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I have this exercise on my electromagnetism course :

Consider that there exist two pairs of fields E and B that satisfy Maxwell's equations, with the same boundary conditions and have the same initial conditions. Using Poynting's theorem (conservation of energy) show that these fields are equal, that is: the solution of the problem is unique.

I already tried using the conservation of the energy theorem and using that the energy of the EM field are constant and consequently the fields I supposed were different are finally the same ones, but I can't prove that for all t. When I use the they have the same initial conditions I can assure that $E_1$ and $E_2$ are the same for $t=0$ as I proposed $E=E_1-E_2$ (same criteria used for B) but can't find the way to get that for all $t$.

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  • $\begingroup$ Include some equations of what you've done so far. This makes it a lot easier for others to help you. $\endgroup$
    – Tobias Fünke
    Commented Feb 20, 2023 at 16:12
  • $\begingroup$ Conservation of energy would dictate that the rate of energy change is zero. I'd probably start by writing an expression for the total energy, differentiate with respect to time, then set the result to zero. That should give an answer with t in it. $\endgroup$
    – Rich006
    Commented Feb 20, 2023 at 18:10
  • $\begingroup$ Hint: how would the Poynting energy of EM field defined as difference of the hypothetical two solutions evolve in time? This leads to violation of the Poynting theorem for such field unless that field is zero everywhere all the time. $\endgroup$
    – Ján Lalinský
    Commented Feb 21, 2023 at 3:34
  • $\begingroup$ However, note that this is not very mathematically satisfactory "proof", as it relies on the Poynting theorem, which has to be proven as well, and that requires some conditions, such as no point and no line charges, so that EM energy converges. So good enough for smooth distributions of charge, but not in general case. $\endgroup$
    – Ján Lalinský
    Commented Feb 21, 2023 at 3:43
  • $\begingroup$ Thinking some more on this, the only case the Poynting theorem needs to be applied to is the subtracted system (field is difference of the two solutions, sources vanish, boundary condition is zero). For such system it is hard to imagine how EM field could be divergent enough for the Poynting theorem not to hold. So maybe this "proof" is actually valid for any distribution of sources... $\endgroup$
    – Ján Lalinský
    Commented Feb 21, 2023 at 4:02

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