Editing Lorentz force (section)

== History ==
[[File:H. A. Lorentz - Lorentz force, div E = ρ, div B = 0 - La théorie electromagnétique de Maxwell et son application aux corps mouvants, Archives néerlandaises, 1892 - p 451 - Eq. I, II, III.png|thumb|Lorentz's theory of electrons. Formulas for the Lorentz force (I, ponderomotive force) and the [[Maxwell equations]] for the [[divergence]] of the [[electrical field]] E (II) and the [[magnetic field]] B (III), {{lang|fr|La théorie electromagnétique de Maxwell et son application aux corps mouvants}}, 1892, p. 451. {{mvar|V}} is the velocity of light.]]
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,<ref>{{cite book | first = Michel | last = Delon | title = Encyclopedia of the Enlightenment | place = Chicago, Illinois | publisher = Fitzroy Dearborn | year = 2001 | page = 538 | isbn = 1-57958-246-X}}</ref> and electrically charged objects, by [[Henry Cavendish]] in 1762,<ref>{{cite book | first = Elliot H. | last = Goodwin | title = The New Cambridge Modern History Volume 8: The American and French Revolutions, 1763–93 | place = Cambridge | publisher = Cambridge University Press | year = 1965 | page = 130 | isbn = 978-0-521-04546-9}}</ref> 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, Connecticut | publisher = Burndy Library | year = 1972 | pages = 30–31 | isbn = 0-262-13070-X | url = https://archive.org/details/AHistoryof_00_Meye}}</ref> Soon after the discovery in 1820 by [[Hans Christian Ørsted]] that a magnetic needle is acted on by a voltaic current, [[André-Marie Ampère]] that same year was able to devise through experimentation the formula for the angular dependence of the force between two current elements.<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 = [https://archive.org/details/hiddenattraction00vers/page/78 78–79] | url = https://archive.org/details/hiddenattraction00vers/page/78}}</ref>{{sfn|Darrigol|2000|pp=9,25}} In all these descriptions, the force was always described in terms of the properties of the matter involved and the distances between two masses or charges rather than in terms of electric and magnetic fields.<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 = [https://archive.org/details/hiddenattraction00vers/page/76 76] | url = https://archive.org/details/hiddenattraction00vers/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]].{{sfn|Darrigol|2000|pp=126-131,139-144}} 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 = 978-0-470-54276-7 | year = 2010 | page = 22 | url = https://books.google.com/books?id=0QsDgdd0MhMC&pg=PA22}}</ref> although 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>{{cite book |first=Paul J. |last=Nahin |url=https://books.google.com/books?id=e9wEntQmA0IC |title=Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age |publisher=JHU Press |year=2002}}</ref><ref>{{cite journal| last=Thomson |first=J. J. | date=1881-04-01|title=XXXIII. On the electric and magnetic effects produced by the motion of electrified bodies|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|volume=11|issue=68|pages=229–249|doi=10.1080/14786448108627008|issn=1941-5982}}</ref>
<math display="block">\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. [[Oliver Heaviside]] invented the modern vector notation and applied it to Maxwell's field equations; he also (in 1885 and 1889) had fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force on a moving charged object.<ref name=Nahin/>{{sfn|Darrigol|2000|pp=200,429-430}}<ref>{{cite journal | last= Heaviside |first=Oliver| title=On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric | journal=Philosophical Magazine |date=April 1889 | volume=27 |page=324 |url=http://en.wikisource.org/wiki/Motion_of_Electrification_through_a_Dielectric}}</ref> Finally, in 1895,<ref name=Dahl>{{cite book | first = Per F. | last = Dahl | title = Flash of the Cathode Rays: A History of J J Thomson's Electron | publisher = CRC Press | year = 1997| page= 10}}</ref><ref>{{cite book |last=Lorentz |first=Hendrik Antoon |title=Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern |language=de |year=1895}}</ref> [[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.{{sfn|Darrigol|2000|p=327}}<ref>{{cite book | last = Whittaker | first = E. T. | author-link=E. T. Whittaker | title = [[A History of the Theories of Aether and Electricity|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 | isbn = 1-143-01208-9}}</ref>