ECE 450
Homework 2
Due: Tue, Sept 8, 2009, 5 PM
1. Consider the timeharmonic charge distribution
ρ
(
r
, t
) =
q
[
δ
(
r

h
ˆ
x
)

δ
(
r
+
h
ˆ
x
) +
δ
(
r

h
ˆ
y
)

δ
(
r
+
h
ˆ
y
)] cos(
ω
t
)
C
m
3
in free space.
a) Write down the phasors for the charge density
ρ
(
r
, t
)
and its time derivative
∂ρ
∂
t
.
b) Calculate the divergence
∇
·
J
of the current density associated with
ρ
(
r
, t
)
above.
Hint:
conti
nuity equation.
c) What is the retarded scalar potential phasor
˜
V
generated by
ρ
(
r
, t
)
above?
2. Give the timeharmonic expressions with the frequency
ω
for the fields
A
(
r
, t
)
with the following
phasors:
a)
˜
A
(
r
) =
j
2ˆ
x

2ˆ
y
b)
˜
A
(
r
) =
j
ω
ˆ
y
+ 2
ω
2
ˆ
z
c)
˜
A
(
r
) = (
j
ˆ
x
+ 2ˆ
y
)
e

jkz
where
k
is a constant.
3. Consider the timeharmonic vector field
A
(
r
, t
) = ˆ
xA
x
(
r
) cos (
ω
t
+
φ
x
(
r
)) + ˆ
yA
y
(
r
) cos (
ω
t
+
φ
y
(
r
)) + ˆ
zA
z
(
r
) cos (
ω
t
+
φ
z
(
r
))
.
a) Show that its complex valued phasor
˜
A
(
r
)
can be written in the form
˜
A
(
r
) =
A
R
(
r
) +
j
A
I
(
r
)
,
where
A
R
(
r
)
and
A
I
(
r
)
are real vectors and identify
A
R
(
r
)
and
A
I
(
r
)
.
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 Fall '09
 FRANKE
 Vector Calculus, Electric charge, Vector field, retarded scalar potential, divergence nuity equation

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