Page:Elementary Text-book of Physics (Anthony, 1897).djvu/82

68 indefinitely. If we consider an infinitely extended plane sheet, it is evident that the lines of force in the region near it are perpendicular to its surface. Take any small area on the surface of the sheet, and consider the closed surface bounded by the lines of force which pass through the boundary of that area and by two cross-sections taken parallel with the sheet on the opposite sides of it. The ilux of force through the sides of the surface thus formed is zero, because the lines of force lie in that surface. The only portions of the surface, therefore, which contribute to the flux of force, are the end cross-sections. Let $$s$$ represent the area of each of these cross-sections, which are equal, $$F$$ the force at one of them, and $$F'$$ that at the other. If $$\sigma$$ represent the surface density of the sheet, $$\sigma s$$ is the mass enclosed within the closed surface. Applying the theorem of the flux of force, we have $$(F + F')s = 4 \pi \sigma s$$ or $$F + F' = 4 \pi \sigma$$. Remembering that the directions of these forces are outward from the closed surface, and that therefore $$+F$$ and $$-F'$$ are forces drawn in the same direction along the lines of force, this equation shows that in passing through a sheet of surface density $$\sigma$$ the force changes by $$4\pi \sigma$$. If the forces in the field be due only to the sheet, it is manifest, from symmetry, that the force $$F$$ and the force $$F'$$ are equal, and that their directions are opposite. We thus have $$F + F' = 2F = 4\pi \sigma$$, or $$F = 2\pi \sigma$$. That is, the force at a point infinitely near a plane sheet of surface density $$\sigma$$ is equal to $$2\pi \sigma$$. This proposition holds, even if the sheet be not plane, for any points so near it that the neighboring lines of force are parallel.

The force within a closed spherical shell of uniform surface density vanishes at every point. For, let us construct a sphere in the region contained by the shell and concentric with it. Since no matter is contained by this sphere, the total flux of force through its surface is zero, and since, by symmetry, the force at any point on the inner sphere must have the same value and the same direction to or from the centre, it follows that $$\sum Fs = F 4 \pi r^2 = 0$$, and hence that $$F = 0$$. The force, therefore, vanishes for all points in the interior of the shell. It manifestly vanishes also within a