116
11
INDUCTION IN ROTATING SPHERES
Rectilinear
currents
3. I shall now apply the formulæ to another example.
and unlim. Suppose that above the rotating disc two wires are stretched
ited disc. parallel to the x-axis and are traversed in opposite directions
by equal currents of unit intensity. For a single current the
currents induced in the unlimited disc would become infinite.
Let the coordinates of the points in which the wires meet
the plane yz be 0, a, c, and 0, a', -c'; we then have for
positive values of z
-
X = tan
-1 y-a
z+c
tan
-13 - a'
2+ c
r, r1
Hence it follows, by means of the formulæ used before, if
denote perpendicular distances from the wires, that
Ω
=
2πω
x log
k
W
= x
Y½-1 log (=)
k
For the potential of the free electricity in the plate we
get
r
$ = wy log
See Fig.
12 a.
so that the equipotential lines are straight lines parallel to
the wires. In Fig. 12 a the lines of flow are drawn for the
case where
c=c' = 10 mm,
α = a' = 20 mm.
--
Since, moreover, at infinity the currents become infinite, we
must suppose 27w/k to be exceedingly small in order to get a
sufficient approximation in a finite region.
Further, as all the currents are closed at infinity we cannot,
from the case of an unlimited disc, directly draw inferences as
to a limited one.
Hence I shall calculate, by the method developed in § 7,
the currents in a limited disc under like conditions. Let the
radius of the disc be R.
The exact solution of the problem requires us to develop
rather complicated functions in series of sines and cosines.
I