Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/235

 sphere, that is to say, they lie in the same radius, and the radius is a mean proportional between their distances from the centre.

Since this spherical surface is at potential zero, if we suppose it constructed of thin metal and connected with the earth, there will be no alteration of the potential at any point either outside or inside, but the electrical action everywhere will remain that due to the two electrified points $$A$$ and $$B$$.

If we now keep the metallic shell in connexion with the earth and remove the point $$B$$, the potential within the sphere will become everywhere zero, but outside it will remain the same as before. For the surface of the sphere still remains at the same potential, and no change has been made in the exterior electrification.

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 that due to the point $$A$$ together with another point $$B$$ within the sphere, which we may call the electrical image of $$A$$.

In the same way we may shew that if $$B$$ is a point placed inside the spherical shell, the electrical action within the sphere is that due to $$B$$, together with its image $$A$$.

157.] Definition of an Electrical Image. An electrical image is an electrified point or system of points on one side of a surface which would produce on the other side of that surface the same electrical action which the actual electrification of that surface really does produce.

In Optics a point or system of points on one side of a mirror or lens which if it existed would emit the system of rays which actually exists on the other side of the mirror or lens, is called a virtual image.

Electrical images correspond to virtual images in optics in being related to the space on the other side of the surface. They do not correspond to them in actual position, or in the merely approximate character of optical foci.

There are no real electrical images, that is, imaginary electrified points which would produce, in the region on the same side of the electrified surface, an effect equivalent to that of the electrified surface.

For if the potential in any region of space is equal to that due to a certain electrification in the same region it must be actually produced by that electrification. In fact, the electrification at any point may be found from the potential near that point by the application of Poisson's equation.