Editing Poynting vector (section)

==Definition==
In Poynting's original paper and in most textbooks, the Poynting vector <math>\mathbf{S}</math> is defined as the [[cross product]]<ref name="Poynting1884">{{cite journal
| last = Poynting
| first = John Henry
| author-link = John Henry Poynting
| title = On the Transfer of Energy in the Electromagnetic Field
| journal = Philosophical Transactions of the Royal Society of London
| volume = 175
| year = 1884
| pages = 343–361
| doi = 10.1098/rstl.1884.0016
| doi-access = free
}}</ref><ref>{{cite book
| last1 = Grant
| first1 = Ian S.
| last2 = Phillips
| first2 = William R.
| title = Electromagnetism
| edition = 2nd
| publisher = John Wiley & Sons
| place = New York
| year = 1990
| isbn = 978-0-471-92712-9
| url = https://books.google.com/books?id=Wi073n5G-8oC
}}</ref><ref>{{cite book
| last = Griffiths
| first = David J. | author-link = David J. Griffiths
| title = Introduction to Electrodynamics
| edition = 3rd
| publisher = Addison-Wesley
| place = Boston
| year = 2012
| isbn = 978-0-321-85656-2
| url = https://books.google.com/books?id=J9ygBwAAQBAJ
}}</ref>
<math display=block>\mathbf{S} = \mathbf{E} \times \mathbf{H},</math>
where bold letters represent [[Euclidean vector|vector]]s and
* '''E''' is the [[electric field]] vector;
* '''H''' is the [[magnetic field]]'s auxiliary field vector or ''[[Magnetic field#The H-field|magnetizing field]]''.
This expression is often called the ''Abraham form'' and is the most widely used.<ref name="Kinsler2009">{{cite journal
| last1 = Kinsler
| first1 = Paul
| last2 = Favaro
| first2 = Alberto
| last3 = McCall
| first3 = Martin W.
| title = Four Poynting Theorems
| journal = European Journal of Physics
| volume = 30
| issue = 5
| year = 2009
| page = 983
| arxiv=0908.1721
| doi = 10.1088/0143-0807/30/5/007
|bibcode = 2009EJPh...30..983K | s2cid = 118508886
}}</ref> The Poynting vector is usually denoted by '''S''' or '''N'''.

In simple terms, the Poynting vector '''S''' depicts the direction and rate of transfer of energy, that is [[Power (physics)|power]], due to electromagnetic fields in a region of space that may or may not be empty. More rigorously, it is the quantity that must be used to make [[Poynting's theorem]] valid. Poynting's theorem essentially says that the difference between the electromagnetic energy entering a region and the electromagnetic energy leaving a region must equal the energy converted or dissipated in that region, that is, turned into a different form of energy (often heat). So if one accepts the validity of the Poynting vector description of electromagnetic energy transfer, then Poynting's theorem is simply a statement of the [[conservation of energy]].

If electromagnetic energy is not gained from or lost to other forms of energy within some region (e.g., mechanical energy, or heat), then electromagnetic energy is [[Conservation law#Global and local conservation laws|locally conserved]] within that region, yielding a [[continuity equation]] as a special case of Poynting's theorem:
<math display="block">\nabla\cdot \mathbf{S} = -\frac{\partial u}{\partial t}</math>
where <math>u</math> is the energy density of the electromagnetic field. This frequent condition holds in the following simple example in which the Poynting vector is calculated and seen to be consistent with the usual computation of power in an electric circuit.