A proton, an electron, and a Helium nucleus, have the same energy.They are in circular orbits in a plane due to magnetic field perpendicular to the plane. Let `r_p,r_e` and `r_(He)` be their respective radii, then,

Options

A. `r^egtr_p=r_(He)`
B. `r^eltr_p=r_(He)`
C. `r^eltr_pltr_(He)`
D. `r^egtr_pgtr_(He)`

Answer

To solve the problem, we need to find the relationship between the radii of the circular orbits of a proton, an electron, and a helium nucleus when they have the same energy and are subjected to a magnetic field perpendicular to their plane of motion.

### Step-by-Step Solution:

1. **Understanding the Motion in a Magnetic Field:**
When a charged particle moves in a magnetic field perpendicular to its velocity, it experiences a centripetal force due to the magnetic field, causing it to move in a circular path. The radius \( r \) of the 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. **Relating Energy to Velocity:**
The kinetic energy \( E \) of a particle is given by:
\[ E = \frac{1}{2} mv^2 \]
From this, we can express \( mv \) in terms of energy:
\[ mv = \sqrt{2mE} \]

3. **Substituting in the Radius Formula:**
Substituting \( mv \) into the radius formula gives:
\[ r = \frac{\sqrt{2mE}}{qB} \]
Since \( E \) and \( B \) are the same for all three particles, we can simplify our analysis to:
\[ r \propto \sqrt{\frac{m}{q}} \]

4. **Comparing the Radii:**
Now we will compare the radii of the three particles:
- For the proton:
- Mass \( m_p \)
- Charge \( q_p \)
- For the electron:
- Mass \( m_e \)
- Charge \( q_e \)
- For the helium nucleus (which is a \( \text{He}^{2+} \) ion):
- Mass \( m_{He} \) (approximately 4 times the mass of a proton)
- Charge \( q_{He} \) (2 times the charge of a proton)

Using the proportionality:
\[ r_p \propto \sqrt{\frac{m_p}{q_p}}, \quad r_e \propto \sqrt{\frac{m_e}{q_e}}, \quad r_{He} \propto \sqrt{\frac{m_{He}}{q_{He}}} \]

5. **Calculating the Ratios:**
- For the proton:
\[ r_p \propto \sqrt{\frac{m_p}{q_p}} \]
- For the electron:
\[ r_e \propto \sqrt{\frac{m_e}{q_e}} = \sqrt{\frac{m_e}{e}} \]
- For the helium nucleus:
\[ r_{He} \propto \sqrt{\frac{m_{He}}{q_{He}}} = \sqrt{\frac{4m_p}{2e}} = \sqrt{\frac{2m_p}{e}} \]

6. **Establishing the Relationships:**
- Since \( m_e \ll m_p \), it follows that \( r_e < r_p \).
- For the helium nucleus, since \( m_{He} = 4m_p \) and \( q_{He} = 2e \):
\[ r_{He} = \sqrt{\frac{2m_p}{e}} > r_p \]

7. **Final Relationships:**
Thus, we can conclude:
\[ r_e < r_p = r_{He} \]

### Conclusion:
The final relationship between the radii is:
\[ r_e < r_p = r_{He} \]