Problems

Note to Instructors: Most of the homework problems in this chapter are available for assignment via an online homework management program such as WileyPLUS or WebAssign, and those marked with the icon are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details.
ssm Solution is in the Student Solutions Manual.
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Section 22.2 Motional Emf

| 1 | The wingspan (tip to tip) of a Boeing 747 jetliner is 59 m. The plane is flying horizontally at a speed of 220 m/s. The vertical component of the earth’s magnetic field is 5.0 × 10−6 T. Find the emf induced between the wing tips.

| 2 | In 1996, NASA performed an experiment called the Tethered Satellite experiment. In this experiment a 2.0 × 104-m length of wire was let out by the space shuttle Atlantis to generate a motional emf. The shuttle had an orbital speed of 7.6 × 103 m/s, and the magnitude of the earth’s magnetic field at the location of the wire was 5.1 × 10−5 T. If the wire had moved perpendicular to the earth’s magnetic field, what would have been the motional emf generated between the ends of the wire?

| 3 | ssm A spark can jump between two nontouching conductors if the potential difference between them is sufficiently large. A potential difference of approximately 940 V is required to produce a spark in an air gap of 1.0 × 10−4 m. Suppose the light bulb in Figure 22-4b is replaced by such a gap. How fast would a 1.3-m rod have to be moving in a magnetic field of 4.8 T to cause a spark to jump across the gap?

| 4 | The drawing shows a type of flow meter that can be used to measure the speed of blood in situations when a blood vessel is sufficiently exposed (e.g., during surgery). Blood is conductive enough that it can be treated as a moving conductor. When it flows perpendicularly with respect to a magnetic field, as in the drawing, electrodes can be used to measure the small voltage that develops across the vessel. Suppose the speed of the blood is 0.30 m/s and the diameter of the vessel is 5.6 mm. In a 0.60-T magnetic field what is the magnitude of the voltage that is measured with the electrodes in the drawing?

| 5 | ssm www The drawing shows three identical rods (A, B, and C) moving in different planes. A constant magnetic field of magnitude 0.45 T is directed along the +y axis. The length of each rod is L = 1.3 m, and the speeds are the same, vA = vB = vC = 2.7 m/s. For each rod, find the magnitude of the motional emf, and indicate which end (1 or 2) of the rod is positive.

* | 6 | Multiple-Concept Example 2 discusses the concepts that are used in this problem. Suppose, in Figure 22-5, that the magnetic field has a magnitude of 1.2 T, the rod has a length of 0.90 m, and the hand keeps the rod moving to the right at a constant speed of 3.5 m/s. If the current in the circuit is 0.040 A, what is the average power being delivered to the circuit by the hand?

* | 7 | ssm Suppose the light bulb in Figure 22-4b is a 60.0-W bulb with a resistance of . The magnetic field has a magnitude of 0.40 T, and the length of the rod is 0.60 m. The only resistance in the circuit is that due to the bulb. Minimally, how long would the rails on which the moving rod slides have to be, in order that the bulb can remain lit for one-half second?

* | 8 | Refer to the drawing that accompanies Conceptual Question 6 (not Problem 6). Suppose that the voltage of the battery in the circuit is 3.0 V, the magnitude of the magnetic field (directed perpendicularly into the plane of the paper) is 0.60 T, and the length of the rod between the rails is 0.20 m. Assuming that the rails are very long and have negligible resistance, find the maximum speed attained by the rod after the switch is closed.

** | 9 | Review Conceptual Example 3 and Figure 22-7b as an aid in solving this problem. A conducting rod slides down between two frictionless vertical copper tracks at a constant speed of 4.0 m/s perpendicular to a 0.50-T magnetic field. The resistance of the rod and tracks is negligible. The rod maintains electrical contact with the tracks at all times and has a length of 1.3 m. A resistor is attached between the tops of the tracks. (a) What is the mass of the rod? (b) Find the change in the gravitational potential energy that occurs in a time of 0.20 s. (c) Find the electrical energy dissipated in the resistor in 0.20 s.
Section 22.3 Magnetic Flux
For problems in this set, assume that the magnetic flux is a positive quantity.

| 10 | A magnetic field has a magnitude of 0.078 T and is uniform over a circular surface whose radius is 0.10 m. The field is oriented at an angle with respect to the normal to the surface. What is the magnetic flux through the surface?

| 11 | A house has a floor area of 112 m2 and an outside wall that has an area of 28 m2. The earth’s magnetic field here has a horizontal component of 2.6 × 10−5 T that points due north and a vertical component of 4.2 × 10−5 T that points straight down, toward the earth. Determine the magnetic flux through the wall if the wall faces (a) north and (b) east. (c) Calculate the magnetic flux that passes through the floor.

| 12 | The drawing shows two surfaces that have the same area. A uniform magnetic field fills the space occupied by these surfaces and is oriented parallel to the yz plane as shown. Find the ratio of the magnetic fluxes that pass through the surfaces.

| 13 | ssm A standard door into a house rotates about a vertical axis through one side, as defined by the door’s hinges. A uniform magnetic field is parallel to the ground and perpendicular to this axis. Through what angle must the door rotate so that the magnetic flux that passes through it decreases from its maximum value to one-third of its maximum value?

| 14 | A loop of wire has the shape shown in the drawing. The top part of the wire is bent into a semicircle of radius r = 0.20 m. The normal to the plane of the loop is parallel to a constant magnetic field of magnitude 0.75 T. What is the change in the magnetic flux that passes through the loop when, starting with the position shown in the drawing, the semicircle is rotated through half a revolution?

* | 15 | ssm www A five-sided object, whose dimensions are shown in the drawing, is placed in a uniform magnetic field. The magnetic field has a magnitude of 0.25 T and points along the positive y direction. Determine the magnetic flux through each of the five sides.

* | 16 | A long and narrow rectangular loop of wire is moving toward the bottom of the page with a speed of 0.020 m/s (see the drawing). The loop is leaving a region in which a 2.4-T magnetic field exists; the magnetic field outside this region is zero. During a time of 2.0 s, what is the magnitude of the change in the magnetic flux?

Section 22.4 Faraday’s Law of Electromagnetic Induction

| 17 | A planar coil of wire has a single turn. The normal to this coil is parallel to a uniform and constant (in time) magnetic field of 1.7 T. An emf that has a magnitude of 2.6 V is induced in this coil because the coil’s area A is shrinking. What is the magnitude of , which is the rate (in m2/s) at which the area changes?

| 18 | A rectangular loop of wire with sides 0.20 and 0.35 m lies in a plane perpendicular to a constant magnetic field (see part a of the drawing). The magnetic field has a magnitude of 0.65 T and is directed parallel to the normal of the loop’s surface. In a time of 0.18 s, one-half of the loop is then folded back onto the other half, as indicated in part b of the drawing. Determine the magnitude of the average emf induced in the loop.

| 19 | ssm A circular coil (950 turns, radius = 0.060 m) is rotating in a uniform magnetic field. At t = 0 s, the normal to the coil is per pendicular to the magnetic field. At t = 0.010 s, the normal makes an angle of with the field because the coil has made one- eighth of a revolution. An average emf of magnitude 0.065 V is induced in the coil. Find the magnitude of the magnetic field at the location of the coil.

| 20 | The magnetic flux that passes through one turn of a 12-turn coil of wire changes to 4.0 from 9.0 Wb in a time of 0.050 s. The average induced current in the coil is 230 A. What is the resistance of the wire?

| 21 | Interactive LearningWare 22.2 at reviews the fundamental approach in problems such as this. A constant magnetic field passes through a single rectangular loop whose dimensions are 0.35 m × 0.55 m. The magnetic field has a magnitude of 2.1 T and is inclined at an angle of 65° with respect to the normal to the plane of the loop. (a) If the magnetic field decreases to zero in a time of 0.45 s, what is the magnitude of the average emf induced in the loop? (b) If the magnetic field remains constant at its initial value of 2.1 T, what is the magnitude of the rate at which the area should change so that the average emf has the same magnitude as in part (a)?

| 22 | A magnetic field is perpendicular to the plane of a single-turn circular coil. The magnitude of the field is changing, so that an emf of 0.80 V and a current of 3.2 A are induced in the coil. The wire is then re-formed into a single-turn square coil, which is used in the same magnetic field (again perpendicular to the plane of the coil and with a magnitude changing at the same rate). What emf and current are induced in the square coil?

* | 23 | Interactive Solution 22.23 at offers some help for this problem. A copper rod is sliding on two conducting rails that make an angle of 19° with respect to each other, as in the drawing. The rod is moving to the right with a constant speed of 0.60 m/s. A 0.38-T uniform magnetic field is perpendicular to the plane of the paper. Determine the magnitude of the average emf induced in the triangle ABC during the 6.0-s period after the rod has passed point A.

* | 24 | A piece of copper wire is formed into a single circular loop of radius 12 cm. A magnetic field is oriented parallel to the normal to the loop, and it increases from 0 to 0.60 T in a time of 0.45 s. The wire has a resistance per unit length of . What is the average electrical energy dissipated in the resistance of the wire?

* | 25 | ssm www A conducting coil of 1850 turns is connected to a galvanometer, and the total resistance of the circuit is . The area of each turn is 4.70 × 10−4 m2. This coil is moved from a region where the magnetic field is zero into a region where it is nonzero, the normal to the coil being kept parallel to the magnetic field. The amount of charge that is induced to flow around the circuit is measured to be 8.87 × 10−3 C. Find the magnitude of the magnetic field. (Such a device can be used to measure the magnetic field strength and is called a flux meter.)

* | 26 | The drawing shows a copper wire (negligible resistance) bent into a circular shape with a radius of 0.50 m. The radial section BC is fixed in place, while the copper bar AC sweeps around at an angular speed of 15 rad/s. The bar makes electrical contact with the wire at all times. The wire and the bar have negligible resistance. A uniform magnetic field exists everywhere, is perpendicular to the plane of the circle, and has a magnitude of 3.8 × 10−3 T. Find the magnitude of the current induced in the loop ABC.

** | 27 | ssm A solenoid has a cross-sectional area of 6.0 × 10−4 m2, consists of 400 turns per meter, and carries a current of 0.40 A. A 10-turn coil is wrapped tightly around the circumference of the solenoid. The ends of the coil are connected to a resistor. Suddenly, a switch is opened, and the current in the solenoid dies to zero in a time of 0.050 s. Find the average current induced in the coil.
Section 22.5 Lenz’s Law

| 28 | The drawing depicts a copper loop lying flat on a table (not shown) and connected to a battery via a closed switch. The current I in the loop generates the magnetic field lines shown in the drawing. The switch is then opened and the current goes to zero. There are also two smaller conducting loops A and B lying flat on the table, but not connected to batteries. Determine the direction of the induced current in (a) loop A and (b) loop B. Specify the direction of each induced current to be clockwise or counterclockwise when viewed from above the table. Provide a reason for each answer.

| 29 | ssm Suppose in Figure 22-1 that the bar magnet is held stationary, but the coil of wire is free to move. Which way will current be directed through the ammeter, left to right or right to left, when the coil is moved (a) to the left and (b) to the right? Explain.

| 30 | The drawing shows that a uniform magnetic field is directed perpendicularly into the plane of the paper and fills the entire region to the left of the y axis. There is no magnetic field to the right of the y axis. A rigid right triangle ABC is made of copper wire. The triangle rotates counterclockwise about the origin at point C. What is the direction (clockwise or counterclockwise) of the induced current when the triangle is crossing (a) the +y axis, (b) the −x axis, (c) the −y axis, and (d) the +x axis? For each case, justify your answer.

| 31 | ssm Review Conceptual Example 9 as an aid in understanding this problem. A long, straight wire lies on a table and carries a current I. As the drawing shows, a small circular loop of wire is pushed across the top of the table from position 1 to position 2. Determine the direction of the induced current, clockwise or counterclockwise, as the loop moves past (a) position 1 and (b) position 2. Justify your answers.

* | 32 | The drawing shows a bar magnet falling through a metal ring. In part a the ring is solid all the way around, but in part b it has been cut through. (a) Explain why the motion of the magnet in part a is retarded when the magnet is above the ring and below the ring as well. Draw any induced currents that appear in the ring. (b) Explain why the motion of the magnet is unaffected by the ring in part b.

* | 33 | Interactive Solution 22.33 at for one approach to this problem. A circular loop of wire rests on a table. A long, straight wire lies on this loop, directly over its center, as the drawing illustrates. The current I in the straight wire is decreasing. In what direction is the induced current, if any, in the loop? Give your reasoning.

** | 34 | A wire loop is suspended from a string that is attached to point P in the drawing. When released, the loop swings downward, from left to right, through a uniform magnetic field, with the plane of the loop remaining perpendicular to the plane of the paper at all times. (a) Determine the direction of the current induced in the loop as it swings past the locations labeled I and II. Specify the direction of the current in terms of the points x, y, and z on the loop (e.g., x → y → z or z → y → x). The points x, y, and z lie behind the plane of the paper, (b) What is the direction of the induced current at the locations II and I when the loop swings back, from right to left? Provide reasons for your answers.

Section 22.7 The Electric Generator

| 35 | ssm A generator has a square coil consisting of 248 turns. The coil rotates at 79.1 rad/s in a 0.170-T magnetic field. The peak out put of the generator is 75.0 V. What is the length of one side of the coil?

| 36 | A 120.0-V motor draws a current of 7.00 A when running at normal speed. The resistance of the armature wire is . (a) Determine the back emf generated by the motor. (b) What is the current at the instant when the motor is just turned on and has not begun to rotate? (c) What series resistance must be added to limit the starting current to 15.0 A?

| 37 | One generator uses a magnetic field of 0.10 T and has a coil area per turn of 0.045 m2. A second generator has a coil area per turn of 0.015 m2. The generator coils have the same number of turns and rotate at the same angular speed. What magnetic field should be used in the second generator so that its peak emf is the same as that of the first generator?

| 38 | A vacuum cleaner is plugged into a 120.0-V socket and uses 3.0 A of current in normal operation when the back emf generated by the electric motor is 72.0 V. Find the coil resistance of the motor.

| 39 | Interactive Solution 22.39 at provides one model for solving this problem. The maximum strength of the earth’s magnetic field is about 6.9 × 10−5 T near the south magnetic pole. In principle, this field could be used with a rotating coil to generate 60.0-Hz ac electricity. What is the minimum number of turns (area per turn = 0.022 m2) that the coil must have to produce an rms voltage of 120 V?

* | 40 | A generator uses a coil that has 100 turns and a 0.50-T magnetic field. The frequency of this generator is 60.0 Hz, and its emf has an rms value of 120 V. Assuming that each turn of the coil is a square (an approximation), determine the length of the wire from which the coil is made.

* | 41 | ssm At its normal operating speed, an electric fan motor draws only 15.0% of the current it draws when it just begins to turn the fan blade. The fan is plugged into a 120.0-V socket. What back emf does the motor generate at its normal operating speed?

* | 42 | Consult Multiple-Concept Example 11 for background material relating to this problem. A small rubber wheel on the shaft of a bicycle generator presses against the bike tire and turns the coil of the generator at an angular speed that is 38 times as great as the angular speed of the tire itself. Each tire has a radius of 0.300 m. The coil consists of 125 turns, has an area of 3.86 × 10−3 m2, and rotates in a 0.0900-T magnetic field. The bicycle starts from rest and has an acceleration of +0.550 m/s2. What is the peak emf produced by the generator at the end of 5.10 s?

** | 43 | A motor is designed to operate on 117 V and draws a current of 12.2 A when it first starts up. At its normal operating speed, the motor draws a current of 2.30 A. Obtain (a) the resistance of the armature coil, (b) the back emf developed at normal speed, and (c) the current drawn by the motor at one-third normal speed.
Section 22.8 Mutual Inductance and Self-Inductance

| 44 | The current through a 3.2-mH inductor varies with time according to the graph shown in the drawing. What is the average induced emf during the time intervals (a) 0–2.0 ms, (b) 2.0–5.0 ms, and (c) 5.0–9.0 ms?

| 45 | ssm The earth’s magnetic field, like any magnetic field, stores energy. The maximum strength of the earth’s field is about 7.0 × 10−5 T. Find the maximum magnetic energy stored in the space above a city if the space occupies an area of 5.0 × 108 m2 and has a height of 1500 m.

| 46 | Two coils of wire are placed close together. Initially, a current of 2.5 A exists in one of the coils, but there is no current in the other. The current is then switched off in a time of 3.7 × 10−2 s. During this time, the average emf induced in the other coil is 1.7 V. What is the mutual inductance of the two-coil system?

| 47 | Mutual induction can be used as the basis for a metal detector. A typical setup uses two large coils that are parallel to each other and have a common axis. Because of mutual induction, the ac generator connected to the primary coil causes an emf of 0.46 V to be induced in the secondary coil. When someone without metal objects walks through the coils, the mutual inductance and, thus, the induced emf do not change much. But when a person carrying a hand gun walks through, the mutual inductance increases. The change in emf can be used to trigger an alarm. If the mutual inductance increases by a factor of three, find the new value of the induced emf.

| 48 | Multiple-Concept Example 13 reviews some of the principles used in this problem. Suppose you wish to make a solenoid whose self-inductance is 1.4 mH. The inductor is to have a cross-sectional area of 1.2 × 10−3 m2 and a length of 0.052 m. How many turns of wire are needed?

| 49 | ssm During a 72-ms interval, a change in the current in a primary coil occurs. This change leads to the appearance of a 6.0-mA current in a nearby secondary coil. The secondary coil is part of a circuit in which the resistance is . The mutual inductance between the two coils is 3.2 mH. What is the change in the primary current?

* | 50 | A capacitor has a voltage of 35 V between its plates. What must be the current in a 5.0-mH inductor, such that the energy stored in the inductor equals the energy stored in the capacitor?

* | 51 | A long, current-carrying solenoid with an air core has 1750 turns per meter of length and a radius of 0.0180 m. A coil of 125 turns is wrapped tightly around the outside of the solenoid. What is the mutual inductance of this system?

* | 52 | Multiple-Concept Example 13 reviews the concepts used in this problem. A long solenoid (cross-sectional area = 1.0 × 10−6 m2, number of turns per unit length = 2400 turns/m) is bent into a circular shape so it looks like a doughnut. This wire-wound doughnut is called a toroid. Assume that the diameter of the solenoid is small compared to the radius of the toroid, which is 0.050 m. Find the emf induced in the toroid when the current decreases to 1.1 A from 2.5 A in a time of 0.15 s.

** | 53 | Coil 1 is a flat circular coil that has N1 turns and a radius R1. At its center is a much smaller flat, circular coil that has N2 turns and radius R2. The planes of the coils are parallel. Assume that coil 2 is so small that the magnetic field due to coil 1 has nearly the same value at all points covered by the area of coil 2. Determine an expression for the mutual inductance between these two coils in terms of , N1, R1, N2, and R2.
Section 22.9 Transformers

| 54 | The batteries in a portable CD player are recharged by a unit that plugs into a wall socket. Inside the unit is a step-down transformer with a turns ratio of 1:13. The wall socket provides 120 V. What voltage does the secondary coil of the transformer provide?

| 55 | Interactive Solution 22.55 at offers one approach to problems such as this one. The secondary coil of a step-up transformer provides the voltage that operates an electrostatic air filter. The turns ratio of the transformer is 50:1. The primary coil is plugged into a standard 120-V outlet. The current in the secondary coil is 1.7 × 10−3 A. Find the power consumed by the air filter.

| 56 | A step-down transformer (turns ratio = 1:8) is used with an electric train to reduce the voltage from the wall receptacle to a value needed to operate the train. When the train is running, the current in the secondary coil is 1.6 A. What is the current in the primary coil?

| 57 | ssm Electric doorbells found in many homes require 10.0 V to operate. To obtain this voltage from the standard 120-V supply, a transformer is used. Is a step-up or a step-down transformer needed, and what is its turns ratio Ns/Np?

| 58 | The resistances of the primary and secondary coils of a transformer are 56 and , respectively. Both coils are made from lengths of the same copper wire. The circular turns of each coil have the same diameter. Find the turns ratio Ns/Np.

* | 59 | ssm A generating station is producing 1.2 × 106 W of power that is to be sent to a small town located 7.0 km away. Each of the two wires that comprise the transmission line has a resistance per kilometer of length of . (a) Find the power used to heat the wires if the power is transmitted at 1200 V. (b) A 100:1 step-up transformer is used to raise the voltage before the power is transmitted. How much power is now used to heat the wires?

* | 60 | Suppose there are two transformers between your house and the high-voltage transmission line that distributes the power. In addition, assume your house is the only one using electric power. At a substation the primary of a step-down transformer (turns ratio = 1:29) receives the voltage from the high-voltage transmission line. Because of your usage, a current of 48 mA exists in the primary of this transformer. The secondary is connected to the primary of another step-down transformer (turns ratio = 1:32) somewhere near your house, perhaps up on a telephone pole. The secondary of this transformer delivers a 240-V emf to your house. How much power is your house using? Remember that the current and voltage given in this problem are rms values.

** | 61 | A generator is connected across the primary coil (Np turns) of a transformer, while a resistance R2 is connected across the secondary coil (Ns turns). This circuit is equivalent to a circuit in which a single resistance R1 is connected directly across the generator, without the transformer. Show that R1 = (Np/Ns)2R2, by starting with Ohm’s law as applied to the secondary coil.

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