magnetic force
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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
(specifically in
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an electric field and a magnetic field experiences a force of \mathbf = q\,\mathbf + q\,\mathbf \times \mathbf (in SI unitsIn SI units, is measured in teslas (symbol: T). In Gaussian-cgs units, is measured in
gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
(symbol: G). See e.g. )
The -field is measured in amperes per metre (A/m) in SI units, and in
oersted The oersted (symbol Oe) is the coherent derived unit of the auxiliary magnetic field H in the centimetre–gram–second system of units (CGS). It is equivalent to 1 dyne per maxwell. Difference between CGS and SI systems In the CGS system, ...
s (Oe) in cgs units.
). It says that the electromagnetic force on a charge is a combination of a force in the direction of the electric field proportional to the magnitude of the field and the quantity of charge, and a force at right angles to the magnetic field and the velocity 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 called Laplace force), the electromotive force in a wire loop moving through a magnetic field (an aspect of Faraday's law of induction), and the force on a moving charged particle. Historians suggest that the law is implicit in a paper by
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
, published in 1865. Hendrik Lorentz arrived at a complete derivation in 1895, identifying the contribution of the electric force a few years after
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed ...
correctly identified the contribution of the magnetic force.


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 and .See, for example, Jackson, pp. 777–8. To be specific, the Lorentz force is understood to be the following empirical statement:
''The electromagnetic force on a
test charge In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...
at a given point and time is a certain function of its charge and velocity , which can be parameterized by exactly two vectors and , in the functional form'': \mathbf = q(\mathbf+\mathbf \times \mathbf)
This is valid, even for particles approaching the speed of light (that is,
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
of , ). So the two vector fields and 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. As a definition of and , the Lorentz force is only a definition in principle because a real particle (as opposed to the hypothetical "test charge" of infinitesimally-small mass and charge) would generate its own finite and fields, which would alter the electromagnetic force that it experiences. In addition, if the charge experiences acceleration, as if forced into a curved trajectory, it emits radiation that causes it to lose kinetic energy. See for example
Bremsstrahlung ''Bremsstrahlung'' (), from "to brake" and "radiation"; i.e., "braking radiation" or "deceleration radiation", is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typicall ...
and synchrotron light. These effects occur through both a direct effect (called the
radiation reaction force In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes: * ''electromagnetic radiation'', such as radio waves, microwaves, infrared, visi ...
) and indirectly (by affecting the motion of nearby charges and currents).


Equation


Charged particle

The force acting on a particle of
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
with instantaneous velocity , due to an external electric field and magnetic field , is given by (in SI units): where is the vector cross product (all boldface quantities are vectors). In terms of Cartesian components, we have: F_x = q \left(E_x + v_y B_z - v_z B_y\right), F_y = q \left(E_y + v_z B_x - v_x B_z\right), F_z = q \left(E_z + v_x B_y - v_y B_x\right). In general, the electric and magnetic fields are functions of the position and time. Therefore, explicitly, the Lorentz force can be written as: \mathbf\left(\mathbf(t),\dot\mathbf(t),t,q\right) = q\left mathbf(\mathbf,t) + \dot\mathbf(t) \times \mathbf(\mathbf,t)\right/math> in which is the position vector of the charged particle, is time, and the overdot is a time derivative. A positively charged particle will be accelerated in the ''same'' linear orientation as the field, but will curve perpendicularly to both the instantaneous velocity vector and the field according to the right-hand rule (in detail, if the fingers of the right hand are extended to point in the direction of and are then curled to point in the direction of , then the extended thumb will point in the direction of ). The term is called the electric force, while the term is called the magnetic force.See Griffiths, page 204. According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force,For example, see th
website of the Lorentz Institute
or Griffiths.
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 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 the Laplace 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 is \mathbf \cdot \mathbf = q \, \mathbf \cdot \mathbf. Notice that the magnetic field does not contribute to the power because the magnetic force is always perpendicular to the velocity of the particle.


Continuous charge distribution

For a continuous charge distribution in motion, the Lorentz force equation becomes: \mathrm\mathbf = \mathrmq\left(\mathbf + \mathbf \times \mathbf\right) where \mathrm\mathbf is the force on a small piece of the charge distribution with charge \mathrmq. If both sides of this equation are divided by the volume of this small piece of the charge distribution \mathrmV, the result is: \mathbf = \rho\left(\mathbf + \mathbf \times \mathbf\right) where \mathbf is the ''force density'' (force per unit volume) and \rho is the
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...
(charge per unit volume). Next, the current density corresponding to the motion of the charge continuum is \mathbf = \rho \mathbf so the continuous analogue to the equation is The total force is the
volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ...
over the charge distribution: \mathbf = \iiint \left ( \rho \mathbf + \mathbf \times \mathbf \right)\mathrmV. By eliminating \rho and \mathbf, using
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
, and manipulating using the theorems of
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
, this form of the equation can be used to derive the
Maxwell stress tensor The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as ...
\boldsymbol, in turn this can be combined with the Poynting vector \mathbf to obtain the electromagnetic stress–energy tensor T used in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. In terms of \boldsymbol and \mathbf, another way to write the Lorentz force (per unit volume) is \mathbf = \nabla\cdot\boldsymbol - \dfrac \dfrac where c is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
and · denotes the divergence of a
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
. Rather than the amount of charge and its velocity in electric and magnetic fields, this equation relates the energy flux (flow of ''energy'' per unit time per unit distance) in the fields to the force exerted on a charge distribution. See
Covariant formulation of classical electromagnetism The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformatio ...
for more details. The density of power associated with the Lorentz force in a material medium is \mathbf \cdot \mathbf. 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 \mathbf = \left(\rho_f - \nabla \cdot \mathbf P\right) \mathbf + \left(\mathbf_f + \nabla\times\mathbf + \frac\right) \times \mathbf. where: \rho_f is the density of free charge; \mathbf is the
polarization density In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...
; \mathbf_f is the density of free current; and \mathbf is the magnetization 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 \left(\mathbf_f + \nabla\times\mathbf + \frac\right) \cdot \mathbf.


Equation in cgs units

The above-mentioned formulae use SI units which are the most common. In older cgs-Gaussian units, which are somewhat more common among some theoretical physicists as well as condensed matter experimentalists, one has instead \mathbf = q_\mathrm \left(\mathbf_\mathrm + \frac \times \mathbf_\mathrm\right). where ''c'' is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. Although this equation looks slightly different, it is completely equivalent, since one has the following relations: q_\mathrm = \frac,\quad \mathbf E_\mathrm = \sqrt\,\mathbf E_\mathrm,\quad \mathbf B_\mathrm = \,, \quad c = \frac. where is the vacuum permittivity and the
vacuum permeability The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
. In practice, the subscripts "cgs" and "SI" are always omitted, and the unit system has to be assessed from context.


History

Early attempts to quantitatively describe the electromagnetic force were made in the mid-18th 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 Henry Cavendish ( ; 10 October 1731 – 24 February 1810) was an English natural philosopher and scientist who was an important experimental and theoretical chemist and physicist. He is noted for his discovery of hydrogen, which he termed "infl ...
in 1762, obeyed an inverse-square law. However, in both cases the experimental proof was neither complete nor conclusive. It was not until 1784 when
Charles-Augustin de Coulomb Charles-Augustin de Coulomb (; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of attra ...
, using a
torsion balance A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. When it is twisted, it exerts a torque in the opposite direction, proportional ...
, was able to definitively show through experiment that this was true. Soon after the discovery in 1820 by Hans 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. 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. The modern concept of electric and magnetic fields first arose in the theories of
Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic inducti ...
, particularly his idea of
lines of force A line of force in Faraday's extended sense is synonymous with Maxwell's line of induction. According to J.J. Thomson, Faraday usually discusses ''lines of force'' as chains of polarized particles in a dielectric, yet sometimes Faraday discusses ...
, later to be given full mathematical description by
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
and
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
. 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, although in the time of Maxwell it was not evident how his equations related to the forces on moving charged objects.
J. J. Thomson Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered. In 1897, Thomson showed that ...
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 ray Cathode rays or electron beam (e-beam) are streams of electrons observed in discharge tubes. If an evacuated glass tube is equipped with two electrodes and a voltage is applied, glass behind the positive electrode is observed to glow, due to el ...
s, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field asPaul J. Nahin
''Oliver Heaviside''
JHU Press, 2002.
\mathbf = \frac\mathbf \times \mathbf. Thomson derived the correct basic form of the formula, but, because of some miscalculations and an incomplete description of the
displacement current In electromagnetism, displacement current density is the quantity appearing in Maxwell's equations that is defined in terms of the rate of change of , the electric displacement field. Displacement current density has the same units as electric ...
, included an incorrect scale-factor of a half in front of the formula.
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed ...
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. Finally, in 1895,Per F. Dahl, ''Flash of the Cathode Rays: A History of J J Thomson's Electron'', CRC Press, 1997, p. 10. 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 the
luminiferous aether Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), so ...
and sought to apply the Maxwell equations at a microscopic scale. Using Heaviside's version of the Maxwell equations for a stationary ether and applying
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
(see below), Lorentz arrived at the correct and complete form of the force law that now bears his name.


Trajectories of particles due to the Lorentz force

In many cases of practical interest, the motion in a magnetic field of an electrically charged particle (such as an
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
or
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
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.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, , which gives the force acting on a point charge ''q'' in the presence of electromagnetic fields."See Griffiths, page 326, which states that Maxwell's equations, "together with the orentzforce law...summarize the entire theoretical content of classical electrodynamics". 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 collective behavior of charged particles, both in principle and as a matter of computation. The charged particles in a material medium not only respond to the E and B fields but also generate these fields. Complex transport equations must be solved to determine the time and spatial response of charges, for example, the
Boltzmann equation The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
or the Fokker–Planck equation or the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
. For example, see
magnetohydrodynamics Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magneto­fluids include plasmas, liquid metals, ...
, fluid dynamics,
electrohydrodynamics Electrohydrodynamics (EHD), also known as electro-fluid-dynamics (EFD) or electrokinetics, is the study of the dynamics of electrically charged fluids. It is the study of the motions of ionized particles or molecules and their interactions with ...
, superconductivity, stellar evolution. An entire physical apparatus for dealing with these matters has developed. See for example,
Green–Kubo relations The Green–Kubo relations ( Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of time correlation functions: :\gamma = \int_0^\infty \left\langle \dot(t) \dot ...
and
Green's function (many-body theory) In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators. The name comes from ...
.


Force on a current-carrying wire

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 the Laplace 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: \mathbf = I \boldsymbol \times \mathbf where is a vector whose magnitude is the length of wire, and whose direction is along the wire, aligned with the direction of
conventional current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving ...
charge flow . 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 \mathrm d \boldsymbol \ell , then adding up all these forces by integration. Formally, the net force on a stationary, rigid wire carrying a steady current is \mathbf = I\int \mathrm\boldsymbol\times \mathbf This is the net force. In addition, there will usually be
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
, plus other effects if the wire is not perfectly rigid. One application of this is
Ampère's force law In magnetostatics, the force of attraction or repulsion between two current-carrying wires (see first figure below) is often called Ampère's force law. The physical origin of this force is that each wire generates a magnetic field, followin ...
, 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. For more information, see the article:
Ampère's force law In magnetostatics, the force of attraction or repulsion between two current-carrying wires (see first figure below) is often called Ampère's force law. The physical origin of this force is that each wire generates a magnetic field, followin ...
.


EMF

The magnetic force () 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 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 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 (''q''E) 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
Maxwell–Faraday equation Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic induct ...
(one of the four modern
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
).See Griffiths, pages 301–3. Both of these EMFs, despite their apparently distinct origins, are 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 below.) Einstein's
special theory of relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
was partially motivated by the desire to better understand this link between the two effects. In fact, the electric and magnetic fields are different facets of the same electromagnetic field, and in moving from one inertial frame to another, the
solenoidal vector field In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf ...
portion of the ''E''-field can change in whole or in part to a ''B''-field or ''vice versa''.


Lorentz force and Faraday's law of induction

Given a loop of wire in a magnetic field, Faraday's law of induction states the induced electromotive force (EMF) in the wire is: \mathcal = -\frac where \Phi_B = \iint_ \mathrm \mathbf \cdot \mathbf(\mathbf, t) is the magnetic flux through the loop, B is the magnetic field, Σ(''t'') is a surface bounded by the closed contour ∂Σ(''t''), at time ''t'', dA is an infinitesimal
vector area In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an ''oriented area'' in three dimensions. Every bounded surface in three dimensions can be associated with ...
element of Σ(''t'') (magnitude is the area of an infinitesimal patch of surface, direction is orthogonal to that surface patch). The ''sign'' of the EMF is determined by
Lenz's law Lenz's law states that the direction of the electric current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes changes in the initial magnetic field. It is named after p ...
. Note that this is valid for not only a ''stationary'' wirebut also for a ''moving'' wire. From Faraday's law of induction (that is valid for a moving wire, for instance in a motor) and the
Maxwell Equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. T ...
, the Lorentz Force can be deduced. The reverse is also true, the Lorentz force and the
Maxwell Equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. T ...
can be used to derive the Faraday Law. Let be the moving wire, moving together without rotation and with constant velocity v and Σ(''t'') be the internal surface of the wire. The EMF around the closed path ∂Σ(''t'') is given by: \mathcal =\oint_ \mathrm \boldsymbol \cdot \mathbf / q where \mathbf = \mathbf / q is the electric field and is an infinitesimal vector element of the contour . NB: Both dℓ and dA have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in the article Kelvin–Stokes 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 ''Maxwell–Faraday equation'': \nabla \times \mathbf = -\frac \, . The Maxwell–Faraday equation also can be written in an ''integral form'' using the Kelvin–Stokes theorem. So we have, the Maxwell Faraday equation: \oint_\mathrm \boldsymbol \cdot \mathbf(\mathbf,\ t) = - \ \iint_ \mathrm \mathbf \cdot and the Faraday Law, \oint_\mathrm \boldsymbol \cdot \mathbf/q(\mathbf,\ t) = - \frac \iint_ \mathrm \mathbf \cdot \mathbf(\mathbf,\ t). The two are equivalent if the wire is not moving. Using the
Leibniz integral rule In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form \int_^ f(x,t)\,dt, where -\infty < a(x), b(x) < \infty and the integral are
and that , results in, \oint_ \mathrm \boldsymbol \cdot \mathbf/q(\mathbf, t) = - \iint_ \mathrm \mathbf \cdot \frac \mathbf(\mathbf, t) + \oint_ \!\!\!\!\mathbf \times \mathbf \,\mathrm \boldsymbol and using the Maxwell Faraday equation, \oint_ \mathrm \boldsymbol \cdot \mathbf/q(\mathbf,\ t) = \oint_ \mathrm \boldsymbol \cdot \mathbf(\mathbf,\ t) + \oint_\!\!\!\! \mathbf \times \mathbf(\mathbf,\ t)\, \mathrm \boldsymbol since this is valid for any wire position it implies that, \mathbf= q\,\mathbf(\mathbf,\ t) + q\,\mathbf \times \mathbf(\mathbf,\ t). 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 magnetic flux linking the loop can change in several ways. For example, if the -field varies with position, and the loop moves to a location with different -field, will change. Alternatively, if the loop changes orientation with respect to the B-field, the differential element will change because of the different angle between and , also changing . As a third example, if a portion of the circuit is swept through a uniform, time-independent -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 time-dependent). In all three cases, Faraday's law of induction then predicts the EMF generated by the change in . Note that the Maxwell Faraday's equation implies that the Electric Field is non conservative when the Magnetic Field varies in time, and is not expressible as the gradient of a scalar field, and not subject to the
gradient theorem The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...
since its rotational is not zero.


Lorentz force in terms of potentials

The and fields can be replaced by the magnetic vector potential and ( scalar)
electrostatic potential Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for ambe ...
by \mathbf = - \nabla \phi - \frac \mathbf = \nabla \times \mathbf where is the gradient, is the divergence, and is the curl. The force becomes \mathbf = q\left \nabla \phi- \frac+\mathbf\times(\nabla\times\mathbf)\right Using an identity for the triple product this can be rewritten as, \mathbf = q\left \nabla \phi- \frac+\nabla\left(\mathbf\cdot \mathbf \right)-\left(\mathbf\cdot \nabla\right)\mathbf\right (Notice that the coordinates and the velocity components should be treated as independent variables, so the del operator acts only on \mathbf, not on \mathbf; thus, there is no need of using Feynman's subscript notation in the equation above). Using the chain rule, the
total derivative In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with res ...
of \mathbf is: \frac = \frac+(\mathbf\cdot\nabla)\mathbf so that the above expression becomes: \mathbf = q\left \nabla (\phi-\mathbf\cdot\mathbf)- \frac\right With , we can put the equation into the convenient Euler–Lagrange form where \nabla_ = \hat \dfrac + \hat \dfrac + \hat \dfrac and \nabla_ = \hat \dfrac + \hat \dfrac + \hat \dfrac.


Lorentz force and analytical mechanics

The
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
for a charged particle of mass and charge in an electromagnetic field equivalently describes the dynamics of the particle in terms of its ''energy'', rather than the force exerted on it. The classical expression is given by: L=\frac\mathbf\cdot\mathbf+q\mathbf\cdot\mathbf-q\phi where and are the potential fields as above. The quantity V = q(\phi - \mathbf\cdot \mathbf) can be thought as a velocity-dependent potential function. Using Lagrange's equations, the equation for the Lorentz force given above can be obtained again. 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 L = -mc^2\sqrt + q \mathbf(\mathbf)\cdot\dot - q \phi(\mathbf) The action is the relativistic arclength of the path of the particle in
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
, minus the potential energy contribution, plus an extra contribution which quantum mechanically is an extra phase a charged particle gets when it is moving along a vector potential.


Relativistic form of the Lorentz force


Covariant form of the Lorentz force


Field tensor

Using the
metric signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
, the Lorentz force for a charge can be written in covariant form: where is the
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
, defined as p^\alpha = \left(p_0, p_1, p_2, p_3 \right) = \left(\gamma m c, p_x, p_y, p_z \right) , the
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
of the particle, the contravariant
electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. T ...
F^ = \begin 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end and is the covariant 4-velocity of the particle, defined as: U_\beta = \left(U_0, U_1, U_2, U_3 \right) = \gamma \left(c, -v_x, -v_y, -v_z \right) , in which \gamma(v)=\frac=\frac is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
. The fields are transformed to a frame moving with constant relative velocity by: F'^ = _ _ F^ \, , where is the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
tensor.


Translation to vector notation

The component (''x''-component) of the force is \frac = q U_\beta F^ = q\left(U_0 F^ + U_1 F^ + U_2 F^ + U_3 F^ \right) . Substituting the components of the covariant electromagnetic tensor ''F'' yields \frac = q \left _0 \left(\frac \right) + U_2 (-B_z) + U_3 (B_y) \right. Using the components of covariant
four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
yields \frac = q \gamma \left \left(\frac \right) + (-v_y) (-B_z) + (-v_z) (B_y) \right= q \gamma \left(E_x + v_y B_z - v_z B_y \right) = q \gamma \left E_x + \left( \mathbf \times \mathbf \right)_x \right\, . The calculation for (force components in the and directions) yields similar results, so collecting the 3 equations into one: \frac = q \gamma\left( \mathbf + \mathbf \times \mathbf \right) , and since differentials in coordinate time and proper time are related by the Lorentz factor, dt=\gamma(v) \, d\tau, so we arrive at \frac = q \left( \mathbf + \mathbf \times \mathbf \right) . This is precisely the Lorentz force law, however, it is important to note that is the relativistic expression, \mathbf = \gamma(v) m_0 \mathbf \,.


Lorentz force in spacetime algebra (STA)

The electric and magnetic fields are dependent 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 \mathcal, and an arbitrary time-direction, \gamma_0. This can be settled through Space-Time Algebra (or the geometric algebra of space-time), a type of Clifford algebra defined on a
pseudo-Euclidean space In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving q(x ...
, as \mathbf = \left(\mathcal \cdot \gamma_0\right) \gamma_0 and i\mathbf = \left(\mathcal \wedge \gamma_0\right) \gamma_0 \mathcal F is a space-time bivector (an oriented plane segment, just like a vector is an oriented line segment), which has six degrees of freedom corresponding to boosts (rotations in space-time planes) and rotations (rotations in space-space planes). The dot product with the vector \gamma_0 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 v = \dot x, where v^2 = 1, (which shows our choice for the metric) and the velocity is \mathbf = cv \wedge \gamma_0 / (v \cdot \gamma_0). The proper (invariant is an inadequate term because no transformation has been defined) form of the Lorentz force law 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.


Lorentz force in general relativity

In the
general theory of relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
the equation of motion for a particle with mass m and charge e, moving in a space with metric tensor g_ and electromagnetic field F_, is given as m\frac-m\fracg_u^au^b=eF_u^b , where u^a= dx^a/ds (dx^a is taken along the trajectory), g_= \partial g_/\partial x^c, and ds^2=g_dx^adx^b. The equation can also be written as m\frac-m\Gamma_u^au^b=eF_u^b , where \Gamma_ is the
Christoffel symbol In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing dist ...
(of the torsion-free metric connection in general relativity), or as m\frac = e F_u^b , where D is the covariant differential in general relativity (metric, torsion-free).


Applications

The Lorentz force occurs in many devices, including: *
Cyclotron A cyclotron is a type of particle accelerator invented by Ernest O. Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: Jan ...
s and other circular path
particle accelerator A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for fundamental research in particle ...
s *
Mass spectrometer Mass spectrometry (MS) is an analytical technique that is used to measure the mass-to-charge ratio of ions. The results are presented as a '' mass spectrum'', a plot of intensity as a function of the mass-to-charge ratio. Mass spectrometry is us ...
s *Velocity Filters *
Magnetron The cavity magnetron is a high-power vacuum tube used in early radar systems and currently in microwave ovens and linear particle accelerators. It generates microwaves using the interaction of a stream of electrons with a magnetic field while ...
s * Lorentz force velocimetry In its manifestation as the Laplace force on an electric current in a conductor, this force occurs in many devices including: *
Electric motor An electric motor is an electrical machine that converts electrical energy into mechanical energy. Most electric motors operate through the interaction between the motor's magnetic field and electric current in a wire winding to generate for ...
s *
Railgun A railgun or rail gun is a linear motor device, typically designed as a weapon, that uses electromagnetic force to launch high velocity projectiles. The projectile normally does not contain explosives, instead relying on the projectile's high ...
s * Linear motors *
Loudspeaker A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or ...
s * Magnetoplasmadynamic thrusters *
Electrical generator In electricity generation, a generator is a device that converts motive power (mechanical energy) or fuel-based power ( chemical energy) into electric power for use in an external circuit. Sources of mechanical energy include steam turbines, g ...
s *
Homopolar generator A homopolar generator is a DC electrical generator comprising an electrically conductive disc or cylinder rotating in a plane perpendicular to a uniform static magnetic field. A potential difference is created between the center of the disc and th ...
s *
Linear alternator A linear alternator is essentially a linear motor used as an electrical generator. An alternator is a type of alternating current (AC) electrical generator. The devices are often physically equivalent. The principal difference is in how they are ...
s


See also

*
Hall effect The Hall effect is the production of a voltage difference (the Hall voltage) across an electrical conductor that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was dis ...
*
Electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
* Gravitomagnetism *
Ampère's force law In magnetostatics, the force of attraction or repulsion between two current-carrying wires (see first figure below) is often called Ampère's force law. The physical origin of this force is that each wire generates a magnetic field, followin ...
* Hendrik Lorentz *
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
*
Formulation of Maxwell's equations in special relativity The covariance and contravariance of vectors, covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifes ...
*
Moving magnet and conductor problem The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant vel ...
* Abraham–Lorentz force *
Larmor formula In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light. When any charged ...
*
Cyclotron radiation Cyclotron radiation is electromagnetic radiation emitted by non-relativistic accelerating charged particles deflected by a magnetic field. The Lorentz force on the particles acts perpendicular to both the magnetic field lines and the particles' mot ...
*
Magnetoresistance Magnetoresistance is the tendency of a material (often ferromagnetic) to change the value of its electrical resistance in an externally-applied magnetic field. There are a variety of effects that can be called magnetoresistance. Some occur in b ...
*
Scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
*
Helmholtz decomposition In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
*
Guiding center In physics, the motion of an electrically charged particle such as an electron or ion in a plasma in a magnetic field can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relativ ...
*
Field line A field line is a graphical visual aid for visualizing vector fields. It consists of an imaginary directed line which is tangent to the field vector at each point along its length. A diagram showing a representative set of neighboring field ...
*
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
* Electromagnetic buoyancy


Footnotes


References

The numbered references refer in part to the list immediately below. *: volume 2. * * * *


External links


Lorentz force (demonstration)Faraday's law: Tankersley and Mosca
see also ttp://hyperphysics.phy-astr.gsu.edu/HBASE/hframe.html home page
Interactive Java applet on the magnetic deflection of a particle beam in a homogeneous magnetic field
by Wolfgang Bauer

{{Authority control Physical phenomena Electromagnetism Maxwell's equations Hendrik Lorentz