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terry (ect328) – homework 24 – Turner – (59130)
1
This printout should have 11 questions.
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beFore answering.
001
10.0 points
A plane loop oF wire oF area A is placed in a
region where the magnetic feld is perpendicu
lar to the plane. The magnitude oF B varies in
time according to the expression
B
=
B
0
e

at
.
That is, at
t
= 0 the feld is
B
0
, and For
t>
0,
the feld decreases exponentially in time.
±ind the induced emF,
E
, in the loop as a
Function oF time.
1.
E
=
aB
0
t
2.
E
=
0
e

at
3.
E
=
a A B
0
e

at
correct
4.
E
=
a A B
0
e

2
at
5.
E
=
a A B
0
6.
E
=
AB
0
e

at
Explanation:
Basic Concepts:
±araday’s Law:
E ≡
±
E
·
ds
=

d
Φ
B
dt
Solution:
Since B is perpendicular to the
plane oF the loop, the magnetic ²ux through
the loop at time
0 is
Φ
B
=
BA
=
0
e

at
Also, since the coe³cient
AB
0
and the pa
rameter a are constants, and ±araday’s Law
says
E
=

d
Φ
B
dt
the induced emF can be calculated the From
Equation above:
E
=

d
Φ
B
dt
=

0
d
dt
e

=
a A B
0
e

That is, the induced emF decays exponentially
in time.
Note:
The maximum emF occurs at
t
=0
,
where
E
=
a A B
0
.
B
=
B
0
e

at
B
0
0
0
±
t
The plot oF
E
versus
t
is similar to the
B
versus
t
curve shown in the fgure above.
002
10.0 points
The magnetic ²ux threading a metal ring
varies with time
t
according to
Φ
B
=3
3

bt
2
,
with
a
=4
.
1s

3
·
m
2
·
T,
and
b
=
7
.
5s

2
·
m
2
·
T.
The resistance oF the ring
is 1
.
7 Ω.
Determine the maximum current induced
in the ring during the interval From
t
1
=

3s
to
t
2
= 2 s.
Correct answer: 0
.
8967 A.
Explanation:
±rom ±araday’s law, the induced
emf
should be
E
=

d
Φ
B
dt
=

(9
2

2
)
,
so the maximum
E
occurs when
d
E
dt
=

18
+2
b
t
=
b
9
a
and the maximum
emf
is
E
max
=

9
a
²
b
9
a
³
2
b
²
b
9
a
³
=

b
2
9
a
+
2
b
2
9
a
=
b
2
9
a
.
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View Full Documentterry (ect328) – homework 24 – Turner – (59130)
2
Thus the maximum current is
I
max
=
E
max
R
=
b
2
9
aR
=
(7
.
5s

2
·
m
2
·
T)
2
9 (4
.
1s

3
·
m
2
·
T) (1
.
7 Ω)
=
0
.
8967 A
.
003
(part 1 of 2) 10.0 points
The resistance of the rectangular current
loop is
R
, and the metal rod is sliding to the
left. The length of the rod is
d
, while the
width of the rails is
±
.
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 Spring '10
 Turner

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