To which of the following quantities, the radius of the circular path of a charged particle moving at right angles to a uniform magnetic field is directly proportional?

Options

A. energy of the particle
B. magnetic field
C. charge of the particle
D. momentum of the particle

Explanation

To determine the quantity to which the radius of the circular path of a charged particle moving at right angles to a uniform magnetic field is directly proportional, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion**: A charged particle moving in a magnetic field experiences a magnetic force that acts as a centripetal force, causing it to move in a circular path. The magnetic force is given by the equation:
\[ F_{\text{magnetic}} = Q \cdot v \cdot B \cdot \sin(\theta) \]
where \( Q \) is the charge of the particle, \( v \) is its velocity, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the velocity vector and the magnetic field vector.

2. **Setting the Angle**: Since the particle is moving at right angles to the magnetic field, \( \theta = 90^\circ \). Therefore, \( \sin(90^\circ) = 1 \), and the equation simplifies to:
\[ F_{\text{magnetic}} = Q \cdot v \cdot B \]

3. **Centripetal Force**: The magnetic force acts as the centripetal force required to keep the particle in circular motion. The centripetal force is given by:
\[ F_{\text{centripetal}} = \frac{m v^2}{r} \]
where \( m \) is the mass of the particle, \( v \) is its velocity, and \( r \) is the radius of the circular path.

4. **Equating Forces**: Setting the magnetic force equal to the centripetal force, we have:
\[ Q \cdot v \cdot B = \frac{m v^2}{r} \]

5. **Rearranging for Radius**: Rearranging this equation to solve for the radius \( r \):
\[ r = \frac{m v}{Q \cdot B} \]

6. **Identifying Proportionalities**: From the equation \( r = \frac{m v}{Q \cdot B} \), we can see that the radius \( r \) is directly proportional to the momentum \( p \) of the particle, where momentum \( p = m \cdot v \). Thus, we can express it as:
\[ r \propto p \]

### Conclusion:
The radius of the circular path \( r \) of a charged particle moving at right angles to a uniform magnetic field is directly proportional to the momentum of the particle.

### Final Answer:
The correct option is that the radius \( r \) is directly proportional to the momentum of the particle.

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