The Lorentz force has two components. Theelectric force acts in the direction of the electric field for positive charges and opposite to it for negative charges, tending to accelerate the particle in a straight line. Themagnetic force is perpendicular to both the particle's velocity and the magnetic field, and it causes the particle to move along a curved trajectory, often circular or helical in form, depending on the directions of the fields.
Together withMaxwell's equations, which describe how electric and magnetic fields are generated by charges and currents, the Lorentz force law forms the foundation ofclassical electrodynamics.[2][3] While the law remains valid inspecial relativity, it breaks down at small scales wherequantum effects become important. In particular, the intrinsicspin of particles gives rise to additional interactions with electromagnetic fields that are not accounted for by the Lorentz force.
Historians suggest that the law is implicit in a paper byJames Clerk Maxwell, published in 1865.[1]Hendrik Lorentz arrived at a complete derivation in 1895,[4] identifying the contribution of the electric force a few years afterOliver Heaviside correctly identified the contribution of the magnetic force.[5]
Lorentz forceF on acharged particle (of chargeq) in motion (instantaneous velocityv). TheE field andB field vary in space and time.
The Lorentz forceF acting on apoint particle withelectric chargeq, moving with velocityv, due to an external electric fieldE and magnetic fieldB, is given by (SI definition of quantities[a]):[2]
Here,× is the vectorcross product, and all quantities in bold are vectors. In component form, the force is written as:
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:in whichr is the position vector of the charged particle,t is time, and theoverdot is a time derivative.
The total electromagnetic force consists of two parts: the electric forceqE, which acts in the direction of the electric field and accelerates the particle linearly, and the magnetic forceq(v ×B), which acts perpendicularly to both the velocity and the magnetic field.[8] 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.[9]
The direction of the magnetic force is often determined using theright-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 ascyclotron motion.[10]
In many practical situations, such as the motion ofelectrons orions in aplasma, the effect of a magnetic field can be approximated as a superposition of two components: a relatively fast circular motion around a point called theguiding 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 orchemical separation.[citation needed]
While the magnetic force affects the direction of a particle's motion, it does nomechanical 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:Here, the magnetic term vanishes because a vector is always perpendicular to its cross product with another vector; thescalar triple product is zero. Thus, only the electric field can transfer energy to or from a particle and change itskinetic energy.[11]
Some textbooks use the Lorentz force law as the fundamental definition of the electric and magnetic fields.[12][13] That is, the fieldsE andB are uniquely defined at each point in space and time by the hypothetical forceF a test particle of chargeq and velocityv would experience there, even if no charge is present. This definition remains valid even for particles approaching thespeed of light (that is,magnitude ofv,|v| ≈c).[14] However, some argue that using the Lorentz force law as the definition of the electric and magnetic fields is not necessarily the most fundamental approach possible.[15][16]
The Lorentz force law also given in terms of continuouscharge distributions, such as those found inconductors orplasmas. For a small element of a moving charge distribution with charge , the infinitesimal force is given by:Dividing both sides by the volume of the charge element gives the force densitywhere is the charge density and is the force per unit volume. Introducing thecurrent density, this can be rewritten as:[17]
The total force is thevolume integral over the charge distribution:
The power density corresponding to the Lorentz force, the rate of energy transfer to the material, is given by:
Inside a material, the total charge and current densities can be separated into free and bound parts. In terms of free charge density, free current density,polarization, andmagnetization, the force density becomes[citation needed]This form accounts for the torque applied to a permanent magnet by the magnetic field. The associated power density is[citation needed]
The above-mentioned formulae use the conventions for the definition of the electric and magnetic field used with theSI, 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 olderCGS-Gaussian units, which are somewhat more common among some theoretical physicists as well as condensed matter experimentalists, one has insteadwherec is thespeed of light. Although this equation looks slightly different, it is equivalent, since one has the following relations:[a]whereε0 is thevacuum permittivity andμ0 thevacuum permeability. In practice, the subscripts "G" and "SI" are omitted, and the used convention (and unit) must be determined from context.
Right-hand rule for the force on a current-carrying wire in a magnetic fieldB
When a wire carrying a steadyelectric current is placed in an external magnetic field, each of the moving charges in the wire experience the Lorentz force. Together, these forces produce a net macroscopic force on the wire. For a straight, stationary wire in a uniform magnetic field, this force is given by:[19]whereI is the current andℓ is a vector whose magnitude is the length of the wire, and whose direction is along the wire, aligned with the direction of the current.
If the wire is not straight or the magnetic field is non-uniform, the total force can be computed by applying the formula to eachinfinitesimal segment of wire, then adding up all these forces byintegration. In this case, the net force on a stationary wire carrying a steady current is[20]
One application of this isAmpère's force law, which describes the attraction or repulsion between two current-carrying wires. Each wire generates a magnetic field, described by theBiot–Savart law, which exerts a Lorentz force on the other wire. If the currents flow in the same direction, the wires attract; if the currents flow in opposite directions, they repel. This interaction provided the basis of the former definition of theampere, as the constant current that produces a force of 2 × 10-7newtons per metre between two straight, parallel wires one metre apart.[21]
Another application is aninduction motor. The stator winding AC current generates a moving magnetic field which induces a current in the rotor. The subsequent Lorentz force acting on the rotor creates a torque, making the motor spin. Hence, though the Lorentz force law does not apply when the magnetic field is generated by the current, it does apply when the current is induced by the movement of magnetic field.
Motional EMF, induced by moving a conductor through a magnetic field.
Transformer EMF, induced by a changing magnetic field.
The Lorentz force acting on electric charges in a conducting loop can produce a current by pushing charges around the circuit. This effect is the fundamental mechanism underlyinginduction motors and generators. It is described in terms ofelectromotive force (emf), a quantity which plays a central role in the theory ofelectromagnetic induction. In a simple circuit with resistance, an emf gives rise to a current according to the Ohm's law.[22]
Both components of the Lorentz force—the electric and the magnetic—can contribute to the emf in a circuit, but through different mechanisms. In both cases, the induced emf is described byFaraday's flux rule, which states that the emf around a closed loop is equal to the negative rate of change of themagnetic flux through the loop:[23]The magnetic flux is defined as thesurface integral of the magnetic fieldB over a surface Σ(t) bounded by the loop:[23]
A conducting rod moving through a uniform magnetic field. The magnetic component of the Lorentz force pushes electrons to one end, resulting in charge separation.
The flux can change either because the loop moves or deforms over time, or because the field itself varies in time. These two possibilities correspond to the two mechanisms described by the flux rule:[23]
Motional emf: The circuit moves through a static but non-uniform magnetic field.
Transformer emf: The circuit remains stationary while the magnetic field changes over time
The sign of the induced emf is given byLenz's law, which states that the induced current produces a magnetic field opposing the change in the original flux.[23]
The flux rule can be derived from theMaxwell–Faraday equation and the Lorentz force law.[22] In some cases, especially in extended systems, the flux rule may be difficult to apply directly or may not provide a complete description, and the full Lorentz force law must be used. (Seeinapplicability of Faraday's law.)[24]
The basic mechanism behind motional emf is illustrated by a conducting rod moving through a magnetic field that is perpendicular to both the rod and its direction of motion. Due to movement in magnetic field, the mobile electrons of the conductor experience the magnetic component (qv ×B) of the Lorentz force that drives them along the length of the rod. This leads to a separation of charge between the two ends of the rod. In the steady state, the electric field from the accumulated charge balances the magnetic force.[25]
The flux rule in three cases: (a) motional emf, with moving circuit and a stationary magnetic field (b) stationary circuit, with the source of the magnetic field moving (c) time-dependent magnetic field strength
If the rod is part of a closed conducting loop moving through a nonuniform magnetic field, the same effect can drive a current around the circuit. For instance, suppose the magnetic field is confined to a limited region of space, and the loop initially lies outside this region. As it moves into the field, the area of the loop that encloses magnetic flux increases, and an emf is induced. From the Lorentz force perspective, this is because the field exerts a magnetic force on charge carriers in the parts of the loop entering the region. Once the entire loop lies in a uniform magnetic field and continues at constant speed, the total enclosed flux remains constant, and the emf vanishes. In this situation, magnetic forces on opposite sides of the loop cancel out.
A complementary case is transformer emf, which occurs when the conducting loop remains stationary but the magnetic flux through it changes due to a time-varying magnetic field. This can happen in two ways: either the source of the magnetic field moves, altering the field distribution through the fixed loop, or the strength of the magnetic field changes over time at a fixed location, as in the case of a powered electromagnet..
In either situation, no magnetic force acts on the charges, and the emf is entirely due to the electric component (qE) of the Lorentz force. According to the Maxwell–Faraday equation, a time-varying magnetic field produces a circulating electric field, which drives current in the loop. This phenomenon underlies the operation ofelectrical machines such assynchronous generators.[26] The electric field induced in this way isnon-conservative, meaning its line integral around a closed loop is not zero.[27][28][29]
From the viewpoint ofspecial relativity, the distinction between motional and transformer emf is frame-dependent. In the laboratory frame, a moving loop in a static field generates emf via magnetic forces. But in a frame moving with the loop, the magnetic field appears time-dependent, and the emf arises from an induced electric field. Einstein'sspecial theory of relativity was partially motivated by the desire to better understand this link between the two effects.[30] In modern terms, electric and magnetic fields are different components of a singleelectromagnetic field tensor, and a transformation betweeninertial frames mixes the two.[31]
Lorentz's theory of electrons. Formulas for the Lorentz force (I, ponderomotive force) and theMaxwell equations for thedivergence of theelectrical field E (II) and themagnetic field B (III),La théorie electromagnétique de Maxwell et son application aux corps mouvants, 1892, p. 451.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, byJohann Tobias Mayer and others in 1760,[32] and electrically charged objects, byHenry Cavendish in 1762,[33] obeyed aninverse-square law. However, in both cases the experimental proof was neither complete nor conclusive. It was not until 1784 whenCharles-Augustin de Coulomb, using atorsion balance, was able to definitively show through experiment that this was true.[34] Soon after the discovery in 1820 byHans 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.[35][36] 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.[37]
The modern concept of electric and magnetic fields first arose in the theories ofMichael Faraday, particularly his idea oflines of force, later to be given full mathematical description byLord Kelvin andJames Clerk Maxwell.[38] 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,[1][39] 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 incathode rays, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as[5][40]Thomson derived the correct basic form of the formula, but, because of some miscalculations and an incomplete description of thedisplacement 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.[5][41][42] Finally, in 1895,[4][43]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 theluminiferous 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 applyingLagrangian mechanics (see below), Lorentz arrived at the correct and complete form of the force law that now bears his name.[44][45]
(Notice that the coordinates and the velocity components should be treated as independent variables, so the del operator acts only on, not on; thus, there is no need of usingFeynman's subscript notation in the equation above.) Using the chain rule, theconvective derivative of is:[46]so that the above expression becomes:
TheLagrangian for a charged particle of massm and chargeq in an electromagnetic field equivalently describes the dynamics of the particle in terms of itsenergy, rather than the force exerted on it. The classical expression is given by:[47]whereA andϕ are the potential fields as above. The quantity can be identified as a generalized, velocity-dependent potential energy and, accordingly, as anon-conservative force.[48] Using the Lagrangian, the equation for the Lorentz force given above can be obtained again.
Derivation of Lorentz force from classical Lagrangian (SI units)
For anA field, a particle moving with velocityv =ṙ haspotential momentum, so its potential energy is. For aϕ field, the particle's potential energy is.
Lagrange's equations are(same fory andz). So calculating the partial derivatives:equating and simplifying:and similarly for they andz directions. Hence the force equation is:
The relativistic Lagrangian is
The action is the relativisticarclength of the path of the particle inspacetime, minus the potential energy contribution, plus an extra contribution whichquantum mechanically is an extraphase a charged particle gets when it is moving along a vector potential.
Derivation of Lorentz force from relativistic Lagrangian (SI units)
The equations of motion derived byextremizing the action (seematrix calculus for the notation):are the same asHamilton's equations of motion:both are equivalent to the noncanonical form:This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.
Substituting the components of the covariant electromagnetic tensorF yields
Using the components of covariantfour-velocity yields
The calculation forα = 2, 3 (force components in they andz directions) yields similar results, so collecting the three equations into one:and since differentials in coordinate timedt and proper timedτ are related by the Lorentz factor,so we arrive at
This is precisely the Lorentz force law, however, it is important to note thatp is the relativistic expression,
The electric and magnetic fields aredependent 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, and an arbitrary time-direction,. This can be settled throughspacetime algebra (or the geometric algebra of spacetime), a type ofClifford algebra defined on apseudo-Euclidean space,[51] asand is a spacetimebivector (an oriented plane segment, just like a vector is anoriented line segment), which has six degrees of freedom corresponding to boosts (rotations in spacetime planes) and rotations (rotations in space-space planes). Thedot product with the vector 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, where(which shows our choice for the metric) and the velocity is
The proper form of the Lorentz force law ('invariant' is an inadequate term because no transformation has been defined) is simply
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 thegeneral theory of relativity the equation of motion for a particle with mass and charge, moving in a space with metric tensor and electromagnetic field, is given as
where ( is taken along the trajectory),, and.
The equation can also be written as
where is theChristoffel symbol (of the torsion-free metric connection in general relativity), or as
Inquantum mechanics, particles are described bywavefunctions whose evolution is governed by theSchrödinger equation. While this formulation does not involveforces explicitly, it extends the framework ofHamiltonian mechanics, by incorporating interactions with electromagnetic fields through potential terms in theHamiltonian. For a non-relativistic particle of massand charge, the Hamiltonian takes the form:This expression is a direct generalization of the classical Hamiltonian that leads to the Lorentz force law. The key difference is that in quantum mechanics, position and momentum are operators that do notcommute. As a result, quantum dynamics incorporate fundamentally different behavior such aswave interference and quantization.[52]
Aharonov–Bohm setup in which the magnetic field is confined to a region that the electrons do not enter. Nevertheless, the interference pattern on the screen is affected by the magnetic flux through the central region.
Unlike in classical physics, where only electric and magnetic fields influence particle motion, quantum mechanics allows the electromagnetic potentials themselves to produce observable effects. This is exemplified by theAharonov–Bohm effect, in which a charged particle passes through a region with zero electric and magnetic fields but encircles a magnetic flux confined to an inaccessible area. Although the classical Lorentz force is zero along the particle's path, the interference pattern observed on a screen shifts depending on the enclosed magnetic flux, revealing the physical significance of the vector potential in quantum mechanics.[53]
Nevertheless, the classical Lorentz force law emerges as an approximation to the quantum dynamics: according to theEhrenfest theorem, the expectation value of the momentum operator evolves according to an equation that resembles the classical Lorentz force law. Even in the Aharonov–Bohm setup, the average motion of a wave packet follows the classical trajectory.[54]
Quantum particles such as electrons also possess intrinsicspin, which introduces additional electromagnetic interactions beyond those described by the classical Lorentz force. In the non-relativistic limit, this is captured by thePauli equation, which includes a spin–magnetic field coupling term:where are thePauli matrices. This term leads to spin-dependent forces absent in the classical theory. A complete relativistic treatment is given by theDirac equation, which incorporates spin and electromagnetic interactions through minimal coupling, and correctly predicts features such as the electron'sgyromagnetic ratio.[55]
In many real-world applications, the Lorentz force is insufficient to accurately describe the collective behavior of charged particles, both in practice and on a fundamental level. Real systems involve many interacting particles that also generate their own fieldsE andB. To account for these collective effects—such as currents, flows, and plasmas—more complex equations are required, such as theBoltzmann equation, theFokker–Planck equation or theNavier–Stokes equations. These models go beyond single-particle dynamics, incorporating particle interactions and self-consistent field generation, and are central to fields likemagnetohydrodynamics,electrohydrodynamics, andplasma physics, as well as to the understanding ofastrophysical andsuperconducting phenomena.
The Lorentz force occurs in many devices, including:
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