The radius of curvature of the path of a charged particle moving in a static uniform magnetic field is

Options

A. Directly propotional to the magnitude of the charge on the particle
B. Directly propotional to the magnitude of the liner momentum of the particle
C. Directly proportional to the kinetic energy of the particle
D. Inversely proportional to the magnitude of the magnetic field

Answer

To find the radius of curvature of the path of a charged particle moving in a static uniform magnetic field, we can use the following steps:

### 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 results in circular motion.

2. **Identify the Relevant Formula**:
The radius of curvature \( r \) of the path of a charged particle in a magnetic field can be expressed by the formula:
\[ r = \frac{mv}{qB} \]
where:
- \( m \) = mass of the particle
- \( v \) = velocity of the particle
- \( q \) = charge of the particle
- \( B \) = magnetic field strength

3. **Analyze the Proportional Relationships**:
- From the formula \( r = \frac{mv}{qB} \):
- The radius \( r \) is **directly proportional** to the momentum \( mv \) of the particle.
- The radius \( r \) is **inversely proportional** to the charge \( q \) of the particle.
- The radius \( r \) is also **inversely proportional** to the magnetic field strength \( B \).

4. **Evaluate the Options**:
- **Option A**: "Directly proportional to the magnitude of charge" - This is incorrect as \( r \) is inversely proportional to \( q \).
- **Option B**: "Directly proportional to the magnitude of linear momentum of the particle" - This is correct since \( r \) is directly proportional to \( mv \).
- **Option C**: "Directly proportional to the kinetic energy of the particle" - This can be evaluated using the relationship \( KE = \frac{p^2}{2m} \). Since \( p = mv \), we can show that \( r \) is also related to kinetic energy.
- **Option D**: "Inversely proportional to the magnetic field strength" - This is correct as \( r \) is inversely proportional to \( B \).

5. **Conclusion**:
Based on the analysis, the correct options are B, C, and D.