Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/112

On Potential Functions.

70.] The quantity $Xdx+Ydy+Zdz$ is an exact differential whenever the force arises from attractions or repulsions whose in tensity is a function of the distance only from any number of points. For if $r_{1}$ be the distance of one of the points from the point $(x,y,z)$, and if $R_{1}$ be the repulsion, then

$X_{1}=R_{1}l=R_{1}{\frac {dr_{1}}{dx}}$,

with similar expressions for $Y_{1}$ and $Z_{1}$, so that

$X_{1}\,dx+Y_{1}\,dy+Z_{1}\,dz=R_{1}\,dr_{1}$;

and since $R_{l}$ is a function of $r_{l}$ only, $R_{l}dr_{1}$ is an exact differential of some function of $r_{1}$, say $V_{1}$.

Similarly for any other force $R_{2}$, acting from a centre at distance $r_{2}$,

$X_{2}\,dx+Y_{2}\,dy+Z_{2}\,dz=R_{2}\,dr_{2}=dV_{2}$.

But $X=X_{1}+X_{2}+{\mbox{etc. and }}Y$ and $Z$ are compounded in the same way, therefore

$X\,dx+Y\,dy+Z\,dz=dV_{1}+dV_{2}+\And \!\!c.=dV$.

$V$, the integral of this quantity, under the condition that $V=0$ at an infinite distance, is called the Potential Function.

The use of this function in the theory of attractions was introduced by Laplace in the calculation of the attraction of the earth. Green, in his essay 'On the Application of Mathematical Analysis to Electricity' gave it the name of the Potential Function. Gauss, working independently of Green, also used the word Potential. Clausius and others have applied the term Potential to the work which would be done if two bodies or systems were removed to an infinite distance from one another. We shall follow the use of the word in recent English works, and avoid ambiguity by adopting the following definition due to Sir W. Thomson.

Definition of Potential. The Potential at a Point is the work which would be done on a unit of positive electricity by the electric forces if it were placed at that point without disturbing the electric distribution, and carried from that point to an infinite distance.

71.] Expressions for the Resultant Force and its components in terms of the Potential.

Since the total electromotive force along any arc $AB$ is

$V_{A}-V_{B}\,\!$,