Lord Kelvin Part 2

Instead of the thermal a.n.a.logy might have been used equally well that of steady flow in an indefinitely extended ma.s.s of h.o.m.ogeneous frictionless and incompressible fluid, into which fluid is being poured at a constant rate by sources and withdrawn by sinks.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.

[Ill.u.s.tration: FIG. 1.]

In the electrical a.n.a.logue, 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 a.n.a.logous 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 sh.e.l.l, coinciding with the surface, which keeps the surface at a potential which is the a.n.a.logue of the temperature at the isothermal surface, while the sh.e.l.l is under the influence of a point-charge of electricity--the a.n.a.logue of the thermal source--is the distribution as affected by the induction of the point-charge. If the sh.e.l.l coincide with the spherical equipotential surface referred to above, and the distribution given by the theorem of replacement be made upon it, the sh.e.l.l will be at zero potential, and the charge will be that which would exist if the sh.e.l.l 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.

[Ill.u.s.tration: 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 a.s.sign.

[Ill.u.s.tration: 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-gla.s.s.

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.

[Ill.u.s.tration: 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 a.n.a.logue of p. 37 above.

Now replace S by a spherical sh.e.l.l 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 sh.e.l.l 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 4ps = 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 4p. 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?4p, to obtain s the surface-density at E.

Calculation of this gradient for the sphere gives 4ps = -q(f - a)?ar.

The surface-density is thus inversely as the cube of the distance PE.

If the influencing point P be situated within the spherical sh.e.l.l, and the sh.e.l.l 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 4ps = -q(a - f)?ar, 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.

[Ill.u.s.tration: 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' = a. 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.

[Ill.u.s.tration: 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 pa.s.sing through the centre of inversion, and of an infinite plane a sphere pa.s.sing 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 pa.s.sing 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.