ON ELECTRIC DENSITY.
Distribution in Three Dimensions.
64.] Definition. The electric volume-density at a given point in space is the limiting ratio of the quantity of electricity within a sphere whose centre is the given point to the volume of the sphere, when its radius is diminished without limit.
We shall denote this ratio by the symbol , which may be positive or negative.
Distribution on a Surface.
It is a result alike of theory and of experiment, that, in certain cases, the electrification of a body is entirely on the surface. The density at a point on the surface, if defined according to the method given above, would be infinite. We therefore adopt a different method for the measurement of surface-density.
Definition. The electric density at a given point on a surface is the limiting ratio of the quantity of electricity within a sphere whose centre is the given point to the area of the surface contained within the sphere, when its radius is diminished without limit.
We shall denote the surface-density by the symbol .
Those writers who supposed electricity to be a material fluid or a collection of particles, were obliged in this case to suppose the electricity distributed on the surface in the form of a stratum of a certain thickness , its density being , or that value of which would result from the particles having the closest contact of which they are capable. It is manifest that on this theory
When is negative, according to this theory, a certain stratum of thickness is left entirely devoid of positive electricity, and filled entirely with negative electricity.
There is, however, no experimental evidence either of the electric stratum having any thickness, or of electricity being a fluid or a collection of particles. We therefore prefer to do without the symbol for the thickness of the stratum, and to use a special symbol for surface-density.
Distribution along a Line.
It is sometimes convenient to suppose electricity distributed on a line, that is, a long narrow body of which we neglect the
