Poynting vector

In physics, the Poynting vector can be thought of as representing the energy flux (in W/m²) of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside independently co-discovered the Poynting vector. In Poynting's original papercite journal
author=Poynting, J. H.
authorlink=John_Henry_Poynting
year=1884
journal=Phil. Trans.
volume=175
pages=277
title=On the Transfer of Energy in the Electromagnetic Field
url=http://www.archive.org/details/collectedscienti00poynuoft] and in many textbookscite book
author=John David Jackson
title=Classical electrodynamics
edition=Third Edition
publisher= Wiley
location=New York
year=1998
isbn=047130932X
url=http://worldcat.org/isbn/047130932X] it is defined as:$mathbf\{S\}\; =\; mathbf\{E\}\; imesmathbf\{H\},$where E is the electric field and H the auxiliary magnetic field. (All bold letters represent vectors.) Sometimes, an alternative definition in terms of electric field E and the magnetic field B is used, which is explained below.

Interpretation

The Poynting vector appears in Poynting's theorem, an energy-conservation law,:$frac\{partial\; u\}\{partial\; t\}\; =\; -\; mathbf\{\; abla\}cdotmathbf\{S\}\; -mathbf\{J\}\_\{f\}\; cdot\; mathbf\{E\},$where J_f is the current density of free charges and u is the electromagnetic energy density,:$u\; =\; frac\{1\}\{2\}left(mathbf\{E\}cdotmathbf\{D\}\; +\; mathbf\{B\}cdotmathbf\{H\}\; ight),$where B is the magnetic field and D the electric displacement field.

The first term in the right-hand side represents the net electromagnetic energy flow into a small volume, while the second term represents the negative of work done by free electrical currents that are not necessarily converted into electromagnetic energy (dissipation, heat). In this definition, bound electrical currents are not included in this term, and instead contribute to S and u.

Note that u can only be given if linear, nondispersive and uniform materials are involved, i.e., if the constitutive relations can be written as :$mathbf\{D\}\; =\; epsilonmathbf\{E\},\; ;;;\; mathbf\{H\}\; =\; mathbf\{B\}/mu$where ε and μ are constants (which depend on the material through which the energy flows), called the permittivity and permeability, respectively, of the material.

This practically limits Poynting's theorem in this form to fields in vacuum. A generalization to dispersive materials is possible under certain circumstances at the cost of additional terms and the loss of their clear physical interpretation.

The Poynting vector is usually interpreted as an energy flux, but this is only strictly correct for electromagnetic radiation. The more general case is described by Poynting's theorem above, where it occurs as a divergence, which means that it can only describe the "change" of energy density in space, rather than the flow.

Formulation in terms of microscopic fields

In some cases, it may be more appropriate to define the Poynting vector S as:$mathbf\{S\}\; =\; frac\{1\}\{mu\_0\}(mathbf\{E\}\; imes\; mathbf\{B\}),$where μ₀ is the magnetic constant. It can be derived directly from and the Lorentz force law only.

The corresponding form of Poynting's theorem is:$frac\{partial\; u\}\{partial\; t\}\; +\; ablacdotmathbf\{S\}\; =\; -\; mathbf\{J\}cdotmathbf\{E\},$where J is the "total" current density and the energy density "u" is :$u\; =\; frac\{1\}\{2\}left(epsilon\_0\; mathbf\{E\}^2\; +\; frac\{mathbf\{B\}^2\}\{mu\_0\}\; ight)$(with the electric constant ε₀).

The two alternative definitions of the Poynting "vector" are equivalent in vacuum or in non-magnetic materials, where $B=mu\_0\; H$. In all other cases, they differ in that S=1/μ₀ (ExB) and the corresponding u are purely radiative, since the dissipation term, -J E, covers the total current, while the definition in terms of H has contributions from bound currents which then lack in the dissipation term.cite journal
author=Richter, F.
coauthors=Florian, M.; Henneberger, K.
year=2008
title=Poynting's theorem and energy conservation in the propagation of light in bounded media
journal=Europhys. Lett.
volume=81
pages=67005
doi=10.1209/0295-5075/81/67005
url=http://arxiv.org/pdf/0710.0515v3
format=reprint]

Since only the microscopic fields E and B are needed in the derivation of S=1/μ₀(ExB), assumptions about any material possibly present can be completely avoided, and Poynting's vector as well as the theorem in this definition are universally valid, in vacuum as in all kinds of material. This is especially true for the electromagnetic energy density, in contrast to the case above.

Invariance to adding a curl of a field

Since the Poynting vector only occurs in Poynting's theorem as a divergence $ablacdotmathbf\{S\}$, the Poynting vector is arbitrary to the extent that the curl $abla\; imesmathbf\; F$ of any field F can be added, because $ablacdot\; abla\; imesmathbf\; F=0$ for any field. Doing so is not common, though, and will lead to inconsistensies in a relativistic description of electromagnetic fields in terms of the stress-energy tensor.

Generalization

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 mechanical energy. The Umov-Poynting vectorcite journal
author=Umov, N. A.
authorlink=Nikolay_Umov
year=1874
journal=Zeitschrift für Mathematik und Physik
volume=XIX
pages=97
title=Ein Theorem über die Wechselwirkungen in Endlichen Entfernungen] discovered by Nikolay Umov in 1874 describes energy flux in liquid and elastic media in a completely generalized view.

Examples and applications

The Poynting vector in a coaxial cable

For example, the Poynting vector within the dielectric insulator of a coaxial cable is nearly parallel to the wire axis (assuming no fields outside the cable) - so electric energy is flowing through the dielectric between the conductors. If the core conductor was replaced by a wire having significant resistance, then the Poynting vector would become tilted toward that wire, indicating that energy flows from the electromagnetic field into the wire, producing resistive Joule heating in the wire.

The Poynting vector in plane waves

In a propagating "sinusoidal" electromagnetic plane wave of a fixed frequency, the Poynting vector oscillates, always pointing in the direction of propagation. The time-averaged magnitude of the Poynting vector is:$langle\; S\; angle\; =\; frac\{1\}\{2\; mu\_0\; c\}\; E\_0^2\; =\; frac\{epsilon\_0\; c\}\{2\}\; E\_0^2,$where $E\_0$ is the maximum amplitude of the electric field and $c$ is the speed of light in free space. This time-averaged value is also called the irradiance or intensity "I".

Derivation

In an electromagnetic plane wave, $mathbf\{E\}$ and $mathbf\{B\}$ are always perpendicular to each other and the direction of propagation. Moreover, their amplitudes are related according to:$B\_0\; =\; frac\{E\_0\}\{c\},$and their time and position dependences are:$Eleft(t,\{mathbf\; r\}\; ight)\; =\; E\_0,cosleft(omega,t-\; \{mathbf\; k\}\; cdot\; \{mathbf\; r\}\; ight),$:$Bleft(t,\{mathbf\; r\}\; ight)\; =\; B\_0,cosleft(omega,t-\; \{mathbf\; k\}\; cdot\; \{mathbf\; r\}\; ight),$where $omega$ is the frequency of the wave and $mathbf\{k\}$ is wave vector. The time-dependent and position magnitude of the Poynting vector is then:$S(t)\; =\; frac\{1\}\{mu\_0\}\; E\_0,B\_0,cos^2left(omega\; t-\{mathbf\; k\}\; cdot\; \{mathbf\; r\}\; ight)\; =\; frac\{1\}\{mu\_0\; c\}\; E\_0^2\; cos^2left(omega\; t-\{mathbf\; k\}\; cdot\; \{mathbf\; r\}\; ight)\; =\; epsilon\_0\; c\; E\_0^2\; cos^2left(omega\; t-\{mathbf\; k\}\; cdot\; \{mathbf\; r\}\; ight).$In the last step, we used the equality $epsilon\_0,mu\_0\; =\; \{c\}^\{-2\}$. Since the time- or space-average of $cos^2left(omega,t-\{mathbf\; k\}\; cdot\; \{mathbf\; r\}\; ight)$ is ½, it follows that:$leftlangle\; S\; ight\; angle\; =\; frac\{epsilon\_0\; c\}\{2\}\; E\_0^2.$

Poynting vector and radiation pressure

"S" divided by the square of the speed of light in free space is the density of the linear momentum of the electromagnetic field. The time-averaged intensity $langlemathbf\{S\}\; angle$ divided by the speed of light in free space is the radiation pressure exerted by an electromagnetic wave on the surface of a target:
:$P\_\{rad\}=frac\{langle\; S\; angle\}\{c\}.$

Problems in certain cases

The common use of the Poynting vector as an energy flux rather than in the context of Poynting's theorem gives rise to controversial interpretions in cases where it is not used to describe electromagnetic radiation. Two examples are given below.

DC Power flow in a concentric cable

Application of Poynting's Theorem to a concentric cable carrying DC current leads to the correct power transfer equation $mathit\{P\}\; =\; mathit\{V\}mathit\{I\}$, where $mathit\{V\}$ is the potential difference between the cable and ground, $mathit\{I\}$ is the current carried by the cable. This power flows through the surrounding dielectric, and not through the cable itself.Citation
last = Jordan
first = Edward
author-link =
last2 = Balmain
first2 = Keith
author2-link =
title = Electromagnetic Waves and Radiating Systems
place = New Jersey
publisher = Prentice-Hall
year = 2003
edition = Second
url = http://worldcat.org/isbn/8120300548
isbn = 81-203-0054-8 ]

However, it is also known that power cannot be radiated without accelerated charges, i.e. time varying currents. Since we are considering DC (time invariant) currents here, radiation is not possible. This has led to speculation that Poynting Vector may not represent the power flow in certain systems.cite paper
author = Jeffries, Clark
title = A New Conservation Law for Classical Electrodynamics
publisher = Society for Industrial and Applied Mathematics (SIAM Review)
date = Sep., 1992
url = http://links.jstor.org/sici?sici=0036-1445%28199209%2934%3A3%3C386%3AANCLFC%3E2.0.CO%3B2-U
format = PDF
accessdate = 2008-03-04 ] cite paper
author = Robinson, F. N. H.
title = Poynting's Vector: Comments on a Recent Paper by Clark Jeffries
publisher = Society for Industrial and Applied Mathematics (SIAM Review)
date = Dec., 1994
url = http://www.jstor.org/pss/2132722 ]

Independent E and B fields

Independent static $mathbf\{E\}$ and $mathbf\{B\}$ fields do not result in power flows along the direction of $mathbf\{E\}\; imes\; mathbf\{B\}$. For example, application of Poynting's Theorem to a bar magnet, on which an electric charge is present, leads to seemingly absurd conclusion that there is a continuous circulation of energy around the magnet. However, there is no divergence of energy flow, or in layman's terms, energy that enters given unit of space equals the energy that leaves that unit of space, so there is no net energy flow into the given unit of space.

References

Further reading

* [http://scienceworld.wolfram.com/physics/PoyntingVector.html "Poynting Vector" from ScienceWorld (A Wolfram Web Resource)] by Eric W. Weisstein
*cite book
author=Richard Becker & Sauter, F
title=Electromagnetic fields and interactions
publisher= Dover
location=New York
year=1964
isbn=0486642909
url=http://worldcat.org/isbn/0486642909
*cite book
author=Joseph Edminister
title=Schaum's outline of theory and problems of electromagnetics
publisher= McGraw-Hill Professional
location=New York
page=p. 225
year=1995
isbn=0070212341
url=http://books.google.com/books?id=xV97IDOqBZIC&pg=PA225&dq=%22Poynting+vector%22&lr=&as_brr=0&sig=3UfNdFoFNJPQi0Ij9oj4WZZn7K0

ee also

*Poynting's theorem