When a charged particle with charge q and mass m enters uniform magnetic field B with velocity v at right angles to B, the force on the moving particle is given by qvB. This force acts as the centripetal force making the charged particle go in a uniform circular motion with radius `r = (mv)/(Bq)`
Now if a hydrogen ion and a deuterium ion enter the magnetic field with velocities in the ratio 2:1 respectively, then the ratio of their radii will be -

Options

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

Explanation

To solve the problem, we need to find the ratio of the radii of the circular motion of a hydrogen ion and a deuterium ion when they enter a uniform magnetic field with given velocities.

### Step-by-Step Solution:

1. **Understanding the Formula for Radius**:
The radius \( r \) of the circular motion 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.

2. **Identifying the Particles**:
We have two particles:
- Hydrogen ion (H\(^+\)) with mass \( m_H \).
- Deuterium ion (D\(^+\)) with mass \( m_D = 2m_H \) (since deuterium has one proton and one neutron).

3. **Given Velocity Ratio**:
The velocities of the hydrogen and deuterium ions are given in the ratio:
\[ v_H : v_D = 2 : 1 \]
This means \( v_H = 2v \) and \( v_D = v \) for some velocity \( v \).

4. **Calculating the Radii**:
- For the hydrogen ion:
\[ r_H = \frac{m_H v_H}{Bq_H} = \frac{m_H (2v)}{Bq_H} = \frac{2m_H v}{Bq_H} \]
- For the deuterium ion:
\[ r_D = \frac{m_D v_D}{Bq_D} = \frac{(2m_H) v}{Bq_D} = \frac{2m_H v}{Bq_D} \]

5. **Finding the Ratio of Radii**:
We need to find the ratio \( \frac{r_H}{r_D} \):
\[ \frac{r_H}{r_D} = \frac{\frac{2m_H v}{Bq_H}}{\frac{2m_H v}{Bq_D}} = \frac{q_D}{q_H} \]
Since both hydrogen and deuterium ions have the same charge (q), we have:
\[ \frac{r_H}{r_D} = 1 \]

6. **Conclusion**:
The ratio of the radii of the hydrogen ion to the deuterium ion is:
\[ \frac{r_H}{r_D} = 1 \]

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