Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/114

in the direction $AB$, that is, positive electricity will tend to pass from the conductor of higher potential to the other.

Potential, in electrical science, has the same relation to Electricity that Pressure, in Hydrostatics, has to Fluid, or that Temperature, in Thermodynamics, has to Heat. Electricity, Fluids, and Heat all tend to pass from one place to another, if the Potential, Pressure, or Temperature is greater in the first place than in the second. A fluid is certainly a substance, heat is as certainly not a substance, so that though we may find assistance from analogies of this kind in forming clear ideas of formal electrical relations, we must be careful not to let the one or the other analogy suggest to us that electricity is either a substance like water, or a state of agitation like heat.

Potential due to any Electrical System.

73.] Let there be a single electrified point charged with a quantity $e$ of electricity, and let $r$ be the distance of the point $x',y',z'$ from it, then

$V=\int _{r}^{\infty }R\,dr=\int _{r}^{\infty }{\frac {e}{r^{2}}}dr={\frac {e}{r}}$.

Let there be any number of electrified points whose coordinates are $(x_{1},y_{1},z_{1})$,$(x_{2},y_{2},z_{2})$ &c. - and their charges $e_{1},e_{2}$ &c., and let their distances from the point $(x',y',z')$ be $r_{1}$, $r_{2}$, &c., then the potential of the system at $x',y',z'$' will be

$V=\sum {\left({\frac {e}{r}}\right)}$.

Let the electric density at any point $(x,y,z)$ within an electrified body be p, then the potential due to the body is

$V=\iiint {\frac {\rho }{r}}dx\,dy\,dz$;

where

$r=\{(x-x')^{2}+(y-y')^{2}+(z-z')^{2}\}^{1 \over 2}$

the integration being extended throughout the body.

On the Proof of the Law of the Inverse Square.

74.] The fact that the force between electrified bodies is inversely as the square of the distance may be considered to be established by direct experiments with the torsion-balance. The results, however, which we derive from such experiments must be regarded as affected by an error depending on the probable error of each experiment, and unless the skill of the operator be very great,