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

$R=-e{\frac {f^{2}-a^{2}}{a}}{\frac {1}{AP^{3}}}$

(3)

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

$R=-e{\frac {a^{2}-f^{2}}{a}}{\frac {1}{AP^{3}}}$

(4)

In all cases we may write

$R=-e{\frac {AD\cdot Ad}{CP}}{\frac {1}{AP^{3}}},$

(5)

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

$\sigma =-e{\frac {AD\cdot Ad}{4\pi \cdot CP}}{\frac {1}{AP^{3}}}$

(6)

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

$\sigma ={\frac {C}{AP^{3}}}$

(7)

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

$C=-e{\frac {AD\cdot Ad}{4\pi a}}$

(8)