Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/380

CHAPTER VII.

CONDUCTION IN THREE DIMENSIONS.

Notation of Electric Currents.

285.] At any point let an element of area $dS$ be taken normal to the axis of $x$, and let $Q$ units of electricity pass across this area from the negative to the positive side in unit of time, then, if ${\frac {Q}{dS}}$ becomes ultimately equal to $u$ when $dS$ is indefinitely diminished, $u$ is said to be the Component of the electric current in the direction of $x$ at the given point.

In the same way we may determine $v$ and $w$, the components of the current in the directions of $y$ and $z$ respectively.

286.] To determine the component of the current in any other direction $OR$ through the given point $O$.

Fig. 22.

Let $l,m,n$ be the direction-cosines of $OR$, then cutting off from the axes of $x,y,z$ portions equal to

${\frac {r}{l}},{\frac {r}{m}},\;\;{\mbox{ and }}\;\;{\frac {r}{n}}$

respectively at $A,B,\mathrm {and} C,$ the triangle $ABC$ will be normal to $OR.$

The area of this triangle $ABC$ will be

$ds={\frac {1}{2}}{\frac {r^{2}}{lmn}}$

and by diminishing $r$ this area may be diminished without limit.

The quantity of electricity which leaves the tetrahedron $ABCO$ by the triangle $ABC$ must be equal to that which enters it through the three triangles $OBC,OCA,$ and $OAB.$

The area of the triangle $OBC$ is ${\frac {1}{2}}{\frac {r^{2}}{mn}},$ and the component of