Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 29

Magnetic Fields Due to Currents - all with Video Answers

Educators

Chapter Questions

Problem 1

In Fig. 29-24, point $P_{1}$ is at distance $R=24.0 \mathrm{~cm}$ 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 are the (a) magnitude and (b) direction of the magnetic field at $P_{1}$ due to $i$ ? (c) If $R$ is in field at $P_{1}$ due to $i ?$ (c) If $R$ is increased, what happens to the magnitude of the field?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 2

Figure 29-25a shows a length of wire carrying a current $i$ and bent into a circular coil of one turn. In Fig. 29-25b 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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 3

In Fig. 29-24, 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 $i=0.500 \mathrm{~A}$. (Note that the wire is not long.) (a) What is the magnitude of the magnetic field at $P_{2} ?$ (b) If the point of measurement is moved from $P_{2}$ to $P_{1}$, does the field magnitude increase, decrease, or remain the same?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 4

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 $3.00 \%$ ? That is, what $L / R$ gives
$$
\frac{(B \text { from Eq. } 29-4)-(B \text { actual })}{(B \text { actual })}(100 \%)=3.00 \% ?
$$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 5

In Fig. $29-26$, four long straight wires are perpendicular to the page, and their cross sections form a square of edge length $a=40 \mathrm{~cm}$. 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 $12 \mathrm{~A}$. In unit-vector notation, what is the net magnetic field at the square's center?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 6

In Fig. 29-27, part of a long insulated wire carrying current $i=5.78 \mathrm{~mA}$ is bent into a circular section of radius $R=1.54 \mathrm{~cm} .$ 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

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 7

In Fig. 29-26, four long straight wires are perpendicular to the page, and their cross sections form a square of edge length $a=13.5 \mathrm{~cm}$
Figure 29-27 Problem $6 .$
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 1,3 , and 4 and into the page in wire 2 . In unit-vector notation, what is the net magnetic force per meter of wire length on wire 4 ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 8

In Fig. 29-26, four long straight wires are perpendicular to the page, and their cross sections form a square of edge length $a=7.00 \mathrm{~cm}$. 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 ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 9

In Fig. 29-28, length $a$ is $2.3 \mathrm{~cm}$ (short) and current $i$ is $18 \mathrm{~A}$. What are the (a) magnitude and (b) direction (into or out of the page) of the magnetic field at point $P ?$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 10

In Fig. $29-29$, five long parallel wires in an $x y$ plane are separated by distance $d=8.00 \mathrm{~cm}$, have lengths of $20.0 \mathrm{~m}$, and carry identical currents of $3.00 \mathrm{~A}$ 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, (c) wire 3, (d) wire 4, and (e) wire 5 ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 11

In Fig. 29-29, five long parallel wires in an $x y$ plane are separated by distance $d=50.0 \mathrm{~cm}$. The currents into the page are $i_{1}=2.00 \mathrm{~A}$, $i_{3}=0.250 \mathrm{~A}, i_{4}=6.00 \mathrm{~A}$, and $i_{5}=2.00 \mathrm{~A}$; 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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 12

Figure $29-30 a$ 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 29-30 $b$ 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 $x_{s}=40.0 \mathrm{~cm}$. (a) At what value of $x>0$ is $B_{y}$ 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 (c) $i_{1}$ and (d) $i_{2}$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 13

In Fig. 29-31, a conductor carries $2.0 \mathrm{~A}$ along the closed path $a b c d e f g h a$ 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 (bcfgb, abgha, 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 $(0,5.0 \mathrm{~m}, 0)$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 14

Figure 29-32 shows two closed paths wrapped around two conducting loops carrying currents $i_{1}=6.0 \mathrm{~A}$ and $i_{2}=3.0 \mathrm{~A}$. What is the value of the integral $\oint \vec{B} \cdot d \vec{s}$ for (a) path 1 and (b) path 2?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 15

Figure $29-33$ shows wire 1 in cross section; the wire is long and straight, carries a current of $2.50 \mathrm{~mA}$ out of the page, and is at distance $d_{1}=4.00 \mathrm{~cm}$ 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 ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 16

An electron is shot into one end of a solenoid. As it enters the uniform magnetic field within the solenoid, its speed is $500 \mathrm{~m} / \mathrm{s}$ 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.)

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 17

A toroid having a square cross section, $5.00 \mathrm{~cm}$ on a side, and an inner radius of $19.0 \mathrm{~cm}$ has 460 turns and carries a current of $0.400 \mathrm{~A}$. (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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 18

Figure 29-34 shows an arrangement known as Helmholtz coil. It consists of two circular coaxial coils, each of 200 turns and radius $R=20.0 \mathrm{~cm}$, separated by a distance $s=R$. The two coils carry equal currents $i=20.2 \mathrm{~mA}$ in the same direction. Find the magnitude of the net magnetic field at $P$, midway between the coils.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 19

Figure 29-35 shows a cross section of a long thin ribbon of width $w=6.20 \mathrm{~cm}$ 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 $d=1.61 \mathrm{~cm}$ from its edge? (Hint: Imagine the ribbon as being constructed from many long, thin, parallel wires.)

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 20

A solenoid $1.30 \mathrm{~m}$ long and
$2.60 \mathrm{~cm}$ in diameter carries a current of $22.0 \mathrm{~A}$. The magnetic field inside the solenoid is $23.0 \mathrm{mT}$. Find the length of the wire forming the solenoid.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 21

In Fig. 29-36, a long straight wire carries a current $i_{1}=30.0 \mathrm{~A}$ and a rectangular loop carries current $i_{2}=20.0 \mathrm{~A}$. Take the dimensions to be $a=1.00 \mathrm{~cm}, b=8.00 \mathrm{~cm}$, and $L=20.0 \mathrm{~cm}$. In unit-vector notation, what is the net force on the loop due to $i_{1}$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 22

Figure 29-37a shows, in cross section, three current-carrying 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 $\vec{F}_{2}$ 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 $29-37 b$ 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 $x_{s}=24.0 \mathrm{~cm}$. What are the (a) size and (b) direction (into or out of the page) of the current in wire $2 ?$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 23

Figure 29-38 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) $6.00 \mathrm{~mm}$, (c) $2.00 \mathrm{~cm}$ (wire's surface), and (d) $5.90 \mathrm{~cm}$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 24

Eight wires cut the page perpendicularly at the points shown in Fig. 29-39. A wire labeled with the integer $k(k=1,2, \ldots, 8)$ carries the current $k i$, where $i=6.00 \mathrm{~mA}$. 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.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 25

Each of the eight conductors
in Fig. 29-40 carries $5.0$ 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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 26

In Fig. 29-41, a long circular pipe with outside radius $R=2.6 \mathrm{~cm}$ carries a (uniformly distributed) current $i=2.12 \mathrm{~mA}$ 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.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 27

A long solenoid with $10.0$ turns/ $\mathrm{cm}$ and a radius of $7.00 \mathrm{~cm}$ carries a current of $35.0 \mathrm{~mA}$. A current of $5.00 \mathrm{~A}$ exists in a straight conductor located along the central axis of the solenoid.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 28

In Fig. 29-42, two semicircular arcs have radii $R_{2}=7.80 \mathrm{~cm}$ and $R_{1}=2.86 \mathrm{~cm}$, 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 $C ?$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 29

A 470 -turn solenoid having a length of $25 \mathrm{~cm}$ and a diameter of $10 \mathrm{~cm}$ carries a current of $0.29 \mathrm{~A}$. Calculate the magnitude of the magnetic field $\vec{B}$ inside the solenoid.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 30

A solenoid that is $95.0 \mathrm{~cm}$ long has a radius of $2.00 \mathrm{~cm}$ and a winding of 1500 turns; it carries a current of $3.60 \mathrm{~A}$. Calculate the magnitude of the magnetic field inside the solenoid.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 31

Figure $29-43$ shows two very long straight wires (in cross section) that each carry a current of $4.00 \mathrm{~A}$ directly out of the page. Distance $d_{1}=6.00 \mathrm{~m}$ and distance $d_{2}=8.00 \mathrm{~m}$. What is the magnitude of the net magnetic field at point $P$, which lies on a perpendicular bisector to the wires?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 32

Figure $29-44 a$ shows an ele- Figure $29-43$ Problem $31 .$
ment of length $d s=1.00 \mu \mathrm{m}$ in a
very long straight wire carrying current. The current in that ele-
ment sets up a differential magnetic field $d \vec{B}$ at points in the
surrounding space. Figure $29-44 b$ gives the magnitude $d B$ of the
field for points $3.5 \mathrm{~cm}$ 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 $d B_{s}=120 \mathrm{pT}$. What is the magnitude of the mag-
netic field set up by the entire wire at perpendicular distance
$3.5 \mathrm{~cm}$ from the wire?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 33

A long solenoid has 123 turns/cm and carries current $i$. 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 $0.0187 c$ ( $c=$ speed of light). Find the current $i$ in the solenoid.

Vysakh M

Numerade Educator

Problem 34

In Fig. 29-45, two long straight wires at separation $d=30.0 \mathrm{~cm}$ carry currents $i_{1}=3.61 \mathrm{~mA}$ and $i_{2}=4.00 i_{1}$ 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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 35

The current density $\vec{J}$ inside a long, solid, cylindrical wire of radius $a=4.5 \mathrm{~mm}$ 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 $J_{0}=420 \mathrm{~A} / \mathrm{m}^{2}$. Find the magnitude of the magnetic field at (a) $r=0$, (b) $r=a / 2$, and (c) $r=a .$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 36

In Fig. 29-46, point $P$ is at perpendicular distance $R=1.50 \mathrm{~cm}$ 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 $(\mathrm{a})$ the greatest contribution to field $\vec{B}$ and (b) $10.0 \%$ of the greatest contribution?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 37

What is the magnitude of the magnetic dipole moment $\vec{\mu}$ of the solenoid described in Problem 29 ?

Vysakh M

Numerade Educator

Problem 38

In a particular region there is a uniform current density of $18 \mathrm{~A} / \mathrm{m}^{2}$ in the positive $z$ direction. What is the value of $\oint \vec{B} \cdot d \vec{s}$ 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 $d=20 \mathrm{~cm} ?$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 39

A circular loop of radius $12 \mathrm{~cm}$ carries a current of $7.2$ A. 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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 40

The current-carrying wire loop in Fig. $29-47 a$ lies all in one plane and consists of a semicircle of radius $25.0 \mathrm{~cm}$, 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. 29-47b). Figure 29-47c gives the magnitude $B$ of the net magnetic field at the center of curvature versus angle $\theta$. The vertical scale is set by $B_{a}=10.0 \mu \mathrm{T}$ and $B_{b}=12.0 \mu \mathrm{T}$. What is the radius of the smaller semicircle?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 41

A surveyor is using a magnetic compass $12.2 \mathrm{~m}$ below a power line in which there is a steady current of $200 \mathrm{~A}$. (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 $20 \mu \mathrm{T}$.

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 42

In Fig. 29-48a, two circular loops, with different currents but the same radius of $2.0 \mathrm{~cm}$, are centered on a $y$ axis. They are initially separated by distance $L=6.0 \mathrm{~cm}$, 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 $29-48 b$ 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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 43

A student makes a short electromagnet by winding 280 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 $3.8 \mathrm{~A}$ 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 $5.0 \mu \mathrm{T}$ (approximately one-tenth that of Earth's magnetic field)?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 44

Figure 29-49 shows, in cross section, two long straight wires held against a plastic cylinder of radius $15.0 \mathrm{~cm}$. 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. wire 2 be positioned such that, at the origin, the net magnetic field due to the two currents has magnitude $80.0 \mathrm{nT}$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 45

In Fig. 29-50, 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 $d_{2}=2.50 \mathrm{~cm}$ from wire $2 ?$ If the current in wire 2 is then Figu reversed, what are the (c) size and (d)

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 46

In Fig. $29-51 a$, wire 1 consists of a circular arc and two radial lengths; it carries current $i_{1}=0.20 \mathrm{~A}$ 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 $29-51 b$ 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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 47

In Fig. 29-52, a current $i=2.2 \mathrm{~A}$
is set up in a long hairpin conductor
formed by bending a wire into a
semicircle of radius $R=8.5 \mathrm{~mm} .$
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 $b$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 48

A current is set up in a wire loop consisting of a semicircle of radius $4.50 \mathrm{~cm}$, a smaller concentric semicircle, and two radial straight lengths, all in the same plane. Figure $29-53 a$ shows the arrange- Figure 29-53 Problem 48. ment but is not drawn to scale. The magnitude of the magnetic field produced at the center of curvature is $47.25 \mu$ T. The smaller semicircle is then flipped over (rotated) until the loop is again entirely in the same plane (Fig. 29-53b). 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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 49

Figure 29-54 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 $i_{2}=3 i_{1}$ and includes a circucurrent $i_{2}=3 i_{1}$ 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) magnitude and (d) direction of $\vec{B}$ if $i_{1}$ is reversed?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 50

Figure 29-55a shows two wires, each carrying a current. Wire 1 consists of a circular arc of radius $R$ and two radial lengths; it carries current $i_{1}=1.5 \mathrm{~A}$ 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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 51

In Fig. 29-56, two long straight wires (shown in cross section) carry the currents $i_{1}=30.0 \mathrm{~mA}$ and $i_{2}=50.0 \mathrm{~mA}$ 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 $\vec{B} 25^{\circ}$ clockwise?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 52

In Fig. 29-57, 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 length $L=13.1 \mathrm{~cm}$. The wire carries $H$ current $i=32.3 \mathrm{~m} \mathrm{~A}$. What are the (a) Figure 2 magnitude and (b) direction (into or magnitude and (b) direction (into or out of the page) of the net magnetic field at the semicircle's center of curvature $C$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 53

Figure 29-58 shows a snapshot of a proton moving at velocity $\vec{v}=(-380 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$ toward a long
straight wire with current $i=470 \mathrm{~mA}$.
At the instant shown, the proton's distance from the wire is $d=2.89 \mathrm{~cm}$.
Figure 29magnetic force on the proton due to the current?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 54

A straight conductor carrying current $i=15 \mathrm{~A}$ splits into identical semicircular arcs as shown in Fig. 29-59. What is the magnetic field at the center $C$ of the resulting circular loop?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 55

One long wire lies along an $x$
Figure 29-59 Problem $54 .$ axis and carries a current of $60 \mathrm{~A}$ 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 (a) the point $(0,2.0 \mathrm{~m}, 0)$ and (b) the point $(2.0 \mathrm{~m}, 4.0 \mathrm{~m}, 0)$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 56

In Fig. $29-60$, 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 $6.00 \mathrm{~mA}$. 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. (a) Through what angle must loop 2 be rotated so that the magnitude of that net field is $200 \mathrm{nT} ?$ (b) What is the least possible magnitude of the net field?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 57

In Fig. 29-61, two circular arcs have radii $a=18.9 \mathrm{~cm}$ 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 $P$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 58

In Fig. 29-62, current $i=56.2 \mathrm{~m} \mathrm{~A}$ is set up in a loop having two radial lengths and two semicircles of radii $a=5.72 \mathrm{~cm}$ and $b=8.57 \mathrm{~cm}$ 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 (c) magnitude and (d) direction of the loop's magnetic dipole moment?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 59

A wire with current $i=9.50 \mathrm{~A}$ is shown in Fig. 29-63. 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 the magnetic field is $B=0$ at the circle's center, what is $\theta$ ?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 60

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 $29-64 a$ 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 $29-64 b$ gives the $x$ component $B_{x}$ of that field as a function of $\theta_{1}$ (the vertical scale is set by $\left.B_{x s}=6.0 \mu \mathrm{T}\right)$, and Fig. $29-64 c$ gives the $y$ component $B_{y}$ (the vertical scale is set by $B_{y s}=4.0 \mu \mathrm{T}$ ). (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

Numerade Educator

Problem 61

Two long straight wires are parallel and $16 \mathrm{~cm}$ apart. They are to carry equal currents such that the magnetic field at a point halfway between them has magnitude $450 \mu \mathrm{T}$. (a) Should the currents be in the same or opposite directions? (b) How much current is needed?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 62

Figure 29-65 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 1 along the $x$ axis in order to rotate $\vec{B}$ counterclockwise by $50^{\circ}$ ? (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?

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 63

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 $2.0 \mathrm{~cm}$ above a long, straight, horizontal wire that carries a constant current. What are the (a) magnitude and (b) direction of the current?

Amit Srivastava

Amit Srivastava

Numerade Educator