II
53
INDUCTION IN ROTATING SPHERES
position previously determined.
tion this angle is proportional to
it approaches to the limit π/2i.
increases in proportion to the
more slowly for larger values and approaches a fixed limit.
2. Finally, when w/k= ∞, 8 π/2, and then
For small velocities of rota-
the velocity; for large ones
The intensity, which at first
velocity of rotation, increases
Ωρ
=
=
Ω, = - Χει
=
n
n + 1 Xn,
2n+1
4π(n + 1)
-Xn •
This result does not hold for those terms of the develop-
ment which are symmetrical about the axis of rotation. For
these i, and therefore also h and , vanish for every velocity
of rotation. These terms produce no currents, but merely a
distribution of free electricity in the sphere.
Hence a spherical shell, rotating with infinite velocity,
only allows those portions of the external potential which are
symmetrical about the axis to produce an effect in its interior.
If such terms are absent, the interior of the shell is com-
pletely screened from outside influences. If the potential is a
spherical harmonic, the current flows along the equipotential
lines.
3. We found, neglecting self-induction, the following
expression for the electric potential corresponding to Xn
The velo-
city is in-
finite
The electric
potential.
= -
ய
n+1
-R sin 0-
Rsin
Xn
до
Taking self-induction into account, we shall have
-
-
W
R sin ex₂+
n+1
до
Hence it follows that the form of the equipotential lines (for
each inducing spherical harmonic) is unaltered by self-induction,
but these lines are turned through the same angle as the lines
of flow. For the parts of the external potential which are