Page:Elementary Text-book of Physics (Anthony, 1897).djvu/81

the surface contains no mass, the total flux of force through it equals zero. But the flux of force through the sphere equals $-4\pi m$, the minus sign being used because the normals to the sphere, when considered as bounding the region enclosing the mass and as bounding the region between it and the closed surface, have opposite directions. Therefore the flux of force through the closed surface must equal $4\pi m$. This proposition holds for each of the masses contained within the closed surface, so that, if the sum of these masses heM,the total flux of force through the closed surface is $4\pi M$.

Let us apply the theorem just proved to the region bounded by a tube of force of very small cross-section and by two rectangular cross-sections of the tube. Since the tube of force is everywhere bounded by lines of force, and since, therefore, the force at a point on the tube has no component normal to the surface of the tube, the only parts of the closed surface under consideration which contribute to the flux of force through it are the two end cross-sections. Represent the areas of the two cross-sections by $s$ and $s'$, and the forces acting at them by $F$ and $F'$ respectively. Then, since the total flux of force equals zero, we have $Fs=F's'=0$ or $Fs=-F's'$. The minus sign appears because the force and the normal to the cross-section are in the same direction at one end of the tube and in opposite directions at the other. If we confine our attention to the numerical value of the product $Fs$, we may say that the flux of force is the same for all cross-sections of the tube of force. This proposition, though here proved only for a tube of force of very small cross-section, manifestly may be generalized for any tube of force whatever.

57. Special Cases.—It is sometimes important, especially in the study of electricity, to know the force which is exerted by a plane sheet of matter at a point near it. We call the quantity of matter which is enclosed by a unit area drawn on such a sheet the surface density of the sheet at the point where the area is taken; more strictly, the surface density is the ratio of the quantity of matter enclosed by the area to the magnitude of the area, as the area di-