82
II
INDUCTION IN ROTATING SPHERES
Summary
of the
result.
Case where
the mag-
nets are
The meaning of the above formula is easily grasped. If
we collect together its result and the results previously obtained
we may describe the phenomena, which would be presented by
a spherical shell rotating with constantly increasing velocity
under the influence of an inducing spherical harmonic function,
in the following terms:-
When self-induction begins to be appreciable, it does not
alter the form of the lines of flow in the various spherical
layers, but these latter commence to undergo an apparent
rotation in the direction of rotation; and then the inner
layers gain on the outer ones. There is no limit to the
rotation of the inner layers; it may increase indefinitely.
The angle of rotation of the outermost layer converges to the
value π/4i; moreover, for spherical shells it may in the first
instance have exceeded this value. If the velocity of rotation
be very great, corresponding points of the different layers lie
on spirals of Archimedes, and the number of turns which
these make in the sphere increases indefinitely with the velocity
of rotation.
At first the intensity increases with the velocity of rota-
tion, but nowhere proportionately to it; more quickly in the
outer than in the inner layers. In the outermost layer it
constantly increases, ultimately as ; in the other layers
it reaches a maximum for some definite velocity and then
decreases. For large velocities it decreases inwards from the
surface in proportion to an exponential, whose argument has
o for a factor.
It is of interest to note also the dependence of the pheno-
menon on the order i (whose square root is involved in µ);
for this I refer to the formulæ.
An apparent contradiction between the theory of an in-
finitely thin spherical shell and that of one of finite thickness
may excite notice; it is easily explained when we consider
that every spherical shell, however thin, may only be regarded
as infinitely thin up to a certain value of the velocity of
rotation.
I shall deal shortly with the case where the inducing
magnets are inside the spherical shell, so that spherical har-
inside the monics of negative degree occur.
shell.