Ionized hydrogen atoms and `alpha-`particle with moments enters perpendicular to a constant megnetic field. B. The ratio of their radii of their paths `r_(H): r_(alpha)` be :

Options

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

Explanation

To solve the problem, we need to determine the ratio of the radii of the paths of ionized hydrogen atoms and alpha particles when they enter a magnetic field perpendicularly.

### Step-by-Step Solution:

1. **Understanding the Motion in a Magnetic Field**:
When charged particles move in a magnetic field, they experience a magnetic force that causes them to move in a circular path. The radius \( r \) of the circular path is given by the formula:
\[ r = \frac{mv}{qB} \]
where:
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle,
- \( q \) is the charge of the particle,
- \( B \) is the magnetic field strength.

2. **Identifying the Particles**:
- For the ionized hydrogen atom (proton), the charge \( q_H = +e \) (where \( e \) is the elementary charge).
- For the alpha particle, which consists of 2 protons and 2 neutrons, the charge \( q_{\alpha} = +2e \).

3. **Assuming Equal Momentum**:
The problem states that both particles have the same momentum \( p \). Therefore, we can express the momentum as:
\[ p = mv \]
This means:
\[ mv_H = mv_{\alpha} \]

4. **Calculating the Radii**:
- For the hydrogen atom:
\[ r_H = \frac{mv_H}{q_H B} = \frac{mv_H}{eB} \]
- For the alpha particle:
\[ r_{\alpha} = \frac{mv_{\alpha}}{q_{\alpha} B} = \frac{mv_{\alpha}}{2eB} \]

5. **Finding the Ratio of the Radii**:
Since \( mv_H = mv_{\alpha} \), we can denote this common momentum as \( p \):
\[ r_H = \frac{p}{eB} \quad \text{and} \quad r_{\alpha} = \frac{p}{2eB} \]
Now, we can find the ratio of the radii:
\[ \frac{r_H}{r_{\alpha}} = \frac{\frac{p}{eB}}{\frac{p}{2eB}} = \frac{2}{1} \]

6. **Conclusion**:
The ratio of the radii of the paths of the ionized hydrogen atom to the alpha particle is:
\[ r_H : r_{\alpha} = 2 : 1 \]

### Final Answer:
The ratio of the radii of their paths is \( 2 : 1 \).