I was wondering if Maxwell equations conserved energy. Jackson is not really convincing about the whole Poynting theorem (neither currently google found articles which just assume a bunch of stuff).

Does anyone have an insightful/intuitive explanation or should I just take it for granted?

Rotational field means the curl is different than $0$.
Conservative force needs $\nabla$x$\vec{F} = 0$
Is energy conserved?

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    $\begingroup$ What's not convincing about the Poynting theorem? $\endgroup$ – Phoenix87 Nov 8 '17 at 22:52
  • $\begingroup$ For a system of electrically charged matter in flat spacetime, the SET is conserved (i.e. $\partial_\mu \left(\Theta^{\mu\nu} (F) + T^{\mu\nu}_{\text{matter}}\right) = 0$, with $\Theta$ the symmetric Belinfante-Rosenfeld tensor). Perhaps you can read the book by Barut on electrodynamics (Barut, A. - Electrodynamics and classical theory of fields and particles) for an alternate proof. $\endgroup$ – DanielC Nov 8 '17 at 23:32
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    $\begingroup$ You don't need any assumptions beyond Maxwell's equations and the Lorentz force law. Perhaps you could elaborate on what you find unsatisfactory. $\endgroup$ – J. Murray Nov 9 '17 at 2:00
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    $\begingroup$ I'm downvoting this question until clarification comes. It's really annoying when someone states that some derivation is not convincing and doesn't point out any problems. What do you expect to get as an answer? $\endgroup$ – OON Nov 9 '17 at 10:02
  • $\begingroup$ I expected to get an answer considering why rotating fields in ME are considered conservative. A physical answer, rather than a mathematical derivation. $\endgroup$ – Dominik Car Nov 9 '17 at 21:49

Maxwell's equations are equivalent to the Lagrangian for classical electrodynamics. According to the Noether theorem, energy is conserved, if the Lagrangian has a continuous symmetry of time transaltion, as it does. Thus Maxwell's equations do conserve energy.

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  • $\begingroup$ Well, that depends whether the current field is dynamical or not. Usually in the Lagrangian framework of EM it's taken to be a background field, in which case if the charges are moving then the Lagrangian isn't time translationally invariant anymore, so conservation of energy isn't obvious. Making the current dynamical requires generalizing to Maxwell-Klein-Gordon theory or spinor electrodynamics, which I think is stretching the OP's question. $\endgroup$ – tparker Nov 9 '17 at 5:26
  • $\begingroup$ @tparker Yeah, The OP just wanted a simple intuitive explanation. In real cases, the math can be complicated, but physically we know that energy is conserved in electromagnetism. Therefore there must be a way to address any discrepancies by potential energy corrections, which may be a bit too deep for me :) $\endgroup$ – safesphere Nov 9 '17 at 5:43
  • $\begingroup$ @tparker why going to matter fields? Can you not use the normal Lagrangian with dynamic electromagnetic field coupled to classical matter. $\endgroup$ – lalala Nov 9 '17 at 8:42
  • $\begingroup$ This is what I mean. You have a circular field, the rotation of the field is $!=$ $0$. If the rotation of the field is $!=$ $0$, the energy is not conserved. $\endgroup$ – Dominik Car Nov 9 '17 at 17:06
  • $\begingroup$ @DominikCar Yeah, I'd say this answer is basically incorrect, because it ignores the EM field's back-reaction on the charges, and any full energy accounting that conserves energy would have to include the mechanical energy that the fields transfer to the charges. $\endgroup$ – tparker Nov 9 '17 at 17:09

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