Page:A Dynamical Theory of the Electromagnetic Field.pdf/33

By the same demonstration as was used in the case of the mechanical action on a magnet, it may be shown that the mechanical force on a small body containing a quantity $e_{2}$ of free electricity placed in a field whose potential arising from other electrified bodies is $\Psi _{1}$, has for components

$\left.{\begin{array}{c}X=e_{2}{\frac {d\Psi _{1}}{dx}}=-P_{1}e_{2},\\\\Y=e_{2}{\frac {d\Psi _{1}}{dy}}=-Q_{1}e_{2},\\\\X=e_{2}{\frac {d\Psi _{1}}{dz}}=-R_{1}e_{2},\end{array}}\right\}$

(D)

So that an electrified body is urged in the direction of the electromotive force with a force equal to the product of the quantity of free electricity and the electromotive force.

If the electrification of the field arises from the presence of a small electrified body containing ex of free electricity, the only solution of $\Psi _{1}$ is

$\Psi _{1}={\frac {k}{4\pi }}{\frac {e_{1}}{r}}$

(43)

where $r$ is the distance from the electrified body.

The repulsion between two electrified bodies $e_{2}$, $e_{2}$ is therefore

$e_{2}{\frac {d\Psi _{1}}{dr}}={\frac {k}{4\pi }}{\frac {e_{1}e_{2}}{r^{2}}}$

(44)

Measurement of Electrical Phenomena by Electrostatic Effects.

(80) The quantities with which we have had to do have been hitherto expressed in terms of the Electromagnetic System of measurement, which is founded on the mechanical action between currents. The electrostatic system of measurement is founded on the mechanical action between electrified bodies, and is independent of, and incompatible with, the electromagnetic system; so that the units of the different kinds of quantity have different values according to the system we adopt, and to pass from the one system to the other, a reduction of all the quantities is required.

According to the electrostatic system, the repulsion between two small bodies charged with quantities $\eta _{1},\eta _{2}$ of electricity is

${\frac {\eta _{1}\eta _{2}}{r^{2}}}$

where $r$ is the distance between them.

Let the relation of the two systems be such that one electromagnetic unit of electricity contains $v$ electrostatic units; then $\eta _{1}=ve_{1}$ and $\eta _{2}=ve_{2}$, and this repulsion becomes

$v^{2}{\frac {e_{1}e_{2}}{r^{2}}}={\frac {k}{4\pi }}{\frac {e_{1}e_{2}}{r^{2}}}$ by equation (44), (45)

(45)

whence $k$, the coefficient of "electric elasticity" in the medium in which the experiments are made, i. e. common air, is related to $v$, the number of electrostatic units in one electromagnetic unit, by the equation

$k=4\pi v^{2}$

(46)