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

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


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


Since we do not at present consider any dielectric except air, K= 1 .

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


\iint R  cos \epsilon dS;


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


\iint R  cos \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 0, and let r be the distance of any point P from 0, the force at that point is R=\frac{e}{r^2} in the direction OP.

Let a line be drawn from O in any direction to an infinite distance. If 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 O is within the surface the line must first