# 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. For a force-free field, the electric current density is either zero or parallel to the magnetic field.

## Definition

When a magnetic field is approximated as force-free, all non-magnetic forces are neglected and the Lorentz force vanishes. For non-magnetic forces to be neglected, it is assumed that the ratio of the plasma pressure to the magnetic pressure—the plasma β—is much less than one, i.e., $\beta \ll 1$. With this assumption, magnetic pressure dominates over plasma pressure such that the latter can be ignored. It is also assumed that the magnetic pressure dominates over other non-magnetic forces, such as gravity, so that these forces can similarly be ignored.
In SI units, the Lorentz force condition for a static magnetic field $\mathbf {B}$ can be expressed as
    $\mathbf {j} \times \mathbf {B} =\mathbf {0} ,$
    $\nabla \cdot \mathbf {B} =0,$
where
    $\mathbf {j} ={\frac {1}{\mu _{0}}}\nabla \times \mathbf {B}$
is the current density and $\mu _{0}$ is the vacuum permeability. Alternatively, this can be written as
    $(\nabla \times \mathbf {B} )\times \mathbf {B} =\mathbf {0} ,$
    $\nabla \cdot \mathbf {B} =0.$
These conditions are fulfilled when the current vanishes or is parallel to the magnetic field.

### Zero current density

If the current density is identically zero, then the magnetic field is the gradient of a magnetic scalar potential $\phi$:
    $\mathbf {B} =-\nabla \phi .$
The substitution of this into $\nabla \cdot \mathbf {B} =0$ results in Laplace's equation, $\nabla ^{2}\phi =0,$ which can often be readily solved, depending on the precise boundary conditions. In this case, the field is referred to as a potential field or vacuum magnetic field.

### Nonzero current density

If the current density is not zero, then it must be parallel to the magnetic field, i.e., $\mu _{0}\mathbf {j} =\alpha \mathbf {B}$ where $\alpha$ is a scalar function known as the force-free parameter or force-free function. This implies that
    $\nabla \times \mathbf {B} =\alpha \mathbf {B} ,$
    $\mathbf {B} \cdot \nabla \alpha =0.$
The force-free parameter can be a function of position but must be constant along field lines.

#### Linear force-free field

When the force-free parameter $\alpha$ is constant everywhere, the field is called a linear force-free field (LFFF). A constant $\alpha$ allows for the derivation of a vector Helmholtz equation
    $\nabla ^{2}\mathbf {B} =-\alpha ^{2}\mathbf {B}$
by taking the curl of the nonzero current density equations above.

#### Nonlinear force-free field

When the force-free parameter $\alpha$ depends on position, the field is called a nonlinear force-free field (NLFFF). In this case, the equations do not possess a general solution, and usually must be solved numerically.$^{:50–54}$

## Physical examples

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.