Digital Signal Processing Reference
In-Depth Information
we can recycle equation (3-2) to derive the boundary conditions of the normal
components of the electric field at the interface between two dielectrics:
(n
·
ε
1
E
1
)
−
(n
·
ε
2
E
2
)
=
ρ
(3-4)
If the surface charge density between the dielectric layers is assumed to be zero
(usually, a valid assumption), the relationships between the electric fields in both
regions is described by
n
·
ε
1
E
1
=
n
·
ε
2
E
2
(3-5)
Equation (3-5) means that
the normal component of the electric field is not con-
tinuous across a dielectric boundary
.
In Section 2.7.2 it was mentioned that the tangential component of the electric
field must remain continuous across a dielectric boundary. This can be shown
with the integral form of Faraday's law for the electrostatic case:
l
E
·
dl
=
0
(3-6)
If we integrate (3-6) around a closed differential contour that encompasses the
dielectric boundary such as that shown in Figure 3-6, we can calculate the tan-
gential components of the electric field:
b
c
d
a
l
E
dl
E
dl
·
=
·
+
E
·
dl
+
E
·
dl
+
E
·
dl
(3-7)
a
b
c
d
Since we are considering the behavior at the surface (
h
→
0), segments
da
and
bc
can be eliminated. Furthermore, the tangential segments
ab
and
cd
are equal
but opposite, which means that (3-7) can be simplified to
(E
1
t
−
E
2
t
)l
=
0
→
E
1
t
=
E
2
t
(3-8)
Equation (3-8) means that
the tangential components of the electric field across
a dielectric boundary must remain continuous
.
Assume that an electric field
E
1
is incident on a boundary between two
dielectrics as shown in Figure 3-7. The change in the orientation of the electric
∆
l
a
b
e
1
e
2
∆
h
d
c
Figure 3-6
Differential contour encompassing a dielectric boundary.
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