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I am currently reading a chapter in a physics book covering the electric energy stored in a parallel plate capacitor. The author derives the formula for the energy density: $u = 1/2 \epsilon E^2 $ and claims that it applies for any electric field.

Now, this seems to me kind of wrong. Imagining that I would place an electron in empty space, I wouldn't do any work to move it there(because there are no other charges that my electron would interact with). However, the electron would generate a field and thus have energy density associated with it. What am I missing?

andrei
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  • Maxwell's equations are a theory of macroscopic electromagnetic fields. They are not a theory of matter and they can't say anything useful about electrons. For that one would need, at least, quantum field theory, but even that can't say much useful about the rest mass of electrons, so the question is as simple as it is open. All we can say is that the simple formula does not apply. – CuriousOne Aug 10 '16 at 07:02
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    Possible duplicate of Electron Electric Field Mass? – CuriousOne Aug 10 '16 at 07:52
  • The fact that an elecron has a field around it, is associated with an inherent energy of the electron and is said to be stored in the field. – Lelouch Aug 10 '16 at 09:13
  • Possible duplicate of The interaction of a charged particle with its own field – John Rennie Aug 10 '16 at 09:29
  • @CuriousOne what do you mean by "macroscopic electromagnetic fields". Are they those generated by macroscopic bodies? – andrei Aug 10 '16 at 19:02
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    @Lelouch the idea of field energy density is that you have to do to work to amass charges together, thus increasing the electric field strength around them. – andrei Aug 10 '16 at 19:04
  • For a macroscopic field we can average out over many microscopic interactions, i.e. the photon count is so large that we only have to care about the average expectation values, which give us the macroscopic electric and magnetic field components. Even this is a bit tricky as the case of black body radiation shows: even though the Planck spectrum is independent of the photon number (except for fluctuations), the shape of the curve can only be calculated by taking field quantization into account, so there is subtlety in the difference between micro- and macroscopic field descriptions. – CuriousOne Aug 10 '16 at 19:58
  • @CuriousOne What you're saying is too hard for me to grasp with my level of physics. I'll leave it like this for now and after I improve on the topics you mentioned, I will check your answer again to see if I figure it out. – andrei Aug 12 '16 at 05:05
  • Don't worry about this. Nobody has figured the correct answer to your question out, yet. Learn to understand the classical fields really well. Get an intuition for quantum mechanics, then try to master quantum field theory, maybe one day you will be among the first to understand how all of this really works. :-) – CuriousOne Aug 12 '16 at 06:39

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