An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii `r_(e),r_(p),r_(alpha)` respectively in a uniform magnetic field B. The relation between `r_(e),r_(p),r_(alpha)` is:

Options

A. `r_(e)gtr_(p)=r_(alpha)`
B. `r_(e)ltr_(p)=r_(alpha)`
C. `r_(e)ltr_(p)ltr_(alpha)`
D. `r_(e)ltr_(alpha)ltr_(p)`

Explanation

To solve the problem, we need to establish the relationship between the radii of the circular orbits of an electron, a proton, and an alpha particle when they have the same kinetic energy and are moving in a uniform magnetic field \( B \).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The radius \( r \) of the circular path of a charged particle moving in a magnetic field is given by the formula:
\[ r = \frac{mv}{qB} \]
where:
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle,
- \( q \) is the charge of the particle,
- \( B \) is the magnetic field strength.

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

3. **Substituting Velocity into the Radius Formula**:
Substituting the expression for \( v \) into the radius formula:
\[ r = \frac{m \cdot \sqrt{\frac{2 \cdot KE}{m}}}{qB} \]
Simplifying this gives:
\[ r = \frac{\sqrt{2 \cdot KE \cdot m}}{qB} \]

4. **Analyzing Each Particle**:
- For the **electron**:
- Mass \( m_e \)
- Charge \( q_e = e \)
- Radius \( r_e = \frac{\sqrt{2 \cdot KE \cdot m_e}}{eB} \)

- For the **proton**:
- Mass \( m_p \)
- Charge \( q_p = e \)
- Radius \( r_p = \frac{\sqrt{2 \cdot KE \cdot m_p}}{eB} \)

- For the **alpha particle** (which consists of 2 protons and 2 neutrons):
- Mass \( m_{\alpha} = 4m_p \)
- Charge \( q_{\alpha} = 2e \)
- Radius \( r_{\alpha} = \frac{\sqrt{2 \cdot KE \cdot 4m_p}}{2eB} = \frac{\sqrt{8 \cdot KE \cdot m_p}}{2eB} = \frac{\sqrt{2 \cdot KE \cdot m_p}}{eB} \sqrt{2} \)

5. **Comparing the Radii**:
Now we can compare the radii:
- For the electron:
\[ r_e \propto \frac{\sqrt{m_e}}{e} \]
- For the proton:
\[ r_p \propto \frac{\sqrt{m_p}}{e} \]
- For the alpha particle:
\[ r_{\alpha} \propto \frac{\sqrt{2m_p}}{2e} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{m_p}}{e} \]

6. **Final Relation**:
Since \( m_e < m_p < 4m_p \), we can conclude:
\[ r_e < r_p < r_{\alpha} \]

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