50
II
INDUCTION IN ROTATING SPHERES
§ 3. COMPLETE SOLUTION FOR INFINITELY THIN SPHERICAL
SHELLS.
We shall now take into account the effect of self-induction,
but in this paragraph we shall confine our considerations to
the case of infinitely thin shells. For simplicity n will be
supposed positive in the calculations.
In accordance with usual views we regard the total in-
duction as compounded of an infinite series of separate
inductions; the current induced by the external magnets
induces a second system of currents, this a third, and so on
ad infinitum. We calculate all these currents and add them
together to form a series which, so long as its sum converges
to a finite limit, certainly represents the current actually
produced.
Let
Xn
x = ( )*x.
P
R
n
Calculation
cessive in-
ductions.
represent a part of the external potential. The potential in-
duced by this part is
Ω =
Ste
Q;=
=
-
4πR W
% (1)Y..
2n+1 k R
4πRn
n+1
w/R
(2n + 1)(n + 1) k P
'
Y'n
In the first place, if inside the spherical shell a second
of the suc- rotate infinitely close to the first and with the same velocity,
the currents of the first order (,) will induce in it a current
system whose internal magnetic potential is
=
Ω, -
4πR w 2
2n+1k R
n
Y":
n'
Secondly, if outside the first shell another rotate with the
same velocity and infinitely close to the first, the currents of
the first order (2) will induce in it a current system whose
potential inside is