Lorentz Force - Wikipedia, The Free Encyclopedia

7/29/2019 Lorentz Force - Wikipedia, The Free Encyclopedia

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Lorentz forceFrom Wikipedia, the free encyclopedia

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 experiencea force

(in SI units). Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called Laplace f orce ), the el ectromotive force in a wire loop moving through a magnetic field (an aspect of Faraday's law of induction), and the f orce on a particle which might be traveling near the speed of light (relativistic form of the Lorentz force).

The first d erivation of the Lorentz force is commonly attributed to Oliver Heaviside in 1889, [1] although other historianssuggest an earlier or igin in an 1865 paper by J ames Clerk Maxwell. [2] Lorentz derived it a few years after Heaviside.

Conte nts

1 Equation (SI units)1.1 Charged particle1.2 Continuous charge distrib ution

2 History3 Traject ories of particles due to the Lorentz force4 Signif icance of the Lorentz for ce5 Lorentz force la w as the definition of E and B6 Force on a current-carrying wire7 EMF

8 Lorentz force and Faraday's law of induction9 Lorentz force in terms of po tentials10 Lorentz force and analytical mechanics11 Equation (cgs units)12 Relativistic form of the Lorentz force

12.1 Covariant form of the Lorentz force12.1.1 Field tensor 12.1.2 Translation to vector notation

12.2 STA form of the Lorentz force13 Applications14 See also15 Footnotes16 References17 External links

Equation (SI units)

See also: SI units

Charged particleThe force F acting on a particle of electric charge q with instantaneous velocity v, due to an external electric field E andmagnetic field B , is given by: [3]
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Lorentz force f on a charged particle (of charge q) in motion(instantaneous velocity v). The Efield and B field vary in space andtime.

Lorentz force (per unit 3-volume)f on a continuous chargedistribution (charge density ) inmotion. The 3-current density Jcorresponds to the motion of thecharge element dq in volumeelement dV and varies throughoutthe continuum.

where is the vector cross product. All boldface quantities are vectors. Moreexplicitly stated:

in which r is the position vector of the charged particle, t is time, and the overdot is atime derivative.

A positively charged particle will be accelerated in the same linear orientation as the Efield, but will curve perpendicularly to both the instantaneous velocity vector v and theB field according to the right-hand rule (in detail, if the thumb of the right hand pointsalong 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 themagnetic force .[4] According to some definitions, the term "Lorentz force" refersspecifically to the formula for the magnetic force, [5] with the total electromagneticforce (including the electric force) given some other (nonstandard) name. This articlewill 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 current-carrying wire in amagnetic 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:

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 chargedistribution dV , the result is:

where f is the force density (force per unit volume) and is the charge density(charge per unit volume). Next, the current density corresponding to the motion of thecharge continuum is

so the continuous analogue to the equation is [6]

The total force is the volume integral over the charge distribution:
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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 sc reen.

Beam of electrons moving in a circle, dueto the presence of a magnetic field. Purplelight is emitted along the electron path, due

to the electrons colliding with gasmolecules in the bulb. Using a Teltron tube.

Charged particles experiencing the Lorentz force.

By eliminating and J , using Maxwell's equations, and manipulating using the theorems of vector calculus, this form of theequation can be used to derive the Maxwell stress tensor , in turn this can be combined with the Poynting vector S to obtainthe electromagnetic stress-energy tensor T used in general relativity. [6]

In terms of and S, another way to write the Lorentz force (per unit 3d volume) is [6]

where c is the speed of light and denotes the divergence of a tensor field. Rather than the amount of charge and itsvelocity in electric and magnetic fields, this equation relates the energy flux (flow of energy per unit time per unit distance) inthe fields to the force exerted on a charge distribution. See Covariant formulation of classical electromagnetism for moredetails.

History

Early attempts to quantitativelydescribe the electromagnetic forcewere made in the mid-18thcentury. It was proposed that theforce on magnetic poles, byJohann Tobias Mayer and othersin 1760 [citation needed ], andelectrically charged objects, byHenry Cavendish in1762 [citation needed ], obeyed aninverse-square law. However, in

both cases the experimental proof was neither complete nor

conclusive. It was not until 1784when Charles-Augustin deCoulomb, using a torsion balance,was able to definitively showthrough experiment that this wastrue. [7] Soon after the discovery in 1820 by H. C. rsted that a magnetic needle is acted on by a voltaic current, Andr-Marie Ampre that same year was able to devise through experimentation the formula for the angular dependence of theforce 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 equationin relation to electric currents, [2] however, in the time of Maxwell it was not evident how his equations related to the forceson moving charged objects. J. J. Thomson was the first to attempt to derive from Maxwell's field equations theelectromagnetic forces on a moving charged object in terms of the object's properties and external fields. Interested indetermining the electromagnetic behavior of the charged particles in cathode rays, Thomson published a paper in 1881wherein he gave the force on the particles due to an external magnetic field as [1]

Thomson derived the correct basic form of the formula, but, because of some miscalculations and an incomplete descriptionof the displacement current, included an incorrect scale-factor of a half in front of the formula. It was Oliver Heaviside, whohad invented the modern vector notation and applied them to Maxwell's field equations, that in 1885 and 1889 fixed themistakes 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
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Charged particle drifts in a homogeneousmagnetic field. (A) No disturbing force (B) With anelectric field, E (C) With an independent force, F(e.g. gravity) (D) In an inhomogeneous magneticfield, grad H

contributions to the total force from both the electric and the magnetic fields. Lorentz began by abandoning the Maxwelliandescriptions of the ether and conduction. Instead, Lorentz made a distinction between matter and the luminiferous aether andsought to apply the Maxwell equations at a microscopic scale. Using the Heaviside's version of the Maxwell equations for astationary ether and applying Lagrangian mechanics (see below), Lorentz arrived at the correct and complete form of theforce 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 slowdrift of this point. The drift speeds may differ for various speciesdepending on their charge states, masses, or temperatures, possiblyresulting in electric currents or chemical separation.

Significance of the Lorentz forceWhile the modern Maxwell's equations describe how electricallycharged particles and currents or moving charged particles give riseto electric and magnetic fields, the Lorentz force law completes that

picture by describing the force acting on a moving point charge q inthe presence of electromagnetic fields. [3][16] The Lorentz force lawdescribes the effect of E and B upon a point charge, but suchelectromagnetic forces are not the entire picture. Charged particlesare 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 currentdensities. The response of a point charge to the Lorentz law is oneaspect; 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 bothrespond to the E and B fields and generate these fields. Complextransport equations must be solved to determine the time and spatialresponse of charges, for example, the Boltzmann equation or theFokkerPlanck equation or the NavierStokes 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, GreenKubo relations and Green'sfunction (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 electricand magnetic fields E and B .[17][18][19] To be specific, the Lorentz force is understood to be the following empiricalstatement:

The electromagnetic force F on a test charge at a given point and time is a certain function of its charge qand velocity v , which can be parameterized by exactly two vectors E and B , in the functional form :
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Right-hand rule for a current-carrying wirein a magnetic field B

This is valid; countless experiments have shown that it is, even for particles approaching the speed of light (that is, magnitudeof 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 forcea 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 opposedto the hypothetical "test charge" of infinitesimally-small mass and charge) would generate its own finite E and B fields, whichwould 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 exampleBremsstrahlung 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 current-carrying wire

When a wire carrying an electrical current is placed in a magnetic field, eachof the moving charges, which comprise the current, experiences the Lorentzforce, and together they can create a macroscopic force on the wire(sometimes called the Laplace force ). By combining the Lorentz force lawabove with the definition of electrical current, the following equation results, inthe case of a straight, stationary wire:

where is a vector whose magnitude is the length of wire, and whosedirection is along the wire, aligned with the direction of conventional currentflow 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 , then adding up all these forces by integration. Formally, the net force on a stationary, rigid wire carrying a steady

current I is

This is the net force. In addition, there will usually be torque, plus other effects if the wire is not perfectly rigid.

One application of this is Ampre's force law, which describes how two current-carrying wires can attract or repel eachother, since each experiences a Lorentz force from the other's magnetic field. For more information, see the article: Ampre'sforce 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, themagnetic 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 electricforce ( qE) term in the Lorentz Force equation. The electric field in question is created by the changing magnetic field,resulting in an induced EMF, as described by the MaxwellFaraday equation (one of the four modern Maxwell's

equations).[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 relativitywas partially motivated by the desire to better understand this link between the two effects. [21] In fact, the electric and
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magnetic fields are different faces of the same electromagnetic field, and in moving from one inertial frame to another, thesolenoidal vector field portion of the E -field can change in whole or in part to a B-field or vice versa .[22]

Lorentz force and Faraday's law of induction

Main article: 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 wireis:

where

is the magnetic flux through the loop, B is the magnetic field, ( t ) is a surface bounded by the closed contour ( t ), at all attime t , dA is an infinitesimal vector area element of ( t ) (magnitude is the area of an infinitesimal patch of surface, direction isorthogonal 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 amoving wire.

From Faraday's law of induction (that is valid for a moving wire, for instance in a motor) and the Maxwell Equations, theLorentz Force can be deduced. The reverse is also true, the Lorentz force and the Maxwell Equations can be used to derivethe Faraday Law.

Let ( t ) be the moving wire, moving together without rotation and with constant velocity v and ( t ) be the internal surface of the wire. The EMF around the closed path ( t ) is given by: [23]

where

is the electric field and d is an infinitesimal vector element of the contour ( t ).

NB: Both d and d A have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in the articleKelvin-Stokes theorem.

The above result can be compared with the version of Faraday's law of induction that appears in the modern Maxwell'sequations, called here the Maxwell-Faraday equation :

The Maxwell-Faraday equation also can be written in an integral form using the Kelvin-Stokes theorem:. [24]

So we have, the Maxwell Faraday equation:
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and the Faraday Law,

The two are equivalent if the wire is not moving. Using the Leibniz integral rule and that div B = 0, results in,

and using the Maxwell Faraday equation,

since this is valid for any wire position it implies that,

Faraday's law of induction holds whether the loop of wire is rigid and stationary, or in motion or in process of deformation,and it holds whether the magnetic field is constant in time or changing. However, there are cases where Faraday's law iseither inadequate or difficult to use, and application of the underlying Lorentz force law is necessary. See inapplicability of Faraday's law.

If the magnetic field is fixed in time and the conducting loop moves through the field, the magnetic flux B linking the loop canchange in several ways. For example, if the B-field varies with position, and the loop moves to a location with different B-field, B will change. Alternatively, if the loop changes orientation with respect to the B-field, the B d A differential elementwill change because of the different angle between B and d A, also changing B. As a third example, if a portion of the circuitis 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 ( t )time-dependent). In all three cases, Faraday's law of induction then predicts the EMF generated by the change in B.

Note that the Maxwell Faraday's equation implies that the Electric Field E is non conservative when the Magnetic Field Bvaries in time, and is not expressible as the gradient of a scalar field, and not subject to the gradient theorem since itsrotational is not zero. See also. [23][25]

Lorentz force in terms of potentials

See also: Mathematical descriptions of the electromagnetic f ield, Maxwell's equations, and Helmholtz decomposition

The E and B fields can be replaced by the magnetic vector potential A and (scalar) electrostatic potential by

where is the gradient, is the divergence, is the curl.

The force becomes

and using an identity for the triple product simplifies to
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using the chain rule, the total derivative of A is:

so the above expression can be rewritten as;

which can take the convenient Euler-Lagrange form

Lorentz force and analytical mechanics

See also: 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:[26]

where A and are the potential fields as above. Using Lagrange's equations, the equation for the Lorentz force can beobtained.

Derivation of Lorentz force from classical Lagrangian (SI units)

For an A field, a particle moving with velocity v = has potential momentum , so its potential energy is. For a field, the particle's potential energy is .

The total potential energy is then:

and the kinetic energy is:

hence the Lagrangian:
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Lagrange's equations are

(same for y and z ). So calculating the partial derivatives:

equating and simplifying:

and similarly for the y and z directions. Hence the force equation is:

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

The action is the relativistic arclength of the path of the particle in space time, minus the potential energy contribution, plus anextra 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 thenotation):
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are the same as Hamilton's equations of motion:

both are equivalent to the noncanonical form:

This formula is the Lorentz force, representing the rate at which the EM field adds relativisticmomentum 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 cgs-Gaussian units, which are somewhat more common among theoretical physicists, one has instead

where c is the speed of light. Although this equation looks slightly different, it is completely equivalent, since one has thefollowing relations:

where 0 is the vacuum permittivity and 0 the vacuum permeability. In practice, the subscripts "cgs" and "SI" are alwaysomitted, 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 articles: Covariant formulation of classical electromagnetism and Mathematical descriptions of theelectromagnetic field

Using the metric signature (-1,1,1,1), The Lorentz force for a charge q can be written in covariant form:
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where p is the four-momentum, defined as:

the proper time of the particle, F the contravariant electromagnetic tensor

and U is the covariant 4-velocity of the particle, defined as:

where is the Lorentz factor defined above.

The fields are transformed to a frame moving with constant relative velocity by:

where is the Lorentz transformation tensor.

Translation to vector notation

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

Substituting the components of the covariant electromagnetic tensor F yields

Using the components of covariant four-velocity yields

The calculation for = 2, 3 (force components in the y and z directions) yields similar results, so collecting the 3 equationsinto one:

which is the Lorentz force.

STA form of the Lorentz force
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The electric and magnetic fields are dependent on the velocity of an observer, so the relativistic form of the Lorentz force lawcan best be exhibited starting from a coordinate-independent expression for the electromagnetic and magnetic fields, [27] ,and an arbitrary time-direction, , where

and

is a space-time bivector (an oriented plane segment, just like a vector is an oriented line segment), which has six degreesof freedom corresponding to boosts (rotations in space-time planes) and rotations (rotations in space-space planes). The dot

product with the vector pulls a vector (in the space algebra) from the translational part, while the wedge-product createsa trivector (in the space algebra) who is dual to a vector which is the usual magnetic field vector. The relativistic velocity isgiven by the (time-like) changes in a time-position vector , where

(which shows our choice for the metric) and the velocity is

The proper (invariant is an inadequate term because no transformation has been defined) form of the Lorentz force law issimply

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:

Cyclotrons and other circular path particle acceleratorsMass spectrometersVelocity FiltersMagnetrons

VelocimetersIn its manifestation as the Laplace force on an electric current in a conductor, this force occurs in many devices including:

Electric motorsRailgunsLinear motorsLoudspeakers

Magnetoplasmadynamic thrustersElectrical generatorsHomopolar generatorsLinear alternators

See also

Hall effectElectromagnetism

Moving magnet and conductor problem

Scalar potentialHelmholtz decomposition
http://en.wikipedia.org/wiki/Helmholtz_decompositionhttp://en.wikipedia.org/wiki/Scalar_potentialhttp://en.wikipedia.org/wiki/Moving_magnet_and_conductor_problemhttp://en.wikipedia.org/wiki/Electromagnetismhttp://en.wikipedia.org/wiki/Hall_effecthttp://en.wikipedia.org/wiki/Linear_alternatorhttp://en.wikipedia.org/wiki/Homopolar_generatorhttp://en.wikipedia.org/wiki/Electrical_generatorhttp://en.wikipedia.org/wiki/Magnetoplasmadynamic_thrusterhttp://en.wikipedia.org/wiki/Loudspeakerhttp://en.wikipedia.org/wiki/Linear_motorhttp://en.wikipedia.org/wiki/Railgunhttp://en.wikipedia.org/wiki/Electric_motorhttp://en.wikipedia.org/wiki/Lorentz_force_velocimetryhttp://en.wikipedia.org/wiki/Magnetronhttp://en.wikipedia.org/wiki/Mass_spectrometerhttp://en.wikipedia.org/wiki/Particle_acceleratorhttp://en.wikipedia.org/wiki/Cyclotronhttp://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity


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GravitomagnetismAmpre's force lawHendrik LorentzMaxwell's equationsFormulation of Maxwell'sequations in special relativity

AbrahamLorentz forceLarmor formulaCyclotron radiationMagnetic potentialMagnetoresistance

Guiding center Field line

Footnotes

1. ^ a b c Oliver Heaviside By Paul J. Nahin, p120 (http://books.google.com/books?id=e9wEntQmA0IC&pg=PA120)2. ^ a b Huray, Paul G. (2009). Maxwell's Equations (http://books.google.com/books?

id=0QsDgdd0MhMC&pg=PA22#v=onepage&q&f=false). Wiley-IEEE. p. 22. ISBN 0-470-54276-4.3. ^ a b See Jackson page 2. The book lists the four modern Maxwell's equations, and then states, "Also essential for

consideration of charged particle motion is the Lorentz force equation, F = q ( E+ v B ), which gives the force acting on a point charge q in the presence of electromagnetic fields."

4. ^ See Griffiths page 204.5. ^ For example, see the website of the "Lorentz Institute": \[1] (http://ilorentz.org/history/lorentz/lorentz.html), or Griffiths.

6. ^ a b c Griffiths, David J. (1999). Introduction to electrodynamics . reprint. with corr. (3rd ed.). Upper Saddle River, NJ[u.a.]: Prentice Hall. ISBN 9780138053260.

7. ^ Meyer, Herbert W. (1972). A History of Electricity and Magnetism . Norwalk, CT: Burndy Library. pp. 3031. ISBN 0-262-13070-X.

8. ^ Verschuur, Gerrit L. (1993). Hidden Attraction : The History And Mystery Of Magnetism . New York: Oxford UniversityPress. pp. 7879. ISBN 0-19-506488-7.

9. ^ Darrigol, Olivier (2000). Electrodynamics from Ampre to Einstein . Oxford, [England]: Oxford University Press. pp. 9, 25.ISBN 0-19-850593-0

10. ^ Verschuur, Gerrit L. (1993). Hidden Attraction : The History And Mystery Of Magnetism . New York: Oxford UniversityPress. p. 76. ISBN 0-19-506488-7.

11. ^ Darrigol, Olivier (2000). Electrodynamics from Ampre to Einstein . Oxford, [England]: Oxford University Press. pp. 126 131, 139144. ISBN 0-19-850593-0

12. ^ Darrigol, Olivier (2000). Electrodynamics from Ampre to Einstein . Oxford, [England]: Oxford University Press. pp. 200,429430. ISBN 0-19-850593-0

13. ^ Heaviside, Oliver. "On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric"(http://en.wikisource.org/wiki/Motion_of_Electrification_through_a_Dielectric). Philosophical Magazine, April 1889, p. 324 .

14. ^ Darrigol, Olivier (2000). Electrodynamics from Ampre to Einstein . Oxford, [England]: Oxford University Press. p. 327.ISBN 0-19-850593-0

15. ^ Whittaker, E. T. (1910). A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century (http://books.google.com/books?id=CGJDAAAAIAAJ&printsec=frontcover#v=onepage&q&f=false).Longmans, Green and Co. pp. 420423. ISBN 1-143-01208-9.

16. ^ See Griffiths page 326, which states that Maxwell's equations, "together with the [Lorentz] force law...summarize theentire theoretical content of classical electrodynamics".

17. ^ See, for example, Jackson p777-8.18. ^ J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation . W.H. Freeman & Co. pp. 7273. 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.19. ^ I.S. Grant, W.R. Phillips, Manchester Physics (2008). Electromagnetism (2nd ed.). John Wiley & Sons. p. 122.

ISBN 978-0-471-92712-9.20. ^ I.S. Grant, W.R. Phillips, Manchester Physics (2008). Electromagnetism (2nd Edition) . John Wiley & Sons. p. 123.

ISBN 978-0-471-92712-9.21. ^ a b See Griffiths pages 3013.22. ^ Tai L. Chow (2006). Electromagnetic theory (http://books.google.com/?

id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271). Sudbury MA: Jones and Bartlett. p. 395. ISBN 0-7637-3827-1.23. ^ a b Landau, L. D., Lifshit s , E. M., & Pitaevski, L. P. (1984). Electrodynamics of continuous media; Volume 8 Course

of Theoretical Physics (http://worldcat.org/search?q=0750626348&qt=owc_search) (Second ed.). Oxford: Butterworth-

Heinemann. p. 63 (49 pp. 205207 in 1960 edition). ISBN 0-7506-2634-8.24. ^ Roger F Harrington (2003). Introduction to electromagnetic engineering (http://books.google.com/?id=ZlC2EV8zvX8C&pg=PA57&dq=%22faraday%27s+law+of+induction%22). Mineola, NY: Dover Publications. p. 56.ISBN 0-486-43241-6.

25. ^ M N O Sadiku (2007). Elements of elctromagnetics (http://books.google.com/?id=w2ITHQAACAAJ&dq=ISBN0-19-530048-3) (Fourth ed.). NY/Oxford: Oxford University Press. p. 391. ISBN 0-19-530048-3.
http://en.wikipedia.org/wiki/Special:BookSources/0-19-530048-3http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/?id=w2ITHQAACAAJ&dq=ISBN0-19-530048-3http://en.wikipedia.org/wiki/Special:BookSources/0-486-43241-6http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/?id=ZlC2EV8zvX8C&pg=PA57&dq=%22faraday%27s+law+of+induction%22http://en.wikipedia.org/wiki/Special:BookSources/0-7506-2634-8http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://worldcat.org/search?q=0750626348&qt=owc_searchhttp://en.wikipedia.org/wiki/Special:BookSources/0-7637-3827-1http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271http://en.wikipedia.org/wiki/Special:BookSources/978-0-471-92712-9http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/978-0-471-92712-9http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Electromagnetic_tensorhttp://en.wikipedia.org/wiki/Special:BookSources/0-7167-0344-0http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/1-143-01208-9http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/books?id=CGJDAAAAIAAJ&printsec=frontcover#v=onepage&q&f=falsehttp://en.wikipedia.org/wiki/Special:BookSources/0-19-850593-0http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Albert_Einsteinhttp://en.wikipedia.org/wiki/Andr%C3%A9_Amp%C3%A8rehttp://en.wikisource.org/wiki/Motion_of_Electrification_through_a_Dielectrichttp://en.wikipedia.org/wiki/Special:BookSources/0-19-850593-0http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Albert_Einsteinhttp://en.wikipedia.org/wiki/Andr%C3%A9_Amp%C3%A8rehttp://en.wikipedia.org/wiki/Special:BookSources/0-19-850593-0http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Albert_Einsteinhttp://en.wikipedia.org/wiki/Andr%C3%A9_Amp%C3%A8rehttp://en.wikipedia.org/wiki/Special:BookSources/0-19-506488-7http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0-19-850593-0http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Albert_Einsteinhttp://en.wikipedia.org/wiki/Andr%C3%A9_Amp%C3%A8rehttp://en.wikipedia.org/wiki/Special:BookSources/0-19-506488-7http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0-262-13070-Xhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/9780138053260http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://ilorentz.org/history/lorentz/lorentz.htmlhttp://en.wikipedia.org/wiki/Special:BookSources/0-470-54276-4http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/books?id=0QsDgdd0MhMC&pg=PA22#v=onepage&q&f=falsehttp://books.google.com/books?id=e9wEntQmA0IC&pg=PA120http://en.wikipedia.org/wiki/Field_linehttp://en.wikipedia.org/wiki/Guiding_centerhttp://en.wikipedia.org/wiki/Magnetoresistancehttp://en.wikipedia.org/wiki/Magnetic_potentialhttp://en.wikipedia.org/wiki/Cyclotron_radiationhttp://en.wikipedia.org/wiki/Larmor_formulahttp://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_forcehttp://en.wikipedia.org/wiki/Formulation_of_Maxwell%27s_equations_in_special_relativityhttp://en.wikipedia.org/wiki/Maxwell%27s_equationshttp://en.wikipedia.org/wiki/Hendrik_Lorentzhttp://en.wikipedia.org/wiki/Amp%C3%A8re%27s_force_lawhttp://en.wikipedia.org/wiki/Gravitomagnetism


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. ass ca ec an cs n on , . . . e, uropean ys cs er es, c raw , , - -0.

27. ^ Hestenes, David. "SpaceTime Calculus" (http://geocalc.clas.asu.edu/html/STC.html).

References

The numbered references refer in part to the list immediately below.

Feynman, Richard Phillips; Leighton, Robert B.; Sands, Matthew L. (2006). The Feynman lectures on physics (3vol.). Pearson / Addison-Wesley. ISBN 0-8053-9047-2: volume 2.

Griffiths, David J. (1999). Introduction to electrodynamics (3rd ed.). Upper Saddle River, [NJ.]: Prentice-Hall.ISBN 0-13-805326-X

Jackson, John David (1999). Classical electrodynamics (3rd ed.). New York, [NY.]: Wiley. ISBN 0-471-30932-X

Serway, Raymond A.; Jewett, John W., Jr. (2004). Physics for scientists and engineers, with modern physics .Belmont, [CA.]: Thomson Brooks/Cole. ISBN 0-534-40846-X

Srednicki, Mark A. (2007). Quantum field theory (http://books.google.com/?id=5OepxIG42B4C&pg=PA315&dq=isbn=9780521864497). Cambridge, [England] ; New York [NY.]:Cambridge University Press. ISBN 978-0-521-86449-7

External links

Interactive Java tutorial on the Lorentz force(http://www.magnet.fsu.edu/education/tutorials/java/lorentzforce/index.html) National High Magnetic Field LaboratoryLorentz force (demonstration) (http://www.youtube.com/watch?v=mxMMqNrm598)

Faraday's law: Tankersley and Mosca (http://www.nadn.navy.mil/Users/physics/tank/Public/FaradaysLaw.pdf) Notes from Physics and Astronomy HyperPhysics at Georgia State University (http://hyperphysics.phy-astr.gsu.edu/HBASE/hframe.html); see also home page (http://hyperphysics.phy-astr.gsu.edu/HBASE/hframe.html)Interactive Java applet on the magnetic deflection of a particle beam in a homogeneous magnetic field(http://chair.pa.msu.edu/applets/Lorentz/a.htm) by Wolfgang Bauer

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