The ratio of radii of trajectories of a-particle and proton moving in plane of paper in a region of uniform magnetic field, normal to the plane of paper and having equal linear momentum:

Options

A. `2:1`
B. `1:1`
C. `2:5`
D. `1:2`

Explanation

To solve the problem of finding the ratio of the radii of the trajectories of an alpha particle and a proton moving in a uniform magnetic field with equal linear momentum, we can follow these steps:

### 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 a centripetal force, causing it to move in a circular path. The radius \( r \) of this circular path is given by the formula:
\[ r = \frac{mv}{qB} \]
where:
- \( m \) = mass of the particle,
- \( v \) = velocity of the particle,
- \( q \) = charge of the particle,
- \( B \) = magnetic field strength.

2. **Equal Linear Momentum**:
- We are given that the linear momentum \( p \) of both the alpha particle and the proton is equal. Linear momentum is defined as:
\[ p = mv \]
Therefore, for both particles:
\[ p_{\alpha} = m_{\alpha} v_{\alpha} \quad \text{and} \quad p_{p} = m_{p} v_{p} \]
Since \( p_{\alpha} = p_{p} \), we have:
\[ m_{\alpha} v_{\alpha} = m_{p} v_{p} \]

3. **Expressing Velocity in Terms of Momentum**:
- From the momentum equation, we can express the velocities as:
\[ v_{\alpha} = \frac{p}{m_{\alpha}} \quad \text{and} \quad v_{p} = \frac{p}{m_{p}} \]

4. **Substituting Velocity into the Radius Formula**:
- Now substituting these expressions for velocity into the radius formula:
\[ r_{\alpha} = \frac{m_{\alpha} v_{\alpha}}{q_{\alpha} B} = \frac{m_{\alpha} \left(\frac{p}{m_{\alpha}}\right)}{q_{\alpha} B} = \frac{p}{q_{\alpha} B} \]
\[ r_{p} = \frac{m_{p} v_{p}}{q_{p} B} = \frac{m_{p} \left(\frac{p}{m_{p}}\right)}{q_{p} B} = \frac{p}{q_{p} B} \]

5. **Finding the Ratio of Radii**:
- Now, we can find the ratio of the radii:
\[ \frac{r_{\alpha}}{r_{p}} = \frac{\frac{p}{q_{\alpha} B}}{\frac{p}{q_{p} B}} = \frac{q_{p}}{q_{\alpha}} \]

6. **Charge of the Particles**:
- The charge of a proton \( q_{p} = e \) (elementary charge).
- The charge of an alpha particle \( q_{\alpha} = 2e \) (since it consists of 2 protons).
- Therefore, the ratio becomes:
\[ \frac{r_{\alpha}}{r_{p}} = \frac{e}{2e} = \frac{1}{2} \]

### Final Answer:
The ratio of the radii of the trajectories of the alpha particle to the proton is:
\[ \frac{r_{\alpha}}{r_{p}} = \frac{1}{2} \]