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Chapter 23 Problems 1, 2, 3 = straightforward, intermediate, challenging = full solution available in Student Solutions Manual/Study Guide = coached solution with hints available at www.pop4e.com = computer useful in solving problem = paired numerical and symbolic problems = biomedical application Section 23.1 Faraday’s Law of Induction Section 23.3 Lenz’s Law 1. A flat loop of wire consisting of a single turn of cross-sectional area 8.00 cm2 is perpendicular to a magnetic field that increases uniformly in magnitude from 0.500 T to 2.50 T in 1.00 s. What is the resulting induced current if the loop has a resistance of 2.00 Ω? reaches zero in 20.0 ms. What is the current induced in the coil? 4. An aluminum ring of radius r1 and resistance R is placed around the top of a long air-core solenoid with n turns per meter and smaller radius r2 as shown in Figure P23.4. Assume that the axial component of the field produced by the solenoid over the area of the end of the solenoid is one-half as strong as at the solenoid’s center. Assume that the solenoid produces negligible field outside its crosssectional area. The current in the solenoid is increasing at a rate of ΔI/Δt. (a) What is the induced current in the ring? (b) At the center of the ring, what is the magnetic field produced by the induced current in the ring? (c) What is the direction of this field? 2. A 25-turn circular coil of wire has diameter 1.00 m. It is placed with its axis along the direction of the Earth’s magnetic field of 50.0 μT, and then in 0.200 s it is flipped 180°. An average emf of what magnitude is generated in the coil? 3. A strong electromagnet produces a uniform magnetic field of 1.60 T over a cross-sectional area of 0.200 m2. We place a coil having 200 turns and a total resistance of 20.0 Ω around the electromagnet. We then smoothly reduce the current in the electromagnet until it Figure P23.4 5. (a) A loop of wire in the shape of a rectangle of width w and length L and a long, straight wire carrying a current I lie on a tabletop as shown in Figure P23.5. (a) Determine the magnetic flux through the loop due to the current I. (b) Suppose the current is changing with time according to I = a + bt, where a and b are constants. Determine the emf that is induced in the loop if b = 10.0 A/s, h = 1.00 cm, w = 10.0 cm, and L = 100 cm. What is the direction of the induced current in the rectangle? Figure P23.6 7. A 30-turn circular coil of radius 4.00 cm and resistance 1.00 Ω is placed in a magnetic field directed perpendicular to the plane of the coil. The magnitude of the magnetic field varies in time according to the expression B = 0.010 0t + 0.040 0t2, where t is in seconds and B is in teslas. Calculate the induced emf in the coil at t = 5.00 s. Figure P23.5 Problems 23.5 and 23.59. 6. A coil of 15 turns and radius 10.0 cm surrounds a long solenoid of radius 2.00 cm and 1.00 × 103 turns/m (Fig. P23.6). The current in the solenoid changes as I = (5.00 A) sin(120t). Find the induced emf in the 15turn coil as a function of time. 8. An instrument based on induced emf has been used to measure projectile speeds up to 6 km/s. A small magnet is imbedded in the projectile as shown in Figure P23.8. The projectile passes through two coils separated by a distance d. As the projectile passes through each coil, a pulse of emf is induced in the coil. The time interval between pulses can be measured accurately with an oscilloscope, and thus the speed can be determined. (a) Sketch a graph of ΔV versus t for the arrangement shown. Consider a current that flows counterclockwise as viewed from the starting point of the projectile as positive. On your graph, indicate which pulse is from coil 1 and which is from coil 2. (b) If the pulse separation is 2.40 ms and d = 1.50 m, what is the projectile speed? Figure P23.8 9. When a wire carries an AC current with a known frequency, you can use a Rogowski coil to determine the amplitude Imax of the current without disconnecting the wire to shunt the current in a meter. The Rogowski coil, shown in Figure P23.9, simply clips around the wire. It consists of a toroidal conductor wrapped around a circular return cord. The toroid has n turns per unit length and a cross-sectional area A. The current to be measured is given by I(t) = Imax sin ωt. (a) Show that the amplitude of the emf induced in the Rogowski coil is ε Figure P23.9 10. A piece of insulated wire is shaped into a figure eight as shown in Figure P23.10. The radius of the upper circle is 5.00 cm and that of the lower circle is 9.00 cm. The wire has a uniform resistance per unit length of 3.00 W/m. A uniform magnetic field is applied perpendicular to the plane of the two circles, in the direction shown. The magnetic field is increasing at a constant rate of 2.00 T/s. Find the magnitude and direction of the induced current in the wire. = μ0nAω Imax. (b) Explain why the wire carrying the unknown current need not be at the center of the Rogowski coil and why the coil will not respond to nearby currents that it does not enclose. max Figure P23.10 Section 23.2 Motional emf Section 23.3 Lenz’s Law Note: Problem 22.62 can be assigned with this section. 11. An automobile has a vertical radio antenna 1.20 m long. The automobile travels at 65.0 km/h on a horizontal road where the Earth’s magnetic field is 50.0 μT directed toward the north and downward at an angle of 65.0° below the horizontal. (a) Specify the direction that the automobile should move to generate the maximum motional emf in the antenna, with the top of the antenna positive relative to the bottom. (b) Calculate the magnitude of this induced emf. 12. Consider the arrangement shown in Figure P23.12. Assume that R = 6.00 Ω, ℓ = 1.20 m, and a uniform 2.50-T magnetic field is directed into the page. At what speed should the bar be moved to produce a current of 0.500 A in the resistor? Figure P23.12 Problems 23.12, 23.13, 23.14, and 23.15. 13. Figure P23.12 shows a top view of a bar that can slide without friction. The resistor is 6.00 Ω, and a 2.50-T magnetic field is directed perpendicularly downward, into the paper. Let ℓ = 1.20 m. (a) Calculate the applied force required to move the bar to the right at a constant speed of 2.00 m/s. (b) At what rate is energy delivered to the resistor? 14. A conducting rod of length ℓ moves on two horizontal, frictionless rails as shown in Figure P23.12. If a constant force of 1.00 N moves the bar at 2.00 m/s through a magnetic field B that is directed into the page, (a) what is the current through the 8.00-Ω resistor R? (b) What is the rate at which energy is delivered to the resistor? (c) What is the mechanical power delivered by the force Fapp ? 15. A metal rod of mass m slides without friction along two parallel horizontal rails, separated by a distance ℓ and connected by a resistor R, as shown in Figure P23.12. A uniform vertical magnetic field of magnitude B is applied perpendicular to the plane of the paper. The applied force shown in the figure acts only for a moment, to give the rod a speed v. In terms of m, ℓ, R, B, and v, find the distance the rod will then slide as it coasts to a stop. 16. Very large magnetic fields can be produced using a procedure called flux compression. A metallic cylindrical tube of radius R is placed coaxially in a long solenoid of somewhat larger radius. The space between the tube and the solenoid is filled with a highly explosive material. When the explosive is set off, it collapses the tube to a cylinder of radius r < R. If the collapse happens very rapidly, induced current in the tube maintains the magnetic flux nearly constant inside the tube. If the initial magnetic field in the solenoid is 2.50 T and R/r = 12.0, what maximum value of magnetic field can be achieved? 17. The homopolar generator, also called the Faraday disk, is a low-voltage, highcurrent electric generator. It consists of a rotating conducting disk with one stationary brush (a sliding electrical contact) at its axle and another at a point on its circumference as shown in Figure P23.17. A magnetic field is applied perpendicular to the plane of the disk. Assume that the field is 0.900 T, the angular speed is 3 200 rev/min, and the radius of the disk is 0.400 m. Find the emf generated between the brushes. When superconducting coils are used to produce a large magnetic field, a homopolar generator can have a power output of several megawatts. Such a generator is useful, for example, in purifying metals by electrolysis. If a voltage is applied to the output terminals of the generator, it runs in reverse as a homopolar motor capable of providing great torque, useful in ship propulsion. Figure P23.17 18. Review problem. A flexible metallic wire with linear density 3.00 × 10–3 kg/m is stretched between two fixed clamps 64.0 cm apart and held under tension 267 N. A magnet is placed near the wire as shown in Figure P23.18. Assume that the magnet produces a uniform field of 4.50 mT over a 2.00-cm length at the center of the wire and a negligible field elsewhere. The wire is set vibrating at its fundamental (lowest) frequency. The section of the wire in the magnetic field moves with a uniform amplitude of 1.50 cm. Find (a) the frequency and (b) the amplitude of the electromotive force induced between the ends of the wire. the current I in Figure P23.20c decreases rapidly to zero? (d) A copper bar is moved to the right while its axis is maintained in a direction perpendicular to a magnetic field as shown in Figure P31.28d. If the top of the bar becomes positive relative to the bottom, what is the direction of the magnetic field? Figure P23.18 19. A helicopter (Fig. P23.19) has blades of length 3.00 m, extending out from a central hub and rotating at 2.00 rev/s. If the vertical component of the Earth’s magnetic field is 50.0 μT, what is the emf induced between the blade tip and the center hub? Figure P23.20 (Ross Harrison/Getty Images) Figure P23.19 20. Use Lenz’s law to answer the following questions concerning the direction of induced currents. (a) What is the direction of the induced current in resistor R in Figure P23.20a when the bar magnet is moved to the left? (b) What is the direction of the current induced in the resistor R immediately after the switch S in Figure P23.20b is closed? (c) What is the direction of the induced current in R when 21. A conducting rectangular loop of mass M, resistance R, and dimensions w by ℓ falls from rest into a magnetic field B as shown in Figure P23.21. During the time interval before the top edge of the loop reaches the field, the loop approaches a terminal speed vT. (a) Show that vT MgR B2w 2 (b) Why is vT proportional to R? (c) Why is it inversely proportional to B2? 23. A coil of area 0.100 m2 is rotating at 60.0 rev/s with the axis of rotation perpendicular to a 0.200-T magnetic field. (a) If the coil has 1 000 turns, what is the maximum emf generated in it? (b) What is the orientation of the coil with respect to the magnetic field when the maximum induced voltage occurs? Figure P23.21 22. A rectangular coil with resistance R has N turns, each of length ℓ and width w as shown in Figure P23.22. The coil moves into a uniform magnetic field B with constant velocity v . What are the magnitude and direction of the total magnetic force on the coil (a) as it enters the magnetic field, (b) as it moves within the field, and (c) as it leaves the field? Figure P23.22 24. A long solenoid, with its axis along the x axis, consists of 200 turns per meter of wire that carries a steady current of 15.0 A. A coil is formed by wrapping 30 turns of thin wire around a circular frame that has a radius of 8.00 cm. The coil is placed inside the solenoid and mounted on an axis that is a diameter of the coil and that coincides with the y axis. The coil is then rotated with an angular speed of 4.00π rad/s. (The plane of the coil is in the yz plane at t = 0.) Determine the emf generated in the coil as a function of time. Section 23.4 Induced emfs and Electric Fields 25. A magnetic field directed into the page changes with time according to B = (0.030 0t2 + 1.40) T, where t is in seconds. The field has a circular cross-section of radius R = 2.50 cm (Fig. P23.25). What are the magnitude and direction of the electric field at point P1 when t = 3.00 s and r1 = 0.020 0 m? 30. An emf of 24.0 mV is induced in a 500-turn coil at an instant when the current is 4.00 A and is changing at the rate of 10.0 A/s. What is the magnetic flux through each turn of the coil? Figure P23.25 Problems 23.25 and 23.26. 26. For the situation shown in Figure P23.25, the magnetic field changes with time according to the expression B = (2.00t3 – 4.00t2 + 0.800) T and r2 = 2R = 5.00 cm. (a) Calculate the magnitude and direction of the force exerted on an electron located at point P2 when t = 2.00 s. (b) At what time is this force equal to zero? Section 23.5 Self-Inductance 27. A coil has an inductance of 3.00 mH, and the current in it changes from 0.200 A to 1.50 A in a time interval of 0.200 s. Find the magnitude of the average induced emf in the coil during this time interval. 28. A coiled telephone cord forms a spiral with 70 turns, a diameter of 1.30 cm, and an unstretched length of 60.0 cm. Determine the self-inductance of one conductor in the unstretched cord. 29. A 10.0-mH inductor carries a current I = Imax sin ωt, with Imax = 5.00 A and ω/2π = 60.0 Hz. What is the selfinduced emf as a function of time? 31. The current in a 90.0-mH inductor changes with time as I = 1.00t2 – 6.00t (in SI units). Find the magnitude of the induced emf at (a) t = 1.00 s and (b) t = 4.00 s. (c) At what time is the emf zero? 32. A toroid has a major radius R and a minor radius r, and is tightly wound with N turns of wire as shown in Figure P23.32. If R >> r, the magnetic field in the region enclosed by the wire of the torus, of crosssectional area A = πr2, is essentially the same as the magnetic field of a solenoid that has been bent into a large circle of radius R. Modeling the field as the uniform field of a long solenoid, show that the selfinductance of such a toroid is approximately L 0 N 2 A 2R (An exact expression of the inductance of a toroid with a rectangular cross-section is derived in Problem 23.60.) Figure P23.32 Section 23.6 RL Circuits 33. A 12.0-V battery is connected into a series circuit containing a 10.0-Ω resistor and a 2.00-H inductor. In what time interval will the current reach (a) 50.0% and (b) 90.0% of its final value? 34. Show that I = Iie–t/τ is a solution of the differential equation IR L dI 0 dt where τ = L/R and Ii is the current at t = 0. 35. Consider the circuit in Figure P23.35, taking ε = 6.00 V, L = 8.00 mH, and R = 4.00 Ω. (a) What is the inductive time constant of the circuit? (b) Calculate the current in the circuit 250 μs after the switch is closed. (c) What is the value of the final steadystate current? (d) How long does it take the current to reach 80.0% of its maximum value? Figure P23.35 Problems 23.35, 23.36, and 23.38. 36. For the RL circuit shown in Figure P23.35, let the inductance be 3.00 H, the resistance 8.00 Ω, and the battery emf 36.0 V. (a) Calculate the ratio of the potential difference across the resistor to the voltage across the inductor when the current is 2.00 A. (b) Calculate the voltage across the inductor when the current is 4.50 A. 37. A circuit consists of a coil, a switch, and a battery, all in series. The internal resistance of the battery is negligible compared with that of the coil. The switch is originally open. It is thrown closed, and after a time interval Δt, the current in the circuit reaches 80.0% of its final value. The switch remains closed for a time interval much longer than Δt. Then the battery is disconnected and the terminals of the coil are connected together to form a short circuit. (a) After an equal additional time interval Δt elapses, the current is what percentage of its maximum value? (b) At the moment 2Δt after the coil is shortcircuited, the current in the coil is what percentage of its maximum value? 38. When the switch in Figure P23.35 is closed, the current takes 3.00 ms to reach 98.0% of its final value. If R = 10.0 Ω, what is the inductance? 39. The switch in Figure P23.39 is open for t < 0 and then closed at time t = 0. Find the current in the inductor and the current in the switch as functions of time thereafter. 41. A 140-mH inductor and a 4.90-Ω resistor are connected with a switch to a 6.00-V battery as shown in Figure P23.41. (a) If the switch is thrown to the left (connecting the battery), how much time elapses before the current reaches 220 mA? (b) What is the current in the inductor 10.0 s after the switch is closed? (c) Now the switch is quickly thrown from a to b. How much time elapses before the current falls to 160 mA? Figure P23.39 40. One application of an RL circuit is the generation of time-varying high voltage from a low-voltage source as shown in Figure P23.40. (a) What is the current in the circuit a long time after the switch has been in position a? (b) Now the switch is thrown quickly from a to b. Compute the initial voltage across each resistor and across the inductor. (c) How much time elapses before the voltage across the inductor drops to 12.0 V? Figure P23.40 Figure P23.41 Section 23.7 Energy Stored in a Magnetic Field 42. The magnetic field inside a superconducting solenoid is 4.50 T. The solenoid has an inner diameter of 6.20 cm and a length of 26.0 cm. Determine (a) the magnetic energy density in the field and (b) the energy stored in the magnetic field within the solenoid. 43. An air-core solenoid with 68 turns is 8.00 cm long and has a diameter of 1.20 cm. How much energy is stored in its magnetic field when it carries a current of 0.770 A? 44. An RL circuit in which the inductance is 4.00 H and the resistance is 5.00 Ω is connected to a 22.0-V battery at t = 0. (a) What energy is stored in the inductor when the current is 0.500 A? (b) At what rate is energy being stored in the inductor when I = 1.00 A? (c) What power is being delivered to the circuit by the battery when I = 0.500 A? 45. On a clear day at a certain location, a 100-V/m vertical electric field exists near the Earth’s surface. At the same place, the Earth’s magnetic field has a magnitude of 0.500 × 10–4 T. Compute the energy densities of the two fields. not induced in the opposite direction to apply a downward force on the vehicle.) The vehicle has a mass of 5 × 104 kg and travels at a speed of 400 km/h. If the vehicle has 100 loops carrying current at any moment, what is the approximate magnitude of the magnetic field required to levitate the vehicle? Assume that the magnetic force acts over the entire 10-cm length of the horizontal wire. Your answer should suggest that this design is impractical for magnetic levitation. Section 23.8 Context Connection—The Repulsive Model for Magnetic Levitation 46. The following is a crude model for levitating a commercial transportation vehicle using Faraday’s law. Assume that magnets are used to establish regions of horizontal magnetic field across the track as shown in Figure P23.46. Rectangular loops of wire are mounted on the vehicle so that the lower 20 cm of each loop passes into these regions of magnetic field. The upper portion of each loop contains a 25-Ω resistor. As the leading edge of a loop passes into the magnetic field, a current is induced in the loop as shown in the figure. The magnetic force on this current in the bottom of the loop, of length 10 cm, results in an upward force on the vehicle. (By electronic timing, a switch is opened in the loop before the loop’s leading edge leaves the region of magnetic field, so a current is Figure P23.46 47. The Meissner effect. Compare this problem with Problem 20.76 on the force attracting a perfect dielectric into a strong electric field. A fundamental property of a type I superconducting material is perfect diamagnetism, or demonstration of the Meissner effect, illustrated in the photograph of the levitating magnet on page 693 and described as follows. The superconducting material has B 0 everywhere inside it. If a sample of the material is placed into an externally produced magnetic field or if it is cooled to become superconducting while it is in a magnetic field, electric currents appear on the surface of the sample. The currents have precisely the strength and orientation required to make the total magnetic field zero throughout the interior of the sample. The following problem will help you understand the magnetic force that can then act on the superconducting sample. A vertical solenoid with a length of 120 cm and a diameter of 2.50 cm consists of 1 400 turns of copper wire carrying a counterclockwise current of 2.00 A as shown in Figure P23.47a. (a) Find the magnetic field in the vacuum inside the solenoid. (b) Find the energy density of the magnetic field. Note that the units J/m3 of energy density are the same as the units N/m2 of pressure. (c) Now a superconducting bar 2.20 cm in diameter is inserted partway into the solenoid. Its upper end is far outside the solenoid, where the magnetic field is negligible. The lower end of the bar is deep inside the solenoid. Identify the direction required for the current on the curved surface of the bar so that the total magnetic field is zero within the bar. The field created by the supercurrents is sketched in Figure P23.47b, and the total field is sketched in Figure P23.47c. (d) The field of the solenoid exerts a force on the current in the superconductor. Identify the direction of the force on the bar. (e) Calculate the magnitude of the force by multiplying the energy density of the solenoid field times the area of the bottom end of the superconducting bar. Figure P23.47 Additional Problems 48. Figure P23.48 is a graph of the induced emf versus time for a coil of N turns rotating with angular speed ω in a uniform magnetic field directed perpendicular to the axis of rotation of the coil. Copy this sketch (on a larger scale) and on the same set of axes show the graph of emf versus t (a) if the number of turns in the coil is doubled, (b) if instead the angular speed is doubled, and (c) if the angular speed is doubled while the number of turns in the coil is halved. Figure P23.48 shown in Figure P23.52. A battery that 49. A steel guitar string vibrates (Figure 23.6). The component of magnetic field perpendicular to the area of a pickup coil nearby is given by B = 50.0 mT + (3.20 mT) sin(2π 523 t/s) The circular pickup coil has 30 turns and radius 2.70 mm. Find the emf induced in the coil as a function of time. 50. Strong magnetic fields are used in such medical procedures as magnetic resonance imaging. A technician wearing a brass bracelet enclosing area 0.005 00 m2 places her hand in a solenoid whose magnetic field is 5.00 T directed perpendicular to the plane of the bracelet. The electrical resistance around the circumference of the bracelet is 0.020 0 Ω. An unexpected power failure causes the field to drop to 1.50 T in a time of 20.0 ms. Find (a) the current induced in the bracelet and (b) the power delivered to the bracelet. (Note: As this problem implies, you should not wear any metal objects when working in regions of strong magnetic fields.) 51. Suppose you wrap wire onto the core from a roll of cellophane tape to make a coil. Describe how you can use a bar magnet to produce an induced voltage in the coil. What is the order of magnitude of the emf you generate? State the quantities you take as data and their values. 52. A bar of mass m, length d, and resistance R slides without friction in a horizontal plane, moving on parallel rails as maintains a constant emf ε is connected between the rails, and a constant magnetic field B is directed perpendicularly to the plane of the page. Assuming that the bar starts from rest, show that at time t it moves with a speed v 1 e Bd B 2 d 2 t / mR Figure P23.52 53. Review problem. A particle with a mass of 2.00 × 10–16 kg and a charge of 30.0 nC starts from rest, is accelerated by a strong electric field, and is fired from a small source inside a region of uniform constant magnetic field 0.600 T. The velocity of the particle is perpendicular to the field. The circular orbit of the particle encloses a magnetic flux of 15.0 μWb. (a) Calculate the speed of the particle. (b) Calculate the potential difference through which the particle accelerated inside the source. 54. An induction furnace uses electromagnetic induction to produce eddy currents in a conductor, thereby raising the conductor’s temperature. Commercial units operate at frequencies ranging from 60 Hz to about 1 MHz and deliver powers from a few watts to several megawatts. Induction heating can be used for warming a metal pan on a kitchen stove. It can also be used for welding in a vacuum enclosure so as to avoid oxidation and contamination of the metal. At high frequencies, induced currents occur only near the surface of the conductor—this phenomenon is the “skin effect.” By creating an induced current for a short time interval at an appropriately high frequency, one can heat a sample down to a controlled depth. For example, the surface of a farm tiller can be tempered to make it hard and brittle for effective cutting while keeping the interior metal soft and ductile to resist breakage. To explore induction heating, consider a flat conducting disk of radius R, thickness b, and resistivity ρ. A sinusoidal magnetic field Bmax cos ωt is applied perpendicular to the disk. Assume that the field is uniform in space and that the frequency is so low that the skin effect is not important. Assume that the eddy currents occur in circles concentric with the disk. (a) Calculate the average power delivered to the disk. By what factor does the power change (b) when the amplitude of the field doubles, (c) when the frequency doubles, and (d) when the radius of the disk doubles? 55. The magnetic flux through a metal ring varies with time t according to ΦB = 3(at3 – bt2)T · m2, with a = 2.00 s–3 and b = 6.00 s–2. The resistance of the ring is 3.00 Ω. Determine the maximum current induced in the ring during the interval from t = 0 to t = 2.00 s. 56. Figure P23.56 shows a stationary conductor whose shape is similar to the letter e. The radius of its circular portion is a = 50.0 cm. It is placed in a constant magnetic field of 0.500 T directed out of the page. A straight conducting rod, 50.0 cm long, is pivoted about point O and rotates with a constant angular speed of 2.00 rad/s. (a) Determine the induced emf in the loop POQ. Note that the area of the loop is θa2/2. (b) If all the conducting material has a resistance per length of 5.00 Ω/m, what is the induced current in the loop POQ at the instant 0.250 s after point P passes point Q? Figure P23.56 57. A betatron accelerates electrons to energies in the MeV range by means of electromagnetic induction. Electrons in a vacuum chamber are held in a circular orbit by a magnetic field perpendicular to the orbital plane. The magnetic field is gradually increased to induce an electric field around the orbit. (a) Show that the electric field is in the correct direction to make the electrons speed up. (b) Assume that the radius of the orbit remains constant. Show that the average magnetic field over the area enclosed by the orbit must be twice as large as the magnetic field at the circumference of the circle. 58. To monitor the breathing of a hospital patient, a thin belt is girded around the patient’s chest. The belt is a 200-turn coil. When the patient inhales, the area encircled by the coil increases by 39.0 cm2. The magnitude of the Earth’s magnetic field is 50.0 μT and makes an angle of 28.0° with the plane of the coil. Assuming that a patient takes 1.80 s to inhale, find the average induced emf in the coil during this time. 59. A long, straight wire carries a current that is given by I = Imax sin(ωt + ). The wire lies in the plane of a rectangular coil of N turns of wire as shown in Figure P23.5. The quantities Imax, ω, and are all constants. Determine the emf induced in the coil by the magnetic field created by the current in the straight wire. Assume that Imax = 50.0 A, ω = 200 πs–1, N = 100, h = w = 5.00 cm, and L = 20.0 cm. 60. The toroid in Figure P23.60 consists of N turns and has a rectangular crosssection. Its inner and outer radii are a and b, respectively. (a) Show that the inductance of the toroid is L 0 N 2 h b ln 2 a (b) Using this result, compute the selfinductance of a 500-turn toroid for which a = 10.0 cm, b = 12.0 cm, and h = 1.00 cm. (c) In Problem 23.32, an approximate expression for the inductance of a toroid with R >> r was derived. To get a feel for the accuracy of that result, use the expression in Problem 23.32 to compute the approximate inductance of the toroid described in part (b). Compare the result with the answer to part (b). Figure P23.60 61. (a) A flat, circular coil does not really produce a uniform magnetic field in the area it encloses. Nonetheless, estimate the self-inductance of a flat, compact, circular coil, with radius R and N turns, by assuming that the field at its center is uniform over its area. (b) A circuit on a laboratory table consists of a 1.5-volt battery, a 270-Ω resistor, a switch, and three 30-cm-long patch cords connecting them. Suppose the circuit is arranged to be circular. Think of it as a flat coil with one turn. Compute the order of magnitude of its self-inductance and (c) of the time constant describing how fast the current increases when you close the switch. 62. To prevent damage from arcing in an electric motor, a discharge resistor is sometimes placed in parallel with the armature. If the motor is suddenly unplugged while running, this resistor limits the voltage that appears across the armature coils. Consider a 12.0-V DC motor with an armature that has a resistance of 7.50 Ω and an inductance of 450 mH. Assume that the magnitude of the selfinduced emf in the armature coils is 10.0 V when the motor is running at normal speed. (The equivalent circuit for the armature is shown in Fig. P23.62.) Calculate the maximum resistance R that limits the voltage across the armature to 80.0 V when the motor is unplugged. Figure P23.62 Review problems. Problems 23.63 through 23.65 apply ideas from this chapter and earlier chapters to some properties of superconductors, which were introduced in Section 21.3. 63. The resistance of a superconductor. In an experiment carried out by S. C. Collins between 1955 and 1958, a current was maintained in a superconducting lead ring for 2.50 yr with no observed loss. If the inductance of the ring was 3.14 × 10–8 H and the sensitivity of the experiment was 1 part in 109, what was the maximum resistance of the ring? (Suggestion: Treat this problem as a decaying current in an RL circuit and recall that e–x ≈ 1 – x for small x.) 64. A novel method of storing energy has been proposed. A huge underground superconducting coil, 1.00 km in diameter, would be fabricated. It would carry a maximum current of 50.0 kA through each winding of a 150-turn Nb3Sn solenoid. (a) If the inductance of this huge coil were 50.0 H, what would be the total energy stored? (b) What would be the compressive force per meter length acting between two adjacent windings 0.250 m apart? 65. Superconducting power transmission. The use of superconductors has been proposed for power transmission lines. A single coaxial cable (Fig. P23.65) could carry 1.00 × 103 MW (the output of a large power plant) at 200 kV, DC, over a distance of 1 000 km without loss. An inner wire of radius 2.00 cm, made from the superconductor Nb3Sn, carries the current I in one direction. A surrounding superconducting cylinder, of radius 5.00 cm, would carry the return current I. In such a system, what is the magnetic field (a) at the surface of the inner conductor and (b) at the inner surface of the outer conductor? (c) How much energy would be stored in the space between the conductors in a 1 000-km superconducting line? (d) What is the pressure exerted on the outer conductor? Figure P23.65 © Copyright 2004 Thomson. All rights reserved.