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 direction. The path of the particle will bea)A straight lineb)A circlec)A helix with uniform pitchd)A helix with nonuniform pitchCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

The path of a charged particle moving in a uniform magnetic field and a parallel, uniform electric field depends on the initial conditions of the particle's velocity and the relative strengths of the magnetic and electric fields. In this case, we are given that the velocity of the particle is perpendicular to the field direction at some instant.

1. Initial Conditions:
At the instant when the velocity of the particle is perpendicular to the field direction, we can assume that the particle's velocity can be resolved into two components:
- A component parallel to the electric field direction
- A component perpendicular to both the magnetic and electric field directions

2. Magnetic Field Effect:
When a charged particle moves in a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field direction. This force is known as the magnetic Lorentz force. In this case, since the particle's velocity is perpendicular to the magnetic field, the Lorentz force will act as a centripetal force on the particle.

3. Electric Field Effect:
The electric field will accelerate the particle in the direction of the field. This acceleration will contribute to the path of the particle.

Based on these effects, we can conclude that the path of the particle will be a helix with nonuniform pitch. Here's a detailed explanation:

- The perpendicular component of the particle's velocity will cause the particle to move in a circular path due to the magnetic Lorentz force acting as a centripetal force.
- The parallel component of the particle's velocity will cause the particle to accelerate in the direction of the electric field.
- As the particle moves in a circular path, the electric field will continue to accelerate it in the direction of the field, causing the pitch of the helix to change.

Since the pitch of the helix is not uniform, the path of the particle will be a helix with nonuniform pitch. This means that the particle's path will have both a circular component due to the magnetic field and an overall linear component due to the electric field.

In conclusion, the path of the particle will be a helix with nonuniform pitch when its velocity is perpendicular to the field direction.

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