Energy density of an electromagnetic field

[PhysicsKT]

The energy density of an electromagnetic field with a linear dielectric is often expressed as

. It is also known that energy can be found by

. Using the latter, the energy density is found to be

, as is well known. If you integrate the latter only over free charge and ignore bound charge, you write

, use integration by parts, and obtain the first result. Does the first result neglect the energy from bound charge? If not, why does

break down (I.e. why can’t one find the energy with a dielectric by treating the bound charge as its own independent charge arrangement and using formulae for a vacuum?)

 

[vanhees71]

You are mixing macroscopic with microscopic electrodynamics. For macroscopic electrodynamics $u=\vec{E} \cdot \vec{D}/2$ is correct. Within Markovian linear-response theory $\vec{D}=\epsilon \vec{E}=\epsilon_0 \epsilon_r \vec{E}$.

 

[TSny]

For some discussion, see Griffith's Electrodynamics.
https://www.amazon.com/gp/product/1108420419/?tag=pfamazon01-20
View the table of contents and click on section 4.4.3 "Energy in Dielectric Systems". See especially the bottom of page 198 and the top of 199.

 

[PhysicsKT]

I see. Makes perfect sense now. Thanks!