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- The distance between the source domain I and the observation domain S is much
larger than maximum radii
r
i
and
r
s
of these domains. Implying that the normal field
rather slowly varies along the Earth's surface, we assume that
,
.
H
y
H
z
H
z
H
x
(5
.
4)
y
z
x
z
Under these conditions, we write the Maxwell equations (5.2) in the form
H
y
H
y
H
x
H
x
−
N
E
x
,
N
E
y
,
=−
=
=
0
,
z
z
x
y
(5
.
5)
E
y
E
y
E
x
E
x
−
o
H
x
,
o
H
y
,
o
H
z
=−
i
=
i
=
i
.
z
z
x
y
Such a field can be expressed in terms of a scalar function
U
(
x
,
y
,
z
):
2
U
2
U
1
1
E
x
=−
z
,
E
y
=
z
,
E
z
=
,
0
y
x
N
N
2
U
2
U
=
U
=
U
1
+
H
x
H
y
H
z
x
,
y
,
=
.
i
o
N
z
x
2
y
2
(5
.
6)
It is easy to verify that the function
U
(
x
,
y
,
z
) meets the one-dimensional
equation
1
N
U
+
i
o
U
=
0
(5
.
7)
z
z
with boundary condition
U
|
z
=
0
=
U
o
(
x
,
y
) at the Earth's surface and continuity
N
U
1
conditions for
U
and
z
at the boundaries between layers.
Proceeding from (5.7), we represent the function
U
as
U(
x
,
y
,
z
)
=
U
o
(
x
,
y
)
(
z
)
,
(5
.
8)
where
(
z
) is the solution of the one-dimensional problem
1
d
dz
d
dz
+
i
o
=
0
,
z
≥
0
(5
.
9)
N
with boundary conditions
|
z
=
0
=
1 at the Earth's surface, continuity conditions for
1
N
d
dz
(
z
) and
at the boundaries between layers and condition
→
0at
z
→∞
.
