99.] The calculation of the distribution of electrification on the surface of a conductor when electrified bodies are placed near it is in general an operation beyond the powers of existing mathematical methods. When the conductor is a sphere, and when the distribution of electricity on external bodies is given, a solution, depending on an infinite series was obtained by Poisson. This solution agrees with that which was afterwards given in a far simpler form by Sir W. Thomson, and which is the foundation of his method of Electric Images.

By this method he has solved problems in electricity which have never been attempted by any other method, and which, even after the solution has been pointed out, no other method seems capable of attacking. This method has the great advantage of being intelligible by the aid of the most elementary mathematical reasoning, especially when it is considered in connection with the diagrams of equipotential surfaces described in Arts. 93-96. 100.] The idea of an image is most easily acquired by considering the optical phenomena on account of which the term image was first introduced into science.

We are accustomed to make use of the visual impressions we receive through our eyes in order to ascertain the positions of distant objects. We are doing this all day long in a manner sufficiently accurate for ordinary purposes. Surveyors and astronomers by means of artificial instruments and mathematical deductions do the same thing with greater exactness. In whatever way, however, we make our deductions, we find that they are consistent with the hypothesis that an object exists in a certain position in space, from which it emits light which travels to our eyes or to our instruments in straight lines.

But if we stand in front of a plane mirror and make observations on the apparent direction of the objects reflected therein, we find that these observa- tions are consistent with the hypothesis that there is no mirror, but that certain objects exist in the region beyond the plane of the mirror.

These hypothetical objects are geometrically related to certain real objects in front of the plane of the mirror, and they are called the images of these objects. We are not provided with a special sense for enabling us to ascertain the presence and the position of distant bodies by means of their electrical effects, but we have instrumental methods by which the distribution of potential and of electric force in any part of the field may be ascertained, and from these data we obtain a certain amount of evidence as to the position and electrification of the distant body.

If an astronomer, for instance, could ascertain the direction and magnitude of the force of gravitation at any desired point in the heavenly spaces, he could deduce the positions and masses of the bodies to which the force is due. When Adams and Leverrier discovered the hitherto unknown planet Neptune, they did so by ascertaining the direction and magnitude of the gravitating force due to the unseen planet at certain points of space. In the electrical problem we employed an electrified pith ball, which we moved about in the field at pleasure. The astronomers employed for a similar purpose the planet Uranus, over which, indeed, they had no control, but which moved of itself into such positions that the alterations of the elements of its orbit served to indicate the position of the unknown disturbing planet.

101.] In one of the electrified systems which we have already investigated, that of a spherical conductor A within a con- centric spherical conducting vessel B, we have one of the simplest cases of the principle of electric images.

CONCENTRIC SPHERES

The electric field is in this case the region which lies between the two concentric spherical surfaces. The electric force at any point P within this region is in the direction of the radius OP and nu- merically equal to the charge of the in- ner sphere, A, divided by the square of Fig. 22. the distance, OP, of the point from the common centre.

The force within this region will be the same if we substitute for the electrified spherical surfaces, A and B, any other two concentric spherical surfaces, C and D, one of them, C, lying within the smaller sphere, A, and the other, D, lying outside of B, the charge of C being equal to that of A in the former case. The electric phenomena in the region between A and B are therefore the same as before, the only differ- ence between the cases is that in the region between A and C and also in the region between B and D we now find electric forces acting according to the same law as in the region between A and B, whereas when the region was bounded by the conducting surfaces A and B there was no electrical force whatever in the regions beyond these surfaces. We may even, for mathemat- ical purposes, suppose the inner sphere C to be reduced to a physical point at O, and the outer sphere D to expand to an infinite size, and thus we as- similate the electric action in the region between A and B to that due to an electrified point at O placed in an infinite region.

It appears, therefore, that when a spherical surface is uniformly electrified, the electric phenomena in the region outside the sphere are exactly the same as if the spherical surface had been removed, and a very small body placed at the centre of the sphere, having the same electric charge as the sphere. This is a simple instance in which the phenomena in a certain region are consistent with a false hypothesis as to what exists beyond that region. The action of a uniformly electrified spherical surface in the region outside that surface is such that the phenomena may be attributed to an imaginary elec- trified point at the centre of the sphere.

The potential, ψ, of a sphere of radius a, placed in infinite space and e charged with a quantity e of electricity, is . Hence if ψ is the potential a of the sphere, the imaginary charge at its centre is ψa. 102.] Now let us calculate the potential at a point P (Fig. 23.) in a spher- ical surface whose centre is C and radius CP, due to two electrified points A and B in the same radius produced, and such that the product of their dis- tances from the centre is equal to the square of the radius. Points thus related to one another are called inverse points with respect to the sphere.

Let a = CP be the radius of the sphere. Let CA = ma, then CB will beIMAGE OF A POINT. 93 a . m Also the triangle AP C is similar to P CB, and AP ∶ P B ∶∶ AC ∶ P C, or AP = mBP. See Euclid vi. prop. E. e m of the opposite kind be placed at B. The potential due to these charges at P will be Now let a charge of electricity equal to e be placed at A and a charge e′ = − V= e e′ , BP e

− , mBP mBP = 0; AP e + or the potential due to the charges at A and B at any point P of the spherical surface is zero. We may now suppose the spherical surface to be a thin shell of metal.

Its potential is already zero at every point, so that if we connect it by a fine wire with the earth there will be no alter- ation of its potential, and therefore the potential at every point, whether within or without the surface, will remain un- altered, and will be that due to the two Fig. 23. electrified points A and B.

If we now keep the metallic shell in connection with the earth and remove the electrified point B, the potential at every point within the sphere will become zero, but outside it will remain as before. For the surface of theELECTRICAL IMAGES. 94 sphere still remains of the same potential, and no change has been made in the distribution of electrified bodies in the region outside the sphere. Hence, if an electrified point A be placed outside a spherical conductor which is at potential zero, the electrical action at all points outside the sphere will be equivalent to that due to the point A together with another point, B, within the sphere, which is the inverse point to A, and whose charge is to that of A as −1 is to m. The point B with its imaginary charge is called the electric image of A.

In the same way by removing A and retaining B, we may shew that if an electrified point B be placed inside a hollow conductor having its inner surface spherical, the electrical action within the hollow is equivalent to that of the point B, together with an imaginary point, A, outside the sphere, whose charge is to that of B as m is to −1.

If the sphere, instead of being in connection with the earth, and therefore at potential zero, is at potential ψ the electrical effects outside the sphere will be the same as if, in addition to the image of the electrified point, another imaginary charge equal to ψa were placed at the centre of the sphere. Within the sphere the potential will simply be increased by ψ.