nguyen (jmn727) – homework 24 – Turner – (59070)
1
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001
10.0 points
A plane loop of wire of area A is placed in a
region where the magnetic field 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 field is
B
0
, and for
t >
0,
the field decreases exponentially in time.
Find the induced emf,
E
, in the loop as a
function of time.
1.
E
=
a A B
0
e

2
at
2.
E
=
a A B
0
3.
E
=
A B
0
e

at
4.
E
=
a B
0
e

at
5.
E
=
a A B
0
e

at
correct
6.
E
=
a B
0
t
Explanation:
Basic Concepts:
Faraday’s Law:
E ≡
contintegraldisplay
E
·
ds
=

d
Φ
B
dt
Solution:
Since B is perpendicular to the
plane of the loop, the magnetic flux through
the loop at time
t >
0 is
Φ
B
=
B A
=
A B
0
e

at
Also, since the coefficient
AB
0
and the pa
rameter a are constants, and Faraday’s Law
says
E
=

d
Φ
B
dt
the induced emf can be calculated the from
Equation above:
E
=

d
Φ
B
dt
=

A B
0
d
dt
e

at
=
a A B
0
e

at
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
vector
t
The plot of
E
versus
t
is similar to the
B
versus
t
curve shown in the figure above.
002
10.0 points
The magnetic flux threading a metal ring
varies with time
t
according to
Φ
B
= 3
a t
3

b t
2
,
with
a
=
4
.
7
s

3
·
m
2
·
T,
and
b
=
3
.
1 s

2
·
m
2
·
T.
The resistance of the ring
is 2
.
5 Ω.
Determine the maximum current induced
in the ring during the interval from
t
1
=

4 s
to
t
2
= 3 s.
Correct answer: 0
.
0908747 A.
Explanation:
From
Faraday’s
law,
the
induced
emf
should be
E
=

d
Φ
B
dt
=

(9
a t
2

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

18
a t
+ 2
b
= 0
t
=
b
9
a
and the maximum
emf
is
E
max
=

9
a
parenleftbigg
b
9
a
parenrightbigg
2
+ 2
b
parenleftbigg
b
9
a
parenrightbigg
=

b
2
9
a
+
2
b
2
9
a
=
b
2
9
a
.
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nguyen (jmn727) – homework 24 – Turner – (59070)
2
Thus the maximum current is
I
max
=
E
max
R
=
b
2
9
a R
=
(3
.
1 s

2
·
m
2
·
T)
2
9 (4
.
7 s

3
·
m
2
·
T) (2
.
5 Ω)
=
0
.
0908747 A
.
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 Spring '08
 Turner
 Magnetism, Work, Magnetic Field, Magnet

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