This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)(Learn how and when to remove this template message)
In electrodynamics, Poynting's theorem is a statement of conservation of energy for the electromagnetic field,[clarification needed] in the form of a partial differential equation, due to the British physicist John Henry Poynting. 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 done on a charge distribution (i.e. an electrically charged object), through energy flux.
In words, the theorem is an energy balance:
- The rate of energy transfer (per unit volume) from a region of space equals the rate of work done on a charge distribution plus the energy flux leaving that region.
A second statement can also explain the theorem - "The decrease in the electromagnetic energy per unit time in a certain volume is equal to the sum of work done by the field forces and the net outward flux per unit time".
Mathematically, this is summarised in differential form as:
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 current density corresponding to the motion of charge, E is the electric field, and • is the dot product). The energy density u, assuming no electric or magnetic polarizability, is given by:
where is the boundary of a volume V. The shape of the volume is arbitrary but fixed for the calculation.
- ε0 is the electric constant and μ0 is the magnetic constant.
- is the density of reactive power driving the build-up of electric field,
- is the density of reactive power driving the build-up of magnetic field, and
- is the density of electric power dissipated by the Lorentz force acting on charge carriers.
While conservation of energy and the Lorentz force law can give 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.
- 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:
- 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:
- The Lorentz force density f on a charge distribution, integrated over the volume to get the total force F, is
So by conservation of energy, the balance equation for the energy flow per unit time is the integral form of the theorem:
and since the volume V is arbitrary, this is true for all volumes, implying
which is Poynting's theorem in differential form.
next taking the dot product of the Maxwell–Ampère equation with E:
Collecting the results so far gives:
then, using the vector calculus identity:
gives an expression for the Poynting vector:
which physically means the energy transfer due to time-varying electric and magnetic fields is perpendicular to the fields.
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.
It is possible to derive alternative versions of Poynting's theorem. Instead of the flux vector E × B as above, it is possible to follow the same style of derivation, but instead choose the Abraham form E × H, the Minkowski form D × B, or perhaps D × H. Each choice represents the response of the propagation medium in its own way: the E × B form above has the property that the response happens only due to electric currents, while the 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.
The mechanical energy counterpart of the above theorem for the electromagnetic energy continuity equation is
where Sm is the flux of their energies, or a "mechanical Poynting vector":
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:
covering both types of energy and the conversion of one into the other.
- Poynting, J. H. "On the Transfer of Energy in the Electromagnetic Field". 175: 343–361. doi:10.1098/rstl.1884.0016. Cite journal requires
- Griffiths, David J. Introduction to electrodynamics. Prentice Hall, 1962.
- Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, p.364, ISBN 81-7758-293-3
- Silveirinha, M. G. (2010). "Poynting vector, heating rate, and stored energy in structured materials: a first principles derivation". Phys. Rev. B. 82: 037104. doi:10.1103/physrevb.82.037104.
- Costa, J. T. , M. G. Silveirinha, A. Alù (2011). "Poynting Vector in Negative-Index Metamaterials". Phys. Rev. B. 83: 165120. doi:10.1103/physrevb.83.165120.
- Kinsler, P.; Favaro, A.; McCall M.W. (2009). "Four Poynting theorems" (PDF). European Journal of Physics. 30 (5): 983. arXiv:0908.1721. Bibcode:2009EJPh...30..983K. doi:10.1088/0143-0807/30/5/007.
- Richter, E.; Florian, M.; Henneberger, K. (2008). "Poynting's theorem and energy conservation in the propagation of light in bounded media". Europhysics Letters. 81 (6): 67005. arXiv:0710.0515. Bibcode:2008EL.....8167005R. doi:10.1209/0295-5075/81/67005.