Let be the radius of the sphere.
Let be the distance of the electrified point from the centre .
Let be the charge of this point.
Then the image of the point is at , on the same radius of the sphere at a distance , and the charge of the image is .
![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/20/A_Treatise_on_Electricity_and_Magnetism_Volume_1_Fig7.png/300px-A_Treatise_on_Electricity_and_Magnetism_Volume_1_Fig7.png)
We have shewn that this image will produce the same effect on the opposite side of the surface as the actual electrification of the surface does. We shall next determine the surface-density of this electrification at any point of the spherical surface, and for this purpose we shall make use of the theorem of Coulomb, Art. 80, that if is the resultant force at the surface of a conductor, and the superficial density,
being measured away from the surface.
We may consider as the resultant of two forces, a repulsion acting along , and an attraction acting along .
Resolving these forces in the directions of and , we find that the components of the repulsion are
along AC, and along CP.
Those of the attraction are
along AC, and along CP.
Now , and , so that the components of the attraction may be written
along AC, and along CP.
The components of the attraction and the repulsion in the direction of are equal and opposite, and therefore the resultant force is entirely in the direction of the radius . This only confirms what we have already proved, that the sphere is an equipotential surface, and therefore a surface to which the resultant force is everywhere perpendicular.