Editing Poynting vector (section)

==Interpretation==
The Poynting vector appears in [[Poynting's theorem]] (see that article for the derivation), an energy-conservation law:
<math display="block">\frac{\partial u}{\partial t} = -\mathbf{\nabla} \cdot \mathbf{S} - \mathbf{J_\mathrm{f}} \cdot \mathbf{E},</math>
where '''J'''<sub>f</sub> is the [[current density]] of [[Maxwell's equations#Formulation in terms of free charge and current|free charges]] and ''u'' is the electromagnetic energy density for linear, [[dispersion (optics)|nondispersive]] materials, given by
<math display="block">u = \frac{1}{2}\! \left(\mathbf{E} \cdot \mathbf{D} + \mathbf{B} \cdot \mathbf{H}\right)\! ,</math>
where
* '''E''' is the electric field;
* '''D''' is the electric displacement field;
* '''B''' is the magnetic flux density;
* '''H''' is the magnetizing field.<ref name="Jackson1998">{{cite book
| last = Jackson
| first = John David | author-link = John David Jackson (physicist)
| title = Classical Electrodynamics
| edition = 3rd
| publisher = John Wiley & Sons
| place = New York
| year = 1998
| isbn = 978-0-471-30932-1
| url = https://books.google.com/books?id=FOBBEAAAQBAJ
}}</ref>{{rp|pp=258–260}}

The first term in the right-hand side represents the electromagnetic energy flow into a small volume, while the second term subtracts the work done by the field on free electrical currents, which thereby exits from electromagnetic energy as [[dissipation]], heat, etc. In this definition, bound electrical currents are not included in this term and instead contribute to '''S''' and ''u''.

For light in free space, the linear momentum density is
<math> \frac{\langle S \rangle}{c^2} </math>

For linear, [[dispersion (optics)|nondispersive]] and isotropic (for simplicity) materials, the [[Maxwell's equations#Constitutive relations|constitutive relations]] can be written as
<math display="block">\mathbf{D} = \varepsilon \mathbf{E},\quad \mathbf{B} = \mu\mathbf{H},</math>
where
* ''ε'' is the [[permittivity]] of the material;
* ''μ'' is the [[permeability (electromagnetism)|permeability]] of the material.<ref name="Jackson1998" />{{rp|pp=258–260}}
Here ''ε'' and ''μ'' are scalar, real-valued constants independent of position, direction, and frequency.

In principle, this limits Poynting's theorem in this form to fields in vacuum and nondispersive{{clarify|date=November 2021}} linear materials.  A generalization to dispersive materials is possible under certain circumstances at the cost of additional terms.<ref name="Jackson1998" />{{rp|pp=262–264}}

One consequence of the Poynting formula is that for the electromagnetic field to do work, both magnetic and electric fields must be present. The magnetic field alone or the electric field alone cannot do any work.<ref>{{Cite web|title=K. McDonald's Physics Examples - Railgun|url=https://physics.princeton.edu//~mcdonald/examples/railgun.pdf|access-date=2021-02-14|website=puhep1.princeton.edu}}</ref>