Energy density of an electromagnetic field

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Main Question or Discussion Point

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?)

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vanhees71
Gold Member
2019 Award
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
Homework Helper
Gold Member
Delta2
I see. Makes perfect sense now. Thanks!