- Force-free magnetic field
A

**force-free magnetic field**is a type of field which arises as a special case from the magnetostaticequation in plasmas. This special case arises when the plasmapressure is so small, relative to themagnetic pressure , that the plasma pressure may be ignored, and so only the magnetic pressure is considered. The name "force-free" comes from being able to neglect the force from the plasma.**Basic Equations**Start with the simplified magnetostatic equations, in which the effects of gravity may be neglected:

$0=-\; abla\; ho+mathbf\{j\}\; imesmathbf\{B\}.$

Supposing that the gas pressure is small compared to the magnetic pressure, i.e.,

$ho<^2\; 2mu\; math>$

then the pressure term can be neglected, and we have:

$mathbf\{j\}\; imesmathbf\{B\}\; =\; 0$.

From

Maxwell's equations :$abla\; imesmathbf\{B\}=mu\_\{0\}mathbf\{j\}$

$ablacdotmathbf\{B\}=0$.

Vector identity:

$ablacdot(\; abla\; imesmathbf\{B\})=0$

The first equation implies that:$mu\_\{0\}mathbf\{j\}=alphamathbf\{B\}$. e.g. the

current density is either zero or parallel to themagnetic field , and where alpha is a spatial-varying function which must be determined. Combing this equation with Maxwell's equations and the vector identity leads to a pair of equations for alpha and B:$mathbf\{B\}cdot\; ablaalpha=0$

$abla\; imesmathbf\{B\}=alphamathbf\{B\}$

**Physical Examples**In the

corona of thesun , 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. thegradient of ascalar magnetic potential .:In particular, if $mathbf\{j\}=0$:then $abla\; imesmathbf\{B\}=0$ which implies, that $mathbf\{B\}=\; ablaphi$.

:The substitution of this into one of

Maxwell's Equations , $ablacdotmathbf\{B\}=0$, results inLaplace's equation ,:$abla^2phi=0$,

:which can often be readily solved, depending on the precise boundary conditions.

::This limit is usually referred to as the potential field case.

*If the current density is not zero, then it must be parallel to the magnetic field, i.e., ::$mumathbf\{j\}=alpha\; mathbf\{B\}$ which implies, that $abla\; imesmathbf\{B\}=alpha\; mathbf\{B\}$, where $alpha$ is some scalar function.

::then we have, from above,

::$mathbf\{B\}cdot\; ablaalpha=0$

::$abla\; imesmathbf\{B\}=alphamathbf\{B\}$ , which implies that

::$abla\; imes(\; abla\; imesmathbf\{B\})=\; abla\; imes(alphamathbf\{B\})$

::There are then two cases::::Case 1: The proportionality between the current density and the magnetic field is constant everywhere .

::::$abla\; imes(alphamathbf\{B\})=\; alpha(\; abla\; imesmathbf\{B\})=alpha^2\; mathbf\{B\}$

::::and also

::::$abla\; imes(\; abla\; imesmathbf\{B\})=\; abla(\; ablacdotmathbf\{B\})\; -\; abla^2mathbf\{B\}=-\; abla^2mathbf\{B\}$,

::::and so

::::$-\; abla^2mathbf\{B\}\; =alpha^2\; mathbf\{B\}$

:::::This is a

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

::::$abla\; imes(alphamathbf\{B\})=\; alpha(\; abla\; imesmathbf\{B\})+\; ablaalpha\; imesmathbf\{B\}=alpha^2\; mathbf\{B\}\; +\; ablaalpha\; imesmathbf\{B\}$

:::: and so the result is coupled equations:

::::$abla^2mathbf\{B\}+alpha^2mathbf\{B\}=\; mathbf\{B\}\; imes\; ablaalpha$

and

::::$mathbf\{B\}cdot\; ablaalpha=\; 0$

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

**ee also***

Laplace's equation

*Helmholtz equation **References*** Low, Boon Chye, " [

*http://eaa.iop.org/index.cfm?action=summary&doc=eaa%2F2221%40eaa-xml Force-Free Magnetic Fields*] ". November 2000.

*Wikimedia Foundation.
2010.*

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