Solved: A square loop of wire consisting of a single turn

Suppose that the coil and the magnet in Figure 22.1a were each moving with the same velocity relative to the earth. Would there be an induced current in the coil?

Suppose that the rod in Figure 22.4b is moving at a speed of 5.0 m/s in a direction perpendicular to a 0.80-T magnetic field. The rod has a length of 1.6 m and a negligible electrical resistance. The rails also have negligible resistance. The light bulb, however, has a resistance of 96 . Find (a) the emf produced by the rod, (b) the current induced in the circuit, (c) the electric power delivered to the bulb, and (d) the energy used by the bulb in 60.0 s.

As we saw in Example 1, an induced current of 0.067 A exists in the circuit due to the moving rod. As Figure 22.5 shows, the hand provides a force hand that keeps the rod moving to the right. Determine the work done by this force in a time of 60.0 s. Assume, as in Example 1, that the magnetic field has a magnitude of 0.80 T and that the rod has a length of 1.6 m and moves at a constant speed of 5.0 m/s.

Figure 22.7a illustrates a conducting rod that is free to slide down between two vertical copper tracks. There is no kinetic friction between the rod and the tracks, although the rod maintains electrical contact with the tracks during its fall. A constant magnetic field is directed perpendicular to the motion of the rod, as the drawing shows. Because there is no friction, the only force that acts on the rod is its weight , so the rod falls with an acceleration equal to the acceleration due to gravity, which has a magnitude of g 9.8 m/s2 . Suppose that a resistance R is connected between the tops of the tracks, as in part b of the drawing. Is the magnitude of the acceleration with which the rod now falls (a) equal to g, (b) greater than g, or (c) less than g?

Consider the induced emf being generated in Figure 22.4. Suppose that the length of the rod is reduced by a factor of four. For the induced emf to be the same, what should be done? (a) Without changing the speed of the rod, increase the strength of the magnetic field by a factor of four. (b) Without changing the magnetic field, increase the speed of the rod by a factor of four. (c) Increase both the speed of the rod and the strength of the magnetic field by a factor of two. (d) All of the previous three methods may be used.

In the discussion concerning Figure 22.5, we saw that a force of 0.086 N from an external agent was required to keep the rod moving at a constant velocity. Suppose that friction is absent and that the light bulb is suddenly removed from its socket while the rod is moving. How much force does the external agent then need to apply to the rod to keep it moving at a constant velocity? (a) 0 N (b) Greater than 0 N but less than 0.086 N (c) More than 0.086 N (d) 0.086 N

Eddy currents are electric currents that can arise in a piece of metal when it moves through a region where the magnetic field is not the same everywhere. The drawing shows, for example, a metal sheet moving to the right at a velocity and a magnetic field that is directed perpendicular to the sheet. At the instant represented, the field only extends over the left half of the sheet. An emf is induced that leads to the eddy current indicated. Such eddy currents cause the velocity of the moving sheet to decrease and are used in various devices as a brake to damp out unwanted motion. Does the eddy current in the drawing circulate (a) counterclockwise or (b) clockwise?

A rectangular coil of wire is situated in a constant magnetic field whose magnitude is 0.50 T. The coil has an area of 2.0 m2 . Determine the magnetic flux for the following three orientations, 0, 60.0, and 90.0, shown in Figure 22.11. Exam

A magnetic field has the same direction and the same magnitude B everywhere. A circular area A is bounded by a loop of wire. Which of the following statements is true concerning the magnitude of the magnetic flux that passes through this area? (a) It is zero. (b) It is BA. (c) Its maximum possible value is BA. (d) Its minimum possible value is BA.

Suppose that a magnetic field is constant everywhere on a flat 1.0-m2 surface and that the magnetic flux through this surface is 2.0 Wb. From these data, which one of the following pieces of information can be determined about the magnetic field? (a) The magnitude of the field (b) The magnitude of the component of the field that is perpendicular to the surface (c) The magnitude of the component of the field that is parallel to the surfac

A coil of wire consists of 20 turns, or loops, each with an area of 1.5 103 m2 . A magnetic field is perpendicular to the surface of each loop at all times, so that 0 0. At time t0 0 s, the magnitude of the field at the location of the coil is B0 0.050 T. At a later time t 0.10 s, the magnitude of the field at the coil has increased to B 0.060 T. (a) Find the average emf induced in the coil during this time. (b) What would be the value of the average induced emf if the magnitude of the magnetic field decreased from 0.060 T to 0.050 T in 0.10 s? Reasonin

A flat coil of wire has an area of 0.020 m2 and consists of 50 turns. At t0 0 s the coil is oriented so the normal to its surface has the same direction (0 0) as a constant magnetic field of magnitude 0.18 T. The coil is then rotated through an angle of 30.0 in a time of 0.10 s (see Figure 22.11). (a) Determine the average induced emf. (b) What would be the induced emf if the coil were returned to its initial orientation in the same time of 0.10 s? Exampl

Figure 22.14 shows two pots of water that were placed on an induction stove at the same time. There are two interesting features in this drawing. First, the stove itself is cool to the touch. Second, the water in the ferromagnetic metal pot is boiling while the water in the glass pot is not. How can such a cool stove boil water, and why isnt the water in the glass pot boiling?

In the most common form of lightning, electric charges flow between the ground and a cloud. The flow changes dramatically over short periods of time. Even without directly striking an electrical appliance in your house, a bolt of lightning that strikes nearby can produce a current in the circuits of the appliance. Note that such circuits typically contain coils or loops of wire. Why can the lightning cause the current to appear?

A solenoid is connected to an ac source. A copper ring and a rubber ring are placed inside the solenoid, with the normal to the plane of each ring parallel to the axis of the solenoid. An induced emf appears _________. (a) in the copper ring but not in the rubber ring (b) in the rubber ring but not in the copper ring (c) in both rings

A magnetic field of magnitude B 0.20 T is reduced to zero in a time interval of t 0.10 s, thereby creating an induced current in a loop of wire. Which one or more of the following choices would cause the same induced current to appear in the same loop of wire? (a) B 0.40 T and t 0.20 s (b) B 0.30 T and t 0.10 s (c) B 0.30 T and t 0.30 s (d) B 0.10 T and t 0.050 s (e) B 0.50 T and t 0.40 s 10. A coil

A coil is placed in a magnetic field, and the normal to the plane of the coil remains parallel to the field. Which one of the following options causes the magnitude of the average emf induced in the coil to be as large as possible? (a) The magnitude of the field is small, and its rate of change is large. (b) The magnitude of the field is large, and its rate of change is small. (c) The magnitude of the field is large, and it does not change.

Determine whether the magnetic flux that penetrates a coil is increasing or decreasing.

Find what the direction of the induced magnetic field must be so that it can oppose the change in flux by adding to or subtracting from the original field.

Having found the direction of the induced magnetic field, use RHR-2 (see Section 21.7) to determine the direction of the induced current. Then the polarity of the induced emf can be assigned because conventional current is directed out of the positive terminal, through the external circuit, and into the negative terminal.

Figure 22.15a shows a permanent magnet approaching a loop of wire. The external circuit attached to the loop consists of the resistance R, which could represent the filament in a light bulb, for instance. In Figure 22.15a, what is the polarity of the induced emf? In other words, (a) is point A positive and point B negative or (b) is point A negative and point B positive?

In Figure 22.16 there is a constant magnetic field in a rectangular region of space. This field is directed perpendicularly into the page. Outside this region there is no magnetic field. A copper ring slides through the region, from position 1 to position 5. Since the field is zero outside the rectangular region, no flux passes through the ring in positions 1 and 5, there is no change in the flux through the ring, and there is no induced emf or current in the ring. Which one of the following options correctly describes the induced current in the ring as it passes through positions 2, 3, and 4? (a) I2 is clockwise, I3 is counterclockwise, I4 is counterclockwise. (b) I2 is counterclockwise, I3 is clockwise, I4 is clockwise. (c) I2 is clockwise, I3 0 A, I4 is counterclockwise. (d) I2 is counterclockwise, I3 0 A, I4 is clockwise.

In Figure 22.3 a coil of wire is being stretched. What would be the direction of the induced current if the direction of the external magnetic field in the figure were reversed? (a) Clockwise (b) Counterclockwise

A circular loop of wire is lying flat on a horizontal table, and you are looking down at it. An external magnetic field has a constant direction that is perpendicular to the table, and there is an induced clockwise current in the loop. Is the external field directed upward toward you or downward away from you, and is its magnitude increasing or decreasing? Note that there are two possible answers.

When the switch in the drawing is closed, the current in the coil increases to its equilibrium value. While the current is increasing there is an induced current in the metal ring. The ring is free to move. What happens to the ring? (a) It does not move. (b) It jumps downward. (c) It jumps upward

A conducting rod is free to slide along a pair of conducting rails, in a region where a uniform and constant (in time) magnetic field is directed into the plane of the paper, as the drawing illustrates. Initially the rod is at rest. There is no friction between the rails and the rod. What happens to the rod after the switch is closed? If any induced emf develops, be sure to account for its effect. (a) The rod accelerates to the right, its velocity increasing without limit. (b) The rod does not move. (c) The rod accelerates to the right for a while and then slows down and comes to a halt. (d) The rod accelerates to the right and eventually reaches a constant velocity at which it continues to move.

The string of an electric guitar vibrates in a standing wave pattern that consists of nodes and antinodes. (Section 17.5 discusses standing waves.) Where should an electromagnetic pickup be located in the standing wave pattern to produce a maximum emf? (a) At a node (b) At an antinode

In Figure 22.21 the coil of the ac generator rotates at a frequency of f 60.0 Hz and develops an emf of 120 V (rms; see Section 20.5). The coil has an area of A 3.0 103 m2 and consists of N 500 turns. Find the magnitude of the magnetic field in which the coil rotates. Re

A bicyclist is traveling at night, and a generator mounted on the bike powers a headlight. A small rubber wheel on the shaft of the generator presses against the bike tire and turns the coil of the generator at an angular speed that is 44 times as great as the angular speed of the tire itself. The tire has a radius of 0.33 m. The coil consists of 75 turns, has an area of 2.6 103 m2 , and rotates in a 0.10-T magnetic field. When the peak emf being generated is 6.0 V, what is the linear speed of the bike?

The coil of an ac motor has a resistance of R 4.1 . The motor is plugged into an outlet where V 120.0 volts (rms), and the coil develops a back emf of 118.0 volts (rms) when rotating at normal constant speed. The motor is turning a wheel. Find (a) the current when the motor first starts up and (b) the current when the motor is operating at normal speed. Rea

In a car, the generator-like action of the alternator occurs while the engine is running and keeps the battery fully charged. The headlights would discharge an old and failing battery quickly if it were not for the alternator. Why does the engine of a parked car run more quietly with the headlights off than with them on when the battery is in bad shape?

You have a fixed length of wire and need to design a generator that will produce the greatest peak emf for a given frequency and magnetic field strength. You should use (a) a one-turn square coil, (b) a two-turn square coil, (c) either a one- or a two-turn square coil because both give the same peak emf for a given frequency and magnetic field strength.

An electric motor in a hair dryer is running at its normal constant operating speed and, thus, is drawing a relatively small current, as in part (b) of Example 12. The wire in the coil of the motor has some resistance. What happens to the temperature of the coil if the shaft of the motor is prevented from turning, so that the back emf is suddenly reduced to zero? (a) Nothing. (b) The temperature decreases. (c) The temperature increases (the coil could even burn up).

A long solenoid of length 8.0  102 m and cross-sectional area 5.0  105 m2 contains 6500 turns per meter of length. Determine the emf induced in the solenoid when the current in the solenoid changes from 0 to 1.5 A during the time interval from 0 to 0.20 s.

A step-down transformer inside a stereo receiver has 330 turns in the primary coil and 25 turns in the secondary coil. The plug connects the primary coil to a 120-V wall socket, and there is a current of 0.83 A in the primary coil while the receiver is turned on. Connected to the secondary coil are the transistor circuits of the receiver. Find (a) the voltage across the secondary coil, (b) the current in the secondary coil, and (c) the average electric power delivered to the transistor circuits.

A transformer changes the 120-V voltage at a wall socket to 12 000 V. The current delivered by the wall socket is (a) stepped up by a factor of 100, (b) stepped down by a factor of 100, (c) neither stepped up nor stepped down.

A transformer that stepped up the voltage and the current simultaneously would (a) produce less power at the secondary coil than was supplied at the primary coil, (b) produce more power at the secondary coil than was supplied at the primary coil, (c) produce the same amount of power at the secondary coil that was supplied at the primary coil, (d) violate the law of conservation of energy. Choose one or more.

A circular coil of radius 0.11 m contains a single turn and is located in a constant magnetic field of magnitude 0.27 T. The magnetic field has the same direction as the normal to the plane of the coil. The radius increases to 0.30 m in a time of 0.080 s. (a) Determine the magnitude of the emf induced in the coil. (b) The coil has a resistance of 0.70 . Find the magnitude of the induced current.

Why is there an emf induced in the coil?

Does the magnitude of the induced emf depend on whether the area is increasing or decreasing?

What determines the amount of current induced in the coil?

If the coil is cut so it is no longer one continuous piece, are there an induced emf and induced current?

The graph in Figure 22.31 shows the emf produced by a generator as a function of time t. The coil of the generator has an area of A 0.15 m2 and consists of N 10 turns. The coil rotates in a magnetic field of magnitude 0.27 T. (a) Determine the period of the motion. (b) What is the angular frequency of the rotating coil? (c) Find the value of the emf when where T denotes the period of the coil motion. (d) What is the emf when t 0.025 s? t 1 4

Can the period of the rotating coil be determined from the graph?

The emf produced by a generator depends on its angular frequency. How is the angular frequency of the rotating coil related to the period?

Starting at t 0 s, how much time is required for the generator to produce its peak emf? Express the answer in terms of the period T of the motion (e

How often does the polarity of the emf change in one cycle?

You have three light bulbs; bulb A has a resistance of 240 bulb B has a resistance of 192 and bulb C has a resistance of 144 Each of these bulbs is used for the same amount of time in a setup like the one in the drawing. In each case the speed of the rod and the magnetic field strength are the same. Rank the setups in descending order, according to how much work the hand in the drawing must do (largest amount of work first). (a) A, B, C (b) A, C, B (c) B, C, A (d) B, A, C (e

The drawing shows a cube. The dashed lines in the drawing are perpendicular to faces 1, 2, and 3 of the cube. Magnetic fields are oriented with respect to these faces as shown, and each of the three fields 1, 2, and 3 has the same magnitude. Note that 2 is parallel to face 2 of the cube. Rank the magnetic fluxes that pass through the faces 1, 2, and 3 of the cube in decreasing order (largest first). (a) 1, 2, 3 (b) 1, 3, 2 (c) 2, 1, 3 (d) 2, 3, 1 (e

The drawing shows three flat coils, one square and two rectangular, that are each being pushed into a region where there is a uniform magnetic field directed into the page. Outside of this region the magnetic field is zero. In each case the magnetic field within the region has the same magnitude, and the coil is being pushed at the same velocity . Each coil begins with one side just at the edge of the field region. Consider the magnitude of the average emf induced as each coil is pushed from the starting position shown in the drawing until the coil is just completely within the field region. Rank the magnitudes of the average emfs in descending order (largest first). (a) A, B, C (b) A, C, B (c) B, A and C (a tie) (d) C, A and B (a tie) 8. A long,

A long, vertical, straight wire carries a current I. The wire is perpendicular to the plane of a circular metal loop and passes through the center of the loop (see the drawing). The loop is allowed to fall and maintains its orientation with respect to the straight wire while doing so. In what direction does the current induced in the loop flow? (a) There is no induced current. (b) It is flowing around the loop from A to B to C to A. (c) It is flowing around the loop from C to B to A to C.

The drawing shows a top view of two circular coils of conducting wire lying on a flat surface. The centers of the coils coincide. In the larger coil there are a switch and a battery. The smaller coil contains no switch and no battery. Describe the induced current that appears in the smaller coil when the switch in the larger coil is closed. (a) It flows counterclockwise forever after the switch is closed. (b) It flows clockwise forever after the switch is closed. (c) It flows counterclockwise, but only for a short period just after the switch is closed. (d) It flows clockwise, but only for a

You have a fixed length of conducting wire. From it you can construct a single-turn flat coil that has the shape of a square, a circle, or a rectangle with the long side twice the length of the short side. Each can be used with the same magnetic field to produce a generator that operates at the same frequency. Rank the peak emfs 0 of the three generators in descending order (largest first). (a) 0, square, 0, circle, 0, rectangle (b) 0, circle, 0, square, 0, rectangle (c) 0, square, 0, rectangle, 0, circle (d) 0, rectangle, 0, square, 0, circle (e) 0, rectangle, 0, circle, 0, square 12. An electric

An electric motor is plugged into a standard wall socket and is running at normal speed. Suddenly, some dirt prevents the shaft of the motor from turning quite so rapidly. What happens to the back emf of the motor, and what happens to the current that the motor draws from the wall socket? (a) The back emf increases, and the current drawn from the socket decreases. (b) The back emf increases, and the current drawn from the socket increases. (c) The back emf decreases, and the current drawn from the socket decreases. (d) The back emf decreases, and the current drawn from the socket increases.

Inductor 1 stores the same amount of energy as inductor 2, although it has only one-half the inductance of inductor 2. What is the ratio I1/I2 of the currents in the two inductors? (a) 2.000 (b) 1.414 (c) 4.000 (d) 0.500 (e) 0.250

The primary coil of a step-up transformer is connected across the terminals of a standard wall socket, and resistor 1 with a resistance R1 is connected across the secondary coil. The current in the resistor is then measured. Next, resistor 2 with a resistance R2 is connected directly across the terminals of the wall socket (without the transformer). The current in this resistor is also measured and found to be the same as the current in resistor 1. How does the resistance R2 compare to the resistance R1? (a) The resistance R2 is less than the resistance R1. (b) The resistance R2 is greater than the resistance R1. (c) The resistance R2 is the same as the resistance R1.

A 0.80-m aluminum bar is held with its length parallel to the eastwest direction and dropped from a bridge. Just before the bar hits the river below, its speed is 22 m/s, and the emf induced across its length is 6.5 104 V. Assuming the horizontal component of the earths magnetic field at the location of the bar points directly north, (a) determine the magnitude of the horizontal component of the earths magnetic field, and (b) state whether the east end or the west end of the bar is positive.

Near San Francisco, where the vertically downward component of the earths magnetic field is 4.8 105 T, a car is traveling forward at 25 m/s. The width of the car is 2.0 m. (a) Find the emf induced between the two sides of the car. (b) Which side of the car is positivethe drivers side or the passengers side?

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 earths magnetic field at the location of the wire was 5.1 105 T. If the wire had moved perpendicular to the earths magnetic field, what would have been the motional emf generated between the ends of the wire?

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?

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 rods each have the same speed, 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. El

Two circuits contain an emf produced by a moving metal rod, like that shown in Figure 22.4b. 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 , and that in circuit 2 is 110 . Determine (a) the ratio 1/2 of the emfs and (b) the ratio I1/I2 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 P1/P2 of the powers in the circuits? *

Refer to the drawing that accompanies Check Your Understanding Question 14. 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.

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 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?

Suppose that 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. 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?

Review Conceptual Example 3 and Figure 22.7b. 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 0.75- 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

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

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 from the horizontal, and a net magnetic flux of 8.4 103 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.

A standard door into a house rotates about a vertical axis through one side, as defined by the doors 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?

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 ( 0) 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

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 of 25 with respect to the normal to the surface. What is the magnetic flux through the surface? *1

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  103 Wb. What is the flux that passes through the circular loop?

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.

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: 03.0 s, 3.06.0 s, and 6.09.0 s. The loop consists of 50 turns of wire and has an area of 0.15 m2 . The magnetic field is oriented parallel to the normal to the loop. For purposes of this problem, this means that in Equation 22.2. (a) For each interval, determine the induced emf. (b) The wire has a resistance of 0.50 . Determine the induced current for the first and third intervals.

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 loops 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.

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 m2 and a normal that is parallel to a magnetic field of 1.5 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 V.

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 perpendicular 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. 22

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?

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)?

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 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?

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.

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 so that the normal becomes perpendicular to the magnetic field. The coil has an area of 1.5 103 m2 , 50 turns, and a resistance of During the time while it is rotating, a charge of 8.5 105 C flows in the coil. What is the magnitude of the magnetic field?

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.

A flat circular coil with 105 turns, a radius of 4.00 102 m, and a resistance of 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 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.

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 T. The resistance of the coil is and the smaller semicircle has a radius of 0.20 m. The angular frequency at which the small semicircle rotates is 1.5 rad/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.

A conducting coil of 1850 turns is connected to a galvanometer, and the total resistance of the circuit is 45.0 . The area of each turn is 4.70 104 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 8.87 103 C. Find the magnitude of the magnetic field.

Two 0.68-m-long conducting rods are rotating at the same speed in opposite directions, and both are perpendicular to a 4.7-T magnetic field. As the drawing shows, the ends of these rods come to within 1.0 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 4.5 103 V is required to cause a 1.0-mm spark in air, what is the angular speed (in rad/s) of the rods when a spark jumps across the gap?

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 positivethe left or the right end? Explain your reasoning.

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.

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 Rleft-to-right or right-to-left?

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.

A conducting coil of 1850 turns is connected to a galvanometer, and the total resistance of the circuit is 45.0 . The area of each turn is 4.70 104 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 8.87 103 C. Find the magnitude of the magnetic field.

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.

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

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?

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 output of the generator is 75.0 V. What is the length of one side of the coil?

You need to design a 60.0-Hz ac generator that has a maximum emf of 5500 V. The generator is to contain a 150-turn coil that has an area per turn of 0.85 m2 . What should be the magnitude of the magnetic field in which the coil rotates?

The maximum strength of the earths magnetic field is about 6.9 105 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?

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.

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

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

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 3.86 103 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? 4

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 of the normal speed.

The earths magnetic field, like any magnetic field, stores energy. The maximum strength of the earths field is about 7.0 105 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. 50. 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) 02.0 ms, (b) 2.05.0 ms, and (c) 5.09.0 ms? 51

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 3.7 102 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? 5

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?

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.

A constant current of exists in a solenoid whose inductance is The current is then reduced to zero in a certain amount of time. (a) If the current goes from 15 to 0 A in a time of 75 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 A in a time of 75 ms? Note that the rate at which energy is removed is the power.

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 1.2 103 m2 and a length of 0.052 m. How many turns of wire are needed?

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, so it has virtually the same radius as the solenoid. What is the mutual inductance of this system?

Multiple-Concept Example 13 provides useful background for this problem. A 5.40 105 H solenoid is constructed by wrapping 65 turns of wire around a cylinder with a cross-sectional area of 9.0 104 m2 . When the solenoid is shortened by squeezing the turns closer together, the inductance increases to 8.60 105 H. Determine the change in the length of the solenoid

Multiple-Concept Example 13 reviews the concepts used in this problem. A long solenoid (cross-sectional area 1.0 106 m2 , 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 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.

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 0, N1, R1, N2, and R2.

The battery charger for an MP3 player contains a step-down transformer 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 pr

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 1.7 103 A. Find the power consumed by the air filter.

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 V, and a current of 225 mA is used to charge them. The wall socket provides a voltage of 120 V. (a) Determine the turns ratio of the 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.

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

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?

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?

In a television set the power needed to operate the picture tube comes from the secondary of a transformer. The primary of the transformer is connected to a 120-V receptacle on a wall. The picture tube of the television set uses 91 W, and there is 5.5 mA of current in the secondary coil of the transformer to which the tube is connected. Find the turns ratio Ns/Np of the transformer.

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 5.0 102 /km. (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?

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 (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 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-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. *

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) 2 R2, by starting with Ohms law as applied to the secondary coil.

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 V. The emf induced in coil 2 is 4.23 V. How many loops does coil 2 have?

m When its coil rotates at a frequency of 280 Hz, a certain generator has a peak emf of 75 V. (a) What is the peak emf of the generator when its coil rotates at a frequency of 45 Hz? (b) Determine the frequency of the coils rotation when the peak emf of the generator is 180 V.

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 coils area A is shrinking. What is the magnitude of A/ t, which is the rate (in m2 /s) at which the area changes?

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

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

A 3.0-F capacitor has a voltage of 35 V between its plates. What must be the current in a 5.0-mH inductor so that the energy stored in the inductor equals the energy stored in the capac

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?

Parts a and b of the drawing show the same uniform and constant (in time) magnetic field 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 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? B

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.

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 3.3 102 /m. What is the average electrical energy dissipated in the resistance of the wire?

The purpose of this problem is to show that the work W needed to establish a final current If in an inductor is (Equation 22.10). In Section 22.8 we saw that the amount of work needed to change the current through an inductor by an amount is where L is the inductance. The drawing shows a graph of LI versus I. Notice that is the area of the shaded vertical rectangle whose height is LI and whose width is Use this fact to show that the total work W needed to establish a current If is 81. A solenoid has a cross-sectional area of 6.0 104 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 1.5- 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. 82. A 60.0-Hz generator delivers an average power of 75 W to a single light bulb. When an induced current exists in the rotating coil of a generator, a torquecalled a countertorqueis 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.) Current, I If LI LIf I W 1 2LI 2 f .

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