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Fundamentals of Physics

David Halliday, Robert Resnick , Jearl Walker

Chapter 29

Magnetic Fields Due to Currents - all with Video Answers

Educators

+ 11 more educators

Chapter Questions

03:07

Problem 1

A surveyor is using a magnetic compass 6.1 $\mathrm{m}$ below a power line
in which there is a steady current of
100 A. (a) What is the magnetic
field at the site of the compass due
to the power line? (b) Will this field interfere seriously with the compass reading? The horizontal component of Earth's magnetic field at
the site is 20$\mu \mathrm{T}$ .

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:20

Problem 2

Figure $29-35 a$ shows an element of length $d s=1.00 \mu \mathrm{m}$ in a
very long straight wire carrying
current. The current in that element sets up a differential magnetic field d\vec at points in the surrounding space. Figure $29-35 b$
gives the magnitude $d B$ of the field
for points 2.5 $\mathrm{cm}$ from the element, as a function of angle $\theta$ between the wire and a straight line to the
point. The vertical scale is set by $d B_{s}=60.0$ pT. What is the magnitude of the magnetic field set up by the entire wire at perpendicular
distance 2.5 $\mathrm{cm}$ from the wire?

Rahul Nikhar
Rahul Nikhar
Numerade Educator
00:57

Problem 3

At a certain location in the Philippines, Earth's magnetic field of 39$\mu \mathrm{T}$ is horizontal and directed due north. Suppose the
net field is zero exactly 8.0 $\mathrm{cm}$ above a long, straight, horizontal
wire that carries a constant current. What are the (a) magnitude
and (b) direction of the current?

Salamat Ali
Salamat Ali
Numerade Educator
03:32

Problem 4

A straight conductor carrying current $i=5.0$ A splits into identical semicircular arcs as shown in Fig. $29-36$ .
What is the magnetic field at the center
Cof the resulting circular loop?

Rahul Nikhar
Rahul Nikhar
Numerade Educator
06:45

Problem 5

In Fig. $29-37,$ a current $i=10 \mathrm{A}$ is set up in a long hairpin conductor
formed by bending a wire into a
semicircle of radius $R=5.0 \mathrm{mm} .$ Point
$b$ is midway between the straight sections and so distant from the semicircle that each straight section can be approximated as being an infinite
wire. What are the (a) magnitude and
(b) direction (into or out of the page)
of $\vec{B}$ at $a$ and the (c) magnitude and
(d) direction of $\vec{B}$ at $b$ ?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
05:25

Problem 6

In Fig. $29-38,$ point $P$ is at perpendicular distance $R=2.00 \mathrm{cm}$ from a very long straight wire carrying a current. The magnetic field
$\vec{B}$ set up at point $P$ is due to contributions from all the identical current-length elements $i d \vec{s}$ along the wire. What is the distance $s$ to the element making (a) the greatest contribution to field $\vec{B}$ and $(b) 10.0 \%$
of the greatest contribution?

Rahul Nikhar
Rahul Nikhar
Numerade Educator
View

Problem 7

In Fig. $29-39,$ two circular arcs have radii $a=13.5 \mathrm{cm}$ and $b=$
$10.7 \mathrm{cm},$ subtend angle $\theta=74.0^{\circ},$
carry current $i=0.411 \mathrm{A},$ and share
the same center of curvature $P .$ What
are the (a) magnitude and (b) direction (into or out of the page) of the
net magnetic field at $P ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
04:33

Problem 8

In Fig. $29-40,$ two semicircular arcs have radii $R_{2}=7.80 \mathrm{cm}$ and
$R_{1}=3.15 \mathrm{cm},$ carry current $i=0.281$
$\mathrm{A},$ and have the same center of curvature $C .$ What are the (a) magnitude
and (b) direction (into or out of the
page) of the net magnetic field at $C ?$

Rahul Nikhar
Rahul Nikhar
Numerade Educator
01:16

Problem 9

Two long straight wires are parallel and 8.0 $\mathrm{cm}$ apart. They are to carry equal currents such that the magnetic field at a point
halfway between them has magnitude 300$\mu$ (a) Should the
currents be in the same or opposite directions? (b) How much
current is needed?

Salamat Ali
Salamat Ali
Numerade Educator
03:07

Problem 10

In Fig. $29-41,$ a wire forms a semicircle of radius $R=9.26 \mathrm{cm}$ and two
(radial) straight segments each of
length $L=13.1 \mathrm{cm} .$ The wire carries
current $i=34.8 \mathrm{m} \mathrm{A} .$ What are the (a)
magnitude and (b) direction (into or out of the page) of the net magnetic field at the semicircle's center of curvature $C ?$

Rahul Nikhar
Rahul Nikhar
Numerade Educator
06:06

Problem 11

In Fig. $29-42,$ two long straight wires are perpendicular to the page and
separated by distance $d_{1}=0.75 \mathrm{cm}$ .
Wire 1 carries 6.5 $\mathrm{A}$ into the page. What
are the (a) magnitude and (b) direction
(into or out of the page) of the current
in wire 2 if the net magnetic field due to the two currents is zero at point $P$ located at distance $d_{2}=1.50 \mathrm{cm}$ from
wire 2$?$

Linda Winkler
Linda Winkler
Numerade Educator
04:08

Problem 12

In Fig. $29-43,$ two long straight wires at separation $d=16.0 \mathrm{cm}$ carry
currents $i_{1}=3.61 \mathrm{m} \mathrm{A}$ and $i_{2}=3.00 i_{1}$
out of the page. (a) Where on the $x$ axis
is the net magnetic field equal to zero?
(b) If the two currents are doubled, is
the zero-field point shifted toward wire
$1,$ shifted toward wire $2,$ or unchanged?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:09

Problem 13

In Fig. $29-44,$ point $P_{1}$ is at distance $R=13.1 \mathrm{cm}$ on the perpendicular bisector of a straight wire of length $L=18.0 \mathrm{cm}$ carrying current $i=58.2 \mathrm{mA}$ . (Note that the wire is not long.) What is the
magnitude of the magnetic field at $P_{1}$ due to $i$ ?

Salamat Ali
Salamat Ali
Numerade Educator
01:21

Problem 14

Equation $29-4$ gives the magnitude $B$ of the magnetic field set up by a current in an infinitely long straight wire, at a point $P$
at perpendicular distance $R$ from the wire. Suppose that point $P$
is actually at perpendicular distance $R$ from the midpoint of a
wire with a finite length $L .$ Using Eq. $29-4$ to calculate $B$ then results in a certain percentage error. What value must the ratio
$L / R$ exceed if the percentage error is to be less than 1.00$\% ?$ That
is, what $L / R$ gives
$$
\frac{(B \text { from Eq. } 29-4)-(B \text { actual) }}{(B \text { actual })}(100 \%)=1.00 \% ?
$$

Salamat Ali
Salamat Ali
Numerade Educator
02:30

Problem 15

Figure $29-45$ shows two current segments. The lower segment
carries a current of $i_{1}=0.40 \mathrm{A}$ and
includes a semicircular arc with
radius $5.0 \mathrm{cm},$ angle $180^{\circ},$ and center
point $P .$ The upper segment carries
current $i_{2}=2 i_{1}$ and includes a circular arc with radius $4.0 \mathrm{cm},$ angle $120^{\circ}$ ,
and the same center point P. What are the (a) magnitude and (b) direction of the net magnetic field $\vec{B}$
at $P$ for the indicated current directions? What are the (c) magni-
tude and (d) direction of $\vec{B}$ if $i_{1}$ is reversed?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
09:57

Problem 16

In Fig. $29-46,$ two concentric circular loops of wire carrying
current in the same direction lie in
the same plane. Loop 1 has radius
1.50 $\mathrm{cm}$ and carries 4.00 $\mathrm{mA} .$ Loop 2
has radius 2.50 $\mathrm{cm}$ and carries 6.00 mA. Loop 2 is to be rotated about a diameter while the net magnetic field $\vec{B}$ set up by the two loops at their common center is
measured. Through what angle must loop 2 be rotated so that the
magnitude of that net field is 100 $\mathrm{nT}$ ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:57

Problem 17

In Fig. $29-44,$ point $P_{2}$ is at perpendicular distance $R=$ 25.1 $\mathrm{cm}$ from one end of a straight wire of length $L=13.6 \mathrm{cm}$ carrying current $i=0.693 \mathrm{A} .$ (Note that the wire is not long.) What is
the magnitude of the magnetic field at $P_{2} ?$

Salamat Ali
Salamat Ali
Numerade Educator
01:39

Problem 18

A current is set up in a wire loop consisting of a semicircle of radius $4.00 \mathrm{cm},$ a smaller concentric
semicircle, and two radial straight
lengths, all in the sadial straight
29-47a shows the arrangement but is
not drawn to scale. The magnitude
of the magnetic field produced at the center of curvature is 47.25$\mu \mathrm{T} .$ The smaller semicircle is then
flipped over (rotated) until the loop is again entirely in the same plane (Fig. $29-47 b )$ . The magnetic field produced at the (same) center of curvature now has magnitude $15.75 \mu \mathrm{T},$ and its direction is reversed from the initial magnetic field. What is the radius of the
smaller semicircle?

Salamat Ali
Salamat Ali
Numerade Educator
01:25

Problem 19

One long wire lies along an $x$ axis and carries a current of 30 A in the positive $x$ direction. A second long wire is perpendicular to
the $x y$ plane, passes through the point $(0,4.0 \mathrm{m}, 0),$ and carries a current of 40 $\mathrm{A}$ in the positive $z$ direction. What is the magnitude of the
resulting magnetic field at the point $(0,2.0 \mathrm{m}, 0) ?$

Salamat Ali
Salamat Ali
Numerade Educator
03:22

Problem 20

In Fig. $29-48,$ part of a long in sulated wire carrying current
$i=5.78 \mathrm{mA}$ is bent into a circular
section of radius $R=1.89 \mathrm{cm} .$ In
unit-vector notation, what is the
magnetic field at the center of curvature $C$ if the circular section (a) lies in the plane of the page as shown
and (b) is perpendicular to the plane
of the page after being rotated $90^{\circ}$
counterclockwise as indicated?

Salamat Ali
Salamat Ali
Numerade Educator
01:26

Problem 21

Figure $29-49$ shows two very long straight wires (in cross section) that each carry a current of
4.00 A directly out of the page-Distance $d_{1}=6.00 \mathrm{m}$ and distance
$d_{2}=4.00 \mathrm{m} .$ What is the magnitude
of the net magnetic field at point $P,$
which lies on a perpendicular bisector to the wires?

Salamat Ali
Salamat Ali
Numerade Educator
04:01

Problem 22

Figure $29-50 a$ shows, in cross section, two long, parallel wires carrying current and separated by distance $L .$ The ratio $i_{1} / i_{2}$
of their currents is $4.00 ;$ the directions of the currents are not indicated. Figure $29-50 b$ shows the $y$ component $B_{y}$ of their net magnetic field along the $x$ axis to the right of wire $2 .$ The vertical scale is set by $B_{y s}=4.0 \mathrm{nT}$ , and the horizontal scale is set by $x_{s}=20.0 \mathrm{cm}$ .
(a) At what value of $x>0$ is $B_{v}$ maximum? (b) If $i_{2}=3 \mathrm{mA}$ , what is the value of that maximum? What is the direction (into or out of the
page $)$ of $(\mathrm{c}) i_{1}$ and $(\mathrm{d}) i_{2}$ ?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:56

Problem 23

Figure $29-51$ shows a snapshot of a proton moving at velocity
$\vec{v}=(-200 \mathrm{m} / \mathrm{s}) \hat{\mathrm{j}}$ toward a long straight
wire with current $i=350 \mathrm{mA}$ . At the
instant shown, the proton's distance
from the wire is $d=2.89 \mathrm{cm} .$ In unit-vector notation, what is the magnetic
force on the proton due to the current?

Umar Sohail Qureshi
Umar Sohail Qureshi
Numerade Educator
16:35

Problem 24

Figure $29-52$ shows, in cross section, four thin wires that are parallel, straight, and very long. They carry
identical currents in the directions indicated. Initially all four wires are at
distance $d=15.0 \mathrm{cm}$ from the origin
of the coordinate system, where they
create a net magnetic field $\vec{B} .$ (a) To
what value of $x$ must you move wire 1along the $x$ axis in order to rotate $\vec{B}$
counterclockwise by $30^{\circ} ?$ (b) With wire
1 in that new position, to what value of $x$ must you move wire 3 along the $x$ axis to rotate $\vec{B}$ by $30^{\circ}$ back to its
initial orientation?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:22

Problem 25

A wire with current $i=3.00 \mathrm{A}$ is shown in Fig. $29-53 .$ Two
semi-infinite straight sections, both tangent to the same circle, are connected
by a circular arc that has a central angle
$\theta$ and runs along the circumference of
the circle. The arc and the two straight
sections all lie in the same plane. If $B=$
0 at the circle's center, what is $\theta$ ?

Salamat Ali
Salamat Ali
Numerade Educator
02:49

Problem 26

In Fig. $29-54 a,$ wire 1 consists of a circular arc and two radial lengths; it carries current $i_{1}=0.50 \mathrm{A}$ in the direction
indicated. Wire $2,$ shown in cross section, is long, straight, and perpendicular to the plane of the figure. Its distance from the center of the arc is equal to the radius $R$ of the arc, and it carries a current $i_{2}$
that can be varied. The two currents set up a net magnetic field $\vec{B}$ at
the center of the arc. Figure $29-54 b$ gives the square of the field's magnitude $B^{2}$ plotted versus the square of the current $i_{2}^{2} .$ The vertical scale is set by $B_{s}^{2}=10.0 \times 10^{-10} \mathrm{T}^{2} .$ What angle is subtended by
the arc?

Salamat Ali
Salamat Ali
Numerade Educator
01:12

Problem 27

In Fig. $29-55,$ two long straight wires (shown in cross section) carry
the currents $i_{1}=30.0 \mathrm{mA}$ and $i_{2}=$
40.0 $\mathrm{mA}$ directly out of the page.
They are equal distances from the
origin, where they set up a magnetic
field $\vec{B} .$ To what value must current $i_{1}$
be changed in order to rotate $\vec{B} 20.0^{\circ}$
clockwise?

Salamat Ali
Salamat Ali
Numerade Educator
00:52

Problem 28

Figure $29-56 a$ shows two wires, each carrying a current. Wire 1 consists of a circular arc of radius $R$ and two radial lengths; it carries current $i_{1}=2.0 \mathrm{A}$ in the
direction indicated. Wire 2 is long and straight; it carries a current $i_{2}$
that can be varied; and it is at distance $R / 2$ from the center of the
arc. The net magnetic field $\vec{B}$ due to the two currents is measured at the center of curvature of the arc. Figure $29-56 b$ is a plot of
the component of $\vec{B}$ in the direction perpendicular to the figure as
a function of current $i_{2}$ . The horizontal scale is set by $i_{2 s}=1.00 \mathrm{A}$ .
What is the angle subtended by the arc?

Salamat Ali
Salamat Ali
Numerade Educator
03:10

Problem 29

In Fig. $29-57,$ four long straight wires are perpendicular to the page, and their cross sections form a square of edge length
$a=20 \mathrm{cm} .$ The currents are out of the page in wires 1 and 4 and
into the page in wires 2 and $3,$ and each wire carries 20 $\mathrm{A}$ . In
unit-vector notation, what is the net magnetic field at the
square's center?

Salamat Ali
Salamat Ali
Numerade Educator
03:16

Problem 30

Two long straight thin wires with current lie against an equally long plastic cylinder, at radius $R=20.0 \mathrm{cm}$ from the cylinder's central axis. Figure $29-58 a$ shows, in cross section, the cylinder
and wire 1 but not wire $2 .$ With wire 2 fixed in place, wire 1 is moved around the cylinder, from angle $\theta_{1}=0^{\circ}$ to angle $\theta_{1}=180^{\circ},$ through
the first and second quadrants of the $x y$ coordinate system. The net
magnetic field $\vec{B}$ at the center of the cylinder is measured as a function of $\theta_{1}$ . Figure $29-58 b$ gives the $x$ component $B_{x}$ of that field as a function of $\theta_{1}$ (the vertical scale is set by $B_{x s}=6.0 \mu \mathrm{T} ),$ and Fig.
$29-58 c$ gives the $y$ component $B_{y}$ (the vertical scale is set by $B_{y s}=4.0$ \muT). (a) At what angle $\theta_{2}$ is wire 2 located? What are the (b) size and (c) direction (into or out of the page) of the current in wire 1 and the
(d) size and ( e) direction of the current in wire 2?

Dading Chen
Dading Chen
Numerade Educator
01:00

Problem 31

In Fig. $29-59$ , length $a$ is 4.7 $\mathrm{cm}$ (short) and current $i$ is 13 A. What are
the (a) magnitude and (b) direction
(into or out of the page) of the magnetic
field at point $P ?$

Salamat Ali
Salamat Ali
Numerade Educator
02:28

Problem 32

The current-carrying wire loop in Fig. $29-60 a$ lies all in one plane
and consists of a semicircle of radius 10.0
$\mathrm{cm},$ a smaller semicircle with the same
center, and two radial lengths. The
smaller semicircle is rotated out of that
plane by angle $\theta$ , until it is perpendicular to the plane (Fig. $29-60 b )$ . Figure $29-60 c$ gives the magnitude of the
net magnetic field at the center of curvature versus angle $\theta$ . The
vertical scale is set by $B_{a}=10.0 \mu$ and $B_{b}=12.0 \mu$ . What is
the radius of the smaller semicircle?

Salamat Ali
Salamat Ali
Numerade Educator
02:23

Problem 33

Figure $29-61$ shows a cross section of a long thin ribbon
of width $w=4.91 \mathrm{cm}$ that is carrying
a uniformly distributed total current
$i=4.61 \mu \mathrm{A}$ into the page. In unit-vector notation, what is the magnetic field $\vec{B}$ at a point $P$ in the plane of the
ribbon at a distance $d=2.16 \mathrm{cm}$ from
its edge? (Hint: Imagine the ribbon as
being constructed from many long,
thin, parallel wires.)

Salamat Ali
Salamat Ali
Numerade Educator
06:48

Problem 34

Figure $29-62$ shows, in cross section, two long straight wires
held against a plastic cylinder of ra-
dius 20.0 $\mathrm{cm} .$ Wire 1 carries current
$i_{1}=60.0 \mathrm{mA}$ out of the page and is
fixed in place at the left side of the
cylinder. Wire 2 carries current $i_{2}=$ 40.0 $\mathrm{mA}$ out of the page and can be
moved around the cylinder. At what
(positive) angle $\theta_{2}$ should wire 2 $\mathrm{be}$
positioned such that, at the origin,
the net magnetic field due to the two
currents has magnitude 80.0 $\mathrm{nT} ?$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:44

Problem 35

Figure $29-63$ shows wire 1 in cross section; the wire is long and straight, carries a current of 4.00 $\mathrm{mA}$ out of the page, and is
at distance $d_{1}=2.40 \mathrm{cm}$ from a surface. Wire $2,$ which is parallel
to wire 1 and also long, is at horizontal distance $d_{2}=5.00 \mathrm{cm}$ from wire 1 and carries a current of 6.80 $\mathrm{mA}$ into the page. What
is the $x$ component of the magnetic force per unit length on wire
2 due to wire 1$?$

Salamat Ali
Salamat Ali
Numerade Educator
01:42

Problem 36

In Fig. $29-64$ , five long parallel wires in an $x y$ plane are separated by
distance $d=8.00 \mathrm{cm},$ have lengths of
$10.0 \mathrm{m},$ and carry identical currents
of 3.00 A out of the page. Each wire
experiences a magnetic force due to
the currents in the other wires. In
unit-vector notation, what is the net magnetic force on (a) wire $1,$ (b) wire $2,(\mathrm{c})$ wire $3,(\mathrm{d})$ wire $4,$ and $(\mathrm{e})$
wire 5 ?

Penny Riley
Penny Riley
Numerade Educator
17:13

Problem 37

In Fig. $29-57,$ four long straight wires are perpendicular to the page, and their cross sections form a square of edge length
$a=13.5 \mathrm{cm} .$ Each wire carries $7.50 \mathrm{A},$ and the currents are out of the page in wires 1 and 4 and into the page in wires 2 and $3 .$ In unit-
vector notation, what is the net magnetic force per meter of wire
length on wire 4 ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
07:59

Problem 38

Figure $29-65 a$ shows, in cross section, three currentcarrying wires that are long, straight, and parallel to one another.
Wires 1 and 2 are fixed in place on an $x$ axis, with separation $d$ .
Wire 1 has a current of $0.750 \mathrm{A},$ but the direction of the current is not given. Wire $3,$ with a current of 0.250 A out of the page, can be
moved along the $x$ axis to the right of wire $2 .$ As wire 3 is moved,
the magnitude of the net magnetic force $\overline{F}_{2}$ on wire 2 due to the
currents in wires 1 and 3 changes. The $x$ component of that force is $F_{2 x}$ and the value per unit length of wire 2 is $F_{2 x} / L_{2} .$ Figure $29-65 b$
gives $F_{2 x} / L_{2}$ versus the position $x$ of wire $3 .$ The plot has an asymptote $F_{2 x} / L_{2}=-0.627 \mu \mathrm{N} / \mathrm{m}$ as $x \rightarrow \infty .$ The horizontal scale is set by
$x_{s}=12.0 \mathrm{cm} .$ What are the (a) size and (b) direction (into or out of
the page) of the current in wire 2 ?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:55

Problem 39

In Fig. $29-64,$ five long parallel wires in an $x y$ plane are separated by distance $d=50.0 \mathrm{cm} .$ The currents into the page
are $i_{1}=2.00 \mathrm{A}, \quad i_{3}=0.250 \mathrm{A}, \quad i_{4}=4.00 \mathrm{A},$ and $\quad i_{5}=2.00 \mathrm{A}$
the current out of the page is $i_{2}=4.00 \mathrm{A}$ . What is the magnitude of the net force per unit length acting on wire 3 due to the currents in the other wires?

Salamat Ali
Salamat Ali
Numerade Educator
02:02

Problem 40

In Fig. $29-57,$ four long straight wires are perpendicular to the page, and their cross sections form a square of edge length
$a=8.50 \mathrm{cm} .$ Each wire carries 15.0 $\mathrm{A}$ , and all the currents are out of
the page. In unit-vector notation, what is the net magnetic force per
meter of wire length on wire 1?

Salamat Ali
Salamat Ali
Numerade Educator
01:36

Problem 41

In Fig. $29-66,$ a long straight wire carries a current $i_{1}=$
30.0 $\mathrm{A}$ and a rectangular loop carries current $i_{2}=20.0$ A. Take the dimensions to be $a=1.00 \mathrm{cm}, b=$
$8.00 \mathrm{cm},$ and $L=30.0 \mathrm{cm} .$ In unit-
vector notation, what is the net force
on the loop due to $i_{1} ?$

Salamat Ali
Salamat Ali
Numerade Educator
00:58

Problem 42

In a particular region there is a uniform current density of 15
$\mathrm{A} / \mathrm{m}^{2}$ in the positive $z$ direction. What is the value of $\ \vec{B} \cdot d \vec{s}$ when that line integral is calculated
along a closed path consisting of the three straight-line segments from $(x, y, z)$ coordinates $(4 d, 0,0)$ to
$(4 d, 3 d, 0)$ to $(0,0,0)$ to $(4 d, 0,0),$ where $d=$
20 $\mathrm{cm} ?$

Salamat Ali
Salamat Ali
Numerade Educator
05:12

Problem 43

Figure $29-67$ shows a cross section across a diameter of a long cylindrical conductor of radius $a=2.00 \mathrm{cm}$ carrying uniform
current 170 $\mathrm{A} .$ What is the magnitude of the
current's magnetic field at radial distance (a) $0,$ (b) $1.00 \mathrm{cm},$ (c) 2.00 $\mathrm{cm}$ (wire's surface), 20 $\mathrm{cm} ?$

Kumar Vaibhav
Kumar Vaibhav
Numerade Educator
06:33

Problem 44

Figure $29-68$ shows two closed paths wrapped around two conducting loops carrying currents $i_{1}=5.0 \mathrm{A}$ and $i_{2}=3.0 \mathrm{A} .$ What is the value of
the integral $\oint \vec{B} \cdot d \vec{s}$ for $(\mathrm{a})$ path
1 and $(\mathrm{b})$ path 2 ?

Stanley Enemuo
Stanley Enemuo
Numerade Educator
01:44

Problem 45

Each of the eight conductors in Fig. $29-69$ carries 2.0 A of current into or out of the page. Two paths are indicated for the line
integral $\oint \vec{B} \cdot d \vec{s}$ . What is the value of the integral for (a) path 1 and
(b) path 2 ?

Salamat Ali
Salamat Ali
Numerade Educator
00:50

Problem 46

Eight wires cut the page perpendicularly at the points shown in
Fig. $29-70 .$ A wire labeled with the
integer $k(k=1,2, \ldots, 8)$ carries
the current $k i,$ where $i=4.50 \mathrm{mA}$
For those wires with odd $k,$ the current is out of the page; for those with even $k,$ it is into the page.
Evaluate $\oint \vec{B} \cdot d \vec{s}$ along the closed
path indicated and in the direction
shown.

Salamat Ali
Salamat Ali
Numerade Educator
02:00

Problem 47

The current density $\vec{J}$ inside a long, solid, cylindrical wire of radius $a=3.1 \mathrm{mm}$ is in the direction of the central axis, and its magnitude varies linearly with
radial distance $r$ from the axis according to $J=J_{0} r / a,$ where $J_{0}=$ 310 $\mathrm{A} / \mathrm{m}^{2} .$ Find the magnitude of the magnetic field at $(\mathrm{a}) r=0,$ (b) $r=a / 2,$ and
(c) $r=a$ .

Salamat Ali
Salamat Ali
Numerade Educator
01:54

Problem 48

In Fig. $29-71,$ a long circular pipe with outside radius $R=2.6 \mathrm{cm}$ carries a
(uniformly distributed) current i=
8.00 $\mathrm{mA}$ into the page. A wire runs parallel
to the pipe at a distance of 3.00$R$ from center to center. Find the (a) magnitude and
(b) direction (into or out of the page) of the
current in the wire such that the net magnetic field at point $P$ has the same magnitude as the net magnetic field at the center
of the pipe but is in the opposite direction.

Salamat Ali
Salamat Ali
Numerade Educator
01:26

Problem 49

A toroid having a square cross section, 5.00 $\mathrm{cm}$ on a side, and an inner radius of 15.0 $\mathrm{cm}$ has 500 turns and carries a current of 0.800
A. (It is made up of a square solenoid- instead of a round one as in
Fig. $29-17-$ bent into a doughnut shape.) What is the magnetic field
inside the toroid at (a) the inner radius and (b) the outer radius?

Salamat Ali
Salamat Ali
Numerade Educator
00:40

Problem 50

A solenoid that is 95.0 $\mathrm{cm}$ long has a radius of 2.00 $\mathrm{cm}$ and a winding of 1200 turns; it carries a current of 3.60 A. Calculate
the magnitude of the magnetic field inside the solenoid.

Salamat Ali
Salamat Ali
Numerade Educator
00:41

Problem 51

A 200 -turn solenoid having a length of 25 $\mathrm{cm}$ and a diameter of 10 $\mathrm{cm}$ carries a current of 0.29 A. Calculate the magnitude of the
magnetic field $\vec{B}$ inside the solenoid.

Salamat Ali
Salamat Ali
Numerade Educator
01:46

Problem 52

A solenoid 1.30 $\mathrm{m}$ long and 2.60 $\mathrm{cm}$ in diameter carries a current of 18.0 A. The magnetic field inside the solenoid is 23.0 $\mathrm{mT}$ .
Find the length of the wire forming the solenoid.

Salamat Ali
Salamat Ali
Numerade Educator
01:31

Problem 53

A long solenoid has 100 turns/cm and carries current $i . \mathrm{An}$ electron moves within the solenoid in a circle of radius 2.30 $\mathrm{cm}$
perpendicular to the solenoid axis. The speed of the electron is
0.0460$c(c=$ speed of light). Find the current $i$ in the solenoid.

Salamat Ali
Salamat Ali
Numerade Educator
01:24

Problem 54

An electron is shot into one end of a solenoid. As it enters the uniform magnetic field within the solenoid, its speed
is 800 $\mathrm{m} / \mathrm{s}$ and its velocity vector makes an angle of $30^{\circ}$ with the
central axis of the solenoid. The solenoid carries 4.0 $\mathrm{A}$ and has
8000 turns along its length. How many revolutions does the electron make along its helical path within the solenoid by the time it
emerges from the solenoid's opposite end? (In a real solenoid,
where the field is not uniform at the two ends, the number of revolutions would be slightly less than the answer here.)

Salamat Ali
Salamat Ali
Numerade Educator
02:10

Problem 55

A long solenoid with 10.0 turns/cm and a radius of 7.00 $\mathrm{cm}$ carries a current of 20.0 $\mathrm{mA}$ . A current of 6.00 $\mathrm{A}$
exists in a straight conductor located along the central axis of the solenoid. (a) At what radial distance from the axis will the direction of the resulting magnetic field be at $45.0^{\circ}$ to
the axial direction? (b) What is the
magnitude of the magnetic field there?

Salamat Ali
Salamat Ali
Numerade Educator
03:40

Problem 56

Figure $29-72$ shows an arrangement known as a Helmholtz coil. It
consists of two circular coaxial coils,
each of 200 turns and radius
$R=25.0 \mathrm{cm},$ separated by a distance $s=R .$ The two coils carry equal currents $i=12.2 \mathrm{mA}$ in the same direction. Find the magnitude of the net magnetic field at $P,$ midway
between the coils.

Abhishek Kumar
Abhishek Kumar
Numerade Educator
01:46

Problem 57

A student makes a short electromagnet by winding 300 turns of wire around a wooden cylinder of diameter $d=5.0 \mathrm{cm}$ .
The coil is connected to a battery producing a current of 4.0 $\mathrm{A}$ in
the wire. (a) What is the magnitude of the magnetic dipole moment of this device? (b) At what axial distance $z \geqslant d$ will the magnetic
field have the magnitude 5.0$\mu$ T (approximately one-tenth that of
Earth's magnetic field)?

Salamat Ali
Salamat Ali
Numerade Educator
02:13

Problem 58

Figure $29-73 a$ shows a length of wire carrying a current $i$ and bent into
a circular coil of one turn. In Fig. $29-$
73$b$ the same length of wire has been
bent to give a coil of two turns, each of
half the original radius. (a) If $B_{a}$ and $B_{b}$
are the magnitudes of the magnetic fields at the centers of the two coils,
what is the ratio $B_{b} / B_{a} ?$ (b) What is the
ratio $\mu_{b} / \mu_{a}$ of the dipole moment magnitudes of the coils?

Salamat Ali
Salamat Ali
Numerade Educator
00:56

Problem 59

What is the magnitude of the magnetic dipole moment $\overrightarrow{\mu}$ of the solenoid described in Problem 51$?$

Salamat Ali
Salamat Ali
Numerade Educator
03:21

Problem 60

In Fig. $29-74 a,$ two circular loops, with different currents but the same radius of $4.0 \mathrm{cm},$ are centered on a $y$ axis.
They are initially separated by distance $L=3.0 \mathrm{cm},$ with loop 2 positioned at the origin of the axis. The currents in the two loops produce a net magnetic field at the origin, with $y$ component $B_{y} .$ That component is to be measured as loop 2 is gradually moved in the
positive direction of the $y$ axis. Figure $29-74 b$ gives $B_{y}$ as a function of the position $y$ of loop $2 .$ The curve approaches an asymptote of
$B_{y}=7.20 \mu \mathrm{T}$ as $y \rightarrow \infty .$ The horizontal scale is set by $y_{s}=10.0 \mathrm{cm} .$
What are (a) current $i_{1}$ in loop 1 and (b) current $i_{2}$ in loop 2 ?

Salamat Ali
Salamat Ali
Numerade Educator
01:30

Problem 61

A circular loop of radius 12 $\mathrm{cm}$ carries a current of 15 $\mathrm{A} . \mathrm{A}$ flat coil of radius $0.82 \mathrm{cm},$ having 50 turns and a current of $1.3 \mathrm{A},$ is
concentric with the loop. The plane of the loop is perpendicular to the plane of the coil. Assume the
loop's magnetic field is uniform
across the coil. What is the magnitude of (a) the magnetic field
produced by the loop at its center
and (b) the torque on the coil due
to the loop?

Salamat Ali
Salamat Ali
Numerade Educator
02:31

Problem 62

In Fig. $29-75,$ current $i=$ 56.2 $\mathrm{mA}$ is set up in a loop having
two radial lengths and two semicircles of radii $a=5.72 \mathrm{cm}$ and $b=9.36 \mathrm{cm}$ with a common center
$P .$ What are the $($ a magnitude and (b) direction (into or out of the page $)$ of the magnetic field at $P$ and the $(\mathrm{c})$ magnitude and
(d) direction of the loop's magnetic dipole moment?

Salamat Ali
Salamat Ali
Numerade Educator
02:02

Problem 63

In Fig. $29-76,$ a conductor carries 6.0 $\mathrm{A}$ along the closed path
abcdefgha running along 8 of the 12
edges of a cube of edge length 10 $\mathrm{cm} .$
(a) Taking the path to be a combination of three square current loops $(b c f g b, a b g h a,$ and $c d e f c),$ find the net
magnetic moment of the path in unit-
vector notation. (b) What is the magnitude of the net magnetic field at the
$x y z$ coordinates of $(0,5.0 \mathrm{m}, 0) ?$

Salamat Ali
Salamat Ali
Numerade Educator
01:34

Problem 64

In Fig. $29-77,$ a closed loop carries current $i=200 \mathrm{mA}$ . The loop consists
of two radial straight wires and two
concentric circular arcs of radii 2.00 $\mathrm{m}$
and 4.00 $\mathrm{m} .$ The angle $\theta$ is $\pi / 4$ rad. What are the (a) magnitude and (b) direction
(into or out of the page) of the net
magnetic field at the center of
curvature $P ?$

Salamat Ali
Salamat Ali
Numerade Educator
00:27

Problem 65

A cylindrical cable of radius 8.00 mm carries a current of 25.0 $\mathrm{A}$ , uniformly spread over its cross-sectional area. At what distance from
the center of the wire is there a point within the wire where the
magnetic field magnitude is 0.100 $\mathrm{mT}$ ?

Salamat Ali
Salamat Ali
Numerade Educator
02:27

Problem 66

Two long wires lie in an xy plane, and each carries a current in the positive direction of the $x$ axis. Wire 1 is at $y=10.0 \mathrm{cm}$ and carries 6.00 $\mathrm{A}$ ; wire 2 is at $y=5.00 \mathrm{cm}$ and carries 10.0 $\mathrm{A}$ . (a) In unit-
vector notation, what is the net magnetic field $\vec{B}$ at the origin? (b)
At what value of $y$ does $\vec{B}=0 ?$ (c) If the current in wire 1 is reversed, at what value of $y$ does $\vec{B}=0 ?$

Salamat Ali
Salamat Ali
Numerade Educator
02:56

Problem 67

Two wires, both of length $L,$ are formed into a circle and a square, and each carries current $i$ Show that the square produces a
greater magnetic field at its center than the circle produces at its
center.

Salamat Ali
Salamat Ali
Numerade Educator
01:49

Problem 68

A long straight wire carries a current of 50 A. An electron, traveling at $1.0 \times 10^{7} \mathrm{m} / \mathrm{s}$ , is 5.0 $\mathrm{cm}$ from the wire. What is the magnitude of the magnetic force on the electron if the electron velocity
is directed (a) toward the wire, (b) parallel to the wire in the direction of the current, and (c) perpendicular to the two directions defined by (a) and (b)?

Salamat Ali
Salamat Ali
Numerade Educator
02:34

Problem 69

Three long wires are parallel to a $z$ axis, and each carries a current of
10 $\mathrm{A}$ in the positive $z$ direction. Their
points of intersection with the $x y$
plane form an equilateral triangle
with sides of $50 \mathrm{cm},$ as shown in Fig.29- $78 .$ A fourth wire (wire $b )$ passes
through the midpoint of the base of
the triangle and is parallel to the
other three wires. If the net magnetic
force on wire $a$ is zero, what are the (a) size and (b) direction $(+z$ or $-z)$ of the current in wire $b ?$

Salamat Ali
Salamat Ali
Numerade Educator
01:20

Problem 70

Figure $29-79$ shows a closed loop with current $i=2.00$ A. The loop consists of a half-circle of radius 4.00 $\mathrm{m}$
two quarter-circle each of radius 2.00
$\mathrm{m},$ and three radial straight wires.
What is the magnitude of the net magnetic field at the common center of the
circular sections?

Salamat Ali
Salamat Ali
Numerade Educator
00:37

Problem 71

A 10 -gauge bare copper wire $(2.6 \mathrm{mm}$ in diameter) can carry a current of 50 $\mathrm{A}$ without overheating. For this current, what is the magnitude of the magnetic field at
the surface of the wire?

Salamat Ali
Salamat Ali
Numerade Educator
00:38

Problem 72

A long vertical wire carries an unknown current. Coaxial with the wire is a long, thin, cylindrical conducting surface that carries a
current of 30 $\mathrm{mA}$ upward. The cylindrical surface has a radius of
3.0 $\mathrm{mm} .$ If the magnitude of the magnetic field at a point 5.0 $\mathrm{mm}$
from the wire is $1.0 \mu \mathrm{T},$ what are the (a) size and (b) direction of
the current in the wire?

Salamat Ali
Salamat Ali
Numerade Educator
07:49

Problem 73

Figure $29-80$ shows a cross section of a long cylindrical conductor of radius $a=4.00 \mathrm{cm}$
containing a long cylindrical hole of radius
$b=1.50 \mathrm{cm} .$ The central axes of the cylinder
and hole are parallel and are distance $d=$
2.00 $\mathrm{cm}$ apart; current $i=5.25 \mathrm{A}$ is uniformly distributed over the tinted area. (a) What is
the magnitude of the magnetic field at the
center of the hole? (b) Discuss the two special cases $b=0$ and $d=0 .$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:20

Problem 74

The magnitude of the magnetic field at a point 88.0 $\mathrm{cm}$ from the central axis of a long straight wire is 7.30$\mu \mathrm{T}$ . What is the current in the wire?

Salamat Ali
Salamat Ali
Numerade Educator
03:24

Problem 75

Figure $29-81$ shows a wire segment of length $\Delta s=3.0 \mathrm{cm},$ centered at the origin, carrying current
$i=2.0$ A in the positive $y$ direction $($ as
part of some complete circuit). To calculate the magnitude of the magnetic
field $\vec{B}$ produced by the segment at a $a$ point several meters from the origin,
we can use $B=\left(\mu_{0} / 4 \pi\right) i \Delta s(\sin \theta) / r^{2}$ as
the Biot- Savart law. This is because $r$
and $\theta$ are essentially constant over the segment. Calculate $\vec{B}$ (in unit-vector notation) at the $(x, y, z)$ coordinates (a) $(0,0,5.0 \mathrm{m}),(\mathrm{b})(0,6.0 \mathrm{m}, 0),(\mathrm{c})(7.0 \mathrm{m}, 7.0 \mathrm{m}, 0),$ and $(\mathrm{d})$
$(-3.0 \mathrm{m},-4.0 \mathrm{m}, 0)$

Salamat Ali
Salamat Ali
Numerade Educator
01:55

Problem 76

Figure $29-82$ shows, in cross section, two long parallel wires spaced by distance $d=10.0 \mathrm{cm}$ ; each carries 100 $\mathrm{A}$ , out of the
page in wire $1 .$ Point $P$ is on a perpendicular bisector of the line connecting the wires. In unit-vector notation, what is the net magnetic
field at $P$ if the current in wire 2 is (a) out of the page and (b) into the
page?

Salamat Ali
Salamat Ali
Numerade Educator
03:19

Problem 77

In Fig. $29-83,$ two infinitely long wires carry equal currents $i .$ Each follows a $90^{\circ}$
arc on the circumference of the same circle of radius $R .$ Show that the magneticfield $\vec{B}$ at the center of the circle is the
same as the field $\vec{B}$ a distance $R$ below an
infinite straight wire carrying a current $i$
to the left.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:33

Problem 78

A long wire carrying 100 $\mathrm{A}$ is perpendicular to the magnetic field lines of
a uniform magnetic field of magnitude 5.0 $\mathrm{mT} .$ At what distance from the wire is the net magnetic field
equal to zero?

Salamat Ali
Salamat Ali
Numerade Educator
02:16

Problem 79

A long, hollow, cylindrical conductor (with inner radius 2.0 $\mathrm{mm}$ and outer radius 4.0 $\mathrm{mm}$ ) carries a current of 24 $\mathrm{A}$ distributed uniformly across its cross section. A long thin wire that is coaxial with the cylinder carries a current of 24 $\mathrm{A}$ in the opposite direction. What is the magnitude of the magnetic field (a) 1.0 $\mathrm{mm}$ ,
(b) $3.0 \mathrm{mm},$ and $(\mathrm{c}) 5.0 \mathrm{mm}$ from the central axis of the wire and
cylinder?

Salamat Ali
Salamat Ali
Numerade Educator
01:05

Problem 80

A long wire is known to have a radius greater than 4.0 $\mathrm{mm}$ and to carry a current that is uniformly distributed over its cross section.
The magnitude of the magnetic field due to that current is 0.28 $\mathrm{mT}$
at a point 4.0 $\mathrm{mm}$ from the axis of the wire, and 0.20 $\mathrm{mT}$ at a point 10
$\mathrm{mm}$ from the axis of the wire. What is the radius of the wire?

Salamat Ali
Salamat Ali
Numerade Educator
08:34

Problem 81

Figure $29-84$ shows a cross section of an infinite conducting
sheet carrying a current per unit
$x$ -length of $\lambda ;$ the current emerges
perpendicularly out of the page. (a)
Use the Biot- Savart law and symmetry to show that for all points $P$
above the sheet and all points $P^{\prime}$ below it, the magnetic field $\vec{B}$ is parallel to the sheet and directed as
shown. (b) Use Ampere's law to prove that $B=\frac{1}{2} \mu_{0} \lambda$ at all
points $P$ and $P^{\prime}$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:16

Problem 82

Figure $29-85$ shows, in cross section, two long parallel wires that are
separated by distance $d=18.6 \mathrm{cm} .$
Each carries $4.23 \mathrm{A},$ out of the page
in wire 1 and into the page in wire $2 .$
In unit-vector notation, what is the
net magnetic field at point $P$ at distance $R=34.2 \mathrm{cm},$ due to the two
currents?

Salamat Ali
Salamat Ali
Numerade Educator
06:46

Problem 83

In unit-vector notation, what is the magnetic field at point
$P$ in Fig. $29-86$ if $i=10 \mathrm{A}$ and $a=$
8.0 $\mathrm{cm} ?$ (Note that the wires are
not long.)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:28

Problem 84

Three long wires all lie in an $x y$ plane parallel to the $x$ axis. They are
spaced equally, 10 $\mathrm{cm}$ apart. The two
outer wires each carry a current of
5.0 $\mathrm{A}$ in the positive $x$ direction.
What is the magnitude of the force
on a 3.0 $\mathrm{m}$ section of either of the
outer wires if the current in the center wire is 3.2 $\mathrm{A}$ (a) in the positive $x$ direction and (b) in the negative $x$ direction?

Salamat Ali
Salamat Ali
Numerade Educator
06:07

Problem 85

Figure $29-87$ shows a cross section of a hollow cylindrical conductor of radii a
and $b$ , carrying a uniformly distributed current $i .($ a) Show that the magnetic field magnitude $B(r)$ for the radial distance $r$ in the range
$b < r < a$ is given by
$$
B=\frac{\mu_{0} i}{2 \pi\left(a^{2}-b^{2}\right)} \frac{r^{2}-b^{2}}{r}
$$
(b) Show that when $r=a$ , this equation gives the magnetic field magnitude $B$ at the surface of a long straight
wire carrying current $i$ ; when $r=b,$ it gives zero magnetic field;
and when $b=0,$ it gives the magnetic field inside a solid conductor of radius $a$ carrying current $i .$ (c) Assume that $a=2.0$
$\mathrm{cm}, b=1.8 \mathrm{cm},$ and $i=100 \mathrm{A},$ and then plot $B(r)$ for the range
$0 < r < 6 \mathrm{cm} .$

Abhishek Kumar
Abhishek Kumar
Numerade Educator
04:03

Problem 86

Show that the magnitude of the magnetic field produced at the center of a rectangular loop of wire of length $L$ and width $W$
carrying a current $i,$ is
$$
B=\frac{2 \mu_{0} i}{\pi} \frac{\left(L^{2}+W^{2}\right)^{1 / 2}}{L W}
$$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:17

Problem 87

Figure $29-88$ shows a cross section of a long conducting coaxial cable and gives its
radii $(a, b, c) .$ Equal but opposite currents $i$ are
uniformly distributed in the two conductors.
Derive expressions for $B(r)$ with radial distance $r$ in the ranges (a) $r < c,(\mathrm{b}) c < r < b, (c)
b < r < a,$ and (d) r > a (e) Test these expressions for all the special cases that occur to you. (f) Assume that $a=2.0 \mathrm{cm}, b=1.8 \mathrm{cm}, c=$
$0.40 \mathrm{cm},$ and $i=120 \mathrm{A}$ and plot the function
$B(r)$ over the range $0 < r < 3 \mathrm{cm} .$

Salamat Ali
Salamat Ali
Numerade Educator
03:30

Problem 88

Figure $29-89$ is an idealized schematic drawing of a rail gun. Projectile $P$ sits between two wide rails of circular cross section;a
source of current sends current through the rails and through the
(conducting) projectile (a fuse is not used). (a) Let $w$ be the distance between the rails, $R$ the radius of each rail, and $i$ the current.
Show that the force on the projectile is directed to the right along
the rails and is given approximately by
$$
F=\frac{i^{2} \mu_{0}}{2 \pi} \ln \frac{w+R}{R}
$$
(b) If the projectile starts from the left end of the rails at rest, find
the speed $v$ at which it is expelled at the right. Assume that $i=$
$450 \mathrm{kA}, w=12 \mathrm{mm}, R=6.7 \mathrm{cm}, L=4.0 \mathrm{m},$ and the projectile
mass is 10 $\mathrm{g} .$

Salamat Ali
Salamat Ali
Numerade Educator