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Basically, the question that you are asking is if Maxwell's Equations are
a correct description of the way electromagnetic fields behave. If you
agree that you have a time-varying electric field between the plates, and
you agree that Maxwell's Equations give a correct description of the fields,
then you reach the conclusion that you have to generate a magnetic field
that also is time-varying. If you are willing to do the calculation for a
simple case (a parallel plate capacitor with circular plates) it is not
hard to show that you generate lines of the B-field that are circular,
and that the amplitude of the B-field decreases linearly with distance
from the axis of the capacitor at any instant of time. That's the condition
you need for radiation!