Geoscience Reference
In-Depth Information
Combining Eqs. (
11.74
) and (
11.79
), we finally obtain the dimensionless magnetic
moment of the plasma ball
0
1
Z
Z
t
t
X
dLJ
2
dt
0
2
n
2
0
p
R
0
LJ
2
dt
00
3LJ.t/
2
1
n
2
@
A
dt
0
:
D
exp
(11.80)
n
D
1
t
0
0
Residual Magnetic Field
First of all consider Eq. (
11.23
) in the region R
c
<r<R
e
. All the values in
Eq. (
11.23
) are independent of azimuthal angle due to the cylindrical symmetry of
the problem. This implies that only azimuthal component of the curl is nonzero.
Therefore, substituting of Eq. (
9.29
)for
J
into Eq. (
11.23
)gives
1
r
Œ@
r
.rB
/
@
B
r
D
0
C
m
Jd
r
s
rr
sin ;
(11.81)
where B
r
and B
are the radial and tangential components of magnetic field and
d
r
denotes derivative with respect to r, that is d
r
s
rr
D
ds
rr
=dr. Maxwell equation
r
B
D
0 can be written in the form
r
2
@
r
r
2
B
r
C
1
1
r sin
@
.sin B
/
D
0:
(11.82)
We seek for the solution of Eqs. (
11.81
) and (
11.82
) in the form B
r
D
B
1
.r/ cos
and B
D
B
2
.r/ sin , where B
1
.r/ and B
2
.r/ are unknown functions. This yields
d
r
.rB
2
/
C
B
1
D
r
0
C
m
Jd
r
s
rr
;
(11.83)
and
d
r
r
2
B
1
C
2rB
2
D
0:
(11.84)
For the inner and outside areas, i.e. at r<R
c
and r>R
e
, the right-
hand side of Eq. (
11.83
) is equal to zero whereas Eq. (
11.84
) is valid in the
whole space. Integrating of Eq. (
11.83
) over short intervals .R
c
";R
c
C
"/ and
.R
e
";R
e
C
"/, where "
!
0, gives the boundary conditions for tangential
component B
2
B
2
.R
c
C
0/
B
2
.R
c
0/
D
0
C
m
Js
rr
.R
c
/;
(11.85)
B
2
.R
e
C
0/
B
2
.R
e
0/
D
0
C
m
Js
rr
.R
e
/:
(11.86)
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