electromagnetism - Induced magnetic field produces electric field and vice versa forever!

So here are the two of Maxwell's laws that I am interested in:

So we have the simple circuit (from google):

So, before the system goes into steady-state we know that charge slowly accumulates on the plates of the conductor. So the charge on the plates gets bigger and bigger while the charges that carry the current get smaller and smaller, so the current gets weaker.
Applying Ampere's law on the wire we find the induced magnetic field due to the current $I(t)$ that penetrates the surface $\Sigma$ (see the integral of $\mathbf{J}\mathrm{d}\mathbf{S}$) and not due to an electric field.
Now, this induced magnetic field is changing with respect to time (because current is changing). But from the Maxwell-Faraday equation we conclude that this changing magnetic field will produce an electric field which again changes with respect to time. And then we have another induced magnetic field due to that changing electric field. And the cycle goes on.
So, am I right? And if I am, when does this stop? And how does it changes the way I calculate each induced field? Does it have to do with electromagnetic waves?