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

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

Now the quantity of electricity transmitted by the current $$C$$ in the time $$t$$ is $$Ct$$ in electromagnetic measure, or $$n Ct$$ 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

Let the distance $$r$$ be so chosen that this repulsion is equal to the attraction of the currents, then 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, $$2\pi u u^\prime$$ for every unit of area of the opposed surfaces.

The electromagnetic attraction between the two current-sheets is, by Art. 653, $$2 \pi u u^\prime$$ 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