# Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/117

76.] ELECTRIC INDUCTION. 77

this condition is shewn to be fulfilled by the electric forces with the most perfect accuracy. Hence the law of electric force is verified to a corresponding degree of accuracy.

### Surface-Integral of Electric Induction, and Electric Displacement through a Surface.

75.] Let ${\displaystyle R}$ be the resultant force at any point of the surface, and ${\displaystyle \epsilon }$ the angle which R makes with the normal drawn towards the positive side of the surface, then ${\displaystyle Rcos\epsilon }$ is the component of the force normal to the surface, and if ${\displaystyle dS}$ is the element of the surface, the electric displacement through ${\displaystyle dS}$ will be, by Art. 68,

 ${\displaystyle {\frac {1}{4\pi }}\,KR\,cos\epsilon \,dS}$

Since we do not at present consider any dielectric except air, ${\displaystyle K=1}$ .

We may, however, avoid introducing at this stage the theory of electric displacement, by calling ${\displaystyle Rcos\epsilon dS}$ the Induction through the element ${\displaystyle dS}$. This quantity is well known in mathematical physics, but the name of induction is borrowed from Faraday. The surface-integral of induction is

 ${\displaystyle \iint Rcos\epsilon dS}$;

and it appears by Art. 21, that if ${\displaystyle X,Y,Z}$ are the components of ${\displaystyle R}$, and if these quantities are continuous within a region bounded by a closed surface ${\displaystyle S}$, the induction reckoned from within outwards is

 ${\displaystyle \iint Rcos\epsilon dS=\iiint \left({\frac {}{}}+{\frac {}{}}+{\frac {}{}}\right)dx\,dy\,dz}$,

the integration being extended through the whole space within the surface.

### Induction through a Finite Closed Surface due to a Single Centre of Force.

76.] Let a quantity e of electricity be supposed to be placed at a point ${\displaystyle 0}$, and let ${\displaystyle r}$ be the distance of any point ${\displaystyle P}$ from ${\displaystyle 0}$, the force at that point is ${\displaystyle R={\frac {e}{r^{2}}}}$ in the direction ${\displaystyle OP}$.

Let a line be drawn from ${\displaystyle O}$ in any direction to an infinite distance. If ${\displaystyle O}$ is without the closed surface this line will either not cut the surface at all, or it will issue from the surface as many times as it enters. If ${\displaystyle O}$ is within the surface the line must first