The Lorentz Force and Equation of Motion
The force law that governs every charged particle and, aggregated, all plasma behavior.
The force on a charge
A charged particle in electric and magnetic fields feels the Lorentz force, giving the equation of motion:
m dv/dt = q (E + v x B)The electric part accelerates the particle along E; the magnetic part is always perpendicular to the velocity, so it does no work but bends the trajectory into a helix around the field line. This single equation, applied to enormous numbers of particles, underlies all of plasma physics.
Gyration and the guiding center
In a uniform field the motion separates into fast circular gyration at the cyclotron frequency plus the steady drift of the guiding center. Averaging over the gyration is the starting point for guiding-center theory, drift kinetics, and gyrokinetics, each removing the fast gyromotion to reach slower, relevant timescales.
How it is solved numerically
- The Boris algorithm is the standard particle pusher: it conserves phase-space volume and does not artificially heat or cool over long runs
- Higher-order symplectic integrators preserve invariants for long-time orbit studies
- Guiding-center pushers integrate the drift equations directly when gyromotion is not needed
The Boris method's stability and energy conservation are why it is used in essentially every particle-in-cell code, despite its simplicity.
From single particles to plasma
Summing the Lorentz force over a distribution gives the fluid momentum equation; keeping the full distribution gives the Vlasov equation. Every plasma model traces back to this equation of motion. Orbit-following with the Lorentz force is used to model fast-ion confinement and losses in devices such as the Hyperion breeder.