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. `45^@+90^@`
D. `(45^@+90^@)/(2)`

Answer

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 its direction by 45 degrees.

### Step-by-Step Solution:

1. **Understanding the Motion in a Magnetic Field**:
- Both the alpha particle and the proton are charged particles moving in a magnetic field. The magnetic field is perpendicular to their initial velocity vectors, causing them to move in circular paths.

2. **Angular Velocity Calculation**:
- The angular velocity (\( \omega \)) of a charged particle in a magnetic field is given by:
\[ \omega = \frac{Q B}{m} \]
- For the alpha particle (charge \( Q = 2e \), mass \( m = 4m_p \)):
\[ \omega_{\alpha} = \frac{2eB}{4m_p} = \frac{eB}{2m_p} \]
- For the proton (charge \( Q = e \), mass \( m = m_p \)):
\[ \omega_{p} = \frac{eB}{m_p} \]

3. **Comparing Angular Velocities**:
- The angular velocity of the proton is twice that of the alpha particle:
\[ \omega_{p} = 2 \omega_{\alpha} \]
- This means the proton moves faster than the alpha particle.

4. **Direction Change of the Alpha Particle**:
- It is given that the alpha particle changes its direction by \( 45^\circ \). This means its velocity vector has rotated \( 45^\circ \) from its original direction.

5. **Determining the Angle of the Proton**:
- Since the proton moves twice as fast, it will cover an angle of:
\[ 2 \times 45^\circ = 90^\circ \]
- Therefore, after the same time interval, the proton's velocity vector will have rotated \( 90^\circ \).

6. **Finding the Angle Between the Two Velocity Vectors**:
- The angle between the velocity vector of the alpha particle (which has rotated \( 45^\circ \)) and the velocity vector of the proton (which has rotated \( 90^\circ \)) is:
\[ \text{Total angle} = 90^\circ + 45^\circ = 135^\circ \]

### Final Answer:
The angle between the velocity vectors of the alpha particle and the proton is \( 135^\circ \).

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