Editing Magnetic field (section)

==Magnetic field of permanent magnets==
{{Main|Magnetic moment#Models}}

''Permanent magnets'' are objects that produce their own persistent magnetic fields. They are made of [[ferromagnetism|ferromagnetic]] materials, such as iron and [[nickel]], that have been magnetized, and they have both a north and a south pole.

The magnetic field of permanent magnets can be quite complicated, especially near the magnet. The magnetic field of a small<ref group="note" name="ex05">Here, "small" means that the observer is sufficiently far away from the magnet, so that the magnet can be considered as infinitesimally small. "Larger" magnets need to include more complicated terms in the mathematical expression of the magnetic field and depend on the entire geometry of the magnet not just {{math|'''m'''}}.</ref> straight magnet is proportional to the magnet's ''strength'' (called its [[magnetic dipole moment]] {{math|'''m'''}}). The [[Dipole#Field of a static magnetic dipole|equations]] are non-trivial and depend on the distance from the magnet and the orientation of the magnet. For simple magnets, {{math|'''m'''}} points in the direction of a line drawn from the south to the north pole of the magnet. Flipping a bar magnet is equivalent to rotating its {{math|'''m'''}} by 180 degrees.

The magnetic field of larger magnets can be obtained by modeling them as a collection of a large number of small magnets called [[dipole]]s each having their own {{math|'''m'''}}. The magnetic field produced by the magnet then is the net magnetic field of these dipoles; any net force on the magnet is a result of adding up the forces on the individual dipoles.

There are two simplified models for the nature of these dipoles: the [[Magnetic field#Magnetic pole model|magnetic pole model]] and the [[Magnetic field#Amperian loop model|Amperian loop model]]. These two models produce two different magnetic fields, {{math|'''H'''}} and {{math|'''B'''}}. Outside a material, though, the two are identical (to a multiplicative constant) so that in many cases the distinction can be ignored. This is particularly true for magnetic fields, such as those due to electric currents, that are not generated by magnetic materials.

A realistic model of magnetism is more complicated than either of these models; neither model fully explains why materials are magnetic. The monopole model has no experimental support. The Amperian loop model explains some, but not all of a material's magnetic moment. The model predicts that the motion of electrons within an atom are connected to those electrons' [[Electron magnetic moment#Orbital magnetic dipole moment|orbital magnetic dipole moment]], and these orbital moments do contribute to the magnetism seen at the macroscopic level. However, the motion of electrons is not classical, and the [[spin magnetic moment]] of electrons (which is not explained by either model) is also a significant contribution to the total moment of magnets.

===Magnetic pole model===
{{See also|Magnetic monopole}}
[[Image:VFPt dipole electric.svg|thumb|200px|upright|The magnetic pole model: two opposing poles, North (+) and South (−), separated by a distance d produce a {{math|'''H'''}}-field (lines).]]

Historically, early physics textbooks would model the force and torques between two magnets as due to magnetic poles repelling or attracting each other in the same manner as the [[Coulomb force]] between electric charges. At the microscopic level, this model contradicts the experimental evidence, and the pole model of magnetism is no longer the typical way to introduce the concept.<ref name="Griffiths3ed"/>{{rp|p=258}} However, it is still sometimes used as a macroscopic model for ferromagnetism due to its mathematical simplicity.<ref>{{cite book|title=Magnetostatic Principles in Ferromagnetism|first=William Fuller|last=Brown | year=1962 |publisher=North Holland publishing company|asin=B0006AY7F8|page=12}}</ref>

In this model, a magnetic {{math|'''H'''}}-field is produced by fictitious ''magnetic charges'' that are spread over the surface of each pole. These ''magnetic charges'' are in fact related to the magnetization field {{math|'''M'''}}. The {{math|'''H'''}}-field, therefore, is analogous to the [[electric field]] {{math|'''E'''}}, which starts at a positive [[electric charge]] and ends at a negative electric charge. Near the north pole, therefore, all {{math|'''H'''}}-field lines point away from the north pole (whether inside the magnet or out) while near the south pole all {{math|'''H'''}}-field lines point toward the south pole (whether inside the magnet or out). Too, a north pole feels a force in the direction of the {{math|'''H'''}}-field while the force on the south pole is opposite to the {{math|'''H'''}}-field.

In the magnetic pole model, the elementary magnetic dipole {{math|'''m'''}} is formed by two opposite magnetic poles of pole strength {{math|''q''{{sub|m}}}} separated by a small distance vector {{math|'''d'''}}, such that {{math|1='''m''' = ''q''{{sub|m}}&thinsp;'''d'''}}. The magnetic pole model predicts correctly the field {{math|'''H'''}} both inside and outside magnetic materials, in particular the fact that {{math|'''H'''}} is opposite to the magnetization field {{math|'''M'''}} inside a permanent magnet.

Since it is based on the fictitious idea of a ''magnetic charge density'', the pole model has limitations. Magnetic poles cannot exist apart from each other as electric charges can, but always come in north–south pairs. If a magnetized object is divided in half, a new pole appears on the surface of each piece, so each has a pair of complementary poles. The magnetic pole model does not account for magnetism that is produced by electric currents, nor the inherent connection between [[angular momentum]] and magnetism.

The pole model usually treats magnetic charge as a mathematical abstraction, rather than a physical property of particles. However, a [[magnetic monopole]] is a hypothetical particle (or class of particles) that physically has only one magnetic pole (either a north pole or a south pole). In other words, it would possess a "magnetic charge" analogous to an electric charge. Magnetic field lines would start or end on magnetic monopoles, so if they exist, they would give exceptions to the rule that magnetic field lines neither start nor end. Some theories (such as [[Grand Unified Theory|Grand Unified Theories]]) have predicted the existence of magnetic monopoles, but so far, none have been observed.

===Amperian loop model===
{{Main|Magnetic dipole}}
{{see also|Spin magnetic moment|Micromagnetism}}

{{Multiple image|header=The Amperian loop model
| align = right | total_width = 320
|image1=VFPt dipole magnetic3.svg | alt1 = |caption1=
|image2=VFPt dipole animation magnetic.gif| alt2 = |caption2=
|footer=A current loop (ring) that goes into the page at the x and comes out at the dot produces a {{math|'''B'''}}-field (lines). As the radius of the current loop shrinks, the fields produced become identical to an abstract "magnetostatic dipole" (represented by an arrow pointing to the right).}}

In the model developed by [[André-Marie Ampère|Ampere]], the elementary magnetic dipole that makes up all magnets is a sufficiently small Amperian loop with current {{math|''I''}} and loop area {{math|''A''}}. The dipole moment of this loop is {{math|1=''m'' = ''IA''}}.

These magnetic dipoles produce a magnetic {{math|'''B'''}}-field.

The magnetic field of a magnetic dipole is depicted in the figure. From outside, the ideal magnetic dipole is identical to that of an ideal electric dipole of the same strength. Unlike the electric dipole, a magnetic dipole is properly modeled as a current loop having a current {{math|''I''}} and an area {{math|''a''}}. Such a current loop has a magnetic moment of
<math display="block">m = Ia ,</math>
where the direction of {{math|'''m'''}} is perpendicular to the area of the loop and depends on the direction of the current using the right-hand rule. An ideal magnetic dipole is modeled as a real magnetic dipole whose area {{math|''a''}} has been reduced to zero and its current {{math|''I''}} increased to infinity such that the product {{math|1=''m'' = ''Ia''}} is finite. This model clarifies the connection between angular momentum and magnetic moment, which is the basis of the [[Einstein–de Haas effect]] ''rotation by magnetization'' and its inverse, the [[Barnett effect]] or ''magnetization by rotation''.<ref name=Graham>See [[Magnetic moment#Magnetic dipoles|magnetic moment]]{{Broken anchor|date=2024-03-25|bot=User:Cewbot/log/20201008/configuration|target_link=Magnetic moment#Magnetic dipoles|reason= The anchor (Magnetic dipoles) [[Special:Diff/839030611|has been deleted]].}} and {{cite book |title=Introduction to Magnetic Materials |author1=B. D. Cullity |author2=C. D. Graham |url=https://books.google.com/books?id=ixAe4qIGEmwC&pg=PA103 |page=103 |isbn=978-0-471-47741-9 |year=2008 |publisher=Wiley-IEEE |edition=2}}</ref> Rotating the loop faster (in the same direction) increases the current and therefore the magnetic moment, for example.