If the ratio of time periods of circular motion of two charged particles in magnetic field in `1//2`, then the ratio of their kinetic energies must be :

Options

A. `1:sqrt(2)`
B. `sqrt(2):1`
C. `2:1`
D. it can have any value

Explanation

To solve the problem, we need to analyze the relationship between the time periods of two charged particles moving in a magnetic field and their kinetic energies.

### Step-by-Step Solution:

1. **Understanding the Time Period Formula**:
The time period \( T \) of a charged particle moving in a magnetic field is given by the formula:
\[ T = \frac{2\pi m}{qB} \]
where:
- \( m \) is the mass of the particle,
- \( q \) is the charge of the particle,
- \( B \) is the magnetic field strength.

2. **Setting Up the Ratios**:
Given that the ratio of the time periods \( \frac{T_1}{T_2} = \frac{1}{2} \), we can express this as:
\[ \frac{T_1}{T_2} = \frac{2\pi m_1}{q_1 B_1} \div \frac{2\pi m_2}{q_2 B_2} = \frac{m_1 q_2 B_2}{m_2 q_1 B_1} \]
Since \( T_1/T_2 = 1/2 \), we can write:
\[ \frac{m_1 q_2 B_2}{m_2 q_1 B_1} = \frac{1}{2} \]

3. **Kinetic Energy of Charged Particles**:
The kinetic energy \( K \) of a charged particle is given by:
\[ K = \frac{1}{2} mv^2 \]
The speed \( v \) of a charged particle in a magnetic field can be expressed as:
\[ v = \frac{qBr}{m} \]
where \( r \) is the radius of the circular path.

4. **Expressing Kinetic Energy in Terms of Time Period**:
We can relate the speed \( v \) to the time period \( T \):
\[ v = \frac{2\pi r}{T} \]
Thus, substituting for \( v \) in the kinetic energy formula:
\[ K = \frac{1}{2} m \left(\frac{2\pi r}{T}\right)^2 = \frac{2\pi^2 m r^2}{T^2} \]

5. **Finding the Ratio of Kinetic Energies**:
Now, we can find the ratio of the kinetic energies \( \frac{K_1}{K_2} \):
\[ \frac{K_1}{K_2} = \frac{\frac{2\pi^2 m_1 r_1^2}{T_1^2}}{\frac{2\pi^2 m_2 r_2^2}{T_2^2}} = \frac{m_1 r_1^2}{m_2 r_2^2} \cdot \frac{T_2^2}{T_1^2} \]
Since \( \frac{T_1}{T_2} = \frac{1}{2} \), we have \( \frac{T_2}{T_1} = 2 \). Thus:
\[ \frac{K_1}{K_2} = \frac{m_1 r_1^2}{m_2 r_2^2} \cdot (2^2) = \frac{m_1 r_1^2}{m_2 r_2^2} \cdot 4 \]

6. **Conclusion**:
Without specific values for \( m_1, m_2, r_1, \) and \( r_2 \), we cannot determine a numerical ratio for the kinetic energies. However, we can conclude that the ratio of kinetic energies is dependent on the masses and radii of the particles, multiplied by 4.

### Final Answer:
The ratio of their kinetic energies cannot be determined without additional information about the masses and radii of the particles.