ECE329
Fall 2008
Homework 5 — Solution
October 23, 2008
1. Let us consider the following four plane waves in free space,
E
1
=
2 cos(
ωt

βz
)ˆ
x
V
m
E
2
=
E
o
(cos(
ωt

βz
)ˆ
x

sin(
ωt

βz
)ˆ
y
)
V
m
H
3
=
cos(
ωt
+
βz
+
π
3
)ˆ
x
+ sin(
ωt
+
βz

π
6
)ˆ
y
A
m
H
4
=
cos(
ωt

βx
)ˆ
z
+ sin(
ωt

βx
)ˆ
y
A
m
.
a) The electric and magnetic fields (
E
and
H
) of uniform plane waves are perpendicular to each
other and to the direction of propagation, therefore, it can be verified that such fields satisfy the
following relation
E
=
η
H
×
ˆ
β,
where
ˆ
β
is the unit vector parallel to the propagation direction and
η
is the intrinsic impedance.
Using this relation, we can find the expressions for
H
or
E
that accompany the waves given
above,
ˆ
β
1
=
ˆ
z
→
H
1
=
2
η
o
cos(
ωt

βz
)ˆ
y
A
m
ˆ
β
2
=
ˆ
z
→
H
2
=
E
o
η
o
(cos(
ωt

βz
)ˆ
y
+ sin(
ωt

βz
)ˆ
x
)
A
m
ˆ
β
3
=

ˆ
z
→
E
3
=
η
o
(
cos(
ωt
+
βz
+
π
3
)ˆ
y

sin(
ωt
+
βz

π
6
)ˆ
x
)
V
m
ˆ
β
4
=
ˆ
x
→
E
4
=
η
o
(cos(
ωt

βx
)ˆ
y

sin(
ωt

βx
)ˆ
z
)
V
m
.
b) The instantaneous power flow density is given by the Poynting vector
P
=
E
×
H
.
Therefore,
the instantaneous power that crosses some surface
S
is given by
P
=
´
S
P ·
d
S
.
In the case of
uniform plane waves, this expression simplifies to
P
=
A
ˆ
n
· P
,
where
ˆ
n
is the vector normal to the flat area
A
. Below, we are considering
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 Fall '08
 FRANKE
 Polarization, Electromagnet, 2W, 2 w, 2 j

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