II
45
INDUCTION IN ROTATING SPHERES
U.(p) = (
V.(p)
R
R2n+1
Ui(p),
2n+1
=
V₁(p),
ρ
R
2n+1
W.(p)
=
Wi(p),
P
From these values of U, V, W we calculate the magnetising
forces inside, viz:
av au
-
ёх ду
>
etc.
we put them
n+1ay
R Əz
etc.,
and so obtain the function (§ 1, 5). We find
4πR w d
X
2n + 1 k dz
JXn - Y
dy
Jxn
4TR wa
--
x
2n+1 k dx ay
4πR ωθ
2n+1 k dy
whence
Vi
=
·
x
-
да
y
ax
JXn - Y
dx
ду
4πR2
ய
(2n+1)(n + 1) k
=
4πR2
n+1a
=
Y
R Əz
=
= -
Xn
dy
(2n + 1) (n + 1 %X'n,
n + 1 ay₁,
R Ox
n+1d
R dy
Jxn
Y ax
and now all the remaining properties of the currents follow at
once from Y. An arbitrary constant may be added to ỵ, but
is of no importance.
of formulæ.
We thus obtain the solution of our problem for a spherical Summary
shell in the following form (§ 1, 5):—
Let
Xn
=
x-(R)*x.
n
P
Y n > 0,
Ꭱ
F
*
•