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Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 30

Sources of the Magnetic Field

Educators


Problem 1

In 1962 , measurements of the magnetic field of a large tornado were made at the Geophysical Observatory in Tulsa, Oklahoma. If the magnitude of the tornado's field was $B=1.50 \times 10^{-8}$ 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? Model the vortex as a long, straight wire carrying a current.

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Problem 2

In each of parts (a) through (c) of Figure $\mathrm{P} 30.2$ , find the direction of the current in the wire that would produce a magnetic field directed as shown.

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Problem 3

Calculate the magnitude of the magnetic field at a point 25.0 $\mathrm{cm}$ from a long, thin conductor carrying a current of $2.00 \mathrm{A} .$

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Problem 4

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 this motion produces at the location of the proton.

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Problem 5

(a) A conducting loop in the shape of a square of edge length $\ell=0.400 \mathrm{m}$ carries a current $I=10.0 \mathrm{A}$ as shown in Figure $\mathrm{P} 30.5 .$ Calculate the magnitude and direction of the magnetic field at the center of the square. (b) What If? If this conductor is reshaped to form a circular loop and carries the same current, what is the value of the magnetic field at the center?

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Problem 6

An infinitely long wire carrying a current I is bent at a right angle as shown in Figure P30.6. Determine the magnetic field at point P, located a distance x from the corner of the wire.

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Problem 7

A conductor consists of a circular loop of radius $R=$ 15.0 $\mathrm{cm}$ and two long, straight sections as shown in Figure P30.7. The wire lies in the plane of the paper and carries a current $I=1.00$ A. Find the magnetic field at the center of the loop.

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Problem 8

A conductor consists of a circular loop of radius $R$ and two long, straight sections as shown in Figure $\mathrm{P} 30.7 .$ The wire lies in the plane of the paper and carries a current $I$ . (a) What is the direction of the magnetic field at the center $r$ . of the loop? (b) Find an expression for the magnitude of the magnetic field at the center of the loop.

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Problem 9

Two long, straight, parallel wires carry currents that are directed perpendicular to the page as shown in Figure $\mathrm{P} 30.9 .$ Wire 1 carries a current $I_{1}$ into the page (in the negative $z$ direction $)$ and passes through the $x$ axis at $x=-2 a$. 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 $2 \mu_{0} I_{1} /(2 \pi a) .$ 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 $I_{2}$ .

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Problem 10

Consider a flat, circular current loop of radius $R$ carrying a current $I$ . Choose the $x$ axis to be along the axis of the loop, with the origin at the loop's center. 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 helpful to use a programmable calculator or a computer to solve this problem.

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Problem 11

A long, straight wire carries a current $I .$ A right-angle bend is made in the middle of the wire. The bend forms an arc of a circle of radius $r$ as shown in Figure $\mathrm{P} 30.11$ . Determine the magnetic field at point $P,$ the center of the arc.

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Problem 12

One long wire carries current 30.0 A to the left along the $x$ axis. A second long wire carries current 50.0 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$\hat{\mathrm{i}} \mathrm{Mm} / \mathrm{s}$ along the line $(y=0.100 \mathrm{m},$ warticle. (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.

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Problem 13

A current path shaped as shown in Figure $\mathrm{P} 30.13$ produces a magnetic field at $P,$ the center of the arc. If the arc subtends an angle of $\theta=30.0^{\circ}$ and the radius of the arc is $0.600 \mathrm{m},$ what are the magnitude and direction of the field produced at $P$ if the current is 3.00 $\mathrm{A}$ ?

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Problem 14

In a long, straight, vertical lightning stroke, electrons move downward and positive ions move upward and constitute a current of magnitude 20.0 kA. At a location 50.0 m east of the middle of the stroke, a free electron drifts through the air toward the west with a speed of 300 m/s. (a) Make a sketch showing the various vectors involved. Ignore the effect of the Earth’s magnetic field. (b) Find the vector force the lightning stroke exerts on the electron. (c) Find the radius of the electron’s path. (d) Is it a good approximation to model the electron as moving in a uniform field? Explain your answer. (e) If it does not collide with any obstacles, how many revolutions will the electron complete during the $60.0-\mu \mathrm{s}$ duration of the lightning stroke?

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Problem 15

Three long, parallel conductors each carry a current of $I=$ 2.00 A. Figure $\mathrm{P} 30.15$ is an end view of the conductors, with each current coming out of the page. Taking $a=1.00 \mathrm{cm},$ determine the magnitude and direction of the magnetic field at (a) point $A,(\mathrm{b})$ point $B,$ and (c) point $C .$

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Problem 16

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? Give a qualitative argument for your answer.

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Problem 17

Determine the magnetic field (in terms of $I, a,$ and $d )$ at the origin due to the current loop in Figure P30.17. The loop extends to infinity above the figure.

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Problem 18

Two long, parallel wires carry currents of $I_{1}=3.00 \mathrm{A}$ and $I_{2}=5.00 \mathrm{A}$ in the directions indicated in Figure $\mathrm{P} 30.18$ . (a) Find the magnitude and direction of the magnetic field at a point midway between the wires. (b) Find the magnitude and direction of the magnetic field at point $P,$ located $d=20.0 \mathrm{cm}$ above the wire carrying the $5.00-\mathrm{A}$ current.

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Problem 19

The two wires shown in Figure $\mathrm{P} 30.19$ are separated by $d=10.0 \mathrm{cm}$ and carry currents of $I=5.00 \mathrm{A}$ in opposite directions. Find the magnitude and direction of the net magnetic field $(\text { a) at a point midway between the wires; }$ (b) at point $P_{1}, 10.0 \mathrm{cm}$ to the right of the wire on the right; and $(\mathrm{c})$ at point $P_{2}, 2 d=20.0 \mathrm{cm}$ to the left of the wire on the left.

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Problem 20

Two parallel wires are separated by 6.00 cm, each carrying 3.00 A of current in the same direction. (a) What is the magnitude of the force per unit length between the wires? (b) Is the force attractive or repulsive?

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Problem 21

Two long, parallel conductors, separated by $10.0 \mathrm{cm},$ carry currents in the same direction. The first wire carries a 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 $I_{2} ?(\text { b) What is the force per unit length }$ 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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Problem 22

Two parallel wires separated by 4.00 cm repel each other with a force per unit length of $2.00 \times 10^{-4} \mathrm{N} / \mathrm{m}$ . The current in one wire is 5.00 $\mathrm{A}$ . ( a) Find the current in the other wire. (b) Are the currents in the same direction or in opposite directions? (c) What would happen if the direction of one current were reversed and doubled?

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Problem 23

In Figure $\mathrm{P} 30.23$ , 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 a current $I_{2}=10.0 \mathrm{A}$ . The dimensions in the figure 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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Problem 24

In Figure $\mathrm{P} 30.23$ , the current in the long, straight wire is $I_{1}$ and the wire lies in the plane of a rectangular loop, which carries a current $I_{2}$ . The loop is of length $\ell$ and width a. Its left end is a distance $c$ from the wire. 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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Problem 25

Two long, parallel wires are attracted to each other by a force per unit length of 320$\mu \mathrm{N} / \mathrm{m}$ . One wire carries a current of 20.0 $\mathrm{A}$ to the right and is located along the line $y=$ 0.500 $\mathrm{m}$ . The second wire lies along the $x$ axis. Determine the value of $y$ for the line in the plane of the two wires along which the total magnetic field is zero.

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Problem 26

Two long wires hang vertically. Wire 1 carries an upward current of 1.50 A. Wire 2, 20.0 cm to the right of wire 1, carries a downward current of 4.00 A. A third wire, wire 3, is to be hung vertically and located such that when it carries a certain current, each wire experiences no net force. (a) Is this situation possible? Is it possible in more than one way? Describe (b) the position of wire 3 and
(c) the magnitude and direction of the current in wire 3.

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Problem 27

The unit of magnetic flux is named for Wilhelm Weber. A practical-size unit of magnetic field is named for Johann Karl Friedrich Gauss. Along with their individual accomplishments, Weber and Gauss built a telegraph in 1833 that consisted of a battery and switch, at one end of a transmission line 3 km long, operating an electromagnet at the other end. Suppose their transmission line was as diagrammed in Figure P30.27. Two long, parallel wires, each having a mass per unit length of 40.0 g/m, are supported in a horizontal plane by strings $\ell=6.00 \mathrm{cm}$ long. When both wires carry the same current, the wires repel each other so that the angle between the supporting strings is $\theta=16.0^{\circ} .$ (a) Are the currents in the same direction or in opposite directions? (b) Find the magnitude of the current. (C) If this transmission line were taken to Mars, would the current required to separate the wires by the same angle be larger or smaller than that required on the Earth? Why?

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Problem 28

Why is the following situation impossible? Two parallel copper conductors each have length $\ell=0.500 \mathrm{m}$ and radius $r=250 \mu \mathrm{m} .$ They carry currents $I=10.0 \mathrm{A}$ in opposite directions and repel each other with a magnetic force $F_{B}=$ $1.00 \mathrm{N} .$

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Problem 29

Figure P30.29 is a cross-sectional view of a coaxial cable. The center conductor is surrounded by a rubber layer, an outer conductor, and another rubber layer. In a particular application, the current in the inner conductor is $I_{1}=$ 1.00 A out of the page and the current in the outer conductor is $I_{2}=3.00$ A into the page. Assuming the distance $d=$ $1.00 \mathrm{mm},$ determine the magnitude and direction of the magnetic field at (a) point $a$ and (b) point $b$ .

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Problem 30

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

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Problem 31

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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Problem 32

Four long, parallel conductors carry equal currents of $I=5.00$ A. Figure $\mathrm{P} 30.32$ is an end view of the conductors. The current direction is into the page at points $A$ and $B$ and out of the page at points $C$ and $D .$ Calculate (a) the magnitude and $(\mathrm{b})$ the direction of the magnetic field at point $P,$ located at the center of the square of edge length $\ell=0.200 \mathrm{m} .$

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Problem 33

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. Ignoring the magnetic field due to the Earth, determine the value of $d .$

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Problem 34

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

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Problem 35

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 $2.00-\mathrm{A}$ 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 the cable?

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Problem 36

A long, cylindrical conductor of radius $R$ carries a current $I$ as shown in Figure $\mathrm{P} 30.36$ . The current density $J$ however, is not uniform over the cross section of the conductor but rather is a function of the radius according to $J=b r,$ where $b$ is a constant. Find an expression for the magnetic field magnitude $B$ (a) at a distance $r_{1}< R$ and (b) at a distance $r_{2}>R,$ measured from the center of the conductor.

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Problem 37

The magnetic field created by a large current passing through plasma (ionized gas) can force current-carrying particles together. This pinch effect has been used in designing fusion reactors. It 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 current, uniformly distributed over the can’s curved wall. Determine the magnetic field (a) just inside the wall and (b) just outside. (c) Determine the pressure on the wall.

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Problem 38

An infinite sheet of current lying in the $y$ z plane carries a surface current of linear density $J .$ The current is in the positive $z$ direction, and $J_{s}$ represents the current per unit length measured along the $y$ axis. Figure $\mathrm{P} 30.38$ is an edge view of the sheet. Prove that the magnetic field near the sheet is parallel to the sheet and perpendicular to the current direction, with magnitude $\mu_{0} J_{s} / 2$.

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Problem 39

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

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Problem 40

A certain superconducting magnet in the form of a solenoid of length 0.500 $\mathrm{m}$ can generate a magnetic field of 9.00 $\mathrm{T}$ in its core when its coils carry a current of 75.0 $\mathrm{A}$ . Find the number of turns in the solenoid.

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Problem 41

A certain superconducting magnet in the form of a solenoid of length 0.500 $\mathrm{m}$ can generate a magnetic field of 9.00 $\mathrm{T}$ in its core when its coils carry a current of 75.0 $\mathrm{A}$ . Find the number of turns in the solenoid.

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Problem 42

You are given a certain volume of copper from which you can make copper wire. To insulate the wire, you can have as much enamel as you like. You will use the wire to make a tightly wound solenoid 20 cm long having the greatest possible magnetic field at the center and using a power supply that can deliver a current of 5 A. The solenoid can be wrapped with wire in one or more layers. (a) Should you make the wire long and thin or shorter and thick? Explain. (b) Should you make the radius of the solenoid small or large? Explain.

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Problem 43

It is desired to construct a solenoid that will have a resistance of 5.00$\Omega\left(\text { at } 20.0^{\circ} \mathrm{C}\right)$ and produce a magnetic field of $4.00 \times 10^{-2} \mathrm{T}$ at its center when it carries a current of 4.00 A. The solenoid is to be constructed from copper wire having a diameter of $0.500 \mathrm{mm} .$ If the radius of the solenoid is to be $1.00 \mathrm{cm},$ determine (a) the number of turns of wire needed and (b) the required length of the solenoid.

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Problem 44

A solenoid 10.0 cm in diameter and 75.0 cm long is made from copper wire of diameter 0.100 cm, with very thin insulation. The wire is wound onto a cardboard tube in a single layer, with adjacent turns touching each other. What power must be delivered to the solenoid if it is to produce a field of 8.00 mT at its center?

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Problem 45

A cube of edge length $\ell=2.50 \mathrm{cm}$ is positioned as shown in Figure $\mathrm{P} 30.45 .$ A uniform magnetic field given by $\overrightarrow{\mathbf{B}}=(5 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \mathrm{T}$ exists throughout the region.
(a) Calculate the magnetic flux through the shaded face.
(b) What is the total flux through the six faces?

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Problem 46

Consider the hemispherical closed surface in Figure P30.46. 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 $S_{2} .$

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Problem 47

A solenoid of radius $r=1.25 \mathrm{cm}$ and length $\ell=30.0 \mathrm{cm}$ has 300 turns and carries 12.0 $\mathrm{A}$ . ( a) Calculate the flux through the surface of a disk-shaped area of radius $R=$ 5.00 $\mathrm{cm}$ that is positioned perpendicular to and centered on the axis of the solenoid as shown in Figure $\mathrm{P} 30.47 \mathrm{a}$ . (b) Figure $\mathrm{P} 30.47 \mathrm{b}$ shows an enlarged end view of the same solenoid. Calculate the flux through the tan area, which is an annulus with an inner radius of $a=0.400 \mathrm{cm}$ and an outer radius of $b=0.800 \mathrm{cm} .$

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Problem 48

At saturation, when nearly all the atoms have their magnetic moments aligned, the magnetic field is equal to the permeability constant $\mu_{0}$ multiplied by the magnetic moment per unit volume. In a sample of iron, where the number density of atoms is approximately $8.50 \times 10^{28}$ atoms $/ \mathrm{m}^{3}$ , the magnetic field can reach 2.00 $\mathrm{T}$ . If each electron con- tributes a magnetic moment of $9.27 \times 10^{-24} \mathrm{A} \cdot \mathrm{m}^{2}$ (1 Bohr magneton), how many electrons per atom contribute to the saturated field of iron?

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Problem 49

The magnetic moment of the Earth is approximately $8.00 \times 10^{22} \mathrm{A} \cdot \mathrm{m}^{2} .$ Imagine that the planetary magnetic field were caused by the complete magnetization of a huge iron deposit with density 7900 $\mathrm{kg} / \mathrm{m}^{3}$ and approximately 8.50 $\times 10^{28}$ iron atoms/m^{3} . (a) How many unpaired electrons, each with a magnetic moment of 9.27 $\times 10^{-24} \mathrm{A} \cdot \mathrm{m}^{2}$ , would participate? (b) At two unpaired electrons per iron atom, how many kilograms of iron would be present in the deposit?

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Problem 50

A 30.0 -turn solenoid of length 6.00 $\mathrm{cm}$ produces a magnetic field of magnitude 2.00 $\mathrm{mT}$ at its center. Find the current in the solenoid.

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Problem 51

Suppose you install a compass on the center of a car’s dash-board. (a) Assuming the dashboard is made mostly of plastic, compute an order-of-magnitude estimate for the magnetic field at this location produced by the current when you switch on the car’s headlights. (b) How does this estimate compare with the Earth’s magnetic field?

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Problem 52

Why is the following situation impossible? The magnitude of the Earth’s magnetic field at either pole is approximately 7.00 $\times 10^{-5}$ T. Suppose the field fades away to zero before its next reversal. Several scientists propose plans for artificially generating a replacement magnetic field to assist with devices that depend on the presence of the field. The plan that is selected is to lay a copper wire around the equator and supply it with a current that would generate a magnetic field of magnitude $7.00 \times 10^{-5} \mathrm{T}$ at the poles. (Ignore magnetization of any materials inside the Earth. The plan is implemented and is highly successful.

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Problem 53

A very long, thin strip of metal of width w carries a current $I$ along its length as shown in Figure $\mathrm{P} 30.53$ . The cur- rent is distributed uniformly across the width of the strip. Find the magnetic field at point $P$ in the diagram. Point $P$ is in the plane of the strip at distance $b$ away from its edge.

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Problem 54

A circular coil of five 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 coil's center is made to deflect $45.0^{\circ}$ from magnetic north by a current of 0.600 $\mathrm{A}$ 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?

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Problem 55

A nonconducting ring of radius 10.0 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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Problem 56

A nonconducting ring of radius R is uniformly charged with a total positive charge q. The ring rotates at a constant angular speed v 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 $\frac{1}{2} R$ from its center?

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Problem 57

A very large parallel-plate capacitor has 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 just above or just below the plates? (c) What are the magnitude and direction of the magnetic force per unit area on the upper plate? (d) At what extrapolated speed $v$ will the plate? Suggestion: Use Ampere's law and choose a path that closes between the plates of the capacitor.

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Problem 58

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 P30.58. (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. We may then conclude that the magnetic field in the region midway between the coils is uniform. Coils in this configuration are called Helmholtz coils.

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Problem 59

Two identical, flat, circular coils of wire each have 100 turns and radius $R=0.500 \mathrm{m}$ . The coils are arranged as a set of Helmholtz coils so that the separation distance between the coils is equal to the radius of the coils (see Fig. $\mathrm{P} 30.58$ ). Each coil carries current $I=10.0$ A. Determine the magnitude of the magnetic field at a point on the common axis of the coils and halfway between them.

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Problem 60

Two circular loops are parallel, coaxial, and almost in contact, with their centers 1.00 mm apart (Fig. P30.60). Each loop is 10.0 $\mathrm{cm}$ in radius. The top loop carries a clockwise current of $I=140 \mathrm{A}$ . The bottom loop carries a counterclockwise current of $I=140 \mathrm{A}$ . (a) Calculate the magnetic force exerted by the bottom loop on the top loop. (b) Suppose a student thinks the first step in solving part (a) is to use Equation 30.7 to find the magnetic field created by one of the loops. How would you argue for or against this idea? (c) The upper loop has a mass of 0.021 0 kg. Calculate its acceleration, assuming the only forces acting on it are the force in part (a) and the gravitational force.

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Problem 61

Two long, straight wires cross each other perpendicularly as shown in Figure P30.61. The wires do not touch. Find (a) the magnitude and (b) the direction of the magnetic field at point $P,$ which is in the same plane as the two wires. (c) Find the magnetic field at a point 30.0 $\mathrm{cm}$ above the point of intersection of the wires along the $z$ axis; that is, 30.0 $\mathrm{cm}$ out of the page, toward you.

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Problem 62

Rail guns have been suggested for launching projectiles into space without chemical rockets. A tabletop model rail gun (Fig. P30.62) consists of two long, parallel, horizontal rails $\ell=3.50 \mathrm{cm}$ apart, bridged by a bar of mass $m=3.00 \mathrm{g}$ that is free to slide without friction. The rails and bar have low electric resistance, and the current is limited to a constant $I=24.0 \mathrm{A}$ by a power supply that is far to the left of the figure, so it has no magnetic effect on the bar. Figure P30.62 shows the bar at rest at the mid-point of the rails at the moment the current is established. We wish to find the speed with which the bar leaves the rails after being released from the midpoint of the rails. (a) Find the magnitude of the magnetic field at a distance of 1.75 cm from a single long wire carrying a current of 2.40 A. (b) For purposes of evaluating the magnetic field, model the rails as infinitely long. Using the result of part (a), find the magnitude and direction of the magnetic field at the midpoint of the bar. (c) Argue that this value of the field will be the same at all positions of the bar to the right of the midpoint of the rails. At other points along the bar, the field is in the same direction as at the midpoint, but is larger in magnitude. Assume the average effective magnetic field along the bar is five times larger than the field at the midpoint. With this assumption, find (d) the magnitude and (e) the direction of the force on the bar. (f) Is the bar properly modeled as a particle under constant acceleration? (g) Find the velocity of the bar after it has traveled a distance $d=130 \mathrm{cm}$ to the end of the rails.

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Problem 63

As seen in previous chapters, any object with electric charge, stationary or moving, other than the charged object that created the field experiences a force in an electric field. Also, any object with electric charge, stationary or moving, can create an electric field (Chapter 23). Similarly, an electric current or a moving electric charge, other than the current or charge that created the field, experiences a force in a magnetic field (Chapter 29), and an electric current creates a magnetic field (Section 30.1). (a) To understand how a moving charge can also create a magnetic field, consider a particle with charge $q$ moving with velocity $\overrightarrow{\mathbf{v}}$ . Define the position vector $\overrightarrow{\mathbf{r}}=r \hat{\mathbf{r}}$ leading from the particle to some location. Show that the magnetic field at that location is
$$\overrightarrow{\mathbf{B}}=\frac{\mu_{0}}{4 \pi} \frac{q \overrightarrow{\mathbf{v}} \times\hat{\mathbf{r}}}{r^{2}}$$
(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.

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Problem 64

Two coplanar and concentric circular loops of wire carry currents of $I_{1}=5.00 \mathrm{A}$ and $I_{2}=3.00 \mathrm{A}$ in opposite directions as in Figure $\mathrm{P} 30.64$ . If $r_{1}=12.0 \mathrm{cm}$ and $r_{2}=9.00 \mathrm{cm},$ what are (a) the magnitude and (b) the direction of the net magnetic field at the center of the two loops? (c) Let $r_{1}$ remain fixed at 12.0 $\mathrm{cm}$ and let $r_{2}$ be a variable. Determine the value of $r_{2}$ such that the net field at the center of the loops is zero.

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Problem 65

Fifty turns of insulated wire 0.100 cm in diameter are tightly wound to form a flat spiral. The spiral fills a disk surrounding a circle of radius 5.00 cm and extending to a radius 10.00 cm at the outer edge. Assume the wire carries a 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 cm, another at 5.15 cm, and so on. Numerically calculate the magnetic field at the center of the coil.

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Problem 66

An infinitely long, straight wire carrying a current $I_{1}$ is partially surrounded by a loop as shown in Figure $\mathrm{P} 30.66$ . The loop has a length $L$ and radius $R,$ and it carries a current $I_{2} .$ The axis of the loop coincides with the wire. Calculate the magnetic force exerted on the loop.

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Problem 67

A thin copper bar of length $\ell=10.0 \mathrm{cm}$ is supported horizontally by two (nonmagnetic) contacts at its ends. The bar carries a current of $I_{1}=100 \mathrm{A}$ in the negative $x$ direction as shown in Figure $\mathrm{P} 30.67$ . At a distance $h=0.500 \mathrm{cm}$ below one end of the bar, a long, straight wire carries a current of $I_{2}=200 \mathrm{A}$ in the positive $z$ direction. Determine the magnetic force exerted on the bar.

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Problem 68

In Figure $\mathrm{P} 30.68$ , both currents in the infinitely long wires are 8.00 $\mathrm{A}$ in the negative $x$ direction. The wires are separated by the distance $2 a=6.00 \mathrm{cm} .$ (a) Sketch the magnetic field pattern in the $y z$ plane. (b) What is the value of the magnetic field at the origin? (c) At $(y=0, z \rightarrow \infty) ?$ (d) Find the magnetic field at points along the $z$ axis as a function of $z .$ (e) At what distance $d$ along the positive $z$ axis is the magnetic field a maximum? (f) What is this maximum value?

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Problem 69

Consider a solenoid of length $\ell$ and radius $a$ 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 position $x$ 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.

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Problem 70

We have seen that a long solenoid produces a uniform magnetic field directed along the axis of a cylindrical region. To produce a uniform magnetic field directed parallel to a diameter of a cylindrical region, however, one can use the saddle coils illustrated in Figure P30.70. The loops are wrapped over a long, somewhat flattened tube. Figure P30.70a shows one wrapping of wire around the tube. This wrapping is continued in this manner until the visible side has many long sections of wire carrying current to the left in Figure P30.70a and the back side has many lengths carrying current to the right. The end view of the tube in Figure P30.70b shows these wires and the currents they carry. By wrapping the wires carefully, the distribution of wires can take the shape suggested in the end view such that the overall current distribution is approximately the superposition of two overlapping, circular cylinders of radius R (shown by the dashed lines) with uniformly distributed current, one toward you and one away from you. The current density $J$ is the same for each cylinder. The center of one cylinder is described by a position vector $\overrightarrow{\mathrm{d}}$ relative to the center of the other cylinder. Prove that the magnetic field inside the hollow tube is $\mu_{0} J d / 2$ downward. Suggestion: The use of vector methods simplifies the calculation.

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Problem 71

The magnitude of the force on a magnetic dipole $\vec{\mu}$ aligned with a nonuniform magnetic field in the positive $x$ direction is $F_{x}=|\vec{\mu}| d B / d x$ . Suppose two flat loops of wire each have radius $R$ and carry a current $I .$ (a) The loops are parallel to each other and share the same axis. They are separated by a variable distance $x>R .$ Show that the magnetic force between them varies as $1 / x^{4} .$ (b) Find the magnitude of this force, taking $I=10.0 \mathrm{A}, R=0.500 \mathrm{cm},$ and $x=5.00 \mathrm{cm} .$

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Problem 72

A wire is formed into the shape of a square of edge length $L(\text { Fig. } \mathrm{P} 30.72)$ . 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
$$B=\frac{\mu_{0} I L^{2}}{2 \pi\left(x^{2}+L^{2} / 4\right) \sqrt{x^{2}+L^{2} / 2}}$$

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Problem 73

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 sug- gested in Figure $\mathrm{P} 30.73$ . To complete a loop, the ends of the spiral are connected by a straight wire along the $x$ axis. (a) The angle $\beta$ between a radial line and its tangent line at any point on a curve $r=f(\theta)$ is related to the function by
$$\tan \beta=\frac{r}{d r / d \theta}$$
Use this fact to show that $\beta=\pi / 4 .$ (b) Find the magnetic field at the origin.

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Problem 74

A sphere of radius R has a uniform volume charge density $\rho .$ When the sphere rotates as a rigid object with angular speed $\omega$ about an axis through its center (Fig. P30.74), determine (a) the magnetic field at the center of the sphere and (b) the magnetic moment of the sphere.

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Problem 75

A long, cylindrical conductor of radius a has two cylindrical cavities each of diameter $a$ through its entire length as shown in the end view of Figure $\mathrm{P} 30.75 .$ A current $I$ is directed out of the page and is uniform through a cross section of the conducting material. 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 $P_{2} .$

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