Standard Reference Method - Revision history

142.157.35.25: /* Measurement Method */

2014-07-08T20:20:10Z

Measurement Method

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	~~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]~~]

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2013-11-07T14:56:49Z

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2012-06-14T18:24:40Z

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