If you have a potential difference between two plates (a capacitor) then you have and electric field between the plates.
Is there also a perpendicular magnetic component to this electric field? I've never heard of this - but I thought electric and magnetic fields must always coexist in phase and orthogonal for every packet of radiation?
If the electric field is not electromagnetic radiation then how does the other plate "know" about the potential from the top plate?
Magnetic fields are created by changes in the Electric Field. If the Voltage difference between the plates is constant, so is the electric field, so there will be no induced magnetic field.
To answer your second question, the bottom plate doesn't need to "know" about the top plate. The potential difference is a consequence of the charge distribution between the two plates, taken together as the system.
Radiation is a self-propagating electromagnetic field - an oscillating in electric field that produces an oscillating magnetic field, that produces an oscillating electric field, ....
Electric field gives a birth to magnetic field only when it changes in time. This is expressed in Maxwell's equations:
$\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$
$\nabla \times \vec{B} = \frac{\partial \vec{E}}{\partial t} + const \cdot \vec{J}$
Static electric (magnetic) fields do not radiate.
Static electric field gives a birth to Coulomb force:
$\vec{F} = const \cdot \frac{q_1 q_2}{r^2} = q \cdot \vec{E}$
That is how the plates "know" about each other.
You confusion probably comes from a fact that EM-waves can be described by particles - photons, wheres in undergraduate physics course there is no word about such thing for static fields. Well, if you want to think about that this way, there are particles which are analogues of photons but for stationary fields - virtual photons.
Is an electric field electromagnetic radiation?
Electromagnetic radiation is a propagating disturbance in the electromagnetic field. This disturbance has an associated energy and momentum and so, electromagnetic waves transport energy and momentum through space.
So no, an electric field is not electromagnetic radiation.
However, static electric and magnetic fields have an energy density and crossed static electric and magnetic fields have momentum. The key is that this energy and momentum is not in transit when the fields are static.
If the electric field is not electromagnetic radiation then how does the other plate "know" about the potential from the top plate?
The electric field is a physical entity. An electrically charged particle here 'feels' (couples to) the electric field here now without 'regard' for the charges elsewhere that produced the electric field.
