92
II
INDUCTION IN ROTATING SPHERES
Couple ex-
erted on the
inducing
magnets.
rotation which can be calculated from the heat generated.
For if we imagine the shell at rest and the inducing magnets
rotating with angular velocity w, then the couple of moment
D, which maintains the rotation, does work 2πwD per unit
time, and this work is equal to the heat generated. Thus
D =
W
2πω
But this couple is equal to that with which in the reverse
case the rotating sphere acts on the magnets at rest. It is
easy to see that for small values of a D increases proportion-
ately to w/x, but for large values decreases proportionately to
x/w, ultimately becoming zero. (This does not prevent work
being done of the order.) On the other hand we have
seen already that for infinitely large values of a the forces
exerted on the inducing magnets are finite, and since now they
produce no couple about the axis of rotation their resultant
must act in a plane through the axis.
In fact for infinite velocities the sphere behaves as regards
the external magnets as a conducting sphere does with regard
to electric charges; but a conducting sphere cannot impart
to inducing charges any rotation about an axis through its
centre.
§ 6. ROTATION OF MAGNETIC SPHERES.
I now assume that the material of the sphere is capable
of magnetisation, but that it is without coercive force.
We must first form the expressions for the electromotive
forces in this case. According to the precedent of § 1, 6, in
order to find the effect of magnetisation, we must in the
general expressions for the electromotive forces replace
әм
ƏM
U by
Əz
-
ƏN
dy'
ƏN ƏL
V by
ax Əz'
ƏL ƏM
W by
dy
მე