Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/387

CHAPTER VIII.

RESISTANCE AND CONDUCTIVITY IN THREE DIMENSIONS.

On the most General Relations between Current and Electromotive Force.

297.] Let the components of the current at any point be $u,\,v,\,w.$

Let the components of the electromotive force be $X,\,Y,\,Z.$

The electromotive force at any point is the resultant force on a unit of positive electricity placed at that point. It may arise (1) from electrostatic action, in which case if $V$ is the potential,

$X=-{\frac {dV}{dx}},\quad Y=-{\frac {dV}{dy}},\quad Z=-{\frac {dV}{dz}};$

(1)

or (2) from electromagnetic induction, the laws of which we shall afterwards examine; or (3) from thermoelectric or electrochemical action at the point itself, tending to produce a current in a given direction.

We shall in general suppose that $X,\,Y,\,Z$ represent the components of the actual electromotive force at the point, whatever be the origin of the force, but we shall occasionally examine the result of supposing it entirely due to variation of potential.

By Ohm's Law the current is proportional to the electromotive force. Hence $X,\,Y,\,Z$ must be linear functions of $u,\,v,\,w.$ We may therefore assume as the equations of Resistance,

$\left.{\begin{matrix}X&=&R_{1}u+Q_{3}v+P_{2}w,\\Y&=&P_{3}u+R_{2}v+Q_{1}w,\\Z&=&Q_{2}u+P_{1}v+R_{3}w.\end{matrix}}\right\}$

(2)

We may call the coefficients $R$ the coefficients of longitudinal resistance in the directions of the axes of coordinates.

The coefficients $P$ and $Q$ may be called the coefficients of transverse resistance. They indicate the electromotive force in one direction required to produce a current in a different direction.