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The problem is solved in the Price-Sheinmann thin-sheet approximation.
On
the
Earth's
surface
the
electric
and
magnetic
fields
E
τ
(
E
x
,
E
y
)
and
H
(
H
x
,
H
y
,
H
z
) are determined from the integral equation
E
N
E
A
τ
E
N
[
G
E
(
r
E
τ
(
r
)
=
τ
+
(
r
)
=
τ
+
|
r
s
)]
S
1
(
r
s
)
E
τ
(
r
s
)
ds
(7
.
99a)
V
and the integral relation
H
N
H
A
τ
H
N
[
G
H
(
r
H
(
r
)
=
τ
+
(
r
)
=
τ
+
|
r
s
)]
S
1
(
r
s
)
E
τ
(
r
s
)
ds
,
(7
.
99b)
V
where
E
N
τ
(
E
x
,
E
y
)
H
N
τ
(
H
x
,
H
y
) and
E
A
τ
(
E
x
,
E
y
)
H
A
τ
(
H
x
,
H
y
,
,
) are the normal
and anomalous electric and magnetic fields,
S
1
is the excessive
conductance, [
G
E
] and [
G
H
] are the electric and magnetic Green tensors for the
horizontally layered normal background.
Let the distance to the anomalous domain
V
be much greater than its maximum
diameter. Turn to (7.99a) and (7.99b) and represent the anomalous electromagnetic
field
E
A
S
1
(
r
s
)
=
S
1
(
r
s
)
−
H
A
observed far avay from the domain
V
as the field of an equivalent
electric dipole located at a point O in the middle of
V
,
.
Taking the Green tensors
outside the integrals, we use cylindrical coordinates
r
,
,
z
with the origin at the
equivalent dipole, and write
[
G
E
]
E
A
[
G
E
]
P
τ
=
S
1
(
r
s
)
E
τ
(
r
s
)
ds
=
V
[
G
H
]
(7
.
100)
H
A
[
G
H
]
P
τ
=
S
1
(
r
s
)
E
τ
(
r
s
)
ds
=
,
V
where
P
=
S
1
(
r
s
)
E
τ
(
r
s
)
ds
(7
.
101)
V
is the moment of the equivalent electric dipole. Note that
E
τ
and
P
tend to zero as
→
.
Using cylindrical coordinates
r
0
,
,
z
with the origin at the point O, we write
Q
1
(
r
)
r
dQ
1
(
r
)
dr
P
r
S
1
dQ
4
(
r
)
dr
P
S
1
Q
4
(
r
)
r
E
r
E
A
=
+
,
=
+
,
1
r
dQ
3
(
r
)
dr
Q
3
(
r
)
r
dr
r
dQ
2
(
r
)
d
Q
2
(
r
)
r
2
H
r
H
A
H
z
=
P
,
=−
P
r
,
=−
P
−
,
dr
.
(7
102)
where
P
r
=
·
1
r
,
P
=
·
1
,
=
/
,
1
=
1
r
×
1
z
,
=
P
r
1
r
+
P
1
.
P
P
1
r
r
r
P