22 Phasor form of Maxwell’s equations and damped
waves in conducting media
•
When the fields and the sources in Maxwell’s equations are all monochro
∇
·
D
=
ρ
∇
·
B
= 0
∇ ×
E
=

∂
B
∂
t
∇ ×
H
=
J
+
∂
D
∂
t
.
matic functions of time expressed in terms of their phasors, Maxwell’s
equations can be transformed into the phasor domain.
–
In the phasor domain all
∂
∂
t
→
j
ω
and all variables
D
,
ρ
, etc. are replaced by their phasors
˜
D
,
˜
ρ
,
and so on. With those changes Maxwell’s equations take the form
shown in the margin.
∇
·
˜
D
=
˜
ρ
∇
·
˜
B
= 0
∇ ×
˜
E
=

j
ω
˜
B
∇ ×
˜
H
=
˜
J
+
j
ω
˜
D
–
Also in these equations it is implied that
˜
D
=
˜
E
˜
B
=
μ
˜
H
˜
J
=
σ
˜
E
where
,
μ
, and
σ
could be a function of frequency
ω
(as, strictly
speaking, they all are as seen in Lecture 11).
–
We can derive from the phasor form Maxwell’s equations shown
in the margin the TEM wave properties obtained earlier on using
the timedomain equations by assuming
˜
ρ
=
˜
J
= 0
.
1
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We will do that, and and after that relax the requirement
˜
J
= 0
with
˜
J
=
σ
˜
E
to examine how TEM waves propagate in conducting media.
•
With
˜
ρ
=
˜
J
= 0
the phasor form Maxwell’s equation take their simpli
fied forms shown in the margin.
∇
·
˜
E
= 0
∇
·
˜
H
= 0
∇ ×
˜
E
=

j
ω
μ
˜
H
∇ ×
˜
H
=
j
ω
˜
E
–
Using
∇ ×
[
∇ ×
˜
E
=

j
ω
μ
˜
H
]
⇒
∇
2
˜
E
=

j
ω
μ
∇ ×
˜
H
which combines with the Ampere’s law to produce
∇
2
˜
E
+
ω
2
μ
˜
E
= 0
.
–
For
x
polarized waves with phasors
˜
E
= ˆ
x
˜
E
x
(
z
)
,
the phasor wave equation above simplifies as
∂
2
∂
z
2
˜
E
x
+
ω
2
μ
˜
E
x
= 0
.
–
Try solutions of the form
˜
E
x
(
z
) =
e

γ
z
or
e
γ
z
where
γ
is to be determined.
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 Spring '08
 Kim
 γ, Wave propagation, e−αz cos, Damped wave snapshot

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