Editing Poynting vector (section)

==Formulation in terms of microscopic fields==
The "microscopic" (differential) version of Maxwell's equations admits only the fundamental fields '''E''' and '''B''', without a built-in model of material media. Only the vacuum permittivity and permeability are used, and there is no '''D''' or '''H'''. When this model is used, the Poynting vector is defined as
<math display="block">\mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B},</math>
where
* ''μ''<sub>0</sub> is the [[vacuum permeability]];
* '''E''' is the electric field vector;
* '''B''' is the magnetic flux.

This is actually the general expression of the Poynting vector{{dubious|date=November 2021}}.<ref>{{Cite book|title=Modern Electrodynamics|last=Zangwill|first=Andrew|publisher=Cambridge University Press|year=2013|isbn=9780521896979|pages=508}}</ref> The corresponding form of [[Poynting's theorem]] is
<math display="block">\frac{\partial u}{\partial t} = -  \nabla \cdot \mathbf{S} -\mathbf{J} \cdot \mathbf{E},</math>
where '''J''' is the ''total'' [[current density]] and the energy density ''u'' is given by
<math display="block">u = \frac{1}{2}\! \left(\varepsilon_0 |\mathbf{E}|^2 + \frac{1}{\mu_0} |\mathbf{B}|^2\right)\! ,</math>
where ''ε''<sub>0</sub> is the [[vacuum permittivity]]. It can be derived directly from [[Maxwell's equations#Formulation in terms of free charge and current|Maxwell's equations in terms of ''total'' charge and current]] and the [[Lorentz force]] law only.

The two alternative definitions of the Poynting ''vector'' are equal in vacuum or in non-magnetic materials, where {{nowrap|1='''B''' = ''μ''<sub>0</sub>'''H'''}}. In all other cases, they differ in that {{nowrap|1='''S''' = (1/''μ''<sub>0</sub>) '''E''' × '''B'''}} and the corresponding ''u'' are purely radiative, since the dissipation term {{nowrap|−'''J''' ⋅ '''E'''}} covers the total current, while the '''E''' × '''H''' definition has contributions from bound currents which are then excluded from the dissipation term.<ref name="Richter2008">{{cite journal
| last1 = Richter
| first1 = Felix
| last2 = Florian
| first2 = Matthias
| last3 = Henneberger
| first3 = Klaus
| title = Poynting's Theorem and Energy Conservation in the Propagation of Light in Bounded Media
| journal = EPL
| volume = 81
| issue = 6
| year = 2008
| pages = 67005
| doi = 10.1209/0295-5075/81/67005
|arxiv = 0710.0515 |bibcode = 2008EL.....8167005R | s2cid = 119243693
}}</ref>

Since only the microscopic fields '''E''' and '''B''' occur in the derivation of {{nowrap|1='''S''' = (1/''μ''<sub>0</sub>) '''E''' × '''B'''}} and the energy density, assumptions about any material present are avoided.  The Poynting vector and theorem and expression for energy density are universally valid in vacuum and all materials.<ref name="Richter2008" />