# Meaning of the Poynting vector

In a book I am studying, the Poynting vector is defined as: $$\mathcal{P} = \mathbb{E} \times\mathbb{H}$$ and it is described the Poynting's theorem, that states that the flux through a surface wrapping a volume is the energy balance on that volume.

In another part of the book, it says that the only thing that has a physical meaning is the flux of the Poynting vector, while the vector by itself hasn't it. This is because, if you add to the Poynting vector the curl of another vector, it will gave the same flux. $$\mathcal{P}' = \mathbb{E} \times\mathbb{H} + \operatorname{curl}( \mathbb{F})$$ $$\Rightarrow\iint_Σ \mathcal{P}' \cdot n \, dS = \iint_Σ \mathbb{E} \times\mathbb{H} \cdot n\, dS = \iint_Σ \mathcal{P} \cdot n \, dS$$ Looking on the web, I found on the Wikipedia page about Poynting vector, in the section "Adding the curl of a vector field" that (for some reason I cannot fully understand since I have never had a course about special relativity) that the expression for the Poynting vector is unique. To me, this seems to give the Poynting vector a local meaning, not only to its flux through a surface.

Is it true? What is this local meaning (if present)? Where can I look to start comprehend it better and more mathematically?

$\mathbb{E}$: electric field

$\mathbb{H}$: magnetic field

$Σ$: generic closed surface

$\mathcal{P}$: Poynting vector

• There is a reference to the Jackson's textbook in the Wikipedia article, so you may wish to look at it. – akhmeteli Dec 26 '16 at 22:39
• Isn't the Poynting vector itself an energy flux? What do you mean by the "flux of the Poynting vector"? – probably_someone Dec 27 '16 at 5:17
• @probably_someone That's the wonted, or at least common, name for the OP's surface integrals, is it not? At least I've read so. – WetSavannaAnimal Dec 27 '16 at 6:50
• @probably_someone by "flux of the Poynting vector" I mean the surface integral. My question is about if I can say that the Poynting vector has a physical meaning just when it is integrated through a surface or if it is physically meaningful as a vector, locally. – JackI Dec 27 '16 at 6:55
• OP = "Original Poster"- i.e. the person who asks the question – WetSavannaAnimal Dec 27 '16 at 7:05

It does not, however, say that the field that you get when adding the curl is still the energy flux density as computed with $\vec{E}\times\vec{H}$. If you add some weird random field, it will not represent the local energy flux density anymore, but it will still satisfy Poynting's theorem.