56
II
INDUCTION IN ROTATING SPHERES
On the opposite side to the magnets the potential is
zero; the currents flow everywhere along the equipotential
lines of the inducing distribution.
Apart from this limiting case the application of the above
solution is very cumbersome; we therefore seek approximate
methods. In the first place we find such methods by intro-
ducing successive inductions. That this method may be
permissible it is necessary that 2πa/k be a proper fraction; if
this condition be satisfied the calculation leads to a convergent
series, as we have already shown in the general case.
We again start from the infinite spherical shell. The
inducing potential X--1 produced in the space outside the
induced potential
Que
4πR @ X-n-1
=
2n+1 k მო
We allow R to become infinite while we replace
then
a
a
მო
by Ro
n by nR,
wR by a,
X-n-1 by X₂ = A„e˜” cos rn cos s§,
Ωμ
2πα 1 όχη
21
=
k
n an
But we have
Xnds = Xn
n
Hence summing for all values of n
∞
Ω+
=
-
2πα (χ αζ.
k
an
Now from this potential we can get in the same way the
potential of the second order; and proceeding in this way we
obtain finally