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
5.4 Poisson's Relation
The gravitational potential is written
£/(P)=7 [ -dv J rR
so that
- f
lR
J r 7p
R
Substituting this last integral into equation 5.11 provides
_ CmMgm
~ ^ — ' ( 5 J 2 )
where gm is the component of gravity in the direction of magnetiza- tion. Equation 5.12 is called Poisson's relation. It states that, if (a) the boundaries of a gravitational and magnetic source are the same and (b) the magnetization and density are uniform, then the magnetic poten- tial is proportional to the component of gravitational attraction in the direction of magnetization (Figure 5.3).
What if the density and magnetization are not uniform? We can view both the gravity and magnetic sources as composed of elemental vol- umes. If the magnetization and density distributions are sufficiently well behaved, the density and magnetization within each elemental volume will approach a constant as the volume becomes arbitrarily small. Pois- son's relation holds for each elemental volume, and by superposition must hold for the entire body. Hence, Poisson's relation is appropriate for any gravity and magnetic source where the intensity of magnetiza- tion is everywhere proportional to density and where the direction of magnetization is uniform.
Poisson's relation is an intriguing observation. With assumptions as stated, the magnetic field can be calculated directly from the gravity field without knowledge about the shape of the body or how magnetization and density are distributed within the body. Carried to its extreme, one might argue that magnetic surveys are unnecessary in geophysical inves- tigations because they can be calculated directly from gravity surveys, or vice versa. In real geologic situations, of course, sources of gravity
5.4 Poisson's Relation 93 anomalies never have magnetization distributions in exact proportion to their density distributions. Nevertheless, Poisson's relation can be useful.
First, it can be used to transform a magnetic anomaly into pseudograv- ity, the gravity anomaly that would be observed if the magnetization were replaced by a density distribution of exact proportions (Baranov [9]). We might wish to do this, not because we believe that such a mass actually exists, but because gravity anomalies have certain properties that simplify the determination of the shape and location of causative bodies. Thus, the pseudogravity transformation can be used to aid in- terpretation of magnetic data, a topic that will be discussed at some length in Chapter 12.
Second, Poisson's relation can be used to derive expressions for the magnetic induction of simple bodies when the expression for gravita- tional attraction is known. For example, the following sections use Pois- son's relation to derive the magnetic induction of some simple bodies, such as spheres, cylinders, and slabs. We could do these derivations the hard way, by integrating equation 5.3. But we already know the grav- itational attraction of these simple bodies because we derived them in Chapter 3. The magnetic expressions are more easily derived by simply applying Poisson's relation to the analogous gravitational expressions.
5.^.1 Example: A Sphere
From Chapter 3, the gravitational attraction of a solid sphere of uniform density is
4 3 1 ~
g = - - ™ 7p—r.
Substituting this into Poisson's relation yields the magnetic potential of a uniformly magnetized sphere (Figure 5.4), that is,
= c
m-
where
m = -7ra M.4 o
This is just the magnetic potential of a single dipole. Therefore, the magnetic potential due to a uniformly magnetized sphere is identical to the magnetic potential of a dipole located at the center of the sphere with
94
* P(x,y,z)
Fig. 5.4. Magnetic potential at point P due to a uniformly magnetized sphere.
dipole moment equal to the magnetization times the volume of the sphere.
It follows that the magnetic field of a uniformly magnetized sphere is proportional to both its volume and its magnetization. Although the location of the center of the sphere can be determined directly from its magnetic field, the size of the sphere cannot be found without first knowing its magnetization. Subroutine B.3 in Appendix B calculates the three components of magnetic induction due to a uniformly magnetized sphere.
5.4-2 Example: Infinite Slab
As shown by equation 3.27, the gravitational attraction of an infinitely extended, uniformly dense layer is in the direction normal to the layer, is proportional to the thickness of the layer, and is independent of distance from the layer. For a horizontal layer,
where t is the thickness of the layer and k is the unit vector directed toward and normal to the layer. If the magnetization is vertical, then Poisson's relation provides
cj
gz IP= -2-KCmMt,
5.4 Poisson's Relation 95 P(x,y,z)
Fig. 5.5. Magnetic potential at point P of a vertically magnetized, infinitely extended slab.
Fig. 5.6. The magnetic field of a spherical cavity in a uniformly magnetized layer is equal to the field of a sphere with opposite magnetization.
and the magnetic potential of a uniformly magnetized slab is constant (Figure 5.5). Consequently, the magnetic field of a uniformly magnetized slab is zero, and the slab cannot be detected through magnetic measure- ments alone.
This remarkable fact can be used with the superposition principle to simplify certain problems. For example, the magnetic field caused by a spherical cavity within an infinitely extended layer with uniform magnetization M is identical to the field of an isolated sphere magnetized in the opposite direction, that is, with magnetization —M (Figure 5.6).
5.1^.3 Example: Horizontal Cylinder
Equation 3.19 shows that the gravitational potential of an infinitely long cylinder with uniform density is given by
U(P) = V 7 g
where a is the radius of the cylinder and r is the perpendicular distance to the axis of the cylinder (Figure 5.7). Applying Poisson's relation to this expression provides the magnetic potential of a uniformly magne- tized cylinder:
V = 2Cm7ra2^-^-. (5.13)
\P
Fig. 5.7. Field at point P caused by a uniformly magnetized cylinder.
Note that, because f is perpendicular to the cylinder, only the perpen- dicular components of M are significant to the cylinder's potential. In fact, the quantity 7ra2M has units of dipole moment per unit length, and equation 5.13 is equivalent to the potential of a line of dipoles located along the axis of the cylinder. Hence, the potential and the magnetic field of a uniformly magnetized cylinder are identical to those of a line
of dipoles; that is, the potential of a line of dipoles is given by
V = 2cJ^- (5.14) where m' is dipole moment per unit length, and applying B = — VV inr cylindrical coordinates to equation 5.14 provides the magnetic field of a line of dipoles,
B = 29™pL [2(m' • f )f - m']. (5.15) Exercise 5.2 What is the magnetic potential of an infinitely long cylinder
magnetized in the direction parallel to its axis?
Note that the magnitude of the magnetic field of a line of dipoles is pro- portional to its dipole moment and inversely proportional to the square of its perpendicular distance from the observation point.
5.5 Two-Dimensional Distributions of Magnetization