In physics, particularly electromagnetism, the Lorentz force is the force on a point charge due to electromagnetic fields. If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field B, then it will experience a force
 $\backslash mathbf\{F\}\; =\; q\backslash left(\backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}\backslash right)$
(in SI units). Variations on this basic formula describe the magnetic force on a currentcarrying 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 particle which might be traveling near the speed of light (relativistic form of the Lorentz force).
The first derivation of the Lorentz force is commonly attributed to Oliver Heaviside in 1889,^{[1]} although other historians suggest an earlier origin in an 1865 paper by James Clerk Maxwell.^{[2]} Hendrik Lorentz derived it a few years after Heaviside.
Equation (SI units)
Charged particle
The force F acting on a particle of electric charge q with instantaneous velocity v, due to an external electric field E and magnetic field B, is given by:^{[3]}
Template:Equation box 1
where × is the vector cross product. All boldface quantities are vectors. More explicitly stated:
 $\backslash mathbf\{F\}(\backslash mathbf\{r\},\backslash mathbf\{\backslash dot\{r\}\},t,q)\; =\; q[\backslash mathbf\{E\}(\backslash mathbf\{r\},t)\; +\; \backslash mathbf\{\backslash dot\{r\}\}\; \backslash times\; \backslash mathbf\{B\}(\backslash mathbf\{r\},t)]$
in which r 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 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.^{[4]} According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force,^{[5]} 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.
Continuous charge distribution
For a continuous charge distribution in motion, the Lorentz force equation becomes:
 $d\backslash mathbf\{F\}\; =\; dq\backslash left(\backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}\backslash right)\backslash ,\backslash !$
where dF is the force on a small piece of the charge distribution with charge dq. If both sides of this equation are divided by the volume of this small piece of the charge distribution dV, the result is:
 $\backslash mathbf\{f\}\; =\; \backslash rho\backslash left(\backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}\backslash right)\backslash ,\backslash !$
where f is the force density (force per unit volume) and ρ is the charge density (charge per unit volume). Next, the current density corresponding to the motion of the charge continuum is
 $\backslash mathbf\{J\}\; =\; \backslash rho\; \backslash mathbf\{v\}\; \backslash ,\backslash !$
so the continuous analogue to the equation is^{[6]}
Template:Equation box 1
The total force is the volume integral over the charge distribution:
 $\backslash mathbf\{F\}\; =\; \backslash int\; dV\; (\; \backslash rho\; \backslash mathbf\{E\}\; +\; \backslash mathbf\{J\}\; \backslash times\; \backslash mathbf\{B\}\; ).\; \backslash ,\backslash !$
By eliminating ρ and J, using Maxwell's equations, and manipulating using the theorems of vector calculus, this form of the equation can be used to derive the Maxwell stress tensor σ, in turn this can be combined with the Poynting vector S to obtain the electromagnetic stressenergy tensor T used in general relativity.^{[6]}
In terms of σ and S, another way to write the Lorentz force (per unit 3d volume) is^{[6]}
 $\backslash mathbf\{f\}\; =\; \backslash nabla\backslash cdot\backslash boldsymbol\{\backslash sigma\}\; \; \backslash dfrac\{1\}\{c^2\}\; \backslash dfrac\{\backslash partial\; \backslash mathbf\{S\}\}\{\backslash partial\; t\}\; \backslash ,\backslash !$
where c is the speed of light and ∇· denotes the divergence of a tensor field. 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 for more details.
History
Trajectory of a particle with a positive or negative charge q under the influence of a magnetic field B, 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 is emitted along the electron path, due to the electrons colliding with gas molecules in the bulb. Using a
Teltron tube.
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.^{[7]} 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.^{[8]}^{[9]} 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.^{[10]}
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.^{[11]} 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,^{[2]} 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^{[1]}
 $\backslash mathbf\{F\}\; =\; \backslash frac\{q\}\{2\}\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}.$
Thomson derived 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.^{[1]}^{[12]}^{[13]} Finally, in 1892, 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 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 (see below), Lorentz arrived at the correct and complete form of the force law that now bears his name.^{[14]}^{[15]}
Trajectories of particles due to the Lorentz force
Main article:
Guiding center
In 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.^{[3]}^{[16]} 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.^{[17]}^{[18]}^{[19]} To be specific, the Lorentz force is understood to be the following empirical statement:
 The electromagnetic force F on a test charge at a given point and time is a certain function of its charge q and velocity v, which can be parameterized by exactly two vectors E and B, in the functional form:
 $\backslash mathbf\{F\}=q(\backslash mathbf\{E\}+\backslash mathbf\{v\}\backslash times\backslash mathbf\{B\})$
This is valid; countless experiments have shown that it is, even for particles approaching the speed of light (that is, magnitude of v = v = c).^{[20]} So the 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 with respect to what force a test charge would receive regardless of whether a charge is present to experience the force.
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, as 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, net force must include gravity, electroweak, and any other forces aside from electromagnetic force.
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:
 $\backslash mathbf\{F\}\; =\; I\; \backslash boldsymbol\{\backslash ell\}\; \backslash times\; \backslash mathbf\{B\}\; \backslash ,\backslash !$
where ℓ is a vector whose magnitude is the length of wire, and whose direction is along the wire, aligned with the direction of conventional current flow I.
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 by integration. Formally, the net force on a stationary, rigid wire carrying a steady current I is
 $\backslash mathbf\{F\}\; =\; I\backslash int\; d\backslash boldsymbol\{\backslash ell\}\backslash times\; \backslash mathbf\{B\}$
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 Maxwell–Faraday equation (one of the four modern Maxwell's equations).^{[21]}
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.^{[21]} 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.^{[22]}
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:
 $\backslash mathcal\{E\}\; =\; \backslash frac\{d\backslash Phi\_B\}\{dt\}$
where
 $\backslash Phi\_B\; =\; \backslash iint\_\{\backslash Sigma(t)\}\; d\; \backslash mathbf\{A\}\; \backslash cdot\; \backslash mathbf\{B\}(\backslash mathbf\{r\},\; t)$
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 all at time t, dA is an infinitesimal vector area 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. Note that this is valid for not only a stationary wire — but 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, 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 Σ(t) 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:^{[23]}
 $\backslash mathcal\{E\}\; =\backslash oint\_\{\backslash part\; \backslash Sigma\; (t)\}\; d\; \backslash boldsymbol\{\backslash ell\}\; \backslash cdot\; \backslash mathbf\{F\}\; /\; q$
where
 $\backslash mathbf\{E\}\; =\; \backslash mathbf\{F\}\; /\; q$
is the electric field and dℓ is an infinitesimal vector element of the contour ∂Σ(t).
NB: 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:
 $\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\{\backslash partial\; t\}\; \backslash \; .$
The MaxwellFaraday equation also can be written in an integral form using the KelvinStokes theorem:.^{[24]}
So we have, the Maxwell Faraday equation:
 $\backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}d\; \backslash boldsymbol\{\backslash ell\}\; \backslash cdot\; \backslash mathbf\{E\}(\backslash mathbf\{r\},\backslash \; t)\; =\; \; \backslash \; \backslash iint\_\{\backslash Sigma(t)\}\; d\; \backslash mathbf\; \{A\}\; \backslash cdot\; \{\backslash partial\; t\}+\backslash mathbf\{v\}\backslash times(\backslash nabla\backslash times\backslash mathbf\{A\})\backslash right]$
and using an identity for the triple product simplifies to
Template:Equation box 1{\partial t}+ \nabla(\mathbf{v}\cdot\mathbf{A})(\mathbf{v}\cdot\nabla)\mathbf{A} \right]
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using the chain rule, the total derivative of A is:
 $\backslash frac\{d\backslash mathbf\{A\}\}\{dt\}\; =\; \backslash frac\{\backslash partial\backslash mathbf\{A\}\}\{\backslash partial\; t\}+(\backslash mathbf\{v\}\backslash cdot\backslash nabla)\backslash mathbf\{A\}$
so the above expression can be rewritten as;
 $\backslash mathbf\{F\}\; =\; q\backslash left[\backslash nabla\; (\backslash phi\backslash mathbf\{v\}\backslash cdot\backslash mathbf\{A\})\; \backslash frac\{d\backslash mathbf\{A\}\}\{dt\}\backslash right]$
which can take the convenient EulerLagrange form
Template:Equation box 1(\phi\dot{\mathbf{x}}\cdot\mathbf{A})+ \frac{d}{dt}\nabla_{\dot{\mathbf{x}}}(\phi\dot{\mathbf{x}}\cdot\mathbf{A})\right]
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Lorentz force and analytical mechanics
The Lagrangian for a charged particle of mass m and charge q 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:^{[26]}
 $L=\backslash frac\{m\}\{2\}\backslash mathbf\{\backslash dot\{r\}\}\backslash cdot\backslash mathbf\{\backslash dot\{r\}\}+q\backslash mathbf\{A\}\backslash cdot\backslash mathbf\{\backslash dot\{r\}\}q\backslash phi$
where A and ϕ are the potential fields as above. Using Lagrange's equations, the equation for the Lorentz force can be obtained.
Derivation of Lorentz force from classical Lagrangian (SI units)

For an A field, a particle moving with velocity v = ṙ has potential momentum $q\backslash mathbf\{A\}(\backslash mathbf\{r\},t)$, so its potential energy is $q\backslash mathbf\{A\}(\backslash mathbf\{r\},t)\backslash cdot\backslash mathbf\{\backslash dot\{r\}\}$. For a ϕ field, the particle's potential energy is $q\backslash phi(\backslash mathbf\{r\},t)$.
The total potential energy is then:
 $V=q\backslash phiq\backslash mathbf\{A\}\backslash cdot\backslash mathbf\{\backslash dot\{r\}\}$
and the kinetic energy is:
 $T=\backslash frac\{m\}\{2\}\backslash mathbf\{\backslash dot\{r\}\}\backslash cdot\backslash mathbf\{\backslash dot\{r\}\}$
hence the Lagrangian:
 $L=TV=\backslash frac\{m\}\{2\}\backslash mathbf\{\backslash dot\{r\}\}\backslash cdot\backslash mathbf\{\backslash dot\{r\}\}+q\backslash mathbf\{A\}\backslash cdot\backslash mathbf\{\backslash dot\{r\}\}q\backslash phi$
 $L=\backslash frac\{m\}\{2\}(\backslash dot\{x\}^2+\backslash dot\{y\}^2+\backslash dot\{z\}^2)\; +\; q(\backslash dot\{x\}A\_x+\backslash dot\{y\}A\_y+\backslash dot\{z\}A\_z)\; \; q\backslash phi$
Lagrange's equations are
 $\backslash frac\{d\}\{dt\}\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\{x\}\}=\backslash frac\{\backslash partial\; L\}\{\backslash partial\; x\}$
(same for y and z). So calculating the partial derivatives:
 $\backslash begin\{align\}\backslash frac\{d\}\{dt\}\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\{x\}\}\; \&\; =m\backslash ddot\{x\}+q\backslash frac\{d\; A\_x\}\{dt\}\; \backslash \backslash $
& = m\ddot{x}+ \frac{q}{dt}\left(\frac{\partial A_x}{\partial t}dt+\frac{\partial A_x}{\partial x}dx+\frac{\partial A_x}{\partial y}dy+\frac{\partial A_x}{\partial z}dz\right) \\
& = m\ddot{x}+ q\left(\frac{\partial A_x}{\partial t}+\frac{\partial A_x}{\partial x}\dot{x}+\frac{\partial A_x}{\partial y}\dot{y}+\frac{\partial A_x}{\partial z}\dot{z}\right)\\
\end{align}
 $\backslash frac\{\backslash partial\; L\}\{\backslash partial\; x\}=\; q\backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; x\}+\; q\backslash left(\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; x\}\backslash dot\{x\}+\backslash frac\{\backslash partial\; A\_y\}\{\backslash partial\; x\}\backslash dot\{y\}+\backslash frac\{\backslash partial\; A\_z\}\{\backslash partial\; x\}\backslash dot\{z\}\backslash right)$
equating and simplifying:
 $m\backslash ddot\{x\}+\; q\backslash left(\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; t\}+\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; x\}\backslash dot\{x\}+\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; y\}\backslash dot\{y\}+\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; z\}\backslash dot\{z\}\backslash right)=\; q\backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; x\}+\; q\backslash left(\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; x\}\backslash dot\{x\}+\backslash frac\{\backslash partial\; A\_y\}\{\backslash partial\; x\}\backslash dot\{y\}+\backslash frac\{\backslash partial\; A\_z\}\{\backslash partial\; x\}\backslash dot\{z\}\backslash right)$
 $\backslash begin\{align\}\; F\_x\; \&\; =\; q\backslash left(\backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; x\}+\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; t\}\backslash right)\; +\; q\backslash left[\backslash dot\{y\}\backslash left(\backslash frac\{\backslash partial\; A\_y\}\{\backslash partial\; x\}\; \; \backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; y\}\backslash right)+\backslash dot\{z\}\backslash left(\backslash frac\{\backslash partial\; A\_z\}\{\backslash partial\; x\}\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; z\}\backslash right)\backslash right]\; \backslash \backslash $
& = qE_x + q[\dot{y}(\nabla\times\mathbf{A})_z\dot{z}(\nabla\times\mathbf{A})_y] \\
& = qE_x + q[\mathbf{\dot{r}}\times(\nabla\times\mathbf{A})]_x \\
& = qE_x + q(\mathbf{\dot{r}}\times\mathbf{B})_x
\end{align}
and similarly for the y and z directions. Hence the force equation is:
 $\backslash mathbf\{F\}=\; q(\backslash mathbf\{E\}\; +\; \backslash mathbf\{\backslash dot\{r\}\}\backslash times\backslash mathbf\{B\})$

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
 $L\; =\; m\backslash sqrt\{1\backslash left(\backslash frac\{\backslash dot\{\backslash mathbf\{r\}\}\}\{c\}\backslash right)^2\}\; +\; e\; \backslash mathbf\{A\}(\backslash mathbf\{r\})\backslash cdot\backslash dot\{\backslash mathbf\{r\}\}\; \; e\; \backslash phi(\backslash mathbf\{r\})\; \backslash ,\backslash !$
The action is the relativistic arclength of the path of the particle in space time, 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.
Derivation of Lorentz force from relativistic Lagrangian (SI units)

The equations of motion derived by extremizing the action (see matrix calculus for the notation):
 $\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{P\}\}\{\backslash mathrm\{d\}t\}\; =\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash mathbf\{r\}\}\; =\; e\; \{\backslash partial\; \backslash mathbf\{A\}\; \backslash over\; \backslash partial\; \backslash mathbf\{r\}\}\backslash cdot\; \backslash dot\{\backslash mathbf\{r\}\}\; \; e\; \{\backslash partial\; \backslash phi\; \backslash over\; \backslash partial\; \backslash mathbf\{r\}\; \}\backslash ,\backslash !$
 $\backslash mathbf\{P\}\; e\backslash mathbf\{A\}\; =\; \backslash frac\{m\backslash dot\{\backslash mathbf\{r\}\}\}\{\backslash sqrt\{1\backslash left(\backslash frac\{\backslash dot\{\backslash mathbf\{r\}\}\}\{c\}\backslash right)^2\}\}\backslash ,$
are the same as Hamilton's equations of motion:
 $\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{r\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash mathbf\{p\}\}\backslash left\; (\; \backslash sqrt\{(\backslash mathbf\{P\}e\backslash mathbf\{A\})^2\; +(mc^2)^2\}\; +\; e\backslash phi\; \backslash right\; )\; \backslash ,\backslash !$
 $\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{p\}\}\{\backslash mathrm\{d\}t\}\; =\; \{\backslash partial\; \backslash over\; \backslash partial\; \backslash mathbf\{r\}\}\backslash left\; (\; \backslash sqrt\{(\backslash mathbf\{P\}e\backslash mathbf\{A\})^2\; +\; (mc^2)^2\}\; +\; e\backslash phi\; \backslash right\; )\; \backslash ,\backslash !$
both are equivalent to the noncanonical form:
 $\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\backslash left\; (\; \{m\backslash dot\{\backslash mathbf\{r\}\}\; \backslash over\; \backslash sqrt\{1\backslash left(\backslash frac\{\backslash dot\{\backslash mathbf\{r\}\}\}\{c\}\backslash right)^2\}\}\; \backslash right\; )\; =\; e\backslash left\; (\; \backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}\; \backslash right\; )\; .\; \backslash ,\backslash !$
This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.

Equation (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
 $\backslash mathbf\{F\}\; =\; q\_\backslash mathrm\{cgs\}\; \backslash left(\backslash mathbf\{E\}\_\backslash mathrm\{cgs\}\; +\; \backslash frac\{\backslash mathbf\{v\}\}\{c\}\; \backslash times\; \backslash mathbf\{B\}\_\backslash mathrm\{cgs\}\backslash right).$
where c is the speed of light. Although this equation looks slightly different, it is completely equivalent, since
one has the following relations:
 $q\_\backslash mathrm\{cgs\}=\backslash frac\{q\_\backslash mathrm\{SI\}\}\{\backslash sqrt\{4\backslash pi\; \backslash epsilon\_0\}\},\backslash quad\; \backslash mathbf\; E\_\backslash mathrm\{cgs\}\; =\backslash sqrt\{4\backslash pi\backslash epsilon\_0\}\backslash ,\backslash mathbf\; E\_\backslash mathrm\{SI\},\backslash quad\; \backslash mathbf\; B\_\backslash mathrm\{cgs\}\; =\{\backslash sqrt\{4\backslash pi\; /\backslash mu\_0\}\}\backslash ,\{\backslash mathbf\; B\_\backslash mathrm\{SI\}\}$
where ε_{0} is the vacuum permittivity and μ_{0} the vacuum permeability. In practice, the subscripts "cgs" and "SI" are always omitted, and the unit system has to be assessed from context.
Relativistic form of the Lorentz force
Covariant form of the Lorentz force
Field tensor
Using the metric signature (1,1,1,1), The Lorentz force for a charge q can be written in covariant form:
Template:Equation box 1
where p^{α} is the fourmomentum, defined as:
 $p^\backslash alpha\; =\; \backslash left(p\_0,\; p\_1,\; p\_2,\; p\_3\; \backslash right)\; =\; \backslash left(m\; c,\; p\_x,\; p\_y,\; p\_z\; \backslash right)\; \backslash ,\; ,$
$\backslash scriptstyle\; \backslash tau$ the proper time of the particle, F^{αβ} the contravariant electromagnetic tensor
 $F^\{\backslash alpha\; \backslash beta\}\; =\; \backslash begin\{pmatrix\}$
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{pmatrix}
and U is the covariant 4velocity of the particle, defined as:
 $U\_\backslash beta\; =\; \backslash left(U\_0,\; U\_1,\; U\_2,\; U\_3\; \backslash right)\; =\; \backslash gamma\; \backslash left(c,\; u\_x,\; u\_y,\; u\_z\; \backslash right)\; \backslash ,\; ,$
where $\backslash scriptstyle\; \backslash gamma$ is the Lorentz factor defined above.
The fields are transformed to a frame moving with constant relative velocity by:
 $\backslash acute\{F\}^\{\backslash mu\; \backslash nu\}\; =\; \{\backslash Lambda^\{\backslash mu\}\}\_\{\backslash alpha\}\; \{\backslash Lambda^\{\backslash nu\}\}\_\{\backslash beta\}\; F^\{\backslash alpha\; \backslash beta\}\; \backslash ,\; ,$
where Λ^{μ}_{α} is the Lorentz transformation tensor.
Translation to vector notation
The α = 1 component (xcomponent) of the force is
 $\backslash frac\{d\; p^1\}\{d\; \backslash tau\}\; =\; q\; U\_\backslash beta\; F^\{1\; \backslash beta\}\; =\; q\backslash left(U\_0\; F^\{10\}\; +\; U\_1\; F^\{11\}\; +\; U\_2\; F^\{12\}\; +\; U\_3\; F^\{13\}\; \backslash right)\; .\backslash ,$
Substituting the components of the covariant electromagnetic tensor F yields
 $\backslash frac\{d\; p^1\}\{d\; \backslash tau\}\; =\; q\; \backslash left[U\_0\; \backslash left(\backslash frac\{E\_x\}\{c\}\; \backslash right)\; +\; U\_2\; (B\_z)\; +\; U\_3\; (B\_y)\; \backslash right].\; \backslash ,$
Using the components of covariant fourvelocity yields
 $\backslash begin\{align\}$
\frac{d p^1}{d \tau} & = q \gamma \left[c \left(\frac{E_x}{c} \right) + u_y B_z + u_z (B_y) \right] \\
&= q \gamma \left(E_x + u_y B_z  u_z B_y \right) \\
& = q \gamma \left[ E_x + \left( \mathbf{u} \times \mathbf{B} \right)_x \right] \, .
\end{align}
The calculation for α = 2, 3 (force components in the y and z directions) yields similar results, so collecting the 3 equations into one:
 $\backslash frac\{d\; \backslash mathbf\{p\}\; \}\{d\; \backslash tau\}\; =\; q\; \backslash gamma\backslash left(\; \backslash mathbf\{E\}\; +\; \backslash mathbf\{u\}\; \backslash times\; \backslash mathbf\{B\}\; \backslash right)\; \backslash ,\; ,$
which is the Lorentz force.
STA form of the Lorentz force
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 coordinateindependent expression for the electromagnetic and magnetic fields,^{[27]} $\backslash mathcal\{F\}$, and an arbitrary timedirection, $\backslash gamma\_0$, where
 $\backslash mathbf\{E\}\; =\; (\backslash mathcal\{F\}\backslash cdot\backslash gamma\_0)\backslash gamma\_0$
and
 $i\backslash mathbf\{B\}\; =\; (\backslash mathcal\{F\}\backslash wedge\backslash gamma\_0)\backslash gamma\_0$
$\backslash mathcal\; F$ is a spacetime 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 spacetime planes) and rotations (rotations in spacespace planes). The dot product with the vector $\backslash gamma\_0$ pulls a vector (in the space algebra) from the translational part, while the wedgeproduct 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 (timelike) changes in a timeposition vector $v=\backslash dot\; x$, where
 $v^2\; =\; 1,$
(which shows our choice for the metric) and the velocity is
 $\backslash mathbf\{v\}\; =\; cv\; \backslash wedge\; \backslash gamma\_0\; /\; (v\; \backslash cdot\; \backslash gamma\_0).$
The proper (invariant is an inadequate term because no transformation has been defined) form of the Lorentz force law is simply
Template:Equation box 1
Note that the order is important because between a bivector and a vector the dot product is antisymmetric. Upon a space time split like one can obtain the velocity, and fields as above yielding the usual expression.
Applications
The Lorentz force occurs in many devices, including:
In its manifestation as the Laplace force on an electric current in a conductor, this force occurs in many devices including:
See also
References
The numbered references refer in part to the list immediately below.
External links
 Interactive Java tutorial on the Lorentz force National High Magnetic Field Laboratory
 Lorentz force (demonstration)
 Faraday's law: Tankersley and Mosca
 home page
 Interactive Java applet on the magnetic deflection of a particle beam in a homogeneous magnetic field by Wolfgang Bauer
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