Lorentz Force¶
The electromagnetic force on a charge from local electric and magnetic fields, with a velocity-dependent magnetic contribution.
Core Idea¶
The Lorentz force is the electromagnetic force on a charge \(q\) moving with velocity \(\mathbf v\) in local electric and magnetic fields \(\mathbf E\) and \(\mathbf B\), given by \(\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B)\).[1]
The electric contribution \(q\mathbf E\) can act even when the charge is stationary.[1][2] The magnetic contribution \(q\mathbf v\times\mathbf B\) depends on motion and is perpendicular to both velocity and field.[1][2] Its magnitude is \(|q|vB\sin\theta\), with \(\theta\) the angle between \(\mathbf v\) and \(\mathbf B\).[1][2] Only the magnetic contribution is necessarily perpendicular to motion and does no work on the charge.[1][2] Describing the entire law as a sideways, speed-preserving force would erase the electric part.[1][2]
The law applies to an individual charged particle and, by summing forces on mobile charge carriers, to a current-carrying conductor.[2][3][4] A free particle can bend in a magnetic field; a wire segment can experience force; and current-loop forces can create motor torque.[2][3][4] Those are different realizations of one charge–field response law, not separate laws named after the apparatus.[2][3][4]
How would you explain it like I'm…
Electric and Magnetic Pushes
Electric Push Plus Magnetic Bend
Electromagnetic Force Law
Structural Signature¶
Sig role-phrases:
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R1 — Charged carrier. A charge \(q\), or many charges whose forces can be aggregated, is the force bearer.[1][3] If \(q=0\), this law gives no force on that carrier.[1][3] A neutral bulk object may still contain charged constituents, so the relevant claim must specify what is being modeled.[1][3]
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R2 — Local electromagnetic fields. Electric and magnetic fields are evaluated at the carrier's location.[1] One contribution can be zero; neither an electric field nor a magnetic field is mandatory in every nonzero instance.[1] If both are absent, the law contributes no electromagnetic force.[1]
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R3 — Motion relative to the chosen field description. Velocity sets the magnetic term's magnitude and direction.[1][2] With \(\mathbf v\) parallel to \(\mathbf B\), the magnetic term vanishes; a stationary charge can still feel \(q\mathbf E\).[1][2] Removing motion therefore removes only the magnetic contribution, not the whole law.[1][2]
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R4 — Vector response. The formula combines the electric and velocity-cross-magnetic contributions into the force on the carrier.[1][2] Changing charge sign reverses the force direction for fixed fields and velocity.[1][2] A hand rule can help evaluate a direction but cannot replace the physical charge–field interaction.[1][2]
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R5 — Aggregation in matter, when relevant. In a current-carrying conductor, forces on carriers can be summed over a path to yield \(d\mathbf F=I\,d\boldsymbol\ell\times\mathbf B\).[3] This is a derived setting, not a defining requirement: the law already applies to one free charge. \(\mathbf F=I\boldsymbol\ell\times\mathbf B\) is a special simplification for a straight segment in a uniform field.[3]
What It Is Not¶
The law is not a rule that every charge moves sideways.[1][2] A charge at rest in a magnetic field has no magnetic force; a charge moving parallel to that field has none either.[1][2] An electric field can push along the motion, change kinetic energy, or act on a stationary charge.[1][2] The zero-force case still lies within the law's domain of evaluation; it is not a magnetic-deflection event.[1][2]
The Lorentz force law is also not a hand mnemonic.[1][2][3][5] Fleming's left-hand rule concerns a conventional-current conductor-force orientation; the right-hand cross-product convention can be used for a positive charge.[1][2][3][5] Reversing charge sign reverses the magnetic-force direction.[1][2][3][5] The screened “catapult effect” article supplies a wire demonstration and says “horizontally away from the magnetic field.”[6] That phrasing cannot stand as a general force direction: the vector cross product, current and field geometry determine that direction.[1][3]
Nor is the law the whole of classical electromagnetism. Given \(\mathbf E\) and \(\mathbf B\), it specifies their force on charged matter; the equation alone does not specify how those fields arise or evolve.[1] A motor further needs a suitable circuit and mechanical arrangement.[1][4] The force law alone does not prove that any named appliance contains a particular motor or has a particular performance.[1][4]
Scope of Application¶
Individual charged particles: In a uniform magnetic field without an electric field, a particle entering perpendicular to \(\mathbf B\) follows a circular path if other forces are negligible.[2] With both parallel and perpendicular velocity components, the path can be helical; with velocity wholly parallel to \(\mathbf B\), the magnetic term is zero.[2] These outcomes require the stated field and motion conditions.[2]
Conductors and machines: A current element in an external magnetic field can experience force.[3][4] A simple motor uses forces on different loop segments to produce torque and uses additional switching hardware to continue rotation.[3][4] The conductor formula follows from carrier forces but must be integrated over a curved conductor or nonuniform field.[3][4]
Transport measurement: In a Hall strip, mobile carriers are deflected across the strip by a magnetic field.[7] Charge accumulation creates a transverse electric field until its force balances magnetic deflection.[7] The observed voltage is therefore a coupled transport and equilibrium result, not a direct synonym for magnetic force.[7]
The positive cases here use the classical charge-motion law with specified fields.[1][2] That equation alone does not supply material parameters, field-generation equations, or the mechanical constraints of a device.[1][4]
Clarity¶
The formula answers which part of the force depends on velocity.[1][2] For a freely moving charge with a nonzero velocity component perpendicular to a pure magnetic field, the magnetic force changes its direction without doing work; a charge moving parallel to the field has no magnetic deflection.[1][2] An electric field can change a charge’s energy.[1][2] Calling both “the Lorentz force” is correct, but assigning the magnetic no-work property to the combined expression is not.[1][2]
It also separates a force law from a direction aid and from a device outcome.[3][4][5] Knowing just the direction of a field cannot determine both current and force directions in a conductor.[3][4][5] An electric motor's rotation depends on how forces on current-carrying segments combine, not on a drawing of a single field line or a one-input mnemonic.[3][4][5]
Manages Complexity¶
Many electromagnetic devices differ in geometry, material and purpose.[1][3][4] The force calculation begins with a smaller set of local inputs—\(q\), \(\mathbf E\), \(\mathbf v\) and \(\mathbf B\)—and then, where needed, sums the result over carriers or a current path.[1][3][4] That separates a universal local response law from a device-specific assembly.[1][3][4] A wire in a uniform field admits the simple \(I\boldsymbol\ell\times\mathbf B\) expression; a loop must be evaluated segment by segment to distinguish net force from torque.[1][3][4]
This reduction does not make field generation, charge transport or mechanical constraints disappear. Their models supply the inputs and determine how a calculated force changes a whole system. The law is a compositional part of an electromagnetic analysis, not a shortcut that predicts every device from a field arrow.
Abstract Reasoning¶
Given a charge and specified fields, the relation permits a conditional prediction: compute \(q\mathbf E\), compute \(q\mathbf v\times\mathbf B\), add them as vectors, then apply the appropriate motion or force-balance model.[1][2] If \(\mathbf E=0\) and \(\mathbf v\perp\mathbf B\) in a uniform region, the magnetic force continuously turns a free particle without changing its speed.[1][2] If \(\mathbf v\parallel\mathbf B\), that magnetic inference is unavailable.[1][2]
The same conditional reasoning explains a Hall balance: magnetic deflection first separates carriers, and the resulting electric field grows until electric and magnetic transverse contributions cancel at steady state.[7] Zero net transverse force in that final state does not imply that neither component exists.[7]
Knowledge Transfer¶
The law transfers literally among electromagnetic settings because the same physical quantities and vector relation recur.[1][2][3][7] It can guide analysis of a particle beam, a current element, a motor loop and a Hall strip while the surrounding field and material models differ.[1][2][3][7] The transferred question is precise: what are the charge, local fields and velocity, and what geometry or aggregation connects carrier force to the observation?[1][2][3][7]
Outside electromagnetism, a sideways response may be analogous but is not the Lorentz force. The Coriolis term in a rotating frame also depends on velocity and can be expressed as a cross product, yet it concerns motion described in a rotating coordinate frame rather than electric charge in a magnetic field.[8][1] The mathematical resemblance does not transfer the electromagnetic identity.
Examples¶
Canonical: one charged particle in a uniform magnetic field¶
OpenStax treats a charged particle entering a uniform field perpendicular to its velocity.[2] With electric field absent and other forces neglected, \(q\mathbf v\times\mathbf B\) points toward the center of the resulting circular path.[2] Because force stays perpendicular to instantaneous velocity, the magnetic field changes the direction of motion without changing speed.[2] An oblique entry adds motion along \(\mathbf B\) and yields a helix under the same idealizations.[2]
Mapped back: R1 is the particle's charge, R2 is the external magnetic field with \(\mathbf E=0\), R3 is its velocity relative to the field, and R4 is the cross-product force.[2] R5 is absent because there is no conductor to aggregate.[2] The circular result requires perpendicular entry; it is not every magnetic trajectory.[2]
Applied/practice: a current loop in a simple motor¶
OpenStax's simple DC motor places a current loop in a magnetic field.[3][4] Forces on differently directed loop segments create torque.[3][4] The example also needs commutator and brushes to sustain rotation through changing loop orientation; the force law alone does not supply those control elements.[3][4]
Mapped back: R1 is the mobile charge in the wire, R2 is the field acting on the loop, R3 is carrier drift along each segment, R4 is the force on each carrier, and R5 aggregates those forces into segment forces and torque.[3][4] The motor's switching hardware affects continuing motion, not the local definition of the force.[3][4]
Further check: a Hall strip¶
A current-carrying strip in a transverse magnetic field accumulates charge at its edges.[7] The induced electric field grows until the forces balance.[7] This case exposes both terms of the full law in one measurement setting, whereas the first example isolates the magnetic term and the motor example aggregates carrier forces.[1][2][3][4][7]
Mapped back: R1 is the mobile carrier population; R2 includes external \(\mathbf B\) and induced transverse \(\mathbf E\); R3 is drift along the strip; R4 sums the electric and magnetic terms; R5 is relevant to current transport, though the measured transverse voltage is an equilibrium readout rather than an aggregate mechanical thrust.[7]
Structural Tensions¶
T1: transverse steering versus straight-through passage in a crossed-field selector. In a setting with electric and magnetic transverse forces opposed, a selected charge velocity can make the components cancel and allow passage without lateral deflection.[7] A charge at another velocity does not generally satisfy that balance and is deflected.[7] The same field setting cannot both cancel and preserve a nonzero lateral force for the same carrier at the same velocity.[7] The trade-off belongs to a particular selector's aim—transmit a target motion while rejecting others—not to every occurrence of the Lorentz force.[7]
Diagnostic: At the carrier's actual velocity and sign of charge, are the two transverse contributions arranged to cancel or to steer?
There is no universal “more force is better” objective in the law. Motor torque, particle steering and Hall equilibrium evaluate different outcomes. Likewise, the accepted physical-field prerequisite clarifies the DAG without replacing \(q\), \(\mathbf E\), \(\mathbf v\) and \(\mathbf B\) when calculating a force.
Structural–Framed Character¶
Lorentz Force is predominantly structural within its physical domain. Its rule relates charged carriers, fields, motion and vector response independently of a user's purpose. Evaluative weight enters when an engineer wants a selected trajectory, torque or signal; no such success threshold defines the force. Human-practice dependence is low for charged-particle motion but high for designing a motor or Hall measurement. Institutional origin concerns the name and teaching conventions; the charged-particle response does not depend on those conventions.
Vocabulary travel is limited: outside electromagnetism, “charge,” electric field and magnetic field cannot be replaced with generic agents without changing the claim. Import versus recognition distinguishes using the law to analyze a newly built apparatus from recognizing the same interaction in an existing particle path. Either way the carrier and fields remain literal. Its character: a structural electromagnetic response law whose domain-bound quantities prevent it from becoming a Prime abstraction merely because its equation has a reusable vector shape.
Structural Core vs. Domain Accent¶
The structural core is an input–response relation: field state and charged motion determine a force, and local forces may aggregate in a larger object.[1][3] That skeleton can suggest a general relation such as coupling, but the domain accent carries the predictive content: electric charge, \(\mathbf E\), \(\mathbf B\), the velocity cross product, and the sign and geometry rules.[1][3] Remove those physical commitments and the Lorentz force itself has been removed.[1][3]
A particle, motor loop and Hall strip each realize this charge–field force law in a different setting.[1][2][3][4][7] A rotating-frame Coriolis effect can also involve a velocity cross product and perpendicular response, but its carrier and cause differ from the electromagnetic force on a charge.[8][1] A Prime may capture the broader pattern; this article remains domain-specific. The accepted physical-field prerequisite follows from the law’s inputs, rather than from equation shape alone.
Instantiates / Related Primes¶
This entry presupposes Field (physics).
Approved prerequisite: Field (physics) supplies the physically interpreted pointwise electric or magnetic field values that the Lorentz law evaluates at a charged carrier. Remove those fields and the terms \(\mathbf E\) and \(\mathbf B\) lose their physical referents. A field can be described without asserting this force law, and the response law is not itself a kind of field. This supports one strict child-to-parent composition/presupposes edge, pending prospective graph validation.
Coupling is too broad for a strict edge because a prescribed field acting on one charge does not require two dynamically interdependent subsystems. Causality is related only at a more general level. Non-Contact Force was also rejected as a strict parent: the live identity's no-contact-between-objects condition was not proven for every carrier-in-matter case.
Classical Electromagnetism contains Lorentz matter coupling alongside Maxwell field equations; an outgoing part_of edge from the force law would reverse the repository's defined direction. The force law can be applied to given fields without solving their evolution. Right-Hand Rule is optional directional representation rather than force generation.
Relationships to Other Abstractions¶
Current abstraction Lorentz Force Domain-specific
Parents (1) — outgoing typed relations
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Lorentz Force presupposes Field (physics) Domain-specific
The Lorentz force law requires pointwise physical electric or magnetic fields to supply E and B at a charged carrier.Every positive Lorentz-force case evaluates at least one local electric or magnetic field on a charge. Without the physical-field assignment, E and B in q(E+v×B) lack their referents; fields can exist without this force-law assertion. The Lorentz force is a response law, not a species of field, and the field is an external required input rather than an internal part of the law. Zero-valued fields remain fields; both fields absent gives no positive electromagnetic-force event.
Hierarchy path (1) — routes to 1 parentless root
- Lorentz Force → presupposes Field (physics) → is a kind of Physical quantity → is a kind of Measurement
Neighborhood in Abstraction Space¶
Lorentz Force sits in a sparse region of the domain-specific corpus (75th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Physical Quantities, Operators & Formulas (34 abstractions)
Nearest neighbors
- Rooted product of graphs — 0.84
- Change of Rings — 0.83
- Non-Contact Force — 0.83
- Central potential — 0.82
- Computational Lagrangian Particle Tracking — 0.82
Computed from structural-signature embeddings · 2026-10-31
Not to Be Confused With¶
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Coriolis force: a rotating-frame term that also has a velocity cross product, but whose operative quantities are frame rotation and relative velocity rather than electromagnetic field and charge.[8][1] See Coriolis Force.
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Hall effect: charge separation and a transverse voltage in a conductor.[7] Lorentz force helps explain it but is not itself the voltage or material-specific Hall response.[7]
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Right-hand rule: a convention for determining a vector or rotation direction, not the interaction that generates force.[1][2] See Right-Hand Rule.
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The “catapult effect” teaching picture: a particular moving-wire illustration whose broad horizontal/away imagery does not define the vector law.[6][9][1] It is a source-member context, not an alias for every electric or magnetic Lorentz-force case.
References¶
[1] Richard P. Feynman, Robert B. Leighton and Matthew Sands, The Feynman Lectures on Physics II (1964), ch.13 (Magnetostatics), §13-1, equation 13.1 and preceding text. Caltech online edition. https://www.feynmanlectures.caltech.edu/II_13.html. First-party full lecture text for the combined Lorentz law and magnetic geometry. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y ↩z ↩27 ↩28 ↩29 ↩30 ↩31 ↩32 ↩33 ↩34 ↩35 ↩36 ↩37 ↩38 ↩39 ↩40 ↩41 ↩42 ↩43 ↩44 ↩45 ↩46 ↩47 ↩48 ↩49 ↩50 ↩51 ↩52 ↩53 ↩54 ↩55
[2] OpenStax, University Physics Volume 2, §11.3, (Motion of a Charged Particle in a Magnetic Field), Figure 11.7 and equations 11.4–11.8. https://openstax.org/books/university-physics-volume-2/pages/11-3-motion-of-a-charged-particle-in-a-magnetic-field. Publisher full text; ideal uniform-field trajectories. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y ↩z ↩27 ↩28 ↩29 ↩30 ↩31 ↩32 ↩33 ↩34 ↩35 ↩36 ↩37 ↩38 ↩39 ↩40 ↩41 ↩42 ↩43 ↩44 ↩45
[3] OpenStax, University Physics Volume 2, §11.4, (Magnetic Force on a Current-Carrying Conductor), equations 11.10–11.13 and Example 11.4. https://openstax.org/books/university-physics-volume-2/pages/11-4-magnetic-force-on-a-current-carrying-conductor. Publisher full text; carrier-force aggregation and geometry limits. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y ↩z ↩27 ↩28 ↩29 ↩30 ↩31 ↩32 ↩33 ↩34 ↩35
[4] OpenStax, University Physics Volume 2, §11.5, (Force and Torque on a Current Loop), opening motor example and Figure 11.15. https://openstax.org/books/university-physics-volume-2/pages/11-5-force-and-torque-on-a-current-loop. Publisher full text; motor setting and required commutation. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w
[5] Odisha Directorate of Technical Education, Engineering Physics coursebook, PDF p.121 (Fleming's Left Hand Rule). https://sctevtservices.nic.in/docs/website/pdf/140201.pdf. Government educational text; conductor-force direction mnemonic only. registry ↩a ↩b ↩c ↩d ↩e ↩f
[6] “Catapult effect,” Wikipedia revision 1292611258, the exact screened provenance page, https://en.wikipedia.org/w/index.php?oldid=1292611258. Used only to document what the screened source says, not as independent authority for the Lorentz force law. registry ↩a ↩b
[7] OpenStax, University Physics Volume 2, §11.6, (The Hall Effect), Figure 11.17, equations 11.24–11.25, and Example 11.8. https://openstax.org/books/university-physics-volume-2/pages/11-6-the-hall-effect. Publisher full text; transverse force balance and velocity-selector context. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s
[8] MIT OpenCourseWare, 8.01SC Classical Mechanics, ch.31, “Non-Inertial Linear and Rotating Reference Frames,” especially the rotating-frame summary equation (1.8) and §31.5.3. https://live.ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/mit8_01scs22_chapter31.pdf. University course text; rotating-frame Coriolis cross product and its distinction from physical electromagnetic force. registry ↩a ↩b ↩c
[9] M. W. McCall, P. Kinsler and V. Tymms (2020), “Catapult Description of Magnetic Fields and Forces,” The Physics Teacher 58(6), 416–417, DOI: 10.1119/10.0001840. Authors' institution abstract questions the physical accuracy and utility of the catapult-field picture. The full accepted manuscript was not independently inspected. registry ↩