11
85
INDUCTION IN ROTATING SPHERES
or since we have in part already neglected quantities of order o
$1 √ − 1 − $₂ = X² = √ -1
as must be the case.
-
$₁ =1, ₂ = 0,
On the other hand, if up be large compared with unity,
and we put for the q's their approximate values for large
arguments (p. 77), we get
$₁+4₂√-1=(2n+1)
n
1
-€
- λ(σ − s).
2
λε
Hence result phenomena similar to those for the spherical
shell; the rotation is π/4i at the innermost layer, and thence
increases indefinitely as p increases to infinity.
For we find
ป
=
2n+1
A
iw
-
(p-r) sin iw
2n
sin(?
-
Kπ
π μ
4
nis
which expression is quite analogous to that obtained for the
spherical shell.
Moreover, we easily see that, even for the smallest
velocities of rotation, p can always be chosen so large in un-
limited space that the approximation made may be permissible: Neglecting
hence even for the smallest values of @ the induction will
pass tion.
through all possible angles, at distances, it is true, where the
intensity is very small.
I here wish to draw attention to the remark I have already
made on p. 61 in regard to neglecting self-induction.
It would be very easy to extend the results obtained for
spherical shells to plane plates of finite thickness; but in
order to avoid complicated calculations I omit the investiga-
tion. The chief part of the phenomena can, in fact, be deduced
without calculation from what has been already discussed.
self-induc-
5. FORCES WHICH ARE EXERTED BY THE INDUCED CURRENTS.
We shall now calculate the forces exerted by the induced
currents and the heat generated by them. The latter is
equivalent to the work which must be done in order to main-
tain the rotation.