A cubical region of space is filled with some uniform electric and magnetic fields. An electron enters the cube across one of its faces with velocity `vecv` and a positron enters via opposite face with velocity `-vecv`. At this instant,

Options

A. the electric forces on both the particles cause identical accelerations
B. the magnetic forces on both the particles cause equal acceleration.
C. Both particles gain or loose energy at the same rate
D. the motion of the centre of mass (CM) is determined by B alone.

Explanation

To solve the problem, we need to analyze the behavior of an electron and a positron entering a cubical region filled with uniform electric and magnetic fields. The electron enters with velocity \(\vec{v}\) and the positron enters with velocity \(-\vec{v}\).

### Step-by-Step Solution:

1. **Understanding the Forces Acting on the Particles**:
- Both the electron and positron will experience forces due to the electric field (\(\vec{E}\)) and the magnetic field (\(\vec{B}\)).
- The electric force on the electron (\(F_e\)) is given by:
\[ \vec{F_e} = q \vec{E} \]
where \(q = -e\) for the electron.
- The electric force on the positron (\(F_p\)) is given by:
\[ \vec{F_p} = q \vec{E} \]
where \(q = +e\) for the positron.

2. **Direction of Electric Forces**:
- Since the electric field is uniform, the direction of the electric force on the electron will be opposite to that on the positron. If we assume the electric field is in the positive \(y\)-direction, then:
- The force on the electron will be in the negative \(y\)-direction.
- The force on the positron will be in the positive \(y\)-direction.

3. **Acceleration of the Particles**:
- The acceleration (\(a\)) of each particle can be calculated using Newton's second law:
\[ a = \frac{F}{m} \]
- Since both particles have the same mass \(m\) and experience forces of equal magnitude (but opposite direction), their accelerations will also be equal in magnitude but opposite in direction. Therefore, they do not have identical acceleration.

4. **Magnetic Forces on the Particles**:
- The magnetic force (\(F_m\)) on a charged particle moving in a magnetic field is given by:
\[ \vec{F_m} = q (\vec{v} \times \vec{B}) \]
- For the electron moving with velocity \(\vec{v}\) and the positron moving with \(-\vec{v}\), the magnetic forces will also be equal in magnitude but will depend on the direction of the velocity and magnetic field.

5. **Direction of Magnetic Forces**:
- Using the right-hand rule, the direction of the magnetic force on the electron will be opposite to that on the positron if they are moving in opposite directions. Therefore, the magnetic forces will also be equal in magnitude and direction for both particles.

6. **Energy Gain or Loss**:
- The work done by the magnetic force is zero since it is always perpendicular to the displacement. Thus, any energy gain or loss will occur due to the electric force.
- Since both particles experience the same electric force and travel the same distance, they will gain or lose energy at the same rate.

7. **Motion of the Center of Mass (COM)**:
- The net electric force acting on the system is zero because the forces on the electron and positron cancel each other out.
- However, the net magnetic force will determine the motion of the center of mass. Since the magnetic forces are equal and opposite, the motion of the center of mass will be influenced by the magnetic field.

### Conclusion:
- The correct statements are:
- The magnetic forces on both particles cause equal acceleration (correct).
- Both particles gain or lose energy at the same rate (correct).
- The motion of the center of mass is determined by the magnetic field alone (correct).

### Final Answer:
The correct options are 2, 3, and 4.