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University Physics with Modern Physics

Hugh D. Young

Chapter 27

Magnetic Field and Magnetic Forces - all with Video Answers

Educators

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Chapter Questions

10:47

Problem 1

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

Meghan Miholics
Meghan Miholics
Numerade Educator
05:30

Problem 2

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
Ceren Uzun
Texas Tech University
01:07

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.

Salamat Ali
Salamat Ali
Numerade Educator
04:47

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 $\vec{\boldsymbol{v}}=(3.00 \times$ $10^{4} \mathrm{m} / \mathrm{s} ) \hat{\boldsymbol{J}}$ . What are the magnitude and direction of the particle's acceleration produced by a uniform magnetic field $\vec{\boldsymbol{B}}=(1.63 \mathrm{T}) \hat{\boldsymbol{\imath}}+$ $(0.980 \mathrm{T}) \hat{\boldsymbol{l}}$ ?

Shahab Ullah
Shahab Ullah
Numerade Educator
02:44

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
Sachin Rao
Numerade Educator
04:13

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? (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?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
06:36

Problem 7

A particle with charge 7.80$\mu \mathrm{C}$ is moving with velocity $\vec{\boldsymbol{v}}=-\left(3.80 \times 10^{3} \mathrm{m} / \mathrm{s}\right) \hat{\boldsymbol{J}}$ . The magnetic force on the particle is measured to be $\vec{\boldsymbol{F}}=+\left(7.60 \times 10^{-3} \mathrm{N}\right) \hat{\boldsymbol{\imath}}-\left(5.20 \times 10^{-3} \mathrm{N}\right) \hat{\boldsymbol{k}}$ (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 $\vec{B} \cdot \vec{F} .$ What is the angle between $\vec{B}$ and $\vec{\boldsymbol{F}} ?$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
08:59

Problem 8

A particle with charge $-5.60 \mathrm{nC}$ is moving in a uniform magnetic field $\vec{\boldsymbol{B}}=-(1.25 \mathrm{T}) \hat{\boldsymbol{k}}$ . The magnetic force on the particle is measured to be $\vec{\boldsymbol{F}}=-\left(3.40 \times 10^{-7} \mathrm{N}\right) \hat{\imath}+(7.40 \times$ $10^{-7} \mathrm{N} ) \hat{J}$ . (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 $\vec{v} \cdot$ What
is the angle between $\vec{v}$ and $\vec{F} ?$

Jacob Schulze
Jacob Schulze
Numerade Educator
14:36

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 $\mathrm{km} / \mathrm{s}$ in 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 $-z$ -direction experiences a force of $8.50 \times$ $10^{-16} \mathrm{N}$ in the $+y$ -direction. (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 3.20 $\mathrm{km} / \mathrm{s}$ ?

Jayashree Behera
Jayashree Behera
Numerade Educator
03:17

Problem 10

A flat, square surface with side length 3.40 $\mathrm{cm}$ is in the $x y$ -plane at $z=0 .$ Calculate the magnitude of the flux through this surface produced by a magnetic field $\vec{\boldsymbol{B}}=(0.200 \mathrm{T}) \hat{\boldsymbol{\imath}}+$ $(0.300 \mathrm{T}) \hat{\boldsymbol{J}}-(0.500 \mathrm{T}) \hat{\boldsymbol{k}} .$

Guilherme Barros
Guilherme Barros
Numerade Educator
04:17

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; $(\mathrm{b})$ at an angle of $53.1^{\circ}$ from the $+z$ -direction; $(\mathrm{c})$ in the $+y$ -direction?

Daniel Matthias
Daniel Matthias
Numerade Educator
03:20

Problem 12

A horizontal rectangular surface has dimensions 2.80 $\mathrm{cm}$ by 3.20 $\mathrm{cm}$ and is in a uniform magnetic field that is directed at an angle of $30.0^{\circ}$ above the horizontal. What must the magnitude of the magnetic field be in order to produce a flux of $4.20 \times 10^{-4} \mathrm{Wb}$ through the surface?

Guilherme Barros
Guilherme Barros
Numerade Educator
01:51

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 -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?

Jayashree Behera
Jayashree Behera
Numerade Educator
04:39

Problem 14

The magnetic field $\vec{B}$ in a certain region is 0.128 $\mathrm{T}$ , and its direction is that of the $+z$ -axis in Fig. $\mathrm{E} 27.14$ . (a) What is the magnetic flux across the surface abcd 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?

Ceren Uzun
Ceren Uzun
Texas Tech University
04:21

Problem 15

An electron at point $A$ in Fig. $\mathrm{E} 27.15$ has a speed $v_{0}$ of $1.41 \times 10^{6} \mathrm{m} / \mathrm{s} .$ Find $(\mathrm{a})$ the magnitude and direction of the magnetic field that will cause the electron to follow the semi-circular path from $A$ to $B,$ and (b) the time required for the electron to move from $A$ to $B$ .

Zhaojie Xu
Zhaojie Xu
Numerade Educator
03:30

Problem 16

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

Ceren Uzun
Ceren Uzun
Texas Tech University
03:45

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 enters the field.

Jayashree Behera
Jayashree Behera
Numerade Educator
03:20

Problem 18

An alpha particle (a He nucleus, containing two protons and two neutrons and having a mass of $6.64 \times 10^{-27}$ kg) traveling horizontally at 35.6 $\mathrm{km} / \mathrm{s}$ enters a uniform, vertical, 1.10 -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
Averell Hause
Carnegie Mellon University
02:12

Problem 19

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
particle? (b) What is the magnitude of the angular momentum $\vec{L}$ of the particle?

Ajay Singhal
Ajay Singhal
Numerade Educator
14:36

Problem 20

(a) An $\mathrm{}^{16} \mathrm{O}$ 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 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
04:42

Problem 21

A deuteron (the nucleus of an isotope of hydrogen) 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 potential difference would the deuteron have to be accelerated to acquire this speed?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
07:33

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 $\mathrm{T}$ and is bent in a semicircle of diameter $95.0 \mathrm{cm},$ as shown in Fig. E27.22. (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
04:40

Problem 23

A physicist wishes to produce electromagnetic waves of frequency 3.0 $\mathrm{THz}$ ($1$ $\mathrm{THz}=1$ terahertz $=10^{12} \mathrm{Hz} )$ 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 T. (b) Would there be any advantage to using protons instead of electrons in the magnetron? Why or why not?

Vishal Gupta
Vishal Gupta
Numerade Educator
02:48

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. E27.24). The beam travels a distance of 1.18 $\mathrm{cm}$ while in the field. What is the magnitude of the magnetic field?

Ceren Uzun
Ceren Uzun
Texas Tech University
03:30

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
04:32

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 dif- ference 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?

Shahab Ullah
Shahab Ullah
Numerade Educator
11:14

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 $\vec{\boldsymbol{B}}=(0.500 \mathrm{T}) \hat{\boldsymbol{l}}$ . At $t=0$ the proton has velocity components $v_{x}=1.50 \times 10^{5} \mathrm{m} / \mathrm{s}, v_{y}=0$ and $v_{z}=2.00 \times 10^{5} \mathrm{m} / \mathrm{s}$ (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 $t=0 ?$

Jacob Schulze
Jacob Schulze
Numerade Educator
03:41

Problem 28

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

Averell Hause
Averell Hause
Carnegie Mellon University
05:21

Problem 29

In designing a velocity selector that uses uniform perpendicular electric and magnetic fields, you want to select positive ions of charge $+5 e$ that are traveling perpendicular to the fields at 8.75 $\mathrm{km} / \mathrm{s}$ . The magnetic field available to you has a magnitude of
0.550 $\mathrm{T}$ (a) What magnitude of electric field do you need? (b) Show how the two fields should be oriented relative to each other and to the velocity of the ions. (c) Will your velocity selector also allow the following ions (having the same velocity as the $+5 e$ ions) to pass through undeflected: (i) negative ions of charge $-5 e,$ (ii) positive ions of charge different from $+5 e ?$

Jacob Schulze
Jacob Schulze
Numerade Educator
05:28

Problem 30

Crossed $\vec{E}$ and $\vec{B}$ Fields. A particle with initial velocity $\vec{\boldsymbol{v}}_{0}=\left(5.85 \times 10^{3} \mathrm{m} / \mathrm{s}\right) \hat{\boldsymbol{J}}$ enters a region of uniform electric and magnetic fields. The magnetic field in the region is $\vec{\boldsymbol{B}}=$ $-(1.35 \mathrm{T}) \hat{k}$ . 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 (a) $+0.640 \mathrm{nC}$ and $(\mathrm{b})-0.320 \mathrm{nC.}$ You can ignore the weight of the particle.

Shahab Ullah
Shahab Ullah
Numerade Educator
04:36

Problem 31

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, as shown in Fig. $\mathrm{E} 27.31 .$ What magnitude and direction of magnetic field are needed so that the alpha particles emerge undeflected from between the plates?

Jayashree Behera
Jayashree Behera
Numerade Educator
02:49

Problem 32

A singly ionized (one electron removed) $^{40} \mathrm{K}$ atom passes through a velocity selector consisting of uniform perpendicular electric and magnetic fields. The selector is adjusted to allow ions having a speed of 4.50 $\mathrm{km} / \mathrm{s}$ to pass through undeflected when the magnetic field is 0.0250 T. The ions next enter a second uniform magnetic field $\left(B^{\prime}\right)$ oriented at right angles to their velocity. 40 contains 19 protons and 21 neutrons and has a mass of $6.64 \times 10^{-26} \mathrm{kg} .$ (a) What is the magnitude of the electric field in the velocity selector? (b) What must be the magnitude of $B^{\prime}$ so that the ions will be bent into a semicircle of radius 12.5 $\mathrm{cm} ?$

Ceren Uzun
Ceren Uzun
Texas Tech University
03:36

Problem 33

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

Jayashree Behera
Jayashree Behera
Numerade Educator
02:23

Problem 34

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 $P^{\prime} ?$

Shahab Ullah
Shahab Ullah
Numerade Educator
03:10

Problem 35

Ancient Meat Eating. The amount of meat in prehistoric diets can be determined by measuring the ratio of the isotopes nitrogen-15 to nitrogen-14 in bone from human remains. Carnivores concentrate $^{15} \mathrm{N},$ so this ratio tells archaeologists how much meat was consumed by ancient people. Use the spectrometer of Exercise 27.34 to find the separation of the $^{14} \mathrm{N}$ and $^{15} \mathrm{N}$ isotopes at the detector. The measured masses of these isotopes are
$2.32 \times 10^{-26} \mathrm{kg}\left(^{14} \mathrm{N}\right)$ and $2.49 \times 10^{-26} \mathrm{kg}\left(^{15} \mathrm{N}\right)$

Ajay Singhal
Ajay Singhal
Numerade Educator
02:44

Problem 36

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

Salamat Ali
Salamat Ali
Numerade Educator
05:32

Problem 37

A straight, 2.00 -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 seem 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.

Guilherme Barros
Guilherme Barros
Numerade Educator
02:32

Problem 38

An electromagnet produces a magnetic field of 0.550 $\mathrm{T}$ in a cylindrical region of radius 2.50 $\mathrm{cm}$ between its poles. A straight wire carrying a current of 10.8 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
Zulfiqar Ali
Numerade Educator
View

Problem 39

A long wire carrying 4.50 A of current makes two $90^{\circ}$ bends, as shown in Fig. $\mathrm{E} 27.39$ . The bent part of the wire passes through a uniform $0.240-\mathrm{T}$ 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
05:10

Problem 40

A straight, vertical wire carries a current of 1.20 A downward in a region between the poles of a large superconductingelectromagnet, 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?

Keshav Singh
Keshav Singh
Numerade Educator
07:39

Problem 41

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. E27. $41 .$ 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.

Shahab Ullah
Shahab Ullah
Numerade Educator
02:31

Problem 42

Magnetic Balance. The circuit shown in Fig. $\mathrm { E } 27.42$
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
uniform magnetic field of 1.50 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 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?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:32

Problem 43

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
Jacob Schulze
Numerade Educator
05:47

Problem 44

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
Ceren Uzun
Texas Tech University
01:44

Problem 45

The 20.0$\mathrm { cm } \times 35.0 \mathrm { cm }$ rectangular circuit shown in Fig. E27.45 is hinged along side ab.
It carries a clockwise $5.00 - \mathrm { A }$
current and is located in a uniform 1.20 - $\mathrm { T }$ magnetic field oriented perpendicular to two of
its sides, as shown. (a) Draw a
clear diagram showing the
direction of the force that the
magnetic field exerts on each
segment of the circuit $( a b , b c ,$
etc.). (b) Of the four forces you
drew in part (a), decide which ones exert a torque about the hinge $a b .$ Then calculate only those
forces that exert this torque. (c) Use your results from part (b) to
calculate the torque that the magnetic field exerts on the circuit
about the hinge axis ab.

Narayan Hari
Narayan Hari
Numerade Educator
02:22

Problem 46

A rectangular coil of wire, 22.0$\mathrm { cm }$ by 35.0$\mathrm { cm }$ and
carrying a current of $1.40 \mathrm { A } ,$ is
oriented with the plane of its
loop perpendicular to a uniform
$1.50 - \mathrm { T }$ magnetic field, as shown
in Fig. E27. 46. (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
Salamat Ali
Numerade Educator
09:34

Problem 47

A uniform rectangular coil of total mass 212$\mathrm { g }$ and dimensions 0.500$\mathrm { m } \times$
1.00$\mathrm { m }$ is oriented with its plane parallel to a uniform $3.00 - \mathrm { T }$
magnetic field (Fig. E27.47). A current of 2.00$\mathrm { A }$ is suddenly started in the coil. (a) About which axis $( A _ { 1 }$ or $A _ { 2 })$ will the coil begin to rotate? Why?
(b) Find the initial angular acceleration of the coil just after the current is started.

Jayashree Behera
Jayashree Behera
Numerade Educator
04:48

Problem 48

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 $\vec { B }$ 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.
E27.48.

Jayashree Behera
Jayashree Behera
Numerade Educator
02:01

Problem 49

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
Ceren Uzun
Texas Tech University
03:32

Problem 50

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?

Keshav Singh
Keshav Singh
Numerade Educator
03:30

Problem 51

In a shunt-wound dc motor with the field coils and
rotor connected in parallel (Fig.
E27.51), the resistance $R _ { \text { f of the } }$
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 mechani-
cal power, the current supplied to it is 4.82 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?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
07:56

Problem 52

A shunt-wound dc motor with the field coils and rotor connected in parallel (see Fig. E27.51) operates from a $120 - \mathrm { V }$ dc
power line. The resistance of the field windings, $R _ { \mathrm { f } } ,$ is 218$\Omega$ . The connected in parallel (see Fig. E27.51) operates from a $120 - \mathrm { V } \mathrm { dc }$ power line. The resistance of the field windings, $R _ { \mathrm { f } } ,$ is 218$\Omega .$ The resistance of the rotor, $R _ { \mathrm { r } } ,$ is 5.9$\Omega .$ 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
Keshav Singh
Numerade Educator
04:07

Problem 53

Figure $\mathrm { E } 27.53$ shows a portion of a silver ribbon with
$z _ { 1 } = 11.8 \mathrm { mm } \quad$ and $\quad y _ { 1 } =$
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$\mathrm { T }$ 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
03:03

Problem 54

Let Fig. E27. 53 represent a strip of an unknown metal of the same dimensions as those of the silver ribbon in Exercise 27.53 .
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?

Zhaojie Xu
Zhaojie Xu
Numerade Educator
04:04

Problem 55

When a particle of charge $q > 0$ moves with a velocity of $\vec { \boldsymbol { v } } _ { 1 }$ at $45.0 ^ { \circ }$ from the $\pm x$ -axis in the $x y$ -plane, a uniform magnetic field exerts a force $F _ { 1 }$ along the $- z$ -axis (Fig. $P 27.55 ) .$ When the same particle moves with a velocity $\vec { \boldsymbol { v } } _ { 2 }$ with the same magnitude as $\vec { \boldsymbol { v } } _ { 1 }$ but along the $+ z$ -zaxis, a force $\vec { \boldsymbol { F } } _ { 2 }$ of magnitude $F _ { 2 }$ 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 $\vec { F } _ { 1 }$ in terms of $F _ { 2 } ?$

Zhaojie Xu
Zhaojie Xu
Numerade Educator
07:40

Problem 56

A particle with charge $9.45 \times 10 ^ { - 8 } \mathrm { C }$ is moving in a region where there is a uniform magnetic field of 0.650 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?

Meghan Miholics
Meghan Miholics
Numerade Educator
03:30

Problem 57

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 them 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? (Assume their
speeds are equal. 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$\mathrm { m } ?$

Jayashree Behera
Jayashree Behera
Numerade Educator
03:50

Problem 58

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?

Shahab Ullah
Shahab Ullah
Numerade Educator
02:42

Problem 59

You wish to hit a target from several meters away with a charged coin having a mass of 4.25$\mathrm { g }$ 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 through-out 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?

Jayashree Behera
Jayashree Behera
Numerade Educator
06:34

Problem 60

A cyclotron is to accelerate protons to an energy of 5.4 MeV. The superconducting electromagnet of the cyclotron produces a 2.9 -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$\mathrm { MeV }$ .

Shahab Ullah
Shahab Ullah
Numerade Educator
09:55

Problem 61

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)? 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?

Ceren Uzun
Ceren Uzun
Texas Tech University
05:02

Problem 62

A particle with charge $q$ is moving with speed $v$ in the $- y$ -direction. It is moving in a uniform magnetic field $\vec { B } =$
$B _ { x } \hat { i } + B _ { y } \hat { J } + B _ { z } \hat { k } .$ (a) What are the components of the force $\vec { \boldsymbol { F } }$ exerted on the particle by the magnetic field? (b) If $q > 0 ,$ what
must the signs of the components of $\vec { B }$ be if the components of $\vec { \boldsymbol { F } }$
are all nonnegative? (c) If $q < 0$ and $B _ { x } = B _ { y } = B _ { z } > 0 ,$ find the
direction of $\vec { \boldsymbol { F } }$ and find the magnitude of $\vec { \boldsymbol { F } }$ in terms of $| q | , v ,$ and $B _ { x }$

Zhaojie Xu
Zhaojie Xu
Numerade Educator
View

Problem 63

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 $\vec { \boldsymbol { B } } = - ( 0.120 \mathrm { T } ) \hat { \boldsymbol { k } }$ . At a particular instant of time the velocity
of the particle is $\vec { \boldsymbol { v } } = \left( 1.05 \times 10 ^ { 6 } \mathrm { m } / \mathrm { s } \right) ( - 3 \hat { \imath } + 4 \hat { \jmath } + 12 \hat { \boldsymbol { k } } )$ and
the force $\vec { \boldsymbol { F } }$ on the particle has a magnitude of 2.45$\mathrm { N }$ . (a) Determine the charge $q .$ (b) Determine the acceleration $\vec { a }$ 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. (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.

Victor Salazar
Victor Salazar
Numerade Educator
03:14

Problem 64

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

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:11

Problem 65

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
Shahab Ullah
Numerade Educator
03:56

Problem 66

A particle of charge $q > 0$ is moving at speed $v$ in the $+ z$ -direction through a region of uniform magnetic field $\vec { \boldsymbol { B } }$ . The
magnetic force on the particle is $\vec { \boldsymbol { F } } = F _ { 0 } ( 3 \hat { \boldsymbol { \imath } } + 4 \hat { \boldsymbol { J } } ) ,$ where $F _ { 0 }$ 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 $\vec { B } .$

Zhaojie Xu
Zhaojie Xu
Numerade Educator
04:33

Problem 67

Suppose the electric field between the plates in Fig. 27.24 is $1.88 \times 10 ^ { 4 } \mathrm { V } / \mathrm { m }$ and the magnetic field in both regions is
0.682$\mathrm { T }$ . 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 particle detector. 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 $= 1 \mathrm { u } = 1.66 \times 10 ^ { - 27 } \mathrm { kg. }$

Jayashree Behera
Jayashree Behera
Numerade Educator
12:43

Problem 68

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 $R .$ A detector 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 $0.150 - \mathrm { T }$ 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 27.67 for the
masses of the ions.

Robert Zaballa
Robert Zaballa
Numerade Educator
05:44

Problem 69

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. P27.69).
There is a uniform, vertical magnetic field $\vec { B }$ at all points (produced by an arrangement of
magnets not shown in the figure). To keep the wire from sliding 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
06:01

Problem 70

A 2.60 -N metal bar, 1.50$\mathrm { m }$ long and having a resistance of 10.0$\Omega$ , rests horizontally on conducting wires connecting
it to the circuit shown in Fig. P27.70. The bar is in a uniform, horizontal, 1.60 -T magnetic field and is not attached to the wires in the
circuit. What is the acceleration of the bar just after the switch $S$ is
closed?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
05:05

Problem 71

Using Gauss's Law for Magnetism. In a certain region of space, the magnetic field $B$ 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
$B _ { r }$ 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
Zhaojie Xu
Numerade Educator
05:44

Problem 72

A plastic circular loop has 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 $\vec { \boldsymbol { B } }$ directed parallel
to the plane of the loop, calculate the magnitude of the magnetic
torque on the loop.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:38

Problem 73

Determining Diet. One method for determining the amount of corn in early Native American diets is the stable isotope ratio analysis (SIRA) technique. As corn photosynthesizes, it
concentrates the isotope carbon-13, whereas most other plants concentrate carbon-12. Overreliance on corn consumption can then be
correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use a mass spectrometer to separate the 12$\mathrm { C }$ and $^ { 13 } \mathrm { C }$ isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing
one electron) atoms of speed $8.50 \mathrm { km } / \mathrm { s } ,$ and 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
$1.99 \times 10 ^ { - 26 } \mathrm { kg } \left( ^ { 12 } \mathrm { C } \right)$ 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 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
04:53

Problem 74

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 $\vec { B }$ fills the region between the rails (Fig. $P 27.74 )$ (a) Find the magnitude and direction of the net force on the con-
ducting 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 $B = 0.80 \mathrm { T } , \quad I = 2.0 \times 10 ^ { 3 } \mathrm { A } , \quad m = 25 \mathrm { kg }$
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.

Keshav Singh
Keshav Singh
Numerade Educator
01:07

Problem 75

A long wire carrying a $6.00 - A$ current reverses direc-
tion by means of two right angle bends, as shown in Fig.
P27.75. The part of the wire
where the bend occurs is in a magnetic field of 0.666$\mathrm { T }$ confined to the circular region of
diameter $75 \mathrm { cm } ,$ as shown. Find
the magnitude and direction of
the net force that the magnetic
field exerts on this wire.

Dading Chen
Dading Chen
Numerade Educator
06:54

Problem 76

A wire 25.0$\mathrm { cm }$ long lies along the $z$ -axis and carries a current of 7.40 A in the $+ z$ -direction. The magnetic field is uniform and has components $B _ { x } = - 0.242$ 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 magnetic force on
the wire?

Shahab Ullah
Shahab Ullah
Numerade Educator
03:23

Problem 77

The rectangular loop of wire shown in Fig. $\mathrm { P } 27.77$
has a mass of 0.15$\mathrm { g }$ per centimeter of length and is pivoted
about side ab 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 cause the
makes an angle of $30.0 ^ { \circ }$ with the
yz-plane.

Ajay Singhal
Ajay Singhal
Numerade Educator
04:02

Problem 78

The rectangular loop shown in Fig. P27.78 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 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?

Ajay Singhal
Ajay Singhal
Numerade Educator
10:33

Problem 79

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.
P27.79). 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
Vishal Gupta
Numerade Educator
05:31

Problem 80

The loop of wire shown in Fig. P27. 80 forms a right triangle and carries a current $I = 5.00$ A in the direction shown. The
loop is in a uniform magnetic field that has magnitude $B = 3.00 \mathrm { T }$
and the same 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 27.79 ). (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?

Ajay Singhal
Ajay Singhal
Numerade Educator
02:01

Problem 81

A uniform, 458 -g metal bar 75.0$\mathrm { cm }$ long carries a current $I$ in a uniform,
horizontal $1.25 -$ T magnetic field as shown in
Fig. P27.81. The directions of $I$ and $\vec { B }$ are shown in the figure. The bar is free to rotate
about a frictionless hinge at point $b$ . The other
end of the bar rests on a conducting support at point $a$ but is not attached there. The bar rests
at an angle of $60.0 ^ { \circ }$ above the horizontal. What is the largest value the current $I$ can have without breaking
the electrical contact at $a ?$ (See Problem $27.77 . )$

Ajay Singhal
Ajay Singhal
Numerade Educator
06:05

Problem 82

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 $2.99 \times 10 ^ { - 26 } \mathrm { kg } ( 8 \mathrm { O } ) .$ If the magnetic field
is $0.050 \mathrm { T } ,$ what must be the accelerating potential $V$ so that these
two isotopes will be separated by 4.00$\mathrm { cm }$ at the detector?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
05:04

Problem 83

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 con-
stant 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. $\mathrm { Pa } 7.83 )$ . 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.

Jayashree Behera
Jayashree Behera
Numerade Educator
04:42

Problem 84

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 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 of three fundamental
particles called quarks: an "up"
(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 clock-
wise circular path, all of radius $r$ and all with the same speed $v$ (Fig. P27.84). (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
Ceren Uzun
Texas Tech University
10:21

Problem 85

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 $\vec { B }$ is not uniform? Figure
P27.85 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 $x$ -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 Eq. (27.20). (c) Find the magnitude
and direction of the net magnetic force on the loop.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
12:08

Problem 86

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 $\vec { B }$ was
uniform. But what if $\vec { B }$ is not uniform? Figure $P 27.85$ shows a
square loop of wire that lies in the $x y$ -plane. The loop has corners at 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: $\vec { \vec { B } } = \left( B _ { 0 } y / L \right) \hat { \boldsymbol { \imath } } + \left( B _ { 0 } x / L \right) \hat { \boldsymbol { J } } ,$ 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

Jayashree Behera
Jayashree Behera
Numerade Educator
09:55

Problem 87

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. P27.87). 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 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, deter-
mine $I .$ (c) Ignoring the resistance of the mercury and the circuit
wires, determine the resistance of the moving wire.

Vishal Gupta
Vishal Gupta
Numerade Educator
08:00

Problem 88

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 external magnetic field $\vec { B }$ is given by $\vec { \tau } = D ( 4 \hat { \imath } - 3 \hat { \jmath } ) ,$ where $D$ is a positive constant, and for this orientation of the loop the magnetic potential energy $U = - \vec { \mu } \cdot B$ is
negative. The magnitude of the magnetic field is $B _ { 0 } = 13 D / I A$ .
(a) Determine the vector magnetic moment of the current loop.
(b) Determine the components $B _ { x } , B _ { y }$ and $B _ { z }$ of $\vec { B } .$

CM
Corbyn Mellinger
Numerade Educator
05:25

Problem 89

A particle with charge 2.15$\mu \mathrm { C }$ and 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. $\mathrm { P } 27.89 .$ The magnitude of the field is 0.420$\mathrm { T }$ . 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 partiole 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
01:50

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.
P27.90). 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 $\Delta p = J / B$ . (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 $l = 35.0 \mathrm { mm } ?$

Ajay Singhal
Ajay Singhal
Numerade Educator
06:50

Problem 91

A Cycloidal Path. A particle with mass $m$ and positive charge $q$ starts from rest at the origin shown in Fig. $P 27.91$ .
There is a uniform electric field $\vec { E }$ in the $+ y$ -direction and a uniform magnetic field $\vec { \boldsymbol { B } }$ 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
Zhaojie Xu
Numerade Educator