Physics

Robert Coleman Richardson; Betty McCarthy Richardson; Alan Giambattista

Chapter 19

Magnetic Forces and Fields - all with Video Answers

Educators

+ 2 more educators

Chapter Questions

Problem 1

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

Noor Aldeen Almusleh

Noor Aldeen Almusleh

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

Sketch some magnetic field lines for two identical bar magnets in the given configuration. Be sure to show field lines inside the magnets as well as outside.
$$
\begin{array}{|c|c||}
\hline \mathbf{N} & \mathbf{S} \\
\hline \mathbf{N} & \mathbf{S} \\
\hline
\end{array}
$$

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 4

Sketch some magnetic field lines for two identical bar magnets in the given configuration. Be sure to show field lines inside the magnets as well as outside.
$$
\begin{array}{|c|c|}
\hline \mathbf{N} & \mathbf{S} \\
\hline \mathbf{S} & \mathbf{N} \\
\hline
\end{array}
$$

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 5

Sketch some magnetic field lines for two identical bar magnets in the given configuration. Be sure to show field lines inside the magnets as well as outside.
$$
\begin{array}{llll}
\text { S } & \text { N } & \text { N } & \text { S }
\end{array}
$$

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 6

Sketch some magnetic field lines for two identical bar magnets in the given configuration. Be sure to show field lines inside the magnets as well as outside.
$$
\begin{array}{|l|l|l|l|l|}
\hline \mathrm{N} & \mathrm{S} & \mathrm{N} & \mathrm{S} \\
\hline
\end{array}
$$

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 7

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 8

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 $2.50 \mathrm{T}$ directed north.

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 9

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?

Nishant Kumar

Numerade Educator

Problem 10

A uniform magnetic field points vertically upward; its magnitude is 0.800 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 11

Rank the electrons in order of the magnitude of the magnetic force on them, from greatest to least.

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 12

Find the magnetic force on the electron at point $a$.

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 13

Find the magnetic force on the electron at point $b$.

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 14

Find the magnetic force on the electron at point $c$.

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 15

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 16

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

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 17

A cosmic ray muon with the same charge as an electron and a mass of $1.9 \times 10^{-28} \mathrm{kg}$ is moving toward the ground at an angle of $25^{\circ}$ from the vertical with a speed of $7.0 \times 10^{7} \mathrm{m} / \mathrm{s}$. As it crosses point $P$, the muon is at a horizontal distance of $85.0 \mathrm{cm}$ from a high-voltage power line. At that moment, the power line has a current of 16.0 A. What is the magnitude and direction of the force on the muon at the point $P$ in the diagram?

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 18

In a CRT, electrons moving at $1.8 \times 10^{7} \mathrm{m} / \mathrm{s}$ pass between the poles of an electromagnet where the magnetic field is $2.0 \mathrm{mT}$ directed upward. (a) What is the radius of their circular path while in the magnetic field?
(b) The time the electrons spend in the magnetic field is $0.41 \mathrm{ns} .$ By what angle does the direction of the beam change while it passes through the magnetic field? (c) In what direction is the beam deflected, as viewed by an observer louking al the screen?

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 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. (If there is more than one possible answer, find all the possibilities.)

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 22

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 23

Six protons move (at speed $v$ ) in magnetic fields (magnitude $B$ ) along circular paths. Rank them in order of the radius of their paths, greatest to smallest.
(a) $v=6 \times 10^{6} \mathrm{m} / \mathrm{s}, B=0.3 \mathrm{T}$
(b) $v=3 \times 10^{6} \mathrm{m} / \mathrm{s}, B=0.6 \mathrm{T}$
(c) $v=3 \times 10^{6} \mathrm{m} / \mathrm{s}, B=0.1 \mathrm{T}$
(d) $v=1.5 \times 10^{6} \mathrm{m} / \mathrm{s}, B=0.15 \mathrm{T}$
(e) $v=2 \times 10^{6} \mathrm{m} / \mathrm{s}, B=0.1 \mathrm{T}$
(f) $v=1 \times 10^{6} \mathrm{m} / \mathrm{s}, B=0.3 \mathrm{T}$

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 24

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 25

The magnetic field in a hospital's 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.

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 26

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

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 27

The magnetic field in a cyclotron used to produce radioactive tracers is $0.50 \mathrm{T}$. What must be the mini-
mum radius of the dees if the maximum proton speed desired is $1.0 \times 10^{7} \mathrm{m} / \mathrm{s} ?$

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 28

A beam of $\alpha$ particles (helium nuclei) is used to treat a tumor located $10.0 \mathrm{cm}$ inside a patient. To penetrate to the tumor, the $\alpha$ particles must be accelerated to a speed of $0.458 c,$ where $c$ is the speed of light. (Ignore relativistic effects.) The mass of an $\alpha$ particle is $4.003 \mathrm{u}$ and its charge is $+2 e .$ The cyclotron used to accelerate the beam has radius $1.00 \mathrm{m}$. What is the magnitude of the magnetic field?

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 29

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 field. What is the mass of the ion?

Anna Zeng

Numerade Educator

Problem 30

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. 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 ions?

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 31

(Naturally occurring carbon consists of two different isotopes (excluding ${ }^{14} \mathrm{C},$ which is present in only trace amounts). The most abundant isotope is ${ }^{12} \mathrm{C}$. When carbon is placed in a mass spectrometer, ${ }^{12} \mathrm{C}^{+}$ ions moved in a circle of radius $15.0 \mathrm{cm},$ whereas ions of the other isotope moved in a circle of radius $15.6 \mathrm{cm} .$ What is the atomic mass of the rarer isotope?

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 32

After being accelerated through a potential difference of $5.0 \mathrm{kV},$ a singly charged ${ }^{12} \mathrm{C}^{+}$ ion moves in a circle of radius $21 \mathrm{cm}$ in the magnetic field of a mass spectrometer. What is the magnitude of the field?

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 33

A sample containing ${ }^{12} \mathrm{C},{ }^{16} \mathrm{O},$ and an unknown isotope is analyzed in a mass spectrometer. As in Fig. $19.17(\mathrm{a}),$ the ions move around a semicircle before striking a photographic plate. The ${ }^{12} \mathrm{C}^{+}$ and ${ }^{16} \mathrm{O}^{+}$ ions are separated by $2.250 \mathrm{cm}$ on the plate, and the unknown isotope strikes the plate $1.160 \mathrm{cm}$ from the ${ }^{12} \mathrm{C}^{+}$ ions. What is the mass of the unknown element?

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 34

A sample containing sulfur (atomic mass 32 u), manganese $(55 \mathrm{u})$, and an unknown element is analyzed in a mass spectrometer. As in Fig. $19.17(a),$ the ions move around half a circle before striking a photographic plate. The sulfur and manganese ions are separated by $3.20 \mathrm{cm}$ on the plate, and the unknown element strikes the plate $1.07 \mathrm{cm}$ from the sulfur line.
(a) What is the mass of the unknown element?
(b) Identify the element.

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 35

Show 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 magnitude.

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 36

Crossed electric and magnetic fields are established over a certain region. The magnetic field is 0.635 T vertically downward. The electric 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 37

A current $I=40.0$ 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. How would you hook up a voltmeter to measure the Hall voltage? Show how the voltmeter is connected on a sketch of the strip. Assuming the carriers are electrons, which lead of your voltmeter is at the higher potential? Mark it with a "+" sign in your sketch. Explain briefly.

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 38

In Problem $37,$ 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?

Bettina Hanlon

Numerade Educator

Problem 39

In Problem $37,$ 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 A. What is the density of carriers (number of carriers per unit volume) in the strip?

Bettina Hanlon

Numerade Educator

Problem 40

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 magnitude? Explain.
(b) What happens if the field is in the plane of the strip?

Bettina Hanlon

Numerade Educator

Problem 41

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 42

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 (not shown) 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 $3,$ does the proton follow path 1 or $2 ?$
(d) What is the radius of the circular path in region $3 ?$

Vishal Gupta

Numerade Educator

Problem 43

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 $\left.\mathrm{m}^{3} / \mathrm{s}\right) ?$ (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 44

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 45

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 46

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 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 magnitude.

Bettina Hanlon

Numerade Educator

Problem 47

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 48

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 47. 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 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?

Vishal Gupta

Numerade Educator

Problem 49

A straight, stiff wire of length $1.00 \mathrm{m}$ and mass 25 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 50

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 51

Repeat Problem 50 if the magnetic field is $2.5 \mathrm{T}$ to the left (in the $-x$ -direction).

Vishal Gupta

Numerade Educator

Problem 52

Repeat Problem 50 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 53

A straight wire is aligned east-west in a region where Earth's magnetic field has magnitude $0.048 \mathrm{mT}$ 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 $I ?$

Vishal Gupta

Numerade Educator

Problem 54

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 $8.00 \mathrm{A}$ 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 $\overline{\mathbf{B}} ?$

Vishal Gupta

Numerade Educator

Problem 55

In each of six electric motors, a cylindrical coil with $N$ turns and radius $r$ is immersed in a magnetic field of magnitude $B$. The current in the coil is $I .$ Rank the motors in order of the maximum torque on the coil, greatest to smallest.
(a) $N=100, r=2 \mathrm{cm}, B=0.4 \mathrm{T}, I=0.5 \mathrm{A}$
(b) $N=100, r=4 \mathrm{cm}, B=0.2 \mathrm{T}, I=0.5 \mathrm{A}$
(c) $N=75, r=2 \mathrm{cm}, B=0.4 \mathrm{T}, I=0.5 \mathrm{A}$
(d) $N=50, r=2 \mathrm{cm}, B=0.8 \mathrm{T}, I=0.5 \mathrm{A}$
(e) $N=100, r=3 \mathrm{cm}, B=0.4 \mathrm{T}, I=0.5 \mathrm{A}$
(f) $N=50, r=2 \mathrm{cm}, B=0.8 \mathrm{T}, I=1 \mathrm{A}$

DM

Debra Mangion

Numerade Educator

Problem 56

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

Vishal Gupta

Numerade Educator

Problem 57

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 magnitude 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?

Bettina Hanlon

Numerade Educator

Problem 58

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 59

The intrinsic magnetic dipole moment of the electron has magnitude $9.3 \times 10^{-24} \mathrm{A} \cdot \mathrm{m}^{2} .$ What is the maximum torque on an electron due to its intrinsic dipole moment in a $1.0 \mathrm{T}$ magnetic field?

Bettina Hanlon

Numerade Educator

Problem 60

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? [Hint: How long does it take the electron to make one revolution?]
(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 61

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 62

Use the following method to show that the torque on an irregularly shaped planar loop due to a perpendicular magnetic field is $\tau=N I A B .$ 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. [Hint: Think of

Dominador Tan

Numerade Educator

Problem 63

Estimate the magnetic field at distances of $1 \mu \mathrm{m}$ and $1 \mathrm{mm}$ produced by a current of $3 \mu \mathrm{A}$ along the medial nerve of the human arm. Model the nerve as a straight current-carrying wire. Compare with the magnitude of Earth's magnetic field near the surface, about $0.05 \mathrm{mT}$.

Vishal Gupta

Numerade Educator

Problem 64

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

Donya Dobbin

Numerade Educator

Problem 65

Kieran measures the magnetic field of an electron beam. The beam strength is such that $1.40 \times 10^{11}$ electrons pass a point every $1.30 \mu$ s. What magnetic field does Kieran measure at a distance of $2.00 \mathrm{cm}$ from the beam center?

Bettina Hanlon

Numerade Educator

Problem 66

Some animals are capable of detecting magnetic fields and use this sense to help them navigate. Suppose a high-voltage direct-current power line carries a current of $5.0 \mathrm{kA}$. (a) How far from the wire would a homing pigeon have to be so the field due to the wire has magnitude $45 \mu \mathrm{T},$ which is comparable to Earth's magnetic field at the surface? (b) On a long-distance flight, the pigeon is flying at an altitude of $700 \mathrm{m}$. What would the magnetic field be at that distance from the power line? If the homing pigeon navigates by sensing the magnetic field, might the power line disrupt its ability to navigate on a long-distance flight?

Vishal Gupta

Numerade Educator

Problem 67

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

Bettina Hanlon

Numerade Educator

Problem 68

In Problem $67,$ what is the magnetic field at a point midway between the wires in the plane of the wires?

Vishal Gupta

Numerade Educator

Problem 69

What is the magnetic field at point $P$ if the currents instead both run to the left in Problem $67 ?$

Vishal Gupta

Numerade Educator

Problem 70

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 71

Repeat Problem 70 if the current in the wire on the east side runs toward the south instead.

Vishal Gupta

Numerade Educator

Problem 72

A long straight rent of 50.0 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?

Vishal Gupta

Numerade Educator

Problem 73

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 74

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 $\mathrm{de}-$ creasing field magnitude.

Vishal Gupta

Numerade Educator

Problem 75

In Problem $74,$ 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 76

In Problem $74,$ find the magnetic field at points $A$ and $B$ when $d=6.75 \mathrm{cm}$ and $I=57.0 \mathrm{mA}$

Vishal Gupta

Numerade Educator

Problem 77

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 78

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 $1.0 \times 10^{-2}$ T directed in the $-y$ -direction. Calculate the current in wire 1 and its direction.

Bettina Hanlon

Numerade Educator

Problem 79

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 $d$. Find the magnitudes and directions of the (a) the magnetic field due to wire 1 at the location of wire $2,$ (b) the magnetic force on wire $2,$ (c) the magnetic field due to wire 2 at the location of wire $1,$ and $(\mathrm{d})$ the magnetic force on wire $1 .$ (e) Do parallel currents in the same direction attract or repel? What about parallel currents in opposite directions?
(f) Are the magnitudes and directions of the forces consistent with Newton's third law?

Dominador Tan

Numerade Educator

Problem 80

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. ( $19-32$ ). $.$

Bettina Hanlon

Numerade Educator

Problem 81

You are designing the main solenoid for an MRI machine. The solenoid should be $1.5 \mathrm{m}$ long. When the current is $80 \mathrm{A},$ the magnetic field inside should be 1.5 T. How many turns should your solenoid have?

Vishal Gupta

Numerade Educator

Problem 82

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

Donya Dobbin

Numerade Educator

Problem 83

Find the magnetic field at the center of the square.

Vishal Gupta

Numerade Educator

Problem 84

Find the magnetic field at point $P,$ the midpoint of the top side of the square.

Vishal Gupta

Numerade Educator

Problem 85

Find the magnetic field at point $R,$ the midpoint of the left side of the square.

Vishal Gupta

Numerade Educator

Problem 86

Four long straight wires, each 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 magnetic field at the center of a circular loop of radius $r$ carrying current $I$ [see Eq. $(19-32)]$

Bettina Hanlon

Numerade Educator

Problem 87

Two parallel long straight wires are suspended by strings of length $L=1.2 \mathrm{m}$. Each wire has mass per unit length $0.050 \mathrm{kg} / \mathrm{m} .$ When one wire carries $25.0 \mathrm{A}$ of current and the other carries $100.0 \mathrm{A}$, the wires swing
apart.
(a) How far apart are the wires in equilibrium? Assume that this distance is small compared with $L$.

Vishal Gupta

Numerade Educator

Problem 88

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) Sketch a graph of the magnetic field magnitude as a function of $r .$
(c) Find the magnetic field for $r>b$.

Dominador Tan

Numerade Educator

Problem 89

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 90

In this problem, use Ampère's law to show that the magnetic field inside a long solenoid is $B=\mu_{0} n I .$ 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 $B_{\|} \Delta l$ 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 carries the same current $I,$ and some number $N$ of loops cut through the path, so the total current is $N I .$ 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 91

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 92

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 $\kappa_{\mathrm{B}}$ of the iron core? (See Section 19.10 for the definition of $\kappa_{\mathrm{B}}$. $)$

Vishal Gupta

Numerade Educator

Problem 93

You want to build a cyclotron to accelerate protons to a speed of $3.0 \times 10^{7} \mathrm{m} / \mathrm{s}$ for use in proton beam therapy. The largest magnetic field 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.

Bettina Hanlon

Numerade Educator

Problem 94

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 $1.993 \times 10^{-26} \mathrm{kg}$ ).
(e) Find the correct values of $\Delta V_{1}$ and $\Delta V_{2}$ in order to count ${ }^{14} \mathrm{C}^{+}$ ions $\left(\mathrm{mass} 2.325 \times 10^{-26} \mathrm{kg}\right)$

Dominador Tan

Numerade Educator

Problem 95

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

Problem 96

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 97

C The figure shows hysteresis curves for three different materials. A hysteresis curve is a plot of the magnetic field magnitude 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? Explain.

Vishal Gupta

Numerade Educator

Problem 98

What is the greatest possible magnetic force on the sodium ion due to Earth's field?

Vishal Gupta

Numerade Educator

Problem 99

If the magnetic force due to Earth's field were the only force on the ion, what would the smallest possible radius of its trajectory be?

Vishal Gupta

Numerade Educator

Problem 100

Magnetic forces cause an excess of positive ions to flow along one side of the artery and negative ions on the opposite side. What is the greatest possible potential difference across the artery?

Vishal Gupta

Numerade Educator

Problem 101

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 102

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 72 .

Vishal Gupta

Numerade Educator

Problem 103

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 104

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

Donya Dobbin

Numerade Educator

Problem 105

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 106

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 107

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. As illustrated, a compass is placed in a horizontal plane at the center of the coil. When no current flows, the compass needle points

Vishal Gupta

Numerade Educator

Problem 108

A rectangular loop of wire, carrying current $I_{1}=2.0 \mathrm{mA},$ is next to a very long wire carrying a current $I_{2}=8.0$ 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. [Hint: The long wire does not produce a uniform magnetic field.] (c) What is the magnetic force on the long wire due to the loop?

Vishal Gupta

Numerade Educator

Problem 109

Two long, straight wires, each with a current of $5.0 \mathrm{A},$ are placed on two corners of an equilateral triangle with sides of length $3.2 \mathrm{cm}$ as shown. One of the wires has a current into the page and one has a current out of the page.
(a) What is the magnetic field at the third corner of the triangle? (b) A proton has a velocity of $1.8 \times 10^{7} \mathrm{m} / \mathrm{s}$ out of the page when it crosses the plane of the page at the third corner of the triangle. What is the magnetic force on the proton at that point due to the two wires?

Vishal Gupta

Numerade Educator

Problem 110

A solenoid with 8500 turns per meter has radius $65 \mathrm{cm}$. The current in the solenoid is 25.0 A. A circular loop of wire with 100 turns and radius $8.00 \mathrm{cm}$ is put inside the solenoid. The current in the circular loop is $2.20 \mathrm{A}$. What is the maximum possible magnetic torque on the loop? What orientation does the loop have if the magnetic torque has its maximum value?

Vishal Gupta

Numerade Educator

Problem 111

Two long, straight wires, each with a current of $12.0 \mathrm{A},$ are placed on two corners of an equilateral triangle with sides of length $2.50 \mathrm{cm}$ as shown. Both of the wires have a current into the page.
(a) What is the magnetic field at the third corner of the triangle? (b) Another wire is placed at the third corner, parallel to the other two wires. In which direction should current flow in the third wire so that the force on it is in the $+y$ -direction?

Brandy Heflin

Numerade Educator

Problem 112

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? [Hint: since the distance between the coils is small relative to the radius of the coils, approximate the setup as two long parallel straight wires.]

Vishal Gupta

Numerade Educator

Problem 113

In a certain region of space, there is a uniform electric field $\overrightarrow{\mathbf{E}}=3.0 \times 10^{4} \mathrm{V} / \mathrm{m}$ directed due east and a uniform magnetic field $\overrightarrow{\mathbf{B}}=0.080 \mathrm{T}$ 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 114

An early 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 T over a region in the shape of a flat cylinder. Two hollow copper dees

Vishal Gupta

Numerade Educator

Problem 115

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 A flows toward the east in wire $B .$ What are 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 116

In Problem $111,$ the wire that goes through the top corner of the triangle has a linear mass density of $0.150 \mathrm{g} / \mathrm{m} .$ What current in this wire would make it "hover" above the other two? [Hint: The sum of the magnetic and gravitational forces on the wire is zero.]

Vishal Gupta

Numerade Educator

Problem 117

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 118

Electrons in an old television's CRT (see Sec. 16.5 ) 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 magnitude electric field would give the electrons the same magnitude acceleration as

Vishal Gupta

Numerade Educator

Problem 119

A square loop of wire with side $0.60 \mathrm{m}$ carries a current of $9.0 \mathrm{A}$ as shown in the side-view diagram. 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. [Hint: Set the sum of the torques from the spring and the magnetic field equal to 0.]

Vishal Gupta

Numerade Educator

Problem 120

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 121

The number density of free electrons in silver is $5.85 \times 10^{28} \mathrm{m}^{-3} .$ 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?

Bettina Hanlon

Numerade Educator

Problem 122

An electromagnetic rail gun can fire a projectile using a magnetic field and an electric current. Consider two horizontal conducting rails that are $0.500 \mathrm{m}$ apart with a $50.0 \mathrm{g}$ conducting projectile that slides along the two rails. A magnetic field of $0.750 \mathrm{T}$ is directed upward. A constant current of $2.00 \mathrm{A}$ passes through the projectile. (a) What direction is the force on the projectile? (b) If the coefficient of kinetic

Vishal Gupta

Numerade Educator

Problem 123

An engineer wants to design a toy racetrack using an electromagnetic rail gun (see Problem 122 ) to accelerate a car of mass 40 g starting from rest. The horizontal rails are to be $1.0 \mathrm{m}$ long and $2.0 \mathrm{cm}$ apart. The magnetic field in the rail gun is to be 0.10 T upward. Leaving the rail gun, the car slides onto a horizontal track and then around a vertical loop-the-loop of radius $15 \mathrm{cm} .$ Ignore friction everywhere. What minimum current must flow in the rails to give the car enough kinetic energy to make it around the loop without losing contact with the track? Is the required current reasonable?

Vishal Gupta

Numerade Educator

Problem 124

The accelerating plates have area $A$ and are a distance $d$ apart. (a) What should the charges on the plates be so the ions emerge at speed $v$, ignoring their initial kinetic energies? Indicate which plate is positive and which negative.
(b) Sketch the electric field lines between the plates.

Vishal Gupta

Numerade Educator

Problem 125

The uniform magnetic field in the velocity selector is directed out of the page and has magnitude $B$.
(a) What should the magnitude and direction of the electric field in the selector be to allow ions with speed $v$ to pass straight through?
(b) Sketch the trajectory inside the velocity selector for ions that enter with speeds slightly less than $v$.

Vishal Gupta

Numerade Educator

Problem 126

Suppose some ${ }^{235} \mathrm{U}^{+}$ ions are present in the beam. They have the same charge as the ${ }^{238} \mathrm{U}^{+}$ ions but a smaller mass (approximately 0.98737m). (a) With what speed do the ${ }^{23} \mathrm{U}^{+}$ ions emerge from the accelerating plates, assuming ${ }^{238} \mathrm{U}^{+}$ ions emerge with speed $v ?$
(b) Sketch the trajectory of ${ }^{235} \mathrm{U}^{+}$ ions inside the velocity selector.
(c) Now the velocity selector is removed. ${ }^{238} \mathrm{U}^{+}$ ions move in a circular path of diameter $D$ in the uniform magnetic field. What is the diameter of the path of the ${ }^{235} \mathrm{U}^{+}$ ions?

Dominador Tan

Numerade Educator

Problem 127

Find the mass of the ${ }^{238} \mathrm{U}^{+}$ ions in terms of $v, B, D,$ and universal constants.

Vishal Gupta

Numerade Educator

Problem 128

Suppose some ${ }^{238} \mathrm{U}^{2+}$ ions are present in the beam. They have the same mass $m$ as the ${ }^{238} \mathrm{U}^{+}$ ions but twice the charge $(+2 e)$. (a) With what speed do the ${ }^{238} \mathrm{U}^{2+}$ ions emerge from the accelerating plates, assuming ${ }^{238} \mathrm{U}^{+}$ ions emerge with speed $v ?$ (b) Sketch the trajectory of ${ }^{238} \mathrm{U}^{2+}$ ions inside the velocity selector.
(c) Now the velocity selector is removed. ${ }^{238} \mathrm{U}^{+}$ ions move in a circular path of diameter $D$ in the uniform magnetic field. What is the diameter of the path of the ${ }^{238} \mathrm{U}^{2+}$ ions?

Vishal Gupta

Numerade Educator