44
II
INDUCTION IN ROTATING SPHERES
tion of the
If we multiply these equations by x, y, z and add we get
ux+vy + wz = 0.
Thus the current is everywhere perpendicular to the
radius and flows in concentric spherical shells about the
origin. This is a consequence of the fact that equation (b) is
satisfied throughout the mass and not merely at its surfaces.
Further we find
V²u =
·
ய
K
▼²v = 0,
-
2
Əxəz
Xn
+22² xxn l
Xn
=
= 0.
Əxǝz
V²w= 0, since also v²x = 0.
In fact u, v, w are homogeneous functions of x, y, z of the
nth degree; thus u, v, w are exhibited as spherical harmonics
of the nth order. We shall soon find simpler expressions
for u, v, w.
Determina- Since the currents are similar in concentric spherical
function . shells, they are also similar to those which occur in an
infinitely thin spherical shell; therefore we first consider such
a shell and determine the values of the integrals U, V, W for
internal space when n is positive, for external space when n is
negative. We shall only work through the first case.
For U, V, W the conditions hold
V²U = 0, V²V = 0, _V²W = 0
replace by k.
throughout space.
At the surface of the shell
-
au, au;
др др
=
-
- Απυ,
K we
and corresponding equations for V and W; and in addition we
have the usual conditions of continuity.
are satisfied by putting
2n+1 k
n+1 ax
1 a
ρ
2
-
Jxn-
4πR w
1 a
U₁ =
-
ρ
=
Xn
Əz
Vi
4πR w
1 a
2
Xn
-
Wi
W₁ =
4πR W
2n+1 k
-
n + 1 dy
2n+1 kl n+1 Əz
Əz
Əz
-
All these conditions
+ 2 x = }
Xn
nzxn+x
+
nzxn+y
Xn) +
xzxx)
Əz
Xn
Əz
nzxn+z
Əz
}
-
[¹ U₁, U, denote respectively U internal, U external.—TR.]
nzxn},