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If there is no source then also there is electromagnetic waves described by Maxwell equation. how if there is no source then existence of EM waves. What gives energy to this EM waves. Is it vacuum fluctuation or something else?

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    $\begingroup$ In classical electromagnetism waves always have a source, it's just not always necessary to describe the source as part of the problem we are interested in. In that case we just write down the idealized free wave and the wave equations guarantee that it will propagate even in absence of a source. $\endgroup$ – CuriousOne Jan 5 '15 at 11:19
  • $\begingroup$ my question is if there is only vacuum in the universe then whether there is generation of electromagnetic waves or not $\endgroup$ – Hare Krishna Jan 5 '15 at 17:26
  • $\begingroup$ @HareKrishna, there isn't only vacuum in the universe, so you're starting off on the wrong foot. And if there was only vacuum (i.e. total quality-less void), then there would be no electromagnetic interaction as we know it, because there would be nothing it could possibly bear on, nothing it could be conceived in relation to, no place (in space or across all time) which it could come from or go to. $\endgroup$ – Steve Feb 3 '18 at 9:51
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The fact that you can solve Maxwell's equations in vacuo means that the vacuum can propagate electromagnetic waves. It cannot generate them, since as you point out there are no sources (charges and currents). The waves were generated somewhere else, where there was a current or a charge moving around, and you are just looking at a region where there's "nothing"*.

The energy of those waves is the energy stored in the oscillating electric and magnetic fields, and it comes from the source. E.g. for dipole radiation (see link later) an accelerating charge slows down and the lost kinetic energy goes into creating EM waves.

Remember that the Maxwell's equations are differential equations, i.e. they require boundary conditions. One of the things to have to fix is the amplitude of a wave $E_0$. You can set it to $0$ and have no electromagnetic waves whatsoever. But if you indeed have a wave propagating, then it means that $E_0 \neq 0$ and its value must be computed with information about how this wave was generated. For example, in the case of dipole radiation, you can compute the pre-factor.

*: Let me just stress that this is in the context of classical electrodyanmics. If you were to venture into the quantum world, then you would have particle-antiparticle pairs and their respective electric fields popping out of nowhere.

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