Back in the bad old days, people used to solve this
problem without computers.
Normally I'm a big fan of straightforward methods.
Students like the straightforward approach. However,
sometimes it pays to use the straightbackward approach.
That is, you write down an alleged answer and then
show that it works.
In this case, the trick is to solve for a uniformly
charged /ellipsoid/ and then take the limit as it
gets reeeeally eccentric, i.e. flat.
There are a bunch of tricks in the same general family,
based on functions of a complex variable. The conformal
mapping technique is useful for lots of things, including
fluid dynamics, not just electrostatics. You can create
lots of shapes, not just ellipsoids. https://en.wikipedia.org/wiki/Zhukovsky_transform
On the other hand, nowadays computers are so ridiculously
powerful that it's easier to do the finite-element
modeling than it is to learn the mathematical methods
of physics.
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On 07/10/2015 03:02 PM, Carl Mungan wrote:
Consider a circular, uniformly charged metal disk
I am ignoring the word "metal". A metal or other
good conductor cannot support a uniform surface charge
density in this situation. I assume it is an insulating
disk.
surprised me a little.
The unstated but obviously implied question is, how
do we reconcile the result with our experience and
intuition about parallel plate capacitors?
I'm not sure I have the perfect answer, but a good
starting point is to realize that most of our experience
applies to capacitors with a very narrow gap. When
the gap becomes huge compared to the other dimensions
in the problem, all bets are off. There are nonidealities
in the charge distribution as well as the field pattern.
In the introductory class, we usually don't look too
closely at the details of the fringing field. It
doesn't contribute much to the I/V characteristic of
the overall capacitor in normal use. HOWEVER in the
real world it matters a great deal. Capacitors are
always limited by breakdown, and a high working voltage
is a big selling point. The high field at the edge
of the electrode is a big deal, because that's where
breakdown is going to occur.