ECE 329
Homework 7
Due: July 13, 2009, 5PM
1. Verify that vector identity
H
· ∇ ×
E

E
· ∇ ×
H
=
∇·
(
E
×
H
)
holds for
E
= 2ˆ
xe

αz
and
H
= 4ˆ
ye

αz
by expanding both sides of the identity. Treat
α
as a real
constant.
You should download the table of vector identities from ECE 329 web site and examine the list to
familiarize yourself with the listed identities — they are widely employed in electromagnetics as well
as in other branches of engineering such as Fuid dynamics.
2.
a) ±or current density
J
= (4
z
2
ˆ
x
+3
x
3
y
ˆ
y
+2
z
(
y

y
o
)
2
ˆ
z
)
A/m
2
, which is time independent, ²nd
the charge density
ρ
(0
,t
)
at the origin (0,0,0) as a function of time
t
, if
ρ
=0
there at time
t
=0
,
y
o
=1
m, and coordinates
x
,
y
, and
z
are speci²ed in meter units.
Hint:
use the continuity
equation
∂ρ
∂t
+
∇·
J
=0
.
b) In part (a), deduce the physical units of the coe³cients
4
,
3
, and
2
used in
J
x
,
J
y
, and
J
z
speci²cations, respectively, by applying dimensional analysis.
3. Copper is a highly conducting metal with a
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 Fall '08
 Kim
 Electromagnet, Electric charge, web site, 1 m, 400 m, 1.5 mm, 1028 m

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