A cubical region of space is filled with some uniform electric and magnetic field. An electron enters the cube across one of its faces with velocity v and a positron enters via opposite face with velocity- v. At this instant, which one of the following is not correct?

Options

A. The electric forces on both the particles cause identical acceleration.
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 is determined by B alone.

Answer

To solve the problem step by step, we will analyze the situation involving the electron and positron entering a cubical region filled with uniform electric and magnetic fields.

### Step-by-Step Solution:

1. **Identify the Charges and Velocities**:
- An electron enters the cube with velocity \( \mathbf{v} \).
- A positron enters the cube with velocity \( -\mathbf{v} \).
- The charges are \( -e \) for the electron and \( +e \) for the positron.

2. **Calculate the Electric Force on Each Particle**:
- The electric force on the electron is given by:
\[ \mathbf{F}_e = -e \mathbf{E} \]
- The electric force on the positron is given by:
\[ \mathbf{F}_p = +e \mathbf{E} \]
- The accelerations can be calculated using \( \mathbf{F} = m \mathbf{a} \):
- For the electron:
\[ \mathbf{a}_e = \frac{-e \mathbf{E}}{m} \]
- For the positron:
\[ \mathbf{a}_p = \frac{+e \mathbf{E}}{m} \]

3. **Compare the Accelerations**:
- Since \( \mathbf{a}_e \) and \( \mathbf{a}_p \) have opposite signs, they are not identical. Thus, the statement that "the electric force on both the particles causes identical acceleration" is incorrect.

4. **Calculate the Magnetic Force on Each Particle**:
- The magnetic force on the electron is given by:
\[ \mathbf{F}_{B_e} = -e (\mathbf{v} \times \mathbf{B}) \]
- The magnetic force on the positron is given by:
\[ \mathbf{F}_{B_p} = +e (-\mathbf{v} \times \mathbf{B}) = -e (-\mathbf{v} \times \mathbf{B}) = e (\mathbf{v} \times \mathbf{B}) \]
- The corresponding accelerations are:
- For the electron:
\[ \mathbf{a}_{B_e} = \frac{-e (\mathbf{v} \times \mathbf{B})}{m} \]
- For the positron:
\[ \mathbf{a}_{B_p} = \frac{e (\mathbf{v} \times \mathbf{B})}{m} \]

5. **Compare the Magnetic Accelerations**:
- The magnitudes of the magnetic forces are equal, thus the accelerations due to the magnetic force are equal in magnitude but opposite in direction. Therefore, the statement that "the magnetic force on both the particles causes equal acceleration" is correct.

6. **Energy Gain or Loss**:
- Both particles have the same charge magnitude and mass. Therefore, they gain or lose energy at the same rate when subjected to the same electric and magnetic fields. This statement is correct.

7. **Motion of the Center of Mass**:
- The net electric force on the system is zero since the forces on the electron and positron cancel each other out. The magnetic forces also cancel out due to equal and opposite forces. Thus, the motion of the center of mass is influenced only by the magnetic field. This statement is also correct.

### Conclusion:
The incorrect statement is that "the electric force on both the particles causes identical acceleration."