Editing Magnetic field (section)

==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|pages=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|1420058266}}.</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>
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|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=978-1107013605}}</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>
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|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 |url-status=dead }}</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=9780387333670 }}</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=978-0070621503|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>
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|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=http://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. -->