Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/154

The actual potential at any point $Q$ due to the electricity at $P$ and on $S$ is

${\frac {1}{r_{pq}}}+G_{pq},$

where $r_{pq}$ denotes the distance between $P$ and $Q$.

At the surface $S$ and at all points on the negative side of $S$, the potential is zero, therefore

$G_{pa}=-{\frac {1}{r_{pa}}},$

(1)

where the suffix $_{a}$ indicates that a point $A$ on the surface $S$ is taken instead of $Q$.

Let $\sigma _{pa'}$ denote the surface-density induced by $P$ at a point $A'$ of the surface $S$, then, since $G_{pq}$ is the potential at $Q$ due to the superficial distribution,

$G_{pq}=\iint {\frac {\sigma _{pa'}}{r_{qa'}}}dS',$

(2)

where $dS'$ is an element of the surface $S$ at $A'$, and the integration is to be extended over the whole surface $S$.

But if unit of electricity had been placed at $Q$, we should have had by equation (1),

${\frac {1}{r_{qa'}}}=-G_{qa'}$

(3)

$=-\iint {\frac {\sigma _{qa}}{r_{aa'}}}dS;$

(4)

where $\sigma _{qa}$ is the density induced by $Q$ on an element $dS$ at $A$, and $r_{aa'}$ is the distance between $A$ and $A'$. Substituting this value of ${\frac {1}{r_{qa'}}}$ in the expression for $G_{pq}$, we find

$G_{pq}=-\iiiint {\frac {\sigma _{qa}\sigma _{pa'}}{r_{aa'}}}dS\ dS'.$

(5)

Since this expression is not altered by changing $_{p}$ into $_{q}$ and $_{q}$ into $_{p}$,we find that

$G_{pq}=G_{qp};$

(6)

a result which we have already shewn to be necessary in Art. 88, but which we now see to be deducible from the mathematical process by which Green’s function may be calculated.

If we assume any distribution of electricity whatever, and place in the field a point charged with unit of electricity, and if the surface of potential zero completely separates the point from the assumed distribution, then if we take this surface for the surface $S$, and the point for $P$, Green’s function, for any point on the same side of the surface as $P$, will be the potential of the assumed distribution on the other side of the surface. In this way we may construct any number of cases in which Green’s function can be