College Physics

Hugh D. Young Philip W. Adams

Chapter 21

Electromagnetic Induction - all with Video Answers

Educators

Chapter Questions

Problem 1

A circular area with a radius of $6.50 \mathrm{~cm}$ lies in the $x$ -y plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field $B=0.230 \mathrm{~T}$ that points
(a) in the $+z$ direction? (b) at an angle of $53.1^{\circ}$ from the $+z$ direction? (c) in the $+y$ direction?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 2

The magnetic field $\overrightarrow{\boldsymbol{B}}$ in a certain region is $0.128 \mathrm{~T},$ and its direction is that of the $+z$ axis in Figure $21.44 .$ (a) What is the magnetic flux across the surface $a b c d$ in the figure? (b) What is the magnetic flux across the surface befc? (c) What is the magnetic flux across the surface aefd? (d) What is the net flux through all five surfaces that enclose the shaded volume?

Vishal Gupta

Numerade Educator

Problem 3

An empty cylindrical food container with a lid on it has a diameter of $5.00 \mathrm{~cm} .$ The container is placed on a table, and a uniform $1.75 \mathrm{~T}$ magnetic field is applied to the container, with the field directed vertically upward. (a) What is the total magnetic flux through the jar with the lid on? (b) What is the total magnetic flux with the lid off?

Linda Winkler

Numerade Educator

Problem 4

A single loop of wire with an area of $0.0900 \mathrm{~m}^{2}$ is in a uniform magnetic field that has an initial value of $3.80 \mathrm{~T},$ is perpendicular to the plane of the loop, and is decreasing at a constant rate of $0.190 \mathrm{~T} / \mathrm{s} .$ (a) What emf is induced in this loop? (b) If the loop has a resistance of $0.600 \Omega$, find the current induced in the loop.

Averell Hause

Carnegie Mellon University

Problem 5

A coil of wire with 200 circular turns of radius $3.00 \mathrm{~cm}$ is in a uniform magnetic field along the axis of the coil. The coil has $R=40.0 \Omega$. At what rate, in teslas per second, must the magnetic field be changing to induce a current of $0.150 \mathrm{~A}$ in the coil?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 6

In a physics laboratory experiment, a coil with 200 turns enclosing an area of $12 \mathrm{~cm}^{2}$ is rotated from a position where its plane is perpendicular to the earth's magnetic field to one where its plane is parallel to the field. The rotation takes $0.040 \mathrm{~s}$. The earth's magnetic field at the location of the laboratory is $6.0 \times 10^{-5} \mathrm{~T}$. (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?

Averell Hause

Carnegie Mellon University

Problem 7

A closely wound rectangular coil of 80 turns has dimensions of $25.0 \mathrm{~cm}$ by $40.0 \mathrm{~cm} .$ The plane of the coil is rotated from a position where it makes an angle of $37.0^{\circ}$ with a magnetic field of $1.10 \mathrm{~T}$ to a position perpendicular to the field. The rotation takes $0.0600 \mathrm{~s}$. What is the average emf induced in the coil?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 8

A very long, straight solenoid with a diameter of $3.00 \mathrm{~cm}$ is wound with 40 turns of wire per centimeter, and the windings carry a current of 0.250 A. A second coil having $N$ turns and a larger diameter is slipped over the solenoid so that the two are coaxial. The current in the solenoid is ramped down to zero over a period of 0.5 s. (a) What average emf is induced in the second coil if it has a diameter of $4.0 \mathrm{~cm}$ and $N=5 ?$ (b) What is the induced emf if the diameter is $8.0 \mathrm{~cm}$ and $N=10 ?$

Vishal Gupta

Numerade Educator

Problem 9

A $30.0 \mathrm{~cm} \times 60.0 \mathrm{~cm}$ rectangular circuit containing a $15 \Omega$ resistor is perpendicular to a uniform magnetic field that starts out at $2.65 \mathrm{~T}$ and steadily decreases at $0.25 \mathrm{~T} / \mathrm{s}$. (See Figure $21.45 .)$ While this field is changing, what does the ammeter read?

Vishal Gupta

Numerade Educator

Problem 10

A circular loop of wire with a radius of $12.0 \mathrm{~cm}$ is lying flat on a tabletop. A magnetic field of $1.5 \mathrm{~T}$ is directed vertically downward through the loop (Figure 21.46).
(a) If the loop is removed from the field region in a time interval of $2.0 \mathrm{~ms}$, find the average emf that will be induced in the wire loop during the extraction process. (b) If the loop is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

Vishal Gupta

Numerade Educator

Problem 11

A cardboard tube is wrapped with windings of insulated wire that are connected to a resistor. The tube is placed into a magnetic field that is changing with time. Determine whether the induced current in the resistor $R$ is directed from left to right or from right to left or is zero for (a) Figure 21.47 a with the magnetic field increasing, (b) Figure $21.47 \mathrm{~b}$ with the field decreasing, and (c) Figure $21.47 \mathrm{c}$ with the field increasing. (Consider only the magnetic flux through the tube.)

Vishal Gupta

Numerade Educator

Problem 12

A circular loop of wire is in a spatially uniform magnetic field, as shown in Figure $21.48 .$ The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) $B$ is increasing; (b) $B$ is decreasing; (c) $B$ is constant with a value of $B_{0}$. Explain your reasoning.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 13

Using Lenz's law, determine the direction of the current in resistor $a b$ of Figure 21.49 when (a) switch $S$ is opened after having been closed for several minutes; (b) coil $B$ is brought closer to coil $A$ with the switch closed; (c) the resistance of $R$ is decreased while the switch remains closed.

Averell Hause

Carnegie Mellon University

Problem 14

A solenoid carrying a current $I$ is moving toward a metal ring, as shown in Figure $21.50 .$ In what direction, clockwise or counterclockwise (as seen from the solenoid), is a current induced in the ring? In what direction will the induced current be if the solenoid now stops moving toward the ring, but the current in it begins to decrease?

Vishal Gupta

Numerade Educator

Problem 15

A metal bar is pulled to the right perpendicular to a uniform magnetic field. The bar rides on parallel metal rails connected through a resistor, as shown in Figure $21.51,$ so the apparatus makes a complete circuit. Find the direction of the current induced in the circuit in two ways: (a) by looking at the magnetic force on the charges in the moving bar and (b) by using Lenz's law.

Averell Hause

Carnegie Mellon University

Problem 16

Two closed loops $A$ and $C$ are close to a long wire carrying a current $I$. (See Figure $21.52 .$ ) Find the direction (clockwise or counterclockwise) of the current induced in each of these loops if $I$ is steadily increasing.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 17

A bar magnet is held above a circular loop of wire as shown in Figure 21.53. Find the direction (clockwise or counterclockwise, as viewed from below the loop) of the current induced in this loop in each of the following cases: (a) The loop is dropped. (b) The magnet is dropped. (c) Both the loop and magnet are dropped at the same instant.

Averell Hause

Carnegie Mellon University

Problem 18

The current in Figure 21.54 obeys the equation $I=I_{0}(1+t / \tau)$, where $t$ is time and $\tau>0$. Find the direction (clockwise or counterclockwise) of the current induced in the round coil for $t>0$

Vishal Gupta

Numerade Educator

Problem 19

A bar magnet is close to a metal loop. When this magnet is suddenly moved to the left away from the loop, as shown in Figure $21.55,$ a counterclockwise current is induced in the coil, as viewed by an observer looking through the coil toward the magnet. Identify the north and south poles of the magnet.

Vishal Gupta

Numerade Educator

Problem 20

A very thin $15.0 \mathrm{~cm}$ copper bar is aligned horizontally along the east-west direction. If it moves horizontally from south to north at $11.5 \mathrm{~m} / \mathrm{s}$ in a vertically upward magnetic field of $1.22 \mathrm{~T},$ (a) what potential difference is induced across its ends, and (b) which end (east or west) is at a higher potential? (c) What would be the potential difference if the bar moved from east to west instead?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 21

I When a thin $12.0 \mathrm{~cm}$ iron rod moves with a constant velocity of $4.50 \mathrm{~m} / \mathrm{s}$ perpendicular to the rod in the direction shown in Figure 21.56 the induced emf across its ends is measured to be $0.450 \mathrm{~V}$. (a) What is the magnitude of the magnetic field? (b) Which point is at a higher potential, $a$ or $b$ ? (c) If the rod is rotated clockwise by $90^{\circ}$ in the plane of the paper, but keeps the same velocity, what is the potential difference induced across its ends?

Vishal Gupta

Numerade Educator

Problem 22

You wish to produce a potential difference of $10 \mathrm{~V}$ along the length of a $1 \mathrm{~m}$ copper rod by moving it through the earth's magnetic field. In a region where the field is $5.5 \times 10^{-5} \mathrm{~T}$, what minimum speed would be needed?

Vishal Gupta

Numerade Educator

Problem 23

A $1.41 \mathrm{~m}$ bar moves through a uniform, 1.20 T magnetic field with a speed of $2.50 \mathrm{~m} / \mathrm{s}$. For each case in Figure $21.57,$ find the emf induced between the ends of this bar, and identify which, if either, end $(L$ or $R)$ is at the higher potential.

Vishal Gupta

Numerade Educator

Problem 24

The conducting rod $a b$ shown in Figure 21.58 makes frictionless contact with metal rails $c a$ and $d b$. The apparatus is in a uniform magnetic field of 0.800 T, perpendicular to the plane of the figure. (a) Find the magnitude of the emf induced in the rod when it is moving toward the right with a speed of $7.50 \mathrm{~m} / \mathrm{s}$. (b) In what direction does the current flow in the rod? (c) If the resistance of the circuit $a b d c$ is a constant $1.50 \Omega$, find the magnitude and direction of the force required to keep the rod moving to the right with a constant speed of $7.50 \mathrm{~m} / \mathrm{s}$.

Vishal Gupta

Numerade Educator

Problem 25

BIO Measuring blood flow. Blood contains positive and negative ions and so is a conductor. A blood vessel, therefore, can be viewed as an electrical wire. We can even picture the flowing blood as a series of parallel conducting slabs whose thickness is the diameter $d$ of the vessel moving with speed $v$. (See Figure $21.59 .$ ) (a) If the blood vessel is placed in a magnetic field $\overrightarrow{\boldsymbol{B}}$ perpendicular to the vessel, as in the figure, show that the motional potential difference induced across it is $\mathcal{E}=v B d$. (b) If you expect that the blood will be flowing at $15 \mathrm{~cm} / \mathrm{s}$ for a vessel $5.0 \mathrm{~mm}$ in diameter, what strength of magnetic field will you need to produce a potential difference of $1.0 \mathrm{mV} ?$ (c) Show that the volume rate of flow $(R)$ of the blood is equal to $R=\pi \mathcal{E} d / 4 B$. (Note:
Although the method developed here is useful in measuring the rate of blood flow in a vessel, it is of limited use in surgery because measurement of the potential $\mathcal{E}$ must be made directly across the vessel.)

Vishal Gupta

Numerade Educator

Problem 26

A toroidal solenoid has a mean radius of $10.0 \mathrm{~cm}$ and a crosssectional area of $4.00 \mathrm{~cm}^{2}$ and is wound uniformly with 100 turns. A second coil with 500 turns is wound uniformly on top of the first. What is the mutual inductance of these coils?

Vishal Gupta

Numerade Educator

Problem 27

A 10.0 -cm-long solenoid of diameter $0.400 \mathrm{~cm}$ is wound uniformly with 800 turns. A second coil with 50 turns is wound around the solenoid at its center. What is the mutual inductance of the combination of the two coils?

Abhishek Jana

Numerade Educator

Problem 28

Two coils are wound around the same cylindrical form, like the coils in Example $21.8 .$ When the current in the first coil is decreasing at a rate of $0.242 \mathrm{~A} / \mathrm{s},$ the induced emf in the second coil has magnitude $1.65 \mathrm{mV}$. (a) What is the mutual inductance of the pair of coils? (b) If the second coil has 25 turns, what is the average magnetic flux through each turn when the current in the first coil equals $1.20 \mathrm{~A} ?$ (c) If the current in the second coil increases at a rate of $0.360 \mathrm{~A} / \mathrm{s},$ what is the magnitude of the induced emf in the first coil?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 29

One solenoid is centered inside another. The outer one has a length of $50.0 \mathrm{~cm}$ and contains 6750 coils, while the coaxial inner solenoid is $3.0 \mathrm{~cm}$ long and $0.120 \mathrm{~cm}$ in diameter and contains 15 coils. The current in the outer solenoid is changing at $37.5 \mathrm{~A} / \mathrm{s}$. (a) What is the mutual inductance of these solenoids?
(b) Find the emf induced in the inner solenoid.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 30

A $4.5 \mathrm{mH}$ toroidal inductor has 125 identical equally spaced coils. (a) If it carries an 11.5 A current, how much magnetic flux passes through each of its coils? (b) If the potential difference across its ends is $1.16 \mathrm{~V}$, at what rate is the current in it changing?

Averell Hause

Carnegie Mellon University

Problem 31

At the instant when the current in an inductor is increasing at a rate of $0.0640 \mathrm{~A} / \mathrm{s}$, the magnitude of the self-induced emf is $0.0160 \mathrm{~V}$. What is the inductance of the inductor?

Averell Hause

Carnegie Mellon University

Problem 32

An inductor has inductance of $0.260 \mathrm{H}$ and carries a current that is decreasing at a uniform rate of $18.0 \mathrm{~mA} / \mathrm{s}$. Find the self-induced $\mathrm{emf}$ in this inductor.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 33

A toroidal solenoid has a self-inductance $L$ and $N=100$ coils. How many more coils would have to be added to the solenoid in order to increase its inductance to $2 L ?$

Averell Hause

Carnegie Mellon University

Problem 34

elf-inductance of a straight solenoid. Suppose you were to take the toroidal solenoid in Example 21.10 and unbend it in order to form a straight solenoid. Show that its inductance would be given by the equation $L=\mu_{0} A N^{2} / l,$ where $l$ is the length of the newly formed straight solenoid. (Your answer is approximate because $B$ is actually smaller at the ends than at the center of the straight solenoid. For this reason, your answer is actually an upper limit on the inductance.)

Linda Winkler

Numerade Educator

Problem 35

When the current in a toroidal solenoid is changing at a rate of $0.0260 \mathrm{~A} / \mathrm{s}$, the magnitude of the induced emf is $12.6 \mathrm{mV}$. When the current equals $1.40 \mathrm{~A}$, the average flux through each turn of the solenoid is 0.00285 Wb. How many turns does the solenoid have?

Vishal Gupta

Numerade Educator

Problem 36

A transformer consists of 275 primary windings and 834 secondary windings. If the potential difference across the primary coil is $25.0 \mathrm{~V}$ (a) what is the voltage across the secondary coil, and (b) what is the current in the secondary coil if it is connected across a $125 \Omega$ resistor?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 37

You need a transformer that will draw $15 \mathrm{~W}$ of power from a $220 \mathrm{~V}$ (rms) power line, stepping the voltage down to $6.0 \mathrm{~V}$ (rms). (a) What will be the current in the secondary coil? (b) What should be the resistance of the secondary circuit? (c) What will be the equivalent resistance of the input circuit?

Vishal Gupta

Numerade Educator

Problem 38

A step-up transformer. A transformer connected to a $120 \mathrm{~V}$ (rms) ac line is to supply $13,000 \mathrm{~V}$ (rms) for a neon sign. To reduce the shock hazard, a fuse is to be inserted in the primary circuit and is to blow when the rms current in the secondary circuit exceeds $8.50 \mathrm{~mA}$. (a) What is the ratio of secondary to primary turns of the transformer? (b) What power must be supplied to the transformer when the rms secondary current is $8.50 \mathrm{~mA} ?$ (c) What current rating should the fuse in the primary circuit have?

Averell Hause

Carnegie Mellon University

Problem 39

An air-filled toroidal solenoid has a mean radius of $15.0 \mathrm{~cm}$ and a cross-sectional area of $5.00 \mathrm{~cm}^{2}$. When the current is $12.0 \mathrm{~A}$, the energy stored is $0.390 \mathrm{~J}$. How many turns does the winding have?

Vishal Gupta

Numerade Educator

Problem 40

Energy in a typical inductor. (a) How much energy is stored in a $10.2 \mathrm{mH}$ inductor carrying a 1.15 A current? (b) How much current would such an inductor have to carry to store $1.0 \mathrm{~J}$ of energy? Is this a reasonable amount of current for ordinary laboratory circuit elements?

Fadil Iqbal

Numerade Educator

Problem 41

(a) What would have to be the self-inductance of a solenoid for it to store $10.0 \mathrm{~J}$ of energy when a $1.50 \mathrm{~A}$ current runs through it? (b) If this solenoid's cross-sectional diameter is $4.00 \mathrm{~cm},$ and if you could wrap its coils to a density of 10 coils $/ \mathrm{mm}$, how long would the solenoid be? (See Problem 34.) Is this a realistic length for ordinary laboratory use?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 42

A solenoid $25.0 \mathrm{~cm}$ long and with a cross-sectional area of $0.500 \mathrm{~cm}^{2}$ contains 400 turns of wire and carries a current of 80.0 A. Calculate (a) the magnetic field in the solenoid; (b) the energy density in the magnetic field if the solenoid is filled with air; (c) the total energy contained in the coil's magnetic field (assume the field is uniform $) ;$ (d) the inductance of the solenoid.

Vishal Gupta

Numerade Educator

Problem 43

Large inductors have been proposed as energy-storage devices. (a) How much electrical energy is converted to light and thermal energy by a $200 \mathrm{~W}$ lightbulb in one day? (b) If the amount of energy calculated in part (a) is stored in an inductor in which the current is $80.0 \mathrm{~A},$ what is the inductance?

Abhishek Jana

Numerade Educator

Problem 44

When a certain inductor carries a current $I$, it stores $3.0 \mathrm{~mJ}$ of magnetic energy. How much current (in terms of $I$ ) would it have to carry to store $9.0 \mathrm{~mJ}$ of energy?

Averell Hause

Carnegie Mellon University

Problem 45

A $12.0 \mathrm{~V}$ dc battery having no appreciable internal resistance, a $150.0 \Omega$ resistor, an $11.0 \mathrm{mH}$ inductor, and an open switch are all connected in series. After the switch is closed, what are (a) the time constant for this circuit, (b) the maximum current that flows through it, (c) the current $73.3 \mu$ s after the switch is closed, and (d) the maximum energy stored in the inductor?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 46

An inductor with an inductance of $2.50 \mathrm{H}$ and a resistor with a resistance of $8.00 \Omega$ are connected to the terminals of a battery with an emf of $6.00 \mathrm{~V}$ and negligible internal resistance. Find (a) the initial rate of increase of the current in the circuit, (b) the initial potential difference across the inductor, (c) the current 0.313 s after the circuit is closed, and (d) the maximum current.

Vishal Gupta

Numerade Educator

Problem 47

In the circuit shown in Figure 21.60 , the battery and the inductor have no appreciable internal resistance and there is no current in the circuit. After the switch is closed, find the readings of the ammeter (A) and voltmeters $\left(\mathrm{V}_{1}\right.$ and $\mathrm{V}_{2}$ ) (a) the instant after the switch is closed; (b) after the switch has been closed for a very long time. (c) Which answers in parts (a) and (b) would change if the inductance were $24.0 \mathrm{mH}$ instead?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 48

A $35.0 \mathrm{~V}$ battery with negligible internal resistance, a $50.0 \mathrm{~V}$ resistor, and a $1.25 \mathrm{mH}$ inductor with negligible resistance are all connected in series with an open switch. The switch is suddenly closed. (a) How long after closing the switch will the current through the inductor reach onehalf of its maximum value? (b) How long after closing the switch will the energy stored in the inductor reach one-half of its maximum value?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 49

A $1.50 \mathrm{mH}$ inductor is connected in series with a dc battery of negligible internal resistance, a $0.750 \mathrm{k} \Omega$ resistor, and an open switch. How long after the switch is closed will it take for (a) the current in the circuit to reach half of its maximum value, (b) the energy stored in the inductor to reach half of its maximum value? (Hint: You will have to solve an exponential equation.)

Vishal Gupta

Numerade Educator

Problem 50

A $12.0 \mu \mathrm{F}$ capacitor and a $5.25 \mathrm{mH}$ inductor are connected in series with an open switch. The capacitor is initially charged to $6.20 \mu \mathrm{C} .$ What is the angular frequency of the charge oscillations in the capacitor after the switch is closed?

Vishal Gupta

Numerade Educator

Problem 51

A $5.00 \mu \mathrm{F}$ capacitor is initially charged to a potential of $16.0 \mathrm{~V}$. It is then connected in series with a $3.75 \mathrm{mH}$ inductor. (a) What is the total energy stored in this circuit? (b) What is the maximum current in the inductor? What is the charge on the capacitor plates at the instant the current in the inductor is maximal?

Averell Hause

Carnegie Mellon University

Problem 52

A $15.0 \mu \mathrm{F}$ capacitor is charged to $175 \mu \mathrm{C}$ and then connected across the ends of a $5.00 \mathrm{mH}$ inductor. (a) Find the maximum current in the inductor. At the instant the current in the inductor is maximal, how much charge is on the capacitor plates? (b) Find the maximum potential across the capacitor. At this instant, what is the current in the inductor? (c) Find the maximum energy stored in the inductor. At this instant, what is the current in the circuit?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 53

An inductor is connected to the terminals of a battery that has an emf of $12.0 \mathrm{~V}$ and negligible internal resistance. The current is $4.86 \mathrm{~mA}$ at $0.725 \mathrm{~ms}$ after the connection is completed. After a long time, the current is $6.45 \mathrm{~mA}$. What are (a) the resistance $R$ of the inductor and (b) the inductance $L$ of the inductor?

Vishal Gupta

Numerade Educator

Problem 54

A rectangular circuit is moved at a constant velocity of $3.0 \mathrm{~m} / \mathrm{s}$ into, through, and then out of a uniform $1.25 \mathrm{~T}$ magnetic field, as shown in Figure $21.61$. The magnetic-field region is considerably wider than $50.0 \mathrm{~cm}$. Find the magnitude and direction (clockwise or counterclockwise) of the current induced in the circuit as it is (a) going into the magnetic field, (b) totally within the magnetic field, but still moving, and (c) moving out of the field. (d) Sketch a graph of the current in this circuit as a function of time, including the preceding three cases.

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 55

DATA A physics student is studying the behavior of an $R-L$ circuit, such as the one shown in Figure $21.30 .$ She measures the potential difference across the inductor $V_{L}$ as a function of time. According to her lecture notes, $V_{L}$ should decay with time as $V_{L}=\mathcal{E} e^{-(t / \tau)},$ where $\mathcal{E}$ is the emf of the battery and $\tau=L / R$ is the time constant of the circuit. After closing the switch, she obtains the following data:
$$
\begin{array}{cc}
\hline \text { Time (s) } & \boldsymbol{V}_{\boldsymbol{L}} \text { ( } \mathbf{V} \text { ) } \\
\hline 0 & 9.0 \\
10.0 & 4.8 \\
18.0 & 2.9 \\
27.0 & 1.7 \\
35.0 & 1.0 \\
41.0 & 0.7 \\
\hline
\end{array}
$$
(a) What is the emf of the battery? (b) Make a semi-log plot of $V_{L}$ as a function of time. Using a "best fit" to the data, determine the time constant $\tau$ of the $R-L$ circuit.

Bruce Edelman

Numerade Educator

Problem 56

A flexible circular loop $6.50 \mathrm{~cm}$ in diameter lies in a magnetic field with magnitude 0.950 $\mathrm{T},$ directed into the plane of the page as shown in Figure 21.62. The loop is pulled at the points indicated by the arrows, forming a loop of zero area in $0.250 \mathrm{~s}$. (a) Find the average induced emf in the circuit. (b) What is the direction of the current in $R:$ from $a$ to $b$ or from $b$ to $a$ ? Explain your reasoning.

Vishal Gupta

Numerade Educator

Problem 57

An electromagnetic car alarm. Your latest invention is a car alarm that produces sound at a particularly annoying frequency of $3500 \mathrm{~Hz}$. To do this, the car-alarm circuitry must produce an alternating electric current of the same frequency. That's why your design includes an inductor and a capacitor in series. The maximum voltage across the capacitor is to be $12.0 \mathrm{~V}$ (the same voltage as the car battery). To produce a sufficiently loud sound, the capacitor must store $0.0160 \mathrm{~J}$ of energy. What values of $\mathrm{ca}-$ pacitance and inductance should you choose for your car-alarm circuit?

Vishal Gupta

Numerade Educator

Problem 58

In the circuit shown in Figure $21.63, S_{1}$ has been closed for a long enough time so that the current reads a steady 3.50 A. Suddenly, $S_{2}$ is closed and $S_{1}$ is opened at the same instant. (a) What is the maximum charge that the capacitor will receive? (b) What is the current in the inductor at this time?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 59

Consider the circuit in Figure $21.64 .$ (a) Just after the switch is closed, what is the current through each of the resistors? (b) After the switch has been closed a long time, what is the current through each resistor? (c) After $S$ has been closed a long time, it is opened again. Just after it is opened, what is the current through the $20.0 \Omega$ resistor?

Averell Hause

Carnegie Mellon University

Problem 60

Quenching an MRI magnet. Magnets carrying very large currents are used to produce the uniform, large-magnitude magnetic fields that are required for magnetic resonance imaging (MRI). A typical MRI magnet may be a solenoid that is $2.0 \mathrm{~m}$ long and $1.0 \mathrm{~m}$ in diameter, has a self-inductance of $4.4 \mathrm{H},$ and carries a current of 750
A. A normal wire carrying that much current would dissipate a great deal of electric power as heat, so most MRI magnets are made with coils of superconducting wire cooled by liquid helium at a temperature just under its boiling point $(4.2 \mathrm{~K})$. After a current is established in the wire, the power supply is disconnected and the magnet leads are shorted together through a piece of superconductor so that the current flows without resistance as long as the liquid helium keeps the magnet cold.
Under rare circumstances, a small segment of the magnet's wire may lose its superconducting properties and develop resistance. In this segment, electrical energy is converted to thermal energy, which can boil off some of the liquid helium. More of the wire then warms up and loses its superconducting properties, thus dissipating even more energy as heat. Because the latent heat of vaporization of liquid helium is quite low $(20.9 \mathrm{~kJ} / \mathrm{kg}),$ once the wire begins to warm up, all of the liquid helium may boil off rapidly. This event, called a quench, can damage the magnet. Also, a large volume of helium gas is generated as the liquid helium boils off, causing an asphyxiation hazard, and the resulting rapid pressure buildup can lead to an explosion. You can see how important it is to keep the wire resistance in an MRI magnet at zero and to have devices that detect a quench and shut down the current immediately.
How many turns does this typical MRI magnet have?
A. 1100
B. 3000
C. 4000
D. 22,000

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 61

Quenching an MRI magnet. Magnets carrying very large currents are used to produce the uniform, large-magnitude magnetic fields that are required for magnetic resonance imaging (MRI). A typical MRI magnet may be a solenoid that is $2.0 \mathrm{~m}$ long and $1.0 \mathrm{~m}$ in diameter, has a self-inductance of $4.4 \mathrm{H},$ and carries a current of 750
A. A normal wire carrying that much current would dissipate a great deal of electric power as heat, so most MRI magnets are made with coils of superconducting wire cooled by liquid helium at a temperature just under its boiling point $(4.2 \mathrm{~K})$. After a current is established in the wire, the power supply is disconnected and the magnet leads are shorted together through a piece of superconductor so that the current flows without resistance as long as the liquid helium keeps the magnet cold.
Under rare circumstances, a small segment of the magnet's wire may lose its superconducting properties and develop resistance. In this segment, electrical energy is converted to thermal energy, which can boil off some of the liquid helium. More of the wire then warms up and loses its superconducting properties, thus dissipating even more energy as heat. Because the latent heat of vaporization of liquid helium is quite low $(20.9 \mathrm{~kJ} / \mathrm{kg}),$ once the wire begins to warm up, all of the liquid helium may boil off rapidly. This event, called a quench, can damage the magnet. Also, a large volume of helium gas is generated as the liquid helium boils off, causing an asphyxiation hazard, and the resulting rapid pressure buildup can lead to an explosion. You can see how important it is to keep the wire resistance in an MRI magnet at zero and to have devices that detect a quench and shut down the current immediately.
If a small part of this magnet loses its superconducting properties and the resistance of the magnet wire suddenly rises from 0 to a constant $0.005 \Omega$, how much time will it take for the current to decrease to half of its initial value?
A. $4.7 \mathrm{~min}$
B. $10 \mathrm{~min}$
C. $15 \mathrm{~min}$
D. $30 \mathrm{~min}$

Dading Chen

Numerade Educator

Problem 62

Quenching an MRI magnet. Magnets carrying very large currents are used to produce the uniform, large-magnitude magnetic fields that are required for magnetic resonance imaging (MRI). A typical MRI magnet may be a solenoid that is $2.0 \mathrm{~m}$ long and $1.0 \mathrm{~m}$ in diameter, has a self-inductance of $4.4 \mathrm{H},$ and carries a current of 750
A. A normal wire carrying that much current would dissipate a great deal of electric power as heat, so most MRI magnets are made with coils of superconducting wire cooled by liquid helium at a temperature just under its boiling point $(4.2 \mathrm{~K})$. After a current is established in the wire, the power supply is disconnected and the magnet leads are shorted together through a piece of superconductor so that the current flows without resistance as long as the liquid helium keeps the magnet cold.
Under rare circumstances, a small segment of the magnet's wire may lose its superconducting properties and develop resistance. In this segment, electrical energy is converted to thermal energy, which can boil off some of the liquid helium. More of the wire then warms up and loses its superconducting properties, thus dissipating even more energy as heat. Because the latent heat of vaporization of liquid helium is quite low $(20.9 \mathrm{~kJ} / \mathrm{kg}),$ once the wire begins to warm up, all of the liquid helium may boil off rapidly. This event, called a quench, can damage the magnet. Also, a large volume of helium gas is generated as the liquid helium boils off, causing an asphyxiation hazard, and the resulting rapid pressure buildup can lead to an explosion. You can see how important it is to keep the wire resistance in an MRI magnet at zero and to have devices that detect a quench and shut down the current immediately.
If part of the magnet develops resistance and liquid helium boils away, rendering more and more of the magnet nonsuperconducting, how will this quench affect the time for the current to drop to half of its initial value?
A. The time will be shorter because the resistance will increase.
B. The time will be longer because the resistance will increase.
C. The time will be the same.
D. Not enough information is given.

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 63

Quenching an MRI magnet. Magnets carrying very large currents are used to produce the uniform, large-magnitude magnetic fields that are required for magnetic resonance imaging (MRI). A typical MRI magnet may be a solenoid that is $2.0 \mathrm{~m}$ long and $1.0 \mathrm{~m}$ in diameter, has a self-inductance of $4.4 \mathrm{H},$ and carries a current of 750
A. A normal wire carrying that much current would dissipate a great deal of electric power as heat, so most MRI magnets are made with coils of superconducting wire cooled by liquid helium at a temperature just under its boiling point $(4.2 \mathrm{~K})$. After a current is established in the wire, the power supply is disconnected and the magnet leads are shorted together through a piece of superconductor so that the current flows without resistance as long as the liquid helium keeps the magnet cold.
Under rare circumstances, a small segment of the magnet's wire may lose its superconducting properties and develop resistance. In this segment, electrical energy is converted to thermal energy, which can boil off some of the liquid helium. More of the wire then warms up and loses its superconducting properties, thus dissipating even more energy as heat. Because the latent heat of vaporization of liquid helium is quite low $(20.9 \mathrm{~kJ} / \mathrm{kg}),$ once the wire begins to warm up, all of the liquid helium may boil off rapidly. This event, called a quench, can damage the magnet. Also, a large volume of helium gas is generated as the liquid helium boils off, causing an asphyxiation hazard, and the resulting rapid pressure buildup can lead to an explosion. You can see how important it is to keep the wire resistance in an MRI magnet at zero and to have devices that detect a quench and shut down the current immediately.
If all of the magnetic energy stored in this MRI magnet is converted to thermal energy, how much liquid helium will boil off?
A. $27 \mathrm{~kg}$
B. $38 \mathrm{~kg}$
C. $60 \mathrm{~kg}$
D. $110 \mathrm{~kg}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 64

BIO Stimulating the brain. Communication in the nervous system is based on the propagation of electrical signals called action potentials along axons, which are extensions of nerve cells (see the MCAT-Style Passage Problems in Chapter 19). Action potentials are generated when the electric potential difference across the membrane of the nerve cell changes: Specifically, the inside of the cell becomes more positive. Researchers in clinical medicine and neurobiology cannot stimulate nerves (even non-invasively) at specific locations in conscious human subjects. Using electrodes to apply current to the skin is painful and requires large currents, which could be dangerous. Anthony Barker and colleagues at the University of Sheffield in England developed a technique called transcranial magnetic stimulation (TMS). In this widely used procedure, a coil positioned near the skull produces a time-varying magnetic field, which induces in the conductive tissue of the brain (Figure 21.65 a) electric currents that are sufficient to cause action potentials in nerve cells. For example, if the coil is placed near the motor cortex (the region of the brain that controls voluntary movement), scientists can monitor muscle contraction and assess the connections between the brain and the muscles. Figure $21.65 \mathrm{~b}$ is a graph of the typical dependence on time $t$ of the magnetic field $\overrightarrow{\boldsymbol{B}}$ produced by the coil.
In Figure $21.65 \mathrm{a}$, a current pulse increases to a peak and then decreases to zero in the direction shown in the stimulating coil. What will be the direction of the induced current (dashed line) in the brain tissue?
A. 1
B. 2
C. 1 while the current increases in the stimulating coil, 2 while the current decreases
D. 2 while the current increases in the stimulating coil, 1 while the current decreases

Jake Steedman

Numerade Educator

Problem 65

BIO Stimulating the brain. Communication in the nervous system is based on the propagation of electrical signals called action potentials along axons, which are extensions of nerve cells (see the MCAT-Style Passage Problems in Chapter 19). Action potentials are generated when the electric potential difference across the membrane of the nerve cell changes: Specifically, the inside of the cell becomes more positive. Researchers in clinical medicine and neurobiology cannot stimulate nerves (even non-invasively) at specific locations in conscious human subjects. Using electrodes to apply current to the skin is painful and requires large currents, which could be dangerous. Anthony Barker and colleagues at the University of Sheffield in England developed a technique called transcranial magnetic stimulation (TMS). In this widely used procedure, a coil positioned near the skull produces a time-varying magnetic field, which induces in the conductive tissue of the brain (Figure 21.65 a) electric currents that are sufficient to cause action potentials in nerve cells. For example, if the coil is placed near the motor cortex (the region of the brain that controls voluntary movement), scientists can monitor muscle contraction and assess the connections between the brain and the muscles. Figure $21.65 \mathrm{~b}$ is a graph of the typical dependence on time $t$ of the magnetic field $\overrightarrow{\boldsymbol{B}}$ produced by the coil.
Consider the brain tissue at the level of the dashed line to be a series of concentric circles, each behaving independently of the others. Where will the induced emf be the greatest?
A. At the center of the dashed line
B. At the periphery of the dashed line
C. Nowhere-it will be the same in all concentric circles
D. At the center while the stimulating current increases, at the periphery while the current decreases

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 66

BIO Stimulating the brain. Communication in the nervous system is based on the propagation of electrical signals called action potentials along axons, which are extensions of nerve cells (see the MCAT-Style Passage Problems in Chapter 19). Action potentials are generated when the electric potential difference across the membrane of the nerve cell changes: Specifically, the inside of the cell becomes more positive. Researchers in clinical medicine and neurobiology cannot stimulate nerves (even non-invasively) at specific locations in conscious human subjects. Using electrodes to apply current to the skin is painful and requires large currents, which could be dangerous. Anthony Barker and colleagues at the University of Sheffield in England developed a technique called transcranial magnetic stimulation (TMS). In this widely used procedure, a coil positioned near the skull produces a time-varying magnetic field, which induces in the conductive tissue of the brain (Figure 21.65 a) electric currents that are sufficient to cause action potentials in nerve cells. For example, if the coil is placed near the motor cortex (the region of the brain that controls voluntary movement), scientists can monitor muscle contraction and assess the connections between the brain and the muscles. Figure $21.65 \mathrm{~b}$ is a graph of the typical dependence on time $t$ of the magnetic field $\overrightarrow{\boldsymbol{B}}$ produced by the coil.
Which time-dependent graph of the magnetic field $\overrightarrow{\boldsymbol{B}}$ in the coil will increase the maximum induced current in the brain tissue? Assume that everything else remains constant.
A. A
B. $B$
C. Either A or B
D. Neither A nor B

Zachary Warner

Numerade Educator

Problem 67

BIO Stimulating the brain. Communication in the nervous system is based on the propagation of electrical signals called action potentials along axons, which are extensions of nerve cells (see the MCAT-Style Passage Problems in Chapter 19). Action potentials are generated when the electric potential difference across the membrane of the nerve cell changes: Specifically, the inside of the cell becomes more positive. Researchers in clinical medicine and neurobiology cannot stimulate nerves (even non-invasively) at specific locations in conscious human subjects. Using electrodes to apply current to the skin is painful and requires large currents, which could be dangerous. Anthony Barker and colleagues at the University of Sheffield in England developed a technique called transcranial magnetic stimulation (TMS). In this widely used procedure, a coil positioned near the skull produces a time-varying magnetic field, which induces in the conductive tissue of the brain (Figure 21.65 a) electric currents that are sufficient to cause action potentials in nerve cells. For example, if the coil is placed near the motor cortex (the region of the brain that controls voluntary movement), scientists can monitor muscle contraction and assess the connections between the brain and the muscles. Figure $21.65 \mathrm{~b}$ is a graph of the typical dependence on time $t$ of the magnetic field $\overrightarrow{\boldsymbol{B}}$ produced by the coil.
Which graph best represents the time $t$ dependence of the current $i$ induced in the brain tissue, assuming that this tissue can be modeled as a resistive circuit? (The units of $i$ are arbitrary.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 67

Which graph best represents the time $t$ dependence of the current $i$ induced in the brain tissue, assuming that this tissue can be modeled as a resistive circuit? (The units of $i$ are arbitrary.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator