Re: capacitance of a disk

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Bob's comment:

> After floundering a bit more through Smythe, I now think that his method of
> "confocal conicoids" is indeed the same, at least in broad outline, as
> Bowman's method of "confocal oblate spheroids".   Perhaps Dave can get at
> Smythe (pg 111-114) and comment.

Unfortunately, I do not have Smythe, and I doubt that our meager College
library has it (I won't be able to check the library until Monday), and
if it is not there I would have to make a special trip to the University
of Kentucky library or order it through interlibrary loan to get at it.

Jack has mentioned that Morse & Feshbach also have it worked out there.
That figures; I should have known.  I think my collegue here might have
a copy of it that I might be able to borrow.

It's interesting that Smythe calls the generic surfaces of revolution
made from the conic sections (i.e. ellipses, and hyperbolas) "conicoids".
I like the sound of that name.

But since there have been some private requests for more of the details
of my calculation, I thought I would post an outline of it, and hope that
the ASCII representation of the equations would not render them
completely unreadable.  *If* I get around to writing it up in a nicely
formatted .pdf, .rtf, .doc, .ps or latex file I will notify the list
when and where it could be accessed.

Consider the surface represented (in Cartesian coordinates) by the
equation:

(x^2 + y^2)/a^2 + z^2/(a^2 - R^2) = 1

where R and a are taken to be positive real constants.  *If* a > R this
equation represents an oblate spheroid (centered on the origin) whose
symmetry axis is the z-axis, whose semimajor axis length is a, and whose
eccentricity is e = R/a.  This spheroid can be constructed by first
drawing an ellipse in the x-z plane centered on the origin which has its
two foci at +/- R on the x-axis and whose semimajor axis length is a.
Its eccentricity is e = R/a.  We get the spheroid by rotating this
ellipse about the z-axis.  When this is done the locus of each of the
foci traces out a circle of radius R in the x-y plane centered on the
origin.  Thus, the two foci become a continuously connected *focal
circle*.  The semi-minor axis length is c = sqrt(a^2 - R^2).  Suppose we
hold R fixed and imagine varying the value of a.  When this is done we
build a nested family of concentric and confocal (i.e. having the same
focal circle) oblate spheroids where the spheroids get larger and less
eccentric as the value of a increases.  OTOH when the value of a
decreases toward R the spheroids shrink and collapse flat toward the
the focal circle and its interior and approach a flat disk in the limit
of a --> R.

Now let's look at the exact same equation again but now suppose that
a < R instead.  In this case the surface is a hyperboloid of revolution
of one sheet made by rotating a hyperbola that was originally in the
x-z plane about the z-axis.  The hyperbola has the same focal points
as the ellipse did, and the hyperboloid has the same focal circle as
the spheroids do.  The focal semi-axis of the hyperbola along the x-axis
(i.e. the axis on which the foci reside) is a.  Since we have a
hyperbola and since a < R we see that the eccentricity e = R/a > 1.
The semi-axis perpendicular to the focal (i.e. x-)axis is
c = sqrt(R^2 - a^2).  Again let's imagine varying the a parameter for the
rotated hyperboloid while holding R fixed.  We again make a nested family
of confocal/concentric hyperboloids.  As a gets very close to R these
hyperboloids tend to fold in half toward the x-y plane and have a very
small asymptotic slope rising along the z direction very slowly
(as x^2 + y^2 increases).  In the limit of a --> R the hyperboloid
completely folds up and coincides with the part of the x-y plane outside
the focal circle.  OTOH, as a increases and approaches R the minimal
separation across the "waist" of the hyperboloids gets very narrow and
the hyperboloids tend to "straighten up" and become more tubular and
less folded.  The asymptotic limit of a --> 0 makes the hyperboloid
collapse to the z-axis.

We now make a coordinate system out of these two families of surfaces.
We choose one coordinate u to have a value larger than R and a surface
of constant u-value is just the corresponding spheroid with u = a.
We choose the other coordinate v to have a value 0 < v < R so that
a surface of constant v value is the corresponding hyperboloid with
a = v.  For our third coordinate we define the usual azimuthal angle A
about the z-axis.  Normally this angle is called [phi] in usual
spherical coordinates or [theta] in cylindrical or polar coordinates.
I'll call it A here since it is easier to type A than [phi] or [theta].

Therefore the coordinate system is defined by the equations:

(x^2 + y^2)/u^2 + z^2/(u^2 - R^2) = 1 where u > R and

(x^2 + y^2)/v^2 - z^2/(R^2 - v^2) = 1 where R > v > 0 and

y/x = tan(A).

The Cartesian coordinates are given explicitly as:

x = u*v*cos(A)/R

y = u*v*sin(A)/R

z = (+/-)sqrt((u^2 - R^2)*(R^2 - v^2))/R

The inverse of this set of equations is:

u = sqrt((1/2)*(x^2 + y^2 + z^2 + R^2 + sqrt((x^2 + y^2 + z^2 - R^2)^2
    + (2*R*z)^2)))

v = sqrt((1/2)*(x^2 + y^2 + z^2 + R^2 - sqrt((x^2 + y^2 + z^2 - R^2)^2
    + (2*R*z)^2)))

A = arctan(y/x)

The metric tensor for our new coordinate system is diagonal because the
confocal hyperboloids are always locally orthogonal to the spheroids,
and the vertical planes of constant angle A are also locally orthogonal
to both of these other sets.  The covariant components of the metric
tensor are:

g_uu = (u^2 - v^2)/(u^2 - R^2)

g_vv = (u^2 - v^2)/(R^2 - v^2)

g_AA = (u*v/R)^2

g_uv = g_vu = g_uA = g_Au = g_vA = g_Av = 0

We make use of these metric coefficients to calculate the formula for
the Laplacian operator in our new coordinate system.  In our new
coordinate system Laplace's equation (i.e. 0 = div(grad(V))) becomes:

0 = (1/(u^2 - v^2))*(sqrt(1 -(R/u)^2)*d/du(u*sqrt(u^2 - R^2)*dV/du)
    + sqrt((R/v)^2 - 1)*d/dv(v*sqrt(R^2 - v^2)*dV/dv))
    + (R/(u*v))^2*d^2V/dA^2

where d/du(...), dV/du, d^2V/dA^2, etc. all signify *partial*
derivatives.

The nice thing about this coordinate system is the focal circle and
its interior disk is represented by the equation u = R so if that set
is supposed to represent a charged conductor, it has a constant
equipotential value *independent* of the other coordinates v and A.
If there are no other charge sources present which would break the
symmetry of the problem the solution for the potential field V(u,v,A)
will have this symmetry everywhere in space so that the potential really
is globally just a function of u alone, i.e. V = V(u).  Using this
symmetry we can make Laplace's equation simplify *greatly* for any
solution which respects that symmetry.  In this case the Laplace equation
eventually boils down to just:

0 = d/du(u*sqrt(u^2 - R^2)*dV/du) .

Since something that has a vanishing derivative must be a constant we
observe (after a little rearranging) that:

dV/du = (constant)/(u*sqrt(u^2 - R^2)) .

Integrating this gives:

V(u) = C + D*arcsin(R/u)/R

where C and D are two integration constants to be chosen so that the
boundary conditions on the problem are met.  Since the u coordinate
represents the semi-major axis of each of the confocal spheroids we
observe that as we get asymptotically far from our u = R disk the
value of u diverges. The asymptotic behavior of the V(u) function at
large u must be V(u) = C + D/u --> C.  Since, by convention, we regard
the potential at "infinity" to vanish this means that the constant C
must also vanish.  From our transformation equation solved for u
above we see that when our distance r = sqrt (x^2 + y^2 + z^2) from the
origin diverges toward infinity the value of u asymoptotically
approaches u --> r.  This means that very far from the charged
conducting disk the potential approaches V (r) = D/r.  But we already
know that any localized charge distribution of total charge Q will
have its large distance potential field dominated by its monopole
contribution and it will act like a point charge when we observe it
from a sufficient distance.  This means that matching the asymptotic
behavior gives:

D/r = Q/(4*[pi]*[epsilon]_0*r)

or cancelling the 1/r factors gives:

D = Q/(4*[pi]*[epsilon]_0) .

Substituting this result back into the expression for the potential
gives the result:

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

To get an explicit expression for the potential field in Cartesian
coordinates we merely substitute the expression for u from our previous
above transformation equation into the argument of the arcsin() function
in the potential.  Since the expression for u takes up 2 lines of
text when written in fully Cartesian coordinates, I will
simplify the resulting expression by replacing the squared radial
distance x^2 + y^2 + z^2 by the spherical coordinate notation r^2 giving
an expression for the potential in terms of r & z:

V(r_vec) = Q*S/(4*[pi]*[epsilon]_0*R)

where we have defined the quantity S by

S = arcsin(sqrt(2/((r/R)^2 + 1 + sqrt(((r/R)^2 - 1)^2 + (2*z/R)^2)))) .

When V(r_vec) is evaluated on the charged disk we have u --> R and
S --> [pi]/2 since the arcsin(1) = [pi]/2.  If we let V be the
potential on the charged conducting disk we get:

V = Q/(8*[epsilon]_0*R) = Q/C .

Thus the capacitance is C = 8*[epsilon]_0*R.

If instead of a conducting charged disk, we have an oblate spheroid of
semimajor axis length a and eccentricity e we can use the same potential
function for it as for the disk case since in the region external to the
spheroid the potential and electric field is identical to that obtained
from a disk of the same radius as the spheroid's focal circle.  This is
because each confocal spheroid is a surface of constant potential, and
any one of them can be thought of as a conducting surface as far as
the world exterior to it is concerned.  The only thing we need to change
about the potential function is to replace the parameter R by a*e which
gives the radius of the focal circle of the conducting spheroid.
The potential is thus:

V(u) = Q*arcsin(a*e/u)/(4*[pi]*[epsilon]_0*a*e) .

In order to find the capacitance of the spheroid we just evaluate the
potential on the spheroid u = a giving:

V = Q*arcsin(e)/(4*[pi]*[epsilon]_0*a*e) = Q/C .

Therefore for the capacitance C we get:

C = 4*[pi]*[epsilon]_0*a*e/arcsin(e) .

Notice that this formula nicely interpolates between the familiar
formula for the capacitance of a sphere when e --> 0, and the formula
for the capacitance of a flat disk when e --> 1.

QED

David Bowman
David_Bowman@georgetowncollege.edu