Editing Lorentz force (section)

=== 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}}

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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>