Fig.1 Magnetic field outside the plates of a charging capacitor
This question could be rephrased as:
Does a non-ideal capacitor have a magnetic field outside its plates after it is fully charged?
It is difficult for me to accept an electrostatic dipole field existing without a magnetic field being present as well (i.e. case of fully charged ideal capacitor).
A non-ideal capacitor will always have a dielectric leak current after it is fully charged and thus a tiny magnetic field will be always present. You could also say that you cannot ever charge "fully" a capacitor.
I don't believe the case of an ideal fully charged capacitor with no magnetostatic field present outside its plates is theoretically realistic and can be analytically proven.
In my opinion, one could say capacitors are possible only because there is no perfect dielectric. Even so if the capacitor was using vacuum space as a dielectric between its plates since there is no perfect dielectric existing in nature.
But I could be wrong?…
Best Answer
Unfortunately, “non-ideal capacitor” is too vague. There are too many different ways that a capacitor could depart from ideal-ness, and only some of them would lead to a steady state magnetic field.
For example, here is a model of typical non-ideal capacitors:
There are many different parameters representing varying specific departures from ideal behavior. Of those, there is only one that would produce a steady state magnetic field. That is $R_{leak}$.
For a finite $R_{leak}$ there would be a steady state current and hence a steady state magnetic field. For any other departure from ideal behavior there would not.