Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/237

 The resultant force measured along $$CP$$, the normal to the surface in the direction towards the side on which $$A$$ is placed, is

If $$A$$ is taken inside the sphere $$f$$ is less than $$a$$, and we must measure $$R$$ inwards. For this case therefore

In all cases we may write

where $$AD, Ad$$ are the segments of any line through $$A$$ cutting the sphere, and their product is to be taken positive in all cases.

158.] From this it follows, by Coulomb's theorem, Art. 80, that the surface-density at $$P$$ is

The density of the electricity at any point of the sphere varies inversely as the cube of its distance from the point $$A$$.

The effect of this superficial distribution, together with that of the point $$A$$, is to produce on the same side of the surface as the point $$A$$ a potential equivalent to that due to $$e$$ at $$A$$, and its image $$-e\tfrac{a}{f}$$ at $$B$$, and on the other side of the surface the potential is everywhere zero. Hence the effect of the superficial distribution by itself is to produce a potential on the side of $$A$$ equivalent to that due to the image $$-e\tfrac{a}{f}$$ at $$B$$, and on the opposite side a potential equal and opposite to that of $$e$$ at $$A$$.

The whole charge on the surface of the sphere is evidently $$-e\tfrac{a}{f}$$ since it is equivalent to the image at $$B$$.

We have therefore arrived at the following theorems on the action of a distribution of electricity on a spherical surface, the surface-density being inversely as the cube of the distance from a point $$A$$ either without or within the sphere.

Let the density be given by the equation

where $$C$$ is some constant quantity, then by equation (6)