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{{short description|Measure of directional electromagnetic energy flux}}
[[File:DipoleRadiation.gif|300px|thumb|[[Dipole radiation]] of a dipole vertically in the page showing electric field strength (colour) and Poynting vector (arrows) in the plane of the page.]]
{{Electromagnetism|Electrodynamics}}

In [[physics]], the '''Poynting vector''' represents the directional [[energy flux]] (the energy transfer per unit area per unit time) of an [[electromagnetic field]]. The [[SI]] unit of the Poynting vector is the [[watt]] per square metre (W/m<sup>2</sup>). It is named after its discoverer [[John Henry Poynting]] who first derived it in 1884.<ref name="Stratton1941">{{cite book
| last = Stratton
| first = Julius Adams
| title = Electromagnetic Theory
| edition = 1st
| publisher = McGraw-Hill
| place = New York
| year = 1941
| isbn = 978-0-470-13153-4
| url = https://books.google.com/books?id=zFeWdS2luE4C
}}</ref>{{rp|132}}  [[Oliver Heaviside]] also discovered it independently.

==Definition==
In Poynting's original paper and in many textbooks, the Poynting vector is defined as<ref name="Poynting1884">{{cite journal
| last = Poynting
| first = John Henry
| authorlink = John Henry Poynting
| title = On the Transfer of Energy in the Electromagnetic Field
| journal = Philosophical Transactions of the Royal Society of London
| volume = 175
| year = 1884
| pages = 343–361
| doi = 10.1098/rstl.1884.0016
}}</ref><ref>{{cite book
| last1 = Grant
| first1 = Ian S.
| last2 = Phillips
| first2 = William R.
| title = Electromagnetism
| edition = 2nd
| publisher = John Wiley & Sons
| place = New York
| year = 1990
| isbn = 978-0-471-92712-9
| url = https://books.google.com/books?id=Wi073n5G-8oC
}}</ref><ref>{{cite book
| last = Griffiths
| first = David J.
| title = Introduction to Electrodynamics
| edition = 3rd
| publisher = Addison-Wesley
| place = Boston
| year = 2012
| isbn = 978-0-321-85656-2
| url = https://books.google.com/books?id=J9ygBwAAQBAJ
}}</ref>
: <math>\mathbf{S} = \mathbf{E} \times \mathbf{H},</math>
where bold letters represent [[Euclidean vector|vector]]s and
* '''E''' is the [[electric field]] vector;
* '''H''' is the [[magnetic field]]'s auxiliary field vector.
This expression is often called the ''Abraham form''.<ref name="Kinsler2009">{{cite journal
| last1 = Kinsler
| first1 = Paul
| last2 = Favaro
| first2 = Alberto
| last3 = McCall
| first3 = Martin W.
| title = Four Poynting Theorems
| journal = European Journal of Physics
| volume = 30
| issue = 5
| year = 2009
| page = 983
| arxiv=0908.1721
| doi = 10.1088/0143-0807/30/5/007
|bibcode = 2009EJPh...30..983K }}</ref> The Poynting vector is usually denoted by '''S''' or '''N'''.

In the "microscopic" version of Maxwell's equations, this definition must be replaced by a [[Poynting vector#Formulation in terms of microscopic fields|definition]] in terms of the electric field '''E''' and the [[magnetic field]] '''B''' (it is described later in the article).

It is also possible to combine the [[electric displacement field]] '''D''' with the magnetic field '''B''' to get the ''Minkowski form'' of the Poynting vector, or use '''D''' and '''H''' to construct yet another version. The choice has been controversial: Pfeifer et al.<ref name="Pfeifer2007">{{cite journal
| last1 = Pfeifer
| first1 = Robert N. C.
| last2 = Nieminen
| first2 = Timo A.
| last3 = Heckenberg
| first3 = Norman R.
| last4 = Rubinsztein-Dunlop
| first4 = Halina
| title = Momentum of an Electromagnetic Wave in Dielectric Media
| journal = Reviews of Modern Physics
| volume = 79
| issue = 4
| year = 2007
| page = 1197
| doi = 10.1103/RevModPhys.79.1197
|arxiv = 0710.0461 |bibcode = 2007RvMP...79.1197P }}</ref> summarize and to a certain extent resolve the century-long dispute between proponents of the Abraham and Minkowski forms (see [[Abraham–Minkowski controversy]]).

The Poynting vector represents the particular case of an energy flux vector for electromagnetic energy. However, any type of energy has its direction of movement in space, as well as its density, so energy flux vectors can be defined for other types of energy as well, e.g., for [[Poynting's theorem#Generalization|mechanical energy]]. The Umov–Poynting vector<ref name="Umov1874">{{cite journal
| last = Umov
| first = Nikolay Alekseevich
| authorlink = Nikolay Alekseevich Umov
| title = Ein Theorem über die Wechselwirkungen in Endlichen Entfernungen
| journal = Zeitschrift für Mathematik und Physik
| volume = 19
| pages = 97–114
| year = 1874
| url = http://resolver.sub.uni-goettingen.de/purl?PPN599415665_0019
}}</ref> discovered by [[Nikolay Umov]] in 1874 describes energy flux in liquid and elastic media in a completely generalized view.

==Interpretation==
[[File:Poynting vectors of DC circuit.svg|thumb|upright=2|A DC circuit consisting of a [[Battery (electricity)|battery]] (''V'') and [[resistor]] (''R''), showing the direction of the Poynting vector (<span style="color:blue;">'''S''', blue arrows</span>) in the space surrounding it, along with the fields it is derived from; the [[electric field]] (<span style="color:red;">'''E''', red arrows</span>) and the [[magnetic field]] (<span style="color:green;">'''H''', green arrows</span>).  In the region around the battery the Poynting vector is directed outward, indicating power flowing out of the battery into the fields; in the region around the resistor the vector is directed inward, indicating field power flowing into the resistor.  Across any plane ''P'' between the battery and resistor, the Poynting flux is in the direction of the resistor.  The magnitudes (lengths) of the vectors are not shown accurately; only the directions are significant. ]]

The Poynting vector appears in [[Poynting's theorem]] (see that article for the derivation), an energy-conservation law:
: <math>\frac{\partial u}{\partial t} = -\mathbf{\nabla} \cdot \mathbf{S} - \mathbf{J_\mathrm{f}} \cdot \mathbf{E},</math>
where '''J'''<sub>f</sub> is the [[current density]] of [[Maxwell's equations#Formulation in terms of free charge and current|free charges]] and ''u'' is the electromagnetic energy density for linear, [[dispersion (optics)|nondispersive]] materials, given by
: <math>u = \frac{1}{2}\! \left(\mathbf{E} \cdot \mathbf{D} + \mathbf{B} \cdot \mathbf{H}\right)\! ,</math>
where
* '''E''' is the electric field;
* '''D''' is the electric displacement field;
* '''B''' is the magnetic field;
* '''H''' is the magnetic auxiliary field.<ref name="Jackson1998">{{cite book
| last = Jackson
| first = John David
| title = Classical Electrodynamics
| edition = 3rd
| publisher = John Wiley & Sons
| place = New York
| year = 1998
| isbn = 978-0-471-30932-1
| url = https://books.google.com/books?id=tYEAkAEACAAJ
}}</ref>{{rp|258–260}}

The first term in the right-hand side represents the electromagnetic energy flow into a small volume, while the second term subtracts the work done by the field on free electrical currents, which thereby exits from electromagnetic energy as [[dissipation]], heat, etc. In this definition, bound electrical currents are not included in this term, and instead contribute to '''S''' and ''u''.

For linear, [[dispersion (optics)|nondispersive]] and isotropic (for simplicity) materials, the [[Maxwell's equations#Constitutive relations|constitutive relations]] can be written as
: <math>\mathbf{D} = \varepsilon \mathbf{E},\quad \mathbf{H} = \frac{1}{\mu}\mathbf{B},</math>
where
* ''ε'' is the [[permittivity]] of the material;
* ''μ'' is the [[permeability (electromagnetism)|permeability]] of the material.<ref name="Jackson1998" />{{rp|258–260}}
Here ''ε'' and ''μ'' are scalar, real-valued constants independent of position, direction, and frequency.

In principle, this limits Poynting's theorem in this form to fields in vacuum and nondispersive linear materials.  A generalization to dispersive materials is possible under certain circumstances at the cost of additional terms.<ref name="Jackson1998" />{{rp|262–264}}

==Formulation in terms of microscopic fields==
The "microscopic" (differential) version of Maxwell's equations admits only the fundamental fields '''E''' and '''B''', without a built-in model of material media. Only the vacuum permittivity and permeability are used, and there is no '''D''' or '''H'''. When this model is used, the Poynting vector is defined as
: <math>\mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B},</math>
where
* ''μ''<sub>0</sub> is the [[vacuum permeability]];
* '''E''' is the electric field;
* '''B''' is the magnetic field.

This is actually the general expression of the Poynting vector.<ref>{{Cite book|title=Modern Electrodynamics|last=Zangwill|first=Andrew|publisher=Cambridge University Press|year=2013|isbn=9780521896979|location=|pages=508}}</ref> The corresponding form of [[Poynting's theorem]] is
: <math>\frac{\partial u}{\partial t} = -  \nabla \cdot \mathbf{S} -\mathbf{J} \cdot \mathbf{E},</math>
where '''J''' is the ''total'' [[current density]] and the energy density ''u'' is given by
: <math>u = \frac{1}{2}\! \left(\varepsilon_0 \mathbf{E}^2 + \frac{1}{\mu_0} \mathbf{B}^2\right)\! ,</math>
where ε<sub>0</sub> is the [[vacuum permittivity]].
It can be derived directly from [[Maxwell's equations#Formulation in terms of free charge and current|Maxwell's equations in terms of ''total'' charge and current]] and the [[Lorentz force]] law only.

The two alternative definitions of the Poynting ''vector'' are equal in vacuum or in non-magnetic materials, where {{nowrap|1='''B''' = ''μ''<sub>0</sub>'''H'''}}. In all other cases, they differ in that {{nowrap|1='''S''' = (1/''μ''<sub>0</sub>) '''E''' × '''B'''}} and the corresponding ''u'' are purely radiative, since the dissipation term {{nowrap|−'''J''' ⋅ '''E'''}} covers the total current, while the '''E''' × '''H''' definition has contributions from bound currents which are then excluded from the dissipation term.<ref name="Richter2008">{{cite journal
| last1 = Richter
| first1 = Felix
| last2 = Florian
| first2 = Matthias
| last3 = Henneberger
| first3 = Klaus
| title = Poynting's Theorem and Energy Conservation in the Propagation of Light in Bounded Media
| journal = Europhysics Letters Association
| volume = 81
| issue = 6
| year = 2008
| pages = 67005
| doi = 10.1209/0295-5075/81/67005
|arxiv = 0710.0515 |bibcode = 2008EL.....8167005R }}</ref>

Since only the microscopic fields '''E''' and '''B''' occur in the derivation of {{nowrap|1='''S''' = (1/''μ''<sub>0</sub>) '''E''' × '''B'''}}, assumptions about any material present are completely avoided, and Poynting vector and theorem are universally valid, in vacuum as in all kinds of material. This is especially true{{clarify|reason=What does "especially true" mean in a scientific claim?"|date=October 2015}} for the electromagnetic energy density, in contrast to the "macroscopic" form {{nowrap|'''E''' × '''H'''}}.<ref name="Richter2008" />

==Time-averaged Poynting vector==
[[File:Dipoles and mirrors.gif|300px|thumb|The field lines of the time-averaged Poynting vector of an electric dipole close to a mirror create complex patterns.]]
The above form for the Poynting vector represents the ''instantaneous'' power flow due to ''instantaneous'' electric and magnetic fields. More commonly, problems in electromagnetics are solved in terms of [[sinusoidal]]ly varying fields at a specified frequency. The results can then be applied more generally, for instance, by representing incoherent radiation as a superposition of such waves at different frequencies and with fluctuating amplitudes.

We would thus not be considering the instantaneous '''E'''(''t'') and '''H'''(''t'') used above, but rather a complex (vector) amplitude for each which describes a coherent wave's phase (as well as amplitude) using [[phasor]] notation. These complex amplitude vectors are ''not'' functions of time, as they are understood to refer to oscillations over all time. A phasor such as <math>\mathbf{E_\mathrm{m}}</math> is understood to signify a sinusoidally varying field whose instantaneous amplitude '''E'''(''t'') follows the real part of <math>\mathbf{E_\mathrm{m}}e^{j\omega t}</math> where ''ω'' is the (radian) frequency of the sinusoidal wave being considered.

In the time domain it will be seen that the instantaneous power flow will be fluctuating at a frequency of 2''ω''. But what is normally of interest is the ''average'' power flow in which those fluctuations are not considered. In the math below, this is accomplished by integrating over a full cycle <math>T=2\pi/\omega</math>. The following quantity, still referred to as a "Poynting vector", is expressed directly in terms of the phasors as:

<math>\mathbf{S}_\mathrm{m} = \tfrac{1}{2} \mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m}^* ,</math>

where <sup>∗</sup> denotes the complex conjugate. The time-averaged power flow (according to the instantaneous  Poynting vector averaged over a full cycle, for instance) is then given by the ''real part'' of <math>\mathbf{S}_\mathrm{m}</math>. The imaginary part is usually ignored, however it signifies "reactive power" such as the interference due to a [[standing wave]] or the [[Electromagnetic radiation#Near and far fields|near field]] of an antenna. In a single electromagnetic [[plane wave]] (rather than a standing wave which can be described as two such waves travelling in opposite directions), '''E''' and '''H''' are exactly in phase, so <math>\mathbf{S}_\mathrm{m}</math> is simply a real number according to the above definition.

The equivalence of <math>\operatorname{Re}(\mathbf{S}_\mathrm{m})</math> to the time-average of the ''instantaneous'' Poynting vector '''S''' can be shown as follows.

: <math>\begin{align}\mathbf{S}(t) &= \mathbf{E}(t) \times \mathbf{H}(t)\\
&= \operatorname{Re}\! \left(\mathbf{E}_\mathrm{m} e^{j\omega t}\right) \times \operatorname{Re}\!\left(\mathbf{H}_\mathrm{m} e^{j\omega t}\right)\\
&= \tfrac{1}{2}\! \left(\mathbf{E}_\mathrm{m} e^{j\omega t} + \mathbf{E}_\mathrm{m}^* e^{-j\omega t}\right) \times \tfrac{1}{2}\! \left(\mathbf{H}_\mathrm{m} e^{j\omega t} + \mathbf{H}_\mathrm{m}^* e^{-j\omega t}\right)\\
&= \tfrac{1}{4}\! \left(\mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m}^* + \mathbf{E}_\mathrm{m}^* \times \mathbf{H}_\mathrm{m} + \mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m} e^{2j\omega t} + \mathbf{E}_\mathrm{m}^* \times \mathbf{H}_\mathrm{m}^* e^{-2j\omega t}\right)\\
&= \tfrac{1}{2} \operatorname{Re}\! \left(\mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m}^*\right) + \tfrac{1}{2}\operatorname{Re}\! \left(\mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m} e^{2j\omega t}\right)\! .
\end{align}</math>

The average of  the instantaneous Poynting vector '''S'''  over time is given by:
: <math>\langle\mathbf{S}\rangle = \frac{1}{T} \int_0^T \mathbf{S}(t)\, dt = \frac{1}{T} \int_0^T\! \left[\tfrac{1}{2} \operatorname{Re}\! \left(\mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m}^*\right) + \tfrac{1}{2} \operatorname{Re}\! \left({\mathbf{E}_\mathrm{m}} \times {\mathbf{H}_\mathrm{m}} e^{2j\omega t}\right)\right]dt.</math>

The second term is the double-frequency component having an average value of zero, so we find:
: <math>\langle \mathbf{S}\rangle = \operatorname{Re}\! \left(\tfrac{1}{2}{\mathbf{E}_\mathrm{m}} \times \mathbf{H}_\mathrm{m}^*\right) =  \operatorname{Re}\! \left(\mathbf{S}_\mathrm{m}\right) </math>

According to some conventions the factor of 1/2 in the above definition may be left out. Multiplication by 1/2 is required to properly describe the power flow since the magnitudes of  <math>\mathbf{E}_\mathrm{m}</math> and <math>\mathbf{H}_\mathrm{m}</math> refer to the ''peak'' fields of the oscillating quantities. If rather the fields are described in terms of their [[root mean square]] (rms) values (which are each smaller by the factor <math>\sqrt{2}/2</math>), then the correct average power flow is obtained without multiplication by 1/2.

==Examples and applications==

===Coaxial cable===
[[File:Poynting vector coaxial cable.svg|300px|right|thumb|Poynting vector in a coaxial cable, shown in red.]]

For example, the Poynting vector within the [[dielectric]] [[Insulator (electrical)|insulator]] of a [[coaxial cable]] is nearly parallel to the wire axis (assuming no fields outside the cable and a wavelength longer than the diameter of the cable, including DC). Electrical energy delivered to the load is flowing entirely through the dielectric between the [[Electrical conductor|conductor]]s. Very little energy flows in the conductors themselves, since the electric field strength is nearly zero. The energy flowing in the conductors flows radially into the conductors and accounts for energy lost to resistive heating of the conductor. No energy flows outside the cable, either, since there the magnetic fields of inner and outer conductors cancel to zero.

===Resistive dissipation===
If a conductor has significant resistance, then, near the surface of that conductor, the Poynting vector would be tilted toward and impinge upon the conductor. Once the Poynting vector enters the conductor, it is bent to a direction that is almost perpendicular to the surface.<ref name="Harrington2001">{{cite book
| last = Harrington
| first = Roger F.
| year = 2001
| title = Time-Harmonic Electromagnetic Fields
| edition = 2nd
| publisher = McGraw-Hill
| isbn = 978-0-471-20806-8
| url = https://books.google.com/books?id=4-6kNAEACAAJ
}}</ref>{{rp|61}} This is a consequence of [[Snell's law]] and the very slow speed of light inside a conductor. The definition and computation of the speed of light in a conductor can be given.<ref name="Hayt2011">{{cite book
| last = Hayt
| first = William
| title = Engineering Electromagnetics
| edition= 4th
| publisher = McGraw-Hill
| place = New York
| year = 2011
| isbn = 978-0-07-338066-7
| url = https://books.google.com/books?id=XeaHcgAACAAJ
}}</ref>{{rp|402}} Inside the conductor, the Poynting vector represents energy flow from the [[electromagnetic field]] into the wire, producing resistive [[Joule heating]] in the wire. For a derivation that starts with Snell's law see Reitz page 454.<ref name="Reitz2008">{{cite book
| last1 = Reitz
| first1 = John R.
| last2 = Milford
| first2 = Frederick J.
| last3 = Christy
| first3 = Robert W.
| title = Foundations of Electromagnetic Theory
| edition = 4th
| publisher = Addison-Wesley
| place = Boston
| year = 2008
| isbn = 978-0-321-58174-7
| url = https://books.google.com/books?id=PYYSplKkBOoC
}}</ref>{{rp|454}}

===Plane waves===
In a propagating ''sinusoidal'' [[Linear polarisation|linearly polarized]] electromagnetic [[plane wave]] of a fixed frequency, the Poynting vector always points in the direction of propagation while oscillating in magnitude. The time-averaged magnitude of the Poynting vector is found as above to be:
:<math>\langle S\rangle = \frac{1}{2\eta}|E_\mathrm{m}|^2 </math>
where ''E''<sub>m</sub> is the complex amplitude of the electric field and η is the characteristic impedance of the transmission medium, or just [[Impedance of free space|η<sub>0</sub>]] <math>\approx</math> 377Ω for a plane wave in free space. This directly follows from the above expression for the average Poynting vector using phasor quantities, and the fact that in a plane wave the magnetic field <math>H_\mathrm{m}</math> is equal to the electric field <math>E_\mathrm{m}</math> divided by η (and thus exactly in phase).

In optics, the time-averaged value of the radiated flux is technically known as the [[irradiance]],  more often simply referred to as the ''[[Intensity (physics)|intensity]]''.

===Radiation pressure===
The density of the linear momentum of the electromagnetic field is ''S''/c<sup>2</sup> where ''S'' is the magnitude of the Poynting vector and c is the speed of light in free space. The [[radiation pressure]] exerted by an electromagnetic wave on the surface of a target is given by
: <math>P_\mathrm{rad} = \frac{\langle S\rangle}{\mathrm{c}}.</math>

===Static fields===
[[File:Poynting-Paradoxon.svg|200px|right|thumb|Poynting vector in a static field, where '''E''' is the electric field, '''H''' the magnetic field, and '''S''' the Poynting vector.]]

The consideration of the Poynting vector in static fields shows the relativistic nature of the Maxwell equations and allows a better understanding of the magnetic component of the [[Lorentz force]], {{nowrap|''q''('''v''' × '''B''')}}. To illustrate, the accompanying picture is considered, which describes the Poynting vector in a cylindrical capacitor, which is located in an '''H''' field (pointing into the page) generated by a permanent magnet. Although there are only static electric and magnetic fields, the calculation of the Poynting vector produces a clockwise circular flow of electromagnetic energy, with no beginning or end.

While the circulating energy flow may seem nonsensical or paradoxical, it is necessary to maintain [[conservation of momentum]]. Momentum density is proportional to energy flow density, so the circulating flow of energy contains an ''angular'' momentum.<ref name="Feynman">{{cite book
| last = Feynman
| first = Richard Phillips
| authorlink = Richard Feynman
| title = The Feynman Lectures on Physics
| volume = Vol. II: Mainly Electromagnetism and Matter
| edition = The New Millennium
| publisher = Basic Books
| place = New York
| year = 2011
| isbn = 978-0-465-02494-0
| url = http://www.feynmanlectures.caltech.edu/II_toc.html
}}</ref> This is the cause of the magnetic component of the Lorentz force which occurs when the capacitor is discharged. During discharge, the angular momentum contained in the energy flow is depleted as it is transferred to the charges of the discharge current crossing the magnetic field.

==Adding the curl of a vector field==
The Poynting vector occurs in Poynting's theorem only through its [[divergence]] {{nowrap|∇ ⋅ '''S'''}}, that is, it is only required that the [[surface integral]] of the Poynting vector around a closed surface describe the net flow of electromagnetic energy into or out of the enclosed volume. This means that adding a [[solenoidal vector field]] (one with zero divergence) to '''S''' will result in another field which satisfies this required property of a Poynting vector field according to Poynting's theorem. Since the  [[vector calculus identities#Divergence of the curl|divergence of any curl is zero]], one can add the [[Curl (mathematics)|curl]] of any vector field to the Poynting vector and the resulting vector field '''S'''' will still satisfy Poynting's theorem.<ref name="Jackson1998" />{{rp|258–260}}

However the theory of [[special relativity]], in which energy and momentum are defined locally and invariantly via the [[stress–energy tensor]], shows that the above-given expression for the Poynting vector is unique.<ref name="Jackson1998" />{{rp|258–260,605–612}}

==References==
{{reflist}}

==Further reading==
{{wikiquote}}
* {{cite book
| last = Becker
| first = Richard
| title = Electromagnetic Fields and Interactions
| edition = 1st
| publisher = Dover Publications
| place = Mineola, New York
| year = 1982
| isbn = 978-0-486-64290-1
| url = https://books.google.com/books?id=TyPCAgAAQBAJ
}}
* {{cite book
| last1 = Edminister
| first1 = Joseph
| last2 = Nahvi
| first2 = Mahmood
| title = Electromagnetics
| edition = 4th
| publisher = McGraw-Hill
| location = New York
| year = 2013
| isbn = 978-0-07-183149-9
| url = https://books.google.com/books?id=lySuAgAAQBAJ
}}

[[Category:Electromagnetic radiation]]
[[Category:Optics]]
[[Category:Vectors (mathematics and physics)]]