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into Eqs. (
11.10
) and (
11.11
) we get the estimate ıB
4 pT which is consistent
in magnitude with the signals observed during underground detonations. At the
distance exceeding approximately 10 km, the amplitude of the signals falls off below
the level of background noise.
The maximal radius R of the explosion cavity is proportional to Y
1=3
where Y
is the TNT equivalent of the explosion (e.g., see Chadwick et al.
1964
; Rodionov
et al.
1971
). This implies that the magnitude of the GMPs in Eqs. (
11.10
) and
(
11.11
) is proportional to Y . The similar relationship holds true for the electric field
variations that contradicts the empirical dependence given by Eq. (
11.2
). This means
that the simplified model considered above does not describe the EMP effect quite
adequately.
To estimate the EMP relaxation time due to return diffusion of the magnetic field
into the plasma, we now suppose that the cooling of the uniformly expanding plasma
follows the adiabatic law. The adiabatic equation of a perfect gas reads
D
T
0
R
3.1/
TR
3.1/
;
(11.12)
0
where stands for the adiabatic exponent and the subscript zero is related to the
initial values of the plasma temperature and the radius of underground cavity.
Considering the moment of the cavity stoppage and substituting the numerical
values R=R
0
D
30 and
D
5=3 into Eqs. (
11.3
), (
11.4
), and (
11.12
), we find
that at this moment the plasma temperature and conductivity are T
1-10 eV and
p
1:5
10
3
-4
10
4
S=m. Substituting these values into Eq. (
11.5
)givesthe
rough estimate
d
0:4-10 s which is compatible with the duration of the EMP
signals shown in Figs.
11.2
and
11.3
.
In the strict sense, the amplitude estimates given by Eqs. (
11.10
) and (
11.11
)
are valid in the extreme case of a perfectly conducting plasma ball. To study the
effect of finite plasma conductivity we consider the expanding uniform plasma ball
situated in the rock at higher depth. The conductivity and radius of the plasma ball
are assumed to be given functions of time; that is
p
D
p
.t/ and R.t/
D
R
0
LJ.t/,
where R
0
is the initial ball radius (Ablyazov et al.
1988
). In this model the rock
conductivity is much smaller than the plasma one. A detailed analysis of this
problem presented in Appendix I has shown that the ULF GMPs outside the ball
can be qualified as magnetic dipole field. The solution of the problem is represented
as a series with respect of eigenfunctions of the problem. The effective magnetic
moment of the plasma ball can be found from Eq. (
11.80
)
0
1
Z
Z
t
t
X
12R
0
B
0
LJ.t/
0
dLJ
2
dt
0
2
n
2
0
p
R
0
LJ
2
dt
00
1
n
2
@
A
dt
0
: (11.13)
M
.t/
D
exp
n
D
1
0
t
0
In the limit
p
!1
we get
R
2
.t/
R
0
1
;
12.2/R
0
R.t/
B
0
0
M
.t/
D
(11.14)
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