Geoscience Reference
In-Depth Information
These currents play a role of the source function which generates the conduction
current
j
D
E
in the medium. The effective magnetic moments,
M
, of the extrinsic
current system is given by (
7.4
). Substituting Eq. (
7.54
)for
j
e
into Eq. (
7.4
)for
M
and taking the triple cross product in the integral (
7.4
), we obtain
Z
Z
2
2
.
r
.
V
B
0
//dV
0
D
f
V
.
r
B
0
/
B
0
.
r
V
/
g
dV
0
:
M
D
(7.55)
V
0
V
0
As before we use the coordinate system whose
z
axis is directed along the
external field
B
0
. In calculating the integral in Eq. (
7.55
), we apply spherical
coordinates r, and , where the polar angle is measured from the
z
axis and
the azimuthal angle is measured from the positive direction of x axis. A small
element of volume has the form dV
0
D
r
2
sin drdd. Taking into account the
symmetry of the problem, only
z
-component of the vector
M
has to be nonzero.
Substituting Eq. (
7.40
)for
V
D
V .r;t/
O
r
into Eq. (
7.55
), integrating over and
over , and taking into account that @
t
f
D
C
l
@
r
f , yields
Z
r
l
r
r@
r
f
@
r
f
dr:
M
D
B
0
SC
l
R
0
3
(7.56)
0
Integrating Eq. (
7.56
) in parts we come to
Z
r
l
fdr
B
0
u
0
SC
l
t:
M
D
B
0
SC
l
R
0
(7.57)
0
As is seen from Eq. (
7.57
) the magnetic moment increases with time directly
proportional to t. To gain better understanding of this result let us recall that the
magnetic moment is proportional, first, to the magnitude of the mass velocity and,
second, to the volume occupied by the extrinsic current. The magnetic field and
current distribution around the acoustic source is sketched in Fig.
7.5
. Actually
the azimuthal currents shown in this figure with dashed lines 3 are predominantly
concentrated in the vicinity of the acoustic wave front shown with line 1. Since the
characteristic size or wavelength,
a
, is supposed to be short, the volume occupied
by the currents is as much as 4r
l
a
. Due to the fact that r
l
D
C
l
t ,thevolume
increases with time as t
2
. On the other hand, far away from the seismic source
the amplitude of mass velocity falls off inversely proportional to distance. Hence,
at the front of seismic wave the velocity magnitude changes as r
l
, i.e., inversely
proportional to t. Taking together these two factors results in an enhancement of the
magnetic moment in time according to Eq. (
7.57
). Notice that this conclusion holds
true for the time interval (
7.49
).
As is evident from Eq. (
7.57
), the effective magnetic moment
M
is directed
oppositely to the vector of external magnetic field. This so-called diamagnetic
effect occurs whenever the conductor moves in the external magnetic field (e.g.,
see Fedorovich
1969
and Landau and Lifshits
1982
).
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