Editing Magnetic field (section)

==== 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/ |url-status=dead }}</ref>

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

The ''rotating magnetic field'' is a key 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 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 orthogonal coils with 90 degrees phase difference in their AC currents. However, in practice such a system would be supplied through a three-wire arrangement with unequal currents.

This inequality would cause serious problems in standardization of the conductor size and so, to overcome it, [[Three-phase electric power|three-phase]] systems are used where the three currents are equal in magnitude and have 120 degrees phase difference. Three similar coils having mutual geometrical angles of 120 degrees create the rotating magnetic field in this case. 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 in the rotating field of the stator, and these currents in turn move the rotor by 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|page=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.