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# College Physics 2017

## Educators

### Problem 1

A uniform magnetic field of magnitude 0.50 T is directed perpendicular to the plane of a rectangular loop having dimensions 8.0 cm by 12 cm. Find the magnetic flux through the loop.

Farhanul H.

### Problem 2

Find the flux of Earth's magnetic field of magnitude $5.00 \times$ $10^{-5} \mathrm{T}$ through a square loop of area 20.0 $\mathrm{cm}^{2}(\mathrm{a})$ when the field is perpendicular to the plane of the loop, (b) when the field makes a $30.0^{\circ}$ angle with the normal to the plane of the loop, and (c) when the field makes a $90.0^{\circ}$ angle with the normal to the plane.

Zachary W.

### Problem 3

Figure $\mathrm{P} 20.3$ shows three edge views of a square loop with sides of length $\ell=0.250 \mathrm{m}$ in a magnetic field of magnitude 2.00 T. Calculate the magnetic flux through the loop oriented (a) perpendicular to the magnetic field, (b) $60.0^{\circ}$ from the magnetic field, and (c) parallel to the magnetic field.

Farhanul H.

### Problem 4

A long, straight wire carrying a current of 2.00 A is placed along the axis of a cylinder of radius 0.500 m and a length of 3.00 m. Determine the total magnetic flux through the cylinder.

Zachary W.

### Problem 5

A long, straight wire lies in the plane of a circular coil with a radius of 0.010 m. The wire carries a current of 2.0 A and is placed along a diameter of the coil. (a) What is the net flux through the coil? (b) If the wire passes through the center of the coil and is perpendicular to the plane of the coil, what is the net flux through the coil ?

Farhanul H.

### Problem 6

A magnetic field of magnitude 0.300 T is oriented perpendicular to the plane of a circular loop. (a) Calculate the loop radius if the magnetic flux through the loop is 2.70 Wb. (b) Calculate the new magnetic flux if the loop radius is doubled.

Zachary W.

### Problem 7

A cube of edge length $\ell=$ 2.5 $\mathrm{cm}$ is positioned as shown in Figure $\mathrm{P} 20.7$ . There is a uniform magnetic field throughout the region with components $B_{x}=+5.0 \mathrm{T}, B_{y}=$ +4.0 $\mathrm{T},$ and $B_{z}=+3.0 \mathrm{T}$ . (a) Calculate the flux through the shaded face of the cube. (b) What is the total flux emerging from the volume enclosed by the cube (i.e., the total flux through all six faces)?

Farhanul H.

### Problem 8

Transcranial magnetic stimulation (TMS) is a noninvasive technique used to stimulate regions of the human brain. A small coil is placed on the scalp, and a brief burst of current in the coil produces a rapidly changing magnetic field inside the brain. The induced emf can be sufficient to stimulate neuronal activity. One such device generates a magnetic field within the brain that rises from zero to 1.5 T in 120 ms. Determine the induced emf within a circle of tissue of radius 1.6 mm and that is perpendicular to the direction of the field.

Zachary W.

### Problem 9

Three loops of wire move near a long straight wire carrying a current as in Figure P20.9. What is the direction of the induced current, if any, in (a) loop A, (b) loop B, and (c) loop C.

Farhanul H.

### Problem 10

The flexible loop in Figure P20.10 has a radius of 12 cm and is in a magnetic field of strength 0.15 T. The loop is grasped at points A and B and stretched until its area is nearly zero. If it takes 0.20 s to close the loop, what is the magnitude of the average induced emf in it during this time?

Zachary W.

### Problem 11

Inductive charging is used to wirelessly charge electronic devices ranging from toothbrushes to cell phones. Suppose the base unit of an inductive charger produces a $1.00 \times$ $10^{-3}-$ T magnetic field. Varying this magnetic field magnitude changes the flux through a 15.0 -turn circular loop in the device, creating an emf that charges its battery. Suppose the loop area is $3.00 \times 10^{-4} \mathrm{m}^{2}$ and the induced emf has an average magnitude of 5.00 $\mathrm{V}$ . Calculate the time required for the magnetic field to decrease to zero from its maximum value.

Farhanul H.

### Problem 12

Medical devices implanted inside the body are often powered using transcutaneous energy transfer (TET), a type of wireless charging using a pair of closely spaced coils. An emf is generated around a coil inside the body by varying the current through a nearby coil outside the body, producing a changing magnetic flux. Calculate the average induced emf if each 10-turn coil has a radius of 1.50 cm and the current in the external coil varies from its maximum value of 10.0 $\mathrm{A}$ to zero in $6.25 \times 10^{-6} \mathrm{s}$ . (Hint: Recall from Topic 19 that the magnetic field at the center of the current-carrying external coil is $B=N \frac{\mu_{0} I}{2 R}$ . Assume this magnetic field is constant and oriented perpendicular to the internal coil.)

Zachary W.

### Problem 13

A technician wearing a circular metal band on his wrist moves his hand into a uniform magnetic field of magnitude 2.5 T in a time of 0.18 s. If the diameter of the band is 6.5 cm and the field is at an angle of $45^{\circ}$ with the plane of the metal band while the hand is in the field, find the magnitude of the average emf induced in the band.

Farhanul H.

### Problem 14

In Figure P20.14, what is the direction of the current induced in the resistor at the instant the switch is closed?

Zachary W.

### Problem 15

A bar magnet is positioned near a coil of wire, as shown in Figure P20.15. What is the direction of the current in the resistor when the magnet is moved (a) to the left and (b) to the right?

Farhanul H.

### Problem 16

Find the direction of the current in the resistor shown in Figure P20.16 (a) at the instant the switch is closed, (b) after the switch has been closed for several minutes, and (c) at the instant the switch is opened.

Zachary W.

### Problem 17

A circular loop of wire lies below a long wire carrying a current that is increasing as in Figure P20.17a. (a) What is the direction of the induced current in the loop, if any? (b) Now suppose the loop is next to the same wire as in Figure P20.17b. What is the direction of the induced current in the loop, if any? Explain your answers.

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### Problem 18

A square, single-turn wire loop $\ell=1.00 \mathrm{cm}$ on a side is placed inside a solenoid that has a circular cross section of radius $r=$ $3.00 \mathrm{cm},$ as shown in the end view of Figure $\mathrm{P} 20.18$ . The solenoid is 20.0 cm long and wound with 100 turns of wire. (a) If the current in the solenoid is 3.00 A, what is the flux through the square loop? (b) If the current in the solenoid is reduced to zero in 3.00 s, what is the magnitude of the average induced emf in the square loop?

Zachary W.

### Problem 19

A 300-turn solenoid with a length of 20.0 cm and a radius of 1.50 cm carries a current of 2.00 A. A second coil of four turns is wrapped tightly around this solenoid, so it can be considered to have the same radius as the solenoid. The current in the 300-turn solenoid increases steadily to 5.00 A in 0.900 s. (a) Use Ampère’s law to calculate the initial magnetic field in the middle of the 300-turn solenoid. (b) Calculate the magnetic field of the 300-turn solenoid after 0.900 s. (c) Calculate the area of the 4-turn coil. (d) Calculate the change in the magnetic flux through the 4-turn coil during the same period. (e) Calculate the average induced emf in the 4-turn coil. Is it equal to the instantaneous induced emf? Explain. (f) Why could contributions to the magnetic field by the current in the 4-turn coil be neglected in this calculation?

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### Problem 20

A circular coil enclosing an area of 100 $\mathrm{cm}^{2}$ is made of 200 turns of copper wire. The wire making up the coil has resistance of 5.0$\Omega$ , and the ends of the wire are connected to form a closed circuit. Initially, a 1.1-T uniform magnetic field points perpendicularly upward through the plane of the coil. The direction of the field then reverses so that the final magnetic field has a magnitude of 1.1 T and points downward through the coil. If the time required for the field to reverse directions is 0.10 s, what is the average current in the coil during that time?

Zachary W.

### Problem 21

To monitor the breathing of a hospital patient, a thin belt is girded around the patient’s chest as in Figure P20.21. The belt is a 200-turn coil. When the patient inhales, the area encircled by the coil increases by 39.0 $\mathrm{cm}^{2} .$ The magnitude of Earth's magnetic field is 50.0$\mu \mathrm{T}$ and makes an angle of $28.0^{\circ}$ with the plane of the coil. Assuming a patient takes 1.80 s to inhale, find the magnitude of the average induced emf in the coil during that time.

Farhanul H.

### Problem 22

An $N$ -turn circular wire coil of radius $r$ lies in the xy-plane (the plane of the page), as in Figure $\mathrm{P} 20.10$ . A uniform magnetic field is turned on, increasing steadily from 0 to $B_{0}$ in the positive $z$ -direction in $t$ seconds. (a) Find a symbolic expression for the emf, $\boldsymbol{\varepsilon},$ induced in the coil in terms of the variables given. (b) Looking down on at the $x y$ -plane from the positive $z$ -axis, is the direction of the induced cur- rent clockwise or counterclockwise? (c) If each loop has resistance $R$ , find an expression for the magnitude of the induced current, $I .$

Zachary W.

### Problem 23

A truck is carrying a steel beam of length 15.0 m on a freeway. An accident causes the beam to be dumped off the truck and slide horizontally along the ground at a speed of 25.0 m/s. The velocity of the center of mass of the beam is north-ward while the length of the beam maintains an east–west orientation. The vertical component of the Earth’s magnetic field at this location has a magnitude of 35.0$\mu \mathrm{T} .$ What is the magnitude of the induced emf between the ends of the beam?

Farhanul H.

### Problem 24

A 2.00 -m length of wire is held in an east-west direction and moves horizontally to the north with a speed of 15.0 $\mathrm{m} / \mathrm{s}$ . The vertical component of Earth's magnetic field in this region is 40.0$\mu \mathrm{t}$ directed downward. Calculate the induced emf between the ends of the wire and determine which end is positive.

Zachary W.

### Problem 25

A pickup truck has a width of 79.8 in. If is traveling north at 37 $\mathrm{m} / \mathrm{s}$ through a magnetic field with vertical component of $35 \mu \mathrm{T},$ what magnitude emf is induced between the driver and passenger sides of the truck?

Farhanul H.

### Problem 26

In one of NASA's space tether experiments, a 20.0 -km-long conducting wire was deployed by the space shuttle as it orbited at $7.86 \times 10^{3} \mathrm{m} / \mathrm{s}$ around Earth and across Earth's magnetic field lines. The resulting motional emf was used as a power source. If the component of Earth's magnetic field perpendicular to the tether was $1.50 \times 10^{-5} \mathrm{T}$ , determine the maximum possible potential difference between the two ends of the tether.

Zachary W.

### Problem 27

An automobile has a vertical radio antenna 1.20 m long. The automobile travels at 65.0 $\mathrm{km} / \mathrm{h}$ on a horizontal road where Earth's magnetic field is 50.0$\mu \mathrm{T}$ , directed toward the north and downward at an angle of $65.0^{\circ}$ below the horizontal. ( a ) Specify the direction the automobile should move so as to generate the maximum motional emf in the antenna, with the top of the antenna positive relative to the bottom. (b) Calculate the magnitude of this induced emf.

Farhanul H.

### Problem 28

An astronaut is connected to her spacecraft by a 25-m-long tether cord as she and the spacecraft orbit Earth in a circular path at a speed of $3.0 \times 10^{3} \mathrm{m} / \mathrm{s}$ . At one instant, the voltage measured between the ends of a wire embedded in the cord is measured to be 0.45 $\mathrm{V}$ . Assume the long dimension of the cord is perpendicular to the vertical component of Earth's magnetic field at that instant. (a) What is the magnitude of the vertical component of Earth's field at this location? (b) Does the measured voltage change as the system moves from one location to another? Explain.

Zachary W.

### Problem 29

Figure $\mathrm{P} 20.29$ shows a bar of mass $m=0.200 \mathrm{kg}$ that can slide without friction on a pair of rails separated by a distance $\ell=$ 1.20 $\mathrm{m}$ and located on an inclined plane that makes an angle $\theta=25.0^{\circ}$ with respect to the ground. The resistance of the resistor is $R=1.00 \Omega,$ and a uniform magnetic field of magnitude $B=0.500 \mathrm{T}$ is directed downward, perpendicular to the ground, over the entire region through which the bar moves. With what constant speed $v$ does the bar slide along the rails?

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### Problem 30

Consider the arrangement shown in Figure P20.30 where $R=6.00 \Omega, \ell=1.20 \mathrm{m},$ and $B=2.50 \mathrm{T}$ . (a) At what constant speed should the bar be moved to produce a current of 1.00 $\mathrm{A}$ in the resistor? (b) What power is delivered to the resistor? $(\mathrm{c})$ What magnetic force is exerted on the moving bar? (d) What instantaneous power is delivered by the force $F_{\mathrm{app}}$ on the moving bar?

Zachary W.

### Problem 31

A square coil of wire of side 2.80 cm is placed in a uniform magnetic field of magnitude 1.25 T directed into the page as in Figure P20.31. The coil has 28.0 turns and a resistance of 0.780$\Omega .$ If the coil is rotated through an angle of $90.0^{\circ}$ about the horizontal axis shown in 0.335 s, find $(a)$ the magnitude of the average emf induced in the coil during this rotation and (b) the average current induced in the coil during this rotation.

Farhanul H.

### Problem 32

A 100-turn square wire coil of area 0.040 $\mathrm{m}^{2}$ rotates about a vertical axis at 1500 $\mathrm{rev} / \mathrm{min}$ , as indicated in Figure $\mathrm{P} 20.32 .$ The horizontal component of Earth's magnetic field at the location of the loop is $2.0 \times 10^{-5} \mathrm{T}$ . Calculate the maximum emf induced in the coil by Earth's field.

Zachary W.

### Problem 33

Considerable scientific work is currently under way to determine whether weak oscillating magnetic fields such as those found near outdoor electric power lines can affect human health. One study indicated that a magnetic field of magnitude $1.0 \times 10^{-3} \mathrm{T}$ , oscillating at $60 . \mathrm{Hz}$ , might stimulate red blood cells to become cancerous. If the diameter of a red blood cell is 8.0$\mu \mathrm{mm}$ , determine the maximum emf that can be generated around the perimeter of the cell.

Farhanul H.

### Problem 34

A flat coil enclosing an area of 0.10 $\mathrm{m}^{2}$ is rotating at 60 $\mathrm{rev} / \mathrm{s}$ , with its axis of rotation perpendicular to a 0.20 $\mathrm{-T}$ magnetic field. (a) If there are 1000 turns on the coil, what is the maximum voltage induced in the coil? (b) When the maximum induced voltage occurs, what is the orientation of the coil with respect to the magnetic field?

Zachary W.

### Problem 35

A generator connected to the wheel or hub of a bicycle can be used to power lights or small electronic devices. A typical bicycle generator supplies 6.00 V when the wheels rotate at $\omega=20.0 \mathrm{rad} / \mathrm{s} .$ (a) If the generator's magnetic field has magnitude $B=0.600 \mathrm{T}$ with $N=100$ turns, find the loop area $A$ . (b) Find the time interval between the maximum emf of $+6.00 \mathrm{V}$ and the minimum emf of $-6.00 \mathrm{V}$ .

Farhanul H.

### Problem 36

A motor has coils with a resistance of $30 . \Omega$ and operates from a voltage of 240 V. When the motor is operating at its maximum speed, the back emf is 145 V. Find the current in the coils (a) when the motor is first turned on and (b) when the motor has reached maximum speed. (c) If the current in the motor is 6.0 A at some instant, what is the back emf at that time?

Zachary W.

### Problem 37

A coil of 10.0 turns is in the shape of an ellipse having a major axis of 10.0 cm and a minor axis of 4.00 cm. The coil rotates at 100. rpm in a region in which the magnitude of Earth's magnetic field is 55.0$\mu \mathrm{T}$ . What is the maximum voltage induced in the coil if the axis of rotation of the coil is along its major axis and is aligned (a) perpendicular to Earth's magnetic field and (b) parallel to Earth's magnetic field? Note: The area of an ellipse is given by $A=\pi a b,$ where $a$ is the length of the semimajor axis and $b$ is the length of the semiminor axis.

Farhanul H.

### Problem 38

A solenoid with 475 turns has a length of 6.00 $\mathrm{cm}$ and a cross-sectional area of $2.80 \times 10^{-9} \mathrm{m}^{2} .$ Find (a) the solenoid's inductance and (b) the average emf around the solenoid if the current changes from $+2.00 \mathrm{A}$ to $-2.00 \mathrm{A}$ in $8.33 \times 10^{-3} \mathrm{s}$

Zachary W.

### Problem 39

The current in a coil drops from 3.5 A to 2.0 A in 0.50 s. If the average emf induced in the coil is 12 mV, what is the self-inductance of the coil?

Farhanul H.

### Problem 40

Show that the two expressions for inductance given by
$$L=\frac{N \Phi_{\mathrm{B}}}{I} \quad \text { and } \quad L=\frac{-\boldsymbol{\varepsilon}}{\Delta I / \Delta t}$$
have the same units.

Zachary W.

### Problem 41

A solenoid of radius 2.5 cm has 400 turns and a length of 20 cm. Find (a) its inductance and (b) the rate at which current must change through it to produce an emf of 75 mV.

Farhanul H.

### Problem 42

An emf of 24.0 mV is induced in a 500-turn coil when the current is changing at a rate of 10.0 A/s. What is the magnetic flux through each turn of the coil at an instant when the current is 4.00 A?

Zachary W.

### Problem 43

An electromagnet can be modeled as an inductor in series with a resistor. Consider a large electromagnet of inductance $L=12.0 \mathrm{H}$ and resistance $R=$ 4.50$\Omega$ connected to a 24.0 $\mathrm{V}$ battery and switch as in Figure $\mathrm{P} 20.43$ . After the switch is closed, find (a) the maximum current carried by the electromagnet, (b) the time constant of the circuit, and (c) the time it takes the current to reach 95.0% of its maximum value.

Farhanul H.

### Problem 44

An $R L$ circuit with $L=3.00 \mathrm{H}$ and an $R C$ circuit with $C=3.00 \mu \mathrm{F}$ have the same time constant. If the two circuits have the same resistance $R,(\mathrm{a})$ what is the value of $R$ and $(\mathrm{b})$ what is this common time constant?

Zachary W.

### Problem 45

The battery terminal voltage in Figure $\mathrm{P} 20.43$ is $\boldsymbol{\varepsilon}=9.00 \mathrm{V}$ and the current I reaches half its maximum value of 2.00 $\mathrm{A}$ at $t=0.100 \mathrm{s}$ after the switch is closed. Calculate $(\mathrm{a})$ the time constant $\tau$ . (b) What is the emf across the inductor at $t=0.100$ s? (c) What is the emf across the inductor in the instant after the switch is closed at $t=0 ?$

Farhanul H.

### Problem 46

A $25-\mathrm{mH}$ inductor, an $8.0-\Omega$ resistor, and a $6.0-\mathrm{V}$ battery are connected in series as in Figure $\mathrm{P} 20.43 .$ The switch is closed at $t=0 .$ Find the voltage drop across the resistor (a) at $t=0$ and (b) after one time constant has passed. Also, find the voltage drop across the inductor $(\mathrm{c})$ at $l=0$ and $(\mathrm{d})$ after one time constant has elapsed.

Zachary W.

### Problem 47

Calculate the resistance in an $R L$ circuit in which $L=2.50 \mathrm{H}$ and the current increases to 90.0$\%$ of its final value in 3.00 $\mathrm{s}$

Farhanul H.

### Problem 48

Consider the circuit shown in Figure P20.43. Take $\varepsilon=6.00 \mathrm{V}, L=8.00 \mathrm{mH},$ and $R=4.00 \Omega$ (a) What is the inductive time constant of the circuit? (b) Calculate the cur- rent in the circuit 250 . $\mu$ s after the switch is closed. (c) What is the value of the final steady-state current? (d) How long does it take the current to reach 80.0$\%$ of its maximum value?

Zachary W.

### Problem 49

(a) If an inductor carrying a 1.70-A current stores an energy of 0.300 mJ, what is its inductance? (b) How much energy does the same inductor store if it carries a 3.00-A current?

Farhanul H.

### Problem 50

A 300-turn solenoid has a radius of 5.00 cm and a length of 20.0 cm. Find (a) the inductance of the solenoid and (b) the energy stored in the solenoid when the current in its windings is 0.500 A.

Zachary W.

### Problem 51

A 24 V battery is connected in series with a resistor and an inductor, with $R=8.0 \Omega$ and $L=4.0 \mathrm{H}$ , respectively. Find the energy stored in the inductor (a) when the current reaches its maximum value and (b) one time constant after the switch is closed.

Farhanul H.

### Problem 52

A 60.0-m length of insulated copper wire is wound to form a solenoid of radius 2.0 cm. The copper wire has a radius of 0.50 mm. (a) What is the resistance of the wire? (b) Treating each turn of the solenoid as a circle, how many turns can be made with the wire? (c) How long is the resulting solenoid? (d) What is the self-inductance of the solenoid? (e) If the solenoid is attached to a battery with an emf of 6.0 V and internal resistance of $350 \mathrm{m} \Omega,$ compute the time constant of the circuit. (f ) What is the maximum current attained? (g) How long would it take to reach 99.9% of its maximum current? (h) What maximum energy is stored in the inductor?

Zachary W.

### Problem 53

Two circular loops of wire surround an insulating rod as in Figure P20.53. Loop 1 carries a current I in the clockwise direction when viewed from the left end. If loop 1 moves toward loop 2, which remains stationary, what is the direction of the induced current in loop 2 when viewed from the left end?

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### Problem 54

A circular loop of wire of resistance $R=0.500 \Omega$ and radius $r=8.00 \mathrm{cm}$ is in a uniform magnetic field directed out of the page as in Figure $\mathrm{P} 20.54 .$ If a clockwise current of $I=2.50$ mA is induced in the loop, (a) is the magnetic field increasing or decreasing in time? (b) Find the rate at which the field is changing with time.

Zachary W.

### Problem 55

A rectangular coil with resistance $R$ has $N$ turns, each of length $\ell$ and width $w,$ as shown in Figure $\mathrm{P} 20.55 .$ The coil moves into a uniform magnetic field $\overrightarrow{\mathbf{B}}$ with constant velocity $\overrightarrow{\mathbf{v}}$ . What are the magnitude and direction of the total magnetic force on the coil (a) as it enters the magnetic field, (b) as it moves within the field, and (c) as it leaves the field?

Vishal G.

### Problem 56

A conducting bar of length $\ell$ moves to the right on two frictionless rails, as shown in Figure $P 20.30 .$ A uniform magnetic field directed into the page has a magnitude of 0.30 $\mathrm{T}$ . Assume $\ell=35 \mathrm{cm}$ and $R=9.0 \Omega .(\text { a) At what constant speed }$ should the bar move to produce an 8.5 $\mathrm{mA}$ current in the resistor? What is the direction of this induced current? (b) At what rate is energy delivered to the resistor? (c) Explain the origin of the energy being delivered to the resistor.

Zachary W.

### Problem 57

An 820 -turn wire coil of resistance 24.0$\Omega$ is placed on top of a 12500 -turn, $7.00-\mathrm{cm}$ -long solenoid, as in Figure $\mathrm{P} 20.57 .$ Both coil and solenoid have cross-sectional areas of $1.00 \times 10^{-4} \mathrm{m}^{2}$ (a) How long does it take the solenoid current to reach 0.632 times its maximum value? (b) Determine the average back emf caused by the self-inductance of the solenoid during this interval. The magnetic field produced by the solenoid at the location of the coil is one-half as strong as the field at the center of the solenoid. (c) Determine the average rate of change in magnetic flux through each turn of the coil during the stated interval. (d) Find the magnitude of the average induced current in the coil.

Ajay S.

### Problem 58

A spacecraft is in a circular orbit of radius equal to $3.0 \times$ $10^{4} \mathrm{km}$ around a $2.0 \times 10^{30} \mathrm{kg}$ pulsar. The magnetic field of the pulsar at that radial distance is $1.0 \times 10^{2} \mathrm{T}$ directed perpendicular to the velocity of the spacecraft. The spacecraft is 0.20 $\mathrm{km}$ long with a radius of 0.040 $\mathrm{km}$ and moves counter-clockwise in the $x$ ylane around the pulsar. (a) What is the speed of the spacecraft: (b) If the magnetic field points in the positive $z$ -direction, is the emf induced from the back to the front of the spacecraft or from side to side? (c) Compute the induced emf. (d) Describe the hazards for astronauts inside any spacecraft moving in the vicinity of a pulsar.

Zachary W.

### Problem 59

A conducting rod of length $\ell$ moves on two horizontal frictionless rails, as in Figure $\mathrm{P} 20.30 .$ A constant force of magnitude 1.00 $\mathrm{N}$ moves the bar at a uniform speed of 2.00 $\mathrm{m} / \mathrm{s}$ through a magnetic field $\overrightarrow{\mathrm{B}}$ that is directed into the page. (a) What is the current in an $8.00-\Omega$ resistor $R ?(\mathrm{b})$ What is the rate of energy dissipation in the resistor? (c) What is the mechanical power delivered by the constant force?

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### Problem 60

A long solenoid of radius $r=2.00 \mathrm{cm}$ is wound with $3.50 \times 10^{3}$ turns/m and carries a current that changes at the rate of 28.5 $\mathrm{A} / \mathrm{s}$ as in Figure $\mathrm{P} 20.60 .$ What is the magnitude of the emf induced in the square conducting loop surrounding the center of the solenoid?

Zachary W.

### Problem 61

The bolt of lightning depicted in Figure $P 20.61$ passes $200 . \mathrm{m}$ from a 100 -turn coil oriented as shown. If the current in the lightning bolt falls from $6.02 \times 10^{6} \mathrm{A}$ to zero in $10.5 \mu \mathrm{s},$ what is the average voltage induced in the coil? Assume the distance to the center of the coil determines the average magnetic field at the coil's position. Treat the lightning bolt as a long, vertical wire.

Farhanul H.

### Problem 62

The square loop in Figure $\mathrm{P} 20.62$ is made of wires with a total series resistance of 10.0$\Omega$ It is placed in a uniform $0.100-\mathrm{T}$ magnetic field directed per- pendicular into the plane of the paper. The loop, which is the paper. The loop, which is hinged at each corner, is pulled as shown until the separation between points $A$ and $B$ is 3.00 $\mathrm{m} .$ If this process takes $0.100 \mathrm{s},$ what is the average current generated in the loop? What is the direction of the current?

Zachary W.

### Problem 63

The magnetic field shown in Figure P20.63 has a uniform magnitude of 25.0 mT directed into the paper. The initial diameter of the kink is 2.00 cm. (a) The wire is quickly pulled taut, and the kink shrinks to a diameter of zero in 50.0 ms. Determine the average voltage induced between endpoints A and B. Include the polarity. (b) Suppose the kink is undisturbed, but the magnetic field increases to 100 $\mathrm{mT}$ in $4.00 \times$ $10^{-3}$ s. Determine the average voltage across terminals $A$ and $B,$ including polarity, during this period.

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### Problem 64

An aluminum ring of radius 5.00 $\mathrm{cm}$ and resistance $3.00 \times$ $10^{-4} \Omega$ is placed around the top of a long air-core solenoid with 1000 turns per meter and a smaller radius of $3.00 \mathrm{cm},$ as in Figure $\mathrm{P} 20.64 .$ If the current in the solenoid is increasing at a constant rate of $270 . \mathrm{A} / \mathrm{s}$ , what is the induced current in the ring? Assume the magnetic field produced by the solenoid over the area at the end of the solenoid is one-half as strong as the field at the center of the solenoid. Assume also the solenoid produces a negligible field outside its cross-sectional area.

Zachary W.

### Problem 65

In Figure $\mathrm{P} 20.65$ the rolling axle of length 1.50 $\mathrm{m}$ is pushed along horizontal rails at a constant speed $v=3.00 \mathrm{m} / \mathrm{s}$ . A resistor $R=0.400 \Omega$ is connected to the rails at points $a$ and $b,$ directly opposite each other. (The wheels make good electrical contact with the rails, so the axle, rails, and $R$ form a closed-loop circuit. The only significant resistance in the circuit is $R$ ) A uniform magnetic field $B=0.800 \mathrm{T}$ is directed vertically downward. (a) Find the induced current I in the resistor. (b) What horizontal force $\overrightarrow{\mathbf{F}}$ is required to keep the axle rolling at constant speed? (c) Which end of the resistor, $a$ or $b,$ is at the higher electric potential? (d) After the axle rolls past the resistor, does the current in $R$ reverse direction? Explain your answer.

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### Problem 66

An $N$ -turn square coil with side $\ell$ and resistance $R$ is pulled to the right at constant speed $v$ in the positive $x$ -direction in the presence of a uniform magnetic field $B$ acting perpendicular to the coil, as shown in Figure $\mathrm{P} 20.66$ . At $t=0$ , the right side of the coil is at the edge of the field. After a time $t$ has elapsed, the entire coil is in the region where $B=0 .$ In terms of the quantities $N, B, \ell, v,$ and $R,$ find symbolic expressions for $(a)$ the magnitude of the induced emf in the loop during the time interval $t,(\mathrm{b})$ the magnitude of the induced current in the coil, (c) the power delivered to the coil, and (d) the force required to remove the coil from the field. (e) What is the direction of the induced current in the loop? (f) What is the direction of the magnetic force on the loop while it is being pulled out of the field?

Zachary W.
A conducting rectangular loop of mass $M,$ resistance $R,$ and dimensions $w$ by $\ell$ falls from rest into a magnetic field $\overrightarrow{\mathbf{B}},$ as shown in Figure $\mathrm{P} 20.67$ . During the time interval before the top edge of the loop reaches the field, the loop approaches a terminal speed $v_{\mathrm{T}}$ (a) Show that
$$v_{T}=\frac{M g R}{B^{2} w^{2}}$$
(b) Why is $v_{T}$ proportional to $R ?$
(c) Why is it inversely proportional to $B^{2} ?$