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{{Electromagnetism|cTopic=Magnetostatics}}
'''Magnetostatics''' is the study of [[magnetic field]]s in systems where the [[electric currents|currents]] are steady (not changing with time). It is the magnetic analogue of [[electrostatic]]s, where the [[electric charge|charges]] are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast [[Magnetization reversal|magnetic switching]] events that occur on time scales of nanoseconds or less.<ref name=Hiebert>{{harvnb|Hiebert|Ballentine|Freeman|2002}}</ref> Magnetostatics is even a good approximation when the currents are not static — as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of [[micromagnetics]] such as models of [[magnetic recording]] devices.

==Applications==
===Magnetostatics as a special case of Maxwell's equations===
Starting from [[Maxwell's equations]] and assuming that charges are either fixed or move as a steady current <math>\scriptstyle\vec{J}</math>, the equations separate into two equations for the [[electric field]] (see [[electrostatics]]) and two for the [[magnetic field]].<ref name=Feynman>{{harvnb|Feynman|Leighton|Sands|2006}}</ref> The fields are independent of time and each other. The magnetostatic equations, in both differential and integral forms, are shown in the table below.

{| class="wikitable"
|- style="background-color: #aaddcc;"
! Name
! [[Partial differential equation|Partial differential]] form
! [[Integral]] form
|-
| [[Gauss's law for magnetism]]:
| <math>\vec{\nabla} \cdot \vec{B} = 0</math>
| <math>\oint_S \vec{B} \cdot \mathrm{d}\vec{S} = 0</math>
|-
| [[Ampère's law]]:
| <math>\vec{\nabla} \times \vec{H} = \vec{J}</math>
| <math>\oint_C \vec{H} \cdot \mathrm{d}\vec{l} = I_{\mathrm{enc}}</math>
|}
The first integral is over a surface <math>S</math> with oriented surface element <math>\scriptstyle d\vec{S}</math>. The second is a line integral around a closed loop <math>C</math> with line element <math>\scriptstyle\vec{l}</math>. The current going through the loop is <math>\scriptstyle I_\text{enc}</math>.

The quality of this approximation may be guessed by comparing the above equations with the full version of [[Maxwell's equations]] and considering the importance of the terms that have been removed. Of particular significance is the comparison of the <math>\scriptstyle \vec{J}</math> term against the <math>\scriptstyle \partial \vec{D} / \partial t</math> term. If the <math>\scriptstyle \vec{J}</math> term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.

===Re-introducing Faraday's law===
A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term <math>\scriptstyle \partial \vec{B} / \partial t</math>. Plugging this result into [[Faraday's law of induction|Faraday's Law]] finds a value for <math>\scriptstyle \vec{E}</math> (which had previously been ignored). This method is not a true solution of [[Maxwell's equations]] but can provide a good approximation for slowly changing fields.{{Citation needed|date=October 2010}}

==Solving for the magnetic field==
===Current sources===

If all currents in a system are known (''i.e.,'' if a complete description of <math>\scriptstyle \vec{J}</math> is available) then
the magnetic field can be determined from the currents by the [[Biot-Savart law|Biot-Savart equation]]:

:<math>\vec{B}= \frac{\mu_{0}}{4\pi}I \int{\frac{\mathrm{d}\vec{l} \times \hat{r}}{r^2}}</math>

This technique works well for problems where the medium is a [[vacuum]] or [[air]] or some similar material with a [[Permeability (electromagnetism)|relative permeability]] of 1. This includes Air core inductors and Air core transformers. One advantage of this technique is that a complex coil geometry can be integrated in sections, or for a very difficult geometry [[numerical integration]] may be used. Since this equation is primarily used to solve [[linear]] problems, the complete answer will be a sum of the integral of each component section.

For problems where the dominant magnetic material is a highly permeable [[magnetic core]] with relatively small air gaps, a [[magnetic circuit]] approach is useful. When the air gaps are large in comparison to the [[magnetic circuit]] length, [[magnetic fringing|fringing]] becomes significant and usually requires a [[finite element]] calculation. The [[finite element]] calculation uses a modified form of the magnetostatic equations above in order to calculate [[magnetic potential]]. The value of <math>\scriptstyle \vec{B}</math> can be found from the [[magnetic potential]].

The magnetic field can be derived from the [[Magnetic vector potential|vector potential]]. Since the divergence of the magnetic flux density is always zero,
::<math> \vec{B} = \nabla \times \vec{A}, </math>
and the relation of the vector potential to current is:
::<math> \vec{A} = \frac{\mu_{0}}{4\pi} \int{ \frac{\vec{J} } {r} dV} </math>
where <math>\scriptstyle \vec{J} </math> is the [[current density]].

===Magnetization===

{{Further2|[[Demagnetizing field]] and [[Micromagnetics]]}}
Strongly magnetic materials (''i.e.,'' [[Ferromagnetic]], [[Ferrimagnetic]] or [[Paramagnetic]]) have a [[magnetization]] that is primarily due to [[spin (physics)|electron spins]]. In such materials the magnetization must be explicitly included using the relation

:<math> \vec{B} = \mu_0(\vec{M}+\vec{H}).</math>

Except in metals, electric currents can be ignored. Then Ampère's law is simply

:<math> \nabla\times\vec{H} = 0.</math>

This has the general solution

:<math> \vec{H} = -\nabla U, </math>

where <math>U</math> is a scalar [[potential]]. Substituting this in Gauss's law gives

:<math> \nabla^2 U = \nabla\cdot\vec{M}.</math>

Thus, the divergence of the magnetization, <math>\scriptstyle \nabla\cdot\vec{M},</math> has a role analogous to the electric charge in electrostatics <ref>{{harvnb|Aharoni|1996}}</ref> and is often referred to as an effective charge density <math>\rho_M</math>.

The vector potential method can also be employed with an effective current density
::<math> \vec{J_M} = \nabla \times \vec{M}. </math>

==See also==
* [[Darwin Lagrangian]]

==Notes==
{{Reflist|2}}

==References==
{{Refbegin}}
*{{cite book
|last = Aharoni
|first = Amikam
|author-link=Amikam Aharoni
|title=Introduction to the Theory of Ferromagnetism
|publisher=[[Clarendon Press]]
|year = 1996
|isbn=0-19-851791-2
|url=http://www.oup.com/us/catalog/general/subject/Physics/ElectricityMagnetism/?view=usa&ci=9780198508090
}}
*{{Cite book
|last1 = Feynman
|first1 = Richard P.
|author-link = Richard Feynman
|first2 = Robert B.
|last2 = Leighton
|author2-link = Robert B. Leighton
|first3 = Matthew
|last3 = Sands
|author3-link = Matthew Sands
|title = [[The Feynman Lectures on Physics]]
|volume = 2
|year = 2006
|isbn = 0-8053-9045-6
|ref=harv
}}
*{{cite article
|last=Hiebert
|first=W
|last2=Ballentine
|first2=G
|last3=Freeman
|first3=M
|title=Comparison of experimental and numerical micromagnetic dynamics in coherent precessional switching and modal oscillations
|journal = [[Physical Review B]]
|volume=65
|number=14
|pages=140404
|year=2002
|doi=10.1103/PhysRevB.65.140404
|ref=harv
}}
{{Refend}}

[[Category:Electric and magnetic fields in matter]]
[[Category:Magnetostatics]]
[[Category:Potentials]]

Mixed-data sampling - Revision history

en>Comp.arch at 11:50, 30 March 2014

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