Geology Reference
In-Depth Information
the magnetic fields is known as
Ampere's law
.It
relates the spatial structure of the magnetic field
B
to the geometry of the electric currents and to
the temporal variations of the electric field.
Assuming that the latter does not change in
time, the differential form of Ampere's law can
be written as follows:
r
B
D
0
j
(3.27)
If
j
D
0
in a region
R
, hence in absence of
currents, we have that
r
B
D
0
in
R
. Therefore,
for any point in a region where the current density
is zero, there exists a scalar field
V
D
V
(
r
)such
that:
Fig. 3.8
Discretization of Laplace's equation. The region
R
is divided into a set of squared grid cells of dimension
h
.
It is assumed that the values of the potential
V
are known
along the boundary (
R
), which is represented by the
grayed cells. The equation is solved calculating iteratively
the values of
V
at each point (
x
,
y
) in the interior of
R
(
white
cells)
B
Dr
V
(3.28)
The field
V
is called
scalar magnetic potential
.
If we combine (
3.28
) with (
3.25
), we obtain the
following fundamental equation, which is valid
in current
free regions:
A numerical solution to this equation can be
found by discretization of the domain
R
through
squared grid cells. In this approach, we subdivide
the region
R
in small grid cells of dimension
h
(Fig.
3.8
) and search for an approximate solution
at the centre of each cell. Let
V
D
V
(
x
,
y
)bea
solution of (
3.30
).
2
V
D
0
r
(3.29)
This second-order differential equation is
called
Laplace's equation
, and its solutions are
harmonic functions
. It represents the fundamental
tool for the study of the Earth's magnetic and
gravity fields, which form a substantial portion
of a branch of geosciences known as
potential
a solution of this equation in a region
R
can be
found if
boundary conditions
have been assigned
along the frontier (
R
)of
R
, that is, if the values
of
V
are known on the closed surface (
R
). Once
a solution
V
D
V
(
r
) has been determined in
R
,we
can use (
3.28
) to determine uniquely the vector
field
B
D
B
(
r
) in that region.
To understand the basic properties of the har-
monic functions, let us consider now the example
of a two-dimensional potential, depending only
from
x
ed
y
. In this instance, Laplace's equation
assumes the form:
If we expand
V
in a Taylor series with respect
to variable
x
, it results:
8
<
2
h
2
@
2
V
V.x
C
h;y/
D
V.x;y/
C
h
@V
1
@x
C
@x
2
6
h
3
@
3
V
1
@x
3
C
:::
V.x
h;y/
D
V.x;y/
h
@V
C
2
h
2
@
2
V
:
1
@x
C
@x
2
6
h
3
@
3
V
1
@x
3
C
::
(3.31)
Summing these two expressions, it results:
V.x
C
h;y/
C
V.x
h;y/
D
2V .x;y/
C
h
2
@
2
V
@x
2
C
O
h
4
(3.32)
@
2
V
@x
2
C
@
2
V
@y
2
D
0
An analog expression can be found by expand-
ing
V
in a Taylor series with respect to variable
y
:
(3.30)
