Laboratory Study of Rock Deformation and Fracture - Ultra and Extremely Low Frequency Electromagnetic Fields - page 341

absent, then the potential difference between plates of the capacitor is equal to zero.
In this case we have

† † 1 C † 2
2" 2

† 2 † 1 †
2" 1

x C .l x/ D 0: (9.19)

Here † is the surface charge density on the SW, l is a thickness of the dielectric,
x D V c t is a distance traversed by the SW and s D 1 C m is a coefficient of shock
compression of the material. Taking into account that † D † 1 C † 2 one can find
the current density in the external circuit:

d† 1
dt D

‚t 0 †
Œ‚t 0 C t .1 ‚/ 2 ;

j D (9.20)

where ‚ D " 2 s=" 1 and t 0 D l=V c is time of the SW propagation through the
dielectric with initial thickness l.
It follows from Eq. ( 9.20 ) that the electric current density shows a maximum
j m D j .0/ D †=.‚t 0 /; that is, the current is not equal to zero at the initial moment.
This result is due to the above approximation whereas, in reality, the current in the
external circuit is formed during a finite time t r RC, where C is the capacitance
of elements of the external circuit and R is the load resistance. In other respects,
Eq. ( 9.20 ) gives a satisfactory approximation of the initial part of the signal shown
in Fig. 9.2 , at least of the interval between the first and second spikes.
Taking account of Eq. ( 9.18 )for† we get an estimate of the current density
magnitude:

2qa 2 M m
" 2 l .1 C m / :

" 1 V c †
" 2 sl D

j m D (9.21)

Using the above parameters as well as m D 0:2 and l D 0.1-1 cm, we find that
j m .0:2 2/ 10 2 A/m 2 which agrees with experimental data within an order of
magnitude.
In the inverse case, if the relaxation time of shock polarization r t 0 ,the
qualitative distinctions of the current behavior will occur. Suppose that the matter
behind the front of the SW becomes conductive with a constant conductivity 2 .In
such a case the charge relaxation time due to the conductivity of matter is of the
order of r D " 2 " 0 = 2 so that the above condition can be written as " 2 " 0 = 2 t 0 .
As before the plane SW front carries a surface charge with density †. To be specific
in Fig. 9.4 this charge is chosen as positive. The rise of conductivity results in the
formation of compensated charges with opposite sign behind the SW front. These
charges are not shown in Fig. 9.4 because this figure displays the case of medium
keeping of dielectric property behind of the SW.
Now we shall estimate a thickness of this oppositely charged layer and a potential
difference between the capacitor sides. We use a reference frame and coordinate
system connected with the moving SW front. The y-axis is directed oppositely to