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BASIC EQUATIONS


Start with the simplified magnetostatic equations, in which the effects of gravity may be neglected:

0=-
abla ho+\mathbf{j} imes\mathbf{B}.

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

ho<

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

\mathbf{j} imes\mathbf{B} = 0.

From Maxwell's Equations :


abla imes\mathbf{B}=\mu_{0}\mathbf{j}


abla\cdot\mathbf{B}=0.

The first equation implies that:
\mu_{0}\mathbf{j}=\alpha\mathbf{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, leads to a pair of equations for alpha and B:

\mathbf{B}\cdot
abla\alpha=0


abla imes\mathbf{B}=\alpha\mathbf{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




:then
abla imes\mathbf{B}=0 which implies, that \mathbf{B}=
abla\phi .

:The substitution of this into one of Maxwell's Equations ,
abla\cdot\mathbf{B}=0 , results in Laplace's Equation ,

:
abla^2\phi=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.,


::\mu\mathbf{j}=\alpha \mathbf{B} which implies, that
abla imes\mathbf{B}=\alpha \mathbf{B} , where \alpha is some scalar function.

::then we have, from above,

:: \mathbf{B}\cdot
abla\alpha=0

::
abla imes\mathbf{B}=\alpha\mathbf{B} , which implies that

::
abla imes(
abla imes\mathbf{B})=
abla imes(\alpha\mathbf{B}

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

::::
abla imes(\alpha\mathbf{B})= \alpha(
abla imes\mathbf{B})=\alpha^2 \mathbf{B})

::::and also

::::
abla imes(
abla imes\mathbf{B})=
abla(
abla\cdot\mathbf{B}) -
abla^2\mathbf{B}=-
abla^2\mathbf{B} ,

::::and so

::::-
abla^2\mathbf{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(\alpha\mathbf{B})= \alpha(
abla imes\mathbf{B})+
abla\alpha imes\mathbf{B}=\alpha^2 \mathbf{B} +
abla\alpha imes\mathbf{B}

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

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

and

::::\mathbf{B}\cdot
abla\alpha= 0

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


SEE ALSO




REFERENCES