Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/124

Since the resultant force within a conductor is zero, the resultant force just outside the conductor is along the normal and is equal to $4\pi \sigma$, acting outwards from the conductor.

81.] If we now suppose an elongated body to be electrified, we may, by diminishing its lateral dimensions, arrive at the conception of an electrified line.

Let $ds$ be the length of a small portion of the elongated body, and let $c$ be its circumference, and $\sigma$ the superficial density of the electricity on its surface; then, if $\lambda$ is the electricity per unit of length, $\lambda =c\sigma$, and the resultant electrical force close to the surface will be

$4\pi \sigma =4\pi {\frac {\lambda }{c}}$

If, while $\lambda$ remains finite,$c$ be diminished indefinitely, the force at the surface will be increased indefinitely. Now in every dielectric there is a limit beyond which the force cannot be increased without a disruptive discharge. Hence a distribution of electricity in which a finite quantity is placed on a finite portion of a line is inconsistent with the conditions existing in nature.

Even if an insulator could be found such that no discharge could be driven through it by an infinite force, it would be impossible to charge a linear conductor with a finite quantity of electricity, for an infinite electromotive force would be required to bring the electricity to the linear conductor.

In the same way it may be shewn that a point charged with a finite quantity of electricity cannot exist in nature. It is convenient, however, in certain cases, to speak of electrified lines and points, and we may suppose these represented by electrified wires, and by small bodies of which the dimensions are negligible com pared with the principal distances concerned.

Since the quantity of electricity on any given portion of a wire diminishes indefinitely when the diameter of the wire is indefinitely diminished, the distribution of electricity on bodies of considerable dimensions will not be sensibly affected by the introduction of very fine metallic wires into the field, so as to form electrical connexions between these bodies and the earth, an electrical machine, or an electrometer.

On Lines of Force.

82.] If a line be drawn whose direction at every point of its course coincides with that of the resultant force at that point, the line is called a Line of Force.