Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/153

where

$\cos \epsilon =ll'+mm'+nn'\,$,

$\epsilon$ being the angle between the directions of R and $R'$.

Now if K is what we have called the coefficient of electric inductive capacity, then KR will be the electric displacement due to the electromotive force R, and the product $KRR'\cos \epsilon$ will represent the work done by the force $R'$ on account of the displacement caused by the force $R'$, or in other words, the amount of intrinsic energy in that part of the field due to the mutual action of $R$ and $R'$.

We therefore conclude that the physical interpretation of Green's theorem is as follows :

If the energy which is known to exist in an electrified system is due to actions which take place in all parts of the field, and not to direct action at a distance between the electrified bodies, then that part of the intrinsic energy of any part of the field upon which the mutual action of two electrified systems depends is $KRR'\cos \epsilon$ per unit of volume.

The energy of an electrified system due to its action on itself is, by Art. 85,

${\frac {1}{2}}\sum (eV)$,

which is by Green's theorem, putting U = V,

$Q={\frac {1}{8\pi }}\iiint {K\left({\frac {\overline {dV}}{dx}}{\Big |}^{2}+{\frac {\overline {dV}}{dy}}{\Big |}^{2}+{\frac {\overline {dV}}{dz}}{\Big |}^{2}\right)dx\,dy\,dz};$

(11)

and this is the unique minimum value of the integral considered in Thomson's theorem.

Green's Function.

101.] Let a closed surface S be maintained at potential zero. Let P and Q be two points on the positive side of the surface S (we may suppose either the inside or the outside positive), and let a small body charged with unit of electricity be placed at P; the potential at the point Q will consist of two parts, of which one is due to the direct action of the electricity on P, while the other is due to the action of the electricity induced on S by P. The latter part of the potential is called Green's Function, and is denoted by G_pq.

This quantity is a function of the positions of the two points P and Q, the form of which depends on that of the surface S. It has been determined in the case in which S is a sphere, and in a very few other cases. It denotes the potential at Q due to the electricity induced on S by unit of electricity at P.