University Physics with Modern Physics

Hugh D. Young, Roger A. Freeman

Chapter 27

Magnetic Field and Magnetic Forces - all with Video Answers

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Chapter Questions

Problem 1

A particle with a charge of $-1.24 \times 10^{-8} \mathrm{C}$ is moving with instantancous $\quad$ velocity $\quad \vec{v}=\left(4.19 \times 10^{4} \mathrm{m} / \mathrm{s}\right) \hat{\imath}+(-3.85 \times$ $10^{4} \mathrm{m} / \mathrm{s}$ ) $\hat{\mathrm{v}}$ . What is the force exerted on this particle by a magnetic field $(a) \vec{B}=(1.40 \mathrm{T}) \hat{\imath}$ and $(b) \vec{B}=(1.40 \mathrm{T}) \hat{k} ?$

Zulfiqar Ali

Zulfiqar Ali

Numerade Educator

Problem 2

A particle of mass 0.195 g carries a charge of $-2.50 \times$ $10^{-8} \mathrm{C}$ . The particle is given an initial horizontal velocity that is due north and has magnitude $4.00 \times 10^{4} \mathrm{m} / \mathrm{s}$ . What are the magnitude and direction of the minimum magnetic field that will keep the particle moving in the earth's gravitational field in the same horizontal, northward direction?

Ceren Uzun

Texas Tech University

Problem 3

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

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 4

A particle with mass $1.81 \times 10^{-3} \mathrm{kg}$ and a charge of $1.22 \times$ $10^{-8} \mathrm{C}$ has, at a given instant, a velocity $\vec{v}=\left(3.00 \times 10^{4} \mathrm{m} / \mathrm{s}\right) \hat{j}$ What are the magnitude and direction of the particle's accoleration produced by a uniform magnetic field $\vec{B}=(1.63 \mathrm{T}) \hat{\imath}+$ $(0.980 \mathrm{T}) \hat{\jmath} ?$

Zulfiqar Ali

Numerade Educator

Problem 5

An electron experiences a magnetic force of magnitude $4.60 \times 10^{-15} \mathrm{N}$ when moving at an angle of $60.0^{\circ}$ with respect to a magnetic field of magnitude $3.50 \times 10^{-3} \mathrm{T}$ . Find the speed of the electron.

Sachin Rao

Numerade Educator

Problem 6

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

Zulfiqar Ali

Numerade Educator

Problem 7

A particle with charge 7.80$\mu \mathrm{C}$ is moving with velocity $\vec{v}=-\left(3.80 \times 10^{3} \mathrm{m} / \mathrm{s}\right) \hat{\jmath} .$ The magnetic force on the particle is measured to be $\overrightarrow{\boldsymbol{F}}=+\left(7.60 \times 10^{-3} \mathrm{N}\right) \hat{\boldsymbol{i}} \left(5.20 \times 10^{-3} \mathrm{N}\right) \hat{\boldsymbol{k}}$ (a) Calculate all the components of the magnetic field you can from this information. (b) Are there components of the magnetic field that are not determined by the measurement of the force? Explain. (c) Calculate the scalar product $\overrightarrow{\boldsymbol{B}} \cdot \overrightarrow{\boldsymbol{F}}$ What is the angle between $\overrightarrow{\boldsymbol{B}}$ and $\overrightarrow{\boldsymbol{F}} ?$

Zulfiqar Ali

Numerade Educator

Problem 8

A particle with charge $-5.60 \mathrm{nC}$ is moving in a uniform magnetic field $\overrightarrow{\boldsymbol{B}}=-(1.25 \mathrm{T}) \hat{\boldsymbol{k}} .$ The magnetic force on the particle is measured to be $\overrightarrow{\boldsymbol{F}}=-\left(3.40 \times 10^{-7} \mathrm{N}\right) \hat{\boldsymbol{i}}+\left(7.40 \times 10^{-7} \mathrm{N}\right) \hat{\boldsymbol{j}}$
(a) Calculate all the components of the velocity of the particle that you can from this information. (b) Are there components of the velocity that are not determined by the measurement of the force? Explain. (c) Calculate the scalar product $\overrightarrow{\boldsymbol{F}} \cdot \overrightarrow{\boldsymbol{F}}$ . What is the angle between $\overrightarrow{\boldsymbol{v}}$ and $\overrightarrow{\boldsymbol{F}} ?$

Zulfiqar Ali

Numerade Educator

Problem 9

A group of particles is traveling in a magnetic field of unknown magnitude and direction. You observe that a proton moving at 1.50 $\mathrm{km} / \mathrm{s}$ im the $+x$ -direction experiences a force of $2.25 \times 10^{-16} \mathrm{N}$ in the $+y$ -direction, and an electron moving at 4.75 $\mathrm{km} / \mathrm{s}$ in the $-\mathrm{z}$ -direction experiences a force of $8.50 \times 10^{-16} \mathrm{N}$ . (a) What are the magnitude and direction of the magnetic field? (b) What are the magnitude and direction of the magnetic force on an electron moving in the $-y$ -direction at 3.2 $\mathrm{km} / \mathrm{s} ?$

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 10

The magnetic flux through one face of a cube is $+0.120 \mathrm{Wb} .$ (a) What must the total magnetic flux through the other five faces of the cube be? (b) Why didn't you need to know the dimensions of the cube in order to answer part (a) 2 (c) Suppose the magnetic flux is due to a permanent magnet like that shown in Fig. $27.11 .$ In a sketch, show where the cube in part (a) might be located relative to the magnet.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

A circular area with a radius of 6.50 $\mathrm{cm}$ lies in the $x y$ -plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field $B=0.230 \mathrm{T}$ (a) in the $+z$ -direction; $(b)$ at an angle of $53.1^{\circ}$ from the $+z$ -direction; $(\mathrm{c})$ in the $+y$ -direction?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 12

The magnetic field $\overrightarrow{\boldsymbol{B}}$ in a certain region is 0.128 $\mathrm{T}$ , and its direction is that of the $+z$ -axis in Fig. 27.45 . (a) What is the magnetic flux across the surface $a b c d$ in the figure? (b) What is the magnetic flux across the surface befc? (c) What is the magnetic flux across the surface aefd? (d) What is the net flux through all five surfaces that enclose the shaded volume?

Vishal Gupta

Numerade Educator

Problem 13

An open plastic soda bottle with an opening diameter of 2.5 $\mathrm{cm}$ is placed on a table. A
uniform $1.75-\mathrm{T}$ magnetic field directed upward and oriented $25^{\circ}$ from vertical encompasses the bottle. What is the total magnetic flux through the plastic of the soda bottle?

Daniel Matthias

Daniel Matthias

Numerade Educator

Problem 14

A particle with charge $6.40 \times 10^{-19} \mathrm{C}$ travels in a circular orbit with radius 4.68 $\mathrm{mm}$ due to the force exerted on it by a magnetic field with magnitude 1.65 $\mathrm{T}$ and perpendicular to the orbit. (a) What is the magnitude of the linear momentum $\vec{p}$ of the partcle? (b) What is the magnitude of the angular momentum $\overrightarrow{\boldsymbol{L}}$ of the particle?

Ajay Singhal

Numerade Educator

Problem 15

An electron at point $A$ in Fig. 27.46 has a speed $v_{0}$ of $1.41 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Find magnitude and direction of the magnetic field that will cause the electron to follow the semicircular path from $A$ to $B,$ and (b) the time required for the electron to move from $A$ to $B$ .

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

Repeat Exercise 27.15 for the case in which the particle is a proton rather than an electron.

Ceren Uzun

Texas Tech University

Problem 17

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

Zulfiqar Ali

Numerade Educator

Problem 18

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

Averell Hause

Carnegie Mellon University

Problem 19

Fusion Reactor. If two deuterium nuclei (charge $+e$ . mass $3.34 \times 10^{-27} \mathrm{kg}$ ) get close enough together, the attraction of the strong nuclear force will fuse to make an isotope of helium, releasing vast amounts of energy. The range of this force is about $10^{-15} \mathrm{m}$ . This is the principle behind the fusion reactor. The deuterium nuclei are moving much too fast to be contained by physical walls, so they are confined magnetically. (a) How fast would two nuclei have to move so that in a head-on collision they would get close enough to fuse? (Treat the nuclei as point charges, and assume that a separation of $1.0 \times 10^{-15}$ is required for fusion, $)$ (b) What strength magnetic field is needed to make deuterium nuclei with this speed travel in a circle of diameter 2.50 $\mathrm{m} ?$

Zulfiqar Ali

Numerade Educator

Problem 20

(a) An 16 nucleus (charge $+8 e )$ moving horizontally from west to east with a speed of 500 $\mathrm{km} / \mathrm{s}$ experiences a magnetic force of 0.00320 $\mathrm{nN}$ vertically downward. Find the magnitude and direction of the weakest magnetic field required to produce this force. Explain how this same force could be caused by a larger magnetic field. (b) An electron moves in a uniform, horizontal, 2.10 - T magnetic field that is toward the west. What must the magnitude and
direction of the minimum velocity of the electron be so that the magnetic force on it will be 4.60 $\mathrm{pN}$ , vertically upward? Explain how the velocity could be greater than this minimum value and the force still have this same magnitude and direction.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 21

A deuteron (the nucleus of an isotope of bydrogen) has a mass of $3.34 \times 10^{-27} \mathrm{kg}$ and a charge of $+e .$ The deuteron travels in a circular path with a radius of 6.96 $\mathrm{mm}$ in a magnetic field with magnitude 2.50 $\mathrm{T}$ . (a) Find the speed of the deuteron. (b) Find the time required for it to make half a revolution. (c) Through what to potential difference would the deuteron have to be accelerated to acquire this speed?

Zulfiqar Ali

Numerade Educator

Problem 22

In an experiment with cosmic rays, a vertical beam of particles that have charge of magnitude 3$e$ and mass 12 times the proton mass enters a uniform horizontal magnetic field of 0.250 . T and is bent in a semicircle of diameter $95.0 \mathrm{cm},$ as shown in Fig. 27.47 . (a) Find the speed of the particles and the sign of their charge. (b) Is it reasonable to ignore the gravity force on the particles? (c) How does the speed of the particles as they enter the field compare to their speed as they exit the field?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 23

A physicist wishes to produce electromagnetic waves of frequency 3.0 THz $\left(1 \mathrm{THz}=1 \text { terahertz }=10^{12} \mathrm{Hz}\right)$ using a magnetron (see Example $27.3 ) .$ (a) What magnetic field would be required? Compare this field with the strongest constant magnetic fields yet produced on earth, about 45 $\mathrm{T}$ . (b) Would there be any advantage to using protons instead of electrons in the magnetron? Why or why not?

Vishal Gupta

Numerade Educator

Problem 24

A beam of protons traveling at 1.20 $\mathrm{km} / \mathrm{s}$ enters a uniform magnetic field, traveling perpendicular to the field. The beam exits the magnetic field, leaving the field in a direction perpendicular- to its original direction (Fig. 27.48$)$ . The beam travels a distance of 1.18 $\mathrm{cm}$ while in the field. What is the magnitude of the magnetic field?

Narayan Hari

Numerade Educator

Problem 25

An electron in the beam of a TV picture tube is accelerated by a potential difference of 2.00 $\mathrm{kV}$ . Then it passes through a region of transverse magnetic field, where it moves in a circular arc with radius 0.180 $\mathrm{m}$ . What is the magnitude of the field?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 26

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

Zulfiqar Ali

Numerade Educator

Problem 27

A proton $\left(q=1.60 \times 10^{-19} \mathrm{C}, m=1.67 \times 10^{-27} \mathrm{kg}\right)$ moves in a uniform magnetic field $\overrightarrow{\boldsymbol{B}}=(0.500 \mathrm{T}) \hat{\boldsymbol{i}} .$ At $t=0$ the proton has velocity components $v_{x}=1.50 \times 10^{5} \mathrm{m} / \mathrm{s}, v_{y}=0,$ and $v_{z}=2.00 \times 10^{5} \mathrm{m} / \mathrm{s}(\text { see Example } 27.4) .$ (a) What are the magnitude and direction of the magnetic force acting on the proton? In addition to the magnetic field there is a uniform electric field in the $+x$ -direction, $\vec{E}=\left(+2.00 \times 10^{4} \mathrm{V} / \mathrm{m}\right) \hat{\imath}$ (b) Will the proton have a component of acceleration in the direction of the electric field?(c) Describe the path of the proton. Does the electric field affect the radius of the helix? Explain. (d) At $t=T / 2$ , where $T$ is the period of the circular motion of the proton, what is the $x$ -component of the displacement of the proton from its position at $t=0 ?$

Zulfiqar Ali

Numerade Educator

Problem 28

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

Vishal Gupta

Numerade Educator

Problem 29

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

Averell Hause

Carnegie Mellon University

Problem 30

Crussed $\vec{E}$ and $\vec{B}$ Fields. A particle with initial velocity $\overrightarrow{\boldsymbol{v}}_{0}=\left(5.85 \times 10^{3} \mathrm{m} / \mathrm{s}\right) \hat{\boldsymbol{j}}$ enters a region of uniform electric and magnetic fields. The magnetic field in the region is $\overrightarrow{\boldsymbol{B}}=$ $-(1.35 \mathrm{T}) \hat{\boldsymbol{k}} .$ Calculate the magnitude and direction of the electric field in the region if the particle is to pass through undeflected, for a particle of charge $(\mathrm{a})+0.640 \mathrm{nC}$ and $(\mathrm{b})-0.320 \mathrm{nC}$ . You can ignore the weight of the particle.

Zulfiqar Ali

Numerade Educator

Problem 31

Determining the Mass of an Isotope. The electric field between the plates of the velocity selector in a Bainbridge mass spectrometer (see Fig. 27.22) is 1.12 $\times 10^{5} \mathrm{V} / \mathrm{m}$ , and the magnetic field in both regions is 0.540 T. A stream of singly charged selenium ions moves in a circular path with a radius of 31.0 $\mathrm{cm}$ in the magnetic field. Determine the mass of one selenium ion and the mass number of this selenium isotope. (The mass number is equal to the mass of the isotope in atomic mass units, rounded to the nearest integer. One atomic mass unit $=1 \mathbf{u}=1.66 \times 10^{-27} \mathrm{kg} .$ .

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 32

In the Bainbridge mass spectrometer (see Fig. 27.24$)$ , the magnetic-field magnitude in the velocity selector is 0.650 $\mathrm{T}$ , and ions having a speed of $1.82 \times 10^{6} \mathrm{m} / \mathrm{s}$ pass through undeflected.
(a) What is the electric-field magnitude in the velocity selector?
(b) If the separation of the plates is $5.20 \mathrm{mm},$ what is the potential difference between plates $P$ and $P^{\prime \prime} ?$

Prabhat Tyagi

Numerade Educator

Problem 33

A straight $2.00-\mathrm{m}, 150-\mathrm{g}$ wire carries a current in a region where the earth's magnetic field is horizontal with a magnitude of 0.55 gauss. (a) What is the minimum value of the current in this wire so that its weight is completely supported by the magnetic force due to earth's field, assuming that no other forces except gravity act on it? Does it scem likely that such a wire could support this size of current? (b) Show how the wire would have to be oriented relative to the earth's magnetic field to be supported in this way.

Zulfiqar Ali

Numerade Educator

Problem 34

An electromagnet produces a magnetic field of 0.550 T in a cylindrical region of radius 2.50 $\mathrm{cm}$ between its poles. A straight wire carrying a current of 10.8 $\mathrm{A}$ passes through the center of this region and is perpendicular to both the axis of the cylindrical region and the magnetic field. What magnitude of force is exerted on the wire?

Zulfiqar Ali

Numerade Educator

Problem 35

A long wire carrying 4.50 $\mathrm{A}$ of current makes two $90^{\circ}$ bends, as shown in Fig. 27.49 . The bent part of the wire passes through a uniform 0.240 - T magnetic field directed as shown in the figure and confined to a limited region of space. Find the magnitude and direction of the force that the magnetic field exerts on the wire.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 36

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

Zulfiqar Ali

Numerade Educator

Problem 37

A horizontal rod 0.200 $\mathrm{m}$ long is mounted on a balance and carries a current. At the location of the rod a uniform horizontal magnetic field has magnitude 0.067 T and direction perpendicular to the rod. The magnetic force on the rod is measured by the balance and is found to be 0.13 $\mathrm{N}$ . What is the current?

Zulfiqar Ali

Numerade Educator

Problem 38

In Fig. $27.50,$ a wire carrying current into the plane of the figure is between the north and south poles of two bar magnets. What is the direction of the force exerted by the magnets on the wire?

Zulfiqar Ali

Numerade Educator

Problem 39

$\mathbf{A}$ thin 50.0 -cm-long metal bar with mass 750 g rests on, but is not attached to, two
metallic supports in a uniform 0.450 - T magnetic field, as shown in Fig. 27.51 . A battery and a $25.0-\Omega$ resistor in series are connected to the supports. (a) What is the highest voltage the battery can have without breaking the circuit at the supports? (b) The battery voltage has the maximum value calculated in part (a). If the resistor suddenly gets partially short-circuited, decreasing its resistance to 2.0$\Omega$ , find the initial acceleration of the bar.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 40

Magnetic Balance. The circuit shown in Fig. 27.52 is used to make a magnetic balance to weigh objects. The mass $m$ to be measured is hung from the center of the bar that is in a uni- form magnetic field of 1.50 $\mathrm{T}$ , directed into the plane of the figure. The battery voltage can be
adjusted to vary the current in the circuit. The horizontal bar is 60.0 $\mathrm{cm}$ long and is made of extremely light-weight material. It is connected to the battery by thin vertical wires that can support no appreciable tension; all the weight of the suspended mass $m$ is supported by the magnetic force on the bar. A resistor with $R=5.00 \Omega$ is in series with the bar; the resistance of the rest of the circuit is much less than this. (a) Which point, $a$ or $b$ , should be the positive terminal of the battery? (b) If the maximum terminal voltage of the battery is 175 $\mathrm{V}$ , what is the greatest mass $m$ that this instrument can measure?

Zulfiqar Ali

Numerade Educator

Problem 41

Consider the conductor and current in Example $27.8,$ but now let the magnetic field be parallel to the $x$ -axis. (a) What are the magnitude and direction of the total magnetic force on the conductor? (b) In Example 27.8 , the total force is the same as if we replaced the semicircle with a straight segment along the $x$ -axis. Is that still true when the magnetic field is in this different direction? Can you explain why, or why not?

Jacob Schulze

Numerade Educator

Problem 42

The plane of a $5.0 \mathrm{cm} \times 8.0 \mathrm{cm}$ rectangular loop of wire is parallel to a $0.19-\mathrm{T}$ magnetic field. The loop carries a current of 6.2 $\mathrm{A}$ . (a) What torque acts on the loop? (b) What is the magnetic moment of the loop? (c) What is the maximum torque that can be obtained with the same total length of wire carrying the same current in this magnetic field?

Ceren Uzun

Texas Tech University

Problem 43

Magnetic Moment of the Hydrogen Atom. In the Bohr model of the hydrogen atom (see Section $38.5 ),$ in the lowest energy state the electron orbits the proton at a speed of $2.2 \times$ $10^{6} \mathrm{m} / \mathrm{s}$ in a circular orbit of radius $5.3 \times 10^{-11} \mathrm{m}$ (a) What is the orbital period of the electron? (b) If the orbiting electron is considered to be a current loop, what is the current $I ?$ (c) What is the magnetic moment of the atom due to the motion of the electron?

Zulfiqar Ali

Numerade Educator

Problem 44

A rectangular coil of wire, 22.0 $\mathrm{cm}$ by 35.0 $\mathrm{cm}$ and carrying a current of $1.40 \mathrm{A},$ is oriented with the plane of its loop perpendicular to a uniform 1.50 - T magnetic field, as shown in Fig. 27.53 . (a) Calculate the net force and torque that the magnetic field exerts on the coil. (b) The coil is rotated through a $30.0^{\circ}$ angle about the axis shown, with the left side coming out of
the plane of the figure and the right side going into the plane. Calculate the net force and torque that the magnetic field now exerts on the coil. (Hint: In order to help visualize this three dimensional problem, make a careful drawing of the coil as viewed along the rotation axis.)

Salamat Ali

Numerade Educator

Problem 45

A uniform rectangular coil of total mass 210 $\mathrm{g}$ and dimensions $0.500 \mathrm{m} \times 1.00 \mathrm{m}$ is oriented perpendicular to a uniform $3.00-\mathrm{T}$ magnetic field $(\mathrm{Fig}, 27.54)$ . A current of 2.00 $\mathrm{A}$ is suddenly started in the coil. (a) About which axis $\left(A_{1} \text { or } \mathrm{A}_{2}\right)$ will the coil begin to rotate? Why? (b) Find the initial angular acceleration of the coil just after the current is started.

Vishal Gupta

Numerade Educator

Problem 46

A circular coil with area $A$ and $N$ turns is free to rotate about a diameter that coincides with the $x$ -axis. Current $I$ is circulating in the coil. There is a uniform magnetic field $\overrightarrow{\boldsymbol{B}}$ in the positive $y$ -direction. Calculate the magnitude and direction of the torque $\vec{\tau}$ and the value of the potential energy $U,$ as given in Eq. $(27.27),$ when the coil is oriented as shown in parts (a) through (d) of Fig. 27.55 .

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 47

A coil with magnetic moment 1.45 $\mathrm{A} \cdot \mathrm{m}^{2}$ is oriented initially with its magnetic moment antiparallel to a uniform $0.835-\mathrm{T}$ magnetic field. What is the change in potential energy of the coil when it is rotated $180^{\circ}$ so that its magnetic moment is parallel to the field?

Ceren Uzun

Texas Tech University

Problem 48

A dc motor with its rotor and field coils connected in series has an internal resistance of 3.2$\Omega$ . When the motor is running at full load on a $120-\mathrm{V}$ line, the emf in the rotor is 105 $\mathrm{V}$ . (a) What is the current drawn by the motor from the line? (b) What is the power delivered to the motor? (c) What is the mechanical power developed by the motor?

Ceren Uzun

Texas Tech University

Problem 49

In a shunt-wound dc motor with the field coils and rotor connected in parallel (Fig. $27.56 ),$ the resistance $R_{t}$ of the field coils is $106 \Omega,$ and the resistance $R_{r}$ of the rotor is 5.9$\Omega$ . When a potential difference of 120 $\mathrm{V}$ is applied to the brushes and the motor is running at full speed delivering mechanical power, the current supplied to it is 4.82 $\mathrm{A}$ (a) What is the current in the field coils? (b) What is the current in the rotor? (c) What is the induced emf developed by the motor? (d) How much mechanical power is developed by this motor?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 50

A shunt-wound de motor with the field coils and rotor connected in parallel (Fig. 27.56 ) operates from a $120-\mathrm{V}$ dc power linc. The resistance of the ficld windings, $R_{f},$ is 218$\Omega$ . The resistance of the rotor, $R_{r}$ is 5.9 . When the motor is running, the rotor develops an emf $\mathcal{E}$ . The motor draws a current of 4.82 A from the line. Friction losses amount to 45.0 W. Compute (a) the field current; $(b)$ the rotor current; $(c)$ the emf $\mathcal{E} ;$ (d) the rate of development of thermal energy in the field windings; (e) the rate of development of thermal energy in the rotor; $(f)$ the power input to the motor; (g) the efficiency of the motor.

Keshav Singh

Numerade Educator

Problem 51

Figure 27.57 shows a portion of a silver ribbon with $z_{1}=11.8 \mathrm{mm}$ $0.23 \mathrm{mm},$ carrying a current of 120 $\mathrm{A}$ in the $+x$ -direction. The ribbon lies in a uniform magnetic
field, in the $y$ -direction, with magnitude 0.95 T. Apply the simplified model of the Hall effect presented in Section $27.9 .$ If there are $5.85 \times 10^{28}$ free electrons per cubic meter, find (a) the magnitude of the drift velocity of the electrons in the $x$ -direction; $(b)$ the magnitude and direction of the electric field in the $z$ -direction due to the Hall effect; (c) the Hall emf.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 52

Let Fig. 27.57 represent a strip of an unknown metal of the same dimensions as those of the silver ribbon in Exercise $27.51 .$ When the magnetic field is 2.29 $\mathrm{T}$ and the current is 78.0 $\mathrm{A}$ , the Hall emf is found to be 131$\mu \mathrm{V}$ . What does the simplified model of the Hall effect presented in Section 27.9 give for the density of free electrons in the unknown metal?

Ceren Uzun

Texas Tech University

Problem 53

When a particle of charge $q>0$ moves with a velocity of $\overrightarrow{\boldsymbol{v}}_{1}$ at $45.0^{\circ}$ from the $+x$ -axis in the $x y$ -plane, a uniform magnetic field exerts a force $\overrightarrow{\boldsymbol{F}}_{1}$ along the $-z$ -axis (Fig. $27.58 ) .$ When the same particle moves with a velocity $\overrightarrow{\boldsymbol{v}}_{2}$ with the same magnitude as $\overrightarrow{\boldsymbol{v}}_{1}$ but along the $+z$ -axis, a force $\overrightarrow{\boldsymbol{F}}_{2}$ of magnitude $\boldsymbol{F}_{2}$ is exerted on it along the $+x$ -axis. (a) What are the magnitude (in terms of $q$ , $v_{1},$ and $F_{2} )$ and direction of the magnetic field? (b) What is the magnitude of $\overrightarrow{\boldsymbol{F}}_{1}$ in terms of $F_{2} ?$

Zhaojie Xu

Numerade Educator

Problem 54

A particle with charge $9.45 \times 10^{-8} \mathrm{C}$ is moving in a region where there is a uniform magnetic field of 0.450 $\mathrm{T}$ in the $+x$ - direction. At a particular instant of time the velocity of the particle has components $v_{x}=-1.68 \times 10^{4} \mathrm{m} / \mathrm{s}, v_{y}=-3.11 \times 10^{4} \mathrm{m} / \mathrm{s}$ and $v_{z}=5.85 \times 10^{4} \mathrm{m} / \mathrm{s}$ . What are the components of the force on the particle at this time?

Zulfiqar Ali

Numerade Educator

Problem 55

You wish to hit a target from several meters away with a charged coin having a mass of 5.0 $\mathrm{g}$ and a charge of $+2500 \mu \mathrm{C}$ . The coin is given an initial velocity of 12.8 $\mathrm{m} / \mathrm{s}$ , and a downward, uniform electric field with field strength 27.5 $\mathrm{N} / \mathrm{C}$ exists throughout the region. If you aim directly at the target and fire the coin horizontally, what magnitude and direction of uniform magnetic field are needed in the region for the coin to hit the target?

Zulfiqar Ali

Numerade Educator

Problem 56

A cycloron is to accelerate protons to an energy of 5.4 MeV. The superconducting electromagnet of the cyclotron produces a $3.5-$ T magnetic field perpendicular to the proton orbits. (a) When the protons have achieved a kinetic energy of 2.7 $\mathrm{MeV}$ , what is the radius of their circular orbit and what is their angular speed? (b) Repeat part (a) when the protons have achieved their final kinetic energy of 5.4 $\mathrm{MeV}$ .

Prashant Bana

Numerade Educator

Problem 57

The magnetic poles of a small cyclotron produce a magnetic field with magnitude 0.85 $\mathrm{T}$ . The poles have a radius of $0.40 \mathrm{m},$ which is the maximum radius of the orbits of the accelerated particles. (a) What is the maximum energy to which protons $\left(q=1.60 \times 10^{-19} \mathrm{C}, m=1.67 \times 10^{-27} \mathrm{kg}\right)$ can be accelerated by this cyclotron? Give your answer in electron volts and in joules. (b) What is the time for one revolution of a proton orbiting at this maximum radius? (c) What would the magnetic-field magnitude have to be for the maximum energy to which a proton can be accelerated to be twice that calculated in part (a)? (d) For $B=0.85 \mathrm{T}$ . what is the maximum energy to which alpha particles $\left(q=3.20 \times 10^{-19} \mathrm{C}, m=6.65 \times 10^{-27} \mathrm{kg}\right)$ can be accelerated by this cyclotron? How does this compare to the maximum energy for protons?

Zulfiqar Ali

Numerade Educator

Problem 58

The force on a charged particle moving in a magnetic field can be computed as the vector sunt of the forces due to each separate component of the magnetic field. As an example, a particle with charge $q$ is moving with speed $v$ in the $-y$ -direction. It is moving in a uniform magnetic field $\overrightarrow{\boldsymbol{B}}=\overrightarrow{\boldsymbol{B}}_{\boldsymbol{x}} \hat{\boldsymbol{i}}+\boldsymbol{B}_{\boldsymbol{y}} \hat{\boldsymbol{j}}+\boldsymbol{B}_{\boldsymbol{z}} \hat{\boldsymbol{k}}$
(a) What are the components of the force $\overrightarrow{\boldsymbol{F}}$ exerted on the particle by the magnetic field? (b) If $q>0,$ what must the signs of the components of $\overrightarrow{\boldsymbol{B}}$ be if the components of $\overrightarrow{\boldsymbol{F}}$ are all nonnegative? (c) If $q<0$ and $B_{x}=B_{y}=B_{z}>0,$ find the direction of $\vec{F}$ and find the magnitude of $\vec{F}$ in terms of $|q|, v,$ and $B_{x}$ .

Zhaojie Xu

Numerade Educator

Problem 59

Auniform, 458 -g metal bar 75.0 $\mathrm{cm}$ long carries a current $I$ in a uniform, horizontal, $1.55-\mathrm{T}$ magnetic field as shown in Fig. 27.59 . The bar is hinged at $b$ but rests unattached at $a$ . What is the largest current that can flow from $a$ to $b$ without breaking the electrical contact at $a ?$

Ajay Singhal

Numerade Educator

Problem 60

In the electron gun of a TV picture tube the electrons (charge $-e,$ mass $m )$ are accelerated by a voltage $V .$ After leaving the electron gun, the electron beam travels a distance $D$ to the screen; in this region there is a transverse magnetic field of magnitude $B$ and no electric field. (a) Sketch the path of the electron beam in the tube. (b) Show that the approximate deflection of the beam due to this magnetic field is
$$
d=\frac{B D^{2}}{2} \sqrt{\frac{e}{2 m V}}
$$
(Hint: Place the origin at the center of the electron beam's arc and compare an undefiected beam's path to the deflected beam's path.) (c) Evaluate this expression for $V=750 \mathrm{V}, D=50 \mathrm{cm},$ and $B=5.0 \times 10^{-5} \mathrm{T}$ (comparable to the earth's field). Is this deflection significant?

Vishal Gupta

Numerade Educator

Problem 61

A particle with negative charge $q$ and mass $m=2.58 \times$ 10 $^{-15} \mathrm{kg}$ is traveling through a region containing a uniform magnetic field $\overrightarrow{\boldsymbol{B}}=-(0.120 \mathrm{T}) \hat{\boldsymbol{k}} .$ At a particular instant of time the velocity of the particle is $\overrightarrow{\boldsymbol{v}}=\left(1.05 \times 10^{6} \mathrm{m} / \mathrm{s}\right)(-3 \hat{\imath}+4 \hat{\jmath}+$ 12$\hat{k} )$ and the force $\vec{F}$ on the particle has a magnitude of 1.25 $\mathrm{N}$ . (a) Determine the charge $q$ . (b) Determine the acceleration $\overrightarrow{\boldsymbol{d}}$ of the particle. (c) Explain why the path of the particle is a helix, and determine the radius of curvature $R$ of the circular component of the helical path. (d) Determine the cyclotron frequency of the particle. (e) Although helical motion is not periodic in the full sense of the word, the $x$ - and $y$ -coordinates do vary in a periodic way. If the coordinates of the particle at $t=0$ are $(x, y, z)=(R, 0,0),$ determine its coordinates at a time $t=2 T,$ where $T$ is the period of the
motion in the $x y$ -plane.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 62

A long, straight wire containing a semicircular region of radius 0.95 $\mathrm{m}$ is placed in a uni-
form magnetic field of magnitude 2.20 $\mathrm{T}$ as shown in Fig. 27.60 . What is the net magnetic force acting on the wire when it carries a current of 3.40 $\mathrm{A} ?$

Dading Chen

Numerade Educator

Problem 63

A magnetic field exerts a torque $\tau$ on a round current-carrying loop of wire. What will be the torque on this loop (in terms of $\tau )$ if its diameter is tripled?

Shahab Ullah

Numerade Educator

Problem 64

A particle of charge $q>0$ is moving at speed $v$ in the $+z$ -direction through a region of uniform magnetic field $\overrightarrow{\boldsymbol{B}} .$ The magnetic force on the particle is $\overrightarrow{\boldsymbol{F}}=F_{0}(3 \hat{\imath}+4 \hat{\jmath}),$ where $\boldsymbol{F}_{0}$ is a positive constant. (a) Determine the components $B_{x}, B_{y},$ and $B_{z},$ or at least as many of the three components as is possible from the information given. (b) If it is given in addition that the magnetic field has magnitude $6 F_{0} / q v,$ determine as much as you can about the remaining components of $\overrightarrow{\boldsymbol{B}}$ .

Zhaojie Xu

Numerade Educator

Problem 65

Suppose the electric field between the plates $P$ and $P$ ' in Fig. 27.24 is $1.88 \times 10^{4} \mathrm{V} / \mathrm{m}$ and the magnetic field in both regions is 0.701 $\mathrm{T}$ . If the source contains the three isotopes of krypton, $82 \mathrm{Kr}, 84 \mathrm{Kr},$ and $^{86} \mathrm{Kr}$ , and the ions are singly charged, find the distance between the lines formed by the three isotopes on the photographic plate. Assume the atomic masses of the isotopes (in atomic mass units) are equal to their mass numbers, $82,84,$ and $86 .$ (One atomic mass unit $=1 \mathbf{u}=1.66 \times 10^{-27} \mathrm{kg}$ .

Lainey Roebuck

Numerade Educator

Problem 66

Mass Spectrograph. A mass spectrograph is used to measure the masses of ions, or to separate ions of different masses (see Section 27.5$)$ . In one design for such an instrument, ions with mass $m$ and charge $q$ are accelerated through a potential difference V. They then enter a uniform magnetic field that is perpendicular to their velocity, and they are deflected in a semicircular path of radius $R$ . Adetector measures where the ions complete the semicircle and from this it is easy to calculate $R$ (a) Derive the equation for calculating the mass of the ion from measurements of $B, V, R,$ and $q .$ (b) What potential difference $V$ is needed so that singly ionized $^{12} \mathrm{C}$ atoms will have $R=50.0 \mathrm{cm}$ in a 0.150 . T magnetic field? (c) Suppose the beam consists of a mixture of $^{12} \mathrm{C}$ and $^{14} \mathrm{C}$ ions. If $V$ and $B$ have the same values as in part $(b),$ calculate the separation of these two isotopes at the detector. Do you think that this beam separation is sufficient for the two ions to be distinguished? (Make the assumption described in Problem 27.65 for the masses of the ions.)

Dading Chen

Numerade Educator

Problem 67

A straight piece of conducting wire with mass $M$ and length $L$ is placed on a friction- less incline tilted at an angle $\theta$ from the horizontal (Fig. 27.61 ) There is a uniform, vertical magnetic field $\overrightarrow{\boldsymbol{B}}$ at all points (produced by an arrangement of magnets not shown in the figure). To keep the wire from shiding down the incline, a voltage source is attached to the ends of the wire. When just the right amount of current flows through the wire, the wire remains at rest. Determine the magnitude and direction of the current in the wire that will cause the wire to remain at rest. Copy the figure and draw the direction of the current on your copy. In addition, show in a free-body diagram all the forces that act on the wire.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 68

$\begin{array}{lllll}{\text { A }} & {3.00-\mathrm{N}} & {\text { metal }} & {\text { bar, }}\end{array}$ 1.50 $\mathrm{m}$ long and having a resistance of $10.0 \Omega,$ rests horizontally on conducting wires
connecting it to the circuit shown in Fig. 27.62 . The bar is in a uniform, horizontal, $1.60-\mathrm{T}$ magnetic field and is not attached to the wires in the circuit. What is the acceleration of the bar just after the switch $\mathrm{S}$ is closed?

Salamat Ali

Numerade Educator

Problem 69

Two positive ions having the same charge $q$ but different masses $m_{1}$ and $m_{2}$ are accelexated horizontally from rest through a potential difference $V$ . They then enter a region where there is a uniform magnetic field $\overrightarrow{\boldsymbol{B}}$ normal to the plane of the trajectory. (a) Show that if the beam entered the magnetic field along the x-axis, the value of the $y$ -coordinate for each ion at any time $t$ is approximately
$$
y=B x^{2}\left(\frac{q}{8 m V}\right)^{1 / 2}
$$
provided $y$ remains much smaller than $x$ (b) Can this arrangement be used for isotope separation? Why or why not?

Keshav Singh

Numerade Educator

Problem 70

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

Sheh Lit Chang

University of Washington

Problem 71

Determining Diet. One method for determining the amount of corn in early Native American diets is the stable isotope ratio analysis (SIRA) technique. As corn photosynthesizes, it concentrates the isotope carbon-13, whereas most other plants concentrate carbon- 12 . Overneliance on corn consumption can then be correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use a mass spectrometer to separate the $^{10} \mathrm{C}$ and $^{13} \mathrm{C}$ isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed 8.50 $\mathrm{km} / \mathrm{s}$ , and you want to bend them within a uniform magnetic field in a semicircle of diameter 25.0 $\mathrm{cm}$ for the 12 $\mathrm{C}$ . The measured masses of these isotopes are $1.99 \times 10^{-25} \mathrm{kg}\left(^{12} \mathrm{C}\right)$ and $2.16 \times 10^{-26} \mathrm{kg}\left(^{13} \mathrm{C}\right) .$ (a) What strength of magnetic field is required? (b) What is the diameter of the $^{13} \mathrm{C}$ semicircle? (c) What is the separation of the $^{12}\mathrm{C}$ and $^{13}\mathrm{C}$ ions at the detector at the end of the semicircle? Is this distance large enough to be easily observed?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 72

An Electromagnetic Rail Gun. A conducting bar with mass $m$ and length $L$ slides over horizontal rails that are connected to a voltage source. The voltage source maintains a constant current $I$ in the rails and bar, and a constant, uniform, vertical magnetic field $\overrightarrow{\boldsymbol{B}}$ fills the region between the rails (Fig. 27.63$)$ . (a) Find the magnitude and direction of the net force on the conducting bar Ignore friction, air resistance, and electrical resistance. (b) If the bar has mass $m$ , find the distance $d$ that the bar must move along the rails from rest to attain speed $v$ . (c) It
has been suggested that rail guns based on this principle could accelerate payloads into earth orbit or beyond. Find the distance the bar must travel along the rails if it is to reach the escape speed for the earth $(11.2 \mathrm{km} / \mathrm{s}) .$ Let $B=0.50 \mathrm{T}, \quad I=2.0 \times 10^{3} \mathrm{A}$ $m=25 \mathrm{kg},$ and $L=50 \mathrm{cm} .$ For simplicity assume the net force on the object is equal to the magnetic force, as in parts $(a)$ and $(b)$ , even though gravity plays an important role in an actual launch in space.

st

Sebastien Tawa

Numerade Educator

Problem 73

A long wire carrying a 6.00 - A current reverses direction by means of two right-angle bends, as shown in Fig. 27.64 . The part of the wire where the bend occurs is in a magnetic field of 0.666 T confined to the= circular region of diameter $75 \mathrm{cm},$ as shown. Find the magnitude and direction of the net force that the magnetic field exerts on this wire.

Dading Chen

Numerade Educator

Problem 74

A wire 25.0 $\mathrm{cm}$ long lies along the $z$ -axis and carries a current of 9.00 $\mathrm{A}$ in the $+z$ -direction. The magnetic field is uniform and has components $B_{x}=-0.242 \mathrm{T}, B_{y}=-0.985 \mathrm{T}$ , and $B_{z}=$ $-0.336 \mathrm{T} .$ (a) Find the components of the magnetic force on the wire. (b) What is the magnitude of the net magnetic force on the wire?

Mayukh Nath

Numerade Educator

Problem 75

The rectangular loop of wire shown in Fig. 27.65 has a mass of 0.15 g per centimeter of length and is pivoted about side $a b$ on a frictionless axis. The current in the wire is 8.2 $\mathrm{A}$ in the direction shown. Find the magnitude and direction of the magnetic field parallel to the $y$ -axis that will cause the loop to swing up until its plane makes an angle of $30.0^{\circ}$ with the $y z$ -plane.

Ajay Singhal

Numerade Educator

Problem 76

The rectangular loop shown in Fig. 27.66 is pivoted about the $y$ -axis and carries a current of 15.0 $\mathrm{A}$ in the direction indicated. (a) If the loop is in a uniform magnetic field with magnitude 0.48 $\mathrm{T}$ in the $+x-$ direction, find the magnitude and direction of the torque required to hold the loop in the position shown. (b) Repeat part (a) for the case in which the field is in the - $z$ direction. (c) For each of the above magnetic fields, what torque would be required if the loop were pivoted about an axis through its center, parallel to the $y$ -axis?

Dading Chen

Numerade Educator

Problem 77

A thin, uniform rod with negligible mass and length 0.200 $\mathrm{m}$ is attached to the floor by a frictionless hinge at point $P$ (Fig. 27.67$)$ . A horizontal spring with force constant $k=4.80 \mathrm{N} / \mathrm{m}$ connects the other end of the rod to a vertical wall. The rod is
in a uniform magnetic field $B=$ 0.340 T directed into the plane of the figure. There is current $I=6.50 \mathrm{A}$ in the rod, in the direction shown. (a) Calculate the torque due to the magnetic force on the rod, for an axis at $P .$ Is it correct to take the total magnetic force to act at the center of gravity of the rod when calculating the torque? Explain. (b) When the rod is in equilibrium and makes an angle of $53.0^{\circ}$ with the floor, is the spring stretched or compressed? (c) How much energy is stored in the spring when the rod is in equilibrium?

Vishal Gupta

Numerade Educator

Problem 78

The triangular loop of wire shown in Fig. 27.68 carries a current $I=5.00 \mathrm{A}$ in the
direction shown. The loop is in a uniform magnetic field that has magnitude $B=3.00 \mathrm{T}$ and the salne direction as the current in side $P Q$ of the loop. (a) Find the force exerted by the magnetic field on each side of the triangle. If the force is not zero, specify its direction. (b) What is the net
force on the loop? (c) The loop is pivoted about an axis that lies along side $P R$ . Use the forces calculated in part (a) to calculate the torque on each side of the loop (see Problem $27.77 ) .$ (d) What is the magnitude of the net torque on the loop? Calculate the net torque from the torques calculated in part (c) and also from Eq. $(27.28) .$ Do these two results agree? (e) Is the net torque directed to rotate point $Q$ into the plane of the figure or out of the plane of the figure?

Keshav Singh

Numerade Educator

Problem 79

A Voice Coil. It was shown in Section 27.7 that the net force on a current loop in a uniform magnetic field is zero. The magnetic force on the voice coil of a loudspeaker (see Fig. 27.28$)$ is nonzero because the magnetic field at the coil is not uniform. A voice coil in a loudspeaker has 50 turns of wire and a diameter of 1.56 $\mathrm{cm}$ . and the current in the coil is 0.950 A. Assume that the magnetic field at each point of the coil has a constant magnitude of 0.220 $\mathrm{T}$ and is directed at an angle of $60.0^{\circ}$ outward from the normal to the plane of the coil (Fig. 27.69 ). Let the axis of the coil be in the $y$ -direction. The current in the coil is in the direction shown (counterclockwise as viewed from a point above the coil on the $y$ -axis). Calculate the magnitude and direction of the net magnetic force on the coil.

Dading Chen

Numerade Educator

Problem 80

Paleoclimate. Climatologists can determine the past temperature of the earth by comparing the ratio of the isotope oxygen-18 to the isotope oxygen- 16 in air trapped in ancient ice sheets, such as those in Greenland. In one method for separating these isotopes, a sample containing both of them is first singly ionized (one electron is removed) and then accelerated from rest through a potential difference $V$ . This beam then enters a magnetic field $B$ at right angles to the field and is bent into a quarter circle. A particle detector at the end of the path measures the amount of each isotope, (a) Show that the separation $\Delta r$ of the two isotopes at the detector is given by
$$
\Delta r=\frac{\sqrt{2 e V}}{e B}\left(\sqrt{m_{18}}-\sqrt{m_{16}}\right)
$$
where $m_{16}$ and $m_{18}$ are the masses of the two oxygen isotopes, (b) The measured masses of the two isotopes are $2.66 \times 10^{-26} \mathrm{kg}$ $\left(^{16} \mathrm{O}\right)$ and $2.99 \times 10^{-25} \mathrm{kg}$ $\left(^{18} \mathrm{O}\right)$. If the magnetic field is 0.050 T, what must be the accelerating potential $V$ so that these two isotopes will be separated by 4.00 $\mathrm{cm}$ at the detector?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 81

Force on a Current Loop in a Nonuniform Magnetic Field. It was shown in Section 27.7 that the net force on a current loop in a uniform magnetic field is zero. But what if $\overrightarrow{\boldsymbol{B}}$ is not uniform? Figure 27.70 shows a square loop of
wire that ties in the $x y$ -plane. The loop has corners at $(0,0),(0, L),(L, 0)$ , and $(L, L)$ and carries a constant current $I$ in the clockwise direction. The magnetic field has no $x$ -component but
has both $y-$ and $z$ -components: $\vec{B}=$ $\left(B_{0} z / L\right) \hat{j}+\left(B_{0} y / L\right) \hat{k},$ where $B_{0}$ is a positive constant. (a) Sketch the magnetic field lines in the $y z$ -plane. (b) Find the magnitude and direction of the magnetic force exerted on each of the sides of the loop
by integrating $\mathrm{Eq} .(27.20) .$ (c) Find the magnitude and direction of the net magnetic force on the loop.

Lainey Roebuck

Numerade Educator

Problem 82

Torque on a Current Loop in a Nonuniform Magnetic Field. In Section 27.7 the expression for the torque on a current loop was derived assuming that the magnetic field $\overrightarrow{\boldsymbol{B}}$ was uniform. But what if $\overrightarrow{\boldsymbol{B}}$ is not uniform? Figure 27.70 shows a square loop of wire that lies in the $x y$ -plane. The loop has corners at $(0,0)$ , $(0, L),(L, 0),$ and $(L, L)$ and carries a constant current $I$ in the clockwise direction. The magnetic field has no $z$ -component but has both $x$ - and $y$ -components: $\overrightarrow{\boldsymbol{B}}=\left(\boldsymbol{B}_{0} y / \boldsymbol{L}\right) \hat{\boldsymbol{i}}+\left(\boldsymbol{B}_{0} x / \boldsymbol{L}\right) \hat{\boldsymbol{j}},$ where $B_{0}$ is a positive constant. (a) Sketch the magnetic field lines in the $x y$ -plane. (b) Find the magnitude and direction of the magnetic force exerted on each of the sides of the loop by integrating Eq. $(27.20)$ . (c) If the loop is free to rotate about the $x$ -axis, find the magnitude and direction of the magnetic torque on the loop. (d) Repeat part (c) for the case in which the loop is free to rotate about the $y$ -axis. (e) Is Eq. $(27.26), \vec{\tau}=\vec{\mu} \times \vec{B},$ an appropriate description of the torque on this loop? Why or why not?

Yaqub Khan

Numerade Educator

Problem 83

An insulated wire with mass $m=5.40 \times 10^{-5} \mathrm{kg}$ is bent into the shape of an inverted U such that the horizontal part has a length $l=15.0 \mathrm{cm} .$ The bent ends of the wire are partially immersed in two pools of mercury, with 2.5 $\mathrm{cm}$ of each end below the mercury's surface. The entire structure is in a region containing a uniform $0.00650-\mathrm{T}$ magnetic field directed into the page (Fig. 27.71$)$ . An electrical connection from the mercury pools is made through the ends of the wires. The mercury pools are connected to a $1.50-\mathrm{V}$ battery and a switch $\mathrm{S}$ . When switch $\mathrm{S}$ is closed, the wire jumps 35.0 $\mathrm{cm}$ into the air, measured from its initial position. (a) Determine the speed $v$ of the wire as it leaves the mercury. (b) Assuming that the current $I$ through the wire was constant from the time the switch was closed until the wire left the mercury, determine $I$ (c) Ignoring the resistance of the mercury and the circuit wires, determine the resistance of the moving wire.

Vishal Gupta

Numerade Educator

Problem 84

Derivation of $\mathrm{Eq} .(27.26 \text { for a Circular Current Loop. }$ A wire ring lies in the $x y$ -plane with its center at the origin. The ring carries a counterclockwise current $I$ (Fig. 27.72$)$ . A uniform magnetic field $\overrightarrow{\boldsymbol{B}}$ is in the $+x$ -direction, $\overrightarrow{\boldsymbol{B}}=\boldsymbol{B}_{x} \hat{\boldsymbol{i}}$ (The result is easily extended to $\overrightarrow{\boldsymbol{B}}$ in an arbitrary direction. (a) In Fig. 27.72 .show that the element $d \vec{l}=$ $R d \theta(-\sin \theta \hat{\imath}+\cos \theta \hat{\jmath}),$ and find $d \vec{F}=I d \vec{I} \times \vec{B} \cdot(b)$ Integrate $d \vec{F}$ around the $\operatorname{loop}$ to show that the net force is zero. (c) From part (a), find $d \vec{\tau}=\vec{r} \times d \vec{K},$ where
$\vec{\tau}=R(\cos \theta \hat{\imath}+\sin \theta \hat{\jmath})$ is the vector from the center of the loop to the element $d \vec{l}$ . (Note that $d \vec{t}$ is perpendicular to $\vec{r} . )$ (d) Integrate $d \vec{\tau}$ over the loop to find the total torque $\vec{\tau}$ on the loop. Show that the result can be written as $\vec{\tau}=\vec{\mu} \times \vec{B},$ where $\mu=I A .$ (Note: $\int \cos ^{2} x d x=\frac{1}{2} x+\frac{1}{4} \sin 2 x,$ $\int \sin ^{2} x d x=\frac{1}{2} x-\frac{1}{4} \sin 2 x,$ and $\int \sin x \cos x d x=\frac{1}{2} \sin ^{2} x )$

Hunza Gilgit

Numerade Educator

Problem 85

A circular loop of wire with area $A$ lies in the $x y$ -plane. As viewed along the $z$ -axis looking in the $-z$ -direction toward the origin, a current $I$ is circulating clockwise around the loop. The
torque produced by an extemal magnetic field $\vec{B}$ is given by $\vec{\tau}=D(4 \hat{z}-3 \hat{y}),$ where $D$ is a positive constant, and for this orientation of the loop the magnetic potential energy $U=-\vec{\mu} \cdot \vec{B}$ is negative. The magnitude of the magnetic field is $B_{0}=13 D / L A$
(a) Determine the vector magnetic moment of the current loop.
(b) Determine the components $B_{x}, B_{y}$ , and $B_{z}$ of $\vec{B}$ .

Vishal Gupta

Numerade Educator

Problem 86

Quark Model of the Neutron. The neutron is a particle with zero charge. Nonetheless, it has a nonzero magnetic moment with $z$ -component $9.66 \times$ $10^{-27} \mathrm{A} \cdot \mathrm{m}^{2}$ . This can be explained by the internal structure of the neutron. A substantial body of evidence indicates that a neutron is composed of three fundamental particles called quarks: an $" u p^{p}(u)$ quark, of charge $+2 e / 3$ and two "down" (d) quarks, each of charge $-e / 3 .$ The combination of the three quarks produces a net charge of $2 e / 3-e / 3-e / 3=0 .$ If the quarks are in motion, they can produce a nonzero magnetic moment. As a very simple model, suppose the $u$ quark moves in a counterclockwise circular path and the $d$ quarks move in a clockwise circular path, all of radius $r$ and all with the same speed $v$ (Fig. 27.73$)$ . (a) Determine the current due to the circulation of the $u$ quark. (b) Determine the magnitude of the magnetic moment due to the circulating $u$ quark.
(c) Determine the magnitude of the magnetic moment of the three-quark system. (Be careful to use the correct magnetic moment directions.) (d) With what speed $v$ must the quarks move if this
model is to reproduce the magnetic moment of the neutron? Use $r=1.20 \times 10^{-15} \mathrm{m}$ (the radius of the neutron) for the radius of the orbits.

Ceren Uzun

Texas Tech University

Problem 87

Using Gauss's Law for Magnetism. In a certain region of space, the magnetic field $\overrightarrow{\boldsymbol{B}}$ is not uniform. The magnetic field has both a $z$ -component and a component that points radially away from or toward the $z$ -axis. The z-component is given by $B_{z (z)=\beta z,$ where $\beta$ is a positive constant. The radial component $B_{x}$ depends only on $r$ , the radial distance from the $z$ -axis. (a) Use Gauss's law for magnetism, Eq. $(27.8),$ to find the radial component $B_{r}$ as a function of $r .$ (Hint Try a cylindrical Gaussian surface of radius $r$ concentric with the $z$ -axis, with one end at $z=0$ and the other at $z=L . )$ (b) Sketch the magnetic field lines.

Zhaojie Xu

Numerade Educator

Problem 88

A circular ring with area $4.45 \mathrm{~cm}^{2}$ is carrying a current of 12.5 A. The ring is free to rotate about a diameter. The ring, initially at rest, is immersed in a region of uniform magnetic field given by $\vec{B}=\left(1.15 \times 10^{-2} \mathrm{~T}\right)(12 \hat{\imath}+3 \hat{\jmath}-4 \hat{k}) .$ The ring is positioned initially such that its magnetic moment is given by $\vec{\mu}_{1}=\mu(-0.800 \hat{\imath}+0.600 \hat{\jmath}),$ where $\mu$ is the (positive) magnitude of the magnetic moment. The ring is released and turns through an angle of $90.0^{\circ},$ at which point its magnetic moment is given by $\vec{\mu}_{f}=-\mu \hat{k} .$ (a) Determine the decrease in potential energy. (b) If the moment of inertia of the ring about a diameter is $8.50 \times 10^{-7} \mathrm{~kg} \cdot \mathrm{m}^{2},$ determine the angular speed of the ring as it passes through the second position.

Sheh Lit Chang

University of Washington

Problem 89

A particle with charge 2.15$\mu \mathrm{C}$ and $\mathrm{mass} 3.20 \times 10^{-11} \mathrm{kg}$ is initially traveling in the $+y$ -direction with a speed $v_{0}=$ $1.45 \times 10^{5} \mathrm{m} / \mathrm{s}$ . It then enters a region containing a uniform magnetic field that is directed into, and perpendicular to, the page in Fig. 27.74 . The magnitude of the field is 0.420 T. The region extends a distance of 25.0 $\mathrm{cm}$ along the initial direction of travel; 75.0 $\mathrm{cm}$ from the point of entry into the magnetic field region is a wall. The length of the field-free region is thus 50.0 $\mathrm{cm}$ . When the charged particle enters the magnetic field, it follows a curved path whose radius of curvature is $R$ . It then leaves the magnetic field after a time $t_{1}$ , having been deflected a distance $\Delta x_{1}$ . The particle then travels in the field-free region and strikes the wall after undergoing a total deflection $\Delta x$ (a) Determine the radius $R$ of the curved part of the path. (b) Determine $t_{1}$ , the time the particle spends in the magnetic field. (c) Determine $\Delta x_{1}$ , the horizontal deflection at the point of exit from the field. (d) Determine $\Delta x$ , the total horizontal deflection.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 90

The Electromagnetic Pump. Magnetic forces acting on conducting fluids provide a convenient means of pumping these fluids. For example, this method can be used to pump blood without the damage to the cells that can be caused by a mechanical pump. A horizontal tube with rectangular cross section (height $h,$ width w) is placed at right angles to a uniform magnetic field with magnitude $B$ so
that a length $l$ is in the field (Fig. 27.75). The tube is filled with a conducting liquid, and an electric current of density $J$ is maintained in the third mutually perpendicular direction. (a) Show that the difference of pressure between a point in the liquid on a vertical plane through $a b$ and a point in the liquid on another vertical plane through $c d,$ under conditions in which the liquid is prevented from flowing, is $\Delta p=J I B$ . (b) What current density is needed to provide a pressure difference of
1.00 atm between these two points if $B=2.20 \mathrm{T}$ and $l=35.0 \mathrm{mm} ?$

Dading Chen

Numerade Educator

Problem 91

A Cycloidal Path. A particle with mass $m$ and positive charge $q$ starts from rest at the origin shown in Fig. 27.76 . There is a uniform electric field $\vec{E}$ in the $+y$ -direction and a uniform magnetic field $\vec{B}$ directed out of the page. It is shown in more advanced books that the path is a cycloid whose radius of curvature at the top points is twice the $y$ -coordinate at that level. (a) Explain why the path has this general shape and why it is repetitive. (b) Prove that the speed at any point is equal to $\sqrt{2 q E y / m}$ . (Hint: Use energy conservation.) (c) Applying Newton's second law at the top point and taking as given that the radius of curvature here equals 2$y$ , prove that the speed at this point is 2$E / B$ .

Zhaojie Xu

Numerade Educator