The Lorentz Force: Magnetism in Motion
When a charged particle moves through a magnetic field, it experiences the Lorentz force: F = qv x B. This force is always perpendicular to both the velocity and the magnetic field, which means it changes the particle's direction without changing its speed. The result is circular motion (if velocity is perpendicular to B) or helical motion (if there is a component along B).
Cyclotron Motion and the Larmor Radius
The radius of circular motion — the cyclotron or Larmor radius — is r = mv/(qB). This formula reveals the key physics: heavier particles trace larger circles, faster particles trace larger circles, and stronger fields produce tighter orbits. The cyclotron frequency ω = qB/m depends only on the charge-to-mass ratio and the field strength, not on the particle's speed.
Helical Trajectories and Pitch Angle
When a particle's velocity has both perpendicular and parallel components relative to the field, it traces a helix. The perpendicular component creates circular motion while the parallel component carries the particle along the field direction. The pitch angle — the angle between velocity and field — determines the tightness of the helix. At 90°, motion is purely circular; at 0°, the particle moves freely along the field.
Applications: From Mass Spectrometry to Aurora
The Lorentz force has countless applications. Mass spectrometers separate ions by their cyclotron radius to determine molecular masses. Cyclotrons and synchrotrons accelerate particles for physics research and medical therapy. Earth's magnetic field traps charged particles from the solar wind in helical paths along field lines, and when these particles strike the atmosphere near the poles, they produce the aurora borealis and australis.