What does Poynting Theorem reveal about electrodynamic losses?

[yungman]

Poynting Theorem stated:

$$\vec E \cdot \vec J = - \frac 1 2 \frac {\partial} {\partial t} ( \epsilon E^2 + \frac 1 {\mu} B^2) - \frac 1 {\mu} \nabla \cdot( \vec E \times \vec B)$$

In pure electrodynamic point of view,

$$\vec E \cdot \vec J = \sigma E^2$$ which is nothing more than conductive loss ( resistance ohmic loss).

But I have a suspicion that is more to it than this...nothing is that simple. Please tell me what I missed.

Thanks

Alan

 

[vanhees71]

That's precisely what it is! Perhaps it's easiest to grasp by bringig the first term of the right-hand side of your equation to the left and the EJ term to the right and integrate the whole expression over a finite volume.

From Noether's theorem you know that

$$\mathcal{E}=\frac{1}{2} (\vec{E} \cdot \vec{D}+\vec{B} \cdot \vec{H})$$

is the energy density of the electromagnetic field (including polarization and magnetization effects of the bound charges/magnetic dipoles in the medium).

The Poynting vector,

$$\vec{S}=\vec{E} \times \vec{H}$$

is the energy flux. So manipulating your equation in the way indicated above, you get

$$\frac{\mathrm{d}}{\mathrm{d} t} E_{\text{field}}=-\int_{V} \mathrm{d}^3 \vec{x} \vec{J} \cdot \vec{E} - \int_{\partial V} \mathrm{d}^2 \vec{F} \cdot \vec{S}.$$

For the term with the Poynting vector, I've used Gauß's integral law. This equation means that the total field energy gets lost by the work done, i.e., to the motion of the charges, upon which the em. field acts and by radiation of em. waves, which transports energy through the surface of the volume under consideration.

 

[yungman]

Thanks, that make me feel a lot better.

 

[chrisbaird]

Yes, the Poynting Theorem is really the electrodynamic work-energy conservation theorem. If the total energy contained in the electromagnetic fields decreases over time it must be because the fields did work on charges and created currents, or because the energy simply flowed out of the volume of interest.

Technically, the Poynting vector just describes energy crossing a surface, which can happen even if we don't have traditional radiated waves.