ECE 450
Homework 3
Due: Tue, Feb 8, 2011, 5 PM
1. A Hertzian radiation source with the current density
J
(
r
, t
) = ˆ
xI
o
Δ
x
δ
(
x

100
λ
)
δ
(
y
)
δ
(
z

100
λ
) cos(
ω
t
)
A
m
2
,
is embedded in free space and its oscillation frequency
f
=
ω
2
π
= 300
MHz. Assuming that
I
o
Δ
x
= 1
A.m, determine the numerical values (in appropriate units) of
a) wavelength
λ
of the radiation field,
b) the corresponding wavenumber
k
,
c) retarded vector potential phasor
˜
A
at
r
= 200
λ
ˆ
z
,
d) the radiation electric field phasor
˜
E
at
r
= 200
λ
ˆ
z
,
e) the radiation magnetic field intensity phasor
˜
H
at
r
= 200
λ
ˆ
z
, and
f) the timeaverage Poynting vector
E
×
H
at the same location.
Note:
the vector quantities in parts (cf) should be specified in terms of unit vectors
ˆ
x
,
ˆ
y
,
ˆ
z
.
2. Three identical Hertzian dipoles
I
Δ
are colocated at the origin aligned with the positive
x
,
y
, and
z
axis, respectively.
a) Determine the total retarded vector potential phasor
˜
A
of the dipoles acting in unison.
b) Express the corresponding
radiation field
phasor
˜
E
of the dipoles at (i)
(
x, y, z
) = (
d,
0
,
0)
, and (ii)
(
x, y, z
) = (
d, d, d
)
in terms of dipole moment
I
Δ
, wavenumber
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 Spring '11
 KUDEKI
 Frequency, Hertz, Fundamental physics concepts, Dipole antenna, Loop antenna, retarded vector

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