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Encyclopedia > Lorentz force
Electrostatics Electromagnetism Electricity · Magnetism Electric charge Coulomb's law Electric field Gauss's law Electric potential Electric dipole moment Ampère's Circuital law Magnetic field Magnetic flux Biot-Savart law Magnetic dipole moment Electric current Lorentz force law Electromotive force (EM) Electromagnetic induction Faraday-Lenz law Displacement current Maxwell's equations (EMF) Electromagnetic field (EM) Electromagnetic radiation Electrical conduction Electrical resistance Capacitance Inductance Impedance Resonant cavities Waveguides This box: view • talk • edit
Lorentz force.

$mathbf{F} = q (mathbf{E} + mathbf{v} times mathbf{B}),$

where

F is the force (in newtons)
E is the electric field (in volts per meter)
B is the magnetic field (in webers per square meter, or equivalently, teslas)
q is the electric charge of the particle (in coulombs)
v is the instantaneous velocity of the particle (in meters per second)
and × is the cross product.

Thus a positively charged particle will be accelerated in the same linear orientation as the E field, but will curve perpendicularly to the B field according to the right-hand rule. In physics, force is an influence that may cause an object to accelerate. ... The newton (symbol: N) is the SI unit of force. ... It has been suggested that optical field be merged into this article or section. ... Josephson junction array chip developed by NIST as a standard volt. ... The metre, or meter (symbol: m) is the SI base unit of length. ... This template is misplaced. ... In physics, the weber (symbol: Wb) is the SI unit of magnetic flux. ... SI unit. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... The coulomb (symbol: C) is the SI unit of electric charge. ... In physics, velocity is defined as the rate of change of displacement or the rate of displacement. ... Look up second in Wiktionary, the free dictionary. ... For the crossed product in algebra and functional analysis, see crossed product. ... The left-handed orientation is shown on the left, and the right-handed on the right. ...

## The Significance of the Lorentz Force GA_googleFillSlot("encyclopedia_square");

The Lorentz force is one of the original eight Maxwell's equations (equation D) and it is the solution to the differential form of Faraday's Law. Nowadays, Faraday's law is used instead of the Lorentz force in Maxwell's equations. Faraday's law and the Lorentz force both express the same physics. The discovery of the Lorentz force was before Lorentz's time. It can be seen at equation (77) in Maxwell's 1861 paper On Physical Lines of Force. 1 July 2007 (UTC)Bold text In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... Faradays law of induction (more generally, the law of electromagnetic induction) states that the induced emf (electromotive force) in a closed loop equals the negative of the time rate of change of magnetic flux through the loop. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ...

## Lorentz force in special relativity

When particle speeds approach the speed of light, the Lorentz force equation must be modified according to special relativity: The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...

${d left ( gamma m mathbf{v} right ) over dt } = mathbf{F} = q (mathbf{E} + mathbf{v} times mathbf{B}),$

where

$gamma stackrel{mathrm{def}}{=} frac{1}{sqrt{1 - frac{|mathbf{v}|^2}{c^2}}}$

is called the Lorentz factor and c is the speed of light in a vacuum. It has been suggested that Lorentz term be merged into this article or section. ... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic...

This relativistic form is identical to the conventional expression of the Lorentz force if the momentum form of Newton's law, F= dp/dt, is used, and the momentum p is assumed to be p = γmv.

The change of energy due to the electric and magnetic fields, in relativistic form, is simply

${d left ( gamma m c^2 right ) over dt } = q mathbf{E} cdot mathbf{v} .$

The change in energy depends only on the electric field, and not on the magnetic field.

## Covariant form of the Lorentz force

The Lorentz force equation can be written in covariant form in terms of the field strength tensor. In special relativity, in order to more clearly express the fact that Maxwells equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwells equations are written in terms of four-vectors and tensors in the manifestly covariant form (cgs units): , and where is... It has been suggested that this article or section be merged with Covariant. ...

$frac{d p^alpha}{d tau} = q u_beta F^{alpha beta}$
where
τ is c times the proper time of the particle,
q is the charge,
u is the 4-velocity of the particle, defined as:
$u_beta = left(u_0, u_1, u_2, u_3 right) = gamma left(c, v_x, v_y, v_z right) ,$and
F is the field strength tensor (or electromagnetic tensor) and is written in terms of fields as:
$F^{alpha beta} = begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c E_x/c & 0 & -B_z & B_y E_y/c & B_z & 0 & -B_x E_z/c & -B_y & B_x & 0 end{bmatrix}$.

The fields are transformed to a frame moving with constant relative velocity by: In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector (vector in four-dimensional spacetime) that replaces classical velocity (a three-dimensional vector). ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

$acute{F}^{mu nu} = {Lambda^{mu}}_{alpha} {Lambda^{nu}}_{beta} F^{alpha beta} ,$

where ${Lambda^{mu}}_{alpha}$ is a Lorentz transformation. A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space, while leaving the origin fixed (=rotation of Minkowski space). ...

### Derivation

The μ = 1 component (x-component) of the force is

$gamma frac{d p^1}{d t} = frac{d p^1}{d tau} = q u_beta F^{1 beta} = qleft(-u^0 F^{10} + u^1 F^{11} + u^2 F^{12} + u^3 F^{13} right) .,$

Here, τ is the proper time of the particle. Substituting the components of the electromagnetic tensor F yields In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. ...

$gamma frac{d p^1}{d t} = q left(-u^0 left(frac{-E_x}{c} right) + u^2 (B_z) + u^3 (-B_y) right) ,$

Writing the four-velocity in terms of the ordinary velocity yields In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector (vector in four-dimensional spacetime) that replaces classical velocity (a three-dimensional vector). ...

$gamma frac{d p^1}{d t} = q gamma left(c left(frac{E_x}{c} right) + v_y B_z - v_z B_y right) ,$
$gamma frac{d p^1}{d t} = q gamma left( E_x + left(mathbf{v} times mathbf{B} right)_x right) .,$

The calculation of the μ = 2 or μ = 3 is similar yielding

$gamma frac{d mathbf{p} }{d t} = frac{d mathbf{p} }{d tau} = q gamma left(mathbf{E} + (mathbf{v} times mathbf{B})right) ,$,

which is the Lorentz force law.

## Applications

The Lorentz force is a principle exploited in many devices including:

The Lorentz force can also act on a current carrying conductor, in this case called Laplace Force, by the interaction of the conduction electrons with the atoms of the conductor material. This force is used in many devices including : A pair of Dee electrodes with loops of coolant pipes on their surface at the Lawrence Hall of Science. ... For the DC Comics Superhero also called Atom Smasher, see Albert Rothstein. ... A homopolar generator, also known as a unipolar generator, acyclic generator, or disk dynamo, is an electrical generator in which the magnetic field has the same polarity at every point, so that the armature passes through the magnetic field lines of force continually in the same direction. ... A cavity magnetron is a high-powered vacuum tube that generates coherent microwaves. ... An MPD thruster during test firing The Magnetoplasmadynamic (MPD) thruster (MPDT) is a form of electric propulsion (a subdivision of spacecraft propulsion) which uses the Lorentz force (a force resulting from the interaction between a magnetic field and an electric current) to generate thrust. ... Mass spectrometry is a technique for separating ions by their mass-to-charge (m/z) ratios. ... The Biot-Savart law is a physical law with applications in both electromagnetics and fluid dynamics. ...

// A railgun is a form of gun that converts electrical energy (rather than the more conventional chemical energy from an explosive propellant) into projectile kinetic energy. ... â€œDynamoâ€ redirects here. ... Rotating magnetic field as a sum of magnetic vectors from 3 phase coils An electric motor converts electrical energy into mechanical energy. ...

Wikimedia Commons has media related to:
Lorentz force

Image File history File links Commons-logo. ... The Wikimedia Commons (also called Wikicommons) is a repository of free content images, sound and other multimedia files. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Hendrik Antoon Lorentz (July 18, 1853, Arnhem â€“ February 4, 1928, Haarlem) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and elucidation of the Zeeman effect. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... In special relativity, in order to more clearly express the fact that Maxwells equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwells equations are written in terms of four-vectors and tensors in the manifestly covariant form (cgs units): , and where is... Conductor moving in a magnetic field. ... The Abraham-Lorentz force is the average force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. ...

## Reference

• Serway and Jewett (2004). Physics for Scientists and Engineers with Modern Physics. Thomson Brooks/Cole. ISBN 0-534-40846-X.
• Feynman, Leighton and Sands (2006). The Feynman Lectures on Physics The Definitive Edition Volume II. Pearson Addison Wesley. ISBN 0-8053-9047-2.

Results from FactBites:

 Spartanburg SC | GoUpstate.com | Spartanburg Herald-Journal (425 words) In physics, the Lorentz force is the force exerted on a charged particle in an electromagnetic field. The Lorentz force is one of the original eight Maxwell's equations (equation D) and it is the solution to the differential form of Faraday's Law. The Lorentz force equation can be written in covariant form in terms of the field strength tensor.
More results at FactBites »

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