Lorentz force#Lorentz force and analytical mechanics

File:Lorentz force particle.svg (of charge {{mvar|q}}) in motion (instantaneous velocity {{math|v}}). The electric field and magnetic field vary in space and time.]]

The Lorentz force {{math|F}} acting on a point particle with electric charge {{mvar|q}}, moving with velocity {{math|v}}, due to an external electric field {{math|E}} and magnetic field {{math|B}}, is given by (SI definition of quantities{{efn|name=units|In SI units, {{math|B}} is measured in teslas (symbol: T). In Gaussian-cgs units, {{math|B}} is measured in gauss (symbol: 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}}) {{math|H}} is measured in amperes per metre (A/m) in SI units, and in oersteds (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}}}}):{{sfn|Jackson|1998|pp=2-3}}

{{Equation box 1

|indent =:

|equation = $\backslash mathbf\{F\}\; =\; q\; \backslash left(\backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}\backslash right)$

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Here, {{math|×}} is the vector cross product, and all quantities in bold are vectors. In component form, the force is written as:

$$\backslash begin\{align\}$$

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}

In general, the electric and magnetic fields depend on both position and time. As a charged particle moves through space, the force acting on it at any given moment depends on its current location, velocity, and the instantaneous values of the fields at that location. Therefore, explicitly, the Lorentz force can be written as:

$$\backslash mathbf\{F\}\backslash left(\backslash mathbf\{r\}(t),\backslash dot\backslash mathbf\{r\}(t),t,q\backslash right)\; =\; q\backslash left[\backslash mathbf\{E\}(\backslash mathbf\{r\},t)\; +\; \backslash dot\backslash mathbf\{r\}(t)\; \backslash times\; \backslash mathbf\{B\}(\backslash mathbf\{r\},t)\backslash right]$$

in which {{math|r}} is the position vector of the charged particle, {{mvar|t}} is time, and the overdot is a time derivative.

The total electromagnetic force consists of two parts: the electric force {{math|qE}}, which acts in the direction of the electric field and accelerates the particle linearly, and the magnetic force {{math|1=q(v × B)}}, which acts perpendicularly to both the velocity and the magnetic field.{{sfn|Griffiths|2023|p=211}} Some sources refer to the Lorentz force as the sum of both components, while others use the term to refer to the magnetic part alone.For example, see the [http://ilorentz.org/history/lorentz/lorentz.html website of the Lorentz Institute].

The direction of the magnetic force is often determined using the right-hand rule: if the index finger points in the direction of the velocity, and the middle finger points in the direction of the magnetic field, then the thumb points in the direction of the force (for a positive charge). In a uniform magnetic field, this results in circular or helical trajectories, known as cyclotron motion.{{cite book |last=Zangwill |first=Andrew |title=Modern Electrodynamics |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-89697-9 |publication-place=Cambridge |page=366-367}}

In many practical situations, such as the motion of electrons or ions in a plasma, the effect of a magnetic field can be approximated as a superposition of two components: 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. These differences may lead to electric currents or chemical separation.{{Cn|date=June 2025}}

While the magnetic force affects the direction of a particle's motion, it does no mechanical work on the particle. The rate at which the energy is transferred from the electromagnetic field to the particle is given by the dot product of the particle’s velocity and the force:

$$\backslash mathbf\{v\}\backslash cdot\backslash mathbf\{F\}\; =\; q\backslash mathbf\{v\}\backslash cdot(\backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\})\; =\; q\; \backslash ,\; \backslash mathbf\{v\}\; \backslash cdot\; \backslash mathbf\{E\}.$$Here, the magnetic term vanishes because a vector is always perpendicular to its cross product with another vector; the scalar triple product $\backslash mathbf\{v\}\backslash cdot\; (\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\})$ is zero. Thus, only the electric field can transfer energy to or from a particle and change its kinetic energy.{{cite book |last=Zangwill |first=Andrew |title=Modern Electrodynamics |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-89697-9 |publication-place=Cambridge |page=366}}

File:Lorentz force continuum.svg (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.]]

The Lorentz force law also applies to continuous charge distributions, such as those found in conductors or plasmas. For a small element of a moving charge distribution with charge $\backslash mathrm\{d\}q$, the infinitesimal force is given by:

$$\backslash mathrm\{d\}\backslash mathbf\{F\}\; =\; \backslash mathrm\{d\}q\backslash left(\backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}\backslash right)$$

Dividing both sides by the volume $\backslash mathrm\{d\}V$ of the charge element gives the force density

$$\backslash mathbf\{f\}\; =\; \backslash rho\backslash left(\backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}\backslash right),$$

where $\backslash rho$ is the charge density and $\backslash mathbf\{f\}$ is the force per unit volume. Introducing the current density $\backslash mathbf\{J\}\; =\; \backslash rho\; \backslash mathbf\{v\}$, this can be rewritten as:{{sfn|Griffiths|2023|pp=219,368}}

{{Equation box 1

|indent =:

|equation = $\backslash mathbf\{f\}\; =\; \backslash rho\; \backslash mathbf\{E\}\; +\; \backslash mathbf\{J\}\; \backslash times\; \backslash mathbf\{B\}$

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The total force is the volume integral over the charge distribution:

$$\backslash mathbf\{F\}\; =\; \backslash int\; \backslash left\; (\; \backslash rho\; \backslash mathbf\{E\}\; +\; \backslash mathbf\{J\}\; \backslash times\; \backslash mathbf\{B\}\; \backslash right)\backslash mathrm\{d\}V.$$

Using Maxwell's equations and vector calculus identities, the force density can be reformulated to eliminate explicit reference to the charge and current densities. The force density can then be written in terms of the electromagnetic fields and their derivatives:$$\backslash mathbf\{f\}\; =\; \backslash nabla\backslash cdot\backslash boldsymbol\{\backslash sigma\}\; -\; \backslash dfrac\{1\}\{c^2\}\; \backslash dfrac\{\backslash partial\; \backslash mathbf\{S\}\}\{\backslash partial\; t\}$$

where $\backslash boldsymbol\{\backslash sigma\}$ is the Maxwell stress tensor, $\backslash nabla\; \backslash cdot$ denotes the tensor divergence, $c$ is the speed of light, and $\backslash mathbf\{S\}$ is the Poynting vector. This form of the force law relates the energy flux in the fields to the force exerted on a charge distribution. (See Covariant formulation of classical electromagnetism for more details.){{sfn|Griffiths|2023|pp=369-370}}

The power density corresponding to the Lorentz force, the rate of energy transfer to the material, is given by:$$\backslash mathbf\{J\}\; \backslash cdot\; \backslash mathbf\{E\}.$$

Inside a material, the total charge and current densities can be separated into free and bound parts. In terms of free charge density $\backslash rho\_\{\backslash rm\; f\}$, free current density $\backslash mathbf\{J\}\_\{\backslash rm\; f\}$, polarization $\backslash mathbf\{P\}$, and magnetization $\backslash mathbf\{M\}$, the force density becomes{{Cn|date=June 2025}}

$$\backslash mathbf\{f\}\; =\; \backslash left(\backslash rho\_\{\backslash rm\; f\}\; -\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; P\backslash right)\; \backslash mathbf\{E\}\; +\; \backslash left(\backslash mathbf\{J\}\_\{\{\backslash rm\; f\}\}\; +\; \backslash nabla\backslash times\backslash mathbf\{M\}\; +\; \backslash frac\{\backslash partial\backslash mathbf\{P\}\}\{\backslash partial\; t\}\backslash right)\; \backslash times\; \backslash mathbf\{B\}.$$This form accounts for the torque applied to a permanent magnet by the magnetic field. The associated power density is{{Cn|date=June 2025}}

$$\backslash left(\backslash mathbf\{J\}\_f\; +\; \backslash nabla\backslash times\backslash mathbf\{M\}\; +\; \backslash frac\{\backslash partial\backslash mathbf\{P\}\}\{\backslash partial\; t\}\backslash right)\; \backslash cdot\; \backslash mathbf\{E\}.$$

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 CGS-Gaussian units, which are somewhat more common among some theoretical physicists as well as condensed matter experimentalists, one has instead

$$\backslash mathbf\{F\}\; =\; q\_\backslash mathrm\{G\}\; \backslash left(\backslash mathbf\{E\}\_\backslash mathrm\{G\}\; +\; \backslash frac\{\backslash mathbf\{v\}\}\{c\}\; \backslash times\; \backslash mathbf\{B\}\_\backslash mathrm\{G\}\backslash right),$$

where {{mvar|c}} is the speed of light. Although this equation looks slightly different, it is equivalent, since one has the following relations:{{efn|name=units}}

$$q\_\backslash mathrm\{G\}\; =\; \backslash frac\{q\_\backslash mathrm\{SI\}\}\{\backslash sqrt\{4\backslash pi\; \backslash varepsilon\_0\}\},\backslash quad$$

\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}}.

where {{math|ε₀}} is the vacuum permittivity and {{math|μ₀}} the vacuum permeability. In practice, the subscripts "G" and "SI" are omitted, and the used convention (and unit) must be determined from context.

Maxwell's equations describe how electric charges and currents generate electric and magnetic fields. The Lorentz force law completes the picture by describing how those fields act on a moving point charge {{mvar|q}}.{{sfn|Jackson|1998|pp=2-3}}{{sfn|Griffiths|2023|p=340}} Together, they form a self-consistent framework: the fields influence the motion of charges, and charges and currents generate the fields. In a broader physical context, charged particles may also be subject to other forces, such as gravity or nuclear interactions, which lie outside the scope of classical electrodynamics.

{{multiple image|position

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| direction = horizontal

| footer = Charged particles 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

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}}

In many textbook treatments of classical electromagnetism, the Lorentz force law is also used to define the electric and magnetic fields {{math|E}} and {{math|B}}. The force {{math|F}} acting on a test particle of charge {{math|q}} and and velocity {{math|v}} is taken to follow a form that uniquely determines the fields:{{sfn|Jackson|1998|pp=777-778}}{{cite book |last1=Wheeler |first1=J. A. |author1-link=John Archibald Wheeler |url=https://archive.org/details/gravitation00misn_003 |title=Gravitation |last2=Misner |first2=C. |author-link2=Charles W. Misner |last3=Thorne |first3=K. S. |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}}{{cite book |last1=Grant |first1=I. S. |title=Electromagnetism |last2=Phillips |first2=W. R. |publisher=John Wiley & Sons |year=1990 |isbn=978-0-471-92712-9 |edition=2nd |series=The Manchester Physics Series |page=122}}

{{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: $$\backslash mathbf\{F\}\; =\; q(\backslash mathbf\{E\}+\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\})$$}}

This definition remains valid even for particles approaching the speed of light (that is, magnitude of {{math|v}}, {{math|1={{abs|v}} ≈ c}}).{{cite book |last1=Grant |first1=I. S. |title=Electromagnetism |last2=Phillips |first2=W. R. |publisher=John Wiley & Sons |year=1990 |isbn=978-0-471-92712-9 |edition=2nd |series=The Manchester Physics Series |page=123}} The fields {{math|E}} and {{math|B}} are thereby defined throughout space and time by the force a test charge would experience, regardless of whether any charge is actually present.

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).{{Cn|date=June 2025}}

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 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,{{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}} and electrically charged objects, by Henry Cavendish in 1762,{{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}} 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.{{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}} 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.{{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}}{{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.{{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}}

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 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,{{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}} 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 rays, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as{{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}}{{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}}

$$\backslash mathbf\{F\}\; =\; \backslash frac\{q\}\{2\}\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}.$$

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.{{sfn|Darrigol|2000|pp=200,429-430}}{{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}} Finally, in 1895,{{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}}{{cite book |last=Lorentz |first=Hendrik Antoon |title=Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern |language=de |year=1895}} 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}}{{cite book | last = Whittaker | first = E. T. | author-link=E. T. Whittaker | title = 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}}

{{see also|Electric motor#Force and torque|Biot–Savart law}}

File:Regla mano derecha Laplace.svg

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}}

$$\backslash mathbf\{F\}\; =\; I\; \backslash boldsymbol\{\backslash ell\}\; \backslash times\; \backslash mathbf\{B\}\; ,$$

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 $\backslash mathrm\; d\; \backslash boldsymbol\; \backslash ell$, then adding up all these forces by 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}}

$$\backslash mathbf\{F\}\; =\; I\backslash int\; (\backslash mathrm\{d\}\backslash boldsymbol\{\backslash ell\}\backslash times\; \backslash mathbf\{B\}).$$

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 $\backslash mathbf\{F\}$ acting on the rotor creates a torque, making the motor spin. Hence, though the Lorentz force law does not apply when the magnetic field $\backslash mathbf\{B\}$ is generated by the current $I$, it does apply when the current $I$ is induced by the movement of magnetic field $\backslash mathbf\{B\}$.

{{main|Electromotive force}}

{{multiple image|position

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| 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.

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The magnetic force component ({{math|qv × B}}) of the Lorentz force is responsible for motional electromotive force (or motional EMF); the fundamental mechanism underlying induction motors and 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 machines such as synchronous generators, the magnetic field moves, while the conductors do not. In this case, the EMF is due to the electric force component ({{math|qE}}) of the Lorentz Force. 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 Maxwell–Faraday equation.{{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 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.{{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 }}

{{main|Faraday's law of induction}}

File:Lorentz force - mural Leiden 1, 2016.jpg

Given a loop of wire in a magnetic field, Faraday's law of induction states the induced electromotive force (EMF) in the wire is:

$$\backslash mathcal\{E\}\; =\; -\backslash frac\{\backslash mathrm\{d\}\backslash Phi\_B\}\{\backslash mathrm\{d\}t\}$$

where

$$\backslash Phi\_B\; =\; \backslash int\_\{\backslash Sigma(t)\}\; \backslash mathbf\{B\}(\backslash mathbf\{r\},\; t)\backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\},$$

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|dA}} 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 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:{{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}}

$$\backslash mathcal\{E\}\; =\; \backslash oint\_\{\backslash partial\; \backslash Sigma\; (t)\}\; \backslash frac\{\backslash mathbf\{F\}\}\{q\}\backslash cdot\; \backslash mathrm\{d\}\; \backslash boldsymbol\{\backslash ell\}$$

where $\backslash mathbf\{E\}\text{'}(\backslash mathbf\{r\},\; t)\; =\; \backslash mathbf\{F\}/q(\backslash mathbf\{r\},\; t)$ is the electric field and {{math|dℓ}} is an infinitesimal vector element of the contour {{math|∂Σ(t)}}.{{sfn|Jackson|1998|p=209}}{{efn|Both {{math|dℓ}} and {{math|dA}} have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in the article Kelvin–Stokes theorem.}} Equating both integrals leads to the field theory form of Faraday's law, given by:{{sfn|Jackson|1998|pp=209-210}}

$$\backslash mathcal\{E\}\; =\; \backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\backslash mathbf\{E\}\text{'}(\backslash mathbf\{r\},\; t)\; \backslash cdot\; \backslash mathrm\{d\}\; \backslash boldsymbol\{\backslash ell\}\; =\; -\; \backslash frac\{\backslash mathrm\{d\}\; \}\{\backslash mathrm\{d\}t\}\; \backslash int\_\{\backslash Sigma(t)\}\; \backslash mathbf\{B\}(\backslash mathbf\{r\},t)\; \backslash cdot\; \backslash mathrm\{d\}\; \backslash mathbf\{A\}.$$

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) Maxwell–Faraday equation:{{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}}

$$\backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash mathbf\{E\}(\backslash mathbf\{r\},t)\; \backslash cdot\; \backslash mathrm\{d\}\; \backslash boldsymbol\{\backslash ell\}\; =\; -\; \backslash int\_\{\backslash Sigma(t)\}\; \backslash frac\{\backslash partial\; \backslash mathbf\; \{B\}(\backslash mathbf\{r\},\; t)\}\{\; \backslash partial\; t\; \}\; \backslash cdot\; \backslash mathrm\{d\}\; \backslash mathbf\{A\}.$$

The two equations are equivalent if the wire is not moving. In case the circuit is moving with a velocity $\backslash mathbf\{v\}$ in some direction, then, using the Leibniz integral rule and that {{math|1=div B = 0}}, gives

$$\backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\backslash mathbf\{E\}\text{'}(\backslash mathbf\{r\},\; t)\; \backslash cdot\; \backslash mathrm\{d\}\; \backslash boldsymbol\{\backslash ell\}=$$

- \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}.

Substituting the Maxwell-Faraday equation then gives

$$\backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash mathbf\{E\}\text{'}(\backslash mathbf\{r\},\; t)\backslash cdot\; \backslash mathrm\{d\}\; \backslash boldsymbol\{\backslash ell\}\; =$$

\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}

since this is valid for any wire position it implies that

$$\backslash mathbf\{F\}\; =\; q\backslash ,\backslash mathbf\{E\}(\backslash mathbf\{r\},\backslash ,\; t)\; +\; q\backslash ,\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}(\backslash mathbf\{r\},\backslash ,\; t).$$

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 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_B}} 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_B}} will change. Alternatively, if the loop changes orientation with respect to the B-field, the {{math|B ⋅ dA}} differential element will change because of the different angle between {{math|B}} and {{math|dA}}, also changing {{math|Φ_B}}. 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_B}}.

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 is not zero.{{sfn|Sadiku|2018|pp=424-425}}

{{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) electrostatic potential {{math|ϕ}} by

$$\backslash begin\{align\}$$

\mathbf{E} &= - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t } \\[1ex]

\mathbf{B} &= \nabla \times \mathbf{A}

\end{align}

where {{math|∇}} is the gradient, {{math|∇⋅}} is the divergence, and {{math|∇×}} is the curl.

The force becomes

$$\backslash mathbf\{F\}\; =\; q\backslash left[-\backslash nabla\; \backslash phi-\; \backslash frac\{\backslash partial\; \backslash mathbf\{A\}\}\{\backslash partial\; t\}+\backslash mathbf\{v\}\backslash times(\backslash nabla\backslash times\backslash mathbf\{A\})\backslash right].$$

Using an identity for the triple product this can be rewritten as

$$\backslash mathbf\{F\}\; =\; q\backslash left[-\backslash nabla\; \backslash phi-\; \backslash frac\{\backslash partial\; \backslash mathbf\{A\}\}\{\backslash partial\; t\}+\backslash nabla\backslash left(\backslash mathbf\{v\}\backslash cdot\; \backslash mathbf\{A\}\; \backslash right)-\backslash left(\backslash mathbf\{v\}\backslash cdot\; \backslash nabla\backslash right)\backslash mathbf\{A\}\backslash right].$$

(Notice that the coordinates and the velocity components should be treated as independent variables, so the del operator acts only on {{nowrap|$\backslash mathbf\{A\}$,}} not on {{nowrap|$\backslash mathbf\{v\}$;}} thus, there is no need of using Feynman's subscript notation in the equation above.) Using the chain rule, the convective derivative of $\backslash mathbf\{A\}$ is:{{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}}

$$\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{A\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash frac\{\backslash partial\backslash mathbf\{A\}\}\{\backslash partial\; t\}+(\backslash mathbf\{v\}\backslash cdot\backslash nabla)\backslash mathbf\{A\}$$

so that the above expression becomes:

$$\backslash mathbf\{F\}\; =\; q\backslash left[-\backslash nabla\; (\backslash phi-\backslash mathbf\{v\}\backslash cdot\backslash mathbf\{A\})-\; \backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{A\}\}\{\backslash mathrm\{d\}t\}\backslash right].$$

With {{math|1=v = ẋ}} and

$$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\backslash left[\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\{\backslash mathbf\{x\}\}\}\backslash left(\backslash phi\; -\; \backslash dot\{\backslash mathbf\{x\}\}\backslash cdot\; \backslash mathbf\{A\}\; \backslash right)\; \backslash right]\; =\; -\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{A\}\}\{\backslash mathrm\{d\}t\},$$

we can put the equation into the convenient Euler–Lagrange form

{{Equation box 1

|indent =:

|equation = $\backslash mathbf\{F\}\; =\; q\backslash left[-\backslash nabla\_\{\backslash mathbf\{x\}\; \}(\backslash phi-\backslash dot\{\backslash mathbf\{x\}\; \}\backslash cdot\backslash mathbf\{A\})\; +\; \backslash frac\{\backslash mathrm\{d\}\; \}\{\backslash mathrm\{d\}t\}\backslash nabla\_\{\backslash dot\{\backslash mathbf\{x\}\; \}\; \}(\backslash phi-\backslash dot\{\backslash mathbf\{x\}\; \}\backslash cdot\backslash mathbf\{A\})\backslash right]$

|cellpadding= 6

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

where $$\backslash nabla\_\{\backslash mathbf\{x\}\; \}\; =\; \backslash hat\{x\}\; \backslash dfrac\{\backslash partial\}\{\backslash partial\; x\}\; +\; \backslash hat\{y\}\; \backslash dfrac\{\backslash partial\}\{\backslash partial\; y\}\; +\; \backslash hat\{z\}\; \backslash dfrac\{\backslash partial\}\{\backslash partial\; z\}$$ and $$\backslash nabla\_\{\backslash dot\{\backslash mathbf\{x\}\; \}\; \}\; =\; \backslash hat\{x\}\; \backslash dfrac\{\backslash partial\}\{\backslash partial\; \backslash dot\{x\}\; \}\; +\; \backslash hat\{y\}\; \backslash dfrac\{\backslash partial\}\{\backslash partial\; \backslash dot\{y\}\; \}\; +\; \backslash hat\{z\}\; \backslash dfrac\{\backslash partial\}\{\backslash partial\; \backslash dot\{z\}\; \}.$$

{{see also|Magnetic vector potential#Interpretation as Potential Momentum}}

The 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:{{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}}

$$L\; =\; \backslash frac\{m\}\{2\}\; \backslash mathbf\{\backslash dot\{r\}\; \}\backslash cdot\backslash mathbf\{\backslash dot\{r\}\; \}\; +\; q\; \backslash mathbf\{A\}\backslash cdot\backslash mathbf\{\backslash dot\{r\}\; \}-q\backslash phi$$

where {{math|A}} and {{math|ϕ}} are the potential fields as above. The quantity $V\; =\; q(\backslash phi\; -\; \backslash mathbf\{A\}\backslash cdot\; \backslash mathbf\{\backslash dot\{r\}\})$ can be identified as a generalized, velocity-dependent potential energy and, accordingly, $\backslash mathbf\{F\}$ as a non-conservative force.{{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}} 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 potential momentum $q\backslash mathbf\{A\}(\backslash mathbf\{r\},\; t)$, so its potential energy is $q\backslash mathbf\{A\}(\backslash mathbf\{r\},t)\backslash cdot\backslash mathbf\{\backslash dot\{r\}\}$. For a ϕ field, the particle's potential energy is $q\backslash phi(\backslash mathbf\{r\},t)$.

The total potential energy is then:

$$V\; =\; q\backslash phi\; -\; q\backslash mathbf\{A\}\backslash cdot\backslash mathbf\{\backslash dot\{r\}\}$$

and the kinetic energy is:

$$T\; =\; \backslash frac\{m\}\{2\}\; \backslash mathbf\{\backslash dot\{r\}\}\backslash cdot\backslash mathbf\{\backslash dot\{r\}\}$$

hence the Lagrangian:

$$\backslash begin\{align\}$$

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}

Lagrange's equations are

$$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\{x\}\}\; =\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; x\}$$

(same for {{math|y}} and {{math|z}}). So calculating the partial derivatives:

$$\backslash begin\{align\}$$

\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}

$$\backslash frac\{\backslash partial\; L\}\{\backslash partial\; x\}=\; -q\backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; x\}+\; q\backslash left(\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; x\}\backslash dot\{x\}+\backslash frac\{\backslash partial\; A\_y\}\{\backslash partial\; x\}\backslash dot\{y\}+\backslash frac\{\backslash partial\; A\_z\}\{\backslash partial\; x\}\backslash dot\{z\}\backslash right)$$

equating and simplifying:

$$m\backslash ddot\{x\}+\; q\backslash left(\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; t\}+\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; x\}\backslash dot\{x\}+\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; y\}\backslash dot\{y\}+\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; z\}\backslash dot\{z\}\backslash right)=\; -q\backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; x\}+\; q\backslash left(\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; x\}\backslash dot\{x\}+\backslash frac\{\backslash partial\; A\_y\}\{\backslash partial\; x\}\backslash dot\{y\}+\backslash frac\{\backslash partial\; A\_z\}\{\backslash partial\; x\}\backslash dot\{z\}\backslash right)$$

$$\backslash begin\{align\}$$

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}

and similarly for the {{math|y}} and {{math|z}} directions. Hence the force equation is:

$$\backslash mathbf\{F\}=\; q(\backslash mathbf\{E\}\; +\; \backslash mathbf\{\backslash dot\{r\}\}\backslash times\backslash mathbf\{B\})$$

}}

The relativistic Lagrangian is

$$L\; =\; -mc^2\backslash sqrt\{1-\backslash left(\backslash frac\{\backslash dot\{\backslash mathbf\{r\}\; \}\; \}\{c\}\backslash right)^2\}\; +\; q\; \backslash mathbf\{A\}(\backslash mathbf\{r\})\; \backslash cdot\; \backslash dot\{\backslash mathbf\{r\}\; \}\; -\; q\; \backslash phi(\backslash mathbf\{r\})$$

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 mechanically is an extra 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 extremizing the action (see matrix calculus for the notation):

$$\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{P\}\}\{\backslash mathrm\{d\}t\}\; =\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash mathbf\{r\}\}\; =\; q\; \{\backslash partial\; \backslash mathbf\{A\}\; \backslash over\; \backslash partial\; \backslash mathbf\{r\}\}\backslash cdot\; \backslash dot\{\backslash mathbf\{r\}\}\; -\; q\; \{\backslash partial\; \backslash phi\; \backslash over\; \backslash partial\; \backslash mathbf\{r\}\; \}$$

$$\backslash mathbf\{P\}\; -q\backslash mathbf\{A\}\; =\; \backslash frac\{m\backslash dot\{\backslash mathbf\{r\}\}\}\{\backslash sqrt\{1-\backslash left(\backslash frac\{\backslash dot\{\backslash mathbf\{r\}\}\}\{c\}\backslash right)^2\}\}$$

are the same as Hamilton's equations of motion:

$$\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{r\}\; \}\{\backslash mathrm\{d\}t\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash mathbf\{p\}\; \}\; \backslash left\; (\; \backslash sqrt\{(\backslash mathbf\{P\}-q\backslash mathbf\{A\})^2\; +\; (mc^2)^2\}\; +\; q\backslash phi\; \backslash right\; )$$

$$\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{p\}\; \}\{\backslash mathrm\{d\}t\}\; =\; -\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash mathbf\{r\}\}\; \backslash left\; (\; \backslash sqrt\{(\backslash mathbf\{P\}-q\backslash mathbf\{A\})^2\; +\; (mc^2)^2\}\; +\; q\backslash phi\; \backslash right\; )$$

both are equivalent to the noncanonical form:

$$\backslash frac\{\backslash mathrm\{d\}\; \}\{\backslash mathrm\{d\}t\}\; \{m\backslash dot\{\backslash mathbf\{r\}\; \}\; \backslash over\; \backslash sqrt\{1-\backslash left(\backslash frac\{\backslash dot\{\backslash mathbf\{r\}\; \}\; \}\{c\}\backslash right)^2\}\; \}\; =\; q\backslash left\; (\; \backslash mathbf\{E\}\; +\; \backslash dot\backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{B\}\; \backslash right\; )\; .$$

This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.

}}

{{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 covariant form:{{sfn|Jackson|1998|loc=chpt. 11}}

{{Equation box 1

|indent =:

|equation = $\backslash frac\{\backslash mathrm\{d\}\; p^\backslash alpha\}\{\backslash mathrm\{d\}\; \backslash tau\}\; =\; q\; F^\{\backslash alpha\; \backslash beta\}\; U\_\backslash beta$

|cellpadding

|border

|border colour = #50C878

|background colour = #ECFCF4}}

where {{mvar|p^α}} is the four-momentum, defined as

$$p^\backslash alpha\; =\; \backslash left(p\_0,\; p\_1,\; p\_2,\; p\_3\; \backslash right)\; =\; \backslash left(\backslash gamma\; m\; c,\; p\_x,\; p\_y,\; p\_z\; \backslash right)\; ,$$

{{mvar|τ}} the proper time of the particle, {{mvar|F^αβ}} the contravariant electromagnetic tensor

$$F^\{\backslash alpha\; \backslash beta\}\; =\; \backslash begin\{pmatrix\}$$

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}

and {{mvar|U}} is the covariant 4-velocity of the particle, defined as:

$$U\_\backslash beta\; =\; \backslash left(U\_0,\; U\_1,\; U\_2,\; U\_3\; \backslash right)\; =\; \backslash gamma\; \backslash left(c,\; -v\_x,\; -v\_y,\; -v\_z\; \backslash right)\; ,$$

in which

$$\backslash gamma(v)=\backslash frac\{1\}\{\backslash sqrt\{1-\; \backslash frac\{v^2\}\{c^2\}\; \}\; \}=\backslash frac\{1\}\{\backslash sqrt\{1-\; \backslash frac\{v\_x^2\; +\; v\_y^2+\; v\_z^2\}\{c^2\}\; \}\; \}$$

is the Lorentz factor.

The fields are transformed to a frame moving with constant relative velocity by:

$$F\text{'}^\{\backslash mu\; \backslash nu\}\; =\; \{\backslash Lambda^\{\backslash mu\}\; \}\_\{\backslash alpha\}\; \{\backslash Lambda^\{\backslash nu\}\; \}\_\{\backslash beta\}\; F^\{\backslash alpha\; \backslash beta\}\; \backslash ,\; ,$$

where {{math|Λ^μ_α}} is the Lorentz transformation tensor.

The {{math|1=α = 1}} component ({{mvar|x}}-component) of the force is

$$\backslash frac\{\backslash mathrm\{d\}\; p^1\}\{\backslash mathrm\{d\}\; \backslash tau\}\; =\; q\; U\_\backslash beta\; F^\{1\; \backslash beta\}\; =\; q\backslash left(U\_0\; F^\{10\}\; +\; U\_1\; F^\{11\}\; +\; U\_2\; F^\{12\}\; +\; U\_3\; F^\{13\}\; \backslash right)\; .$$

Substituting the components of the covariant electromagnetic tensor F yields

$$\backslash frac\{\backslash mathrm\{d\}\; p^1\}\{\backslash mathrm\{d\}\; \backslash tau\}\; =\; q\; \backslash left[U\_0\; \backslash left(\backslash frac\{E\_x\}\{c\}\; \backslash right)\; +\; U\_2\; (-B\_z)\; +\; U\_3\; (B\_y)\; \backslash right]\; .$$

Using the components of covariant four-velocity yields

$$\backslash frac\{\backslash mathrm\{d\}\; p^1\}\{\backslash mathrm\{d\}\; \backslash tau\}$$

= 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] \, .

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:

$$\backslash frac\{\backslash mathrm\{d\}\; \backslash mathbf\{p\}\; \}\{\backslash mathrm\{d\}\; \backslash tau\}\; =\; q\; \backslash gamma\backslash left(\; \backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}\; \backslash right)\; ,$$

and since differentials in coordinate time {{mvar|dt}} and proper time {{mvar|dτ}} are related by the Lorentz factor,

$$dt=\backslash gamma(v)\; \backslash ,\; d\backslash tau,$$

so we arrive at

$$\backslash frac\{\backslash mathrm\{d\}\; \backslash mathbf\{p\}\; \}\{\backslash mathrm\{d\}\; t\}\; =\; q\; \backslash left(\; \backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}\; \backslash right)\; .$$

This is precisely the Lorentz force law, however, it is important to note that {{math|p}} is the relativistic expression,

$$\backslash mathbf\{p\}\; =\; \backslash gamma(v)\; m\_0\; \backslash mathbf\{v\}\; \backslash ,.$$

The electric and magnetic fields are 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 $\backslash mathcal\{F\}$, and an arbitrary time-direction, $\backslash gamma\_0$. This can be settled through spacetime algebra (or the geometric algebra of spacetime), a type of Clifford algebra defined on a pseudo-Euclidean space,{{cite web|last=Hestenes|first=David|author-link=David Hestenes|title=SpaceTime Calculus|url=https://davidhestenes.net/geocalc/html/STC.html}} as

$$\backslash mathbf\{E\}\; =\; \backslash left(\backslash mathcal\{F\}\; \backslash cdot\; \backslash gamma\_0\backslash right)\; \backslash gamma\_0$$

and

$$i\backslash mathbf\{B\}\; =\; \backslash left(\backslash mathcal\{F\}\; \backslash wedge\; \backslash gamma\_0\backslash right)\; \backslash gamma\_0$$

$\backslash mathcal\; F$ 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 $\backslash gamma\_0$ 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|$v\; =\; \backslash dot\; x$,}} where

$$v^2\; =\; 1,$$

(which shows our choice for the metric) and the velocity is

$$\backslash mathbf\{v\}\; =\; cv\; \backslash wedge\; \backslash gamma\_0\; /\; (v\; \backslash cdot\; \backslash gamma\_0).$$

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 = $F\; =\; q\backslash mathcal\{F\}\backslash cdot\; v$

|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.

In the general theory of relativity the equation of motion for a particle with mass $m$ and charge $e$, moving in a space with metric tensor $g\_\{ab\}$ and electromagnetic field $F\_\{ab\}$, is given as

$$m\backslash frac\{du\_c\}\{ds\}\; -\; m\; \backslash frac\{1\}\{2\}\; g\_\{ab,c\}\; u^a\; u^b\; =\; e\; F\_\{cb\}u^b\; ,$$

where $u^a=\; dx^a/ds$ ($dx^a$ is taken along the trajectory), $g\_\{ab,c\}\; =\; \backslash partial\; g\_\{ab\}/\backslash partial\; x^c$, and $ds^2\; =\; g\_\{ab\}\; dx^a\; dx^b$.

The equation can also be written as

$$m\backslash frac\{du\_c\}\{ds\}-m\backslash Gamma\_\{abc\}u^a\; u^b\; =\; eF\_\{cb\}u^b\; ,$$

where $\backslash Gamma\_\{abc\}$ is the Christoffel symbol (of the torsion-free metric connection in general relativity), or as

$$m\backslash frac\{Du\_c\}\{ds\}\; =\; e\; F\_\{cb\}u^b\; ,$$

where $D$ is the covariant differential in general relativity.

The Lorentz force occurs in many devices, including:

• Cyclotrons and other circular path particle accelerators
• Mass spectrometers
• Velocity filters
• Magnetrons
• 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 motors
• Railguns
• Linear motors
• Loudspeakers
• Magnetoplasmadynamic thrusters
• Electrical generators
• Homopolar generators
• Linear alternators

{{cols|colwidth=26em}}

• Hall effect
• Gravitomagnetism
• 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
• Field line
• Coulomb's law
• Electromagnetic buoyancy

{{colend}}

{{notelist|30em}}

{{reflist|30em}}

• {{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 }}

{{Commons|Lorentz force}}

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• [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

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Category:Physical phenomena

Category:Electromagnetism

Category:Maxwell's equations

Category:Hendrik Lorentz