Force acting on charged particles in electric and magnetic fields
Lorentz force acting on fast-moving chargedparticles in abubble chamber. Positive and negative charge trajectories curve in opposite directions.
Inphysics, specifically inelectromagnetism, theLorentz force law is the combination of electric and magneticforce on apoint charge due toelectromagnetic fields. TheLorentz force, on the other hand, is aphysical effect that occurs in the vicinity of electrically neutral, current-carrying conductors causing moving electrical charges to experience amagnetic force.
TheLorentz force law states that a particle of chargeq moving with a velocityv in anelectric fieldE and amagnetic fieldB experiences a force (inSI units[1][2]) ofIt says that the electromagnetic force on a chargeq is a combination of (1) a force in the direction of the electric fieldE (proportional to the magnitude of the field and the quantity of charge), and (2) a force at right angles to both the magnetic fieldB and the velocityv of the charge (proportional to the magnitude of the field, the charge, and the velocity).
Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes calledLaplace force), theelectromotive force in a wire loop moving through a magnetic field (an aspect ofFaraday's law of induction), and the force on a moving charged particle.[3]
Historians suggest that the law is implicit in a paper byJames Clerk Maxwell, published in 1865.[4]Hendrik Lorentz arrived at a complete derivation in 1895,[5] identifying the contribution of the electric force a few years afterOliver Heaviside correctly identified the contribution of the magnetic force.[6]
Trajectory of a particle with a positive or negative chargeq under the influence of a magnetic fieldB, which is directed perpendicularly out of the screen
Beam of electrons moving in a circle, due to the presence of a magnetic field. Purple light revealing the electron's path in thisTeltron tube is created by the electrons colliding with gas molecules.
In many textbook treatments of classical electromagnetism, the Lorentz force law is used as thedefinition of the electric and magnetic fieldsE andB.[7][8][9] To be specific, the Lorentz force is understood to be the following empirical statement:
The electromagnetic forceF on atest charge at a given point and time is a certain function of its chargeq and velocityv, which can be parameterized by exactly two vectorsE andB, in the functional form:
This is valid, even for particles approaching the speed of light (that is,magnitude ofv,|v| ≈c).[10] So the twovector fieldsE andB are thereby defined throughout space and time, and these are called the "electric field" and "magnetic field". The fields are defined everywhere in space and time with respect to what force a test charge would receive regardless of whether a charge is present to experience the force.
Coulomb's law is only valid for point charges at rest. In fact, the electromagnetic force between two point charges depends not only on the distance but also on therelative velocity. For small relative velocities and very small accelerations, instead of the Coulomb force, theWeber force can be applied. The sum of the Weber forces of all charge carriers in a closed DC loop on a single test charge produces – regardless of the shape of the current loop – the Lorentz force.
The interpretation of magnetism by means of a modified Coulomb law was first proposed byCarl Friedrich Gauss. In 1835, Gauss assumed that each segment of a DC loop contains an equal number of negative and positive point charges that move at different speeds.[11] If Coulomb's law were completely correct, no force should act between any two short segments of such current loops. However, around 1825,André-Marie Ampère demonstrated experimentally that this is not the case. Ampère also formulated aforce law. Based on this law, Gauss concluded that the electromagnetic force between two point charges depends not only on the distance but also on the relative velocity.
Lorentz forceF on acharged particle (of chargeq) in motion (instantaneous velocityv). TheE field andB field vary in space and time.
The forceF acting on a particle ofelectric chargeq with instantaneous velocityv, due to an external electric fieldE and magnetic fieldB, is given by (SI definition of quantities[1]):[12]
where× is the vector cross product (all boldface quantities are vectors). In terms of Cartesian components, we have:
In general, the electric and magnetic fields are functions of the position and time. Therefore, explicitly, the Lorentz force can be written as:in whichr is the position vector of the charged particle,t is time, and the overdot is a time derivative.
A positively charged particle will be accelerated in thesame linear orientation as theE field, but will curve perpendicularly to both the instantaneous velocity vectorv and theB field according to theright-hand rule (in detail, if the fingers of the right hand are extended to point in the direction ofv and are then curled to point in the direction ofB, then the extended thumb will point in the direction ofF).
The termqE is called theelectric force, while the termq(v ×B) is called themagnetic force.[13] According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force,[14] with thetotal electromagnetic force (including the electric force) given some other (nonstandard) name. This article willnot follow this nomenclature: in what follows, the termLorentz force will refer to the expression for the total force.
The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wire in a magnetic field. In that context, it is also called theLaplace force.
The Lorentz force is a force exerted by the electromagnetic field on the charged particle, that is, it is the rate at which linear momentum is transferred from the electromagnetic field to the particle. Associated with it is the power which is the rate at which energy is transferred from the electromagnetic field to the particle. That power isNotice that the magnetic field does not contribute to the power because the magnetic force is always perpendicular to the velocity of the particle.
For a continuouscharge distribution in motion, the Lorentz force equation becomes:where is the force on a small piece of the charge distribution with charge. If both sides of this equation are divided by the volume of this small piece of the charge distribution, the result is:where is theforce density (force per unit volume) and is thecharge density (charge per unit volume). Next, thecurrent density corresponding to the motion of the charge continuum isso the continuous analogue to the equation is[15]
The total force is thevolume integral over the charge distribution:
In terms of and, another way to write the Lorentz force (per unit volume) is[15]where is thespeed of light and ∇· (nabla followed by a middle dot) denotes the divergence of atensor field. Rather than the amount of charge and its velocity in electric and magnetic fields, this equation relates theenergy flux (flow ofenergy per unit time per unit distance) in the fields to the force exerted on a charge distribution. SeeCovariant formulation of classical electromagnetism for more details.
The density of power associated with the Lorentz force in a material medium is
If we separate the total charge and total current into their free and bound parts, we get that the density of the Lorentz force is
where: is the density of free charge; is thepolarization density; is the density of free current; and is themagnetization density. In this way, the Lorentz force can explain the torque applied to a permanent magnet by the magnetic field. The density of the associated power is
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:[1]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.
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,[16] and electrically charged objects, byHenry Cavendish in 1762,[17] 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.[18] 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.[19][20] 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.[21]
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.[22] 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,[4] 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[6][23]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.[6][24][25] Finally, in 1895,[5][26]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.[27][28]
Trajectories of particles due to the Lorentz force
Charged particle drifts in a homogeneous magnetic field. (A) No disturbing force. (B) With an electric field,E. (C) With an independent force,F (e.g. gravity). (D) In an inhomogeneous magnetic field,gradH.
In many cases of practical interest, the motion in amagnetic field of anelectrically charged particle (such as anelectron orion in aplasma) can be treated as thesuperposition of a relatively fast circular motion around a point called theguiding center and a relatively slowdrift of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.
While the modern Maxwell's equations describe how electrically charged particles and currents or moving charged particles give rise to electric and magnetic fields, the Lorentz force law completes that picture by describing the force acting on a moving point chargeq in the presence of electromagnetic fields.[12][29] The Lorentz force law describes the effect ofE andB upon a point charge, but such electromagnetic forces are not the entire picture. Charged particles are possibly coupled to other forces, notably gravity and nuclear forces. Thus, Maxwell's equations do not stand separate from other physical laws, but are coupled to them via the charge and current densities. The response of a point charge to the Lorentz law is one aspect; the generation ofE andB by currents and charges is another.
Right-hand rule for a current-carrying wire in a magnetic fieldB
When a wire carrying an electric current is placed in a 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 theLaplace 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:[30]whereℓ is a vector whose magnitude is the length of the wire, and whose direction is along the wire, aligned with the direction of theconventional currentI.
If the wire is not straight, the force on it can be computed by applying this formula to eachinfinitesimal segment of wire, then adding up all these forces byintegration. This results in the same formal expression, butℓ 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 nettorque.
If, in addition, the magnetic field is inhomogeneous, the net force on a stationary rigid wire carrying a steady currentI is given by integration along the wire,
One application of this isAmpère'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 magnetic field.
The magnetic force (qv ×B) component of the Lorentz force is responsible formotionalelectromotive force (ormotional EMF), the phenomenon underlying many electrical generators. When a conductor is moved through a magnetic field, the magnetic field exerts opposite forces on electrons and nuclei in the wire, and this creates the EMF. The term "motional EMF" is applied to this phenomenon, since the EMF is due to themotion of the wire.
In other electrical generators, the magnets move, while the conductors do not. In this case, the EMF is due to the electric force (qE) term in the Lorentz Force equation. The electric field in question is created by the changing magnetic field, resulting in aninduced EMF, as described by theMaxwell–Faraday equation (one of the four modernMaxwell's equations).[31]
Both of these EMFs, despite their apparently distinct origins, are described by the same equation, namely, the EMF is the rate of change ofmagnetic flux through the wire. (This is Faraday's law of induction, seebelow.) Einstein'sspecial theory of relativity was partially motivated by the desire to better understand this link between the two effects.[31] In fact, the electric and magnetic fields are different facets of the same electromagnetic field, and in moving from one inertial frame to another, thesolenoidal vector field portion of theE-field can change in whole or in part to aB-field orvice versa.[32]
Given a loop of wire in amagnetic field, Faraday's law of induction states the inducedelectromotive force (EMF) in the wire is:whereis themagnetic flux through the loop,B is the magnetic field,Σ(t) is a surface bounded by the closed contour∂Σ(t), at timet,dA is an infinitesimalvector area element ofΣ(t) (magnitude is the area of an infinitesimal patch of surface, direction isorthogonal to that surface patch).
Thesign of the EMF is determined byLenz's law. Note that this is valid for not only astationary wire – but also for amoving wire.
LetΣ(t) be the moving wire, moving together without rotation and with constant velocityv andΣ(t) be the internal surface of the wire. The EMF around the closed path∂Σ(t) is given by:[33]where is the electric field anddℓ is aninfinitesimal vector element of the contour∂Σ(t).
The above result can be compared with the version of Faraday's law of induction that appears in the modern Maxwell's equations, called here theMaxwell–Faraday equation:
The Maxwell–Faraday equation also can be written in anintegral form using theKelvin–Stokes theorem.[34]
So we have, the Maxwell–Faraday equation:and the Faraday law,
The two are equivalent if the wire is not moving. Using theLeibniz integral rule and thatdivB = 0, results in,and using the Maxwell Faraday equation,since this is valid for any wire position it implies that
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. Seeinapplicability of Faraday's law.
If the magnetic field is fixed in time and the conducting loop moves through the field, the magnetic fluxΦB linking the loop can change in several ways. For example, if theB-field varies with position, and the loop moves to a location with different B-field,ΦB will change. Alternatively, if the loop changes orientation with respect to the B-field, theB ⋅ dA differential element will change because of the different angle betweenB anddA, also changingΦB. As a third example, if a portion of the circuit is swept through a uniform, time-independentB-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∂Σ(t) time-dependent). In all three cases, Faraday's law of induction then predicts the EMF generated by the change inΦB.
Note that the Maxwell Faraday's equation implies that the Electric FieldE is non conservative when the Magnetic FieldB varies in time, and is not expressible as the gradient of ascalar field, and not subject to thegradient theorem since itscurl is not zero.[33][35]
(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, thetotal derivative of is:so that the above expression becomes:
Withv =ẋ, we can put the equation into the convenient Euler–Lagrange form
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:[36]whereA andϕ are the potential fields as above. The quantity can be thought as a velocity-dependent potential function.[37] UsingLagrange's equations, 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 potential energy depends on the velocity of the particle, so the force is velocity dependent, so it is not conservative.
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,[39] 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 aswhere ( is taken along the trajectory),, and.
The equation can also be written aswhere is theChristoffel symbol (of the torsion-free metric connection in general relativity), or aswhere is thecovariant differential in general relativity (metric, torsion-free).
^H is measured inamperes per metre (A/m) in SI units, and inoersteds (Oe) in cgs units."International system of units (SI)".NIST reference on constants, units, and uncertainty. National Institute of Standards and Technology. 12 April 2010. Retrieved9 May 2012.
^abSee Jackson, page 2. The book lists the four modern Maxwell's equations, and then states, "Also essential for consideration of charged particle motion is the Lorentz force equation,F =q(E +v ×B), which gives the force acting on a point chargeq in the presence of electromagnetic fields."
^Delon, Michel (2001).Encyclopedia of the Enlightenment. Chicago, Illinois: Fitzroy Dearborn. p. 538.ISBN1-57958-246-X.
^Goodwin, Elliot H. (1965).The New Cambridge Modern History Volume 8: The American and French Revolutions, 1763–93. Cambridge: Cambridge University Press. p. 130.ISBN978-0-521-04546-9.
^Thomson, J. J. (1881-04-01). "XXXIII. On the electric and magnetic effects produced by the motion of electrified bodies".The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.11 (68):229–249.doi:10.1080/14786448108627008.ISSN1941-5982.
^See Griffiths, page 326, which states that Maxwell's equations, "together with the [Lorentz] force law ... summarize the entire theoretical content of classical electrodynamics".
^abLandau, L. D.; Lifshitz, E. M.; Pitaevskiĭ, L. P. (1984).Electrodynamics of continuous media. Course of Theoretical Physics. Vol. 8 (2nd ed.). Oxford: Butterworth-Heinemann. §63 (§49 pp. 205–207 in 1960 edition).ISBN0-7506-2634-8.
Jackson, John David (1999).Classical electrodynamics (3rd ed.). New York City: Wiley.ISBN0-471-30932-X.
Serway, Raymond A.; Jewett, John W. Jr. (2004).Physics for scientists and engineers, with modern physics. Belmont, California: Thomson Brooks/Cole.ISBN0-534-40846-X.
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