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*From*: David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>*Date*: Sun, 11 Feb 2001 12:59:25 -0500

One thing I neglected to mention in my prevous long post about the

nitty gritty of the exact solution was what the electric field and the

surface charge density are for the problem. Since the potential function

depends only on the one coordinate u, that means that the electric

field points everywhere locally perpendicular to the surface of each u = a

spheroid. The electric field lines run parallel to the u-hat direction

of locally increasing u-value. The potential is:

V(r_vec) = V(u,v,A) = V(u) = Q*arcsin(R/u)/(4*[pi]*[epsilon]_0*R)

The only component of E_vec is its u-component E_u. Its behavior is

independent of coordinate A:

E_u(u,v,A) = E_u(u,v) = Q/(4*[pi]*[epsilon]_0*u*sqrt(u^2 - v^2))

If we have a conductor whose surface coincides with some spheroid

with u = a, then the electric field discontinuously jumps to zero as

we enter the conductor. The amount of the discontinuity gives the

strength of the surface charge density [sigma] = [epsilon]_0* /_\ E_u

on the conductor's surface. That charge density is:

[sigma](v) = Q/(4*[pi]*a*sqrt(a^2 - v^2)).

If we write [sigma] in terms of the radial distance r that a particular

part on the conducting surface is from the origin, we get:

[sigma](r) = Q/(4*[pi]*a^2*sqrt(2 - e^2 - (r/a)^2)) .

Since this formula is only for the u = a surface of the conducting

spheroid (whose eccentricity is e), we need to remember that the radial

ratio (r/a)^2 ranges over the interval: 1 - e^2 <= (r/a)^2 <= 1 . The

extremal values of the surface charge density occur on the spheroid's

"equator" where

[sigma]_max = Q/(4*[pi]*a^2*sqrt(1 - e^2))

and on the "poles" of its symmetry axis where

[sigma]_min = Q/(4*[pi]*a^2) .

If we take the limit of e --> 0 in the above formula for [sigma](r) we

get the proper uniform surface charge density for a charged conducting

sphere. Also, if we take the opposite limit of e --> 1 in that above

formula, we get the case of the flat circular disk:

[sigma](r) = Q/(4*[pi]*a^2*sqrt(1 - (r/a)^2)) .

But because this result was taken as the limit of the top and bottom

edges of the spheroid collapse to a disk this means that the surface

charge density on just one top *or* bottom surface includes 1/2 of the

total charge present. When these two surfaces merge to make a flat disk

we need to double the value of the formula and then not consider the

top and bottom anymore as having separate surface charge densites.

After doubling (and setting the disk radius R to the semi-major axis a)

we get:

[sigma](r) = Q/(2*[pi]*R^2*sqrt(1 - (r/R)^2)) .

If we integrate this (circularly symmetric) surface charge density

over the surface of the disk (but now on only one side) we obtain the

total charge on the disk, which agrees with the value Q as it should.

Note that there is a divergent 1/sqrt singularity in the value of

[sigma] when r --> R on the edge of the (zero-thickness) disk. It seems

that the charges cluster so strongly there because their mutual repulsion

tends to drive them outward as far as they can get.

BTW, in case people are having some trouble visualizing the lines of

the coordinate system and the E-field it might help if it is mentioned

that this confocal "conicoidal" coordinate system u,v,A is sort of a

distorted version of a coordinate system analogous to spherical

coordinates, and the analogy becomes ever closer at ever greater

distances from the central focal circle. The u coordinate is like the

radial coordinate and sort of measures the "distance" away from the

center. The coordinate v is sort a directional coordinate that

asymptotically acts like the polar coordinate [theta] in spherical

coordinates in that it (actually its asymyptote) sort of distinguishes

the (slope of) directions more along the z-axis from those more along

the x-y plane. Asymptotically the large distance mapping

v/R --> sin([theta]) connects sort of how v labels polar directions. Of

course, the azimuthal coordinate A is identically equal (everywhere) to

the azimuthal coordinate [phi] in spherical coordinates.

David Bowman

David_Bowman@georgetowncollege.edu

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