ECE329
Spring 2009
Homework 3 — Solution
February 19, 2009
1. Faraday’s law,
˛
C
E
·
d
l
=

d
dt
ˆ
S
B
·
d
S
,
states that the electromotive force
E
=
¸
C
E
·
d
l
around any closed loop
C
equals the time rate of
change of the magnetic flux
Ψ =
´
S
B
·
d
S
through any surface
S
bounded by the loop.
a) If
B
=
0
at all times, then the magnetic flux is zero (
Ψ = 0
), and therefore, according to
Faraday’s law, the electromotive force over any closed loop
C
is also zero (
E
= 0
).
b) If
B
6
=
0
but timeindependent, then the magnetic flux
Ψ
through a surface bounded by a
fixed loop
C
is also timeindependent and therefore
d
dt
Ψ = 0
.
Because of this, and according to
Faraday’s law, the electromotive force over the loop
C
will be zero (
E
= 0
).
c) Let us define a closed loop
C
passing through the fixed points
P
1
and
P
2
(see the next figure).
P
2
P
1
dl
dl
path
A
path
B
C
Since
B
is timeindependent, the corresponding magnetic flux
Ψ
is also time independent, and
therefore,
˛
C
E
·
d
l
=

d
Ψ
dt
= 0
.
Breaking the closed path integral into two parts, we have
˛
C
E
·
d
l
=
ˆ
P
1
→
P
2
path
A
E
·
d
l
+
ˆ
P
2
→
P
1
path
B
E
·
d
l
= 0
.
Reversing the direction of integration of the second integral, it can be shown that
ˆ
P
1
→
P
2
path
A
E
·
d
l
=
ˆ
P
1
→
P
2
path
B
E
·
d
l
.
In consequence, the line integral
´
12
E
·
d
l
does not depend on the path taken from
P
1
to
P
2
.
d) If
B
6
=
0
but timeindependent, it is still possible for the emf
E
to be nonzero if the path
C
is
disturbed or displaced in such a way that the magnetic flux
Ψ =
´
S
B
·
d
S
varies in time.
1
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ECE329
Spring 2009
2. Given
B
=
B
0
(
t
cos(
ωt
) ˆ
x
+ sin(
ωt
) ˆ
z
)
Wb
/
m
2
, we can apply Faraday’s law to compute the emf
E
around the following closed paths. Since the closed paths are not varying in time and the magnetic
field
B
is independent of position, we can rewrite Faraday’s law as follows
E
=
˛
C
E
·
d
l
=

d
B
dt
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 Spring '08
 FRANKE
 Electromagnet, Electromotive Force, Magnetic Field, Faraday, Faraday's law of induction

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