WHAT WE HAVE LEARNED

EXAM STUDY GUIDE

The magnetic permeability of free space,

*μ*_{0}, is given by 4*π*· 10^{−7}T m/A.The Biot-Savart Law, , describes the differential magnetic field, , caused by a current element, , at position relative to the current element.

The magnitude of the magnetic field at distance

*r*_{┴}from a long, straight wire carrying current*i*is*B*=*μ*_{0}*i*/2*πr*_{┴}.The magnetic field magnitude at the center of a loop with radius

*R*carrying current*i*is*B*=*μ*_{0}*i*/2*R*.Ampere's Law is given by , where is the integration path and

*i*_{enc}is the current enclosed in a chosen Amperian loop.The magnitude of the magnetic field inside a solenoid carrying current

*i*and having*n*turns per unit length is*B*=*μ*_{0}*ni*.The magnitude of the magnetic field inside a toroid having

*N*turns and carrying current*i*at radius*r*is given by*B*=*μ*_{0}*Ni*/2*πr*.For an electron with charge –

*e*and mass*m*moving in a circular orbit, the magnetic dipole moment can be related to the orbital angular momentum through .For diamagnetic and paramagnetic materials, magnetization is proportional to the magnetic field strength: . Ferromagnetic materials follow a hysteresis loop and thus deviate from this linear relationship.

The magnetic field inside a diamagnetic or paramagnetic material is due to the external magnetic field strength and the magnetization: , where

*κ*_{m}is the relative magnetic permeability.The four right-hand rules related to magnetic fields are shown in Figure 28.31. Right-hand rule 1 gives the direction of the magnetic force on a charged particle moving in a magnetic field. Right-hand rule 2 gives the direction of the unit normal vector for a current-carrying loop. Right-hand rule 3 gives the direction of the magnetic field from a current-carrying wire. Right-hand rule 4 gives the direction of the magnetic field inside a toroidal magnet.

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KEY TERMS

Biot-Savart Law, p. 894
magnetic permeability of free space, p. 894
Ampere's Law, p. 903
Amperian loop, p. 903
Helmholtz coil, p. 904
solenoid, p. 905
toroid, p. 907
magnetization, p. 910
magnetic field strength, p. 911
magnetic susceptibility, p. 911
diamagnetism, p. 912
paramagnetism, p. 912
relative magnetic permeability, p. 912
ferromagnetism, p. 912
domain, p. 912

NEW SYMBOLS AND EQUATIONS

*μ*_{0} = 4*π* · 10^{−7} T m/A, magnetic permeability of free space

, vector direction of integration in Ampere's Law

*i*_{enc}, enclosed current inside an Amperian loop

, Ampere's Law

, orbital magnetic dipole moment for an electron in circular orbit

, orbital angular momentum for an electron moving in a circular orbit in an atom

, magnetization

, magnetic field strength

*χ*_{m}, magnetic susceptibility

*κ*_{m}, relative magnetic permeability

ANSWERS TO SELF-TEST OPPORTUNITIES

**28.1**The magnetic field at point*P*_{1}points in the positive*y*-direction. The magnetic field at point*P*_{2}points in the negative*x*-direction.**28.2**Two parallel wires carrying current in the same direction attract each other. Two parallel wires carrying current in the opposite direction repel each other.**28.3**

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PROBLEM-SOLVING PRACTICE

Problem-Solving Guidelines

When using the Biot-Savart Law, you should always draw a diagram of the situation, with the current element highlighted. Check for simplifying symmetries before proceeding with calculations; you can save yourself a significant amount of work.

When applying Ampere's Law, choose an Amperian loop that has some geometrical symmetry, in order to simplify the evaluation of the integral. Often, you can use right-hand rule 3 to choose the direction of integration along the loop: Point your thumb in the direction of the net current through the loop and your fingers curl in the direction of integration. This method will also remind you to sum the currents through the Amperian loop to determine the enclosed current.

Remember the superposition principle for magnetic fields: The net magnetic field at any point in space is the vector sum of the individual magnetic fields generated by different objects. Make sure you do not simply add the magnitudes. Instead, you generally need to add the spatial components of the different sources of magnetic field separately.

All of the principles governing motion of charged particles in magnetic fields and all of the problem-solving guidelines presented in Chapter 27 still apply. It does not matter if the magnetic field is due to a permanent magnet or an electromagnet.

In order to calculate the magnetic field in a material, you can use the formulas derived from Ampere's Law and Biot-Savart's Law, but you have to replace

*μ*_{0}with*μ*≡*κ*_{m}*μ*_{0}≡ (1+*χ*_{m})*μ*_{0}.

SOLVED PROBLEM28.4
Magnetic Field from Four Wires

Four wires are each carrying a current of magnitude *i* =1.00 A. The wires are located at the four corners of a square with side *a* = 3.70 cm. Two of the wires are carrying current into the page, and the other two are carrying current out of the page (Figure 28.32).

PROBLEM

What is the *y*-component of the magnetic field at the center of the square?

SOLUTION

THINK

The magnetic field at the center of the square is the vector sum of the magnetic fields from the four current-carrying wires. The magnitude of the magnetic field from all four wires is the same. The direction of the magnetic field from each wire is determined using right-hand rule 3.

SKETCH

Figure 28.33 shows the magnetic fields from the four wires: is the magnetic field from wire 1, is the magnetic field from wire 2, is the magnetic field from wire 3, and is the magnetic field from wire 4. Note that and are equal and and are equal.

RESEARCH

The magnitude of the magnetic field from each of the four wires is given by

where is the distance from each wire to the center of the square.

Right-hand rule 3 gives us the directions of the magnetic fields, which are shown in Figure 28.33. The *y*-component of each of the magnetic fields is given by

SIMPLIFY

The sum of the *y-*components of the four magnetic fields is

where we have used .

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CALCULATE

Putting in the numerical values gives us

ROUND

We report our result to three significant figures:

DOUBLE-CHECK

To double-check our result, we calculate the magnitude of the magnetic field from one wire at the center of the square:

The sum of the *y*-components is then

which agrees with our result.

SOLVED PROBLEM28.5
Electron Motion in a Solenoid

An ideal solenoid has 200.0 turns/cm. An electron inside the coil of a solenoid moves in a circle with radius *r* = 3.00 cm perpendicular to the solenoid's axis. The electron moves with a speed of *v* = 0.0500*c*, where *c* is the speed of light.

PROBLEM

What is the current in the solenoid?

SOLUTION

THINK

The solenoid produces a uniform magnetic field, which is proportional to the current flowing in the solenoid. The radius of circular motion of the electron is related to the speed of the electron and the magnetic field inside the solenoid.

SKETCH

Figure 28.34 shows the circular path of the electron in the uniform magnetic field of the solenoid.

RESEARCH

The magnitude of the magnetic field inside the solenoid is given by

(i)

where *i* is the current in the solenoid and *n* is the number of turns per unit length. The magnetic force provides the centripetal force needed for the electron to move in a circle and so the radius of the electron's path can be related to *B*:

(ii)

where *m* is the electron's mass, *v* is its speed, and *e* is the magnitude of the electron's charge.

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SIMPLIFY

Combining equations (i) and (ii), we have

Solving this equation for the current in the solenoid, we obtain

(iii)

CALCULATE

The speed of the electron was specified in terms of the speed of light:

Putting this and the other numerical values into equation (iii), we get

ROUND

We report our result to three significant figures:

DOUBLE-CHECK

To double-check our result, we use it to calculate the magnitude of the magnetic field inside the solenoid:

This magnitude of magnetic field seems reasonable. Thus, our calculated value for the current in the solenoid seems reasonable.

MULTIPLE-CHOICE QUESTIONS

**28.1**Two long, straight wires are parallel to each other. The wires carry currents of different magnitudes. If the amount of current flowing in each wire is doubled, the magnitude of the force between the wires will betwice the magnitude of the original force.

four times the magnitude of the original force.

the same as the magnitude of the original force.

half of the magnitude of the original force.

**28.2**A current element produces a magnetic field in the region surrounding it. At any point in space, the magnetic field produced by this current element points in a direction that isradial from the current element to the point in space.

parallel to the current element.

perpendicular to the current element and to the radial direction.

**28.3**The number of turns in a solenoid is doubled, and its length is halved. How does its magnetic field change?it doubles

it is halved

it quadruples

it remains unchanged

**28.4**The magnetic force cannot do work on a charged particle since the force is always perpendicular to the velocity. How then can magnets pick up nails? Consider two parallel current-carrying wires. The magnetic fields cause attractive forces between the wires, so it appears that the magnetic field due to one wire is doing work on the other wire. How is this explained?The magnetic force can do no work on isolated charges; this says nothing about the work it can do on charges confined in a conductor.

Since only an electric field can do work on charges, it is actually the electric fields doing the work here.

This apparent work is due to another type of force.

**28.5**In a solenoid in which the wires are wound such that each loop touches the adjacent ones, which of the following will increase the magnetic field inside the magnet?making the radius of the loops smaller

increasing the radius of the wire

increasing the radius of the solenoid

decreasing the radius of the wire

immersion of the solenoid in gasoline

**28.6**Two insulated wires cross at a 90° angle. Currents are sent through the two wires. Which one of the figures best represents the configuration of the wires, if the current in the horizontal wire flows in the positive*x*-direction and the current in the vertical wire flows in the positive*y*-direction?**28.7**What is a good rule of thumb for designing a simple magnetic coil? Specifically, given a circular coil of radius ~1 cm, what is the approximate magnitude of the magnetic field, in gauss per amp per turn? (*Note*: 1 G = 0.0001 T.)0.0001 G/(A-turn)

0.01 G/(A-turn)

1 G/(A-turn)

100 G/(A-turn)

**28.8**A solid cylinder carries a current that is uniform over its cross section. Where is the magnitude of the magnetic field the greatest?at the center of the cylinder's cross section

in the middle of the cylinder

at the surface

none of the above

**28.9**Two long, straight wires have currents flowing in them in the same direction as shown in the figure. The force between the wires isattractive.

repulsive.

zero.

**28.10**In a magneto-optic experiment, a liquid sample in a 10-mL spherical vial is placed in a highly uniform magnetic field, and a laser beam is directed through the sample. Which of the following should be used to create the uniform magnetic field required by the experiment?a 5-cm-diameter flat coil consisting of one turn of 4-gauge wire

a 10-cm-diameter, 20 turn, single layer, tightly wound coil made of 18-gauge wire

a 2-cm-diameter, 10-cm long, tightly wound solenoid made of 18-gauge wire

a set of two coaxial 10-cm-diameter coils at a distance of 5 cm apart, each consisting of one turn of 4-gauge wire

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QUESTIONS

**28.11**Many electrical applications use twisted-pair cables in which the ground and signal wires spiral about each other. Why?**28.12**Discuss how the accuracy of a compass needle in showing the true direction of north can be affected by the magnetic field due to currents in wires and appliances in a residential building.**28.13**Can an ideal solenoid, one with no magnetic field outside the solenoid, exist? If not, does that render the derivation of the magnetic field inside the solenoid (Section 28.4) void?**28.14**Conservative forces tend to act on objects in such a way as to minimize the system's potential energy. Use this principle to explain the direction of the force on the current-carrying loop described in Example 28.1.**28.15**Two particles, each with charge*q*and mass*m*, are traveling in a vacuum on parallel trajectories a distance*d*apart, both at speed*v*(much less than the speed of light). Calculate the ratio of the magnitude of the magnetic force that each exerts on the other to the magnitude of the electric force that each exerts on the other:*F*_{m}/F_{e}.**28.16**A long, straight cylindrical tube of inner radius*a*and outer radius*b*, carries a total current*i*uniformly across its cross section. Determine the magnitude of the magnetic field from the tube at the midpoint between the inner and outer radii.**28.17**Three identical straight wires are connected in a T, as shown in the figure. If current*i*flows into the junction, what is the magnetic field at point*P*, a distance*d*from the junction?**28.18**In a certain region, there is a constant and uniform magnetic field, . Any electric field in the region is also unchanging in time. Find the current density, , in this region.**28.19**The magnetic character of bulk matter is determined largely by electron spin magnetic moments, rather than by orbital dipole moments. (Nuclear contributions are negligible, as the proton's spin magnetic moment is about 658 times smaller than that of the electron.) If the atoms or molecules of a substance have unpaired electron spins, the associated magnetic moments give rise to paramagnetic behavior or to ferromagnetic behavior if the interactions between atoms or molecules are strong enough to align them in domains. If the atoms or molecules have no net unpaired spins, then magnetic perturbations of the electron orbits give rise to diamagnetic behavior.Molecular hydrogen gas (H

_{2}) is weakly diamagnetic. What does this imply about the spins of the two electrons in the hydrogen molecule?What would you expect the magnetic behavior of atomic hydrogen gas (H) to be?

**28.20**Exposed to sufficiently high magnetic fields, materials*saturate*, or approach a maximum magnetization. Would you expect the saturation (maximum) magnetization of paramagnetic materials to be much less than, roughly the same as, or much greater than that of ferromagnetic materials? Explain why.**28.21**A long, straight wire carries a current, as shown in the figure. A single electron is shot directly toward the wire from above. The trajectory of the electron and the wire are in the same plane. Will the electron be deflected from its initial path, and if so, in which direction?

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PROBLEMS

A blue problem number indicates a worked-out solution is available in the Student Solutions Manual. One **•** and two **••** indicate increasing level of problem difficulty.

Sections 28.1 and 28.2

**28.22**Two long parallel wires are separated by 3.0 mm. The current flowing in one of the wires is twice that in the other wire. If the magnitude of the force on a 1.0-m length of one of the wires is 7.0 μN, what are the values of the two currents?**28.23**An electron is shot from an electron gun with a speed of 4.0 · 10^{5}m/s and moves parallel to and a distance of 5.0 cm above a long, straight wire carrying a current of 15 A. Determine the magnitude and the direction of the acceleration of the electron the instant it leaves the electron gun.**28.24**An electron moves in a straight line at a speed of 5 · 10^{6}m/s. What is the magnitude and the direction of the magnetic field created by the moving electron at a distance*d*= 5 m ahead of it on its line of motion? How does the answer change if the moving particle is a proton?**28.25**Suppose that the magnetic field of the Earth were due to a single current moving in a circle of radius 2.00 · 10^{3}km through the Earth's molten core. The strength of the Earth's magnetic field on the surface near a magnetic pole is about 6.00 · 10^{−5}T. About how large a current would be required to produce such a field?**28.26**A square ammeter has sides of length 3.00 cm. The sides of the ammeter are capable of measuring the magnetic field they are subject to. When the ammeter is clamped around a wire carrying a DC current, as shown in the figure, the average value of the magnetic field measured in the sides is 3.00 G. What is the current in the wire?**•27.27**A long, straight wire carrying a 2.00-A current lies along the*x*-axis. A particle with charge*q*= –3.00 μC passes parallel to the*z*-axis through the point (*x,y,z*) = (0,2,0). Where in the*xy*-plane should another long, straight wire be placed so that there is no magnetic force on the particle at the point where it crosses the plane?**•****28.28**Find the magnetic field in the center of a wire semicircle like that shown in the figure, with radius*R*= 10.0 cm, if the current is*i*= 12.0 A.**•****28.29**Two very long wires run parallel to the*z*-axis, as shown in the figure. They each carry a current,*i*_{1}=*i*_{2}= 25.0 A, in the direction of the positive*z*-axis. The magnetic field of the Earth is given by T (in the*xy*-plane and pointing due north). A magnetic compass needle is placed at the origin. Determine the angle*θ*between the compass needle and the*x*-axis. (*Hint*: The compass needle will align its axis along the direction of the net magnetic field.)**•****28.30**Two identical coaxial coils of wire of radius 20.0 cm are directly on top of each other, separated by a 2.00-mm gap. The lower coil is on a flat table and has a current*i*in the clockwise direction; the upper coil carries an identical current and has a mass of 0.0500 kg. Determine the magnitude and the direction that the current in the upper coil has to have to keep the coil levitated at its current height.**•28.31**A long, straight wire lying along the*x*-axis carries a current,*i*, flowing in the positive*x*-direction. A second long, straight wire lies along the*y*-axis and has a current*i*in the positive*y*-direction. What is the magnitude and the direction of the magnetic field at point*z*=*b*on the*z*-axis?**•****28.32**A square loop of wire with a side length of 10.0 cm carries a current of 0.300 A. What is the magnetic field in the center of the square loop?**•****28.33**The figure shows the cross section through three long wires with a linear mass distribution of 100. g/m. They carry currents*i*_{1},*i*_{2}, and*i*_{3}in the directions shown. Wires 2 and 3 are 10.0 cm apart and are attached to a vertical surface, and each carries a current of 600. A. What current,*i*_{1}, will allow wire 1 to “float” at a perpendicular distance*d*from the vertical surface of 10.0 cm? (Neglect the thickness of the wires.)**•****28.34**A hairpin configuration is formed of two semi-infinite straight wires that are 2.00 cm apart and joined by a semicircular piece of wire (whose radius must be 1.00 cm and whose center is at the origin of*xyz*-coordinates). The top straight wire is along the line*y*= 1.00 cm, and the bottom straight wire is along the line*y*= –1.00 cm; these two wires are in the left side (*x*< 0) of the*xy*-plane. The current in the hairpin is 3.00 A, and it is directed toward the right in the top wire, clockwise around the semicircle, and to the left in the bottom wire. Find the magnetic field at the origin of the coordinate system.**•28.35**A long, straight wire is located along the*x*-axis (*y*= 0 and*z*= 0). The wire carries a current of 7.00 A in the positive*x*-direction. What is the magnitude and the direction of the force on a particle with a charge of 9.00 C located at (+1.00 m,+2.00 m,0), when it has a velocity of 3000. m/s in each of the following directions?the positive

*x*-directionthe positive

*y*-directionthe negative

*z*-direction

**•****28.36**A long, straight wire has a 10.0-A current flowing in the positive*x*-direction, as shown in the figure. Close to the wire is a square loop of copper wire that carries a 2.00-A current in the direction shown. The near side of the loop is*d*= 0.50 m away from the wire. The length of each side of the square is*a*= 1.00 m.Find the net force between the two current-carrying objects.

Find the net torque on the loop.

**28.37**A square box with sides of length 1.00 m has one corner at the origin of a coordinate system, as shown in the figure. Two coils are attached to the outside of the box. One coil is on the box face that is in the*xz*-plane at*y*= 0, and the second is on the box face in the*yz*-plane at*x*= 1.00 m. Each of the coils has a diameter of 1.00 m and contains 30.0 turns of wire carrying a current of 5.00 A in each turn. The current in each coil is clockwise when the coil is viewed from outside the box. What is the magnitude and the direction of the magnetic field at the center of the box?

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Section 28.3

**28.38**A square loop, with sides of length*L*, carries current*i*. Find the magnitude of the magnetic field from the loop at the center of the loop, as a function of*i*and*L*.**28.39**The figure shows a cross section across the diameter of a long, solid, cylindrical conductor. The radius of the cylinder is*R*= 10.0 cm. A current of 1.35 A is uniformly distributed through the conductor and is flowing out of the page. Calculate the direction and the magnitude of the magnetic field at positions*r*_{a}= 0.0 cm,*r*_{b}= 4.00 cm,*r*_{c}= 10.00 cm, and*r*_{d}= 16.0 cm.**28.40**Parallel wires, a distance*D*apart, carry a current,*i*, in opposite directions as shown in the figure. A circular loop, of radius*R*=*D*/2, has the same current flowing in a counterclockwise direction. Determine the magnitude and the direction of the magnetic field from the loop and the parallel wires at the center of the loop as a function of*i*and*R*.**28.41**A current of constant density,*J*_{0}, flows through a very long cylindrical conducting shell with inner radius*a*and outer radius*b*. What is the magnetic field in the regions*r*<*a, a*<*r*<*b*, and*r*>*b?*Does*B*_{a<r<b}=*B*_{r>b}for*r*=*b?***28.42**The current density in a cylindrical conductor of radius*R*, varies as*J*(*r*) =*J*_{0}*r*/*R*(in the region from zero to*R*). Express the magnitude of the magnetic field in the regions*r*<*R*and*r*>*R*. Produce a sketch of the radial dependence,*B*(*r*).**•28.43**A very large sheet of a conductor located in the*xy*-plane, as shown in the figure, has a uniform current flowing in the*y*-direction. The current density is 1.5 A/cm. Use Ampere's Law to calculate the direction and the magnitude of the magnetic field just above the center of the sheet (not close to any edges).**••****28.44**A coaxial wire consists of a copper core of radius 1.00 mm surrounded by a copper sheath of inside radius 1.50 mm and outside radius 2.00 mm. A current*i*, flows in one direction in the core and in the opposite direction in the sheath. Graph the magnitude of the magnetic field as a function of the distance from the center of the wire.**••****28.45**The current density of a cylindrical conductor of radius*R*varies as*J*(*r*) =*J*_{0}*e*^{−r/R}(in the region from zero to*R*). Express the magnitude of the magnetic field in the regions*r*<*R*and*r*>*R*. Produce a sketch of the radial dependence,*B*(*r*).

Section 28.4

**28.46**A current of 2.00 A is flowing through a 1000-turn solenoid of length*L*= 40.0 cm. What is the magnitude of the magnetic field inside the solenoid?**28.47**Solenoid A has twice the diameter, three times the length, and four times the number of turns of solenoid B. The two solenoids have currents of equal magnitudes flowing through them. Find the ratio of the magnitude of the magnetic field in the interior of solenoid A to that of solenoid B.**28.48**A long solenoid (diameter of 6.00 cm) is wound with 1000 turns per meter of thin wire through which a current of 0.250 A is maintained. A wire carrying a current of 10.0 A is inserted along the axis of the solenoid. What is the magnitude of the magnetic field at a point 1.00 cm from the axis?**28.49**A long, straight wire carries a current of 2.5 A.What is the strength of the magnetic field at a distance of 3.9 cm from the wire?

If the wire still carries 2.5 A, but is used to form a long solenoid with 32 turns per centimeter and a radius of 3.9 cm, what is the strength of the magnetic field at the center of the solenoid?

**28.50**Figure 28.18a shows a Helmholtz coil used to generate uniform magnetic fields. Suppose the Helmholtz coil consists of two sets of coaxial wire loops with 15 turns of radius*R*= 75.0 cm, which are separated by*R*, and each coil carries a current of 0.123 A flowing in the same direction. Calculate the magnitude and the direction of magnetic field in the center between the coils.**•****28.51**A particle detector utilizes a solenoid that has 550 turns of wire per centimeter. The wire carries a current of 22 A. A cylindrical detector that lies within the solenoid has an inner radius of 0.80 m. Electron and positron beams are directed into the solenoid parallel to its axis. What is the minimum momentum perpendicular to the solenoid axis that a particle can have if it is to be able to enter the detector?

Page 922

Sections 28.5 through 28.7

**28.52**An electron has a spin magnetic moment of magnitude*μ*= 9.285 · 10^{−24}A m^{2}. Consequently, it has energy associated with its orientation in a magnetic field. If the difference between the energy of an electron that is “spin up” in a magnetic field of magnitude*B*and the energy of one that is “spin down” in the same magnetic field (where “up” and “down” refer to the direction of the magnetic field) is 9.460 · 10^{−25}J, what is the field magnitude,*B?***28.53**When a magnetic dipole is placed in a magnetic field, it has a natural tendency to minimize its potential energy by aligning itself with the field. If there is sufficient thermal energy present, however, the dipole may rotate so that it is no longer aligned with the field. Using*k*_{B}T as a measure of the thermal energy, where*k*_{B}is Boltzmann's constant and T is the temperature in kelvins, determine the temperature at which there is sufficient thermal energy to rotate the magnetic dipole associated with a hydrogen atom from an orientation parallel to an applied magnetic field to one that is antiparallel to the applied field. Assume that the strength of the field is 0.15 T.**28.54**Aluminum becomes superconducting at a temperature around 1.0 K if exposed to a magnetic field of magnitude less than 0.0105 T. Determine the maximum current that can flow in an aluminum superconducting wire with radius*R*= 1.0 mm.**28.55**If you want to construct an electromagnet by running a current of 3.0 A through a solenoid with 500 windings and length 3.5 cm and you want the magnetic field inside the solenoid to have magnitude*B*= 2.96 T, you can insert a ferrite core into the solenoid. What value of the relative magnetic permeability should this ferrite core have in order to make this work?**28.56**What is the magnitude of the magnetic field inside a long, straight tungsten wire of circular cross section with diameter 2.4 mm and carrying a current of 3.5 A, at a distance of 0.60 mm from its central axis?**•****28.57**You charge up a small rubber ball of mass 200. g by rubbing it over your hair. The ball acquires a charge of 2.00 μC. You then tie a 1.00-m-long string to it and swing it in a horizontal circle, providing a centripetal force of 25.0 N. What is the magnetic moment of the system?**•****28.58**Consider a model of the hydrogen atom in which an electron orbits a proton in the plane perpendicular to the proton's spin angular momentum (and magnetic dipole moment) at a distance equal to the Bohr radius,*a*_{0}= 5.292 · 10^{−11}m. (This is an oversimplified classical model.) The spin of the electron is allowed to be either parallel to the proton's spin or antiparallel to it; the orbit is the same in either case. But since the proton produces a magnetic field at the electron's location, and the electron has its own intrinsic magnetic dipole moment, the energy of the electron differs depending on its spin. The magnetic field produced by the proton's spin may be modeled as a dipole field, like the electric field due to an electric dipole discussed in Chapter 22. Calculate the energy difference between the two electron-spin configurations. Consider only the interaction between the magnetic dipole moment associated with the electron's spin and the field produced by the proton's spin.**••28.59**Consider an electron to be a uniformly dense sphere of charge, with a total charge of –*e*= –1.60 · 10^{−19}C, spinning at an angular frequency,*ω*.Write an expression for its classical angular momentum of rotation,

*L*.Write an expression for its magnetic dipole moment,

*μ*.Find the ratio,

*γ*=_{e}*μ*/*L*, known as the*gyromagnetic ratio*.

Additional Problems

**28.60**Two 50-turn coils, each with a diameter of 4.00 m, are placed 1.00 m apart, as shown in the figure. A current of 7.00 A is flowing in the wires of both coils; the direction of the current is clockwise for both coils when viewed from the left. What is the magnitude of the magnetic field in the center between the two coils?**28.61**The wires in the figure are separated by a vertical distance*d*. Point*B*is at the midpoint between the two wires; point*A*is a distance*d*/2 from the lower wire. The horizontal distance between*A*and*B*is much larger than*d*. Both wires carry the same current,*i*. The strength of magnetic field at point*A*is 2.00 mT. What is the strength of the field at point*B*?**28.62**You are standing at a spot where the magnetic field of the Earth is horizontal, points due northward, and has magnitude 40.0 μT. Directly above your head, at a height of 12.0 m, a long, horizontal cable carries a steady DC current of 500.0 A due northward. Calculate the angle*θ*by which your magnetic compass needle is deflected from true magnetic north by the effect of the cable. Don't forget the*sign*of*θ*—is the deflection eastward or westward?**28.63**The magnetic dipole moment of the Earth is approximately 8.0 · 10^{22}A m^{2}. The source of the Earth's magnetic field is not known; one possibility might be the circulation of ions in the Earth's molten outer core. Assume that the circulating ions move a circular loop of radius 2500 km. What “current” must they produce to yield the observed field?**28.64**A circular wire loop has radius*R*= 0.12 m and carries current*i*= 0.10 A. The loop is placed in the*xy*-plane in a uniform magnetic field given by , as shown in the figure. Determine the direction and the magnitude of the loop's magnetic moment and calculate the potential energy of the loop in the position shown. If the wire loop can move freely, how will it orient itself to minimize its potential energy, and what is the value of the lowest potential energy?**28.65**A 0.90 m-long solenoid has a radius of 5.0 mm. When the wire carries a 0.20-A current, the magnetic field in the solenoid is 5.0 mT. How many turns of wire are there in the solenoid?**28.66**In a coaxial cable, the solid core carries a current*i*. The sheath also carries a current*i*but in the opposite direction and has an inner radius*a*and an outer radius*b*. The current density is equally distributed over each conductor. Find an expression for the magnetic field at a distance*a < r < b*from the center of the core.**•28.67**A 50-turn rectangular coil of wire of dimensions 10.0 cm by 20.0 cm lies in a horizontal plane, as shown in the figure. The axis of rotation of the coil is aligned north and south. It carries a current*i*= 1.00 A, and is in a magnetic field pointing from west to east. A mass of 50.0 g hangs from one side of the loop. Determine the strength the magnetic field has to have to keep the loop in the horizontal orientation.**•****28.68**Two long, straight parallel wires are separated by a distance of 20.0 cm. Each wire carries a current of 10.0 A in the same direction. What is the magnitude of the resulting magnetic field at a point that is 12.0 cm from each wire?**•****28.69**A particle with a mass of 1.00 mg and a charge of*q*is moving at a speed of 1000. m/s along a horizontal path 10.0 cm below and parallel to a straight current-carrying wire. Determine*q*if the magnitude of the current in the wire is 10.0 A.**•****28.70**A conducting coil consisting of*n*turns of wire is placed in a uniform magnetic field given by , as shown in the figure. The radius of the coil is*R*= 5.00 cm, and the angle between the magnetic field vector and the unit normal vector to the coil is*θ*= 60.0°. The current through the coil is*i*= 5.00 A.Specify the direction of the current in the coil, given the direction of the magnetic dipole moment, , in the figure.

Calculate the number of turns,

*n*, the coil must have for the torque on the loop to be 3.40 N m.If the radius of the loop is decreased to

*R*= 2.50 cm, what should the number of turns,*N*, be for the torque to remain unchanged? Assume that*i, B*, and*θ*stay the same.

**•28.71**A loop of wire of radius*R*= 25.0 cm has a smaller loop of radius*r*= 0.900 cm at its center such that the planes of the two loops are perpendicular to each other. When a current of 14.0 A is passed through both loops, the smaller loop experiences a torque due to the magnetic field produced by the larger loop. Determine this torque assuming that the smaller loop is sufficiently small that the magnetic field due to the larger loop is same across the entire surface.**•****28.72**Two wires, each 25.0 cm long, are connected to two separate 9.00-V batteries as shown in the figure. The resistance of the first wire is 5.00 Ω, and that of the other wire is unknown (*R*). If the separation between the wires is 4.00 mm, what value of*R*will produce a force of magnitude 4.00 · 10^{−5}N between them? Is the force attractive or repulsive?**•****28.73**A proton is moving under the combined influence of an electric field (*E*= 1000. V/m) and a magnetic field (*B*= 1.20 T), as shown in the figure.What is the acceleration of the proton at the instant it enters the crossed fields?

What would the acceleration be if the direction of the proton's motion was reversed?

**•****28.74**A toy airplane of mass 0.175 kg, with a charge of 36 mC, is flying at a speed of 2.8 m/s at a height of 17.2 cm above and parallel to a wire, which is carrying a 25-A current; the airplane experiences some acceleration. Determine this acceleration.**•28.75**An electromagnetic doorbell has been constructed by wrapping 70 turns of wire around a long, thin rod, as shown in the figure. The rod has a mass of 30.0 g, a length of 8.00 cm, and a cross-sectional area of 0.200 cm^{2}. The rod is free to pivot about an axis through its center, which is also the center of the coil. Initially, the rod makes an angle of*θ*= 25.0° with the horizontal. When*θ*= 0.00°, the rod strikes a bell. There is a uniform magnetic field of 900.0 G directed along*θ*= 0.00°.If a current of 2.00 A is flowing in the coil, what is the torque on the rod when

*θ*= 25.0°?What is the angular velocity of the rod when it strikes the bell?

**•****28.76**Two long, parallel wires separated by a distance,*d*, carry currents in opposite directions. If the left-hand wire carries a current*i*/2, and the right-hand wire carries a current*i*, determine where the magnetic field is zero.**•****28.77**A horizontally oriented coil of wire of radius 5.00 cm and carrying a current,*i*, is being levitated by the south pole of a vertically oriented bar magnet suspended above the center of the coil. If the magnetic field on all parts of the coil makes an angle*θ*of 45.0° with the vertical, determine the magnitude and the direction of the current needed to keep the coil floating in midair. The magnitude of the magnetic field is*B*= 0.0100 T, the number of turns in the coil is*N*= 10.0, and the total coil mass is 10.0 g.**•****28.78**As shown in the figure, a long, hollow, conducting cylinder of inner radius*a*and outer radius*b*carries a current that is flowing out of the page. Suppose that*a*= 5.00 cm,*b*= 7.00 cm, and the current*i*= 100. mA, uniformly distributed over the cylinder wall (between*a*and*b*). Find the magnitude of the magnetic field at each of the following distances*r*from the center of the cylinder:*r*= 4.00 cm*r*= 6.50 cm*r*= 9.00 cm

**•28.79**A wire of radius*R*carries current*i*. The current density is given by*J*=*J*_{0}(1 –*r*/*R*), where*r*is measured from the center of the wire and*J*_{0}is a constant. Use Ampere's Law to find the magnetic field inside the wire at a distance*r*<*R*from the central axis.**•****28.80**A circular wire of radius 5.0 cm has a current of 3.0 A flowing in it. The wire is placed in a uniform magnetic field of 5.0 mT.Determine the maximum torque on the wire.

Determine the range of the magnetic potential energy of the wire.

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