86
II
INDUCTION IN ROTATING SPHERES
Its value in
external
space.
A. Potential of the Induced Currents.
1. We first calculate it for external space. The part of
it due to the spherical layer between pa and p=a+da is
dne
=
4πη
2n+1
n+1
a
Yni(a)da,
when we consider the term ni of the whole current-function .
Now
n
Yni(a)da
W a
= -
A
KC R
n+1
(ƒ₁(a) sin iw+ƒ½(a) cos iw)Pnida.
Substituting this value in dî, and attempting to perform
the integrations, we meet with the integrals
R
a²n+2f (a)da.
But we have
R
f(a)da
(2n+1)R2n+1
=
F₁(R)
4π
(2n+1)K
4ria (R). R2n+1
Απίω
according to the definition of F (p. 68), and the equations
satisfied by f1f2, F1, F2 (p. 70); and similarly
R
a²n+2f2(a)da =
-
( 2 n + 1 ) x ( 1
(1 −ƒ₁(R)). R²n+1
Απίω
Using these expressions we find
n
R\n+1
Ω,
=
A
-
n+1 ρ
[ƒ½(R) sin iw+ {1 −ƒ₁(R)} cos iw]Pni
For very small angular velocities f₂ = 0, f₁ = 1, and thus
Ωρ = 0. For very large ones f₁ =f2 = 0, and thus at the
surface of the spherical shell
Ω
=-
n
n + 1Xn•