100
II
INDUCTION IN ROTATING SPHERES
The solu-
tion.
sphere, we must only retain those solutions which are finite
at the centre, and may put
$1(0) = Ap₂(mo)+Вp₂(λ₂o),
Þ₂(0)= − x}{Ap„(Mo) – λ¿Âµ‚(^σ),
λ=~= √}(1+ √ −1),
λ = √(1 - √ − 1).
-
The constants are determined in precisely the same way
as above.
The integrals to be formed are not different from
those got before, but the calculation is somewhat more in-
tricate, owing to the complicated constants. The result,
however, is comparatively simple, namely
f1(p)+f2(p) √ −1 =
(2n+1)(1 +4π0)Î₂(λµp)
2ηγη-1(λμ) + 4πθηρ,(λμR)
We first verify this result.
For vanishing it gives
2
fi+ƒ₂√ −1 =
2n+1 Ρ (λμρ)
2n Pn-1(μR)'
Compari-
son with
previous
results.
Small vel-
ocities of
rotation.
which agrees with the result already obtained for a non-
magnetic solid sphere (p. 75).
Further, for vanishing w it gives, since
2n+1|n
1.3... (2n+1)
Pr(0):
=
ƒ₁ +ƒ½ √ − 1
-
=
"
(2n+1)(1 + 4π0)
2n+1+4πOn
which result also we have found (p. 97).
In general it appears that the form of the currents in a
magnetic sphere is the same as for a non-magnetic sphere of
equal resistance which is rotating (1+470) times as fast
as the magnetic sphere. But in addition the two current-
systems differ in that they are turned as a whole through a
certain angle relatively to one another, and that their in-
tensities are different.
I apply the formula to two special cases.
1. Let 470 be very great, but w sufficiently small that