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