Three paticles, an electron (e), a proton (p) and a helium atom (He) are moving in circular paths with constant speeds in the x-y plane in a region where a uniform magnetic field B exists along z-axis. The times taken by e, p and He inside the field to complete one revolution are `t_e, t_p and t_He` respectively. Then,

Options

A. `t_(He)gtt_p=t_e`
B. `t_(He)gtt_pgtt_e`
C. `t_(He)=t_p=t_e`
D. None of these

Explanation

To solve the problem, we need to analyze the time periods of the three particles (electron, proton, and helium atom) moving in a magnetic field. The time period of a charged particle moving in a magnetic field is given by the formula:

\[ T = \frac{2\pi m}{qB} \]

where:
- \( T \) is the time period,
- \( m \) is the mass of the particle,
- \( q \) is the charge of the particle,
- \( B \) is the magnetic field strength.

### Step-by-Step Solution:

1. **Identify the particles and their properties:**
- **Electron (e)**:
- Mass (\( m_e \)) = \( 9.11 \times 10^{-31} \) kg
- Charge (\( q_e \)) = \( -1.6 \times 10^{-19} \) C
- **Proton (p)**:
- Mass (\( m_p \)) = \( 1.67 \times 10^{-27} \) kg
- Charge (\( q_p \)) = \( +1.6 \times 10^{-19} \) C
- **Helium atom (He)**:
- Mass (\( m_{He} \)) = \( 4 \times m_p \) (approximately)
- Charge (\( q_{He} \)) = \( +2 \times q_p \)

2. **Calculate the time period for each particle:**
- **For the electron**:
\[ T_e = \frac{2\pi m_e}{|q_e|B} = \frac{2\pi m_e}{eB} \]
- **For the proton**:
\[ T_p = \frac{2\pi m_p}{|q_p|B} = \frac{2\pi m_p}{eB} \]
- **For the helium atom**:
\[ T_{He} = \frac{2\pi m_{He}}{|q_{He}|B} = \frac{2\pi (4m_p)}{2eB} = \frac{2\pi m_p}{eB} \]

3. **Compare the time periods**:
- From the calculations:
- \( T_e = \frac{2\pi m_e}{eB} \)
- \( T_p = \frac{2\pi m_p}{eB} \)
- \( T_{He} = \frac{2\pi m_p}{eB} \)

Since \( m_e < m_p \), we have:
\[ T_e < T_p = T_{He} \]

4. **Conclusion**:
The order of the time periods is:
\[ T_e < T_p = T_{He} \]
Thus, the correct option that matches this sequence is option B.