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Z
G
k
1
t
0
G
k
2
t
0
C
dt
0
k
1
k
2
./
D
(4.79)
1
and the constants h
k
1
k
2
are given by
Z
G
k
1
;k
2
t
0
dt
0
:
h
k
1
k
2
D
(4.80)
1
In practice, the mean value of the magnetic variations generated by the lightning
discharge is of little importance since it is close to zero. This means that integrals in
Eq. (
4.80
) tend to zero and thus the constants h
and h
'
are negligible.
Using the statistical independence of random values M
m
yields
h
M
m
1
ih
M
m
2
iDh
M
m
i
2
; if m
1
¤
m
2
ǝ
M
m
1
Ǜ
D
ǝ
M
m
Ǜ
; if m
1
D
m
2
h
M
m
1
M
m
2
iD
(4.81)
Taking into account Eqs. (
4.78
)-(
4.81
) and making allowance for
h
M
m
iDh
M
i
and
ǝ
M
m
Ǜ
D
ǝ
M
2
Ǜ
,wearriveat
n
ǝ
M
2
Ǜ
T
D
.n/
b
x
.t/b
x
.t
C
/
E
./ cos
2
'
C
''
./ sin
2
'
c
./ sin 2'
D
n
2
n
h
M
i
2
h
cos '
h
'
sin '
2
:
C
(4.82)
T
2
where
2
ǚ
'
./
C
'
./
1
c
./
D
(4.83)
Taking the notice of a useful relation followed from the Poisson distribution:
ǝ
n
2
n
Ǜ
Dh
n
i
2
D
2
T
2
;
(4.84)
and substituting Eq. (
4.82
) into Eq. (
4.75
) yields
ǝ
b
x;
.t/b
x;
.t
C
/
Ǜ
D
D
M
2
E
r
;
cos
2
'
C
''
r
;
sin
2
'
c
r
;
sin 2'
C
ǝ
b
x;
r
;t
Ǜ
2
:
(4.85)
Here we have restored the index and arguments of the functions.
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