II
67
INDUCTION IN ROTATING SPHERES
It is not difficult to see the connection between this
theorem and the results obtained in preceding paragraphs.
2. The U, V, W which are of the above form are due
to currents in concentric spherical shells. For we have
x▼²U+y▼²V+zV²W = 0.
And vice versa the U, V, W of such currents may always
be expressed in the above form. For if Xf (p) is the term
involving the nth spherical harmonic in the development of
the current function, then the U, V, W belonging to this
term are at once seen to have the above form.
On the other hand, the induced currents also flow in con-
centric spherical shells. For we have
ux+vy+wz = 0.
The flow is
concentric
shells in- always in
Further- spherical
rest in a
Hence we deduce the following conclusion:-
A current which flows in concentric spherical
duces a current system possessing the same property.
more, the currents which are induced by magnets at
rotating spherical shell always flow in concentric spherical
shells about the origin.
3. We find that
$ = w(xV - yU)
whenever U, V, W have the above form, and the inducing
currents the property discussed. This we shall have to make
use of in § 8.
shells.
cessive
There is now no further difficulty in calculating the Calculation
successive inductions produced by a given external potential. of the suc
Xn denote the nth term in its development. We found for inductions.
the currents of the first induction
Let
1 wax
U₁ =
=
n+ 1 x dw₂'
1
wax'
1 w dx
V1
=
W1
n + 1 x dwy
n + 1 x dwz
The corresponding values of U, V, W are
2π ωοχ
R2
U₁ =
=
n+1 x dw, 2n+1
p²
22n+3
2n+3 (2n+1)(2n+3)p²n+1
L
SUJIENUE EINSTA