I find the following exercise to be interesting without being overly
complicated.
Let's start with a slightly messy real-world application, and then
simplify it. My motivation was that I wanted an explanation of how
an electrophorus works. I found a whole bunch of bogus explanations.
There may be some good explanations out there, but they seem to be
hopelessly outnumbered by bogus explanations. This version is typical: http://en.wikipedia.org/wiki/Electrophorus
In particular, the diagram shows an unbalanced negative charge on
the dielectric ... but it does not show the corresponding positive
charge. There must be some electric field lines that end on the
dielectric, but it is impossible to guess how those lines are
arranged because we have no idea where the counterelectrode is.
There is a room with conducting walls (shown in blue). The floor
of the room is X by X. The height of the room is Y. Centered over
the middle of the floor is a thin metallic disk of radius r (shown
in red). The disk is horizontal at a height h above the floor.
There is a charge Q on the disk ... and correspondingly a charge -Q
on the walls of the room. The disk is not very large, and the room
is much wider than it is tall, so that X ≫ Y ≫ r.
Your mission, should you decide to accept it, is to find an algebraic
expression for the voltage on the disk, as a function of h. Voltage
is measured relative to the walls, which we take to be effectively
"chassis ground".
The disk has an insulating handle, so you can change h arbitrarily
without changing Q.
Make whatever reasonable simplifying assumptions you like. (Replacing
the conducting walls with non-conducting walls would not be considered
reasonable.)
As a specific numerical instance, set
X = 10 m
Y = 2 m
r = 0.1 m
h = 1e-5 m initially
Charge up the disk to 1.5 volts using a battery, and then disconnect
the battery. Raise the disk (maintaining constant Q during phase of
the process). What happens to the voltage?
The usual jsd puzzle rules apply: Everything I have said is (to the
best of my knowledge) true and non-misleading, although some details
may be irrelevant. OTOH I have not told you everything I know. I
have not told you the answer or even the method of solution ... and
more importantly, the problem may be underspecified.
Again, the point is that after doing this warm-up exercise you should
have a fighting chance of figuring out how an electrophorus works.