Question

A charged particle \(q\) moving with a velocity \( \mathbf{v} \) is subjected to a uniform magnetic field \( \mathbf{B} \) acting perpendicular to \( \mathbf{v} \). If a uniform electric field \( \mathbf{E} \) is also set up in the region along the direction of \( \mathbf{B} \), describe the path followed by the particle and draw its shape.

Hint:

When both electric and magnetic fields are present, the particle moves in a helical path. The magnetic field causes circular motion, while the electric field causes acceleration along the direction of the field.

Answer 1

In the presence of both electric and magnetic fields, the charged particle experiences forces from both fields. The total force on the charged particle is given by the Lorentz force equation: \[ \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \] where: - \( q \) is the charge of the particle, - \( \mathbf{E} \) is the electric field, - \( \mathbf{B} \) is the magnetic field, - \( \mathbf{v} \) is the velocity of the particle. Direction of Forces: - The magnetic force \( \mathbf{F}_B = q (\mathbf{v} \times \mathbf{B}) \) is perpendicular to the velocity \( \mathbf{v} \) and the magnetic field \( \mathbf{B} \), resulting in a circular motion of the particle. The magnetic force provides the centripetal force needed for circular motion.
- The electric force \( \mathbf{F}_E = q \mathbf{E} \) is in the direction of the electric field \( \mathbf{E} \). Path of the Particle: Since the magnetic force is perpendicular to the velocity of the particle, it causes the particle to move in a circular or helical path. The presence of the electric field \( \mathbf{E} \) along the direction of \( \mathbf{B} \) adds a constant acceleration along the direction of \( \mathbf{B} \). Thus, the charged particle will follow a helical path. The motion of the particle will be a combination of:
1. Circular motion due to the magnetic force, perpendicular to the velocity.
2. Linear acceleration along the direction of \( \mathbf{E} \) (which is also along \( \mathbf{B} \)). Shape of the Path: The shape of the path is a helix moving along the direction of the magnetic field \( \mathbf{B} \), with the radius of the helix determined by the velocity and the magnetic field, and the pitch of the helix determined by the electric field. The radius \( r \) of the circular path in the plane perpendicular to the magnetic field is given by: \[ r = \frac{mv_{\perp}}{qB} \] where: - \( m \) is the mass of the particle, - \( v_{\perp} \) is the component of the velocity perpendicular to the magnetic field, - \( q \) is the charge of the particle, - \( B \) is the magnetic field strength. The velocity component parallel to the electric field \( v_{\parallel} \) increases linearly due to the constant force \( F_E = qE \) in the direction of \( \mathbf{B} \). The velocity along this direction changes according to the equation: \[ v_{\parallel}(t) = v_{\parallel}(0) + \frac{qE}{m} t \] Thus, the overall path is helical, with the radius of the helix determined by the velocity perpendicular to the magnetic field, and the pitch of the helix determined by the electric field.