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In many books, for a linear dielectric medium in which we have Maxwell's equations and the relationships $${\mathbf D}=\varepsilon({\mathbf x}){\mathbf E}$$ and $${\mathbf B}=\mu({\mathbf x}){\mathbf H}$$ we find formulae for the electromagnetic energy as $$U=\int d{\mathbf x}\frac{1}{2}({\mathbf E}\cdot{\mathbf D}+{\mathbf B}\cdot{\mathbf H})$$ in e.g. J.D. Jackson's book.

Suppose, instead, we have Maxwell's equations and the ${\it general}$ linear relationship $${\mathbf D}=\varepsilon({\mathbf x}){\mathbf E}+{\mathbf P}$$ and $${\mathbf B}=\mu({\mathbf x})({\mathbf H}+{\mathbf M}).$$ We are willing to suppose that we know what the energy for making the polarization $\mathbf P$ and magnetization $\mathbf M$ appear are - but want to know what the electromagnetic energy is. Think for example that we are only moving a bunch of permanent magnets around and want to know how the energy changes when we do. We don't care about the condensed matter cost of making the magnets and the medium has no hysteresis and is linear as above. What then is the formula for the electromagnetic energy? Do you have a reference for this (likely) more than century old result?

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  • $\begingroup$ I believe that your questions is about, in a more general setting, 'how do I calculate the Energy-Momentum tensor of and EM + medium system?', if so, it's not totally understood, and is a very complicated subject. I don't know if in this specific case is still viable to calculate the energy density, but it's not obvious how to do it in a generic situation. $\endgroup$ – Hydro Guy May 29 '14 at 14:18
  • $\begingroup$ Oh, I understand that in the general non-linear case, life is hard. But, in the specific case given above, e.g. "I am moving permanent magnets around a medium with finite susceptibility" I would think the answer was knowable and known. $\endgroup$ – user47505 May 29 '14 at 14:49
  • $\begingroup$ Why do you want to introduce strange relations like that? Material constants (functions of $\mathbf x$) $\epsilon,\mu$ for expression of linear relations are usually introduced in a simpler way : $\mathbf D = \epsilon \mathbf E$, $\mathbf B = \mu \mathbf H$. You can take these and try to derive some analogue to the Poynting theorem for the case $\epsilon,\mu$ depend on position. $\endgroup$ – Ján Lalinský May 29 '14 at 19:29
  • $\begingroup$ There may be cases where you can't take $\vec B = \mu \vec H$ and similar, e.g., you may have a linear medium, but there is some kind of memory effect. $\endgroup$ – Hydro Guy May 29 '14 at 22:10
  • $\begingroup$ The formulae I used for the magnetization $\vec M$ and polarization $\vec P$ are quite standard (can be found in many textbooks) and describe e.g. permanent magnets and ferroelectrics. I am not interested in hysteresis, which is common in ferromagnets and ferroelectrics. I am just interested in moving ferromagnets (mostly) around in magnetic fields. $\endgroup$ – user47505 May 30 '14 at 3:14

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