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Re: [Phys-L] amusing electrostatics exercise

In the context of the self-capacitance and/or mutual capacitance
of a disk ... on 02/25/2013 02:18 PM, Carl Mungan wrote:

I'm still wondering whether the method of images can't be used to do
better?

Nifty question. That's even more amusing than my original question.

This can be rather well solved using "qualitative reasoning". Some
people are really good at qualitative reasoning, but others not so
much. It is a skill that can be taught (and learned). This example
can serve as nice teachable moment.

Specifically, the situation is this: We have an asymmetric capacitor
consisting of a moderate-sized flat disk somewhere near an enormous
flat slab.

____ +Q
_________________________________________________ -Q


Step 1: My physics intuition says that when the disk is very near
the slab, the capacitance is the same as it would be for a simple
symmetric two-disk parallel-plate capacitor.

In this case "near" means g is very small compared to (π/8) r,
where g is the capacitor gap and r is the radius of the disk.

Rationale: Draw the picture. Energy principles say that virtually
all the relevant field lines will be in the gap. This accounts for
virtually all the charge and virtually all the field lines. Experience
playing with field-calculating software helps with this:
http://www.av8n.com/physics/laplace.htm

Step 2: When the disk is far away from the slab, the capacitance
of the setup as a whole must be the same as the self-capacitance of
the disk.

Rationale: For capacitances in series, we should be able
to add the elastances (i.e. inverse capacitances). This is
analogous to the rule for adding resistances in series. The
slab has zero elastance (infinite capacitance) so the
elastance of the overall situation is dominated by the
elastance of the disk.

Step 3: Even if you don't know how to find the exact result for an
arbitrary-sized gap, you can get a good approximation by interpolating
between the small-gap asymptote and the large-gap asymptote, as worked
out in the previous steps. A preliminary version of this is discussed
and diagrammed at
http://www.av8n.com/physics/electrophorus.htm#fig-pre-electrophorus-plot

Step 4: The formula for a symmetrical capacitor is simple and
well known for the case of a small gap. It is neither simple nor
super-widely known -- but still known /1/ -- for an arbitrary-sized
gap. We should be able to (a) invoke symmetry and (b) invoke the
method of images to reduce the asymmetrical case to the corresponding
symmetrical case ... /provided/ we keep track of various factors of
two. The original situation should have half the elastance of the
following situation:

____ +Q
_________________________________________________ (zero charge)
____ -Q

The factor of 2 arises for a prosaic reason, namely that we are
now measuring the voltage from the bottom disk to the top disk,
rather from slab to disk. Twice as much V per unit Q means twice
as much elastance.

By symmetry and/or by the method of images, that must be the same
as
____ +Q
(no slab at all)
____ -Q (but double gap)


In the large-gap limit, the elastance here is just the series
combination of the self-elastance of two disks. From this we
obtain the final result, namely that the capacitance of the
original asymmetrical situation (disk plus slab) should be double
the capacitance of the corresponding symmetrical situation
/with a double gap/.

In the small-gap case the two factors of two cancel out, so that
disk + slab has the same capacitance as disk + disk (with the
same gap) as expected. As the gap gets larger, the asymmetrical
case starts to approach the large-gap asymptote twice as soon as
you might have guessed by naively looking at the symmetrical formula.

Step 5: There is a piece of intuition that is implicit in all
the previous steps. Consider the extreme large-gap limit. Zoom
out your point of view so that the disk becomes essentially a
single point. You know how the electric field of a point charge
falls off with distance. You know that the Poyinting field energy
goes like the square of the field. If you integrate this energy
over all space, starting some not-too-small distance R away from
the disk, then the integral converges. So the intuition is that
whatever is going on in the far field never matters very much for
electrostatics problems.

It's a bit messy in that it's like an infinite hall of mirrors, with
an infinite sequence of images behind all 4 walls.

I'm not worried too much about that, given that the overall charge
is neutral. The images we see in the "hall of mirrors" can be
paired to form dipoles, and the dipole field falls off even faster
than the monopole field. The lattice sum converges quite quickly,
even with an infinitude of sources. Sure, it contributes a
correction term in principle, but under a wide range of practical
situations (electrophorus disk small compared to the size of the
room) this won't change the story in any qualitative way.

====================

Tangents, pedagogical and otherwise:

I call this sort of thing "qualitative reasoning". A more
pretentious name that means more-or-less the same thing is
/Fermi problems/. Enrico Fermi was famous for being super-
good at finding ways to apply this sort of reasoning. Also
G I Taylor, Viki Weisskopf, and Richard Feynman.

However, this particular problem is so easy that any student
who wants to grow up to be a physicist should learn how to do
it. Problems like this come up all the time in physics. Turn
this problem over in your mind, again and again, until you have
N different ways of solving it.

This example illustrates the importance of drawing the diagram.
Without the diagram, it would have been rather easy to blow off
a factor of 2. On the other hand, once you see how to solve it,
do it again, solving it in your mind without a written diagram,
so you get better at visualizing things in your mind's eye and
better at keeping track of factors of 2. I'm not saying this
is easy, but it is a good skill to have.

Note that getting any traction at all on this problem requires
pulling together a large number of disparate ideas and combining
them in not-entirely-obvious ways.

This is what real physics looks like to me. The reasoning involved
in this sort of problem is exceedingly different from the sort of
reasoning involved in (say) the FCI. I don't know whether to laugh
or cry when people write PER papers judging the effectiveness of
this-or-that teaching method based on FCI scores.

I often complain about multiple-guess tests, but the fact that
the FCI (and similar instruments) use the multiple-guess format
is not even the main issue here. The deeper problem is that for
some reason (perhaps ease of interpretation) the tests check
only one idea at a time. There is no check of reasoning, i.e.
pulling together a large number of disparate ideas. Yeah, maybe
this makes the test easier to design, easier to administer, and
easier to interpret ... but that strikes me as quite an extreme
form of looking under the lamp-post, when most of what we
*should* be looking for is elsewhere.

To say the same thing in more positive terms: We should collect
a bunch of physics/reasoning problems like this. We should judge
teaching effectiveness on how well students acquire /and retain/
the ability to do problems like this.

For more about the physics of capacitance, including odd-shaped
capacitors, see
http://www.av8n.com/physics/capacitance.htm


Reference:
/1/ G. T. Carlson and B. L. Illman
“The circular disk parallel plate capacitor”
American Journal of Physics 62 12, pp. 1099 (December 1994)
http://ajp.aapt.org/resource/1/ajpias/v62/i12/p1099_s1