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⎧
⎨
)
sin
1
−
1
1
+
a
3
r
2
E
y
(
−
r
+
cos
for
r
≥
a
1
2
U
(
r
,
,
,
)
=
(7
.
7)
1
1
+
3
⎩
E
y
(
−
)
r
sin
cos
for
r
≤
a
,
1
2
where
E
y
(
E
y
(
x
) is the normal electric field on the Earth's sur-
face. The function
U
satisfies the boundary conditions
)
=
,
y
,
z
=
0
,
=
π/
2
−
0
=
r
=
a
+
0
=
r
=
a
−
0
.
U
1
U
1
1
U
|
r
=
a
+
0
=
|
r
=
a
−
0
0
U
U
r
r
On differentiating
U
, we get the electric field along the
y
-axis:
=
/
2
=
0
)
x
=
0
=−
U
(
r
,
,
,
)
E
y
(
y
,
)
=
E
y
(
x
,
y
,
z
=
0
,
r
⎧
⎨
⎩
1
E
y
(
2
1
−
1
1
+
a
3
(7
.
8)
+
)f r
|
y
| ≥
a
1
2
3
|
y
|
=
1
1
+
3
E
y
(
|
y
| ≤
.
)
for
a
1
2
The corresponding magnetic field can be determined by the Bio-Savart law (inte-
grating excess currents inside and outside the hemisphere). The estimation shows
that at
a
<<
h
1
the magnetic effect of the hemisphere is negligibly small within the
S
1
- and
h
-intervals. So, we can write
H
x
(
x
,
y
,
z
=
0
,
)
=
H
x
, where
H
x
is the
normal magnetic field at the Earth surface. On simplest mathematics we get
⎧
⎨
1
Z
N
(
2
1
−
1
1
+
a
3
+
)f r
|
y
| ≥
a
E
y
(
y
,
)
1
2
3
|
y
|
Z
yx
(
y
,
)
=−
=
(7
.
9)
1
1
+
3
⎩
H
x
Z
N
(
)
for
|
y
| ≤
a
1
2
and
Z
yx
(
y
)
2
,
yx
(
y
,
=
)
o
⎧
⎨
1
2
2
1
−
1
1
+
a
3
.
(7
10)
=
N
(
+
N
(
)
)f r
y
≥
a
1
2
3
|
y
|
=
3
2
⎩
1
1
+
=
N
(
)
N
(
)
for
y
≤
a
,
1
2
2
= |
|
/
o
is the normal apparent resistivity and
,
are the real
where
Z
N
N
frequency-independent distortion factors
