Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/401

 

netic measure, and $b$ the distance between them. If we make $b=2a$, then

$F=CC^{\prime }$.

Now the quantity of electricity transmitted by the current $C$ in the time $t$ is $Ct$ in electromagnetic measure, or $nCt$ in electrostatic measure, if $n$ is the number of electrostatic units in one electromagnetic unit.

Let two small conductors be charged with the quantities of electricity transmitted by the two currents in the time $t$, and placed at a distance $r$ from each other. The repulsion between them will be

$F^{\prime }={\frac {CC^{\prime }n^{2}t^{2}}{r^{2}}}$.

Let the distance $r$ be so chosen that this repulsion is equal to the attraction of the currents, then

${\frac {CC^{\prime }n^{2}t^{2}}{r^{2}}}=CC^{\prime }$.

Hence

$r=nt$;

or the distance $r$ must increase with the time $t$ at the rate $n$. Hence $n$ is a velocity, the absolute magnitude of which is the same, whatever units we assume.

769.] To obtain a physical conception of this velocity, let us imagine a plane surface charged with electricity to the electrostatic surface-density $\sigma$, and moving in its own plane with a velocity $\nu$. This moving electrified surface will be equivalent to an electric current-sheet, the strength of the current flowing through unit of breadth of the surface being $\sigma \nu$ in electrostatic measure, or ${\frac {1}{n}}\sigma \nu$ in electromagnetic measure, if $n$ is the number of electrostatic units in one electromagnetic unit. If another plane surface, parallel to the first, is electrified to the surface-density $\sigma ^{\prime }$, and moves in the same direction with the velocity $\nu ^{\prime }$, it will be equivalent to a second current-sheet.

The electrostatic repulsion between the two electrified surfaces is, by Art. 124, $2\pi uu^{\prime }$ for every unit of area of the opposed surfaces.

The electromagnetic attraction between the two current-sheets is, by Art. 653, $2\pi uu^{\prime }$ for every unit of area, $u$ and $u^{\prime }$ being the surface-densities of the currents in electromagnetic measure.

But $u=-{\frac {1}{n}}\sigma \nu$, and $u^{\prime }={\frac {1}{n}}\sigma ^{\prime }\nu ^{\prime }$, so that the attraction is

$2\pi \sigma \sigma ^{\prime }{\frac {\nu \nu ^{\prime }}{n^{2}}}$.