🚀 go-pugleaf

   The change in potential between two points in space is
proportional to the integral of the electric field in that
region of space.  If the field is zero in some region of space,
the potential is constant.  (Not necessarily zero, as students
often claim.)  If the charge distribution on the van de Graaf's
dome is really static, then essentially all the charge is on
the outer surface of the dome.  Gauss' law tells us that the
field outside some region of space depends on the total charge
contained within that region.  If there is no charge, the field
outside the region must be zero, and the electric potential in
that region will be constant (again, not necessarily zero).
So if all the charge on the van de Graaf is on the outer surface
of the dome and there is no charge anwhere inside the dome, then
the field inside the dome is zero and the entire volume of the
dome is at the same electric potential, and that potential is
the same as the potential on the dome itself.  (That potential
depends on the size of the dome, of course.  Most of the
demonstration van de Graaf generators I've seen over the years
have 10-inch domes, which implies a theoretical maximum of about
375 kV in dry air.)
   So you're right--the field inside the dome is essentially
zero, but the potential isn't zero unless you have defined the
dome itself to be at zero potential.  If you adopt the zero
of potential to be at infinity, then the region inside the
dome is at a large potential.  (Incidentally, I sort of recall
reading somewhere that van de Graafs usually end up putting
excess electrons on the dome, so the potential is actually
negative.)

Steve Luzader
sluzader@fre.fsu.umd.edu