College Physics With an Integrated Approach to Forces and Kinematics

Alan Giambattista, Betty McCarthy Richardson , Robert C. Richardson

Chapter 19

Magnetic Forces and Fields - all with Video Answers

Educators

Chapter Questions

Problem 1

At which point in the diagram is the magnetic field strength (a) the smallest and (b) the largest? Explain.

Anna Zeng

Anna Zeng

Numerade Educator

Problem 2

Draw vector arrows to indicate the direction and relative magnitude of the magnetic field at each of the points $A-F$

Anna Zeng

Numerade Educator

Problem 3

Two identical bar magnets lie next to one another on a table. Sketch the magnetic field lines if the north poles are at the same end.

Vishal Gupta

Numerade Educator

Problem 4

Two identical bar magnets lic next to one another on a table. Sketch the magnetic field lines if the north poles are at opposite ends.

Anna Zeng

Numerade Educator

Problem 5

Two identical bar magnet lie on a table along a straight line with their north poles facing each other. Sketch the magnetic field lines.

Vishal Gupta

Numerade Educator

Problem 6

Two identical bar magnets lie on a table along a straight line withopposite poles facing each other. Sketch the magnetic field lines.

Anna Zeng

Numerade Educator

Problem 7

The magnetic forces on a magnetic dipole result in a torque that tends to make the dipole line up with the magnetic field. In this problem we show that the electric forces on an electric dipole result in a torque that tends to make the electric dipole line up with the electric field.
(a) For each orientation of the dipole shown in the diagram, sketch the electric forces and determine the direction of the torque-clockwise or counterclockwiseabout an axis perpendicular to the page through the center of the dipole. (b) The torque always tends to make the dipole rotate toward what orientation?

Anna Zeng

Numerade Educator

Problem 8

Find the magnetic force exerted on an electron moving vertically upward at a speed of $2.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$ by a horizontal magnetic field of $0.50 \mathrm{~T}$ directed north.

Anna Zeng

Numerade Educator

Problem 9

Find the magnetic force exerted on a proton moving east at a speed of $6.0 \times 10^{6} \mathrm{~m} / \mathrm{s}$ by a horizontal magnetic field of $2.50$ T directed north.

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 10

A uniform magnetic field points north; its magnitude is $1.5 \mathrm{~T}$. A proton with kinetic energy $8.0 \times 10^{-13} \mathrm{~J}$ is moving vertically downward in this field. What is the magnetic force acting on it?

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 11

A uniform magnetic field points vertically upward; its magnitude is $0.800 \mathrm{~T}$. An electron with kinetic energy $7.2 \times 10^{-18} \mathrm{~J}$ is moving horizontally eastward in this field. What is the magnetic force acting on it?

Anna Zeng

Numerade Educator

Problem 12

Several electrons move at speed $8.0 \times$ $10^{5} \mathrm{~m} / \mathrm{s}$ in a uniform magnetic field with magnitude $B=0.40 \mathrm{~T}$ directed downward.
Find the magnetic force on the electron at point $a$.

Anna Zeng

Numerade Educator

Problem 13

Several electrons move at speed $8.0 \times$ $10^{5} \mathrm{~m} / \mathrm{s}$ in a uniform magnetic field with magnitude $B=0.40 \mathrm{~T}$ directed downward.
Find the magnetic force on the electron at point $b$.

Anna Zeng

Numerade Educator

Problem 14

Several electrons move at speed $8.0 \times$ $10^{5} \mathrm{~m} / \mathrm{s}$ in a uniform magnetic field with magnitude $B=0.40 \mathrm{~T}$ directed downward.
Find the magnetic force on the electron at point $c$.

Anna Zeng

Numerade Educator

Problem 15

Electrons in a television's CRT are accelerated from rest by an electric field through a potential difference of $2.5 \mathrm{kV}$. In contrast to an oscilloscope, where the electron beam is deflected by an electric field, the beam is deflected by a magnetic field. (a) What is the speed of the electrons? (b) The beam is deflected by a perpendicular magnetic field of magnitude $0.80 \mathrm{~T}$. What is the magnitude of the acceleration of the electrons while in the field? (c) What is the speed of the electrons after they travel $4.0 \mathrm{~mm}$ through the magnetic field? (d) What strength electric field would give the electrons the same magnitude acceleration as in (b)? (e) Why do we have to use an electric field in the first place to get the electrons up to speed? Why not use the large acceleration due to a magnetic field for that purpose?

Donya Dobbin

Numerade Educator

Problem 16

A magnet produces a 0.30-T field between its poles, directed to the east. A dust particle with charge $q=-8.0 \times 10^{-18} \mathrm{C}$ is moving straight down at $0.30 \mathrm{~cm} / \mathrm{s}$
in this field. What is the magnitude and direction of the magnetic force on the dust particle?

Anna Zeng

Numerade Educator

Problem 17

At a certain point on Earth's surface in the southern hemisphere, the magnetic field has a magnitude of $5.0 \times 10^{-5} \mathrm{~T}$ and points upward and toward the north at an angle of $55^{\circ}$ above the horizontal. A cosmic ray muon with the same charge as an electron and a mass of $1.9 \times 10^{-28} \mathrm{~kg}$ is moving directly down toward Earth's surface with a speed of $4.5 \times 10^{7} \mathrm{~m} / \mathrm{s}$. What is the magnitude and direction of the force on the muon?

Anna Zeng

Numerade Educator

Problem 18

An electron beam in vacuum moving at $1.8 \times 10^{7} \mathrm{~m} / \mathrm{s}$ passes between the poles of an electromagnet. The diameter of the magnet pole faces is $2.4 \mathrm{~cm}$ and the field between them is $0.20 \times 10^{-2} \mathrm{~T}$. How far and in what direction is the beam deflected when it hits the screen, which is $25 \mathrm{~cm}$ past the magnet?

Vishal Gupta

Numerade Educator

Problem 19

A positron $(q=+e)$ moves at $5.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$ in a magnetic field of magnitude $0.47 \mathrm{~T}$. The magnetic force on the positron has magnitude $2.3 \times 10^{-12} \mathrm{~N}$. (a) What is the component of the positron's velocity perpendicular to the magnetic field? (b) What is the component of the positron's velocity parallel to the magnetic field?
(c) What is the angle between the velocity and the field?

Anna Zeng

Numerade Educator

Problem 20

An electron moves with speed $2.0 \times 10^{5} \mathrm{~m} / \mathrm{s}$ in a $1.2-\mathrm{T}$ uniform magnetic field. At one instant, the electron is moving due west and experiences an upward magnetic force of $3.2 \times 10^{-14} \mathrm{~N}$. What is the direction of the magnetic field? Be specific: give the angle(s) with respect to $\mathrm{N}, \mathrm{S}, \mathrm{E}, \mathrm{W}$, up, down. (If there is more than one possible answer, find all the possibilities.)

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 21

An electron moves with speed $2.0 \times 10^{5} \mathrm{~m} / \mathrm{s}$ in a uniform magnetic field of $1.4 \mathrm{~T}$, pointing south. At one instant, the electron experiences an upward magnetic force of $1.6 \times 10^{-14} \mathrm{~N}$. In what direction is the electron moving at that instant? Be specific: give the angle(s) with respect to $\mathrm{N}, \mathrm{S}, \mathrm{E}, \mathrm{W}$, up, down.

Anna Zeng

Numerade Educator

Problem 22

The magnetic field in a cyclotron is $0.50 \mathrm{~T}$. Find the magnitude of the magnetic force on a proton with speed $1.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$ moving in a plane perpendicular to the field.

Anna Zeng

Numerade Educator

Problem 23

An electron moves at speed $8.0 \times 10^{5} \mathrm{~m} / \mathrm{s}$ in a plane perpendicular to a cyclotron's magnetic field. The magnitude of the magnetic force on the electron is $1.0 \times 10^{-13} \mathrm{~N}$. What is the magnitude of the magnetic field?

Anna Zeng

Numerade Educator

Problem 24

When two particles travel through a region of uniform magnetic field pointing out of the plane of the paper, they follow the trajectories shown. What are the signs of the charges of each particle?

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 25

The magnetic field in a cyclotron is $0.360 \mathrm{~T}$. The dees have radius $82.0 \mathrm{~cm}$. What maximum speed can a proton achieve in this cyclotron?

Anna Zeng

Numerade Educator

Problem 26

The magnetic field in a cyclotron is $0.50 \mathrm{~T}$. What must be the minimum radius of the dees if the maximum proton speed desired is $1.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$ ?

Anna Zeng

Numerade Educator

Problem 27

A singly charged ion of unknown mass moves in a circle of radius $12.5 \mathrm{~cm}$ in a magnetic field of $1.2 \mathrm{~T}$. The ion. was accelerated through a potential difference of $7.0 \mathrm{kV}$ before it entered the magnetic ficld. What is the mass of the ion?

Anna Zeng

Numerade Educator

Problem 28

The conversion between atomic mass units and kilograms is $$1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{~kg}$$.
Natural carbon consists of two different isotopes (excluding ${ }^{14} \mathrm{C}$, which is present in only trace amounts). The isotopes have different masses, which is due to different numbers of neutrons in the nucleus; however, the number of protons is the same, and subsequently the chemical properties are the same. The most abundant isotope has an atomic mass of $12.00 \mathrm{u}$. When natural carbon is placed in a mass spectrometer, two lines are formed on the photographic plate. The lines show that the more abundant isotope moved in a circle of radius $15.0 \mathrm{~cm}$, while the rarer isotope moved in a circle of radius $15.6 \mathrm{~cm}$. What is the atomic mass of the rarer isotope? (The ions have the same charge and are accelerated through the same potential difference before entering the magnetic field.)

Donya Dobbin

Numerade Educator

Problem 29

The conversion between atomic mass units and kilograms is $$1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{~kg}$$.
After being accelerated through a potential difference of $5.0 \mathrm{kV}$, a singly charged carbon ion $\left({ }^{12} \mathrm{C}^{+}\right)$ moves in a circle of radius $21 \mathrm{~cm}$ in the magnetic field of a mass spectrometer. What is the magnitude of the field?

Donya Dobbin

Numerade Educator

Problem 30

The conversion between atomic mass units and kilograms is $$1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{~kg}$$.
A sample containing carbon (atomic mass $12 \mathrm{u}$ ), oxygen (16 u), and an unknown element is placed in a mass spectrometer. The ions all have the same charge and are accelerated through the same potential difference before entering the magnetic field. The carbon and oxygen lines are separated by $2.250 \mathrm{~cm}$ on the photographic plate, and the unknown element makes a line between them that is $1.160 \mathrm{~cm}$ from the carbon line. (a) What is the mass of the unknown element? (b) Identify the element.

Donya Dobbin

Numerade Educator

Problem 31

The conversion between atomic mass units and kilograms is $$1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{~kg}$$.
A sample containing sulfur (atomic mass $32 \mathrm{u}$ ), manganese $(55 \mathrm{u})$, and an unknown element is placed in a mass spectrometer. The ions have the same charge and are accelerated through the same potential difference before entering the magnetic field. The sulfur and manganese lines are separated by $3.20 \mathrm{~cm}$, and the unknown element makes a line between them that is $1.07 \mathrm{~cm}$ from the sulfur line. (a) What is the mass of the unknown element? (b) Identify the element.

Donya Dobbin

Numerade Educator

Problem 32

The conversion between atomic mass units and kilograms is $$1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{~kg}$$.
In one type of mass spectrometer, ions having the same velocity move through a uniform magnetic field. The spectrometer is being used to distinguish ${ }^{12} \mathrm{C}^{+}$ and ${ }^{14} \mathrm{C}^{+}$ ions that have the same charge. The ${ }^{12} \mathrm{C}^{+}$ ions move in a circle of diameter $25 \mathrm{~cm} .$ (a) What is the diameter of the orbit of ${ }^{14} \mathrm{C}^{+}$ ions? (b) What is the ratio of the frequencies of revolution for the two types of ion?

Donya Dobbin

Numerade Educator

Problem 33

Prove that the time for one revolution of a charged particle moving perpendicular to a uniform magnetic field is independent of its speed. (This is the principle on which the cyclotron operates.) In doing so, write an expression that gives the period $T$ (the time for one revolution) in terms of the mass of the particle, the charge of the particle, and the magnetic field strength.

Donya Dobbin

Numerade Educator

Problem 34

Crossed electric and magnetic fields are established over a certain region. The magnetic field is $0.635 \mathrm{~T}$ vertically downward. The clectric field is $2.68 \times 10^{6} \mathrm{~V} / \mathrm{m}$ horizontally east. An electron, traveling horizontally northward, experiences zero net force from these fields and so continues moving in a straight line. What is the electron's speed?

Anna Zeng

Numerade Educator

Problem 35

A current $I=40.0 \mathrm{~A}$ flows through a strip of metal. An electromagnet is switched on so that there is a uniform magnetic field of magnitude $0.30 \mathrm{~T}$ directed into the page. (a) How would you hook up a voltmeter to measure the Hall voltage? Show how the voltmeter is connected on a sketch of the strip. (b) Assuming the carriers are electrons, which lead of your voltmeter is at the higher potential? Mark it with $\mathrm{a}^{\text {"' }}+$ " sign in your sketch. Explain briefly.

Donya Dobbin

Numerade Educator

Problem 36

In Problem 35 , if the width of the strip is $3.5 \mathrm{~cm}$, the magnetic field is $0.43 \mathrm{~T}$, and the Hall voltage is measured to be $7.2 \mu \mathrm{V}$, what is the drift velocity of the carriers in the strip?

Donya Dobbin

Numerade Educator

Problem 37

In Problem 35 , the width of the strip is $3.5 \mathrm{~cm}$, the magnetic field is $0.43 \mathrm{~T}$, the Hall voltage is measured to be $7.2 \mu \mathrm{V}$, the thickness of the strip is $0.24 \mathrm{~mm}$, and the current in the wire is $54 \mathrm{~A}$. What is the density of carriers (number of carriers per unit volume) in the strip?

Donya Dobbin

Numerade Educator

Problem 38

The strip in the diagram is used as a Hall probe to measure magnetic fields. (a) What happens if the strip is not perpendicular to the field? Does the Hall probe still read the correct field strength? Explain. (b) What happens if the field is in the plane of the strip?

Bettina Hanlon

Numerade Educator

Problem 39

A strip of copper $2.0 \mathrm{~cm}$ wide carries a current $I=$ $30.0 \mathrm{~A}$ to the right. The strip is in a magnetic field $B=5.0 \mathrm{~T}$ into the page. (a) What is the direction of the average magnetic force on the conduction electrons?
(b) The Hall voltage is $20.0 \mu \mathrm{V}$. What is the drift velocity?

Donya Dobbin

Numerade Educator

Problem 40

A proton is initially at rest and moves through three different regions as shown in the figure. In region 1 , the proton accelerates across a potential difference of $3330 \mathrm{~V}$. In region 2, there is a magnetic field of $1.20 \mathrm{~T}$ pointing out of the page and an electric field pointing perpendicular to the magnetic field and perpendicular to the proton's velocity. Finally, in region 3, there is no electric field. but just a $1.20-\mathrm{T}$ magnetic field pointing out of the page. (a) What is the speed of the proton as it leaves region 1 and enters region $2 ?$ (b) If the proton travels in a straight line through region 2, what is the magnitude and direction of the electric field? (c) In region 3 , will the proton follow path 1 or $2 ?$ (d) What will be the radius of the circular path the proton travels in region 3 ?

Vishal Gupta

Numerade Educator

Problem 41

An electromagnetic flowmeter is used to measure blood flow rates during surgery. Blood containing $\mathrm{Na}^{+}$ ions flows due south through an artery with a diameter of $0.40 \mathrm{~cm}$. The artery is in a downward magnetic field of $0.25 \mathrm{~T}$ and develops a Hall voltage of $0.35 \mathrm{mV}$ across its diameter. (a) What is the blood speed (in $\mathrm{m} / \mathrm{s}$ )?
(b) What is the flow rate (in $\mathrm{m}^{3} / \mathrm{s}$ )? (c) The leads of a voltmeter are attached to diametrically opposed points on the artery to measure the Hall voltage. Which of the two leads is at the higher potential?

Vishal Gupta

Numerade Educator

Problem 42

An electromagnetic flowmeter is used to measure blood flow rates during surgery. Blood containing ions (primarily $\mathrm{Na}^{+}$ ) flows through an artery with a diameter of $0.50 \mathrm{~cm}$. The artery is in a magnetic field of $0.35 \mathrm{~T}$ and develops a Hall voltage of $0.60 \mathrm{mV}$ across its diameter.
(a) What is the blood speed (in $\mathrm{m} / \mathrm{s}$ )? (b) What is the flow rate (in $\mathrm{m}^{3} / \mathrm{s}$ )? (c) If the magnetic field points west and the blood flow is north, is the top or bottom of the artery at the higher potential?

Vishal Gupta

Numerade Educator

Problem 43

A charged particle is accelerated from rest through a potential difference $\Delta V$. The particle then passes straight through a velocity selector (field magnitudes $E$ and $B$ ). Derive an expression for the charge-to-mass ratio $(q / m)$ of the particle in terms of $\Delta V, E$, and $B$.

Bettina Hanlon

Numerade Educator

Problem 44

A straight wire segment of length $0.60 \mathrm{~m}$ carries a current of $18.0 \mathrm{~A}$ and is immersed in a uniform external magnetic field of magnitude $0.20 \mathrm{~T}$. (a) What is the magnitude of the maximum possible magnetic force on the wire segment? (b) Explain why the given information enables you to calculate only the maximum possible force.

Bettina Hanlon

Numerade Educator

Problem 45

A straight wire segment of length $25 \mathrm{~cm}$ carries a current of $33.0 \mathrm{~A}$ and is immersed in a uniform external magnetic field. The magnetic force on the wire segment has magnitude $4.12 \mathrm{~N}$. (a) What is the minimum possible magnitude of the magnetic field? (b) Explain why the given information enables you to calculate only the minimum possible field strength.

Donya Dobbin

Numerade Educator

Problem 46

Parallel conducting tracks, separated by $2.0 \mathrm{~cm}$, run north and south. There is a uniform magnetic field of $1.2 \mathrm{~T}$ pointing upward (out of the page). A $0.040-\mathrm{kg}$ cylindrical metal rod is placed across the tracks and a battery is connected between the tracks, with its positive terminal connected to the east track. If the current through the rod is $3.0 \mathrm{~A}$, find the magnitude and direction of the magnetic force on the rod.-

Vishal Gupta

Numerade Educator

Problem 47

An electromagnetic rail gun can fire a projectile using a magnetic field and an electric current. Consider two conducting rails that are $0.500 \mathrm{~m}$ apart with a $50.0-\mathrm{g}$ conducting rod connecting the two rails as in the figure with Problem 46. A magnetic field of magnitude $0.750 \mathrm{~T}$ is directed perpendicular to the plane of the rails and rod. A current of $2.00 \mathrm{~A}$ passes through the rod.
(a) What direction is the force on the rod? (b) If there is no friction between the rails and the rod, how fast is the rod moving after it has traveled $8.00 \mathrm{~m}$ down the rails?

Donya Dobbin

Numerade Educator

Problem 48

A straight, stiff wire of length $1.00 \mathrm{~m}$ and mass $25 \mathrm{~g}$ is suspended in a magnetic field $B=0.75 \mathrm{~T}$. The
wire is connected to an emf. How much current must flow in the wire and in what direction so that the wire is suspended and the tension in the supporting wires is zero?

Donya Dobbin

Numerade Educator

Problem 49

A $20.0 \mathrm{~cm} \times 30.0 \mathrm{~cm}$ rectangular loop of wire carries $1.0 \mathrm{~A}$ of current clockwise around the
loop. (a) Find the magnetic force on each side of the loop if the magnetic field is $2.5 \mathrm{~T}$ out of the page.
(b) What is the net magnetic force on the loop?

Bettina Hanlon

Numerade Educator

Problem 50

Repeat Problem 49 if the magnetic field is $2.5 \mathrm{~T}$ to the left (in the $-x$ -direction).

Donya Dobbin

Numerade Educator

Problem 51

A straight wire is aligned east-west in a region where Earth's magnetic field has magnitude $0.48 \mathrm{mT}$ and direction $72^{\circ}$ below the horizontal, with the horizontal component directed due north. The wire carries a current $I$ toward the west. The magnetic force on the wire per unit length of wire has magnitude $0.020 \mathrm{~N} / \mathrm{m}$.
(a) What is the direction of the magnetic force on the wire? (b) What is the current $I$ ?

Donya Dobbin

Numerade Educator

Problem 52

A straight wire is aligned north-south in a region where Earth's magnetic field $\overrightarrow{\mathbf{B}}$ is directed $58.0^{\circ}$ above the horizontal, with the horizontal component directed due north. The wire carries a current of $8.00$ A toward the south. The magnetic force on the wire per unit length of wire has magnitude $2.80 \times 10^{-3} \mathrm{~N} / \mathrm{m} .$ (a) What is the direction of the magnetic force on the wire? (b) What is the magnitude of $\mathbf{B}$ ?

Vishal Gupta

Numerade Educator

Problem 53

In an electric motor, a circular coil with 100 turns of radius $2.0 \mathrm{~cm}$ can rotate between the poles of a magnet. When the current through the coil is $75 \mathrm{~mA}$, the maximum torque that the motor can deliver is $0.0020 \mathrm{~N} \cdot \mathrm{m}$. (a) What is the strength of the magnetic ficld? (b) Is the torque on the coil clockwise or counterclockwise as viewed from the front at the instant shown in the figure?

Donya Dobbin

Numerade Educator

Problem 54

In an electric motor, a coil with 100 turns of radius $2.0 \mathrm{~cm}$ can rotate between the poles of a magnet. The magnetic field strength is $0.20 \mathrm{~T}$. When the current through the coil is $50.0 \mathrm{~mA}$, what is the maximum torque that the motor can deliver?

Donya Dobbin

Numerade Educator

Problem 55

A square loop of wire of side $3.0 \mathrm{~cm}$ carries $3.0 \mathrm{~A}$ of current. A uniform magnetic field of magnitude $0.67 \mathrm{~T}$ makes an angle of $37^{\circ}$ with the plane of the loop.
(a) What is the magnitude of the torque on the loop?
(b) What is the net magnetic force on the loop?

Donya Dobbin

Numerade Educator

Problem 56

The torque on a loop of wire (a magnetic dipo uniform magnetic field is $t=N I A B \sin \theta$, where angle between $\overrightarrow{\mathbf{B}}$ and a line perpendicular to the loop of wire. Suppose an electric dipole, consisting of two charges $\pm q$ a fixed distance $d$ apart is in a uniform electric field $\overrightarrow{\mathbf{E}}$. (a) Show that the net electric force on the dipole is zero. (b) Let $\theta$ be the angle between $\overrightarrow{\mathbf{E}}$ and a line running from the negative to the positive charge. Show that the torque on the electric dipole is $\tau=q d E \sin \theta$ for all angles $-180^{\circ} \leq \theta \leq 180^{\circ}$. (Thus, for both electric and magnetic dipoles, the torque is the product of the dipole moment times the field strength times $\sin \theta$. The quantity $q d$ is the electric dipole moment; the quantity $N I A$ is the magnetic dipole moment.)

Dominador Tan

Numerade Educator

Problem 57

A certain fixed length $L$ of wire carries a current $I .$
(a) Show that if the wire is formed into a square coil, then the maximum torque in a given magnetic field $B$ is developed when the coil has just one turn. (b) Show that the magnitude of this torque is $\tau=\frac{1}{16} L_{.}^{2} I B$.

Donya Dobbin

Numerade Educator

Problem 58

Use the following method to show that the torque on an irregularly shaped planar loop is given by Eq. (1913a). The irregular loop of current in part (a) of the figure carries current $I$. There is a perpendicular magnetic field $B$. To find the torque on the irregular loop, sum up the torques on each of the smaller loops shown in part (b) of the figure. The pairs of imaginary currents flowing across carry equal currents in opposite directions, so the magnetic forces on them would be equal and opposite; they would therefore contribute nothing to the net torque. Now generalize this argument to a loop of any shape.

Dominador Tan

Numerade Educator

Problem 59

A square loop of wire with side $0.60 \mathrm{~m}$ carries a current of $9.0 \mathrm{~A}$ as shown in the figure. When there is no applied magnetic field, the plane of the loop is horizontal and the nonconducting, nonmagnetic spring $(k=550 \mathrm{~N} / \mathrm{m})$ is unstretched. A horizontal magnetic field of magnitude $1.3 \mathrm{~T}$ is now applied. At what angle $\theta$ is the wire loop's new equilibrium position? Assume the spring remains vertical because $\theta$ is small.

Vishal Gupta

Numerade Educator

Problem 60

Imagine a long straight wire perpendicular to the pagg and carrying a current $I$ into the page. Sketch some $\overline{\mathbf{B}}$ field lines with arrowheads to indicate directions.

Donya Dobbin

Numerade Educator

Problem 61

Two wires each carry $10.0 \mathrm{~A}$ of current (in opposit. directions) and are $3.0 \mathrm{~mm}$ apart. Calculate the mag netic field $25 \mathrm{~cm}$ away at point $P$, in the plane of the wires.

Bettina Hanlon

Numerade Educator

Problem 62

What is the magnetic field at point $P$ if the currents instead both run to the left in Problem 61 ?

Vishal Gupta

Numerade Educator

Problem 63

Point $P$ is midway between two long, straight, parallel wires that run north-south in a horizontal plane. The distance between the wires is $1.0 \mathrm{~cm}$. Each wire carries a current of $1.0 \mathrm{~A}$ toward the north. Find the magnitude and direction of the magnetic field at point $P$.

Vishal Gupta

Numerade Educator

Problem 64

Repeat Problem 63 if the current in the wire on the east side runs toward the south instead.

Donya Dobbin

Numerade Educator

Problem 65

A long straight wire carries a current of $50.0 \mathrm{~A}$. An electron, traveling at $1.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$, is $5.0 \mathrm{~cm}$ from the wire. What force (magnitude and direction) acts on the electron if the electron's velocity is directed toward the wire?

Donya Dobbin

Numerade Educator

Problem 66

A long straight wire carries a current of $3.2 \mathrm{~A}$ in the positive $x$ -direction. An electron, traveling at $6.8 \times 10^{6} \mathrm{~m} / \mathrm{s}$ in the positive $x$ direction, is $4.6 \mathrm{~cm}$ from the wire. What force acts on the electron?

Donya Dobbin

Numerade Educator

Problem 67

Two long straight wires carry the same amount of current in the directions indicated. The wires cross each other in the plane of the paper. Rank points $A, B$. $C$, and $D$ in order of decreasing field strength.

Vishal Gupta

Numerade Educator

Problem 68

In Problem 67 , find the magnetic field at points $C$ and $D$ when $d=3.3 \mathrm{~cm}$
and $I=6.50 \mathrm{~A}$.

Vishal Gupta

Numerade Educator

Problem 69

In Problem 67, find the Problems 67 and 68 magnetic field at points $A$ and $B$ when $d=6.75 \mathrm{~cm}$ and $I=57.0 \mathrm{~mA} .$

Vishal Gupta

Numerade Educator

Problem 70

A solenoid of length $0.256 \mathrm{~m}$ and radius $2.0 \mathrm{~cm}$ has 244 turns of wire. What is the magnitude of the magnetic field well inside the solenoid when there is a current of $4.5 \mathrm{~A}$ in the wire?

Donya Dobbin

Numerade Educator

Problem 71

Two long straight parallel wires separated by $8.0 \mathrm{~cm}$ carry currents of equal magnitude but heading in opposite directions. The wires are shown perpendicular to the plane of this page. Point $P$ is $2.0 \mathrm{~cm}$ from wire 1 . and the magnetic field at point $P$ is $1.0 \times 10^{-2} \mathrm{~T}$ directed in the $-y$ -direction. Calculate the current in wire 1 and its direction.

Bettina Hanlon

Numerade Educator

Problem 72

Two parallel wires in a horizontal plane carry currents $I_{1}$ and $I_{2}$ to the right. The wires each have length $L$ and are separated by a distance $d$. (a) What are the magnitude and direction of the field due to wire 1 at the location of wire $2 ?$ (b) What are the magnitude and direction of the magnetic force on wire 2 due to this field?
(c) What are the magnitude and direction of the field due to wire 2 at the location of wire $1 ?$ (d) What are the magnitude and direction of the magnetic force on wire 1 due to this field? (e) Do parallel currents in the same direction attract or repel? (f) What about parallel currents in opposite directions?

Vishal Gupta

Numerade Educator

Problem 73

Two concentric circular wire loops in the same plane each carry a current. The larger loop has a current of $8.46$ A circulating clockwise and has a radius of $6.20 \mathrm{~cm}$. The smaller loop has a radius of $4.42 \mathrm{~cm} .$ What is the current in the smaller loop if the total magnetic field at the center of the system is zero? [See Eq. (19-16).]

Donya Dobbin

Numerade Educator

Problem 74

A solenoid has 4850 turns per meter and radius $3.3 \mathrm{~cm}$. The magnetic field inside has magnitude $0.24 \mathrm{~T}$. What is the current in the solenoid?

Donya Dobbin

Numerade Educator

Problem 75

Four long parallel wires pass through the corners of a square with side $0.10 \mathrm{~m}$. All four wires carry the same magnitude of current $I=10.0 \mathrm{~A}$ in the directions indicated.
Find the magnetic field at the center of the square.

Donya Dobbin

Numerade Educator

Problem 76

Four long parallel wires pass through the corners of a square with side $0.10 \mathrm{~m}$. All four wires carry the same magnitude of current $I=10.0 \mathrm{~A}$ in the directions indicated.
Find the magnetic field at point $P$, the midpoint of the top side of the square.

Donya Dobbin

Numerade Educator

Problem 77

Four long parallel wires pass through the corners of a square with side $0.10 \mathrm{~m}$. All four wires carry the same magnitude of current $I=10.0 \mathrm{~A}$ in the directions indicated.
Find the magnetic field at point $R$, the midpoint of the left side of the square.

Donya Dobbin

Numerade Educator

Problem 78

Four long straight wires, cach with current $I$ overlap to form a square with side $2 r$ (a) Find the magnetic field at the center of the square. (b) Compare your answer with the magnctic field at the center of a circular loop of radius $r$ carrying current $I$ [see Fq. (19-16)].

Bettina Hanlon

Numerade Educator

Problem 79

Two parallel long straight wires are suspended by strings of length $L=1.2 \mathrm{~m} .$ Each wire has mass per unit length $\quad 0.050 \mathrm{~kg} / \mathrm{m}$. When the wires each carry $50.0 \mathrm{~A}$ of cur-rent, the wires swing apart. (a) How far apart are the wires in equilibrium? (Assume that this distance small compared with L.) (b) Are the wires carrying current i the same or opposite directions?

Vishal Gupta

Numerade Educator

Problem 80

A number of wires carry currents into or out of the page as indicated in the figure. (a) Using loop 1 for Ampère's law, what is the net current through the interior of the loop? (b) Repeat for loop 2.

Vishal Gupta

Numerade Educator

Problem 81

An infinitely long, thick cylindrical shell of inner radius $a$ and outer radius $b$ carries a current $I$ uniformly distributed across a cross section of the shell. (a) On a sketch of a cross section of the shell, draw some magnetic field lines. The current flows out of the page. Consider all regions $(r \leq a, a \leq r \leq b, b \leq r) .$ (b) Find the magnetic field for $r>b$.

Donya Dobbin

Numerade Educator

Problem 82

In this problem, use Ampère's law to show that the magnetic field inside a long solenoid is $B=\mu_{0} n l .$ Assume that the field inside the solenoid is uniform and parallel to the axis and that the field outside is zero. Choose a rectangular path for Ampère's law. (a) Write down $B_{1} \Delta l$ for each of the four sides of the path, in terms of $B, a$, (the short side) and $b$ (the long side).
(b) Sum these to form the circulation. (c) Now, to find the current cutting through the path: each loop carrics the same current $I$, and some number $N$ of loops cut through the path, so the total $\begin{array}{lll}\text { current } & \text { is } & N I \text { . }\end{array}$ Rewrite $N$ in terms of the number of
turns per unit length $(n)$ and the physical dimensions of the path.
(d) Solve for $B$.

Dominador Tan

Numerade Educator

Problem 83

A toroid is like a solenoid that has been bent around in a circle until its ends meet. The field lines are circular, as shown in the figure. What is the magnitude of the magnetic field inside a toroid of $N$ turns carrying current $I$ ? Apply Ampère's law, following a field line at a distance $r$ from
the center of the toroid. Work in terms of the total number of turns $N$, rather than the number of turns per unit length (why?). Is the field uniform, as it is for a long solenoid? Explain.

Vishal Gupta

Numerade Educator

Problem 84

A bar magnet is broken into two parts. If care is taken not to disturb the magnetic domains, what are the polarities of the new ends $c$ and $d$, respectively?

Vishal Gupta

Numerade Educator

Problem 85

The intrinsic magnetic dipole moment of the electron has magnitude $9.3 \times 10^{-24} \mathrm{~A} \cdot \mathrm{m}^{2} .$ In other words, the electron acts as though it were a tiny current loop with NIA $=9.3 \times 10^{-24} \mathrm{~A} \cdot \mathrm{m}^{2}$. What is the maximum torque on an electron due to its intrinsic dipole moment in a 1.0-T magnetic field?

Donya Dobbin

Numerade Educator

Problem 86

An electromagnet is made by inserting a soft iron core into a solenoid. The solenoid has 1800 turns, radius $2.0 \mathrm{~cm}$, and length $15 \mathrm{~cm}$. When $2.0 \mathrm{~A}$ of current flows through the solenoid, the magnetic field inside the iron core has magnitude $0.42 \mathrm{~T}$. What is the relative permeability $k_{\mathrm{B}}$ of the iron core? (See Section $19.10$ for the definition of $\left.\kappa_{\mathrm{B}}^{-}\right)$

Vishal Gupta

Numerade Educator

Problem 87

The figure shows hysteresis curves for three different materials. A hysteresis curve is a plot of the magnetic field strength inside the material $(B)$ as a function of the externally applied field $\left(B_{0}\right)$. (a) Which material would make the best permanent magnet? Explain. (b) Which would make the best core for an electromagnet? Fxplain.

Vishal Gupta

Numerade Educator

Problem 88

In a simple model, the electron in a hydrogen atom orbits the proton at a radius of $53 \mathrm{pm}$ and at a constant speed of $2.2 \times 10^{6} \mathrm{~m} / \mathrm{s}$. The orbital motion of the electron gives it an orbital magnetic dipole moment. (a) What is the current $I$ in this current loop? (b) What is the orbital dipole moment $I A$ ? (c) Compare the orbital dipole moment with the intrinsic magnetic dipole moment of the electron $\left(9.3 \times 10^{-24} \mathrm{~A} \cdot \mathrm{m}^{2}\right)$.

Bettina Hanlon

Numerade Educator

Problem 89

A compass is placed directly on top of a wire (needle not shown). The current in the wire flows to the right. Which way does the north end of the needle point? Explain. (Ignore Earth's magnetic field.)

Bettina Hanlon

Numerade Educator

Problem 90

You want to build a cyclotron to accelerate protons to a speed of $3.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$. The largest magnetic field strength you can attain is $1.5 \mathrm{~T}$. What must be the minimum radius of the dees in your cyclotron? Show how your answer comes from Newton's second law.

Donya Dobbin

Numerade Educator

Problem 91

A long straight wire carries a $4.70$ -A current in the positive $x$ -direction. At a particular instant, an electron moving at $1.00 \times 10^{7} \mathrm{~m} / \mathrm{s}$ in the positive $y$ -direction is $0.120 \mathrm{~m}$ from the wire. Determine the magnetic force on the electron at this instant. See the figure with Problem 65 .

Donya Dobbin

Numerade Educator

Problem 92

A uniform magnetic field of $0.50 \mathrm{~T}$ is directed to the north. At some instant, a particle with charge $+0.020 \mu \mathrm{C}$ is moving with velocity $2.0 \mathrm{~m} / \mathrm{s}$ in a direction $30^{\circ}$ north of east. (a) What is the magnitude of the magnetic force on the charged particle? (b) What is the direction of the magnetic force?

Donya Dobbin

Numerade Educator

Problem 93

Two identical long straight conducting wires with a mass per unit length of $25.0 \mathrm{~g} / \mathrm{m}$ are resting parallel to each other on a table. The wires are separated by $2.5 \mathrm{~mm}$ and are carrying currents in opposite directions. (a) If the coefficient of static friction between the wires and the table is $0.035$, what minimum current is necessary to make the wires start to move? (b) Do the wires move closer together or farther apart?

Donya Dobbin

Numerade Educator

Problem 94

Two long insulated wires lie in the same horizontal plane. A current of $20.0 \mathrm{~A}$ flows toward the north in wire $A$ and a current of $10.0 \mathrm{~A}$ flows toward the east in wire $B$. What is the magnitude and direction of the magnetic field at a point that is $5.00 \mathrm{~cm}$ above the point where the wires cross?

Bettina Hanlon

Numerade Educator

Problem 95

(a) A proton moves with uniform circular motion in a magnetic field of magnitude $0.80 \mathrm{~T}$. At what frequency $f$ does it circulate? (b) Repeat for an electron.

Donya Dobbin

Numerade Educator

Problem 96

The concentration of free electrons in silver is $5.85 \times 10^{28}$ per $\mathrm{m}^{3}$. A strip of silver of thickness $0.050 \mathrm{~mm}$ and width $20.0 \mathrm{~mm}$ is placed in a magnetic field of $0.80 \mathrm{~T}$. A current of $10.0 \mathrm{~A}$ is sent down the strip. (a) What is the drift velocity of the electrons?
(b) What is the Hall voltage measured by the meter?
(c) Which side of the voltmeter is at the higher potential?

Donya Dobbin

Numerade Educator

Problem 97

An electromagnetic flowmeter is to be used to measure blood speed. A magnetic field of $0.115 \mathrm{~T}$ is applied across an artery of inner diameter $3.80 \mathrm{~mm}$. The Hall voltage is measured to be $88.0 \mu \mathrm{V}$. What is the average speed of the blood flowing in the artery?

Donya Dobbin

Numerade Educator

Problem 98

Sketch the magnetic field as it would appear inside the coil of wire to an observer, looking into the coil from the position shown.

Vishal Gupta

Numerade Educator

Problem 99

Two conducting wires perpendicular to the page are shown in cross section as gray dots in the figure. They each carry $10.0 \mathrm{~A}$ out of the page. What is the magnetic field at point $P ?$

Vishal Gupta

Numerade Educator

Problem 100

A strip of copper carries current in the $+x$ -direction. There is an external magnetic field directed out of the page. What is the direction of the Hall electric field?

Donya Dobbin

Numerade Educator

Problem 101

A bar magnet is held near the electron beam in an oscilloscope. The beam passes directly below the south pole of the magnet. In what direction will the beam move on the screen? (I)on't try this with a color TV tube. There is a metal mask just behind the screen that separates the pixels for red, green, and blue. If you succeed in magnetizing the mask, the picture will be permanently distorted.)

Donya Dobbin

Numerade Educator

Problem 102

The strength of Earth's magnetic field, as measured on the surface, is approximately $6.0 \times 10^{-5} \mathrm{~T}$ at the poles and $3.0 \times 10^{-5} \mathrm{~T}$ at the equator. Suppose an alien from outer space were at the North Pole with a single loop of wire of the same circumference as his space helmet. The diameter of his helmet is $20.0 \mathrm{~cm}$. The space invader wishes to cancel Earth's magnetic field at his location. (a) What is the current required to produce a magnetic field (due to the current alone) at the center of his loop of the same size as that of Earth's field at the North Pole? (b) In what direction does the current circulate in the loop, $\mathrm{CW}$ or $\mathrm{CCW}$, as viewed from above, if it is to cancel Earth's field?

Donya Dobbin

Numerade Educator

Problem 103

A tangent galvanometer is an instrument, developed in the nineteenth century, designed to measure current based on the deflection of a compass needle. A coil of wire in a vertical plane is aligned in the magnetic north-south direction. A compass is placed in a horizontal plane at the center of the coil. When no current flows, the compass needle points directly toward the north side of the coil. When a current is sent through the coil, the compass needle rotates through an angle $\theta$. Derive an equation for $\theta$ in terms of the number of coil turns $N$, the coil radius $r$, the coil current $I$, and the horizontal component of Earth's field $B_{\mathrm{H}}$.

Vishal Gupta

Numerade Educator

Problem 104

In the mass spectrometer of the diagram, neon ions $(q=+e)$ come from the ion source and are accelerated through a potential difference $V$. The ions then pass through an aperture in a metal plate into a uniform magnetic field where they travel in semicircular paths until exiting into the detector. Neon ions having a mass of $20.0 \mathrm{u}$ leave the field at a distance of $50.0 \mathrm{~cm}$ from the aperture. At what distance from the aperture do neon ions having a mass of $22.0$ u leave the field? $\left(1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{~kg} .\right)$

Donya Dobbin

Numerade Educator

Problem 105

A rectangular loop of wire, carrying current $I_{1}=2.0 \mathrm{~mA}$, is next to a very long wire carrying a current $I_{2}=8.0 \mathrm{~A}$.
(a) What is the direction of the magnetic force on each of the four sides of the rectangle due to the long wire's magnetic field? (b) Calculate the net magnetic force on the rectangular loop due to the long wire's magnetic field.

Vishal Gupta

Numerade Educator

Problem 106

In a carbon-dating experiment, a particular type of mass spectrometer is used to separate ${ }^{14} \mathrm{C}$ from ${ }^{12} \mathrm{C}$. Carbon ions from a sample are first accelerated through a potential difference $\Delta V_{1}$ between the charged accelerating plates. Then the ions enter a region of uniform vertical magnetic field $B=0.200 \mathrm{~T}$. The ions pass between deflection plates spaced $1.00 \mathrm{~cm}$ apart. By adjusting the potential difference $\Delta V_{2}$ between these plates, only one of the two isotopes $\left({ }^{12} \mathrm{C}\right.$ or $\left.{ }^{14} \mathrm{C}\right)$ is allowed to pass through to the next stage of the mass spectrometer. The distance from the entrance to the ion detector is a fixed $0.200 \mathrm{~m}$. By suitably adjusting $\Delta V_{1}$ and $\Delta V_{2}$, the detector counts only one type of ion, so the relative abundances can be determined. (a) Are the ions positively or negatively charged? (b) Which of the accelerating plates (east or west) is positively charged? (c) Which of the deflection plates (north or south) is positively charged? (d) Find the correct values of $\Delta V_{1}$ and $\Delta V_{2}$ in order to count ${ }^{12} \mathrm{C}^{+}$ ions (mass $\left.1.993 \times 10^{-26} \mathrm{~kg}\right)$.
(e) Find the correct values of $\Delta V_{1}$ and $\Delta V_{2}$ in order to count ${ }^{14} \mathrm{C}^{+}$ ions (mass $2.325 \times 10^{-26} \mathrm{~kg}$ ).

Dominador Tan

Numerade Educator

Problem 107

Repeat Problem 49 if the magnetic field is $2.5 \mathrm{~T}$ in the plane of the loop, $60.0^{\circ}$ below the $+x$ -axis.

Vishal Gupta

Numerade Educator

Problem 108

A current balance is a device to measure magnetic forces. It is constructed from two parallel coils, each with an average radius of $12.5 \mathrm{~cm}$. The lower coil rests on a balance; it has 20 turns and carries a constant current of $4.0 \mathrm{~A}$. The upper coil, suspended $0.314 \mathrm{~cm}$ above the lower coil, has 50 turns and a current that can be varied. The reading of the balance changes as the magnetic force on the lower coil changes. What current is needed in the upper coil to exert a force of $1.0 \mathrm{~N}$ on the bottom coil?

Vishal Gupta

Numerade Educator

Problem 109

In a certain region of space, there is a uniform clectric field $\overrightarrow{\mathbf{E}}=3.0 \times 10^{4} \mathrm{~V} / \mathrm{m}$ directed due east and a uniform magnetic field $\overrightarrow{\mathbf{B}}=0.080$ T also directed due east. What is the electromagnetic force on an electron moving due south at $5.0 \times 10^{6} \mathrm{~m} / \mathrm{s}$ ?

Donya Dobbin

Numerade Educator

Problem 110

An carly cyclotron at Cornell University was used from the 1930 s to the 1950 s to accelerate protons, which would then bombard various nuclei. The cyclotron used a large electromagnet with an iron yoke to produce a uniform magnetic field of $1.3 \mathrm{~T}$ over a region in the shape of a flat cylinder. Two hollow copper dees of inside radius $16 \mathrm{~cm}$ were located in a vacuum chamber in this region. (a) What is the frequency of oscillation necessary for the alternating voltage difference between the dees? (b) What is the kinetic energy of a proton by the time it reaches the outside of the dees? (c) What would be the equivalent voltage necessary to accelerate protons to this energy from rest in one step (say between parallel plates)? (d) If the potential difference between the dees has a magnitude of $10.0 \mathrm{kV}$ each time the protons cross the gap, what is the minimum number of revolutions each proton has to make in the cyclotron?

Donya Dobbin

Numerade Educator

Problem 111

In a certain region of space, there is a uniform electric field $\overrightarrow{\mathbf{E}}=2.0 \times 10^{4} \mathrm{~V} / \mathrm{m}$ to the east and a uniform magnetic field $\overrightarrow{\mathbf{B}}=0.0050 \mathrm{~T}$ to the west. (a) What is the electromagnetic force on an electron moving north at $1.0 \times 10^{7} \mathrm{~m} / \mathrm{s} ?$ (b) With the electric and magnetic fields as specified, is there some velocity such that the net electromagnetic force on the electron would be zero? If so, give the magnitude and direction of that velocity. If not, explain briefly why not.

Donya Dobbin

Numerade Educator

Problem 112

An electron moves in a circle of radius $R$ in a uniform $m a g-$ netic field $\overrightarrow{\mathbf{B}}$. The field is into the page. (a) Does the electron move clockwise or counterclockwise? (b) How much time does the electron take to make one complete revolution? Derive an expression for the time, starting with the magnetic force on the electron. Your answer may include $R, B$, and any fundamental constants.

Donya Dobbin

Numerade Educator

Problem 113

A proton moves in a helical path at speed $v=4.0$ $\times 10^{7} \mathrm{~m} / \mathrm{s}$ high above the atmosphere, where Earth's magnetic field has magnitude $B=1.0 \times 10^{-6} \mathrm{~T}$. The proton's velocity makes an angle of $25^{\circ}$ with the magnetic field. (a) Find the radius of the helix. [Hint:
Use the perpendicular component of the velocity.]
(b) Find the pitch of the helix-the distance between adjacent "coils." [Hint: Find the time for one revolution; then find how far the proton moves along a field line during that time interval.]

Donya Dobbin

Numerade Educator