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These two equations are true in electrostatics/magnetostatics: $$\nabla \cdot \vec{E}= {\rho \over \epsilon_0},$$ $$\nabla \cdot \vec{B}=0.$$

I have learned that they are also true in electrodynamics. But I did not manage to find a convincing proof to why they are also true in electrodynamics.

Intuitively, I thought that the divergence of $\vec{E}$ and $\vec{B}$ in electrodynamics should not be the same as that in electrostatics. In electrodynamics, beside the static charge producing a $\vec{E}$ field, the changing magnetic field also produces an additional contribution to the $\vec{E}$ field. Why then should the $\nabla \cdot \vec{E}$ still be zero, when it is initially derived for only the $\vec{E}$ field of the static charge?

The same argument goes for $\nabla \cdot \vec{B}=0$, where a changing $\vec{E}$ field produces additional contributions to the $\vec{B}$ field in the magnetostatics case. Why is $\nabla \cdot \vec{B}=0$ still true in the electrodynamics case?

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    $\begingroup$ Proof based on what axioms? $\endgroup$ – AccidentalFourierTransform Oct 23 '18 at 21:47
  • $\begingroup$ @AccidentalFourierTransform I would guess that a satisfactory answer could invoke the principle of superposition and the effects of the Coulomb field of a single point charge, plus, perhaps, special relativity. I do not have the time together now to give such an answer but I would not expect it to be too hard to derive that the divergence remains zero if it was zero, and that the induced magnetic field due to this change in reference frame is divergence-free. $\endgroup$ – CR Drost Oct 24 '18 at 0:05
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Divergence is the net flux per unit volume. A positive (negative) divergence indicates field lines beginning (ending) within an infinitesimal volume.

A changing magnetic field acts as a source of curling electric field. The field lines due to such a source have no beginning or end and as such contribute nothing to the divergence. Electric field lines only begin and end on charges. Only a charge density can be responsible for a non-zero electric field divergence.

A similar explanation applies to conduction and displacement currents as sources of magnetic field.

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The Maxwell equations are the basic (most fundamental) laws of classical electrodynamics. This means that you cannot really derive them from anything more basic--otherwise, the more basic thing you derive them from would be called the most basic laws of classical electrodynamics. So, there isn't really an answer to why the divergence laws are true except for saying that we found them to be true experimentally.

But, having said that, I would like to point out how your worry about the electric and magnetic fields produced by the dynamical situations ruining the "divergencelessness" of these fields is avoidable. (Again, this is not a reasoning behind the Maxwell laws, rather, this is an explanation as to how the Maxwell laws are self-consistent.) Notice that the additional electric and the magnetic fields produced due to time-variations in the fields don't have to contribute to the divergences of the field. And, in fact, that is the case. They only contribute to the curl of the overall electric and magnetic field. They simply do not contribute to the divergences. As you can see, this is really just a re-statement of the fact that the curl equations in the Maxwell laws involve the $\dfrac{\partial \vec{E}}{\partial t}$ and $\dfrac{\partial \vec{B}}{\partial t}$ terms and not the divergence equations.

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  • $\begingroup$ The Maxwell equations are derivable from a somewhat smaller set of assumptions. For example if one assumes (1) we have a theory involving a field which can push on things without changing their rest mass (2) it obeys principles of relativity (i.e. is Lorentz covariant) (3) it is as simple as we can make it under these assumptions. This is not quite a full derivation, more a way of showing that electromagnetism is among the most simple and natural field theories that respects relativity. $\endgroup$ – Andrew Steane Oct 23 '18 at 22:04

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