Force-free magnetic field

Force-free magnetic field

A force-free magnetic field is a type of field which arises as a special case from the magnetostatic equation in plasmas. This special case arises when the plasma pressure is so small, relative to the magnetic 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.,


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}


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 the magnetic 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 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 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 in Laplace'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


::::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


* Low, Boon Chye, " [ Force-Free Magnetic Fields] ". November 2000.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Magnetic field — This article is about a scientific description of the magnetic influence of an electric current or magnetic material. For the physics of magnetic materials, see magnetism. For information about objects that create magnetic fields, see magnet. For …   Wikipedia

  • Magnetic pressure — is an energy density associated with the magnetic field. It is identical to any other physical pressure except that it is carried by the magnetic field rather than (in the case of gas) kinetic energy of the gas molecules. Interplay between… …   Wikipedia

  • Magnetic Surveying in Archaeology (book) — Magnetic Surveying in Archaeology (Wormianum, 2008, ISBN 978 87 89531 29 8) is a book written by Russian archaeologist T. N. Smekalova together with O. Voss and S. L. Smekalov. In the book researches collected information about magnetic… …   Wikipedia

  • Magnetic circuit — Magnetic Circuits Conventional Magnetic Circuits Magnetomotive force Magnetic flux Φ Magnetic reluctance Phasor Magnetic Circuits Complex reluctance Zμ …   Wikipedia

  • Magnetic shark repellent — Magnetic shark repellents utilize permanent magnets, which exploit the sensitivity of the Ampullae of Lorenzini in sharks and rays (electrosense). This organ is not found on bony fishes (teleosts), therefore, this type of shark repellent is… …   Wikipedia

  • Magnetic reluctance — Magnetic reluctance, or magnetic resistance, is a concept used in the analysis of magnetic circuits. It is analogous to resistance in an electrical circuit, but rather than dissipating magnetic energy it stores magnetic energy. In likeness to the …   Wikipedia

  • Magnetic shape-memory alloy — Magnetic shape memory alloys (MSMAs), or ferromagnetic shape memory alloys (FSMAs), are ferromagnetic materials which exhibit large strains under the influence of an applied magnetic field due to martensitic phase transformation. Magnetic shape… …   Wikipedia

  • Force — For other uses, see Force (disambiguation). See also: Forcing (disambiguation) Forces are also described as a push or pull on an object. They can be due to phenomena such as gravity, magnetism, or anything that might cause a mass to accelerate …   Wikipedia

  • Magnetic monopole — It is impossible to make magnetic monopoles from a bar magnet. If a bar magnet is cut in half, it is not the case that one half has the north pole and the other half has the south pole. Inst …   Wikipedia

  • Magnetic levitation — This article is about magnetic levitation. For trains based on this effect, see Maglev. For the Ruby interpreter, see MagLev (Ruby interpreter). Levitating pyrolytic carbon Magnetic levitation, maglev, or magnetic suspension is a method by which… …   Wikipedia