III
133
DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS
force of that potential. The free electricity, which by the
currents is brought to the boundary, is by the rotation of the
sphere carried back to its starting-point so quickly that the
density remains infinitely small. A screening of the internal
space no longer takes place.
In particular cases the calculation itself becomes very
simple. In the first place, for a solid sphere = 0. If we
put tan 8=(2n+1)ai/n, then Ɛ/i is the angle through which
the distribution appears to be turned, and the density of the
charge is to that induced on the sphere at rest as cos 8: 1.
When the sphere rotates under the influence of a uniform
force perpendicular to the axis of rotation, the distribution on
it is represented by a spherical harmonic of the first degree.
The lines of flow are parallel straight lines whose direction for
small velocities of rotation is perpendicular to the axis and
to the direction of the force, but for large velocities appears
turned from the latter direction through an angle whose
tangent 3a/T. For a rotating cylinder the circumstances
are quite similar; the angle of rotation is here found to be
2a = x/T.
=
Secondly, suppose e nearly unity, that is, the thickness d
of the spherical shell infinitely small. We must then suppose
the specific resistance x to be so small that x/α = k, the specific
superficial resistance, may be a finite quantity. With this
assumption the tangent of the angle of rotation becomes
(2n+1)i R
generally tand
=
2n(n+1) T
=3
a uniform force tand = 2R/T.
=
and in the particular case of
Under similar circumstances we find for a thin hollow
cylinder tand R/T, so that in this case the rotation is
greater for the cylinder than for the sphere, although for a
solid cylinder it was less. The density in the last case also
is to that for the sphere in the ratio cos 8:1.
As an illustration of the results of the calculation I have
in the accompanying diagram represented the flow of electricity
in a rotating hollow cylinder, whose internal is one-half its
external radius. The time of one turn is twice the specific
resistance of the material. The arrow A marks the direction
of the external inducing force, the arrow B that of the force
in the inside space; the remaining two arrows indicate the