Editing Lorentz force

{{short description|Force acting on charged particles in electric and magnetic fields}}
[[File:Lorentz force on charged particles in bubble chamber - HD.6D.635 (12000265314).svg|thumb|Lorentz force acting on fast-moving charged [[Elementary particle|particles]] in a [[bubble chamber]]. Positive and negative charge trajectories curve in opposite directions.]]
{{electromagnetism|cTopic=Electrodynamics}}

In [[physics]], specifically in [[electromagnetism]], the '''Lorentz force law''' is the combination of electric and magnetic [[force]] on a [[point charge]] due to [[electromagnetic field]]s. The '''Lorentz force''', on the other hand, is a [[Phenomenon|physical effect]] that occurs in the vicinity of electrically neutral, current-carrying conductors causing moving electrical charges to experience a '''magnetic force'''.<!--"Magnetic force" redirects here-->

The '''Lorentz force law''' states that a particle of charge {{mvar|q}} moving with a velocity {{math|'''v'''}} in an [[electric field]] {{math|'''E'''}} and a [[magnetic field]] {{math|'''B'''}} experiences a force (in [[International System of Units|SI units]]<ref group="nb" name="units">In SI units, {{math|'''B'''}} is measured in [[tesla (unit)|teslas]] (symbol: T). In [[Gaussian units|Gaussian-cgs units]], {{math|'''B'''}} is measured in [[gauss (unit)|gauss]] (symbol: G). See e.g. {{cite web | url=https://www.ncei.noaa.gov/products/geomagnetism-frequently-asked-questions | title=Geomagnetism Frequently Asked Questions | publisher=National Geophysical Data Center | access-date=21 October 2013}})</ref><ref group="nb" name="units2">{{math|'''H'''}} is measured in [[ampere]]s per metre (A/m) in SI units, and in [[oersted]]s (Oe) in cgs units. {{cite web | title=International system of units (SI) |url=http://physics.nist.gov/cuu/Units/units.html | work=NIST reference on constants, units, and uncertainty |date=12 April 2010 | publisher=National Institute of Standards and Technology | access-date=9 May 2012}}</ref>) of
<math display="block">\mathbf{F} = q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right).</math>
It says that the electromagnetic force on a charge {{mvar|q}} is a combination of (1) a force in the direction of the electric field {{math|'''E'''}} (proportional to the magnitude of the field and the quantity of charge), and (2) a force at right angles to both the magnetic field {{math|'''B'''}} and the velocity {{math|'''v'''}} of the charge (proportional to the magnitude of the field, the charge, and the velocity).

Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called [[#Force on a current-carrying wire|Laplace force]]), the [[electromotive force]] in a wire loop moving through a magnetic field (an aspect of [[Faraday's law of induction]]), and the force on a moving charged particle.<ref>{{cite book |last=Huray |first=Paul G. |url=https://books.google.com/books?id=0QsDgdd0MhMC&pg=PA22 |title=Maxwell's Equations |date=2009-11-16 |publisher=John Wiley & Sons |isbn=978-0-470-54276-7 |language=en}}</ref>

Historians suggest that the law is implicit in a paper by [[James Clerk Maxwell]], published in 1865.<ref name="Huray" /> [[Hendrik Lorentz]] arrived at a complete derivation in 1895,<ref name="Dahl" /> identifying the contribution of the electric force a few years after [[Oliver Heaviside]] correctly identified the contribution of the magnetic force.<ref name="Nahin" />

== Lorentz force law as the definition of E and B ==

{{multiple image|position
| align             = right
| direction         = horizontal
| footer            = [[Charged particle]]s experiencing the Lorentz force
| image1            = Lorentz force.svg
| caption1          = Trajectory of a particle with a positive or negative charge {{mvar|q}} under the influence of a magnetic field {{mvar|B}}, which is directed perpendicularly out of the screen
| image2            = Cyclotron motion.jpg
| caption2          = Beam of electrons moving in a circle, due to the presence of a magnetic field. Purple light revealing the electron's path in this [[Teltron tube]] is created by the electrons colliding with gas molecules.
| total_width       = 400
| alt1              = 
}}

In many textbook treatments of classical electromagnetism, the Lorentz force law is used as the ''definition'' of the electric and magnetic fields {{math|'''E'''}} and {{math|'''B'''}}.{{sfn|Jackson|1998|pp=777-778}}<ref>{{cite book| first1=J. A. |last1=Wheeler |author1-link=John Archibald Wheeler |url=https://archive.org/details/gravitation00misn_003|title=Gravitation |first2=C. |last2=Misner |author-link2=Charles W. Misner | first3=K. S. |last3=Thorne |author-link3=Kip Thorne | publisher=W. H. Freeman & Co|year=1973|isbn=0-7167-0344-0 | pages=[https://archive.org/details/gravitation00misn_003/page/n96 72]–73 | url-access=limited}} These authors use the Lorentz force in tensor form as definer of the [[electromagnetic tensor]] {{math|''F''}}, in turn the fields {{math|'''E'''}} and {{math|'''B'''}}.</ref><ref>{{cite book|first1=I. S. |last1=Grant|title=Electromagnetism| first2=W. R. |last2=Phillips|series=The Manchester Physics Series|publisher=John Wiley & Sons|year=1990| isbn=978-0-471-92712-9| edition=2nd | page=122}}</ref> To be specific, the Lorentz force is understood to be the following empirical statement:

{{quote|The electromagnetic force {{math|'''F'''}} on a [[test charge]] at a given point and time is a certain function of its charge {{math|''q''}} and velocity {{math|'''v'''}}, which can be parameterized by exactly two vectors {{math|'''E'''}} and {{math|'''B'''}}, in the functional form: <math display="block">\mathbf{F} = q(\mathbf{E}+\mathbf{v} \times \mathbf{B})</math>}}

This is valid, even for particles approaching the speed of light (that is, [[Norm (mathematics)#Euclidean norm|magnitude]] of {{math|'''v'''}}, {{math|1={{abs|'''v'''}} ≈ ''c''}}).<ref>{{cite book|first1=I. S. |last1=Grant|title=Electromagnetism|first2=W. R. |last2=Phillips| series=The Manchester Physics Series |publisher=John Wiley & Sons|year=1990|isbn=978-0-471-92712-9|edition=2nd|page=123}}</ref> So the two [[vector field]]s {{math|'''E'''}} and {{math|'''B'''}} are thereby defined throughout space and time, and these are called the "electric field" and "magnetic field". The fields are defined everywhere in space and time with respect to what force a test charge would receive regardless of whether a charge is present to experience the force.

== Physical interpretation of the Lorentz force ==

[[Coulomb's law]] is only valid for point charges at rest. In fact, the electromagnetic force between two point charges depends not only on the distance but also on the [[relative velocity]]. For small relative velocities and very small accelerations, instead of the Coulomb force, the [[Weber electrodynamics|Weber force]] can be applied. The sum of the Weber forces of all charge carriers in a closed DC loop on a single test charge produces – regardless of the shape of the current loop – the Lorentz force.

The interpretation of magnetism by means of a modified Coulomb law was first proposed by [[Carl Friedrich Gauss]]. In 1835, Gauss assumed that each segment of a DC loop contains an equal number of negative and positive point charges that move at different speeds.<ref>{{cite book | last = Gauss | first = Carl Friedrich | title = Carl Friedrich Gauss Werke |volume=5 | publisher = [[Göttingen Academy of Sciences and Humanities|Königliche Gesellschaft der Wissenschaften zu Göttingen]] | year = 1867 | page = 617}}</ref> If Coulomb's law were completely correct, no force should act between any two short segments of such current loops. However, around 1825, [[André-Marie Ampère]] demonstrated experimentally that this is not the case. Ampère also formulated a [[Ampère's force law|force law]]. Based on this law, Gauss concluded that the electromagnetic force between two point charges depends not only on the distance but also on the relative velocity.

The Weber force is a [[central force]] and complies with [[Newton's laws of motion|Newton's third law]]. This demonstrates not only the [[Momentum|conservation of momentum]] but also that the [[conservation of energy]] and the [[Angular momentum|conservation of angular momentum]] apply. Weber electrodynamics is only a [[quasistatic approximation]], i.e. it should not be used for higher velocities and accelerations. However, the Weber force illustrates that the Lorentz force can be traced back to central forces between numerous point-like charge carriers.

== Equation ==

=== Charged particle ===

[[File:Lorentz force particle.svg|thumb|Lorentz force {{math|'''F'''}} on a [[charged particle]] (of charge {{mvar|q}}) in motion (instantaneous velocity {{math|'''v'''}}). The [[electric field|{{math|'''E'''}} field]] and [[magnetic field|{{math|'''B'''}} field]] vary in space and time.]]

The force {{math|'''F'''}} acting on a particle of [[electric charge]] {{mvar|q}} with instantaneous velocity {{math|'''v'''}}, due to an external electric field {{math|'''E'''}} and magnetic field {{math|'''B'''}}, is given by ([[SI]] definition of quantities<ref group="nb" name="units" />):{{sfn|Jackson|1998|pp=2-3}}
{{Equation box 1
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|equation = <math>\mathbf{F} = q \left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)</math>
|cellpadding
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|border colour = #50C878
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where {{math|×}} is the vector [[cross product]] (all boldface quantities are vectors). In terms of Cartesian components, we have:
<math display="block">\begin{align}
F_x &= q \left(E_x + v_y B_z - v_z B_y\right), \\[0.5ex]
F_y &= q \left(E_y + v_z B_x - v_x B_z\right), \\[0.5ex]
F_z &= q \left(E_z + v_x B_y - v_y B_x\right).
\end{align}</math>

In general, the electric and magnetic fields are functions of the position and time. Therefore, explicitly, the Lorentz force can be written as:
<math display="block">\mathbf{F}\left(\mathbf{r}(t),\dot\mathbf{r}(t),t,q\right) = q\left[\mathbf{E}(\mathbf{r},t) + \dot\mathbf{r}(t) \times \mathbf{B}(\mathbf{r},t)\right]</math>
in which {{math|'''r'''}} is the position vector of the charged particle, {{mvar|t}} is time, and the overdot is a time derivative.

A positively charged particle will be accelerated in the ''same'' linear orientation as the {{math|'''E'''}} field, but will curve perpendicularly to both the instantaneous velocity vector {{math|'''v'''}} and the {{math|'''B'''}} field according to the [[right-hand rule]] (in detail, if the fingers of the right hand are extended to point in the direction of {{math|'''v'''}} and are then curled to point in the direction of {{math|'''B'''}}, then the extended thumb will point in the direction of {{math|'''F'''}}).

The term {{math|''q'''''E'''}} is called the '''electric force''', while the term {{math|1=''q''('''v''' × '''B''')}} is called the '''magnetic force'''.{{sfn|Griffiths|2023|p=211}} According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force,<ref name="Griffiths2">For example, see the [http://ilorentz.org/history/lorentz/lorentz.html website of the Lorentz Institute].</ref> with the ''total'' electromagnetic force (including the electric force) given some other (nonstandard) name. This article will ''not'' follow this nomenclature: in what follows, the term ''Lorentz force'' will refer to the expression for the total force.

The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wire in a magnetic field. In that context, it is also called the [[#Force on a current-carrying wire|Laplace force]].

The Lorentz force is a force exerted by the electromagnetic field on the charged particle, that is, it is the rate at which linear momentum is transferred from the electromagnetic field to the particle. Associated with it is the power which is the rate at which energy is transferred from the electromagnetic field to the particle. That power is
<math display="block">\mathbf{v} \cdot \mathbf{F} = q \, \mathbf{v} \cdot \mathbf{E}.</math>
Notice that the magnetic field does not contribute to the power because the magnetic force is always perpendicular to the velocity of the particle.

=== Continuous charge distribution ===

[[File:Lorentz force continuum.svg|thumb|Lorentz force (per unit 3-volume) {{math|'''f'''}} on a continuous [[charge distribution]] ([[charge density]] {{math|''ρ''}}) in motion. The 3-[[current density]] {{math|'''J'''}} corresponds to the motion of the charge element {{math|''dq''}} in [[volume element]] {{math|''dV''}} and varies throughout the continuum.]]

For a continuous [[charge distribution]] in motion, the Lorentz force equation becomes:
<math display="block">\mathrm{d}\mathbf{F} = \mathrm{d}q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)</math>
where <math>\mathrm{d}\mathbf{F}</math> is the force on a small piece of the charge distribution with charge <math>\mathrm{d}q</math>. If both sides of this equation are divided by the volume of this small piece of the charge distribution <math>\mathrm{d}V</math>, the result is:
<math display="block">\mathbf{f} = \rho\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)</math>
where <math>\mathbf{f}</math> is the ''force density'' (force per unit volume) and <math>\rho</math> is the [[charge density]] (charge per unit volume). Next, the [[current density]] corresponding to the motion of the charge continuum is{{sfn|Griffiths|2023|p=219}}
<math display="block">\mathbf{J} = \rho \mathbf{v} </math>
so the continuous analogue to the equation is{{sfn|Griffiths|2023|p=368}}

{{Equation box 1
|indent =:
|equation = <math>\mathbf{f} = \rho \mathbf{E} + \mathbf{J} \times \mathbf{B}</math>
|cellpadding= 6
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|border colour = #0073CF
|background colour=#F5FFFA}}

The total force is the [[volume integral]] over the charge distribution:
<math display="block"> \mathbf{F} = \int \left ( \rho \mathbf{E} + \mathbf{J} \times \mathbf{B} \right)\mathrm{d}V.</math>

By eliminating <math>\rho</math> and <math>\mathbf{J}</math>, using [[Maxwell's equations]], and manipulating using the theorems of [[vector calculus]], this form of the equation can be used to derive the [[Maxwell stress tensor]] <math>\boldsymbol{\sigma}</math>, in turn this can be combined with the [[Poynting vector]] <math>\mathbf{S}</math> to obtain the [[electromagnetic stress–energy tensor]] {{math|'''T'''}} used in [[general relativity]].{{sfn|Griffiths|2023|pp=369-370}}

In terms of <math>\boldsymbol{\sigma}</math> and <math>\mathbf{S}</math>, another way to write the Lorentz force (per unit volume) is
<math display="block"> \mathbf{f} = \nabla\cdot\boldsymbol{\sigma} - \dfrac{1}{c^2} \dfrac{\partial \mathbf{S}}{\partial t} </math>
where <math>\nabla \cdot</math> denotes the [[Divergence#Definition_in_coordinates|divergence]] of the [[tensor field]] and <math>c</math> is the [[speed of light]]. Rather than the amount of charge and its velocity in electric and magnetic fields, this equation relates the [[energy flux]] (flow of ''energy'' per unit time per unit distance) in the fields to the force exerted on a charge distribution. See [[Covariant formulation of classical electromagnetism#Charge continuum|Covariant formulation of classical electromagnetism]] for more details.

The density of power associated with the Lorentz force in a material medium is
<math display="block">\mathbf{J} \cdot \mathbf{E}.</math>

If we separate the total charge and total current into their free and bound parts, we get that the density of the Lorentz force is
<math display="block">\mathbf{f} = \left(\rho_f - \nabla \cdot \mathbf P\right) \mathbf{E} + \left(\mathbf{J}_f + \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}\right) \times \mathbf{B}.</math>

where: <math>\rho_f</math> is the density of free charge; <math>\mathbf{P}</math> is the [[polarization density]]; <math>\mathbf{J}_f</math> is the density of free current; and <math>\mathbf{M}</math> is the [[magnetization]] density. In this way, the Lorentz force can explain the torque applied to a permanent magnet by the magnetic field. The density of the associated power is
<math display="block">\left(\mathbf{J}_f + \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}\right) \cdot \mathbf{E}.</math>

=== Formulation in the Gaussian system ===
The above-mentioned formulae use the conventions for the definition of the electric and magnetic field used with the [[SI]], which is the most common. However, other conventions with the same physics (i.e. forces on e.g. an electron) are possible and used. In the conventions used with the older [[Gaussian units|CGS-Gaussian units]], which are somewhat more common among some theoretical physicists as well as condensed matter experimentalists, one has instead
<math display="block">\mathbf{F} = q_\mathrm{G} \left(\mathbf{E}_\mathrm{G} + \frac{\mathbf{v}}{c} \times \mathbf{B}_\mathrm{G}\right),</math>
where {{mvar|c}} is the [[speed of light]]. Although this equation looks slightly different, it is equivalent, since one has the following relations:<ref group="nb" name="units" />
<math display="block">q_\mathrm{G} = \frac{q_\mathrm{SI}}{\sqrt{4\pi \varepsilon_0}},\quad
\mathbf E_\mathrm{G} = \sqrt{4\pi\varepsilon_0}\,\mathbf E_\mathrm{SI},\quad
\mathbf B_\mathrm{G} = {\sqrt{4\pi /\mu_0}}\,{\mathbf B_\mathrm{SI}}, \quad
c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}.</math>
where {{math|''ε''<sub>0</sub>}} is the [[vacuum permittivity]] and {{math|''μ''<sub>0</sub>}} the [[vacuum permeability]]. In practice, the subscripts "G" and "SI" are omitted, and the used convention (and unit)  must be determined from context.

== History ==
[[File:H. A. Lorentz - Lorentz force, div E = ρ, div B = 0 - La théorie electromagnétique de Maxwell et son application aux corps mouvants, Archives néerlandaises, 1892 - p 451 - Eq. I, II, III.png|thumb|Lorentz's theory of electrons. Formulas for the Lorentz force (I, ponderomotive force) and the [[Maxwell equations]] for the [[divergence]] of the [[electrical field]] E (II) and the [[magnetic field]] B (III), {{lang|fr|La théorie electromagnétique de Maxwell et son application aux corps mouvants}}, 1892, p. 451. {{mvar|V}} is the velocity of light.]]
Early attempts to quantitatively describe the electromagnetic force were made in the mid-18th century. It was proposed that the force on magnetic poles, by [[Johann Tobias Mayer]] and others in 1760,<ref>{{cite book | first = Michel | last = Delon | title = Encyclopedia of the Enlightenment | place = Chicago, Illinois | publisher = Fitzroy Dearborn | year = 2001 | page = 538 | isbn = 1-57958-246-X}}</ref> and electrically charged objects, by [[Henry Cavendish]] in 1762,<ref>{{cite book | first = Elliot H. | last = Goodwin | title = The New Cambridge Modern History Volume 8: The American and French Revolutions, 1763–93 | place = Cambridge | publisher = Cambridge University Press | year = 1965 | page = 130 | isbn = 978-0-521-04546-9}}</ref> obeyed an [[inverse-square law]]. However, in both cases the experimental proof was neither complete nor conclusive. It was not until 1784 when [[Charles-Augustin de Coulomb]], using a [[torsion balance]], was able to definitively show through experiment that this was true.<ref>{{cite book | first = Herbert W. | last = Meyer | title = A History of Electricity and Magnetism | place = Norwalk, Connecticut | publisher = Burndy Library | year = 1972 | pages = 30–31 | isbn = 0-262-13070-X | url = https://archive.org/details/AHistoryof_00_Meye}}</ref> Soon after the discovery in 1820 by [[Hans Christian Ørsted]] that a magnetic needle is acted on by a voltaic current, [[André-Marie Ampère]] that same year was able to devise through experimentation the formula for the angular dependence of the force between two current elements.<ref>{{cite book | first = Gerrit L. | last = Verschuur | title = Hidden Attraction: The History and Mystery of Magnetism | place = New York | publisher = Oxford University Press | isbn = 0-19-506488-7 | year = 1993 | pages = [https://archive.org/details/hiddenattraction00vers/page/78 78–79] | url = https://archive.org/details/hiddenattraction00vers/page/78}}</ref>{{sfn|Darrigol|2000|pp=9,25}} In all these descriptions, the force was always described in terms of the properties of the matter involved and the distances between two masses or charges rather than in terms of electric and magnetic fields.<ref>{{cite book | first = Gerrit L. | last = Verschuur | title = Hidden Attraction: The History and Mystery of Magnetism | place = New York | publisher = Oxford University Press | isbn = 0-19-506488-7 | year = 1993 | page = [https://archive.org/details/hiddenattraction00vers/page/76 76] | url = https://archive.org/details/hiddenattraction00vers/page/76}}</ref>

The modern concept of electric and magnetic fields first arose in the theories of [[Michael Faraday]], particularly his idea of [[lines of force]], later to be given full mathematical description by [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] and [[James Clerk Maxwell]].{{sfn|Darrigol|2000|pp=126-131,139-144}} From a modern perspective it is possible to identify in Maxwell's 1865 formulation of his field equations a form of the Lorentz force equation in relation to electric currents,<ref name=Huray>{{cite book | first = Paul G. | last = Huray | title = Maxwell's Equations | publisher = Wiley-IEEE | isbn = 978-0-470-54276-7 | year = 2010 | page = 22 | url = https://books.google.com/books?id=0QsDgdd0MhMC&pg=PA22}}</ref> although in the time of Maxwell it was not evident how his equations related to the forces on moving charged objects. [[J. J. Thomson]] was the first to attempt to derive from Maxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's properties and external fields. Interested in determining the electromagnetic behavior of the charged particles in [[cathode ray]]s, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as<ref name=Nahin>{{cite book |first=Paul J. |last=Nahin |url=https://books.google.com/books?id=e9wEntQmA0IC |title=Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age |publisher=JHU Press |year=2002}}</ref><ref>{{cite journal| last=Thomson |first=J. J. | date=1881-04-01|title=XXXIII. On the electric and magnetic effects produced by the motion of electrified bodies|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|volume=11|issue=68|pages=229–249|doi=10.1080/14786448108627008|issn=1941-5982}}</ref>
<math display="block">\mathbf{F} = \frac{q}{2}\mathbf{v} \times \mathbf{B}.</math>
Thomson derived the correct basic form of the formula, but, because of some miscalculations and an incomplete description of the [[displacement current]], included an incorrect scale-factor of a half in front of the formula. [[Oliver Heaviside]] invented the modern vector notation and applied it to Maxwell's field equations; he also (in 1885 and 1889) had fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force on a moving charged object.<ref name=Nahin/>{{sfn|Darrigol|2000|pp=200,429-430}}<ref>{{cite journal | last= Heaviside |first=Oliver| title=On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric | journal=Philosophical Magazine |date=April 1889 | volume=27 |page=324 |url=http://en.wikisource.org/wiki/Motion_of_Electrification_through_a_Dielectric}}</ref> Finally, in 1895,<ref name=Dahl>{{cite book | first = Per F. | last = Dahl | title = Flash of the Cathode Rays: A History of J J Thomson's Electron | publisher = CRC Press | year = 1997| page= 10}}</ref><ref>{{cite book |last=Lorentz |first=Hendrik Antoon |title=Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern |language=de |year=1895}}</ref> [[Hendrik Lorentz]] derived the modern form of the formula for the electromagnetic force which includes the contributions to the total force from both the electric and the magnetic fields. Lorentz began by abandoning the Maxwellian descriptions of the ether and conduction. Instead, Lorentz made a distinction between matter and the [[luminiferous aether]] and sought to apply the Maxwell equations at a microscopic scale. Using Heaviside's version of the Maxwell equations for a stationary ether and applying [[Lagrangian mechanics]] (see below), Lorentz arrived at the correct and complete form of the force law that now bears his name.{{sfn|Darrigol|2000|p=327}}<ref>{{cite book | last = Whittaker | first = E. T. | author-link=E. T. Whittaker | title = [[A History of the Theories of Aether and Electricity|A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century]] | publisher = Longmans, Green and Co. | year = 1910 | pages = 420–423 | isbn = 1-143-01208-9}}</ref>

== Trajectories of particles due to the Lorentz force ==
{{Main|Guiding center}}
[[File:charged-particle-drifts.svg|upright=1.4|thumbnail|right|Charged particle drifts in a homogeneous magnetic field. (A) No disturbing force. (B) With an electric field, {{math|'''E'''}}. (C) With an independent force, {{math|'''F'''}} (e.g. gravity). (D) In an inhomogeneous magnetic field, {{math|grad '''H'''}}.]]
In many cases of practical interest, the motion in a [[magnetic field]] of an [[electric charge|electrically charged]] particle (such as an [[electron]] or [[ion]] in a [[Plasma (physics)|plasma]]) can be treated as the [[Quantum superposition|superposition]] of a relatively fast circular motion around a point called the '''guiding center''' and a relatively slow '''drift''' of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.

== Significance of the Lorentz force ==

While the modern Maxwell's equations describe how electrically charged particles and currents or moving charged particles give rise to electric and magnetic fields, the Lorentz force law completes that picture by describing the force acting on a moving point charge {{mvar|q}} in the presence of electromagnetic fields.{{sfn|Jackson|1998|pp=2-3}}{{sfn|Griffiths|2023|p=340}} The Lorentz force law describes the effect of {{math|'''E'''}} and {{math|'''B'''}} upon a point charge, but such electromagnetic forces are not the entire picture. Charged particles are possibly coupled to other forces, notably gravity and nuclear forces. Thus, Maxwell's equations do not stand separate from other physical laws, but are coupled to them via the charge and current densities. The response of a point charge to the Lorentz law is one aspect; the generation of {{math|'''E'''}} and {{math|'''B'''}} by currents and charges is another.

In real materials the Lorentz force is inadequate to describe the collective behavior of charged particles, both in principle and as a matter of computation. The charged particles in a material medium not only respond to the {{math|'''E'''}} and {{math|'''B'''}} fields but also generate these fields. Complex transport equations must be solved to determine the time and spatial response of charges, for example, the [[Boltzmann equation]] or the [[Fokker–Planck equation]] or the [[Navier–Stokes equations]]. For example, see [[magnetohydrodynamics]], [[fluid dynamics]], [[electrohydrodynamics]], [[superconductivity]], [[stellar evolution]]. An entire physical apparatus for dealing with these matters has developed. See for example, [[Green–Kubo relations]] and [[Green's function (many-body theory)]].

== Force on a current-carrying wire ==
{{see also|Electric motor#Force and torque|Biot–Savart law}}
[[File:Regla mano derecha Laplace.svg|right|thumb|Right-hand rule for a current-carrying wire in a magnetic field {{mvar|B}}]]
When a wire carrying an electric current is placed in an external magnetic field, each of the moving charges, which comprise the current, experiences the Lorentz force, and together they can create a macroscopic force on the wire (sometimes called the '''Laplace force'''). By combining the Lorentz force law above with the definition of electric current, the following equation results, in the case of a straight stationary wire in a homogeneous field:{{sfn|Purcell|Morin|2013|p=284}}
<math display="block">\mathbf{F} = I \boldsymbol{\ell} \times \mathbf{B} ,</math>
where {{math|'''ℓ'''}} is a vector whose magnitude is the length of the wire, and whose direction is along the wire, aligned with the direction of the [[conventional current]] {{mvar|I}}.

If the wire is not straight, the force on it can be computed by applying this formula to each [[infinitesimal]] segment of wire <math> \mathrm d \boldsymbol \ell </math>, then adding up all these forces by [[integration (calculus)|integration]]. This results in the same formal expression, but {{math|'''ℓ'''}} should now be understood as the vector connecting the end points of the curved wire with direction from starting to end point of conventional current. Usually, there will also be a net [[torque]].

If, in addition, the magnetic field is inhomogeneous, the net force on a stationary rigid wire carrying a steady current {{mvar|I}} is given by integration along the wire,{{sfn|Griffiths|2023|p=216}}
<math display="block">\mathbf{F} = I\int (\mathrm{d}\boldsymbol{\ell}\times \mathbf{B}).</math>
One application of this is [[Ampère's force law]], which describes how two current-carrying wires can attract or repel each other, since each experiences a Lorentz force from the other's generated magnetic field.

Another application is an [[induction motor]]. The stator winding AC current generates a moving magnetic field which induces a current in the rotor. The subsequent Lorentz force <math>\mathbf{F}</math> acting on the rotor creates a torque, making the motor spin. Hence, though the Lorentz force law does not apply when the magnetic field <math>\mathbf{B}</math> is generated by the current <math>I</math>, it does apply when the current <math>I</math> is induced by the movement of magnetic field <math>\mathbf{B}</math>.

== Electromotive force ==
{{main|Electromotive force}}
{{multiple image|position
| align             = right
| direction         = horizontal
| image1            = Elementary generator.svg
| caption1          = Motional EMF, induced by moving a conductor through a magnetic field.
| image2            = Alternator 1.svg
| caption2          = Transformer EMF, induced by a changing magnetic field.
| total_width       = 400
| alt1              = 
}}

The magnetic force ({{math|''q'''''v''' × '''B'''}}) component of the Lorentz force is responsible for [[Electromotive_force#Electromagnetic_induction|motional electromotive force]] (or ''motional EMF''), the phenomenon underlying many electrical generators. When a conductor is moved through a magnetic field, the magnetic field exerts opposite forces on electrons and nuclei in the wire, and this creates the EMF. The term "motional EMF" is applied to this phenomenon, since the EMF is due to the ''motion'' of the wire.{{sfn|Griffiths|2023|p=307}}

In other electrical generators, the magnets move, while the conductors do not. In this case, the EMF is due to the electric force ({{math|''q'''''E'''}}) term in the Lorentz Force equation. The electric field in question is created by the changing magnetic field, resulting in an induced EMF called the ''transformer EMF'', as described by the [[Electromagnetic induction#Maxwell–Faraday equation|Maxwell–Faraday equation]] (one of the four modern [[Maxwell's equations]]).{{sfn|Sadiku|2018|pp=424-427}}

Both of these EMFs, despite their apparently distinct origins, are described by the same equation, namely, the EMF is the rate of change of [[magnetic flux]] through the wire. (This is Faraday's law of induction, see [[Lorentz force#Lorentz force and Faraday.27s law of induction|below]].) Einstein's [[special theory of relativity]] was partially motivated by the desire to better understand this link between the two effects.{{sfn|Griffiths|2023|pp=316-318}} In fact, the electric and magnetic fields are different facets of the same electromagnetic field, and in moving from one inertial frame to another, the [[solenoidal vector field]] portion of the {{math|'''E'''}}-field can change in whole or in part to a {{math|'''B'''}}-field or ''vice versa''.<ref name=Chow>{{cite book | author=Tai L. Chow | title=Electromagnetic theory | year= 2006 | page = 395 | publisher = Jones and Bartlett | location=Sudbury, Massachusetts | isbn=0-7637-3827-1 | url=https://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153 }}</ref>

== Lorentz force and Faraday's law of induction ==
{{main|Faraday's law of induction}}
[[File:Lorentz force - mural Leiden 1, 2016.jpg|upright=1.35|thumb|Lorentz force image on a wall in Leiden]]
Given a loop of wire in a [[magnetic field]], Faraday's law of induction states the induced [[electromotive force]] (EMF) in the wire is:
<math display="block">\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}</math>
where
<math display="block"> \Phi_B = \int_{\Sigma(t)} \mathbf{B}(\mathbf{r}, t)\cdot \mathrm{d}\mathbf{A},</math>
is the [[magnetic flux]] through the loop, {{math|'''B'''}} is the magnetic field, {{math|Σ(''t'')}} is a surface bounded by the closed contour {{math|∂Σ(''t'')}}, at time {{mvar|t}}, {{math|d'''A'''}} is an infinitesimal [[vector area]] element of {{math|Σ(''t'')}} (magnitude is the area of an infinitesimal patch of surface, direction is [[orthogonal]] to that surface patch).

The ''sign'' of the EMF is determined by [[Lenz's law]]. Note that this is valid for not only a ''stationary'' wire{{snd}}but also for a ''moving'' wire.

From [[Faraday's law of induction]] (that is valid for a moving wire, for instance in a motor) and the [[Maxwell Equations]], the Lorentz Force can be deduced. The reverse is also true, the Lorentz force and the [[Maxwell Equations]] can be used to derive the [[Faraday's law of induction|Faraday Law]].

Let {{math|∂Σ(''t'')}} be the moving wire, moving together without rotation and with constant velocity {{math|'''v'''}} and {{math|Σ(''t'')}} be the internal surface of the wire. The EMF around the closed path {{math|∂Σ(''t'')}} is given by:<ref name=Landau>{{cite book | last1=Landau | first1= L. D. | last2= Lifshitz | first2 = E. M. | last3 = Pitaevskiĭ | first3 = L. P. | title=Electrodynamics of continuous media |volume=8 |series=Course of Theoretical Physics | year= 1984 | at =§63 (§49 pp. 205–207 in 1960 edition) | edition=2nd | publisher=Butterworth-Heinemann | location=Oxford | isbn=0-7506-2634-8 | url=http://worldcat.org/search?q=0750626348&qt=owc_search}}</ref>
<math display="block">\mathcal{E} = \oint_{\partial \Sigma (t)} \frac{\mathbf{F}}{q}\cdot \mathrm{d} \boldsymbol{\ell} </math>
where <math>\mathbf{E}'(\mathbf{r}, t) =  \mathbf{F}/q(\mathbf{r}, t)</math> is the electric field and {{math|d'''ℓ'''}} is an [[infinitesimal]] vector element of the contour {{math|∂Σ(''t'')}}.{{sfn|Jackson|1998|p=209}}<ref group=nb>Both {{math|d'''''ℓ'''''}} and {{math|d'''A'''}} have a sign ambiguity; to get the correct sign, the [[right-hand rule]] is used, as explained in the article [[Kelvin–Stokes theorem]].</ref> Equating both integrals leads to the field theory form of Faraday's law, given by:{{sfn|Jackson|1998|pp=209-210}}
<math display="block"> \mathcal{E} = \oint_{\partial \Sigma(t)}\mathbf{E}'(\mathbf{r}, t)  \cdot \mathrm{d} \boldsymbol{\ell} = - \frac{\mathrm{d} }{\mathrm{d}t} \int_{\Sigma(t)} \mathbf{B}(\mathbf{r},t) \cdot \mathrm{d} \mathbf{A}. </math>
This result can be compared with the version of Faraday's law of induction that appears in the modern Maxwell's equations, called the (integral form of) [[Faraday%27s_law_of_induction#Maxwell–Faraday_equation|Maxwell–Faraday equation]]:<ref name=Harrington>{{cite book | first = Roger F. |last=Harrington | author-link = Roger F. Harrington | title = Introduction to electromagnetic engineering | year = 2003 | page = 56 | publisher = Dover Publications | location = Mineola, New York | isbn = 0-486-43241-6 | url = https://books.google.com/books?id=ZlC2EV8zvX8C&q=%22faraday%27s+law+of+induction%22&pg=PA57}}</ref>
<math display="block"> \oint_{\partial \Sigma(t)} \mathbf{E}(\mathbf{r},t) \cdot \mathrm{d} \boldsymbol{\ell}  = - \int_{\Sigma(t)} \frac{\partial \mathbf {B}(\mathbf{r}, t)}{ \partial t } \cdot \mathrm{d} \mathbf{A}.</math>

The two equations are equivalent if the wire is not moving. In case the circuit is moving with a velocity <math>\mathbf{v}</math> in some direction, then, using the [[Leibniz integral rule]] and that {{math|1=div '''B''' = 0}}, gives
<math display="block"> \oint_{\partial \Sigma(t)}\mathbf{E}'(\mathbf{r}, t) \cdot \mathrm{d} \boldsymbol{\ell}= 
- \int_{\Sigma(t)}  \frac{\partial \mathbf{B}(\mathbf{r}, t)}{\partial t} \cdot \mathrm{d}\mathbf{A}  +
\oint_{\partial \Sigma(t)} \left(\mathbf{v} \times \mathbf{B}(\mathbf{r}, t)\right)\cdot \mathrm{d} \boldsymbol{\ell}.
</math>
Substituting the Maxwell-Faraday equation then gives
<math display="block"> \oint_{\partial \Sigma(t)} \mathbf{E}'(\mathbf{r}, t)\cdot \mathrm{d} \boldsymbol{\ell} = 
\oint_{\partial \Sigma(t)} \mathbf{E}(\mathbf{r}, t) \cdot \mathrm{d} \boldsymbol{\ell} +
\oint_{\partial \Sigma(t)} \left(\mathbf{v} \times \mathbf{B}(\mathbf{r}, t)\right) \mathrm{d} \boldsymbol{\ell}
</math>
since this is valid for any wire position it implies that
<math display="block"> \mathbf{F} = q\,\mathbf{E}(\mathbf{r},\, t) + q\,\mathbf{v} \times \mathbf{B}(\mathbf{r},\, t).</math>

Faraday's law of induction holds whether the loop of wire is rigid and stationary, or in motion or in process of deformation, and it holds whether the magnetic field is constant in time or changing. However, there are cases where Faraday's law is either inadequate or difficult to use, and application of the underlying Lorentz force law is necessary. See [[Faraday paradox#Inapplicability of Faraday's law|inapplicability of Faraday's law]].

If the magnetic field is fixed in time and the conducting loop moves through the field, the magnetic flux {{math|Φ<sub>''B''</sub>}} linking the loop can change in several ways. For example, if the {{math|'''B'''}}-field varies with position, and the loop moves to a location with different B-field, {{math|Φ<sub>''B''</sub>}} will change. Alternatively, if the loop changes orientation with respect to the B-field, the {{math|'''B''' ⋅ d'''A'''}} differential element will change because of the different angle between {{math|'''B'''}} and {{math|d'''A'''}}, also changing {{math|Φ<sub>''B''</sub>}}. As a third example, if a portion of the circuit is swept through a uniform, time-independent {{math|'''B'''}}-field, and another portion of the circuit is held stationary, the flux linking the entire closed circuit can change due to the shift in relative position of the circuit's component parts with time (surface {{math|∂Σ(''t'')}} time-dependent). In all three cases, Faraday's law of induction then predicts the EMF generated by the change in {{math|Φ<sub>''B''</sub>}}.

Note that the Maxwell Faraday's equation implies that the Electric Field {{math|'''E'''}} is non conservative when the Magnetic Field {{math|'''B'''}} varies in time, and is not expressible as the gradient of a [[scalar field]], and not subject to the [[gradient theorem]] since its [[Curl (mathematics)|curl]] is not zero.<ref name="Landau"/>{{sfn|Sadiku|2018|pp=424-425}}

== Lorentz force in terms of potentials ==
{{see also|Mathematical descriptions of the electromagnetic field|Maxwell's equations|Helmholtz decomposition}}

The {{math|'''E'''}} and {{math|'''B'''}} fields can be replaced by the [[magnetic vector potential]] {{math|'''A'''}} and ([[Scalar (mathematics)|scalar]]) [[electrostatic potential]] {{math|''ϕ''}} by
<math display="block">\begin{align}
 \mathbf{E} &= - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t } \\[1ex]
 \mathbf{B} &= \nabla \times \mathbf{A}
\end{align}</math>
where {{math|∇}} is the gradient, {{math|∇⋅}} is the divergence, and {{math|∇×}} is the [[Curl (mathematics)|curl]].

The force becomes
<math display="block">\mathbf{F} = q\left[-\nabla \phi- \frac{\partial \mathbf{A}}{\partial t}+\mathbf{v}\times(\nabla\times\mathbf{A})\right].</math>

Using an [[Triple product#Vector triple product|identity for the triple product]] this can be rewritten as
<math display="block">\mathbf{F} = q\left[-\nabla \phi- \frac{\partial \mathbf{A}}{\partial t}+\nabla\left(\mathbf{v}\cdot \mathbf{A} \right)-\left(\mathbf{v}\cdot \nabla\right)\mathbf{A}\right].</math>

(Notice that the coordinates and the velocity components should be treated as independent variables, so the del operator acts only on {{nowrap|<math>\mathbf{A}</math>,}} not on {{nowrap|<math>\mathbf{v}</math>;}} thus, there is no need of using [[Vector calculus identities#Special notations|Feynman's subscript notation]] in the equation above.) Using the chain rule, the [[convective derivative]] of <math>\mathbf{A}</math> is:<ref>{{cite book | last=Klausen | first=Kristján Óttar | title=A Treatise on the Magnetic Vector Potential | publisher=Springer International Publishing | publication-place=Cham | date=2020 | isbn=978-3-030-52221-6 | doi=10.1007/978-3-030-52222-3 | page=95}}</ref>
<math display="block">\frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t} = \frac{\partial\mathbf{A}}{\partial t}+(\mathbf{v}\cdot\nabla)\mathbf{A} </math>
so that the above expression becomes:
<math display="block">\mathbf{F} = q\left[-\nabla (\phi-\mathbf{v}\cdot\mathbf{A})- \frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t}\right].</math>

With {{math|1='''v''' = '''ẋ'''}} and 
<math display="block">\frac{\mathrm{d}}{\mathrm{d}t}\left[\frac{\partial}{\partial \dot{\mathbf{x}}}\left(\phi - \dot{\mathbf{x}}\cdot \mathbf{A} \right) \right] = -\frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t},</math>
we can put the equation into the convenient [[Euler–Lagrange equation|Euler–Lagrange form]]<ref name="Kibble"></ref>
{{Equation box 1
|indent =:
|equation = <math>\mathbf{F} = q\left[-\nabla_{\mathbf{x} }(\phi-\dot{\mathbf{x} }\cdot\mathbf{A}) + \frac{\mathrm{d} }{\mathrm{d}t}\nabla_{\dot{\mathbf{x} } }(\phi-\dot{\mathbf{x} }\cdot\mathbf{A})\right]</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
where <math display="block">\nabla_{\mathbf{x} } = \hat{x} \dfrac{\partial}{\partial x} + \hat{y} \dfrac{\partial}{\partial y} + \hat{z} \dfrac{\partial}{\partial z}</math> and <math display="block">\nabla_{\dot{\mathbf{x} } } = \hat{x} \dfrac{\partial}{\partial \dot{x} } + \hat{y} \dfrac{\partial}{\partial \dot{y} } + \hat{z} \dfrac{\partial}{\partial \dot{z} }.</math>

== Lorentz force and analytical mechanics ==

{{see also|Magnetic vector potential#Interpretation as Potential Momentum}}
The [[Lagrangian_mechanics#Electromagnetism|Lagrangian]] for a charged particle of mass {{math|''m''}} and charge {{math|''q''}} in an electromagnetic field equivalently describes the dynamics of the particle in terms of its ''energy'', rather than the force exerted on it. The classical expression is given by:<ref name="Kibble">{{cite book | last=Kibble | first=T. W. B. | last2=Berkshire | first2=Frank H. | title=Classical Mechanics | publisher=World Scientific Publishing Company | publication-place=London : River Edge, NJ | date=2004 | isbn=1-86094-424-8 | oclc=54415965 | chapter=10.5 Charged Particle in an Electromagnetic Field}}</ref>
<math display="block">L = \frac{m}{2} \mathbf{\dot{r} }\cdot\mathbf{\dot{r} } + q \mathbf{A}\cdot\mathbf{\dot{r} }-q\phi</math>
where {{math|'''A'''}} and {{math|''ϕ''}} are the potential fields as above. The quantity <math>V = q(\phi - \mathbf{A}\cdot \mathbf{\dot{r}})</math> can be identified as a generalized, velocity-dependent potential energy and, accordingly, <math>\mathbf{F}</math> as a [[Conservative_force#Non-conservative_force|non-conservative force]].<ref>{{cite journal | last=Semon | first=Mark D. | last2=Taylor | first2=John R. | title=Thoughts on the magnetic vector potential | journal=American Journal of Physics | volume=64 | issue=11 | date=1996 | issn=0002-9505 | doi=10.1119/1.18400 | pages=1361–1369}}</ref> Using the Lagrangian, the equation for the Lorentz force given above can be obtained again.

{{math proof|title=Derivation of Lorentz force from classical Lagrangian (SI units)| proof =
For an {{math|1='''A'''}} field, a particle moving with velocity {{math|1='''v''' = '''ṙ'''}} has [[Momentum#In_electromagnetics|potential momentum]] <math>q\mathbf{A}(\mathbf{r}, t)</math>, so its potential energy is <math>q\mathbf{A}(\mathbf{r},t)\cdot\mathbf{\dot{r}}</math>. For a ''ϕ'' field, the particle's potential energy is <math>q\phi(\mathbf{r},t)</math>.

The total [[potential energy]] is then:
<math display="block">V = q\phi - q\mathbf{A}\cdot\mathbf{\dot{r}}</math>
and the [[kinetic energy]] is:
<math display="block">T = \frac{m}{2} \mathbf{\dot{r}}\cdot\mathbf{\dot{r}}</math>
hence the Lagrangian:
<math display="block">\begin{align}
L &= T - V \\[1ex]
&= \frac{m}{2} \mathbf{\dot{r} } \cdot \mathbf{\dot{r} } + q \mathbf{A} \cdot \mathbf{\dot{r} } - q\phi \\[1ex]
&= \frac{m}{2} \left(\dot{x}^2 + \dot{y}^2 + \dot{z}^2\right) + q \left(\dot{x} A_x + \dot{y} A_y + \dot{z} A_z\right) - q\phi
\end{align}</math>

Lagrange's equations are
<math display="block">\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial \dot{x}} = \frac{\partial L}{\partial x}</math>
(same for {{math|''y''}} and {{math|''z''}}). So calculating the partial derivatives:
<math display="block">\begin{align}
\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{x} } &= m\ddot{x} + q\frac{\mathrm{d} A_x}{\mathrm{d}t} \\
& = m\ddot{x} + q \left[\frac{\partial A_x}{\partial t} + \frac{\partial A_x}{\partial x}\frac{dx}{dt} + \frac{\partial A_x}{\partial y}\frac{dy}{dt} + \frac{\partial A_x}{\partial z}\frac{dz}{dt}\right] \\[1ex]
& = m\ddot{x} + q\left[\frac{\partial A_x}{\partial t} + \frac{\partial A_x}{\partial x}\dot{x} + \frac{\partial A_x}{\partial y}\dot{y} + \frac{\partial A_x}{\partial z}\dot{z}\right]\\
\end{align}</math>
<math display="block">\frac{\partial L}{\partial x}= -q\frac{\partial \phi}{\partial x}+ q\left(\frac{\partial A_x}{\partial x}\dot{x}+\frac{\partial A_y}{\partial x}\dot{y}+\frac{\partial A_z}{\partial x}\dot{z}\right)</math>
equating and simplifying:
<math display="block">m\ddot{x}+ q\left(\frac{\partial A_x}{\partial t}+\frac{\partial A_x}{\partial x}\dot{x}+\frac{\partial A_x}{\partial y}\dot{y}+\frac{\partial A_x}{\partial z}\dot{z}\right)= -q\frac{\partial \phi}{\partial x}+ q\left(\frac{\partial A_x}{\partial x}\dot{x}+\frac{\partial A_y}{\partial x}\dot{y}+\frac{\partial A_z}{\partial x}\dot{z}\right)</math>
<math display="block">\begin{align}
F_x & = -q\left(\frac{\partial \phi}{\partial x}+\frac{\partial A_x}{\partial t}\right) + q\left[\dot{y}\left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right)+\dot{z}\left(\frac{\partial A_z}{\partial x}-\frac{\partial A_x}{\partial z}\right)\right] \\[1ex]
& = qE_x + q[\dot{y}(\nabla\times\mathbf{A})_z-\dot{z}(\nabla\times\mathbf{A})_y] \\[1ex]
& = qE_x + q[\mathbf{\dot{r}}\times(\nabla\times\mathbf{A})]_x \\[1ex]
& = qE_x + q(\mathbf{\dot{r}}\times\mathbf{B})_x
\end{align}</math>
and similarly for the {{math|''y''}} and {{math|''z''}} directions. Hence the force equation is:
<math display="block">\mathbf{F}= q(\mathbf{E} + \mathbf{\dot{r}}\times\mathbf{B})</math>
}}

The relativistic Lagrangian is
<math display="block">L = -mc^2\sqrt{1-\left(\frac{\dot{\mathbf{r} } }{c}\right)^2} + q \mathbf{A}(\mathbf{r}) \cdot \dot{\mathbf{r} } - q \phi(\mathbf{r}) </math>

The action is the relativistic [[arclength]] of the path of the particle in [[spacetime]], minus the potential energy contribution, plus an extra contribution which [[Quantum Mechanics|quantum mechanically]] is an extra [[phase (waves)|phase]] a charged particle gets when it is moving along a vector potential.

{{math proof
|title=Derivation of Lorentz force from relativistic Lagrangian (SI units)
|proof=
The equations of motion derived by [[calculus of variations|extremizing]] the action (see [[matrix calculus]] for the notation):
<math display="block"> \frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t} =\frac{\partial L}{\partial \mathbf{r}} = q {\partial \mathbf{A} \over \partial \mathbf{r}}\cdot \dot{\mathbf{r}} - q {\partial \phi \over \partial \mathbf{r} }</math>
<math display="block">\mathbf{P} -q\mathbf{A} = \frac{m\dot{\mathbf{r}}}{\sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2}}</math>
are the same as [[Hamiltonian mechanics|Hamilton's equations of motion]]:
<math display="block"> \frac{\mathrm{d}\mathbf{r} }{\mathrm{d}t} = \frac{\partial}{\partial \mathbf{p} } \left ( \sqrt{(\mathbf{P}-q\mathbf{A})^2 + (mc^2)^2} + q\phi \right ) </math>
<math display="block"> \frac{\mathrm{d}\mathbf{p} }{\mathrm{d}t} = -\frac{\partial}{\partial \mathbf{r}} \left ( \sqrt{(\mathbf{P}-q\mathbf{A})^2 + (mc^2)^2} + q\phi \right ) </math>
both are equivalent to the noncanonical form:
<math display="block"> \frac{\mathrm{d} }{\mathrm{d}t} {m\dot{\mathbf{r} } \over \sqrt{1-\left(\frac{\dot{\mathbf{r} } }{c}\right)^2} } = q\left ( \mathbf{E} + \dot\mathbf{r} \times \mathbf{B} \right ) . </math>
This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.
}}

== Relativistic form of the Lorentz force ==

=== Covariant form of the Lorentz force ===

==== Field tensor ====

{{main|Covariant formulation of classical electromagnetism|Mathematical descriptions of the electromagnetic field}}

Using the [[metric signature]] {{math|(1, −1, −1, −1)}}, the Lorentz force for a charge {{mvar|q}} can be written in [[Lorentz covariance|covariant form]]:{{sfn|Jackson|1998|loc=chpt. 11}}
{{Equation box 1
|indent =:
|equation = <math> \frac{\mathrm{d} p^\alpha}{\mathrm{d} \tau} = q F^{\alpha \beta} U_\beta </math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}
where {{mvar|p<sup>α</sup>}} is the [[four-momentum]], defined as
<math display="block">p^\alpha = \left(p_0, p_1, p_2, p_3 \right) = \left(\gamma m c, p_x, p_y, p_z \right) ,</math>
{{mvar|τ}} the [[proper time]] of the particle, {{mvar|F<sup>αβ</sup>}} the contravariant [[electromagnetic tensor]]
<math display="block">F^{\alpha \beta} = \begin{pmatrix}
0 & -E_x/c & -E_y/c & -E_z/c \\
E_x/c & 0 & -B_z & B_y \\
E_y/c & B_z & 0 & -B_x \\
E_z/c & -B_y & B_x & 0
\end{pmatrix}
</math>
and {{mvar|U}} is the covariant [[four-velocity|4-velocity]] of the particle, defined as:
<math display="block">U_\beta = \left(U_0, U_1, U_2, U_3 \right) = \gamma \left(c, -v_x, -v_y, -v_z \right) ,</math>
in which
<math display="block">\gamma(v)=\frac{1}{\sqrt{1- \frac{v^2}{c^2} } }=\frac{1}{\sqrt{1- \frac{v_x^2 + v_y^2+ v_z^2}{c^2} } }</math>
is the [[Lorentz factor]].

The fields are transformed to a frame moving with constant relative velocity by:
<math display="block"> F'^{\mu \nu} = {\Lambda^{\mu} }_{\alpha} {\Lambda^{\nu} }_{\beta} F^{\alpha \beta} \, ,</math>
where {{math|Λ<sup>''μ''</sup><sub>''α''</sub>}} is the [[Lorentz transformation]] tensor.

==== Translation to vector notation ====

The {{math|1=''α'' = 1}} component ({{mvar|x}}-component) of the force is
<math display="block"> \frac{\mathrm{d} p^1}{\mathrm{d} \tau} = q U_\beta F^{1 \beta} = q\left(U_0 F^{10} + U_1 F^{11} + U_2 F^{12} + U_3 F^{13} \right) .</math>

Substituting the components of the covariant electromagnetic tensor ''F'' yields
<math display="block"> \frac{\mathrm{d} p^1}{\mathrm{d} \tau} = q \left[U_0 \left(\frac{E_x}{c} \right) + U_2 (-B_z) + U_3 (B_y) \right] .</math>

Using the components of covariant [[four-velocity]] yields
<math display="block"> \frac{\mathrm{d} p^1}{\mathrm{d} \tau}
= q \gamma \left[c \left(\frac{E_x}{c} \right) + (-v_y) (-B_z) + (-v_z) (B_y) \right]
= q \gamma \left(E_x + v_y B_z - v_z B_y \right)
= q \gamma \left[ E_x + \left( \mathbf{v} \times \mathbf{B} \right)_x \right] \, .
 </math>

The calculation for {{math|1=''α'' = 2, 3}} (force components in the {{mvar|y}} and {{mvar|z}} directions) yields similar results, so collecting the three equations into one:
<math display="block"> \frac{\mathrm{d} \mathbf{p} }{\mathrm{d} \tau} = q \gamma\left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right) , </math>
and since differentials in coordinate time {{mvar|dt}} and proper time {{mvar|dτ}} are related by the Lorentz factor,
<math display="block">dt=\gamma(v) \, d\tau,</math>
so we arrive at
<math display="block"> \frac{\mathrm{d} \mathbf{p} }{\mathrm{d} t} = q \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right) .</math>

This is precisely the Lorentz force law, however, it is important to note that {{math|'''p'''}} is the relativistic expression,
<math display="block">\mathbf{p} = \gamma(v) m_0 \mathbf{v} \,.</math>

=== Lorentz force in spacetime algebra (STA) ===

The electric and magnetic fields are [[Classical electromagnetism and special relativity|dependent on the velocity of an observer]], so the relativistic form of the Lorentz force law can best be exhibited starting from a coordinate-independent expression for the electromagnetic and magnetic fields <math>\mathcal{F}</math>, and an arbitrary time-direction, <math>\gamma_0</math>. This can be settled through [[spacetime algebra]] (or the geometric algebra of spacetime), a type of [[Clifford algebra]] defined on a [[pseudo-Euclidean space]],<ref>{{cite web|last=Hestenes|first=David|author-link=David Hestenes|title=SpaceTime Calculus|url=https://davidhestenes.net/geocalc/html/STC.html}}</ref> as
<math display="block">\mathbf{E} = \left(\mathcal{F} \cdot \gamma_0\right) \gamma_0</math>
and
<math display="block">i\mathbf{B} = \left(\mathcal{F} \wedge \gamma_0\right) \gamma_0</math>
<math>\mathcal F</math> is a spacetime [[bivector]] (an oriented plane segment, just like a vector is an [[oriented line segment]]), which has six degrees of freedom corresponding to boosts (rotations in spacetime planes) and rotations (rotations in space-space planes). The [[dot product]] with the vector <math>\gamma_0</math> pulls a vector (in the space algebra) from the translational part, while the wedge-product creates a trivector (in the space algebra) who is dual to a vector which is the usual magnetic field vector. The relativistic velocity is given by the (time-like) changes in a time-position vector {{nowrap|<math>v = \dot x</math>,}} where
<math display="block">v^2 = 1,</math>
(which shows our choice for the metric) and the velocity is
<math display="block">\mathbf{v} = cv \wedge \gamma_0 / (v \cdot \gamma_0).</math>

The proper form of the Lorentz force law ('invariant' is an inadequate term because no transformation has been defined) is simply
{{Equation box 1
|indent =:
|equation = <math> F = q\mathcal{F}\cdot v</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

Note that the order is important because between a bivector and a vector the dot product is anti-symmetric. Upon a spacetime split like one can obtain the velocity, and fields as above yielding the usual expression.

=== Lorentz force in general relativity ===
In the [[general theory of relativity]] the equation of motion for a particle with mass <math>m</math> and charge <math>e</math>, moving in a space with metric tensor <math>g_{ab}</math> and electromagnetic field <math>F_{ab}</math>, is given as

<math display="block">m\frac{du_c}{ds} - m \frac{1}{2} g_{ab,c} u^a u^b = e F_{cb}u^b ,
</math>

where <math>u^a= dx^a/ds</math> (<math>dx^a</math> is taken along the trajectory), <math>g_{ab,c} = \partial g_{ab}/\partial x^c</math>, and <math>ds^2 = g_{ab} dx^a dx^b</math>.

The equation can also be written as

<math display="block">m\frac{du_c}{ds}-m\Gamma_{abc}u^a u^b = eF_{cb}u^b ,</math>

where <math>\Gamma_{abc}</math> is the [[Levi-Civita connection#Christoffel symbols|Christoffel symbol]] (of the torsion-free metric connection in general relativity), or as

<math display="block">m\frac{Du_c}{ds} = e F_{cb}u^b ,</math>

where <math>D</math> is the [[covariant differential]] in general relativity.

== Applications ==

The Lorentz force occurs in many devices, including:
* [[Cyclotron]]s and other circular path [[particle accelerator]]s
* [[Mass spectrometer]]s
* [[Velocity filter]]s
* [[Magnetron]]s
* [[Lorentz force velocimetry]]

In its manifestation as the Laplace force on an electric current in a conductor, this force occurs in many devices, including:

* [[Electric motor]]s
* [[Railgun]]s
* [[Linear motor]]s
* [[Loudspeaker]]s
* [[Magnetoplasmadynamic thruster]]s
* [[Electrical generator]]s
* [[Homopolar generator]]s
* [[Linear alternator]]s

== See also ==
{{cols|colwidth=26em}}
* [[Hall effect]]
* [[Electromagnetism]]
* [[Gravitomagnetism]]
* [[Ampère's force law]]
* [[Hendrik Lorentz]]
* [[Maxwell's equations]]
* [[Formulation of Maxwell's equations in special relativity]]
* [[Moving magnet and conductor problem]]
* [[Abraham–Lorentz force]]
* [[Larmor formula]]
* [[Cyclotron radiation]]
* [[Magnetoresistance]]
* [[Scalar potential]]
* [[Helmholtz decomposition]]
* [[Guiding center]]
* [[Field line]]
* [[Coulomb's law]]
* [[Electromagnetic buoyancy]]
{{colend}}

== Notes ==
===Remarks===
{{reflist|group=nb|30em}}

===Citations===
{{reflist|30em}}

== References ==
* {{cite book | last=Darrigol | first=Olivier | title=Electrodynamics from Ampère to Einstein | publisher=Clarendon Press | publication-place=Oxford ; New York | date=2000 | isbn=0-19-850594-9}}
* {{cite book |first1 = Richard Phillips |last1 = Feynman |author-link = Richard Feynman |first2 = Robert B. |last2 = Leighton | first3 = Matthew L. |last3 = Sands |title = The Feynman lectures on physics |publisher = Pearson / Addison-Wesley | year = 2006 |isbn = 0-8053-9047-2  |volume=2}}
* {{cite book | last=Griffiths | first=David J. | title=Introduction to Electrodynamics | publisher=Cambridge University Press | date=2023 | isbn=978-1-009-39773-5 | doi=10.1017/9781009397735 | url=https://www.cambridge.org/highereducation/product/9781009397735/book}}
* {{cite book | last=Jackson | first=John David | title=Classical Electrodynamics | publisher=John Wiley & Sons | publication-place=New York | date=1998 | isbn=978-0-471-30932-1|url=https://www.wiley.com/en-us/Classical+Electrodynamics%2C+3rd+Edition-p-9780471309321}}
* {{cite book | last=Purcell | first=Edward M. | last2=Morin | first2=David J. | title=Electricity and Magnetism: | publisher=Cambridge University Press | date=2013 | isbn=978-1-139-01297-3 | doi=10.1017/cbo9781139012973 | url=https://www.cambridge.org/core/product/identifier/9781139012973/type/book}}
* {{cite book | first=Matthew N. O. |last=Sadiku | title=Elements of electromagnetics | year= 2018 | edition=7th | publisher=Oxford University Press | location=New York/Oxford | isbn = 978-0-19-069861-4 | url=https://lccn.loc.gov/2017046497}}
* {{cite book |first1 = Raymond A. |last1 = Serway | first2 = John W. Jr. |last2 = Jewett |title = Physics for scientists and engineers, with modern physics |place = Belmont, California | publisher = Thomson Brooks/Cole |year = 2004 |isbn = 0-534-40846-X }}
* {{cite book |first = Mark A. |last = Srednicki |title= Quantum field theory |url=https://books.google.com/books?id=5OepxIG42B4C&pg=PA315 |place = Cambridge, England; New York City |publisher = Cambridge University Press | year=2007 | isbn = 978-0-521-86449-7 }}

== External links ==
{{Commons|Lorentz force}}
{{wikiquote}}
* [https://web.archive.org/web/20150713153934/https://www.youtube.com/watch?v=mxMMqNrm598 Lorentz force (demonstration)]
* [http://chair.pa.msu.edu/applets/Lorentz/a.htm Interactive Java applet on the magnetic deflection of a particle beam in a homogeneous magnetic field] {{Webarchive|url=https://web.archive.org/web/20110813101606/http://chair.pa.msu.edu/applets/Lorentz/a.htm |date=2011-08-13 }} by Wolfgang Bauer

{{Authority control}}

[[Category:Physical phenomena]]
[[Category:Electromagnetism]]
[[Category:Maxwell's equations]]
[[Category:Hendrik Lorentz]]