College Physics

Hugh D. Young

Chapter 20

Magnetic Field and Magnetic Force - all with Video Answers

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Chapter Questions

Problem 1

$\bullet$ In a 1.25 T magnetic field directed vertically upward, a particle having a charge of magnitude 8.50$\mu \mathrm{C}$ and initially moving northward at 4.75 $\mathrm{km} / \mathrm{s}$ is deflected toward the east. (a) What is the sign of the charge of this particle? Make a sketch to illustrate how you found your answer. (b) Find the magnetic force on the particle.

Salamat Ali

Salamat Ali

Numerade Educator

Problem 2

An ion having charge $+6 e$ is traveling horizontally to the left at 8.50 $\mathrm{km} / \mathrm{s}$ when it enters a magnetic field that is perpendicular to its velocity and deflects it downward with an initial magnetic force of $6.94 \times 10^{-15} \mathrm{N} .$ What are the direction and magnitude of this field? Illustrate your method of solving this problem with a diagram.

Averell Hause

Carnegie Mellon University

Problem 3

$\bullet$ A proton traveling at 3.60 $\mathrm{km} / \mathrm{s}$ suddenly enters a uniform magnetic
field of 0.750 $\mathrm{T}$ , traveling at an angle of $55.0^{\circ}$ with the field lines (Figure 20.57$)$ . (a) Find the magnitude and direction of the force this magnetic field exerts on the proton. (b) If you can vary the direction of the proton's velocity, find the magnitude of the maximum and minimum forces you could achieve, and show how the velocity should be oriented to achieve these forces. (c) What would the answers to part (a) be if the proton were replaced by an electron traveling in the same way as the proton?

Salamat Ali

Numerade Educator

Problem 4

$\cdot$ A particle having a mass of 0.195 g carries a charge of $-2.50 \times 10^{-8} \mathrm{C} .$ The particle is given an initial horizontal northward velocity of $4.00 \times 10^{4} \mathrm{m} / \mathrm{s} .$ What are the magnitude and direction of the minimum magnetic field that will balance the earth's gravitational pull on the particle?

Averell Hause

Carnegie Mellon University

Problem 5

At a given instant, a particle with a mass of $5.00 \times 10^{-3} \mathrm{kg}$ and a charge of $3.50 \times 10^{-8} \mathrm{C}$ has a velocity with a magnitude of $2.00 \times 10^{5} \mathrm{m} / \mathrm{s}$ in the $+y$ direction. It is moving in a uniform magnetic field that has magnitude 0.8 $\mathrm{T}$ and is in the $-x$ direction. What are (a) the magnitude and direction of the magnetic force on the particle and (b) its resulting acceleration?

Salamat Ali

Numerade Educator

Problem 6

If the magnitude of the magnetic force on a proton is $F$ when it is moving at $15.0^{\circ}$ with respect to the field, what is the magnitude of the force (in terms of $F$ ) when this charge is moving at $30.0^{\circ}$ with respect to the field?

Averell Hause

Carnegie Mellon University

Problem 7

$\bullet \mathrm{A}^{9}$ Be nucleus containing four protons and five neutrons has a mass of $1.50 \times 10^{-26} \mathrm{kg}$ and is traveling vertically upward at 1.35 $\mathrm{km} / \mathrm{s} .$ If this particle suddenly enters a horizontal magnetic field of 1.12 T pointing from west to east, find the
magnitude and direction of its acceleration vector the instant after it enters the field.

Salamat Ali

Numerade Educator

Problem 8

A particle with a charge of $-2.50 \times 10^{-8} \mathrm{C}$ is moving with an instantaneous velocity of magnitude 40.0 $\mathrm{km} / \mathrm{s}$ in the $x y-$ plane at an angle of $50^{\circ}$ counterclockwise from the $+x$ axis. What are the magnitude and direction of the force exerted on
this particle by a magnetic field with magnitude 2.00 T in the (a) $-x$ direction, and (b) $+z$ direction?

Averell Hause

Carnegie Mellon University

Problem 9

\bullet A particle with mass $1.81 \times 10^{-3} \mathrm{kg}$ and charge of $+1.22 \times 10^{-8} \mathrm{Chas},$ at a given instant, a velocity of $3.00 \times 10^{4} \mathrm{m} / \mathrm{s}$ along the $+y$ -axis, as shown in Figure $20.58 .$ What are the magnitude and direction of the particle's acceleration produced by a magnetic field of magnitude 1.25 $\mathrm{T}$ in the $x y-$ plane, directed at an angle of $45.0^{\circ}$ counterclockwise from the $+x$ -axis?

Salamat Ali

Numerade Educator

Problem 10

$\bullet$ A 150 $\mathrm{V}$ battery is connected across two parallel metal plates of area 28.5 $\mathrm{cm}^{2}$ and separation 8.20 $\mathrm{mm} .$ A beam of alpha particles (charge $+2 e,$ mass $6.64 \times 10^{-27} \mathrm{kg} )$ is accelerated from rest through a potential difference of 1.75 $\mathrm{kV}$ and enters the region between the plates perpendicular to the electric field. What magnitude and direction of magnetic field are needed so that the alpha particles emerge undeflected from between the plates?

Averell Hause

Carnegie Mellon University

Problem 11

A velocity selector having uniform perpendicular electric and magnetic fields is shown in Figure $20.59 .$ The electric field is provided by a 150 $\mathrm{V} \mathrm{DC}$ battery connected across two large parallel metal plates that are 4.50 $\mathrm{cm}$ apart. (a) What must be the magnitude of the magnetic field so that charges having a velocity of 3.25 $\mathrm{km} / \mathrm{s}$ perpendicular to the fields will pass through undeflected? (b) Show how the magnetic field should point in the region between the plates.

Rashmi Sinha

Numerade Educator

Problem 12

An electron moves at $2.50 \times 10^{6} \mathrm{m} / \mathrm{s}$ through a region in which there is a magnetic field of unspecified direction and magnitude $7.40 \times 10^{-2} \mathrm{T}$ (a) What are the largest and smallest possible magnitudes of the acceleration of the electron due to the magnetic field? (b) If the actual acceleration of the electron is one-fourth of the largest magnitude in part (a), what is the angle between the electron velocity and the magnetic field?

Narayan Hari

Numerade Educator

Problem 13

$\bullet$ In a cloud chamber experiment, a proton enters a uniform 0.250 T magnetic field directed perpendicular to its motion. You measure the proton's path on a photograph and find that it follows a circular arc of radius 6.13 $\mathrm{cm} .$ How fast was the proton moving?

Salamat Ali

Numerade Educator

Problem 14

An alpha particle (a He nucleus, containing two protons and two neutrons and having a mass of $6.64 \times 10^{-27}$ kg traveling horizontally at 35.6 $\mathrm{km} / \mathrm{s}$ enters a uniform, vertical, 1.10 $\mathrm{T}$ magnetic field. (a) What is the diameter of the path followed by this alpha particle? (b) What effect does the magnetic field have on the speed of the particle? (c) What are the magnitude and direction of the acceleration of the alpha particle while it is in the magnetic field? (d) Explain why the speed of the particle does not change even though an unbalanced external force acts on it.

Averell Hause

Carnegie Mellon University

Problem 15

A deuteron particle (the nucleus of an isotope of hydrogen consisting of one proton and one neutron and having a mass of $3.34 \times 10^{-27} \mathrm{kg}$ ) moving horizontally enters a uniform, vertical, 0.500 T magnetic field and follows a circular arc of radius 55.6 $\mathrm{cm} .$ (a) How fast was this deuteron moving just before it entered the magnetic field and just after it came out of the field? (b) What would be the radius of the arc followed by a proton that entered the field with the same velocity as the deuteron?

Salamat Ali

Numerade Educator

Problem 16

$\bullet$ A beam of protons traveling at 1.20 $\mathrm{km} / \mathrm{s}$ enters a uniform magnetic field, traveling perpendicular to the field. The beam exits the magnetic field in a direction perpendicular to its original direction (Fig. $20.60 ) .$ The beam travels a distance of 1.18 cm while in the field. What is the magnitude of the magnetic field?

Averell Hause

Carnegie Mellon University

Problem 17

A uniform magnetic field bends an electron in a circular arc of radius $R .$ What will be the radius of the arc (in terms of $R )$ if the field is tripled?

Salamat Ali

Numerade Educator

Problem 18

\bullet An electron at point $A$ in Figure 20.61 has a speed $v_{0}$ of $1.41 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Find (a) the magnitude and direction of the magnetic field that will cause the electron to follow the semicircular path from $A$ to $B$ and (b) the time required for the electron to move from $A$ to $B$ . (c) What magnetic field would be needed if the particle were a proton instead of an electron?

Averell Hause

Carnegie Mellon University

Problem 19

$\bullet$ A beam of protons is accelerated through a potential dif- ference of 0.745 $\mathrm{kV}$ and then enters a uniform magnetic field traveling perpendicular to the field. (a) What magnitude of field is needed to bend these protons in a circular arc of diameter 1.75 $\mathrm{m} ?$ (b) What magnetic field would be needed to produce a path with the same diameter if the particles were electrons having the same speed as the protons?

Salamat Ali

Numerade Educator

Problem 20

$\bullet$ A 3.25 g bullet picks up an electric charge of 1.65$\mu C$ as it travels down the barrel of a rifle. It leaves the barrel at a speed of 425 $\mathrm{m} / \mathrm{s}$ , traveling perpendicular to the earth's magnetic field, which has a magnitude of $5.50 \times 10^{-4} \mathrm{T} .$ Calculate (a) the magnitude of the magnetic force on the bullet and (b) the magnitude of the bullet's acceleration due to the magnetic force at the instant it leaves the rifle barrel.

Averell Hause

Carnegie Mellon University

Problem 21

An electron in the beam of a TV picture tube is accelerated through a potential difference of 2.00 $\mathrm{kV}$ . It then passes into a magnetic field perpendicular to its path, where it moves in a circular arc of diameter 0.360 $\mathrm{m} .$ What is the magnitude of this field?

Salamat Ali

Numerade Educator

Problem 22

(a) What is the speed of a beam of electrons when the simultaneous influence of an electric field of $1.56 \times 10^{4} \mathrm{V} / \mathrm{m}$ and a magnetic field of $4.62 \times 10^{-3} \mathrm{T}$ , with both fields normal to the beam and to each other, produces no deflection of the electrons? (b) In a diagram, show the relative orientation of the vectors $\vec{\boldsymbol{v}}$ , $\vec{\boldsymbol{E}}$ and $\vec{\boldsymbol{B}}$ . (c) When the electric field is removed, what is the radius of the electron orbit? What is the period of the orbit?

Averell Hause

Carnegie Mellon University

Problem 23

$\bullet$ Singly ionized (onne electron removed) atoms are accelerated and then passed through a velocity selector consisting of perpendicular electric and magnetic fields. The electric field is 155 $\mathrm{V} / \mathrm{m}$ and the magnetic field is 0.0315 $\mathrm{T}$ . The ions next enter a uniform magnetic field of magnitude 0.0175 $\mathrm{T}$ that is oriented perpendicular to their velocity. (a) How fast are the ions moving when they emerge from the velocity selector? (b) If the radius of the path of the ions in the second magnetic field is $17.5 \mathrm{cm},$ what is their mass?

Salamat Ali

Numerade Educator

Problem 24

$\bullet$ Determining diet. One method for determining the amount of corn in early Native American diets is the stable isotope ratio analysis (SIRA) technique. As corn photosynthesizes, it concentrates the isotope carbon-13, whereas most other plants concentrate carbon-12. Overreliance on corn consumption can then be correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use a mass spectrometer to separate the $^{12} \mathrm{C}$ and $^{13} \mathrm{C}$ isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed 8.50 $\mathrm{km} / \mathrm{s}$ and want to bend them within a uniform magnetic field in a semicircle of diameter 25.0 $\mathrm{cm}$ for the 12 $\mathrm{C}$ . The measured masses of these isotopes are $1.99 \times 10^{-26} \mathrm{kg}(12 \mathrm{C})$ and $2.16 \times 10^{-26} \mathrm{kg}\left(^{13} \mathrm{C}\right) .$ (a) What strength of magnetic field is required? (b) What is the diameter of the $^{13} \mathrm{C}$ semicircle? (c) What is the separation of the $^{12} \mathrm{C}$ and $^{13} \mathrm{Cions}$ at the detec- tor at the end of the semicircle? Is this distance large enough to be easily observed?

Averell Hause

Carnegie Mellon University

Problem 25

$\bullet$ Ancient meat eating. The amount of meat in prehistoric diets can be determined by measuring the ratio of the isotopes nitrogen-15 to nitrogen-l4 in bone from human remains. Carnivores concentrate $^{15} \mathrm{N},$ so this ratio tells archaeologists how much meat was consumed by ancient people. Use the spectrometer of the previous problem to find the separation of the 14 $\mathrm{N}$ and 15 $\mathrm{N}$ isotopes at the detector. The measured masses of these isotopes are $2.32 \times 10^{-26} \mathrm{kg}(14 \mathrm{N})$ and $2.49 \times 10^{-26} \mathrm{kg}$ $\left(^{15} \mathrm{N}\right) .$

Ajay Singhal

Numerade Educator

Problem 26

A straight vertical wire carries a current of 1.20 $\mathrm{A}$ down- ward in a region between the poles of a large electromagnet where the field strength is 0.588 $\mathrm{T}$ and is horizontal. What are the magnitude and direction of the magnetic force on a 1.00 $\mathrm{cm}$ section of this wire if the magnetic-field direction is (a) toward the east, (b) toward the south, (c) $30.0^{\circ}$ south of west?

Averell Hause

Carnegie Mellon University

Problem 27

Magnetic force on a lightning bolt. Currents during lightning strikes can be up to $50,000$ A (or more!). We can model such a strike as a $50,000$ A vertical current perpendicular to the earth's magnetic field, which is about $\frac{1}{2}$ gauss. What is the force on each meter of this current due to the earth's magnetic field?

Salamat Ali

Numerade Educator

Problem 28

A horizontal rod 0.200 $\mathrm{m}$ long carries a current through a uniform horizontal magnetic field of magnitude 0.067 T that points perpendicular to the rod. If the magnetic force on this rod is measured to be $0.13 \mathrm{N},$ what is the current flowing through the rod?

Averell Hause

Carnegie Mellon University

Problem 29

A straight 2.5 $\mathrm{m}$ wire carries a typical household current of 1.5 $\mathrm{A}$ (in one direction) at a location where the earth's magnetic field is 0.55 gauss from south to north. Find the magnitude and direction of the force that our planet's magnetic field exerts on this wire if is oriented so that the current in it is running (a) from west to east, (b) vertically upward, (c) from north to south. (d) Is the magnetic force ever large enough to cause significant effects under normal household conditions?

Salamat Ali

Numerade Educator

Problem 30

$\bullet$ Between the poles of a powerful magnet is a cylindrical uniform magnetic field with a diameter of 3.50 $\mathrm{cm}$ and a strength of 1.40 $\mathrm{T}$ . A wire carries a current through the center of the field at an angle of $65.0^{\circ}$ to the magnetic field lines. If the wire experiences a magnetic force of $0.0514 \mathrm{N},$ what is the current flowing in it?

Averell Hause

Carnegie Mellon University

Problem 31

A rectangular 10.0 $\mathrm{cm}$ by 20.0 $\mathrm{cm}$ circuit carrying an 8.00 $\mathrm{A}$ current is oriented with its plane parallel to a uniform 0.750 T magnetic field (Figure 20.62$)$ . (a) Find the magnitude and direction of the magnetic force on each segment
$(a b, b c,$ etc. $)$ of this circuit. Illustrate your answers with clear diagrams. (b) Find the magnitude of the net force on the entire circuit.

Salamat Ali

Numerade Educator

Problem 32

\bulletA long wire carrying a 6.00 A current reverses direction by means of two right- angle bends, as shown in Figure $20.63 .$ The part of the wire where the bend occurs is in a magnetic field of 0.666 $\mathrm{T}$ confined to the circular region of diameter $75 \mathrm{cm},$ as shown. Find the magnitude and direction of the net force that the magnetic field exerts on this wire.

Ankur S

Numerade Educator

Problem 33

e. A long wire carrying 4.50 A of current makes two $90^{\circ}$ bends, as shown in Figure 20.64 . The bent part of the wire passes through a uniform 0.240 T magnetic field directed as shown in the figure and confined to a limited region of space. Find the magnitude and direction of the force that the magnetic field exerts on the wire.

James Kiss

Numerade Educator

Problem 34

$\cdot$ The 20.0 $\mathrm{cm}$ by 35.0 $\mathrm{cm}$ rectangular circuit shown in Figure 20.65 is hinged along side ab. It carries a clockwise 5.00 A current and is located in a uniform 1.20 $\mathrm{T}$ magnetic field oriented perpendicular to two of its sides, as shown. (a) Make a clear diagram showing the direction of the force that the magnetic field exerts on each segment of the circuit $(a b, b c,$ etc. $) .$ (b) Of the four forces you drew in part (a), decide which ones exert a torque about the hinge ab. Then calculate only those forces that exert this torque. (c) Use your results from part (b) to calculate the torque that the magnetic field exerts on the circuit about the hinge axis $a b .$

Averell Hause

Carnegie Mellon University

Problem 35

$\cdot$ The plane of a 5.0 $\mathrm{cm}$ by 8.0 $\mathrm{cm}$ rectangular loop of wire is parallel to a 0.19 T magnetic field, and the loop carries a cur- rent of 6.2 $\mathrm{A}$ . (a) What torque acts on the loop? (b) What is the magnetic moment of the loop?

Salamat Ali

Numerade Educator

Problem 36

A circular coil of wire 8.6 $\mathrm{cm}$ in diameter has 15 turns and carries a current of 2.7 $\mathrm{A}$ . The coil is in a region where the magnetic field is 0.56 $\mathrm{T}$ (a) What orientation of the coil gives the maximum torque on the coil, and what is this maximum torque? (b) For what orientation of the coil is the magnitude of the torque 71$\%$ of the maximum found in part (a)?

Averell Hause

Carnegie Mellon University

Problem 37

A rectangular coil of wire 22.0 $\mathrm{cm}$ by 35.0 $\mathrm{cm}$ and carrying a current of 1.40 $\mathrm{A}$ is oriented with the plane of its loop perpendicular to a uniform 1.50 $\mathrm{T}$ magnetic field, as shown in Figure 20.66 (a) Calculate the net force and torque that the magnetic field exerts on this coil. (b) The coil is now rotated through a $30.0^{\circ}$ angle about the axis shown, the left side coming out of the plane and the right side going into the plane. Calculate the net force and torque that the magnetic field exerts on the coil. (Hint: In order to help visualize this three-dimensional problem, make a careful drawing of the coil as viewed along its axis of rotation.)

Salamat Ali

Numerade Educator

Problem 38

A solenoid having 165 turns and a cross-sectional area of 6.75 $\mathrm{cm}^{2}$ carries a current of 1.20 A. If it is placed in a uniform 1.12 T magnetic field, find the torque this field exerts on the solenoid if its axis is oriented (a) perpendicular to the field, (b) parallel to the field, (c) at 35. $0^{\circ}$ with the field.

Averell Hause

Carnegie Mellon University

Problem 39

\bullet A circular coil of 50 loops and diameter 20.0 $\mathrm{cm}$ is lying flat on a tabletop, and carries a clockwise current of 2.50 A. A magnetic field of 0.450 $\mathrm{T}$ , directed to the north and at an angle of $45.0^{\circ}$ from the vertical down through the coil and into the tabletop is turned on. (a) What is the torque on the coil, and (b) which side of the coil (north or south) will tend to rise from the tabletop?

Vishal Gupta

Numerade Educator

Problem 40

You want to produce a magnetic field of magnitude $5.50 \times 10^{-4} \mathrm{T}$ at a distance of 0.040 $\mathrm{m}$ from a long, straight wire's center. (a) What current is required to produce this field? (b) With the current found in part (a), how strong is the magnetic field 8.00 $\mathrm{cm}$ from the wire's center?

Averell Hause

Carnegie Mellon University

Problem 41

Household magnetic fields. Home circuit breakers typically have current capacities of around 10 A. How large a magnetic field would such a current produce 5.0 $\mathrm{cm}$ from a long- wire's center? How does this field compare with the strength of the earth's magnetic field?

Salamat Ali

Numerade Educator

Problem 42

$\bullet$ (a) How large a current would a very long, straight wire have to carry so that the magnetic field 2.00 $\mathrm{cm}$ from the wire is equal to 1.00 $\mathrm{G}$ (comparable to the earth's northward-pointing magnetic field)? (b) If the wire is horizontal with the current running from east to west, at what locations would the magnetic field of the wire point in the same direction as the horizontal component
of the earth's magnetic field? (c) Repeat part (b) except with the wire vertical and the current going upward.

Averell Hause

Carnegie Mellon University

Problem 43

$\bullet$ Currents in the heart. The body contains many small currents caused by the motion of ions in the organs and cells. Measurements of the magnetic field around the chest due to currents in the heart give values of about 1.0 $\mu$ . Although the actual currents are rather complicated, we can gain a rough understanding of their magnitude if we model them as a long, straight wire. If the surface of the chest is 5.0 $\mathrm{cm}$ from this current, how large is the current in the heart?

Salamat Ali

Numerade Educator

Problem 44

Magnetic sensitivity of electric fish. In a problem dealing with electric fish in Chapter 19, we saw that these fish navigate by responding to changes in the current in seawater. This current is due to a potential difference of around 3.0 $\mathrm{V}$ generated by the fish and is about 12 $\mathrm{mA}$ within a centimeter or so from the fish. Receptor cells in the fish are sensitive to the current. since the current is at some distance from the fish, the sensitivity of these cells suggests that they might be responding to the magnetic field created by the current. To get some estimate of how sensitive the cells are, we can model the current as that of a long, straight wire with the receptor cells 2.0 $\mathrm{cm}$ away. What is the strength of the magnetic field at the receptor cells?

Averell Hause

Carnegie Mellon University

Problem 45

$\cdot$ In a conventional cheap flashlight, a straight copper strip runs along the tube of the flashlight to connect the bulb to the negative terminal of the battery at the bottom of the tube. If this strip carries a current of 0.65 A while you're holding the flashlight, what is the magnitude of the magnetic field at the
surface of your hand, 0.30 $\mathrm{cm}$ from the strip? (You can treat the strip as a long, thin, straight wire.) How does your answer compare to the earth's magnetic field?

Salamat Ali

Numerade Educator

Problem 46

$\bullet$ If the magnetic field due to a long, straight current-carrying wire has a magnitude $B$ at a distance $R$ from the wire's center, how far away must you be (in terms of $R$ ) for the magnetic field to decrease to $B / 3 ?$

Averell Hause

Carnegie Mellon University

Problem 47

$\bullet$ A current in a long, straight wire produces a magnetic field of 8.0$\mu$ t at 2.0 $\mathrm{cm}$ from the wire's center. Answer the following questions without finding the current: (a) What is the magnetic field strength 4.0 $\mathrm{cm}$ from the wire's center? (b) How far from the wire's center will the field be 1.0$\mu \mathrm{T} ?$ (c) If the current were doubled, what would the field be 2.0 $\mathrm{cm}$ from the wire's center?

Salamat Ali

Numerade Educator

Problem 48

$\bullet$ EMF. Currents in DC transmission lines can be 100 A or more. Some people have expressed concern that the electromagnetic fields (EMFs) from such lines near their homes could cause health dangers. Using your own observations, estimate how high such lines are above the ground. Then use your estimate to calculate the strength of the magnetic field these lines produce at ground level. Express your answer in teslas and as a percent of the earth's magnetic field (which is 0.50 gauss). Does it seem that there is cause for worry?

Averell Hause

Carnegie Mellon University

Problem 49

A long, straight telephone cable contains six wires, each carrying a current of 0.300 A. The distances between wires can be neglected. (a) If the currents in all six wires are in the same direction, what is the magnitude of the magnetic field 2.50 $\mathrm{m}$ from the cable? (b) If four wires carry currents in one direction and the other two carry currents in the opposite direction, what is the magnitude of the field 2.50 m from the cable?

Salamat Ali

Numerade Educator

Problem 50

" Two insulated wires perpendicular to each other in the same plane carry cur- rents as shown in Figure $20.67 .$ Find the magnitude of the net magnetic field these wires produce at points $P$ and $Q$ if the 10.0 A current is (a) to the right or (b) to the left.

Averell Hause

Carnegie Mellon University

Problem 51

Two long, straight parallel wires are 10.0 $\mathrm{cm}$ apart and carry 4.00 $\mathrm{A}$ currents in the same direction (Figure 20.68). Find the magnitude and direction of the magnetic field at (a) point $P_{1}$ midway between the wires, (b) point $P_{2}, 25.0 \mathrm{cm}$ to the right of $P_{1}$

Vishal Gupta

Numerade Educator

Problem 52

Two long parallel transmission lines 40.0 $\mathrm{cm}$ apart carry 25.0 $\mathrm{A}$ and 75.0 A currents. Find all locations where the net magnetic field of the two wires is zero if these currents are in (a) the same direction, (b) opposite directions.

Averell Hause

Carnegie Mellon University

Problem 53

$\cdot$ Two high-current transmission lines carry currents of 25 $\mathrm{A}$ and 75 $\mathrm{A}$ in the same direction and are suspended parallel to each other 35 $\mathrm{cm}$ apart. If the vertical posts supporting these wires divide the lines into straight 15 $\mathrm{m}$ segments, what magnetic force does each segment exert on the other? Is this force attractive or repulsive?

Salamat Ali

Numerade Educator

Problem 54

. Two long current-carrying wires run parallel to each other. Show that if the currents run in the same direction, these wires attract each other, whereas if they run in opposite directions, the wires repel.

Averell Hause

Carnegie Mellon University

Problem 55

\bullet A $\mathrm{A} 2.0 \mathrm{m}$ ordinary lamp extension cord carries a 5.0 $\mathrm{A}$ current. Such a cord typically consists of two parallel wires carrying equal currents in opposite directions. Find the magnitude and direction (attractive or repulsive) that the two segments of this cord exert on each other. (You will need to inspect an actual lamp cord at home and measure or reasonably estimate the quantities needed to do this calculation.)

Salamat Ali

Numerade Educator

Problem 56

$\bullet$ An electric bus operates by drawing current from two parallel overhead cables, at a potential difference of $600 \mathrm{V},$ and spaced 55 $\mathrm{cm}$ apart. When the power input to the bus's motor is at its maximum power of $65 \mathrm{hp},($ a) what current does it draw and (b) what is the attractive force per unit length between the cables?

Averell Hause

Carnegie Mellon University

Problem 57

A circular metal loop is 22 $\mathrm{cm}$ in diameter. (a) How large a current must flow through this metal so that the magnetic field at its center is equal to the earth's magnetic field of $0.50 \times$ $10^{-4} \mathrm{T}$ (b) Show how the loop should be oriented so that it can cancel the earth's magnetic field at its center.

Salamat Ali

Numerade Educator

Problem 58

A closely wound circular coil with a diameter of 4.00 $\mathrm{cm}$ has 600 turns and carries a current of 0.500 A. What is the magnetic field at the center of the coil?

Averell Hause

Carnegie Mellon University

Problem 59

$\bullet$ A closely wound circular coil has a radius of 6.00 $\mathrm{cm}$ and carries a current of 2.50 A. How many turns must it have if the magnetic field at its center is $6.39 \times 10^{-4} \mathrm{T} ?$

Salamat Ali

Numerade Educator

Problem 60

Currents in the brain. The magnetic field around the head has been measured to be approximately $3.0 \times 10^{-8}$ gauss. Although the currents that cause this field are quite complicated, we can get a rough estimate of their size by modeling them as a single circular current loop 16 $\mathrm{cm}$ (the width of a typical head) in diameter. What is the current needed to produce such a field at the center of the loop?

Averell Hause

Carnegie Mellon University

Problem 61

A closely wound, circular coil with radius 2.40 $\mathrm{cm}$ has 800 turns. What must the current in the coil be if the magnetic field at the center of the coil is 0.0580 $\mathrm{T}$ ?

Salamat Ali

Numerade Educator

Problem 62

$\bullet$ Two circular concentric loops of wire lie on a tabletop, one inside the other. The inner loop has a diameter of 20.0 $\mathrm{cm}$ and carries a clockwise current of 12.0 $\mathrm{A}$ , as viewed from above, and the outer wire has a diameter of 30.0 $\mathrm{cm} .$ What must be the magnitude and direction (as viewed from above) of the current in the outer loop so that the net magnetic field due to this combination of loops is zero at the common center of the loops?

Averell Hause

Carnegie Mellon University

Problem 63

$\bullet$ Calculate the magnitude and direction of the magnetic field at point $P$ due to the current in the semicircular section of wire shown in Figure 20.69 . (Hint: The current in the long, straight section of wire produces no field at $P .$ Can you relate the semicircle to a current loop?)

Salamat Ali

Numerade Educator

Problem 64

$\cdot$ A solenoid contains 750 coils of very thin wire evenly wrapped over a length of 15.0 $\mathrm{cm} .$ Each coil is 0.800 $\mathrm{cm}$ in diameter. If this solenoid carries a current of $7.00 \mathrm{A},$ what is the magnetic field at its center?

Averell Hause

Carnegie Mellon University

Problem 65

$\bullet$ As a new electrical technician, you are designing a large solenoid to produce a uniform 0.150 T magnetic field near its center. You have enough wire for 4000 circular turns, and the solenoid must be 1.40 m long and 2.00 $\mathrm{cm}$ in diameter. What current will you need to produce the necessary field?

Salamat Ali

Numerade Educator

Problem 66

$\cdot$ A solenoid is designed to produce a 0.0279 T magnetic field near its center. It has a radius of 1.40 $\mathrm{cm}$ and a length of $40.0 \mathrm{cm},$ and the wire carries a current of 12.0 A. (a) How many turns must the solenoid have? (b) What total length of wire is required to make this solenoid?

Averell Hause

Carnegie Mellon University

Problem 67

\bullet A single circular current loop 10.0 $\mathrm{cm}$ in diameter carries a 2.00 A current. (a) What is the magnetic field at the center of this loop? (b) Suppose that we now connect 1000 of these loops in series within a 500 $\mathrm{cm}$ length to make a solenoid 500 $\mathrm{cm}$ long. What is the magnetic field at the center of this solenoid? Is it 1000 times the field at the center of the loop in part (a)? Why or why not?

Salamat Ali

Numerade Educator

Problem 68

A solenoid that is 35 $\mathrm{cm}$ long and contains 450 circular coils 2.0 $\mathrm{cm}$ in diameter carries a 1.75 A current. (a) What is the magnetic field at the center of the solenoid, 1.0 $\mathrm{cm}$ from the coils? (b) Suppose we now stretch out the coils to make a very long wire carrying the same current as before. What is the magnetic field 1.0 $\mathrm{cm}$ from the wire's center? Is it the same as you found in part (a)? Why or why not?

Averell Hause

Carnegie Mellon University

Problem 69

. You have 25 $\mathrm{m}$ of wire, which you want to use to construct a 44 $\mathrm{cm}$ diameter coil whose magnetic field at its center will exactly cancel the earth's field of 0.55 gauss. What current will your coil require?

Salamat Ali

Numerade Educator

Problem 70

$\bullet$ A toroidal solenoid (see Figure 20.42$)$ has inner radius $r_{1}=15.0 \mathrm{cm}$ and outer radius $r_{2}=18.0 \mathrm{cm} .$ The solenoid has 250 turns and carries a current of 8.50 A. What is the magnitude of the magnetic field at the following distances from the center of the torus: (a) $12.0 \mathrm{cm} ;$ (b) $16.0 \mathrm{cm} ;$ (c) 20.0 $\mathrm{cm} ?$

Averell Hause

Carnegie Mellon University

Problem 71

A long, straight wire carries a current of 10.0 $\mathrm{A}$ , as shown in Figure $20.70 .$ Use the law of Biot and Savart to find the magnitude and direction of the magnetic field at point $P$ due to each of the following 2.00 $\mathrm{mm}$ segments of this wire: (a) segment $A$ and (b) segment $C .$

Salamat Ali

Numerade Educator

Problem 72

A long wire carrying a 5.00 A current makes an abrupt right-angle bend as shown in Figure 20.71 Use the law of Biot and Savart to determine the magnitude and direction of the magnetic field at point $P$ due to the 1.50 $\mathrm{cm}$ bent segment if $P$ is 15.0 $\mathrm{cm}$ from the midpoint of that segment.

Averell Hause

Carnegie Mellon University

Problem 73

$\bullet$ Three long, straight electrical cables, running north and south, are tightly enclosed in an insulating sheath. One of the cables carries a 23.0 A current southward; the other two carry currents of 17.5 $\mathrm{A}$ and 11.3 A northward. Use Ampere's law to calculate the magnitude of the magnetic field at a distance of 10.0 $\mathrm{m}$ from the cables.

Salamat Ali

Numerade Educator

Problem 74

A long, straight, cylindrical wire of radius $R$ carries a cur- rent uniformly distributed over its cross section. At what location is the magnetic field produced by this current equal to half of its largest value? Use Ampere's law and consider points inside and outside the wire.

Averell Hause

Carnegie Mellon University

Problem 75

$\cdot$ Platinum is a paramagnetic metal having a relative perme- ability of 1.00026 (a) What is the magnetic permeability of platinum? (b) If a thin rod of platinum is placed in an external magnetic field of 1.3500 $\mathrm{T}$ , with its axis parallel to that field, what will be the magnetic field inside the rod?

Salamat Ali

Numerade Educator

Problem 76

$\bullet$ When a certain paramagnetic material is placed in an external magnetic field of 1.5000 $\mathrm{T}$ , the field inside the material is measured to be 1.5023 $\mathrm{T}$ . Find (a) the relative permeability and (b) the magnetic permeability of this material.

Averell Hause

Carnegie Mellon University

Problem 77

\bullet A 150 g ball containing $4.00 \times 10^{8}$ excess electrons is dropped into a 125 vertical shaft. At the bottom of the shaft, the ball suddenly enters a uniform horizontal 0.250 T magnetic field directed from east to west. If air resistance is negligibly small, find the magnitude and direction of the force that this magnetic field exerts on the ball just as it enters the field.

Salamat Ali

Numerade Educator

Problem 78

. Magnetic balance. The circuit shown in Figure 20.72 is used to make a magnetic balance to weigh objects. The mass $m$ to be measured is hung from the center of the bar, which is in a uniform magnetic field of 1.50 $\mathrm{T}$ directed into the plane of the figure. The battery volt- age can be adjusted to vary the current in the circuit. The horizontal bar is 60.0 $\mathrm{cm}$ long and is made of extremely light-weight material, so its mass can be neglected. It is connected to the battery by thin vertical wires that can support no appreciable tension; all the weight of the mass $m$ is supported by the magnetic force on the bar. A 5.00$\Omega$ resistor is in series with the bar, and the resistance of the rest of the circuit is negligibly small. (a) Which point, $a$ or $b,$ should be the positive terminal of the battery? (b) If the maximum terminal voltage of the battery is 175 $\mathrm{V}$ , what is the greatest mass $m$ that this instrument can measure?

Averell Hause

Carnegie Mellon University

Problem 79

\bullet A thin 50.0 -cm-long metal bar with mass 750 g rests on, but is not attached to, two metal supports in a uniform 0.450 T magnetic field, as shown in Figure $20.73 .$ A battery and a 25.0$\Omega$ resistor in series are connected to the supports. What is the largest terminal voltage the battery can have without breaking the circuit at the supports?

Salamat Ali

Numerade Educator

Problem 80

\bullet A long, straight wire containing a semicircular region of radius 0.95 m is placed in a uni-form magnetic field of magnitude 2.20 $\mathrm{T}$ as shown in Figure 20.74 . What is the net magnetic force acting on the wire when it carries a current of 3.40 A? (Hint: In Figure $20.74,$ what does symmetry tell you about the forces on the upper and lower halves of the semicircular region?

Averell Hause

Carnegie Mellon University

Problem 81

$\bullet$ A singly charged ion of $^{7} \mathrm{Li}$ (an isotope of lithium containing three protons and four neutrons) has a mass of $1.16 \times$ $10^{-26} \mathrm{kg} .$ It is accelerated through a potential difference of 220 $\mathrm{V}$ and then enters a 0.723 T magnetic field perpendicular to the ion's path. What is the radius of the path of this ion in the magnetic field?

Salamat Ali

Numerade Educator

Problem 82

An insulated circular ring of diameter 6.50 $\mathrm{cm}$ carries a 12.0 A current and is tangent to a very long, straight insulated wire carrying 10.0 $\mathrm{A}$ of current, as shown in Figure $20.75 .$ Find the magnitude and direction of the magnetic field at the center of the ring due to this combination of wires.

Averell Hause

Carnegie Mellon University

Problem 83

The effect of transmission lines. Two hikers are reading a compass under an overhead transmission line that is 5.50 $\mathrm{m}$ above the ground and carries a current of 0.800 $\mathrm{kA}$ in a horizontal direction from north to south. (a) Find the magnitude and direction of the magnetic field at a point on the ground directly under the transmission line. (b) One hiker suggests that they walk 50 $\mathrm{m}$ away from the lines to avoid inaccurate compass readings due to the current. Considering that the earth's magnetic field is on the order of $0.5 \times 10^{-4} \mathrm{T},$ is the current really a problem?

Vishal Gupta

Numerade Educator

Problem 84

A long, straight horizontal wire carries a current of 2.50 $\mathrm{A}$ directed toward the right. An electron is traveling in the vicinity of this wire. (a) At the instant the electron is 4.50 $\mathrm{cm}$ above the wire's center and moving with a speed of $6.00 \times 10^{4} \mathrm{m} / \mathrm{s}$ directly toward it, what are the magnitude and direction of the force that the magnetic field of the current exerts on the electron? (b) What would be the magnitude and direction of the magnetic force if the electron were instead moving parallel to the wire in the same direction as the current?

Averell Hause

Carnegie Mellon University

Problem 85

$\bullet$ Two very long, straught wires carry currents as shown in Figure $20.76 .$ For each case shown, find all locations where the net magnetic field due to these wires is zero.

Vishal Gupta

Numerade Educator

Problem 86

\bullet Bubble chamber, I. Certain types of bubble chambers are filled with liquid hydrogen. When a particle (such as an electron or a proton) passes through the liquid, it leaves a track of bubbles, which can be photographed to show the path of the particle. The apparatus is immersed in a known magnetic field, which causes the particle to curve. Figure 20.77 is a trace of a bubble chamber image showing the path of an electron. (a) How could you determine the sign of the charge of a particle from a photograph of its path? (b) How can physicists determine the momentum and the speed of this electron by using measurements made on the photograph, given that the magnetic field is known and is perpendicular to the plane of the figure? (c) The electron is obviously spiraling into smaller and smaller circles. What properties of the electron must be changing to cause this behavior? Why does this happen? (d) What would be the path of a neutron in a bubble chamber? Why?

Averell Hause

Carnegie Mellon University

Problem 87

\bullet A 3.00 N metal bar, 1.50 $\mathrm{m}$ long and having a resistance of 10.0$\Omega$ , rests horizontally on conducting wires connecting it to the circuit shown in a Figure $20.78 .$ The bar is in a uniform, horizontal, 1.60 $\mathrm{T}$ magnetic field and is not attached to the wires in the circuit. What is the acceleration of the bar just after the switch $S$ is closed?

Salamat Ali

Numerade Educator

Problem 88

A pair of long, rigid metal rods, each of length $L$ , lie parallel to each other on a perfectly smooth table. Their ends are connected by identical, very light conducting springs of force constant $k$ (Figure 20.79$)$ and negligible unstretched length. If a current $I$ runs through this circuit, the springs will stretch. At what separation will stretch. At remain at rest? Assume that $k$ is large enough so that the separation of the rods will be much less than $L .$

Averell Hause

Carnegie Mellon University

Problem 89

$\bullet$ Atom smashers! A cyclotron particle accelerator (sometimes called an "atom smasher" in the popular press) is a device for accelerating charged particles, such as electrons and protons, to speeds close to the speed of light. The basic design is quite simple. The particle is bent in a circular path by a uniform magnetic field. An electric field is pulsed periodically to increase the speed of the particle. The charged particle (or ion) of mass $m$ and charge $q$ is introduced into the cyclotron so that it is moving perpendicular to a uniform magnetic field $\vec{B}$ (a) Starting with the radius of the circular path of a charge moving in a uniform magnetic field, show that the time $T$ for this particle to make one complete circle is $T=\frac{2 \pi m}{|q| B}$ . (Hint: You can express the speed $v$ in terms of $R$ and $T$ because the particle travels through one circumference of the circle in time $T$ . (b) Which would take longer to complete one circle, an ion moving in a large circle or one moving in a small circle? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 90

Medical uses of cyclotrons. The largest cyclotron (see the previous problem) in the United States is the Tevatron at Fermilab, near Chicago, Illinois. It is called a Tevatron be- cause it can accelerate particles to energies in the TeV range (1 tera-eV $=10^{12}$ eV ). Its circumference is $6.4 \mathrm{km},$ and it currently can produce a maximum energy of 2.0 TeV. In a certain medical experiment, protons will be accelerated to energies of 1.25 MeV and aimed at a tumor to destroy its cells. (a) How fast are these protons moving when they hit the tumor? (b) How strong must the magnetic field be to bend the protons in the circle indicated?

Averell Hause

Carnegie Mellon University

Problem 91

\bullet A plastic circular loop has radius $R,$ and a positive charge $q$ is distributed uniformly around the circumference of the loop. The loop is now rotated around its central axis, perpendicular to the plane of the loop, with angular speed $\omega .$ If the loop is in a region where there is a uniform magnetic field $\frac{1}{B}$ directed parallel to the plane of the loop, calculate the magnitude of the magnetic torque on the loop.

Salamat Ali

Numerade Educator

Problem 92

\bulletA long wire carrying 6.50 A of current makes two bends, as shown in Figure 20.80 . The bent part of the wire passes through a uni-form 0.280 T magnetic field directed as shown in the figure and confined to a limited region of space. Find the magnitude and direction of the force that the magnetic field exerts on the wire.

Averell Hause

Carnegie Mellon University

Problem 93

If a proton is exposed to an external magnetic field of 2 $\mathrm{T}$ that has a direction perpendicular to the axis of the spin of the proton, what will be the torque on the proton?
A. 0
B. $1.4 \times 10^{-266} \mathrm{N} \cdot \mathrm{m}$
C. $2.8 \times 10^{-26} \mathrm{N} \cdot \mathrm{m}$
D. $0.7 \times 10^{-26} \mathrm{N} \cdot \mathrm{m}$

Salamat Ali

Numerade Educator

Problem 94

Which of following elements is a candidate for MRI?
$\begin{array}{ll}{\text { A. }^{12} \mathrm{C}_{6}} & {\text { B. }^{16} \mathrm{O}_{8}} \\ {\text { C. }^{40} \mathrm{Ca}_{20}} & {\text { D. }^{31} \mathrm{P}_{15}} \\ \hline\end{array}$

Averell Hause

Carnegie Mellon University