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a
b
x
z
x
B
0
d
B
v
r
ʸ
E
j
y
d
eff
y
a
a
d
B
Fig. 10.11
A model of conducting magma upwelling along the volcano vent. (
a
) The magma
velocity vector
V
is positive parallel to
z
axis while the effective magnetic moment
d
eff
is
perpendicular to both
V
and the Earth's magnetic field
B
0
.(
b
) A configuration of total current
j
flowing across the vent. The
circles with point and cross
inside indicate the directions of magnetic
field perturbation ı
B
where
d
eff
is the effective magnetic moment per unit length and a is the radius of the
cylinder. Let y axis be positive parallel to the vector
d
eff
. Thus, the vectors
V
and
B
0
must be in x
z
plane as shown in Fig.
10.11
a. We use a polar coordinate system in
which r is the distance from the
z
-axis and .0
</ is the polar angle between
the x-axis and the vector
r
. Taking into account the symmetry of the problem and
the fact that the moment (
10.50
) is normal to the cylinder axis, one can find that the
GMPs have only
z
-projection
0
d
eff
r cos
2a
2
ıB
z
D
; .r<a/
I
(10.51)
and
0
d
eff
cos
2r
ıB
z
D
; .r>a/;
(10.52)
where d
eff
Dj
d
eff
j
. On account of
m
r
,Eqs.(
10.51
)-(
10.52
) can be simplified.
For example, in the region r>awe obtain
ıB
z
D
0
r
a
2
B
0x
V
cos
r
:
(10.53)
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