Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/396

the whole conductor will be increased if that of the portion is increased, and diminished if that of the portion be diminished.

This principle may be regarded as self-evident, but it may easily be shewn that the value of the expression for the resistance of a system of conductors between two points selected as electrodes, increases as the resistance of each member of the system increases.

It follows from this that if a surface of any form be described in the substance of the conductor, and if we further suppose this surface to be an infinitely thin sheet of a perfectly conducting substance, the resistance of the conductor as a whole will be diminished unless the surface is one of the equipotential surfaces in the natural state of the conductor, in which case no effect will be produced by making it a perfect conductor, as it is already in electrical equilibrium.

If therefore we draw within the conductor a series of surfaces, the first of which coincides with the first electrode, and the last with the second, while the intermediate surfaces are bounded by the non-conducting surface and do not intersect each other, and if we suppose each of these surfaces to be an infinitely thin sheet of perfectly conducting matter, we shall have obtained a system the resistance of which is certainly not greater than that of the original conductor, and is equal to it only when the surfaces we have chosen are the natural equipotential surfaces.

To calculate the resistance of the artificial system is an operation of much less difficulty than the original problem. For the resistance of the whole is the sum of the resistances of all the strata contained between the consecutive surfaces, and the resistance of each stratum can be found thus:

Let $dS$ be an element of the surface of the stratum, $\nu$ the thickness of the stratum perpendicular to the element, $\rho$ the specific resistance, $E$ the difference of potential of the perfectly conducting surfaces, and $dC$ the current through $dS,$ then

$dC=E{\frac {1}{\rho \nu }}\;dS,$

(1)

and the whole current through the stratum is

$C=E\iint {\frac {1}{\rho \nu }}\;dS,$

(2)

the integration being extended over the whole stratum bounded by the non-conducting surface of the conductor.