Maxwell's Equations in Plasmas
The four field equations that, coupled to charged-particle motion, govern all electromagnetic behavior of a plasma.
The four equations
Every plasma model ultimately rests on Maxwell's equations for the electric field E and magnetic field B, with charge density rho_c and current density J as sources.
- Gauss: div E = rho_c / epsilon0
- No monopoles: div B = 0
- Faraday: curl E = -dB/dt
- Ampere-Maxwell: curl B = mu0 J + mu0 epsilon0 dE/dt
The self-consistent loop
What makes plasma hard is that rho_c and J are not given: they are produced by the same charged particles the fields push around. Maxwell's equations feed the fields into the equation of motion or a kinetic equation, which returns the sources. Solving this loop self-consistently is the central task of plasma theory.
Common approximations
At the low frequencies relevant to confinement, the displacement current mu0 epsilon0 dE/dt is dropped, giving the magnetostatic Ampere law mu0 J = curl B used in MHD. Quasi-neutrality (rho_c approximately 0 over scales larger than the Debye length) replaces Gauss's law for the bulk. Full Maxwell is retained for wave heating, where the displacement current and finite frequency matter.
How they are solved numerically
Particle-in-cell codes solve the full Maxwell system on a Yee staggered grid using the finite-difference time-domain method, which naturally preserves div B = 0 and div E consistency. Fields are interpolated to particles, particles are pushed (Boris algorithm), and their charge and current are scattered back to the grid.
- FDTD on a Yee lattice keeps the fields divergence-consistent to machine precision
- Spectral Maxwell solvers reduce numerical dispersion for wave problems
- Implicit solvers relax the Courant limit for stiff, magnetized regimes
Radio-frequency and electron-cyclotron heating design, including auxiliary heating for the Hyperion breeder, requires solving the wave form of Maxwell's equations in a hot, inhomogeneous plasma dielectric.