A particle moves in a region having a uniform magnetic field and a parallel, uniform electric field. At some instant, the velocity of the particle is perpendicular to the field directjion. The path of the particle will be

Options

A. a straight line
B. a circle
C. a helix with uniform pitch
D. a helix with nonuniform pitch

Answer

To solve the problem of a particle moving in a uniform magnetic field and a parallel, uniform electric field, we will analyze the forces acting on the particle and the resulting motion.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Particle**:
- The particle experiences two forces: the electric force (\( \vec{F}_E \)) due to the electric field (\( \vec{E} \)) and the magnetic force (\( \vec{F}_B \)) due to the magnetic field (\( \vec{B} \)).
- The electric force is given by \( \vec{F}_E = q \vec{E} \), where \( q \) is the charge of the particle.
- The magnetic force is given by \( \vec{F}_B = q (\vec{v} \times \vec{B}) \), where \( \vec{v} \) is the velocity of the particle.

2. **Determine the Direction of Forces**:
- The electric field is uniform and parallel, which means it will exert a constant force in its direction, causing the particle to accelerate in that direction.
- The magnetic field is uniform and perpendicular to the velocity of the particle, which will cause the particle to experience a centripetal force, resulting in circular motion.

3. **Analyze the Motion**:
- Since the velocity of the particle is initially perpendicular to the magnetic field, the magnetic force will cause the particle to move in a circular path.
- Simultaneously, the electric field will cause the particle to accelerate in a straight line along the direction of the electric field.

4. **Combine the Motions**:
- The combination of the circular motion (due to the magnetic field) and the linear motion (due to the electric field) results in a helical path.
- As the particle moves, the speed increases due to the electric field, which means the radius of the circular motion will also change, leading to a non-uniform helical path.

5. **Conclusion**:
- The path of the particle will be a helix with a non-uniform pitch because the speed of the particle is increasing as it moves through the electric field.

### Final Answer:
The path of the particle will be a **helix with non-uniform pitch**.