An electron, a proton, a deuteron and an alpha particle, each having the same speed are in a region of constant magnetic field perpendicular to the direction of the velocities of the particles. The radius of the circular orbits of these particles are respectively `R_(e), R_(p), R_(d)` and `R_(alpha)` It follows that

Options

A. `R_e = R_p`
B. `R_p = R_d`
C. `R_d = R_(alpha)`
D. `R_p = R_alpha`

Answer

To solve the problem, we need to determine the relationship between the radii of the circular orbits of an electron, a proton, a deuteron, and an alpha particle when they are moving in a magnetic field with the same speed.

### Step-by-Step Solution:

1. **Understanding the Motion in a Magnetic Field**:
When a charged particle moves in a magnetic field, it experiences a magnetic force that acts as a centripetal force, causing it to move in a circular path. 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.

2. **Equating Forces**:
The magnetic force provides the centripetal force necessary for circular motion, which is given by:
\[ F = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle and \( r \) is the radius of the circular path. Setting these two expressions for force equal gives:
\[ qvB = \frac{mv^2}{r} \]

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

4. **Analyzing Charge-to-Mass Ratios**:
Since all particles have the same speed \( v \) and are in the same magnetic field \( B \), the radius \( r \) depends on the charge-to-mass ratio \( \frac{q}{m} \):
\[ r \propto \frac{m}{q} \]

5. **Calculating Charge and Mass for Each Particle**:
- **Electron**:
- Charge \( q_e = -e \) (magnitude \( e \))
- Mass \( m_e \approx 9.11 \times 10^{-31} \) kg
- Charge-to-mass ratio \( \frac{q_e}{m_e} = \frac{e}{m_e} \)
- **Proton**:
- Charge \( q_p = +e \)
- Mass \( m_p \approx 1.67 \times 10^{-27} \) kg
- Charge-to-mass ratio \( \frac{q_p}{m_p} = \frac{e}{m_p} \)
- **Deuteron**:
- Charge \( q_d = +e \)
- Mass \( m_d \approx 2 \times 1.67 \times 10^{-27} \) kg (roughly twice the proton mass)
- Charge-to-mass ratio \( \frac{q_d}{m_d} = \frac{e}{2m_p} \)
- **Alpha Particle**:
- Charge \( q_{\alpha} = +2e \)
- Mass \( m_{\alpha} \approx 4 \times 1.67 \times 10^{-27} \) kg (roughly four times the proton mass)
- Charge-to-mass ratio \( \frac{q_{\alpha}}{m_{\alpha}} = \frac{2e}{4m_p} = \frac{e}{2m_p} \)

6. **Comparing the Ratios**:
- For the electron, \( \frac{q_e}{m_e} \) is much larger than for the proton, deuteron, and alpha particle.
- For the proton and deuteron, the charge-to-mass ratios are the same.
- For the alpha particle, it has the same charge-to-mass ratio as the deuteron.

7. **Conclusion**:
The radii of the circular orbits are related as follows:
\[ R_e > R_p = R_d = R_{\alpha} \]
Therefore, the correct option is \( R_p = R_d = R_{\alpha} \).

### Final Answer:
The relationship is \( R_e > R_p = R_d = R_{\alpha} \).