THE MATHEMATICAL THEORY OF ELECTRICITY IN EQUILIBRIUM. ELECTRIC IMAGES.
ELECTRIC INVERSION
In describing Thomson's early electrical researches we shall not enter
into detailed calculations, but merely explain the methods employed. The
meaning of certain technical terms may be recalled in the first place.
The whole space in which a distribution of electricity produces any
action on electrified bodies is called the _electrical field_ of the
distribution. The force exerted on a very small insulated trial
conductor, on which is an electric charge of amount equal to that taken
as the unit quantity of electricity, measures the _field-intensity_ at
any point at which the conductor is placed. The direction of the
field-intensity at the point is that in which the small conductor is
there urged. If the charge on the small conductor were a negative unit,
instead of a positive, the direction of the force would be reversed; the
magnitude of the force would remain the same. To make the
field-intensity quite definite, a positive unit is chosen for its
specification. For a charge on the trial-conductor consisting of any
number of units, the force is that number of times the field-intensity.
The field-intensity is often specified by its components, X, Y, Z in
three chosen directions at right angles to one another.
Now in all cases in which the action, whether attraction or repulsion,
between two unit quantities of matter concentrated at points is
inversely as the square of the distance between the charges, the
field-intensity, or its components, can be found from a certain function
V of the charges forming the acting distribution [which is always
capable of being regarded for mathematical purposes as a system of small
charges existing at points of space, _point-charges_ we shall call
them], their positions, and the position of the point at which the
field-intensity is to be found. If q₁, q₂, ... be the point-charges, and
be positive when the charges are positive and negative when the charges
are negative, and r₁, r₂, ... be their distances from the point P,
V is q₁⧸r₁ + q₂⧸r₂ + ... The field-intensity is the rate of diminution
of the value of V at P, taken along the specified direction. The three
gradients parallel to the three chosen coordinate directions are
X, Y, Z; but for their calculation it is necessary to insert the values
of r₁, r₂, ... in terms of the coordinates which specify the positions
of the point-charges, and the coordinates x, y, z which specify the
position of P. Once this is done, X, Y, Z are obtained by a simple
systematic process of calculation, namely, differentiation of the
function V with respect to x, y, z.
This function V seems to have been first used by Laplace for
gravitational matter in the _Mécanique Céleste_; its importance for
electricity and magnetism was recognised by Green, who named it the
potential. It has an important physical signification. It represents the
work which would have to be done to bring a unit of positive
electricity, against the electrical repulsion of the distribution, up to
the point P from a point at an infinite distance from every part of the
distribution; or, in other words, what we now call the _potential
energy_ of a charge q situated at P is qV. The excess of the potential
at P, over the potential at any other point Q in the field, is the work
which must be spent in carrying a positive unit from Q to P against
electrical repulsion. Of course, if the force to be overcome from Q to P
is on the whole an attraction, work has not been spent in effecting the
transference, but gained by allowing it to take place. The difference of
potential is then negative, that is, the potential of Q is higher than
that of P.
The difference of potential depends only on the points P and Q, and not
at all on the path pursued between them. Thus, if a unit of electricity
be carried from P to Q by any path, and back by any other, no work is
done on the whole by the agent carrying the unit. This simple fact
precludes the possibility of obtaining a so-called perpetual motion (a
self-acting machine doing useful work) by means of electrical action.
The same thing is true _mutatis mutandis_ of gravitational action.
In the thermal analogy explained by Thomson in his first paper, the
positive point-charges are point-sources of heat, which is there poured
at constant rate into the medium (supposed of uniform quality) to be
drawn off in part from the medium at constant rate where there are sinks
(or negative sources),--the negative point-charges in the electrical
case,--while the remainder is conducted away to more and more distant
parts of the conducting medium supposed infinitely extended. Whenever a
point-source, or a point-sink, exists at a distance from other sources
or sinks, the flow in the vicinity is in straight lines from or to the
point, and these straight lines would be indefinitely extended if either
source or sink existed by itself. As it is, the direction and amount of
flow everywhere depends on the flow resulting from the whole arrangement
of sources and sinks. Lines can be drawn in the medium which show the
direction of the resultant flow from point to point, and these lines of
flow can be so spaced as to indicate, by their closeness together or
their distance apart, where the rate of flow is greater or smaller; and
such lines start from sources, and either end in sinks or continue their
course to infinity. In the electrical case these lines are the analogues
of the lines of electric force (or field-intensity) in the insulating
medium, which start from positive charges and end in negative, or are
prolonged to infinity.
Across such lines of flow can be drawn a family of surfaces, to each of
which the lines met by the surface are perpendicular. These surfaces are
the equitemperature surfaces, or, as they are usually called, the
isothermal surfaces. They can be drawn more closely crowded together, or
more widely separated, so as to indicate where the rate of falling off
of temperature (the "temperature slope") is greater or less, just as the
contour lines in a map show the slopes on a hill-side.
Instead of the thermal analogy might have been used equally well that of
steady flow in an indefinitely extended mass of homogeneous frictionless
and incompressible fluid, into which fluid is being poured at a constant
rate by sources and withdrawn by sinks. The isothermal surfaces are
replaced by surfaces of equal pressure, while lines of flow in one are
also lines of flow in the other.
Now let heat be poured into the medium at constant rate by a single
point-source P (Fig. 1), and drawn off at a smaller rate by a single
point-sink P', while the remainder flows to more and more remote parts
of the medium, supposed infinite in extent in every direction. After a
sufficient time from the beginning of the flow a definite system of
lines of flow and isothermal surfaces can be traced for this case in the
manner described above. One of the isothermal surfaces will be a sphere
S surrounding the sink, which, however, will not be at the centre of the
sphere, but so situated that the source, sink, and centre are in line,
and that the radius of the sphere is a mean proportional between the
distances of the source and sink from the centre. If a be the radius of
the sphere and f the distance of the source from the centre of the
sphere, the heat carried off by the sink is the fraction a⧸f of that
given out by the source.
[Illustration: FIG. 1.]
In the electrical analogue, the source and sink are respectively a
point-charge and what is called the "electric image" of that charge with
respect to the sphere, which is in this case an equipotential surface.
And just as the lines of flow of heat meet the spherical isothermal
surface at right angles, so the lines of force in the electrical case
meet the equipotential surface also at right angles. Now obviously in
the thermal case a spherical sink could be arranged coinciding with the
spherical surface so as to receive the flow there arriving and carry
off the heat from the medium, without in the least disturbing the flow
outside the sphere. The whole amount of heat arriving would be the same:
the amount received per unit area at any point on the sphere would
evidently be proportional to the gradient of temperature there towards
the surface. Of course the same thing could be done at any isothermal
surface, and the same proportionality would hold in that case.
Similarly the source could be replaced by a surface-distribution of
sources over any surrounding isothermal surface; and the condition to be
fulfilled in that case would be that the amount of heat given out per
unit area anywhere should be exactly that which flows out along the
lines of flow there in the actual case. Outside the surface the field of
flow would not be affected by this replacement. It is obvious that in
this case the outflow per unit area must be proportional to the
temperature slope outward from the surface.
The same statements hold for any complex system of sources and sinks.
There must be the same outflow from the isothermal surface or inflow
towards it, as there is in the actual case, and the proportionality to
temperature slope must hold.
This is exactly analogous to the replacement by a distribution on an
equipotential surface of the electrical charge or charges within the
surface, by a distribution over the surface, with fulfilment of
Coulomb's theorem (p. 43 below) at the surface. Thomson's paper on the
"Uniform Motion of Heat" gave an intuitive proof of this great theorem
of electrostatics, which the statements above may help to make clear to
those who have, or are willing to acquire, some elementary knowledge of
electricity.
Returning to the distribution on any isothermal surface surrounding the
sink (or sinks) we see that it represents a surface-sink in equilibrium
with the flow in the field. The distribution on a metal shell,
coinciding with the surface, which keeps the surface at a potential
which is the analogue of the temperature at the isothermal surface,
while the shell is under the influence of a point-charge of
electricity--the analogue of the thermal source--is the distribution as
affected by the induction of the point-charge. If the shell coincide
with the spherical equipotential surface referred to above, and the
distribution given by the theorem of replacement be made upon it, the
shell will be at zero potential, and the charge will be that which would
exist if the shell were uninsulated, that is, the "induced charge."
The consideration of the following simple problem will serve to make
clear the meaning of an electric image, and form a suitable introduction
to a description of the application of the method to the electrification
of spherical surfaces. Imagine a very large plane sheet of tinfoil
connected by a conducting wire with the earth. If there are no
electrified bodies near, the sheet will be unelectrified. But let a very
small metallic ball with a charge of positive electricity upon it be
brought moderately close to one face of the tinfoil. The tinfoil will be
electrified negatively by induction, and the distribution of the
negative charge will depend on the position of the ball. Now, it can be
shown that the field of electric force, on the same side of the tinfoil
as the ball, is precisely the same as would be produced if the foil (and
everything behind it) were removed, and an equal negative charge of
electricity placed behind the tinfoil on the prolonged perpendicular
from the ball to the foil, and as far from the foil behind as the ball
is from it in front. Such a negative charge behind the tinfoil sheet is
called an electric image of the positive charge in front. It is
situated, as will be seen at what would be, if the tinfoil were a
mirror, the optical image of the ball in the mirror.
[Illustration: FIG. 2.]
Now, suppose a second very large sheet of tinfoil to be placed parallel
to the first sheet, so that the small electrified sphere is between the
two sheets, and that this second sheet is also connected to the earth.
The charge on the ball induces negative electricity on both sheets, but
besides this each sheet by its charge influences the other. The problem
of distribution is much more complicated than in the case of a single
sheet, but its solution is capable of very simple statement. Let us call
the two sheets A and B (Fig. 2), and regard them for the moment as
mirrors. A first image of an object P between the two mirrors is
produced directly by each, but the image I₁ in A is virtually an object
in front of B, and the image J₁ in B an object in front of A, so that
a second image more remote from the mirror than the first is produced in
each case. These second images I₂ and J₂ in the same way produce third
images still more remote, and so on. The positions are determined just
as for an object and a single mirror. There is thus an infinite trail of
images behind each mirror, the places of which any one can assign.
[Illustration: FIG. 3.]
Every one may see the realisation of this arrangement in a shop window,
the two sides of which are covered by parallel sheets of mirror-glass.
An infinite succession of the objects in the window is apparently seen
on both sides. When the objects displayed are glittering new bicycles in
a row the effect is very striking; but what we are concerned with here
is a single small object like the little ball, and its two trails of
images. The electric force at any point between the two sheets of
tinfoil is exactly the same as if the sheets were removed and charges
alternately negative and positive were placed at the image-points,
negative at the first images, positive at the second images, and so on,
each charge being the same in amount as that on the ball. We have an
"electric kaleidoscope" with parallel mirrors. When the angle between
the conducting planes is an aliquot part of 360°, let us say 60°, the
electrified point and the images are situated, just as are the object
and its image in Brewster's kaleidoscope, namely at the angular points
of a hexagon, the sides of which are alternately (as shown in Fig. 3) of
lengths twice the distance of the electrified point from A and from B.
[Illustration: FIG. 4.]
Now consider the spherical surface referred to at p. 37, which is kept
at uniform potential by a charge at the external point P, and a charge
q' at the inverse point P' within the sphere. If E (Fig. 4) be any point
whatever on the surface, and r, r' be its distances from P and P', it is
easy to prove by geometry that the two triangles CPE and CEP' are
similar, and therefore r' = ra⧸f. [Here a⧸f is used to mean a divided
by f. The mark ⧸ is adopted instead of the usual bar of the fraction,
for convenience of printing.] Now, by the explanation given above, the
potential produced at any point by a charge q at another point, is equal
to the ratio of the charge q to the distance between the points. Thus
the potential at E due to the charge q at P is q⧸r, and that at E due to
a charge q' at P' is q'⧸r'. Thus if q' = －qa⧸f, q' at P' will produce
a potential at E = －qa⧸fr' = －q⧸r, by the value of r. Hence q at P
and -qa⧸f at P' coexisting will give potential q⧸r + －q⧸r or zero,
at E. Thus the charge －qa⧸f, at the internal point P' will in presence
of +q at P keep all points of the spherical surface at zero potential.
These two charges represent the source and sink in the thermal analogue
of p. 37 above.
Now replace S by a spherical shell of metal connected to the earth
by a long fine wire, and imagine all other conductors to be at a great
distance from it. If this be under the influence of the charge q at P
alone, a charge is induced upon it which, in presence of P, maintains
it at zero potential. The internal charge －qa⧸f, and the induced
distribution on the shell are thus equivalent as regards the potential
produced by either at the spherical surface; for each counteracts then
the potential produced by q at P. But it can be proved that if a
distribution over an equipotential surface can be made to produce the
same potential over that surface as a given internal distribution does,
they produce the same potentials at all external points, or, as it is
usually put, the external fields are the same. This is part of the
statement of what has been called the "theorem of replacement"
discovered by Green, Gauss, Thomson, and Chasles as described above.
Another part of the statement of the theorem may now be formulated.
Coulomb showed long ago that the surface-density of electricity at any
point on a conductor is proportional to the resultant field-intensity
just outside the surface at that point. Since the surface is throughout
at one potential this intensity is normal to the surface. Let it be
denoted by N, and s be the surface-density: then according to the
system of units usually adopted 4πs = N.
Let now the rate of diminution of potential per unit of distance
outwards (or downward gradient of potential) from the equipotential
surface be determined for every point of the surface, and let
electricity be distributed over the surface, so that the amount per unit
area at each point (the surface-density) is made numerically equal to
the gradient there divided by 4π. This, by Coulomb's law, stated
above, gives that field-intensity just outside the surface which exists
for the actual distribution, and therefore, as can be proved, gives the
same field everywhere else outside the surface. The external fields will
therefore be equivalent, and further, the amount of electricity on the
surface will be the same as that situated within it in the actual
distribution.
Thus it is only necessary to find for －qa⧸f at P' and q at P, the
falling off gradient N of potential outside the spherical surface at
any point E, and to take N⧸4π, to obtain s the surface-density at E.
Calculation of this gradient for the sphere gives 4πs = －q(f² － a²)⧸ar³.
The surface-density is thus inversely as the cube of the distance PE.
If the influencing point P be situated within the spherical shell, and
the shell be connected to earth as before, the induced distribution
will be on its interior surface. The corresponding point P will now
be outside, but given by the same relation. And a will now be greater
than f, and the density will be given by 4πs = －q(a² － f²)⧸ar³,
where, f and r have the same meanings with regard to E and P
as before.
P' is in each case called the image of P in the sphere S, and the
charge －qa⧸f there supposed situated is the _electric image_ of the
charge q at P. It will be seen that an electric image is a charge, or
system of charges, on one side of an electrified surface which produces
on the other side of that surface the same electrical field as is
produced by the actual electrification of the surface.
While by the theorem of replacement there is only one distribution over
a surface which produces at all points on one side of a surface the same
field as does a distribution D on the other side of the surface, this
surface distribution may be equivalent to several different arrangements
of D. Thus the point-charge at P' is only one of various
image-distributions equivalent to the surface-distribution in the sense
explained. For example, a uniform distribution over any spherical
surface with centre at P' (Fig. 4) would do as well, provided this
spherical surface were not large enough to extend beyond the surface S.
In order to find the potential of the sphere (Fig. 4) when insulated
with a charge Q upon it, in presence of the influencing charge q at the
external point P, it is only necessary to imagine uniformly distributed
over the sphere, already electrified in the manner just explained, the
charge Q + aq⧸f. Then the whole charge will be Q, and the uniformity of
distribution will be disturbed, as required by the action of the
influencing point-charge. The potential will be Q⧸a + q⧸f. For a
given potential V of the sphere, the total charge is aV － aq⧸f,
that is the charge is aV over and above the induced charge.
If instead of a single influencing point-charge at P there be a system
of influencing point-charges at different external points, each of these
has an image-charge to be found in amount and situation by the method
just described, and the induced distribution is that obtained by
superimposing all the surface distributions found for the different
influencing points.
The force of repulsion between the point-charge q and the sphere
(with total charge Q) can be found at once by calculating the sum
of the forces between q at P and the charges Q + aq⧸f at C
and －aq⧸f at P'.
This can be found also by calculating the energy of the system, which
will be found to consist of three terms, one representing the energy of
the sphere with charge Q uninfluenced by an external charge, one
representing the energy on a small conductor (not a point) at P existing
alone, and a third representing the mutual energy of the electrification
on the sphere and the charge q at P existing in presence of one another.
By a known theorem the energy of a system of conductors is one half of
the sum obtained by multiplying the potential of each conductor by its
charge and adding the products together. It is only necessary then to
find the variation of the last term caused by increasing f by a small
amount df. This will be the product F.df of the force F required and the
displacement.
Either method may be applied to find the forces of attraction and
repulsion for the systems of electrified spheres described below.
The problem of two mutually influencing non-intersecting spheres, S₁, S₂
(Fig. 5), insulated with given charges, q₁, q₂, may now be dealt with in
the following manner. Let each be supposed at first charged uniformly.
By the known theorem referred to above, the external field of each is
the same as if its whole charge were situated at the centre. Now if the
distribution on S₂, say, be kept unaltered, while that on S₁ is allowed
to change, the action of S₂ on S₁ is the same as if the charge q₂ were
at the centre C₂ of S₂. Thus if f be the distance between the centres
C₁, C₂, and a₁ be the radius of S₁, the distribution will be that
corresponding to q₁ + a₁q₂⧸f uniformly distributed on S₁ together with
the induced charge －a₁q₂⧸f, which corresponds to the image-charge at
the point I₁ (within S₁), the inverse of C₂ with respect to S₁. Now
let the charge on S₁ be fixed in the state just supposed while that
on S₂ is freed. The charge on S₂ will rearrange itself under the
influence of q₁ + a₁q₂⧸f ( = q') and －a₁q₂⧸f, considered as at C₁
and I₁ respectively. The former of these will give a distribution
equivalent to q₂ + a₂q'⧸f uniformly distributed over S₂, and an
induced distribution of amount a₂q'⧸f at J₁, the inverse point of C₁
with regard to S₂. The image-charge －a₁q₂⧸f at I₁ in S₁ will react
on S₂ and give an induced distribution －a₂(－a₁q₂⧸f)f', (I₁C₂ = f')
corresponding to an image-charge a₂a₁q₂⧸ff' at the inverse point J₂
of P₁ with respect to C₂S₂. Thus the distribution on S₂ is equivalent
to q₂ + a₂q'⧸f － a₂a₁q₂⧸ff' distributed uniformly over it, together with
the two induced distributions just described.
[Illustration: FIG. 5.]
In the same way these two induced distributions on S₂ may now be
regarded as reacting on the distribution on S₁ as would point-charges
－a₂q₁⧸f and a₂a₁q₂⧸ff', situated at J₁ and J₂ respectively, and would
give two induced distributions on S₁ corresponding to their images
in S₁.
Thus by partial influences in unending succession the equilibrium state
of the two spheres could be approximated to as nearly as may be desired.
An infinite trail of electric images within each of the two spheres is
thus obtained, and the final state of each conductor can be calculated
by summation of the effects of each set of images.
If the final potentials, V₁, V₂, say, of the spheres are given the
process is somewhat simpler. Let first the charges be supposed to
exist uniformly distributed over each sphere, and to be of amount a₁V₁,
a₂V₂ in the two cases. The uniform distribution on S₁ will raise the
potential of S₂ above V₂, and to bring the potential down to V₂ in
presence of this distribution we must place an induced distribution
over S₂, represented as regards the external field by the image-charge
－a₂a₁V₁⧸f (at the image of C₁ in S₂) where f is the distance
between the centres. The charge a₂V₂ on S₂ will similarly have an
action on S₁ to be compensated in the same way by an image-charge
－a₁a₂V₂⧸f at the image of C₂ in S₁. Now these two image-charges
will react on the spheres S₁ and S₂ respectively, and will have to be
balanced by induced distributions represented by second image-charges,
to be found in the manner just exemplified. These will again react on
the spheres and will have to be compensated as before, and so on
indefinitely. The charges diminish in amount, and their positions
approximate more and more, according to definite laws, and the final
state is to be found by summation as before.
The force of repulsion is to be found by summing the forces between all
the different pairs of charges which can be formed by taking one charge
of each system at its proper point: or it can be obtained by calculating
the energy of the system.
The method of successive influences was given originally by Murphy, but
the mode of representing the effects of the successive induced charges
by image-charges is due to Thomson. Quite another solution of this
problem is, however, possible by Thomson's method of electrical
inversion.
A similar process to that just explained for two charged and mutually
influencing spheres will give the distribution on two concentric
conducting spheres, under the influence of a point-charge q at P between
the inner surface of the outer and the outer surface of the inner, as
shown in Fig. 7. There the influence of q at P, and of the induced
distributions on one another, is represented by two series of images,
one within the inner sphere and one outside the outer. These charges and
positions can be calculated from the result for a single sphere and
point-charge.
Thomson's method of electrical inversion, referred to above, enabled the
solutions of unsolved problems to be inferred from known solutions of
simpler cases of distribution. We give here a brief account of the
method, and some of its results. First we have to recall the meaning of
geometrical inversion. In Fig. 6 the distances OP, OP, OQ, OQ' fulfil
the relation OP.OP' = OQ.OQ' = a². Thus P' is (see p. 37) the inverse
of the point P with respect to a sphere of radius a and centre O
(indicated by the dotted line in Fig. 6), and similarly Q' is the
inverse of Q with respect to the same sphere and centre. O is called the
centre of inversion, and the sphere of radius a is called the sphere of
inversion. Thus the sphere of Figs. 1 and 4 is the sphere of inversion
for the points P and P', which are inverse points of one another. For
any system of points P, Q, ..., another system P', Q', ... of inverse
points can be found, and if the first system form a definite locus, the
second will form a derived locus, which is called the inverse of the
former. Also if P', Q', ... be regarded as the direct system, P, Q, ...
will be the corresponding inverse system with regard to the same sphere
and centre. P' is the image of P, and P is the image of P', and so on,
with regard to the same sphere and centre of inversion.
[Illustration: FIG. 6.]
The inverse of a circle is another circle, and therefore the inverse of
a sphere is another sphere, and the inverse of a straight line is a
circle passing through the centre of inversion, and of an infinite plane
a sphere passing through the centre of inversion. Obviously the inverse
of a sphere concentric with the sphere of inversion is a concentric
sphere.
The line P'Q' is of course not the inverse of the line PQ, which has
for its inverse the circle passing through the three points O, P', Q',
as indicated in Fig. 6.
The following results are easily proved.
A locus and its inverse cut any line OP at the same angle.
To a system of point-charges q₁, q₂, ... at points P₁, P₂, ... on
one side of the surface of the sphere of inversion there is a system
of charges aq₁⧸f₁, aq₂⧸f₂, ... on the other side of the spherical
surface [OP₁ = f₁, OP₂ = f₂]. This inverse system, as we shall call
it, produces the same potential at any point of the sphere of inversion,
as does the direct system from which it is derived.
If V, V' be the potentials produced by the whole direct system at Q,
and by the whole inverse system at Q', V'⧸V = r⧸a = a⧸r', where OQ = r,
OQ' = r'.
Thus if V is constant over any surface S', V' is not a constant over the
inverse surface S', unless r is a constant, that is, unless the surface
S' is a sphere concentric with the sphere of inversion, in which case
the inverse surface is concentric with it and is an equipotential
surface of the inverse distribution.
Further, if q be distributed over an element dS of a surface, the
inverse charge aq⧸f will be distributed over the corresponding element
dS' of the inverse surface. But dS'⧸dS = a⁴⧸f⁴ = f'⁴⧸a⁴ where f, f'
are the distances of O from dS and dS'. Thus if s be the density on dS
and s' the inverse density on dS' we have s'⧸s = a³⧸f'³ = f³⧸a³.
When V is constant over the direct surface, while r has different
values for different directions of OQ, the different points of the
inverse surface may be brought to zero potential by placing at O a
charge －aV. For this will produce at Q' a potential －aV⧸r' which
with V' will give at Q' a potential zero. This shows that V' is the
potential of the induced distribution on S' due to a charge －aV at O,
or that －V' is the potential due to the induced charge on S' produced
by the charge aV at O.
Thus we have the conclusion that by the process of inversion we get from
a distribution in equilibrium, on a conductor of any form, an induced
distribution on the inverse surface supposed insulated and conducting;
and conversely we obtain from a given induced distribution on an
insulated conducting surface, a natural equilibrium distribution on the
inverse surface. In each case the inducing charge is situated at the
centre of inversion. The charges on the conductor (or conductors) after
inversion are always obtainable at once from the fact that they are the
inverses of the charges on the conductor (or conductors) in the direct
case, and the surface-densities or volume-densities can be found from
the relations stated above.
[Illustration: FIG. 7.]
Now take the case of two concentric spheres insulated and influenced by
a point-charge q placed at a point P between them as shown in Fig. 7. We
have seen at p. 49 how the induced distribution, and the amount of the
charge, on each sphere is obtained from the two convergent series of
images, one outside the outer sphere, the other inside the inner sphere.
We do not here calculate the density of distribution at any point, as
our object is only to explain the method; but the quantities on the
spheres S₁ and S₂, are respectively －q.OA.PB⧸(OP.AB), －q.OB.AP⧸(OP.AB).
It may be noticed that the sum of the induced charges is －q, and that
as the radii of the spheres are both made indefinitely great, while
the distance AB is kept finite, the ratios OA⧸OP, OB⧸OP approximate
to unity, and the charges to －q.PB⧸AB, －q.AP⧸AB, that is, the
charges are inversely as the distances of P from the nearest points of
the two surfaces. But when the radii are made indefinitely great we have
the case of two infinite plane conducting surfaces with a point-charge
between them, which we have described above.
Now let this induced distribution, on the two concentric spheres, be
inverted from P as centre of inversion. We obtain two non-intersecting
spheres, as in Fig. 5, for the inverse geometrical system, and for the
inverse electrical system an equilibrium distribution on these two
spheres in presence of one another, and charged with the charges which
are the inverses of the induced charges. These maintain the system of
two spheres at one potential. From this inversion it is possible to
proceed as shown by Maxwell in his _Electricity and Magnetism_, vol. i,
§ 173, to the distribution on two spheres at two different potentials;
but we have shown above how the problem may be dealt with directly by
the method of images.
[Illustration: FIG. 8.]
Again take the case of two parallel infinite planes under the influence
of a point-charge between them. This system inverted from P as centre
gives the equilibrium distribution on two charged insulated spheres in
contact (Fig. 8); for this system is the inverse of the planes and the
charges upon them. Another interesting case is that of the "electric
kaleidoscope" referred to above. Here the two infinite conducting planes
are inclined at an angle 360°⧸n, where n is a whole number, and are
therefore bounded in one direction by the straight line which is their
intersection. The image points I₁, J₁, ..., of P placed in the angle
between the planes are situated as shown in Fig. 3, and are n － 1 in
number. This system inverted from P as centre gives two spherical
surfaces which cut one another at the same angle as do the planes. This
system is one of electrical equilibrium in free space, and therefore the
problem of the distribution on two intersecting spheres is solved, for
the case at least in which the angle of intersection is an aliquot part
of 360°. When the planes are at right angles the result is that for two
perpendicularly intersecting planes, for which Fig. 9 gives a diagram.
[Illustration: FIG. 9.]
But the greatest achievement of the method was the determination of the
distribution on a segment of a thin spherical shell with edge in one
plane. The solution of this problem was communicated to M. Liouville in
the letter of date September 16, 1846, referred to above, but without
proof, which Thomson stated he had not time to write out owing to
preparation for the commencement of his duties as Professor of Natural
Philosophy at Glasgow on November 1, 1846. It was not supplied until
December 1868 and January 1869; and in the meantime the problem had not
been solved by any other mathematician.
As a starting point for this investigation the distribution on a thin
plane circular disk of radius a is required. This can be obtained by
considering the disk as a limiting case of an oblate ellipsoid of
revolution, charged to potential V, say. If Fig. 10 represent the disk
and P the point at which the density is sought, so that CP = r, and
CA = a, the density is V⧸{2π²√(a² － r²)}.
The ratio q⧸V, of charge to potential, which is called the electrostatic
capacity of the conductor, is thus 2a⧸π, that is a⧸1.571. It is, as
Thomson notes in his paper, very remarkable that the Hon. Henry
Cavendish should have found long ago by experiment with the rudest
apparatus the electrostatic capacity of a disk to be 1⧸1.57 of that
of a sphere of the same radius.
[Illustration: FIG. 10.]
[Illustration: FIG. 11.]
Now invert this disk distribution with any point Q as centre of
inversion, and with radius of inversion a. The geometrical inverse is
a segment of a spherical surface which passes through Q. The inverse
distribution is the induced distribution on a conducting shell
uninsulated and coincident with the segment, and under the influence of
a charge －aV situated at Q (Fig. 11). Call this conducting shell the
"bowl." If the surface-densities at corresponding points on the disk and
on the inverse, say points P and P', be s and s', then, as on page 51,
s' = sa³⧸QP'³. If we put in the value of s given above, that of s' can
be put in a form given by Thomson, which it is important to remark is
independent of the radius of the spherical surface. This expression is
applicable to the other side of the bowl, inasmuch as the densities at
near points on opposite sides of the plane disk are equal.
If v, v' be the potentials at any point R of space, due to the disk
and to its image respectively, －v' = av⧸QR. If then R be coincident
with a point P' on the spherical segment we have (since then v = V)
V' = aV⧸QP', which is the potential due to the induced distribution
caused by the charge －aV at Q as already stated.
The fact that the value of s' does not involve the radius makes it
possible to suppose the radius infinite, in which case we have the
solution for a circular disk uninsulated and under the influence of a
charge of electricity at a point Q in the same plane but outside the
bounding circle.
Now consider the two parts of the spherical surface, the bowl B, and the
remainder S of the spherical surface. Q with the charge －aV may be
regarded as situated on the latter part of the surface. Any other
influencing charges situated on S will give distributions on the bowl to
be found as described above, and the resulting induced electrification
can be found from these by summation. If S be uniformly electrified to
density s, and held so electrified, the inducing distribution will be
one given by integration over the whole of S, and the bowl B will be at
zero potential under the influence of this electrification of S, just as
if B were replaced by a shell of metal connected to the earth by a long
fine wire. The densities are equal at infinitely near points on the two
sides of B.
Let the bowl be a thin metal shell connected with the earth by a long
thin wire and be surrounded by a concentric and complete shell of
diameter f greater than that of the spherical surface, and let this
shell be rigidly electrified with surface density －s. There will be no
force within this shell due to its own electrification, and hence it
will produce no change of the distribution in the interior. But the
potential within will be －2πfs, for the charge is －πf²s, and the
capacity of the shell is ½f. The potential of the bowl will now be
zero, and its electrification will just neutralise the potential
-2πfs, that is, will be exactly the free electrification required to
produce potential 2πfs.
To find this electrification let the value of f be only infinitesimally
greater than the diameter of the spherical surface of which B is a part;
then the bowl is under the influence (1) of a uniform electrification of
density －s infinitely close to its outer surface, and (2) of a uniform
electrification of the same density, which may be regarded as upon the
surface which has been called S above. It is obvious that by (1) a
density s is produced on the outer surface of the bowl, and no other
effect; by (2) an equal density at infinitely near points on the
opposite sides of the bowl is produced which we have seen how to
calculate. Thus the distribution on the bowl freely electrified is
completely determined and the density can easily be calculated. The
value will be found in Thomson's paper.
Interesting results are obtained by diminishing S more and more until
the shell is a complete sphere with a circular hole in it. Tabulated
results for different relative dimensions of S will be found in
Thomson's paper, "Reprint of Papers," Articles V, XIV, XV. Also the
reader will there find full particulars of the mathematical calculations
indicated in this chapter, and an extension of the method to the case of
an influencing point not on the spherical surface of which the shell
forms part. Further developments of the problem have been worked out by
other writers, and further information with references will be found in
Maxwell's _Electricity and Magnetism_, loc. cit.
It is not quite clear whether Thomson discovered geometrical inversion
independently or not: very likely he did. His letter to Liouville of
date October 8, 1845, certainly reads as if he claimed the geometrical
transformation as well as the application to electricity. Liouville,
however, in his Note in which he dwells on the analytical theory of the
transformation says, "La transformation dont il s'agit est bien connue,
du reste, et des plus simples; c'est celle que M. Thomson lui-même a
jadis employée sous le nom de principe des images." In Thomson and
Tail's _Natural Philosophy_, § 513, the reference to the method is as
follows: "Irrespectively of the special electric application, the method
of images gives a remarkable kind of transformation which is often
useful. It suggests for mere geometry what has been called the
transformation by reciprocal radius-rectors, that is to say...." Then
Maxwell, in his review of the "Reprint of Papers" (Nature, vol. vii),
after referring to the fact that the solution of the problem of the
spherical bowl remained undemonstrated from 1846 to 1869, says that the
geometrical idea of inversion had probably been discovered and
rediscovered repeatedly, but that in his opinion most of these
discoveries were later than 1845, the date of Thomson's first paper.[10]
A very general method of finding the potential at any point of a region
of space enclosed by a given boundary was stated by Green in his 'Essay'
for the case in which the potential is known for every point of the
boundary. The success of the method depends on finding a certain
function, now called Green's function. When this is known the potential
at any point is at once obtained by an integration over the surface.
Thomson's method of images amounts to finding for the case of a region
bounded by one spherical surface or more the proper value of Green's
function. Green's method has been successfully employed in more
complicated cases, and is now a powerful method of attack for a large
range of problems in other departments of physical mathematics. Thomson
only obtained a copy of Green's paper in January 1845, and probably
worked out his solutions quite independently of any ideas derived from
Green's general theory.