University Physics with Modern Physics

Hugh D. Young

Chapter 29

Electromagnetic Induction - all with Video Answers

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Chapter Questions

Problem 1

A single loop of wire with an area of 0.0900 m$^2$ is in a uniform magnetic field that has an initial value of 3.80 T, is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 T/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.

Kathleen Tatem

Kathleen Tatem

Numerade Educator

Problem 2

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 cm$^2$ is rotated in 0.040 s from a position where its plane is perpendicular to the earth's magnetic field to a position where its plane is parallel to the field. The earth's magnetic field at the lab location is 6.0 $\times$ 10$^{-5}$ 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?

Farhanul Hasan

Numerade Educator

Problem 3

One practical way to measure magnetic field strength uses a small, closely wound coil called a $search$ $coil$. The coil is initially held with its plane perpendicular to a magnetic field. The coil is then either quickly rotated a quarter-turn about a diameter or quickly pulled out of the field. (a) Derive the equation relating the total charge $Q$ that flows through a search coil to the magnetic-field magnitude B. The search coil has N turns, each with area A, and the flux through the coil is decreased from its initial maximum value to zero in a time $\Delta{t}$ . The resistance of the coil is $R$, and the total charge is $Q = \Delta{t}$, where $I$ is the average current induced by the change in flux. (b) In a credit card reader, the magnetic strip on the back of a credit card is rapidly "swiped" past a coil within the reader. Explain, using the same ideas that underlie the operation of a search coil, how the reader can decode the information stored in the pattern of magnetization on the strip. (c) Is it necessary that the credit card be "swiped" through the reader at exactly the right speed? Why or why not?

Bruce Edelman

Numerade Educator

Problem 4

A closely wound search coil (see Exercise 29.3) has an area of 3.20 cm$^2$, 120 turns, and a resistance of 60.0 $\Omega$. It is connected to a charge-measuring instrument whose resistance is 45.0 $\Omega$. When the coil is rotated quickly from a position parallel to a uniform magnetic field to a position perpendicular to the field, the instrument indicates a charge of 3.56 $\times$ 10$^{-5}$ C. What is the magnitude of the field?

Farhanul Hasan

Numerade Educator

Problem 5

A circular loop of wire with a radius of 12.0 cm and oriented in the horizontal $xy$-plane is located in a region of uniform magnetic field. A field of 1.5 T is directed along the positive z-direction, which is upward. (a) If the loop is removed from the field region in a time interval of 2.0 ms, find the average emf that will be induced in the wire loop during the extraction process. (b) If the coil is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

Bruce Edelman

Numerade Educator

Problem 6

A coil 4.00 cm in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to $B =$ (0.0120 T/s)$t$ + (3.00 $\times$ 10$^{-5}$ T/s$^4)t^4$. The coil is connected to a 600-$\Omega$ resistor, and its plane is perpendicular to the magnetic field. You can ignore the resistance of the coil. (a) Find the magnitude of the induced emf in the coil as a function of time. (b) What is the current in the resistor at time $t =$ 5.00 s?

Farhanul Hasan

Numerade Educator

Problem 7

The current in the long, straight wire $AB$ shown in $\textbf{Fig. E29.7}$ is upward and is increasing steadily at a rate $di/dt$. (a) At an instant when the current is $i$, what are the magnitude and direction of the field $\overrightarrow{B}$ at a distance $r$ to the right of the wire? (b) What is the flux $d{\Phi}_B$ through the narrow, shaded strip? (c) What is the total flux through the loop? (d) What is the induced emf in the loop? (e) Evaluate the numerical value of the induced emf if $a =$ 12.0 cm, $b =$ 36.0 cm, $L =$ 24.0 cm, and $di/dt =$ 9.60 A/s.

SB

Suman Baral

Numerade Educator

Problem 8

A flat, circular, steel loop of radius 75 cm is at rest in a uniform magnetic field, as shown in an edge-on view in $\textbf{Fig. E29.8}$. The field is changing with time, according to $B(t2) =$ (1.4 T)$e^{-(0.057 s-1)}t$. (a) Find the emf induced in the loop as a function of time. (b) When is the induced emf equal to $\frac{1}{10}$ of its initial value? (c) Find the direction of the current induced in the loop, as viewed from above the loop.

Hafiz Sheriff

Numerade Educator

Problem 9

A circular loop of flexible iron wire has an initial circumference of 165.0 cm, but its circumference is decreasing at a constant rate of 12.0 cm/s due to a tangential pull on the wire. The loop is in a constant, uniform magnetic field oriented perpendicular to the plane of the loop and with magnitude 0.500 T. (a) Find the emf induced in the loop at the instant when 9.0 s have passed. (b) Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

Bruce Edelman

Numerade Educator

Problem 10

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 $1.70 \mathrm{~T}$ to a position perpendicular to the field. The rotation takes $0.0600 \mathrm{~s}$. What is the average emf induced in the coil?

Farhanul Hasan

Numerade Educator

Problem 11

In a region of space, a magnetic field points in the +$x$-direction (toward the right). Its magnitude varies with position according to the formula $B_x = B_0 + bx$, where $B_0$ and $b$ are positive constants, for $x \geq$ 0. A flat coil of area $A$ moves with uniform speed $v$ from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?

Bruce Edelman

Numerade Educator

Problem 12

In many magnetic resonance imaging (MRI) systems, the magnetic field is produced by a superconducting magnet that must be kept cooled below the superconducting transition temperature. If the cryogenic cooling system fails, the magnet coils may lose their superconductivity and the strength of the magnetic field will rapidly decrease, or $quench$. The dissipation of energy as heat in the now-nonsuperconducting magnet coils can cause a rapid boil-off of the cryogenic liquid (usually liquid helium) that is used for cooling. Consider a superconducting MRI magnet for which the magnetic field decreases from 8.0 T to nearly 0 in 20 s. What is the average emf induced in a circular wedding ring of diameter 2.2 cm if the ring is at the center of the MRI magnet coils and the original magnetic field is perpendicular to the plane that is encircled by the ring?

Farhanul Hasan

Numerade Educator

Problem 13

The armature of a small generator consists of a flat, square coil with 120 turns and sides with a length of 1.60 cm. The coil rotates in a magnetic field of 0.0750 T. What is the angular speed of the coil if the maximum emf produced is 24.0 mV?

Bruce Edelman

Numerade Educator

Problem 14

A flat, rectangular coil of dimensions $l$ and $w$ is pulled with uniform speed $v$ through a uniform magnetic field $B$ with the plane of its area perpendicular to the field ($\textbf{Fig. E29.14}$). (a) Find the emf induced in this coil. (b) If the speed and magnetic field are both tripled, what is the induced emf?

Farhanul Hasan

Numerade Educator

Problem 15

A circular loop of wire is in a region of spatially uniform magnetic field, as shown in $\textbf{Fig. E29.15}$. 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 value $B_0$ . Explain your reasoning.

Abhishek Jana

Numerade Educator

Problem 16

The current $I$ in a long, straight wire is constant and is directed toward the right as in $\textbf{Fig. E29.16}$. Conducting loops $A$, $B$, $C$, and $D$ are moving, in the directions shown, near the wire. (a) For each loop, is the direction of the induced current clockwise or counterclockwise, or is the induced current zero? (b) For each loop, what is the direction of the net force that the wire exerts on the loop? Give your reasoning for each answer.

Salamat Ali

Numerade Educator

Problem 17

Two closed loops $A$ and $C$ are close to a long wire carrying a current $I$ ($\textbf{Fig. E29.17}$). (a) Find the direction (clockwise or counterclockwise) of the current induced in each loop if $I$ is steadily decreasing. (b) While $I$ is decreasing, what is the direction of the net force that the wire exerts on each loop? Explain how you obtain your answer.

Bruce Edelman

Numerade Educator

Problem 18

The current in $\textbf{Fig. E29.18}$ obeys the equation $I(t) = I_0 e^{-bt}$, where $b > 0$. Find the direction (clockwise or counterclockwise) of the current induced in the round coil for $t > 0$.

Farhanul Hasan

Numerade Educator

Problem 19

Using Lenz's law, determine the direction of the current in resistor $ab$ of $\textbf{Fig. E29.19}$ 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.

Bruce Edelman

Numerade Educator

Problem 20

A cardboard tube is wrapped with two windings of insulated wire wound in opposite directions, as shown in $\textbf{Fig. E29.20}$. Terminals $a$ and $b$ of winding $A$ may be connected to a battery through a reversing switch. State whether the induced current in the resistor $R$ is from left to right or from right to left in the following circumstances: (a) the current in winding $A$ is from $a$ to $b$ and is increasing; (b) the current in winding $A$ is from $b$ to $a$ and is decreasing; (c) the current in winding $A$ is from $b$ to $a$ and is increasing.

Dading Chen

Numerade Educator

Problem 21

A small, circular ring is inside a larger loop that is connected to a battery and a switch ($\textbf{Fig. E29.21}$). Use Lenz's law to find the direction of the current induced in the small ring (a) just after switch $S$ is closed; (b) after $S$ has been closed a long time; (c) just after $S$ has been reopened after it was closed for a long time.

Bruce Edelman

Numerade Educator

Problem 22

A circular loop of wire with radius $r =$ 0.0480 m and resistance $R =$ 0.160 $\Omega$ is in a region of spatially uniform magnetic field, as shown in $\textbf{Fig. E29.22}$. The magnetic field is directed out of the plane of the figure. The magnetic field has an initial value of 8.00 T and is decreasing at a rate of $dB/dt =$ -0.680 T>s. (a) Is the induced current in the loop clockwise or counterclockwise? (b) What is the rate at which electrical energy is being dissipated by the resistance of the loop?

Farhanul Hasan

Numerade Educator

Problem 23

A circular loop of wire with radius $r =$ 0.0250 m and resistance $R =$ 0.390 $\Omega$ is in a region of spatially uniform magnetic field, as shown in $\textbf{Fig. E29.23}$. The magnetic field is directed into the plane of the figure. At $t = 0, B = 0$. The magnetic field then begins increasing, with $B(t) =$ (0.380 T/$s^3)t^3$. What is the current in the loop (magnitude and direction) at the instant when $B =$ 1.33 T?

Bruce Edelman

Numerade Educator

Problem 24

A rectangular loop of wire with dimensions 1.50 cm by 8.00 cm and resistance $R$ = 0.600 $\Omega$ is being pulled to the right out of a region of uniform magnetic field. The magnetic field has magnitude $B =$ 2.40 T and is directed into the plane of $\textbf{Fig. E29.24}$. At the instant when the speed of the loop is 3.00 m/s and it is still partially in the field region, what force (magnitude and direction)
does the magnetic field exert on the loop?

Vishal Gupta

Numerade Educator

Problem 25

In $\textbf{Fig. E29.25}$ a conducting rod of length $L =$ 30.0 cm moves in a magnetic field $\overrightarrow{B}$ of magnitude 0.450 T directed into the plane of the figure. The rod moves with speed $v =$ 5.00 m/s in the direction shown. (a) What is the potential difference between the ends of the rod? (b) Which point, $a$ or $b$, is at higher potential? (c) When the charges in the rod are in equilibrium, what are the magnitude and direction of the electric field within the rod? (d) When the charges in the rod are in equilibrium, which point, $a$ or $b$, has an excess of positive charge? (e) What is the potential difference across the rod if it moves (i) parallel to $ab$ and (ii) directly out of the page?

Bruce Edelman

Numerade Educator

Problem 26

A rectangle measuring 30.0 cm by 40.0 cm is located inside a region of a spatially uniform magnetic field of 1.25 T, with the field perpendicular to the plane of the coil ($\textbf{Fig. E29.26}$). The coil is pulled out at a steady rate of 2.00 cm/s traveling perpendicular to the field lines. The region of the field ends abruptly as shown. Find the emf induced in this coil when it is (a) all inside the field; (b) partly inside the field; (c) all outside the field.

Farhanul Hasan

Numerade Educator

Problem 27

How fast (in m/s and mph) would a 5.00-cm copper bar have to move at right angles to a 0.650-T magnetic field to generate 1.50 V (the same as a AA battery) across its ends? Does this seem like a practical way to generate electricity?

Bruce Edelman

Numerade Educator

Problem 28

Airplanes and trains move through the earth's magnetic field at rather high speeds, so it is reasonable to wonder whether this field can have a substantial effect on them. We shall use a typical value of 0.50 G for the earth's field. (a) The French TGV train and the Japanese "bullet train" reach speeds of up to 180 mph moving on tracks about 1.5 m apart. At top speed moving perpendicular to the earth's magnetic field, what potential difference is induced across the tracks as the wheels roll? Does this seem large enough to produce noticeable effects? (b) The Boeing 747-400 aircraft has a wingspan of 64.4 m and a cruising speed of 565 mph. If there is no wind blowing (so that this is also their speed relative to the ground), what is the maximum potential difference that could be induced between the opposite tips of the wings? Does this seem large enough to cause problems with the plane?

Farhanul Hasan

Numerade Educator

Problem 29

The conducting rod ab shown in $\textbf{Fig. E29.29}$ makes contact with metal rails ca and $db$. 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 7.50 m/s. (b) In what direction does the current flow in the rod? (c) If the resistance of the circuit $abdc$ is 1.50 $\Omega$ (assumed to be constant), find the force (magnitude and direction) required to keep the rod
moving to the right with a constant speed of 7.50 m/s. You can ignore friction. (d) Compare the rate at which mechanical work is done by the force ($Fv$) with the rate at which thermal energy is developed in the circuit ($I^2R$).

Bruce Edelman

Numerade Educator

Problem 30

A 0.650-m-long metal bar is pulled to the right at a steady 5.0 m/s perpendicular to a uniform, 0.750 T magnetic field. The bar rides on parallel metal rails connected through a 25.0-$\Omega$ resistor ($\textbf{Fig. E29.30}$), so the apparatus makes a complete circuit. Ignore the resistance of the bar and the rails. (a) Calculate the magnitude of the emf induced in the circuit. (b) Find the direction of the current induced in the circuit by using (i) the magnetic force on the charges in the moving bar; (ii) Faraday's law; (iii) Lenz's law. (c) Calculate the current through the resistor.

Vishal Gupta

Numerade Educator

Problem 31

A 0.360-m-long metal bar is pulled to the left by an applied force $F$. The bar rides on parallel metal rails connected through a 45.0-$\Omega$ resistor, as shown in $\textbf{Fig. E29.31}$, so the apparatus makes a complete circuit. You can ignore the resistance of the bar and rails. The circuit is in a uniform 0.650-T magnetic field that is directed out of the plane of the figure. At the instant when the bar is moving to the left at 5.90 m/s, (a) is the induced current in the circuit clockwise or counterclockwise and (b) what is the rate at which the applied force is doing work on the bar?

Bruce Edelman

Numerade Educator

Problem 32

Consider the circuit shown in Fig. E29.31, but with the bar moving to the right with speed v. As in Exercise 29.31, the bar has length 0.360 m, $R$ = 45.0 $\Omega$, and $B =$ 0.650 T. (a) Is the induced current in the circuit clockwise or counterclockwise? (b) At an instant when the 45.0-$\Omega$ resistor is dissipating electrical energy at a rate of 0.840 J/s, what is the speed of the bar?

Zachary Warner

Numerade Educator

Problem 33

A 0.250-m-long bar moves on parallel rails that are connected through a 6.00-$\Omega$ resistor, as shown in $\textbf{Fig. E29.33}$, so the apparatus makes a complete circuit. You can ignore the resistance of the bar and rails. The circuit is in a uniform magnetic field $B =$ 1.20 T that is directed into the plane of the figure. At an instant when the induced current in the circuit is counterclockwise and equal to 1.75 A, what is the velocity of the bar (magnitude and direction)?

Bruce Edelman

Numerade Educator

Problem 34

Blood contains positive and negative ions and thus 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 $\textbf{Fig. E29.34}$.) (a) If the blood vessel is placed in a magnetic field $B$ perpendicular to the vessel, as in the figure, show that the motional potential difference induced across it is $\varepsilon = vBd$. (b) If you expect that the blood will be flowing at 15 cm/s for a vessel 5.0 mm in diameter, what strength of magnetic field will you need to produce a potential difference of 1.0 mV? (c) Show that the volume rate of flow ($R$) of the blood is equal to $R = \pi\varepsilon{d}/4B$. (Note: Although the method developed here is useful in measuring the rate of blood flow in a vessel, it is limited to use in surgery because measurement of the potential $\varepsilon$ must be made directly across the vessel.)

Stanley Enemuo

Numerade Educator

Problem 35

A rectangular circuit is moved at a constant velocity of 3.0 m/s into, through, and then out of a uniform 1.25-T magnetic field, as shown in $\textbf{Fig. E29.35}$. The magnetic-field region is considerably wider than 50.0 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.

Keshav Singh

Numerade Educator

Problem 36

A metal ring 4.50 cm in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic field. These magnets produce an initial uniform field of 1.12 T between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.250 T/s. (a) What is the magnitude of the electric field induced in the ring? (b) In which direction (clockwise or counterclockwise) does the current flow as viewed by someone on the south pole of the magnet?

Salamat Ali

Numerade Educator

Problem 37

The magnetic field within a long, straight solenoid with a circular cross section and radius $R$ is increasing at a rate of $dB/dt$. (a) What is the rate of change of flux through a circle with radius $r_1$ inside the solenoid, normal to the axis of the solenoid, and with center on the solenoid axis? (b) Find the magnitude of the induced electric field inside the solenoid, at a distance $r_1$ from its axis. Show the direction of this field in a diagram. (c) What is the magnitude of the induced electric field $outside$ the solenoid, at a distance $r_2$ from the axis? (d) Graph the magnitude of the induced electric field as a function of the distance $r$ from the axis from $r =$ 0 to $r = 2R$. (e) What is the magnitude of the induced emf in a circular turn of radius R/2 that has its center on the solenoid axis? (f) What is the magnitude of the induced emf if the radius in part (e) is $R$? (g) What is the induced emf if the radius in part (e) is 2$R$?

Keshav Singh

Numerade Educator

Problem 38

A long, thin solenoid has 900 turns per meter and radius 2.50 cm. The current in the solenoid is increasing at a uniform rate of 36.0 A/s. What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 cm from the axis of the solenoid; (b) 1.00 cm from the axis of the solenoid?

Zachary Warner

Numerade Educator

Problem 39

A long, thin solenoid has 400 turns per meter and radius 1.10 cm. The current in the solenoid is increasing at a uniform rate $di/dt$. The induced electric field at a point near the center of the solenoid and 3.50 cm from its axis is 8.00 $\times$ 10$^{-6}$ V/m. Calculate $di/dt$.

Keshav Singh

Numerade Educator

Problem 40

The magnetic field $\overrightarrow{B}$ at all points within the colored circle shown in Fig. E29.15 has an initial magnitude of 0.750 T. (The circle could represent approximately the space inside a long, thin solenoid.) The magnetic field is directed into the plane of the diagram and is decreasing at the rate of -0.0350 T/s. (a) What is the shape of the field lines of the induced electric field shown in Fig. E29.15, within the colored circle? (b) What are the magnitude and direction of this field at any point on the circular conducting ring with radius 0.100 m? (c) What is the current in the ring if its resistance is 4.00 $\Omega$? (d) What is the emf between points $a$ and $b$ on the ring? (e) If the ring is cut at some point and the ends are separated slightly, what will be the emf between the ends?

Salamat Ali

Numerade Educator

Problem 41

A long, straight solenoid with a cross-sectional area of 8.00 cm$^2$ is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 A. A second winding of 12 turns encircles the solenoid at its center. The current in the solenoid is turned off such that the magnetic field of the solenoid becomes zero in 0.0400 s. What is the average induced emf in the second winding?

Keshav Singh

Numerade Educator

Problem 42

A parallel-plate, air-filled capacitor is being charged as in Fig. 29.23. The circular plates have radius 4.00 cm, and at a particular instant the conduction current in the wires is 0.520 A. (a) What is the displacement current density $j_D$ in the air space between the plates? (b) What is the rate at which the electric field between the plates is changing? (c) What is the induced magnetic field between the plates at a distance of 2.00 cm from the axis? (d) At 1.00 cm from the axis?

Stanley Enemuo

Numerade Educator

Problem 43

Suppose that the parallel plates in Fig. 29.23 have an area of 3.00 cm$^2$ and are separated by a 2.50-mm-thick sheet of dielectric that completely fills the volume between the plates. The dielectric has dielectric constant 4.70. (You can ignore fringing effects.) At a certain instant, the potential difference between the plates is 120 V and the conduction current $i_C$ equals 6.00 mA. At this instant, what are (a) the charge $q$ on each plate; (b) the rate of change of charge on the plates; (c) the displacement current in the dielectric?

Keshav Singh

Numerade Educator

Problem 44

In Fig. 29.23 the capacitor plates have area 5.00 cm2 and separation 2.00 mm. The plates are in vacuum. The charging current $i_C$ has a $constant$ value of 1.80 mA. At $t$ = 0 the charge on the plates is zero. (a) Calculate the charge on the plates, the electric field between the plates, and the potential difference between the plates when $t =$ 0.500 $\mu$s. (b) Calculate $dE/dt$, the time rate of change of the electric field between the plates. Does $dE/dt$ vary in time? (c) Calculate the displacement current density $j_D$ between the plates, and from this the total displacement current $i_D$. How do $i_C$ and $i_D$ compare?

Salamat Ali

Numerade Educator

Problem 45

At temperatures near absolute zero, $B_c$ approaches 0.142 T for vanadium, a type-I superconductor. The normal phase of vanadium has a magnetic susceptibility close to zero. Consider a long, thin vanadium cylinder with its axis parallel to an external magnetic field $\overrightarrow{B}_0$ in the +$x$-direction. At points far from the ends of the cylinder, by symmetry, all the magnetic vectors are parallel to the x-axis. At temperatures near absolute zero, what are the resultant magnetic field $\overrightarrow{B}$ and the magnetization $\overrightarrow{M}$ inside and outside the cylinder (far from the ends) for (a) $\overrightarrow{B}_0$ = (0.130 T)$\hat{\imath}$ and (b) $\overrightarrow{B}_0$ = (0.260 T)$\hat{\imath}$ ?

Bruce Edelman

Numerade Educator

Problem 46

A very long, rectangular loop of wire can slide without friction on a horizontal surface. Initially the loop has part of its area in a region of uniform magnetic field that has magnitude $B =$ 2.90 T and is perpendicular to the plane of the loop. The loop has dimensions 4.00 cm by 60.0 cm, mass 24.0 g, and resistance $R =$ 5.00 $\times$ 10$^{-3} \Omega$. The loop is initially at rest; then a constant force $F_{ext}$ = 0.180 N is applied to the loop to pull it out of the field (Fig. P29.46). (a) What is the acceleration of the loop when $v =$ 3.00 cm/s? (b) What are the loop's terminal speed and acceleration when the loop is moving at that terminal speed? (c) What is the acceleration of the loop when it is completely out of the magnetic field?

Zachary Warner

Numerade Educator

Problem 47

In the circuit shown in $\textbf{Fig. P29.47}$, the capacitor has capacitance $C =$ 20 $\mu$F and is initially charged to 100 V with the polarity shown. The resistor $R_0$ has resistance 10 $\Omega$. At time $t =$ 0 the switch $S$ is closed. The small circuit is not connected in any way to the large one. The wire of the small circuit has a resistance of 1.0 $\Omega$/m and contains 25 loops. The large circuit is a rectangle 2.0 m by 4.0 m, while the small one has dimensions $a =$ 10.0 cm and $b =$ 20.0 cm. The distance c is 5.0 cm. (The figure is not drawn to scale.) Both circuits are held stationary. Assume that only the wire nearest the small circuit produces an appreciable magnetic field through it. (a) Find the current in the large circuit 200 $\mu$s after S is closed. (b) Find the current in the small circuit 200 $\mu$s after S is closed. ($Hint:$ See Exercise 29.7.) (c) Find the direction of the current in the small circuit. (d) Justify why we can ignore the magnetic field from all the wires of the large circuit except for the wire closest to the small circuit.

Keshav Singh

Numerade Educator

Problem 48

In the circuit in Fig. P29.47, an emf of 90.0 V is added in series with the capacitor and the resistor, and the capacitor is initially uncharged. The emf is placed between the capacitor and switch $S$, with the positive terminal of the emf adjacent to the capacitor. Otherwise, the two circuits are the same as in Problem 29.47. The switch is closed at $t =$ 0. When the current in the large circuit is 5.00 A, what are the magnitude and direction of the induced current in the small circuit?

Zachary Warner

Numerade Educator

Problem 49

A very long, straight solenoid with a crosssectional area of 2.00 cm$^2$ is wound with 90.0 turns of wire per centimeter. Starting at t = 0, the current in the solenoid is increasing according to $i(t) = (0.160 A/s^2)t^2$. A secondary winding of 5 turns encircles the solenoid at its center, such that the secondary winding has the same cross-sectional area as the solenoid. What is the magnitude of the emf induced in the secondary winding at the instant that the current in the solenoid is 3.20 A?

Keshav Singh

Numerade Educator

Problem 50

Suppose the loop in $\textbf{Fig. P29.50}$ is (a) rotated about the $y$-axis; (b) rotated about the x-axis; (c) rotated about an edge parallel to the $z$-axis. What is the maximum induced emf in each case if $A =$ 600 cm$^2$, $\omega$ = 35.0 rad/s, and $B =$ 0.320 T?

Zachary Warner

Numerade Educator

Problem 51

In $\textbf{Fig. P29.51}$ the loop is being pulled to the right at constant speed $v$. A constant current $I$ flows in the long wire, in the direction shown. (a) Calculate the magnitude of the net emf $\varepsilon$ induced in the loop. Do this two ways: (i) by using Faraday's law of induction ($Hint:$ See Exercise 29.7) and (ii) by looking at the emf induced in each segment of the loop due to its motion. (b) Find the direction (clockwise or counterclockwise) of the current induced in the loop. Do this two ways: (i) using Lenz's law and (ii) using the magnetic force on charges in the loop. (c) Check your answer for the emf in part (a) in the following special cases to see whether it is physically reasonable: (i) The loop is stationary; (ii) the loop is very thin, so $a \rightarrow$ 0; (iii) the loop gets very far from the wire.

Bruce Edelman

Numerade Educator

Problem 52

You are shipwrecked on a deserted tropical island. You have some electrical devices that you could operate using a generator but you have no magnets. The earth's magnetic field at your location is horizontal and has magnitude 8.0 $\times$ 10$^{-5}$ T, and you decide to try to use this field for a generator by rotating a large circular coil of wire at a high rate. You need to produce a peak emf of 9.0 V and estimate that you can rotate the coil at 30 rpm by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum number of turns the coil can have is 2000. (a) What area must the coil have? (b) If the coil is circular, what is the maximum translational speed of a point on the coil as it rotates? Do you think this device is feasible? Explain.

Farhanul Hasan

Numerade Educator

Problem 53

A flexible circular loop 6.50 cm in diameter lies in a magnetic field with magnitude 1.35 T, directed into the plane of the page as shown in $\textbf{Fig. P29.53}$. The loop is pulled at the points indicated by the arrows, forming a loop of zero area in 0.250 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.

Bruce Edelman

Numerade Educator

Problem 54

A conducting rod with length $L =$ 0.200 m, mass $m =$ 0.120 kg, and resistance $R =$ 80.0 $\Omega$ moves without friction on metal rails as shown in Fig. 29.11. A uniform magnetic field with magnitude $B =$ 1.50 T is directed into the plane of the figure. The rod is initially at rest, and then a constant force with magnitude $F =$ 1.90 N and directed to the right is applied to the rod. How many seconds after the force is applied does the rod reach a speed of 25.0 m/s?

Salamat Ali

Numerade Educator

Problem 55

A very long, cylindrical wire of radius $R$ carries a current $I_0$ uniformly distributed across the cross
section of the wire. Calculate the magnetic flux through a rectangle that has one side of length $W$ running down the center of the wire and another side of length $R$, as shown in $\textbf{Fig. P29.55}$ (see Exercise 29.7).

Bruce Edelman

Numerade Educator

Problem 56

A bar of length $L =$ 0.36 m is free to slide without friction on horizontal rails as shown in $\textbf{Fig. P29.56}$. A uniform magnetic field $B =$ 2.4 T is directed into the plane of the figure. At one end of the rails there is a battery with emf $\varepsilon =$ 12 V and a switch S. The bar has mass 0.90 kg and resistance 5.0 $\Omega$; ignore all other resistance in the circuit. The
switch is closed at time $t =$ 0. (a) Sketch the bar's speed as a function of time. (b) Just after the switch is closed, what is the acceleration of the bar? (c) What is the acceleration of the bar when its speed is 2.0 m/s? (d) What is the bar's terminal speed?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

The long, straight wire shown in $\textbf{Fig. P29.57a}$ carries constant current $I$. A metal bar with length $L$ is moving at constant velocity $\vec{v}$, as shown in the figure. Point a is a distance $d$ from the wire. (a) Calculate the emf induced in the bar. (b) Which point, $a$ or $b$, is at higher potential? (c) If the bar is replaced by a rectangular wire loop of resistance $R$ (Fig. P29.57b), what is the magnitude of the current induced in the loop?

Bruce Edelman

Numerade Educator

Problem 58

A circular conducting ring with radius $r_0 =$ 0.0420 m lies in the xy-plane in a region of uniform magnetic field $\overrightarrow{B} = B_0 [1 - 3(t/t_0)^2 + 2(t/t_0)^3]\hat{k}$. In this expression, $t_0 =$ 0.0100 s and is constant, $t$ is time, $\hat{k}$ is the unit vector in the +$z$-direction, and $B_0$ = 0.0800 T and is constant. At points $a$ and $b$ (Fig. P29.58) there is a small gap in the ring with wires leading to an external circuit of resistance $R =$ 12.0 $\Omega$. There is no magnetic field at the location of the external circuit. (a) Derive an expression, as a function of time, for the total magnetic flux $\Phi_B$ through the ring. (b) Determine the emf induced in the ring at time $t =$ 5.00 $\times$ 10$^{-3}$ s. What is the polarity of the emf? (c) Because of the internal resistance of the ring, the current through $R$ at the time given in part (b) is only 3.00 mA. Determine the internal resistance of the ring. (d) Determine the emf in the ring at a time $t =$ 1.21 $\times$ 10$^{-2}$ s. What is the polarity of the emf? (e) Determine the time at which the current through $R$ reverses its direction.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 59

A slender rod, 0.240 m long, rotates with an angular speed of 8.80 rad/s about an axis through one end and perpendicular to the rod. The plane of rotation of the rod is perpendicular to a uniform magnetic field with a magnitude of 0.650 T. (a) What is the induced emf in the rod? (b) What is the potential difference between its ends? (c) Suppose instead the rod rotates at 8.80 rad/s about an axis through its center and perpendicular to the rod. In this case, what is the potential difference between the ends of the rod? Between the center of the rod and one end?

Bruce Edelman

Numerade Educator

Problem 60

A 25.0-cm-long metal rod lies in the $xy$-plane and makes an angle of 36.9$^\circ$ with the positive $x$-axis and an angle of 53.1$^\circ$ with the positive $y$-axis. The rod is moving in the $+x$-direction with a speed of 6.80 m/s. The rod is in a uniform magnetic field $\overrightarrow{B} =$ (0.120 T)$\hat{\imath}$ - (0.220 T)$\hat{\jmath}$ - (0.0900 T)$\hat{k}$. (a) What is the magnitude of the emf induced in the rod? (b) Indicate in a sketch which end of the rod is at higher potential.

Sandro Maludze

Numerade Educator

Problem 61

A rectangular loop with width $L$ and a slide wire with mass m are as shown in $\textbf{Fig. P29.61}$. A uniform magnetic field $\overrightarrow{B}$ is directed perpendicular to the plane of the loop into the plane of the figure. The slide wire is given an initial speed of $v_0$ and then released. There is no friction between the slide wire and the loop, and the resistance of the loop is negligible in comparison to the resistance $R$ of the slide wire. (a) Obtain an expression for $F$, the magnitude of the force exerted on the wire while it is moving at speed $v$. (b) Show that the distance $x$ that the wire moves before coming to rest is $x = mv_0R/L^2B^2$.

Bruce Edelman

Numerade Educator

Problem 62

An airplane propeller of total length $L$ rotates around its center with angular speed $\omega$ in a magnetic field that is perpendicular to the plane of rotation. Modeling the propeller as a thin, uniform bar, find the potential difference between (a) the center and either end of the propeller and (b) the two ends. (c) If the field is the earth's field of 0.50 G and the propeller turns at 220 rpm and is 2.0 m long, what is the potential difference between the middle and either end? It this large enough to be concerned about?

Farhanul Hasan

Numerade Educator

Problem 63

The magnetic field $\overrightarrow{B}$, at all points within a circular region of radius $R$, is uniform in space and directed into the plane of the page as shown in $\textbf{Fig. P29.63}$. (The region could be a cross section inside the windings of a long, straight solenoid.) If the magnetic field is increasing at a rate $dB/dt$, what are the magnitude and direction of the force on a stationary positive point charge $q$ located at points $a$, $b$, and $c$? (Point a is a distance r above the center of the region, point $b$ is a distance $r$ to the right of the center, and point $c$ is at the center of the region.)

Bruce Edelman

Numerade Educator

Problem 64

A capacitor has two parallel plates with area A separated by a distance $d$. The space between plates is filled with a material having dielectric constant $K$. The material is not a perfect insulator but has resistivity $\rho$. The capacitor is initially charged with charge of magnitude Q0 on each plate that gradually discharges by conduction through the dielectric. (a) Calculate the conduction current density $j_C(t)$ in the dielectric. (b) Show that at any instant the displacement current density in the dielectric is equal in magnitude to the conduction current density but opposite in direction, so the total current density is zero at every instant.

Salamat Ali

Numerade Educator

Problem 65

A dielectric of permittivity 3.5 $\times$ 10$^{-11}$ F/m completely fills the volume between two capacitor plates. For t 7 0 the electric flux through the dielectric is (8.0 $\times$ 10$^3$ V $\cdot$ m$/s^3)t^3$. The dielectric is ideal and nonmagnetic; the conduction current in the dielectric is zero. At what time does the displacement current in the dielectric equal 21 $\mu$A ?

Bruce Edelman

Numerade Educator

Problem 66

You are evaluating the performance of a large electromagnet. The magnetic field of the electromagnet is zero at $t =$ 0 and increases as the current through the windings of the electromagnet is increased. You determine the magnetic field as a function of time by measuring the time dependence of the current induced in a small coil that you insert between the poles of the electromagnet, with the plane of the coil parallel to the pole faces as in Fig. 29.5. The coil has 4 turns, a radius of 0.800 cm, and a resistance of 0.250 $\Omega$. You measure the current i in the coil as a function of time $t$. Your results are shown in $\textbf{Fig. P29.66}$. Throughout your measurements, the current induced in the coil remains in the same direction. Calculate the magnetic field at the location of the coil for (a) $t =$ 2.00 s, (b) $t =$ 5.00 s, and (c) $t =$ 6.00 s.

Salamat Ali

Numerade Educator

Problem 67

You are conducting an experiment in which a metal bar of length 6.00 cm and mass 0.200 kg slides without friction on two parallel metal rails ($\textbf{Fig. P29.67}$). A resistor with resistance $R =$ 0.800 $\Omega$ is connected across one end of the rails so that the bar, rails, and resistor form a complete conducting path. The resistances of the rails and of the bar are much less than $R$ and can be ignored. The entire apparatus is in a uniform magnetic field $\overrightarrow{B}$ that is directed into the plane of the figure. You give the bar an initial velocity $v =$ 20.0 cm/s to the right and then release it, so that the only force on the bar then is the force exerted by the magnetic field. Using high-speed photography, you measure the magnitude of the acceleration of the bar as a function of its speed. Your results are given in the table:

(a) Plot the data as $a$ graph of a versus $v$. Explain why the data points plotted this way lie close to a straight line, and determine the slope of the best-fit straight line for the data. (b) Use your graph from part (a) to calculate the magnitude $B$ of the magnetic field. (c) While the bar is moving, which end of the resistor, $a$ or $b$, is at higher potential? (d) How many seconds does it take the speed of the bar to decrease from 20.0 cm/s to 10.0 cm/s?

Bruce Edelman

Numerade Educator

Problem 68

You measure the magnitude of the external force $\overrightarrow{F}$ that must be applied to a rectangular conducting loop to pull it at constant speed v out of a region of uniform magnetic field $\overrightarrow{B}$ that is directed out of the plane of $\textbf{Fig. P29.68}$. The loop has dimensions 14.0 cm by 8.00 cm and resistance 4.00 $\times$ 10$^{-3}$ $\Omega$; it does not change shape as it moves. The measurements you collect are listed in the table.

(a) Plot the data as a graph of $F$ versus $v$. Explain why the data points plotted this way lie close to a straight line, and determine the slope of the best-fit straight line for the data. (b) Use your graph from part (a) to calculate the magnitude $B$ of the uniform magnetic field. (c) In Fig. P29.68, is the current induced in the loop clockwise or counterclockwise? (d) At what rate is electrical energy being dissipated in the loop when the speed of the loop is
5.00 cm/s?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 69

A metal bar with length $L$, mass $m$, and resistance $R$ is placed on frictionless metal rails that are inclined at an angle $\phi$ above the horizontal. The rails have negligible resistance. A uniform magnetic field of magnitude $B$ is directed downward as shown in $\textbf{Fig. P29.69}$. The bar is released from rest and slides down the rails. (a) Is the direction of the current induced in the bar from $a$ to $b$ or from $b$ to $a$? (b) What is the terminal speed of the bar? (c) What is the induced current in the bar when the terminal speed has been reached? (d) After the terminal speed has been reached, at what rate is electrical energy being converted to thermal energy in the resistance of the bar? (e) After the terminal speed has been reached, at what rate is work being done on the bar by gravity? Compare your answer to that in part (d).

Bruce Edelman

Numerade Educator

Problem 70

A square, conducting, wire loop of side L, total mass m, and total resistance R initially lies in the horizontal xy-plane, with corners at ($x, y, z$) = (0, 0, 0), (0, $L$, 0), ($L$, 0, 0), and ($L, L$, 0). There is a uniform, upward magnetic field $\overrightarrow{B}$ = B$\hat{k}$ in the space within and around the loop. The side of the loop that extends from (0, 0, 0) to ($L$, 0, 0) is held in place on the $x$-axis;
the rest of the loop is free to pivot around this axis. When the loop is released, it begins to rotate due to the gravitational torque. (a) Find the $net$ torque (magnitude and direction) that acts on the loop when it has rotated through an angle $\phi$ from its original orientation and is rotating downward at an angular speed $\omega$. (b) Find the angular acceleration of the loop at the instant described in part (a). (c) Compared to the case with zero magnetic field, does it take the loop a longer or shorter time to rotate through 90$^\circ$ ? Explain. (d) Is mechanical energy conserved as the loop rotates
downward? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 71

In part (a) of the figure, 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.

Bruce Edelman

Numerade Educator

Problem 72

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.

Zachary Warner

Numerade Educator

Problem 73

It may be desirable to increase the maximum induced current in the brain tissue. In $\textbf{Fig. P29.73}$, which time-dependent graph of the magnetic field $B$ in the coil achieves that goal? Assume that everything else remains constant. (a) A; (b) B; (c) either A or B; (d) neither A nor B.

Bruce Edelman

Numerade Educator

Problem 74

Which graph in $\textbf{Fig. P29.74}$ 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.) (a) A; (b) B; (c) C; (d) D.

Zachary Warner

Numerade Educator