Potential of an ellipsoid.
A more difficult case is presented by the ellipsoid[5]. We have first
to determine the mode in which electricity distributes itself on a
conducting ellipsoid in free space. It must be such a distribution
that the potential in the interior will be constant, since the
electric force must be zero. It is a well-known theorem in attractions
that if a shell is made of gravitative matter whose inner and outer
surfaces are similar ellipsoids, it exercises no attraction on a
particle of matter in its interior[6]. Consider then an ellipsoidal
shell the axes of whose bounding surfaces are (a, b, c) and (a + da),
(b + db), (c + dc), where da/a = db/b = dc/c = [mu]. The potential of
such a shell at any internal point is constant, and the equipotential
surfaces for external space are ellipsoids confocal with the
ellipsoidal shell. Hence if we distribute electricity over an
ellipsoid, so that its density is everywhere proportional to the
thickness of a shell formed by describing round the ellipsoid a
similar and slightly larger one, that distribution will be in
equilibrium and will produce a constant potential throughout the
interior. Thus if [sigma] is the surface density, [delta] the
thickness of the shell at any point, and [rho] the assumed volume
density of the matter of the shell, we have [sigma] = A[delta][rho].
Then the quantity of electricity on any element of surface dS is A
times the mass of the corresponding element of the shell; and if Q is
the whole quantity of electricity on the ellipsoid, Q = A times the
whole mass of the shell. This mass is equal to 4[pi]abc[rho][mu];
therefore Q = A4[pi]abc[rho][mu] and [delta] = [mu]p, where p is the
length of the perpendicular let fall from the centre of the ellipsoid
on the tangent plane. Hence
[sigma] = Qp/4[pi]abc (3).
Capacity of an ellipsoid.
Accordingly for a given ellipsoid the surface density of free
distribution of electricity on it is everywhere proportional to the
length of the perpendicular let fall from the centre on the tangent
plane at that point. From this we can determine the capacity of the
ellipsoid as follows: Let p be the length of the perpendicular from
the centre of the ellipsoid, whose equation is x^2/a^2 + y^2/b^2 +
z^2/c^2 = 1 to the tangent plane at x, y, z. Then it can be shown that
1/p^2 = x^2/a^4 + y^2/b^4 + z^2/c^4 (see Frost's _Solid Geometry_, p.
172). Hence the density [sigma] is given by
Q 1
[sigma] = -------- -----------------------------------,
4[pi]abc [root](x^2/a^4 + y^2/b^4 + z^2/c^4)
and the potential at the centre of the ellipsoid, and therefore its
potential as a whole is given by the expression,
_ _
/ [sigma]dS Q / dS
V = | --------- = -------- | ------------------------------------ (4).
_/ r 4[pi]abc _/ r[root](x^2/a^4 + y^2/b^4 + z^2/c^4)
Accordingly the capacity C of the ellipsoid is given by the equation
_
1 1 / dS
-- = -------- | ---------------------------------------------------------- (5).
C 4[pi]abc _/ [root](x^2 + y^2 + z^2)[root](x^2/a^4 + y^2/b^4 + z^2/c^4)
It has been shown by Professor Chrystal that the above integral may
also be presented in the form,[7]
_
1 /[oo] d[lambda]
-- = 1/2 | -------------------------------------------------------- (6).
C _/0 [root]{(a^2 + [lambda])(b^2 + [lambda])(c^2 + [lambda])}
The above expressions for the capacity of an ellipsoid of three
unequal axes are in general elliptic integrals, but they can be
evaluated for the reduced cases when the ellipsoid is one of
revolution, and hence in the limit either takes the form of a long rod
or of a circular disk.
Thus if the ellipsoid is one of revolution, and ds is an element of
arc which sweeps out the element of surface dS, we have
/dx\ /py\ 2[pi]b^2
dS = 2[pi]yds = 2[pi]ydx / ( -- ) = 2[pi]ydx / ( -- ) = -------- dx.
\ds/ \ b/ p
Hence, since [sigma] = Qp/4[pi]ab^2, [sigma]dS = Qdx/2a.
Accordingly the distribution of electricity is such that equal
parallel slices of the ellipsoid of revolution taken normal to the
axis of revolution carry equal charges on their curved surface.
The capacity C of the ellipsoid of revolution is therefore given by
the expression
_
1 1 / dx
-- = -- | ----------------- (7).
C 2a _/ [root](x^2 + y^2)
If the ellipsoid is one of revolution round the major axis a (prolate)
and of eccentricity e, then the above formula reduces to
1 1 /1 + e\
-- = --- log_[epsilon]( ----- ) (8).
C1 2ae \1 - e/
Whereas if it is an ellipsoid of revolution round the minor axis b
(oblate), we have
1 sin^-1 ae
--- = ---------- (9).
C^2 ae
In each case we have C = a when e = 0, and the ellipsoid thus becomes
a sphere.
In the extreme case when e = 1, the prolate ellipsoid becomes a long
thin rod, and then the capacity is given by
C1 = a/log_[epsilon] 2a/b (10),
which is identical with the formula (2) already obtained. In the other
extreme case the oblate spheroid becomes a circular disk when e = 1,
and then the capacity C2 = 2a/[pi]. This last result shows that the
capacity of a thin disk is 2/[pi] = 1/1.571 of that of a sphere of the
same radius. Cavendish (_Elec. Res._ pp. 137 and 347) determined in
1773 experimentally that the capacity of a sphere was 1.541 times that
of a disk of the same radius, a truly remarkable result for that date.
Three other cases of practical interest present themselves, viz. the
capacity of two concentric spheres, of two coaxial cylinders and of
two parallel planes.
Capacity of two concentric spheres.
Consider the case of two concentric spheres, a solid one enclosed in a
hollow one. Let R1 be the radius of the inner sphere, R2 the inside
radius of the outer sphere, and R2 the outside radius of the outer
spherical shell. Let a charge +Q be given to the inner sphere. Then
this produces a charge -Q on the inside of the enclosing spherical
shell, and a charge +Q on the outside of the shell. Hence the
potential V at the centre of the inner sphere is given by V =
Q/R1 - Q/R2 + Q/R3. If the outer shell is connected to the earth, the
charge +Q on it disappears, and we have the capacity C of the inner
sphere given by