II
103
INDUCTION IN ROTATING SPHERES
are the currents sought in S. Hence the problem is reduced
to this simpler one :—
To determine such a function, that inside S V2, = 0,
and at its surface op/on = N, a given function.
bounded by
a straight
edge.
1. As an example, suppose a plate bounded by the straight Plate
line = b to move parallel to a given straight line. Suppose
the external potential expanded, and let a term of it be
Ae-5" cos rn cos s§.
Then we found for the current in the infinite plate
α
₁ =A. .sin rn. cos s§.
n K
Thus the current perpendicular to the boundary is
-
θη
=
7,2 a
A
n K
cos rn. cos sb.
Hence we get for 4, the conditions
and for b
=
242
=
2² 42+
ə² 2
ô× 10-
p² a
A
N K
•
=
0
"
cos rn. cos sb.
We have
૦૬
Φ
$2
=
rα-b) cos rn. cos sb.
A-
n K
To corresponds the current-function
=
-
τα
Ara.e-b) sin rŋ. cos sb,
Α
N K
and thus the total current-function becomes
¥₁ +₂ = A².
α
η κ
.erb sin rn(e. cos ser. cos sb).
By summing for all the terms we get the complete solu-
tion. The solution for a band bounded on both sides is
similar.