102
II
INDUCTION IN ROTATING SPHERES
becomes π/4i at the outer surface. The phenomenon is
identical with that which occurs in a non-magnetic sphere
with (1+470) times the velocity. The heat generated is
1+40 times that generated in a non-magnetic sphere
rotating with equal velocity.
§ 7. RELATED PROBLEMS.
In this paragraph we shall consider some problems which
stand in very close relation to those already treated.
I.
Any solid
tion.
If we neglect self-induction we may apply our knowledge
of revolu- of the currents in a sphere to find those in a solid of revolution
of
any form whatever, or at least to reduce their determination
to a simpler problem.
Let S be the surface of revolution bounding the solid, n
its inward normal. Describe about it a sphere of any radius.
Let u₁, v₁, w₁ be the currents which would flow in the latter,
and let
N = u₁ cos a + v₁ cos b + w₁ cos C
be the current in the direction of n at the surface S.
determine U2, V2, W2 so that
2'
If we
then clearly
KU₂ =
дх
KV 2
=
KW 2
Ju2+
дх
=
ay
--
Dz
aw 2
2v2+ dz
ду
2
=
0,
ucos a + v₂ cos b+w₂ cos c =
U₁ + U₂, VI+V2, W₁ + W₂
-N,