Energy density of an electromagnetic field

  • #1
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The energy density of an electromagnetic field with a linear dielectric is often expressed as
1994848e8909b58aaa7dfa748264681c15b04cdb.png
. It is also known that energy can be found by
90ce12f273329132bc0a22e77cabd6fadd9317ec.png
. Using the latter, the energy density is found to be
2b95635ceca0346d915aadc5ef5f3f8047d12dd6.png
, as is well known. If you integrate the latter only over free charge and ignore bound charge, you write
a3e754ebda6b4b4d609f6ac85bb3d8b3f6fa3516.png
, use integration by parts, and obtain the first result. Does the first result neglect the energy from bound charge? If not, why does
2b95635ceca0346d915aadc5ef5f3f8047d12dd6.png
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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  • #2
vanhees71
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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}##.
 
  • #3
TSny
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I see. Makes perfect sense now. Thanks!
 

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