II
95
INDUCTION IN ROTATING SPHERES
According to what precedes, the magnetic spherical shell
is in exactly the same condition as a non-magnetisable one
of equal resistance which is subject to the influence of a
potential
2n+1
r
(1+470) A+B -
1+ 4π0) ( A + B ( * ) ***) *~*~
Since this potential consists of two spherical harmonics,
the currents may by what precedes be regarded as known.
For the current-function we get
W
↓ = − 2 (1 + 4π0)|
-
K
A
2n+1
B/r
Xn
-
-
n+1 n\p
Under like conditions, only with = 0, we should have got
the current-function
40=
W P Jxn
k n + 1 dw
n+1 r
B
2n+1
n
By division we get
↓ =(1+4π0) A-
1+4=0) (A
The form of the currents in the various layers is unaltered,
but the intensity is differently distributed. It is convenient
to describe the phenomenon by comparison of and y。. The
quantities A, B are given by equations (6), (7); if we write
e, they are found to be
r
R
A
=
B
=
-
(2n+1){(2n + 1)(1 + 4π0) − 4π0n}
n(n+1)16π²0²(1 − e²n+¹) + (2n+1)²(1 +4π0)'
-
4πθη(2η + 1)
n(n+1)16202(1 − €²n+¹) + (2n + 1)²(1 +4π0)
As the interpretation of these expressions is not very Special
obvious, we shall apply them to some simple cases.
1. is very small. Expanding we get
cases.
n
n
A = 1
-
4π0, В:
=
Απθ,
2n+1
2n+1
e very
small.