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College Physics

lan Giambattista, Betty McCarthy Richardson, Robert C. Richardson

Chapter 19

Magnetic Forces and Fields - all with Video Answers

Educators

Chapter Questions

01:31

Problem 1

The electric field is defined as the electric force per unit charge. Explain why the magnetic field cannot be defined as the magnetic force per unit charge.

Ajay Singhal
Ajay Singhal
Numerade Educator
00:46

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
01:45

Problem 2

A charged particle moves through a region of space at constant velocity. Ignore gravity. In the region, is it possible that there is (a) a magnetic field but no electric field? (b) an electric field but no magnetic field? (c) a magnetic field and an electric field? For each possibility, what must be true about the direction(s) of the field(s)?

Ajay Singhal
Ajay Singhal
Numerade Educator
03:54

Problem 2

Draw vector arrows to Problems 1 and 2 indicate the direction and relative magnitude of the magnetic field at each of the points $A-F$.

Vishal Gupta
Vishal Gupta
Numerade Educator
03:43

Problem 3

Suppose that a horizontal electron beam is deflected to the right by a uniform magnetic field. What is the direction of the magnetic field? If there is more than one possibility, what can you say about the direction of the field?

Stylianos Gregoriou
Stylianos Gregoriou
Numerade Educator
03:04

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
Vishal Gupta
Numerade Educator
01:24

Problem 4

A circular metal loop carries a current $I$ as shown. The points are all in the plane of the page and the loop is perpendicular to the page. Sketch the loop, and draw vector arrows at the points $A, B, C, D$, and $E$ to show the direction of the magnetic field at those points.

Ankur S
Ankur S
Numerade Educator
04:24

Problem 4

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

Anna Zeng
Anna Zeng
Numerade Educator
03:05

Problem 5

In a TV or computer monitor, a constant electric field accelerates the electrons to high speed; then a magnetic field is used to deflect the electrons to the side. Why can't a constant magnetic field be used to speed up the electrons?

Averell Hause
Averell Hause
Carnegie Mellon University
02:52

Problem 5

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

Vishal Gupta
Vishal Gupta
Numerade Educator
02:31

Problem 6

A uniform magnetic field directed upward exists in some region of space. In what direction(s) could an electron be moving if its trajectory is (a) a straight line?
(b) a circle? Assume that the electron is subject only to magnetic forces.

LA
Lazarus Arnau
Numerade Educator
View

Problem 6

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

Anna Zeng
Anna Zeng
Numerade Educator
01:24

Problem 7

In a velocity selector, the electric and magnetic forces cancel if $\overrightarrow{\mathbf{E}}+\overrightarrow{\mathbf{v}} \times \overrightarrow{\mathbf{B}}=0 .$ Show that $\overrightarrow{\mathbf{v}}$ must be in the same direction as $\overrightarrow{\mathbf{E}} \times \overrightarrow{\mathbf{B}}$. [Hint: Since $\overrightarrow{\mathbf{v}}$ is perpendicular to both $\overrightarrow{\mathbf{E}}$ and $\overrightarrow{\mathbf{B}}$ in a velocity selector, there are only two possibilities for the direction of $\overrightarrow{\mathbf{v}}$ : the direction of $\hat{\mathbf{E}} \times \overrightarrow{\mathbf{B}}$ or the direction of $-\overrightarrow{\mathbf{E}} \times \overrightarrow{\mathbf{B}} .]$

Joseph Petrullo
Joseph Petrullo
Numerade Educator
06:04

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 counterclockwise - about 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
Anna Zeng
Numerade Educator
04:40

Problem 8

Two ions with the same velocity and mass but different charges enter the magnetic field of a mass spectrometer. One is singly charged $(q=+e)$ and the other is doubly charged $(q=+2 e)$. Is the radius of their circular paths the same? If not, which is larger? By what factor?

Donald Albin
Donald Albin
Numerade Educator
02:47

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 $0.50$ T directed north.

Anna Zeng
Anna Zeng
Numerade Educator
00:55

Problem 9

The mayor of a city proposes a new law to require that magnetic fields generated by the power lines running through the city be zero outside of the electric company's right of way. What would you say at a public discussion of the proposed law?

Mayukh Banik
Mayukh Banik
Numerade Educator
03:44

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?

Noor Aldeen Almusleh
Noor Aldeen Almusleh
Numerade Educator
03:47

Problem 10

A horizontal wire that runs east-west carries a steady current to the east. A C-shaped magnet (see Fig. $19.3 \mathrm{a}$ ) is placed so that the wire runs between the poles, with the north pole above the wire and the south pole below. What is the direction of the magnetic force on the wire between the poles?

Vishal Gupta
Vishal Gupta
Numerade Educator
08:30

Problem 10

Find the magnetic force (magnitude and direction) on an electron moving at speed $8.0$ $\times 10^{5} \mathrm{~m} / \mathrm{s}$ for each of the
directions shown. The magnetic field has magnitude $B=0.40 \mathrm{~T}$

Vishal Gupta
Vishal Gupta
Numerade Educator
01:37

Problem 11

The magnetic field due to a long straight wire carrying steady current is measured at two points, $P$ and $Q .$ Where is the wire and in what direction does the current flow?

Mayukh Banik
Mayukh Banik
Numerade Educator
06:37

Problem 11

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 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
Donya Dobbin
Numerade Educator
01:08

Problem 12

A circular loop of current carries a steady current. (a) Sketch the magnetic field lines in a plane perpendicular to the plane of the loop. (b) Which side of the loop is the north pole of the magnetic dipole and which is the south pole?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:35

Problem 12

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
Anna Zeng
Numerade Educator
01:08

Problem 13

Computer speakers that are intended to be placed near a computer monitor are magnetically shielded - either they don't use magnets or they are designed so that their magnets produce only a small magnetic field nearby. Why is the shielding important? What might happen if an ordinary speaker (not intended for use near a monitor) is placed next to a computer monitor?

Ajay Singhal
Ajay Singhal
Numerade Educator
09:07

Problem 13

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? [Hint: The electron velocity changes relatively little, so approximate the magnetic force as a constant force acting during a $2.4-\mathrm{cm}$ displacement to the right.]

Vishal Gupta
Vishal Gupta
Numerade Educator
00:51

Problem 14

One iron nail does not necessarily attract another iron nail, although both are attracted by a magnet. Explain.

Salamat Ali
Salamat Ali
Numerade Educator
04:14

Problem 14

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
Anna Zeng
Numerade Educator
06:10

Problem 15

Two wires at right angles in a plane carry equal currents. At what points in the plane is the magnetic field zero?

Dading Chen
Dading Chen
Numerade Educator
03:33

Problem 15

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
00:40

Problem 16

If a magnet is held near the screen of a TV, computer monitor, or oscilloscope, the picture is distorted. [Don't try this - see part (b).] (a) Why is the picture distorted? (b) With a color TV or monitor, the distortion remains even after the magnet is removed. Explain. (A color CRT has a metal mask just behind the screen with holes to line up the electrons from different guns with the red, green, and blue phosphors. Of what kind of metal is the mask made?)

Suzanne W.
Suzanne W.
Numerade Educator
03:20

Problem 16

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
01:04

Problem 17

A metal bar is shown at two different times. The arrows represent the alignment of the dipoles within each magnetic domain. (a) What happened between $t_{1}$ and $t_{2}$ to cause the change? (b) Is the metal a paramagnet, diamagnet, or ferromagnet? Explain.

Narayan Hari
Narayan Hari
Numerade Educator
07:25

Problem 17

At a certain point on the surface of the Earth in the southern hemisphere, the Earth's 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
Anna Zeng
Numerade Educator
03:05

Problem 18

Explain why a constant magnetic field does no work on a point charge moving through the field. Since the field does no work, what can we say about the speed of a point charge acted on only by a magnetic field?

Averell Hause
Averell Hause
Carnegie Mellon University
01:38

Problem 18

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

Vishal Gupta
Vishal Gupta
Numerade Educator
01:49

Problem 19

Refer to the bubble chamber tracks in Fig. 19.15a. Suppose that particle 2 moves in a smaller circle than particle 1. Can we conclude that $\left|q_{2}\right|>\left|q_{1}\right|$ ? Explain.

Yaqub Khan
Yaqub Khan
Numerade Educator
01:32

Problem 19

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
01:37

Problem 20

The trajectory of a charged particle in a uniform magnetic field is a helix if $\overrightarrow{\mathbf{v}}$ has components both parallel to and perpendicular to the field. Explain how the two other cases (circular motion for $v_{\|}=0$ and straight line motion for $v_{\perp}=0$ ) can each be considered to be special cases of helical motion.

Ajay Singhal
Ajay Singhal
Numerade Educator
02:06

Problem 20

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

Vishal Gupta
Vishal Gupta
Numerade Educator
09:09

Problem 21

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
Anna Zeng
Numerade Educator
02:16

Problem 22

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 $12.0000 \mathrm{u}$. 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 are accelerated through the same potential difference before entering the magnetic field.)

Noor Aldeen Almusleh
Noor Aldeen Almusleh
Numerade Educator
03:18

Problem 23

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?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:52

Problem 24

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
Donya Dobbin
Numerade Educator
09:43

Problem 25

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 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.

Noor Aldeen Almusleh
Noor Aldeen Almusleh
Numerade Educator
06:32

Problem 26

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?
(c) Repeat parts (a) and (b) if the ions are given the same kinetic energy rather than the same velocity.

Noor Aldeen Almusleh
Noor Aldeen Almusleh
Numerade Educator
07:26

Problem 27

Crossed electric and magnetic fields are established over a certain region. The magnetic field is $0.635 \mathrm{~T}$ 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
Anna Zeng
Numerade Educator
01:51

Problem 28

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
Bettina Hanlon
Numerade Educator
02:22

Problem 29

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 " "'" sign in your sketch. Explain briefly.

Noor Aldeen Almusleh
Noor Aldeen Almusleh
Numerade Educator
01:22

Problem 30

In Problem 29 , 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
Bettina Hanlon
Numerade Educator
01:11

Problem 31

In Problem 29, 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?

Bettina Hanlon
Bettina Hanlon
Numerade Educator
00:54

Problem 32

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
Bettina Hanlon
Numerade Educator
02:20

Problem 33

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
Donya Dobbin
Numerade Educator
09:18

Problem 34

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 $1.20$ -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
Vishal Gupta
Numerade Educator
07:28

Problem 35

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
Vishal Gupta
Numerade Educator
07:39

Problem 36

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 $\left(\mathrm{in} \mathrm{m}^{3} / \mathrm{s}\right) ?(\mathrm{c})$ 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
Vishal Gupta
Numerade Educator
01:21

Problem 37

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
Bettina Hanlon
Numerade Educator
01:45

Problem 38

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
Donya Dobbin
Numerade Educator
04:46

Problem 39

Parallel conducting tracks, separated by $2.0 \mathrm{~cm}$, run north and south. There is a uniform magnetic field of 1.2 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
Vishal Gupta
Numerade Educator
06:54

Problem 40

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 $39 .$ 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
Vishal Gupta
Numerade Educator
02:47

Problem 41

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
Donya Dobbin
Numerade Educator
04:34

Problem 42

A $20.0 \mathrm{~cm} \times 30.0 \mathrm{~cm}$ rectangular loop of wire carries $1.0 \mathrm{~A}$ of current clockwise around the $-30$ loop. (a) Find the magnetic force on each side $\operatorname{Pr}$ 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?

Vishal Gupta
Vishal Gupta
Numerade Educator
07:22

Problem 43

Repeat Problem 42 if the magnetic field is $2.5 \mathrm{~T}$ to the left (in the $-x$ -direction).

Vishal Gupta
Vishal Gupta
Numerade Educator
03:20

Problem 44

A straight wire is aligned east-west in a region where the 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
Donya Dobbin
Numerade Educator
02:27

Problem 45

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 field? (b) Is the torque on the coil clockwise or counterclockwise as viewed from the front at the instant shown in the figure?

Donya Dobbin
Donya Dobbin
Numerade Educator
01:16

Problem 46

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
Donya Dobbin
Numerade Educator
03:08

Problem 47

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
Donya Dobbin
Numerade Educator
03:08

Problem 48

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
Donya Dobbin
Numerade Educator
07:52

Problem 49

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. [Hint: Set the sum of the torques from the spring and the magnetic field equal to $0 .$ ]

Vishal Gupta
Vishal Gupta
Numerade Educator
02:38

Problem 50

The torque on a loop of wire (a magnetic dipole) in a uniform magnetic field is $\tau=N I A B \sin \theta$, where $\theta$ is the 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 NIA is the magnetic dipole moment.)

Mayukh Banik
Mayukh Banik
Numerade Educator
01:52

Problem 51

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. [Hint: Think of a curved loop as a series of tiny, straight, perpendicular segments.]

Dominador Tan
Dominador Tan
Numerade Educator
09:09

Problem 52

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

Vishal Gupta
Vishal Gupta
Numerade Educator
02:02

Problem 53

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
Donya Dobbin
Numerade Educator
05:57

Problem 54

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
Vishal Gupta
Numerade Educator
03:39

Problem 55

Two wires each carry $10.0 \mathrm{~A}$ of current (in opposite directions) and are $3.0 \mathrm{~mm}$ apart. (a) Calculate the magnetic field $25 \mathrm{~cm}$ away at point $P$, in the plane of the wires. (b) What is the magnetic field at the same point if the currents instead both run to the left?

Ivan Kochetkov
Ivan Kochetkov
Numerade Educator
03:13

Problem 56

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 $1.0$ A toward the north. (a) Find the magnitude and direction of the magnetic field at point $P$. (b) Repeat the question if the current in the wire on the east side runs toward the south instead.

Vishal Gupta
Vishal Gupta
Numerade Educator
01:57

Problem 57

A long straight wire carries a current 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?

Donya Dobbin
Donya Dobbin
Numerade Educator
04:14

Problem 58

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
Donya Dobbin
Numerade Educator
06:03

Problem 59

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
Vishal Gupta
Numerade Educator
01:22

Problem 60

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
Donya Dobbin
Numerade Educator
04:09

Problem 61

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
Bettina Hanlon
Numerade Educator
05:38

Problem 62

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
Vishal Gupta
Numerade Educator
03:02

Problem 63

The derivation of Eq. (19-16) for the magnetic field at the center of a circular current loop, $B_{\text {loop }}=\mu_{0} I /(2 r)$, requires calculus. We can find an approximate value by arranging four long straight wires, each with current $I$, so they intersect to form a square loop with side $2 r$. Find the magnetic field at the center of the square. Express your answer in terms of $B_{\text {loop }}$.

Bettina Hanlon
Bettina Hanlon
Numerade Educator
02:48

Problem 64

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
Donya Dobbin
Numerade Educator
03:14

Problem 65

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
Vishal Gupta
Numerade Educator
01:35

Problem 66

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
Donya Dobbin
Numerade Educator
01:25

Problem 67

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 pei unit length $(n)$ and the physical dimensions of the path. (d) Solve for $B$.

Dominador Tan
Dominador Tan
Numerade Educator
04:54

Problem 68

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
Vishal Gupta
Numerade Educator
03:12

Problem 69

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
Vishal Gupta
Numerade Educator
00:57

Problem 70

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

Donya Dobbin
Donya Dobbin
Numerade Educator
03:43

Problem 71

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 to the intrinsic magnetic dipole moment of the electron $\left(9.3 \times 10^{-24} \mathrm{~A} \cdot \mathrm{m}^{2}\right)$.

Bettina Hanlon
Bettina Hanlon
Numerade Educator
02:47

Problem 72

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 of the iron core?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:12

Problem 73

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? Explain.

Vishal Gupta
Vishal Gupta
Numerade Educator
02:30

Problem 74

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. (Neglect the Earth's magnetic field.)

Vishal Gupta
Vishal Gupta
Numerade Educator
01:55

Problem 75

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
Donya Dobbin
Numerade Educator
05:58

Problem 76

A long straight wire carries a 4.70-A current in the positive $x$ -direction. At a particular instant, an electron moying 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 57 .

Vishal Gupta
Vishal Gupta
Numerade Educator
02:04

Problem 77

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
Donya Dobbin
Numerade Educator
03:30

Problem 78

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
Donya Dobbin
Numerade Educator
04:39

Problem 79

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 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
Bettina Hanlon
Numerade Educator
02:43

Problem 80

An electron moves in a circle of radius $R$ in a uniform magnetic 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
Donya Dobbin
Numerade Educator
02:49

Problem 81

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
Donya Dobbin
Numerade Educator
03:54

Problem 82

(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
Donya Dobbin
Numerade Educator
03:22

Problem 83

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
Donya Dobbin
Numerade Educator
01:16

Problem 84

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
Donya Dobbin
Numerade Educator
02:55

Problem 85

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 $B=0.20 \mathrm{~T}$. The ions pass between deflection plates spaced $1.0 \mathrm{~cm}$ 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 $0.20 \mathrm{~m}$. 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. (e) Find the correct values of $\Delta V_{1}$ and $\Delta V_{2}$ in order to count ${ }^{14} \mathrm{C}^{+}$ ions.

Dominador Tan
Dominador Tan
Numerade Educator
03:58

Problem 86

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
Vishal Gupta
Numerade Educator
07:31

Problem 87

Repeat Problem 42 if the magnetic field is $2.5 \mathrm{~T}$ in the plane of the loop, $60.0^{\circ}$ below the $+x$ -axis.

Vishal Gupta
Vishal Gupta
Numerade Educator
06:37

Problem 88

In Problem 59, find the magnetic field at points $C$ and $D$ when $d=3.3 \mathrm{~cm}$ and $I=6.50 \mathrm{~A}$.

Vishal Gupta
Vishal Gupta
Numerade Educator
09:28

Problem 89

Four long parallel wires pass through the four corners of a square. All four wires carry the same amount of current and the current directions are as indicated. (a) What is the direction of the magnetic field at the center of the square? (b) The current is $10.0 \mathrm{~A}$ in each wire and the square is $0.10 \mathrm{~m}$ on a side. What is the magnitude of the magnetic field?

Vishal Gupta
Vishal Gupta
Numerade Educator
06:52

Problem 90

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 compared to the radius of the coils, approximate the setup as two long parallel straight wires.]

Vishal Gupta
Vishal Gupta
Numerade Educator
09:14

Problem 91

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.]

Vishal Gupta
Vishal Gupta
Numerade Educator
01:06

Problem 92

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
Donya Dobbin
Numerade Educator
01:00

Problem 93

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? (Don'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
Donya Dobbin
Numerade Educator
01:59

Problem 94

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
Donya Dobbin
Numerade Educator
03:02

Problem 95

The strength of the 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 the 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 the 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 the Earth's field?

Donya Dobbin
Donya Dobbin
Numerade Educator
06:18

Problem 96

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. $\mathrm{A}$ 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 $B_{\mathrm{H}}$. [Hint: The name of the instrument is a clue to the result.]

Vishal Gupta
Vishal Gupta
Numerade Educator
04:21

Problem 97

An early cyclotron at Cornell University was used from the $1930 \mathrm{~s}$ to the $1950 \mathrm{~s}$ 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
Donya Dobbin
Numerade Educator
03:40

Problem 98

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
Donya Dobbin
Numerade Educator
02:57

Problem 99

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 $20.0$ 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 $22.0 \mathrm{u}$ leave the field?

Donya Dobbin
Donya Dobbin
Numerade Educator
03:20

Problem 100

A proton moves in a helical path at speed $v=4.0 \times$ $10^{7} \mathrm{~m} / \mathrm{s}$ high in the atmosphere, where the 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
Donya Dobbin
Numerade Educator