Two `alpha`-particles have the ratio of their velocities as 3:2 on entering the field. If they move in different circular paths, then the ratio of the radii of their paths is

Options

A. `2:3`
B. `3:2`
C. `9:4`
D. `4:9`

Explanation

To solve the problem of finding the ratio of the radii of the circular paths of two alpha particles moving in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion of Charged Particles in a Magnetic Field**:
When charged particles like alpha particles move in a magnetic field, they experience a magnetic force that acts perpendicular to their velocity. This force causes them to move in a circular path.

2. **Formula for Radius of Circular Motion**:
The radius \( r \) of the circular path of a charged particle in a magnetic field is given by the formula:
\[ r = \frac{mv}{Bq} \]
where:
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle,
- \( B \) is the magnetic field strength,
- \( q \) is the charge of the particle.

3. **Setting Up the Problem**:
Let’s denote the first alpha particle with:
- Velocity \( v_1 \) and radius \( r_1 \)
- The second alpha particle with:
- Velocity \( v_2 \) and radius \( r_2 \)

According to the problem, the ratio of their velocities is:
\[ \frac{v_1}{v_2} = \frac{3}{2} \]

4. **Expressing the Radii**:
From the formula for the radius, we can express \( r_1 \) and \( r_2 \):
\[ r_1 = \frac{m v_1}{Bq} \quad \text{and} \quad r_2 = \frac{m v_2}{Bq} \]

5. **Finding the Ratio of the Radii**:
To find the ratio \( \frac{r_1}{r_2} \):
\[ \frac{r_1}{r_2} = \frac{\frac{m v_1}{Bq}}{\frac{m v_2}{Bq}} = \frac{v_1}{v_2} \]
Here, the mass \( m \) and charge \( q \) cancel out because they are the same for both alpha particles.

6. **Substituting the Velocity Ratio**:
Now, substituting the given ratio of velocities:
\[ \frac{r_1}{r_2} = \frac{v_1}{v_2} = \frac{3}{2} \]

### Final Answer:
Thus, the ratio of the radii of their paths is:
\[ \frac{r_1}{r_2} = \frac{3}{2} \]