Physics for Scientists and Engineers with Modern Physics

Paul Tipler, Gene Mosca

Chapter 30

Sources of the Magnetic Field - all with Video Answers

Educators

+ 4 more educators

Chapter Questions

Problem 1

In Niels Bohr's 1913 model of the hydrogen atom, an electron circles the proton at a distance of $5.29 \times 10^{-11} \mathrm{m}$ with a speed of $2.19 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Compute the magnitude of the magnetic field that this motion produces at the location of the proton.

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Numerade Educator

Problem 2

A lightning bolt may carry a current of $1.00 \times 10^{4} \mathrm{A}$ for a short period of time. What is the resulting magnetic field 100 $\mathrm{m}$ from the bolt? Suppose that the bolt extends far above and below the point of observation.

Surjit Tewari

Numerade Educator

Problem 3

(a) A conductor in the shape of a square loop of edge length $\ell=0.400 \mathrm{m}$ carries a current $I=10.0 \mathrm{A}$ as in Fig. P30.3. Calculate the magnitude and direction of the magnetic field at the center of the square. (b) What If? If this conductor is formed into a single circular turn and carries the same current, what is the value of the magnetic field at the center?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 4

Calculate the magnitude of the magnetic field at a point 100 $\mathrm{cm}$ from a long, thin conductor carrying a current of $1.00 \mathrm{A} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 5

Determine the magnetic field at a point $P$ located a distance $x$ from the corner of an infinitely long wire bent at a right angle, as shown in Figure $\mathrm{P} 30.5 .$ The wire carries a steady current $I$ .

CF

Carmen Ferrone

Numerade Educator

Problem 6

A conductor consists of a circular loop of radius $R$ and two straight, long sections, as shown in Figure $\mathrm{P} 30.6 .$ The wire lies in the plane of the paper and carries a current $I .$ Find an expression for the vector magnetic field at the center of the loop.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

The segment of wire in Figure $\mathrm{P} 30.7$ carries a current of $I=5.00 \mathrm{A},$ where the radius of the circular arc is $R=3.00 \mathrm{cm} .$ Determine the magnitude and direction of the magnetic field at the origin.

Donya Dobbin

Numerade Educator

Problem 8

Consider a flat circular current loop of radius $R$ carrying current $I$ . Choose the $x$ axis to be along the axis of the loop, with the origin at the center of the loop. Plot a graph of the ratio of the magnitude of the magnetic field at coordinate $x$ to that at the origin, for $x=0$ to $x=5 R$ . It may be useful to use a programmable calculator or a computer to solve this problem.

Donya Dobbin

Numerade Educator

Problem 9

Two very long, straight, parallel wires carry currents that are directed perpendicular to the page, as in Figure P30.9. Wire 1 carries a current $I_{1}$ into the page (in the - z direction) and passes through the $x$ axis at $x=+a$ . Wire 2 passes through the $x$ axis at $x=-2 a$ and carries an unknown current $I_{2}$ . The total magnetic field at the origin due to the current-carrying wires has the magnitude $2 \mu_{0} I /(2 \pi a) .$ The current $I_{2}$ can have either of two possible values. (a) Find the value of $I_{2}$ with the smaller magnitude, stating it in terms of $I_{1}$ and giving its direction. (b) Find the other possible value of $I_{2}$ .

Vishal Gupta

Numerade Educator

Problem 10

A very long straight wire carries current $I$ . In the middle of the wire a right-angle bend is made. The bend forms an arc of a circle of radius $r,$ as shown in Figure $\mathrm{P} 30.10$ . Determine the magnetic field at the center of the arc.

Prashant Bana

Numerade Educator

Problem 11

One very long wire carries current 30.0 $\mathrm{A}$ to the left along the $x$ axis. A second very long wire carries current 50.0 $\mathrm{A}$ to the right along the line $(y=0.280 \mathrm{m}, z=0)$ .
(a) Where in the plane of the two wires is the total magnetic field equal to zero? (b) A particle with a charge of $-2.00 \mu \mathrm{C}$ is moving with a velocity of 150$\hat{\mathrm{i}} \quad \mathrm{Mm} / \mathrm{s}$ along the line $(y=0.100 \mathrm{m}, z=0)$ Calculate the vector magnetic force acting on the particle. (c) What If? A uniform electric field is applied to allow this particle to pass through this region undeflected. Calculate the required vector electric field.

Keshav Singh

Numerade Educator

Problem 12

Consider the current-carrying loop shown in Figure P30.12, formed of radial lines and segments of circles whose centers are at point $P .$ Find the magnitude and direction of $\mathbf{B}$ at $P .$

Prashant Bana

Numerade Educator

Problem 13

A wire carrying a current $I$ is bent into the shape of an equilateral triangle of side $L .$ (a) Find the magnitude of the magnetic field at the center of the triangle. (b) At a point halfway between the center and any vertex, is the field stronger or weaker than at the center?

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Numerade Educator

Problem 14

Determine the magnetic field (in terms of $I, a,$ and $d )$ at the origin due to the current loop in Figure $\mathrm{P} 30.14$ .

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 15

Two long, parallel conductors carry currents $I_{1}=3.00 \mathrm{A}$ and $I_{2}=3.00 \mathrm{A},$ both directed into the page in Figure $\mathrm{P} 30.15 .$ Determine the magnitude and direction of the resultant magnetic field at $P .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

Two long, parallel conductors, separated by $10.0 \mathrm{cm},$ carry currents in the same direction. The first wire carries current $I_{1}=5.00 \mathrm{A}$ and the second carries $I_{2}=8.00 \mathrm{A}$ . (a) What is the magnitude of the magnetic field created by $I_{1}$ at the location of $I_{2} ?$ (b) What is the force per unit length exerted by $I_{1}$ on $I_{2} ?(c)$ What is the magnitude of the magnetic field created by $I_{2}$ at the location of $I_{1}$ ? (d) What is the force per length exerted by $I_{2}$ on $I_{1} ?$

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Numerade Educator

Problem 17

In Figure $P 30.17$ , the current in the long, straight wire is $I_{1}=5.00 \mathrm{A}$ and the wire lies in the plane of the rectangular loop, which carries the current $I_{2}=10.0 \mathrm{A}$ . The dimensions are $c=0.100 \mathrm{m}, a=0.150 \mathrm{m},$ and $\ell=0.450 \mathrm{m} .$ Find the magnitude and direction of the net force exerted on the loop by the magnetic field created by the wire.

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Numerade Educator

Problem 18

Two long, parallel wires are attracted to each other by a force per unit length of 320$\mu \mathrm{N} / \mathrm{m}$ when they are separated by a vertical distance of 0.500 $\mathrm{m}$ . The current in the upper wire is 20.0 $\mathrm{A}$ to the right. Determine the location of the line in the plane of the two wires along which the total magnetic field is zero.

Donya Dobbin

Numerade Educator

Problem 19

Three long wires (wire $1,$ wire $2,$ and wire 3 ) hang vertically. The distance between wire 1 and wire 2 is $20.0 \mathrm{cm} .$ On the left, wire 1 carries an upward current of 1.50 $\mathrm{A}$ . To the right, wire 2 carries a downward current of 4.00 $\mathrm{A}$ . Wire 3 is located such that when it carries a certain current, each wire experiences no net force. Find (a) the position of wire $3,$ and $(b)$ the magnitude and direction of the current in wire 3 .

Narayan Hari

Numerade Educator

Problem 20

The unit of magnetic flux is named for Wilhelm Weber. The practical-size unit of magnetic field is named for Johann Karl Friedrich Gauss. Both were scientists at Gottingen, Germany. Along with their individual accomplishments, together they built a telegraph in 1833 . It consisted of a battery and switch, at one end of a transmission line 3 $\mathrm{km}$ long, operating an electromagnet at the other end. (André Ampere suggested electrical signaling in $1821 ;$ Samuel Morse built a telegraph line between Baltimore and Washington in $1844 .$ ) Suppose that Weber and Gauss's transmission line was as diagrammed in Figure P30.20. Two long, parallel wires, each having a mass per unit length of $40.0 \mathrm{g} / \mathrm{m},$ are supported in a horizontal plane by strings 6.00 $\mathrm{cm}$ long. When both wires carry the same current $I,$ the wires repel each other so that the angle $\theta$ between the supporting strings is $16.0^{\circ} .$ (a) Are the currents in the same direction or in opposite directions? (b) Find the magnitude of the current.

Vishal Gupta

Numerade Educator

Problem 21

Four long, parallel conductors carry equal currents of $I=5.00$ A. Figure $\mathrm{P} 30.21$ is an end view of the conductors. The current direction is into the page at points $A$ and $B$ (indicated by the crosse) and out of the page at $C$ and $D$ (indicated by the dots). Calculate the magnitude and
direction of the magnetic field at point $P,$ located at the center of the square of edge length $0.200 \mathrm{m} .$

Vishal Gupta

Numerade Educator

Problem 22

A long straight wire lies on a horizontal table and carries a current of 1.20$\mu A$ . In a vacuum, a proton moves parallel to the wire (opposite the current) with a constant speed of $2.30 \times 10^{4} \mathrm{m} / \mathrm{s}$ at a distance $d$ above the wire. Determine the value of $d .$ You may ignore the magnetic field due to the Earth.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 23

Figure $\mathrm{P} 30.23$ is a cross-sectional view of a coaxial cable. The center conductor is surrounded by a rubber layer, which is surrounded by an outer conductor, which is surrounded by another rubber layer. In a particular application, the current in the inner conductor is 1.00 $\mathrm{A}$ out of the page and the current in the outer conductor is 3.00 $\mathrm{A}$ into the page. Determine the magnitude and direction of the magnetic field at points $a$ and $b .$

Donya Dobbin

Numerade Educator

Problem 24

The magnetic field 40.0 $\mathrm{cm}$ away from a long straight wire carrying current 2.00 $\mathrm{A}$ is 1.00$\mu \mathrm{T}$ . (a) At what distance is it 0.100$\mu \mathrm{T}$ ? (b) What If? At one instant, the two conductors in a long household extension cord carry equal 2.00 -A currents in opposite directions. The two wires are 3.00 $\mathrm{mm}$ apart. Find the magnetic field 40.0 $\mathrm{cm}$ away from the middle of the straight cord, in the plane of the two wires. (c) At what distance is it one tenth as large? (d) The center wire in a coaxial cable carries current 2.00 $\mathrm{A}$ in one direction and the sheath around it carries current 2.00 $\mathrm{A}$ in the opposite direction. What magnetic field does the cable create at points outside?

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Numerade Educator

Problem 25

A packed bundle of 100 long, straight, insulated wires forms a cylinder of radius $R=0.500 \mathrm{cm} .$ (a) If each wire carries 2.00 $\mathrm{A}$ , what are the magnitude and direction of the magnetic force per unit length acting on a wire located 0.200 $\mathrm{cm}$ from the center of the bundle? (b) What If? Would a wire on the outer edge of the bundle experience a force greater or smaller than the value calculated in part (a)?

Donya Dobbin

Numerade Educator

Problem 26

The magnetic coils of a tokamak fusion reactor are in the shape of a toroid having an inner radius of 0.700 $\mathrm{m}$ and an outer radius of 1.30 $\mathrm{m}$ . The toroid has 900 turns of large- diameter wire, each of which carries a current of 14.0 $\mathrm{kA}$ . Find the magnitude of the magnetic field inside the toroid along (a) the inner radius and (b) the outer radius.

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Numerade Educator

Problem 27

Consider a column of electric current passing through plasma (ionized gas). Filaments of current within the column are magnetically attracted to one another. They can crowd together to yield a very great current density and a very strong magnetic field in a small region. Sometimes the current can be cut off momentarily by this pinch effect. (In a metallic wire a pinch effect is not important, because the current-carrying electrons repel one another with electric forces.) The pinch effect can be demonstrated by making an empty aluminum can carry a large current parallel to its axis. Let $R$ represent the radius of the can and $I$ the upward current, uniformly distributed over its curved wall. Determine the magnetic field (a) just inside the wall and (b) just outside. (c) Determine the pressure on the wall.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 28

Niobium metal becomes a superconductor when cooled below 9 $\mathrm{K}$ . Its superconductivity is destroyed when thesurface magnetic field exceeds 0.100 $\mathrm{T}$ . Determine the maximum current a 2.00 -mm-diameter niobium wire can carry and remain superconducting, in the absence of any external magnetic field.

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Numerade Educator

Problem 29

A long cylindrical conductor of radius $R$ carries a current $I$ as shown in Figure $\mathrm{P} 30.29$ . The current density $J$ , however, is not uniform over the cross section of the conductor but
is a function of the radius according to $J=b r,$ where $b$ is a constant. Find an expression for the magnetic field $B$ (a) at a distance $r_{1}<R$ and $(b)$ at a distance $r_{2}>R,$ measured from the axis.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 30

In Figure $\mathrm{P} 30.30$ , both currents in the infinitely long wires are in the negative $x$ direction. (a) Sketch the magnetic field pattern in the $y$ z plane. (b) At what distance $d$ along the $z$ axis is the magnetic field a maximum?

Keshav Singh

Numerade Educator

Problem 31

What current is required in the windings of a long solenoid that has 1000 turns uniformly distributed over a length of 0.400 $\mathrm{m}$ , to produce at the center of the solenoid a magnetic field of magnitude $1.00 \times 10^{-4} \mathrm{T}$ ?

Donya Dobbin

Numerade Educator

Problem 32

Consider a solenoid of length $\ell$ and radius $R,$ containing $N$ closely spaced turns and carrying a steady current $I$ . (a) In terms of these parameters, find the magnetic field at a point along the axis as a function of distance $a$ from the end of the solenoid. (b) Show that as $\ell$ becomes very long, $B$ approaches $\mu_{0} N I / 2 \ell$ at each end of the solenoid.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

A single-turn square loop of wire, 2.00 $\mathrm{cm}$ on each edge, carries a clockwise current of 0.200 $\mathrm{A}$ . The loop is inside a solenoid, with the plane of the loop perpendicular to the magnetic field of the solenoid. The solenoid has 30 turns/cm and carries a clockwise current of 15.0 $\mathrm{A}$ . Find the force on each side of the loop and the torque acting on the loop.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 34

Consider the hemispherical closed surface in Figure P30.34. The hemisphere is in a uniform magnetic field that makes an angle $\theta$ with the vertical. Calculate the magnetic flux through (a) the flat surface $S_{1}$ and (b) the hemispherical surface $S_{2}$ .

Shoukat Ali

Other Schools

Problem 35

A cube of edge length $\ell=2.50 \mathrm{cm}$ is positioned as shown in Figure $\mathrm{P} 30.35 .$ A uniform magnetic field given by $\mathbf{B}=(5 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \mathrm{T}$ exists throughout the region. (a) Calculate the flux through the shaded face. (b) What is the total flux through the six faces?

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Numerade Educator

Problem 36

A solenoid 2.50 $\mathrm{cm}$ in diameter and 30.0 $\mathrm{cm}$ long has 300 turns and carries 12.0 $\mathrm{A}$ . (a) Calculate the flux through the surface of a disk of radius 5.00 $\mathrm{cm}$ that is positioned perpendicular to and centered on the axis of the solenoid, as shown in Figure P30.36a. (b) Figure P30.36b shows an enlarged end view of the same solenoid. Calculate the flux through the blue area, which is defined by an annulus that has an inner radius of 0.400 $\mathrm{cm}$ and outer radius of $0.800 \mathrm{cm} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 37

A $0.100-\mathrm{A}$ current is charging a capacitor that has square plates 5.00 $\mathrm{cm}$ on each side. The plate separation is 4.00 $\mathrm{mm}$ . Find $(\mathrm{a})$ the time rate of change of electric flux between the plates and $(\mathrm{b})$ the displacement current between the plates.

Narayan Hari

Numerade Educator

Problem 38

A 0.200 -A current is charging a capacitor that has circular plates 10.0 $\mathrm{cm}$ in radius. If the plate separation is $4.00 \mathrm{mm},$ (a) what is the time rate of increase of electric field between the plates? (b) What is the magnetic field between the plates 5.00 $\mathrm{cm}$ from the center?

Keshav Singh

Numerade Educator

Problem 39

In Bohr's 1913 model of the hydrogen atom, the electron is in a circular orbit of radius $5.29 \times 10^{-11} \mathrm{m}$ and its speed is $2.19 \times 10^{6} \mathrm{m} / \mathrm{s}$ . (a) What is the magnitude of the magnetic moment due to the electron's motion? (b) If the electron moves in a horizontal circle, counterclockwise as seen from above, what is the direction of this magnetic moment vector?

Donya Dobbin

Numerade Educator

Problem 40

A magnetic field of 1.30 $\mathrm{T}$ is to be set up in an iron-core toroid. The toroid has a mean radius of $10.0 \mathrm{cm},$ and magnetic permeability of $5000 \mu_{0} .$ What current is required if the winding has 470 turns of wire? The thickness of the iron ring is small compared to $10 \mathrm{cm},$ so the field in the material is nearly uniform.

Rashmi Sinha

Numerade Educator

Problem 41

A toroid with a mean radius of 20.0 $\mathrm{cm}$ and 630 turns (see Fig. 30.30 ) is filled with powdered steel whose magnetic susceptibility $\chi$ is $100 .$ The current in the windings is 3.00 A. Find $B$ (assumed uniform) inside the toroid.

Rashmi Sinha

Numerade Educator

Problem 42

A particular paramagnetic substance achieves 10.0$\%$ of its saturation magnetization when placed in a magnetic field of 5.00 $\mathrm{T}$ at a temperature of 4.00 $\mathrm{K}$ . The density of magnetic atoms in the sample is $8.00 \times 10^{27}$ atoms/m $^{3}$ , and the magnetic moment per atom is 5.00 Bohr magnetons. Calculate the Curie constant for this substance.

Aja S

Numerade Educator

Problem 43

Calculate the magnetic field strength $H$ of a magnetized substance in which the magnetization is $0.880 \times 10^{6} \mathrm{A} / \mathrm{m}$ and the magnetic field has magnitude $4.40 \mathrm{T} .$

Donya Dobbin

Numerade Educator

Problem 44

At saturation, when nearly all of the atoms have their magnetic moments aligned, the magnetic field in a sample of iron can be 2.00 T. If each electron contributes a magnetic moment of $9.27 \times 10^{-24} \mathrm{A} \cdot \mathrm{m}^{2}$ (one Bohr magneton), how many electrons per atom contribute to the saturated field of iron? Iron contains approximately $8.50 \times 10^{28}$ atoms $/ \mathrm{m}^{3}$ .

Donya Dobbin

Numerade Educator

Problem 45

(a) Show that Curie's law can be stated in the following way: The magnetic susceptibility of a paramagnetic substance is inversely proportional to the absolute temperature, according to $\chi=C \mu_{0} / T,$ where $C$ is Curie's constant. (b) Evaluate Curie's constant for chromium.

Narayan Hari

Numerade Educator

Problem 46

A circular coil of 5 turns and a diameter of 30.0 $\mathrm{cm}$ is oriented in a vertical plane with its axis perpendicular to the horizontal component of the Earth's magnetic field. A horizontal compass placed at the center of the coil is made to deflect $45.0^{\circ}$ from magnetic north by a current of 0.600 A in the coil. (a) What is the horizontal component of the Earth's magnetic field? (b) The current in the coil is
switched off. A "dip needle" is a magnetic compass mounted so that it can rotate in a vertical north-south
plane. At this location a dip needle makes an angle of $13.0^{\circ}$ from the vertical. What is the total magnitude of the Earth's magnetic field at this location?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 47

The magnetic moment of the Earth is approximately $8.00 \times 10^{22} \mathrm{A} \cdot \mathrm{m}^{2}$ . (a) If this were caused by the complete magnetization of a huge iron deposit, how many unpaired electrons would this correspond to? (b) At two unpaired electrons per iron atom, how many kilograms of iron would this correspond to? (Iron has a density of $7900 \mathrm{kg} / \mathrm{m}^{3},$ and approximately $8.50 \times 10^{28}$ iron atoms $/ \mathrm{m}^{3} . )$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 48

The magnitude of the Earth's magnetic field at either pole is approximately $7.00 \times 10^{-5} \mathrm{T}$ . Suppose that the field fades away, before its next reversal. Scouts, sailors, and conservative politicians around the world join together in a program to replace the field. One plan is to use a current loop around the equator, without relying on magnetization of any materials inside the Earth.
Determine the current that would generate such a field if this plan were carried out. (Take the radius of the Earth as $R_{E}=6.37 \times 10^{6} \mathrm{m} . )$

Donya Dobbin

Numerade Educator

Problem 49

A very long, thin strip of metal of width $w$ carries a current $I$ along its length as shown in Figure $\mathrm{P} 30.49$ . Find the magnetic field at the point $P$ in the diagram. The point $P$
is in the plane of the strip at distance $b$ away from it.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 50

Suppose you install a compass on the center of the dashboard of a car. Compute an order-of-magnitude estimate for the magnetic field at this location produced by the current when you switch on the headlights. How does it compare with the Earth's magnetic field? You may suppose the dashboard is made mostly of plastic.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

For a research project, a student needs a solenoid that produces an interior magnetic field of 0.0300 $\mathrm{T}$ . She decides to use a current of 1.00 $\mathrm{A}$ and a wire 0.500 $\mathrm{mm}$ in
diameter. She winds the solenoid in layers on an insulating form 1.00 $\mathrm{cm}$ in diameter and 10.0 $\mathrm{cm}$ long. Determine the number of layers of wire needed and the total length of the wire.

Vishal Gupta

Numerade Educator

Problem 52

A thin copper bar of length $\ell=10.0 \mathrm{cm}$ is supported horizontally by two (nonmagnetic) contacts. The bar carries current $I_{1}=100 \mathrm{A}$ in the $-x$ direction, as shown in Figure $\mathrm{P} 30.52$ . At a distance $h=0.500 \mathrm{cm}$ below one end of the bar, a long straight wire carries a current $I_{2}=200 \mathrm{A}$ in the $z$ direction. Determine the magnetic force exerted on the bar.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 53

A nonconducting ring of radius 10.0 $\mathrm{cm}$ is uniformly charged with a total positive charge 10.0$\mu \mathrm{C}$ . The ring rotates at a constant angular speed 20.0 $\mathrm{rad} / \mathrm{s}$ about an axis through its center, perpendicular to the plane of the ring. What is the magnitude of the magnetic field on the axis of the ring 5.00 $\mathrm{cm}$ from its center?

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Numerade Educator

Problem 54

A nonconducting ring of radius $R$ is uniformly charged with a total positive charge $q$ . The ring rotates at a constant angular speed $\omega$ about an axis through its center, perpendicular to the plane of the ring. What is the magnitude of the magnetic field on the axis of the ring a distance $R / 2$ from its center?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 55

Two circular coils of radius $R,$ each with $N$ turns, are perpendicular to a common axis. The coil centers are a distance $R$ apart. Each coil carries a steady current $I$ in the same direction, as shown in Figure $\mathrm{P} 30.55$ . (a) Show that the magnetic field on the axis at a distance $x$ from the center of one coil is
$$B=\frac{N \mu_{0} I R^{2}}{2}\left[\frac{1}{\left(R^{2}+x^{2}\right)^{3 / 2}}+\frac{1}{\left(2 R^{2}+x^{2}-2 R x\right)^{3 / 2}}\right]$$
(b) Show that $d B / d x$ and $d^{2} B / d x^{2}$ are both zero at the point midway between the coils. This means the magnetic field in the region midway between the coils is uniform. Coils in this configuration are called Helmholtz coils.

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Numerade Educator

Problem 56

Two identical, flat, circular coils of wire each have 100 turns and a radius of 0.500 $\mathrm{m}$ . The coils are arranged as a set of Helmholtz coils (see Fig. P30.55), parallel and with separation 0.500 $\mathrm{m}$ . Each coil carries a current of 10.0 A. Determine the magnitude of the magnetic field at a point on the common axis of the coils and halfway between them.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

We have seen that a long solenoid produces a uniform magnetic field directed along the axis of a cylindrical region. However, to produce a uniform magnetic field directed parallel to a diameter of a cylindrical region, one can use the saddle coils illustrated in Figure P30.57. The loops are wrapped over a somewhat flattened tube. Assume the straight sections of wire are very long. The end view of the tube shows how the windings are applied. The overall current distribution is the superposition of two overlapping circular cylinders of uniformly distributed current, one toward you and one away from you. The current density $J$ is the same for each cylinder. The position of the axis of one cylinder is described by a position vector a relative to the other cylinder. Prove that the magnetic field inside the hollow tube is $\mu_{0} J a / 2$ downward. Suggestion: The use of vector methods simplifies the calculation.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 58

A very large parallel-plate capacitor carries charge with uniform charge per unit area $+\sigma$ on the upper plate and $-\sigma$ on the lower plate. The plates are horizontal and both move horizontally with speed $v$ to the right. (a) What is the magnetic field between the plates? (b) What is the magnetic field close to the plates but outside of the capacitor? (c) What is the magnitude and direction of the magnetic force per unit area on the upper plate? (d) At what extrapolated speed $v$ will the magnetic force on a plate balance the electric force on the plate? Calculate this speed numerically.

Victor Salazar

Numerade Educator

Problem 59

Two circular loops are parallel, coaxial, and almost in contact, 1.00 $\mathrm{mm}$ apart (Fig. P30.59). Each loop is 10.0 $\mathrm{cm}$ in radius. The top loop carries a clockwise current of 140 $\mathrm{A}$ . The bottom loop carries a counterclockwise current of 140 $\mathrm{A}$ . (a) Calculate the magnetic force exerted by the bottom loop on the top loop. (b) The upper loop has a mass of 0.0210 $\mathrm{kg}$ . Calculate its acceleration, assuming that the only forces acting on it are the force in part (a) and the gravitational force. Suggestion: Think about how one loop looks to a bug perched on the other loop.

Vishal Gupta

Numerade Educator

Problem 60

What objects experience a force in an electric field? Chapter 23 gives the answer: any electric charge, stationary or moving, other than the charge that created the field. What creates an electric field? Any electric charge, stationary or moving, as you studied in Chapter $23 .$ What objects experience a force in a magnetic field? An electric current or a moving electric charge, other than the current or charge that created the field, as discussed in Chapter $29 .$ What creates a magnetic field? An electric current, as you studied in Section $30.1,$ or a moving electric charge, as shown in this problem. (a) To display how a moving charge creates a magnetic field, consider a charge $q$ moving with velocity $\mathbf{v} .$ Define the vector $\mathbf{r}=r \hat{\mathbf{r}}$ to lead from the charge to some location. Show that the magnetic field at that location is
$$\mathbf{B}=\frac{\mu_{0}}{4 \pi} \frac{q \mathbf{v} \times \hat{\mathbf{r}}}{r^{2}}$$
(b) Find the magnitude of the magnetic field 1.00 $\mathrm{mm}$ to the side of a proton moving at $2.00 \times 10^{7} \mathrm{m} / \mathrm{s}$ . (c) Find the magnetic force on a second proton at this point, moving with the same speed in the opposite direction. (d) Find the electric force on the second proton.

Vishal Gupta

Numerade Educator

Problem 61

Rail guns have been suggested for launching projectiles into space without chemical rockets, and for ground-to-air antimissile weapons of war. A tabletop model rail gun (Fig. P30.61) consists of two long parallel horizontal rails 3.50 $\mathrm{cm}$ apart, bridged by a bar $B D$ of mass 3.00 g. The bar is originally at rest at the midpoint of the rails and is free to slide without friction. When the switch is closed, electric current is quickly established in the circuit $A B C D E A$ . The rails and bar have low electric resistance, and the current is limited to a constant 24.0 $\mathrm{A}$ by the power supply. (a) Find the magnitude of the magnetic field 1.75 $\mathrm{cm}$ from a single very long straight wire carrying current 24.0 A. (b) Find the magnitude and direction of the magnetic field at point $C$ in the diagram, the midpoint of the bar, immediately after the switch is closed. Suggestion: Consider what conclusions you can draw from the Biot-Savart law. (c) At other points along the bar $B D,$ the field is in the same direction as at point $C,$ but larger in magnitude. Assume that the average effective magnetic field along $B D$ is five times larger than the field at $C .$ With this assumption, find the magnitude and direction of the force on the bar. (d) Find the acceleration of the bar when it is in motion. (e) Does the bar move with constant acceleration? (f) Find the velocity of the bar after it has traveled 130 $\mathrm{cm}$ to the end of the rails.

Donya Dobbin

Numerade Educator

Problem 62

Fifty turns of insulated wire 0.100 $\mathrm{cm}$ in diameter are tightly wound to form a flat spiral. The spiral fills a disk surrounding a circle of radius 5.00 $\mathrm{cm}$ and extending to a radius 10.00 $\mathrm{cm}$ at the outer edge. Assume the wire carries current $I$ at the center of its cross section. Approximate each turn of wire as a circle. Then a loop of current exists at radius $5.05 \mathrm{cm},$ another at $5.15 \mathrm{cm},$ and so on. Numerically calculate the magnetic field at the center of the coil.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 63

Two long, parallel conductors carry currents in the same direction as shown in Figure P30.63. Conductor A carries a current of 150 $\mathrm{A}$ and is held firmly in position. Conductor B carries a current $I_{\mathrm{B}}$ and is allowed to slide freely up and down (parallel to A) between a set of nonconducting guides. If the mass per unit length of conductor B is $0.100 \mathrm{g} / \mathrm{cm},$ what value of current $I_{\mathrm{B}}$ will result in equilibrium when the distance between the two conductors is 2.50 $\mathrm{cm}$ ?

Donya Dobbin

Numerade Educator

Problem 64

Charge is sprayed onto a large nonconducting belt above the left-hand roller in Figure P30.64. The belt carries the charge with a uniform surface charge density $\sigma$ as it moves with a speed $v$ between the rollers as shown. The charge is removed by a wiper at the right-hand roller. Consider a point just above the surface of the moving belt. (a) Find an expression for the magnitude of the magnetic field $\mathbf{B}$ at this point. (b) If the belt is positively charged, what is the direction of $\mathbf{B}$ ? (Note that the belt may be considered as an infinite sheet.)

Linda Winkler

Numerade Educator

Problem 65

An infinitely long straight wire carrying a current $I_{1}$ is partially surrounded by a loop as shown in Figure $\mathrm{P} 30.65$ . The loop has a length $L,$ radius $R,$ and carries a current $I_{2}$ . The axis of the loop coincides with the wire. Calculate the force exerted on the loop.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 66

Measurements of the magnetic field of a large tornado were made at the Geophysical Observatory in Tulsa, Oklahoma, in 1962 . The tornado's field was measured to be $B=1.50 \times 10^{-8}$ T pointing north when the tornado was 9.00 $\mathrm{km}$ east of the observatory. What current was carried up or down the funnel of the tornado, modeled as a long straight wire?

Donya Dobbin

Numerade Educator

Problem 67

A wire is formed into the shape of a square of edge length $L(\text { Fig. } \mathrm{P} 30.67)$ . Show that when the current in the loop is $I,$ the magnetic field at point $P,$ a distance $x$ from the center of the square along its axis is
$$B=\frac{\mu_{0} I L^{2}}{2 \pi\left(x^{2}+L^{2} / 4\right) \sqrt{x^{2}+L^{2} / 2}}$$

Rashmi Sinha

Numerade Educator

Problem 68

The force on a magnetic dipole $\mu$ aligned with a nonuniform magnetic field in the $x$ direction is given by $F_{x}=|\mu| d B / d x .$ Suppose that two flat loops of wire each have radius $R$ and carry current $I .$ (a) The loops are arranged coaxially and separated by a variable distance $x,$ large compared to $R .$ Show that the magnetic force between them varies as $1 / x^{4} .$ (b) Evaluate the magnitude of this force if $I=10.0 \mathrm{A}, R=0.500 \mathrm{cm},$ and $x=5.00 \mathrm{cm} .$

Keshav Singh

Numerade Educator

Problem 69

A wire carrying a current $I$ is bent into the shape of an exponential spiral, $r=e^{\theta},$ from $\theta=0$ to $\theta=2 \pi$ as suggested in Figure $\mathrm{P} 30.69$ . To complete a loop, the ends of the spiral are connected by a straight wire along the $x$ axis. Find the magnitude and direction of $\mathbf{B}$ at the origin. Suggestions: Use the Biot-Savart law. The angle $\beta$ between a
radial line and its tangent line at any point on the curve $r=f(\theta)$ is related to the function in the following way:
$$\tan \beta=\frac{r}{d r / d \theta}$$
Thus in this case $r=e^{\theta}, \tan \beta=1$ and $\beta=\pi / 4$
Therefore, the angle between $d \mathbf{s}$ and $\hat{\mathbf{r}}$ is $\pi-\beta=3 \pi / 4$ . Also
$$d s=\frac{d r}{\sin (\pi / 4)}=\sqrt{2} d r$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

Table $\mathrm{P} 30.70$ contains data taken for a ferromagnetic material. (a) Construct a magnetization curve from the data. Remember that $\mathbf{B}=\mathbf{B}_{0}+\mu_{0} \mathbf{M} .$ (b) Determine the ratio $B / B_{0}$ for each pair of values of $B$ and $B_{0},$ and construct a graph of $B / B_{0}$ versus $B_{0} .$ (The fraction $B / B_{0}$ is called the relative permeability, and it is a measure of the induced magnetic field.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 71

A sphere of radius $R$ has a uniform volume charge density $\rho .$ Determine the magnetic field at the center of the sphere when it rotates as a rigid object with angular speed $\omega$ about an axis through its center (Fig. P30.71).

Prashant Bana

Numerade Educator

Problem 72

A sphere of radius $R$ has a uniform volume charge density $\rho .$ Determine the magnetic dipole moment of the sphere when it rotates as a rigid body with angular speed $\omega$ about an axis through its center (Fig. P30.71).

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

A long cylindrical conductor of radius $a$ has two cylindrical cavities of diameter $a$ through its entire length, as shown in Figure $\mathrm{P} 30.73$ . A current $I$ is directed out of the page and is uniform through a cross section of the conductor Find the magnitude and direction of the magnetic field in terms of $\mu_{0}, I, r,$ and $a$ at (a) point $P_{1}$ and (b) point $P_{2}$ .

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator