Jackson 1.12 & 1.13: Prove Thomson's & Insulated Conductor Theorem

[nctweg]

Homework Statement

1.12 -
Prove Thomson's theorem : If a number of conducting surfaces are fixed in
position and a given total charge is placed on each surface, then the electrostatic
energy in the region bounded by the surfaces is a minimum when the
charges are placed so that every surface is an equipotential.

1.13 -
Prove the following theorem: If a number of conducting surfaces are
fixed in position with a given total charge on each, the introduction of an
uncharged, insulated conductor into the region bounded by the surfaces
lowers the electrostatic energy.

The Attempt at a Solution

So I haven't actually started them yet because I don't quite understand the geometry I'm being asked about. Note - I am not really looking for help on how to do the problems (at least not yet, I want to give it at least a week before giving in).

My questions are simple and perhaps dumb; are these conducting surfaces connected? It says that there is a region bounded by the surfaces but if that were the case and they were conducting, wouldn't they just end up forming some kind of closed shape with the charge spread throughout (rather than, as the problem indicates, each surface having Q total charge). My other thought is that Jackson's just referring to a general region that's between the surfaces but not necessarily closed but I really can't tell which is what the question is asking about.

Also in problem 1.13, is the insulated conductor a surface or does it have volume? (Or does it not make a difference?). Are we just approximating the insulation as being thin enough that it makes no difference?

Thanks for the help!

 

[Orodruin]

The surfaces are not connected.

 

[nctweg]

Okay, thanks for confirming my stupid question. I figured as much but I did want to check first.