A very thin line antenna with very small section area. But it can receive the energy nearly same as a much thick line antenna. If we use the antenna section area multiply the Poynting vector obtained some energy. We define this energy as 1. The thin wire line antenna can receive energy as high as 1000. This looks like the Poynting theorem is wrong.
It is seams that the energy transfer between a transmitting antenna and a receiving antenna should not described by Poynting theorem, perhaps some other theorem for example reciprocity theorem or mutual energy theorem?
The reciprocity theorem describes transmission and reception of power by antennas. It gives the result that the gain of a receiving antenna is the same as its gain when transmitting. Antennas seem to have a minimum capture aperture equal to a circle of radius=wavelength/(2 pi), and a thin wire captures energy from about this distance on both sides. This makes sense, because, for example, the electric fields can't change from minimum strength (close to the wire) to maximum (in the passing plane wave) over a distance much shorter than this without there being evanescent waves. This has some common maths with the rule-of-thumb risetime of a pulse being 0.35/bandwidth.
The Poynting theorem is applied to the effective area of linear antennas but not their physical area which is negligible or difficult to define. Reciprocity relates antenna gain, which is normally defined for transmitting mode, to the effective are of receiving antenna: Eff_Area = Gain x lambda^2/(4 pi). Gain (radiation gain or directivity) is defined by antenna pattern shapes.
Antennae and in particular wire antennas can have a much biher scattering cross section than their physical one..in particular when they are small compared to a wavelength AND at resonance..no reason to worry about limits of validity of the Poynting vector concept..for such kind of receiving antennae the radiation is kind of sucked in...
An answer is written in he added file, because of a problem in the upload on the site.
I taught antennas for forty years as well as designing them professionally. The first thing I said when talking about receiving antennas isthat their receiving aperture is not related to any physical area on the antenna structure. It is defined as the raio of incoming power density (Watts/square metre) to the power received from the antenna (Watts) under conditions of conjugate impedance match and polarisation match. Even for a dish antenna the receiving aperture is not equal to the area of the dish.
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