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University Physics with Modern Physics

Hugh D. Young

Chapter 28

Sources of Magnetic Field - all with Video Answers

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Chapter Questions

07:42

Problem 1

A $+6.00-\mu \mathrm{C}$ point charge is moving at a constant $8.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ in the $+y$ -direction, relative to a reference frame. At the instant when the point charge is at the origin of this reference frame, what is the magnetic-field vector $\vec{B}$ it produces at the following points: (a) $x=0.500 \mathrm{m}, y=0, \quad z=0 ;$ (b) $x=0$ $y=-0.500 \mathrm{m}, z=0 ; \quad(\mathrm{c}) x=0, \quad y=0, z=+0.500 \mathrm{m} ;$ (d) $x=0, y=-0.500 \mathrm{m}, z=+0.500 \mathrm{m} ?$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
04:30

Problem 2

Fields Within the Atom. In the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius $5.3 \times 10^{-11} \mathrm{m}$ with a speed of $2.2 \times 10^{6} \mathrm{m} / \mathrm{s} .$ If we are viewing the atom in such a way that the electron's orbit is in the plane of the paper with the electron moving clockwise, find the magnitude and direction of the electric and magnetic fields that the electron produces at the location of the nucleus (treated as a point).

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
04:49

Problem 3

An electron moves at 0.100$c$ as shown in Fig. E28.3. Find the magnitude and direction of the magnetic field this electon produces at the following points, each 2.00$\mu \mathrm{m}$ from the electron: (a) points $A$ and $B ;(\mathrm{b})$ point $C ;(\mathrm{c})$ point $D .$

CJ
Cameron Johnson
Numerade Educator
04:12

Problem 4

An alpha particle (charge $+2 e )$ and an electron move in opposite directions from the same point, each with the speed of $2.50 \times 10^{5} \mathrm{m} / \mathrm{s}$ (Fig. E28.4). Find the magnitude and direction of the total magnetic field these charges produce at point $P,$ which is 1.75 nm from each of them.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
05:02

Problem 5

A $-4.80-\mu \mathrm{C}$ charge is moving at a constant speed of $6.80 \times 10^{5} \mathrm{m} / \mathrm{s}$ in the $+x$ -direction relative to a reference frame. At the instant when the point charge
is at the origin, what is the magnetic-field vector it produces at the following points: (a) $x=0.500 \mathrm{m}, y=0, z=0 ;$ (b) $x=0$ $y=0.500 \mathrm{m}, \quad z=0 ; \quad$ (c) $x=0.500 \mathrm{m}, \quad y=0.500 \mathrm{m}, \quad z=0$ (d) $x=0, y=0, z=0.500 \mathrm{m} ?$

Salamat Ali
Salamat Ali
Numerade Educator
11:00

Problem 6

Positive point charges $q=$ $+8.00 \mu C$ and $q^{\prime}=+3.00 \mu C$ are moving relative to an observer at point $P,$ as shown in Fig. E28.6. The distance $d$ is $0.120 \mathrm{m}, v=$
$4.50 \times 10^{6} \mathrm{m} / \mathrm{s},$ and $v^{\prime}=9.00 \times$ $10^{6} \mathrm{m} / \mathrm{s} .$ (a) When the two charges are at the locations shown in the figure, what are the magnitude and direction of the net magnetic field they produce at point $P ?$ (b) What are the magnitude and direction of the electric and magnetic forces that each charge exerts on the other, and what is the ratio of the magnitude of the electric force to the magnitude of the magnetic force? (c) If the direction of $\vec{v}^{\prime}$ is reversed, so both charges are moving in the same direction, what are the magnitude and direction of the magnetic forces that the two charges exert on each other?

Stephen Place
Stephen Place
University of California, Irvine
14:02

Problem 7

Figure E28.6 shows two point charges, $q$ and $q^{\prime},$ moving relative to an observer at point $P .$ Suppose that the lower charge is actually negative, with $q^{\prime}=-q .$ (a) Find the magnetic field
(magnitude and direction) produced by the two charges at point $P$ if (i) $v^{\prime}=v / 2 ;$ (ii) $v^{\prime}=v ;$ (iii) $v^{\prime}=2 v$ . (b) Find the direction of the magnetic force that $q$ exerts on $q^{\prime},$ and find the direction of the magnetic force that $q^{\prime}$ exerts on $q .$ (c) If $v=v^{\prime}=3.00 \times$ $10^{5} \mathrm{m} / \mathrm{s},$ what is the ratio of the magnitude of the magnetic force acting on each charge to that of the Coulomb force acting on each charge?

Guilherme Barros
Guilherme Barros
Numerade Educator
17:18

Problem 8

An electron and a proton are each moving at 845 $\mathrm{km} / \mathrm{s}$ in perpendicular paths as shown in Fig. E28.8. At the instant when they are at the positions shown in the figure, find the magnitude and direction of (a) the total magnetic field they produce at the origin; (b) the magnetic field the electron
produces at the location of the proton; (c) the total electric force and the total magnetic force that the electron exerts on the proton.

Bradley Nordell
Bradley Nordell
Numerade Educator
05:17

Problem 9

A negative charge $q=-3.60 \times 10^{-6} \mathrm{C}$ is located at the origin and has velocity $\vec{v}=\left(7.50 \times 10^{4} \mathrm{m} / \mathrm{s}\right) \hat{\imath}+(-4.90 \times$ $10^{4} \mathrm{m} / \mathrm{s} ) \hat{\boldsymbol{J}}$ . At this instant what are the magnitude and direction of the magnetic field produced by this charge at the point $x=$ $0.200 \mathrm{m}, y=-0.300 \mathrm{m}, z=0 ?$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
08:01

Problem 10

A short current element $d \vec{l}=(0.500 \mathrm{mm}) \hat{\jmath}$ carries a current of 8.20 $\mathrm{A}$ in the same direction as $d \vec{l} .$ Point $P$ is located at $\vec{r}=(-0.730 \mathrm{m}) \hat{\imath}+(0.390 \mathrm{m}) \hat{k} .$ Use unit vectors to express the magnetic field at $P$ produced by this current element.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
05:16

Problem 11

A straight wire carries a 10.0 -A current (Fig. E28.11). $A B C D$ is a rectangle with point $D$
in the middle of a $1.10-$ mm segment of the wire and point $C$ in the wire. Find the magnitude and
direction of the magnetic field due to this segment at (a) point $A ;$ (b) point $B ;$ (c) point $C .$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
06:05

Problem 12

A long, straight wire, carrying a current of 200 $\mathrm{A}$ , runs through a cubical wooden
box, entering and leaving through holes in the centers of opposite faces (Fig. E28.12). The length of each side of the box is 20.0 $\mathrm{cm} .$ Consider an element $d l$ of the wire 0.100 $\mathrm{cm}$ long at the center of the box. Compute the magnitude $d B$ of the magnetic field produced by this element at the points $a, b, c, d,$ and $e$ in Fig. Fig. .28 $.12 .$ Points $a, c,$ and $d$ are at the centers of the faces of the cube; point $b$ is at the midpoint of one edge; and point $e$ is at a corner. Copy the figure and show the directions and relative magnitudes of the field vectors. (Note: Assume that the length $d l$ is small in comparison to the distances from the current element to the points where the magnetic field is to be calculated.)

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:01

Problem 13

A long, straight wire lies along the $z$ -axis and carries a 4.00 -A current in the $+z$ -direction. Find the magnetic field (magnitude and direction) produced at the following points by a $0.500-\mathrm{mm}$
segment of the wire centered at the origin: (a) $x=2.00 \mathrm{m}, y=0$ , $z=0 ;($ b) $x=0, y=2.00 \mathrm{m}, z=0 ;(\mathrm{c}) x=2.00 \mathrm{m}, y=2.00 \mathrm{m}$ $z=0 ;(\mathrm{d}) x=0, y=0, z=2.00 \mathrm{m}$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
05:06

Problem 14

Two parallel wires are 5.00 $\mathrm{cm}$ apart and carry currents in opposite directions, as shown in Fig. E28.14. Find the magnitude and direction of the magnetic field at point $P$ due to two 1.50 -mm segments of wire that are opposite each other and each 8.00 $\mathrm{cm}$ from $P .$

Carolyn Shasha
Carolyn Shasha
Numerade Educator
01:37

Problem 15

A wire carrying a $28.0-\mathrm{A}$ current bends through a right angle. Consider two 2.00 -mm segments of wire, each 3.00 cm from the bend (Fig. $E 28.15 ) .$ Find the magnitude and direction of the magnetic field these two segments produce at point $P,$ which is midway between them.

Salamat Ali
Salamat Ali
Numerade Educator
View

Problem 16

A square wire loop 10.0 $\mathrm{cm}$ on each side carries a clockwise current of 15.0 A. Find the magnitude and direction of the magnetic field at its center due to the four 1.20 -mm wire segments at the midpoint of each side.

Victor Salazar
Victor Salazar
Numerade Educator
01:10

Problem 17

The Magnetic Field from a Lightning Bolt. Lightning bolts can carry currents up to approximately 20 kA. We can model such a current as the equivalent of a very long, straight wire. (a) If you were unfortunate enough to be 5.0 m away from such a lightning bolt, how large a magnetic field would you experience? (b) How does this field compare to one you would experience by being 5.0 $\mathrm{cm}$ from a long, straight household current of 10 A ?

Salamat Ali
Salamat Ali
Numerade Educator
02:23

Problem 18

A very long, straight horizontal wire carries a current such that $3.50 \times 10^{18}$ electrons per second pass any given point going from west to east. What are the magnitude and direction of the
magnetic field this wire produces at a point 4.00 $\mathrm{cm}$ directly above it?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
01:53

Problem 19

BIO Currents in the Heart. The body contains many small currents caused by the motion of ions in the organs and cells. Measurements of the magnetic field around the chest due to currents in the heart give values of about 10$\mu \mathrm{G}$ . Although the actual currents are rather complicated, we can gain a rough understanding of their magnitude if we model them as a long, straight wire. If the surface of the chest is 5.0 $\mathrm{cm}$ from this current, how large is the current in the heart?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
04:04

Problem 20

BIO Bacteria Navigation. Certain bacteria (such as Aquaspirillum magnetotacticum) tend to swim toward the earth's geographic north pole because they contain tiny particles, called magnetosomes, that are sensitive to a magnetic field. If a transmission line carrying 100 $\mathrm{A}$ is laid underwater,
at what range of distances would the magnetic field from this line be great enough to interfere with the migration of these bacteria? (Assume that a field less than 5 percent of the earth's field would have little effect on the bacteria. Take the earth's field to be $5.0 \times 10^{-5} \mathrm{T}$ and ignore the
effects of the seawater.)

Brandy Heflin
Brandy Heflin
Numerade Educator
02:33

Problem 21

(a) How large a current would a very long, straight wire have to carry so that the magnetic field 2.00 $\mathrm{cm}$ from the wire is equal to 1.00 G (comparable to the earth's northward-pointing
magnetic field)? (b) If the wire is horizontal with the current running from east to west, at what locations would the magnetic field of the wire point in the same direction as the horizontal component of the earth's magnetic field? (c) Repeat part (b) except the wire is vertical with the current going upward.

Salamat Ali
Salamat Ali
Numerade Educator
06:16

Problem 22

Two long, straight wires, one above the other, are separated by a distance 2$a$ and are parallel to the $x$ -axis. Let the $+y$ -axis be in the plane of the wires in the direction from the lower wire to
the upper wire. Each wire carries current $I$ in the $+x$ - x-direction. What are the magnitude and direction of the net magnetic field of the two wires at a point in the plane of the wires (a) midway
between them; (b) at a distance a above the upper wire; $(\mathrm{c})$ at a distance $a$ below the lower wire?

Carolyn Shasha
Carolyn Shasha
Numerade Educator
04:37

Problem 23

A long, straight wire lies along the $y$ -axis and carries a current $I=8.00 \mathrm{A}$ in the
$-y$ -direction (Fig. $\mathrm{E} 28.23$ ). In addition to the magnetic field due to the current in the wire, a uniform magnetic field $\vec{\boldsymbol{B}}_{0}$ with magnitude $1.50 \times 10^{-6} \mathrm{T}$
is in the $+x$ -direction What is the total field (magnitude and direction) at the following points
in the $x z$ -plane: (a) $x=0, z=$ $1.00 \mathrm{m} ;$ (b) $x=1.00 \mathrm{m}, \quad z=0$ (c) $x=0, z=-0.25 \mathrm{m} ?$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
01:16

Problem 24

BIO EMF. Currents in de transmission lines can be 100 A or more. Some people have expressed concern that the electromagnetic fields (EMFs) from such lines near their homes could cause health dangers. For a line with current 150 $\mathrm{A}$ and at a height of 8.0 $\mathrm{m}$ above the ground, what magnetic field does the line produce at ground level? Express your answer in teslas and as a percent of the earth's magnetic field, which is 0.50 gauss. Does this seem to be cause for worry?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
07:57

Problem 25

Two long, straight, parallel wires, 10.0 $\mathrm{cm}$ apart, carry equal $4.00-\mathrm{A}$
currents in the same direction, as shown in Fig. $\mathrm{E} 28.25 .$ Find the magnitude and
direction of the magnetic field at (a) point $P_{1},$ midway between the wires; (b) point $P_{2}, 25.0 \mathrm{cm}$ to the right of $P_{1} ;(\mathrm{c})$ point $P_{3}, 20.0 \mathrm{cm}$ directly above $P_{1}$

Salamat Ali
Salamat Ali
Numerade Educator
07:54

Problem 26

A rectangular loop with dimensions 4.20 $\mathrm{cm}$ by 9.50 $\mathrm{cm}$ carries current $1 .$ The current in the loop produces a magnetic field at the center of the loop that has magnitude $5.50 \times 10^{-5} \mathrm{T}$ and direction away from you as you view the plane of the loop. What
are the magnitude and direction (clockwise or counterclockwise) of the current in the loop?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
04:33

Problem 27

Four, long, parallel power lines each carry 100 -A currents. A cross-sectional diagram of these lines is a square, 20.0 $\mathrm{cm}$ on each side. For each of the three cases shown in Fig. $\mathrm{E} 28.27,$ calculate the magnetic field at the center of the square.

Keshav Singh
Keshav Singh
Numerade Educator
06:06

Problem 28

Four very long, current-carrying wires in the same plane intersect to form a square 40.0 $\mathrm{cm}$ on each side, as shown in Fig. E28.28. Find the magnitude and direction of the current $I$ so that the magnetic field at the center of the square is zero.

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
11:10

Problem 29

Two insulated wires perpendicular to each other in the same plane carry currents as shown in Fig. E28.29. Find the magnitude of the net magnetic field these wires produce at points $P$ and $Q$ if the 10.0 A-current is (a) to the right or (b) to the left.

Yaqub Khan
Yaqub Khan
Numerade Educator
03:20

Problem 30

Three parallel wires each carry current $I$ in the directions shown in Fig. E28.30. If the separation between adjacent wires is $d$ , calculate the magnitude and direction of the net magnetic force per unit length on each wire.

Keshav Singh
Keshav Singh
Numerade Educator
03:06

Problem 31

Two long, parallel wires are separated by a distance of 0.400 $\mathrm{m}($ Fig. $\mathrm{E} 28.31) .$
The currents $I_{1}$ and $I_{2}$ have the directions shown. (a) Calculate the magnitude of the force exerted by each wire on a 1.20 -m length of the other. Is the force attractive or repulsive? (b) Each current is doubled, so that $I_{1}$ becomes 10.0 A and $I_{2}$ becomes 4.00 A. Now what is the magnitude of the force that each wire exerts on a $1.20-\mathrm{m}$ length of the other?

Keshav Singh
Keshav Singh
Numerade Educator
02:54

Problem 32

Two long, parallel wires are separated by a distance of 2.50 $\mathrm{cm} .$ The force per unit length that each wire exerts on the other is $4.00 \times 10^{-5} \mathrm{N} / \mathrm{m},$ and the wires repel each other. The current in one wire is 0.600 A. (a) What is the current in the second wire?
(b) Are the two currents in the same direction or in opposite directions?

Carolyn Shasha
Carolyn Shasha
Numerade Educator
02:45

Problem 33

Lamp Cord Wires. The wires in a household lamp cord are typically 3.0 $\mathrm{mm}$ apart center to center and carry equal currents in opposite directions. If the cord carries current to a $100-\mathrm{W}$
light bulb connected across a $120-\mathrm{V}$ potential difference, what force per meter does each wire of the cord exert on the other? Is the force attractive or repulsive? Is this force large enough so it
should be considered in the design of the lamp cord? (Model the lamp cord as a very long straight wire.)

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
04:06

Problem 34

A long, horizontal wire $A B$ rests on the surface of a table and carries a current $I$ Horizontal wire $C D$ is vertically above wire $A B$ and is free to slide up and down on the two vertical metal guides
$C$ and $D$ (Fig. E28.34). Wire $C D$ is connected through the sliding contacts to another wire that
also carries a current $I$ opposite in direction to the current in wire AB. The mass per unit length of the wire $C D$ is $\lambda .$ To what equilibrium height $h$ will the wire $C D$ rise, assuming that the magnetic force on it is due entirely to the current in the wire $A B ?$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
01:52

Problem 35

BIO Currents in the Brain. The magnetic field around the head has been measured to be approximately $3.0 \times 10^{-8}$ G. Although the currents that cause this field are quite complicated, we can get a rough estimate of their size by modeling them as a single circular current loop 16 $\mathrm{cm}$ (the width of a typical head) in diameter. What is the current needed to produce such a field at the center of the loop?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
02:11

Problem 36

Calculate the magnitude and direction of the magnetic field at point $P$ due to the current in the semicircular section of wire shown in Fig. E28. 36. (Hint: Does the current in the long, straight section of the wire produce any field at $P ?$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
02:42

Problem 37

Calculate the magnitude of the magnetic field at point $P$ of Fig. E28.37 in terms of $R, I_{1},$ and $I_{2} .$ What does your expression give when $I_{1}=I_{2} ?$

Salamat Ali
Salamat Ali
Numerade Educator
04:54

Problem 38

A closely wound, circular coil with radius 2.40 $\mathrm{cm}$ has 800 turns. (a) What must the current in the coil be if the magnetic field at the center of the coil is 0.0580 $\mathrm{T}$ ? (b) At what distance $x$ from the center of the coil, on the axis of the coil, is the magnetic field half its value at the center?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
03:01

Problem 39

A closely wound, circular coil with a diameter of 4.00 $\mathrm{cm}$ has 600 turns and carries a current of 0.500 A. What is the magnitude of the magnetic field (a) at the center of the coil and (b) at a point on the axis of the coil 8.00 $\mathrm{cm}$ from its center?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
02:41

Problem 40

A closely wound coil has a radius of 6.00 $\mathrm{cm}$ and carries a current of 2.50 A. How many turns must it have if, at a point on the coil axis 6.00 $\mathrm{cm}$ from the center of the coil, the magnetic field is $6.39 \times 10^{-4} \mathrm{T} ?$

Carolyn Shasha
Carolyn Shasha
Numerade Educator
02:08

Problem 41

Two concentric circular loops of wire lie on a tabletop, one inside the other. The inner wire has a diameter of 20.0 $\mathrm{cm}$ and carries a clockwise current of $12.0 \mathrm{A},$ as viewed from above, and the outer wire has a diameter of 30.0 $\mathrm{cm} .$ What must be the magnitude and direction (as viewed from above) of the current in the outer wire so that the net magnetic field due to this combination of wires is zero at the common center of the wires?

Salamat Ali
Salamat Ali
Numerade Educator
04:57

Problem 42

Figure E28.42 shows, in cross section, several conductors that cartion, surrents through the plane of the figure. The currents have the magnitudes $I_{1}=4.0 \mathrm{A}$ $I_{2}=6.0 \mathrm{A},$ and $I_{3}=2.0 \mathrm{A},$ and the directions shown. Four paths, labeled a through $d,$ are shown.
What is the line integral $\oint \vec{\boldsymbol{B}} \cdot \vec{d \vec{l}}$ for each path? Each integral involves going around the path in the counterclockwise direction. Explain your answers.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
02:48

Problem 43

A closed curve encircles several conductors. The line integral $\oint \vec{B} \cdot d \vec{l}$ around this curve is $3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m}$ . (a) What is the net current in the conductors? (b) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral? Explain.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
00:55

Problem 44

As a new electrical technician, you are designing a large solenoid to produce a uniform $0.150-$ T magnetic field near the center of the solenoid. You have enough wire for 4000 circular turns. This solenoid must be 1.40 $\mathrm{m}$ long and 2.80 $\mathrm{cm}$ in diameter. What current will you need to produce the necessary field?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
02:32

Problem 45

Coaxial Cable. A solid conductor with radius $a$ is supported by insulating disks on the axis of a conducting tube with inner radius $b$ and outer radius $c$ (Fig. $\mathrm{E} 28.45$ ). The central conductor and tube carry equal currents $I$ in opposite directions. The currents are distributed uniformly over the cross sections of each conductor. Derive an expression for the magnitude of the
magnetic field (a) at points outside the central, solid conductor but inside the tube $(a<r<b)$ and (b) at points outside the tube $(r>c) .$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
04:11

Problem 46

Repeat Exercise 28.45 for the case in which the current in the central, solid conductor is $I_{1},$ the current in the tube is $I_{2},$ and these currents are in the same direction rather than in opposite directions.

Carolyn Shasha
Carolyn Shasha
Numerade Educator
02:36

Problem 47

A long, straight, cylindrical wire of radius $R$ carries a current uniformly distributed over its cross section. At what locations is the magnetic field produced by this current equal to half of its largest value? Consider points inside and outside the wire.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
01:50

Problem 48

A 15.0 -cm-long solenoid with radius 0.750 $\mathrm{cm}$ is closely wound with 600 turns of wire. The current in the windings is 8.00 $\mathrm{A}$ . Compute the magnetic field at a point near the center of the solenoid.

Carolyn Shasha
Carolyn Shasha
Numerade Educator
View

Problem 49

A solenoid is designed to produce a magnetic field of 0.0270 T at its center. It has radius 1.40 $\mathrm{cm}$ and length $40.0 \mathrm{cm},$ and the wire can carry a maximum current of 12.0 A. (a) What minimum number of turns per unit length must the solenoid have?
(b) What total length of wire is required?

Yaqub Khan
Yaqub Khan
Numerade Educator
02:01

Problem 50

A toroidal solenoid has an inner radius of 12.0 $\mathrm{cm}$ and an outer radius of 15.0 $\mathrm{cm} .$ It carries a current of 1.50 A. How many equally spaced turns must it have so that it will produce a magnetic field of 3.75 mT at points within the coils 14.0 $\mathrm{cm}$ from its center?

Carolyn Shasha
Carolyn Shasha
Numerade Educator
02:48

Problem 51

A magnetic field of 37.2 $\mathrm{T}$ has been achieved at the MIT Francis Bitter National Magnetic Laboratory. Find the current needed to achieve such a field (a) 2.00 $\mathrm{cm}$ from a long, straight
wire; (b) at the center of a circular coil of radius 42.0 $\mathrm{cm}$ that has 100 turns; $(\mathrm{c})$ near the center of a solenoid with radius $2.40 \mathrm{cm},$ length $32.0 \mathrm{cm},$ and $40,000$ turns.

Salamat Ali
Salamat Ali
Numerade Educator
02:01

Problem 52

A toroidal solenoid (see Example 28.10$)$ has inner radius $r_{1}=15.0 \mathrm{cm}$ and outer radius $r_{2}=18.0 \mathrm{cm} .$ The solenoid has 250 turns and carries a current of 8.50 A. What is the magnitude of the magnetic field at the following distances from the center of the torus: (a) $12.0 \mathrm{cm} ;$ (b) $16.0 \mathrm{cm} ;$ (c) 20.0 $\mathrm{cm} ?$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
01:01

Problem 53

A wooden ring whose mean diameter is 14.0 $\mathrm{cm}$ is wound with a closely spaced toroidal winding of 600 turns. Compute the magnitude of the magnetic field at the center of the cross section of the windings when the current in the windings is 0.650 $\mathrm{A}$ .

Salamat Ali
Salamat Ali
Numerade Educator
03:28

Problem 54

A toroidal solenoid with 400 turns of wire and a mean radius of 6.0 $\mathrm{cm}$ carries a current of 0.25 A. The relative permeability of the core is $80 .$ (a) What is the magnetic field in the core?
(b) What part of the magnetic field is due to atomic currents?

Carolyn Shasha
Carolyn Shasha
Numerade Educator
00:53

Problem 55

A toroidal solenoid with 500 turns is wound on a ring with a mean radius of 2.90 $\mathrm{cm} .$ Find the current in the winding that is required to set up a magnetic field of 0.350 T in the ring (a) if the
ring is made of annealed iron $\left(K_{\mathrm{m}}=1400\right)$ and $(\mathrm{b})$ if the ring is made of silicon steel $\left(K_{\mathrm{m}}=5200\right)$

Christopher Dzorkpata
Christopher Dzorkpata
Numerade Educator
02:30

Problem 56

The current in the windings of a toroidal solenoid is 2.400 A. There are 500 turns, and the mean radius is 25.00 $\mathrm{cm} .$ The toroidal solenoid is filled with a magnetic material. The magnetic field inside the windings is found to be 1.940 T. Calculate (a) the relative permeability and (b) the magnetic susceptibility of the material that fills the toroid.

Carolyn Shasha
Carolyn Shasha
Numerade Educator
03:56

Problem 57

A long solenoid with 60 turns of wire per centimeter carries a current of 0.15 A. The wire that makes up the solenoid is wrapped around a solid core of silicon steel $\left(K_{\mathrm{m}}=5200\right) .$ (The
wire of the solenoid is jacketed with an insulator so that none of the current flows into the core.) (a) For a point inside the core, find the magnitudes of (i) the magnetic field $\vec{B}_{0}$ due to the solenoid current; (ii) the magnetization $\vec{M} ;$ (iii) the total magnetic field $\vec{\boldsymbol{B}}$ . (b) In
a sketch of the solenoid and core, show the directions of the vectors $\vec{\boldsymbol{B}}, \vec{\boldsymbol{B}}_{0},$ and $\vec{M}$ inside the core.

Salamat Ali
Salamat Ali
Numerade Educator
02:22

Problem 58

When a certain paramagnetic material is placed in an external magnetic field of 1.5000 T, the field inside the material is measured to be 1.5023 T. Find (a) the relative permeability and (b) the magnetic permeability of this material.

Averell Hause
Averell Hause
Carnegie Mellon University
06:24

Problem 59

A pair of point charges, $q=+8.00 \mu \mathrm{C}$ and $q^{\prime}=-5.00$ $\mu \mathrm{C},$ are moving as shown in Fig. $\mathrm{P} 28.59$ with speeds $v=$ $9.00 \times 10^{4} \mathrm{m} / \mathrm{s}$ and $v^{\prime}=6.50 \times$ $10^{4} \mathrm{m} / \mathrm{s}$ . When the charges are at the locations shown in the figure, what are the magnitude and direction of (a) the magnetic field produced at the origin and (b) the magnetic force that $q^{\prime}$ exerts on $q ?$

Keshav Singh
Keshav Singh
Numerade Educator
06:41

Problem 60

At a particular instant, charge $q_{1}=+4.80 \times 10^{-6} \mathrm{C}$ is at the point $(0,0.250 \mathrm{m}, 0)$ and has velocity $\vec{\boldsymbol{v}}_{1}=\left(9.20 \times 10^{5} \mathrm{m} / \mathrm{s}\right) \hat{\imath}$ Charge $q_{2}=-2.90 \times 10^{-6} \mathrm{C}$ is at the point $(0.150 \mathrm{m}, 0,0)$ and has velocity $\vec{v}_{2}=\left(-5.30 \times 10^{5} \mathrm{m} / \mathrm{s}\right) \hat{\boldsymbol{J}}$ . At this instant, what are the magnitude and direction of the magnetic force that $q_{1}$ exerts on $q_{2} ?$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
04:20

Problem 61

Two long, parallel transmission lines, 40.0 $\mathrm{cm}$ apart, carry $25.0-\mathrm{A}$ and 75.0 -A currents. Find all locations where the net magnetic field of the two wires is zero if these currents are in (a) the same direction and (b) the opposite direction.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
03:23

Problem 62

A long, straight wire carries a current of 5.20 A. An electron is traveling in the vicinity of the wire. At the instant when the electron is 4.50 $\mathrm{cm}$ from the wire and traveling with a speed of $6.00 \times 10^{4} \mathrm{m} / \mathrm{s}$ directly toward the wire, what are the magnitude and direction (relative to the direction of the current) of the force that the magnetic field of the current exerts on the
electron?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
06:45

Problem 63

A long, straight wire carries a 13.0 -A current. An electron is fired parallel to this wire with a velocity of 250 $\mathrm{km} / \mathrm{s}$ in the same direction as the current, 2.00 $\mathrm{cm}$ from the wire. (a) Find the magnitude and direction of the electron's initial acceleration. (b) What should be the magnitude and direction of a uniform electric field that will allow the electron to continue to travel parallel to the wire? (c) Is it necessary to include the effects of gravity? Justify your answer.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
06:27

Problem 64

Two very long, straight wires carry currents as shown in Fig. P28.64. For each case, find all locations where the net magnetic field is zero.

Zhaojie Xu
Zhaojie Xu
Numerade Educator
03:59

Problem 65

Two identical circular, wire loops 40.0 $\mathrm{cm}$ in diameter each carry a current of 3.80 $\mathrm{A}$ in the same direction. These loops are parallel to each other and are 25.0 $\mathrm{cm}$ apart. Line ab is normal to the plane of the loops and passes through their centers. A proton is fired at 2400 $\mathrm{km} / \mathrm{s}$ perpendicular to line $a b$ from a point midway between the centers of the loops. Find the magnitude of the magnetic force these loops exert on the proton just after it is
fired.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
05:52

Problem 66

A negative point charge $q=-7.20 \mathrm{mC}$ is moving in a reference frame. When the point charge is at the origin, the magnetic field it produces at the point $x=25.0 \mathrm{cm}, y=0, z=0$ is
$\vec{\boldsymbol{B}}=(6.00 \mu \mathrm{T}) \hat{\boldsymbol{J}},$ and its speed is 800 $\mathrm{m} / \mathrm{s} .(\mathrm{a})$ What are the $x_{-}, y-$ and $z$ -components of the velocity $\vec{\boldsymbol{v}}_{0}$ of the charge? (b) At this same instant, what is the magnitude of the magnetic field that the charge produces at the point $x=0, y=25.0 \mathrm{cm}, z=0 ?$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
06:37

Problem 67

Two long, straight, parallel wires are 1.00 $\mathrm{m}$ apart (Fig. $\mathrm{P} 28.67$ . The wire on the left carries a current $I_{1}$ of 6.00 $\mathrm{A}$ into the plane of the paper. (a) What must the magnitude and direction of the current $I_{2}$ be for the net field at point $P$ to be zero? (b) Then
what are the magnitude and direction of the net field at $Q ?$ (c) Then what is the magnitude of the net field at $S ?$

Dading Chen
Dading Chen
Numerade Educator
05:16

Problem 68

Figure $P 28.68$ shows an end view of two long, parallel wires perpendicular to the $x y-$
plane, each carrying a current $/$ but in opposite directions. (a) Copy the diagram, and draw vectors to show the $\vec{B}$ field of each wire and the net $\vec{B}$ field at point $P .$ (b) Derive the expression for the magnitude of $\vec{B}$ at any point on the $x$ -axis in terms of the $x$ -coordinate of the point. What is the direction of $\vec{B} ?(\mathrm{c})$ Graph the magnitude of $\vec{\boldsymbol{B}}$ at points on the $x$ -axis. (d) At what value of $x$ is the magnitude of $\vec{\boldsymbol{B}}$ a maximum? (e) What is the magnitude of $\vec{\boldsymbol{B}}$ when $x>a$ ?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
00:55

Problem 69

Refer to the situation in Problem $28.68 .$ Suppose that a third long, straight wire, parallel to the other two, passes through point $P$ (see Fig. $P 28.68 )$ and that each wire carries a current $I=6.00$ A. Let $a=40.0 \mathrm{cm}$ and $x=60.0 \mathrm{cm} .$ Find the magnitude and direction of the force per unit length on the third wire, (a) if the current in it is directed into the plane of the figure, and (b) if the current in it is directed out of the plane of the figure.

Salamat Ali
Salamat Ali
Numerade Educator
01:37

Problem 70

A pair of long, rigid metal rods, each of length $L,$ lie parallel to each other on a perfectly smooth table. Their ends are connected by identical, very light conducting springs of force constant $k$ (Fig. P28.70) and negligible unstretched length. If a current I runs through this circuit, the springs will stretch. At what separation will the rods remain at rest? Assume that $k$ is large enough so that the separation of the rods will be much less than $L .$

Averell Hause
Averell Hause
Carnegie Mellon University
03:44

Problem 71

Two long, parallel wires hang by 4.00 -cm-long cords from a common axis (Fig. P28.71). The wires have a mass per unit length of 0.0125 $\mathrm{kg} / \mathrm{m}$ and carry the same current in opposite directions. What is the current in each wire if the cords hang at an angle of $6.00^{\circ}$ with the vertical?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
03:24

Problem 72

The long, straight wire $A B$ shown in Fig. $P 28.72$ carries a current of 14.0 A. The rectangular loop whose long edges are parallel to the wire carries a current of 5.00 A. Find the magnitude and direction
of the net force exerted on the loop by the magnetic field of the wire.

Keshav Singh
Keshav Singh
Numerade Educator
02:54

Problem 73

A flat, round iron ring 5.00 $\mathrm{cm}$ in diameter has a current running through it that produces a magnetic field of 75.4$\mu \mathrm{T}$ at its center. This ring is placed in a uniform external magnetic field of 0.375 T. What is the maximum torque the external field can exert on the ring? Show how the ring should be oriented relative to the field for the torque to have its maximum value.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
07:32

Problem 74

The wire semicircles shown in Fig. $\mathrm{P} 28.74$ have radii $a$ and $b$ . Calculate the net magnetic field (magnitude and direction) that the current in the wires produces at point $P .$

Carolyn Shasha
Carolyn Shasha
Numerade Educator
15:01

Problem 75

Helmholtz Coils. Figure 28.75 is a sectional view of two circular coils with radius $a$ , each wound with $N$ turns of wire carrying a current $I,$ circulating in the same direction in both coils. The coils are separated by a distance $a$ equal to their radii. In this configuration the coils are called Helmholtz coils; they produce a very uniform magnetic field in the region between them. (a) Derive the expression for the magnitude $B$ of the magnetic field at a point on the axis a distance
$x$ to the right of point $P,$ which is midway between the coils. (b) Graph $B$ versus $x$ for $x=0$ to $x=a / 2 .$ Compare this graph to one for the magnetic field due to the right-hand coil alone. (c) From part (a), obtain an expression for the magnitude of the magnetic field at point $P .$ (d) Calculate the magnitude of the magnetic field at $P$ if $N=300$ turns, $I=6.00 \mathrm{A},$ and $a=8.00 \mathrm{cm} .$ (e) Calculate $d B / d x$ and $d^{2} B / d x^{2}$ at $P(x=0) .$ Discuss how your results show that the field is very uniform in the vicinity of $P .$

Keshav Singh
Keshav Singh
Numerade Educator
07:26

Problem 76

A circular wire of diameter $D$ lies on a horizontal table and carries a current $I .$ In Fig. $P 28.76$ point $A$ marks the center of the circle and point $C$ is on its rim. (a) Find the magnitude and direction of the magnetic field at point $A .($ b) The wire is now unwrapped so it is straight, centered on point $C,$ and perpendicular to the line $A C,$ but the same current is maintained in it. Now find the magnetic field at point $A .(\mathrm{c})$ Which field is greater: the one in part (a) or in part (b)? By what factor? Why is this result physically reasonable?

Zhaojie Xu
Zhaojie Xu
Numerade Educator
04:24

Problem 77

A long, straight wire with a circular cross section of radius $R$ carries a current $I .$ Assume that the current density is not constant across the cross section of the wire, but rather varies as $J=\alpha r,$ where $\alpha$ is a constant. (a) By the requirement that $J$ integrated over the cross section of the wire gives the total current I $I$ , calculate the constant $\alpha$ in terms of $I$ and $R .$ (b) Use Ampere's law to calculate the magnetic field $B(r)$ for (i) $r \leq R$ and (ii) $r \geq R$ . Express your answers in terms of $I .$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
03:29

Problem 78

The wire shown in Fig. P28.78 is infinitely long and carries a current I. Calculate the magnitude and direction of the magnetic field that this current produces at point $P .$

Carolyn Shasha
Carolyn Shasha
Numerade Educator
02:33

Problem 79

A conductor is made in the form of a hollow cylinder with inner and outer radii $a$ and $b$
respectively. It carries a current $I$ uniformly distributed over its cross section. Derive expressions for the magnitude of the magnetic field in the regions (a) $r < a ;$ (b) $a < r < b ;(c) r > b$ .

Narayan Hari
Narayan Hari
Numerade Educator
04:54

Problem 80

A circular loop has radius $R$ and carries current $I_{2}$ in a clockwise direction (Fig. P28.80). The center of the loop is a distance $D$ above a long, straight wire. What are the magnitude and direction
of the current $I_{1}$ in the wire if the magnetic field at the center of the loop is zero?

Carolyn Shasha
Carolyn Shasha
Numerade Educator
06:40

Problem 81

A long, straight, solid cylinder, oriented with its axis in the $z$ -direction, carries a current whose current density is $\vec{J}$ . The current density, although symmetric about the cylinder axis, is
not constant but varies according to the relationship $$ \begin{aligned} \vec{J} &=\frac{2 I_{0}}{\pi a^{2}}\left[1-\left(\frac{r}{a}\right)^{2}\right] \hat{k} & \text { for } r \leq a \\ &=0 \quad \text { for } r \geq a \end{aligned} $$ where $a$ is the radius of the cylinder, $r$ is the radial distance from the cylinder axis, and $I_{0}$ is a constant having units of amperes. (a) Show that $I_{0}$ is the total current passing through the entire cross section of the wire. (b) Using Ampere's law, derive an expression for the magnitude of the magnetic field $\vec{B}$ in the region $r \geq a$ .
(c) Obtain an expression for the current $I$ contained in a circular cross section of radius $r \leq a$ and centered at the cylinder axis. (d) Using Ampere's law, derive an expression for the magnitude of
the magnetic field $\vec{B}$ in the region $r \leq a .$ How do your results in parts (b) and (d) compare for $r=a ?$

Ajay Singhal
Ajay Singhal
Numerade Educator
15:09

Problem 82

A long, straight, solid cylinder, oriented with its axis in the $z$ -direction, carries a current whose current density is $\vec{J}$ . The current density, although symmetric about the cylinder axis, is not
constant and varies according to the relationship $$ \begin{aligned} \vec{J} &=\left(\frac{b}{r}\right) e^{(r-a) / \delta} \hat{k} & \text { for } r \leq a \\ &=0 \quad \quad \text { for } r \geq a \end{aligned}
$$ where the radius of the cylinder is $a=5.00 \mathrm{cm}, r$ is the radial distance from the cylinder axis, $b$ is a constant equal to $600 \mathrm{A} / \mathrm{m},$ and $\delta$ is a constant equal to 2.50 $\mathrm{cm} .$ (a) Let $I_{0}$ be the total current passing through the entire cross section of the wire. Obtain an expression for $I_{0}$ in terms of $b, \delta,$ and $a$ . Evaluate your expression to obtain a numerical value for $I_{0}$ . (b) Using Ampere's law, derive an expression for the magnetic field $\vec{\boldsymbol{B}}$ in the region $r \geq a .$ Express your answer in terms of $I_{0}$ rather than $b$ . (c) Obtain an expression for the current $I$ contained in a circular cross section of radius $r \leq a$ and centered at the cylinder axis. Express your answer in terms of $I_{0}$
rather than $b .$ (d) Using Ampere's law, derive an expression for the magnetic field $\vec{\boldsymbol{B}}$ in the region $r \leq a .$ (e) Evaluate the magnitude of the magnetic field at $r=\delta, r=a,$ and $r=2 a .$

Linda Winkler
Linda Winkler
Numerade Educator
01:16

Problem 83

An Infinite Current Sheet. Long, straight conductors with square cross sections and each carrying current $I$ are laid side by side to form an infinite current sheet (Fig. $P 28.83$ ). The conductors lie in the $x y$ -plane, are parallel to the $y$ -axis, and carry current in the $+y$ -direction There are $n$ conductors per unit length measured along the $x$ -axis. (a) What are the magnitude and direction of the magnetic field a distance $a$ below the current sheet? (b) What are the magnitude and direction of the magnetic field a distance $a$ above the current sheet?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
04:06

Problem 84

Long, straight conductors with square cross section, each carrying current $I$ are laid side by side
to form an infinite current sheet with current directed out of the plane of the page (Fig. $\mathrm{P28.84} )$ . A second infinite current sheet is a distance $d$ below the first and is
parallel to it. The second sheet carries current into the plane of the page. Each sheet has $n$ conductors
per unit length. (Refer to Problem $28.83 . )$ Calculate the magnitude and direction of the net magnetic field at (a) point $P$ (above the upper sheet); (b) point $R($ midway between the two sheets); (c) point
$S$ (below the lower sheet).

Carolyn Shasha
Carolyn Shasha
Numerade Educator
04:24

Problem 85

A piece of iron has magnetization $M=6.50 \times$ $10^{4} \mathrm{A} / \mathrm{m} .$ Find the average magnetic dipole moment per atom in this piece of iron. Express your answer both in $\mathrm{A} \cdot \mathrm{m}^{2}$ and in Bohr magnetons. The density of iron is given in Table $14.1,$ and the atomic mass of iron (in grams per mole) is given in Appendix D. The chemical symbol for iron is Fe.

Aja S
Aja S
Numerade Educator
03:34

Problem 86

A wide, long, insulating belt has a uniform positive charge per unit area $\sigma$ on its upper surface. Rollers at each end move the belt to the right at a constant speed $v$ . Calculate the magnitude and direction of the magnetic field produced by the moving belt at a point just above its surface. (Hint: At points near the surface and far from its edges or ends, the moving belt can be considered to be an infinite current sheet like that in Problem $28.83 . )$

Carolyn Shasha
Carolyn Shasha
Numerade Educator
06:06

Problem 87

Two long, straight conducting wires with linear mass density $\lambda$ are suspended from cords so that they are each horizontal, parallel to each other, and a distance $d$ apart. The back
ends of the wires are connected to each other by a slack, low-resistance connecting wire. A charged capacitor (capacitance $C )$ is now added to the system; the positive plate of the capacitor (initial
charge $+Q_{0}$ ) is connected to the front end of one of the wires, and the negative plate of the capacitor (initial charge $-Q_{0} )$ is connected to the front end of the other wire (Fig. P28.87). Both of these connections are also made by slack, low-resistance wires. When the connection is made,
the wires are pushed aside by the repulsive force between the wires, and each wire has an initial horizontal velocity of magnitude $v_{0} .$ Assume that the time constant for the capacitor to discharge is negligible compared to the time it takes for any appreciable displacement in the position of the wires to occur. (a) Show that the initial speed $v_{0}$ of either wire is given by $$ v_{0}=\frac{\mu_{0} Q_{0}^{2}}{4 \pi \lambda R C d} $$ where $R$ is the total resistance of the circuit. (b) To what height $h$ will each wire rise as a result of the circuit connection?

Keshav Singh
Keshav Singh
Numerade Educator
41:11

Problem 88

A wire in the shape of a semicircle with radius $a$ is oriented in the yz-plane with its center of curvature at the origin (Fig. P28.88). If the current in the wire is $I,$ calculate the magnetic-field components
produced at point $P,$ a distance $x$ out along the $x$ -axis. (Note: Do not forget the contribution from the straight wire at the bottom of the semicircle that runs from $z=-a$ to $z=+a .$ You may use the fact that the fields of the two antiparallel currents at $z>a$ cancel, but you must explain why they cancel.)

TM
Thomas M
Numerade Educator