Poynting's theorem

Poynting's theorem is a statement due to John Henry Poyntingcite 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] about the conservation of energy for the electromagnetic field. It relates the time derivative of the energy density, "u" to the energy flow and the rate at which the fields do work. It is summarised by the following formula:

:$frac\{partial\; u\}\{partial\; t\}\; +\; ablacdotmathbf\{S\}\; =\; -mathbf\{J\}cdotmathbf\{E\}$

where S is the Poynting vector representing the flow of energy, J is the current density and E is the electric field. Energy density "u" is (symbol ε₀ is the electric constant and μ₀ is the magnetic constant):

:$u\; =\; frac\{1\}\{2\}left(epsilon\_0\; mathbf\{E\}^2\; +\; frac\{mathbf\{B\}^2\}\{mu\_0\}\; ight).$

Since the magnetic field does no work, the right hand side gives the negative of the total work done by the electromagnetic field per second·meter³.

Poynting's theorem in integral form::$frac\{partial\}\{partial\; t\}\; int\_V\; u\; dV\; +\; oint\_\{partial\; V\}mathbf\{S\}\; dmathbf\{A\}\; =\; -int\_Vmathbf\{J\}cdotmathbf\{E\}\; dV$

Where $partial\; V\; !$ is the surface which bounds (encloses) volume $V\; !$.

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:

$ablacdotmathbf\{S\}\; +\; epsilon\_0\; mathbf\{E\}cdotfrac\{partial\; mathbf\{E\{partial\; t\}\; +\; frac\{mathbf\{B\{mu\_0\}cdotfrac\{partialmathbf\{B\{partial\; t\}\; +mathbf\{J\}cdotmathbf\{E\}\; =\; 0$

Where $mathbf\{S\}$ is the energy flow of the electromagnetic wave, $epsilon\_0\; mathbf\{E\}cdotfrac\{partial\; mathbf\{E\{partial\; t\}$ is the power consumed for the build-up of electric field, $frac\{mathbf\{B\{mu\_0\}cdotfrac\{partialmathbf\{B\{partial\; t\}$ is the power consumed for the build-up of magnetic field and $mathbf\{J\}cdotmathbf\{E\}$ is the power consumed by the Lorentz force acting on charge carriers.

Derivation

The theorem can be derived from two of Maxwell's Equations. First consider Faraday's Law::$abla\; imes\; mathbf\{E\}\; =\; -\; frac\{partial\; mathbf\{B\{partial\; t\}$Taking the dot product of this equation with $mathbf\{B\}$ yields::$mathbf\{B\}\; cdot\; (\; abla\; imes\; mathbf\{E\})\; =\; -\; mathbf\{B\}\; cdot\; frac\{partial\; mathbf\{B\{partial\; t\}$Next consider the equation::$abla\; imes\; mathbf\{B\}\; =\; mu\_0\; mathbf\{J\}\; +\; epsilon\_0\; mu\_0\; frac\{partial\; mathbf\{E\{partial\; t\}$Taking the dot product of this equation with $mathbf\{E\}$ yields::$mathbf\{E\}\; cdot\; (\; abla\; imes\; mathbf\{B\})\; =\; mathbf\{E\}\; cdot\; mu\_0\; mathbf\{J\}\; +\; mathbf\{E\}\; cdot\; epsilon\_0\; mu\_0\; frac\{partial\; mathbf\{E\{partial\; t\}$Subtracting the first dot product from the second yields::$mathbf\{E\}\; cdot\; (\; abla\; imes\; mathbf\{B\})\; -\; mathbf\{B\}\; cdot\; (\; abla\; imes\; mathbf\{E\})\; =\; mu\_0\; mathbf\{E\}\; cdot\; mathbf\{J\}\; +\; epsilon\_0\; mu\_0\; mathbf\{E\}\; cdot\; frac\{partial\; mathbf\{E\{partial\; t\}\; +\; mathbf\{B\}\; cdot\; frac\{partial\; mathbf\{B\{partial\; t\}$Finally, by the product rule, as applied to the divergence operator over the cross product (described here)::$-\; ablacdot\; (\; mathbf\{E\}\; imes\; mathbf\{B\}\; )\; =\; mu\_0\; mathbf\{E\}\; cdot\; mathbf\{J\}\; +\; epsilon\_0\; mu\_0\; mathbf\{E\}\; cdot\; frac\{partial\; mathbf\{E\{partial\; t\}\; +\; mathbf\{B\}\; cdot\; frac\{partial\; mathbf\{B\{partial\; t\}$Since the Poynting vector $mathbf\{S\}$ is defined as::$mathbf\{S\}\; =\; frac\{1\}\{mu\_0\}\; mathbf\{E\}\; imes\; mathbf\{B\}$This is equivalent to::$ablacdotmathbf\{S\}\; +\; epsilon\_0\; mathbf\{E\}cdotfrac\{partial\; mathbf\{E\{partial\; t\}\; +\; frac\{mathbf\{B\{mu\_0\}cdotfrac\{partialmathbf\{B\{partial\; t\}\; +mathbf\{J\}cdotmathbf\{E\}\; =\; 0$

Generalization

The "mechanical" energy counterpart of the above theorem for the "electromagnetical" energy continuity equation is:$frac\{partial\}\{partial\; t\}\; u\_m(mathbf\{r\},t)\; +\; ablacdot\; mathbf\{S\}\_m\; (mathbf\{r\},t)\; =mathbf\{J\}(mathbf\{r\},t)cdotmathbf\{E\}(mathbf\{r\},t),$where "u_m" 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 $mathbf\{r\}\_\{alpha\}(t)$::$u\_m(mathbf\{r\},t)\; =\; sum\_\{alpha\}\; frac\{m\_\{alpha\{2\}\; dot\{r\}^2\_\{alpha\}delta(mathbf\{r\}-mathbf\{r\}\_\{alpha\}(t)),$$mathbf\{S\_m\}$ is the flux of their energies, or a "mechanical Poynting vector"::$mathbf\{S\}\_m\; (mathbf\{r\},t)\; =\; sum\_\{alpha\}\; frac\{m\_\{alpha\{2\}\; dot\{r\}^2\_\{alpha\}dot\{mathbf\{r\_\{alpha\}delta(mathbf\{r\}-mathbf\{r\}\_\{alpha\}(t)).$Both can be combined via the Lorentz force, which the electromagnetical fields exert on the moving charged particles (see above), to the following energy continuity equation or energy conservation law 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] ::$frac\{partial\}\{partial\; t\}left(u\_e\; +\; u\_m\; ight)\; +\; ablacdot\; left(\; mathbf\{S\}\_e\; +mathbf\{S\}\_m\; ight)\; =\; 0,$covering both types of energy and the conversion of one into the other.

ee also

*Poynting vector

References

External links

* [http://scienceworld.wolfram.com/physics/PoyntingTheorem.html Eric W. Weisstein "Poynting Theorem" From ScienceWorld--A Wolfram Web Resource.]