Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/414

If $r=\infty$, that is, if the parallelepipeds are perfect conductors,

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

In every case, provided $k_{1}=k_{2}=k_{3},$ it may be shewn that $r_{l},\,r_{2}$ and $r_{3}$ are in ascending order of magnitude, so that the greatest conductivity is in the direction of the longest dimensions of the parallelepipeds, and the greatest resistance in the direction of their shortest dimensions.

323.] In a rectangular parallelepiped of a conducting solid, let there be a conducting channel made from one angle to the opposite, the channel being a wire covered with insulating material, and let the lateral dimensions of the channel be so small that the conductivity of the solid is not affected except on account of the current conveyed along the wire.

Let the dimensions of the parallelepiped in the directions of the coordinate axes be $a,\,b,\,c,$ and let the conductivity of the channel, extending from the origin to the point $(abc),$ be $abcK.$

The electromotive force acting between the extremities of the channel is

$aX+bY+cZ,$

and if $C'$ be the current along the channel

$C'=Kabc(aX+bY+cZ).$

The current across the face be of the parallelepiped is $bcu,$ and this is made up of that due to the conductivity of the solid and of that due to the conductivity of the channel, or

or

${\begin{array}{rl}bcu=&bc(r_{1}X+p_{3}Y+q_{2}Z)+Kabc(aX+bY+cZ),\\u=&(r_{1}+Ka^{2})X+(p_{3}+Kab)Y+(q_{2}+Kca)Z.\end{array}}$

In the same way we may find the values of $v$ and $w.$ The coefficients of conductivity as altered by the effect of the channel will be

${\begin{array}{lll}r_{1}+Ka^{2},&r_{2}+Kb^{2},&r_{3}+Kc^{2},\\p_{1}+Kbc,&p_{2}+Kca,&p_{3}+Kab,\\q_{1}+Kbc,&q_{2}+Kca,&q_{3}+Kab.\end{array}}$

In these expressions, the additions to the values of $p_{1},$ &c., due to the effect of the channel, are equal to the additions to the values of $q_{1},$ &c. Hence the values of $p_{1}$ and $q_{1}$ cannot be rendered unequal by the introduction of linear channels into every element of volume of the solid, and therefore the rotatory property of Art. 303, if it does not exist previously in a solid, cannot be introduced by such means.