Force free magnetic field

Basic Equations

Neglecting the effects of gravity, the Navier-Stokes equation for a plasma, in steady state, reads

0 = − ∇ p + j × B ,

where p is the thermal pressure, B is the magnetic field and j is the electric current. Assuming that the gas pressure p is small compared to the magnetic pressure, i.e.,

p ≪ B 2 / 2 μ

then the pressure term can be neglected. Here μ is the magnetic permeability of the plasma. Therefore,

j × B = 0 .

This equation implies that: μ 0 j = α B . e.g. the current density is either zero or parallel to the magnetic field, and where α is a spatial-varying function to be determined. Combining this equation with Maxwell's equations:

∇ × B = μ j

∇ ⋅ B = 0

and the vector identity:

∇ ⋅ ( ∇ × B ) = 0

leads to a pair of equations for α and B :

B ⋅ ∇ α = 0 ,

∇ × B = α B .

Physical Examples

In the corona of the sun, the ratio of the gas pressure to the magnetic pressure is ~0.004, and so there the magnetic field is force-free.

Mathematical Limits

If the current density is identically zero, then the magnetic field is potential, i.e. the gradient of a scalar magnetic potential.

In particular, if j = 0 then ∇ × B = 0 which implies that B = ∇ ϕ . The substitution of this into one of Maxwell's Equations, ∇ ⋅ B = 0 , results in Laplace's equation, ∇ 2 ϕ = 0 , which can often be readily solved, depending on the precise boundary conditions.

If the current density is not zero, then it must be parallel to the magnetic field, i.e.,

Case 2: The proportionality between the current density and the magnetic field is a function of position.

and