View source for Magnetic field

{{short description|Distribution of magnetic force}}
{{Other uses}}{{pp-semi-indef}}
{{Multiple issues|
{{Technical|date=September 2010}}
{{Lead too long|date=October 2025}}
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{{Use dmy dates|date=October 2019}}

{{multiple image
| align = right
| total_width = 300
| direction = horizontal
| header   =
| image1   = Magnetic field of horseshoe magnet.png|
| caption1 = A [[permanent magnet]], made of a magnetized metal alloy.
| image2   = Magnetic field around solenoid.jpg 
| caption2 = A [[solenoid]] (or [[electromagnet]]), a coil of wire carrying an electric current.
| footer   = The shape of the magnetic fields around the permanent magnet and the electromagnet are revealed by the orientation of the iron filings.
}}
A '''magnetic field''' (sometimes called '''B-field'''<ref>{{cite web |last1=Nave |first1=Rod |title=Magnetic Field |url=http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfie.html |website=HyperPhysics |access-date=20 May 2024}}</ref>) is a [[physical field]] that describes the magnetic influence on moving [[electric charge]]s, [[electric currents]],<ref name="feynman">{{cite book|last1=Feynman|first1=Richard P.|url=https://feynmanlectures.caltech.edu | title=The Feynman Lectures on Physics|last2=Leighton|first2=Robert B.|last3=Sands|first3=Matthew|date=1963|publisher=California Institute of Technology | volume=2}}</ref>{{Rp|at=ch1}}<ref>{{cite book |last1 = Young |first1 = Hugh D. |last2 = Freedman |first2 = Roger A. |last3 = Ford |first3 = A. Lewis |title = Sears and Zemansky's university physics: with modern physics | publisher = Pearson Addison-Wesley |volume = 2 |date = 2008 |pages = 918–919 |isbn = 978-0-321-50121-9}}</ref> and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own [[velocity]] and to the magnetic field.<ref name="feynman" />{{Rp|at=ch13}}<ref name="Purcell3ed">{{Cite book|last1=Purcell |first1=Edward M.|author-link1=Edward Mills Purcell|last2=Morin |first2=David J.|year=2013 |title=[[Electricity and Magnetism (book)|Electricity and Magnetism]] |edition=3rd |publisher=Cambridge University Press |isbn=978-1-107-01402-2}}</ref>{{rp|p=278}} A [[permanent magnet]]'s magnetic field pulls on [[ferromagnetic material]]s such as [[iron]], and attracts or repels other magnets. In addition, a nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: [[paramagnetism]], [[diamagnetism]], and [[antiferromagnetism]], although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, electric currents, and [[electric field]]s varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a [[function (mathematics)|function]] assigning a [[Euclidean vector|vector]] to each point of space, called a [[vector field]] (more precisely, a [[pseudovector]] field).

== Electromagnetics ==

In [[electromagnetics]], the term ''magnetic field'' is used for two distinct but closely related vector fields denoted by the symbols {{math|'''B'''}} and {{math|'''H'''}}. In the [[International System of Units]], the unit of {{math|'''B'''}}, [[magnetic flux]] density, is the [[tesla (unit)|tesla]] (in SI base units: kilogram per second squared per ampere),<ref name=":1"/>{{rp|p=21}} which is equivalent to [[newton (unit)|newton]] per meter per ampere. The unit of {{math|'''H'''}}, magnetic field strength, is [[ampere]] per meter (A/m).<ref name=":1">{{SIbrochure9th}}</ref>{{rp|p=22}} {{math|'''B'''}} and {{math|'''H'''}} differ in how they take the medium and/or magnetization into account. In [[vacuum]], the two fields are related through the [[vacuum permeability]], <math>\mathbf{B}/\mu_0 = \mathbf{H}</math>; in a magnetized material, the quantities on each side of this equation differ by the [[magnetization]] field of the material.

Magnetic fields are produced by moving electric charges and the intrinsic [[magnetic moment]]s of [[elementary particle]]s associated with a fundamental quantum property, their [[Spin (physics)|spin]].<ref name="Jiles">{{cite book | last = Jiles | first = David C. | title = Introduction to Magnetism and Magnetic Materials | publisher = CRC | edition = 2 | date = 1998 | page = 3 | url = https://books.google.com/books?id=axyWXjsdorMC&pg=PA3 | isbn = 978-0-412-79860-3 }}</ref><ref name="feynman" />{{Rp|at=ch1}} Magnetic fields and [[electric field]]s are interrelated and are both components of the [[electromagnetic force]], one of the four [[fundamental force]]s of nature.

Magnetic fields are used throughout modern technology, particularly in [[electrical engineering]] and [[electromechanics]]. Rotating magnetic fields are used in both [[electric motor]]s and [[electric generator|generators]]. The interaction of magnetic fields in electric devices such as transformers is conceptualized and investigated as [[magnetic circuit]]s. Magnetic forces give information about the charge carriers in a material through the [[Hall effect]]. The Earth produces [[Earth's magnetic field|its own magnetic field]], which shields the Earth's ozone layer from the [[solar wind]] and is important in [[navigation]] using a [[compass]].
{{TOC limit|limit=3}}

==Description== <!-- note that [[Hans Christian Ørsted]] links here -->

The force on an electric charge depends on its location, speed, and direction; two vector fields are used to describe this force.<ref name="feynman" />{{Rp|at=ch1}} The first is the [[electric field]], which describes the force acting on a stationary charge and gives the component of the force that is independent of motion. The magnetic field, in contrast, describes the component of the force that is proportional to both the speed and direction of charged particles.<ref name="feynman" />{{Rp|at=ch13}} The field is defined by the [[Lorentz force law]] and is, at each instant, perpendicular to both the motion of the charge and the force it experiences.

There are two different, but closely related vector fields which are both sometimes called the "magnetic field" written {{math|'''B'''}} and {{math|'''H'''}}.<ref group="note">The letters B and H were originally chosen by Maxwell in his ''[[Treatise on Electricity and Magnetism]]'' (Vol. II, pp. 236–237). For many quantities, he simply started choosing letters from the beginning of the alphabet. See {{cite journal|author=Ralph Baierlein|title=Answer to Question #73. S is for entropy, Q is for charge|journal=American Journal of Physics|year=2000|volume=68|issue=8|page=691|doi=10.1119/1.19524|bibcode = 2000AmJPh..68..691B }}</ref> While both the best names for these fields and exact interpretation of what these fields represent has been the subject of long running debate, there is wide agreement about how the underlying physics work.<ref>{{cite journal|doi=10.1119/1.19459|title=B and H, the intensity vectors of magnetism: A new approach to resolving a century-old controversy |author=John J. Roche|year=2000|volume=68|issue=5|page=438|journal=American Journal of Physics|bibcode=2000AmJPh..68..438R }}</ref> Historically, the term "magnetic field" was reserved for {{math|'''H'''}} while using other terms for {{math|'''B'''}}, but many recent textbooks use the term "magnetic field" to describe {{math|'''B'''}} as well as or in place of {{math|'''H'''}}.<ref group="note" name="ex03">[[Edward Mills Purcell|Edward Purcell]], in Electricity and Magnetism, McGraw-Hill, 1963, writes, ''Even some modern writers who treat {{math|'''B'''}} as the primary field feel obliged to call it the magnetic induction because the name magnetic field was historically preempted by {{math|'''H'''}}. This seems clumsy and pedantic. If you go into the laboratory and ask a physicist what causes the pion trajectories in his bubble chamber to curve, he'll probably answer "magnetic field", not "magnetic induction." You will seldom hear a geophysicist refer to the Earth's magnetic induction, or an astrophysicist talk about the magnetic induction of the galaxy. We propose to keep on calling {{math|'''B'''}} the magnetic field. As for {{math|'''H'''}}, although other names have been invented for it, we shall call it "the field {{math|'''H'''}}" or even "the magnetic field {{math|'''H'''}}."'' In a similar vein, {{cite book |author=M Gerloch |title=Magnetism and Ligand-field Analysis |url=https://books.google.com/books?id=Ovo8AAAAIAAJ&pg=PA110 |page=110 |isbn=978-0-521-24939-3 |publisher=Cambridge University Press |year=1983}} says: "So we may think of both {{math|'''B'''}} and {{math|'''H'''}} as magnetic fields, but drop the word 'magnetic' from {{math|'''H'''}} so as to maintain the distinction ... As Purcell points out, 'it is only the names that give trouble, not the symbols'."</ref>
There are many alternative names for both (see sidebars).

=== The B-field ===
{{Multiple image|header=Finding the magnetic force
| align = right | total_width = 400px
|image1=FuerzaCentripetaLorentzP.svg | alt1 = |caption1=A charged particle that is moving with velocity '''v''' in a magnetic field '''B''' will feel a magnetic force '''F'''. Since the magnetic force always pulls sideways to the direction of motion, the particle moves in a circle.
|image2=Mano-2.svg| alt2 = |caption2=Since these three vectors are related to each other by a [[cross product]], the direction of this force can be found using the [[right hand rule]].
|footer=}}

{| class="wikitable" style="float:right;"
|-
!Alternative names for '''B'''<ref name=Electromagnetics>E. J. Rothwell and M. J. Cloud (2010) [https://books.google.com/books?id=7AHLBQAAQBAJ&pg=PA23 ''Electromagnetics'']. Taylor & Francis. p. 23. {{ISBN|1-4200-5826-6}}.</ref>
|-
|
* Magnetic flux density<ref name=":1"/>{{rp|p=138}}
* Magnetic induction<ref name=Stratton />
* Magnetic field (ambiguous)
|}

The magnetic field vector {{math|'''B'''}} at any point can be defined as the vector that, when used in the [[Lorentz force law]], correctly predicts the force on a charged particle at that point:<ref name="purcell2ed" /><ref name="Griffiths3ed">{{cite book | title=Introduction to Electrodynamics | first=David J. | last=Griffiths | author-link = David J. Griffiths | year=1999 | edition=3rd|isbn=0-13-805326-X | publisher=Pearson}}</ref>{{rp|p=204}}

{{Equation box 1
|indent=:
|title='''Lorentz force law'''  (''[[Euclidean vector|vector]] form, [[International System of Units|SI units]]'')
|equation=<math>\mathbf{F} = q\mathbf{E} + q(\mathbf{v} \times \mathbf{B})</math>
|cellpadding
|border
|border colour = rgb(0,115,207)
|background colour = rgb(0,115,207,10%)}}

Here {{math|'''F'''}} is the force on the particle, {{math|''q''}} is the particle's [[electric charge]], {{math|'''E'''}} is the external electric field, {{math|'''v'''}}, is the particle's [[velocity]], and × denotes the [[cross product]]. The direction of force on the charge can be determined by a [[mnemonic]] known as the ''right-hand rule'' (see the figure).<ref group="note">An alternative mnemonic to the right hand rule is [[Fleming's left-hand rule for motors|Fleming's left-hand rule]].</ref> Using the right hand, pointing the thumb in the direction of the current, and the fingers in the direction of the magnetic field, the resulting force on the charge points outwards from the palm. The force on a negatively charged particle is in the opposite direction. If both the speed and the charge are reversed then the direction of the force remains the same. For that reason a magnetic field measurement (by itself) cannot distinguish whether there is a positive charge moving to the right or a negative charge moving to the left. (Both of these cases produce the same current.) On the other hand, a magnetic field combined with an electric field ''can'' distinguish between these, see [[#Hall effect|Hall effect]] below.

The first term in the Lorentz equation is from the theory of [[electrostatics]], and says that a particle of charge {{math|''q''}} in an electric field {{math|'''E'''}} experiences an electric force:
<math display="block">\mathbf{F}_{\text{electric}} = q \mathbf{E}.</math>

The second term is the magnetic force:<ref name="Griffiths3ed" />
<math display="block">\mathbf{F}_{\text{magnetic}} = q(\mathbf{v} \times \mathbf{B}).</math>

Using the definition of the cross product, the magnetic force can also be written as a [[Scalar (physics)|scalar]] equation:<ref name="purcell2ed"/>{{rp|p=357}}
<math display="block">F_{\text{magnetic}} = q v B \sin(\theta)</math>
where {{math|''F''<sub>magnetic</sub>}}, {{mvar|v}}, and {{mvar|B}} are the [[Norm (mathematics)|scalar magnitude]] of their respective vectors, and {{mvar|θ}} is the angle between the velocity of the particle and the magnetic field. The vector {{math|'''B'''}} is ''defined'' as the vector field necessary to make the Lorentz force law correctly describe the motion of a charged particle. In other words,<ref name="purcell2ed">{{cite book|last=Purcell | first = E. | title=Electricity and Magnetism | url=https://archive.org/details/electricitymagne00purc_621|url-access=limited|year=2011|edition=2nd | publisher=Cambridge University Press|isbn=978-1-107-01360-5}}</ref>{{rp|pages=[https://archive.org/details/electricitymagne00purc_621/page/n192 173]–4}}

{{blockquote|[T]he command, "Measure the direction and magnitude of the vector {{math|'''B'''}} at such and such a place," calls for the following operations: Take a particle of known charge {{math|''q''}}. Measure the force on {{math|''q''}} at rest, to determine {{math|'''E'''}}. Then measure the force on the particle when its velocity is {{math|'''v'''}}; repeat with {{math|'''v'''}} in some other direction. Now find a {{math|'''B'''}} that makes the Lorentz force law fit all these results—that is the magnetic field at the place in question.}}

The {{math|'''B'''}} field can also be defined by the torque on a magnetic dipole, {{math|'''m'''}}.<ref name="jackson3ed">{{cite book | last1=Jackson | first1=John David | author-link = John David Jackson (physicist) | title=Classical electrodynamics | date=1998 | publisher=Wiley | location=New York | isbn=0-471-30932-X | edition=3rd}}</ref>{{rp|p=174}}

{{Equation box 1
|indent=:
|title='''Magnetic torque'''  (''[[Euclidean vector|vector]] form, [[International System of Units|SI units]]'')
|equation=<math>\boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}</math>
|cellpadding
|border
|border colour = rgb(0,115,207)
|background colour = rgb(0,115,207,10%)}}

The [[SI]] unit of {{math|'''B'''}} is [[tesla (unit)|tesla]] (symbol: T).<ref group="note">The SI unit of {{math|Φ<sub>''B''</sub>}} ([[magnetic flux]]) is the [[Weber (unit)|weber]] (symbol: Wb), related to the [[tesla (unit)|tesla]] by 1&nbsp;Wb/m<sup>2</sup> = 1&nbsp;T. The SI unit tesla is equal to ([[newton (unit)|newton]]·[[second]])/([[coulomb (unit)|coulomb]]·[[metre]]). This can be seen from the magnetic part of the Lorentz force law.</ref> The [[Gaussian units|Gaussian-cgs unit]] of {{math|'''B'''}} is the [[gauss (unit)|gauss]] (symbol: G). (The conversion is 1&nbsp;T ≘ 10000&nbsp;G.<ref name=BIPMTab9>{{cite web |url=https://www.bipm.org/en/publications/si-brochure/table9.html |title=Non-SI units accepted for use with the SI, and units based on fundamental constants (contd.) |website=SI Brochure: The International System of Units (SI) [8th edition, 2006; updated in 2014] |publisher=Bureau International des Poids et Mesures |access-date=19 April 2018 |archive-date=8 June 2019 |archive-url=https://web.archive.org/web/20190608123210/https://www.bipm.org/en/publications/si-brochure/table9.html }}</ref><ref name=KLang/>) One nanotesla corresponds to 1 gamma (symbol: γ).<ref name=KLang>{{cite book |title=A Companion to Astronomy and Astrophysics|url=https://books.google.com/books?id=aUjkKuaVIloC&pg=PA176 |last=Lang|first=Kenneth R.|publisher=Springer |access-date=19 April 2018|date=2006|page=176|isbn=978-0-387-33367-0 }}</ref>

=== The H-field ===

{| class="wikitable" style="float:right;"
|-
!Alternative names for '''H'''<ref name=Electromagnetics />
|-
|
* Magnetic field intensity<ref name=Stratton>{{cite book|title=Electromagnetic Theory|first=Julius Adams|last=Stratton|year=1941|publisher=McGraw-Hill|page=1|isbn=978-0-07-062150-3|edition=1st}}</ref>
* Magnetic field strength<ref name=":1"/>{{rp|p=139}}
* Magnetic field
* Magnetizing field
* Auxiliary magnetic field
|}

The magnetic {{math|'''H'''}} field is defined:<ref name="Griffiths3ed"/>{{rp|p=269}}<ref name="jackson3ed"/>{{rp|p=192}}<ref name="feynman" />{{Rp|at=ch36}}

{{Equation box 1
|indent=:
|title='''Definition of the {{math|H}} field''' ''([[Euclidean vector|vector]] form, [[International System of Units|SI units]])''
|equation=<math>\mathbf{H}\equiv\frac{1}{\mu_0}\mathbf{B}-\mathbf{M}</math>
|cellpadding
|border
|border colour = rgb(0,115,207)
|background colour = rgb(0,115,207,10%)}}

where <math>\mu_0</math> is the [[vacuum permeability]], and {{math|'''M'''}} is the [[Magnetization|magnetization vector]]. In a vacuum, {{math|'''B'''}} and {{math|'''H'''}} are proportional to each other. Inside a material they are different (see [[#H-field and magnetic materials|H and B inside and outside magnetic materials]]). The SI unit of the {{math|'''H'''}}-field is the [[ampere]] per metre (A/m),<ref>{{cite web|title=International system of units (SI) |url=https://physics.nist.gov/cuu/Units/units.html |website=NIST reference on constants, units, and uncertainty |date=12 April 2010 |publisher=National Institute of Standards and Technology |access-date=9 May 2012}}</ref> and the CGS unit is the [[oersted]] (Oe).<ref name=BIPMTab9/><ref name="purcell2ed"/>{{rp|p=286|quote= Units: Tesla for describing a large magnetic force; gauss (tesla/10000) for describing a small magnetic force as that at the surface of earth.}}

===Measurement===
{{main|Magnetometer}}

An instrument used to measure the local magnetic field is known as a [[magnetometer]]. Important classes of magnetometers include using [[Search coil|induction magnetometers]] (or search-coil magnetometers) which measure only varying magnetic fields, [[Magnetometer#Rotating coil magnetometer|rotating coil magnetometers]], [[Hall effect]] magnetometers, [[proton magnetometer|NMR magnetometers]], [[SQUID|SQUID magnetometers]], and [[Magnetometer#Fluxgate magnetometer|fluxgate magnetometers]]. The magnetic fields of distant [[astronomical object]]s are measured through their effects on local charged particles. For instance, electrons spiraling around a field line produce [[synchrotron radiation]] that is detectable in [[radio waves]]. The finest precision for a magnetic field measurement was attained by [[Gravity Probe B]] at {{val|5|u=aT}} ({{val|5|e=-18|u=T}}).<ref>{{cite web | url=http://www.nasa.gov/pdf/168808main_gp-b_pfar_cvr-pref-execsum.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.nasa.gov/pdf/168808main_gp-b_pfar_cvr-pref-execsum.pdf |archive-date=2022-10-09 |url-status=live|title=Gravity Probe B Executive Summary|pages=10, 21}}</ref>

=== Visualization{{anchor|Magnetic field line}}===
{{Main|Field line}}

{{Multiple image|header=Visualizing magnetic fields
| align = right | total_width = 350
|image1=Magnet0873.png | alt1= |caption1=
|image2=Magnetic field near pole.svg | alt2= |caption2=
|footer=Left: the direction of magnetic [[field line]]s represented by [[iron filings]] sprinkled on paper placed above a bar magnet.<br />
Right: [[compass]] needles point in the direction of the local magnetic field, towards a magnet's south pole and away from its north pole.
}}

The field can be visualized by a set of ''magnetic field lines'', that follow the direction of the field at each point. The lines can be constructed by measuring the strength and direction of the magnetic field at a large number of points (or at every point in space). Then, mark each location with an arrow (called a [[vector (geometry)|vector]]) pointing in the direction of the local magnetic field with its magnitude proportional to the strength of the magnetic field. Connecting these arrows then forms a set of magnetic field lines. The direction of the magnetic field at any point is parallel to the direction of nearby field lines, and the local density of field lines can be made proportional to its strength. Magnetic field lines are like [[Streamlines, streaklines, and pathlines|streamlines]] in [[Fluid dynamics|fluid flow]], in that they represent a continuous distribution, and a different resolution would show more or fewer lines.

An advantage of using magnetic field lines as a representation is that many laws of magnetism (and electromagnetism) can be stated completely and concisely using simple concepts such as the "number" of field lines through a surface. These concepts can be quickly "translated" to their mathematical form. For example, the number of field lines through a given surface is the [[surface integral]] of the magnetic field.<ref name="purcell2ed"/>{{rp|p=237}}

Various phenomena "display" magnetic field lines as though the field lines were physical phenomena. For example, iron filings placed in a magnetic field form lines that correspond to "field lines".<ref group="note" name="ex07">The use of iron filings to display a field presents something of an exception to this picture; the filings alter the magnetic field so that it is much larger along the "lines" of iron, because of the large [[magnetic permeability|permeability]] of iron relative to air.</ref> Magnetic field "lines" are also visually displayed in [[Aurora (astronomy)|polar auroras]], in which [[Plasma (physics)|plasma]] particle dipole interactions create visible streaks of light that line up with the local direction of Earth's magnetic field.

Field lines can be used as a qualitative tool to visualize magnetic forces. In [[ferromagnetic]] substances like [[iron]] and in plasmas, magnetic forces can be understood by imagining that the field lines exert a [[Maxwell stress tensor|tension]], (like a rubber band) along their length, and a pressure perpendicular to their length on neighboring field lines. "Unlike" poles of magnets attract because they are linked by many field lines; "like" poles repel because their field lines do not meet, but run parallel, pushing on each other.<!-- Note: this last "explanation" is purely pedagogical, without technical substance and should be improved. -->

==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#Relation to angular momentum|magnetic moment]] 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.

==Interactions with magnets==
===Force between magnets===
{{Main|Force between magnets}}

Specifying the [[magnetic moment#Forces between two magnetic dipoles|force between two small magnets]] is quite complicated because it depends on the strength and [[Orientation (geometry)|orientation]] of both magnets and their distance and direction relative to each other. The force is particularly sensitive to rotations of the magnets due to magnetic torque. The force on each magnet depends on its magnetic moment and the magnetic field<ref group="note" name="ex04">Either {{math|'''B'''}} or {{math|'''H'''}} may be used for the magnetic field outside the magnet.</ref> of the other.

To understand the force between magnets, it is useful to examine the ''magnetic pole model'' given above. In this model, the ''{{math|'''H'''}}-field'' of one magnet pushes and pulls on ''both'' poles of a second magnet. If this {{math|'''H'''}}-field is the same at both poles of the second magnet then there is no net force on that magnet since the force is opposite for opposite poles. If, however, the magnetic field of the first magnet is ''nonuniform'' (such as the {{math|'''H'''}} near one of its poles), each pole of the second magnet sees a different field and is subject to a different force. This difference in the two forces moves the magnet in the direction of increasing magnetic field and may also cause a net torque.

This is a specific example of a general rule that magnets are attracted (or repulsed depending on the orientation of the magnet) into regions of higher magnetic field. Any non-uniform magnetic field, whether caused by permanent magnets or electric currents, exerts a force on a small magnet in this way.

The details of the Amperian loop model are different and more complicated but yield the same result: that magnetic dipoles are attracted/repelled into regions of higher magnetic field. Mathematically, the force on a small magnet having a magnetic moment {{math|'''m'''}} due to a magnetic field {{math|'''B'''}} is:<ref name=Trigg>{{cite book |title=AIP physics desk reference |author1=E. Richard Cohen |author2=David R. Lide |author3=George L. Trigg |url=https://books.google.com/books?id=JStYf6WlXpgC&pg=PA381 |page=381 |isbn=978-0-387-98973-0 |publisher=Birkhäuser |year=2003 |edition=3}}</ref>{{rp|at=Eq. 11.42}}

<math display="block">\mathbf{F} = \boldsymbol{\nabla} \left(\mathbf{m}\cdot\mathbf{B}\right),</math>

where the [[gradient]] {{math|'''∇'''}} is the change of the quantity {{math|'''m''' · '''B'''}} per unit distance and the direction is that of maximum increase of {{math|'''m''' · '''B'''}}. The [[dot product]] {{math|1='''m''' · '''B''' = ''mB''cos(''θ'')}}, where {{math|''m''}} and {{math|''B''}} represent the [[magnitude (vector)|magnitude]] of the {{math|'''m'''}} and {{math|'''B'''}} vectors and {{math|''θ''}} is the angle between them. If {{math|'''m'''}} is in the same direction as {{math|'''B'''}} then the dot product is positive and the gradient points "uphill" pulling the magnet into regions of higher {{math|'''B'''}}-field (more strictly larger {{math|'''m''' · '''B'''}}). This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions each having their own {{math|'''m'''}} then [[Integral|summing up the forces on each of these very small regions]].

===Magnetic torque on permanent magnets===
{{Main|Magnetic torque}}
If two like poles of two separate magnets are brought near each other, and one of the magnets is allowed to turn, it promptly rotates to align itself with the first. In this example, the magnetic field of the stationary magnet creates a ''magnetic torque'' on the magnet that is free to rotate. This magnetic torque {{math|'''τ'''}} tends to align a magnet's poles with the magnetic field lines. A compass, therefore, turns to align itself with Earth's magnetic field.

{{Multiple image|header=Torque on a dipole
| align = right | total_width =320
|image1=Dipole in uniform H field.svg | alt1 = |caption1=In the pole model of a dipole, an {{math|'''H'''}} field (to right) causes equal but opposite forces on a N pole ({{math|+''q''}}) and a S pole ({{math|−''q''}}) creating a torque.
|image2=Torque-current-loop.svg| alt2 = |caption2=Equivalently, a {{math|'''B'''}} field induces the same torque on a current loop with the same magnetic dipole moment.
|footer=}} In terms of the pole model, two equal and opposite magnetic charges experiencing the same {{math|'''H'''}} also experience equal and opposite forces. Since these equal and opposite forces are in different locations, this produces a torque proportional to the distance (perpendicular to the force) between them. With the definition of {{math|'''m'''}} as the pole strength times the distance between the poles, this leads to {{math|1=''τ'' = ''μ''<sub>0</sub> ''m H'' sin&thinsp;''θ''}}, where {{math|''μ''<sub>0</sub>}} is a constant called the [[vacuum permeability]], measuring {{val|4|end= π|e=-7}} [[Volt|V]]·[[Second|s]]/([[Ampere|A]]·[[meter|m]]) and {{mvar|θ}} is the angle between {{math|'''H'''}} and {{math|'''m'''}}.

Mathematically, the torque {{math|'''τ'''}} on a small magnet is proportional both to the applied magnetic field and to the magnetic moment {{math|'''m'''}} of the magnet:

<math display="block">\boldsymbol{\tau}=\mathbf{m}\times\mathbf{B} = \mu_0\mathbf{m}\times\mathbf{H}, \,</math>

where × represents the vector [[cross product]]. This equation includes all of the qualitative information included above. There is no torque on a magnet if {{math|'''m'''}} is in the same direction as the magnetic field, since the cross product is zero for two vectors that are in the same direction. Further, all other orientations feel a torque that twists them toward the direction of magnetic field.

==Interactions with electric currents==
Currents of electric charges both generate a magnetic field and feel a force due to magnetic B-fields.

===Magnetic field due to moving charges and electric currents===
{{Main|Electromagnet|Biot–Savart law|Ampère's circuital law|l3=Ampère's law}}
[[Image:Manoderecha.svg|thumb|right|[[Right hand grip rule]]: a current flowing in the direction of the white arrow produces a magnetic field shown by the red arrows.]]

All moving charged particles produce magnetic fields. Moving [[point particle|point]] charges, such as [[electron]]s, produce complicated but well known magnetic fields that depend on the charge, velocity, and acceleration of the particles.<ref>{{harvnb|Griffiths|1999|p=438}}</ref>

Magnetic field lines form in [[concentric]] circles around a [[Cylinder (geometry)|cylindrical]] current-carrying conductor, such as a length of wire. The direction of such a magnetic field can be determined by using the "[[right-hand grip rule]]" (see figure at right). The strength of the magnetic field decreases with distance from the wire. (For an infinite length wire the strength is inversely proportional to the distance.)
[[File:VFPt cylindrical tightly-wound coil-and-bar-magnet-comparison stacked.svg|150 px|thumb|left|A [[Solenoid]] with electric current running through it behaves like a magnet.]]

Bending a current-carrying wire into a loop concentrates the magnetic field inside the loop while weakening it outside. Bending a wire into multiple closely spaced loops to form a coil or "[[solenoid]]" enhances this effect. A device so formed around an iron [[Magnetic core|core]] may act as an ''electromagnet'', generating a strong, well-controlled magnetic field. An infinitely long cylindrical electromagnet has a uniform magnetic field inside, and no magnetic field outside. A finite length electromagnet produces a magnetic field that looks similar to that produced by a uniform permanent magnet, with its strength and polarity determined by the current flowing through the coil.

The magnetic field generated by a steady current {{math|I}} (a constant flow of electric charges, in which charge neither accumulates nor is depleted at any point){{refn|group="note" |name="ex12"|In practice, the Biot–Savart law and other laws of magnetostatics are often used even when a current change in time, as long as it does not change too quickly. It is often used, for instance, for standard household currents, which oscillate sixty times per second.<ref name="Griffiths4ed"/>{{rp|p=223}}}} is described by the ''[[Biot–Savart law]]'':<ref name="Griffiths4ed">{{cite book |last1=Griffiths |first1=David J. | author-link = David J. Griffiths |title=[[Introduction to Electrodynamics]] |date=2017 |publisher=Cambridge University Press | isbn=978-1-108-35714-2 |edition=4th}}</ref>{{rp|p=224}}
<math display="block"> \mathbf{B} = \frac{\mu_0I}{4\pi}\int_{\mathrm{wire}}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2},</math>
where the integral sums over the wire length where vector {{math|d'''ℓ'''}} is the vector [[line element]] with direction in the same sense as the current {{math|''I''}}, {{math|''μ''<sub>0</sub>}} is the [[magnetic constant]], {{math|''r''}} is the distance between the location of {{math|d'''ℓ'''}} and the location where the magnetic field is calculated, and {{math|'''r̂'''}} is a unit vector in the direction of {{math|'''r'''}}. For example, in the case of a sufficiently long, straight wire, this becomes:
<math display="block"> |\mathbf{B}| = \frac{\mu_0}{2\pi r}I</math>
where {{math|1=''r'' = {{abs|'''r'''}}}}. The direction is tangent to a circle perpendicular to the wire according to the right hand rule.<ref name="Griffiths4ed"/>{{rp|p=225}}

A slightly more general<ref>{{harvnb|Griffiths|1999|pp=222–225}}</ref><ref group="note" name="ex13">
The Biot–Savart law contains the additional restriction (boundary condition) that the B-field must go to zero fast enough at infinity. It also depends on the divergence of {{math|'''B'''}} being zero, which is always valid.  (There are no magnetic charges.)</ref> way of relating the current <math>I</math> to the {{math|'''B'''}}-field is through [[Ampère's circuital law|Ampère's law]]:
<math display="block">\oint \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 I_{\mathrm{enc}},</math>
where the [[line integral]] is over any arbitrary loop and <math>I_\text{enc}</math> is the current enclosed by that loop. Ampère's law is always valid for steady currents and can be used to calculate the {{math|'''B'''}}-field for certain highly symmetric situations such as an infinite wire or an infinite solenoid.

In a modified form that accounts for time varying electric fields, Ampère's law is one of four [[Maxwell's equations]] that describe electricity and magnetism.

===Force on moving charges and current===

====Force on a charged particle====
{{Main|Lorentz force}}

A [[charged particle]] moving in a {{math|'''B'''}}-field experiences a ''sideways'' force that is proportional to the strength of the magnetic field, the component of the velocity that is perpendicular to the magnetic field and the charge of the particle. This force is known as the ''Lorentz force'', and is given by
<math display="block">\mathbf{F} = q\mathbf{E} + q \mathbf{v} \times \mathbf{B},</math>
where {{math|'''F'''}} is the [[force]], {{math|''q''}} is the [[electric charge]] of the particle, {{math|'''v'''}} is the instantaneous [[velocity]] of the particle, and {{math|'''B'''}} is the magnetic field (in [[tesla (unit)|teslas]]).

The Lorentz force is always perpendicular to both the velocity of the particle and the magnetic field that created it. When a charged particle moves in a static magnetic field, it traces a helical path in which the helix axis is parallel to the magnetic field, and in which the speed of the particle remains constant. Because the magnetic force is always perpendicular to the motion, the magnetic field can do no [[mechanical work|work]] on an isolated charge.<ref>{{Cite web|title=K. McDonald's Physics Examples - Disk|url=https://puhep1.princeton.edu/~kirkmcd/examples/disk.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://puhep1.princeton.edu/~kirkmcd/examples/disk.pdf |archive-date=2022-10-09 |url-status=live|access-date=2021-02-13|website=puhep1.princeton.edu}}</ref><ref>{{Cite web|title=K. McDonald's Physics Examples - Railgun|url=https://physics.princeton.edu//~mcdonald/examples/railgun.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://physics.princeton.edu//~mcdonald/examples/railgun.pdf |archive-date=2022-10-09 |url-status=live | access-date=2021-02-13|website=puhep1.princeton.edu}}</ref> It can only do work indirectly, via the electric field generated by a changing magnetic field. It is often claimed that the magnetic force can do work to a non-elementary [[magnetic dipole]], or to charged particles whose motion is constrained by other forces, but this is incorrect<ref name=Deissler>
{{cite journal
 |last = Deissler | first = R.J.
 |year=2008
 |title=Dipole in a magnetic field, work, and quantum spin
 |url=http://academic.csuohio.edu/deissler/PhysRevE_77_036609.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://academic.csuohio.edu/deissler/PhysRevE_77_036609.pdf |archive-date=2022-10-09 |url-status=live
 |journal=[[Physical Review E]]
 |volume=77 |issue=3, pt 2
 |article-number=036609
 |pmid=18517545
 |doi=10.1103/PhysRevE.77.036609
 |bibcode = 2008PhRvE..77c6609D }}</ref> because the work in those cases is performed by the electric forces of the charges deflected by the magnetic field.

====Force on current-carrying wire====
{{Main|Laplace force}}

The force on a current carrying wire is similar to that of a moving charge as expected since a current carrying wire is a collection of moving charges. A current-carrying wire feels a force in the presence of a magnetic field. The Lorentz force on a macroscopic current is often referred to as the ''Laplace force''.
Consider a conductor of length {{math|''ℓ''}}, cross section {{math|''A''}}, and charge {{math|''q''}} due to electric current {{math|''i''}}. If this conductor is placed in a magnetic field of magnitude {{math|''B''}} that makes an angle {{math|θ}} with the velocity of charges in the conductor, the force exerted on a single charge {{math|''q''}} is
<math display="block">F = qvB \sin\theta,</math>
so, for {{math|''N''}} charges where
<math display="block">N = n \ell A ,</math>
the force exerted on the conductor is
<math display="block">f = F N = q v B n\ell A \sin\theta = Bi\ell \sin\theta, </math>
where {{math|1=''i'' = ''nqvA''}}.

==Relation between H and B==
The formulas derived for the magnetic field above are correct when dealing with the entire current. A magnetic material placed inside a magnetic field, though, generates its own [[bound current]], which can be a challenge to calculate. (This bound current is due to the sum of atomic sized current loops and the [[Spin (physics)|spin]] of the subatomic particles such as electrons that make up the material.) The {{math|'''H'''}}-field as defined above helps factor out this bound current; but to see how, it helps to introduce the concept of ''magnetization'' first.

===Magnetization===
{{Main|Magnetization}}

The ''magnetization'' vector field {{math|'''M'''}} represents how strongly a region of material is magnetized. It is defined as the net [[magnetic dipole moment]] per unit volume of that region. The magnetization of a uniform magnet is therefore a material constant, equal to the magnetic moment {{math|'''m'''}} of the magnet divided by its volume. Since the SI unit of magnetic moment is A⋅m<sup>2</sup>, the SI unit of magnetization {{math|'''M'''}} is ampere per meter, identical to that of the {{math|'''H'''}}-field.

The magnetization {{math|'''M'''}} field of a region points in the direction of the average magnetic dipole moment in that region. Magnetization field lines, therefore, begin near the magnetic south pole and ends near the magnetic north pole. (Magnetization does not exist outside the magnet.)

In the Amperian loop model, the magnetization is due to combining many tiny Amperian loops to form a resultant current called ''[[bound current]]''. This bound current, then, is the source of the magnetic {{math|'''B'''}} field due to the magnet. Given the definition of the magnetic dipole, the magnetization field follows a similar law to that of Ampere's law:<ref>{{harvnb|Griffiths|1999|pp=266–268}}</ref>
<math display="block">\oint \mathbf{M} \cdot \mathrm{d}\boldsymbol{\ell} = I_\mathrm{b},</math>
where the integral is a line integral over any closed loop and {{math|''I''<sub>b</sub>}} is the bound current enclosed by that closed loop.

In the magnetic pole model, magnetization begins at and ends at magnetic poles. If a given region, therefore, has a net positive "magnetic pole strength" (corresponding to a north pole) then it has more magnetization field lines entering it than leaving it. Mathematically this is equivalent to:
<math display="block">\oint_S \mu_0 \mathbf{M} \cdot \mathrm{d}\mathbf{A} = - q_\mathrm{M},</math>
where the integral is a closed surface integral over the closed surface {{math|''S''}} and {{math|''q''<sub>M</sub>}} is the "magnetic charge" (in units of [[magnetic flux]]) enclosed by {{math|''S''}}. (A closed surface completely surrounds a region with no holes to let any field lines escape.) The negative sign occurs because the magnetization field moves from south to north.

===H-field and magnetic materials===
[[File:VFPt magnets BHM.svg|thumb|Comparison of {{math|'''B'''}}, {{math|'''H'''}} and {{math|'''M'''}} inside and outside a cylindrical bar magnet.]]
{{see also|Demagnetizing field}}
In SI units, the H-field is related to the B-field by
<math display="block">\mathbf{H}\ \equiv \ \frac{\mathbf{B}}{\mu_0} - \mathbf{M}.</math>

In terms of the H-field, Ampere's law is
<math display="block">\oint \mathbf{H} \cdot \mathrm{d}\boldsymbol{\ell} = \oint \left(\frac{\mathbf{B}}{\mu_0} - \mathbf{M}\right) \cdot \mathrm{d}\boldsymbol{\ell} = I_\mathrm{tot} - I_\mathrm{b} = I_\mathrm{f},</math>
where {{math|I<sub>f</sub>}} represents the 'free current' enclosed by the loop so that the line integral of {{math|'''H'''}} does not depend at all on the bound currents.<ref name=Slater>{{cite book |title=Electromagnetism |author1=John Clarke Slater |author2=Nathaniel Herman Frank |url=https://books.google.com/books?id=GYsphnFwUuUC&pg=PA69 |page=69 |isbn=978-0-486-62263-7 |year=1969 |publisher=Courier Dover Publications |edition=first published in 1947 }}</ref>

For the differential equivalent of this equation see [[#Maxwell's equations|Maxwell's equations]]. Ampere's law leads to the boundary condition
<math display="block">\left(\mathbf{H_1^\parallel} - \mathbf{H_2^\parallel}\right) = \mathbf{K}_\mathrm{f} \times \hat{\mathbf{n}},</math>
where {{math|'''K'''<sub>f</sub>}} is the surface free current density and the unit normal <math>\hat{\mathbf{n}}</math> points in the direction from medium 2 to medium 1.<ref>{{harvnb|Griffiths|1999|p=332}}</ref>

Similarly, a [[surface integral]] of {{math|'''H'''}} over any [[closed surface]] is independent of the free currents and picks out the "magnetic charges" within that closed surface:
<math display="block">\oint_S \mu_0 \mathbf{H} \cdot \mathrm{d}\mathbf{A} = \oint_S (\mathbf{B} - \mu_0 \mathbf{M}) \cdot \mathrm{d}\mathbf{A} = 0 - (-q_\mathrm{M}) = q_\mathrm{M},</math>

which does not depend on the free currents.

The {{math|'''H'''}}-field, therefore, can be separated into two<ref group="note">A third term is needed for changing electric fields and polarization currents; this displacement current term is covered in Maxwell's equations below.</ref> independent parts:
<math display="block">\mathbf{H} = \mathbf{H}_0 + \mathbf{H}_\mathrm{d}, </math>

where {{math|'''H'''<sub>0</sub>}} is the applied magnetic field due only to the free currents and {{math|'''H'''<sub>d</sub>}} is the [[demagnetizing field]] due only to the bound currents.

The magnetic {{math|'''H'''}}-field, therefore, re-factors the bound current in terms of "magnetic charges". The {{math|'''H'''}} field lines loop only around "free current" and, unlike the magnetic {{math|'''B'''}} field, begins and ends near magnetic poles as well.

===Magnetism===
{{Main|Magnetism}}
Most materials respond to an applied {{math|'''B'''}}-field by producing their own magnetization {{math|'''M'''}} and therefore their own {{math|'''B'''}}-fields. Typically, the response is weak and exists only when the magnetic field is applied. The term ''magnetism'' describes how materials respond on the microscopic level to an applied magnetic field and is used to categorize the magnetic [[phase (matter)|phase]] of a material. Materials are divided into groups based upon their magnetic behavior:
* [[Diamagnetism|Diamagnetic materials]]<ref name=Tilley>{{cite book |title=Understanding Solids |author=RJD Tilley |page=[https://archive.org/details/understandingsol0000till/page/368 368] |isbn=978-0-470-85275-0 |publisher=Wiley |year=2004 |url=https://archive.org/details/understandingsol0000till |url-access=registration }}</ref> produce a magnetization that opposes the magnetic field.
* [[Paramagnetism|Paramagnetic materials]]<ref name=Tilley/> produce a magnetization in the same direction as the applied magnetic field.
* [[Ferromagnetism|Ferromagnetic materials]] and the closely related [[Ferrimagnetism|ferrimagnetic materials]] and [[Antiferromagnetism|antiferromagnetic materials]]<ref name=Chikazumi>{{cite book |title=Physics of ferromagnetism|author1=Sōshin Chikazumi |author2=Chad D. Graham |url=https://books.google.com/books?id=AZVfuxXF2GsC |page=118 |edition=2|year=1997 |publisher=Oxford University Press |isbn=978-0-19-851776-4}}</ref><ref name=Aharoni>{{cite book |title=Introduction to the theory of ferromagnetism |page=27 |author=Amikam Aharoni |url=https://books.google.com/books?id=9RvNuIDh0qMC&pg=PA27 |edition=2 |year=2000 |publisher=Oxford University Press |isbn=978-0-19-850808-3}}</ref> can have a magnetization independent of an applied B-field with a complex relationship between the two fields.
* [[Superconductor]]s (and [[ferromagnetic superconductor]]s)<ref name=Bennemann>{{cite book |title=Superconductivity |editor1=K. H. Bennemann |editor2=John B. Ketterson |chapter-url=https://books.google.com/books?id=PguAgEQTiQwC&pg=PA640 |page=640 |isbn=978-3-540-73252-5 |year=2008 |publisher=Springer |chapter=Unconventional superconductivity in novel materials |author=M Brian Maple |display-authors=etal }}</ref><ref name=Lewis>{{cite book |title=Superconductivity research at the leading edge |author=Naoum Karchev |editor1=Paul S. Lewis |editor2=D. Di (CON) Castro |chapter-url=https://books.google.com/books?id=3AFo_yxBkD0C&pg=PA169 |page=169 |isbn=978-1-59033-861-2 |publisher=Nova Publishers |year=2003 |chapter=Itinerant ferromagnetism and superconductivity}}</ref> are materials that are characterized by perfect conductivity below a critical temperature and magnetic field. They also are highly magnetic and can be perfect diamagnets below a lower critical magnetic field. Superconductors often have a broad range of temperatures and magnetic fields (the so-named [[Type II superconductor#Mixed state|mixed state]]) under which they exhibit a complex hysteretic dependence of {{math|'''M'''}} on {{math|'''B'''}}.

In the case of paramagnetism and diamagnetism, the magnetization {{math|'''M'''}} is often proportional to the applied magnetic field such that:
<math display="block">\mathbf{B} = \mu \mathbf{H},</math>
where {{math|''μ''}} is a material dependent parameter called the [[permeability (electromagnetism)|permeability]]. In some cases the permeability may be a second rank [[tensor]] so that {{math|'''H'''}} may not point in the same direction as {{math|'''B'''}}. These relations between {{math|'''B'''}} and {{math|'''H'''}} are examples of [[constitutive equation]]s. However, superconductors and ferromagnets have a more complex {{math|'''B'''}}-to-{{math|'''H'''}} relation; see [[hysteresis#Magnetic hysteresis|magnetic hysteresis]].

==Stored energy==
{{Main|Magnetic energy}}
{{see also|Magnetic hysteresis|l1=Magnetic hysteresis}}

Energy is needed to generate a magnetic field both to work against the electric field that a changing magnetic field creates and to change the magnetization of any material within the magnetic field. For non-dispersive materials, this same energy is released when the magnetic field is destroyed so that the energy can be modeled as being stored in the magnetic field.

For linear, non-dispersive, materials (such that {{math|1='''B''' = ''μ'''''H'''}} where {{math|''μ''}} is frequency-independent), the [[energy density]] is:

<math display="block"> u = \frac{\mathbf{B} \cdot \mathbf{H}}{2}= \frac{\mathbf{B} \cdot \mathbf{B}}{2\mu} = \frac{\mu\mathbf{H} \cdot \mathbf{H}}{2}.</math>

If there are no magnetic materials around then {{math|''μ''}} can be replaced by {{math|''μ''<sub>0</sub>}}. The above equation cannot be used for nonlinear materials, though; a more general expression given below must be used.

In general, the incremental amount of work per unit volume {{math|''δW''}} needed to cause a small change of magnetic field {{math|''δ'''''B'''}} is:

<math display="block">\delta W = \mathbf{H}\cdot\delta\mathbf{B}.</math>

Once the relationship between {{math|'''H'''}} and {{math|'''B'''}} is known this equation is used to determine the work needed to reach a given magnetic state. For [[Hysteresis|hysteretic materials]] such as ferromagnets and superconductors, the work needed also depends on how the magnetic field is created. For linear non-dispersive materials, though, the general equation leads directly to the simpler energy density equation given above.

==Appearance in Maxwell's equations==
{{Main|Maxwell's equations}}
{{See also|Electromagnetism}}

Like all vector fields, a magnetic field has two important mathematical properties that relates it to its ''sources''. (For {{math|'''B'''}} the ''sources'' are currents and changing electric fields.) These two properties, along with the two corresponding properties of the electric field, make up ''Maxwell's Equations''. Maxwell's Equations together with the Lorentz force law form a complete description of [[Classical electromagnetism|classical electrodynamics]] including both electricity and magnetism.

The first property is the [[divergence]] of a vector field {{math|'''A'''}}, {{math|'''∇''' · '''A'''}}, which represents how {{math|'''A'''}} "flows" outward from a given point. As discussed above, a {{math|'''B'''}}-field line never starts or ends at a point but instead forms a complete loop. This is mathematically equivalent to saying that the divergence of {{math|'''B'''}} is zero. (Such vector fields are called [[solenoidal vector field]]s.) This property is called [[Gauss's law for magnetism]] and is equivalent to the statement that there are no isolated magnetic poles or [[magnetic monopole]]s.

The second mathematical property is called the [[curl (mathematics)|curl]], such that {{math|'''∇''' × '''A'''}} represents how {{math|'''A'''}} curls or "circulates" around a given point. The result of the curl is called a "circulation source". The equations for the curl of {{math|'''B'''}} and of {{math|'''E'''}} are called the [[Ampère–Maxwell equation]] and [[Faraday's law of induction|Faraday's law]] respectively.

===Gauss' law for magnetism===
{{main|Gauss's law for magnetism}}

One important property of the {{math|'''B'''}}-field produced this way is that magnetic {{math|'''B'''}}-field lines neither start nor end (mathematically, {{math|'''B'''}} is a [[solenoidal vector field]]); a field line may only extend to infinity, or wrap around to form a closed curve, or follow a never-ending (possibly chaotic) path.<ref name="lieb">{{cite journal | last1=Lieberherr | first1=Martin | date=6 July 2010|title=The magnetic field lines of a helical coil are not simple loops | journal=American Journal of Physics|volume=78|issue=11|pages=1117–1119|bibcode=2010AmJPh..78.1117L|doi=10.1119/1.3471233|doi-access=free}}</ref> Magnetic field lines exit a magnet near its north pole and enter near its south pole, but inside the magnet {{math|'''B'''}}-field lines continue through the magnet from the south pole back to the north.<ref group="note" name="ex08">To see that this must be true imagine placing a compass inside a magnet. There, the north pole of the compass points toward the north pole of the magnet since magnets stacked on each other point in the same direction.</ref> If a {{math|'''B'''}}-field line enters a magnet somewhere it has to leave somewhere else; it is not allowed to have an end point.

More formally, since all the magnetic field lines that enter any given region must also leave that region, subtracting the "number"<ref group="note" name="ex09">As discussed above, magnetic field lines are primarily a conceptual tool used to represent the mathematics behind magnetic fields. The total "number" of field lines is dependent on how the field lines are drawn. In practice, integral equations such as the one that follows in the main text are used instead.</ref> of field lines that enter the region from the number that exit gives identically zero. Mathematically this is equivalent to [[Gauss's law for magnetism]]:
<math display="block">\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math>
where the integral is a [[surface integral]] over the [[closed surface]] {{math|''S''}} (a closed surface is one that completely surrounds a region with no holes to let any field lines escape). Since {{math|d'''A'''}} points outward, the dot product in the integral is positive for {{math|'''B'''}}-field pointing out and negative for {{math|'''B'''}}-field pointing in.

===Faraday's Law===
{{Main|Faraday's law of induction}}

A changing magnetic field, such as a magnet moving through a conducting coil, generates an [[electric field]] (and therefore tends to drive a current in such a coil). This is known as ''Faraday's law'' and forms the basis of many [[electrical generator]]s and [[electric motor]]s. Mathematically, Faraday's law is:
<math display="block">\mathcal{E} = - \frac{\mathrm{d}\Phi}{\mathrm{d}t}</math>

where <math>\mathcal{E}</math> is the [[electromotive force]] (or ''EMF'', the [[voltage]] generated around a closed loop) and {{math|Φ}} is the [[magnetic flux]]—the product of the area times the magnetic field [[Tangential and normal components|normal]] to that area. (This definition of magnetic flux is why {{math|'''B'''}} is often referred to as ''magnetic flux density''.)<ref>{{cite book|last1=Jackson | first1=John David | author-link = John David Jackson (physicist) | title=Classical electrodynamics | date=1975 | publisher=Wiley|location=New York | isbn=978-0-471-43132-9|edition=2nd|url=https://archive.org/details/classicalelectro00jack_0}}</ref>{{rp|p=210}} The negative sign represents the fact that any current generated by a changing magnetic field in a coil produces a magnetic field that ''opposes'' the ''change'' in the magnetic field that induced it. This phenomenon is known as [[Lenz's law]]. This integral formulation of Faraday's law can be converted<ref group="note" name="ex14">
A complete expression for Faraday's law of induction in terms of the electric {{math|'''E'''}} and magnetic fields can be written as:
<math display="block">\mathcal{E} = - \frac{d\Phi}{dt} = \oint_{\partial \Sigma (t)} \left( \mathbf{E}( \mathbf{r},\ t) +\mathbf{ v} \times \mathbf{B}(\mathbf{r},\ t)\right) \cdot d\boldsymbol{\ell}\ =-\frac {d} {dt} \iint_{\Sigma (t)} d \boldsymbol {A} \cdot \mathbf {B} (\mathbf{r},\ t)</math>
where {{math|'''∂Σ'''(''t'')}} is the moving closed path bounding the moving surface {{math|'''Σ'''(''t'')}}, and {{math|d'''A'''}} is an element of surface area of {{math|'''Σ'''(''t'')}}. The first integral calculates the work done moving a charge a distance {{math|d'''ℓ'''}} based upon the Lorentz force law. In the case where the bounding surface is stationary, the [[Kelvin–Stokes theorem]] can be used to show this equation is equivalent to the Maxwell–Faraday equation.
</ref> into a differential form, which applies under slightly different conditions.

<math display="block">
  \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}
</math>

===Ampère's Law and Maxwell's correction===
{{Main|Ampère's circuital law}}

Similar to the way that a changing magnetic field generates an electric field, a changing electric field generates a magnetic field. This fact is known as ''Maxwell's correction to Ampère's law'' and is applied as an additive term to Ampere's law as given above. This additional term is proportional to the time rate of change of the electric flux and is similar to Faraday's law above but with a different and positive constant out front. (The electric flux through an area is proportional to the area times the perpendicular part of the electric field.)

The full law including the correction term is known as the Maxwell–Ampère equation. It is not commonly given in integral form because the effect is so small that it can typically be ignored in most cases where the integral form is used.

The Maxwell term ''is'' critically important in the creation and propagation of electromagnetic waves. Maxwell's correction to Ampère's Law together with Faraday's law of induction describes how mutually changing electric and magnetic fields interact to sustain each other and thus to form [[electromagnetic waves]], such as light: a changing electric field generates a changing magnetic field, which generates a changing electric field again. These, though, are usually described using the differential form of this equation given below.

<math display="block">
  \nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}
</math>

where {{math|'''J'''}} is the complete microscopic [[current density]], and {{mvar|ε{{sub|0}}}} is the [[vacuum permittivity]].

As discussed above, materials respond to an applied electric {{math|'''E'''}} field and an applied magnetic {{math|'''B'''}} field by producing their own internal "bound" charge and current distributions that contribute to {{math|'''E'''}} and {{math|'''B'''}} but are difficult to calculate. To circumvent this problem, {{math|'''H'''}} and {{math|'''D'''}} fields are used to re-factor Maxwell's equations in terms of the ''free current density'' {{math|'''J'''<sub>f</sub>}}:

<math display="block">\nabla \times \mathbf{H} = \mathbf{J}_\mathrm{f} + \frac{\partial \mathbf{D}} {\partial t}</math>

These equations are not any more general than the original equations (if the "bound" charges and currents in the material are known). They also must be supplemented by the relationship between {{math|'''B'''}} and {{math|'''H'''}} as well as that between {{math|'''E'''}} and {{math|'''D'''}}. On the other hand, for simple relationships between these quantities this form of Maxwell's equations can circumvent the need to calculate the bound charges and currents.

==Formulation in special relativity and quantum electrodynamics==
===Relativistic electrodynamics===
{{Main|Relativistic electromagnetism}}

==== As different aspects of the same phenomenon ====
According to [[special relativity|the special theory of relativity]], the partition of the [[electromagnetic force]] into separate electric and magnetic components is not fundamental, but varies with the [[Frame of reference#Observational frames of reference|observational frame of reference]]: An electric force perceived by one observer may be perceived by another (in a different frame of reference) as a magnetic force, or a mixture of electric and magnetic forces.

The magnetic field existing as electric field in other frames can be shown by consistency of equations obtained from [[Lorentz transformation]] of four force from [[Coulomb's law|Coulomb's Law]] in particle's rest frame with [[Maxwell's equations|Maxwell's laws]] considering definition of fields from [[Lorentz force#Definition and properties|Lorentz force]] and for non accelerating condition. The form of magnetic field hence obtained by [[Lorentz transformation]] of [[four-force]] from the form of [[Coulomb's law]] in source's initial frame is given by:<ref name="Rosser1968">{{Cite book |last1=Rosser |first1=W. G. V. |url=https://link.springer.com/book/10.1007/978-1-4899-6559-2 |title=Classical Electromagnetism via Relativity |year=1968 |isbn=978-1-4899-6258-4 |publisher=Springer |location=Boston, MA |language=en |doi=10.1007/978-1-4899-6559-2}}</ref>{{rp|pages=29–42}}

<math display="block">\mathbf{B} = \frac q {4 \pi \varepsilon_0 r^3}   \frac {1- \beta^2} {(1- \beta^2 \sin^2 \theta)^{3/2}} \frac{\mathbf{v} \times \mathbf{r}}{c^2} = \frac{\mathbf{v} \times \mathbf{E}}{c^2}</math>

where <math>q</math> is the charge of the point source, <math>\varepsilon_0</math> is the [[vacuum permittivity]], <math>\mathbf{r}</math> is the position vector from the point source to the point in space, <math>\mathbf{v}</math> is the velocity vector of the charged particle, <math>\beta</math> is the ratio of speed of the charged particle divided by the speed of light and <math>\theta</math> is the angle between <math>\mathbf{r}</math> and <math>\mathbf{v}</math>. This form of magnetic field can be shown to satisfy Maxwell's laws within the constraint of particle being non accelerating.<ref>{{Cite book |last=Purcell |first=Edward |title=Electricity and Magnetism |date=2011-09-22 |publisher=Cambridge University Press |isbn=978-1-107-01360-5 |doi=10.1017/cbo9781139005043}}</ref> The above reduces to [[Biot–Savart law|Biot-Savart law]] for non relativistic stream of current (<math>\beta\ll 1</math>).

Formally, special relativity combines the electric and magnetic fields into a rank-2 [[tensor]], called the ''[[electromagnetic tensor]]''. Changing reference frames ''mixes'' these components. This is analogous to the way that special relativity ''mixes'' space and time into [[spacetime]], and mass, momentum, and energy into [[four-momentum]].<ref>C. Doran and A. Lasenby (2003) ''Geometric Algebra for Physicists'', Cambridge University Press, p. 233. {{ISBN|0-521-71595-4}}.</ref> Similarly, the [[Magnetic energy|energy stored in a magnetic field]] is mixed with the energy stored in an electric field in the [[electromagnetic stress–energy tensor]].

==== Magnetic vector potential ====
{{Main|Magnetic vector potential}}

In advanced topics such as [[quantum mechanics]] and [[Theory of relativity|relativity]] it is often easier to work with a potential formulation of electrodynamics rather than in terms of the electric and magnetic fields. In this representation, the ''[[magnetic vector potential]]'' {{math|'''A'''}}, and the [[electric potential|electric scalar potential]] {{math|''φ''}}, are defined using [[gauge fixing]] such that:

<math display="block">\begin{align}
  \mathbf{B} &=  \nabla \times \mathbf{A}, \\
  \mathbf{E} &= -\nabla \varphi - \frac{ \partial \mathbf{A} }{ \partial t }.
\end{align}</math>

The vector potential, ''{{math|'''A'''}}'' given by this form may be interpreted as a ''generalized potential [[momentum]] per unit charge'' <ref>{{cite journal | author = E. J. Konopinski | author-link = Emil Konopinski | year =  1978 | title = What the electromagnetic vector potential describes | journal = Am. J. Phys. | volume = 46 | issue = 5| pages = 499–502 | doi = 10.1119/1.11298 | bibcode = 1978AmJPh..46..499K }}</ref> just as {{math|''φ''}} is interpreted as a ''generalized [[potential energy]] per unit charge''. There are multiple choices one can make for the potential fields that satisfy the above condition. However, the choice of potentials is represented by its respective gauge condition.

Maxwell's equations when expressed in terms of the potentials in [[Lorenz gauge condition|Lorenz gauge]] can be cast into a form that agrees with [[special relativity]].<ref>{{harvnb|Griffiths|1999|p=422}}</ref> In relativity, {{math|'''A'''}} together with {{math|''φ''}} forms a [[four-potential]] regardless of the gauge condition, analogous to the [[Four-vector#Four-momentum|four-momentum]] that combines the momentum and energy of a particle. Using the four potential instead of the electromagnetic tensor has the advantage of being much simpler—and it can be easily modified to work with quantum mechanics.

==== Propagation of Electric and Magnetic fields ====
[[Special relativity|Special theory of relativity]] imposes the condition for events related by [[Causality|cause and effect]] to be time-like separated, that is that causal efficacy propagates no faster than light.<ref>{{Cite book |author=Naber, Gregory L. |title=The Geometry of Minkowski spacetime: an introduction to the mathematics of the special theory of relativity |date=2012 |publisher=Springer |isbn=978-1-4419-7837-0 |pages=4–5 |oclc=804823303}}</ref> [[Maxwell's equations]] for electromagnetism are found to be in favor of this as electric and magnetic disturbances are found to travel at the speed of light in space. Electric and magnetic fields from classical electrodynamics obey the [[principle of locality]] in physics and are expressed in terms of retarded time or the time at which the cause of a measured field originated given that the influence of field travelled at speed of light. The retarded time for a point particle is given as solution of:

<math display="block">t_r = \mathbf{t} - \frac{\left|\mathbf{r} - \mathbf{r}_s(t_r)\right|}{c}</math>

where <math display="inline">t_r</math> is [[retarded time]] or the time at which the source's contribution of the field originated, <math display="inline">r_s(t)</math> is the position vector of the particle as function of time, <math display="inline">\mathbf{r}</math> is the point in space, <math display="inline">\mathbf{t}</math> is the time at which fields are measured and <math display="inline">c</math> is the speed of light. The equation subtracts the time taken for light to travel from particle to the point in space from the time of measurement to find time of origin of the fields. The uniqueness of solution for <math display="inline">t_r</math> for given <math>\mathbf{t}</math>, <math>\mathbf{r}</math> and <math>r_s(t)</math> is valid for charged particles moving slower than speed of light.<ref name="Rosser1968"/>

==== Magnetic field of arbitrary moving point charge ====
{{Main|Liénard–Wiechert potential}}
The solution of maxwell's equations for electric and magnetic field of a point charge is expressed in terms of [[retarded time]] or the time at which the particle in the past causes the field at the point, given that the influence travels across space at the speed of light.

Any arbitrary motion of point charge causes electric and magnetic fields found by solving maxwell's equations using green's function for retarded potentials and hence finding the fields to be as follows:

<math display="block">\begin{align}
\mathbf{A}(\mathbf{r},\mathbf{t}) &= \frac{\mu_0c}{4 \pi} \left[\frac{q \boldsymbol{\beta}_s}{(1 - \mathbf{n}_s \cdot \boldsymbol{\beta}_s)|\mathbf{r} - \mathbf{r}_s|} \right]_{t=t_r} \\[1ex]
&= \frac{\boldsymbol{\beta}_s(t_r)}{c} \varphi(\mathbf{r}, \mathbf{t})
\end{align}</math>
<math display="block">\begin{align}
\mathbf{B}(\mathbf{r}, \mathbf{t}) &= \frac{\mu_0}{4 \pi} \left[
    \frac{q c(\boldsymbol{\beta}_s \times \mathbf{n}_s)}{\gamma^2 {\left(1 - \mathbf{n}_s \cdot \boldsymbol{\beta}_s\right)}^3 {\left|\mathbf{r} - \mathbf{r}_s\right|}^2}
    + \frac{q \mathbf{n}_s \times \left(\mathbf{n}_s \times \left(\left(\mathbf{n}_s - \boldsymbol{\beta}_s\right) \times \dot{\boldsymbol{\beta}_s}\right) \right)}{{\left(1 - \mathbf{n}_s \cdot \boldsymbol{\beta}_s\right)}^3 \left|\mathbf{r} - \mathbf{r}_s\right|} \right]_{t=t_r} \\[1ex]
&= \frac{\mathbf{n}_s(t_r)}{c} \times \mathbf{E}(\mathbf{r}, \mathbf{t})
\end{align}</math>

where <math display="inline">\varphi(\mathbf{r}, \mathbf{t})</math>and <math display="inline">\mathbf{A}(\mathbf{r},\mathbf{t})</math> are electric scalar potential and magnetic vector potential in Lorentz gauge, <math>q</math> is the charge of the point source, <math display="inline">n_s(\mathbf{r},t)</math> is a unit vector pointing from charged particle to the point in space, <math display="inline"> \boldsymbol{\beta}_s(t)</math> is the velocity of the particle divided by the speed of light and <math display="inline">\gamma (t)</math> is the corresponding [[Lorentz factor]]. Hence by the [[Superposition principle|principle of superposition]], the fields of a system of charges also obey [[principle of locality]].

===Quantum electrodynamics===
{{See also|Standard Model|quantum electrodynamics}}
The classical electromagnetic field incorporated into quantum mechanics forms what is known as the semi-classical theory of radiation. However, it is not able to make experimentally observed predictions such as [[Spontaneous emission|spontaneous emission process]] or [[Lamb shift]] implying the need for quantization of fields. In modern physics, the electromagnetic field is understood to be not a ''[[classical physics|classical]]'' [[field (physics)|field]], but rather a [[quantum field]]; it is represented not as a vector of three [[real number|numbers]] at each point, but as a vector of three [[operator (physics)|quantum operators]] at each point. The most accurate modern description of the electromagnetic interaction (and much else) is ''quantum electrodynamics'' (QED),<ref>
For a good qualitative introduction see:
{{Cite book|author=Richard Feynman|author-link=Richard Feynman|title=QED: the strange theory of light and matter|year=2006| publisher=[[Princeton University Press]]|isbn=978-0-691-12575-6|title-link=QED (book)}}
</ref> which is incorporated into a more complete theory known as the ''Standard Model of particle physics''.

In QED, the magnitude of the electromagnetic interactions between charged particles (and their [[antiparticle]]s) is computed using [[perturbation theory (quantum mechanics)|perturbation theory]]. These rather complex formulas produce a remarkable pictorial representation as [[Feynman diagram]]s in which [[virtual photon]]s are exchanged.

Predictions of QED agree with experiments to an extremely high degree of accuracy: currently about 10<sup>−12</sup> (and limited by experimental errors); for details see [[precision tests of QED]]. This makes QED one of the most accurate physical theories constructed thus far.

All equations in this article are in the [[classical electromagnetism|classical approximation]], which is less accurate than the quantum description mentioned here. However, under most everyday circumstances, the difference between the two theories is negligible.

==Uses and examples==

===Earth's magnetic field===
{{Main|Earth's magnetic field}}
[[File:VFPt Earths Magnetic Field Confusion.svg|thumb|A sketch of Earth's magnetic field representing the source of the field as a magnet. The south pole of the magnetic field is near the geographic north pole of the Earth.]]

The Earth's magnetic field is produced by [[convection]] of a liquid iron alloy in the [[outer core]]. In a [[Dynamo theory|dynamo process]], the movements drive a feedback process in which electric currents create electric and magnetic fields that in turn act on the currents.<ref name="Weiss">{{cite journal |last=Weiss |first=Nigel |title=Dynamos in planets, stars and galaxies |journal=Astronomy and Geophysics |year=2002 |volume=43 |issue=3 |pages=3.09–3.15 |doi=10.1046/j.1468-4004.2002.43309.x | bibcode=2002A&G....43c...9W|doi-access=free }}</ref>

The field at the surface of the Earth is approximately the same as if a giant bar magnet were positioned at the center of the Earth and tilted at an angle of about 11° off the rotational axis of the Earth (see the figure).<ref>{{cite web |url=https://www.ncei.noaa.gov/products/geomagnetism-frequently-asked-questions |title=What is the Earth's magnetic field? |website=Geomagnetism Frequently Asked Questions |publisher=National Centers for Environmental Information, National Oceanic and Atmospheric Administration |access-date=19 April 2018}}</ref> The north pole of a magnetic compass needle points roughly north, toward the [[North Magnetic Pole]]. However, because a magnetic pole is attracted to its opposite, the North Magnetic Pole is actually the south pole of the geomagnetic field. This confusion in terminology arises because the pole of a magnet is defined by the geographical direction it points.<ref>{{cite book|author=Raymond A. Serway|title=College physics|url=https://archive.org/details/collegephysics00serw_139|url-access=limited|year=2009|publisher=Brooks/Cole, Cengage Learning|location=Belmont, CA | isbn=978-0-495-38693-3|edition=8th |author2=Chris Vuille |author3=Jerry S. Faughn |page=[https://archive.org/details/collegephysics00serw_139/page/n659 628]}}</ref>

Earth's magnetic field is not constant—the strength of the field and the location of its poles vary.<ref>{{cite book |last1=Merrill|first1=Ronald T.|last2=McElhinny|first2=Michael W.|last3=McFadden|first3=Phillip L.|chapter=2. The present geomagnetic field: analysis and description from historical observations|title=The magnetic field of the earth: paleomagnetism, the core, and the deep mantle|publisher=[[Academic Press]]|year=1996|isbn=978-0-12-491246-5}}</ref> Moreover, the poles periodically reverse their orientation in a process called [[geomagnetic reversal]]. The [[Brunhes–Matuyama reversal|most recent reversal]] occurred 780,000 years ago.<ref>{{cite news |url=https://science.nasa.gov/science-news/science-at-nasa/2003/29dec_magneticfield/ |title=Earth's Inconstant Magnetic Field |work=Science@Nasa |last=Phillips |first=Tony |date=29 December 2003 |access-date=27 December 2009 |archive-date=1 November 2022 |archive-url=https://web.archive.org/web/20221101165248/https://science.nasa.gov/science-news/science-at-nasa/2003/29dec_magneticfield/ }}</ref>

===Rotating magnetic fields===
{{Main|Rotating magnetic field|Alternator}}

The ''rotating magnetic field'' is a common design principle in the operation of [[electric motor#AC motors|alternating-current motors]]. A permanent magnet in such a field rotates so as to maintain its alignment with the external field.

Magnetic torque is used to drive [[electric motor]]s. In one simple motor design, a magnet is fixed to a freely rotating shaft and is subjected to a magnetic field from an array of [[electromagnet]]s. By continuously switching the electric current through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are kept next to the rotor; the resultant torque is transferred to the shaft.

A rotating magnetic field can be constructed using two coils at right angles with a phase difference of 90 degrees between their AC currents. In practice, [[Three-phase electric power|three-phase]] systems are used where the three currents are equal in magnitude and have a phase difference of 120 degrees. Three similar coils at mutual geometrical angles of 120 degrees create the rotating magnetic field. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the main reasons why three-phase systems dominate the world's [[electrical power]] supply systems.

[[Synchronous motor]]s use DC-voltage-fed rotor windings, which lets the excitation of the machine be controlled—and [[induction motor]]s use short-circuited [[Rotor (electric)|rotors]] (instead of a magnet) following the rotating magnetic field of a multicoiled [[stator]]. The short-circuited turns of the rotor develop [[eddy current]]s induced by the rotating field of the stator, and these currents in turn produce a torque on the rotor through the Lorentz force.

The Italian physicist [[Galileo Ferraris]] and the Serbian-American [[electrical engineer]] [[Nikola Tesla]] independently researched the use of rotating magnetic fields in electric motors. In 1888, Ferraris published his research in a paper to the ''Royal Academy of Sciences'' in [[Turin]] and Tesla gained {{US patent|381968}} for his work.

===Hall effect===
{{Main|Hall effect}}

The charge carriers of a current-carrying conductor placed in a transverse magnetic field experience a sideways Lorentz force; this results in a charge separation in a direction perpendicular to the current and to the magnetic field. The resultant voltage in that direction is proportional to the applied magnetic field. This is known as the ''Hall effect''.

The ''Hall effect'' is often used to measure the magnitude of a magnetic field. It is used as well to find the sign of the dominant charge carriers in materials such as semiconductors (negative electrons or positive holes).

===Magnetic circuits===
{{Main|Magnetic circuit}}
An important use of {{math|'''H'''}} is in ''magnetic circuits'' where {{math|1='''B''' = ''μ'''''H'''}} inside a linear material. Here, {{math|''μ''}} is the [[magnetic permeability]] of the material. This result is similar in form to [[Ohm's law]] {{math|1='''J''' = ''σ'''''E'''}}, where {{math|'''J'''}} is the current density, {{math|''σ''}} is the conductance and {{math|'''E'''}} is the electric field. Extending this analogy, the counterpart to the macroscopic Ohm's law ({{math|1=''I'' = ''V''⁄''R''}}) is:

<math display="block">\Phi = \frac F R_\mathrm{m},</math>

where <math display="inline">\Phi = \int \mathbf{B}\cdot \mathrm{d}\mathbf{A}</math> is the magnetic flux in the circuit, <math display="inline">F = \int \mathbf{H}\cdot \mathrm{d}\boldsymbol{\ell}</math> is the [[magnetomotive force]] applied to the circuit, and {{math|''R''<sub>m</sub>}} is the [[reluctance]] of the circuit. Here the reluctance {{math|''R''<sub>m</sub>}} is a quantity similar in nature to [[Electrical resistance|resistance]] for the flux. Using this analogy it is straightforward to calculate the magnetic flux of complicated magnetic field geometries, by using all the available techniques of [[circuit theory]].

===Largest magnitude magnetic fields===
{{update|section|October 2018|date=July 2021}}

{{As of|October 2018}}, the largest magnitude magnetic field produced over a macroscopic volume outside a lab setting is 2.8&nbsp;kT ([[VNIIEF]] in [[Sarov]], [[Russia]], 1998).<ref>{{cite book|chapter=With record magnetic fields to the 21st Century |doi=10.1109/PPC.1999.823621|title=Digest of Technical Papers. 12th IEEE International Pulsed Power Conference. (Cat. No.99CH36358)|volume=2| pages=746–749 | year=1999 | last1=Boyko | first1=B.A.| last2=Bykov | first2=A.I. | last3=Dolotenko | first3=M.I. | last4=Kolokolchikov|first4=N.P. | last5=Markevtsev|first5=I.M.| last6=Tatsenko|first6=O.M.| last7=Shuvalov | first7=K. | isbn=0-7803-5498-2|s2cid=42588549}}</ref><ref name=smithsonianMagnetRecord>{{Cite web| last=Daley| first=Jason | title=Watch the Strongest Indoor Magnetic Field Blast Doors of Tokyo Lab Wide Open|url=https://www.smithsonianmag.com/smart-news/strongest-indoor-magnetic-field-blows-doors-tokyo-lab-180970436/|access-date=8 September 2020|website=Smithsonian Magazine| language=en}}</ref> As of October 2018, the largest magnitude magnetic field produced in a laboratory over a macroscopic volume was 1.2&nbsp;kT by researchers at the [[University of Tokyo]] in 2018.<ref name=smithsonianMagnetRecord/>
The largest magnitude magnetic fields produced in a laboratory occur in particle accelerators, such as [[Relativistic Heavy Ion Collider|RHIC]], inside the collisions of heavy ions, where microscopic fields reach 10<sup>14</sup>&nbsp;T.<ref>{{cite journal| last1=Tuchin|first1=Kirill|title=Particle production in strong electromagnetic fields in relativistic heavy-ion collisions| journal=Adv. High Energy Phys.|date=2013|volume=2013|article-number=490495|doi=10.1155/2013/490495| arxiv=1301.0099| bibcode=2013arXiv1301.0099T| s2cid=4877952 |doi-access=free}}</ref><ref>{{cite journal|last1=Bzdak|first1=Adam | last2=Skokov | first2=Vladimir | title=Event-by-event fluctuations of magnetic and electric fields in heavy ion collisions|journal=Physics Letters B|date=29 March 2012|volume=710|issue=1 | pages=171–174|doi=10.1016/j.physletb.2012.02.065 |arxiv=1111.1949 |bibcode=2012PhLB..710..171B |s2cid=118462584 }}</ref> [[Magnetar]]s have the strongest known magnetic fields of any naturally occurring object, ranging from 0.1 to 100&nbsp;GT (10<sup>8</sup> to 10<sup>11</sup>&nbsp;T).<ref>Kouveliotou, C.; Duncan, R. C.; Thompson, C. (February 2003). "[http://solomon.as.utexas.edu/~duncan/sciam.pdf Magnetars] {{webarchive|url=https://web.archive.org/web/20070611144829/http://solomon.as.utexas.edu/~duncan/sciam.pdf|date=11 June 2007}}". ''[[Scientific American]]''; Page 36.</ref>

== Common formulæ==
{| class="wikitable"
|+
! Current configuration
!Figure
! colspan="2" |Magnetic field
|-
|Finite beam of current
|[[File:Finite beam of current.svg|frameless|class=skin-invert|235x235px]]
| colspan="2" | <math>B=\frac{\mu_0I}{4\pi x}(\cos \theta_1+\cos\theta_2)</math>

where <math>I</math> is the uniform current throughout the beam, with the direction of magnetic field as shown.
|-
|Infinite wire
| [[File:Infinite_current_carrying_wire.svg|frameless|class=skin-invert|235x235px]]
| colspan="2" | <math> B = \frac{\mu_0I}{2\pi x}</math>

where <math>I</math> is the uniform current flowing through the wire with the direction of magnetic field as shown.
|-
|Infinite cylindrical wire
|[[File:Infinite current carrying cylinder.svg|frameless|class=skin-invert|235x235px]]
|<math> B = \frac{\mu_0I}{2\pi x}</math>

outside the wire carrying a current <math>I</math> uniformly, with the direction of magnetic field as shown.
|<math> B = \frac{\mu_0Ix}{2\pi R^2}</math>

inside the wire carrying a current <math>I</math> uniformly, with the direction of magnetic field as shown.
|-
|Circular loop
|[[File:Current carrying ring.svg|frameless|class=skin-invert|235x235px]]
| colspan="2" | <math>\mathbf B = \frac{\mu_0IR^2}{2(x^2+R^2)^{3/2}}\hat\mathbf x</math>

along the axis of the loop, where <math>I</math> is the uniform current flowing through the loop. 
|-
|Solenoid
|[[File:Solenoid segment.svg|frameless|class=skin-invert|235x235px]]
|colspan="2" | <math>B=\frac{\mu_0nI}2(cos\theta_1+\cos\theta_2)</math>

along the axis of the solenoid carrying current <math>I</math> with <math>n</math>, uniform number of loops of currents per length of solenoid; and the direction of magnetic field as shown.
|-
|Infinite solenoid
|[[File:Infinite solenoid.svg|frameless|class=skin-invert|235x235px]]
| <math>\mathbf B=0</math>

outside the solenoid carrying current <math>I</math> with <math>n</math>, uniform number of loops of currents per length of solenoid. 
| <math> B = \mu_0 n I</math>

inside the solenoid carrying current <math>I</math> with <math>n</math>, uniform number of loops of currents per length of solenoid, with the direction of magnetic field as shown.
|-
|Circular Toroid
|[[File:Circular toroidal inductor.svg|frameless|class=skin-invert|235x235px]]
| colspan="2" | <math> B = \frac{\mu_0NI}{2\pi R}</math>

along the bulk of the circular toroid carrying uniform current <math>I</math> through <math>N</math> number of uniformly distributed poloidal loops, with the direction of magnetic field as indicated.
|-
|Magnetic Dipole
|[[File:Magnetic dipole.svg|frameless|class=skin-invert|235x235px]]
| <math>\mathbf B = -\frac{\mu_0\mathbf m}{4\pi r^3},</math>

on the equatorial plane, where <math>\mathbf m</math> is the [[magnetic dipole moment]].
| <math>\mathbf B = \frac{\mu_0\mathbf m}{2\pi {|x|}^3},</math>

on the axial plane (given that <math>x \gg R</math>), where <math>x</math> can also be negative to indicate position at the opposite direction on the axis, and <math>\mathbf m</math> is the [[magnetic dipole moment]].
|}

Additional magnetic field values can be found through the magnetic field of a finite beam, for example, that the magnetic field of an arc of angle <math>\theta</math> and radius <math>R</math> at the center is <math>B={\mu_0\theta I\over 4\pi R}</math>, or that the magnetic field at the center of a N-sided regular polygon of side <math>a</math> is <math>B= {\mu_0NI\over\pi a} \sin{\pi\over N}\tan{\pi\over N}</math>, both outside of the plane with proper directions as inferred by right hand thumb rule.

==History==
{{Main|History of electromagnetic theory}}
{{see also|Timeline of electromagnetism and classical optics}}
[[Image:Descartes magnetic field.jpg|thumb|upright=1.2|One of the first drawings of a magnetic field, by [[René Descartes]], 1644, showing the Earth attracting [[lodestone]]s. It illustrated his theory that magnetism was caused by the circulation of tiny helical particles, "threaded parts", through threaded pores in magnets.]]

===Early developments===
While magnets and some properties of magnetism were known to ancient societies, the research of magnetic fields began in 1269 when French scholar [[Petrus Peregrinus de Maricourt]] mapped out the magnetic field on the surface of a spherical magnet using iron needles. Noting the resulting field lines crossed at two points he named those points "poles" in analogy to Earth's poles. He also articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them.<ref>{{cite book | doi=10.1007/978-1-4020-4423-6_261 | chapter=Peregrinus, Petrus (Flourished 1269) | title=Encyclopedia of Geomagnetism and Paleomagnetism | date=2007 | last1=Chapman | first1=Allan | pages=808–809 | publisher=Springer | location=Dordrecht | isbn=978-1-4020-3992-8 }}</ref>{{refn|group="note"|His ''Epistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete'', which is often shortened to ''Epistola de magnete'', is dated 1269 C.E.}}

Almost three centuries later, [[William Gilbert (astronomer)|William Gilbert]] of [[Colchester]] replicated Petrus Peregrinus' work and was the first to state explicitly that Earth is a magnet.<ref name=Whittaker1910>{{cite book|last=Whittaker|first=E. T.| author-link=E. T. Whittaker|title=[[A History of the Theories of Aether and Electricity]]|year=1910|publisher=[[Dover Publications]] | isbn=978-0-486-26126-3}}</ref>{{rp|34}} Published in 1600, Gilbert's work, ''[[De Magnete]]'', helped to establish magnetism as a science.

===Mathematical development===
[[File:Hans Christian Ørsted, Der Geist in der Natur, 1854.tiff|thumb|[[Hans Christian Ørsted]], ''Der Geist in der Natur'', 1854]]

In 1750, [[John Michell]] stated that magnetic poles attract and repel in accordance with an [[inverse square law]]<ref name=Whittaker1910 />{{rp|p=56}} [[Charles-Augustin de Coulomb]] experimentally verified this in 1785 and stated explicitly that north and south poles cannot be separated.<ref name=Whittaker1910 />{{rp|p=59}} Building on this force between poles, [[Siméon Denis Poisson]] (1781–1840) created the first successful model of the magnetic field, which he presented in 1824.<ref name=Whittaker1910 />{{rp|p=64}} In this model, a magnetic {{math|'''H'''}}-field is produced by ''magnetic poles'' and magnetism is due to small pairs of north–south magnetic poles.

Three discoveries in 1820 challenged this foundation of magnetism. [[Hans Christian Ørsted]] demonstrated that a current-carrying wire is surrounded by a circular magnetic field.{{refn|group="note"|During a lecture demonstration on the effects of a current on a campus needle, Ørsted showed that when a current-carrying wire is placed at a right angle with the compass, nothing happens. When he tried to orient the wire parallel to the compass needle, however, it produced a pronounced deflection of the compass needle. By placing the compass on different sides of the wire, he was able to determine the field forms perfect circles around the wire.<ref name=Whittaker1910 />{{rp|p=85}}}}<ref>{{cite encyclopedia|last1=Williams|first1=L. Pearce|date=1974|page=185| location=New York|title=Oersted, Hans Christian|url=https://archive.org/details/dictionaryofscie10gill/page/184|encyclopedia=Dictionary of Scientific Biography|editor-last1=Gillespie|editor-first1=C. C.|publisher=Charles Scribner's Sons}}</ref> Then [[André-Marie Ampère]] showed that parallel wires with currents attract one another if the currents are in the same direction and repel if they are in opposite directions.<ref name=Whittaker1910 />{{rp|p=87}}<ref>{{cite book|last1=Blundell|first1=Stephen J.|title=Magnetism: A Very Short Introduction|date=2012|publisher=OUP Oxford|isbn=978-0-19-163372-0|page=31}}</ref> Finally, [[Jean-Baptiste Biot]] and [[Félix Savart]] announced empirical results about the forces that a current-carrying long, straight wire exerted on a small magnet, determining the forces were inversely proportional to the perpendicular distance from the wire to the magnet.<ref name=Tricker23>{{cite book|last1=Tricker|first1=R. A. R.|title=Early electrodynamics|url=https://archive.org/details/earlyelectrodyna0000tric|url-access=registration|date=1965|publisher=Pergamon|location=Oxford|page=[https://archive.org/details/earlyelectrodyna0000tric/page/23 23]}}</ref><ref name=Whittaker1910 />{{rp|p=86}} [[Laplace]] later deduced a law of force based on the differential action of a differential section of the wire,<ref name=Tricker23/><ref>{{cite journal| last1=Erlichson| first1=Herman|title=The experiments of Biot and Savart concerning the force exerted by a current on a magnetic needle| journal=American Journal of Physics|date=1998|volume=66|issue=5|page=389| doi=10.1119/1.18878| bibcode=1998AmJPh..66..385E |doi-access=free}}</ref> which became known as the [[Biot–Savart law]], as Laplace did not publish his findings.<ref>{{cite book|last1=Frankel| first1=Eugene|title=Jean-Baptiste Biot: The career of a physicist in nineteenth-century France|date=1972|publisher=Doctoral dissertation| location=Princeton University|page=334}}</ref>

Extending these experiments, Ampère published his own successful model of magnetism in 1825. In it, he showed the equivalence of electrical currents to magnets<ref name=Whittaker1910 />{{rp|p=88}} and proposed that magnetism is due to perpetually flowing loops of current instead of the dipoles of magnetic charge in Poisson's model.{{refn|group="note"|From the outside, the field of a dipole of magnetic charge has exactly the same form as a current loop when both are sufficiently small. Therefore, the two models differ only for magnetism inside magnetic material.}} Further, Ampère derived both [[Ampère's force law]] describing the force between two currents and [[Ampère's law]], which, like the Biot–Savart law, correctly described the magnetic field generated by a steady current. Also in this work, Ampère introduced the term [[electrodynamics]] to describe the relationship between electricity and magnetism.<ref name=Whittaker1910 />{{rp|pp=88–92}}

In 1831, [[Michael Faraday]] discovered [[electromagnetic induction]] when he found that a changing magnetic field generates an encircling electric field, formulating what is now known as [[Faraday's law of induction]].<ref name=Whittaker1910 />{{rp|pp=189–192}} Later, [[Franz Ernst Neumann]] proved that, for a moving conductor in a magnetic field, induction is a consequence of Ampère's force law.<ref name=Whittaker1910 />{{rp|p=222}} In the process, he introduced the magnetic vector potential, which was later shown to be equivalent to the underlying mechanism proposed by Faraday.<ref name=Whittaker1910 />{{rp|p=225}}

In 1850, [[Lord Kelvin]], then known as William Thomson, distinguished between two magnetic fields now denoted {{math|'''H'''}} and {{math|'''B'''}}. The former applied to Poisson's model and the latter to Ampère's model and induction.<ref name=Whittaker1910 />{{rp|p=224}} Further, he derived how {{math|'''H'''}} and {{math|'''B'''}} relate to each other and coined the term ''permeability''.<ref name=Whittaker1910 />{{rp|p=245}}<ref>[https://www.physik.uni-augsburg.de/lehrstuehle/did/Physik/Schulservice/Physikerkalender/06_26_Lord-Kelvin-of-Largs.pdf Lord Kelvin of Largs]. physik.uni-augsburg.de. 26 June 1824</ref>

Between 1861 and 1865, [[James Clerk Maxwell]] developed and published [[Maxwell's equations]], which explained and united all of [[classical theory|classical]] electricity and magnetism. The first set of these equations was published in a paper entitled ''[[:CCommons:File:On Physical Lines of Force.pdf|On Physical Lines of Force]]'' in 1861. These equations were valid but incomplete. Maxwell completed his set of equations in his later 1865 paper ''[[A Dynamical Theory of the Electromagnetic Field]]'' and demonstrated the fact that light is an [[electromagnetic wave]]. [[Heinrich Hertz]] published papers in 1887 and 1888 experimentally confirming this fact.<ref name=h202>Huurdeman, Anton A. (2003) ''The Worldwide History of Telecommunications''. Wiley. {{ISBN|0-471-20505-2}}. p. 202</ref><ref>{{cite web|url=https://www.hhi.fraunhofer.de/fraunhofer-hhi-the-institute/about-us/history-of-hhi/the-most-important-experiments.html|publisher=Fraunhofer Heinrich Hertz Institute|title=The most important Experiments – The most important Experiments and their Publication between 1886 and 1889|access-date=19 February 2016}}</ref>

===Modern developments===

In 1887, Tesla developed an [[induction motor]] that ran on [[alternating current]]. The motor used [[Polyphase system|polyphase]] current, which generated a [[rotating magnetic field]] to turn the motor (a principle that Tesla claimed to have conceived in 1882).<ref>{{cite book|title=Networks of Power: Electrification in Western Society, 1880–1930|publisher=JHU Press|page=117|url=https://books.google.com/books?id=g07Q9M4agp4C&pg=PA117|isbn=978-0-8018-4614-4|date=March 1993 }}</ref><ref>Thomas Parke Hughes, ''Networks of Power: Electrification in Western Society, 1880–1930'', pp. 115–118</ref><ref>{{cite book | url=https://books.google.com/books?id=1AsFdUxOwu8C&pg=PA204|title=Robert Bud, Instruments of Science: An Historical Encyclopedia | page=204|access-date=18 March 2013|isbn=978-0-8153-1561-2|last1=Ltd|first1=Nmsi Trading|author2 = Smithsonian Institution | year=1998 |publisher=Taylor & Francis }}</ref> Tesla received a patent for his electric motor in May 1888.<ref>{{US patent|381968}}</ref><ref>{{Cite journal | last1=Porter|first1=H. F. J.|last2=Prout|first2=Henry G.|date=January 1924|title=A Life of George Westinghouse|journal=The American Historical Review| volume=29| issue=2|page=129|doi=10.2307/1838546| jstor=1838546|hdl=2027/coo1.ark:/13960/t15m6rz0r|issn=0002-8762|hdl-access=free}}</ref> In 1885, [[Galileo Ferraris]] independently researched rotating magnetic fields and subsequently published his research in a paper to the ''Royal Academy of Sciences'' in [[Turin]], just two months before Tesla was awarded his patent, in March 1888.<ref>{{Cite web |title=Galileo Ferraris (March 1888) ''Rotazioni elettrodinamiche prodotte per mezzo di correnti alternate'' (Electrodynamic rotations by means of alternating currents), memory read at Accademia delle Scienze, Torino, in ''Opere di Galileo Ferraris'', Hoepli, Milano, 1902 vol I pages 333 to 348 |url=http://ieeemilestones.ethw.org/images/a/a0/1-Galileo_Ferraris_Rotating-field--Rotazioni_Elettrodinamiche.pdf |archive-url=https://web.archive.org/web/20210709182151/http://ieeemilestones.ethw.org/images/a/a0/1-Galileo_Ferraris_Rotating-field--Rotazioni_Elettrodinamiche.pdf |archive-date=9 July 2021 |access-date=2 July 2021}}</ref>

The twentieth century showed that classical electrodynamics is already consistent with special relativity, and extended classical electrodynamics to work with quantum mechanics. [[Albert Einstein]], in his paper of 1905 that established relativity, showed that both the electric and magnetic fields are part of the same phenomena viewed from different reference frames. Finally, the emergent field of [[quantum mechanics]] was merged with electrodynamics to form [[quantum electrodynamics]], which first formalized the notion that electromagnetic field energy is quantized in the form of photons.

==See also==

===General===
* [[Magnetohydrodynamics]]&nbsp;– the study of the dynamics of electrically conducting fluids
* [[Magnetic hysteresis]]&nbsp;– application to [[ferromagnetism]]
* [[Magnetic nanoparticles]]&nbsp;– extremely small magnetic particles that are tens of atoms wide
* [[Magnetic reconnection]]&nbsp;– an effect that causes [[solar flare]]s and auroras
* [[Magnetic scalar potential]]
* [[SI electromagnetism units]]&nbsp;– common units used in electromagnetism
* [[Orders of magnitude (magnetic field)]]&nbsp;– list of magnetic field sources and measurement devices from smallest magnetic fields to largest detected
* [[Upward continuation]]
* [[Moses Effect]]

===Mathematics===
* [[Magnetic helicity]]&nbsp;– extent to which a magnetic field wraps around itself

===Applications===
* [[Dynamo theory]]&nbsp;– a proposed mechanism for the creation of the Earth's magnetic field
* [[Helmholtz coil]]&nbsp;– a device for producing a region of nearly uniform magnetic field
* [[Magnetic field viewing film]]&nbsp;– Film used to view the magnetic field of an area
* [[Magnetic pistol]]&nbsp;– a device on torpedoes or naval mines that detect the magnetic field of their target
* [[Maxwell coil]]&nbsp;– a device for producing a large volume of an almost constant magnetic field
* [[Stellar magnetic field]]&nbsp;– a discussion of the magnetic field of stars
* [[Teltron tube]]&nbsp;– device used to display an electron beam and demonstrates effect of electric and magnetic fields on moving charges

==Notes==
<references group="note"/>

==References==
{{Reflist|30em}}

==Further reading==
{{Refbegin}}
* {{Cite book|last=Jiles |first=David |year=1994 |title=Introduction to Electronic Properties of Materials |edition=1st |publisher=Springer |isbn=978-0-412-49580-9}}
* {{Cite book | last=Tipler | first = Paul | title=Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.) | publisher=W. H. Freeman | year=2004 | isbn=978-0-7167-0810-0 | oclc=51095685}}
{{Refend}}

==External links==
*{{Commons category-inline|Magnetic fields}}
* Crowell, B., "''[http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html Electromagnetism] {{Webarchive|url=https://web.archive.org/web/20100430152732/http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html |date=30 April 2010 }}''".
* Nave, R., "''[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfie.html Magnetic Field]''". HyperPhysics.
* "''Magnetism''", [https://web.archive.org/web/20060709195126/http://theory.uwinnipeg.ca/physics/mag/node2.html#SECTION00110000000000000000 The Magnetic Field] (archived 9 July 2006). theory.uwinnipeg.ca.
* Hoadley, Rick, "''[http://my.execpc.com/~rhoadley/magfield.htm What do magnetic fields look like] {{Webarchive|url=https://web.archive.org/web/20110219102314/http://my.execpc.com/~rhoadley/magfield.htm |date=19 February 2011 }}?''" 17 July 2005.

{{magnetic states}}

{{Authority control}}

{{DEFAULTSORT:Magnetic Field}}
[[Category:Magnetism]]
[[Category:Electromagnetic quantities]]