Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/413

If we begin with both substances isotropic, but of different conductivities, then the result of stratification will be to make the resistance greatest in the direction of a normal to the strata, and the resistance in all directions in the plane of the strata will be equal.

322.] Take an isotropic substance of conductivity $r,$ cut it into exceedingly thin slices of thickness $a,$ and place them alternately with slices of a substance whose conductivity is $s,$ and thickness $k_{1}a.$

Let these slices be normal to $x.$ Then cut this compound conductor into thicker slices, of thickness $b,$ normal to $y,$ and alternate these with slices whose conductivity is $s$ and thickness $k_{2}b.$

Lastly, cut the new conductor into still thicker slices, of thickness $c,$ normal to $z.$ and alternate them with slices whose conductivity is $s$ and thickness $k_{3}c.$

The result of the three operations will be to cut the substance whose conductivity is $r$ into rectangular parallelepipeds whose dimensions are $a,\,b$ and $c,$ where $b$ is exceedingly small compared with $c,$ and $a$ is exceedingly small compared with $b,$ and to embed these parallelepipeds in the substance whose conductivity is $s,$ so that they are separated from each other $k_{1}a$ in the direction of $x,\,k_{2}b$ in that of $y,$ and $k_{3}c$ in that of $z.$ The conductivities of the conductor so formed in the directions of $x,\,y$ and $z$ are

${\begin{array}{l}r_{1}={\frac {\{1+k_{1}(1+k_{2})(1+k_{3})\}r+(k_{2}+k_{3}+k_{2}k_{3})s}{(1+k_{2})(1+k_{3})(k_{1}r+s)}}s,\\r_{2}={\frac {(1+k_{2}+k_{2}k_{3})r+(k_{1}+k_{3}+k_{1}k_{2}+k_{1}k_{3}+k_{1}k_{2}k_{3})s}{(1+k_{3})\{k_{2}r+(1+k_{1}+k_{1}k_{2})s\}}}s,\\r_{3}={\frac {(1+k_{3})(r+(k_{1}+k_{2}+k_{1}k_{2})s)}{k_{3}r+(1+k_{1}+k_{2}+k_{2}k_{3}+k_{3}k_{1}+k_{1}k_{2}+k_{1}k_{2}k_{3})s}}s.\end{array}}$

The accuracy of this investigation depends upon the three dimensions of the parallelepipeds being of different orders of magnitude, so that we may neglect the conditions to be fulfilled at their edges and angles. If we make $k_{l},\,k_{2}$ and $k_{3}$ each unity, then

$r_{1}={\frac {5r+3s}{4r+4s}}s,\quad \quad r_{2}={\frac {3r+5s}{2r+6s}}s,\quad \quad r_{3}={\frac {2r+6s}{r+7s}}s.$

If $r=0,$ that is, if the medium of which the parallelepipeds are made is a perfect insulator, then

$r_{1}={\frac {3}{4}}s,\quad \quad r_{2}={\frac {5}{6}}s,\quad \quad r_{3}={\frac {6}{7}}s.$