80
II
INDUCTION IN ROTATING SPHERES
angle through which the lines of flow are turned. Retaining
only lowest powers we find
tan
-1f2=8=
fi
2πωί
R2
p²
-
K
2n+1 2n +3
so that the angle in question is
8
=
2πω R2
K 2n+1
2
p²
2n +3
Thus all the layers appear rotated: the rotation is least at
the surface of the sphere and increases continuously inwards.
If we imagine a plane section taken through the equator of
the sphere and join corresponding points of the different
layers we get a system of congruent curves, which is very
suitable for representing the state of the sphere. The equa-
tion of one of these curves clearly is
y = x tan
¿› p= √x² + y²,
or very approximately
2πω
R2
y =
x2
-
K
2n+1
2n+3
In Fig. 8 these curves are drawn for a copper sphere for
which R = 50 mm., n = 1, when it makes 1, 2, 3, 4 revolutions
per second.
Large
velocities
FIG. 8.
=
3. Thirdly, let us assume that μ is so large that for q(Sλ)
of rotation. and g(sλ) we may put their approximate values. Further,
assume that the ratio r/R is neither very nearly 1 nor very
nearly = 0.
= 0. The former case has been considered already;
the latter requires special consideration. Substituting the
approximate values in the exact formula we find
2n+1 S"
(0-8)+€-λ(0-8)
$1 + $2
--
1
=
λ
+1
(S−8). -λ(S− s)*
-Є