Motion Of A Charged Particle In A Magnetic Field

A charged particle moving perpendicular to a uniform magnetic field experiences a magnetic force that is always perpendicular to its velocity. Since the force does no work on the particle, its speed remains constant while only its direction changes. This perpendicular force acts as a centripetal force, causing the particle to follow a circular path.

The radius of the circular path is determined by equating the magnetic force to the centripetal force using Newton's second law. The magnetic force on a moving charge is F = QvB, and the centripetal force is F = mv²/r. Solving for r gives the radius in terms of the particle's mass, charge, velocity, and magnetic field strength.

The period is the time for one complete circular orbit, calculated as the circumference divided by the particle's speed. Interestingly, the period is independent of the particle's speed and depends only on its mass, charge, and the magnetic field strength. This means particles with different speeds take the same time to complete one cycle.

When velocity has both perpendicular and parallel components, each behaves differently. The perpendicular component produces circular motion, while the parallel component creates constant motion along the magnetic field direction. These motions combine to produce helical motion, where the particle spirals along the field lines.

The pitch is the distance between adjacent turns of the helix, calculated as the product of the parallel component of velocity and the period of circular motion. Since the period remains constant regardless of speed, the pitch depends only on the parallel velocity component and the magnetic field strength.

A magnetic bottle forms when a charged particle travels between regions of varying magnetic field strength. As the particle moves from a weaker field to a stronger field region, it reflects back before entering the stronger field, similar to a wave reflecting off a wall. If reflection occurs at both ends, the particle becomes trapped in the magnetic bottle region.

The magnetic force is always perpendicular to the particle's velocity, so it performs no work on the particle. Since work equals the change in kinetic energy, and no work is done, the particle's kinetic energy and speed remain constant. Only the direction of motion changes, not the magnitude.