Editing Magnetostatics

{{Short description|Branch of physics about magnetism in systems with steady electric currents}}
{{Use American English|date = February 2019}}
{{more footnotes needed|date=September 2016}}
{{Electromagnetism|cTopic=Magnetostatics}}
'''Magnetostatics''' is the study of [[magnetic field]]s in systems where the [[electric currents|currents]] are [[steady current|steady]] (not changing with time). It is the magnetic analogue of [[electrostatics]], 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">{{cite journal |last1=Hiebert |first1=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 |bibcode=2002PhRvB..65n0404H }}</ref> Magnetostatics is even a good approximation when the currents are not static &ndash; as long as the currents do not [[alternating current|alternate]] rapidly. Magnetostatics is widely used in applications of [[micromagnetics]] such as models of [[magnetic storage]] devices as in [[computer memory]].

==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>\mathbf{J}</math>, the equations separate into two equations for the [[electric field]] (see [[electrostatics]]) and two for the [[magnetic field]].<ref>[https://feynmanlectures.caltech.edu/II_13.html The Feynman Lectures on Physics Vol. II Ch. 13: Magnetostatics]</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;"
! rowspan=2 | Name
! colspan=2 | Form
|-
! [[Partial differential equation|Differential]]
! [[Integral]]
|-
| [[Gauss's law for magnetism|Gauss's law <br/>for magnetism]]
| <math>\mathbf{\nabla} \cdot \mathbf{B} = 0</math> 
| <math>\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{S} = 0</math>
|-
| [[Ampère's law]]
| <math>\mathbf{\nabla} \times \mathbf{H} = \mathbf{J}</math>
| <math>\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I_{\mathrm{enc}}</math>
|}

Where ∇ with the dot denotes [[divergence]], and '''B''' is the [[magnetic flux density]], the first integral is over a surface <math> S</math> with oriented surface element <math> d\mathbf{S}</math>. Where ∇ with the cross denotes [[Curl (mathematics)|curl]], '''J''' is the [[current density]] and {{math|'''H'''}} is the [[magnetic field intensity]], the second integral is a line integral around a closed loop <math> C</math> with line element <math>\mathbf{l}</math>. The current going through the loop is <math> 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> \mathbf{J}</math> term against the <math> \partial \mathbf{D} / \partial t</math> term. If the <math>\mathbf{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> \partial \mathbf{B} / \partial t</math>. Plugging this result into [[Faraday's law of induction|Faraday's Law]] finds a value for <math> \mathbf{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===
[[File:Magnetostatics_relation_triangle.svg|thumb|Summary of magnetostatic relations between magnetic vector potential, magnetic field and current density. Here, <math>\mathbf r = \mathbf x - \mathbf{x'}</math>.]]
If all currents in a system are known (i.e., if a complete description of the current density <math> \mathbf{J}(\mathbf{r})</math> is available) then the magnetic field can be determined, at a position '''r''', from the currents by the [[Biot–Savart law|Biot–Savart equation]]:<ref name="jackson75">{{cite book|last1=Jackson|first1=John David|title=Classical electrodynamics | date=1975 | publisher=Wiley | location=New York | isbn=047143132X | edition = 2nd | url=https://archive.org/details/classicalelectro00jack_0 }}</ref>{{rp|174}}
<math display="block">\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int{\frac{\mathbf{J}(\mathbf{r}') \times \left(\mathbf{r} - \mathbf{r}'\right)}{|\mathbf{r} - \mathbf{r}'|^3} \mathrm{d}^3\mathbf{r}'}</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 [[Inductor#Air-core inductor|air-core inductor]]s and [[air-core transformer]]s. One advantage of this technique is that, if a coil has a complex geometry, it can be divided into sections and the integral evaluated for each section. Since this equation is primarily used to solve [[linear]] problems, the contributions can be added. For a very difficult geometry, [[numerical integration]] may be used.

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 scalar potential|magnetic potential]]. The value of <math> \mathbf{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 display="block"> \mathbf{B} = \nabla \times \mathbf{A}, </math>
and the relation of the vector potential to current is:<ref name="jackson75"/>{{rp|176}}
<math display="block"> \mathbf{A}(\mathbf{r}) = \frac{\mu_{0}}{4\pi} \int{ \frac{\mathbf{J(\mathbf{r}')} } {|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3\mathbf{r}'}. </math>

===Magnetization===
{{further|Demagnetizing field|Micromagnetics}}
Strongly magnetic materials (i.e., [[Ferromagnetism|ferromagnetic]], [[Ferrimagnetism|ferrimagnetic]] or [[Paramagnetism|paramagnetic]]) have a [[magnetization]] that is primarily due to [[Spin (physics)|electron spin]]. In such materials the magnetization must be explicitly included using the relation
<math display="block"> \mathbf{B} = \mu_0(\mathbf{M}+\mathbf{H}).</math>

Except in the case of conductors, electric currents can be ignored. Then Ampère's law is simply
<math display="block"> \nabla\times\mathbf{H} = 0.</math>

This has the general solution
<math display="block"> \mathbf{H} = -\nabla \Phi_M, </math>
where <math>\Phi_M</math> is a scalar [[potential]].<ref name="jackson75"/>{{rp|192}} Substituting this in Gauss's law gives
<math display="block"> \nabla^2 \Phi_M = \nabla\cdot\mathbf{M}.</math>

Thus, the divergence of the magnetization, <math> \nabla\cdot\mathbf{M},</math> has a role analogous to the electric charge in electrostatics<ref>{{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 = https://archive.org/details/introductiontoth00ahar }}</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 display="block"> \mathbf{J_M} = \nabla \times \mathbf{M}. </math>

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

==References==
{{Reflist|2}}

==External links==
*{{Commons category-inline}}

{{Branches of physics}}
{{Authority control}}

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