To solve the question regarding the motion of a charged particle in a gravity-free space without a change in velocity, we need to analyze each option provided.
### Step-by-Step Solution:
1. **Understanding the Problem**:
- A charged particle is moving in a gravity-free space.
- Its velocity is constant, meaning there is no net force acting on it (according to Newton's first law).
2. **Identifying Forces**:
- The forces acting on a charged particle can be due to an electric field (E) and a magnetic field (B).
- The force due to an electric field is given by \( F_E = Q \cdot E \).
- The force due to a magnetic field is given by \( F_B = Q \cdot (V \times B) \), where \( V \) is the velocity of the particle.
3. **Analyzing Each Option**:
- **Option A: \( E = 0 \) and \( B = 0 \)**:
- If both electric and magnetic fields are zero, then \( F_E = 0 \) and \( F_B = 0 \).
- Therefore, the net force \( F = F_E + F_B = 0 \).
- The particle will continue to move with constant velocity. This option is **possible**.
- **Option B: \( E = 0 \) and \( B \neq 0 \)**:
- Here, \( F_E = 0 \) and \( F_B \neq 0 \).
- The magnetic force \( F_B \) can be zero if the velocity vector \( V \) is parallel to the magnetic field \( B \) (i.e., \( V \times B = 0 \)).
- Thus, this option is **possible** under the condition that \( V \) is parallel to \( B \).
- **Option C: \( E \neq 0 \) and \( B = 0 \)**:
- In this case, \( F_E \neq 0 \) and \( F_B = 0 \).
- Since there is a non-zero electric field, the force \( F_E \) will cause the particle to accelerate, changing its velocity. This option is **not possible**.
- **Option D: \( E \neq 0 \) and \( B \neq 0 \)**:
- Here, both forces \( F_E \) and \( F_B \) are non-zero.
- The particle can still maintain a constant velocity if the forces are equal in magnitude but opposite in direction (i.e., \( F_E + F_B = 0 \)). This is possible under specific conditions. Thus, this option is **possible**.
4. **Conclusion**:
- The possible options for the charged particle moving without a change in velocity are:
- **Option A**: \( E = 0 \), \( B = 0 \)
- **Option B**: \( E = 0 \), \( B \neq 0 \) (if \( V \) is parallel to \( B \))
- **Option D**: \( E \neq 0 \), \( B \neq 0 \) (if forces balance each other)
### Final Answer:
The correct options are **A, B, and D**.
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