98
I[
INDUCTION IN ROTATING SPHERES
Self-induc-
tion taken
into
account.
Limitation
to solid
spheres.
hold; the calculations are easily performed, but since they
give no very simple results they have been omitted here.
We shall now take into account self-induction, but shall
only perform the calculations for a solid sphere. Spherical
shells do not offer analytical difficulties of any special kind,
but the calculations become exceedingly complicated.
We find the currents by the following reasoning:-
Let the inducing potential be
Xni
น
P
Α
R
=A(f) "cos
cos iw Pri
Let be the current-function directly induced by Xni
then we have
=
-
1 + 4πθ
n
1+
Απθ
2n+1
n
i
sin iwPni
Rn+1
·
W ρ
Apl
K
Let the actual current-function be
n
W P
1 -
K
Ꭱ
i
n+1
(f, sin iw+f, cos iw)Pni
2
For this
Xe
induced
We have to find the currents induced by this.
purpose it is necessary first to know the potential
by in the magnetic mass.
The current-function
n
* = p(f)' J (p) X
produces a radial magnetic force
Np = 2 (3W _ OY) + 2(37
=
аду
P
(p. 63)
P
n
aw
au
+
дх
მე:
Əy
n(n+1)F(p). (£) Y. (p. 65),
P
R
where F, ƒ are connected by the equation of p. 68.