Maxwell's Equations

The set of equations describing the behaviour of the electric field, E, magnetic field, B (actually B is magnetic induction but is commonly referred to as magnetic field in astrophysics), and current density, j, were first combined by Maxwell (1873). These are as follows. Firstly, Poisson's equation,

where ρ^c is charge density and ε₀ is the permittivity of free space. This states that a net charge will act as a source for an electric field. Secondly, Faraday's law,

which states that a change in magnetic field with time will produce an electric field. Thirdly, Ampere's law,

which states that either a current, or time-varying electric field will produce a magnetic field. In this equation μ₀ is the magnetic permeability of free space, related to the speed of light, c, by,

Under the MHD approximation, whereby plasma velocities, v, are much less than the speed of light, the second term on the RHS of Eqn. 6 is ignored. Finally, Gauss' law,

such that there are no magnetic monopoles. In addition to the four Maxwell's equations above, Ohm's law states that, in a frame of reference moving with the plasma, the current density is proportional to the sum of the electric field due to the movement of the plasma (v X B) plus the electric field which would act on it at rest, E, i.e.,

where σ is electrical conductivity.