Re: confocal electrodes

["John S. Denker" <jsd@MONMOUTH.COM>]

At 02:32 PM 2/10/01 -0500, David Bowman wrote:
> This makes the capacitance of the disk 27.32395...% greater than half
> that of a sphere of the same radius.
...
> nested family of confocal oblate spheroids

This has been driving me nuts for 24 hours.  I figured there was some
physics hiding in this, obscured by the math.  I think I now see a way to
apply some nice qualitative reasoning to this.

The key word here is confocal.  That is conFOCAL.  What does it mean to
focus something?  Why does that matter?

1a) Consider a spherical whispering gallery.  If you clap your hands in the
center, all the sound is focused back to you.  Now imagine "inflating" the
sphere.  The new sphere has the same focus as the old sphere.  That is, it
is confocal.  In the new situation, the sound still comes back to you, just
a bit later.

1b) We know what the field likes look like for a charged conducting
sphere.  They are radial lines.  The field lines of the small sphere
coincide with the field lines of the larger confocal sphere.

2a) Let's consider parallel planes.  They can be considered confocal
mirrors also, sharing a focal point at infinity.

2b) Again, the field lines of one charged conducting plane line up with the
field lines of another parallel (i.e. confocal) one.  Of course, if you
have two non-parallel planes, their field lines don't line up.

3a) What about oblate spheroids?  In particular, these are elliptical in
cross-section.  Now consider "inflating" this cross-sectional ellipse.  We
could replace it with some generic larger ellipse, but let's replace it
with one that is confocal.  That means that sound from one focus still gets
reflected to the other focus, just a wee bit later because of the inflation.

3b) What about the field lines near our ellipse?  They are curved, but
let's consider an infinitesimal inflation, so that they are only curved to
second order.  To first order, they are straight lines that are
perpendicular to the inner ellipse (because it is an equipotential;  field
likes are always perpendicular to nearby equipotentials, by Maxwell's
equation (curl E = 0)).

Now we wonder whether the inflated ellipse might be an equipotential
also.  Perhaps it is also perpendicular to these little lines.  Indeed it
must be, because of the FOCUSING property of ellipses.  There are several
ways to see this.  For one, consider the equality of angle-of-incidence and
angle-of-reflection.  If one ellipse were tilted relative to the other, it
wouldn't focus.  Equivalently, consider Fermat's least-time principle;  if
one surface were tilted relative to the other, one of the rays could
shorten its journey by choosing a different path.

So we can see, using qualitative reasoning with essentially no equations,
that infinitesimal "inflation" of one equipotential object creates another
equipotential object of the same family with the same focus.  By iterating
this process, we can exhibit the whole family of confocal equipotentials.

4a) You can "inflate" any object.  If you inflate a tetrahedron, you will
get something that resembles a tetrahedron with rounded-off corners.

4b) If the original object has a focus, the inflated version will be
confocal. (Tetrahedrons don't have a focus.)

5) Now that we know these ellipsoids are equipotentials, we would like to
know what potential goes with which ellipsoid.  I don't see a qualitative
way to find this out;  maybe it's time to break down and do a quantitative
calculation.  We can however say a few qualitative things;  for one thing,
the places where the new ellipsoid comes closest to the old ellipsoid
correspond to regions of highest charge density.  It may be possible to use
this, plus conservation of flux, to obtain an equation for potential as a
function of size.

6) It is fun to visualize putting two thumbtacks in a board.  Put a loop of
string over them and draw an ellipse.  Then do it again with a sequence of
longer and shorter strings.  Rotate the resulting ellipses around the minor
axis to generate the ellipsoids.  The extreme case is just a line joining
the two thumbtacks;  this generates the disk that motivated this whole
discussion.

An ellipse is a locus of constant sum (summing the distance to the two
foci).  A hyperbola is a locus of constant difference (subtracting the
distance to the two foci).  It's hardly surprising that you can make a
coordinate system out of them, with contours that meet at right angles at
each point.

Its clear that we can flip this problem around:  If an ellipsoidal
electrode has hyperbolic field lines, hyperboloidal electrodes will have
elliptical field lines.