CollegePhysics2e-SSM-Ch23

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OpenStax College Physics 2e Student Solutions Manual Chapter 23 CHAPTER 23: ELECTROMAGNETIC INDUCTION, AC CIRCUITS, AND ELECTRICAL TECHNOLOGIES 23.1 INDUCED EMF AND MAGNETIC FLUX 1. What is the value of the magnetic flux at coil 2 in Figure 23.53 due to coil 1? Solution Using the equation , we can calculate the flux through coil 2, since the coils are perpendicular: 23.2 FARADAY’S LAW OF INDUCTION: LENZ’S LAW 3. Referring to Figure 23.54 (a), what is the direction of the current induced in coil 2: (a) If the current in coil 1 increases? (b) If the current in coil 1 decreases? (c) If the current in coil 1 is constant? Explicitly show how you follow the steps in the Problem-Solving Strategy for Lenz's Law . Solution (a) CCW (b) CW (c) no current in coil 2 5. Referring to Figure 23.55 , what are the directions of the currents in coils 1, 2, and 3 (assume that the coils are lying in the plane of the circuit): (a) When the switch is first closed? (b) When the switch has been closed for a long time? (c) Just after the switch is opened? Solution (a) (i) CCW (ii) CCW (iii) CW (b) (i) no current (ii) no current (iii) no current (c) (i) CW (ii) CW (iii) CCW 9. (a) An MRI technician moves his hand from a region of very low magnetic field strength into an MRI scanner’s 2.00 T field with his fingers pointing in the direction of the field. Find the average emf induced in his wedding ring, given its diameter is 2.20 cm and assuming it takes 0.250 s to move it into the field. (b) Discuss whether this current would significantly change the temperature of the ring.

OpenStax College Physics 2e Student Solutions Manual Chapter 23 Solution (a) (b) Since power goes as , the smaller the resistance the larger the power and the more heat transferred to the ring. So, as a lower limit on the ring, estimate . In this case, the power would be: . This power exists for of a second, so the heat transferred will be: . This is not a significant amount of heat. 11. An emf is induced by rotating a 1000-turn, 20.0 cm diameter coil in the Earth’s magnetic field. What average emf is induced, given the plane of the coil is originally perpendicular to the Earth’s field and is rotated to be parallel to the field in 10.0 ms? Solution 13. Approximately how does the emf induced in the loop in Figure 23.54 (b) depend on the distance of the center of the loop from the wire? Solution The magnetic field due to an infinite straight wire is , so the flux and hence the emf induced in the loop will be proportional to , where r is the distance from the wire. 23.3 MOTIONAL EMF 17. (a) A jet airplane with a 75.0 m wingspan is flying at 280 m/s. What emf is induced between wing tips if the vertical component of the Earth’s field is ? (b) Is an emf of this magnitude likely to have any consequences? Explain. Solution (a) (b) No , this is a very small emf. 18. (a) A nonferrous screwdriver is being used in a 2.00 T magnetic field. What maximum emf can be induced along its 12.0 cm length when it moves at 6.00 m/s? (b) Is it likely that this emf will have any consequences or even be noticed? Solution (a) (b) No. 19. At what speed must the sliding rod in Figure 23.10 move to produce an emf of 1.00 V in a 1.50 T field, given the rod’s length is 30.0 cm? Solution

OpenStax College Physics 2e Student Solutions Manual Chapter 23 25. Integrated Concepts The Tethered Satellite discussed in this module is producing 5.00 kV, and a current of 10.0 A flows. (a) What magnetic drag force does this produce if the system is moving at 7.80 km/s? (b) How much kinetic energy is removed from the system in 1.00 h, neglecting any change in altitude or velocity during that time? (c) What is the change in velocity if the mass of the system is 100,000 kg? (d) Discuss the long term consequences (say, a week-long mission) on the space shuttle’s orbit, noting what effect a decrease in velocity has and assessing the magnitude of the effect. Solution (a) (b) (c) Note that five digits are needed to see the difference between and . (d) For a week-long mission (168 hours), the change in velocity will be 60 m/s, or approximately 1%. In general, a decrease in velocity would cause the orbit to start spiraling inward because the velocity would no longer be sufficient to keep the circular orbit. The long term consequences are that the shuttle would require a little more fuel to maintain the desired speed, otherwise the orbit would spiral slightly inward. 23.5 ELECTRIC GENERATORS 28. Calculate the peak voltage of a generator that rotates its 200-turn, 0.100 m diameter coil at 3600 rpm in a 0.800 T field. Solution 30. What is the peak emf generated by rotating a 1000-turn, 20.0 cm diameter coil in the Earth’s magnetic field, given the plane of the coil is originally perpendicular to the Earth’s field and is rotated to be parallel to the field in 10.0 ms? Solution

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OpenStax College Physics 2e Student Solutions Manual Chapter 23 32. (a) A bicycle generator rotates at 1875 rad/s, producing an 18.0 V peak emf. It has a 1.00 by 3.00 cm rectangular coil in a 0.640 T field. How many turns are in the coil? (b) Is this number of turns of wire practical for a 1.00 by 3.00 cm coil? Solution (a) (b) Yes , 50 turns is practical for a coil of this size. 34. (a) A car generator turns at 400 rpm when the engine is idling. Its 300-turn, 5.00 by 8.00 cm rectangular coil rotates in an adjustable magnetic field so that it can produce sufficient voltage even at low rpms. What is the field strength needed to produce a 24.0 V peak emf? (b) Discuss how this required field strength compares to those available in permanent and electromagnets. Solution (a) (b) This field strength is small enough that it can be obtained using either a permanent magnet or an electromagnet. 36. A 75-turn, 10.0 cm diameter coil rotates at an angular velocity of 8.00 rad/s in a 1.25 T field, starting with the plane of the coil parallel to the field. (a) What is the peak emf? (b) At what time is the peak emf first reached? (c) At what time is the emf first at its most negative? (d) What is the period of the AC voltage output? Solution (a) (b) We know this is the correct expression for since must change sign when peak emf is when (just after the generator is turned on). (c)

OpenStax College Physics 2e Student Solutions Manual Chapter 23 (d) 38. Unreasonable Results A 500-turn coil with a area is spun in the Earth’s field, producing a 12.0 kV maximum emf. (a) At what angular velocity must the coil be spun? (b) What is unreasonable about this result? (c) Which assumption or premise is responsible? Solution (a) (b) This angular velocity is unreasonably high, higher than can be obtained for any mechanical system. (c) The assumption that a voltage as great as 12.0 kV could be obtained is unreasonable. 23.6 BACK EMF 39. Suppose a motor connected to a 120 V source draws 10.0 A when it first starts. (a) What is its resistance? (b) What current does it draw at its normal operating speed when it develops a 100 V back emf? Solution (a) (b) 41. What is the back emf of a 120 V motor that draws 8.00 A at its normal speed and 20.0 A when first starting? Solution 43. Integrated Concepts The motor in a toy car is powered by four batteries in series, which produce a total emf of 6.00 V. The motor draws 3.00 A and develops a 4.50 V back emf at normal speed. Each battery has a internal resistance. What is the resistance of the motor? Solution Since the resistors are in series, we know the total internal resistance of the batteries is . Therefore, 23.7 TRANSFORMERS

OpenStax College Physics 2e Student Solutions Manual Chapter 23 44. A plug-in transformer, like that in Figure 23.25 , supplies 9.00 V to a video game system. (a) How many turns are in its secondary coil, if its input voltage is 120 V and the primary coil has 400 turns? (b) What is its input current when its output is 1.30 A? Solution (a) (b) 46. A cassette recorder uses a plug-in transformer to convert 120 V to 12.0 V, with a maximum current output of 200 mA. (a) What is the current input? (b) What is the power input? (c) Is this amount of power reasonable for a small appliance? Solution (a) (b) (c) Yes , this amount of power is quite reasonable for a small appliance. 48. (a) The plug-in transformer for a laptop computer puts out 7.50 V and can supply a maximum current of 2.00 A. What is the maximum input current if the input voltage is 240 V? Assume 100% efficiency. (b) If the actual efficiency is less than 100%, would the input current need to be greater or smaller? Explain. Solution (a) (b) If the efficiency is less than 100%, the input current would need to be greater because more power would need to be input into the transformer to produce the required power output. 50. A large power plant generates electricity at 12.0 kV. Its old transformer once converted the voltage to 335 kV. The secondary of this transformer is being replaced so that its output can be 750 kV for more efficient cross-country transmission on upgraded transmission lines. (a) What is the ratio of turns in the new secondary compared with the old secondary? (b) What is the ratio of new current output to old output (at 335 kV) for the same power? (c) If the upgraded transmission lines have the same resistance, what is the ratio of new line power loss to old?

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OpenStax College Physics 2e Student Solutions Manual Chapter 23 Solution (a) (b) (c) 52. Unreasonable Results The 335 kV AC electricity from a power transmission line is fed into the primary coil of a transformer. The ratio of the number of turns in the secondary to the number in the primary is . (a) What voltage is induced in the secondary? (b) What is unreasonable about this result? (c) Which assumption or premise is responsible? Solution (a) (b) The result is way too high, well beyond the breakdown voltage of air over reasonable distances. (c) The input voltage is too high. 23.8 ELECTRICAL SAFETY: SYSTEMS AND DEVICES 54. Integrated Concepts A short circuit to the grounded metal case of an appliance occurs as shown in Figure 23.57 . The person touching the case is wet and only has a resistance to earth/ground. (a) What is the voltage on the case if 5.00 mA flows through the person? (b) What is the current in the short circuit if the resistance of the earth/ground wire is ? (c) Will this trigger the 20.0 A circuit breaker supplying the appliance? Solution (a) (b) (c) Yes. 23.9 INDUCTANCE 55. Two coils are placed close together in a physics lab to demonstrate Faraday’s law of induction. A current of 5.00 A in one is switched off in 1.00 ms, inducing a 9.00 V emf in the other. What is their mutual inductance? Solution Using the equation where the minus sign is an expression of Lenz’s law, we can calculate the mutual inductance between the two coils:

OpenStax College Physics 2e Student Solutions Manual Chapter 23 57. The 4.00 A current through a 7.50 mH inductor is switched off in 8.33 ms. What is the emf induced opposing this? Solution 61. A large research solenoid has a self-inductance of 25.0 H. (a) What induced emf opposes shutting it off when 100 A of current through it is switched off in 80.0 ms? (b) How much energy is stored in the inductor at full current? (c) At what rate in watts must energy be dissipated to switch the current off in 80.0 ms? (d) In view of the answer to the last part, is it surprising that shutting it down this quickly is difficult? Solution (a) Using the equation , we have (b) Using (c) Using the equation ,we have (d) No , it is not surprising since this power is very high. 63. A precision laboratory resistor is made of a coil of wire 1.50 cm in diameter and 4.00 cm long, and it has 500 turns. (a) What is its self-inductance? (b) What average emf is induced if the 12.0 A current through it is turned on in 5.00 ms (one-fourth of a cycle for 50 Hz AC)? (c) What is its inductance if it is shortened to half its length and counter-wound (two layers of 250 turns in opposite directions)? Solution (a) (b) (c) . Two 250 turn coils in opposite directions will cancel each other. 65. When the 20.0 A current through an inductor is turned off in 1.50 ms, an 800 V emf is induced, opposing the change. What is the value of the self-inductance? Solution

OpenStax College Physics 2e Student Solutions Manual Chapter 23 67. Integrated Concepts A very large, superconducting solenoid such as one used in MRI scans, stores 1.00 MJ of energy in its magnetic field when 100 A flows. (a) Find its self- inductance. (b) If the coils “go normal,” they gain resistance and start to dissipate thermal energy. What temperature increase is produced if all the stored energy goes into heating the 1000 kg magnet, given its average specific heat is ? Solution (a) (b) 23.10 RL CIRCUITS 69. If you want a characteristic RL time constant of 1.00 s, and you have a resistor, what value of self-inductance is needed? Solution Using the equation we know 71. A large superconducting magnet, used for magnetic resonance imaging, has a 50.0 H inductance. If you want current through it to be adjustable with a 1.00 s characteristic time constant, what is the minimum resistance of system? Solution 73. Suppose you have a supply of inductors ranging from 1.00 nH to 10.0 H, and resistors ranging from to . What is the range of characteristic RL time constants you can produce by connecting a single resistor to a single inductor? Solution 75. What percentage of the final current flows through an inductor in series with a resistor , three time constants after the circuit is completed? Solution The current is 95.0% of the final current after 3 time constants. 77. (a) Use the exact exponential treatment to find how much time is required to bring the current through an 80.0 mH inductor in series with a resistor to 99.0% of its final value, starting from zero. (b) Compare your answer to the approximate treatment using integral numbers of . (c) Discuss how significant the difference is.

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OpenStax College Physics 2e Student Solutions Manual Chapter 23 Solution (a) (b) Using the result from part (a), (c) So, using integer values is off by about 9% which is greater than the inherent uncertainty in the given parameters. 23.11 REACTANCE, INDUCTIVE AND CAPACITIVE 79. At what frequency will a 30.0 mH inductor have a reactance of ? Solution 81. What capacitance should be used to produce a reactance at 60.0 Hz? Solution 83. (a) Find the current through a 0.500 H inductor connected to a 60.0 Hz, 480 V AC source. (b) What would the current be at 100 kHz? Solution (a) (b) 85. A 20.0 kHz, 16.0 V source connected to an inductor produces a 2.00 A current. What is the inductance? Solution

OpenStax College Physics 2e Student Solutions Manual Chapter 23 87. (a) An inductor designed to filter high-frequency noise from power supplied to a personal computer is placed in series with the computer. What minimum inductance should it have to produce a reactance for 15.0 kHz noise? (b) What is its reactance at 60.0 Hz? Solution (a) Using the equation , (b) Again using , 89. The capacitor in Figure 23.52 (b) will filter high-frequency signals by shorting them to earth/ground. (a) What capacitance is needed to produce a reactance of for a 5.00 kHz signal? (b) What would its reactance be at 3.00 Hz? (c) Discuss the implications of your answers to (a) and (b). Solution (a) (b) (c) The reactance from the capacitor at high frequencies is much smaller than that at lower frequencies, so the capacitor does indeed allow the high frequencies to short to ground much better than the low frequencies, therefore working as it is designed to. 23.12 RLC SERIES AC CIRCUITS 92. An RL circuit consists of a resistor and a 3.00 mH inductor. (a) Find its impedance at 60.0 Hz and 10.0 kHz. (b) Compare these values of with those found in Example 23.12 , in which there was also a capacitor. Solution (a) (b) At 60 Hz, with a capacitor, , over 13 times as high as without the capacitor. The capacitor makes a large difference at low frequencies. At 10 kHz, with a capacitor , about the same as without the capacitor. The capacitor has a smaller effect at high frequencies.

OpenStax College Physics 2e Student Solutions Manual Chapter 23 94. An LC circuit consists of a inductor and a capacitor. (a) Find its impedance at 60.0 Hz and 10.0 kHz. (b) Compare these values of with those found in Example 23.12 , in which there was also a resistor. Solution (a) (b) These values are close to those obtained in Example 23.12 because at low frequency the capacitor dominates and at high frequency the inductor dominates. So in both cases the resistor makes little contribution to the total impedance. 96. To receive AM radio, you want an RLC circuit that can be made to resonate at any frequency between 500 and 1650 kHz. This is accomplished with a fixed inductor connected to a variable capacitor. What range of capacitance is needed? Solution 98. What capacitance do you need to produce a resonant frequency of 1.00 GHz, when using an 8.00 nH inductor? Solution 100. The lowest frequency in the FM radio band is 88.0 MHz. (a) What inductance is needed to produce this resonant frequency if it is connected to a 2.50 pF capacitor? (b) The capacitor is variable, to allow the resonant frequency to be adjusted to as high as 108 MHz. What must the capacitance be at this frequency?

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OpenStax College Physics 2e Student Solutions Manual Chapter 23 Solution (a) (b) 102. An RLC series circuit has a resistor, a inductor, and a 25.0 nF capacitor. (a) Find the circuit’s impedance at 500 Hz. (b) Find the circuit’s impedance at 7.50 kHz. (c) If the voltage source has , what is at each frequency? (d) What is the resonant frequency of the circuit? (e) What is at resonance? Solution (a) (b) (c) (i) (ii) (d) (e) 104. An RLC series circuit has a resistor, a inductor, and a 25.0 nF capacitor. (a) Find the power factor at . (b) What is the phase angle at this frequency? (c) What is the average power at this frequency? (d) Find the average power at the circuit’s resonant frequency.

OpenStax College Physics 2e Student Solutions Manual Chapter 23 Solution (a) From Exercise 23.102 , we know : (b) (c) (d) 106. Referring to Example 23.14 , find the average power at 10.0 kHz. Solution 107. Critical Thinking A length of 4.000 m of wire is to be used to detect a magnetic field. The wire is made into a single square loop and rotated at a rate of 400 cycles per second. (a) If the magnetic field is 0.02000 T, what is the magnitude of the average emf that can be generated in the first quarter cycle, provided the loop is initially oriented in a plane perpendicular to the magnetic field? (b) Is there a difference in the magnitude of the average emf generated if the wire is made into two square loops and rotated at the same rate, starting with the same orientation of the loops as that of the loop in part a? If so, what is the average emf possible for the first quarter cycle for two loops being rotated at 400 cycles per second in the magnetic field? (c) If the wire is made into a figure eight, what is the average emf for the first quarter cycle that can be generated by rotating it at 400 cycles per second in the magnetic field, again starting with the same orientation of the oops with respect to the magnetic field? The wire crosses itself in this arrangement. (d) Does the shape of the loop matter? Solution (a) | emf | = 1 Δ ϕ Δt = 1.000 ( 0.02000 ) 100 V = 0.0002000 V (b) | emf | = 2 Δ ϕ Δt = 2 ( 0.2500 )( 0.02000 ) 100 V = 0.0001000 V (c) The average emf is 0 since the loops are made by a wire that crosses, so the voltage from each loop cancels the emf from the other.

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