Page:Elementary Text-book of Physics (Anthony, 1897).djvu/322

308 throughout the tube in such a way that each unit of length of the tube contains the energy $$\tfrac{1}{2}qF.$$ If the tube be a unit tube so that $$q = 1,$$ each unit of length of this tube will have in it a quantity of energy equal to $$\tfrac{1}{2}F.$$

To find the energy in unit volume of the dielectric, we consider a small cylinder, its height $$l$$ being taken along the lines of force and its base $$s$$ normal to them. The number of unit tubes which pass through the base is $$Ns.$$ Since the energy in unit length of each of these tubes is $$\tfrac{1}{2}F,$$ and since therefore the energy in the length $$l$$ is $$\tfrac{1}{2}Fl,$$ we have the energy in the volume $$ls$$ equal to $$\tfrac{1}{2}FNls,$$ or the energy in unit volume equal to $$\tfrac{1}{2} FN.$$ Now we have $$F = \frac{4\pi N}{K},$$ so that the energy in unit volume is $$\frac{2\pi N^2}{K} = \frac{KF^2}{8\pi}\cdot$$

By comparing this result with the value obtained for the tension across unit area it appears that the tension across unit area and the energy of unit volume are numerically equal. They both vary from point to point in the dielectric, depending upon the electrical force at each point. Unless the force is appreciably constant for all points of a finite region, the actual tension across a unit area and the actual energy of unit volume will not be given accurately by these expressions: they are more strictly the limits of the ratios between the tension and the area on which it acts, and the energy and the volume containing it.

268. Forces on Electrified Bodies.—It has already been stated that the stresses between charges may be represented by supposing that the tubes of force exert a tension along the lines of force and an equal pressure in all directions perpendicular to the lines of force, or as may be said, the lines of force tend to diminish in length and to repel each other. This mode of conceiving the stresses between charged bodies may be illustrated in some simple cases without the aid of diagrams of lines of force. The lines of force around a uniformly charged sphere are radial and the tubes of force are similar cones; if the sphere be charged positively, the force is directed outward from it, and if charged negatively, is