Force-free magnetic field

In plasma physics, a force-free magnetic field is a magnetic field in which the Lorentz force is equal to zero and the magnetic pressure greatly exceeds the plasma pressure such that non-magnetic forces can be neglected.

When a magnetic field is approximated as force-free, all non-magnetic forces are neglected and the Lorentz force vanishes.

In SI units, the Lorentz force condition for a static magnetic field

Alternatively, this can be written as These conditions are fulfilled when the current vanishes or is parallel to the magnetic field.

[1] If the current density is identically zero, then the magnetic field is the gradient of a magnetic scalar potential

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.,

This implies that The force-free parameter can be a function of position but must be constant along field lines.

allows for the derivation of a vector Helmholtz equation by taking the curl of the nonzero current density equations above.

In this case, the equations do not possess a general solution, and usually must be solved numerically.

[1][2][3]: 50–54 In the Sun's upper chromosphere and lower corona, the plasma β can locally be of order 0.01 or lower allowing for the magnetic field to be approximated as force-free.