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After reading your note (even the parts not cited above) I need more
information before I can help you.
1) Most importantly, can you say what physical principle or process you
are trying to demonstrate?
This is important because in the scenario you describe, there are many
processes that could lead to a net charge, and I can't tell which you
would consider interesting and which you wouldn't.
Specifically:
1.1) Suppose there was a huge amount of net charge on the dielectric
*before* it were put into the capacitor. To a good approximation that
charge would be expected to remain. So measuring the charge after it sits
in the capacitor for a while doesn't tell us much about the nature of
capacitors.
1.2) Similarly, you wrote that
We now have a power supply whose terminals are floating.Well, if the system is floating it means that the whole system (power
supply, capacitor plates, dielectric, etc.) is free to pick up stray
charges during operation. Some of these net charges could be distributed
to the dielectric, but I'm not sure why anybody would be interested in
the details.
1.3) You mentioned opening the capacitor, and indicated this was
problematic. I would say "problematic indeed", including an interesting
problem you didn't mention. Unless special precautions are taken
(including, among others, short-circuiting the capacitor terminals),
opening the capacitor illustrates the same physical principles as an
electrophorus. You can create a huuuge amount of electrostatic energy
and voltage.
So.... Step one is to figure out what physical principle or process you
are trying to demonstrate. Then it should be possible to design an
experiment that controls for other factors.