A proton and an `alpha-`particle, moving with the same velocity, enter a uniform magnetic field, acting normal to the plane of their motion. The ratio of the radii of the circular paths descirbed by the proton and `alpha`-particle is

Options

A. `1:2`
B. `1:4`
C. `1:16`
D. `4:1`

Explanation

To solve the problem, we need to find the ratio of the radii of the circular paths described by a proton and an alpha particle when they move with the same velocity in a uniform magnetic field.

### Step-by-Step Solution:

1. **Understanding the Motion in a Magnetic Field**:
When a charged particle moves in a magnetic field, it experiences a magnetic force that acts as the centripetal force, causing it to move in a circular path. The magnetic force \( F \) on a charged particle is given by:
\[ F = qvB \]
where \( q \) is the charge of the particle, \( v \) is its velocity, and \( B \) is the magnetic field strength.

2. **Centripetal Force**:
The centripetal force required to keep a particle moving in a circular path of radius \( r \) is given by:
\[ F = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle.

3. **Setting the Forces Equal**:
For the particle to move in a circular path, the magnetic force must equal the centripetal force:
\[ qvB = \frac{mv^2}{r} \]

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

5. **Finding the Ratio of Radii**:
We need to find the ratio of the radii \( r_p \) (for the proton) and \( r_\alpha \) (for the alpha particle):
\[ \frac{r_p}{r_\alpha} = \frac{\frac{m_p v}{q_p B}}{\frac{m_\alpha v}{q_\alpha B}} = \frac{m_p}{m_\alpha} \cdot \frac{q_\alpha}{q_p} \]

6. **Substituting Mass and Charge Values**:
- The mass of the alpha particle \( m_\alpha \) is approximately 4 times the mass of the proton \( m_p \):
\[ m_\alpha = 4m_p \]
- The charge of the alpha particle \( q_\alpha \) is approximately 2 times the charge of the proton \( q_p \):
\[ q_\alpha = 2q_p \]

7. **Calculating the Ratio**:
Substituting these values into the ratio:
\[ \frac{r_p}{r_\alpha} = \frac{m_p}{4m_p} \cdot \frac{2q_p}{q_p} = \frac{1}{4} \cdot 2 = \frac{1}{2} \]

8. **Final Result**:
Therefore, the ratio of the radii of the circular paths described by the proton and the alpha particle is:
\[ \frac{r_p}{r_\alpha} = \frac{1}{2} \]

### Conclusion:
The ratio of the radii of the circular paths described by the proton and the alpha particle is \( 1:2 \).