Physics

John D. Cutnell, Kenneth W. Johnson

Chapter 22

Electromagnetic Induction - all with Video Answers

Educators

Chapter Questions

Problem 1

A 0.80 -m aluminum bar is held with its length parallel to the east-west direction and dropped from a bridge. Just before the bar hits the river below, its speed is 22 $\mathrm{m} / \mathrm{s}$ , and the emf induced across its length is $6.5 \times 10^{-4} \mathrm{V}$ . Assuming the horizontal component of the earth's magnetic field at the location of the bar points directly north, (a) determine the magnitude of the horizontal component of the earth's magnetic field, and (b) state whether the east end or the west end of the bar is positive.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 2

Near San Francisco, where the vertically downward component of the earth's magnetic field is $4.8 \times 10^{-5} \mathrm{T}$ , a car is traveling forward at 25 $\mathrm{m} / \mathrm{s}$ . The width of the car is 2.0 $\mathrm{m}$ . (a) Find the emf induced between the two sides of the car. $(\mathrm{b})$ Which side of the car is positive-the driver's side or the passenger's side?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 3

In 1996 , NASA performed an experiment called the Tethered Satellite experiment. In this experiment a $2.0 \times 10^{4}-\mathrm{m}$ length of wire was let out by the space shuttle Atlantis to generate a motional emf. The shuttle had an orbital speed of $7.6 \times 10^{3} \mathrm{m} / \mathrm{s}$ , and the magnitude of the earth's magnetic field at the location of the wire was $5.1 \times 10^{-5} \mathrm{T}$ 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?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 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 that 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?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 5

ssm 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 $L=1.3 \mathrm{m},$ and the rods each have the same speed, $v_{\mathrm{A}}=v_{\mathrm{B}}=v_{\mathrm{C}}=2.7 \mathrm{m} / \mathrm{s}$ . For each rod, find the magnitude of the motional emf, and indicate which end (1 or 2 ) of the rod is positive.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 6

Two circuits contain an emf produced by a moving metal rod, like that shown in Figure 22.4$b$ . The speed of the rod is the same in each circuit, but the bulb in circuit 1 has one-half the resistance of the bulb in circuit 2. The circuits are otherwise identical. The resistance of the light bulb in circuit 1 is $55 \Omega,$ and that in circuit 2 is 110 \Omega. Determine (a) the ratio $\&_{1} / 8_{2}$ of the emfs and $(\mathbf{b})$ the ratio $I_{1} / I_{2}$ of the currents in the circuits. (c) If the speed of the rod in circuit 1 were twice that in circuit $2,$ what would be the ratio $P_{1} / P_{2}$ of the powers in the circuits?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 7

Refer to the drawing that accompanies Check Your Understanding Question 14. Suppose that the voltage of the battery in the circuit is $3.0 \mathrm{V},$ the magnitude of the magnetic field (directed perpendicularly into the plane of the paper) is 0.60 $\mathrm{T}$ , and the length of the rod between the rails is 0.20 $\mathrm{m}$ . Assuming that the rails are very long and have negligible resistance, find the
maximum speed attained by the rod after the switch is closed.

Jack Hou

Numerade Educator

Problem 8

Multiple-Concept Example 2 discusses the concepts that are used in this problem. Suppose that the magnetic field in Figure 22.5 has a magnitude of 1.2 $\mathrm{T}$ , the rod has a length of $0.90 \mathrm{m},$ and the hand keeps the rod moving to the right at a constant speed of 3.5 $\mathrm{m} / \mathrm{s}$ . If the current in the circuit is $0.040 \mathrm{A},$ what is the average power being delivered to the circuit by the hand?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 9

ssm Suppose that the light bulb in Figure 22.4$b$ is a 60.0 -W bulb with a resistance of 240$\Omega .$ The magnetic field has a magnitude of 0.40 $\mathrm{T}$ , and the length of the rod is 0.60 $\mathrm{m}$ . The only resistance in the circuit is that due to the bulb. What is the shortest distance along the rails that the rod would have to slide for the bulb to remain lit for one-half second?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 10

Review Conceptual Example 3 and Figure 22.7$b$ . A conducting rod slides down between two frictionless vertical copper tracks at a constant speed of 4.0 $\mathrm{m} / \mathrm{s}$ perpendicular to a $0.50-\mathrm{T}$ magnetic field. The resistance of the rod and tracks is negligible. The rod maintains electrical contact with the 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 $\mathrm{s}$ . (c) Find the electrical energy dissipated in the resistor in 0.20 $\mathrm{s}$

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 11

ssm The drawing shows two surfaces that have the same area. A uniform magnetic field $\overrightarrow{\mathbf{B}}$ fills the space occupied by these surfaces, and it is oriented parallel to the $y z$ plane as shown. Find the ratio $\Phi_{x z} / \Phi_{x y}$ of the magnetic fluxes that pass through the surfaces.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 12

Two flat surfaces are exposed to a uniform, horizontal magnetic field of magnitude 0.47 T. When
viewed edge-on, the first surface is tilted at an angle of $12^{\circ}$ from the horizontal, and a net magnetic flux of $8.4 \times 10^{-3}$ Wb passes through it. The same net magnetic flux passes
through the second surface. (a) Determine the area of the first surface. (b) Find the smallest possible value for the area of the second surface.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 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?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 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 $r=0.20 \mathrm{m}$ . The normal to the plane of the loop is parallel to a constant magnetic
field $\left(\phi=0^{\circ}\right)$ of magnitude 0.75 $\mathrm{T}$ . What is the change $\Delta \Phi$ 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?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 15

A magnetic field has a magnitude of 0.078 $\mathrm{T}$ and is uniform over a circular surface whose radius is 0.10 $\mathrm{m}$ . The field is oriented at an angle of $\phi=25^{\circ}$ with respect to the normal to the surface. What is the magnetic flux through the surface?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 16

A square loop of wire consisting of a single turn is perpendicular to a uniform magnetic field. The square loop is then re-formed into a circular loop, which also consists of a single turn and is also perpendicular to the same magnetic field. The magnetic flux that passes- through the square loop is $7.0 \times 10^{-3}$ Wb. What is the flux that passes through the circular loop?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 17

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 $\mathrm{T}$ and points along the positive $y$ direction. Determine the magnetic flux through each of the five sides.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 18

A magnetic field passes through a stationary wire loop, and its magnitude changes in time according to the graph in the drawing. The direction of the field remains constant, however. There are three equal time intervals indicated in the graph: $0-3.0 \mathrm{s}, 3.0-6.0 \mathrm{s},$ and $6.0-9.0 \mathrm{s}$ . The loop consists of 50 turns of wire and has an area of 0.15 $\mathrm{m}^{2}$ . The magnetic field is oriented parallel to the normal to the loop. For purposes of this problem, this means that $\phi=0^{\circ}$ in Equation $22.2 . \quad(\text { a ) For each }$ interval, determine the induced emf. (b) The wire has a resistance of 0.50$\Omega$ . Determine the induced current for the first and third intervals.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 19

A rectangular loop of wire with sides 0.20 and 0.35 $\mathrm{m}$ 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 $\mathrm{T}$ 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.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 20

Magnetic resonance imaging (MRI) is a medical technique for producing pictures of the interior of the body. The patient is placed within a strong magnetic field. One safety concern is what would happen to the positively and negatively charged particles in the body fluids if an equipment failure caused the magnetic field to be shut off suddenly. An induced emf could cause these particles to flow, producing
an electric current within the body. Suppose the largest surface of the body through which flux passes has an area of 0.032 $\mathrm{m}^{2}$ and a normal that is parallel to a magnetic field of 1.5 $\mathrm{T}$ . Determine the smallest time period during which the field can be allowed to vanish if the magnitude of the average induced emf is to be kept less than 0.010 $\mathrm{V}$ .

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 21

ssm A circular coil $(950 \text { turns, radius }=0.060 \mathrm{m})$ is rotating in a uniform magnetic field. At $t=0$ s, the normal to the coil is perpendicular to the magnetic field. At $t=0.010$ s, the normal makes an angle of $\phi=45^{\circ}$ with the field because the coil has made one-eighth of a revolution.
An average emf of magnitude 0.065 $\mathrm{V}$ is induced in the coil. Find the magnitude of the magnetic field at the location of the coil.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 22

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

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 23

mmh A constant magnetic field passes through a single rectangular loop whose dimensions are 0.35 $\mathrm{m} \times 0.55 \mathrm{m}$ . The magnetic field has a magnitude of 2.1 $\mathrm{T}$ and is inclined at an angle of $65^{\circ}$ 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 $\mathrm{s}$ , 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 $\mathrm{T}$ , what is the magnitude of the rate $\Delta A / \Delta t$ which the area should change so that the average emf has the same magnitude as in part (a)?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 24

A uniform 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 $\mathrm{V}$ and a current of 3.2 $\mathrm{A}$ are induced in the coil. The wire is then reformed 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?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 25

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 $\mathrm{m} / \mathrm{s} .$ A $0.38-\mathrm{T}$ uniform magnetic field is perpendicular to the plane of the paper. Determine the magnitude of the average emf induced in the triangle $A B C$ during the 6.0 -s period after the rod has passed point $A$ .

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 26

A flat coil of wire has an area $A, N$ turns, and a resistance $R .$ It is situated in a magnetic field, such that the normal to the coil is parallel to the magnetic field. The coil is then rotated through an angle of $90^{\circ},$ so that the normal becomes perpendicular to the magnetic field. The coil has an area of $1.5 \times 10^{-3} \mathrm{m}^{2}, 50$ turns, and a resistance of 140$\Omega .$ During the time while it is rotating, a charge of $8.5 \times 10^{-5}$ C flows in the coil. What is the magnitude of the magnetic field?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 27

*27. ssm A magnetic field is passing through a loop of wire whose area is 0.018 m2 The direction of the magnetic field is parallel to the normal to the loop, and the magnitude of the field is increasing at the rate of 0.20 T/s. (a) Determine the magnitude of the emf induced in the loop. (b) Suppose that the area of the loop can be enlarged or shrunk. If the magnetic field is increasing as in part (a), at what rate (in m2
/s) should the area be changed at the instant when B 1.8 T if the induced emf is to be zero? Explain whether the area is to be enlarged or shrunk.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 28

A flat circular coil with 105 turns, a radius of $4.00 \times 10^{-2} \mathrm{m}$ and a resistance of 0.480$\Omega$ is exposed to an external magnetic field that is directed perpendicular to the plane of the coil. The magnitude of the external magnetic field is changing at a rate of $\Delta B / \Delta t=0.783 \mathrm{T} / \mathrm{s}$ , thereby inducing a current in the coil. Find the magnitude of the magnetic
field at the center of the coil that is produced by the induced current.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 29

The drawing shows a coil of copper wire that consists of two semicircles joined by straight sections of wire. In part a the coil is lying flat on a horizontal surface. The dashed line also lies in the plane of the horizontal surface. Starting from the orientation in part a, the smaller semicircle rotates at an angular frequency about the dashed line, until its plane becomes perpendicular to the horizontal surface, as shown in part b. A uniform magnetic field is constant in time and is directed upward,
perpendicular to the horizontal surface. The field completely fills the region occupied by the coil in either part of the drawing. The magnitude of the magnetic field is 0.35 $\mathrm{T}$ . The resistance of the coil is $0.025 \Omega,$ and the smaller semicircle has a radius of 0.20 $\mathrm{m}$ . The angular frequency at which the small semicircle rotates is 1.5 $\mathrm{rad} / \mathrm{s}$ . Determine the average current $I,$ if any, induced in the coil as the coil changes shape from that in part $a$ of the drawing to that in part $b$ . Be sure to include an explicit
plus or minus sign along with your answer. ends are at the same electric potential. If a potential difference of $4.5 \times 10^{3} \mathrm{V}$ is required to cause a $1.0-\mathrm{mm}$ spark in air, what is the angular speed (in rad/s) of the rods when a spark jumps across the gap?

Vishal Gupta

Numerade Educator

Problem 30

A conducting coil of 1850 turns is connected to a galvanometer, and the total resistance of the circuit is $45.0 \Omega .$ The area of each turn is $4.70 \times 10^{-4} \mathrm{~m}^{2} .$ 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 $8.87 \times 10^{-3} \mathrm{C}$. Find the magnitude of the magnetic field.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 31

ssm Two 0.68 -m-long conducting rods are rotating at the same speed in opposite directions, and both are perpendicular to a $4.7-\mathrm{T}$ magnetic field. As the drawing
shows, the ends of these rods come to within 1.0 $\mathrm{mm}$ of each other as they rotate. Moreover, the fixed ends about which the rods are rotating are connected by a wire, so these ends are at the same electric potential. If a potential difference of $4.5 \times 10^{3} \mathrm{V}$ is required to cause a $1.0-\mathrm{mm}$ spark in air, what is the angular speed (in rad/s) of the rods when a spark jumps across the gap?

Vishal Gupta

Numerade Educator

Problem 32

Starting from the position indicated in the drawing, the semicircular piece of wire rotates through half a revolution in the direction shown. Which end of the resistor is positive-the left or the right end? Explain your reasoning.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 33

ssm The plane of a flat, circular loop of wire is horizontal. An external magnetic field is directed perpendicular to the plane of the loop. The magnitude of the external magnetic field is increasing with time. Because of this increasing magnetic field, an induced current is flowing clockwise in the loop, as viewed from above. What is the direction of the external magnetic field? Justify your conclusion.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 34

The drawing shows a straight wire carrying a current I. Above the wire is a rectangular loop that contains a resistor R. If the current I is decreasing in time, what is the direction of the induced current through the
resistor R—left-to-right or right-to-left?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 35

ssm 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.

Vishal Gupta

Numerade Educator

Problem 36

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 counterclock- wise about the origin at point C. What is the direction (clock-wise or counterclockwise) of the induced current when the triangle is crossing $(\mathrm{a})$ the $+y$ axis, (b) the $-x$ axis, $(\mathrm{c})$ the $-y$ axis, an
(d) the $+x$ axis? For each case, justify your answer.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 37

mmh 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.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 38

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.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 39

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., $x \rightarrow y \rightarrow z$ or $z \rightarrow y \rightarrow x )$ . 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.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 40

A 120.0 -V motor draws a current of 7.00 $\mathrm{A}$ when running at normal speed. The resistance of the armature wire is 0.720$\Omega .$ (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 $\mathrm{A}$ ?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 41

ssm A generator has a square coil consisting of 248 turns. The coil rotates at 79.1 $\mathrm{rad} / \mathrm{s}$ in a $0.170-\mathrm{T}$ magnetic field. The peak output of the generator is 75.0 $\mathrm{V}$ . What is the length of one side of the coil?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 42

You need to design a $60.0-\mathrm{Hz}$ ac generator that has a maximum emf of 5500 $\mathrm{V}$ . The generator is to contain a 150 -turn coil that has an area per turn of 0.85 $\mathrm{m}^{2}$ . What should be the magnitude of the magnetic field in which the coil rotates?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 43

mmh The maximum strength of the earth's magnetic field is about $6.9 \times 10^{-5} \mathrm{T}$ near the south magnetic pole. In principle, this field could be used with a rotating coil to generate $60.0 . \mathrm{-Hz}$ ac electricity. What is the minimum number of turns (area per turn $=0.022 \mathrm{m}^{2}$ ) that the coil must have to produce an rms voltage of 120 $\mathrm{V}$ ?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 44

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

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 45

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

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 46

The coil of a generator has a radius of 0.14 $\mathrm{m}$ . When this coil is unwound, the wire from which it is made has a length of 5.7 $\mathrm{m}$ . The magnetic field of the generator is $0.20 \mathrm{T},$ and the coil rotates at an angular speed of 25 $\mathrm{rad} / \mathrm{s}$ . What is the peak emf of this generator?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 47

ssm 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 $\mathrm{m}$ . The coil consists of 125 turns, has an area of $3.86 \times 10^{-3} \mathrm{m}^{2}$ , and rotates in a $0.0900-\mathrm{T}$ magnetic field. The bicycle starts from rest and has an acceleration of $+0.550 \mathrm{m} / \mathrm{s}^{2} .$ What is the peak emf produced by the generator at the end of 5.10 $\mathrm{s} ?$

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 48

A motor is designed to operate on 117 $\mathrm{V}$ and draws a current of 12.2 $\mathrm{A}$ 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 of the normal speed.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 49

ssm The earth's magnetic field, like any magnetic field, stores energy. The maximum strength of the earth's field is about $7.0 \times 10^{-5} \mathrm{T}$ . Find the maximum magnetic energy stored in the space above a city if the space occupies an area of $5.0 \times 10^{8} \mathrm{m}^{2}$ and has a height of 1500 $\mathrm{m}$ .

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 50

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

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 51

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

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 52

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 12$\Omega .$ The mutual inductance between the two coils is 3.2 $\mathrm{mH}$ . What is the change in the primary current?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 53

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 handgun 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.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 54

A constant current of $I=15$ A exists in a solenoid whose inductance is $L=3.1 \mathrm{H}$ . The current is then reduced to zero in a certain amount of time. $(\mathrm{a})$ If the current goes from 15 to 0 $\mathrm{A}$ in a time of $75 \mathrm{ms},$ what is the emf induced in the solenoid? (b) How much electrical energy is stored in the solenoid? (c) At what rate must the electrical energy be removed from the solenoid when the current is reduced to 0 $\mathrm{A}$ in a time of 75 $\mathrm{ms}$ ? Note that the rate at which energy is removed is the power.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 55

ssm 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 $\mathrm{mH}$ . The inductor is to have a cross-sectional area of $1.2 \times 10^{-3} \mathrm{m}^{2}$ and a length of 0.052 $\mathrm{m}$ . How many turns of wire are needed?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 56

A long, current-carrying solenoid with an air core has 1750 turns per meter of length and a radius of 0.0180 $\mathrm{m}$ . A coil of 125 turns is wrapped tightly around the outside of the solenoid, so it has virtually the same radius as the solenoid. What is the mutual inductance of this system?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 57

Multiple-Concept Example 13 provides useful background for this problem. A $5.40 \times 10^{-5} \mathrm{H}$ solenoid is constructed by wrapping 65 turns of wire around a cylinder with a cross-sectional area of $9.0 \times 10^{-4} \mathrm{m}^{2}$ When the solenoid is shortened by squeezing the turns closer together, the inductance increases to $8.60 \times 10^{-5} \mathrm{H}$ . Determine the change in the length of the solenoid.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 58

Multiple-Concept Example 13 reviews the concepts used in this problem. A long solenoid (cross-sectional area $=1.0 \times 10^{-6} \mathrm{m}^{2}$ , number of turns per unit length $=2400$ turns/m) is bent into a circular shape so it looks like a donut. This wire-wound donut is called a toroid. Assume that the diameter of the solenoid is small compared to the radius of the toroid, which is 0.050 $\mathrm{m}$ . 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.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 59

Coil 1 is a flat circular coil that has $N_{1}$ turns and a radius $R_{1}$ . At its center is a much smaller flat, circular coil that has $N_{2}$ turns and radius $R_{2}$ . 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 $\mu_{0}, N_{1}, R_{1}, N_{2},$ and $R_{2}$ .

Vishal Gupta

Numerade Educator

Problem 60

The battery charger for an MP3 player contains a step-down trans- former with a turns ratio of 1: 32, so that the voltage of 120 V available at a wall socket can be used to charge the battery pack or operate the
player. What voltage does the secondary coil of the transformer provide?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 61

mmh 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 $120-\mathrm{V}$ outlet. The current in the secondary coil is $1.7 \times 10^{-3}$ A. Find the power consumed by the air filter.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 62

The rechargeable batteries for a laptop computer need a much smaller voltage than what a wall socket provides. Therefore, a transformer is plugged into the wall socket and produces the necessary voltage for charging the batteries. The batteries are rated at $9.0 \mathrm{V},$ and a current of 225 $\mathrm{mA}$ is used to charge them. The wall socket provides a voltage of 120 $\mathrm{V}$ (a) Determine the turns ratio of transformer. (b) What is the current coming from the wall socket? (c) Find the average power delivered by the wall socket and the average power sent to the batteries.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 63

The resistances of the primary and secondary coils of a transformer are 56 and $14 \Omega,$ respectively. Both coils are made from length of the same copper wire. The circular turns of each coil have the same diameter. Find the turns ratio $N_{s} / N_{p} .$

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 64

A transformer consisting of two coils wrapped around an iron core is connected to a generator and a resistor, as shown in the drawing. There are 11 turns in the primary coil and 18 turns in the secondary
coil. The peak voltage across the resistor is 67 V. What is the peak emf of the generator?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 65

ssm 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 $\mathrm{A}$ . What is the current in the primary coil?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 66

In a television set the power needed to operate the picture tube comes from the secondary of a transformer. The primary of the trans- former is connected to a $120-\mathrm{V}$ receptacle on a wall. The picture tube of the television set uses 91 $\mathrm{W}$ , and there is 5.5 $\mathrm{mA}$ of current in the secondary coil of the transformer to which the tube is connected. Find the turns ratio $N_{s} / N_{\mathrm{p}}$ of the transformer.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 67

ssm A generating station is producing $1.2 \times 10^{6} \mathrm{W}$ of power that is to be sent to a small town located 7.0 $\mathrm{km}$ away. Each of the two wires that comprise the transmission line has a resistance per kilometer of $5.0 \times 10^{-2} \Omega / \mathrm{km}$ (a) Find the power used to heat the wires if the power is transmitted at 1200 $\mathrm{V}$ . (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?

Vishal Gupta

Numerade Educator

Problem 68

Suppose there are two transformers between your house and the high-voltage transmission line that distributes the power. In addition, assume that your house is the only one using electric power. At a substation the primary coil of a step-down transformer (turms 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 coil of this transformer. The secondary coil 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 coil of this transformer delivers a $240-\mathrm{V}$ emf to your house. How much power is your house using? Remember that the current and voltage given in this problem are rms values.

Vishal Gupta

Numerade Educator

Problem 69

A generator is connected across the primary coil ( $N_{p}$, turns) of a transformer, while a resistance $R_{2}$ is connected across the secondary coil $\left(N_{\mathrm{s}}\right.$ turns). This circuit is equivalent to a circuit in which a single resistance $R_{1}$ is connected directly across the generator, without the transformer. Show that $R_{1}=\left(N_{\mathrm{p}} / N_{\mathrm{s}}\right)^{2} R_{2},$ by starting with $\mathrm{Ohm}^{\prime}$ s law as applied to the secondary coil.

Vishal Gupta

Numerade Educator

Problem 70

In each of two coils the rate of change of the magnetic flux in a single loop is the same. The emf induced in coil 1 , which has 184 loops, is 2.82 $\mathrm{V}$ . The emf induced in coil 2 is 4.23 $\mathrm{V}$ . How many loops does coil 2 have?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 71

ssm When its coil rotates at a frequency of 280 $\mathrm{Hz}$ , a certain generator has a peak emf of 75 $\mathrm{V}$ . (a) What is the peak emf of the generator when its coil rotates at a frequency of 45 $\mathrm{Hz}$ ? (b) Determine the frequency of the coil's rotation when the peak emf of the generator
is 180 $\mathrm{V} .$

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 72

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 $\mathrm{T}$ . An emf that has a magnitude of 2.6 $\mathrm{V}$ is induced in this coil because the coil's area $A$ is shrinking. What is the magnitude of $\Delta A / \Delta t,$ which is the rate (in $\mathrm{m}^{2} / \mathrm{s} )$ at which the area changes?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 73

ssm Review Conceptual Example 9 as an aid in under- standing 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 $(\mathbf{b})$ position 2 . Justify your answers.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 74

In some places, insect "zappers," with their blue lights, are a familiar sight on a summer's night. These devices use a high voltage to electrocute insects. One such device uses an ac voltage of 4320 $\mathrm{V}$ , which is obtained from a standard $120.0-\mathrm{V}$ outlet by means of a transformer. If the primary coil has 21 turns, how many turns are in the secondary coil?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 75

ssm A $3.0-\mu \mathrm{F}$ capacitor has a voltage of 35 $\mathrm{V}$ between its plates.
What must be the current in a $5.0-\mathrm{mH}$ inductor so that the energy stored
in the inductor equals the energy stored in the capacitor?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 76

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 $120.0-\mathrm{V}$ socket. What back emf does the motor generate at its normal operating speed?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 77

Parts $a$ and $b$ of the drawing show the same uniform and constant (in time) magnetic field $\overrightarrow{\mathbf{B}}$ directed perpendicularly into the paper over a rectangular region. Outside this region, there is no field. Also shown is a rectangular coil (one turn), which lies in the plane of the paper. In part $a$ the long side of the coil (length $=L )$ is just at the edge of the field region, while in part $b$ the short side (width $=W$ is just at the edge. It is known that $L / W=3.0 .$ In both parts of the drawing the coil is pushed into the field with the same velocity $\overrightarrow{\mathbf{v}}$ until it is completely within the field region. The magnitude of the average emf induced in the coil in
part $a$ is 0.15 V. What is its magnitude in part $b$ ?

Vishal Gupta

Numerade Educator

Problem 78

Indicate the direction of the electric field between the plates of the parallel plate capacitor shown in the
drawing if the magnetic field is decreasing in time. Give your reasoning.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 79

ssm A piece of copper wire is formed into a single circular loop of radius 12 $\mathrm{cm} .$ A magnetic field is oriented parallel to the normal to the loop, and it increases from 0 to 0.60 $\mathrm{T}$ in a time of 0.45 s. The wire has a resistance per unit length of $3.3 \times 10^{-2} \Omega / \mathrm{m} .$ What is the average electrical energy dissipated in the resistance of the wire?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 80

The purpose of this problem is to show that the work $W$ needed to establish a final current $I_{\mathrm{f}}$ in an inductor is $W=\frac{1}{2} L I_{\mathrm{f}}^{2}(\text { Equation } 22.10)$ In Section 22.8 we saw that the amount of work $\Delta W$ needed to change the current through an inductor by an amount $\Delta I$ is $\Delta W=L I(\Delta I)$ , where $L$ is the inductance. The drawing shows a graph of $L I$ versus $I$ . Notice that $L I(\Delta I)$ is the area of the shaded vertical rectangle whose height is $L I$ and whose width is $\Delta I .$ Use this fact to show that the total work $W$ needed to establish a current $I_{f}$ is $W=\frac{1}{2} L I_{\mathrm{f}}^{2}$

Vishal Gupta

Numerade Educator

Problem 81

A solenoid has a cross-sectional area of $6.0 \times 10^{-4} \mathrm{m}^{2},$ consists of 400 turns per meter, and carries a current of 0.40 $\mathrm{A}$ . A 10 -turn coil is wrapped tightly around the circumference of the solenoid. The ends of the coil are connected to a $1.5-\Omega$ 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.

Vishal Gupta

Numerade Educator

Problem 82

A 60.0 -Hz generator delivers an average power of 75 $\mathrm{W}$ to a single light bulb. When an induced current exists in the rotating coil of a generator, a torque- called a countertorque-is exerted on the coil. Determine the maximum countertorque in the generator coil. (Hint: The peak current, peak emf, and maximum countertorque all occur at the same instant.)

Mahnoor Amin

Numerade Educator