Force-free magnetic field
A force-free magnetic field is a
Definition
When a magnetic field is approximated as force-free, all non-magnetic forces are neglected and the
In
where
is the current density and is the vacuum permeability. Alternatively, this can be written as
These conditions are fulfilled when the current vanishes or is parallel to the magnetic field.[1]
Zero current density
If the current density is identically zero, then the magnetic field is the gradient of a magnetic scalar potential :
The substitution of this into results in Laplace's equation, 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., where is a scalar function known as the force-free parameter or force-free function. This implies that
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 is constant everywhere, the field is called a linear force-free field (LFFF). A constant allows for the derivation of a vector Helmholtz equation
by taking the curl of the nonzero current density equations above.
Nonlinear force-free field
When the force-free parameter 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.[1][2][3]: 50–54
Physical examples
In the
See also
References
- ^ S2CID 232107294. Retrieved 18 May 2022.
- ISBN 0521528003.
- ISBN 978-0-19-882996-6.
- .
- doi:10.1086/168541.
- .