II
101
INDUCTION IN ROTATING SPHERES
μ²R² may be neglected in comparison with unity. We must
expand the expression, retaining only the first power of that
quantity. We have
=
2η + 1 γ (μρλ)
n Pn(μRλ)
ƒ₁+f₂√ = 1
2n+1
2(2n+3)+µ² p² √
=
(p. 77).
N 2(2n+3)+µ²R² / -1
We get for the angle of rotation, neglecting higher powers
than the first
fi 2(2n+3)
Stan
-1f2_
με
= -
(R² - p²),
င်
=
(R² - p²).
i
16π²w0
2(2n+3)x
Hence the rotation vanishes at the outer surface;¹ gener-
ally it is considerably increased, compared with that for a
non-magnetic sphere nearly in the ratio 470: 1.
In Fig. 9 are given the curves for an iron sphere cor-
responding to those for a copper sphere represented on p. 80.
FIG. 9.
The resistance of iron is taken to be six times that of copper,
and 40 is put = 200. The velocities represented are exceed-
ingly small ones, namely one revolution in five and one in ten
seconds; even here the effect of self-induction is well marked
(cf. Fig. 15 b, p. 123).
velocities.
2. If a become very great, whilst retains a finite but Large
otherwise arbitrary value, the phenomenon becomes very
similar to that in non-magnetic spheres, as may be easily
deduced from the formulæ. Here also the angle of rotation
1 A consequence of the fact that at this surface for large values of 9, according
to the equations for xe,
дхо
N, =
Ər