104
II
INDUCTION IN ROTATING SPHERES
Limited
disc.
2. In order to determine the currents in a limited
rotating disc, let a term of the external potential be
Then we had
Ae-n cos iwJ(np).
wi
Y₁ = A-- sin iwJ,(np).
K n
Thus the inward radial current is at the boundary where
P=R
=
Row
w 22
A
K n
cos iw Ji(nR)
R
Hence we find as above
()
i
Φ -- AJ, (nR) cos ia.
Þ₂ =
2
K n
R
Determining the corresponding current-function we get
for the total current-function
=
A
w i sin iw
K N Ri
'{R¿J¿(np) — p¹J¿(nR)}.
We again get the complete solution by summing for the
various terms. In the same way the currents may be deter-
mined in rings bounded by concentric circles.
In general the solution of the problem requires neither the
development in a series of separate terms nor the determination
of the potential 2; it is sufficient to determine, so that
inside the plate
2²√42=0,
2242
Əx² + dy²
and at its boundary
₂ = -
Some simple examples will be
given in § 9.
II.
Dielectric
spheres.
In conductors electromotive forces of electromagnetic
origin produce the same effects as numerically equal electro-
static forces. If this is true also of dielectrics, then spheres
of dielectric material must become polarised when rotating in
a magnetic field.