II
63
INDUCTION IN ROTATING SPHERES
We have in the treatment of plane plates all along assumed
that inducing magnets exist only on one side of the plate.
This assumption is unnecessary. If it is not true, we divide
the total potential into two parts according to its origin, and
treat each part separately, as we have shown above for one of
the parts.
4. COMPLETE SOLUTION FOR SPHERES AND SPHERICAL
SHELLS OF FINITE THICKNESS.
We now turn to the consideration of the induction in a
spherical shell of finite thickness. To avoid complication we
shall at first suppose inducing magnets to exist only outside
the shell.
Let U, V, W be the components of a vector potential due
to closed currents, wholly or partly inside the shell. The
currents u', v', w' induced by U, V, W are given by the
equations
ки =
-
аф
дх
KV =
-
Kw
= -
аф
+wx
24+ wyl
dy
Эф
Əz
-
av au
Jx Əy
av au
aw av
dy Dz
Further, inside we have
and at the surface
ди
Ən əv' Əw'
+ +
дх ay Əz
Differential
equations.
wy
aw
Əz Əx
0,
u'x+v'y+w'z = 0.
We write for shortness
aw av
au aw
av au
0=x
+y
+
dy дъ
Əz
Əx
Əx
dy
VLIENUE ENVINCERIAS LE