Here's another amusing way to look at the self-capacitance of a disk:
Consider the self-capacitance of a sphere: 4 π є0 r.
As a useful mnemonic, you can think of that as the capacitance
of parallel-plate capacitor, with an appropriate area (the area
of the sphere) and an appropriate "effective gap" (equal to the
radius of the sphere). This is spelled out at http://www.av8n.com/physics/electrophorus.htm#sec-sphere
This mnemonic gives the right answer, exactly.
Moving on to the self-capacitance of a disk, it follows the same
pattern as the parallel-plate case and the spherical-plate case:
The capacitance can be thought of as proportional to the area
divided by some "effective gap". This is spelled out at http://www.av8n.com/physics/electrophorus.htm#sec-est-disk
This mnemonic gives the the right answer to within a factor
of 4/π i.e. to within 30%.
====================
This is what physicists call playing around with the physics.
It involves finding new, slightly off-beat ways of looking at
things.
Again let me emphasize the value of this. Every new way of
looking at things strengthens your memory and your understanding
of what's going on. As always, when I speak of "memory" that
includes but is *not* limited to rote memory. An expert has
a much more powerful memory that is based on understanding,
not on rote. This has been understood in the literature for
many decades (James, 1898). http://ebooks.adelaide.edu.au/j/james/william/talks/chapter12.html
(search for "network")
This has also been understood for a long time by the people
who hire scientists. In some places, it is a traditional
part of the job interview to ask Fermi problems. For example,
the interviewer might take off his wedding ring and ask the
candidate to estimate the inductance thereof. Similarly,
imaging cutting the bottom off a trash barrel to form a
metal cylinder about 1m long and 1m in diameter. Estimate
the inductance thereof. Or, suppose I push to sugar cubes
together to form a capacitor about 1cm by 1cm with a gap
of 1mm. Pretend they are good conductors. Estimate the
capacitance. There are three types of candidates:
a) Some of them just know the answer.
b) Some of them don't know but can figure it out
faster than you can finish asking the question.
c) Some of them just sit there gasping like fish
out of water.
Case (b) can be distinguished from case (a) by asking them to
explain how they figured it out. In case (a) the interviewer
just asks more questions until we get to case (b), which is of
course the point of the exercise.
Let's be clear: Most job interviews don't work this way.
Maybe only 0.001% of the jobs are in this category. And
not everybody in the world wants this kind of job. That is,
not everybody enjoys figuring stuff out. Still, the point
remains, for this kind of job, this is a good contribution
to the process of finding the right candidate for the job.
It's a relevant skill, partly because the /process/ used to
answer such questions looks a lot like the process used to
answer much more challenging questions ... and partly because
in the course of attacking a truly hard problem there are
about 500 little Fermi problems that stand in the way, so
the quicker you can deal with them the more productive you
are.