In words, the theorem is an energy balance: 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 and J•E is the rate at which the fields do work on a charged object. The energy densityu, assuming no electric or magnetic polarizability, is given by: in which B is the magnetic flux density. Using the divergence theorem, Poynting's theorem can be rewritten in integral form: where is the boundary of a volume V. The shape of the volume is arbitrary but fixed for the calculation.
Electrical Engineering
In electrical engineering context the theorem is usually written with the energy density term u expanded in the following ways, which resembles the continuity equation: where
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.
Poynting's theorem
Considering the statement in words above - there are three elements to the theorem, which involve writing energy transfer as volume integrals: 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.
Poynting vector
From the theorem, the actual form of the Poynting vector S can be found. The time derivative of the energy density is using the constitutive relations The partial time derivatives suggest using two of Maxwell's Equations. Taking the dot product of the Maxwell–Faraday equation with H: next taking the dot product of the 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 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.
Alternative forms
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 magnetic monopole currents. The other two forms use complementary combinations of electric and magnetic currents to represent the polarization and magnetization responses of the medium.
Generalization
The mechanical energy counterpart of the above theorem for the electromagnetic energy continuity equation is where um is the kinetic energy density in the system. It can be described as the sum of kinetic energies of particles α, whose trajectory is given by rα: 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, to the following energy continuity equation or energy conservation law: covering both types of energy and the conversion of one into the other.
