Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/113

if we put $ds$ for the arc $AB$ we shall have for the force resolved in the direction of $ds$,

$R\,cos\epsilon \,=-{\frac {dV}{ds}}$;

whence, by assuming $ds$ parallel to each of the axes in succession, we get

$X=-{\frac {dV}{dx}},{\color {White}xxxx}Y=-{\frac {dV}{dy}},{\color {White}xxxx}Z=-{\frac {dV}{dx}}$;

$R={\left({\left.{\frac {\overline {dV}}{dx}}\right|}^{2}+{\left.{\frac {\overline {dV}}{dy}}\right|}^{2}+{\left.{\frac {\overline {dV}}{dz}}\right|}^{2}\right)}^{1 \over 2}$

We shall denote the force itself, whose magnitude is $R$ and whose components are $X,Y,Z$, by the German letter ${\mathfrak {C}}$, as in Arts. 17 and 68.

The Potential at all Points within a Conductor is the same.

72.] A conductor is a body which allows the electricity within it to move from one part of the body to any other when acted on by electromotive force. When the electricity is in equilibrium there can be no electromotive force acting within the conductor. Hence $R=0$ throughout the whole space occupied by the conductor. From this it follows that

${\frac {dV}{dx}}=0,{\color {White}xxx}{\frac {dV}{dy}}=0,{\color {White}xxx}{\frac {dV}{dz}}=0$;

and therefore for every point of the conductor

$V\;=\;C$,

where $C$ is a constant quantity.

Potential of a Conductor.

Since the potential at all points within the substance of the conductor is C, the quantity C is called the Potential of the conductor. C may be defined as the work which must be done by external agency in order to bring a unit of electricity from an infinite distance to the conductor, the distribution of electricity being supposed not to be disturbed by the presence of the unit.

If two conductors have equal potentials, and are connected by a wire so fine that the electricity on the wire itself may be neglected, the total electromotive force along the wire will be zero, and no electricity will pass from the one conductor to the other.

If the potentials of the conductors $A$ and $B$ be $V_{A}$ and $V_{B}$ then the electromotive force along any wire joining $A$ and $B$ will be

$V_{A}\;-\;V_{B}$