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Notes 9 fall 2004.doc
Conductive materials
Ampere’s law in free space:
E
j
H
o
G
G
G
ωε
=
×
∇
In a conductive medium it becomes:
( )
E
j
E
E
j
j
H
G
G
G
G
G
ε′
ω
+
ε ′
′
ω
=
ε ′
′
−
ε′
ω
=
×
∇
Alternatively we’ve written it as
E
j
E
J
J
H
d
c
G
G
G
G
G
G
ε′
ω
+
σ
=
+
=
×
∇
Comparing the first terms of the last two equations we can see that
ε′
′
ω
=
σ
, or
ω
σ
=
ε ′
′
We can then write the permittivity for conductive media as
ω
σ
−
ε′
=
ε′
′
−
ε′
=
ε
j
j
.
Dividing the boxed equation by
we find the loss tangent that we mentioned earlier:
ε′
tangent
loss
⇒
ε′
ω
σ
=
ε′
ε′
′
Note that the ratio of the magnitudes of the phasor conduction current density to the phasor
displacement current density is
°
−
∠
ε′
ω
σ
=
ε′
ω
σ
=
ε′
ω
σ
=
90
J
J
j
E
j
E
J
J
d
c
d
c
which shows that the displacement current density
leads
the conduction current density by
(think ICE)
°
90
1
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Slightly conducting media
We can make a useful approximation if we recall that for
1
x
<<
,
.
The
approximation gets better as x gets smaller (try it yourself).
()
nx
1
n
x
1
±
≅
±
A slightly conducting
medium is defined by a loss tangent that is much less than one, i.e.,
1
<<
ε′
ω
σ
or
.
In this case the complex propagation constant,
, can be
c
d
J
J
>>
jk
approximated
by
()
()
ε′
µ
ω
+
ε′
µ
σ
=
ε′
µ
ω
+
ε′
ω
ε′
µ
ωσ
=
⎟
⎠
⎞
⎜
⎝
⎛
ε′
ω
σ
−
ε′
µ
ω
≅
⎟
⎠
⎞
⎜
⎝
⎛
ε′
ω
σ
−
ε′
µ
ω
=
−
ε′
µ
ω
=
ε′
′
−
ε′
µ
ω
=
µε
ω
=
ω
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This note was uploaded on 01/20/2012 for the course EE 4460 taught by Professor Czarnecki during the Fall '10 term at LSU.
 Fall '10
 Czarnecki

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