In electrodynamics, Poynting's theorem is a statement of conservation of energy for electromagnetic fields developed by British physicist John Henry Poynting.
[1] It states that in a given volume, the stored energy changes at a rate given by the work done on the charges within the volume, minus the rate at which energy leaves the volume.
[2] The theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation.
Poynting's theorem states that the rate of energy transfer per unit volume from a region of space equals the rate of work done on the charge distribution in the region, plus the energy flux leaving that region.
where In an electrical engineering context the theorem is sometimes written with the energy density term u expanded as shown.
[citation needed] This form resembles the continuity equation: where The rate of work done by the electromagnetic field on the infinitesimal charge
is given by the Lorentz Force Law as:
is the volume charge density and
is the current density at the point and time where
is the velocity of the charge dq.
(Note that the H and D forms of the magnetic and electric fields are used here.
)[3] Substituting this into the expression for rate of work gives:
Continuing the derivation requires the following assumptions:[2] It can be shown[4] that:
Returning to the equation for rate of work,
Since the volume is arbitrary, this can be cast in differential form as:
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.
[5][6] It is possible to derive alternative versions of Poynting's theorem.
[7] Instead of the flux vector E × H as above, it is possible to follow the same style of derivation, but instead choose E × B, 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.
[7] The derivation of the statement is dependent on the assumption that the materials the equation models can be described by a set of susceptibility properties that are linear, isotropic, homogenous and independent of frequency.
[8] 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 absorption in a material, which can be calculated by a simplified approximation based on the Drude model.
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,
Then the following mathematical identity holds: where
Note that in free space,
are real, thus, taking the real part of the above formula, it expresses the fact that the averaged radiated power flowing through
is equal to the work on the charges.