The Poynting vector, power flow, and the CFA
This extract is from the classic text, "The Mathematical Theory of Electricity and Magnetism", by Sir James Jeans, published by Cambridge University Press in the fifth edition of 1933. Page 519.
quote..
"The integral of the Poynting Flux over a closed surface gives the total flow of energy into or out of a surface, but it has not been proved, and we are not entitled to assume, that there is an actual flow of energy at every point equal to the Poynting Flux. For instance, if an electrified sphere is placed near to a bar magnet, this latter assumption would require a perpetual flow of energy at every point in the field except the special points at which the electric and magnetic lines of force are tangential to one another. It is difficult to believe that this predicted circulation of energy can have any physical reality. On the other hand, it is to be noticed that such a circulation of energy is almost meaningless. The circulation of a fluid is a definite conception because it is possible to identify the different particles of a fluid; we can say for instance whether or not the particles entering a small element of volume are identical or not with an equal number of particles coming out, but the same is not true of energy."
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The Poynting theorem actually is a relationship between the divergence (div S) of the Poynting vector flux S and the sources of energy at a point. For a region containing no sources or sinks of energy, the Poynting theorem merely states that div S = 0, and therefore by the divergence theorem, that the integral of the normal component of S over a closed surface surrounding a region containing no sources or sinks of energy must also be zero. There is, as Sir James points out, no reason at all to identify S with a local flow of energy. His remarks about "circulation" merely say that if div S = 0 then S, if it exists, (that is to say, it is not zero everywhere) must be the curl of another vector (say T), since if S = curl T then div S is then identically zero by the well known vector identity div (curl[a vector field]) = 0. Thus we can envisage a region containing a Poynting flux S where there is no power flow, provided S = curl T.
D.Jefferies 22nd November 1999