A force-free magnetic field is a type of field

Field (physics)

In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

 which arises as a special case from the magnetostatic

Magnetostatics

Magnetostatics is the study of magnetic fields in systems where the currents are steady . It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that...

 equation

Equation

An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

 in plasmas

Plasma (physics)

In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...

. This special case arises when the plasma pressure

Pressure

Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

 is so small, relative to the magnetic pressure

Magnetic pressure

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 kinetic energy of the gas molecules. Interplay between magnetic pressure and ordinary gas pressure is important to...

, 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:



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



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

.

This equation implies that:
. e.g. the current density

Current density

Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...

 is either
zero or parallel to the magnetic field

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

, and where is a spatial-varying function
to be determined. Combining this equation with Maxwell's equations

Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

:



.

... and the vector identity:



... leads to a pair of equations for and :

Physical Examples

In the corona

Corona

A corona is a type of plasma "atmosphere" of the Sun or other celestial body, extending millions of kilometers into space, most easily seen during a total solar eclipse, but also observable in a coronagraph...

 of the sun

Sun

The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

, 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
  Potential
  *In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
  
  , i.e. the gradient
  Gradient
  In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
  
   of a scalar
  Scalar (physics)
  In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...
  
   magnetic potential
  Magnetic potential
  The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...
  
  .

In particular, if

then which implies, that .

The substitution of this into one of Maxwell's Equations

Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

, , results in Laplace's equation

Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

,

,

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.,

which implies, that , where is some scalar function.

then we have, from above,

, which implies that

There are then two cases:

Case 1: The proportionality between the current density and the magnetic field is constant everywhere .

and also

,

and so

This is a Helmholtz equation

Helmholtz equation

The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...

.

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

and so the result is coupled equations:

and

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