Editing Lorentz force (section)

== Lorentz force law as the definition of E and B ==

{{multiple image|position
| align             = right
| 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 {{mvar|q}} under the influence of a magnetic field {{mvar|B}}, which is directed perpendicularly out of the screen
| image2            = Cyclotron motion.jpg
| caption2          = Beam of electrons moving in a circle, due to the presence of a magnetic field. Purple light revealing the electron's path in this [[Teltron tube]] is created by the electrons colliding with gas molecules.
| total_width       = 400
| alt1              = 
}}

In many textbook treatments of classical electromagnetism, the Lorentz force law is used as the ''definition'' of the electric and magnetic fields {{math|'''E'''}} and {{math|'''B'''}}.{{sfn|Jackson|1998|pp=777-778}}<ref>{{cite book| first1=J. A. |last1=Wheeler |author1-link=John Archibald Wheeler |url=https://archive.org/details/gravitation00misn_003|title=Gravitation |first2=C. |last2=Misner |author-link2=Charles W. Misner | first3=K. S. |last3=Thorne |author-link3=Kip Thorne | publisher=W. H. Freeman & Co|year=1973|isbn=0-7167-0344-0 | pages=[https://archive.org/details/gravitation00misn_003/page/n96 72]–73 | url-access=limited}} These authors use the Lorentz force in tensor form as definer of the [[electromagnetic tensor]] {{math|''F''}}, in turn the fields {{math|'''E'''}} and {{math|'''B'''}}.</ref><ref>{{cite book|first1=I. S. |last1=Grant|title=Electromagnetism| first2=W. R. |last2=Phillips|series=The Manchester Physics Series|publisher=John Wiley & Sons|year=1990| isbn=978-0-471-92712-9| edition=2nd | page=122}}</ref> To be specific, the Lorentz force is understood to be the following empirical statement:

{{quote|The electromagnetic force {{math|'''F'''}} on a [[test charge]] at a given point and time is a certain function of its charge {{math|''q''}} and velocity {{math|'''v'''}}, which can be parameterized by exactly two vectors {{math|'''E'''}} and {{math|'''B'''}}, in the functional form: <math display="block">\mathbf{F} = q(\mathbf{E}+\mathbf{v} \times \mathbf{B})</math>}}

This is valid, even for particles approaching the speed of light (that is, [[Norm (mathematics)#Euclidean norm|magnitude]] of {{math|'''v'''}}, {{math|1={{abs|'''v'''}} ≈ ''c''}}).<ref>{{cite book|first1=I. S. |last1=Grant|title=Electromagnetism|first2=W. R. |last2=Phillips| series=The Manchester Physics Series |publisher=John Wiley & Sons|year=1990|isbn=978-0-471-92712-9|edition=2nd|page=123}}</ref> So the two [[vector field]]s {{math|'''E'''}} and {{math|'''B'''}} are thereby defined throughout space and time, and these are called the "electric field" and "magnetic field". 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.