To determine the conditions under which a charged particle continues to move with a constant velocity, we need to analyze the forces acting on the particle in different scenarios involving electric (E) and magnetic (B) fields.
### Step-by-Step Solution:
1. **Understanding Constant Velocity**:
A charged particle moves with constant velocity when the net force acting on it is zero. According to Newton's first law of motion, if no net force acts on an object, it will maintain its state of motion.
2. **Analyzing the First Option (E = 0, B ≠ 0)**:
- If the electric field (E) is zero, the electric force (F_e = qE) is also zero.
- The magnetic force (F_m = qvBsinθ) depends on the velocity (v) and the magnetic field (B). If θ = 0 (the angle between velocity and magnetic field), then sin(0) = 0, resulting in F_m = 0.
- Since both forces are zero, the net force is zero, and the particle continues to move with constant velocity.
- **Conclusion**: This option is valid.
3. **Analyzing the Second Option (E ≠ 0, B ≠ 0)**:
- Here, both electric and magnetic fields are present, leading to non-zero forces.
- The electric force is F_e = qE, and the magnetic force is F_m = qvBsinθ.
- For the particle to move with constant velocity, these forces must balance each other: F_e = F_m. This can happen if v = E/B, which is a condition known as a velocity selector.
- **Conclusion**: This option is also valid.
4. **Analyzing the Third Option (E ≠ 0, B = 0)**:
- With B = 0, the magnetic force F_m = 0.
- However, since E ≠ 0, the electric force F_e = qE is non-zero.
- A non-zero force implies that the particle will experience an acceleration, changing its velocity.
- **Conclusion**: This option is not valid.
5. **Analyzing the Fourth Option (E = 0, B = 0)**:
- In this scenario, both electric and magnetic forces are zero (F_e = 0 and F_m = 0).
- With no net force acting on the particle, it will maintain its constant velocity.
- **Conclusion**: This option is valid.
### Final Answer:
The charged particle would continue to move with a constant velocity in regions described by options 1, 2, and 4 (E = 0, B ≠ 0; E ≠ 0, B ≠ 0; E = 0, B = 0).
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