View source for Magnetic dipole

{{short description|Magnetic analogue of the electric dipole}}
[[File:VFPt_dipoles_magnetic.svg|thumb|350px|The '''[[magnetic field]]''' due to natural magnetic dipoles (upper left), [[magnetic monopole]]s (upper right), an [[electric current]] in a circular loop (lower left) or in a [[solenoid]] (lower right). All generate the same field profile when the arrangement is infinitesimally small.<ref>{{cite book|author=I.S. Grant, W.R. Phillips|title=Electromagnetism|url=https://archive.org/details/electromagnetism0000gran|url-access=registration|edition=2nd|publisher=Manchester Physics, John Wiley & Sons|year=2008|isbn=978-0-471-92712-9}}</ref>]]

In [[electromagnetism]], a '''magnetic dipole''' is the limit of either a closed loop of [[electric current]] or a pair of poles as the size of the source is reduced to zero while keeping the [[magnetic moment]] constant. 

It is a magnetic analogue of the [[Electric dipole moment|electric dipole]], but the analogy is not perfect. In particular, a true [[magnetic monopole]], the magnetic analogue of an [[electric charge]], has never been observed in nature. However, magnetic monopole [[quasiparticles]] have been observed as emergent properties of certain condensed matter systems.<ref>[https://physicsworld.com/a/magnetic-monopoles-spotted-in-spin-ices/ Magnetic monopoles spotted in spin ices], September 3, 2009.</ref> Moreover, one form of magnetic dipole moment is associated with a fundamental quantum property—the [[Spin (physics)|spin]] of [[elementary particles]].

Because magnetic monopoles do not exist, the magnetic field at a large distance from any static magnetic source looks like the field of a dipole with the same dipole moment. For higher-order sources (e.g. [[Quadrupole magnet|quadrupoles]]) with no dipole moment, their field decays towards zero with distance faster than a dipole field does.

== External magnetic field produced by a magnetic dipole moment ==
[[Image:VFPt dipole electric.svg|thumb|200px|upright|An electrostatic analogue for a magnetic moment: two opposing charges separated by a finite distance. Each arrow represents the direction of the field vector at that point.]]
[[Image:VFPt dipole magnetic3.svg|thumbnail|200px|right|The magnetic field of a current loop. The ring represents the current loop, which goes into the page at the x and comes out at the dot.]]

In [[classical physics]], the magnetic field of a dipole is calculated as the limit of either a current loop or a pair of charges as the source shrinks to a point while keeping the [[magnetic moment]] {{math|'''m''' }}constant. For the current loop, this limit is most easily derived from the [[Magnetic vector potential|vector potential]]:<ref name=Chow146>{{harvnb|Chow|2006|pages=146&ndash;150}}</ref>

: <math>{\mathbf{A}}({\mathbf{r}})=\frac{\mu_{0}}{4\pi r^{2}}\frac{{\mathbf{m}}\times{\mathbf{r}}}{r}=\frac{\mu_{0}}{4\pi}\frac{{\mathbf{m}}\times{\mathbf{r}}}{r^{3}},</math>

where ''μ''<sub>0</sub> is the [[vacuum permeability]] constant and {{math|4''&pi; r''<sup>2</sup>}} is the surface of a sphere of radius {{math|''r''}}.
The magnetic flux density (strength of the B-field) is then<ref name=Chow146/>

:<math>\mathbf{B}({\mathbf{r}})=\nabla\times{\mathbf{A}}=\frac{\mu_{0}}{4\pi}\left[\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{r^{5}}-\frac{{\mathbf{m}}}{r^{3}}\right].</math>

Alternatively one can obtain the [[Magnetic scalar potential|scalar potential]] first from the magnetic pole limit,
:<math>\psi({\mathbf{r}})=\frac{{\mathbf{m}}\cdot{\mathbf{r}}}{4\pi r^{3}},</math>

and hence the magnetic field strength (or strength of the H-field) is
:<math>{\mathbf{H}}({\mathbf{r}})=-\nabla\psi=\frac{1}{4\pi}\left[\frac{3\mathbf{\hat{r}}(\mathbf{m}\cdot\mathbf{\hat{r}})-\mathbf{m}}{r^{3}}\right] = \frac{\mathbf{B}}{\mu_0}.</math>

The magnetic field strength is symmetric under rotations about the axis of the magnetic moment.
In spherical coordinates, with <math>\mathbf{\hat{z}} = \mathbf{\hat{r}}\cos\theta - \boldsymbol{\hat{\theta}}\sin\theta</math>, and with the magnetic moment aligned with the z-axis, then the field strength can more simply be expressed as

:<math>\mathbf{H}({\mathbf{r}})=\frac{|\mathbf{m}|}{4\pi r^3} \left (
2 \cos \theta \, \mathbf{\hat{r}} + \sin \theta \, \boldsymbol{\hat{\theta}} \right ) .
</math>

== Internal magnetic field of a dipole ==
{{See also|Magnetic moment#Magnetic pole definition}}

The two models for a dipole (current loop and magnetic poles), give the same predictions for the magnetic field far from the source. However, inside the source region they give different predictions. The magnetic field between poles is in the opposite direction to the magnetic moment (which points from the negative charge to the positive charge), while inside a current loop it is in the same direction (see the figure to the right (above for mobile users)). Clearly, the limits of these fields must also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used to calculate fields inside a magnetic material.

If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current and area constant, the limiting field is
:<math>\mathbf{B}(\mathbf{r})=\frac{\mu_0}{4\pi}\left[\frac{3\mathbf{\hat{r}}(\mathbf{\hat{r}}\cdot \mathbf{m})-\mathbf{m}}{|\mathbf{r}|^3} + \frac{8\pi}{3}\mathbf{m}\delta(\mathbf{r})\right],</math>
where {{math|''δ''('''r''')}} is the [[Dirac delta function]] in three dimensions. Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.

If a magnetic dipole is formed by taking a "north pole" and a "south pole", bringing them closer and closer together but keeping the product of magnetic pole-charge and distance constant, the limiting field is
:<math>\mathbf{H}(\mathbf{r}) =\frac{1}{4\pi}\left[\frac{3\mathbf{\hat{r}}(\mathbf{\hat{r}}\cdot \mathbf{m})-\mathbf{m}}{|\mathbf{r}|^3} - \frac{4\pi}{3}\mathbf{m}\delta(\mathbf{r})\right].</math>

These fields are related by {{math|'''B''' {{=}} ''&mu;''<sub>0</sub>('''H''' + '''M''')}}, where
:<math>\mathbf{M}(\mathbf{r}) = \mathbf{m}\delta(\mathbf{r})</math>
is the [[magnetization]].

== Forces between two magnetic dipoles ==
{{See also|Force between magnets#Magnetic dipole-dipole interaction}}

The force {{math|'''F'''}} exerted by one dipole moment {{math|'''m'''<sub>1</sub>}} on another {{math|'''m'''<sub>2</sub>}} separated in space by a vector {{math|'''r'''}} can be calculated using:<ref>{{cite book|title=Introduction to Electrodynamics|edition=3rd |author=D.J. Griffiths|publisher=Pearson Education|page=276|year=2007|isbn=978-81-7758-293-2}}</ref>

:<math> \mathbf{F} = \nabla\left(\mathbf{m}_2\cdot\mathbf{B}_1\right), </math>

or<ref>{{harvnb|Furlani|2001|p=140}}</ref><ref>{{cite journal |year=1998 |title= An Analytic Solution for the Force between Two Magnetic Dipoles|author1=K.W. Yung |author2=P.B. Landecker |author3=D.D. Villani |url=http://downloads.hindawi.com/archive/1998/079537.pdf|access-date=November 24, 2012 }}</ref>

: <math>

\mathbf{F}(\mathbf{r}, \mathbf{m}_1, \mathbf{m}_2) = \dfrac{3 \mu_0}{4 \pi r^5}\left[(\mathbf{m}_1\cdot\mathbf{r})\mathbf{m}_2 + (\mathbf{m}_2\cdot\mathbf{r})\mathbf{m}_1 + (\mathbf{m}_1\cdot\mathbf{m}_2)\mathbf{r} - \dfrac{5(\mathbf{m}_1\cdot\mathbf{r})(\mathbf{m}_2\cdot\mathbf{r})}{r^2}\mathbf{r}\right],
</math>

where {{math|r}} is the distance between dipoles. The force acting on {{math|'''m'''<sub>1</sub>}} is in the opposite direction.

The torque can be obtained from the formula

: <math>\boldsymbol{\tau}=\mathbf{m}_2 \times \mathbf{B}_1.</math>

== Dipolar fields from finite sources ==
{{See also|Near and far field}}

The [[magnetic scalar potential]] {{math|<var>&psi;</var>}} produced by a finite source, but external to it, can be represented by a [[multipole expansion]]. Each term in the expansion is associated with a characteristic [[Multipole moment|moment]] and a potential having a characteristic rate of decrease with distance {{math|<var>r</var>}} from the source. Monopole moments have a {{math|1/<var>r</var>}} rate of decrease, dipole moments have a {{math|1/<var>r</var><sup>2</sup>}} rate, quadrupole moments have a {{math|1/<var>r</var><sup>3</sup>}} rate, and so on. The higher the order, the faster the potential drops off. Since the lowest-order term observed in magnetic sources is the dipole term, it dominates at large distances. Therefore, at large distances any magnetic source looks like a dipole of the same [[magnetic moment]].

== Notes ==
{{Reflist|30 em}}

== References ==
{{Refbegin|2}}
*{{cite book
|last=Chow
|first=Tai L.
|title=Introduction to electromagnetic theory: a modern perspective
|publisher = [[Jones & Bartlett Learning]]
|year=2006
|isbn=978-0-7637-3827-3
|url=https://books.google.com/books?id=dpnpMhw1zo8C&q=%22magnetic+dipole%22
}}
*{{Cite book | last1=Jackson | first1=John D. | title=Classical Electrodynamics | edition=2nd | publisher=[[John Wiley & Sons|Wiley]] | year=1975 | isbn=0-471-43132-X | url-access=registration | url=https://archive.org/details/classicalelectro00jack_0 }}
*{{cite book
  |last = Furlani
  |first = Edward P.
  |title = Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications
  |publisher = [[Academic Press]]
  |year = 2001
  |url  = https://books.google.com/books?id=irsdLnC5SrsC&q=3.130
  |isbn = 0-12-269951-3
}}
*{{cite journal
| author = Schill, R. A.
| title = General relation for the vector magnetic field of a circular current loop: A closer look
| year = 2003
| journal = [[IEEE Transactions on Magnetics]]
| volume = 39
| issue = 2
| pages = 961–967
| doi = 10.1109/TMAG.2003.808597|bibcode = 2003ITM....39..961S }}
{{Refend}}

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