 Lorentz force

This article is about the equation governing the electromagnetic force. For a qualitative overview of the electromagnetic force, see Electromagnetism. For magnetic force of one magnet on another, see force between magnets.
Electromagnetism Electricity · Magnetism Lorentz force law · emf · Electromagnetic induction · Faraday’s law · Lenz's law · Displacement current · Maxwell's equations · EM field · Electromagnetic radiation · Liénard–Wiechert potential · Maxwell tensor · Eddy currentIn physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:^{[1]}
where
 F is the force (in newtons)
 E is the electric field (in volts per metre)
 B is the magnetic field (in teslas)
 q is the electric charge of the particle (in coulombs)
 v is the instantaneous velocity of the particle (in metres per second)
 × is the vector cross product operator
All the quantities written in boldface are vectors.
The Lorentz force law has a close relationship with Faraday's law of induction.
A positively charged particle will be accelerated in the same linear orientation as the E field, but will curve perpendicularly to both the instantaneous velocity vector v and the B field according to the righthand rule (in detail, if the thumb of the right hand points along v and the index finger along B, then the middle finger points along F).
The term qE is called the electric force, while the term qv × B is called the magnetic force.^{[2]} According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force,^{[3]} with the total electromagnetic force (including the electric force) given some other (nonstandard) name. This article will not follow this nomenclature: In what follows, the term "Lorentz force" will refer only to the expression for the total force.
The magnetic force component of the Lorentz force manifests itself as the force that acts on a currentcarrying wire in a magnetic field. In that context, it is also called the Laplace force.
History
Early attempts to quantitatively describe the electromagnetic force were made in the mid18th century. It was proposed that the force on magnetic poles, by Johann Tobias Mayer and others in 1760, and electrically charged objects, by Henry Cavendish in 1762, obeyed an inversesquare law. However, in both cases the experimental proof was neither complete nor conclusive. It was not until 1784 when CharlesAugustin de Coulomb, using a torsion balance, was able to definitively show through experiment that this was true.^{[4]} Soon after the discovery in 1820 by H. C. Ørsted that a magnetic needle is acted on by a voltaic current, AndréMarie Ampère that same year was able to devise through experimentation the formula for the angular dependence of the force between two current elements.^{[5]}^{[6]} In all these descriptions, the force was always given in terms of the properties of the objects involved and the distances between them rather than in terms of electric and magnetic fields.^{[7]}
The modern concept of electric and magnetic fields first arose in the theories of Michael Faraday, particularly his idea of lines of force, later to be given full mathematical description by Lord Kelvin and James Clerk Maxwell.^{[8]} 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,^{[9]} however, in the time of Maxwell it was not evident how his equations related to the forces on moving charged objects. J. J. Thomson was the first to attempt to derive from Maxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's properties and external fields. Interested in determining the electromagnetic behavior of the charged particles in cathode rays, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as . Thomson was able to arrive at the correct basic form of the formula, but, because of some miscalculations and an incomplete description of the displacement current, included an incorrect scalefactor of a half in front of the formula. It was Oliver Heaviside, who had invented the modern vector notation and applied them to Maxwell's field equations, that in 1885 and 1889 fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force on a moving charged object.^{[10]} Finally, in 1892, Hendrik Lorentz derived the modern day form of the formula for the electromagnetic force which includes the contributions to the total force from both the electric and the magnetic fields. Lorentz began by abandoning the Maxwellian descriptions of the ether and conduction. Instead, Lorentz made a distinction between matter and the luminiferous aether and sought to apply the Maxwell equations at a microscopic scale. Using the Heaviside's version of the Maxwell equations for a stationary ether and applying Lagrangian mechanics, Lorentz arrived at the correct and complete form of the force law that now bears his name.^{[11]}^{[12]}
Trajectories of particles in a Lorentz force
Main article: Guiding centerIn many cases of practical interest, the motion in a magnetic field of an electrically charged particle (such as an electron or ion in a plasma) can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relatively slow drift of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.
Significance of the Lorentz force
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 charge q in the presence of electromagnetic fields.^{[1]}^{[13]} The Lorentz force law describes the effect of E and B 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 of E and B by currents and charges is another.
In real materials the Lorentz force is inadequate to describe the behavior of charged particles, both in principle and as a matter of computation. The charged particles in a material medium both respond to the E and B fields and generate these fields. Complex transport equations must be solved to determine the time and spatial response of charges, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier–Stokes equations. For example, see magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, stellar evolution. An entire physical apparatus for dealing with these matters has developed. See for example, Green–Kubo relations and Green's function (manybody theory).
Lorentz force law as the definition of E and B
In many textbook treatments of classical electromagnetism, the Lorentz Force Law is used as the definition of the electric and magnetic fields E and B.^{[14]} To be specific, the Lorentz Force is understood to be the following empirical statement:
 The electromagnetic force on a test charge at a given point and time is a certain function of its charge and velocity, which can be parameterized by exactly two vectors E and B, in the functional form:
If this empirical statement is valid (and, of course, countless experiments have shown that it is), then two vector fields E and B are thereby defined throughout space and time, and these are called the "electric field" and "magnetic field".
Note that the fields are defined everywhere in space and time, regardless of whether or not a charge is present to experience the force. In particular, the fields are defined with respect to what force a test charge would feel, if it were hypothetically placed there.
Note also that as a definition of E and B, the Lorentz force is only a definition in principle because a real particle (as opposed to the hypothetical "test charge" of infinitesimallysmall mass and charge) would generate its own finite E and B fields, which would alter the electromagnetic force that it experiences. In addition, if the charge experiences acceleration, for example, if forced into a curved trajectory by some external agency, it emits radiation that causes braking of its motion. See, for example, Bremsstrahlung and synchrotron light. These effects occur through both a direct effect (called the radiation reaction force) and indirectly (by affecting the motion of nearby charges and currents).
Moreover, the electromagnetic force is not in general the same as the net force, due to gravity, electroweak and other forces, and any extra forces would have to be taken into account in a real measurement.
Lorentz force and Faraday's law of induction
Main article: Faraday's law of inductionGiven a loop of wire in a magnetic field, Faraday's law of induction states:
where:
 is the magnetic flux through the loop,
 is the electromotive force (EMF) experienced,
 t is time
 The sign of the EMF is determined by Lenz's law.
Note that this is valid for not only a stationary wire but also for a moving wire. From the Faraday Law (that is valid for a moving wire, for instance in a motor) and the Maxwell Equations, the Lorentz Force can be deduced. The reverse is also true, the Lorentz force and the Maxwell Equations can be used to derive the Faraday Law.
Let be the moving wire, moving together without rotation and with constant velocity and be the internal surface of the wire. The EMF around the closed path is given by:^{[15]}
where dℓ is an element of the curve . The flux Φ_{B} in Faraday's law of induction can be expressed explicitly as:
where
 is a surface bounded by the closed contour
 E is the electric field,
 dℓ is an infinitesimal vector element of the contour ,
 v is the velocity of the infinitesimal contour element dℓ,
 B is the magnetic field.
 dA is an infinitesimal vector element of surface , whose magnitude is the area of an infinitesimal patch of surface, and whose direction is orthogonal to that surface patch.
 Both dℓ and dA have a sign ambiguity; to get the correct sign, the righthand rule is used, as explained in the article KelvinStokes theorem.
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 the MaxwellFaraday equation:
The MaxwellFaraday equation also can be written in an integral form using the KelvinStokes theorem:.^{[16]}
So we have, the Maxwell Faraday equation:
and the Faraday Law,
The two are equivalent if the wire is not moving. Using the Leibniz integral rule and that div B = 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. See inapplicability of Faraday's law.
If the magnetic field is fixed in time and the conducting loop moves through the field, the flux magnetic flux Φ_{B} linking the loop can change in several ways. For example, if the Bfield varies with position, and the loop moves to a location with different Bfield, Φ_{B} will change. Alternatively, if the loop changes orientation with respect to the Bfield, the B•dA differential element will change because of the different angle between B and dA, also changing Φ_{B}. As a third example, if a portion of the circuit is swept through a uniform, timeindependent Bfield, 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 timedependent). In all three cases, Faraday's law of induction then predicts the EMF generated by the change in Φ_{B}.
Note that the Maxwell Faraday's equation implies that the Electric Field (E) is non conservative when the Magnetic Field (B) varies in time, and is not expressible as the gradient of a scalar field, and not subject to the gradient theorem since its rotational is not zero.
See Landau, L. D., Lifshit︠s︡, E. M., & Pitaevskiĭ, L. P. (1984). Electrodynamics of continuous media; Volume 8 Course of Theoretical Physics (Second ed.). Oxford: ButterworthHeinemann. p. §63 (§49 pp. 205–207 in 1960 edition). ISBN 0750626348. http://worldcat.org/search?q=0750626348&qt=owc_search. M N O Sadiku (2007). Elements of elctromagnetics (Fourth ed.). NY/Oxford: Oxford University Press. p. 391. ISBN 0195300483. http://books.google.com/?id=w2ITHQAACAAJ&dq=isbn:0195300483.
Lorentz force in terms of potentials
If the scalar potential and vector potential replace E and B (see Helmholtz decomposition), the force becomes:
or, equivalently (making use of the fact that v is a constant; see triple product),
where
 A is the magnetic vector potential
 ϕ is the electrostatic potential
 The symbols denote gradient, curl, and divergence, respectively.
The potentials are related to E and B by
Lorentz force in cgs units
The abovementioned formulae use SI units which are the most common among experimentalists, technicians, and engineers. In cgsGaussian units, which are somewhat more common among theoretical physicists, one has instead
where c is the speed of light. Although this equation looks slightly different, it is completely equivalent, since one has the following relations:
, , and
where ε_{0} and μ_{0} are the vacuum permittivity and vacuum permeability, respectively. In practice, unfortunately, the subscripts "cgs" and "SI" are always omitted, and the unit system has to be assessed from context.
Covariant form of the Lorentz force
Main article: Formulation of Maxwell's equations in special relativityNewton's law of motion can be written in covariant form in terms of the field strength tensor.

 where
 τ is c times the proper time of the particle,
 q is the charge,
 U is the covariant 4velocity of the particle, defined as:
 under the metric signature (1,1,1,1)
 with γ = Lorentz factor defined above, and F is the contravariant electromagnetic tensor written in terms of fields as:

 .
The fields are transformed to a frame moving with constant relative velocity by:
where is a Lorentz transformation. Alternatively, using the four vector:
related to the electric and magnetic fields by:
the field tensor becomes:^{[17]}
where:
Translation to vector notation
The α = 1 component (xcomponent) of the force is
Here, τ is the proper time of the particle. Substituting the components of the covariant electromagnetic tensor F yields
Using the components of covariant fourvelocity yields
The calculation of the α = 2 or α = 3 is similar yielding
or, in terms of the vector and scalar potentials A and φ,
which are the relativistic forms of Newton's law of motion when the Lorentz force is the only force present.
Force on a currentcarrying wire
When a wire carrying an electrical 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 the Laplace force). By combining the Lorentz force law above with the definition of electrical current, the following equation results, in the case of a straight, stationary wire:
where
 F = Force, measured in newtons
 I = current in wire, measured in amperes
 B = magnetic field vector, measured in teslas
 = vector cross product
 L = a vector, whose magnitude is the length of wire (measured in metres), and whose direction is along the wire, aligned with the direction of conventional current flow.
Alternatively, some authors write
where the vector direction is now associated with the current variable, instead of the length variable. The two forms are equivalent.
If the wire is not straight but curved, the force on it can be computed by applying this formula to each infinitesimal segment of wire dℓ, then adding up all these forces via integration. Formally, the net force on a stationary, rigid wire carrying a current I is
(This is the net force. In addition, there will usually be torque, plus other effects if the wire is not perfectly rigid.)
One application of this is Ampère's force law, which describes how two currentcarrying wires can attract or repel each other, since each experiences a Lorentz force from the other's magnetic field. For more information, see the article: Ampère's force law.
EMF
The magnetic force (q v × B) component of the Lorentz force is responsible for motional electromotive force (or motional EMF), the phenomenon underlying many electrical generators. When a conductor is moved through a magnetic field, the magnetic force tries to push electrons through the wire, and this creates the EMF. The term "motional EMF" is applied to this phenomenon, since the EMF is due to the motion of the wire.
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 an induced EMF, as described by the MaxwellFaraday equation (one of the four modern Maxwell's equations).^{[18]}
Both of these EMF's, despite their different origins, can be described by the same equation, namely, the EMF is the rate of change of magnetic flux through the wire. (This is Faraday's law of induction, see above.) Einstein's theory of special relativity was partially motivated by the desire to better understand this link between the two effects.^{[18]} In fact, the electric and magnetic fields are different faces of the same electromagnetic field, and in moving from one inertial frame to another, the solenoidal vector field portion of the Efield can change in whole or in part to a Bfield or vice versa.^{[19]}
General references
The numbered references refer in part to the list immediately below.
 Feynman, Richard Phillips; Leighton, Robert B.; Sands, Matthew L. (2006). The Feynman lectures on physics (3 vol.). Pearson / AddisonWesley. ISBN 0805390472: volume 2.
 Griffiths, David J. (1999). Introduction to electrodynamics (3rd ed.). Upper Saddle River, [NJ.]: PrenticeHall. ISBN 013805326X
 Jackson, John David (1999). Classical electrodynamics (3rd ed.). New York, [NY.]: Wiley. ISBN 047130932X
 Serway, Raymond A.; Jewett, John W., Jr. (2004). Physics for scientists and engineers, with modern physics. Belmont, [CA.]: Thomson Brooks/Cole. ISBN 053440846X
 Srednicki, Mark A. (2007). Quantum field theory. Cambridge, [England] ; New York [NY.]: Cambridge University Press. ISBN 9780521864497. http://books.google.com/?id=5OepxIG42B4C&pg=PA315&dq=isbn=9780521864497
Numbered footnotes and references
 ^ ^{a} ^{b} See 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 charge q in the presence of electromagnetic fields."
 ^ See Griffiths page 204.
 ^ For example, see the website of the "Lorentz Institute": [1], or Griffiths.
 ^ Meyer, Herbert W. (1972). A History of Electricity and Magnetism. Norwalk, CT: Burndy Library. pp. 30–31. ISBN 026213070X.
 ^ Verschuur, Gerrit L. (1993). Hidden Attraction : The History And Mystery Of Magnetism. New York: Oxford University Press. pp. 78–79. ISBN 0195064887.
 ^ Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. pp. 9, 25. ISBN 0198505930
 ^ Verschuur, Gerrit L. (1993). Hidden Attraction : The History And Mystery Of Magnetism. New York: Oxford University Press. p. 76. ISBN 0195064887.
 ^ Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. pp. 126–131, 139–144. ISBN 0198505930
 ^ Huray, Paul G. (2009). Maxwell's Equations. WileyIEEE. p. 22. ISBN 0470542764. http://books.google.com/books?id=0QsDgdd0MhMC&pg=PA22#v=onepage&q&f=false.
 ^ Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. pp. 200, 429–430. ISBN 0198505930
 ^ Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. p. 327. ISBN 0198505930
 ^ Whittaker, E. T. (1910). A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century. Longmans, Green and Co.. pp. 420–423. ISBN 1143012089. http://books.google.com/books?id=CGJDAAAAIAAJ&printsec=frontcover#v=onepage&q&f=false.
 ^ See Griffiths page 326, which states that Maxwell's equations, "together with the [Lorentz] force law...summarize the entire theoretical content of classical electrodynamics".
 ^ See, for example, Jackson p7778.
 ^ Landau, L. D., Lifshit︠s︡, E. M., & Pitaevskiĭ, L. P. (1984). Electrodynamics of continuous media; Volume 8 Course of Theoretical Physics (Second ed.). Oxford: ButterworthHeinemann. p. §63 (§49 pp. 205–207 in 1960 edition). ISBN 0750626348. http://worldcat.org/search?q=0750626348&qt=owc_search.
 ^ Roger F Harrington (2003). Introduction to electromagnetic engineering. Mineola, NY: Dover Publications. p. 56. ISBN 0486432416. http://books.google.com/?id=ZlC2EV8zvX8C&pg=PA57&dq=%22faraday%27s+law+of+induction%22.
 ^ DJ Griffiths (1999). Introduction to electrodynamics. Saddle River NJ: Pearson/AddisonWesley. p. 541. ISBN 013805326X.
 ^ ^{a} ^{b} See Griffiths pages 301–3.
 ^ Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett. p. 395. ISBN 0763738271. http://books.google.com/?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271.
Applications
The Lorentz force occurs in many devices, including:
 Cyclotrons and other circular path particle accelerators
 Mass spectrometers
 Velocity Filters
 Magnetrons
In its manifestation as the Laplace force on an electric current in a conductor, this force occurs in many devices including:
See also
 Hall effect
 Electromagnetism
 Gravitomagnetism
 Ampère's force law
 Hendrik Lorentz
 Maxwell's equations
 Formulation of Maxwell's equations in special relativity
 Moving magnet and conductor problem
 Abraham–Lorentz force
 Larmor formula
 Cyclotron radiation
 Magnetic potential
 Magnetoresistance
External links
 Interactive Java tutorial on the Lorentz force National High Magnetic Field Laboratory
 Lorentz force (demonstration)
 Faraday's law: Tankersley and Mosca
 Notes from Physics and Astronomy HyperPhysics at Georgia State University; see also home page
 Interactive Java applet on the magnetic deflection of a particle beam in a homogeneous magnetic field by Wolfgang Bauer
Categories: Fundamental physics concepts
 Electromagnetism
 Introductory physics
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