Editing Lorentz force (section)

== Equation ==

=== 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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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.

=== Continuous charge distribution ===

[[File:Lorentz force continuum.svg|thumb|Lorentz force (per unit 3-volume) {{math|'''f'''}} on a continuous [[charge distribution]] ([[charge density]] {{math|''ρ''}}) in motion. The 3-[[current density]] {{math|'''J'''}} corresponds to the motion of the charge element {{math|''dq''}} in [[volume element]] {{math|''dV''}} and varies throughout the continuum.]]

For a continuous [[charge distribution]] in motion, the Lorentz force equation becomes:
<math display="block">\mathrm{d}\mathbf{F} = \mathrm{d}q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)</math>
where <math>\mathrm{d}\mathbf{F}</math> is the force on a small piece of the charge distribution with charge <math>\mathrm{d}q</math>. If both sides of this equation are divided by the volume of this small piece of the charge distribution <math>\mathrm{d}V</math>, the result is:
<math display="block">\mathbf{f} = \rho\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)</math>
where <math>\mathbf{f}</math> is the ''force density'' (force per unit volume) and <math>\rho</math> is the [[charge density]] (charge per unit volume). Next, the [[current density]] corresponding to the motion of the charge continuum is{{sfn|Griffiths|2023|p=219}}
<math display="block">\mathbf{J} = \rho \mathbf{v} </math>
so the continuous analogue to the equation is{{sfn|Griffiths|2023|p=368}}

{{Equation box 1
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|equation = <math>\mathbf{f} = \rho \mathbf{E} + \mathbf{J} \times \mathbf{B}</math>
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The total force is the [[volume integral]] over the charge distribution:
<math display="block"> \mathbf{F} = \int \left ( \rho \mathbf{E} + \mathbf{J} \times \mathbf{B} \right)\mathrm{d}V.</math>

By eliminating <math>\rho</math> and <math>\mathbf{J}</math>, using [[Maxwell's equations]], and manipulating using the theorems of [[vector calculus]], this form of the equation can be used to derive the [[Maxwell stress tensor]] <math>\boldsymbol{\sigma}</math>, in turn this can be combined with the [[Poynting vector]] <math>\mathbf{S}</math> to obtain the [[electromagnetic stress–energy tensor]] {{math|'''T'''}} used in [[general relativity]].{{sfn|Griffiths|2023|pp=369-370}}

In terms of <math>\boldsymbol{\sigma}</math> and <math>\mathbf{S}</math>, another way to write the Lorentz force (per unit volume) is
<math display="block"> \mathbf{f} = \nabla\cdot\boldsymbol{\sigma} - \dfrac{1}{c^2} \dfrac{\partial \mathbf{S}}{\partial t} </math>
where <math>\nabla \cdot</math> denotes the [[Divergence#Definition_in_coordinates|divergence]] of the [[tensor field]] and <math>c</math> is the [[speed of light]]. Rather than the amount of charge and its velocity in electric and magnetic fields, this equation relates the [[energy flux]] (flow of ''energy'' per unit time per unit distance) in the fields to the force exerted on a charge distribution. See [[Covariant formulation of classical electromagnetism#Charge continuum|Covariant formulation of classical electromagnetism]] for more details.

The density of power associated with the Lorentz force in a material medium is
<math display="block">\mathbf{J} \cdot \mathbf{E}.</math>

If we separate the total charge and total current into their free and bound parts, we get that the density of the Lorentz force is
<math display="block">\mathbf{f} = \left(\rho_f - \nabla \cdot \mathbf P\right) \mathbf{E} + \left(\mathbf{J}_f + \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}\right) \times \mathbf{B}.</math>

where: <math>\rho_f</math> is the density of free charge; <math>\mathbf{P}</math> is the [[polarization density]]; <math>\mathbf{J}_f</math> is the density of free current; and <math>\mathbf{M}</math> is the [[magnetization]] density. In this way, the Lorentz force can explain the torque applied to a permanent magnet by the magnetic field. The density of the associated power is
<math display="block">\left(\mathbf{J}_f + \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}\right) \cdot \mathbf{E}.</math>

=== Formulation in the Gaussian system ===
The above-mentioned formulae use the conventions for the definition of the electric and magnetic field used with the [[SI]], which is the most common. However, other conventions with the same physics (i.e. forces on e.g. an electron) are possible and used. In the conventions used with the older [[Gaussian units|CGS-Gaussian units]], which are somewhat more common among some theoretical physicists as well as condensed matter experimentalists, one has instead
<math display="block">\mathbf{F} = q_\mathrm{G} \left(\mathbf{E}_\mathrm{G} + \frac{\mathbf{v}}{c} \times \mathbf{B}_\mathrm{G}\right),</math>
where {{mvar|c}} is the [[speed of light]]. Although this equation looks slightly different, it is equivalent, since one has the following relations:<ref group="nb" name="units" />
<math display="block">q_\mathrm{G} = \frac{q_\mathrm{SI}}{\sqrt{4\pi \varepsilon_0}},\quad
\mathbf E_\mathrm{G} = \sqrt{4\pi\varepsilon_0}\,\mathbf E_\mathrm{SI},\quad
\mathbf B_\mathrm{G} = {\sqrt{4\pi /\mu_0}}\,{\mathbf B_\mathrm{SI}}, \quad
c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}.</math>
where {{math|''ε''<sub>0</sub>}} is the [[vacuum permittivity]] and {{math|''μ''<sub>0</sub>}} the [[vacuum permeability]]. In practice, the subscripts "G" and "SI" are omitted, and the used convention (and unit)  must be determined from context.