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UNIVERSITY OF CALIFORNIA
Santa Barbara
Department of Electrical and Computer Engineering
Problem Set No. 4
Issued:
October 22, 2008
Fall 2008
Due:
October 29, 2008
ECE 201A
Midterm:
Midterm will be on November 12, 2008 between 10:00 AM 11:50 AM in Phelps 3515.
All
the material covered till the end the lecture on 10/29/2008 and homework 1  4 will be
included in the exam.
The exam will be closed book.
Formulas and equations too hard to
remember or derive will be provided.
Please bring a blue book for the exam.
1.
Consider the small loop of constant current
I
as shown in the Figure below.
Show that the
magnetic vector potential is
y
I
a
x
z
φ
θ
′
φ
r
AA
Ia
fd
y
φφ
π
==
=
∫
0
0
2
4
cos
''
where
f
jk r
a
ra
ra
r
a
=
−+
−
+−
exp(
sin cos
sin cos
'
'
22
2
2
θφ
Expand
f
in a Maclaurin series about
a
= 0 and show that
A
Ia
e
jk
r
r
a
jkr
→
−
⎯→
⎯
⎯+
0
2
2
4
1
()
s
i
n
which is the vector potential of an infinitesimal loop of electric current.
The quantity
I
a
2
=
IS
where
S
is the area of the loop is called the magnetic moment of the loop.
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