What is the connection between the Poynting vector (EM transverse wave) and the description of photons with $E=h*f$ and $c=f*\lambda$? I think for radio antenna transmitters, if the AC current's frequency is at the frequency of radiowaves, EM waves in the radio frequency will be emitted? Then the frequency has something to do with the way the photons were generated ?
Anyways I have not been able to make the connection between the interpretation of photons as packets of energy and EM radiation, if there's a difference at all.
 A: The Poynting vector gives the energy flux per unit of area and time. Suppose $\Sigma$ is a closed surface that encloses an antenna. Then the total power radiated by the antenna is
$$P = \int_{\Sigma}\mathbf S\cdot\hat{\mathbf n}\ \text d\Sigma$$
where $\mathbf S$ is the Poynting vector. Assuming that the antenna is transmitting at a frequency $\nu$, each emitted photon has an energy of
$$E_\nu = h\nu$$
Knowing the power $P$ of the antenna and the frequency $\nu$ we can estimate the photon flux $\dot N$ by means of the equation
$$P = E_\nu\dot N,$$
which is just a statement of the law of energy conservation.
More generally, if an antenna is transmitting with a spectral power density $\pi(\nu)$, we would have a spectral photon flux $\dot n(\nu)$ given by
$$\dot n(\nu)=\frac{\pi(\nu)}{h\nu}.$$
The total photon flux is then given by the integral over all frequencies, viz.
$$\dot N = \int_0^\infty\frac{\pi(\nu)}{h\nu}\ \text d\nu.$$
Indeed, for an antenna transmitting at a pure frequency $\nu^*$ we would have $\pi(\nu) = P^*\delta(\nu-\nu^*)$, which yields
$$\dot N = \frac{P^*}{h\nu^*}.$$
A: There is another photon energy law close to the Poynting theorem, which is the real energy conservation law in electromagnetic field theory.
Energy conservation law
If in an abstract empty space, there is no thing (no stars, no sun, no earth and moon) else, but two charges: charge 1 and charge 2. If one charge 1 offers charge 2 some energy, the energy of charge 2 will increase, but the energy of charge 1 will decrease. However the total energy will not change. Similarly, if there are N charges in the abstract empty space, the total energy of these $N$ charges will also not charge. Hence, it is clear there are following energy conservation laws for N charges. Each charge has a current element.
In electromagnetic field theory the energy conservation law is
$$
\sum_{i=1}^{N}\sum_{j=1,j\neq i}^{N} \iiint_{V}\boldsymbol{E}_i \cdot \boldsymbol{J}^{\ast}_j dV=0$$
Mutual energy theorem
If $N=2$, the above energy conservation law become the mutual energy theorem:
$$- \iiint_{V_1}(\boldsymbol{E}^\ast_2 \cdot \boldsymbol{J}_1)dV=\iiint_{V_2} (\boldsymbol{E}_1 \cdot \boldsymbol{J}^\ast_2)dV$$
This theorem can be applied to the primary coil and secondary coil of a transformer. It can be applied to the transmitting antenna and receiving antenna. It also can be applied to the photon system which includes an emitter and absorber.
Mutual energy principle
The mutual energy principle can be applied to N current elements in the abstract empty space.
$$
-\sum_{i=1}^{N}\sum_{j=1,j\neq i}^{N}\iint_{\Gamma}(\boldsymbol{E}_i\times\boldsymbol{H}_j)\cdot\hat{n}d\Gamma$$
$$
=\sum_{i=1}^{N}\sum_{j=1,j\neq i}^{N}\iiint_{V}(\boldsymbol{E}_i\cdot\boldsymbol{J}_j
+\boldsymbol{E}_i\cdot\frac{\partial}{\partial t}\boldsymbol{D}_j+\boldsymbol{H}_i\cdot\frac{\partial}{\partial t}\boldsymbol{B}_j)dV$$
$\Gamma$ is the boundary of the volume $V$. Assume all current elements are inside of the volume $V$.
Mutual energy flow theorem
$$-\iiint_{V_{1}} \boldsymbol{E}^*_2 \cdot \boldsymbol{J}_1 dV =(\xi_1,\xi_2)=\iiint_{V_{2}}\boldsymbol{E}_1\cdot \boldsymbol{J}^*_2 dV$$
where $(\xi_{1},\xi_{2})$ is the mutual energy flow, which can be calculated according to,
$$
(\xi_{1},\xi_{2})
=\iint_{\Gamma}(\boldsymbol{E}_1\times\boldsymbol{H}_2^{\ast}+\boldsymbol{E}_2^\ast\times\boldsymbol{H}_1)\cdot\hat{n}d\Gamma$$
$\Gamma$ is any surface separate the two current elements. It can be a closed sphere surrounding one element, or a infinite plane between the two current elements.
$\xi_1=[\boldsymbol{E}_1,\boldsymbol{H}_1]^T$, $\xi_2=[\boldsymbol{E}_2,\boldsymbol{H}_2]^T$, one is retarded wave and another is the advanced wave. $T$ is the Matrix transfer.
The mutual energy flow has all properties of photons, we can say the mutual energy flow is the photon or photon is the mutual energy flow. The mutual energy flow actually is the energy flow. It is the energy flow from emitter to the absorber. It is the energy flow from the transmitting antenna to the receiving antenna. It is also the energy from the primary coil of a transformer to the secondary coil of a transformer.
Axioms of the mutual energy theory
The above can be called the mutual energy theory, which includes two axioms: the energy conservation law, the mutual energy principle. These axioms can be added on the top of Maxwell equations. The energy conservation law is self-explanatory. This axiom will lead to the existence of the advanced wave.
This axiom will reduce the solutions of Maxwell's equations. Most solution of Maxwell's equations are no physical effect, only the pair solutions include one retarded wave sent from the source and the advanced wave sent from the sink is the real physical solution. The pair solution needs to be synchronized. In this situation, a mutual energy flow can be produced which transmitted the energy from the emitter to the absorber.
There is another axiom, the radiation energy cannot overflow to the outside of our universe. This law is also self-exoplanetary. If energy can overflow to the outside of the universe, our universe will continue to lose the energy that is not possible.
Poynting theorem and Poynting vector
In the mutual energy theory, the energy flow corresponding to the Poynting vector belongs to the self-energy flow which doesn't transfer the energy. The power of self-energy flow is reactive. It sends the energy to space but the energy comes back immediately through a time reversal process.
The energy is transferred only through the mutual energy flow. The mutual energy flow is the only energy flow in the electromagnetic field theory. The mutual energy flow can be normalized to the photon energy $\omega \hbar$.
How to learn the Mutual energy theory
The mutual energy flow theory can be found by search the keyword "mutual energy flow" or "mutual energy principle"
