Faraday’s Law of Induction Chapter 31 Directions: Put down all diagrams, equations used in the problem, and all of your work. 1. The plane of a rectangular coil of dimensions 5.0 cm by 8.0 cm is perpendicular to the direction of a magnetic field B. If the coil has 75 turns and a total resistance of 8.0 , at what rate must the magnitude of B change in order to induce a current of 0.10 A in the windings of the coil? 2. A long solenoid has n turns per meter and carries a current I = Io(1 - e-t), with Io = 30.A and =1.6s-1. Inside the solenoid and coaxial with it is a loop that has a radius R = 6.0cm and consists of a total of N turns of fine wire. What emf is induced in the loop by the changing current? Take n = 400 turns/m and N = 250 turns. 3. A long solenoid has 200. turns/cm and carries a current of 1.5 A; Its diameter is 3.0 cm. At its center we place a 100. turn, close packed coil of diameter 2.0 cm. This coil is arranged so that B at the center of the solenoid is parallel to its axis. The current in the solenoid is reduced to zero and then raised to 1.5 A in the other direction at a steady rate over a period of .050 s. What induced emf appears in the coil while the current is being changed? (4.74 x 10-2 V) 4. A 100. turns of insulated copper wire are wrapped around an iron cylinder of cross-sectional area 0.001 m2 and are connected to a resistor. The total resistance in the circuit is 10.0 . If the longitudinal magnetic field in the iron changes from 1.00 T in one direction to 1.0 T in the opposite direction, how much charge flows through the circuit? (2.00 x 10-2 C) 5. A magnetic field of 0.20 T exists within a solenoid of 500 turns and a diameter of 10.0 cm. How rapidly (that is, within what period of time) must the field be reduced to zero magnitude if the average magnitude of the induced emf within the coil during this time interval is to be 10. kV? (7.85 x 10-5 s) B is into the paper 6. A bar of mass m and length l moves on two frictionless parallel rails of resistance R in the presence of a uniform magnetic field directed into the paper. The bar is given an initial velocity vo to the right and is released. Find the velocity of the bar as a function of time. Fm vo L R 7. A long solenoid of radius R has n turns per unit length and carries a time-varying current that varies sinusoidally as I = Io cos t, where Io is the maximum current and is the angular frequency of the current source. a. Determine the electric field outside the solenoid, a distance r from its axis. b. What is the electric field inside the solenoid, a distance r from its axis? 8. Consider the diagram in problem #6. Assume that R = 6.0 , = 1.2 m, and that a uniform 2.5 T magnetic field is directed into the page. At what speed should the bar be moved to produce a current of 0.50 A in the resistor? (Physics for Scientists and Engineers by Serway, 4th Edition, Ch 31# 18) 9. Use Lenz’s law to answer the following questions concerning the direction of induced currents (Physics for Scientists and Engineers by Serway, 4th Edition, Ch 31# 27) a. What is the direction of the induced current in the resistor R in Figure 31.27a when the bar magnet is moved to the left? b. What is the direction of the current induced in the resistor R right after the switch S in the circuit of Figure 31.27b is closed? c. What is the direction of the induced current in R when the current I in Figure 31.27c decreases rapidly to zero? d. A copper bar is moved to the right while its axis is maintained perpendicular to a magnetic field, as in figure 31.27d. If the top of the bar becomes positive relative to the bottom, what is the direction of the magnetic field? 10. A horizontal wire is free to slide on the vertical rails of a conducting frame L as in the figure to the right. The wire has mass m and length L, and the m resistance of the circuit is R. If a uniform magnetic field is directed perpendicular to the frame, what is the terminal velocity of the wire as it falls under the force of gravity? (Neglect mechanical friction) B out 11. A thin metal strip is allowed to slide down parallel frictionless B rails of negligible resistance connected at the bottom end and A elevated to an angle above the horizontal. A uniform magnetic field B is directed vertically upward throughout the region. The strip has mass m = 35 g, resistance R = 0.20 , and length between the rails L = 0.3 m. a. Derive a general expression for the terminal speed of the v strip. b. Calculate the terminal speed achieved by the strip sliding along the incline if = 30.o and B=1.5 T. c. Derive a general expression for the speed of the strip at L any time t. (Physics for Scientists and Engineers by Serway, 4th Edition, Ch 31# 69) 12. The current in a solenoid is increasing at a rate of 10 A/s. The cross-sectional area of the solenoid is cm² and there are 300 turns on its 15 cm length. What is the induced emf which acts to oppose the increasing current? (Physics for Scientists and Engineers by Serway, 4th Edition, Ch 31# 30) 13. A magnetic field directed into the page changes with time according to B = (0.030t2 + 1.4) T, where t is in seconds. The field has a circular cross-section of radius R = 2.5 cm. What are the magnitude and direction of the electric field at point P1 when t = 3.0 s and r1 = 0.020 m? (Physics for Scientists and Engineers by Serway, 4th Edition, Ch 31# 33) 14. A circular coil, enclosing an area of 100.0 cm2, is made of 200 turns of copper wire as shown in Figure 31.34 of the text. Initially, a 1.1 T uniform magnetic field points perpendicularly upward through the plane of the coil. The direction of the field then reverses so that the final magnetic field has a magnitude of 1.1 T pointing downward through the coil. During the time the field is changing, how much charge flows through the coil if the coil is connected to a 5.0 resistor as shown in figure 31.34 of the text? (Physics for Scientists and Engineers by Serway, 4th Edition, Ch 31# 38) 15. A 2.0 H inductor carries a steady current of 0.5 A. When a switch in the circuit is opened, the current disappears in 0.010 s. What is the induced emf that appears in the inductor during this time? (Physics for Scientists and Engineers by Serway, 4th Edition, Ch 32# 1) 16. What is the inductance of a 510 turn solenoid that has a radius of 8.0 cm and an overall length of 1.4 m? 17. The current in a 0.020 H inductor varies in time as I = 3t2 - 4t, where I is in amperes and t is in seconds. a. Calculate the magnitude of the induced emf at t = 1 and t = 5 s. b. For what value of t will the induced emf be zero? 18. A current I = Iosint, with Io = 5.0 A and /2 = 60.0 HZ, flows through in inductor whose inductance is 10.0 mH. What is the back emf as a function of time? 19. A 50.0 V potential difference is suddenly applied to a coil with L = 50.0 mH and R = 180.0 ohms. At what rate is the current increasing after .00100s? 20. In the circuit shown at the right find the following quantities when the switch is just closed and after it has been closed for a long time. a. Current i1 b. Current i2 c. Total current i d. Voltage across R2 e. Voltage across inductor L f. Rate of change in i2 21. Calculate the resistance of an RL circuit in which L = 2.5 H and the current increases to 90% of its final value in 3.0 s 22. A 50.0 V potential difference is suddenly applied to a coil with L = 50.0 mH and R = 180.0 ohms. At what rate is the current increasing after .00100s?