118
11
INDUCTION IN ROTATING SPHERES
See Fig.
12 b.
The corresponding
2 is (§ 7, I., conclusion)
¥2
=
c² + a2
ap² sin 2w
Hence we have
a=
=
-
(3c² – a²)a
6(c² + a²)so² (R² sin 2w - §p² sin 4w).
a(3c² - a²)
6(c²+a²)s
p² sin 2w(R² - p²).
Hence the form of the lines of flow is independent of the
ratio a:c, but the current is greatly dependent upon it. If
0 or a=c√3, it vanishes. If a<c/3, the direction of
the current is the same as in the unlimited plate; if a>c/3,
it is the opposite. When we consider closely the distribution
of the forces which act, this at first sight astonishing result is
explicable. The form of the distribution is shown in Fig. 12 b.
In the same way the problem may be solved for any de-
sired position of the wires. When one of them moves off to
FIG. 12 b.-Rotating disc and rectilinear currents, } nat. size.
infinity, the currents remain finite in the limited disc, and we
find on retaining the first two powers of the dimensions of
the disc
ป
=
c² - a²
8(c²+a²) 20
-
ặp cos w(R² — p²) — α(3c² – a²)
-
-
12(c² + a²)s² sin 2w(R² – p²).
The connection with the previous result is easily seen.