54
11
INDUCTION IN ROTATING SPHERES
symmetrical about the axis, & increases indefinitely as the
velocity increases; for the other parts it approaches a finite
limit which is easily calculated.
Plane plate.
Plates
moving in
a straight
line.
LIMITING FORMS OF THE SPHERICAL SHELL.
We now make the radius of the spherical shell infinite,
but keep the variations of the inducing potential finite, and
then we examine more closely the electric currents at the
equator and at the pole. We thus obtain the theory of plane
plates, both rotating and moving in a straight line. The
latter may be considered as a special case of the former, but
for several reasons it is advisable to treat
these cases separately.
y
FIG. 7.
A. Plates moving in a Straight Line.
We introduce the co-ordinates &, n,,
whose connection with x, y, z is shown
Sin Fig. 7.
The direction of n is the positive
direction of motion. We shall suppose
the inducing magnets inside the sphere,
i.e. on the side of negative. We must examine what form
in §, n, is assumed by the spherical harmonic
R\n+1
Ani
cos iw. Pri.
P
In order to obtain finite variations we must make n and
i ∞ of order R.
We put
for n,
nR,
for i,
TR,
P, w, o
η
R+S,
+
Ꭱ 2
م
امید
ફ્
R
and further replace
by