|
|
| Line 1: |
Line 1: |
| {{electromagnetism|cTopic=Electrodynamics}}
| | The ac1st16.dll error is annoying and quite common with all types of Windows computers. Not just does it make a computer run slower, but it may equally prevent you from utilizing a variety of programs, including AutoCAD. To fix this problem, you should employ a simple way to cure all of the potential difficulties which cause it. Here's what you have to do...<br><br>Document files let the consumer to input information, images, tables plus alternative ingredients to improve the presentation. The only issue with this formatting compared to additional file kinds including .pdf for example is its ability to be readily editable. This signifies that anybody viewing the file will change it by accident. Also, this file format will be opened by different programs yet it does not guarantee which what you see in the Microsoft Word application might nevertheless become the same when you see it using another system. However, it happens to be still preferred by most computer users for its ease of use plus features.<br><br>Over time a disk may moreover get fragmented. Fragmentation causes your computer to slow down because it takes windows much longer to locate a files location. Fortunately, the PC has a built in disk defragmenter. You are able to run this system by clicking "Start" - "All Programs" - "Accessories" - "System Tools" - "Disk Defragmenter". You may have the choice to choose that drives or partition we want to defragment. This action could take we certain time thus it's advised to do this on a regular basis so because to avoid further fragmentation plus to accelerate your windows XP computer.<br><br>There can be several reasons why your computer may lose speed. Normal computer utilize, such as surfing the Internet will get your running system in a condition where it has no choice however to slow down. The continual entering plus deleting of temporary files which occur whenever you surf the Web leave the registries with thousands of false indicators inside our running system's registry.<br><br>Many [http://bestregistrycleanerfix.com/registry-reviver registry reviver] s allow you to download their product for free, to scan the computer oneself. That method you are able to see how countless errors it finds, where it finds them, and how it will fix them. A remarkable registry cleaner usually remove your registry issues, and optimize and accelerate your PC, with small effort on a part.<br><br>S/w associated error handling - If the blue screen physical memory dump occurs following the installation of s/w application or a driver it could be which there is system incompatibility. By booting into safe mode plus removing the software you can swiftly fix this error. You may equally try out a "system restore" to revert to an earlier state.<br><br>Maybe you may be asking how come these windows XP error messages appear. Well, for we to be capable to recognize the fix, you need to first know where those errors come from. There is this software called registry. A registry is software which shops everything on the PC from a normal configuration, setting, info, plus logs of escapades from installing to UN-installing, saving to deleting, along with a lot more alterations you do inside the system pass by it plus gets 'tagged' plus saved as a easy file for recovery purposes. Imagine it as a big recorder, a registrar, of all your records in a PC.<br><br>Frequently the best technique is to read reviews on them plus if many users remark about its efficiency, it is likely to be function. The ideal piece is the fact that there are numerous top registry products which work; we really have to take the pick. |
| | |
| In [[physics]], particularly [[electromagnetism]], the '''Lorentz force''' is the combination of electric and magnetic [[force]] on a [[point charge]] due to [[electromagnetic field]]s. 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. For any produced force there will be an opposite reactive force. In the case of the magnetic field, the reactive force may be obscure, but it must be accounted for.
| |
| : <math>\mathbf{F} = q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)</math>
| |
| (in [[International System of Units|SI units]]). 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 charged particle which might be traveling near the [[speed of light]] ([[special relativity|relativistic]] form of the Lorentz force).
| |
| | |
| The first derivation of the Lorentz force is commonly attributed to [[Oliver Heaviside]] in 1889,<ref name=Nahin/> although other historians suggest an earlier origin in an 1865 paper by [[James Clerk Maxwell]].<ref name=Huray/> [[Hendrik Lorentz]] derived it a few years after Heaviside.{{citation needed|date=October 2013}}
| |
| | |
| ==Equation (SI units)==
| |
| | |
| {{See also|SI units}}
| |
| | |
| ===Charged particle===
| |
| | |
| [[File:Lorentz force particle.svg|200px|thumb|Lorentz force '''F''' on a [[charged particle]] (of [[electric charge|charge]] ''q'') in motion (instantaneous velocity '''v'''). The [[electric field|'''E''' field]] and [[magnetic field|'''B''' field]] vary in space and time.]]
| |
| | |
| 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:<ref name=Jackson2/>
| |
| | |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math>\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})</math>
| |
| |cellpadding
| |
| |border
| |
| |border colour = #50C878
| |
| |background colour = #ECFCF4}}
| |
| | |
| where <big>×</big> is the [[vector cross product]]. All [[boldface]] quantities are [[Vector (geometric)|vectors]]. More explicitly stated:
| |
| | |
| :<math>\mathbf{F}(\mathbf{r},\mathbf{\dot{r}},t,q) = q[\mathbf{E}(\mathbf{r},t) + \mathbf{\dot{r}} \times \mathbf{B}(\mathbf{r},t)]</math>
| |
| | |
| 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 [[right-hand 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 ''q'''''E''' is called the '''electric force''', while the term ''q'''''v''' <big>×</big> '''B''' is called the '''magnetic force'''.<ref name=Griffiths1>See Griffiths page 204.</ref> According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force,<ref name=Griffiths2>For example, see the website of the "Lorentz Institute": \[http://ilorentz.org/history/lorentz/lorentz.html], or Griffiths.</ref> 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 [[electrical current|current]]-carrying [[wire]] in a [[magnetic field]]. In that context, it is also called the '''Laplace force'''.
| |
| | |
| ===Continuous charge distribution===
| |
| | |
| [[File:Lorentz force continuum.svg|200px|thumb|Lorentz force (per unit 3-volume) '''f''' on a continuous [[charge distribution]] ([[charge density]] ρ) in motion. The 3-[[current density]] '''J''' corresponds to the motion of the charge element ''dq'' in [[volume element]] ''dV'' and varies throughout the continuum.]]
| |
| | |
| For a continuous [[charge distribution]] in motion, the Lorentz force equation becomes:
| |
| | |
| :<math>\mathrm{d}\mathbf{F} = \mathrm{d}q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)\,\!</math>
| |
| | |
| where ''d'''''F''' 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:
| |
| | |
| :<math>\mathbf{f} = \rho\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)\,\!</math>
| |
| | |
| 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
| |
| | |
| :<math>\mathbf{J} = \rho \mathbf{v} \,\!</math>
| |
| | |
| so the continuous analogue to the equation is<ref name="Electrodynamics 1999">{{cite book|last=Griffiths|first=David J.|title=Introduction to electrodynamics|year=1999|publisher=Prentice Hall|location=Upper Saddle River, NJ [u.a.]|isbn=9780138053260|edition=3rd |others= reprint. with corr.}}</ref>
| |
| | |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math>\mathbf{f} = \rho \mathbf{E} + \mathbf{J} \times \mathbf{B}\,\!</math>
| |
| |cellpadding= 6
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F5FFFA}}
| |
| | |
| The total force is the [[volume integral]] over the charge distribution:
| |
| | |
| :<math> \mathbf{F} = \iiint \! ( \rho \mathbf{E} + \mathbf{J} \times \mathbf{B} )\,\mathrm{d}V. \,\!</math>
| |
| | |
| 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 stress-energy tensor]] '''T''' used in [[general relativity]].<ref name="Electrodynamics 1999"/>
| |
| | |
| In terms of '''σ''' and '''S''', another way to write the Lorentz force (per unit [[three-dimensional space|3d]] volume) is<ref name="Electrodynamics 1999"/>
| |
| | |
| :<math> \mathbf{f} = \nabla\cdot\boldsymbol{\sigma} - \dfrac{1}{c^2} \dfrac{\partial \mathbf{S}}{\partial t} \,\!</math>
| |
| | |
| 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#Charge continuum|Covariant formulation of classical electromagnetism]] for more details.
| |
| | |
| ==History==
| |
| {{multiple image
| |
| | align = right
| |
| | position
| |
| | direction = horizontal
| |
| | footer = [[Charged particle]]s experiencing the Lorentz force.
| |
| | image1 = Lorentz force.svg
| |
| | caption1 = 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.
| |
| | width1 = 250
| |
| | image2 = Cyclotron motion.jpg
| |
| | caption2 = 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]].
| |
| | width2 = 250
| |
| }}
| |
| 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{{citation needed|date=November 2011}}, and electrically charged objects, by [[Henry Cavendish]] in 1762{{citation needed|date=November 2011}}, 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]], using a [[torsion balance]], was able to definitively show through experiment that this was true.<ref>
| |
| {{Cite book
| |
| |first = Herbert W.
| |
| |last = Meyer
| |
| |title = A History of Electricity and Magnetism
| |
| |place = Norwalk, CT
| |
| |publisher = Burndy Library
| |
| |year = 1972
| |
| |pages = 30–31
| |
| |isbn = 0-262-13070-X}}</ref> Soon after the discovery in 1820 by [[H. C. 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.<ref>
| |
| {{Cite book
| |
| |first = Gerrit L.
| |
| |last = Verschuur
| |
| |title = Hidden Attraction : The History And Mystery Of Magnetism
| |
| |place = New York
| |
| |publisher = Oxford University Press
| |
| |isbn = 0-19-506488-7
| |
| |year = 1993
| |
| |pages = 78–79}}</ref><ref>
| |
| {{Cite book
| |
| |first = Olivier
| |
| |last = Darrigol
| |
| |title = Electrodynamics from [[André Ampère|Ampère]] to [[Albert Einstein|Einstein]]
| |
| |place = Oxford, [England]
| |
| |publisher = Oxford University Press
| |
| |isbn = 0-19-850593-0
| |
| |year = 2000
| |
| |pages = 9, 25
| |
| |postscript = <!--None-->}}</ref> 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.<ref>
| |
| {{Cite book
| |
| |first = Gerrit L.
| |
| |last = Verschuur
| |
| |title = Hidden Attraction : The History And Mystery Of Magnetism
| |
| |place = New York
| |
| |publisher = Oxford University Press
| |
| |isbn = 0-19-506488-7
| |
| |year = 1993
| |
| |page = 76}}</ref>
| |
| | |
| 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 [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] and [[James Clerk Maxwell]].<ref>
| |
| {{Cite book
| |
| |first = Olivier
| |
| |last = Darrigol
| |
| |title = Electrodynamics from [[André Ampère|Ampère]] to [[Albert Einstein|Einstein]]
| |
| |place = Oxford, [England]
| |
| |publisher = Oxford University Press
| |
| |isbn = 0-19-850593-0
| |
| |year = 2000
| |
| |pages = 126–131, 139–144
| |
| |postscript = <!--None-->}}</ref> 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,<ref name=Huray>
| |
| {{Cite book
| |
| |first = Paul G.
| |
| |last = Huray
| |
| |title = Maxwell's Equations
| |
| |publisher = Wiley-IEEE
| |
| |isbn = 0-470-54276-4
| |
| |year = 2009
| |
| |page = 22
| |
| |url = http://books.google.com/books?id=0QsDgdd0MhMC&pg=PA22#v=onepage&q&f=false}}</ref> 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 ray]]s, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as<ref name=Nahin>[http://books.google.com/books?id=e9wEntQmA0IC&pg=PA120 Oliver Heaviside By Paul J. Nahin, p120]</ref>
| |
| :<math>\mathbf{F} = \frac{q}{2}\mathbf{v} \times \mathbf{B}.</math>
| |
| 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 scale-factor 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.<ref name=Nahin/><ref>
| |
| {{Cite book
| |
| |first = Olivier
| |
| |last = Darrigol
| |
| |title = Electrodynamics from [[André Ampère|Ampère]] to [[Albert Einstein|Einstein]]
| |
| |place = Oxford, [England]
| |
| |publisher = Oxford University Press
| |
| |isbn = 0-19-850593-0
| |
| |year = 2000
| |
| |pages = 200, 429–430
| |
| |postscript = <!--None-->}}</ref><ref>{{cite paper | author= Heaviside, Oliver| title=On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric | journal=Philosophical Magazine, April 1889, p. 324 |url=http://en.wikisource.org/wiki/Motion_of_Electrification_through_a_Dielectric}}</ref> 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 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.<ref>
| |
| {{Cite book
| |
| |first = Olivier
| |
| |last = Darrigol
| |
| |title = Electrodynamics from [[André Ampère|Ampère]] to [[Albert Einstein|Einstein]]
| |
| |place = Oxford, [England]
| |
| |publisher = Oxford University Press
| |
| |isbn = 0-19-850593-0
| |
| |year = 2000
| |
| |page = 327
| |
| |postscript = <!--None-->}}</ref><ref>
| |
| {{cite book
| |
| |last = Whittaker
| |
| |first = E. T.
| |
| |title = A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century
| |
| |publisher = Longmans, Green and Co.
| |
| |year = 1910
| |
| |pages = 420–423
| |
| |url = http://books.google.com/books?id=CGJDAAAAIAAJ&printsec=frontcover#v=onepage&q&f=false
| |
| |isbn = 1-143-01208-9}}</ref>
| |
| | |
| ==Trajectories of particles due to the Lorentz force==
| |
| {{Main|Guiding center}}
| |
| [[File:charged-particle-drifts.svg|300px|thumbnail|right|'''Charged particle drifts''' in a homogeneous magnetic field. (A) No disturbing force (B) With an electric field, E (C) With an independent force, F (e.g. gravity) (D) In an inhomogeneous magnetic field, grad H]]
| |
| In many cases of practical interest, the motion in a [[magnetic field]] of an [[electric charge|electrically charged]] particle (such as an [[electron]] or [[ion]] in a [[Plasma (physics)|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.<ref name=Jackson2>See Jackson page 2. The book lists the four modern Maxwell's equations, and then states, "Also essential for consideration of charged particle motion is the Lorentz force equation, '''F''' = ''q'' ( '''E'''+ '''v <big>×</big> B''' ), which gives the force acting on a point charge ''q'' in the presence of electromagnetic fields."</ref><ref name=Griffiths50>See Griffiths page 326, which states that Maxwell's equations, "together with the [Lorentz] force law...summarize the entire theoretical content of classical electrodynamics".</ref> 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 (many-body 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'''.<ref name=Jackson20>See, for example, Jackson p777-8.</ref><ref>{{cite book|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|year=1973|pages=72–73|isbn=0-7167-0344-0}}. These authors use the Lorentz force in tensor form as definer of the [[electromagnetic tensor]] ''F'', in turn the fields '''E''' and '''B'''.</ref><ref>{{cite book|title=Electromagnetism|edition=2nd|author=I.S. Grant, W.R. Phillips, Manchester Physics|publisher=John Wiley & Sons|year=2008|page=122|isbn=978-0-471-92712-9}}</ref> 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'':
| |
| | |
| ::<math>\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})</math>
| |
| | |
| This ''is'' valid; countless experiments have shown that it is, even for particles approaching the [[speed of light]] (that is, [[Norm (mathematics)#Euclidean norm|magnitude]] of '''v''' = |'''v'''| = ''c'').<ref>{{cite book|title=Electromagnetism (2nd Edition)|author=I.S. Grant, W.R. Phillips, Manchester Physics|publisher=John Wiley & Sons|year=2008|page=123|isbn=978-0-471-92712-9}}</ref> So the two [[vector field]]s '''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 infinitesimally-small 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 [[Abraham-Lorentz force|radiation reaction force]]) and indirectly (by affecting the motion of nearby charges and currents). Moreover, net force must include [[gravity]], [[Electroweak interaction|electroweak]], and any other forces aside from electromagnetic force.
| |
| | |
| ==Force on a current-carrying wire==
| |
| [[File:Regla mano derecha Laplace.svg|right|thumb|250px|Right-hand rule for a current-carrying wire in a magnetic field B]]
| |
| 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:
| |
| :<math>\mathbf{F} = I \boldsymbol{\ell} \times \mathbf{B} \,\!</math>
| |
| | |
| 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 ''d'''''ℓ''', then adding up all these forces by [[integration (calculus)|integration]]. Formally, the net force on a stationary, rigid wire carrying a steady current ''I'' is
| |
| | |
| :<math>\mathbf{F} = I\int \mathrm{d}\boldsymbol{\ell}\times \mathbf{B}</math>
| |
| | |
| 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 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]].
| |
| | |
| ==EMF==
| |
| | |
| The magnetic force (''q'' '''v''' <big>×</big> '''B''') component of the Lorentz force is responsible for ''motional'' [[electromotive force]] (or ''motional EMF''), the phenomenon underlying many [[electrical generator]]s. When a [[Electrical conductor|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 (''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 [[Electromagnetic induction#Maxwell–Faraday equation|Maxwell–Faraday equation]] (one of the four modern [[Maxwell's equations]]).<ref name=Griffiths302>See Griffiths pages 301–3.</ref>
| |
| | |
| 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 [[Lorentz force#Lorentz force and Faraday.27s law of induction|above]].) Einstein's [[theory of special relativity]] was partially motivated by the desire to better understand this link between the two effects.<ref name=Griffiths302/> 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 ''E''-field can change in whole or in part to a ''B''-field or ''vice versa''.<ref name=Chow>
| |
| {{cite book
| |
| |author=Tai L. Chow
| |
| |title=Electromagnetic theory
| |
| |year= 2006
| |
| |page =395
| |
| |publisher=Jones and Bartlett
| |
| |location=Sudbury MA
| |
| |isbn=0-7637-3827-1
| |
| |url=http://books.google.com/?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271}}
| |
| </ref>
| |
| | |
| ==Lorentz force and Faraday's law of induction==
| |
| {{main|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:
| |
| | |
| :<math>\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}</math>
| |
| | |
| where
| |
| | |
| :<math> \Phi_B = \iint_{\Sigma(t)} \mathrm{d} \mathbf{A} \cdot \mathbf{B}(\mathbf{r}, t)</math>
| |
| | |
| 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'', d'''A''' 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's law of induction|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:<ref name=Landau>
| |
| {{cite book
| |
| |author=Landau, L. D., Lifshit︠s︡, E. M., & Pitaevskiĭ, L. P.
| |
| |title=Electrodynamics of continuous media; Volume 8 ''Course of Theoretical Physics''
| |
| |year= 1984
| |
| |page =§63 (§49 pp. 205–207 in 1960 edition)
| |
| |edition=Second
| |
| |publisher=Butterworth-Heinemann
| |
| |location=Oxford
| |
| |isbn=0-7506-2634-8
| |
| |url=http://worldcat.org/search?q=0750626348&qt=owc_search}}
| |
| </ref>
| |
| | |
| :<math>\mathcal{E} =\oint_{\part \Sigma (t)} \mathrm{d} \boldsymbol{\ell} \cdot \mathbf{F} / q</math>
| |
| | |
| where
| |
| | |
| :<math>\mathbf{E} = \mathbf{F} / q</math>
| |
| | |
| is the electric field and d'''ℓ''' is an [[infinitesimal]] vector element of the contour ∂Σ(''t'').
| |
| | |
| NB: Both d'''ℓ''' and d'''A''' 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'':
| |
| | |
| :<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \ .</math>
| |
| | |
| The Maxwell-Faraday equation also can be written in an ''integral form'' using the [[Kelvin-Stokes theorem]]:.<ref name=Harrington>
| |
| {{cite book
| |
| |author=Roger F Harrington
| |
| |title=Introduction to electromagnetic engineering
| |
| |year= 2003
| |
| |page =56
| |
| |publisher=Dover Publications
| |
| |location=Mineola, NY
| |
| |isbn=0-486-43241-6
| |
| |url=http://books.google.com/?id=ZlC2EV8zvX8C&pg=PA57&dq=%22faraday%27s+law+of+induction%22}}
| |
| </ref>
| |
| | |
| So we have, the Maxwell Faraday equation:
| |
| | |
| :<math> \oint_{\partial \Sigma(t)}\mathrm{d} \boldsymbol{\ell} \cdot \mathbf{E}(\mathbf{r},\ t) = - \ \iint_{\Sigma(t)} \mathrm{d} \mathbf {A} \cdot {{\mathrm{d} \,\mathbf {B}(\mathbf{r},\ t)} \over \mathrm{d}t } </math>
| |
| | |
| and the Faraday Law,
| |
| | |
| :<math> \oint_{\partial \Sigma(t)}\mathrm{d} \boldsymbol{\ell} \cdot \mathbf{F}/q(\mathbf{r},\ t) = - \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma(t)} \mathrm{d} \mathbf {A} \cdot \mathbf{B}(\mathbf{r},\ t). </math>
| |
| | |
| The two are equivalent if the wire is not moving. Using the [[Leibniz integral rule]] and that ''div'' '''B''' = 0, results in,
| |
| | |
| :<math> \oint_{\partial \Sigma(t)} \mathrm{d} \boldsymbol{\ell} \cdot \mathbf{F}/q(\mathbf{r}, t) =
| |
| - \iint_{\Sigma(t)} \mathrm{d} \mathbf{A} \cdot \frac{\partial}{\partial t} \mathbf{B}(\mathbf{r}, t) +
| |
| \oint_{\partial \Sigma(t)} \!\!\!\!\mathbf{v} \times \mathbf{B} \,\mathrm{d} \boldsymbol{\ell}
| |
| </math>
| |
| | |
| and using the Maxwell Faraday equation,
| |
| | |
| :<math> \oint_{\partial \Sigma(t)} \mathrm{d} \boldsymbol{\ell} \cdot \mathbf{F}/q(\mathbf{r},\ t) =
| |
| \oint_{\partial \Sigma(t)} \mathrm{d} \boldsymbol{\ell} \cdot \mathbf{E}(\mathbf{r},\ t) +
| |
| \oint_{\partial \Sigma(t)}\!\!\!\! \mathbf{v} \times \mathbf{B}(\mathbf{r},\ t)\, \mathrm{d} \boldsymbol{\ell}
| |
| </math>
| |
| | |
| since this is valid for any wire position it implies that,
| |
| | |
| :<math> \mathbf{F}= q\,\mathbf{E}(\mathbf{r},\ t) + q\,\mathbf{v} \times \mathbf{B}(\mathbf{r},\ t).</math>
| |
| | |
| 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 [[Faraday paradox#Inapplicability of Faraday's law|inapplicability of Faraday's law]].
| |
| | |
| If the magnetic field is fixed in time and the conducting loop moves through the field, the magnetic flux Φ<sub>''B''</sub> linking the loop can change in several ways. For example, if the '''B'''-field varies with position, and the loop moves to a location with different '''B'''-field, Φ<sub>''B''</sub> will change. Alternatively, if the loop changes orientation with respect to the '''B'''-field, the '''B''' • d'''A''' differential element will change because of the different angle between '''B''' and d'''A''', also changing Φ<sub>''B''</sub>. As a third example, if a portion of the circuit is swept through a uniform, time-independent '''B'''-field, and another portion of the circuit is held stationary, the flux linking the entire closed circuit can change due to the shift in relative position of the circuit's component parts with time (surface ∂Σ(''t'') time-dependent). In all three cases, Faraday's law of induction then predicts the EMF generated by the change in Φ<sub>''B''</sub>.
| |
| | |
| Note that the Maxwell Faraday's equation implies that the Electric Field '''E''' is non conservative when the Magnetic Field '''B''' varies in time, and is not expressible as the gradient of a [[scalar field]], and not subject to the [[gradient theorem]] since its rotational is not zero. See also.<ref name="Landau"/><ref>{{cite book
| |
| |author=M N O Sadiku
| |
| |title=Elements of elctromagnetics
| |
| |year= 2007
| |
| |page =391
| |
| |edition=Fourth
| |
| |publisher=Oxford University Press
| |
| |location=NY/Oxford
| |
| |isbn=0-19-530048-3
| |
| |url=http://books.google.com/?id=w2ITHQAACAAJ&dq=ISBN0-19-530048-3}}</ref>
| |
| | |
| ==Lorentz force in terms of potentials==
| |
| | |
| {{see also|Mathematical descriptions of the electromagnetic field|Maxwell's equations|Helmholtz decomposition}}
| |
| | |
| The '''E''' and '''B''' fields can be replaced by the [[magnetic vector potential]] '''A''' and ([[Scalar (mathematics)|scalar]]) [[electrostatic potential]] ''ϕ'' by
| |
| | |
| :<math> \mathbf{E} = - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t }</math>
| |
| :<math>\mathbf{B} = \nabla \times \mathbf{A}</math>
| |
| | |
| where ∇ is the [[gradient]], ∇• is the [[divergence]], ∇ <big>×</big> is the [[Curl (mathematics)|curl]].
| |
| | |
| The force becomes
| |
| | |
| :<math>\mathbf{F} = q\left[-\nabla \phi- \frac{\partial \mathbf{A}}{\partial t}+\mathbf{v}\times(\nabla\times\mathbf{A})\right]</math>
| |
| | |
| and using an identity for the [[Triple product#Vector triple product|triple product]] simplifies to
| |
| | |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math>\mathbf{F} = q\left[-\nabla \phi- \frac{\partial \mathbf{A}}{\partial t}+ \nabla(\mathbf{v}\cdot\mathbf{A})-(\mathbf{v}\cdot\nabla)\mathbf{A} \right]</math>
| |
| |cellpadding= 6
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F5FFFA}}
| |
| | |
| using the [[chain rule]], the [[total derivative]] of '''A''' is:
| |
| | |
| :<math>\frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t} = \frac{\partial\mathbf{A}}{\partial t}+(\mathbf{v}\cdot\nabla)\mathbf{A} </math>
| |
| | |
| so the above expression can be rewritten as;
| |
| | |
| :<math>\mathbf{F} = q\left[-\nabla (\phi-\mathbf{v}\cdot\mathbf{A})- \frac{d\mathbf{A}}{\mathrm{d}t}\right]</math>
| |
| | |
| which can take the convenient Euler-Lagrange form
| |
| | |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math>\mathbf{F} = q\left[-\nabla_{\mathbf{x}}(\phi-\dot{\mathbf{x}}\cdot\mathbf{A})+ \frac{\mathrm{d}}{\mathrm{d}t}\nabla_{\dot{\mathbf{x}}}(\phi-\dot{\mathbf{x}}\cdot\mathbf{A})\right]</math>
| |
| |cellpadding= 6
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F5FFFA}}
| |
| | |
| ==Lorentz force and analytical mechanics==<!--this section links from [[Momentum#Lagrangian and Hamiltonian formulation]]-->
| |
| {{see also|Momentum#Lagrangian and Hamiltonian formulation|l1=Momentum}}
| |
| 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:<ref>Classical Mechanics (2nd Edition), T.W.B. Kibble, European Physics Series, Mc Graw Hill (UK), 1973, ISBN 07-084018-0.</ref>
| |
| | |
| :<math>L=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+q\mathbf{A}\cdot\mathbf{\dot{r}}-q\phi</math>
| |
| | |
| where '''A''' and ''ϕ'' are the potential fields as above. Using [[Lagrangian mechanics|Lagrange's equations]], the equation for the Lorentz force can be obtained.
| |
| | |
| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
| |
| !Derivation of Lorentz force from classical Lagrangian (SI units)
| |
| |-
| |
| |For an '''A''' field, a particle moving with velocity '''v''' = '''ṙ''' has [[potential momentum]] <math>q\mathbf{A}(\mathbf{r},t)</math>, so its potential energy is <math>q\mathbf{A}(\mathbf{r},t)\cdot\mathbf{\dot{r}}</math>. For a ''ϕ'' field, the particle's potential energy is <math>q\phi(\mathbf{r},t)</math>.
| |
| | |
| The total [[potential energy]] is then:
| |
| | |
| :<math>V=q\phi-q\mathbf{A}\cdot\mathbf{\dot{r}}</math>
| |
| | |
| and the [[kinetic energy]] is:
| |
| | |
| :<math>T=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}</math>
| |
| | |
| hence the Lagrangian:
| |
| | |
| :<math>L=T-V=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+q\mathbf{A}\cdot\mathbf{\dot{r}}-q\phi</math>
| |
| | |
| :<math>L=\frac{m}{2}(\dot{x}^2+\dot{y}^2+\dot{z}^2) + q(\dot{x}A_x+\dot{y}A_y+\dot{z}A_z) - q\phi</math>
| |
| | |
| Lagrange's equations are
| |
| | |
| :<math>\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{x}}=\frac{\partial L}{\partial x}</math>
| |
| | |
| (same for ''y'' and ''z''). So calculating the partial derivatives:
| |
| | |
| :<math>\begin{align}\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{x}} & =m\ddot{x}+q\frac{\mathrm{d} A_x}{\mathrm{d}t} \\
| |
| & = m\ddot{x}+ \frac{q}{\mathrm{d}t}\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}</math>
| |
| | |
| :<math>\frac{\partial L}{\partial x}= -q\frac{\partial \phi}{\partial x}+ q\left(\frac{\partial A_x}{\partial x}\dot{x}+\frac{\partial A_y}{\partial x}\dot{y}+\frac{\partial A_z}{\partial x}\dot{z}\right)</math>
| |
| | |
| equating and simplifying:
| |
| | |
| :<math>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)= -q\frac{\partial \phi}{\partial x}+ q\left(\frac{\partial A_x}{\partial x}\dot{x}+\frac{\partial A_y}{\partial x}\dot{y}+\frac{\partial A_z}{\partial x}\dot{z}\right)</math>
| |
| | |
| :<math>\begin{align} F_x & = -q\left(\frac{\partial \phi}{\partial x}+\frac{\partial A_x}{\partial t}\right) + q\left[\dot{y}\left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right)+\dot{z}\left(\frac{\partial A_z}{\partial x}-\frac{\partial A_x}{\partial z}\right)\right] \\
| |
| & = 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}</math>
| |
| | |
| and similarly for the ''y'' and ''z'' directions. Hence the force equation is:
| |
| | |
| :<math>\mathbf{F}= q(\mathbf{E} + \mathbf{\dot{r}}\times\mathbf{B})</math>
| |
| |}
| |
| | |
| 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
| |
| | |
| :<math>L = -m\sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2} + e \mathbf{A}(\mathbf{r})\cdot\dot{\mathbf{r}} - e \phi(\mathbf{r}) \,\!</math>
| |
| | |
| 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 Mechanics|quantum mechanically]] is an extra [[phase (waves)|phase]] a charged particle gets when it is moving along a vector potential.
| |
| | |
| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
| |
| !Derivation of Lorentz force from relativistic Lagrangian (SI units)
| |
| |-
| |
| |
| |
| The equations of motion derived by [[calculus of variations|extremizing]] the action (see [[matrix calculus]] for the notation):
| |
| | |
| :<math> \frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t} =\frac{\partial L}{\partial \mathbf{r}} = e {\partial \mathbf{A} \over \partial \mathbf{r}}\cdot \dot{\mathbf{r}} - e {\partial \phi \over \partial \mathbf{r} }\,\!</math>
| |
| | |
| :<math>\mathbf{P} -e\mathbf{A} = \frac{m\dot{\mathbf{r}}}{\sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2}}\,</math>
| |
| | |
| are the same as [[Hamiltonian mechanics|Hamilton's equations of motion]]:
| |
| | |
| :<math> \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = \frac{\partial}{\partial \mathbf{p}}\left ( \sqrt{(\mathbf{P}-e\mathbf{A})^2 +(mc^2)^2} + e\phi \right ) \,\!</math>
| |
| :<math> \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = -{\partial \over \partial \mathbf{r}}\left ( \sqrt{(\mathbf{P}-e\mathbf{A})^2 + (mc^2)^2} + e\phi \right ) \,\!</math>
| |
| | |
| both are equivalent to the noncanonical form:
| |
| | |
| :<math> \frac{\mathrm{d}}{\mathrm{d}t}\left ( {m\dot{\mathbf{r}} \over \sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2}} \right ) = e\left ( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right ) . \,\!</math>
| |
| | |
| This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.
| |
| |}
| |
| | |
| ==Equation (cgs units)==
| |
| | |
| {{see also|cgs units}}
| |
| | |
| The above-mentioned formulae use [[SI units]] which are the most common among experimentalists, technicians, and engineers. In [[Gaussian units|cgs-Gaussian units]], which are somewhat more common among theoretical physicists, one has instead
| |
| : <math>\mathbf{F} = q_\mathrm{cgs} \left(\mathbf{E}_\mathrm{cgs} + \frac{\mathbf{v}}{c} \times \mathbf{B}_\mathrm{cgs}\right).</math>
| |
| where ''c'' is the [[speed of light]]. Although this equation looks slightly different, it is completely equivalent, since
| |
| one has the following relations:
| |
| | |
| :<math>q_\mathrm{cgs}=\frac{q_\mathrm{SI}}{\sqrt{4\pi \epsilon_0}},\quad \mathbf E_\mathrm{cgs} =\sqrt{4\pi\epsilon_0}\,\mathbf E_\mathrm{SI},\quad \mathbf B_\mathrm{cgs} ={\sqrt{4\pi /\mu_0}}\,{\mathbf B_\mathrm{SI}}</math>
| |
| | |
| where ε<sub>0</sub> is the [[vacuum permittivity]] and μ<sub>0</sub> 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====
| |
| | |
| {{main|Covariant formulation of classical electromagnetism|Mathematical descriptions of the electromagnetic field}}
| |
| | |
| Using the [[metric signature]] (-1,1,1,1), The Lorentz force for a charge ''q'' can be written in [[Lorentz covariance|covariant form]]:
| |
| | |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math> \frac{\mathrm{d} p^\alpha}{\mathrm{d} \tau} = q U_\beta F^{\alpha \beta} </math>
| |
| |cellpadding
| |
| |border
| |
| |border colour = #50C878
| |
| |background colour = #ECFCF4}}
| |
| | |
| where ''p<sup>α</sup>'' is the [[four-momentum]], defined as:
| |
| | |
| :<math>p^\alpha = \left(p_0, p_1, p_2, p_3 \right) = \left(m c, p_x, p_y, p_z \right) \, ,</math>
| |
| | |
| <math> \scriptstyle \tau</math> the [[proper time]] of the particle, ''F<sup>αβ</sup>'' the contravariant [[electromagnetic tensor]]
| |
| | |
| :<math>F^{\alpha \beta} = \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}
| |
| </math>
| |
| | |
| and ''U'' is the covariant [[four-velocity|4-velocity]] of the particle, defined as:
| |
| | |
| :<math>U_\beta = \left(U_0, U_1, U_2, U_3 \right) = \gamma \left(-c, u_x, u_y, u_z \right) \, ,</math>
| |
| | |
| where <math>\scriptstyle \gamma </math> is the Lorentz factor defined above.
| |
| | |
| The fields are transformed to a frame moving with constant relative velocity by:
| |
| | |
| :<math> \acute{F}^{\mu \nu} = {\Lambda^{\mu}}_{\alpha} {\Lambda^{\nu}}_{\beta} F^{\alpha \beta} \, ,</math>
| |
| | |
| where Λ''<sup>μ</sup><sub>α</sub>'' is the [[Lorentz transformation]] tensor.
| |
| | |
| ====Translation to vector notation====
| |
| | |
| The α = 1 component (''x''-component) of the force is
| |
| :<math> \frac{\mathrm{d} p^1}{\mathrm{d} \tau} = q U_\beta F^{1 \beta} = q\left(U_0 F^{10} + U_1 F^{11} + U_2 F^{12} + U_3 F^{13} \right) .\,</math>
| |
| | |
| Substituting the components of the covariant electromagnetic tensor ''F'' yields
| |
| | |
| :<math> \frac{\mathrm{d} p^1}{\mathrm{d} \tau} = q \left[U_0 \left(\frac{-E_x}{c} \right) + U_2 (B_z) + U_3 (-B_y) \right]. \,</math>
| |
| | |
| Using the components of covariant [[four-velocity]] yields
| |
| | |
| :<math> \begin{align}
| |
| \frac{\mathrm{d} p^1}{\mathrm{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} </math>
| |
| | |
| The calculation for α = 2, 3 (force components in the ''y'' and ''z'' directions) yields similar results, so collecting the 3 equations into one:
| |
| | |
| :<math> \frac{\mathrm{d} \mathbf{p} }{\mathrm{d} \tau} = q \gamma\left( \mathbf{E} + \mathbf{u} \times \mathbf{B} \right) \, , </math>
| |
| | |
| which is the Lorentz force.
| |
| | |
| ===STA form of the Lorentz force===
| |
| | |
| The electric and magnetic fields are [[Classical electromagnetism and special relativity|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,<ref>{{cite web|last=Hestenes|first=David|authorlink=David Hestenes|title=SpaceTime Calculus|url=http://geocalc.clas.asu.edu/html/STC.html}}</ref> <math>\mathcal{F}</math>, and an arbitrary time-direction, <math>\gamma_0</math>, where
| |
| | |
| : <math>\mathbf{E} = (\mathcal{F}\cdot\gamma_0)\gamma_0</math>
| |
| | |
| and
| |
| | |
| : <math>i\mathbf{B} = (\mathcal{F}\wedge\gamma_0)\gamma_0</math>
| |
| | |
| <math>\mathcal F</math> 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 <math>\gamma_0</math> 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 <math>v=\dot x</math>, where
| |
| | |
| : <math>v^2 = 1,</math>
| |
| | |
| (which shows our choice for the metric) and the velocity is
| |
| | |
| : <math>\mathbf{v} = cv \wedge \gamma_0 / (v \cdot \gamma_0).</math>
| |
| | |
| The proper (invariant is an inadequate term because no transformation has been defined) form of the Lorentz force law is simply
| |
| | |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math> F = q\mathcal{F}\cdot v</math>
| |
| |cellpadding
| |
| |border
| |
| |border colour = #50C878
| |
| |background colour = #ECFCF4}}
| |
| | |
| Note that the order is important because between a bivector and a vector the dot product is anti-symmetric. 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:
| |
| *[[Cyclotron]]s and other circular path [[particle accelerator]]s
| |
| *[[Mass spectrometer]]s
| |
| *Velocity Filters
| |
| *[[Magnetron]]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:
| |
| {{multicol}}
| |
| *[[Electric motor]]s
| |
| *[[Railgun]]s
| |
| *[[Linear motor]]s
| |
| *[[Loudspeaker]]s
| |
| {{multicol-break}}
| |
| *[[Magnetoplasmadynamic thruster]]s
| |
| *[[Electrical generator]]s
| |
| *[[Homopolar generator]]s
| |
| *[[Linear alternator]]s
| |
| {{multicol-end}}
| |
| | |
| ==See also==
| |
| {{multicol}}
| |
| * [[Hall effect]]
| |
| * [[Electromagnetism]]
| |
| * [[Gravitomagnetism]]
| |
| * [[Ampère's force law]]
| |
| * [[Hendrik Lorentz]]
| |
| * [[Maxwell's equations]]
| |
| * [[Formulation of Maxwell's equations in special relativity]]
| |
| {{multicol-break}}
| |
| * [[Moving magnet and conductor problem]]
| |
| * [[Abraham–Lorentz force]]
| |
| * [[Larmor formula]]
| |
| * [[Cyclotron radiation]]
| |
| * [[Magnetic potential]]
| |
| * [[Magnetoresistance]]
| |
| {{multicol-break}}
| |
| * [[Scalar potential]]
| |
| * [[Helmholtz decomposition]]
| |
| * [[Guiding center]]
| |
| * [[Field line]]
| |
| {{multicol-end}}
| |
| {{Commons|Lorentz force}}
| |
| | |
| ==Footnotes==
| |
| {{Reflist|30em}}
| |
| | |
| ==References==
| |
| | |
| The numbered references refer in part to the list immediately below.
| |
| | |
| *{{Cite book |first1 = Richard Phillips |last1 = Feynman |author-link = Richard Feynman |first2 = Robert B. |last2 = Leighton |first3 = Matthew L. |last3 = Sands |title = The Feynman lectures on physics (3 vol.) |publisher = Pearson / Addison-Wesley | year= 2006 |isbn = 0-8053-9047-2 |postscript = <!--None-->}}: volume 2.
| |
| | |
| *{{Cite book |first = David J. |last = Griffiths |title = Introduction to electrodynamics |edition = 3rd |place = Upper Saddle River, [NJ.] |publisher = Prentice-Hall |year = 1999 |isbn = 0-13-805326-X |postscript = <!--None-->}}
| |
| | |
| *{{Cite book |first = John David |last = Jackson |title = Classical electrodynamics |edition = 3rd
| |
| |location = New York, [NY.] |publisher = Wiley | year = 1999 |isbn = 0-471-30932-X |postscript = <!--None-->}}
| |
| | |
| *{{Cite book |first1 = Raymond A. |last1 = Serway |first2 = John W., Jr. |last2 = Jewett |title = Physics for scientists and engineers, with modern physics |place = Belmont, [CA.] |publisher = Thomson Brooks/Cole |year = 2004 |isbn = 0-534-40846-X |postscript = <!--None-->}}
| |
| | |
| *{{Cite book |first = Mark A. |last = Srednicki |title= Quantum field theory |url=http://books.google.com/?id=5OepxIG42B4C&pg=PA315&dq=isbn=9780521864497 |place = Cambridge, [England] ; New York [NY.] |publisher = Cambridge University Press | year=2007 |isbn = 978-0-521-86449-7 |postscript = <!--None-->}}
| |
| | |
| ==External links==
| |
| *[http://www.magnet.fsu.edu/education/tutorials/java/lorentzforce/index.html Interactive Java tutorial on the Lorentz force] National High Magnetic Field Laboratory
| |
| *[http://www.youtube.com/watch?v=mxMMqNrm598 Lorentz force (demonstration)]
| |
| *[http://www.nadn.navy.mil/Users/physics/tank/Public/FaradaysLaw.pdf Faraday's law: Tankersley and Mosca]
| |
| *[http://hyperphysics.phy-astr.gsu.edu/HBASE/hframe.html Notes from Physics and Astronomy HyperPhysics at Georgia State University]; see also [http://hyperphysics.phy-astr.gsu.edu/HBASE/hframe.html home page]
| |
| * [http://chair.pa.msu.edu/applets/Lorentz/a.htm Interactive Java applet on the magnetic deflection of a particle beam in a homogeneous magnetic field] by Wolfgang Bauer
| |
| | |
| [[Category:Concepts in physics]]
| |
| [[Category:Electromagnetism]]
| |
| [[Category:Maxwell's equations]]
| |