University Physics with Modern Physics In SI Units

Hugh D Young; Roger A Freedman

Chapter 29

Electromagnetic Induction - all with Video Answers

Educators

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Chapter Questions

Problem 1

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

Averell Hause

Averell Hause

Carnegie Mellon University

Problem 2

In a physics laboratory experiment, a coil with 200 turns enclosing an area of $12 \mathrm{~cm}^{2}$ is rotated in $0.040 \mathrm{~s}$ 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 $6.0 \times 10^{-5} \mathrm{~T}$. (a) What is the magnetic flux through each turn of the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?

Vishal Gupta

Numerade Educator

Problem 3

The magnetic flux through a coil is given by $\Phi_{B}=\alpha t-\beta t^{3}$, where $\alpha$ and $\beta$ are constants. (a) What are the units of $\alpha$ and $\beta ?$ (b) If the induced emf is zero at $t=0.500 \mathrm{~s}$, how is $\alpha$ related to $\beta ?$ (c) If the emf at $t=0$ is $-1.60 \mathrm{~V}$, what is the emf at $t=0.250 \mathrm{~s}$ ?

Vishal Gupta

Numerade Educator

Problem 4

A small, closely wound coil has $N$ turns, area $A$, and resistance $R$. The coil is initially in a uniform magnetic field that has magnitude $B$ and a direction perpendicular to the plane of the loop. The coil is then rapidly pulled out of the field so that the flux through the coil is reduced to zero in time $\Delta t$. (a) What are the magnitude of the average $\operatorname{emf} \mathcal{E}_{\text {av }}$ and average current $I_{\mathrm{av}}$ induced in the coil? (b) The total charge $Q$ that flows through the coil is given by $Q=I_{\mathrm{av}} \Delta t .$ Derive an expression for $Q$ in terms of $N, A, B,$ and $R .$ Note that $Q$ does not depend on $\Delta t$. (c) What is $Q$ if $N=150$ turns, $A=4.50 \mathrm{~cm}^{2}, R=30.0 \Omega,$ and $B=0.200 \mathrm{~T} ?$

Vishal Gupta

Numerade Educator

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

Numerade Educator

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

Numerade Educator

Problem 7

The current in the long, straight wire $A B$ shown in Fig. $\mathbf{E} 29.7$ is upward and is increasing steadily at a rate $d i / d t$.
(a) At an instant when the current is $i$, what are the magnitude and direction of the field $\overrightarrow{\boldsymbol{B}}$ 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 $\mathrm{emf}$ if $\quad a=12.0 \mathrm{~cm}, b=36.0 \mathrm{~cm}$ $L=24.0 \mathrm{~cm},$ and $d i / d t=9.60 \mathrm{~A} / \mathrm{s}$.

Khaled Yasein

Numerade Educator

Problem 8

A flat, circular, steel loop of radius $75 \mathrm{~cm}$ is at rest in a uniform magnetic field, as shown in an edge-on view in Fig. E29.8. The field is changing with time, according to $B(t)=(1.4 \mathrm{~T}) e^{-\left(0.057 \mathrm{~s}^{-1}\right) t} .$ (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.

Vishal Gupta

Numerade Educator

Problem 9

A circular loop of flexible iron wire has an initial circumference of $165.0 \mathrm{~cm}$, but its circumference 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, uniform 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 $9.0 \mathrm{~s}$ 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

A circular loop of wire with radius $2.00 \mathrm{~cm}$ and resistance $0.600 \Omega$ is in a region of a spatially uniform magnetic field $\vec{B}$ that is perpendicular to the plane of the loop. At $t=0$ the magnetic field has magnitude $B_{0}=3.00 \mathrm{~T}$. The magnetic field then decreases according to the equation $B(t)=B_{0} e^{-t / \tau}$, where $\tau=0.500 \mathrm{~s}$. (a) What is the maximum magnitude of the current $I$ induced in the loop? (b) What is the induced current $I$ when $t=1.50 \mathrm{~s}$ ?

Vishal Gupta

Numerade Educator

Problem 12

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 (Fig. E29.12).
(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 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 field of $0.0750 \mathrm{~T}$. What is the angular speed of the coil if the maximum emf produced is $24.0 \mathrm{mV} ?$

Bruce Edelman

Numerade Educator

Problem 14

A circular loop of wire with radius $r=0.0480 \mathrm{~m}$ and resistance $R=0.160 \Omega$ is in a region of spatially uniform magnetic field, as shown in Fig. E29.14. The magnetic field is directed out of the plane of the figure. The magnetic field has an initial value of $8.00 \mathrm{~T}$ and is decreasing at a rate of $d B / d t=-0.680 \mathrm{~T} / \mathrm{s}$.
(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?

Vishal Gupta

Numerade Educator

Problem 15

A circular loop of wire is in a region of spatially uniform magnetic field, as shown in 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 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 $I$ (Fig. $\mathbf{E} 29.17)$ (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 Fig. E29.18 obeys the equation $I(t)=I_{0} e^{-b t}$, where $b>0 .$ Find the direction (clockwise or counterclockwise) of the current induced in the round coil for $t>0$.

Vishal Gupta

Numerade Educator

Problem 19

Using Lenz's law, determine the direction of the current in resistor $a b$ of Fig. $\mathbf{E} 29.19$ 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.

Averell Hause

Carnegie Mellon University

Problem 20

A cardboard tube is wrapped with two windings of insulated wire wound in opposite directions, as shown in 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 (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 $r=0.0250 \mathrm{~m}$ and resistance $R=0.390 \Omega$ is in a region of spatially uniform magnetic field, as shown in Fig. E29.22. The magnetic field is directed into the plane of the figure. At $t=0, B=0$. The magnetic field then begins increasing, with $B(t)=\left(0.380 \mathrm{~T} / \mathrm{s}^{3}\right) t^{3} .$ What is the current in the loop (magnitude and direction) at the instant when $B=1.33 \mathrm{~T}$ ?

Bruce Edelman

Numerade Educator

Problem 23

A magnetic field of $0.080 \mathrm{~T}$ is in the $y$ -direction. The velocity of wire segment $S$ has a magnitude of $78 \mathrm{~m} / \mathrm{s}$ and components of $18 \mathrm{~m} / \mathrm{s}$ in the $x$ -direction, $24 \mathrm{~m} / \mathrm{s}$ in the $y$ -direction, and $72 \mathrm{~m} / \mathrm{s}$ in the $z$ -direction. The segment has length $0.50 \mathrm{~m}$ and is parallel to the $z$ -axis as it moves. (a) Find the motional emf induced between the ends of the segment. (b) What would the motional emf be if the wire segment was parallel to the $y$ -axis?

Vishal Gupta

Numerade Educator

Problem 24

A rectangular loop of wire with dimensions $1.50 \mathrm{~cm}$ by $8.00 \mathrm{~cm}$ 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 $B=2.40 \mathrm{~T}$ and is directed into the plane of Fig. E29.24. At the instant when the speed of the loop is $3.00 \mathrm{~m} / \mathrm{s}$ and it is still partially in the field region, what force (magnitude and direction) does the magnetic field exert on the loop?

Salamat Ali

Numerade Educator

Problem 25

In Fig. E29.25 a conducting rod of length $L=30.0 \mathrm{~cm}$ moves in a magnetic field $\boldsymbol{B}$ of magnitude $0.450 \mathrm{~T}$ directed into the plane of the figure. The rod 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, $a$ 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 $a b$ and (ii) directly out of the page?

Vishal Gupta

Numerade Educator

Problem 26

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. E29.26). 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

Numerade Educator

Problem 27

The conducting rod $a b$ shown in Fig. E29.27 makes contact with metal rails $c a$ and $d b$. The apparatus is in a uniform magnetic field of $0.800 \mathrm{~T}$, 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 energy is developed in the circuit $\left(I^{2} R\right)$.

Vishal Gupta

Numerade Educator

Problem 28

A 0.650 -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 (Fig. E29.28), so the apparatus makes a complete circuit. Ignore the resistance of the bar and the rails. 29.28 - A 0.650 -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 (Fig. E29.28), so the apparatus makes a complete circuit. Ignore the resistance of the bar and the rails.

Vishal Gupta

Numerade Educator

Problem 29

A $0.360-$ m-long $\quad$ 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 Fig. E29.29, so the apparatus makes a complete circuit. You can ignore the resistance of the bar and rails. The circuit is in a uniform $0.650 \mathrm{~T}$ magnetic field that is directed out of the plane of the figure. At the instant when the bar is moving to the left at $5.90 \mathrm{~m} / \mathrm{s},$ (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 30

Consider the circuit shown in Fig. E29.29, but with the bar moving to the right with speed $v$. As in Exercise 29.29 , the bar has length $0.360 \mathrm{~m}, R=45.0 \Omega,$ and $B=0.650 \mathrm{~T}$. (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 $0.840 \mathrm{~J} / \mathrm{s},$ what is the speed of the bar?

Vishal Gupta

Numerade Educator

Problem 31

A 0.250 -m-long bar moves on parallel rails that are connected through a $6.00 \Omega$ resistor, as shown in 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 magnetic field $B=1.20 \mathrm{~T}$ that is directed into the plane of the figure. At an instant when the induced current in the circuit is counterclockwise and equal to $1.75 \mathrm{~A},$ what is the velocity of the bar (magnitude and direction)?

Bruce Edelman

Numerade Educator

Problem 32

Blood Flow. 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 Fig. E29.32.) (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 $\mathcal{E}=v B d .$ (b) If you expect that the blood will be flowing at $15 \mathrm{~cm} /$ s for a vessel $5.0 \mathrm{~mm}$ in diameter, what strength of magnetic field will you need to produce a potential difference of $1.0 \mathrm{mV} ?$ (c) Show that the volume rate of flow $(R)$ of the blood is equal to $R=\pi \mathcal{E} d / 4 B .$ (Note: Although the method developed here is useful in measuring the rate of blood flow in a vessel, it is limited to use in surgery because measurement of the potential $\mathcal{E}$ must be made directly across the vessel.)

Vishal Gupta

Numerade Educator

Problem 33

A rectangular circuit is moved at a constant velocity of $3.0 \mathrm{~m} / \mathrm{s}$ into, through, and then out of a uniform 1.25 T magnetic field, as shown in Fig. E29.33. The magnetic-field region is considerably wider than $50.0 \mathrm{~cm}$. Find the magnitude and direction (clockwise or counterclockwise) of the current induced in the circuit as it is (a) going into the magnetic field; (b) totally within the magnetic field, but still moving; and (c) moving out of the field. (d) Sketch a graph of the current in this circuit as a function of time, including the preceding three cases.

Bruce Edelman

Numerade Educator

Problem 34

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 field. These magnets produce an initial uniform field of 1.12 $\mathrm{T}$ 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?

Salamat Ali

Numerade Educator

Problem 35

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 $d i / d t$. 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 $d i / d t$.

Vishal Gupta

Numerade Educator

Problem 36

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 $36.0 \mathrm{~A} / \mathrm{s}$. 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?

Zachary Warner

Numerade Educator

Problem 37

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 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 $0.0400 \mathrm{~s}$. What is the average induced emf in the second winding?

Bruce Edelman

Numerade Educator

Problem 38

A parallel-plate, air-filled capacitor 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 $0.520 \mathrm{~A}$. (a) What is the displacement current density $j_{\mathrm{D}}$ 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 $2.00 \mathrm{~cm}$ from the axis? (d) At $1.00 \mathrm{~cm}$ from the axis?

Salamat Ali

Numerade Educator

Problem 39

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 $2.50-\mathrm{mm}$ -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 $230 \mathrm{~V}$ and the conduction current $i_{\mathrm{C}}$ equals $6.00 \mathrm{~mA}$. 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?

Penny Riley

Numerade Educator

Problem 40

In Fig. 29.23 the capacitor plates have area $5.00 \mathrm{~cm}^{2}$ and separation $2.00 \mathrm{~mm}$. 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 $t=0.500 \mu \mathrm{s}$. (b) Calculate $d E / d t,$ the time rate of change of the electric field between the plates. Does $d E / d t$ 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

Numerade Educator

Problem 41

The electric flux is $\left(4.0 \mathrm{~V} \cdot \mathrm{m} / \mathrm{s}^{5}\right) t^{5}$ through a certain area of a dielectric that has dielectric constant $2.5 .$ (a) Find the displacement current through that area at $t=1.5 \mathrm{~s}$. (b) At what time was the displacement current $\frac{1}{6}$ as much?

Eduard Sanchez

Numerade Educator

Problem 42

At temperatures near absolute zero, $B_{\mathrm{c}}$ 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 $x$ -axis. At temperatures near absolute zero, what are the resultant magnetic field $\overrightarrow{\boldsymbol{B}}$ and the magnetization $\overrightarrow{\boldsymbol{M}}$ inside and outside the cylinder (far from the ends) for
(a) $\overrightarrow{\boldsymbol{B}}_{0}=(0.130 \mathrm{~T}) \hat{\imath}$ and
(b) $\overrightarrow{\boldsymbol{B}}_{0}=(0.260 \mathrm{~T}) \hat{\imath} ?$

Bruce Edelman

Numerade Educator

Problem 43

A motor vehicle generates electrical power using an alternator, which employs electromagnetic induction to convert mechanical energy to electrical energy. The alternator acts as a dc generator (Example 29.4 ). The alternator maintains and replenishes charge on the car's battery and operates headlights, radiator fans, windshield wipers, power windows, computer systems, sensors, sound systems, and other components. (a) A typical car battery provides 70 amp-hours of charge. How many coulombs is that? (b) If headlights each draw $20 \mathrm{~A}$ of current, a radiator fan draws $10 \mathrm{~A},$ and windshield wipers each draw $5 \mathrm{~A}$, estimate the peak current needed for a car to operate on a rainy night. (c) A car's alternator supplies an average emf of $14 \mathrm{~V}$ as emf induced in a sequence of stator coils in the presence of a magnetic field created by rotor coil electromagnets turned by a pulley system. A stator coil may have 42 windings and a cross-sectional diameter of $5.0 \mathrm{~cm},$ and it rotates at $400 \mathrm{~Hz}$. Estimate the strength of the magnetic field generated by a rotor coil.

Penny Riley

Numerade Educator

Problem 44

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 $B=2.90 \mathrm{~T}$ and is perpendicular to the plane of the loop. The loop has dimensions $4.00 \mathrm{~cm}$ by $60.0 \mathrm{~cm}$, mass $24.0 \mathrm{~g},$ and resistance $R=5.00 \times 10^{-3} \Omega .$ The loop is initially at rest; then a constant force $F_{\text {ext }}=0.180 \mathrm{~N}$ is applied to the loop to pull it out of the field (Fig. P29.44). (a) What is the acceleration of the loop when $v=3.00 \mathrm{~cm} / \mathrm{s} ?$ (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 45

In the circuit shown in Fig. $\mathbf{P} 29 . \mathbf{4 5}$, the capacitor has capacitance $C=20 \mu \mathrm{F}$ 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 $S$ 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 $1.0 \Omega / \mathrm{m}$ 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 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.

Linda Winkler

Numerade Educator

Problem 46

In the circuit in Fig. $\mathrm{P} 29.45,$ an emf of $90.0 \mathrm{~V}$ 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 $29.45 .$ The switch is closed at $t=0 .$ When the current in the large circuit is $5.00 \mathrm{~A},$ what are the magnitude and direction of the induced current in the small circuit?

Zachary Warner

Numerade Educator

Problem 47

A very long, straight solenoid with a cross-sectional area of $2.00 \mathrm{~cm}^{2}$ is wound with 90.0 turns of wire per centimeter. Starting at $t=0$ the current in the solenoid is increasing according to $i(t)=\left(0.160 \mathrm{~A} / \mathrm{s}^{2}\right) t^{2}$. 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 $3.20 \mathrm{~A}$ ?

Vishal Gupta

Numerade Educator

Problem 48

Suppose the loop in Fig. $\mathbf{P} 29.48$ 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 $A=600 \mathrm{~cm}^{2}, \omega=35.0 \mathrm{rad} / \mathrm{s},$ and $B=0.320 \mathrm{~T} ?$

Zachary Warner

Numerade Educator

Problem 49

In Fig. $\mathbf{P 2 9} . \mathbf{4 9}$ 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 $\operatorname{emf} \mathcal{E}$ 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 reason-
(iii) the able: (i) The loop is stationary; (ii) the loop is very thin, so $a \rightarrow 0$; loop gets very far from the wire.

Lainey Roebuck

Numerade Educator

Problem 50

If you secure a refrigerator magnet about $2 \mathrm{~mm}$ from the metallic surface of a refrigerator door and then move the magnet sideways, you can feel a resistive force, indicating the presence of eddy currents in the surface. (a) Estimate the magnetic field strength $B$ of the magnet to be $5 \mathrm{mT}$ (Problem 28.53 ) and assume the magnet is rectangular with dimensions $4 \mathrm{~cm}$ wide by $2 \mathrm{~cm}$ high, so its area $A$ is $8 \mathrm{~cm}^{2}$. Now estimate the magnetic flux $\Phi_{B}$ into the refrigerator door behind the magnet. (b) If you move the magnet sideways at a speed of $2 \mathrm{~cm} / \mathrm{s}$, what is a corresponding estimate of the time rate at which the magnetic flux through an area $A$ fixed on the refrigerator is changing as the magnet passes over? Use this estimate to estimate the emf induced under the rectangle on the door's surface.

Lainey Roebuck

Numerade Educator

Problem 51

A flexible circular loop $6.50 \mathrm{~cm}$ in diameter lies in a magnetic field with magnitude $1.35 \mathrm{~T}$ directed into the plane of the page as shown in Fig. P29.51. The loop is pulled at the points indicated by the arrows, forming a loop of zero area in $0.250 \mathrm{~s}$. (a) Find the average induced emf in the circuit. (b) What is the direction of the current in $R$ : from $a$ to $b$ or from $b$ to $a$ ? Explain your reasoning.

Bruce Edelman

Numerade Educator

Problem 52

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

Salamat Ali

Numerade Educator

Problem 53

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 Fig. $\mathbf{P} 29.53$ (see Exercise 29.7 ).

Bruce Edelman

Numerade Educator

Problem 54

A bar of length $L=0.36 \mathrm{~m}$ is free to slide without friction on horizontal rails as shown in Fig. P29.54. A uniform magnetic field $B=2.4 \mathrm{~T}$ is directed into the plane of the figure. At one end of the rails there is a battery with $\operatorname{emf} \mathcal{E}=12 \mathrm{~V}$ and a switch $S$. The bar has mass $0.90 \mathrm{~kg}$ 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 $2.0 \mathrm{~m} / \mathrm{s} ?$
(d) What is the bar's terminal speed?

Eduard Sanchez

Numerade Educator

Problem 55

The long, straight wire shown in Fig. P29.55a carries constant current $I$. A metal bar with length $L$ is moving at constant velocity $\overrightarrow{\boldsymbol{v}},$ 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. P29.55b), what is the magnitude of the current induced in the loop?

Bruce Edelman

Numerade Educator

Problem 56

A circular conducting ring with radius $r_{0}=0.0420 \mathrm{~m}$ lies in the $x y-$ plane in a region of uniform magnetic field $\overrightarrow{\boldsymbol{B}}=B_{0}\left[1-3\left(t / t_{0}\right)^{2}+2\left(t / t_{0}\right)^{3}\right] \hat{\boldsymbol{k}} .$ In this expression, $t_{0}=0.0100 \mathrm{~s}$ and is constant, $t$ is time, $\hat{k}$ is the unit vector in the $+z$ -direction, and $B_{0}=0.0800 \mathrm{~T}$ and is constant. At points $a$ and $b$ (Fig. $\mathbf{P 2 9 . 5 6}$ ) there is a small gap in the ring with wires leading to an external circuit of resistance $R=12.0 \Omega$. There is $\mathrm{no}$ 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 $t=5.00 \times 10^{-3} \mathrm{~s}$. 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 $3.00 \mathrm{~mA}$. 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.

Lainey Roebuck

Numerade Educator

Problem 57

A slender rod, $0.240 \mathrm{~m}$ long, rotates 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

Numerade Educator

Problem 58

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 $6.80 \mathrm{~m} / \mathrm{s}$. The rod is in a uniform magnetic field $\overrightarrow{\boldsymbol{B}}=(0.120 \mathrm{~T}) \hat{\imath}-(0.220 \mathrm{~T}) \hat{\jmath}-(0.0900 \mathrm{~T}) \hat{\boldsymbol{k}}$. (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.

Vishal Gupta

Numerade Educator

Problem 59

A rectangular loop with width $L$ and a slidewire with mass $m$ are as shown in Fig. $\mathbf{P} 29.59 . \mathrm{A}$ uniform magnetic field $\overrightarrow{\boldsymbol{B}}$ is directed perpendicular to the plane of the loop into the plane of the figure. The slidewire is given an initial speed of $v_{0}$ and then released. There is no friction between the slidewire and the loop, and the resistance of the loop is negligible in comparison to the resistance $R$ of the slidewire. (a) Obtain an expression for $F$, the magnitude of the force exerted on the wire while it is moving at speed $\underline{v}$. (b) Show that the distance $x$ that the wire moves before coming to rest is $x=m v_{0} R / L^{2} B^{2}$.

Lainey Roebuck

Numerade Educator

Problem 60

A circular coil with $N_{1}=5000$ turns is made of a conducting material with resistance $0.0100 \Omega / \mathrm{m}$ and radius $a=40.0 \mathrm{~cm} .$ The coil is attached to a $C=10.00 \mu \mathrm{F}$ capacitor as shown in Fig. $\mathbf{P} 29.60$. A second coil with radius $b=4.00 \mathrm{~cm},$ made of the same wire, with $N_{2}=100$ turns, is concentric with the first coil and parallel to it. The capacitor has a charge of $+100 \mu \mathrm{C}$ on its upper plate, and the switch $S$ is open. At time $t=0$ the switch is closed. (a) What is the magnitude of the current in the larger coil immediately after the switch is closed? (b) What is the magnetic flux through each turn of the smaller coil immediately after the switch is closed? (Since $b<<a$, we may treat the magnetic field in the smaller coil due to the larger coil as uniform.) (c) What is the direction of the current in the smaller coil immediately after the switch is closed? (d) What is the direction of the current in the smaller coil at $t=1.26 \mathrm{~ms}$ ? (e) What is the magnitude of the current in the smaller coil at $t=1.26 \mathrm{~ms}$ ?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 61

The magnetic field $\boldsymbol{B}$, 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 Fig. P29.61. (The region could be a cross section inside the windings 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 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.)

Zhaojie Xu

Numerade Educator

Problem 62

A bar with mass $M=1.20 \quad \mathrm{~kg} \quad$ and $\quad$ resistance $R=0.500 \Omega$ slides without friction on a horizontal U-shaped rail with width $W=40.0 \mathrm{~cm}$ and negligible resistance. The bar is attached to a spring with spring constant $k=90.0 \mathrm{~N} / \mathrm{m}$ as shown in Fig. P29.62. A constant magnetic field with magnitude $1.00 \mathrm{~T}$ points into the plane everywhere in the vicinity. At time $t=0$ the bar is stretched beyond its equilibrium position by an amount $x=10.0 \mathrm{~cm}$ and released from rest.
(a) This system behaves like a damped oscillator, described by Eq. (14.41). What is the damping coefficient $b ?$ (b) With what frequency does the bar oscillate around its equilibrium position?
(c) What is the amplitude of the motion at time $t=5.00 \mathrm{~s} ?$ (d) What is the magnitude of the current in the bar when it passes the equilibrium position for the first time? (e) What is the direction of that current?

Lainey Roebuck

Numerade Educator

Problem 63

A dielectric of permittivity $3.5 \times 10^{-11} \mathrm{~F} / \mathrm{m}$ 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 $21 \mu \mathrm{A} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 64

DATA 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 $0.800 \mathrm{~cm},$ 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 Fig. P29.64. 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) $t=2.00 \mathrm{~s},$ (b) $t=5.00 \mathrm{~s}$, and (c) $t=6.00 \mathrm{~s}$.

Lainey Roebuck

Numerade Educator

Problem 65

You are conducting an experiment in which a metal bar of length $6.00 \mathrm{~cm}$ and mass $0.200 \mathrm{~kg}$ slides without friction on two parallel metal rails (Fig. P29.65). 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 $\overrightarrow{\boldsymbol{B}}$ that is directed into the plane of the figure. You give the bar an initial velocity $v=20.0 \mathrm{~cm} / \mathrm{s}$ 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 (next page): $$\begin{array}{l|rrrrrr}
\boldsymbol{v}(\mathrm{cm} / \mathrm{s}) & 20.0 & 16.0 & 14.0 & 12.0 & 10.0 & 8.0 \\
\hline a\left(\mathrm{~cm} / \mathrm{s}^{2}\right) & 6.2 & 4.9 & 4.3 & 3.7 & 3.1 & 2.5
\end{array}$$ (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 bestfit 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 $20.0 \mathrm{~cm} / \mathrm{s}$ to $10.0 \mathrm{~cm} / \mathrm{s} ?$

Lainey Roebuck

Numerade Educator

Problem 66

You measure the magnitude of the external force $\overrightarrow{\boldsymbol{F}}$ that must be applied to a rectangular conducting loop to pull it at constant speed $v$ out of a region of uniform magnetic field $\vec{B}$ that is directed out of the plane of Fig. P29.66. The loop has dimensions $14.0 \mathrm{~cm}$ by $8.00 \mathrm{~cm}$ 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.
$$\begin{array}{l|lllll}
\boldsymbol{F}(\mathbf{N}) & 0.10 & 0.21 & 0.31 & 0.41 & 0.52 \\
\hline \boldsymbol{v}(\mathrm{cm} / \mathrm{s}) & 2.0 & 4.0 & 6.0 & 8.0 & 10.0
\end{array}$$
(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. $\mathrm{P} 29.66,$ 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 $5.00 \mathrm{~cm} / \mathrm{s} ?$

Lainey Roebuck

Numerade Educator

Problem 67

A conducting spherical shell with radius $10.0 \mathrm{~cm}$ spins about a vertical axis twice every second in the presence of a constant magnetic field $\overrightarrow{\boldsymbol{B}}$ with magnitude $1.00 \mathrm{~T}$ that points downward. Two conducting rods supported by a frame contact the sphere with conducting brushes and extend away from the sphere radially. One rod extends from the top of the sphere and the other forms a $60.0^{\circ}$ angle with vertical, as shown in Fig. $\mathbf{P} 29.67$. The outer ends of the rods are connected to each other by a conducting wire that includes a $10.0 \Omega$ resistor. (a) Construct a Cartesian coordinate system with the origin at the center of the sphere, the $z$ -axis pointing upward, and the $y$ -axis pointing rightward so that both rods lie in the $y z$ -plane. What is the velocity $\overrightarrow{\boldsymbol{v}}$ of a point on the sphere in the $y z$ -plane at angle $\theta$ measured from the positive $z$ -axis? (b) What is the vector product $\overrightarrow{\boldsymbol{v}} \times \overrightarrow{\boldsymbol{B}}$ ? (c) What is the line element $d \overrightarrow{\boldsymbol{l}}$ along the shortest path on the sphere from the upper rod to the angled rod at angle $\theta ?$ (d) What is the magnitude of the current in the wire?
(e) What direction does the current flow in the vertical rod?

Khaled Yasein

Numerade Educator

Problem 68

A uniform electric field is directed axially in a cylindrical region that includes a rectangular loop of wire with total resistance $R$. This loop has radially oriented width $a$ and axially oriented length $b$, and sits tight against the cylinder axis, as shown in Fig. $\mathbf{P} 29.68$. The electric field is zero at time $t=0$ and then increases in time according to $\overrightarrow{\boldsymbol{E}}=\eta t^{2} \hat{\boldsymbol{k}},$ where $\eta$ is a constant with units of $\mathrm{V} /\left(\mathrm{m} \cdot \mathrm{s}^{2}\right) .$ (a) What is the magnitude of the displacement current through a circular loop centered on the cylinder axis with radius $r \leq a$, at time $t$ ? (b) Use Ampere's law to determine the magnitude of the magnetic field a distance $r \leq a$ from the cylinder axis at time $t .$ (c) What is the magnetic flux at time $t$ through the rectangular wire loop? (d) What magnitude of current flows in the wire? (e) Does the current flow clockwise or counterclockwise from the perspective shown in the figure?

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 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 $x y$ -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 $\overrightarrow{\boldsymbol{B}}=B \hat{k}$ 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) $2 ;(\mathrm{c}) 1 \mathrm{while}$ 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 Fig. $\mathbf{P} 29.73$, which time-dependent graph of the magnetic field $B$ in the coil achieves that goal? Assume that everything else remains constant. (a) $\mathrm{A} ;(\mathrm{b}) \mathrm{B} ;(\mathrm{c})$ either $\mathrm{A}$ or $\mathrm{B} ;(\mathrm{d})$ neither $\mathrm{A}$ nor $\mathrm{B}$.

Bruce Edelman

Numerade Educator

Problem 74

Which graph in Fig. $\mathbf{P} 29.74$ 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) $\mathrm{A} ;(\mathrm{b}) \mathrm{B} ;(\mathrm{c}) \mathrm{C} ;$
(d) $\underline{D}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator