Poynting theory apply to both static and time varying fields?

[yungman]

Poynting vector is flow of energy per unit area. Dose it apply for both static field where E and B are decoupled, AND time varying EM field where E and B are coupled?

 

[yungman]

The reason I ask is referring to page 346-349 of Griffiths. The Poynting theorem was derived using time varying relation where

$$\nabla \times \vec B= \mu\vec J -\mu\frac {\partial \vec D}{\partial t} \;\hbox { and }\; \nabla \times \vec E=-\frac{\partial \vec B}{\partial t}$$

But then in Example 8.1 on page 348, it gave an example of a steady current I flow down a wire and calculate the power flow down the wire ( Poynting vector S). Where is use E= (voltage across wire) divided by the length of wire. B is calculated by current I.

 

[vanhees71]

The Poynting theorem follows from the complete Maxwell equations and thus is valid always.

E.g., it is interesting to calculate the energy flow of a DC conducting coaxial cable (I choose this as an example, because this is a very simple to solve stationary problem). Calculate both, the electric and magnetic fields and then the Poynting vector. Then think about, what this means concerning energy transport.

 

[yungman]

Thanks
What you suggested is very similar to problem 8.1 in Griffiths and I worked it out already.

 

[PJC01]

Yes, Poynting's theory applies to both static and time-varying fields. In the case of a static field, where the electric and magnetic fields are decoupled, the Poynting vector represents the flow of energy in the form of electromagnetic radiation. This is commonly seen in situations such as the propagation of radio waves or the energy transfer in an electric circuit.

In the case of a time-varying electromagnetic field, where the electric and magnetic fields are coupled, the Poynting vector still represents the flow of energy, but it also takes into account the changing nature of the fields. This is important in understanding phenomena such as electromagnetic induction, where the changing magnetic field induces an electric field and results in the transfer of energy.

In both cases, the Poynting vector is a useful tool for quantifying and analyzing the flow of electromagnetic energy. However, it is important to note that in the case of a time-varying field, the Poynting vector may not always be constant and can vary over time. Overall, Poynting's theory is applicable to a wide range of electromagnetic phenomena, providing valuable insights into the behavior of energy in the form of electromagnetic radiation.