The radius of curvature of the path of the charged particle in a uniform magnetic field is directly proportional to

Options

A. The charge on the particle
B. The momentum of the particle
C. The energy of the particle
D. The intensity of the field

Answer

To find the relationship between the radius of curvature (R) of a charged particle moving in a uniform magnetic field and other parameters, we can derive the formula and analyze the dependencies step by step.

### Step-by-Step Solution:

1. **Understand the Motion of Charged Particles in a Magnetic Field**:
- When a charged particle moves in a magnetic field, it experiences a magnetic force that acts perpendicular to its velocity. This force causes the particle to move in a circular path.

2. **Magnetic Force on a Charged Particle**:
- The magnetic force \( F \) acting on a charged particle with charge \( Q \) moving with velocity \( V \) in a magnetic field \( B \) is given by:
\[ F = QV B \sin(\theta) \]
- For uniform magnetic fields, we can assume \( \theta = 90^\circ \) (the angle between velocity and magnetic field), so:
\[ F = QVB \]

3. **Centripetal Force Requirement**:
- The magnetic force acts as the centripetal force required to keep the particle in circular motion. The centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{R} \]
- Here, \( m \) is the mass of the particle, \( v \) is its velocity, and \( R \) is the radius of curvature.

4. **Setting the Forces Equal**:
- For circular motion, we set the magnetic force equal to the centripetal force:
\[ QVB = \frac{mv^2}{R} \]

5. **Rearranging for Radius of Curvature**:
- Rearranging the equation to solve for \( R \):
\[ R = \frac{mv}{QB} \]

6. **Identifying Proportional Relationships**:
- From the equation \( R = \frac{mv}{QB} \), we can see that:
- \( R \) is directly proportional to the mass \( m \) of the particle.
- \( R \) is directly proportional to the velocity \( v \) of the particle.
- \( R \) is inversely proportional to the charge \( Q \) of the particle.
- \( R \) is inversely proportional to the magnetic field strength \( B \).

### Conclusion:
The radius of curvature \( R \) of the path of a charged particle in a uniform magnetic field is directly proportional to the mass \( m \) and velocity \( v \) of the particle, and inversely proportional to the charge \( Q \) and the magnetic field strength \( B \).