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UNIVERSITY OF CALIFORNIA
Santa Barbara
Department of Electrical and Computer Engineering
Problem Set No. 3
Issued:
October 17, 2007
Fall 2007
Due:
October 24, 2007
ECE 201A
1.
A general plane wave in an isotropic, homogenous, uniform and a source free medium can be
described as
0
jk r
EE
e
−
=
G
G
i
G
G
(a)
Using Gauss' law show that
0
kE
=
G
G
i
.
(b)
Substitute the given solution
0
jk r
e
−
=
G
G
i
G
G
into the homogenous wave equation
22
0
ωμε
∇+
=
G
G
and show that general dispersion relationship for a plane wave can be written as
2
kk
=
G G
i
.
(c)
If the medium is lossy
μ
or
ε
or both can be complex hence
k
G
is complex.
Therefore
'"
kkj
k
=−
G
GG
where
'
k
G
and
"
k
G
are real vectors that describe the real and imaginary parts of the
k
G
vector.
Assuming that
k
G
is in the
xz
plane write
'
k
G
and
"
k
G
in terms of their components.
Write an
expression for the general plane wave having this complex
k
G
vector.
Comment on the directions
of
'
k
G
and
"
k
G
.
(d)
Show that the dispersion relationship found in part b can be written as two real equations
given below.
()
2
R
e
−=
G
G
and
2
1
I
m
2
G
G
i
(e)
Now assume that a plane wave is incident on a planar interface between two lossy media.
Let's define the plane of incidence as the
xz
plane
x
being along the interface and
z
normal to the
interface.
Then the
k
G
vector of the incident wave is known and can be expressed as
'"'
"
'
"
ˆˆ
i
i
i
ix
ix
x
iz
iz
z
k
k
j
k
a
k
j
k
a
=− = −
+ −
G
Write similar equations for the
k
G
vector of the reflected and transmitted waves.
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 Spring '09
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