130
III
DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS
It is easy to prove in our
the surface, of all the conductors.
particular case that the equations used agree with the principle
of the conservation of energy, which has, however, been proved
true of them in general.
If x be very small, & may be expanded in ascending powers
of K.
The individual terms of this expansion may be found
in the following way, if we regard the ordinary electrostatic
problem as solved.
=
=
Let be for all time the potential corresponding to the
state of equilibrium for the existing charges and the positions
of the bodies at each instant, and let h, be the density cor-
responding to . Then let o be determined so that V²₂ = 0,
that at the surfaces of the inductors ap₂/ən, kəh,/at, that the
conditions of continuity are satisfied, and that the sum of the
free electricity may vanish for each conductor. In the same
way in which is formed from ₁, let ø, be formed from 2,
from 3, and so on; then clearly ₁+ 2+ 3+... re-
presents exactly the potential, provided the series converges.
The convergence of the series depends on the relation between
K, the dimensions of the conductors, and their velocities; for
any value of we can imagine velocities sufficiently small to
ensure convergence. For metallic conductors and terrestrial
velocities each term vanishes in comparison with the preced-
ing one. The special phenomena due to electrical resistance
are here inappreciable, and the form of the currents alone is
of interest. Since is constant inside a conductor, and s
vanishes in comparison with 42, all the currents flow along
the lines of force of the potential 42, and we have
ки=
ap2
дх
KV =
-
$
მყ
2
Әфг
KW =
>
We shall now confine ourselves to the case in which only
one conductor is in motion, and shall assume this to be a solid
of revolution rotating about its axis. We refer our investiga-
tion to a system of coordinates fixed in space, of which the
z-axis is the axis of rotation. In addition, we employ polar
coordinates p, w, with the same axis. Let T be the time
of one turn. The conditions which must satisfy in the
conductor are in this case: (1) inside V² = 0; (2) at the