Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/397

Hence the conductivity of the stratum is

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

(3)

and the resistance of the stratum is the reciprocal of this quantity.

If the stratum be that bounded by the two surfaces for which the function $F$ has the values $F$ and $F+dF$ respectively, then

${\frac {dF}{\nu }}=\nabla F=\left[\left({\frac {dF}{dx}}\right)^{2}+\left({\frac {dF}{dy}}\right)^{2}+\left({\frac {dF}{dz}}\right)^{2}\right]^{\frac {1}{2}},$

(4)

and the resistance of the stratum is

${\frac {dF}{\iint {\frac {1}{\rho }}\nabla F\,ds}}.$

(5)

To find the resistance of the whole artificial conductor, we have only to integrate with respect to $F,$ and we find

$R_{1}=\int {\frac {dF}{\iint {\frac {1}{\rho }}\nabla F\,ds}}.$

(6)

The resistance $R$ of the conductor in its natural state is greater than the value thus obtained, unless all the surfaces we have chosen are the natural equipotential surfaces. Also, since the true value of $R$ is the absolute maximum of the values of $R_{1}$ which can thus be obtained, a small deviation of the chosen surfaces from the true equipotential surfaces will produce an error of $R$ which is comparatively small.

This method of determining a lower limit of the value of the resistance is evidently perfectly general, and may be applied to conductors of any form, even when $\rho ,$ the specific resistance, varies in any manner within the conductor.

The most familiar example is the ordinary method of determining the resistance of a straight wire of variable section. In this case the surfaces chosen are planes perpendicular to the axis of the wire, the strata have parallel faces, and the resistance of a stratum of section $S$ and thickness $ds$ is

$dR_{1}={\frac {\rho ds}{S}},$

(7)

and that of the whole wire of length $s$ is

$dR_{1}=\int {\frac {\rho ds}{S}},$

(8)

where $S'$ is the transverse section and is a function of $s.$