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*From*: "John S. Denker" <jsd@MONMOUTH.COM>*Date*: Sun, 11 Feb 2001 16:38:11 -0500

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.

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