70
II
INDUCTION IN ROTATING SPHERES
we get
iw –
*'- (~) "Ap({})" — 1 (F,(p) cos in – F₂(p) sin is} Pri-
=
Αρ
R n+1
=
The equation + is satisfied if ƒ₁ and ƒ½ satisfy
the equations
(p)=1+F(p).
K
Treatment
of the equa-
tions for
the f's.
ίω
ƒ₂(p) = _ ¹@F₁(p),
KC
by which f₁ and f, are completely determined.
If we regard f₁ and ƒ, as known, the result of the investi-
gation may be expressed in the following form:-
-
Self-induction leaves the form of the lines of flow un-
altered (for each separate term of the development). Its effect
is :-
Firstly, to turn the system in the direction of rotation
through an angle S/i, where the angle & is different for different
layers and is given by the equation tan d =ƒ½ƒr
Secondly, to change the intensity of the current differently
in different layers. The ratio of the intensity actually
occurring to that found without taking account of self-induc-
tion is fi²+f2:1.
We shall have to occupy ourselves for some time with the
determination of the functions f₁ and f
We introduce the following contractions.
Απίω
=
με,
Let
2
K
μr = 8, μR = S,
ƒ1(p) = $1(µp) = $10,
ƒ2(P) = $2(µp) = $20.
In the equations which give f, and f₂ we put for F, and
F₂ their values, transform the equations so as to involve o
and σ, and thus obtain