Editing Poynting's theorem

{{Short description|Theorem in physics showing the conservation of energy for the electromagnetic field}}

In [[electrodynamics]], '''Poynting's theorem''' is a statement of [[conservation of energy]] for [[electromagnetic field]]s that was developed by British [[physicist]] [[John Henry Poynting]].<ref name="Poynting">
{{cite journal
 |author=Poynting, J. H.
 |date=December 1884
 |title-link=s:On the Transfer of Energy in the Electromagnetic Field |title=On the Transfer of Energy in the Electromagnetic Field
 |journal=Philosophical Transactions of the Royal Society of London |volume=175 |pages=343–361
 |doi=10.1098/rstl.1884.0016| doi-access=free
}}</ref> It states that in a given volume, the stored energy changes at a rate given by the [[work (physics)|work]] done on the [[charge distribution|charges]] within the volume, minus the rate at which energy leaves the volume. It is only strictly true in media that is not [[dispersive medium|dispersive]], but can be extended for the dispersive case.<ref name=Jackson>{{cite book |last1=Jackson |first1=John David |title=Classical Electrodynamics |date=1999 |publisher=John WIley & Sons |isbn=978-0-471-30932-1 |pages=258–267 |edition=3rd |ref=Jackson}}</ref>
The theorem is analogous to the [[work-energy theorem]] in [[classical mechanics]], and mathematically similar to the [[continuity equation]].

== Definition ==

Poynting's theorem states that the rate of energy transfer per unit volume from a region of space equals the rate of [[work (physics)|work]] done on the charge distribution in the region, plus the [[energy flux]] leaving that region.

Mathematically:

{{Equation box 1
|indent =:
|equation = <math>-\frac{\partial u}{\partial t} = \nabla\cdot\mathbf{S}+\mathbf{J}\cdot\mathbf{E}</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where:
* <math>-\frac{\partial u}{\partial t}</math> is the rate of change of the energy density in the volume.
* ∇ ⋅ '''S''' is the energy flow out of the volume, given by the [[divergence]] of the [[Poynting vector]] '''S'''.
* '''J''' ⋅ '''E''' is the power density of the field doing work charges ('''J''' is the [[current density]] corresponding to the motion of charge, '''E''' is the [[electric field]], and ⋅ is the [[dot product]]).

=== Integral form ===

Using the [[divergence theorem]], Poynting's theorem can also be written in integral form:

{{Equation box 1
| indent =:
| equation = 
  {{oiint
  | preintegral = <math>-\frac{d}{dt} \int_V u ~ \mathrm{d}V=</math>
  | intsubscpt = <math>\scriptstyle \partial V</math>
  | integrand = <math>\mathbf{S}\cdot \mathrm{d}\mathbf{A} + \int_V\mathbf{J}\cdot\mathbf{E} ~ \mathrm{d}V</math>
  }}
| cellpadding= 6
| border
| border colour = #0073CF
| background colour=#F5FFFA
}}
where
* '''S''' is the energy flow, given by the Poynting Vector
* ''u'' is the energy density
* <math>\partial V \!</math> is the boundary of the volume. The shape of the volume is arbitrary but fixed.

=== Continuity equation analog ===
In an [[electrical engineering]] context the theorem is sometimes written with the energy density term ''u'' expanded as shown.{{citation needed|date=December 2021}} This form resembles the [[continuity equation]]:
: <math>
\nabla\cdot\mathbf{S} +
\epsilon_0 \mathbf{E}\cdot\frac{\partial \mathbf{E}}{\partial t} + \frac{\mathbf{B}}{\mu_0}\cdot\frac{\partial\mathbf{B}}{\partial t} +
\mathbf{J}\cdot\mathbf{E} = 0
,</math>
where
* ''ε''<sub>0</sub> is the [[vacuum permittivity]] and ''μ''<sub>0</sub> is the [[vacuum permeability]].
* <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,
* <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
* <math>\mathbf{J}\cdot\mathbf{E}</math> is the density of [[electric power]] dissipated by the [[Lorentz force]] acting on charge carriers.

== Derivation ==
The rate of work done by the electromagnetic field on the infinitesimal charge
<math display="block"> dq=\rho d^3x </math>
is given by the [[Lorentz Force Law]] as:
<math display="block">dP=d\mathbf{F}\cdot\mathbf{v}= (\mathbf{E}+\mathbf{v}\times\mathbf{B})dq\cdot\mathbf{v} = \mathbf{E}\cdot \rho\mathbf{v}d^3x+0=\mathbf{E}\cdot\mathbf{J}d^3x</math> (the dot product <math> (\mathbf{v}\times\mathbf{B})\cdot \mathbf{v}=0 </math> because from the definition of cross product the cross product of '''v''' and '''B''' is perpendicular to '''v''').
Where ''ρ'' is the volume charge density and {{nowrap|1='''J''' = ''ρ'''''v'''}} is the current density at the point and time where '''v''' is the velocity of the charge dq.
The rate of work done on the whole charges in the volume ''V'' will be the volume integral
<math display="block">P=\int_V dP = \int_V \mathbf{J} \cdot \mathbf{E} ~\mathrm d^{3}x</math>

By [[Ampère's circuital law]]:
<math display="block">\mathbf{J} = \nabla \times \mathbf{H} - \frac{\partial\mathbf{D}}{\partial t}</math>
(Note that the '''H''' and '''D''' forms of the magnetic and electric fields are used here.  The '''B''' and '''E''' forms could also be used in an equivalent derivation.)<ref>{{cite book |last1=Griffiths |first1=David J. |title=Introduction to electrodynamics |date=1989 |publisher=Prentice Hall |location=Englewood Cliffs, N.J. |isbn=0-13-481367-7 |pages=322–324 |edition=2nd}}</ref>

Substituting this into the expression for rate of work gives:
<math display="block">\int_V \mathbf{J} \cdot \mathbf{E} ~\mathrm d^{3}x = \int_V \left [ \mathbf{E} \cdot (\nabla \times \mathbf{H}) - \mathbf{E} \cdot \frac{\partial\mathbf{D}}{\partial t}\right ] ~ \mathrm d^{3}x</math>

Using the [[vector identity]] <math>\nabla \cdot (\mathbf{E} \times \mathbf{H}) =\ (\nabla {\times} \mathbf{E}) \cdot \mathbf{H} \,-\, \mathbf{E} \cdot (\nabla {\times} \mathbf{H})</math>:
<math display="block"> \int_V \mathbf{J} \cdot \mathbf{E} ~ \mathrm d^{3}x = - \int_V \left [ \nabla \cdot (\mathbf{E} \times \mathbf{H}) - \mathbf{H} \cdot (\nabla \times \mathbf{E}) + \mathbf{E} \cdot \frac{\partial\mathbf{D}}{\partial t}\right ] ~ \mathrm d^{3}x</math>

By [[Maxwell–Faraday equation|Faraday's Law]]:
<math display="block">\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
giving:
<math display="block"> \int_V \mathbf{J} \cdot \mathbf{E} ~ \mathrm d^{3}x = - \int_V \left [ \nabla \cdot (\mathbf{E} \times \mathbf{H})  + \mathbf{E} \cdot \frac{\partial\mathbf{D}}{\partial t} + \mathbf{H} \cdot \frac{\partial \mathbf{B}} {\partial t}\right ] ~ \mathrm d^{3}x</math>

Continuing the derivation requires the following assumptions:<ref name=Jackson />
* the charges are moving in a medium that is not [[dispersive medium|dispersive]].
* the total electromagnetic energy density, even for time-varying fields, is given by <math display="block">u = \frac{1}{2} (\mathbf{E} \cdot \mathbf{D} + \mathbf{B} \cdot \mathbf{H})</math>

It can be shown<ref>{{cite web |last1=Ellingson |first1=Steven |title=Poynting's Theorem |url=https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Book%3A_Electromagnetics_II_(Ellingson)/03%3A_Wave_Propagation_in_General_Media/3.01%3A_Poynting%E2%80%99s_Theorem |website=LibreTexts |date=9 May 2020 |access-date=3 December 2021}}</ref> that:
<math display="block">\frac{\partial}{\partial t}(\mathbf{E} \cdot \mathbf{D}) = 2 \mathbf{E} \cdot \frac{\partial}{\partial t} \mathbf{D}</math> and <math display="block">\frac{\partial}{\partial t}(\mathbf{H} \cdot \mathbf{B}) = 2 \mathbf{H} \cdot \frac{\partial}{\partial t} \mathbf{B}</math>
and so:
<math display="block">\frac{\partial u}{\partial t} = \mathbf{E} \cdot \frac{\partial\mathbf{D}}{\partial t} + \mathbf{H} \cdot \frac{\partial \mathbf{B}} {\partial t} </math>

Returning to the equation for rate of work,
<math display="block"> \int_V \mathbf{J} \cdot \mathbf{E} ~ \mathrm d^{3}x = - \int_V \left [ \frac{\partial u}{\partial t} + \nabla \cdot (\mathbf{E} \times \mathbf{H})\right ] ~ \mathrm d^{3}x</math>

Since the volume is arbitrary, this can be cast in differential form as:
<math display="block">-\frac{\partial u}{\partial t} = \nabla\cdot\mathbf{S}+\mathbf{J}\cdot\mathbf{E}</math>
where <math>\mathbf{S} = \mathbf{E} \times \mathbf{H}</math> is the Poynting vector.

== Poynting vector in macroscopic media == 
In a macroscopic medium, electromagnetic effects are described by spatially averaged (macroscopic) fields. The Poynting vector in a macroscopic medium can be defined self-consistently with microscopic theory, in such a way that the spatially averaged microscopic Poynting vector is exactly predicted by a macroscopic formalism. This result is strictly valid in the limit of low-loss and allows for the unambiguous identification of the Poynting vector form in macroscopic electrodynamics.<ref name="Poynting_first_principles">{{cite journal|last1=Silveirinha|first1=M. G. |title=Poynting vector, heating rate, and stored energy in structured materials: a first principles derivation|journal=Phys. Rev. B|date=2010|volume=82|page=037104|doi=10.1103/physrevb.82.037104}}</ref><ref>{{cite journal|last1=Costa|first1=J. T., M. G. Silveirinha, A. Alù|title=Poynting Vector in Negative-Index Metamaterials|journal=Phys. Rev. B|date=2011|volume=83|issue=16 |page=165120|doi=10.1103/physrevb.83.165120|bibcode=2011PhRvB..83p5120C }}</ref>

== Alternative forms ==
It is possible to derive alternative versions of Poynting's theorem.<ref name=kinslerfavaromccall>
{{cite journal
 | author=Kinsler, P.
 |author2=Favaro, A. |author3=McCall M.W.
 | year=2009
 | title=Four Poynting theorems
 | journal=European Journal of Physics
 | volume=30
 | issue=5
 | pages=983
 | doi=10.1088/0143-0807/30/5/007
 | arxiv=0908.1721
 |bibcode = 2009EJPh...30..983K
 | url=http://spiral.imperial.ac.uk/bitstream/10044/1/18907/2/European%20Journal%20of%20Physics_30_5_2009.pdf
}}</ref> Instead of the flux vector {{nowrap|'''E''' × '''H'''}} as above, it is possible to follow the same style of derivation, but instead choose {{nowrap|'''E''' × '''B'''}}, the [[Abraham–Minkowski controversy|Minkowski]] form {{nowrap|'''D''' × '''B'''}}, or perhaps {{nowrap|'''D''' × '''H'''}}. Each choice represents the response of the propagation medium in its own way: the {{nowrap|'''E''' × '''B'''}} form above has the property that the response happens only due to electric currents, while the {{nowrap|'''D''' × '''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.<ref name=kinslerfavaromccall/>

== Modification ==
The derivation of the statement is dependent on the assumption that the materials the equation models can be described by a set of [[Electric susceptibility|susceptibility]] properties that are [[Linearity|linear]], [[Isotropy|isotropic]], homogenous and independent of [[frequency]].<ref name=FreemanKingLafyatis>{{Citation |last1=Freeman|first1=Richard |title=Essentials of Electricity and Magnetism |date=2019 |url=https://oxford.universitypressscholarship.com/10.1093/oso/9780198726500.001.0001/oso-9780198726500-chapter-1 |work=Electromagnetic Radiation |place=Oxford |publisher=Oxford University Press |doi=10.1093/oso/9780198726500.003.0001 |isbn=978-0-19-872650-0 |access-date=2022-02-18 |last2=King|first2=James |last3=Lafyatis|first3=Gregory|url-access=subscription }}</ref> The assumption that the materials have no absorption must also be made. A modification to Poynting's theorem to account for variations includes a term for the rate of [[Non-ohmic resistance|non-Ohmic]] [[Absorption (electromagnetic radiation)|absorption]] in a material, which can be calculated by a simplified approximation based on the [[Drude model]].<ref name=FreemanKingLafyatis/>
<math display="block">\frac{\partial}{\partial t} \mathcal{U} + \nabla \cdot \mathbf{S} + \mathbf{E} \cdot \mathbf{J}_\text{free} + \mathcal{R}_{\dashv\int} = 0</math>

== Complex Poynting vector theorem ==

This form of the theorem is useful in Antenna theory, where one has often to consider harmonic fields propagating in the space.
In this case, using phasor notation, <math>E(t) = E e^{j\omega t}</math> and <math>H(t) = H e^{j\omega t}</math>.
Then the following mathematical identity holds:
: <math>{1\over 2} \int_{\partial \Omega} E\times H^* \cdot d{\mathbf a} = {j\omega \over 2}\int_\Omega (\varepsilon E E^* - \mu H H^*) dv  - {1\over 2} \int_\Omega EJ^* dv,</math>
where <math>J</math> is the current density.

Note that in free space, <math>\varepsilon</math> and <math>\mu</math> are real, thus,
taking the real part of the above formula, it expresses the fact that the [[Poynting vector#Time-averaged Poynting vector|averaged radiated power]] flowing through <math>\partial \Omega</math> is equal to the work on the charges.

== References ==
{{reflist|25em}}

== External links ==
{{Sister project|project = wikiversity|text = [[Wikiversity]] has a lesson on  '''''[[v:Poynting's theorem|Poynting's theorem]]'''''}}
* [http://scienceworld.wolfram.com/physics/PoyntingTheorem.html Eric W. Weisstein "Poynting Theorem" From ScienceWorld – A Wolfram Web Resource.]

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[[Category:Electrodynamics]]
[[Category:Eponymous theorems of physics]]
[[Category:Circuit theorems]]