Download Example 22-2 An Electric Field Due to a Changing Magnetic Field

Example 22-2 An Electric Field Due to a Changing Magnetic Field
The uniform magnetic field shown in Figure 22-7b decreases in magnitude from 1.50 T to zero in a time t, inducing an electric field. What is the magnitude of this electric field around a loop 3.50 cm in diameter if (a) t = 10.0 s and (b) t = 0.100 s?
Set Up
Equation 20-1 tells us the magnetic flux
through the loop of diameter 3.50 cm.
s is perpendicular to
The magnetic field B
the plane of the loop, so u (the angle
s and the
between the direction of B
perpendicular to the loop) is u = 0.
The electric field in this case has the
same magnitude E all the way around the
circle and is tangent to the circle, so E } in
Equation 22-7 is equal to E.
Magnetic flux:
electromagnet
B = AB# = AB cos u
(20-1)
Faraday’s law in terms of circulation:
a E  / = -
B
t
(22-7)
S
B
E
decreasing
magnetic
field
E
Area of a circle of radius r:
A = pr2
E
E
loop
Circumference of a circle of radius r:
C = 2pr
Solve
(a) First determine the initial and final values
of magnetic flux and the change in magnetic
flux for a circle of diameter 3.50 cm.
N
The radius of the loop is half the diameter:
r =
1
13.50 cm2 = 1.75 cm = 1.75 * 10-2 m
2
The area of the loop is
A = pr2 = p (1.75 * 1022 m)2 = 9.62 * 1024 m2
From Equation 20-1, the initial magnetic flux is
B = AB cos 0 = (9.62 * 1024 m2)(1.50 T)(1)
= 1.44 * 1023 T · m2
The final magnetic field is zero, so the final magnetic flux is zero as well.
The change in magnetic flux is
B = (final flux) 2 (initial flux) = 0 2 1.44 * 1023 T · m2
= 21.44 * 1023 T · m2
Use Equation 22-7 to calculate the
circulation of the electric field in the
case where t = 10.0 s. Since the field
magnitude E has the same value around
the circle and E } = E, we can use this to
calculate E.
The circulation of the electric field is equal to - B > t:
B
Circulation of the electric field = a E } / = t
= -
-1.44 * 10-3 T # m2
T # m2
= 1.44 * 10-4
s
10.0 s
In Example 22-1 (Section 22-2) we saw that
1T = 1
V#s
so the magnitude of the circulation is
m2
1.44 * 10-4
T # m2
V # s m2
b = 1.44 * 10-4 V
= 1.44 * 10-4 a 2 b a
s
s
m
Since E } = E and E has the same value at all points around the circle,
we can write the circulation as
a E } / = a E/ = E a /
The sum a / is the total distance around the loop, equal to the loop
circumference 2pr. So
E (2pr) = 1.44 * 1024 V
1.44 * 10-4 V
1.44 * 10-4 V
=
= 1.31 * 10-3 V>m
E =
2pr
2p 11.75 * 10-2 m2
(b) Repeat the calculation for the case
where t = 0.100 s
Equation 22-7 tells us that the circulation of the electric field, and hence
the electric field itself, is inversely proportional to the time t over which
the magnetic flux changes. If the magnetic field drops to zero in t =
0.100 s rather than t = 10.0 s, the elapsed time is smaller by a factor of
0.100 s
= 1.00 * 10-2
10.0 s
and so the induced field is larger by a factor of
1
= 1.00 * 102
1.00 * 10-2
Therefore the electric field in the case where t = 0.100 s is
Reflect
E = 11.00 * 102 2 11.31 * 10-3 V>m2 = 0.131 V>m
Our results show that the more rapid the change in magnetic field, the greater the magnitude
of the electric field that is induced.
If a circular loop of conducting wire were placed along the circular path of diameter
3.50 cm, a current would be generated so as to produce a magnetic field that would oppose
the change in magnetic flux. The upward magnetic field in the figure decreases, so the magnetic
field produced in this way would have to be upward. The induced current that produces this
magnetic field is in the same direction as the circulating electric field. So the electric field and
current must both have the direction shown.
B decreasing
E
E
E
E
If a wire loop were placed here,
current would be induced to
produce an upward B.