Ludwig,
Let me try a pictorial, semi-quantitative, "Faradayesque"
model for you to chew on:
In an electrostatic situation:
1.) At EVERY space point an electric field line can be drawn by
following the direction of the electric field.
2.) Each of these field lines will begin on a positive charge
and terminate on a negative charge.
3.) No two field lines can intersect (else the E field direction
would be double valued at the point of intersection).
Apply these to your pair of parallel, concentric, circular conducting
plates (of finite radii) carrying unequal and opposite charges.
Say Plate 1 has a positive charge on its inside surface and Plate 2
has a negative charge on its inside surface (at least in the center
region). Certainly, there must be field lines originating on this
inside surface of Plate 1 and heading "straight" across to terminate
on the inside surface of Plate 2. If, in the same region there are
field lines which originate on an inside surface and subsequently
take a 90 degree turn toward infinity, the two families of lines
would intersect (VERBOTEN!).
If there is a family of lines which take this right angle turn to
Valhalla they must begin at some minimum radius and there must
be no "straight across" lines beyond that radius (the two families
of lines cannot co-exist in the same region).