Proton , deuton and alpha particle of same kinetic energy are moving in circular trajectories in a constant magnetic field. The radii of proton , deuteron and alpha particle are respectively `r_(p), r_(d) and r_(alpha)`. Which one of the following relation is correct?

Options

A. `r_(e ) lt r_(alpha) lt r_(p)`
B. `r_(e ) gt r_(p)=r_(alpha)`
C. `r_(e )lt r_(p) lt=r_(alpha)`
D. `r_(e ) lt r_(p) lt r_(alpha)`

Answer

To solve the problem, we need to analyze the motion of a proton, deuteron, and alpha particle in a magnetic field, given that they all have the same kinetic energy.

### Step-by-Step Solution:

1. **Understanding the Forces**:
The centripetal force required for circular motion is provided by the magnetic force. The magnetic force acting on a charged particle moving in a magnetic field is given by:
\[ F_B = qvB \]
where \( q \) is the charge, \( v \) is the velocity, and \( B \) is the magnetic field strength.

2. **Setting Up the Equation**:
For circular motion, the centripetal force can also be expressed as:
\[ F_C = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle, \( v \) is its velocity, and \( r \) is the radius of the circular path. Setting these two forces equal gives:
\[ qvB = \frac{mv^2}{r} \]

3. **Rearranging the Equation**:
Rearranging the above equation to solve for the radius \( r \):
\[ r = \frac{mv}{qB} \]

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

5. **Substituting Velocity into the Radius Equation**:
Substituting \( v \) into the radius equation:
\[ r = \frac{m \sqrt{\frac{2K}{m}}}{qB} = \frac{\sqrt{2Km}}{qB} \]

6. **Finding the Relation Between Radii**:
Since \( K \) and \( B \) are constants for all three particles, the radius \( r \) is proportional to \( \sqrt{\frac{m}{q}} \):
\[ r \propto \sqrt{\frac{m}{q}} \]

7. **Calculating for Each Particle**:
- For the proton: \( m_p = 1 \, \text{u}, q_p = 1 \, e \)
- For the deuteron: \( m_d = 2 \, \text{u}, q_d = 1 \, e \)
- For the alpha particle: \( m_\alpha = 4 \, \text{u}, q_\alpha = 2 \, e \)

Now we can express the ratios of the radii:
\[ \frac{r_p}{r_d} = \frac{\sqrt{\frac{m_p}{q_p}}}{\sqrt{\frac{m_d}{q_d}}} = \frac{\sqrt{\frac{1}{1}}}{\sqrt{\frac{2}{1}}} = \frac{1}{\sqrt{2}} \]
\[ \frac{r_p}{r_\alpha} = \frac{\sqrt{\frac{m_p}{q_p}}}{\sqrt{\frac{m_\alpha}{q_\alpha}}} = \frac{\sqrt{\frac{1}{1}}}{\sqrt{\frac{4}{2}}} = \frac{1}{\sqrt{2}} \]
\[ \frac{r_d}{r_\alpha} = \frac{\sqrt{\frac{m_d}{q_d}}}{\sqrt{\frac{m_\alpha}{q_\alpha}}} = \frac{\sqrt{\frac{2}{1}}}{\sqrt{\frac{4}{2}}} = 1 \]

8. **Final Relationships**:
From the calculations, we find:
\[ r_p : r_d : r_\alpha = 1 : \sqrt{2} : \frac{1}{\sqrt{2}} \]
Thus, the correct relationship is:
\[ r_p < r_d < r_\alpha \]

### Conclusion:
The correct relationship among the radii of the proton, deuteron, and alpha particle is that the radius of the proton is the smallest, followed by the deuteron, and the alpha particle has the largest radius.