Geoscience Reference
In-Depth Information
Circular disk
Let the thickness
b
of the cylinder go to zero such that the product
κ
=
b
(3-17)
remains finite. The quantity
κ
may then be considered as the surface density
with which matter is concentrated on the surface of a circle of radius
a
.We
need potential and attraction for an exterior point. By setting
=
κ
b
(3-18)
in (3-9) and (3-11) and then letting
b
→
0, we get by well-known methods
of the calculus
U
e
=2
πGκ
√
a
2
+
c
2
c
,
−
A
e
=2
πGκ
1
.
(3-19)
c
√
a
2
+
c
2
−
Sectors and compartments
For a sector of radius
a
and angle
α
=
2
π
n
,
(3-20)
we must divide the above formulas by
n
. For a compartment subtending the
same angle and bounded by the radii
a
1
and
a
2
(Fig. 3.4), we get, in an
obvious notation,
n
U
(
a
2
)
− U
(
a
1
)
,
∆
A
=
1
∆
U
=
1
(3-21)
n
A
(
a
2
)
A
(
a
1
)
.
−
P
®
a=
1
a=
2
Fig. 3.4. Template compartment