INDUCTION IN ROTATING SPHERES
51
N';=
=
4πRn w 4πR(n+1)w/p
(2n+1)(n + 1) k (2n+1)n k\R
4TR w
2
n
(2n + 1 + 1 ) ( R ) "Y"..
k Ꭱ
n'
The two expressions for ', are the same.
22
Y" พ
Hence this is
the potential of the current system, which is induced in the
spherical shell itself by the currents of the first order. If in
the same way we calculate the succeeding inductions and add
them together, we get for the whole inductive action
Ω;
=
n
ρ
R
-
= -
4πRO mamyn
(2n+1)k/ wm
m
-
M
>
4πRW mamy
(2n+1)k) dwm
n
n R\"+1
n+1 P
2n+1
4π(n+1).
4πRw Ᏹ" Y,
n
m
-
(2n+1)k/w"
The expressions obtained may be developed still further by
analysing Y, further.
We have
n
Ani
Y₂ = {A,¡ cos iw + B₁¡ sin i∞)P„i -
ni
ni
We confine ourselves to one term only of this series.
Thus let
Then we have
Y₁ = Ani cos iwPni
n
n
Q;= Ani
P
P.
ni
R
4 Rwi
(2n+1)k
siniw
4πRai
(2n+1)k
(2n+1))
2
cos iw
-(2
siniw +
4πRwi 4
(2n+1)h,
cos ίω +
}
4πRwi 3
(2n+1))
Put for shortness
then we find
Li
=
Ani
P
R
4πRwi
(2n+1)k
n
=h (h is a pure number);
Pni(sin iw-h cos iw).h. (1—h² + h¹ — h® + …..).
The
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7
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