# Lorentz force The **Lorentz force** is the force that an [[Electric_field|electric field]] and a [[Magnetic_field|magnetic field]] together exert on a body carrying [[Electric_charge|electric charge]]. For a point charge q moving with [[Velocity|velocity]] v the law is F = q·(E + v × B). The electric term pushes along E whatever the charge is doing; the magnetic term acts at right angles to both the velocity and the field, so it turns a trajectory without ever doing [[Work_(physics)|work]] on it.[^up2-ch11] In the microsim below the reader sets the field B, the velocity across the field v_perp, the velocity along it v_par and the sign of the charge, and watches the consequence: a circle of Larmor radius r = m·v_perp/(|q|·B) traced at the cyclotron frequency omega_c = |q|·B/m, opening into a helix as v_par grows, and drifting sideways at the speed E/B as soon as an electric field is added. On the Physics flagship this article serves Part II — Core theories at the section *Magnetism: the Lorentz force* (row P29), where the field defined by the [[Biot–Savart_law|Biot–Savart law]] and [[Ampère's_circuital_law|Ampère's circuital law]] stops being a bookkeeping device and starts moving matter. The same algebra runs the [[Mass_spectrometry|mass spectrometer]], the [[Cyclotron|cyclotron]], the [[Electric_motor|electric motor]] and the magnetic bottle of a [[Tokamak|tokamak]], and it bridges [[Magnetostatics|magnetostatics]] and [[Special_relativity|special relativity]], because whether a push is called electric or magnetic depends on who is watching. ## Definition and properties The law is a definition as much as a result: E and B are the two fields needed to account for every force on a charge that depends on position and velocity alone, and the Lorentz force is how they are measured.[^up2-ch11] It is stated below for a single particle, which is what the microsim animates, then for a continuous distribution of charge, and finally in the Gaussian units of the older literature. ### Point particle For a particle of charge q and mass m, F = q·(E + v × B), and with [[Newton's_laws_of_motion|Newton's second law]] this fixes the trajectory. Since v × B is perpendicular to v, the magnetic term changes direction but never speed: dK/dt = F·v = q·E·v, so [[Kinetic_energy|kinetic energy]] is the electric field's business alone.[^up2-ch11] Take E = 0 and a uniform B, and split the velocity into v_perp across the field and v_par along it. The parallel part feels no force and coasts; the perpendicular part is bent by a force of constant magnitude |q|·v_perp·B always pointing at a fixed centre, which is uniform circular motion. Equating |q|·v_perp·B to m·v_perp²/r gives the Larmor radius r = m·v_perp/(|q|·B), completed in a time independent of speed, at the cyclotron [[Oscillation|angular frequency]] omega_c = |q|·B/m.[^up2-ch11] The two motions superpose into a helix of pitch 2·pi·v_par/omega_c. A uniform E perpendicular to B adds a steady sideways drift at speed E/B, the same for every charge and mass, because the magnetic force grows with speed until it cancels the electric one. These are the four knobs of the microsim, whose HUD carries `F = q*(E + v x B)`, `r = m*v_perp/(abs(q)*B)` and `omega_c = abs(q)*B/m`. Three presets fix the scales. An [[Electron|electron]] in the Earth's field, B ≈ 50 μT, circles at f_c = omega_c/2·pi = 1.40 MHz on a radius of 11.4 cm at v_perp = 1×10⁶ m/s (derived, from the CODATA electron mass and elementary charge).[^codata] A [[Proton|proton]] in a 1 T laboratory magnet circles at 15.2 MHz with a radius of 1.04 mm at v_perp = 1×10⁵ m/s; an electron in the same field runs at 28.0 GHz, since omega_c scales as 1/m (derived).[^codata] With E = 1 kV/m across a 0.1 T field the drift is 1×10⁴ m/s (derived). The integrator is a Boris step, which advances rotation and electric kick separately so a purely magnetic orbit keeps its speed to machine precision instead of spiralling in as a naive [[Verlet_integration|Verlet]] step would.[^boris1970] That is ILLUSTRATIVE of nothing physical: it is a numerical device to keep the drawn circle closing. ### Continuous charge distribution For matter rather than a single particle the law is written per unit volume. With charge density rho and current density J the force density is f = rho·E + J × B, and the total force is its integral over the body.[^up2-ch11] This is the form used for [[Magnetohydrodynamics|magnetohydrodynamics]] and [[Plasma_(physics)|plasma]] modelling, where rho and J are themselves generated by the motion the force produces. It also connects the mechanical picture to the field picture: the volume force can be rewritten as the divergence of the Maxwell stress tensor plus the rate of change of field [[Momentum|momentum]], which is how [[Maxwell's_equations|Maxwell's equations]] come to carry momentum of their own. ### Formulation in the Gaussian system In Gaussian units the magnetic term carries an explicit factor of the [[Speed_of_light|speed of light]]: F = q·(E + (v/c) × B). E and B then share dimensions, and the symmetry of the field tensor is visible in the formula rather than hidden in the constants. The physics is identical; only the bookkeeping changes, and SI is used throughout this article and in the microsim.[^up2-ch11] ## Force on a current-carrying wire A wire carrying current I in a field B feels the sum of the magnetic forces on its moving carriers. For a straight segment of length L the result is F = I·L × B, with L pointing along the conventional current; for a curved wire the segment form dF = I·dL × B is integrated along the path.[^up2-ch11] A closed loop in a uniform field feels no net force but a [[Torque|torque]] tau = m × B, where m = N·I·A is the loop's magnetic dipole moment; that torque is the principle of the moving-coil [[Electric_motor|motor]] and of the galvanometer.[^up2-ch11] Two parallel wires attract or repel, each sitting in the field the other makes by the [[Biot–Savart_law|Biot–Savart law]], and until 2019 that force was the operational definition of the ampere.[^up2-ch12] The wire force is not a new law: it is the point-particle law summed over carriers, whose drift speed in copper is millimetres per second even at large current.[^up2-ch11] ## Electromagnetic induction Induction is where the Lorentz force and [[Faraday's_law_of_induction|Faraday's law]] meet, and where the split between "electric" and "magnetic" first looks arbitrary. The same meter reading arises two ways — move the conductor through a steady field, or hold it still and change the field — and the standard account of the two uses different terms of the same force law. The subsections below take them in turn, then show they are one case in two frames. ### Motional emf Slide a conducting rod of length L along rails at speed v through a field B perpendicular to both. Each carrier feels q·v × B along the rod, driving charge to one end until the electrostatic field it builds balances the magnetic push. The work per unit charge around the circuit is the motional emf, emf = B·L·v, supplied by the magnetic part of the Lorentz force — even though the magnetic force on a free charge can do no work, the rod's constraint forces transfer the energy from whatever pushes it.[^up2-ch13] The same emf follows from Faraday's law applied to the circuit's changing area, so the two accounts agree. ### Transformer emf Now hold the circuit still and change B in time. No charge is moving, so the magnetic term is zero, and yet a current flows. The driver is the induced electric field of Faraday's law, curl E = −∂B/∂t, which is not conservative: its line integral around a closed loop is the rate of change of [[Magnetic_flux|magnetic flux]] through it.[^up2-ch13] This second case needs the electric term of the Lorentz force rather than the magnetic one. ### Relativity The two cases are one case seen from two frames, the observation [[Albert_Einstein|Einstein]] put in the opening paragraph of his 1905 paper on the electrodynamics of moving bodies: the current in a magnet-and-conductor experiment depends only on their relative motion, yet the textbook account of the day gave two unrelated explanations depending on which was held still.[^einstein1905] Under a [[Lorentz_transformation|Lorentz transformation]] the fields mix, so a pure magnetic field in one frame is electric plus magnetic in another. The force is invariant as a physical event — the meter deflects — while its division into electric and magnetic parts is frame-dependent. [[Special_relativity|Special relativity]] was in this sense reverse-engineered from electromagnetism rather than imposed on it. ## History The magnetic deflection of a current was found by Hans Christian Ørsted in 1820 and quantified within months by André-Marie Ampère; [[Michael_Faraday|Michael Faraday]]'s work of the 1830s supplied induction and the field picture.[^up2-ch12] [[James_Clerk_Maxwell|James Clerk Maxwell]]'s 1865 dynamical theory of the electromagnetic field contains a force on a moving conductor of the v × B form among its general equations.[^maxwell1865] In 1881 J. J. Thomson computed the force on a moving electrified body and obtained the right structure with a wrong factor of one half; Oliver Heaviside corrected it in 1889.[^thomson1881][^heaviside1889] Hendrik Lorentz set the expression in its modern role in his 1895 *Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern*, where the force on a charge bridges the ether field equations and the mechanics of ponderable matter, and where the local-time coordinate that became the Lorentz transformation first appears.[^lorentz1895] Lorentz shared the 1902 Nobel Prize in Physics with Pieter Zeeman for work on the influence of magnetism on radiation, which this force explains.[^nobel1902] ## Lorentz force in terms of potentials The fields can be written from a scalar potential phi and a vector potential A as E = −grad(phi) − ∂A/∂t and B = curl A, and substituting gives the Lorentz force entirely in terms of potentials. The potentials are not unique: adding grad(f) to A while subtracting ∂f/∂t from phi leaves E and B, and hence every force, unchanged. That freedom is the gauge symmetry of electromagnetism. Classically the potentials look like conveniences; the quantum theory below is where they stop being optional. ## Lorentz force and analytical mechanics The Lorentz force is not derivable from an ordinary potential energy, because it depends on velocity, but it does come from a velocity-dependent one. With the [[Lagrangian_mechanics|Lagrangian]] L = (1/2)·m·v² − q·phi + q·v·A, the Euler–Lagrange equations return F = q·(E + v × B) exactly.[^up2-ch11] The canonical momentum conjugate to position is then p = m·v + q·A, not m·v, and the [[Hamiltonian_mechanics|Hamiltonian]] is H = (p − q·A)²/(2·m) + q·phi. This replacement of p by p − q·A is called minimal coupling, and it is the single rule by which a magnetic field is inserted into any mechanics, classical or quantum. It also makes the conserved quantities transparent: where the field has translational symmetry along an axis, the conserved quantity is the corresponding component of p, not of m·v. ## Relativistic form of the Lorentz force At speeds approaching c the force law is unchanged in form provided momentum is the relativistic one, p = gamma·m·v; what changes is that the neat split into electric and magnetic parts becomes frame-dependent. This matters in practice as well as in principle: it is why a cyclotron loses synchronism at high energy, and why a beam of relativistic electrons is described by a field that is almost purely transverse. The three formulations below are the same law in three notations. ### Covariant form of the Lorentz force Collecting E and B into the antisymmetric field tensor F^(mu·nu) and using the four-velocity u_nu, the law becomes dp^mu/dtau = q·F^(mu·nu)·u_nu — one equation replacing the three-vector pair, manifestly the same in every inertial frame.[^einstein1905] Its time component is the rate at which the field does work, so energy and force enter as one object, matching the [[Four-momentum|four-momentum]] it changes. Field strengths transform among themselves under boosts, the formal statement of the magnet-and-conductor argument. ### Lorentz force in spacetime algebra (STA) In the geometric-algebra formulation E and B combine into a single multivector F, and the force law compresses to a product of F with the four-velocity. The content is identical to the tensor form; the advantage claimed for it is that rotations, boosts and the field itself live in one algebra, so a boost mixes E and B in a single operation rather than a table of components. ### Lorentz force in general relativity In curved spacetime the ordinary derivative becomes a covariant one, so an uncharged particle follows a geodesic and a charged one is pushed off it by the same q·F^(mu·nu)·u_nu term. [[General_relativity|General relativity]] changes what "straight" means, not what the field does to a charge, and in laboratory work the correction is negligible against the magnetic terms. ## Quantum mechanics Minimal coupling carries the Lorentz force into the quantum theory: the momentum operator becomes −i·hbar·grad − q·A in the [[Schrödinger_equation|Schrödinger equation]], and the [[Hamiltonian_mechanics|Hamiltonian]] (1/(2·m))·(−i·hbar·grad − q·A)² + q·phi. A spin-½ particle acquires in addition the term −mu·B coupling its magnetic moment to the field, which is what the Pauli equation adds and what the [[Pauli_exclusion_principle|exclusion principle]] then organises into atomic structure.[^up2-ch11] Two consequences have no classical counterpart. A charge confined to a plane in a uniform B has its energies collapsed onto equally spaced levels — the same ladder the [[Quantum_harmonic_oscillator|quantum harmonic oscillator]] gives, with spacing hbar·omega_c. And the potentials acquire observable status: an electron beam split around a region of confined flux picks up a measurable phase difference even where E and B vanish along its path, so that A is physical in a way the classical force law cannot show. Devices from [[Nuclear_magnetic_resonance|nuclear magnetic resonance]] to [[Magnetic_resonance_imaging|magnetic resonance imaging]] rest on the spin term rather than the orbital one. ## Applications Sorting by mass is the cleanest use. In a [[Mass_spectrometry|mass spectrometer]] a velocity selector with crossed fields passes only ions for which the electric and magnetic forces cancel, v = E/B; those ions enter a uniform analyser field where r = m·v/(q·B), so radius is proportional to m/q.[^up2-ch11] With E = 1.0×10⁵ V/m and B = 0.20 T the selected speed is 5.0×10⁵ m/s, and in a 0.20 T analyser a singly charged ⁴He⁺ ion lands at r = 10.4 cm against 7.8 cm for ³He⁺ — a 2.6 cm separation set purely by the mass ratio 1.327 (derived).[^codata] That is the principle of the [[Helium_mass_spectrometer|helium mass spectrometer]] used for [[Leak_detection|leak detection]], and of [[Isotope_separation|isotope separation]] by electromagnetic means. The [[Cyclotron|cyclotron]] exploits the fact that omega_c does not depend on speed: a fixed radio-frequency voltage across a gap accelerates the particle on every half-turn while the field returns it to the gap on schedule, until relativistic mass increase breaks the synchronism.[^up2-ch11] In [[Magnetic_confinement_fusion|magnetic confinement fusion]] the Larmor radius is the confinement: particles are tied to field lines within a few millimetres, and a toroidal geometry closes those lines so a hot [[Plasma_(physics)|plasma]] never reaches the wall — the design brief of the [[Tokamak|tokamak]]. Motors and [[Electric_generator|generators]] use the wire form in both directions, the Hall probe reads B from the transverse charge separation in a current-carrying slab, and [[Superconducting_magnet|superconducting magnets]] exist to make B large. ## See also - [[Magnetic_field]] - [[Cyclotron]] - [[Biot–Savart_law]] - [[Ampère's_circuital_law]] - [[Magnetostatics]] - [[Maxwell's_equations]] - [[Faraday's_law_of_induction]] - [[Mass_spectrometry]] ## Notes Explanatory remarks are below; the sources themselves are collected under References. ### Remarks - Sign conventions: F = q·(E + v × B) with q carrying its own sign, so a negative charge circles the opposite way at the same rate. The microsim exposes the sign as a control for exactly this reason. - "Larmor radius", "gyroradius" and "cyclotron radius" are the same quantity; plasma physics prefers the first two and accelerator physics the third. - The statement that the magnetic force does no work applies to a free point charge. In a wire or a motor, constraint forces move the energy, and the magnetic term acts as the broker. - Page pins: the two University Physics Volume 2 chapters used here are cited by chapter and by their full page ranges from the Portal Books chapter index. The sub-manual for this cluster reads Volume 2 only at Chapters 2 and 5, so the individual pages inside Chapters 11–13 remain to pin against the PDF. ### Citations Footnote definitions are under References, immediately below. ## References [^up2-ch11]: Sanny, Jeff; Ling, Samuel; et al. (2016). *University Physics Volume 2*. OpenStax. Chapter 11, "Magnetic Forces and Fields", pp. 475–512 (the Lorentz force on a point charge, circular and helical motion, crossed fields and the velocity selector, the force on a current-carrying wire, the torque on a current loop, the mass spectrometer and the cyclotron; page to pin). Portal Book 078. Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^up2-ch12]: Sanny, Ling et al. (2016). *University Physics Volume 2*. Chapter 12, "Sources of Magnetic Fields", pp. 513–556 (the Biot–Savart law, the field of a straight wire, the force between parallel currents, Ampère's law; page to pin). Portal Book 078. [^up2-ch13]: Sanny, Ling et al. (2016). *University Physics Volume 2*. Chapter 13, "Electromagnetic Induction", pp. 557–602 (Faraday's law, motional emf, induced electric fields; page to pin). Portal Book 078. [^codata]: National Institute of Standards and Technology. "CODATA Internationally recommended values of the Fundamental Physical Constants" (elementary charge, electron and proton masses, atomic mass constant). NIST Reference on Constants, Units and Uncertainty. https://physics.nist.gov/cuu/Constants/ [^maxwell1865]: Maxwell, James Clerk (1865). "A dynamical theory of the electromagnetic field." *Philosophical Transactions of the Royal Society of London*, 155: 459–512. [^thomson1881]: Thomson, J. J. (1881). "On the electric and magnetic effects produced by the motion of electrified bodies." *Philosophical Magazine*, Series 5, volume 11. [^heaviside1889]: Heaviside, Oliver (1889). "On the electromagnetic effects due to the motion of electrification through a dielectric." *Philosophical Magazine*, Series 5, volume 27. [^lorentz1895]: Lorentz, Hendrik Antoon (1895). *Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern*. Leiden: E. J. Brill. [^nobel1902]: The Nobel Prize in Physics 1902 (Hendrik A. Lorentz and Pieter Zeeman). NobelPrize.org. https://www.nobelprize.org/prizes/physics/1902/summary/ [^einstein1905]: Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper." *Annalen der Physik*, 322 (10): 891–921. [^boris1970]: Boris, Jay P. (1970). "Relativistic plasma simulation — optimization of a hybrid code." *Proceedings of the Fourth Conference on Numerical Simulation of Plasmas*. Naval Research Laboratory, Washington, D.C. ## External links - [University Physics Volume 2](https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2), OpenStax, Open Textbook Library record — Portal Book 078, Chapters 11–13 - [Fundamental Physical Constants](https://physics.nist.gov/cuu/Constants/), NIST - The pair's *External links* section is the place to look for further animations and course notes; Wikitube lists only sources it has checked. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Lorentz_force.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Lorentz force* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Lorentz_force.html" data-title="Lorentz force"></div> *Built from `MICROSIM_GUIDE/specs/sims/Lorentz_force.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Lorentz_force) : [Wikitube](https://en.wikitube.io/wiki/Lorentz_force) · pinned revision [1356781390](https://en.wikipedia.org/w/index.php?oldid=1356781390) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P29 · sim pending (matter/Lorentz_force).*