An `(alpha)`-particle and a proton are both simultaneously projected in opposite direction into a region of constant magnetic field perpendicular to the direction of the field. After some time it is found that the velocity of the `(alpha)`-particle has changed in a direction by `45^(@)`. Then at this time, the angle between velocity vectors of `(alpha)`-particle and proton is

Options

A. `90^(@)`
B. `45^(@)`
C. `135^(@)`
D. none

Explanation

To solve the problem, we need to analyze the motion of the alpha particle and the proton in a magnetic field and determine the angle between their velocity vectors after the alpha particle has changed direction by 45 degrees.

### Step-by-Step Solution:

1. **Understand the Motion of Charged Particles in a Magnetic Field:**
- When a charged particle moves in a magnetic field, it experiences a magnetic force that is perpendicular to both its velocity and the magnetic field. This causes the particle to move in a circular path.

2. **Determine the Angular Frequency (ω) for Both Particles:**
- The angular frequency (ω) for a charged particle in a magnetic field is given by the formula:
\[ \omega = \frac{Q B}{m} \]
- For the proton:
\[ \omega_p = \frac{eB}{m_p} \]
- For the alpha particle (which has a charge of \(2e\) and mass \(4m_p\)):
\[ \omega_{\alpha} = \frac{2eB}{4m_p} = \frac{eB}{2m_p} \]
- This shows that:
\[ \omega_{\alpha} = \frac{1}{2} \omega_p \]
- This means the proton rotates faster than the alpha particle.

3. **Relate the Angle of Rotation:**
- If the alpha particle changes its direction by \(45^\circ\), we can find the corresponding angle for the proton:
\[ \text{Angle rotated by proton} = 2 \times 45^\circ = 90^\circ \]

4. **Visualize the Motion:**
- Let's visualize the situation:
- The alpha particle starts moving in one direction and after rotating \(45^\circ\), its new velocity vector is at \(45^\circ\) to its original direction.
- The proton, which rotates \(90^\circ\), will have its velocity vector perpendicular to its original direction.

5. **Determine the Angle Between the Two Velocity Vectors:**
- The original direction of the alpha particle is opposite to that of the proton.
- After the rotation:
- The alpha particle's velocity vector is at \(45^\circ\) from its original direction.
- The proton's velocity vector is at \(90^\circ\) from its original direction.
- Therefore, the angle between the two velocity vectors can be calculated as:
\[ \text{Angle between the two vectors} = 180^\circ - (90^\circ + 45^\circ) = 45^\circ \]

### Final Answer:
The angle between the velocity vectors of the alpha particle and the proton is \(45^\circ\).