A proton an 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`

Answer

To solve the problem of finding the ratio of the radii of the circular paths described by a proton and an alpha particle moving with the same velocity in a uniform magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Radius in a Magnetic Field**:
The radius \( r \) of the circular path of a charged particle moving in a magnetic field 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. **Identify the Variables for Proton and Alpha Particle**:
- For the proton:
- Mass = \( m_p \)
- Charge = \( q_p \)
- For the alpha particle (which consists of 2 protons and 2 neutrons):
- Mass = \( m_{\alpha} = 4m_p \) (since it has 4 nucleons)
- Charge = \( q_{\alpha} = 2q_p \) (since it has 2 protons)

3. **Write the Radius for Each Particle**:
- Radius of the proton:
\[ r_p = \frac{m_p v}{q_p B} \]
- Radius of the alpha particle:
\[ r_{\alpha} = \frac{m_{\alpha} v}{q_{\alpha} B} = \frac{4m_p v}{2q_p B} = \frac{2m_p v}{q_p B} \]

4. **Take the Ratio of the Radii**:
Now, we can find the ratio of the radii of the proton to the alpha particle:
\[ \frac{r_p}{r_{\alpha}} = \frac{\frac{m_p v}{q_p B}}{\frac{2m_p v}{q_p B}} = \frac{m_p v}{q_p B} \cdot \frac{q_p B}{2m_p v} = \frac{1}{2} \]

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

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

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