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| In [[electrodynamics]], '''Poynting's theorem''' is a statement of [[conservation of energy]] for the [[electromagnetic field]], in the form of a [[partial differential equation]], due to the British [[physicist]] [[John Henry Poynting]].<ref name="Poynting">{{cite journal
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| | author=Poynting, J. H.
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| | authorlink=John_Henry_Poynting
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| | year=1884
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| | journal=Philosophical Transactions of the Royal Society of London
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| | volume=175
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| | pages=343–361
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| | title=[[s:On the Transfer of Energy in the Electromagnetic Field|On the Transfer of Energy in the Electromagnetic Field]]
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| | doi=10.1098/rstl.1884.0016}}</ref> Poynting's theorem is analogous to the [[work-energy theorem]] in [[classical mechanics]], and mathematically similar to the [[continuity equation]], because it relates the energy stored in the electromagnetic field to the [[work (physics)|work]] done on a [[charge distribution]] (i.e. an electrically charged object), through [[energy flux]].
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| ==Statement==
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| ===General===
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| In words, the theorem is an energy balance:<ref name="Electrodynamics 2007, p.364">Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, p.364, ISBN 81-7758-293-3</ref>
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| :''The '''rate of energy transfer (per unit volume)''' from a region of space equals the '''rate of [[work (physics)|work]] done''' on a charge distribution plus the '''[[energy flux]]''' leaving that region.''
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| Mathematically, this is summarised in '''differential form''' as:
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| {{Equation box 1
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| |indent =:
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| |equation = <math>-\frac{\partial u}{\partial t} = \nabla\cdot\mathbf{S}+\mathbf{J}\cdot\mathbf{E}</math>
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| |cellpadding
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| |border
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| where ∇•'''S''' is the [[divergence]] of the [[Poynting vector]] (energy flow) and '''J'''•'''E''' is the rate at which the fields do work on a charged object ('''J''' is the ''[[free current|free]]'' [[current density]] corresponding to the motion of charge, '''E''' is the [[electric field]], and • is the [[dot product]]). The [[energy density#Energy density of electric and magnetic fields|energy density]] ''u'' is given by:<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, chapters 2 and 6, ISBN 9-780471-927129</ref> | |
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| :<math>u = \frac{1}{2}\left(\mathbf{E}\cdot\mathbf{D} + \mathbf{B}\cdot\mathbf{H}\right).</math>
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| in which '''D''' is the [[electric displacement field]], '''B''' is the [[magnetic flux density]] and '''H''' the [[magnetic field strength]], ''ε''<sub>0</sub> is the [[electric constant]] and ''μ''<sub>0</sub> is the [[magnetic constant]]. <!-- Next sentence is emphasis, not repetition. --> Since the charges are free to move, and the '''D''' and '''H''' fields bypass any bound charges and currents in the charge distribution (by their definition), '''J''' is the ''[[free current|free]]'' [[current density]], ''not'' the ''total''.
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| Using the [[divergence theorem]], Poynting's theorem can be rewritten in '''integral form''':
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| {{Equation box 1
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| |indent =:
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| |equation = {{oiint
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| | preintegral = <math>-\frac{\partial}{\partial t} \int_V u dV=</math>
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| | intsubscpt = <math>\scriptstyle \partial V</math>
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| | integrand = <math>\mathbf{S}\cdot d\mathbf{A} + \int_V\mathbf{J}\cdot\mathbf{E}dV</math>
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| }}
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| |cellpadding= 6
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| |border
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| |border colour = #0073CF
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| |background colour=#F5FFFA}}
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| where <math>\partial V \!</math> is the boundary of a volume ''V''. The shape of the volume is arbitrary but fixed for the calculation.
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| === Electrical engineering ===
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| In [[electrical engineering]] context the theorem is usually written with the energy density term ''u'' expanded in the following way, which resembles the [[continuity equation]]:
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| :<math> | |
| \nabla\cdot\mathbf{S} +
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| \epsilon_0 \mathbf{E}\cdot\frac{\partial \mathbf{E}}{\partial t} + \frac{\mathbf{B}}{\mu_0}\cdot\frac{\partial\mathbf{B}}{\partial t} +
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| \mathbf{J}\cdot\mathbf{E} = 0,
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| </math>
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| where
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| *<math>\epsilon_0 \mathbf{E}\cdot\frac{\partial \mathbf{E}}{\partial t}</math> is the density of [[Electric power|reactive power]] driving the build-up of electric field,
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| *<math>\frac{\mathbf{B}}{\mu_0}\cdot\frac{\partial\mathbf{B}}{\partial t}</math> is the density of [[Electric power|reactive power]] driving the build-up of magnetic field, and
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| *<math>\mathbf{J}\cdot\mathbf{E}</math> is the density of [[Electric power]] dissipated by the [[Lorentz force]] acting on charge carriers.
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| ==Derivation==
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| While [[conservation of energy]] and the [[Lorentz force]] law can derive the general form of the theorem, [[Maxwell's equations]] are additionally required to derive the expression for the Poynting vector and hence complete the statement.
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| ===Poynting's theorem===
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| Considering the statement in words above - there are three elements to the theorem, which involve writing energy transfer (per unit time) as [[volume integral]]s:<ref name="Electrodynamics 2007, p.364"/>
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| <ol>
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| <li>Since ''u'' is the energy density, integrating over the volume of the region gives the total energy ''U'' stored in the region, then taking the (partial) time derivative gives the rate of change of energy:
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| :<math> U=\int_V u dV \ \rightarrow \ \frac{\partial U}{\partial t} = \frac{\partial}{\partial t} \int_V u dV = \int_V \frac{\partial u}{\partial t} dV .</math></li>
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| <li>The energy flux leaving the region is the [[surface integral]] of the Poynting vector, and using the [[divergence theorem]] this can be written as a volume integral:
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| :{{oiint
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| | preintegral =
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| | intsubscpt = <math>\scriptstyle \partial V</math>
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| | integrand = <math>\mathbf{S}\cdot d\mathbf{A}=\int_V\nabla\cdot \mathbf{S} dV .</math>
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| }}</li>
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| <li>The [[Lorentz force#Continuous charge distribution|Lorentz force]] density '''f''' on a charge distribution, integrated over the volume to get the total force '''F''', is
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| :<math> \mathbf{f} = \rho\mathbf{E}+\mathbf{J}\times\mathbf{B} \ \rightarrow \ \int_V \mathbf{f} dV = \mathbf{F} = \int_V (\rho\mathbf{E}+\mathbf{J}\times\mathbf{B} )dV ,</math>
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| where ''ρ'' is the [[charge density]] of the distribution and '''v''' its [[velocity]]. Since <math> \mathbf{J} = \rho \mathbf{v}</math>, the rate of work done by the force is
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| :<math> \mathbf{F}\cdot\frac{d \mathbf{r}}{dt} = \mathbf{F}\cdot \mathbf{v} = \int_V (\rho\mathbf{E}\cdot\mathbf{v}+\rho\mathbf{v}\times\mathbf{B}\cdot \mathbf{v} )dV \ \rightarrow \ \mathbf{F}\cdot \mathbf{v} = \int_V \mathbf{E}\cdot \mathbf{J}dV .</math></li>
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| </ol>
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| So by conservation of energy, the balance equation for the energy flow per unit time is the integral form of the theorem:
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| :<math>-\int_V\frac{\partial u}{\partial t}dV = \int_V\nabla\cdot\mathbf{S}dV+\int_V\mathbf{J}\cdot\mathbf{E}dV,</math>
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| and since the volume ''V'' is arbitrary, this is true for all volumes, implying
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| :<math>- \frac{\partial u}{\partial t} = \nabla\cdot\mathbf{S} + \mathbf{J}\cdot\mathbf{E},</math>
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| which is Poynting's theorem in differential form.
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| ===Poynting vector===
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| <!--please leave the derivation here, don't move/delete, it follows streight from the above-->
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| {{main|Poynting vector}}
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| From the theorem, the actual form of the Poynting vector '''S''' can be found. The time derivative of the energy density (using the [[Product rule#For vector functions|product rule]] for vector [[dot product]]s) is
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| :<math> \frac{\partial u}{\partial t} = \frac{1}{2}
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| \left(\mathbf{E}\cdot\frac{\partial \mathbf{D}}{\partial t}
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| + \mathbf{D}\cdot\frac{\partial \mathbf{E}}{\partial t}
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| + \mathbf{H}\cdot\frac{\partial \mathbf{B}}{\partial t}
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| + \mathbf{B}\cdot\frac{\partial \mathbf{H}}{\partial t}\right)=
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| \mathbf{E}\cdot\frac{\partial \mathbf{D}}{\partial t}
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| + \mathbf{H}\cdot\frac{\partial \mathbf{B}}{\partial t},
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| </math>
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| using the [[constitutive relations]]
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| :<math>\mathbf{D} = \epsilon_0 \mathbf{E},\quad \mathbf{B} = \mu_0 \mathbf{H}. </math>
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| The partial time derivatives suggest using two of [[Maxwell's Equations]]. Taking the [[dot product]] of [[Faraday's law of induction|Faraday's Law]] with '''H''':
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| :<math>\frac{\partial \mathbf{B}}{\partial t} = - \nabla \times \mathbf{E} \ \rightarrow \ \mathbf{H}\cdot\frac{\partial \mathbf{B}}{\partial t} = - \mathbf{H}\cdot\nabla \times \mathbf{E},</math>
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| next taking the dot product of the [[Amp%C3%A8re%27s circuital law#Corrected Ampère's circuital law: the Ampère-Maxwell equation|Ampère-Maxwell law]] equation with '''E''':
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| :<math>\frac{\partial \mathbf{D}}{\partial t} + \mathbf{J} = \nabla \times \mathbf{H} \ \rightarrow \ \mathbf{E}\cdot\frac{\partial \mathbf{D}}{\partial t} + \mathbf{E}\cdot\mathbf{J} = \mathbf{E}\cdot\nabla \times \mathbf{H}.</math>
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| Collecting the results so far gives:
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| :<math> \begin{align}
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| -\nabla\cdot\mathbf{S} & = \frac{\partial u}{\partial t} +\mathbf{J}\cdot\mathbf{E} \\
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| & = \left(\mathbf{H}\cdot\frac{\partial \mathbf{B}}{\partial t} + \mathbf{E}\cdot\frac{\partial \mathbf{D}}{\partial t}\right) + \mathbf{J}\cdot\mathbf{E} \\
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| & = \mathbf{E}\cdot\nabla \times \mathbf{H} - \mathbf{H}\cdot\nabla \times \mathbf{E}, \\
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| \end{align}</math>
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| then, using the [[Vector calculus identities#Vector cross product|vector calculus identity]]:
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| :<math>\nabla \cdot\mathbf{E} \times \mathbf{H}=\mathbf{H}\cdot\nabla \times \mathbf{E} - \mathbf{E}\cdot\nabla \times \mathbf{H}, </math>
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| gives an expression for the Poynting vector:
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| :<math>\mathbf{S} =\mathbf{E} \times \mathbf{H} ,</math>
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| which physically means the energy transfer due to time-varying electric and magnetic fields is perpendicular to the fields.
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| ==Alternative forms==
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| It is possible to derive alternative versions of Poynting's theorem.<ref name=kinslerfavaromccall>
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| {{cite journal
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| | author=Kinsler, P.
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| | coauthors=Favaro, A.; McCall M.W.
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| | year=2009
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| | title=Four Poynting theorems
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| | journal=European Journal of Physics
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| | volume=30
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| | issue=5
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| | pages=983
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| | doi=10.1088/0143-0807/30/5/007
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| | arxiv=0908.1721
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| |bibcode = 2009EJPh...30..983K }}</ref> Instead of the flux vector '''E''' <big>×</big> '''B''' as above, it is possible to follow the same style of derivation, but instead choose the Abraham form '''E''' <big>×</big> '''H''', the [[Minkowski]] form '''D''' <big>×</big> '''B''', or perhaps '''D''' <big>×</big> '''H'''. Each choice represents the response of the propagation medium in its own way: the '''E''' <big>×</big> '''B''' form above has the property that the response happens only due to electric currents, while the '''D''' <big>×</big> '''H''' form uses only (fictitious) [[magnetic monopole]] currents. The other two forms (Abraham and Minkowski) use complementary combinations of electric and magnetic currents to represent the polarization and magnetization responses of the medium.
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| ==Generalization==
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| The ''mechanical'' energy counterpart of the above theorem for the ''electromagnetic'' energy continuity equation is
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| :<math>
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| \frac{\partial}{\partial t} u_m(\mathbf{r},t) + \nabla\cdot \mathbf{S}_m (\mathbf{r},t) = \mathbf{J}(\mathbf{r},t)\cdot\mathbf{E}(\mathbf{r},t),
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| </math>
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| where ''u<sub>m</sub>'' is the (mechanical) [[kinetic energy]] density in the system. It can be described as the sum of kinetic energies of particles ''α'' (e.g., electrons in a wire), whose [[trajectory]] is given by ''r<sub>α</sub>''(''t''):
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| :<math> u_m(\mathbf{r},t) = \sum_{\alpha} \frac{m_{\alpha}}{2} \dot{r}^2_{\alpha} \delta(\mathbf{r}-\mathbf{r}_{\alpha}(t)), </math>
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| where '''S'''''<sub>m</sub>'' is the flux of their energies, or a "mechanical Poynting vector":
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| :<math>
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| \mathbf{S}_m (\mathbf{r},t) = \sum_{\alpha} \frac{m_{\alpha}}{2} \dot{r}^2_{\alpha}\dot{\mathbf{r}}_{\alpha} \delta(\mathbf{r}-\mathbf{r}_{\alpha}(t)).
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| </math>
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| Both can be combined via the [[Lorentz force]], which the electromagnetic fields exert on the moving charged particles (see above), to the following energy [[continuity equation]] or energy [[conservation law]]:<ref name=richter>
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| {{cite journal
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| | author=Richter, E.
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| | coauthors=Florian, M.; Henneberger, K.
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| | year=2008
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| | title=Poynting's theorem and energy conservation in the propagation of light in bounded media
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| | journal=Europhysics Letters
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| | volume=81
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| | issue=6
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| | pages=67005
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| | doi=10.1209/0295-5075/81/67005
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| | arxiv=0710.0515
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| |bibcode = 2008EL.....8167005R }}</ref>
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| :<math>
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| \frac{\partial}{\partial t}\left(u_e + u_m\right) + \nabla\cdot \left( \mathbf{S}_e + \mathbf{S}_m\right) = 0,
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| </math>
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| covering both types of energy and the conversion of one into the other.
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| == Notes ==
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| {{Reflist}}
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| ==External links==
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| *[http://scienceworld.wolfram.com/physics/PoyntingTheorem.html Eric W. Weisstein "Poynting Theorem" From ScienceWorld – A Wolfram Web Resource.]
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| [[Category:Electrodynamics]]
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| [[Category:Physics theorems]]
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| [[Category:Circuit theorems]]
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