Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/108

thickness. In this case we may define the line-density at any point to be the limiting ratio of the electricity on an element of the line to the length of that element when the element is diminished without limit.

If $\lambda$ denotes the line-density, then the whole quantity of electricity on a curve is $e=\int \lambda \,ds$, where dS is the element of the curve.

Similarly, if $\rho$ is the surface-density, the whole quantity of electricity on the surface is

$e=\iint \sigma \,dS$,

where dS is the element of surface.

If $\rho$ is the volume-density at any point of space, then the whole electricity within a certain volume is

$e=\iiint \rho \;dx\;dy\;dz$

where $dxdydz$ is the element of volume. The limits of integration in each case are those of the curve, the surface, or the portion of space considered.

It is manifest that e, $\lambda$, $\sigma$ and $\rho$ are quantities differing in kind, each being one dimension in space lower than the preceding, so that if $a$ be a line, the quantities e, $a\lambda ,a^{2}\sigma$ and $a^{3}\rho$ will be all of the same kind, and if a be the unit of length, and $\lambda ,\sigma ,\rho$ each the unit of the different kinds of density, $a\lambda ,a^{2}\sigma$ and $a^{3}\rho$will each denote one unit of electricity.

Definition of the Unit of Electricity.

65.] Let $A$ and $B$ be two points the distance between which is the unit of length. Let two bodies, whose dimensions are small compared with the distance $AB$, be charged with equal quantities of positive electricity and placed at $A$ and $B$ respectively, and let the charges be such that the force with which they repel each other is the unit of force, measured as in Art. 6. Then the charge of either body is said to be the unit of electricity. If the charge of the body at $B$ were a unit of negative electricity, then, since the action between the bodies would be reversed, we should have an attraction equal to the unit of force.

If the charge of $A$ were also negative, and equal to unity, the force would be repulsive, and equal to unity.

Since the action between any two portions of electricity is not