Editing Momentum (section)

====Particle in a field====

In [[Maxwell's equations]], the forces between particles are mediated by electric and magnetic fields. The electromagnetic force (''[[Lorentz force]]'') on a particle with charge {{mvar|q}} due to a combination of [[electric field]] {{math|'''E'''}} and [[magnetic field]] {{math|'''B'''}} is

<math qid=Q849919 display="block">\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}).</math>

(in [[International System of Units|SI units]]).<ref>{{harvnb|Jackson|1975}}</ref>{{rp|2}}
It has an [[electric potential]] {{math|{{var|φ}}('''r''', {{var|t}})}} and [[magnetic vector potential]] {{math|'''A'''('''r''', {{var|t}})}}.<ref name=Lerner />
In the non-relativistic regime, its generalized momentum is

<math display="block">\mathbf{P} = m\mathbf{\mathbf{v}} + q\mathbf{A}, </math>

while in relativistic mechanics this becomes

<math display="block">\mathbf{P} = \gamma m\mathbf{\mathbf{v}} + q\mathbf{A}. </math>

The quantity {{math|{{var|V}} {{=}} {{var|q}}'''A'''}} is sometimes called the ''potential momentum''.<ref>{{Cite journal|last1=Semon|first1=Mark D.|last2=Taylor|first2=John R.|date=November 1996|title=Thoughts on the magnetic vector potential|journal=American Journal of Physics|volume=64|issue=11|pages=1361–1369|doi=10.1119/1.18400|bibcode=1996AmJPh..64.1361S|issn=0002-9505}}</ref><ref>{{Cite book|last=Griffiths |first=David J. |title=Introduction to Electrodynamics |date=29 June 2017 |isbn=978-1-108-42041-9 |edition=4th |location=Cambridge, United Kingdom |oclc=1021068059 |publisher=Cambridge University Press }}</ref><ref>{{Cite journal |last1=Vieira |first1=R. S. |last2=Brentan |first2=H. B. |date=April 2018 |title=Covariant theory of gravitation in the framework of special relativity |journal=The European Physical Journal Plus |volume=133 |issue=4 |page=165 |doi=10.1140/epjp/i2018-11988-9 |arxiv=1608.00815 |bibcode=2018EPJP..133..165V |s2cid=16691128 |issn=2190-5444}}</ref> It is the momentum due to the interaction of the particle with the electromagnetic fields. The name is an analogy with the potential energy {{math|{{var|U}} {{=}} {{var|q}}{{var|φ}}}}, which is the energy due to the interaction of the particle with the electromagnetic fields. These quantities form a four-vector, so the analogy is consistent; besides, the concept of potential momentum is important in explaining the so-called [[hidden momentum]] of the electromagnetic fields.<ref>{{Cite journal|last1=Babson|first1=David|last2=Reynolds|first2=Stephen P.|last3=Bjorkquist|first3=Robin|last4=Griffiths|first4=David J.|date=September 2009|title=Hidden momentum, field momentum, and electromagnetic impulse|journal=American Journal of Physics|volume=77|issue=9|pages=826–833|doi=10.1119/1.3152712|bibcode=2009AmJPh..77..826B|issn=0002-9505}}</ref>