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, we need to find the ratio of the radii of the circular paths of two alpha particles moving in a magnetic field, given the ratio of their velocities.

### 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 as the centripetal force, causing it to move in a circular path. The centripetal force (Fc) is given by:
\[ F_c = \frac{mv^2}{r} \]
where \(m\) is the mass of the particle, \(v\) is its velocity, and \(r\) is the radius of the circular path.

2. **Magnetic Force on the Charged Particle**:
The magnetic force (Fm) acting on the charged particle is given by:
\[ F_m = qvB \]
where \(q\) is the charge of the particle, \(v\) is its velocity, and \(B\) is the magnetic field strength.

3. **Equating the Forces**:
For a charged particle moving in a magnetic field, the centripetal force is provided by the magnetic force:
\[ \frac{mv^2}{r} = qvB \]

4. **Deriving the Expression for Radius**:
Rearranging the above equation gives us the expression for the radius \(r\):
\[ r = \frac{mv}{qB} \]

5. **Finding the Ratio of Radii**:
Let’s denote the two alpha particles as particle 1 and particle 2. The ratio of their radii \(r_1\) and \(r_2\) can be expressed as:
\[ \frac{r_1}{r_2} = \frac{\frac{m v_1}{qB}}{\frac{m v_2}{qB}} = \frac{v_1}{v_2} \]
Here, we see that the mass \(m\), charge \(q\), and magnetic field \(B\) are the same for both alpha particles, so they cancel out.

6. **Substituting the Given Velocity Ratio**:
According to the problem, the ratio of their velocities is given as:
\[ \frac{v_1}{v_2} = \frac{3}{2} \]
Therefore, substituting this into the ratio of the radii gives:
\[ \frac{r_1}{r_2} = \frac{3}{2} \]

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