II
81
INDUCTION IN ROTATING SPHERES
Since S is not nearly equal to s, and both are very great,
the second term in the denominator vanishes compared with
the first, and we get
$1 + $₂√ −1 =
$2
-
2n+1/R\n+1
λS
Є
The second term in the bracket vanishes in comparison
with the first except when p = r; if then we are content with
an approximate knowledge of the current at the inner surface
we may write
$1 + $₂ √√ − 1 =
2n+1/R\"+1
€-λμ(R-p)
as
Since s or r has disappeared from this equation, we may
assume that it holds also for a solid sphere. In fact it is
easily deduced from the exact formula which hold for a solid
sphere if we make similar approximations to those used above,
and do not require an exact knowledge of the currents at the
centre (where, as a matter of fact, the current intensity is
very small).
In the expressions obtained
λ= √½(1 + √ −1);
without performing the separation into real and imaginary
parts we easily find
1f2
tan
fr
= -
π μ
-
(R − p),
√2
2n+1 R" - (R-p)
√ƒi+ƒ½= μ
Substituting these values in we find
=
-
2n+1 iw -
2(n+1)
A
€
(R-P) sin (iw-
(iw -
π
кп
-
μ
√2
4
(R − p) ) Pnis
which is the current-function produced for very large velocities
of rotation by the external potential
M. P.
n
Xn A
P
cos iw P
R
ni.
G