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University Physics with Modern Physics

Hugh D. Young, Roger A. Freeman

Chapter 29

Electromagnetic Induction - all with Video Answers

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Chapter Questions

01:44

Problem 1

A flat, rectangular coil consisting of 50 tums measures 25.0 $\mathrm{cm}$ by 30.0 $\mathrm{cm}$ . It is in a uniform, $1.20-\mathrm{T}$ , magnetic field, with the plane of the coil parallel to the field. In 0.222 s, it is rotated so that the plane of the coil is perpendicular to the field. (a) What is the change in the magnetic flux through the coil due to this rotation? (b) Find the magnitude of the average emf induced in the coil during this rotation.

Keshav Singh
Keshav Singh
Numerade Educator
04:16

Problem 2

In a physics laboratory experiment, a coil with 200 tums enclosing an area of 12 $\mathrm{cm}^{2}$ 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 $6.0 \times 0^{-5} \mathrm{T}$ . (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?

Keshav Singh
Keshav Singh
Numerade Educator
03:50

Problem 3

Search Coils and Credit Cards. (a) Derive the equation relating the total charge $Q$ that flows through a search coil (Conceptual Example 29.3$)$ to the magnetic-field magnitude $B$ . The search

Keshav Singh
Keshav Singh
Numerade Educator
02:07

Problem 4

A closely wound search coil (Exercise 29.3) has an area of $3.20 \mathrm{cm}^{2}, 120$ 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 $3.56 \times 10^{-5} \mathrm{C}$ . What is the magnitude of the field?

Keshav Singh
Keshav Singh
Numerade Educator
04:54

Problem 5

A circular loop of wire with a radius of 12.0 $\mathrm{cm}$ and oriented in the horizontal $x y$ -plane is located in a region of uniform magnetic field. A field of 1.5 $\mathrm{T}$ 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 $\mathrm{ms}$ , 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?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:02

Problem 6

A coil 4.00 $\mathrm{cm}$ in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to $B=(0.0120 \mathrm{T} / \mathrm{s}) t+\left(3.00 \times 10^{-5} \mathrm{T} / \mathrm{s}^{4}\right) t^{4} .$ 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 $t=5.00 \mathrm{s} ?$

Farhanul Hasan
Farhanul Hasan
Numerade Educator
04:31

Problem 7

The current in the long, straight wire $A B$ shown in Fig. 29.27 is upward and is increasing steadily at a rate di/dt. (a) At an instant when the cur- rent is $i$ , what are the magnitude and direction of the field $\vec{B}$ 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 induceemfif $a=12.0 \mathrm{cm}, b=36.0 \mathrm{cm},$
$L=24.0 \mathrm{cm},$ and $d i / d t=9.60 \mathrm{A} / \mathrm{s}$ .

Keshav Singh
Keshav Singh
Numerade Educator
06:45

Problem 8

A flat, circular, steel loop of radius 75 $\mathrm{cm}$ is at rest in a uniform magnetic fleld, as shown in an cdgeon view in Fig, 29.28 . The field is changing with time, according to $B(t)=(1.4 \mathrm{T}) e^{-\left(0.057 \mathrm{s}^{-1}\right)} \cdot(\mathrm{a})$ Find zhe emf induced in the loop as a
function of time. (b) When is the induced emf equal to 10 of its initial value? (c) Find the direction of
the current induced in the loop, as viewed from above the loop.

Keshav Singh
Keshav Singh
Numerade Educator
04:12

Problem 9

Shrinking Loop. A circular loop of flexible iron wire has an initial circunference of $165.0 \mathrm{cm},$ but its circunference is decreasing at a constant rate of 12.0 $\mathrm{cm} / \mathrm{s}$ due to a tangential pull on the wire. The loop is in a constant, viniform magnetic field oriented perpendicular to the plane of the loop and with magnitude 0.500 $\mathrm{T}$ . (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.

Keshav Singh
Keshav Singh
Numerade Educator
03:14

Problem 10

A rectangle measuring 30.0 $\mathrm{cm}$ by 40.0 $\mathrm{cm}$ is located inside a region of a spatially uniform magnetic field of 1.25 $\mathrm{T}$ , with the field perpendicular to the plane of the coil
(Fig. 29.29 ). The coil is pulled out at a steady rate of 2.00 $\mathrm{cm} / \mathrm{s}$ 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
Farhanul Hasan
Numerade Educator
02:07

Problem 11

In a region of space, a magnetic ficld points in the $+x$ -direction (toward the right). Its magnitude varies with position according to the formula $B_{x}=B_{0}+b x,$ 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)?

Keshav Singh
Keshav Singh
Numerade Educator
02:59

Problem 12

Back emf. A motor with a brush-and-commutator arrangement, as described in Example $29.5,$ has a circular coil with radius 2.5 $\mathrm{cm}$ and 150 turns of wire. The magnetic field has magnitude
0.060 $\mathrm{T}$ , and the coil rotates at 440 $\mathrm{rev} / \mathrm{min}$ (a) What is the maximum emf induced in the coil? (b) What is the average back emf?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:44

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 $\mathrm{cm}$ . The coil rotates in a magnetic fleld of 0.0750 T. What is the angular speed
of the coil if the maximum emf produced is 24.0 $\mathrm{mV} ?$

Keshav Singh
Keshav Singh
Numerade Educator
01:18

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 induced in this coil. (b) If the spced and
magnetic field ure both tripled, what is the induced emf?to the field (Fig. 29.30$)$ . (a) Find the emf

Keshav Singh
Keshav Singh
Numerade Educator
02:12

Problem 15

A circular loop of wire is in a region of spatially uniform magnetic field, as shown in Fig. 29.31 . The magnetic field is directed into the plane of the figure. Determine the directioure. Determine the direction
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.

Keshav Singh
Keshav Singh
Numerade Educator
00:22

Problem 16

The current in Fig. 29.32 obeys the equation $I(t)=I_{0} e^{-\omega t},$ where $b>0 .$ Find the dircction (clockwise or counterclockwise) of the current induced in the round coil for $t>0$ .

Keshav Singh
Keshav Singh
Numerade Educator
05:33

Problem 17

Using Lenz's law, determine direction of the current in resistor ab Fig. 29.33 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
Bruce Edelman
Numerade Educator
01:07

Problem 18

A cardboard tube is wrapped with two windings of insulated wire wound in oppositc dircctions, as shown in Fig. 29.34 . Terminals $a$ and $b$ of winding $A$ may be connected to a battery through a reversing switch. State whether

Keshav Singh
Keshav Singh
Numerade Educator
02:29

Problem 19

A small, circular ring is inside a larger loop that is connected to a battery and a switch, as shown in Fig 29.35 . Use Len'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 being
closed a long time.

Keshav Singh
Keshav Singh
Numerade Educator
03:45

Problem 20

A 1.50 -m-long metal bar is pulled to the right at a steady 5.0 $\mathrm{m} / \mathrm{s}$ perpendicular to a uniform, 0.750 - $\mathrm{T}$ magnetic field. The bar rides on parallel metal rails connected through a $25.0-\Omega$ resistor, as shown in $\mathrm{Fig}$ . 29.36 , so the apparatus makes a complete circuli. You can 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 (i) using the magnetic force on the charges in the moving bar; (i) using Faraday's law; (iii) using Lenz's law. (c) Calculate the current through the resistor.

Keshav Singh
Keshav Singh
Numerade Educator
05:59

Problem 21

In Fig. 29.37 a conducting rod of length $L=30.0 \mathrm{cm}$ moves in a magnetic field $\vec{B}$ of magnitude 0.450 T directed into the plane of the figure. The md moves with speed $v=5.00 \mathrm{m} / \mathrm{s}$ in the direction shown. (a) what is the potential difference between the ends of the rod?
(b) Which point, $a$ to $b,$ 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 change? (c) What is the potential difference arruss the rud if it moves (i) parallel to $a b$ and (ii) directly out of the page?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:25

Problem 22

For the situation in Exercise 29.20 , find (a) the motional emf in the bar and $(b)$ the current through the resistor.

Keshav Singh
Keshav Singh
Numerade Educator
01:58

Problem 23

Are Motional emfs a Practical Source of Electricity? How fast (in $\mathrm{m} / \mathrm{s}$ and $\mathrm{mph}$ ) would a $5.00-\mathrm{cm}$ copper bar have to move at right angles to a 0.650 - $T$ magnetic field to generate 1.50 $\mathrm{V}$ (the same as a AA battery) across its cnds? Does this scem like a practical way to generate electricity?

Keshav Singh
Keshav Singh
Numerade Educator
08:49

Problem 24

Motional emfs in Transportation. 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 $\mathrm{G}$ for the earth's field (a) The French TGV train and the Japanese "bullet train" reach speeds of up to 180 $\mathrm{mph}$ moving on tracks about 1.5 $\mathrm{m}$ 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 $\mathrm{m}$ and a cruising speed of 565 $\mathrm{mph}$ . 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?

JK
Jacob Kim
Numerade Educator
05:35

Problem 25

The conducting rod ab shown in Fig. 298 makes 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 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 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 $\mathrm{m} / \mathrm{s}$ . You can ignore friction. (d) Compare the rate at which mechanical work is done by the force $(F v)$ with the rate at which thermal cnergy is developed in the circuit $\left(I^{2} R\right) .$

Keshav Singh
Keshav Singh
Numerade Educator
04:11

Problem 26

A square loop of wire with side length $L$ and resistance $R$ is moved at constant speed $v$ across a uniform magnetic field confined to a square region whose sides are twice the length of those of the square loop (Fig. 29.39$)$ . (a) Graph the external force $F$ needed to move the loop at constant speed as a function of the coordinate $x$ from $x=-2 L$ to $x=+2 L$ . (The coordinate $x$ is measured from the center of the magnetic-field region to the center of the loop. It is negative when the center of the loop is to the left of the center of the magnetic-field region. Take positive force to be w the right. (b) Graph the induced current in the loop as a function of $x$ . Take counterclockwise currents to be positive.

Suhas Katkar
Suhas Katkar
Numerade Educator
04:02

Problem 27

A $1.41-\mathrm{m}$ bar moves through a uniform, 1.20 . $T$ magnetic field with a speed of 2.50 $\mathrm{m} / \mathrm{s}$ (Fig, 29.40$)$ . In cach case, find the emf induced between the ends of this bar and identify which, if any, end $(a \text { or } b)$ is at the higher potential. The bar moves in the direction of (a) the $+x$ -axis; (b) the $-y$ -axis; (c) the $+z$ -axis. (d) How should this bar move so that the emf across its ends has the greatest possible value with $b$ at a higher potential than $a$ , and what is this maximum emf?

Keshav Singh
Keshav Singh
Numerade Educator
View

Problem 28

A long, thin solenoid has 900 turns per meter and radius $2.50 \mathrm{cm} .$ The current in the solenoid is increasing at a uniform rate of 60.0 $\mathrm{A} / \mathrm{s}$ . What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 $\mathrm{cm}$ from the axis of the solenoid; (b) 1.00 $\mathrm{cm}$ from the axis of the solenoid?

Gage Bonner
Gage Bonner
Numerade Educator
14:11

Problem 29

The magnetic field within a long, straight solenoid with a eireular cross section and radius $R$ is increasing at a rate of $d B / d t .$ (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 $r=2 R$ . (e) What is the magnitude of the induced emf in circular turn of radius $R / 2$ that has its center on the solenoid axis? (f) What is the magnitude of the indnced emf if the radins in part (e) is $R ?(g)$ What is the induced emf if the radius in part $(e)$ is 2$R ?$

Bradley Nordell
Bradley Nordell
Numerade Educator
06:16

Problem 30

The magnetic field $\overrightarrow{\boldsymbol{B}}$ at all points within the colored circle shown in Fig. 29.31 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 $-0.0350 \mathrm{T} / \mathrm{s}$ . (a) What is the shape of the field lines of the induced electric field shown in Fig. $29.31,$ 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 $\mathrm{m} ?$ (c) What is the current in the ring if its resistance is 4.00$\Omega ?(\mathrm{d})$ 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?

Ajay Singhal
Ajay Singhal
Numerade Educator
03:09

Problem 31

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

Keshav Singh
Keshav Singh
Numerade Educator
02:29

Problem 32

A metal ring 4.50 $\mathrm{cm}$ in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic ficld. 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 $\mathrm{T} / \mathrm{s}$ (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?

Keshav Singh
Keshav Singh
Numerade Educator
03:06

Problem 33

A long, straight solenoid with a cross-sectional area of 8.00 $\mathrm{cm}^{2}$ is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 $\mathrm{A}$ . A second winding of 12 tums 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 indnced emf in the second winding?

Keshav Singh
Keshav Singh
Numerade Educator
01:54

Problem 34

A diclectric of permitivity $3.5 \times 10^{-11} \mathrm{F} / \mathrm{m}$ completely fills the volume between two capacitor plates. For $t > 0$ the electric flux through the dielectric is $\left(8.0 \times 10^{3} \mathrm{V} \cdot \mathrm{m} / \mathrm{s}^{3}\right) t^{3}$ . 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$\mu \mathrm{A} ?$

Keshav Singh
Keshav Singh
Numerade Educator
02:49

Problem 35

The electric flux through a certain area of a dielectric is $\left(8.76 \times 10^{3} \mathbf{V} \cdot \mathrm{m} / \mathrm{s}^{4}\right) t^{4} .$ The displacement current through that area is 12.9 $\mathrm{pA}$ at time $t=26.1 \mathrm{ms}$ . Calculate the dielectric constant for the diclectric.

Keshav Singh
Keshav Singh
Numerade Educator
06:54

Problem 36

A parallel-plate, air-filled capacitur is being charged as in Fig. 29.23 . The circular plates have radius $4.00 \mathrm{cm},$ and at a particular instant the conduction current in the wires is 0.280 $\mathrm{A}$ . (a) What- is the displacement current density $j_{V}$ in the air space between the plates is changing? (c) What is the induced magnetic field between the plates at a distance of 2.00 $\mathrm{cm}$ from the axis? (d) At 1.00 $\mathrm{cm}$ from the axis?

Keshav Singh
Keshav Singh
Numerade Educator
03:28

Problem 37

Displacement Current in a Dielectric. Suppose that the parallel plates in Fig, 29.23 have an area of 3.00 $\mathrm{cm}^{2}$ 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 $\mathrm{V}$ and the con- duction current $t_{\mathrm{C}}$ equals 6.00 $\mathrm{mA}$ . At this instant, what are (a) the
charge $q$ on each plate; $(\mathrm{b})$ the rate of change of charge on the plates; (c) the displacement current in the dielectric?

Keshav Singh
Keshav Singh
Numerade Educator
09:15

Problem 38

In Fig. 29.23 the capacitor plates have area 5.00 $\mathrm{cm}^{2}$ and scparation 2.00 $\mathrm{mm}$ . The plates are in vacuum. The charging current $i_{\mathrm{C}}$ has a constant value of 1.80 $\mathrm{mA}$ . 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 $d E / d t$ , the time rate of change of the electric field between the plates. Does $d E / d$ vary in time? (c) Calculate the displacement current density $j_{\mathrm{D}}$ between the plates, and from this the total displacement current $i_{\mathrm{D}}$ . How do $i_{\mathrm{C}}$ and $i_{\mathrm{D}}$ compare?

Vishal Gupta
Vishal Gupta
Numerade Educator
06:19

Problem 39

Displacement Current in a Wire. A long, straight, copper wire with a circular cross-scctional area of 2.1 $\mathrm{mm}^{2}$ carries a current of 16 $\mathrm{A}$ . The resistivity of the material is $20 \times 10^{-8} \Omega \cdot \mathrm{m}$ . (a) What is the uniform electric field in the material? (b) If the cur-
rent is changing at the rate of 4000 $\mathrm{A} / \mathrm{s}$ , at what rate is the electric field in the material changing? (c) What is the displacement current density in the material in part (b)? (Hint: Since $K$ for copper is very close to $1,$ use $\epsilon=\epsilon_{0} . )$ (d) If the current is changing as in part (b), what is the magnitude of the magnetic field 6.0 $\mathrm{cm}$ from the center of the wire? Note that both the conduction current and the displacement current should be included in the calculation of $B$ . Is
the contribution from the displacement current significant?

Keshav Singh
Keshav Singh
Numerade Educator
02:02

Problem 40

A long, straight wire made of a type-I superconductor carries a constant current $I$ along its length. Show that the current cannot be uniformly spread over the wire's cross section but instead must all be at the surface.

Keshav Singh
Keshav Singh
Numerade Educator
00:39

Problem 41

A type-II superconductor in an external field between $B_{\mathrm{el}}$ and $B_{c_{2} \text { has regions that contain magnetic flux and have resistance, }}$ and also has superconducting regions. What is the resistance of a long, thin cylinder of such material?

Keshav Singh
Keshav Singh
Numerade Educator
02:35

Problem 42

At temperatures near absolute $\mathrm{zero}, B_{\mathrm{e}}$ approaches 0.142 $\mathrm{T}$ 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 $\overrightarrow{\boldsymbol{B}}_{0}$ in the $+x$ direction. At points far from the ends of the cylinder, by symmetry, all the magnetic vectors are parallel to the cylinder (far from the ends) for (a) $\overrightarrow{\boldsymbol{B}}_{0}=(0.130 \mathrm{T}) \hat{\imath}$ and (b) $\overrightarrow{\boldsymbol{B}}_{0}=(0.260 \mathrm{T}) \hat{\boldsymbol{\imath}} ?$

Keshav Singh
Keshav Singh
Numerade Educator
02:34

Problem 43

The compound $\mathrm{SiV}_{3}$ is a type-II superconductor. At temperatures near absolute zero the two critical fields are $B_{c 1}=$ 55.0 $\mathrm{mT}$ and $B_{c 2}=15.0 \mathrm{T}$ . The normal phase of $\mathrm{SiV}_{3}$ has a magnetic susceptibility close to zero. A long, thin $\mathrm{SiV}_{3}$ cylinder has its axis parallel to an extermal magnetic field $\vec{B}_{0}$ 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 a temperature near absolute zero the external magnetic field is slowly increased from zero. What are the resultant magnetic field $\overrightarrow{\boldsymbol{B}}$ and the magnetization $\vec{M}$ inside the cylinder at points far from its ends (a) just before the magnetic flux begins to penetrate the material, and (b) just after the material becomes completely normal?

Keshav Singh
Keshav Singh
Numerade Educator
11:30

Problem 44

A Changing Magnetic Field. You are testing a new dataacquisition system. This system allows you to record a graph of the current in a circuit as a function of time. As part of the test, you are using a circuit made up of a 4.00 -cm-radius, 500 -urn coil of copper wire connceted in scrics to a $600-\Omega$ resistor. Copper has resistivity $1.72 \times 10^{-8} \Omega \cdot \mathrm{m},$ and the wire used for the coil has diameter 0.0300 $\mathrm{mm}$ . You place the coil on a table that is tilted $30.0^{\circ}$ from the horizontal and that lies between the poles of an electromagnet. The electromagnet generates a vertically upward magnetic field that is zero for $t<0$ , equal to $(0.120 \mathrm{T}) \times$ $(1-\cos \pi t)$ for $0 \leq t \leq 1.00 \mathrm{s}$ , and equal to 0.240 $\mathrm{T}$ for $t > 1.00 \mathrm{s}$ . (a) Draw the graph that should be produced by your data-acquisition system. (This is a full-featured system, so the graph will include labels and numerical values on its axes.) (b) If you were looking vertically downward at the coil, would the current be flowing clockwise or counterclockwise?

Gage Bonner
Gage Bonner
Numerade Educator
08:06

Problem 45

In the circuit shown in Fig, 29.41 the capacitor has capacitance $C=20 \mu F$ and is initially charged to 100 $\mathrm{V}$ with the polarity shown. The resistor $R_{0}$ has resistance $10 \Omega .$ At time $t=0$ the switch 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 $\Omega / \mathrm{m}$ and contains 25 loops. The large circuit is a rectangle 2.0 $\mathrm{m}$ by 4.0 $\mathrm{m}$ , while the small one has dimensions $a=10.0 \mathrm{cm}$ and $b=20.0 \mathrm{cm}$ . The distance $c$ is $5.0 \mathrm{cm} .$ (The figure is not drawn to scale. ) Both circuits are held stationary. Assume that only the wire nearest the small circuit produces an appreciable magnetlefield 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 Problem 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
Keshav Singh
Numerade Educator
06:05

Problem 46

A flat coil is oriented with the plane of its area at right angles to a spatially uniform magnetic field. The magnitude of this field varies with time according to the graph in Fig. 29.42 . Sketch a qualitative (but accuratel) graph of the emf induced in the coil as a function of time. Be sure to idcntify the timcs $t_{1}, t_{2},$ and $t_{3}$ on your graph.

Yaqub Khan
Yaqub Khan
Numerade Educator
06:50

Problem 47

A circular wire loop of radius $a$ and resistance $R$ initially has a magnetic flux through it due to an external magnetic field. The extermal field then decreases to zero. A current is induced in the loop while the external field is changing; however, this current does not stop at the instant that the external field stops changing. The reason is that the current itself generates a magnetic field, which gives rise to a flux through the loop. If the current changes, the flux through the loop changes as well, and an induced emf
appears in the loop to oppose the change. (a) The magnetic field at the center of the loop of radius a produced by a current i in the loop is given by $B=\mu_{0} i / 2 a$ . If we use the crode approximation that the field has this same value at all points within the loop, what is the flux of this field through the loop? (b) By using Faraday's law, Eq. $(29.3),$ and the relationship $\mathcal{E}=i R,$ show that after the external field has stopped changing, the current in the loop obeys the differential equation
$$
\frac{d i}{d t}=-\left(\frac{2 R}{\pi \mu_{0} a}\right) i
$$
(c) If the current has the value $i_{0}$ at $t=0$ , the instant that the external field stops changing. snive the equation in part $(b)$ to find $i$ as a function of time for $t>0 .$ (Hint: In Section 26.4 we encountered a similar differential equation, Eq. $(26.15),$ for the quantity $q .$ This equation for $i$ may be solved in the same way. (d) If the loop has radius $a=50 \mathrm{cm}$ and resistance $R=0.10 \Omega,$ how long after the external field stops changing will the current be equal to 0.010 $\mathrm{o}$ (that is, $\frac{1}{100}$ of its initial value)? (e) In solving the examples in this
chapter, we ignored the effects described in this problem. Explain why this is a good approximation.

Keshav Singh
Keshav Singh
Numerade Educator
02:35

Problem 48

A coil is stationary in a spatially unifunu, exlernal, time-varying magnetic field. The emf induced in this coil as a function of time is shown if Fig. 29.43 . Sketch a clear qualitative graph of the external magnetic field as a function of time, given that it started from zero. Include the points $t_{1}, t_{2}, t_{3}$ and $t_{4}$ on
your graph.

Ajay Singhal
Ajay Singhal
Numerade Educator
06:06

Problem 49

In Fig. 29.44 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 $\mathcal{E}$ induced in the loop. Do this two ways: (i) by using Faraday's law of induction (Hint: See Problem 29.7$)$ and (ii) by looking at the emf induced in cach 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 magnctic 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.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:01

Problem 50

Suppose the loop in Fig. 29.45 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 cach case if
$A=600 \mathrm{cm}^{2}, \omega=35.0 \mathrm{rad} / \mathrm{s}$ , and $B=0.450 \mathrm{T} ?$

Keshav Singh
Keshav Singh
Numerade Educator
03:07

Problem 51

As a new electrical engineer for the local power company, you are assigned the project of designing a generator of sinusoidal ac voltage with a maximum voltage of 120 $\mathrm{V}$ . Resides plenty of wire, you have two strong magnets that can produce a constant uniform magnetic field of 1.5 T over a square area of 10.0 $\mathrm{cm}$ on a side when they are 12.0 $\mathrm{cm}$ apart. The basic design should consist of a square coil turning in the uniform magnetic field. To have an acceptable coil resistance, the coil coil can have at most 400 loops. What is the minimum rotation rate (in $\mathrm{rpm}$ ) of the coil so it will produce the required voltage?

Vishal Gupta
Vishal Gupta
Numerade Educator
04:45

Problem 52

Make a Generator? You are shipwrecked on a deserted tropical island. You have some electrical devices that you could uperate using a generalor but you have nu maguels. The eardis magnetic field at your location is horizontal and has magnitude 8.0 $\times 10^{-5} \mathrm{T}$ , 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 $\mathrm{V}$ and estimate that you can rotate the coil at 30 $\mathrm{rpm}$ by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum mumber 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 this device is feasible? Explain.

Vishal Gupta
Vishal Gupta
Numerade Educator
03:03

Problem 53

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 Fig. 29.46 . 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.

Averell Hause
Averell Hause
Carnegie Mellon University
00:08

Problem 54

A Circuit Within a Circuit. Fig. 29.47 shows a small circuit within a larger one, both lying on the surface of a table. The switch is closed at $t=0$ with the capacitor initially uncharged. Assume that the small circuit has no appreciable effect on the larger one. (a) What is the direction $(a \text { to } b \text { or to } a)$ of the current in the resistor $r$ (i) the instant after the switch is closed and (ii) one time constant after the switch is closed? (b) Sketch a graph of the current in the small circuit as a function of time, calling clockwise positive.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:47

Problem 55

Terminal Speced. A conducting rod with length $L,$ mass $m,$ and resistance $R$ moves without friction on metal rails as shown in Fig. $29.11 .$ A uniform magnetic field $\vec{B}$ is directed into the plane of the figure. The rod starts from rest and is acted on by a constant force $\vec{F}$ directed to the right. The rails are infinitely long and have negligible resistance. (a) Graph the speed of the rod as a function of time. (b) Find an expression for the terminal speed (the speed when the acceleration of the rod is zero).

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
13:52

Problem 56

Terminal Speed. A bar of length $L=0.8 \mathrm{m}$ is free to shide without friction on horizontal rails, as shown in Fig. 29.48 . There is a uniform magnetic ficld $B=1.5 \mathrm{T}$ directed into the plane of the figure. At one end of the rails there is a battery with emf $\mathcal{E}=12 \mathrm{V}$ and a switch. The bar has mass 0.90 $\mathrm{kg}$ and resistance $5.0 \Omega,$ and all other resistance in the circuit can be ignored. The switch is closed at time $t=0$ . (a) Sketch the speed of the bar as a function of time.
(b) Just aner 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 $\mathrm{m} / \mathrm{s} ?$ (d) What is the terminal speed of the bar?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:58

Problem 57

Antenna emf. A satellite, orbiting the earth at the equator at an altitude of 400 $\mathrm{km}$ , has an antenna that can be modeled as a $2.0-\mathrm{m}-$ long rod. The antenna is oriented perpendicular to the earth's surface. At the cquator, the earth's magnetic field is cssentially horizontal and has a value of $8.0 \times 10^{-5} \mathrm{T}$ ; ignore any changes in $B$ with altitude. Assuming the orbit is circular, determine the induced emf between the tips of the antenna.

Keshav Singh
Keshav Singh
Numerade Educator
04:38

Problem 58

emf in a Bullet. At the equator, the earth's magnetic field is approximately horizontal, is directed towand the north, and has a value of $8 \times 10^{-5} \mathrm{T}$ . (a) Estimate the emf induced between the top and bottom of a bullet shot horizontally at a target on the equator if the bullet is shot toward the east. Assume the bullet has a length of 1 $\mathrm{cm}$ and a diamcter of 0.4 $\mathrm{cm}$ and is traveling at 300 $\mathrm{m} / \mathrm{s}$ . Which is at higher potential: the top or boutom of the bullet? (b) What is the emfif the bullet travels south?(c) What is the emf induced between the front and back of the bullet for any horizontal velocity?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:18

Problem 59

A very long, cylindrical wire of radius $R$ carries a current $I_{0}$ uniformly distributed across the cross scction 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 Fig. 29.49 (see Problem 29.7$)$ .

Keshav Singh
Keshav Singh
Numerade Educator
10:03

Problem 60

A circular conducting ring with radius $r_{0}=0.0420 \mathrm{m}$ hes in the $\mathrm{xy}$ -plane
in a region of uniform magnetic field $\vec{B}=B_{0}\left[1-3\left(t / t_{0}\right)^{2}+2\left(t / t_{0}\right)^{3}\right] \hat{k} .$ In this expression, $t_{0}=0.0100 \mathrm{s}$ and is constant, $t$ is time, $k$ is the unit vector in the $+$ z-dircction, and $B_{0}=0.0800 \mathrm{T}$ and is constant. At points $a$ and $b$ (Fig. 29.50 ) there is a small gap in the ring with wires leading to an external circuit of resistance $K=12.0 \Omega$ . 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 given in part $(b)$ is only 3.00 $\mathrm{mA}$ . Determine the internal resistance of the ring. (d) Determine the emf in the ring at a time $t=1.21 \times 10^{-2} \mathrm{s}$ . 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
11:21

Problem 61

The long, straight wire shown in Fig. 29.51 a 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 $R(\text { Fig, } 29.51 b),$ what is the magnitude of the current inducod in the loup?

Brandy Heflin
Brandy Heflin
Numerade Educator
03:42

Problem 62

The cube shown in $\mathrm{FAg}$ $29.52,50.0 \mathrm{cm}$ on a side, is in a uni- form magnetic field of 0.120 $\mathrm{T}$ , directed along the positive $y$ -axis. Wires $A, C,$ and $D$ move in the
directions indicated, cach with a speed of 0.350 $\mathrm{m} / \mathrm{s}$ . (Wire $A$ moves parallel to the $x y$ -plane, $C$ moves at an angle of $45.0^{\circ}$ below the $x y-$ plane, and $D$ moves parallel to the $x z$ plane.) What is the potential difference between the ends of each wire?

Keshav Singh
Keshav Singh
Numerade Educator
04:13

Problem 63

A slender rod, 0.240 $\mathrm{m}$ long, reates with an angular speed of 8.80 $\mathrm{rad} / \mathrm{s}$ 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 $\mathrm{T}$ (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 $\mathrm{rad} / \mathrm{s}$ 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
Bruce Edelman
Numerade Educator
03:08

Problem 64

A Magnetic Exercise Machine. You have designed a new type of exercise machine with an extremely simple mechanism (Fig, 29.36 ). A vertical bar of silver (chosen for its low resistivity and because it makes the machine look cool) with length $L=3.0 \mathrm{m}$ is free to move left or right without friction on silver rails. The contire apparatus is placed in a horizontal, uniform magnetic field of strength 0.25 T. When you push the bar to the left or right, the bar's motion sets up a current in the circuit that includes the bar. The resistance of the bar and the rails can be neglected. The magnetic field exerts a force on the current-carrying bar, and this force opposes the bar's motion. The health benefit is from the exercise that you do in working against this force. (a) Yourdesign goal is that the person doing the exercise is to do work at the rate of 25 watts when moving the bar at a steady 2.0 $\mathrm{m} / \mathrm{s}$ . What should be the resistance $R ?(\mathrm{b})$ You decide you
want to be able to vary the power required from the person, to adapt the machine to the person's strength and fitness. If the power is to be increased to 50 $\mathrm{W}$ by altering $R$ while leaving the other design parameters constant, should $R$ be increased or decreased? Calculate the value of $R$ for 50 $\mathrm{W}$ . (c) When you start to construct a prototype machine, you find it is difficult to produce a 0.25 . T magnetic field over such a large area. If you decrease the length of the bar to 0.20 $\mathrm{m}$ while leaving $B, v,$ and $R$ the same as in part $(a),$ what will be the power required of the person?

Vishal Gupta
Vishal Gupta
Numerade Educator
06:40

Problem 65

A rectangular loop with width $L$ and a slide wire with mass $m$ are us shown in Fig, 29.53 . A unifunu magnetic fixlu $\overrightarrow{\boldsymbol{B}}$ is directed perpendicular to the plane of the loop into the plane of the figure. The shide 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 shide 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 $x=m v_{0} R / a^{2} B^{2}$ .

Bruce Edelman
Bruce Edelman
Numerade Educator
06:42

Problem 66

A 25.0 -cm-long metal rod lies in the $x y$ -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 4.20 $\mathrm{m} / \mathrm{s}$ . The rod is in a uniform magnetic field $\overrightarrow{\boldsymbol{B}}=(0.120 \mathrm{T}) \hat{\boldsymbol{z}}-$ induced in the rod? (b) Indicate in a sketch which end of the rod is at higher potential.

Keshav Singh
Keshav Singh
Numerade Educator
03:43

Problem 67

The magnetic field $\overrightarrow{\boldsymbol{B}},$ at all points within a circular region of $\mathrm{F}$ radius $R,$ is uniform in space and directed into the plane of the page as shown in Fig. 29.54 . (The region could be a cross section inside thwindings of a long, straight solenoid.) If the magnetic field is increasing at a rate $d B / d t,$ what are the magnitude and direction of the force on a station-
ary 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.)

Zhaojie Xu
Zhaojie Xu
Numerade Educator
04:14

Problem 68

An airplane propeller of total length $L$ rotates around its center with angular spced $\omega$ in a magnctic ficld that is perpcndicular 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 $\mathrm{G}$ and the propeller turns at 220 $\mathrm{rpm}$ and is 2.0 $\mathrm{m}$ long, what is the potential difference between the middle and either end? It this
large enough to be concemed about?

Keshav Singh
Keshav Singh
Numerade Educator
03:40

Problem 69

It is impossible to have a uniform electric field that abruptly drops to zero in a region of space in which the magnetic field is constant and in which there are no electric charges. To prove this statement, use the method of contradiction: Assume that such a case is possible and then show that your assumption contradicts a law of nature. (a) In the bottom half of a piece of paper, draw evenly spaced horizontal lines representing a uniform electric field to your right. Use dashed lines to draw a rectangle abcda with horizontal side ab in the electric-field region and horizontal side $c d$ in the top half of your paper where $E=0 .$ (b) Show that integration around your rectangle contradicts Faraday's law, Eq. $(29.21) .$

Keshav Singh
Keshav Singh
Numerade Educator
03:22

Problem 70

Falling Square Loop. A vertically oricnted, square loop of copper wire falls from a region where the field $\overrightarrow{\boldsymbol{B}}$ is horizontal. uniform. where the field is zero. The loop is released from rest and initially is entirely within the magnetic-field region. Let the side length of the loop be $s$ and let the diameter of the wire be $d$ . The resistivity of copper is $\rho_{R}$ and the density of copper is $\rho_{m}$ . If the loop reaches its terminal speed while its upper segment is still in the magnetic- field region, find an expression for the terminal speed.

Narayan Hari
Narayan Hari
Numerade Educator
02:57

Problem 71

In a region of space where there are no conduction or displacement currents, it is impossible to have a uniform magnetic field that abruptly drops to zero. To prove this statement, use the method of contradiction: Assume that such a case is possible, and then show that your assumption contradicts a law of nature. (a) In the bottom half of a piece of paper, draw evenly spaced horizontal lines representing a uniform magnetic field to your right. Use dashed lines to draw a rectangle abcda with horizontal side $a b$ in the magnetic-leld region and horizontal side $c d$ in the top half of your paper where $B=0 .$ (b) Show that integration around your rectangle contradicts Ampere's law, Eq. $(29.15) .$

Keshav Singh
Keshav Singh
Numerade Educator
05:14

Problem 72

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 $Q_{0}$ on each plate that gradually discharges by conduction through the dielectric. (a) Calculate the conduction current density $j_{\mathrm{C}}(t)$ in the dielectric. (b) Show that at any instant the dis-placement current density in the diclectric is equal in magnitude to the oonduotion current density but opposite in direction, so the total current density is zero at every instant.

Salamat Ali
Salamat Ali
Numerade Educator
04:28

Problem 73

A rod of pure silicon (resistivity $\rho=2300 \Omega \cdot \mathrm{m} )$ is carry-ing a current. The electric field varies sinusoidally with time according to $E=E_{0} \sin \omega t,$ where $E_{0}=0.450 \mathrm{V} / \mathrm{m}, \omega=2 \pi f,$ and the frequency $f=120 \mathrm{Hz}$ (a) Find the magnitude of the maximum conduction current density in the wire. (b) Assuming $\epsilon=\epsilon_{0}$ ,
find the maximum displacement current density in the wire, and compare with the result of part (a). (c) At what frequency $f$ would the maximum conduction and displacement densitics become equal if $\epsilon=\epsilon_{0}$ (which is not actually the case)? (d) At the frequency determined in part (c), what is the relative phase of the conduction and displacement currents?

Ajay Singhal
Ajay Singhal
Numerade Educator
02:17

Problem 74

A square, conducting, wire loop of side $L,$ total mass $m,$ and total resistance $R$ initially lies in the horizontal $x y$ -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 $\dot{B}=B \hat{k}$ 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 ficld, does it take the loop a longer or shorer time to rotate through $90^{\circ} \%$ Explain. (d) Is mechanical energy conserved as the loop rotates downward? Explain.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:24

Problem 75

A square conducting loop, 20.0 $\mathrm{cm}$ on a side, is placed in the same magnetic field as shown in Exervise $29.30 .$ ( Sce Fig. $29.55 ;$ the center of the magnetic-field region.) (a) Copy Fig. $29.55,$ and draw vectors to show the directions and relative magnitudes of the induced electric field $\overrightarrow{\boldsymbol{E}}$ at points $a, b,$ and $c .$ (b) Prove that the component of $\overrightarrow{\boldsymbol{B}}$ along the loop has the same value at every point of the loop and is equal to that of the ring shown in Fig. 29.31 (see Excrise $29.30 ) .$ (c) What current is induced in the loop if its resistance is 1.90$\Omega ?$ (d) What is the potential difference between points $a$ and $b ?$

Penny Riley
Penny Riley
Numerade Educator
06:32

Problem 76

A uniform, square, conducting $100 p, 20.0 \mathrm{cm}$ on a side, is placed in the same magnetic fleld as shown in Exercise 29.30 , with side ac along a diameter and with point $b$ at the center of the field $(\mathrm{Fg}, 29.56)$ (a) Copy Fig. $29.56,$ and draw veetors to show the direction and relative magnitude of the induced electric field $\overrightarrow{\boldsymbol{E}}$ at the lettered points. (b) What is the induced emf in side ac?(c) What is the induced emf in the loop? (d) What is the current in the loop if its resistance is 1.90 $\Omega$ ? (e) What is the potential difference between points $a$ and c? Which is at higher potential?

Vishal Gupta
Vishal Gupta
Numerade Educator
07:13

Problem 77

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 Fig, 29.57 . 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 spced 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
Bruce Edelman
Numerade Educator
View

Problem 78

Consider a uniform metal disk rotating through a perpendicular magnetic field $\overrightarrow{\boldsymbol{B}}$ , as shown in Fig. 29.19 a The disk has mass $m,$ radius $R,$ and thickness $t,$ is made of a material with resistivity \rho, and is rotating clockwise in Fig. 29.19 a with angular speed $\omega$ . The magnetic field is directed into the plane of the disk. Suppose that the region to which the magnetic field is confined is not circular, as shown in Fig. 29.19 $\mathrm{a}$ , but is a small square with sides of length $L(L \ll R)$ centered a distance $d$ from the point $O$ (the center of the disk). The sides of this square are horizontal and vertical In Fig, 29.19$a$ . (a) Show that the current induced within the square is approximately equal to $I=\omega d B L t / \rho$ . In which direction does this current flow? (Hint: Assume that the resistance to the current is confined to the region of the square. The current also encounters resistance as it flows outside the region to which the magnetic field is confined, as shown in Fig. 29.196 ; however, this resistance is relatively small, since the current can flow through such a wide area. Recall Eq. $(25.10)$ for resistance, given in Section $25.3 .$ ) (b) Show that the induced current gives rise to a torque of approximate magnitude $\tau=\omega d^{2} B^{2} L^{2} t / \rho$ that opposes the rotation of the disk (that is, a counterclockwise torque). (c) What would be the magnitudes and directions of the induced current and torque if the diroutiva uf $\vec{B}$ were still intu the plane of the disk but the disk rotated counterclockwise? What if the direction of $\overrightarrow{\boldsymbol{B}}$ were out of
the plane and the disk rotated counterclockwise?

Lainey Roebuck
Lainey Roebuck
Numerade Educator