11
87
INDUCTION IN ROTATING SPHERES
2. In precisely the same way we may perform the investi- Value in
gation for the space inside the spherical shell; we find
-
Q₁ = − (£) *A[ƒ:(r) sin iw + {1 − fi(r)} cos iw]Pnt
Ω;
the inside
space.
R
Hence for the whole potential
+x₁ = A{( {) "[ƒ¡(r) cos iw - ƒ½(r) sin iœ]P„-
Li Xn
R
For vanishing angular velocities this expression reduces
to X, for large ones to zero; more exactly for large values of
we find by means of a formula which we have employed
previously (p. 81)
1
2(2n+1) (2) ----
fi(r) +ƒ½(r) √ −1 = 2(2n+1) (R\"
--
λμη
-
-
€
λ(S-s)
=
R
2(n+1)=
n
1
7/ λμπ
- dµ(R − r).
-€-Aμ(R-r)
Hence it follows that
Li + Xn
T
μ
- 5 (R-1 con (ias - - ~ - (R − r) Par
€ √2
2(2n+1)/p
=
A
-
(2)".
n
μη
COS ίω
4
ni.
behaviour
Thus the internal potential diminishes with exceedingly
great rapidity as the velocity increases. At the same time its
equipotential surfaces exhibit the peculiarity of appearing
turned through an angle proportional to the [square root of Peculiar
the] angular velocity. As the velocity gradually increases of the mag-
the forces conditioned by the potential take up successively netic forces
all the directions of the compass; and this can be repeated side space.
any number of times as the velocity goes on increasing.
in the in-
B. Heat Generated.
Let R be the radius of a very thin spherical shell, and
suppose that in it exists the current-function
=
Σψη