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How to go from the wave equations of electric and magnetic field and $$ \boldsymbol{\nabla}\cdot \mathbf E = 0 \quad \text{ and } \quad \ 0 = \boldsymbol{\nabla}\cdot\mathbf B, $$

to the remaining two Maxwell's equation in free space?

I am unable to do this, for a long time now. Please help. :) Is it even possible to do it?

The question arose from the following: Are Wave equations equivalent to Maxwell equations in free space?

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  • $\begingroup$ Can volts per square meter be equal to teslas per meter, even if both are zero? $\endgroup$ – JEB Mar 7 at 19:01
  • $\begingroup$ You need the Lorentz force. Without or without any equation implying it, E and B are undefined. $\endgroup$ – my2cts Mar 7 at 21:46
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No, this is impossible.

A simple counter-example is the fields \begin{align} \mathbf E(\mathbf r,t) & = E_0 \hat{\mathbf e}_x \cos(kz-\omega t)\\ \mathbf B(\mathbf r,t) & = 0, \end{align} i.e. a plane-wave electric field and a vanishing magnetic field. This satisfies both force-field wave equations as well as both transversality conditions, but it breaks both the Faraday-Lenz and the Ampere-Maxwell laws.

The core intuition that this counter-example captures is that the basis that you've laid out simply does not include enough information that relates the electric to the magnetic field (specifically: it doesn't provide any such information at all) for the curl equations to be reconstructed.

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