Yes, I have enjoyed the excellent discussion of this and the related
thread of electrostatic shielding. Your roundup was very good
(although we should now add Bowman's exact solution), but I wonder if
folks could summarize for me two key points that I'm still not
completely sure about:
(1) When charge is dumped onto a conducting disk, why does more
charge accumulate at the center than at the edges? I'm looking for an
intuitive answer, either in terms of the repulsive forces between the
free electrons or in terms of how more charge nearer the center
"clearly" minimizes the potential energy of the system.
(2) On the other hand, regarding the approach using Laplace's
equation, I want to be sure I understand that too. (I suspect some
things I'm about to say are wrong....) Is the idea to fix the
potential on the disk at say 100 V and on some boundary cells far
away at 0 V, and then use the relaxation averaging method to find the
potential everywhere else? In other words, I want to be sure I
understand that the disk potential is kept fixed, as though it were
attached to a battery, rather than keeping the disk charge fixed. (On
the disk itself, it is Poisson's equation not Laplace's equation that
applies, right?) The local potential gradient then gives the field
which in turn gives the local charge density on the disk. Integrating
gives the total charge. If a different total charge is wanted, the
calculated potential can be rescaled to a different value since its
capacitance is of course constant.