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Fields and potential due to a surface electric dipole layer
A surface electric dipole layer is a neutral charge layer with an electric dipole moment per
unit area directed perpendicular to the surface. It can be modeled as two surface charge layers,
(
r
,
)
and
−
(
r
,
)
, lying on each side of the surface defined by
F
(
r
)
=
0. The unit vector
n
=
∇
F
(
r
)/∇
F
(
r
)
is directed from the negative surface charge density to the positive surface
charge density (it’s sufficient to replace
F
(
r
)
by
−
F
(
r
)
in order to adjust the sense of
n
). The
charge layers lie on the surfaces
F
(
r
±
n
/
2
)
and the surface dipole moment density is
d
(
r
)
=
→
0
lim
n
(
r
,
)
=
d
(
r
)
n
→
0
lim
(
r
,
)
=
0
→
0
lim
(
r
,
)
=
d
(
r
)
1.103
Since the surface is neutral (total charge
=
0), in the limit that
→
0 with
(
r
,
)
fixed
,
n
(
r
)
(
1
E
1
(
r
) −
2
E
2
(
r
))
=
o
for electric dipole surface layer
1.104
From Gauss’ law one can determine the electric field contributions,
E
+
and
E
−
, and from
Eq. (1.99) the potential field contributions,
+
and,
−
, from each individual layer . The latter
contributions are shown in the schematics below. The total
E
1
and
1
(above the dipole layer)
and total
E
2
and
2
(below the dipole layer) are given by
E
+
+
E
−
and
+
+
−
.in each region
Schematic of the
E
field contributions
(
r
,
)
n
1
/
2
↑
++++++++++
−
(
r
,
)
n
1
/
2
↓
−
(
r
,
)
n
2
/
2
↓
−−−−−−−−−
(
r
,
)
n
2
/
2
↑
where
n
1
=
n
2
and the negative sign on the field due to the lower layer comes from the
negative charge ”surface layer”.
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View Full DocumentSchematic of the
field contributions
− 
z

(
r
,
)/
2
++++++++++
− 
z

(
r
,
)/
2

z

(
r
,
)/
2
−−−−−−−−−

z

(
r
,
)/
2
Thus the total electric fields are:
E
1,
total
(
r
,
)
=
(
r
,
)
n
1
/
2
−
(
r
,
)
n
2
/
2
=
0 above the dipole layer
E
2,
total
(
r
,
)=−
(
r
,
)
n
1
/
2
+
(
r
,
)
n
2
/
2
=
0 below the dipole layer
giving
n
1
[
E
1,
total
−
E
2,
total
]=
0
Between the two layers for finite
the electric field is constant, directed downward and equal
to,
E
3,
total
(
r
,
(
r
,
)
n
1
/
2
−
(
r
,
)
n
2
/
2
=−
(
r
,
)
n
1
/
.
The total potential above the dipole layer (where

z
2

=
+ 
z
1

)is
1,
total
(
r
,
)
=
+
(
r
,
) +
−
(
r
,
)=−
z
1

(
r
,
)/
2
+ 
z
2

(
r
,
)/
2
=
(
r
,
)/
2
Below the dipole layer (where

z
1

=
+ 
z
2

)
.
2,
total
(
r
,
)
=

z
2

(
r
,
)/
2
− 
z
1

(
r
,
)/
2
(
r
,
)/
2
,
and for finite
between the two charge layers where

z
2

=
− 
z
1

the total potential is
3,
total
(
r
,
z
1

(
r
,
)/
2
+ 
z
2

(
r
,
)/
2
=
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 Spring '10
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