Editing Lorentz force (section)

=== Charged particle ===

[[File:Lorentz force particle.svg|thumb|Lorentz force {{math|'''F'''}} on a [[charged particle]] (of charge {{mvar|q}}) in motion (instantaneous velocity {{math|'''v'''}}). The [[electric field|{{math|'''E'''}} field]] and [[magnetic field|{{math|'''B'''}} field]] vary in space and time.]]

The force {{math|'''F'''}} acting on a particle of [[electric charge]] {{mvar|q}} with instantaneous velocity {{math|'''v'''}}, due to an external electric field {{math|'''E'''}} and magnetic field {{math|'''B'''}}, is given by ([[SI]] definition of quantities<ref group="nb" name="units" />):{{sfn|Jackson|1998|pp=2-3}}
{{Equation box 1
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|equation = <math>\mathbf{F} = q \left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)</math>
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|border colour = #50C878
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where {{math|×}} is the vector [[cross product]] (all boldface quantities are vectors). In terms of Cartesian components, we have:
<math display="block">\begin{align}
F_x &= q \left(E_x + v_y B_z - v_z B_y\right), \\[0.5ex]
F_y &= q \left(E_y + v_z B_x - v_x B_z\right), \\[0.5ex]
F_z &= q \left(E_z + v_x B_y - v_y B_x\right).
\end{align}</math>

In general, the electric and magnetic fields are functions of the position and time. Therefore, explicitly, the Lorentz force can be written as:
<math display="block">\mathbf{F}\left(\mathbf{r}(t),\dot\mathbf{r}(t),t,q\right) = q\left[\mathbf{E}(\mathbf{r},t) + \dot\mathbf{r}(t) \times \mathbf{B}(\mathbf{r},t)\right]</math>
in which {{math|'''r'''}} is the position vector of the charged particle, {{mvar|t}} is time, and the overdot is a time derivative.

A positively charged particle will be accelerated in the ''same'' linear orientation as the {{math|'''E'''}} field, but will curve perpendicularly to both the instantaneous velocity vector {{math|'''v'''}} and the {{math|'''B'''}} field according to the [[right-hand rule]] (in detail, if the fingers of the right hand are extended to point in the direction of {{math|'''v'''}} and are then curled to point in the direction of {{math|'''B'''}}, then the extended thumb will point in the direction of {{math|'''F'''}}).

The term {{math|''q'''''E'''}} is called the '''electric force''', while the term {{math|1=''q''('''v''' × '''B''')}} is called the '''magnetic force'''.{{sfn|Griffiths|2023|p=211}} According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force,<ref name="Griffiths2">For example, see the [http://ilorentz.org/history/lorentz/lorentz.html website of the Lorentz Institute].</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 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 a magnetic field. In that context, it is also called the [[#Force on a current-carrying wire|Laplace force]].

The Lorentz force is a force exerted by the electromagnetic field on the charged particle, that is, it is the rate at which linear momentum is transferred from the electromagnetic field to the particle. Associated with it is the power which is the rate at which energy is transferred from the electromagnetic field to the particle. That power is
<math display="block">\mathbf{v} \cdot \mathbf{F} = q \, \mathbf{v} \cdot \mathbf{E}.</math>
Notice that the magnetic field does not contribute to the power because the magnetic force is always perpendicular to the velocity of the particle.