Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/241

Fig. 8

Hence, if $A$ be an electrified point whose charge is $e$, and $AD$ a perpendicular on the plane, produce $AD$ to $B$ so that $DB=AB$, and place at $B$ a charge equal to $-e$, then this charge at $B$ will be the image of $A$, and will produce at all points on the same side of the plane as $A$, an effect equal to that of the actual electrification of the plane. For the potential on the side of $A$ due to $A$ and $B$ fulfils the conditions that $\nabla ^{2}V=0$ everywhere except at $A$, and that $V=0$ at the plane, and there is only one form of $V$ which can fulfil these conditions.

To determine the resultant force at the point $P$ of the plane, we observe that it is compounded of two forces each equal to ${\tfrac {e}{AP^{2}}}$, one acting along $AP$ and the other along $PB$. Hence the resultant of these forces is in a direction parallel to $AB$ and equal to

${\frac {e}{AP^{2}}}\cdot {\frac {AB}{AP}}$

Hence $R$, the resultant force measured from the surface towards the space in which $A$ lies, is

$R=-{\frac {2eAD}{AP^{3}}}$

(15)

and the density at the point $P$ is

$\sigma =-{\frac {eAD}{2\pi AP^{3}}}.$

(16)

On Electrical Inversion.

162.] The method of electrical images leads directly to a method of transformation by which we may derive from any electrical problem of which we know the solution any number of other problems with their solutions.

We have seen that the image of a point at a distance $r$ from the centre of a sphere of radius $R$ is in the same radius and at a distance $r'$ such that $rr'=R^{2}$. Hence the image of a system of points, lines, or surfaces is obtained from the original system by the method known in pure geometry as the method of inversion, and described by Chasles, Salmon, and other mathematicians.