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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
 
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