Two protons enter a region of transverse magnetic field. What will be ratio of time period of revolution if the ratio of energy is `2 sqrt2 : sqrt3` ?

Answer

To solve the problem, we need to find the ratio of the time periods of revolution of two protons entering a transverse magnetic field, given that the ratio of their kinetic energies is \( \frac{2\sqrt{2}}{\sqrt{3}} \).

### Step-by-Step Solution:

1. **Understanding the Relationship Between Kinetic Energy and Time Period**:
The kinetic energy (KE) of a charged particle moving in a magnetic field is given by:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass and \( v \) is the velocity of the particle.

2. **Time Period in a Magnetic Field**:
The time period \( T \) of a charged particle moving in a magnetic field is given by:
\[ T = \frac{2\pi m}{qB} \]
where \( q \) is the charge of the particle and \( B \) is the magnetic field strength.

3. **Ratio of Time Periods**:
Since both particles are protons, they have the same mass \( m \) and charge \( q \). Therefore, the time periods \( T_1 \) and \( T_2 \) can be expressed as:
\[ T_1 = \frac{2\pi m}{qB} \quad \text{and} \quad T_2 = \frac{2\pi m}{qB} \]
Thus, the ratio of the time periods is:
\[ \frac{T_1}{T_2} = \frac{T_1}{T_1} = 1 \]

4. **Using Kinetic Energy Ratio**:
The ratio of the kinetic energies is given as:
\[ \frac{KE_1}{KE_2} = \frac{2\sqrt{2}}{\sqrt{3}} \]
Since the kinetic energy is related to the velocity, we can express the velocities in terms of kinetic energy:
\[ KE_1 = \frac{1}{2} m v_1^2 \quad \text{and} \quad KE_2 = \frac{1}{2} m v_2^2 \]
Therefore, the ratio of the velocities can be derived from the ratio of kinetic energies:
\[ \frac{v_1^2}{v_2^2} = \frac{KE_1}{KE_2} = \frac{2\sqrt{2}}{\sqrt{3}} \]

5. **Conclusion**:
However, since the mass and charge are constant for both protons, the time periods remain the same regardless of the kinetic energy ratio. Therefore, the final ratio of the time periods is:
\[ \frac{T_1}{T_2} = 1 \]

### Final Answer:
The ratio of the time periods of revolution of the two protons is \( 1:1 \).