Exercise 3.5 Graphically describe the potential and attraction of a uniform, thick-walled shell (inner radius a\ and outer radius CI2) along a line
3.3 Potential of Two-Dimensional Distributions
to derive
= _i2* f
x J cos OdO
*27A
= —l srn0o
x
(3.16) Hence, the gravitational attraction of a finite length of wire viewed along a line perpendicular to the midpoint of the wire is directed toward the center of the wire.
3.3 Potential of Two-Dimensional Distributions
Masses that are infinitely extended in one dimension are said to be two- dimensional, for reasons that soon will become obvious. We begin by investigating the potential and attraction of infinite wires and apply these results to bodies of arbitrary cross-sectional shape.
3.3.1 Potential of an Infinite Wire
First consider the attraction of an infinitely long wire. As a —• oo in equation 3.16, the attraction becomes
Hence, the attraction of an infinitely long wire is inversely proportional to and in the direction of the perpendicular distance to the wire. A general relationship is seen if P is moved to an arbitrary point of the x, y plane:
^ , (3.17) where r is directed from the wire to P and is understood to lie in the x, y plane; that is, r2 = (x — x')2 + (y — yf)2.
Although the gravitational attraction of an infinite wire is straightfor- ward, the potential of an infinite wire is something of a problem. First
consider the potential of a finite wire of length 2a (Figure 3.5):
a
U(P)=j\ f ^dz'
sec 0 dO
= 7 A log
1 — sin 6O
= 7 A log
As a —» 00, the potential also approaches infinity and obviously violates our requirements that the potential should vanish at infinity. This in- convenience is handled by redefining the meaning of the potential for infinitely extended bodies. The potential of an infinite wire is defined so that it vanishes at a unit distance from the wire. This is accomplished by adding a constant to the previous equation:
U(P) = 7 A Now, as a 0 0 ,
C / ( P ) = 27A l o g - ,
and moving P to an arbitrary point of the x, y plane provides the general result
U{P) = 27Alog-,
r (3.18)
where r is the perpendicular distance from P to the wire. Notice that the potential does not vanish at infinity, but rather at r = 1.
Hence, the potential of an infinite wire decreases logarithmically as the point of observation recedes from the wire, a property that will extend to infinitely extended bodies of any cross-sectional shape. Such potentials are called logarithmic potentials for obvious reasons. It can be verified easily that equations 3.17 and 3.18 satisfy
g(P) =
3.3 Potential of Two-Dimensional Distributions 57 and
2 O , r/0.
The attraction of an infinite wire (equation 3.17) can be regarded in two ways. First, of course, it represents the Newtonian attraction of a wire of great length. It also can be regarded as a new kind of point source located at the intersection of the wire and the x, y plane. The attraction of the point source is proportional to the density of the wire A and inversely proportional to the distance from the wire to the point of observation.
It is easily shown by integration of equation 3.18 that the Newtonian potential of an infinitely long, uniformly dense cylinder of radius a is given by
2 (3.19)
r
where p is density and r is the perpendicular distance to the axis of the cylinder. Hence, the potential of an infinitely long, uniform cylinder is identical to the potential of an infinitely long wire located at the axis of the cylinder. Likewise, it follows from equation 3.17 that the gravitational attraction of an infinitely long cylinder is given by
(3.20) where f is directed from the axis of the cylinder to P. Computer sub- routine B.2 in Appendix B provides a Fortran subroutine to calculate the two components of gravitational attraction at external points of an infinitely extended cylinder.
3.3.2 General Two-Dimensional Distributions
The density of a two-dimensional source, by definition, does not vary in the direction parallel to its long axis, and p is a function only of the two dimensions perpendicular to the long axis of the body, that is, p(x,y,z) = p(x,y). Starting with equation 3.5 and referring to Fig- ure 3.6, we write
P(x,y,O)
Fig. 3.6. Gravitational effects observed at point P due to infinitely extended body.
where S in this case represents the cross-sectional area of the two- dimensional source. As a —* oo, the inner integral becomes the loga- rithmic potential of a wire with 7A = 1, and the potential of the two- dimensional distribution is given by
C/(P) = 27 Jp(S) log ^ (3.21)
The gradient of equation 3.21 provides the gravitational attraction
(3.22)
which is perpendicular to the body. Because density is independent of the long dimension of the body, it is sometimes expressed as mass per cross-sectional area c(S), where a/p has dimensions of length.
Equations 3.21 and 3.22 represent the Newtonian potential and at- traction, respectively, of an infinitely long body, uniform in the direc- tion parallel to the long dimension of the body. The attraction also can be considered as originating from a special kind of source: a two- dimensional wafer corresponding to the intersection of the body with the x, y plane (Figure 3.7). The attraction due to each element dS of the wafer is proportional to p(S) and inversely proportional to distance.
\4 Gauss's Law for Gravity Fields / P(x,y,O)
59
Fig. 3.7. Gravitational attraction of a two-dimensional body can be considered to originate from a special kind of source located in the x, y plane. Each element of the body has an attraction inversely proportional to distance.
Two-dimensional objects are generally easier to visualize than three- dimensional ones. Happily, certain geologic features, such as fault con- tacts and synclines, sometimes can be approximated by two-dimensional shapes thereby simplifying the interpretive process. In Chapter 9, we will describe the computation of the gravitational attraction of two-dimen- sional models with known cross section.