II
91
INDUCTION IN ROTATING SPHERES
rotating sphere by the term Xni of the inducing potential.
To Xni corresponds
Yni
=
W
Ap
n
i
(f, sin iw+f₂ cos iw)Pri
K R n+1
and hence the heat
W w² 42 i²(n,i)
ni
=
KC
-A2
R
P
(n+1)² R
2n
"(ƒ} +ƒ³½)p²dp.
The integration can be performed for small and for large Small velo-
angular velocities. For the former f₁ = 1, f₂ = 0, and thus cities.
the heat generated becomes in this case
W ni
2
=
A₂R³w²
K
i²(n,i)
(n+1)²(2n+3)
1
-
− ( 1 )
2n+3
R
For very large angular velocities we had
(2n+1)2 R2n
2n+2
=
με
2
ρ
p²n+q€¯µVXR - p.
The integral Wni may be taken from r = 0, and becomes
Large velo-
cities.
R
(2n+1)2
- - µ√2(R − p)dp =
(2n+1)²
=
XR-P)dp
for Ru may be regarded as infinite.
large values of w
W
ni
=
Α
μ³ №2
Hence we get for very
κων
2(2n+1)²(n,i)
8(n+1)2 N 2
and W depends on R, in so far as A involves R.
Hence the heat generated increases indefinitely as o in-
creases,' and indeed proportionately tow. The same holds
good for the work which has to be done in order to maintain
the rotation. If the inducing magnets form a rigidly con-
nected system, they are subject to a couple about the axis of
1 As regards the apparent contradiction with the result got for infinitely thin
spherical shells the remark on p. 82 holds.