An electron, a proton and a `He^(+)` ion projected into a magnetic field with same kinetic energy, with velocities being perpendicular to the magnetic field. The order of the radii of circles traced by them is :

Options

A. `r_(p)gtf_(He+)gtr_(e)`
B. `r_(He+)gtr_(p)gtr_(e)`
C. `r_(p)=r_(He+)ltr_(e)`
D. None

Answer

To solve the problem, we need to find the order of the radii of the circular paths traced by an electron, a proton, and a Helium ion (He⁺) when they are projected into a magnetic field with the same kinetic energy and their velocities are perpendicular to the magnetic field.

### 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 magnetic force that acts as a centripetal force, causing it to move in a circular path.

2. **Magnetic Force and Centripetal Force**:
The magnetic force \( F \) on a charged particle is given by:
\[ F = qvB \]
where \( q \) is the charge of the particle, \( v \) is its velocity, and \( B \) is the magnetic field strength.

This force provides the centripetal force required for circular motion:
\[ F = \frac{mv^2}{R} \]
where \( m \) is the mass of the particle and \( R \) is the radius of the circular path.

3. **Equating Forces**:
Setting the magnetic force equal to the centripetal force gives us:
\[ qvB = \frac{mv^2}{R} \]

4. **Solving for Radius \( R \)**:
Rearranging the equation to solve for the radius \( R \):
\[ R = \frac{mv}{qB} \]

5. **Kinetic Energy Relation**:
The kinetic energy \( K \) of a particle is given by:
\[ K = \frac{1}{2} mv^2 \]
Since we are given that all three particles have the same kinetic energy, we can express \( v \) in terms of \( K \):
\[ v = \sqrt{\frac{2K}{m}} \]

6. **Substituting \( v \) back into the Radius Equation**:
Substitute \( v \) into the radius equation:
\[ R = \frac{m \sqrt{\frac{2K}{m}}}{qB} = \frac{\sqrt{2Km}}{qB} \]

7. **Analyzing the Radius**:
Since \( K \), \( q \), and \( B \) are constants for all three particles, we find that the radius \( R \) is proportional to the square root of the mass \( m \):
\[ R \propto \sqrt{m} \]

8. **Comparing the Masses**:
- Mass of electron \( m_e \)
- Mass of proton \( m_p \) (approximately 1836 times the mass of an electron)
- Mass of He⁺ ion \( m_{He} \) (approximately 4 times the mass of an electron)

Therefore, the order of masses is:
\[ m_{He} > m_p > m_e \]

9. **Determining the Order of Radii**:
Since \( R \propto \sqrt{m} \), the order of the radii will be the same as the order of the masses:
\[ R_{He} > R_{p} > R_{e} \]

### Final Answer:
The order of the radii of the circles traced by the electron, proton, and He⁺ ion is:
\[ R_{He^+} > R_{p} > R_{e} \]