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 radii of the circular orbits of an electron, a proton, a deuteron, and an alpha particle when they are moving in a magnetic field. The formula for the radius \( R \) of the circular path of a charged particle in a magnetic field is given by:

\[ R = \frac{mv}{qB} \]

where:
- \( m \) is the mass of the particle,
- \( v \) is the speed of the particle,
- \( q \) is the charge of the particle,
- \( B \) is the magnetic field strength.

Since all particles have the same speed \( v \) and are in the same magnetic field \( B \), we can express the radius in terms of the mass-to-charge ratio \( \frac{m}{q} \):

\[ R \propto \frac{m}{q} \]

Now, we will analyze each particle:

1. **Electron**:
- Mass: \( m_e \)
- Charge: \( q_e = -e \) (magnitude)
- Radius:
\[ R_e = \frac{m_e v}{eB} \]

2. **Proton**:
- Mass: \( m_p \)
- Charge: \( q_p = +e \)
- Radius:
\[ R_p = \frac{m_p v}{eB} \]

3. **Deuteron**:
- Mass: \( m_d = 2m_p \) (approximately twice the mass of a proton)
- Charge: \( q_d = +e \)
- Radius:
\[ R_d = \frac{2m_p v}{eB} \]

4. **Alpha Particle**:
- Mass: \( m_{\alpha} = 4m_p \) (approximately four times the mass of a proton)
- Charge: \( q_{\alpha} = +2e \)
- Radius:
\[ R_{\alpha} = \frac{4m_p v}{2eB} = \frac{2m_p v}{eB} \]

Now, we can summarize the ratios of the radii:

- For the electron:
\[ R_e \propto \frac{m_e}{e} \]

- For the proton:
\[ R_p \propto \frac{m_p}{e} \]

- For the deuteron:
\[ R_d \propto \frac{2m_p}{e} \]

- For the alpha particle:
\[ R_{\alpha} \propto \frac{2m_p}{e} \]

From the above, we can conclude:
- \( R_d = R_{\alpha} \) (since both have the same mass-to-charge ratio)
- \( R_e < R_p < R_d = R_{\alpha} \)

Thus, the final relationship among the radii is:
\[ R_e < R_p < R_d = R_{\alpha} \]