Page:On Faraday's Lines of Force.pdf/36

In order to express mathematically the electrical currents in any conductor, we must have a definition, not only of the entire flow across a complete section, but also of the flow at a given point in a given direction.

DEF. The quantity of a current at a given point and in a given direction is measured, when uniform, by the quantity of electricity which flows across unit of area taken at that point perpendicular to the given direction, and when variable by the quantity which would flow across this area, supposing the flow uniformly the same as at the given point.

In the following investigation, the quantity of electric current at the point $(xyz)$ estimated in the directions of the axes $x,y,z$ respectively will be denoted by $a_{2},b_{2},c_{2}$.

The quantity of electricity which flows in unit of time through the elementary area $dS$

$=dS(la_{2}+mb_{2}+nc_{2})$,

where $l,m,n$ are the direction-cosines of the normal to $dS$.

This flow of electricity at any point of a conductor is due to the electromotive forces which act at that point. These may be either external or internal.

External electro-motive forces arise either from the relative motion of currents and magnets, or from changes in their intensity, or from other causes acting at a distance.

Internal electro-motive forces arise principally from difference of electric tension at points of the conductor in the immediate neighbourhood of the point in question. The other causes are variations of chemical composition or of temperature in contiguous parts of the conductor.

Let $p_{2}$ represent the electric tension at any point, and $X_{2},Y_{2},Z_{2}$ the sums of the parts of all the electro-motive forces arising from other causes resolved parallel to the co-ordinate axes, then if $\alpha _{2},\beta _{2},\gamma _{2}$, be the effective electro-motive forces

$\alpha _{2}=X_{2}-{\frac {dp_{2}}{dx}}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ ....................(A).
$\beta _{2}=Y_{2}-{\frac {dp_{2}}{dy}}$
$\gamma _{2}=Z_{2}-{\frac {dp_{2}}{dz}}$