960 CHAPTE R 3 0 • Sources of the Magnetic Field

of the can and I the upward current, uniformly distributedover its curved wall. Determine the magnetic field (a) justinside the wall and (b) just outside. (c) Determine thepressure on the wall.

28. Niobium metal becomes a superconductor when cooledbelow 9 K. Its superconductivity is destroyed when thesurface magnetic field exceeds 0.100 T. Determine themaximum current a 2.00-mm-diameter niobium wire cancarry and remain superconducting, in the absence of anyexternal magnetic field.

A long cylindrical conductor of radius R carries a current Ias shown in Figure P30.29. The current density J , however,is not uniform over the cross section of the conductor butis a function of the radius according to J ! br, where b is aconstant. Find an expression for the magnetic field B(a) at a distance r1 " R and (b) at a distance r 2 # R,measured from the axis.

29.

23. Figure P30.23 is a cross-sectional view of a coaxial cable.The center conductor is surrounded by a rubber layer,which is surrounded by an outer conductor, which issurrounded by another rubber layer. In a particularapplication, the current in the inner conductor is 1.00 Aout of the page and the current in the outer conductor is3.00 A into the page. Determine the magnitude anddirection of the magnetic field at points a and b.

30. In Figure P30.30, both currents in the infinitely long wiresare in the negative x direction. (a) Sketch the magneticfield pattern in the yz plane. (b) At what distance d alongthe z axis is the magnetic field a maximum?

Section 30.4 The Magnetic Field of a SolenoidWhat current is required in the windings of a long

solenoid that has 1 000 turns uniformly distributed over alength of 0.400 m, to produce at the center of the solenoida magnetic field of magnitude 1.00 $ 10%4 T?

32. Consider a solenoid of length ! and radius R , containingN closely spaced turns and carrying a steady currentI. (a) In terms of these parameters, find the magneticfield at a point along the axis as a function of distancea from the end of the solenoid. (b) Show that as !becomes very long, B approaches &0NI/2! at each end ofthe solenoid.

31.

ba1.00 A

1 mm 1 mm 1 mm

3.00 A

. .

!!

!

!!

!

!

!

Figure P30.23

Figure P30.29

Figure P30.30

Rr1

I

r2

xy

a

a

I

I

z

24. The magnetic field 40.0 cm away from a long straight wirecarrying current 2.00 A is 1.00 &T. (a) At what distance isit 0.100 &T? (b) What If? At one instant, the twoconductors in a long household extension cord carryequal 2.00-A currents in opposite directions. The two wiresare 3.00 mm apart. Find the magnetic field 40.0 cm awayfrom the middle of the straight cord, in the plane of thetwo wires. (c) At what distance is it one tenth as large?(d) The center wire in a coaxial cable carries current2.00 A in one direction and the sheath around it carriescurrent 2.00 A in the opposite direction. What magneticfield does the cable create at points outside?

A packed bundle of 100 long, straight, insulated wiresforms a cylinder of radius R ! 0.500 cm. (a) If each wirecarries 2.00 A, what are the magnitude and direction of themagnetic force per unit length acting on a wire located0.200 cm from the center of the bundle? (b) What If ? Woulda wire on the outer edge of the bundle experience a forcegreater or smaller than the value calculated in part (a)?

26. The magnetic coils of a tokamak fusion reactor are in theshape of a toroid having an inner radius of 0.700 m and anouter radius of 1.30 m. The toroid has 900 turns of large-diameter wire, each of which carries a current of 14.0 kA.Find the magnitude of the magnetic field inside the toroidalong (a) the inner radius and (b) the outer radius.

27. Consider a column of electric current passing throughplasma (ionized gas). Filaments of current within thecolumn are magnetically attracted to one another. Theycan crowd together to yield a very great current densityand a very strong magnetic field in a small region.Sometimes the current can be cut off momentarily by thispinch effect. (In a metallic wire a pinch effect is notimportant, because the current-carrying electrons repelone another with electric forces.) The pinch effect can bedemonstrated by making an empty aluminum can carry alarge current parallel to its axis. Let R represent the radius

25.

958 CHAPTE R 3 0 • Sources of the Magnetic Field

10. A very long straight wire carries current I. In the middle ofthe wire a right-angle bend is made. The bend forms

Figure P30.7

Figure P30.9

Figure P30.10

Figure P30.12

I

R

x!I2 I1

2a–2a 0

r

I

60°

b

aP

I

lies in the plane of the paper and carries a current I. Findan expression for the vector magnetic field at the center ofthe loop.

7. The segment of wire in Figure P30.7 carries a currentof I ! 5.00 A, where the radius of the circular arc isR ! 3.00 cm. Determine the magnitude and direction ofthe magnetic field at the origin.

8. Consider a flat circular current loop of radius R carry-ing current I. Choose the x axis to be along the axis of theloop, with the origin at the center of the loop. Plot agraph of the ratio of the magnitude of the magnetic fieldat coordinate x to that at the origin, for x ! 0 to x ! 5R. Itmay be useful to use a programmable calculator or acomputer to solve this problem.

9. Two very long, straight, parallel wires carry currents thatare directed perpendicular to the page, as in Figure P30.9.Wire 1 carries a current I1 into the page (in the " zdirection) and passes through the x axis at x ! # a. Wire 2passes through the x axis at x ! " 2a and carries anunknown current I2. The total magnetic field at the origindue to the current-carrying wires has the magnitude2$0I1/(2%a). The current I2 can have either of twopossible values. (a) Find the value of I2 with the smallermagnitude, stating it in terms of I1 and giving its direction.(b) Find the other possible value of I2.

an arc of a circle of radius r, as shown in Figure P30.10.Determine the magnetic field at the center of the arc.

11. One very long wire carries current 30.0 A to the leftalong the x axis. A second very long wire carries current50.0 A to the right along the line (y ! 0.280 m, z ! 0).(a) Where in the plane of the two wires is the totalmagnetic field equal to zero? (b) A particle witha charge of " 2.00 $C is moving with a velocity of150 i Mm/s along the line (y ! 0.100 m, z ! 0).Calculate the vector magnetic force acting on theparticle. (c) What If? A uniform electric field is appliedto allow this particle to pass through this regionundeflected. Calculate the required vector electric field.

12. Consider the current-carrying loop shown in FigureP30.12, formed of radial lines and segments of circleswhose centers are at point P. Find the magnitude anddirection of B at P.

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

14. Determine the magnetic field (in terms of I, a, and d) atthe origin due to the current loop in Figure P30.14.

15. Two long, parallel conductors carry currents I1 ! 3.00 Aand I2 ! 3.00 A, both directed into the page in FigureP30.15. Determine the magnitude and direction of theresultant magnetic field at P.

– a + aO

d

I

I

y

x

Figure P30.14

Problems 965

B (T) B0 (T)

0.2 4.8 ! 10"5

0.4 7.0 ! 10"5

0.6 8.8 ! 10"5

0.8 1.2 ! 10"4

1.0 1.8 ! 10"4

1.2 3.1 ! 10"4

1.4 8.7 ! 10"4

1.6 3.4 ! 10"3

1.8 1.2 ! 10"1

Table P30.70

Figure P30.69

Figure P30.67

x

PI

L

L

r = e!

y

x

r dr

d s

!

r

= /4"#

I

I

The loop has a length L, radius R, and carries a current I2.The axis of the loop coincides with the wire. Calculate theforce exerted on the loop.

66. Measurements of the magnetic field of a large tornadowere made at the Geophysical Observatory in Tulsa,Oklahoma, in 1962. The tornado’s field was measured tobe B # 1.50 ! 10"8 T pointing north when the tornadowas 9.00 km east of the observatory. What current wascarried up or down the funnel of the tornado, modeled asa long straight wire?

A wire is formed into the shape of a square of edge lengthL (Fig. P30.67). Show that when the current in the loop isI, the magnetic field at point P, a distance x from thecenter of the square along its axis is

B #$0IL2

2%(x 2 & L2/4)$x

2 & L2/2

67.

68. The force on a magnetic dipole ! aligned with a nonuni-form magnetic field in the x direction is given byFx # !!!dB/dx. Suppose that two flat loops of wire eachhave radius R and carry current I. (a) The loopsare arranged coaxially and separated by a variabledistance x, large compared to R. Show that the magneticforce between them varies as 1/x4. (b) Evaluate themagnitude of this force if I # 10.0 A, R # 0.500 cm, andx # 5.00 cm.

69. A wire carrying a current I is bent into the shape of anexponential spiral, r # e', from ' # 0 to ' # 2% as sug-gested in Figure P30.69. To complete a loop, the ends ofthe spiral are connected by a straight wire along the x axis.Find the magnitude and direction of B at the origin.Suggestions: Use the Biot–Savart law. The angle ( between aradial line and its tangent line at any point on the curver # f (') is related to the function in the following way:

Thus in this case r # e ', tan ( # 1 and ( # %/4.Therefore, the angle between ds and r is % " ( # 3%/4.Also

ds #dr

sin(%/4)# $2 dr

tan ( #r

dr/d'

70. Table P30.70 contains data taken for a ferromagneticmaterial. (a) Construct a magnetization curve from thedata. Remember that B # B0 & $0M. (b) Determine theratio B/B0 for each pair of values of B and B0, andconstruct a graph of B/B0 versus B0. (The fraction B/B0 iscalled the relative permeability, and it is a measure of theinduced magnetic field.)

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

Figure P30.71 Problems 71 and 72.

R

*

Problems 965

B (T) B0 (T)

0.2 4.8 ! 10"5

0.4 7.0 ! 10"5

0.6 8.8 ! 10"5

0.8 1.2 ! 10"4

1.0 1.8 ! 10"4

1.2 3.1 ! 10"4

1.4 8.7 ! 10"4

1.6 3.4 ! 10"3

1.8 1.2 ! 10"1

Table P30.70

Figure P30.69

Figure P30.67

x

PI

L

L

r = e!

y

x

r dr

d s

!

r

= /4"#

I

I

The loop has a length L, radius R, and carries a current I2.The axis of the loop coincides with the wire. Calculate theforce exerted on the loop.

66. Measurements of the magnetic field of a large tornadowere made at the Geophysical Observatory in Tulsa,Oklahoma, in 1962. The tornado’s field was measured tobe B # 1.50 ! 10"8 T pointing north when the tornadowas 9.00 km east of the observatory. What current wascarried up or down the funnel of the tornado, modeled asa long straight wire?

A wire is formed into the shape of a square of edge lengthL (Fig. P30.67). Show that when the current in the loop isI, the magnetic field at point P, a distance x from thecenter of the square along its axis is

B #$0IL2

2%(x 2 & L2/4)$x

2 & L2/2

67.

68. The force on a magnetic dipole ! aligned with a nonuni-form magnetic field in the x direction is given byFx # !!!dB/dx. Suppose that two flat loops of wire eachhave radius R and carry current I. (a) The loopsare arranged coaxially and separated by a variabledistance x, large compared to R. Show that the magneticforce between them varies as 1/x4. (b) Evaluate themagnitude of this force if I # 10.0 A, R # 0.500 cm, andx # 5.00 cm.

69. A wire carrying a current I is bent into the shape of anexponential spiral, r # e', from ' # 0 to ' # 2% as sug-gested in Figure P30.69. To complete a loop, the ends ofthe spiral are connected by a straight wire along the x axis.Find the magnitude and direction of B at the origin.Suggestions: Use the Biot–Savart law. The angle ( between aradial line and its tangent line at any point on the curver # f (') is related to the function in the following way:

Thus in this case r # e ', tan ( # 1 and ( # %/4.Therefore, the angle between ds and r is % " ( # 3%/4.Also

ds #dr

sin(%/4)# $2 dr

tan ( #r

dr/d'

70. Table P30.70 contains data taken for a ferromagneticmaterial. (a) Construct a magnetization curve from thedata. Remember that B # B0 & $0M. (b) Determine theratio B/B0 for each pair of values of B and B0, andconstruct a graph of B/B0 versus B0. (The fraction B/B0 iscalled the relative permeability, and it is a measure of theinduced magnetic field.)

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

Figure P30.71 Problems 71 and 72.

R

*

960 CHAPTE R 3 0 • Sources of the Magnetic Field

of the can and I the upward current, uniformly distributedover its curved wall. Determine the magnetic field (a) justinside the wall and (b) just outside. (c) Determine thepressure on the wall.

28. Niobium metal becomes a superconductor when cooledbelow 9 K. Its superconductivity is destroyed when thesurface magnetic field exceeds 0.100 T. Determine themaximum current a 2.00-mm-diameter niobium wire cancarry and remain superconducting, in the absence of anyexternal magnetic field.

A long cylindrical conductor of radius R carries a current Ias shown in Figure P30.29. The current density J , however,is not uniform over the cross section of the conductor butis a function of the radius according to J ! br, where b is aconstant. Find an expression for the magnetic field B(a) at a distance r1 " R and (b) at a distance r 2 # R,measured from the axis.

29.

23. Figure P30.23 is a cross-sectional view of a coaxial cable.The center conductor is surrounded by a rubber layer,which is surrounded by an outer conductor, which issurrounded by another rubber layer. In a particularapplication, the current in the inner conductor is 1.00 Aout of the page and the current in the outer conductor is3.00 A into the page. Determine the magnitude anddirection of the magnetic field at points a and b.

30. In Figure P30.30, both currents in the infinitely long wiresare in the negative x direction. (a) Sketch the magneticfield pattern in the yz plane. (b) At what distance d alongthe z axis is the magnetic field a maximum?

Section 30.4 The Magnetic Field of a SolenoidWhat current is required in the windings of a long

solenoid that has 1 000 turns uniformly distributed over alength of 0.400 m, to produce at the center of the solenoida magnetic field of magnitude 1.00 $ 10%4 T?

32. Consider a solenoid of length ! and radius R , containingN closely spaced turns and carrying a steady currentI. (a) In terms of these parameters, find the magneticfield at a point along the axis as a function of distancea from the end of the solenoid. (b) Show that as !becomes very long, B approaches &0NI/2! at each end ofthe solenoid.

31.

ba1.00 A

1 mm 1 mm 1 mm

3.00 A

. .

!!

!

!!

!

!

!

Figure P30.23

Figure P30.29

Figure P30.30

Rr1

I

r2

xy

a

a

I

I

z

24. The magnetic field 40.0 cm away from a long straight wirecarrying current 2.00 A is 1.00 &T. (a) At what distance isit 0.100 &T? (b) What If? At one instant, the twoconductors in a long household extension cord carryequal 2.00-A currents in opposite directions. The two wiresare 3.00 mm apart. Find the magnetic field 40.0 cm awayfrom the middle of the straight cord, in the plane of thetwo wires. (c) At what distance is it one tenth as large?(d) The center wire in a coaxial cable carries current2.00 A in one direction and the sheath around it carriescurrent 2.00 A in the opposite direction. What magneticfield does the cable create at points outside?

A packed bundle of 100 long, straight, insulated wiresforms a cylinder of radius R ! 0.500 cm. (a) If each wirecarries 2.00 A, what are the magnitude and direction of themagnetic force per unit length acting on a wire located0.200 cm from the center of the bundle? (b) What If ? Woulda wire on the outer edge of the bundle experience a forcegreater or smaller than the value calculated in part (a)?

26. The magnetic coils of a tokamak fusion reactor are in theshape of a toroid having an inner radius of 0.700 m and anouter radius of 1.30 m. The toroid has 900 turns of large-diameter wire, each of which carries a current of 14.0 kA.Find the magnitude of the magnetic field inside the toroidalong (a) the inner radius and (b) the outer radius.

27. Consider a column of electric current passing throughplasma (ionized gas). Filaments of current within thecolumn are magnetically attracted to one another. Theycan crowd together to yield a very great current densityand a very strong magnetic field in a small region.Sometimes the current can be cut off momentarily by thispinch effect. (In a metallic wire a pinch effect is notimportant, because the current-carrying electrons repelone another with electric forces.) The pinch effect can bedemonstrated by making an empty aluminum can carry alarge current parallel to its axis. Let R represent the radius

25.

Cha pter 29 I Magnetic Fields Due to Currents

.59 A student makes a short electromagnet by winding300 turns of wire around a wooden cylinder of diamet er d -5.0 cm. The coil is connected to a battery producing a currentof 4.0 A in the wire. (a) What is the magnitude of the magneticdipole moment of this device? (b) At what axial distance z *r/ will the magnetic field have the magnitude 5.0 p,T (approx-imately one-tenth that of Earth's magnetic field)? ssM

0.60 In Fig. 29-74, current i: 56.2 mA is set up in a loophaving two radial lengths andtwo semicircles of radii a -5.72 cm and b - 9.36 cm with acommon center P. What arethe (u) magnitude and (b) di-rection (into or out of thepage) of the magnetic field at Pand the (c) magnitude and (d)direction of the loop's mag-netic dipole moment?oo61 In Fig. 29-75, a conduc-tor carries 6.0 A along theclosed path abcdefgha runningalong 8 of the 12 edges of acube of edge length 10 cm. (a)Taking the path to be a combi-nation of three square currentloops (bcfgb ., abgha, and cdefc),find the net magnetic moment z

of the path in unit-vector nota-tion. (b) What is the magnitudeof the net magnetic field at thexyz coordinates of (0,5.0 m,0)?

d

FtG. 29-75 Problem 61.

..62 In Fig. 29-76a, two circular loops, with different cur-rents but the same radius of 4.0 cffi, are centered on a y axis.They are initially separated by distance L - 3.0 cm, with loop2 positioned at the origin of the axis. The currents in the twoloops produce a net magnetic field at the origin, with y com-ponent ^B,,. That component is to be measured as loop 2 isgradually moved in the positive direction of the y axis. Figure29-76b gives B,, as a function of the position y of loop Z.Thecurve approaches an asymptote of Bu : 7.20 pT as y + m.Thehorizontal scale is set by y, = 10.0 cm. What are (a) current i1 inloop 1 and (b) current i2inloop2?

20

-40) (cm)

(D)

FlG. 29-76 Problem 62.

..63 A circular loop of radius 12 cm carries a current of 15A. A flat coil of radius 0.82 cffi, having 50 turns and a currentof 1.3 A' is concentric with the loop. The plane of the loop isperpendicular to the plane of the coil. Assume the loop's mag-

FlG. 29-74 Problem 60.

netic field is uniform across the coil. What is the magnitudeof (a) the magnetic field produced by the loop at its centerand (b) the torque on the coil due to the loop?

Additional Problems64 Figure 29-77 shows aclosed loop with current i -2.00 A. The loop consists of ahalf-circle of radius 4.00 m, twoquarter-circles each of radius2.00 m, and three radial straightwires. What is the magnitude ofthe net magnetic field at thecommon center of the circularsections?

65 Figure 29-78 shows a cross section ofa long cylindrical conductor of radius a -4.00 cm containing a long cylindrical holeof radius b : 1.50 cm. The central axes ofthe cylinder and hole are parallel and aredistance d - 2.00 cm apart; current i -5.25 A is uniformly distributed over thetinted area. (a) What is the magnitude ofthe magnetic field at the center of thehole? (b) Discuss the two special cases b-0andd:0.

FlG" 29'-77 Problem64.

FrG. 2q-78Problem 65.

66 The magnitude of the magnetic field 88.0 cm from theaxis of a long straight wire is 7.30 p,T. What is the current inthe wire?67 Three long wlres are paral-lel to a z axis, and each c ,Rocurrent of 10 A in the po .r' t,,

direction. Their poi .,' t,intersection with the xy plane ,' \form an equilateral triangle ,i tt.

twith sides of 50 cffi, as shown"in .,' "'. iFig. 2g-7g. A fourth wire 6- 3-----b

I n

(wire b) passes through themidpoint of the base of ;h" t+ FIG' 29-79 Problem 67'

angle and is parallel to theother three wires. If the net magnetic force on wire a ls zero,what are the (a) size and (b) direction (+z or -z) of the cur-rent in wire b?

68 Figure 29-80 shows, incross section, two long parallelwires spaced by distance d -10.0 cm; each carries 100 A, outof the page in wire 1. Point P ison a perpendicular bisector ofthe line connecting the wires. Inunit-vector notation, what is thenet magnetic field at P if the current in wire 2 is (a) out of thepage and (b) into the page?69 A 1O-gauge bare copper wire (2.6 mm in diameter) cancarry a current of 50 A without overheating. For this current,what is the magnitude of the magnetic field at the surface ofthe wire?70 A long vertical wire carries an unknown current. Coaxialwith the wire is a long, thin, cylindrical conducting surface that

)L_" ,

P,A

-z-\,/\/\0F

ea-

/\/\/\/\l.r' t.2*d#

FlG. ?9-8A Problem 68.

(a)

Chapte r 29 I Magnetic Fields Due to Currents

Show that the magnetic field B at the center of the circle is thesame as the field B a distance R below an infinite straight wirecarrying a current i to the left.

84 A long wire is known to have a radius greater than4.0 mm and to carry a current that is uniformly distributedover its cross section. The magnitude of the magnetic field dueto that current is 0.28 mT at a point 4.0 mm from the axis ofthe wire, and 0.20 mT at a point 10 mm from the axis of thewire. What is the radius of the wire?

85 A long, hollow, cylindrical conductor (inner radius 2.0ffiffi, outer radius 4.0 mm) carries a current of 24 A distributeduniformly across its cross section. A long thin wire that is co-axial with the cylinder carries a current of 24 A in the oppositedirection. What is the magnitude of the magnetic field (a) 1.0ffiffi, (b) 3.0 mm, and (c) 5.0 mm from the central axis of thewire and cylinder?

86 In Fig. 29-73, an arrangement known as Helmholtz coilsconsists of two circular coaxial coils, each of N turns andradius R, separated by distance s. The two coils carry equalcurrents i in the same direction. (a) Show that the first deriva-tive of the magnitude of the net magnetic field of the coils(dBldx) vanishes at the midpoint P regardless of the valueof s. Why would you expect this to be true from symmetry?(b) Show that the second derivative (d2Bldxz) also vanishesat P, provided s - R. This accounts for the uniformity of Bnear P for this particular coil separatlon.

87 A square loop of wire of edge length a carries current i.Show that, at the center of the loop, the magnitude of the mag-netic field produced by the current rs

B _ ZtEpuiTrA

88 Show that the magnitude of the magnetic field producedat the center of a rectangular loop of wire of length L andwidth W,carrying a current i, is

B _ 2pri (L' + 1ryz1trz

7T LW

89 A square loop of wire of edge length a carrres current i.Show that the magnitude of the magnetic field produced ata point on the central perpendicular axis of the loop and adistance x from its center is

B(x) - 4 p"11ia2

rr (4x2 + az)(4xz + 2oz7t rz

Prove that this result is consistent with the result shown inProblem 87. ssM

9CI Figure 29-88 is an idealized schematic drawing of a railgun. Projectile P sits between two wide rails of circular crosssection; a source of current sends current through the rails

and through the (conducting) projectile (a fuse is not used).(a) Let w be the distance between the rails, R the radiusof each rail, and i the current. Show that the force on theprojectile is directed to the right along the rails and is givenapproximately by

(b) If the projectile starts from the left end of the rails at rest,find the speed y at which it is expelled at the right. Assumethat i : 450 kA' w - 12 mm., R - 6.7 cm, L : 4.0 m, and theprojectile mass is 10 g.

Source

FlG" 29-fiS Problem 90.

91 Show that a uniform magneticfield E cannot drop abruptly tozero (as is suggested by the lack offield lines to the right of point a inFig. 29-82) as one moves perpendic-ular to B, say along the horizontalarrow in the figure. (Hint: ApplyAmpere's law to the rectangularpath shown by the dashed lines.) Inactual magnets, "fringing" of themagnetic field lines always occurs,which means that E approacheszero in a gradual manner. Modify the field lines in the figure toindicate a more realistic situation. ssM

93 Show that if the thickness of a toroid is much smaller thanits radius of curvature (a very skinny toroid), then Eq. 29-24forthe field inside a toroid reduces to Eq. 29-23 for the field insidea solenoid. Explain why this result is to be expected.

93 Figure 29-90 shows a cross section ofa long conducting coaxial cable and givesits radii (a, b, ,). Equal but opposite cur-rents i are uniformly distributed in thetwo conductors. Derive expressions forB(r) with radial distance r in the ranges(u) r 1c, (b) c 1r 1b,(r) b < r 1a; and(d) t > a. (e) Test these expressions for all FIG" ?9-90the special cases that occur to you. (f) problem 93.Assume that a - 2.0 cm, b - 1.8 cm, c -0.40 cffi, and i : \20 A and plot the function B(r) over therange0< r.-.-3 cm.

r- i't"r',, w+Rr--tit- 2nR

,l

FlG. ?E-Bq Problem 91

out of the page. They are equal distances from the origin,where they set up a magnetic field d. fo what value mustcurrent i 1 be changed in order to rotateE 20.0" clockwise?."26 Figure 29-55a shows two wires, each carrying a current.Wire I consists of a circular arc of radius R and two radiallengths; it carries current i 1 : 2.0 A in the direction indicated.Wire 2 is long and straight; it carries a current i2 that can bevaried; and it is at distance Rlz from the center of the arc. Thenet magnetic field F du. to the two currents is measured atthe center of curvature of the arc. Figure 29-55b is a plot ofthe component of E in the direction perpendicular to thefigure as a function of current i2. The horizontal scale is set byi?, = 1.00 A. What is the angle subtended by the arc? €D

iz (A)

(b)

FlG. 29-55 Problem 26.

at27 One long wire lies along an x axis and carries a currentof 30 A in the positive -r direction. A second long wire is per-pendicular to the xy plane, passes through the point (0, 4.0 m,0)' and carries a current of 40 A in the positive z direction.What is the magnitude of the resulting magnetic field at thepoint (0'2.0 m,0)?..28 In Fig. 29-56, part of along insulated wire carrying cur-rent r : 5.78 mA is bent into acircular section of radius R -1.89 cm. In unit-vector notation,what is the magnetic field at thecerrter of curvature C it the cir-cular section (a) lies in the plane of the page as shown and (b)is perpendicular to the plane of the page after being rotated90o counterclockwise as indicated?e.29 Figure 29-57 shows twovery long straight wires (in crosssection) that each carry a cur-rent of 4.00 A directly out ofthe page. Distance dt : 6.00 mand distance dt :4.00 m. Whatis the magnitude of the net mag-netic field at point P, which lieson a perpendicular bisector tothe wires?rrrSQ The current-carrying wire loop in Fig. 29-58a lies allin one plane and consists of a semicircle of radius 10.0 cffi,a smaller semicircle with the same center, and two radiallengths. The smaller semicircle is rotated out of that plane byangle 0, until it is perpendicular to the plane (Fig. 29-58b).Figure 29-58c gives the magnitude of the net magnetic field at

ror$l Figure 29-59 shows across section of a long thin ribbonof width w - 49I cm that is carry-irg a uniformly distributed totalcurrent i : 4.67 pA into the page.In unit-vector notation, what is themagnetic field E ata point P in theplane of the ribbon at a distance d - 2.16 cm from its edge?(Hint: Imagine the ribbon as being constructed from many long,thin, parallel wires.) ssM rLw

oootl Figure 29-60 shows, incross section, two long straightwires held against a plastic cylin-der of radius 20.0 cm. Wire \ car-ries current it:60.0 mA out ofthe page and is fixed in place at wire

the left side of the cylinder. Wire2 carries current i.t :40.0 mA

Fa

rRr----J l.<_t2l

kr)

FlG. 29-56 Problem 28.out of the page and can be moved FlG. ZV-4,Oaround the cylinder. At what(positive) angle 92 should wire 2be positioned such that, at the ori-gin,the net magnetic field due to

?the two currents has magnitude80.0 nT?rcrtJ In Fig. 29-61, length a is4.7 cm (short) and current i is 13A. What are the (a) magnitudeand (b) direction (into or out ofthe page) of the magnetic field atpoint P?ero${ Two long straight thinwires with current lie against anequally long plastic cylinder, at radius R - 20.0 cm from thecylinder's central axis. Figure 29-62a shows, in cross section,,the cylinder and wire I but not wire 2. With wire 2 fixed inplace, wire 1 is moved around the cylinder, from angle 9r : 0oto angle 0r : 180o, through the first and second quadrants ofthe xy coordinate system. The net magnetic field E at thecenter of the cylinder is measured as a function of 91. Figure

dl

FlG. 29-57 Problem 29.

6

Problems

the center of curvature versus angle 0. The vertical scale is setby Bn - 10.0 pT and BL, =I2.0 pT. What is the radius of thesmaller semicircle?

rc/4 rc/20 (racl)(.)

FlG. 29-58 Problem 30.

))

I

lp!- x x x x x x x -JC

l--- a ._|--- ru----.-----.----- l

FlS, 2q-59 Problem 31.

Wire 2

Problem 32.

Flffi" 2q-6'l Problem 33.

Chapte r 29 I Magnetic Fields Due to Currents

29-62b gives the x component B..- of that field as a function of91 (the vertical scale is set by 8.,, = 6.0 l/.T), and Fig. 29-62cgives the y component 8,, (the vertical scale is set by Bu, - 4.0pT). (a) At what angle 02 is wire 2located? What are the (b) sizeand (c) direction (into or out of the page) of the current in wire 1

and the (d) size and (e) direction of the current in wire 2?

sec. 29-3 Force Between Two Parallel Currents.35 Figure 29-63 shows wire 1 in cross section; the wire islong and straight, carries a current of 4.00 mA out of the page,and is at distance dt - 2.40 cmfrom a surface. Wira 2, which isparallel to wire 1 and also long,is at horizontal distance dz:5.00 cm from wire I and carries acurrent of 6.80 mA into the page.What is the x component of themagnetic force per unit length onwire 2 due to wire 1? ssM

oc36 In Fig. 29-49,four long straight wires are perpendicularto the page, and their cross sections form a square of edgelength o - 8.50 cm. Each wire carries 15.0 A, and all the cur-rents are out of the page. In unit-vector notation, what is thenet magnetic force per meter of wire length on wire 1?

..37 In Fig. 29-64, five longparallel wires in an xy plane areseparated by distance d - 50.0cm. The currents into the pageare i1 :2.00 A, ir : 0.250 A,iq:4.00 A, and is : 2.00 A; the cur-rent out of the page is iz:4.00A. What is the magnitude of thenet force per unit length acting onthe other wires? S

separated by distance d - 8.00 cffi,have lengths of 10.0 m'andcarry identical currents of 3.00 A out of the page. Each wireexperiences a magnetic force due to the other wires. In unit-vector notation, what is the net magnetic force on (a) wire 1,(b) wire2,(c) wire 3, (d) wire 4, and (e) wire 5?

r.39 In Fig. 29-49,four long straight wires are perpendicularto the page, and their cross sections form a square of edgelength a - 13.5 cm. Each wire carries I .50 A, and the currentsare out of the page in wires 1 and 4 and into the page in wires2 and 3. In unit-vector notation, what is the net magnetic forceper meter of wire length on wire 4? S..40 Figure 29-65a shows, in cross section, three current-carrying wires that are long, straight, and parallel to oneanother. Wires 1 and 2 are fixed in place on an x axis, withseparation d. Wire t has a current of 0.750 A' but the directionof the current is not given. Wire 3, with a current of 0.250 Aout of the page, can be moved along the x axis to the right ofwire 2. As wire 3 is moved, the magnitude of the net magneticforce Ft onwire 2 due to the currents in wires 1 and 3 changes.The y component of that force is F2, and the value per unitlength of wire 2 is F2r,lLt. Figure 29-65b gives FzrlLzversus theposition x of wire 3. The plot has an asymptote F2rlL2:-0.627 pN/m as "r + co. The horizontal scale is set by ", - 12.0cm. What are the (a) size and (b) direction (into or out of thepage) of the current in wire 2?

-0.5x (cm)

(b)

Problem 40.

b

j

FlG. 29-h6 Problem 41.

(")

FrG. 2q-65

orr{'f In Fig. 29-66,, a longstraight wire carries a currentit:30.0A and a rectangularloop carries current iz -- 20.0A. Take a - 1.00 cffi,, b - 8.00cffi, and L - 30.0 cm. In unit-vector notation, what is thenet force on the loop due toi t? rLw

sec. 294 Ampere's Law.42 Figure 29-67 shows twoclosed paths wrapped aroundtwo conducting loops carryingcurrents it : 5.0 A and iz :3.0 A. What is the value of theintegral f E . df for (u) path1 and (b) pathz?.43 Each of the eight con-ductors in Fig. 29-68 carries 2.0A of current into or out of thepage. Two paths are indicated

8,',

oSOa-

Fee

90"e,l

(r)

1900 .'00

FlG. ?q-62 Problem 34.

90" I 8000l

(r)

FlG" 29-63 Problem 35.

l-- a*l* ,t-*l* ,t ---l* d---*l

FlG. 29-64 Problems 37and 38.

wire 3 due to the currents in

l0E

zv{-M--!\,{0 \

. t\l\

oo38 In Fig.29-64,five long parallel wires in an xy plane are FlG. 29-6V Problem 42.