Geoscience Reference
In-Depth Information
The continuity of normal component of the magnetic induction results in
B
1
.R
c
0/
D
B
1
.R
c
C
0/; B
1
.R
e
0/
D
B
1
.R
e
C
0/;
(11.87)
Eliminating the function B
2
from the set of Eqs. (
11.83
) and (
11.84
), we obtain
rd
r
B
1
C
4d
r
B
1
D
2
0
C
m
Jd
r
s
rr
(11.88)
The general solution of Eq.(
11.88
) is given by
Z
r
r
0
2
s
rr
r
0
dr
0
C
2
0
C
m
J
r
3
c
1
r
3
C
c
2
;
B
1
D
(11.89)
R
c
where c
1
and c
2
are arbitrary constants.
At the regions r<R
c
and r>R
e
the solution of problem should be limited
as r
!
0 and r
! 1
. So, we obtain that B
1
D
c
3
if r<R
c
and B
1
D
c
4
=r
3
if r>R
e
where c
3
and c
4
are the arbitrary constants. These solutions should fit
Eq. (
11.89
) at the boundaries r
D
R
c
and r
D
R
e
. Taking into account the boundary
conditions given by Eqs. (
11.85
)-(
11.87
) one can find the constants c
1
c
4
. Whence
it follows that c
1
D
c
2
D
c
3
D
0. So the magnetic field is equal to zero in the inner
area at r<R
c
. For the region R
c
<r<R
e
one can find
Z
r
r
02
s
rr
r
0
dr
0
;
2
0
C
m
J cos
r
3
B
r
D
(11.90)
R
c
0
1
Z
r
r
02
s
rr
r
0
dr
0
s
rr
.
r
/
1
r
3
@
A
:
B
D
0
C
m
J sin
(11.91)
R
c
Note that these formulas are more correct than that obtained by Surkov (
1989
)
in the framework of simplified approach which leaves out of account the boundary
conditions and thereby the contribution of the surface magnetization currents at r
D
R
c
and r
D
R
e
.
References
Ablyazov MK, Surkov VV, Chernov AS (1988) Distortion of an external magnetic field by an
expanding plasma sphere located in a slightly conductive semispace. J Appl Mech Tech Phys
29(6):778-784
Adushkin VV, Soloviev SP (1988) Low-frequency electric fields in the atmospheric surface layer
during underground explosion. Rep USSR Acad Sci (Dokl Akad Nauk SSSR) 299(4):840-844
(in Russian)
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