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| {{Electromagnetism|cTopic=Magnetostatics}}
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| '''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.
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| ==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.
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| {| class="wikitable"
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| |- style="background-color: #aaddcc;"
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| ! Name
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| ! [[Partial differential equation|Partial differential]] form
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| ! [[Integral]] form
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| |-
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| | [[Gauss's law for magnetism]]:
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| | <math>\vec{\nabla} \cdot \vec{B} = 0</math>
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| | <math>\oint_S \vec{B} \cdot \mathrm{d}\vec{S} = 0</math>
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| |-
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| | [[Ampère's law]]:
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| | <math>\vec{\nabla} \times \vec{H} = \vec{J}</math>
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| | <math>\oint_C \vec{H} \cdot \mathrm{d}\vec{l} = I_{\mathrm{enc}}</math>
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| |}
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| 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>.
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| 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.
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| ===Re-introducing Faraday's law===
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| 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}}
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| ==Solving for the magnetic field==
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| ===Current sources===
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| If all currents in a system are known (''i.e.,'' if a complete description of <math>\scriptstyle \vec{J}</math> is available) then
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| the magnetic field can be determined from the currents by the [[Biot-Savart law|Biot-Savart equation]]:
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| :<math>\vec{B}= \frac{\mu_{0}}{4\pi}I \int{\frac{\mathrm{d}\vec{l} \times \hat{r}}{r^2}}</math> | |
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| 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.
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| 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]].
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| The magnetic field can be derived from the [[Magnetic vector potential|vector potential]]. Since the divergence of the magnetic flux density is always zero,
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| ::<math> \vec{B} = \nabla \times \vec{A}, </math>
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| and the relation of the vector potential to current is:
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| ::<math> \vec{A} = \frac{\mu_{0}}{4\pi} \int{ \frac{\vec{J} } {r} dV} </math>
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| where <math>\scriptstyle \vec{J} </math> is the [[current density]].
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| ===Magnetization=== | |
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| {{Further2|[[Demagnetizing field]] and [[Micromagnetics]]}}
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| 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
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| :<math> \vec{B} = \mu_0(\vec{M}+\vec{H}).</math>
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| Except in metals, electric currents can be ignored. Then Ampère's law is simply
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| :<math> \nabla\times\vec{H} = 0.</math>
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| This has the general solution
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| :<math> \vec{H} = -\nabla U, </math>
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| where <math>U</math> is a scalar [[potential]]. Substituting this in Gauss's law gives
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| :<math> \nabla^2 U = \nabla\cdot\vec{M}.</math>
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| 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>.
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| The vector potential method can also be employed with an effective current density
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| ::<math> \vec{J_M} = \nabla \times \vec{M}. </math>
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| ==See also==
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| * [[Darwin Lagrangian]]
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| ==Notes==
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| {{Reflist|2}}
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| ==References==
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| {{Refbegin}}
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| *{{cite book
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| |last = Aharoni
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| |first = Amikam
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| |author-link=Amikam Aharoni
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| |title=Introduction to the Theory of Ferromagnetism
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| |publisher=[[Clarendon Press]]
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| |year = 1996
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| |isbn=0-19-851791-2
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| |url=http://www.oup.com/us/catalog/general/subject/Physics/ElectricityMagnetism/?view=usa&ci=9780198508090
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| }}
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| *{{Cite book
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| |last1 = Feynman
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| |first1 = Richard P.
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| |author-link = Richard Feynman
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| |first2 = Robert B.
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| |last2 = Leighton
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| |author2-link = Robert B. Leighton
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| |first3 = Matthew
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| |last3 = Sands
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| |author3-link = Matthew Sands
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| |title = [[The Feynman Lectures on Physics]]
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| |volume = 2
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| |year = 2006
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| |isbn = 0-8053-9045-6
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| |ref=harv
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| }}
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| *{{cite article
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| |last=Hiebert
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| |first=W
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| |last2=Ballentine
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| |first2=G
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| |last3=Freeman
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| |first3=M
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| |title=Comparison of experimental and numerical micromagnetic dynamics in coherent precessional switching and modal oscillations
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| |journal = [[Physical Review B]]
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| |volume=65
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| |number=14
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| |pages=140404
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| |year=2002
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| |doi=10.1103/PhysRevB.65.140404
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| |ref=harv
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| }}
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| {{Refend}}
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| [[Category:Electric and magnetic fields in matter]]
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| [[Category:Magnetostatics]]
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| [[Category:Potentials]]
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And if you are not pleased with their services you have a 45 days money back guarantee. It's not a full time job but it certainly helps if your saving for a particular thing. Well, your search for Hostgator tutorials on how to install Wordpress ends here. You will not receive the invoice generated notification if you have a credit card on file which automatically pays your hostgator billing invoice. So anyway, after reading that book, I started looking for anything and everything having to do with the subject of flipping websites. If you ever run into problems it's easy enough to contact them through chat or through a toll free number and they will assist you quickly. And right now, people today throughout from a variety of components of this globe have started taking their skilled service. What does HostGator offer that other webhosting platforms don't?