **Finding the Time Period \( T \)**:
The time period \( T \) is the time taken to complete one full circular motion. The distance traveled in one complete revolution is the circumference of the circle, which is \( 2\pi R \). Therefore, the time period can be expressed as:
\[ T = \frac{\text{Distance}}{\text{Speed}} = \frac{2\pi R}{v} \]

6. **Substituting for \( R \)**:
Now substitute the expression for \( R \) into the equation for \( T \):
\[ T = \frac{2\pi \left(\frac{mv}{QB}\right)}{v} \]
The \( v \) in the numerator and denominator cancels out:
\[ T = \frac{2\pi m}{QB} \]

7. **Conclusion**:
The final expression for the time period \( T \) of the charged particle moving in a magnetic field is:
\[ T = \frac{2\pi m}{QB} \]
This shows that the time period is independent of the velocity \( v \) and the radius \( R \).