III
131
DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS
=
surface, op/on, x(Əh/ǝt)+(2π/T)(ah/dw), where h now refers
to a point fixed in space.
When the conductor rotates with
uniform velocity under the influence of a potential independent
of the time, after the lapse of a certain time a stationary state
is reached, the condition for which is ǝh/t0; and thus
ƏÞ¡/ən¸ = (2π/T)/(Əh/Əw).
As an example we shall consider the case of a spherical
shell rotating with constant velocity about a diameter. Let
its external radius be R, its internal radius r. Suppose the
external potential , under whose influence the motion takes
place, developed in a series of spherical harmonics inside the
spherical shell. The actions produced by the separate terms
may be added, so that we may limit the investigation to one
term. Let Þ= Ani(p/R)" cos iwPni(0). Denote by the
potential of the electrical charge itself, which is induced on
the spherical shell; in particular denote it by 4, in the inside
space, by 2 in the substance of the shell, by 3 in the outside
space. In addition to the general conditions for the potential
of electrical charges, & must satisfy the condition that for
p=r and p = R
$1
=
R
Φι
=
R
ӘФ, Әфі
+ = -
K
ә/дфі
афе
-
др др 2Tow ap ap
All these requirements are fulfilled when we put
n
Acosiw+Bsiniw)P(0)+(2)(A'cosiw+B'siniw)P„(0),
r
n
n+1
(Acosia+Bsiniw)Pri(0) +
T
(A'cosio + B'siniw)Pni(0),
=
$37
R2+1
n+1
(Acosiw+Bsiniw)Pri(0) + ((A'cosiw+B'siniw)Pri(0).
For the general conditions are at once satisfied, and the