806 CHA P TER 24 Magnetic Fields and Fo rces

SUMMARY The goal of Chapter 24 has been to learn about magnetic fields and how magnetic fields exert forces on currents and moving charges.

GENERAL PRINCIPLES

Sources of Magnetism At its most fundamental level. magnetism is an interaction between moving charges. Magnetic fi elds can be created by e ither:

• Electric curren ts or • Permanent magnets

~MUCroscop ;c movement of chnrges as u current ~

--\ . ~ Microscopic

S mugoctism of clccutmii

The most bas ic uni t of mag­neti sm is the magnetic dipole, which consists of a north and a south pole.

Three bas ic kinds of dipole...;; are:

I~~ Current Permanent Atomic

loop magnet magnet

IMPORTANT CONCEPTS

Magnetic Fields The direc tion of the magnetic fie ld

is the direction in which the north pole or a compass need le points.

due to a current can be found from the right-hand rule for fields.

The strength of the magnetic field is

proportional to the torque on a compass needle when turned slightly from the fie ld direc tion.

measured in tesla (T)

APPLICATIONS

Consequences of Magnetism Magnet ic fie lds exert long-range forces on magneti c materials and on moving charges (or currents).

Unl ike poles of magnets attract each other; like poles repel each other.

Parallel wires with eUITen ts in the same direct ion attract each other; when the currents are in oppos ite directions, the wires repel each other.

II t t! •

J- ~I • ..l- [j t J-.

Magneti c fie lds exert torques on magneti c dipoles, lining them up with the field.

If two or more sources of magnetic fi eld are presen t, the principle of superposition applies.

Magnetic Forces and Torques The magnitude of the magnetic force on a moving charge depends on its charge q. its speed v, and the angle a. between the velocity and the field:

F ~ IqlvBs ;n a

The direction of thi s fo rce on a positive charge is given by the righ t-hand rule for forces.

The mag ni tude of the force on a cllrrenf-corl),ing wire perpendicular (0 the magnetic fi eld depends on the current and the length of the wire: F = ILB.

The torq ue on a cu rreri/loop in a magnetic field depends on the curren t, the loop's area, and how the loop is ori en ted in the fi eld: T ~ (iA)Bs ;n 8.

Fields due to common currents Charged-particle motion Stability of magnetic dipoles Long straight wire Current toop

Solenoid

®@@®®®®®®®®®®®

®®®®®®®®®®®®®®

No force if li is parallel to B.

x x x x x , x

x

x x x x x

If li is perpendicular to 8, the part i­cle undergoes uni form circular Illa tion with radius r = mV/lqlB.

A magnetic dipole is stable (in a lower energy state) when aligned with the external magnetic field. It is unstable (i n a higher energy state) when aligned oppos ite to the field.

The probe field of an MRI scanner measures the fl ipping of magnet ic dipoles be tween these two orien tations.

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Problem difficulty is labeled as I (straightforward) to 11111 (challenging),

QUESTIONS

Conceptual Questions

I. T he nonh pole of a bar magnet is brought near the center of another bar magnet, as shown in Figure Q24. 1. Will the force between the magnets be attractive, repulsive. or zero? Why?

Is

FIGURE Q24.1

2. You have a bar mag net whose poles are not marked. How can you fi nd which pole is north and which is south by using only a piece of str ing?

3. Whe n YOLI are in the southern hemisphere, does a compass point north or south?

4. Green turtles use the earth 's mag netic fi eld to navigate. They BIO seem to use the ri e ld to te U them the ir lat itude-how far north

or south of the equator they are. Ex plain how knowing the direction of the eillth 's fi eld could give thi s in formation.

5. Ahorsesfloe magnet consists of a bill magnet ben t into aU-shape, as shown in Figure Q24.5. Sketch the magneti c field li nes fo r a horseshoe magnet.

FIGURE 024.5

6. What is the current d irection in the wire of Figure Q24.6? Explain .

, ,

® 0 ,

, ,

FIGURE 024.6 FIGURE 024.7

7. What is the c urren t d irection in the wire of Figure Q24.7? 8. Since the wires in the wall s of your house carry current, you

might expect that you could use a compass to detect the positions of the wires . In fact, a compass will experi ence no de nec tion when brought near a c urrent-carry ing wire because the curre nt is AC (mean ing "alternat ing current"-the current switc hes direction 120 times each second). Explain why a compass doesn '( react to an AC current.

Questions 807

Problems labeled IN! integrate significant material from earlier

chapters; BID are of biological or medical interest.

9. Two wires carry currents in oppos ite directions, as in Figure Q24.9. The rield is 2.0 mT at a po int be low the lower wire. What are the streng th and d irection of the fi eld at point 1 (midway between the two wires) and at point 2 (the same distance above the upper wire as the 2.0 mT po in t is below the lower wire)?

· 2 ., FIGURE 024.9 - 2.0 mT out of page

10. As shown in Figure Q24. 10, a uniform magnet ic field points upward, in the plane of the paper. A long wire perpendicular to the paper ini tiall y carri es no current. When a c urren t is tu rned on in the wire in the direct ion shown, the magnetic fi e ld at point I is found to be zero. Draw the magneti c fi eld vector at point 2 whe n tbe current is on.

• 2·

ii

,.

FIGURE 024.10 FIGURE 024.11

II. Two long wires carry curre nts in the d irections shown in Figure Q24. 11 . One wire is 10 cm above the other. In which d irection is the mag netic fie ld at a po in t halfway between them?

12. If an electron is not moving, is it poss ible to set it in motion using a mag neti c field? Explain .

13. Figure Q24 .1 3 shows a so lenoid as seen in cross sect ion. Com­passes are placed at po in ts I and 2. In which direction will each compass point when the re is a large curre nt in the direction shown? Explain.

2 ·

.............•. ,.

FIGURE 024.13

808 CHAPTER 24 Magnetic Fields and Forces

14. One long so lenoid is placed in side anothe r soleno id with twice the dia· meter but Lhe same length. Each soleno id carries the same current but in opposite direclions, as shown in Figme Q24. 14. If they also have the same number of turns, in which direction does the mag· netic field in the cen ter point? Explain.

IS. What is the inilia/ direction of deflec. FIGURE 024.14

tion for the charged particles e ntering the magnet ic fi elds shown in Figure Q24. IS?

(,) (b)

• Ii •

FIGURE 024.15

16. What is the il/ilia/ direct ion of defl ect ion for the charged parti· c1es enteri ng the magnetic fie lds shown in Figure Q24. 16?

(a) - j

e-!-j j

FIGURE 024.16

j j j

(b)

17. Determine the magneti c fi eld direction that causes the charged partic les shown in Figure Q24. 17 to ex peri ence the indicated magneti c forces.

(,) (b)

~ , r , FIGURE 024.17 F f. into page

18. Determine the magnet ic field di rection that causes the charged partic les shown in Fi gure Q24. IS to experie nce the indicated magnet ic force s.

19.

(a) (b)

FIGURE 024.18 F out of page Ii inlo page

An e lectron is moving near a long, curre nt·carry ing wire, as shown in Figure Q24. 19. What is the direction of the magnet ic force on the electron ?

FIGURE 024.19

, --

20. Two posit ive charges are moving in a uniform magnetic field with veloc ities , as shown in Figure Q24.20. The magnetic force on each charge is also shown. In which d irect ion does the mag· netic field point?

f --t-~f'--'-. - y

x FIGURE 024.20

2 1. An electron is moving in a circu lar orbit in the earth' s magnetic field directly above the north magneti c pole. Viewed from above, is the rotation clockwise or counterclockwise?

22. A proton moves in a region of uniform magnetic fie ld, as shown in Figure Q24.22. The ve locity at one instant is shown. Will the subsequent motion be a clockwise or counterclockwise orbit?

Ions

. -,; . measured

- -1-- - - A B

- -+---'---~---L-~B (T) o 0.15 0.30

FIGURE 024.22 FIGURE 024.23

23 . The detector in a mass spectrometer records the number of ions measured at a fixed position as the fie ld is varied. For a sample consist ing of a single atomic species, two peaks were found where one was expected, as shown in Figure Q24.23. The most like ly explanation is that the atoms received d ifferent charges when ionized. If the two peaks correspond to ions wi th charges +e and +2e. which peak is which? Explain.

24. A proton is moving near a long, current·canying wire. When the proton is at the point shown in Figure Q24.24, in which direc· tion is the force on it?

I' @

I' Wire

@ Wire Proton Proton

FIGURE 024.24 FIGURE 024.25

2S. A proton is moving near a long, curren t·carrying wire . When the proton is at the point shown in Figure Q24.2S, in wh ic h d irection is the force on it?

26. A long wire and a square loop lie in the plane of the paper. Both carry a current in the d irect ion shown in Figure Q24.26. In wh ich d irection is Lhe net force on the loop? Explain .

FIGURE 024.26 L

y

fIl L

27. The computers that contro l MRI machines cannot have CRT monitors. Explain why this is so.

c 28. A Slinky is a child's lOy thal is a long coil spring.

Suppose yo u take a Slinky and let it hang down and stretch out so that the coils do nOllo Lich each other, as in Figure Q24.28. Now you connect the Slinky to a power supply and pass a large DC cur~ rent through il. Think about the current in the coils. Will the Slinky expand or contract?

FIGURE Q24.28

29. A solenoid carries a c urrent that produces a fie ld ins ide il. A wire carryil,1g a c urrent lies inside the solenoid , at the cente r, carrying a current along the so lenoid 's axis. Is there a force on this wire due to the fie ld ofthe solenoid? Explain.

30. You want to make an electromagnet by wrapping wire around a nail. Should you use bare copper wire or wire coaled with inslI­lating plasti c? Explain.

3 1. The moo n does not have a molten iron core like the earth , but the moon does have a small magnetic fi eld. What might be the source of thi s fi eld?

32. Arc haeolog ists can use instruments that measure small varia­tions in magnet ic field to locate buried walls made of fired brick, as shown in Figure Q24.32. When fired , the mag­neti c mome nts in the clay become ran­do ml y aligned; as the clay cools, the magnet ic mo ments line up with the earth ' s field and retain thi s alignment even if the bricks are subsequen tly moved. Explain how thi s leads to a mea­surable mag neti c fi e ld variation over a buried wall. FIGURE 024.32

Multiple-Choice Questions

33. 11 An unmag neti zed metal sphere hangs by a thread. When the north pole of a bar magnet is brought near, the sphere is strong ly attracted to the mag­net, as shown in Figure Q24.33. Then the magnet is re versed S

and its south pole is bro ught FIGURE 024.33

near the sphere. Ho w does the sphere respond? A. It is strongly attracted to the magnet. B. It is wcakJy attracted to the magnet. C. Il does not respo nd. D. It is weak.ly repelled by the magnet. E. It is strongly repelled by the magnet.

PROBLEMS

Section 24.1 Magnetism

Section 24.2 The Magnetic Field

Section 24.3 Electric Currents Also Create Magnetic Fields

Section 24.4 Calculating the Magnetic Field Due to a Current

I . I What c urrents are needed to generate the magnetic fi eld strengths of Table 24 .1 at a point 1.0 em from a long, straig ht wire?

Problems 809

34. II If a compass is placed above a current-carry ing wi.re , as in Figure Q24.34, the needle will line up with the field o f the wire. Which o f the views shows the correct ori e n­tatio n of the need le for the noted current direction?

A.~

B.~

c.~

D.~

FIGURE 024.34

35 . I Two wires carry equal and oppo- 0 • ® site curre nts, as show n in Fi gure

FIGURE 024.35 Q24.35. At a point directly between the two wires, the field is A. Directed up. toward the top of the page. S. Directed down, toward the bottom of the page. C. Directed to the lefl. D. Directed to the right. E. Zero.

36. I Figure Q24.36 shows four particles moving to the ri ght as they enter a reg io n of uniform magneti c field , directed into the paper as noted . All par­ticles move at the same speed and ha ve the same charge. Which particle has the largest mass?

FIGURE 024.36

37 . I Four particles of identi cal charge and mass enter a region o f uniform magnet ic field and fo llow the trajectori es show n in Figure Q24.37. Which particle has the highest veloc ity?

38.

39.

FIGURE 024.37 A. B. C. D.

I If aU of the partic les shown in Figure Q24.37 are electrons, what is the direction of the magnet ic fi eld that produced the indicated deflection? A. Up (toward the top of the page). B. Down (toward the bottom o f the page) . C. Out of the plane of the paper. D. Into the plane of the paper. I If two compasses are brought near enough to each other, the mag net ic fields of the com­passes the mselves will be larger than the field o f the earth, and the needles will line up with each other. Which o f the arrangements of two compasses shown in Figure Q24.39 is a possi-ble stable arrangement?

A.@@ B @@ e @@ D.@ @

FIGURE Q24.39

2. I At what di stances from a very thin , straight wire carrying a lOA curre nt wo uld the magnetic ficld strengths of Table 24. 1 be generated?

3. II The magnetic fi eld at the center of a I .G-cm-diameter loop is 2.5 mT. a. What is the current in the loop? b. A long , straight wire carries the same c urrent you found in

part a. At what distance from the wire is the magnet ic field 2.5 mT?

810 CHAPTER 24 Magnetic Fields and Forces

4. 111 For a particular sc ientific experiment, it is important to be complc tely iso lated from any magneti c fi e ld, including the earth 's fie ld . A I.OO-m-diametercurrent loop with 200 turns of wire is set up so that the field at the center is exact ly equal to the earth's field in magnitude but opposite in direction. What is the current in the current loop?

5. I What are the magnetic field I ·

strength and direc tion 2.0cm at points 1 to 3 in Figure P24.5? lOA

2· 4.0cm lOA

FIGURE P24.S 3· 2.0 Clll

6. 1 Although the ev idence is weak, there has been concern in BID recent yea rs over poss ible health effects from the magneti c:

fi e lds generated by transmiss ion lines. A typical high-voltage transmission line is 20 m off the ground and carries a current of 200 A. Estimate the magnetic field st rength on the ground underneath such a line. What percentage of the earth's magneti c fi eld does this represent?

7. 1 Some consumer groups urge pregnant women not to useelec­BK) tri c blankets, in case there is a health risk from the magnet ic

fields from the approximately I A current in the heater wires. a. Estimate, stat ing any assumptions yo u make, the magnct ic

fi eld strength a fetus might experience. What percentage of the earth 's magnetic field is thi s?

b. It is becoming standard practice to make elec tri c blankets with minimal ex ternal magnetic field. Each wire is paired with another wire that carr ies curre nt in the oppos ite direc­lion . How does thi s reduce the external magneti c fi e ld?

8. III A long wire carrying a 5.0 A current perpendicular to the .\')'-plane inte rsec ts the x-ax is at x = -2.0 cm. A second , paral ­lel wire carrying a 3.0 A current intersects the x-ax is at x = +2.0 cm. At what point or points on the x-ax is is the mag­net ic fi eld zero if (a) the two currents are in the same di rection and (b) the two curren ts are in opposite direc ti ons?

9. II The element niobium, which is a me tal, is a superconductor (i.e. , no e lec trical res istance) at temperatures below 9 K. How­ever, the superconductivity is destroyed if the magnetic field at the surface of the wire of the me tal reaches or exceeds 0. 10 T. What is the max imum current in a stra ight, 3.0-mm-diameter superconducting niobium wire? Hillt: You can assume that all the current nows in the center of the wire.

10. I The small currents in axons corresponding to nerve impulses BID produce measurable magnetic fi elds. A typical axon carries a

peak current of 0.040 J.LA . What is the strength of the fie ld at a di stance of 1.0 mm?

II . II A solenoid used to produce magnetic fields for research pu!'­poses is 2.0 m long, with an inner rad ius of 30 cm and 1000 turns of wire. When running, the solenoid produces a field of 1.0 T in the center. Given thi s, how large a current does it carry?

12. I Two concentric eUITCnt loops lie in the same plane. The smal ler loop has a radius of 3.0 cm and a current of 12 A. The bigger loop has a current of20A. The magnetic field at the center of the loops is found to be zero. What is the radius of the bigger loop?

13. I The magneti c field of the brain has been measured to be BID approx imate ly 3.0 X 10- 12 T . Although the currents that cause

thi s fie ld are quite complicated, we can get a rough es timate of their size by modeling them as a single circular currcnt loop 16cm (the width of a typica l head) in diameter. What current is needed to produce such a fi eld at the center of the loop?

14. II A researcher would like to perform an experiment in zero magnetic fie ld, which means that the field of the earth must be cancelled. Suppose the experiment is done inside a solenoid of diameter 1.0 m, length 4.0 m, with a total of 5000 turns of wire.

15.

The solenoid is oriented to produce a fi e ld that opposes and exac tly cancels the fie ld of the earth . What current is needed in the solenoid 's wire? III What is the magneti c fi e ld at the center of the loop in Figure P24.15? a

FIGURE P24.1S

5.DA

16. I Experimental tests have shown that hammerhead sharks can BIO detect magneti c fi elds. In one such test, 100 turns of wire were

wrapped around a 7.0-m-diamete r cy lind rical shark tank. A magnctic field was created inside the ta nk when thi s co il of wire carri ed a current of 1.5 A. Sharks trained by gelling a food reward when the field was prese nt wou ld late r unambiguously respond when the field was turned on. a. What was the magnet ic fi eld strength in the center of the

tank due to the current in the co il? b. Is the strength of the co il' s fi eld at the center of the tank

larger or smaller than thaI of the earth? 17. I We have seen that the heart produces a magnetic field that BID can be used to diagnose problems with the heart. The magnetic

field of the heart is a dipole field produced by a loop curre nt in the outcr layers of the. heart. Suppose the fie ld at the cen ter of the heart is 90 pT (a pT is 10 .... 12 T) and that the heart has a diam­ete r of approximately 12 Col. What current circu lates around the heart to produce this field?

18. 11111 You have a l.O-m-long copper wire. You want LO make an N-turn current loop that gene rates a 1.0 mT magnetic fi eld at the cente r when the current is 1.0 A. You must use the entire wire. What wilJ be the diameter of your coil?

19. lUll In the Bohr model of the hydrogen atom, the e lecu'on moves in a c ircular orbi t of radius 5.3 X 10- 11 m with speed 2.2 X

106 m/s. According to this mode l, what is the magnetic fi eld at the center of a hydrogen atom due to the motion of the electron? Hint: Determine the avemge current of the orbiting electron.

Sect ion 24.5 Magnetic Fields Exert Forces on Moving Charges

20. I A proton moves with a speed of l.0 X 107 m/s in the direc­tions shown in Figure P24.20. A 0.50 T magnetic fi eld points in the pos itive x-direct ion. For each. what is magneti c force on the proton? Give your answers as a magnitude and a direction.

(. ) (b)

y !' , x x

li Ii

FIGURE P24.20

2 1. n An electron moves with a speed of 1 .. 0 X 107 m/s in the di rec­

tions shown in Figure P24.2 1. A 0.50 T magnet ic field points in the positive x-d irection. For each, what is magnetic force F on the electron? G ive your answers as a magn itude and a d irecti on.

(a) (h)

)'

------,J..:F-/-y~x ii

JiiF----~-x jj

FIGURE P24 .21

22. I An electromagnetic flowmeter applies a magne tic field 0[0.20 T

BKJ to blood flowing through a coronary anery at a speed o f 15 em/s. What force is fell by a chlorine ion with a s ingle negative charge?

23. I The aurora is caused when electrons and protons, movi ng in the earth 's magnet ic field of ;::;:: 5.0 X 10- 5 T , collide w ith mole­cules of the atmosphere and cause them to glow. What is the radius of the c ircu lar orbit for a. An electron with speed 1.0 X 106 m/s? b. A prOlo n with speed 5.0 X 10"" m/s?

24. III Problem 24.23 describes two particles that orbit the earth's

mag netic fie ld li nes. What is thejreqllellcy of the c ircul ar orbit for a. An electron w ith speed 1.0 X lOb Ill /s? b. A proton with speed 5 .0 X 104 m/s?

25. 1111 The microwaves in a microwave oven are produced in a spec ial

lube called a maglleflvll. The electrons orbit in a magnet ic fie ld at a frequency of2.4 GHz, and as they do so they emit 2.4 GHz elec·

lromagnetic waves. W hat is the strength o f the magnetic field? 26. III A mass spectrometer similar to the one in Figure 24.36 is

des igned to separate protein fragme nts. The fragmen ts are ion· ized by the removal of a si ngle e lectron, then they ente r a 0.80 T

uni fo rm magnetic fie ld at a speed o f 2.3 X 105 m/s. If a frag­

ment has a mass 85 times the mass of the proton. what w ill be the d is tance between the points where the ion enters and exits

27.

28.

the magneti c field? I In a certain mass spectrometer, particles with a charge of +e are sen t into the spectrometer w ith a veloc ity of 2.5 X 105 m/s. They are found to move in a circul a r path w ith a radius of

0.2 1 m. If the mag netic fi e ld of the spec tro meter is 0 .050 T,

what kind o f particles are these like ly to be? II I At I = 0 s, a proton is moving

with a speed of 5 .5 X 10 5 mls at an ang le o f 30° from the x-ax is,

as shown in Figure P24.28.

A uniform magnetic fie ld of magnitude 1.50 T is pointing in

the pos itive y-d irection. W hat will be the .v-coordinate of the

proton 10 J..Ls later?

y

ii

- ---1.L--L---- x

FIGURE P24.28

29. III Early black-and-white television sets used an e lectron beam

lNT to draw a picture on the screen. The electrons in the beam were acce leraled by a voltage of 3.0 kV; the beam was the n steered to

different points on the screen by co il s of wire that produced a

magne ti c fi eld of up to 0.65 T. a. What is the speed of electrons in the beam?

Problems 811

b. What accelerati on do they experience due to the magneti c

field, assuming that it is perpendicu lar to the ir path? What is

thi s accelerat io n in units of g? c. If the elecu'ons were to complete a full c ircular orbit, w hat

would be the radius? d. A magneti c fi e ld can be used to redirect the beam, but tbe

electrons are brought to hi gh speed by an e lectric field. Why can ' t we use a magnetic fie ld for this task?

Section 24.6 Magnetic Fields Exert Forces on Currents

30. I What magnetic fie ld strength and d irec ti on will lev it ate the

2.0 g wi re in Figure P24.30?

8-field region 1.5 A

FIGURE P24.30 tOem

2.0g wire

I

3 1. I What is the ne t force (magnitude and direct ion) on each wire

in Figure P24.31?

lOA

lOA 2 2.0em

3 2.0cm

lOA FIGURE P24.31 50cm

32. I The unit of curre nt, the ampere, is defined in terms of the force betwee n curre nts. I f two I.O·meter. long sect ions of very

lo ng wires a d istance 1.0 m apart each carry a curre nl of 1.0 A ,

what is the force between them? ( Ir the force between two actual wires has thi s value, the current is defined to be exactly I A .)

33. I A unifonn 2.5 T magnetic field points to the right. A 3.0-m.long

wire, carry ing 15 A , is placed a t an angle o f 30° to the fie ld, as

shown in Figure 24.33. What is the force (magnitude and di rec­

tion) on the w ire?

ii

FtGURE P24.33

34. I The four w ires in Figure P24.34 are lilted at 20° w ith respect

to a uniform 0 .35 T fi e ld. If each ca rri es 4.5 A and is 0.35 m long, what is the force (di rect ion and magn itude) o n each?

FIGURE P24 .34

812 CHAPTER 24 Magnetic Fields and Forces

35. INT

II Magnetic information on hard drives is accessed by a read head that must move rapidly back and forth across the disk. The force to move the head is generally created with a voice coil actuator, a nat coil aI' fine wire that moves between the poles of a strong magnet, as in Figure P24.35. Assume that the co il is a square 1.0 cm on a side made of 200 turns of fine wire with total resistance 1.5 n. The field between the poles of the magnet is 0.30 1~ assume that the field does not extend beyond the edge of the magnet. The co il and the mount that it rides on have a total mass of 12 g. a. If a voltage of 5.0 V is applied to the co il , what is the cur­

rent? b. If the current is clockwise viewed from above, what are the

magnitude and direction of the net force on the coil? c. What is the magnitude of the accelerat ion of the coil?

s 7 FIGURE P24.35 Side view Top view

Section 24.7 Magnetic Fields Exert Torques on Dipoles

36. II A current loop in a molOr has an area of 0.85 cm2. Il carries a

240 rnA current in a uniform fie ld of 0.62 T. What is the magni­tude of the maximum torque on the current loop?

37. II A square current loop 5.0 cm on each side carries a 500 rnA current. The loop is in a 1.2 T uniform magnet ic field. The axis of the loop, perpendicular to the plane of the loop, is 30° away from the field direct ion. What is the magnitude of the torque on the current loop?

38. I Figure P24.38 shows two square current loops. The loops are far apart and do not interact with each other. a. Use force diagrams to show that both loops are in equilib­

rium, having a net force of zero and no torque. b. One of the loop positions is stable. That is, the forces will

return it to equilibrium if it is rotated slightly. The other position is unstable, like an upside-down pendulum: If rotated sli ghtly, it will not return to the position shown . Which is which? Expla in.

Loop I Loop 2

Ii

FIGURE P24.38

39. 1111 The earth' s magnetic dipole moment of 8.0 X 1022 A· m2 is generated by currents within the molten iron of the earth's outer core . (The inner core is solid iron.) As a simple model, consider the outer core to be a 3000-km-diameter current loop. What is the curren t in the curren t loop?

40. II a. What is the magnitude of the torque on the c ircular cur­rent loop in Figure P24.40?

b. What is the loop 's equilibrium position?

2.0C1l1

® - Wirc 2.0A

o [20"" FIGURE P24 .40 0.20 A

Section 24.8 Magnets and Magnetic Materials

41. II A computer di skette is a plasti c disk with a ferromagnetic coat­ing. A single magnetic domain can have its magnet ic moment oriented to point ei ther up or down, and these two orientations can be interpreted as a binary 0 (up) or I (down). Each 0 or I is called a bir of information. A diskeue stores roughJy 500,000 bytes of data on one side, and each byte contains e ight bits. Esti­mate the width of a magnetic domain, and compare your answer to the typical domain size given in the text. List any assump­tions you use in your est imate.

42. I All ferromagnetic materials have a Curie remperalllre, a tem­perature above which they wiU cease to be magnetic. Explain in some detail why you might expect this to be so.

General Problems

43 . II In Figure P24.43, a compass sits 1.0 cm above a wire in a cir­[NT cuit con taining a 1.0 F capacitor charged to 5.0 V, a 1.0 n resis­

tor, and an open switch. The compass is lined up with the earth's magneti c rield. The switch is then closed, so there is a current in the c ircuit, and the switch remains closed until the capacitor has completely discharged. a. At the position of the compass, what is the magnitude of the

magneti c field due to the current in the wire right after the switch is closed? How does thi s compare with the magni­tude of the field orthe em'lh?

b. Describe how the compass ori en tation changes ri ght after the switch is closed, and how the comp;:t<o;s orientation changes as time goes on.

FIGURE P24.43

44. I The ri ght edge of the c ircuit in Figure P24.44 extends into a [NT 50 mT uniform magnetic field. What are the magnitude and

direction of the net force on the c ircu il?

3.0n

I5V=€;n •

FIGURE P24.44 B = 50mT

45. II The IWO 10-em-long parallel wires in Figure P24.45 are sepa­INT rated by 5.0 mm. For what val ue of the resistor R will the force

between the IWO wires be 5.4 X 10- 5 N?

2.0n R

9.0V (?c,,[ [ [3:90V

FIGURE P24.4S 5.0 111m

46. III The capac itor in Figure P24.46 is charged to 50 Y. The switch INT closes at 1 = 0 s. Draw a graph showing the magnet ic fi eld

strength as a funct ion of time at the pos ition of the dol. On your grap h indicate the max imum field strength and prov ide an appropriate numerical scale on the horizontal ax is.

47. INT

48. INT

49.

INT

50.

INT

I.Oem 1++++++++++1 5.0n _ -_, 1.0c",1

I

FIGURE P24 .46 FIGURE PZ4.47

II An elec tron trave ls with speed 1.0 X 107 mls between the two parallel charged plates shown in Figure P24.47. The plates are separated by 1.0 cm and are charged by a 200 V battery. What magnet ic field strength and direction will allow the elec· tron to pass between the plates without being deflec ted? II The two springs in Figure P24.48 each have a spring constant of 10 N/m. They are stretched by 1.0 cm when a current passes through the wire. How big is the curren t?

20cm -=-- 1.2 V !5em x

"\" B = O.SOT

8=O.SO T

FIGURE P24.48 FIGURE PZ4.49

II A device called a rai/guf/ uses the mag netic force on currents to launch projecti les at very high speeds. An ideali zed model of a railgun is illustrated in Figure 24.49. A 1.2 V power supply is connected to two conducting rails. A segment of copper wire, in a reg ion of uni form magnetic field, sl ides freely on the ra il s. The wire has a 0.85 mn resistance and a mass of 5.0 g. Ignore the res istance of the rails. When the power supply is switched on, a. What is the current? b. What are the mag ni tude and direct ion of the force on the

wire? c. What wiLl be the wire's speed after it has s lid a di stance of

6.0 em? An antiproton (which has

the same prope rti es as a pro­ton except that it s charge is -e) is moving in the com­bined electri c and magneti c fields of Figure P24.50. a. What are the magn itude

and direct ion of the ant iproton's accelerat ion at thi s instant?

J£25T ; I I / ,.~

E = tOOOV/rn

FIGURE P24 .50

b. What would be the magnitude and direction of the accelera­tion if V' were reversed?

Problems 813

5 1. I Typical blood veloc ities in the coronary arteries range from INT 10 to 30 cm/s. An electromagnetic flowmeter applies a magnet ic BIO field of 0.25 T to a coronary artery wi th a blood veloc ity of 15

cm/s. As we saw in Figure 24.38, thi s field exerts a force on ions in the blood, which will separate. The ions will separate un til they make an electric fi eld that exac tly ba lances the magnetic force. This electric tield produces a voltage that can be measured. a. What force is fell by a singly ioni zed (pos itive) sodium ion? b. Charges in the blood will separate unt il they produce an

electric field that cancels this mag net ic force. What will be the resulting electric fie ld?

c. What vo ltage wi ll thi s elec tric field produce across an artery with a diameter of 3.0 mm?

52. I A power line consists of two wires, each carry ing a current of 400 A in the same direc tion. The lines are perpendicular to the earth's magneti c fie ld, and are separated by a distance of 5.0 m. Which is larger: the force of the earth' s magnetic field on each wi re or the magneti c fo rce between the wires?

53. III Bats are capable of nav igating using the earth 's fi eld-a plus BID for an an imal that may fl y great distances from its roost at night.

If, whi le sleeping duri ng the day, bats are exposed to a field of a similar mag nitude but differe nt direc tion than the earth 's fi eld, they are more likely to lose their way during the ir nex t lengthy night fli ght. Suppose YOll are a researcher doing such an experi ­ment in a locat ion where the earth 's fi eld is 50,u.T at a 60° angle below horizonta l. You make a 50-em-diameter, 100-turn co il around a roosting box; the sleepi ng bats are at the center of the coil. You wish to pass a curren t thro ugh the coil to produce a field that, when combi ned with the earth's fi eld, creates a net field with the same strength and dip angle (60° be low hori zon­tal) as the earth's field bu t with a horizontal componen t that points south rather than north. W hat arc (he proper ori entation of the coi I and the necessary current?

54. III At the equator, the earth 's field is essen tially horizon tal; near BID the north pole, it is nearly vertical In between, the angle varies.

As you move farther north, the dip angle, the angle of the earth 's field below horizontal , stead ily inc reases. Green tu rtles seem to use thi s dip angle to determi ne the ir lat itude. Suppose you are a researcher wan ti ng to test thi s idea. You have gathered green turt le hatchli ngs from a beach where the mag ne tic fie ld strength is 50,u.T and the di p angle is 56°. You then put the tur­tl es in a 1.2-m-diameter c ircular tank and monitor the direct ion in which they swi m as you vary the magnetic field in the tank. You change the fi eld by passing a current through a 100-turn hori zontal co il wrapped around the tank. This creates a field that adds to that of the earth. What current should you pass through the coil, and in what direction, to produce a net fiel d in the center of the tank that has a dip angle of 62°?

55 . II Internal components of cathode-ray-tube te lev isions and computer monitors can become magnetized; the res ulting mag­neti c field can de flect the e lectro n beam and distort the colors on the screen. Demagneti zation can be accomplished with a co il of wire whose current switches direct ion rapidly and gradua ll y decreases in ampl itude. Explain what effect thi s will have on the magnet ic moments of the magnet ic materi als in the dev ice, and how thi s might el iminate any magneti c ordering.

56. III A 1.0-m-long, 1.0-mm-diameter copper wire carri es a current INT of 50.0 A to the east. Suppose we crea te a magne tic fi eld that

prod uces an upward force on the wire exact ly equal in magn i­tude to the wire's weight, causing the wire to " lev itate." What are the fi eld 's direct ion and magn itude?

814 CHAPTER 24 Magnetic Fields and Forces

57. 11 An insulated copper wire is wrapped around an iron nail. The resulting co il of wire cons ists of 240 turns of wire that cover 1.8 em of the nail, as shown in Figure P24.57. A current of 0.60 A passes through the wire. If the ferromagnetic propert ies of the na il increase the field by a factor of 100, what is the magnetic field strength ins ide the nail?

FIGURE P24 .57

58. 1111 Figure P24.58 is a cross sec tion through INT three long wires with linear mass density

50 g/m. They each carry equal CUlTents in the directions shown. The lower two wires are 4.0 em apart and are attached to a table. What current ! will allow the upper wire to '; fl oat" so as to form an equil ateral triangle with the lower wires?

59.

INT 111 A long, straight wire with a linear mass density of 50 glm is suspended by threads, as shown in Figure P24.59. There is a uniform magnetic field pointing veltically downward. F A 10 A current in the wire

09 09

4.0Clll

FIGURE P24 .S8

10"

" experi ences a horizontal mag­netic force that deflects it to an equilibrium angle of 100. 'What is the strength of the magnetic field B?

FIGURE P24 .59

60. 11 A mass spectrometer is designed to sep<ll'<lte atoms of carbon to determine the frac tion of different isotopes. (Isotopes of an

a. Make a sketch showing the direction of the magnetic field from the solenoid. On your sketch, label the induced north magnetic pole and the induced south magnetic pole in the iron .

b. Will the force on the iron be attrac ti ve or repulsive? c. Suppose thi s force moves the iron. Which way will the iron

move?

Passage Problems

The Velocity Selector INT Actj"V physcs 13.8

In experiments where all the charged particles in a beam are required to have the same veloc ity (for example, when entering a mass spec­trometer), sc ienti sts lise a velocity seleClOl: A velocity selector has a region of uniform electric and magnetic fi elds that are perpendicular to each other and perpendicular to the motion of the charged particles. Both the electric and magnetic fields exert a force on the charged parti­cles. If a particle has precisely the right velocity, the two forces exactly cancel and the particle is not deflected. Equating the forces due to the electri c field and the magnetic field gives the following equation:

qE= qvB

Solving for the ve loc ity, we get:

E )'= -

B

A particle movi ng at this veloci ty will pass through the reg ion of uniform fields with no deflection, as shown in Figure P24.62. For higher or lower velocities than thi s, the particles will feel a net force and will be deflected. A slit at the end of the region allows only the

element, as we will see in Chapter 30. have the same atomic particles with the correct velocity to pass.

61.

number but different atomic mass, due to different numbers of neutrons.) There are three main isotopes of carbon. with the fol-lowing atomic masses:

Atomic masses

1.99 X 10- 26 kg

2.16 X 10- 26 kg

2.33 X 10- 26 kg

The atoms of carbon are si ngly ionized and enter a mass spec­trometer with magneti c fie ld strength B = 0.200 T at a speed of 1.50 X 105 m/s. The ions move along a semic ircular path and exit through an exit sl iL How far from the en trance will the beams of the different isotope ions end up? I A solenoid is near a piece of iron, as shown in Figure P24.61. When a curren t is present in the solenoid, a magnetic field is created. This magnetic field will mag netize the iron, and there wili be a net force between the solenoid and the iron.

FtGURE P24 .61

Curren! in solenoid is clockwise :IS

viewed from the right end.· .... ..... ....

•

Piece of iron is lined up with the ax is of the solenoid .

if

, ~ ------~--+----r---1"

FIGURE P24.62

62. I Assuming the particle in Figure P24.62 is positively charged, what are the direct ions of the forces due to the electric field and to the magnet ic field? A. The force due to the e lectric field is directed up (toward the

top of the page); the force due to the magne ti c field is directed down (toward the bottom of the page).

B. The force due to the elec tri c field is directed down (toward the bOllom of the page); the force due to the magnetic field is directed up (toward the top of the page).

C. The force due to the electric field is directed out of the plane of the paper; the force due to the magnet ic field is directed into the plane of the paper .

D. The force due to the electric field is directed into the plane of the paper; the force due to the magneti c field is directed out of the plane of the paper.

63. I How does the kinetic energy of the part icle in Figure P24.62 change as it traverses the velocity selector? A. The kinetic energy increases. B. The kinetic energy does not change. C. The kinetic energy decreases.

64. I Suppose a panicle with twice the velocity of the particle in Figure P24.62 enters the velocity selector. The path of thi s par­ticle will curve A. Upward (toward the top of the page). B. Downward (toward the bOllom of the page). C. Out of the plane of the paper. D. Into the plane of the paper.

65. I Next, a particle with the same mass and veloc ity as the pani­cle in Figure P24.62 enters the velocity selec tor. This particle has a charge of 2q-twice the charge of the particle in Figure P24.62. In thi s case, we can say that A. The force of the electr ic field on the particle is greater than

the force of the magnetic field. B. The force of the magnetic field on the particle is greater

than the force of the electric fi eld. C. The forces of the electric and magnetic fields on the particle

are sti II equal.

Ocean Potentials INT

The ocean is salty because it contains many dissolved ions. As these charged panicles move with the water in strong ocean cUlTen ts, they feel a force from the earth' s magnetic field. Positive and negative charges are separated unti I an e lectri c fi eld develops that balances this magnetic force. This field produces measurable potential differ­ences that can be monitored by ocean researchers.

Stop to Think 24.1: C. The compass needle will not rotate since there is no force between the stationary charges on the rod and the magnetic poles of the compass needle.

Stop to Think 24.2: A. The compass needle will rotate to line up with the field of the magnet, which goes from the north to the south pole.

Stop to Think 24.3: D. The compass needle will rotate to line up with the field cirding lhe wire. The ri ght-hand rule for fields shows this to be toward the top of the paper in the fi gure.

Stop to Think 24.4: A, C. The force to produce these circular orbits is directed toward the center of the circle. Using the ri ght-hand rule for forces, we see that thi s will be true for the situations in A and C if the particles are negati vely charged.

Problems 815

The Gulf Stream moves northward off the east coas t of the United States at a speed of up 1.0 3.5 m/s. Assume that the current fl ows at this maximum speed and that the earth ' s field is SO J..LT tipped 60° below hori zon tal.

66. I What is t.he direction of the magnetic force on a singly ionized negati ve chlorine ion moving in this ocean current? A East 8 . West C. Up D. Down

67. I What is the magnitude of the force on this ion? A. 2.8 X 10- 23 N B. 2.4 X 10- 23 N C. 1.6 X 10- 23 N

D. 1.4 X 10 23 N 68. I What magnitude electric field is necessary to exactly balance

thi s magnet ic force? A. 1.8 X 10- ' N/C

B. 1.5 X 10-' NlC C. 1.0 X 10-' N/C D. 0.9 X 10-4 NlC

69. I The electric field produces a potential difference. If YO LI place one elec trode 10m below the surface of the water, you will measure the greatest potential difference if you place the second electrode A At the surface. 8. Aladepthof20m. C. At the same depth 10m to the north. D. At the same depth 10 m to the east.

Stop to Think 24.5: C. The ri ght-hand rule for forces gives the direction of the force. With the field into the paper, the force is to the left if the current is toward the top of the paper.

Stop to Think 24.6: B. Looking at the forces on the top and the bot­tom of the loop, we can see that the loop will rotate counterclock­wise. Alternatively, we can look at the dipole structure of the loop: With a north pole on the left and a south pole on the ri ght, the loop will rotate counterclockwise.

Stop to Think 24.7: B. All of the induced dipoles will be aligned with the field of the bar magnet.