|
|
| Line 1: |
Line 1: |
| A '''force-free magnetic field''' is a type of [[field (physics)|field]] which arises as a special case from the [[magnetostatics|magnetostatic]] [[equation]] in [[plasma (physics)|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.
| | 46 year old Medical Laboratory Technician Catanzaro from Chambly, has numerous hobbies which include country music, new launch property singapore and educational courses. Is a travel enthusiast and recently made a trip to Historic Town of Guanajuato and Adjacent Mines.<br><br>Feel free to surf to my website :: [http://www.Reo02.com/node/3521 The Skywoods youtube] |
| | |
| ==Basic Equations ==
| |
| | |
| Start with the simplified magnetostatic equations, in which the effects of gravity may be neglected:
| |
| | |
| <math>0=-\nabla p+\mathbf{j}\times\mathbf{B}. </math>
| |
| | |
| Supposing that the gas pressure <math>p</math> is small compared to the magnetic pressure, i.e.,
| |
| | |
| <math>p\ll B^2/2\mu</math>
| |
| | |
| then the pressure term can be neglected, and we have:
| |
| | |
| <math>\mathbf{j}\times\mathbf{B} = 0</math>.
| |
| | |
| This equation implies that:
| |
| <math>\mu_{0}\mathbf{j}=\alpha\mathbf{B}</math>. e.g. the [[current density]] is either
| |
| zero or parallel to the [[magnetic field]], and where <math>\alpha</math> is a spatial-varying function
| |
| to be determined. Combining this equation with [[Maxwell's equations]]:
| |
| | |
| <math>\nabla\times\mathbf{B}=\mu_{0}\mathbf{j}</math>
| |
| | |
| <math>\nabla\cdot\mathbf{B}=0</math>.
| |
| | |
| ... and the vector identity:
| |
| | |
| <math>\nabla\cdot(\nabla\times\mathbf{B})=0</math>
| |
| | |
| ... leads to a pair of equations for <math>\alpha</math> and <math>\mathbf{B}</math>:
| |
| | |
| <math> \mathbf{B}\cdot\nabla\alpha=0 </math>
| |
| | |
| <math>\nabla\times\mathbf{B}=\alpha\mathbf{B} </math>
| |
| | |
| ==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 (physics)|scalar]] [[magnetic potential]].
| |
| :In particular, if <math>\mathbf{j}=0</math>
| |
| | |
| :then <math>\nabla\times\mathbf{B}=0 </math> which implies, that <math>\mathbf{B}=\nabla\phi </math>.
| |
| | |
| :The substitution of this into one of [[Maxwell's Equations]], <math>\nabla\cdot\mathbf{B}=0 </math>, results in [[Laplace's equation]],
| |
| | |
| :<math>\nabla^2\phi=0 </math>,
| |
| | |
| :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.,
| |
|
| |
| ::<math>\mu\mathbf{j}=\alpha \mathbf{B}</math> which implies, that <math>\nabla\times\mathbf{B}=\alpha \mathbf{B} </math>, where <math>\alpha</math> is some scalar function. | |
| | |
| ::then we have, from above,
| |
| | |
| ::<math> \mathbf{B}\cdot\nabla\alpha=0 </math>
| |
| | |
| ::<math>\nabla\times\mathbf{B}=\alpha\mathbf{B} </math> , which implies that
| |
| | |
| ::<math>\nabla\times(\nabla\times\mathbf{B})=\nabla\times(\alpha\mathbf{B}) </math>
| |
| | |
| ::There are then two cases:
| |
| :::Case 1: The proportionality between the current density and the magnetic field is constant everywhere .
| |
| | |
| ::::<math>\nabla\times(\alpha\mathbf{B})= \alpha(\nabla\times\mathbf{B})=\alpha^2 \mathbf{B} </math>
| |
| | |
| ::::and also
| |
| | |
| ::::<math>\nabla\times(\nabla\times\mathbf{B})=\nabla(\nabla\cdot\mathbf{B}) -\nabla^2\mathbf{B}=-\nabla^2\mathbf{B} </math>,
| |
| | |
| ::::and so
| |
| | |
| ::::<math>-\nabla^2\mathbf{B} =\alpha^2 \mathbf{B} </math>
| |
| | |
| :::::This is a [[Helmholtz equation]].
| |
| | |
| **Case 2: The proportionality between the current density and the magnetic field is a function of position.
| |
| | |
| ::::<math>\nabla\times(\alpha\mathbf{B})= \alpha(\nabla\times\mathbf{B})+\nabla\alpha\times\mathbf{B}=\alpha^2 \mathbf{B} +\nabla\alpha\times\mathbf{B}</math>
| |
| | |
| :::: and so the result is coupled equations:
| |
| | |
| ::::<math>\nabla^2\mathbf{B}+\alpha^2\mathbf{B}= \mathbf{B}\times\nabla\alpha </math>
| |
| | |
| and
| |
| | |
| ::::<math>\mathbf{B}\cdot\nabla\alpha= 0 </math>
| |
| | |
| :::::In this case, the equations do not possess a general solution, and usually must be solved numerically.
| |
| | |
| ==See 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.
| |
| | |
| [[Category:Plasma physics]]
| |
46 year old Medical Laboratory Technician Catanzaro from Chambly, has numerous hobbies which include country music, new launch property singapore and educational courses. Is a travel enthusiast and recently made a trip to Historic Town of Guanajuato and Adjacent Mines.
Feel free to surf to my website :: The Skywoods youtube