Assignment #5General Physics II

May 4th, 2016

731PROB LE M SPART 3

HALLIDAY REVISED

••55 In Fig. 27-61, Rs is to be ad-justed in value by moving the slid-ing contact across it until points aand b are brought to the same po-tential. (One tests for this condi-tion by momentarily connecting asensitive ammeter between a andb; if these points are at the samepotential, the ammeter will not de-flect.) Show that when this adjust-ment is made, the following rela-tion holds: Rx ! RsR2/R1. Anunknown resistance (Rx) can bemeasured in terms of a standard(Rs) using this device, which iscalled a Wheatstone bridge.

••56 In Fig. 27-62, a voltmeter ofresistance RV ! 300 " and an amme-ter of resistance RA ! 3.00 " are be-ing used to measure a resistance R ina circuit that also contains a resis-tance R0 ! 100 " and an ideal bat-tery of emf ! ! 12.0 V. Resistance Ris given by R ! V/i, where V is thevoltmeter reading and i is the cur-rent in resistance R. However, theammeter reading is not i but ratheri#, which is i plus the current through the voltmeter. Thus, the ratioof the two meter readings is not R but only an apparent resistanceR# ! V/i#. If R ! 85.0 ", what are (a) the ammeter reading, (b) thevoltmeter reading, and (c) R#? (d) If RV is increased, does the dif-ference between R# and R increase, decrease, or remain the same?

sec. 27-9 RC Circuits•57 Switch S in Fig. 27-63 is closed attime t ! 0, to begin charging an initiallyuncharged capacitor of capacitance C !15.0 mF through a resistor of resistance R ! 20.0 ". At what time is the potentialacross the capacitor equal to that acrossthe resistor?

•58 In an RC series circuit, emf ! ! 12.0 V, resistance R ! 1.40M", and capacitance C ! 1.80 mF. (a) Calculate the time constant.(b) Find the maximum charge that will appear on the capacitorduring charging. (c) How long does it take for the charge to buildup to 16.0 mC?

•59 What multiple of the time constant t gives the timetaken by an initially uncharged capacitor in an RC series circuit tobe charged to 99.0% of its final charge?

•60 A capacitor with initial charge q0 is discharged through a re-sistor. What multiple of the time constant t gives the time the ca-pacitor takes to lose (a) the first one-third of its charge and (b)two-thirds of its charge?

•61 A 15.0 k" resistor and a capacitor are connected in se-ries, and then a 12.0 V potential difference is suddenly appliedacross them. The potential difference across the capacitor rises to5.00 V in 1.30 ms. (a) Calculate the time constant of the circuit. (b)Find the capacitance of the capacitor.

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••62 Figure 27-64 shows the circuit ofa flashing lamp, like those attached tobarrels at highway construction sites.The fluorescent lamp L (of negligiblecapacitance) is connected in parallelacross the capacitor C of an RC circuit.There is a current through the lamponly when the potential differenceacross it reaches the breakdown volt-age VL; then the capacitor dischargescompletely through the lamp and the lamp flashes briefly. For alamp with breakdown voltage VL ! 72.0 V, wired to a 95.0 V idealbattery and a 0.150 mF capacitor, what resistance R is needed fortwo flashes per second?

••63 In the circuit ofFig. 27-65, ! ! 1.2 kV, C ! 6.5 mF, R1 !R2 ! R3 ! 0.73 M". With C com-pletely uncharged, switch S is sud-denly closed (at t ! 0). At t ! 0, whatare (a) current i1 in resistor 1, (b) cur-rent i2 in resistor 2, and (c) current i3

in resistor 3? At t ! $ (that is, aftermany time constants), what are (d) i1,(e) i2, and (f) i3? What is the potential difference V2 across resistor2 at (g) t ! 0 and (h) t ! $? (i) Sketch V2 versus t between thesetwo extreme times.

••64 A capacitor with an initial potential difference of 100 V isdischarged through a resistor when a switch between them isclosed at t ! 0. At t ! 10.0 s, the potential difference across the ca-pacitor is 1.00 V. (a) What is the time constant of the circuit? (b)What is the potential difference across the capacitor at t ! 17.0 s?

••65 In Fig. 27-66, R1 ! 10.0 k",R2 ! 15.0 k", C ! 0.400 mF, and theideal battery has emf ! ! 20.0 V.First, the switch is closed a long timeso that the steady state is reached.Then the switch is opened at time t !0. What is the current in resistor 2 at t ! 4.00 ms?

••66 Figure 27-67 displays two circuits with a charged capacitorthat is to be discharged through a resistor when a switch isclosed. In Fig. 27-67a, R1 ! 20.0 " and C1 ! 5.00 mF. In Fig.27-67b, R2 ! 10.0 " and C2 ! 8.00 mF. The ratio of the initialcharges on the two capacitors is q02/q01 ! 1.50. At time t ! 0,both switches are closed. At what time t do the two capacitorshave the same charge?

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+–

R

C L

Fig. 27-64Problem 62.

C

+–

SR3

R2

R1

Fig. 27-65Problem 63.

+– R2

R1

C

Fig. 27-66Problems 65 and 99.

+ –R0b

a

Rs Rx

R1R2

Sliding contact

Fig. 27-61Problem 55.

+ –

R

R0

V

A

Fig. 27-62Problem 56.

CR

S+–

Fig. 27-63Problems 57 and 96.

C2R1 R2C1

(a) (b)

Fig. 27-67 Problem 66.

••67 The potential difference between the plates of a leaky(meaning that charge leaks from one plate to the other) 2.0 mF ca-pacitor drops to one-fourth its initial value in 2.0 s. What is theequivalent resistance between the capacitor plates?

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•••28 In Fig. 30-49, a rectangularloop of wire with length a ! 2.2 cm,width b ! 0.80 cm, and resistanceR ! 0.40 m" is placed near an infi-nitely long wire carrying current i ! 4.7 A. The loop is then movedaway from the wire at constantspeed v ! 3.2 mm/s. When the cen-ter of the loop is at distancer ! 1.5b, what are (a) the magnitude of the magnetic flux throughthe loop and (b) the current induced in the loop?

sec. 30-5 Induction and Energy Transfers•29 In Fig. 30-50, a metal rod is forced to move with constant ve-locity along two parallel metal rails, connected with a strip ofmetal at one end. A magnetic field of magnitude B ! 0.350 Tpoints out of the page. (a) If the rails are separated by L = 25.0 cmand the speed of the rod is 55.0 cm/s, what emf is generated? (b) Ifthe rod has a resistance of 18.0 " and the rails and connector have

v:

•••27 As seen in Fig. 30-48, a square loop of wire has sides oflength 2.0 cm. A magnetic field is directed out of the page; its mag-nitude is given by B ! 4.0t2y, where B is in teslas, t is in seconds,and y is in meters. At t ! 2.5 s, what are the (a) magnitude and (b)direction of the emf induced in the loop?

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820 CHAPTE R 30 I N DUCTION AN D I N DUCTANCE

is placed entirely in a uniform magnetic field with magnitude B !3.50 T and with initially perpendicular to the coil’s plane. Whatis the maximum value of the emf produced when the coil is spun at1000 rev/min about an axis perpendicular to ?

••20 At a certain place, Earth’s magnetic field has magnitude B ! 0.590 gauss and is inclined downward at an angle of 70.0° tothe horizontal. A flat horizontal circular coil of wire with a radiusof 10.0 cm has 1000 turns and a total resistance of 85.0 ". It isconnected in series to a meter with 140 " resistance. The coil isflipped through a half-revolution about a diameter, so that it isagain horizontal. How much charge flows through the meter dur-ing the flip?

••21 In Fig. 30-44, a stiff wire bentinto a semicircle of radius a ! 2.0 cmis rotated at constant angular speed 40rev/s in a uniform 20 mT magneticfield. What are the (a) frequency and(b) amplitude of the emf induced inthe loop?

••22 A rectangular loop (area !0.15 m2) turns in a uniform mag-netic field, B ! 0.20 T.When the an-gle between the field and the normal to the plane of the loop is p/2rad and increasing at 0.60 rad/s,what emf is induced in the loop?

••23 Figure 30-45 showstwo parallel loops of wire havinga common axis. The smaller loop(radius r) is above the larger loop(radius R) by a distance x # R.Consequently, the magnetic fielddue to the counterclockwise currenti in the larger loop is nearly uniformthroughout the smaller loop.Suppose that x is increasing at theconstant rate dx/dt ! v. (a) Find anexpression for the magnetic flux through the area of the smallerloop as a function of x. (Hint: See Eq. 29-27.) In the smaller loop,find (b) an expression for the induced emf and (c) the direction ofthe induced current.

••24 A wire is bent into three cir-cular segments, each of radius r !10 cm, as shown in Fig. 30-46. Eachsegment is a quadrant of a circle, ablying in the xy plane, bc lying in theyz plane, and ca lying in the zxplane. (a) If a uniform magneticfield points in the positive x di-rection, what is the magnitude ofthe emf developed in the wire whenB increases at the rate of 3.0 mT/s?(b) What is the direction of the cur-rent in segment bc?

•••25 Two long, parallel copper wires of diameter 2.5 mmcarry currents of 10 A in opposite directions. (a) Assuming thattheir central axes are 20 mm apart, calculate the magnetic flux permeter of wire that exists in the space between those axes. (b) Whatpercentage of this flux lies inside the wires? (c) Repeat part (a) forparallel currents.

B:

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B:

B:

•••26 For the wire arrangement in Fig. 30-47, a ! 12.0 cm and b !16.0 cm. The current in the long straight wire is i ! 4.50t2 $ 10.0t,where i is in amperes and t is in seconds. (a) Find the emf in thesquare loop at t ! 3.00 s. (b) What is the direction of the inducedcurrent in the loop?

Fig. 30-44 Problem 21.

B

R

a

Fig. 30-45 Problem 23.

i

r

R

x

Fig. 30-46 Problem 24.

r

r

r

a

b

c

z

y

x

Fig. 30-47 Problem 26.

ba

b

i

y

x

B

Fig. 30-48 Problem 27.

b

a

r

i

v

Fig. 30-49 Problem 28.

Fig. 30-50 Problems 29 and 35.

L

B

v

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•••28 In Fig. 30-49, a rectangularloop of wire with length a ! 2.2 cm,width b ! 0.80 cm, and resistanceR ! 0.40 m" is placed near an infi-nitely long wire carrying current i ! 4.7 A. The loop is then movedaway from the wire at constantspeed v ! 3.2 mm/s. When the cen-ter of the loop is at distancer ! 1.5b, what are (a) the magnitude of the magnetic flux throughthe loop and (b) the current induced in the loop?

sec. 30-5 Induction and Energy Transfers•29 In Fig. 30-50, a metal rod is forced to move with constant ve-locity along two parallel metal rails, connected with a strip ofmetal at one end. A magnetic field of magnitude B ! 0.350 Tpoints out of the page. (a) If the rails are separated by L = 25.0 cmand the speed of the rod is 55.0 cm/s, what emf is generated? (b) Ifthe rod has a resistance of 18.0 " and the rails and connector have

v:

•••27 As seen in Fig. 30-48, a square loop of wire has sides oflength 2.0 cm. A magnetic field is directed out of the page; its mag-nitude is given by B ! 4.0t2y, where B is in teslas, t is in seconds,and y is in meters. At t ! 2.5 s, what are the (a) magnitude and (b)direction of the emf induced in the loop?

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820 CHAPTE R 30 I N DUCTION AN D I N DUCTANCE

is placed entirely in a uniform magnetic field with magnitude B !3.50 T and with initially perpendicular to the coil’s plane. Whatis the maximum value of the emf produced when the coil is spun at1000 rev/min about an axis perpendicular to ?

••20 At a certain place, Earth’s magnetic field has magnitude B ! 0.590 gauss and is inclined downward at an angle of 70.0° tothe horizontal. A flat horizontal circular coil of wire with a radiusof 10.0 cm has 1000 turns and a total resistance of 85.0 ". It isconnected in series to a meter with 140 " resistance. The coil isflipped through a half-revolution about a diameter, so that it isagain horizontal. How much charge flows through the meter dur-ing the flip?

••21 In Fig. 30-44, a stiff wire bentinto a semicircle of radius a ! 2.0 cmis rotated at constant angular speed 40rev/s in a uniform 20 mT magneticfield. What are the (a) frequency and(b) amplitude of the emf induced inthe loop?

••22 A rectangular loop (area !0.15 m2) turns in a uniform mag-netic field, B ! 0.20 T.When the an-gle between the field and the normal to the plane of the loop is p/2rad and increasing at 0.60 rad/s,what emf is induced in the loop?

••23 Figure 30-45 showstwo parallel loops of wire havinga common axis. The smaller loop(radius r) is above the larger loop(radius R) by a distance x # R.Consequently, the magnetic fielddue to the counterclockwise currenti in the larger loop is nearly uniformthroughout the smaller loop.Suppose that x is increasing at theconstant rate dx/dt ! v. (a) Find anexpression for the magnetic flux through the area of the smallerloop as a function of x. (Hint: See Eq. 29-27.) In the smaller loop,find (b) an expression for the induced emf and (c) the direction ofthe induced current.

••24 A wire is bent into three cir-cular segments, each of radius r !10 cm, as shown in Fig. 30-46. Eachsegment is a quadrant of a circle, ablying in the xy plane, bc lying in theyz plane, and ca lying in the zxplane. (a) If a uniform magneticfield points in the positive x di-rection, what is the magnitude ofthe emf developed in the wire whenB increases at the rate of 3.0 mT/s?(b) What is the direction of the cur-rent in segment bc?

•••25 Two long, parallel copper wires of diameter 2.5 mmcarry currents of 10 A in opposite directions. (a) Assuming thattheir central axes are 20 mm apart, calculate the magnetic flux permeter of wire that exists in the space between those axes. (b) Whatpercentage of this flux lies inside the wires? (c) Repeat part (a) forparallel currents.

B:

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B:

B:

•••26 For the wire arrangement in Fig. 30-47, a ! 12.0 cm and b !16.0 cm. The current in the long straight wire is i ! 4.50t2 $ 10.0t,where i is in amperes and t is in seconds. (a) Find the emf in thesquare loop at t ! 3.00 s. (b) What is the direction of the inducedcurrent in the loop?

Fig. 30-44 Problem 21.

B

R

a

Fig. 30-45 Problem 23.

i

r

R

x

Fig. 30-46 Problem 24.

r

r

r

a

b

c

z

y

x

Fig. 30-47 Problem 26.

ba

b

i

y

x

B

Fig. 30-48 Problem 27.

b

a

r

i

v

Fig. 30-49 Problem 28.

Fig. 30-50 Problems 29 and 35.

L

B

v

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824 CHAPTE R 30 I N DUCTION AN D I N DUCTANCE

sec. 30-12 Mutual Induction•72 Coil 1 has L1 ! 25 mH and N1 ! 100 turns. Coil 2 has L2 ! 40mH and N2 ! 200 turns. The coils are fixed in place; their mutual in-ductance M is 3.0 mH. A 6.0 mA current in coil 1 is changing at therate of 4.0 A/s. (a) What magnetic flux "12 links coil 1, and (b) whatself-induced emf appears in that coil? (c) What magnetic flux "21 linkscoil 2,and (d) what mutually induced emf appears in that coil?

•73 Two coils are at fixed locations. When coil 1 has nocurrent and the current in coil 2 increases at the rate 15.0 A/s, theemf in coil 1 is 25.0 mV. (a) What is their mutual inductance? (b)When coil 2 has no current and coil 1 has a current of 3.60 A, whatis the flux linkage in coil 2?

•74 Two solenoids are part of the spark coil of an automobile.When the current in one solenoid falls from 6.0 A to zero in 2.5 ms,an emf of 30 kV is induced in the other solenoid. What is the mu-tual inductance M of the solenoids?

••75 A rectangular loop of Nclosely packed turns is positionednear a long straight wire as shown inFig. 30-66. What is the mutual induc-tance M for the loop–wire combina-tion if N ! 100, a ! 1.0 cm, b ! 8.0cm, and l ! 30 cm?

••76 A coil C of N turns is placedaround a long solenoid S of radius Rand n turns per unit length, as in Fig.30-67. (a) Show that the mutual in-ductance for the coil–solenoid com-bination is given by M ! m0pR2nN.(b) Explain why M does not dependon the shape, size, or possible lack ofclose packing of the coil.

••77 Two coils connected asshown in Fig. 30-68 separately have inductances L1 and L2. Theirmutual inductance is M. (a) Show that this combination can be re-placed by a single coil of equivalent inductance given by

Leq ! L1 # L2 # 2M.

(b) How could the coils in Fig. 30-68 be reconnected to yield anequivalent inductance of

Leq ! L1 # L2 $ 2M?

(This problem is an extension of Problem 47, but the requirementthat the coils be far apart has been removed.)

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sistance R. At time t ! 0.150 ms, the current through the inductoris changing at the rate of 280 A/s. Evaluate R.

79 In Fig. 30-69, the batteryis ideal and ! ! 10 V, R1 ! 5.0 %,R2 ! 10 %, and L ! 5.0 H. Switch Sis closed at time t ! 0. Justafterwards, what are (a) i1, (b) i2, (c)the current iS through the switch, (d)the potential difference V2 acrossresistor 2, (e) the potential differ-ence VL across the inductor, and (f)the rate of change di2/dt? A longtime later, what are (g) i1, (h) i2, (i) iS,( j) V2, (k) VL, and (1) di2/dt?

80 In Fig. 30-61, R ! 4.0 k%, L ! 8.0 mH, and the ideal battery has! ! 20 V. How long after switch S is closed is the current 2.0 mA?

81 Figure 30-70a shows arectangular conducting loop of resis-tance R ! 0.020 %, height H ! 1.5cm, and length D ! 2.5 cm beingpulled at constant speed v ! 40 cm/sthrough two regions of uniform mag-netic field. Figure 30-70b gives thecurrent i induced in the loop as afunction of the position x of the rightside of the loop. The vertical axisscale is set by is ! 3.0 mA. For exam-ple, a current equal to is is inducedclockwise as the loop enters region 1.What are the (a) magnitude and (b)direction (into or out of the page) ofthe magnetic field in region 1? What are the (c) magnitude and (d)direction of the magnetic field in region 2?

82 A uniform magnetic field is perpendicular to the plane of acircular wire loop of radius r.The magnitude of the field varies withtime according to B ! B0e$t/t, where B0 and t are constants. Findan expression for the emf in the loop as a function of time.

83 Switch S in Fig. 30-61 isclosed at time t ! 0, initiating thebuildup of current in the 15.0 mHinductor and the 20.0 % resistor.At what time is the emf across theinductor equal to the potentialdifference across the resistor?

84 Figure 30-71a shows twoconcentric circular regions inwhich uniform magnetic fieldscan change. Region 1, with radiusr1 ! 1.0 cm, has an outward mag-netic field that is increasing inmagnitude. Region 2, with radiusr2 ! 2.0 cm, has an outward mag-netic field that may also bechanging. Imagine that a conduct-ing ring of radius R is centered onthe two regions and then the emf! around the ring is determined.Figure 30-71b gives emf ! as a

B:

2

B:

1

B:

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Fig. 30-68 Problem 77.

ii

ML1

L2

N1 N2

Fig. 30-69 Problem 79.

+–

S

i2

R2

i

R1i1

L

Fig. 30-71 Problem 84.

r1

r2

s

0 2R2 (cm2)

4

(n

V)

(a)

(b)

Fig. 30-67 Problem 76.

CS

R

Fig. 30-70 Problem 81.

HD

1 2

is

0i (

A)

µ x

(a)

(b)

Fig. 30-66 Problem 75.

N turnsi

l

b

a

Additional Problems78 At time t ! 0, a 12.0 V potential difference is suddenly ap-plied to the leads of a coil of inductance 23.0 mH and a certain re-

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886 CHAPTE R 32 MAXWE LL’S EQUATION S; MAG N ETI S M OF MATTE R

18.0 cm is connected to a source of emf ! ! !m sin vt, where !m !220 V and v ! 130 rad/s. The maximum value of the displacementcurrent is id ! 7.60 mA. Neglect fringing of the electric field at theedges of the plates. (a) What is the maximum value of the current iin the circuit? (b) What is the maximum value of d"E /dt, where "E

is the electric flux through the region between the plates? (c) Whatis the separation d between the plates? (d) Find the maximumvalue of the magnitude of between the plates at a distance r !11.0 cm from the center.

sec. 32-6 Magnets•30 Assume the average value of the vertical component of Earth’smagnetic field is 43 mT (downward) for all of Arizona, which has anarea of 2.95 # 105 km2. What then are the (a) magnitude and (b) di-rection (inward or outward) of the net magnetic flux through the restof Earth’s surface (the entire surface excluding Arizona)?

•31 In New Hampshire the average horizontal component ofEarth’s magnetic field in 1912 was 16 mT, and the average inclina-tion or “dip” was 73°. What was the corresponding magnitude ofEarth’s magnetic field?

sec. 32-7 Magnetism and Electrons•32 Figure 32-36a is a one-axisgraph along which two of the al-lowed energy values (levels) of anatom are plotted. When the atom isplaced in a magnetic field of 0.500 T,the graph changes to that of Fig. 32-36b because of the energy associ-ated with . (We neglect .)Level E1 is unchanged, but level E2

splits into a (closely spaced) tripletof levels. What are the allowed val-ues of associated with (a) energylevel E1 and (b) energy level E2? (c) In joules, what amount of en-ergy is represented by the spacing between the triplet levels?

•33 If an electron in an atom has an orbital angularmomentum with , what are the components (a) Lorb,z and(b) morb,z? If the atom is in an external magnetic field that hasmagnitude 35 mT and is directed along the z axis, what are (c) theenergy Uorb associated with and (d) the energy Uspin associatedwith ? If, instead, the electron has , what are (e) Lorb,z, (f)morb,z, (g) Uorb, and (h) Uspin?

•34 What is the energy difference between parallel andantiparallel alignment of the z component of an electron’s spinmagnetic dipole moment with an external magnetic field of magni-tude 0.25 T, directed parallel to the z axis?

•35 What is the measured component of the orbital magnetic di-pole moment of an electron with (a) and (b) ?

•36 An electron is placed in a magnetic field that is directedalong a z axis. The energy difference between parallel and antipar-allel alignments of the z component of the electron’s spin magneticmoment with is 6.00 # 10$25 J.What is the magnitude of ?

sec. 32-9 Diamagnetism•37 Figure 32-37 shows a loopmodel (loop L) for a diamagneticmaterial. (a) Sketch the magneticfield lines within and about the mate-rial due to the bar magnet. What is

B:

B:

B:

m" ! $2m" ! 1

m ! $3%: s

%: orb

B:m ! 0

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m"

%:s%:orb ! B:

B:

the direction of (b) the loop’s net magnetic dipole moment ,(c) the conventional current i in the loop (clockwise or counter-clockwise in the figure), and (d) the magnetic force on the loop?

•••38 Assume that an electron of mass m and charge magnitude emoves in a circular orbit of radius r about a nucleus. A uniformmagnetic field is then established perpendicular to the plane ofthe orbit. Assuming also that the radius of the orbit does notchange and that the change in the speed of the electron due to field

is small, find an expression for the change in the orbital magneticdipole moment of the electron due to the field.

sec. 32-10 Paramagnetism•39 A sample of the paramagnetic salt to which the mag-netization curve of Fig. 32-14 applies is to be tested to see whether itobeys Curie’s law.The sample is placed in a uniform 0.50 T magneticfield that remains constant throughout the experiment.The magneti-zation M is then measured at temperatures ranging from 10 to 300K. Will it be found that Curie’s law is valid under these conditions?

•40 A sample of the paramagnetic salt to which the magnetiza-tion curve of Fig. 32-14 applies is held at room temperature (300K). At what applied magnetic field will the degree of magnetic sat-uration of the sample be (a) 50% and (b) 90%? (c) Are these fieldsattainable in the laboratory?

•41 A magnet in the form of a cylindrical rod has alength of 5.00 cm and a diameter of 1.00 cm. It has a uniform mag-netization of 5.30 # 103 A/m.What is its magnetic dipole moment?

•42 A 0.50 T magnetic field is applied to a paramagnetic gaswhose atoms have an intrinsic magnetic dipole moment of 1.0 #10$23 J/T. At what temperature will the mean kinetic energy oftranslation of the atoms equal the energy required to reverse sucha dipole end for end in this magnetic field?

••43 An electron with kinetic energy Ke travels in a circular paththat is perpendicular to a uniform magnetic field, which is in thepositive direction of a z axis. The electron’s motion is subject onlyto the force due to the field. (a) Show that the magnetic dipole mo-ment of the electron due to its orbital motion has magnitude m !Ke /B and that it is in the direction opposite that of . What are the(b) magnitude and (c) direction of the magnetic dipole moment ofa positive ion with kinetic energy Ki under the same circum-stances? (d) An ionized gas consists of 5.3 # 1021 electrons/m3 andthe same number density of ions. Take the average electron kineticenergy to be 6.2 # 10$20 J and the average ion kinetic energy to be7.6 # 10$21 J. Calculate the magnetization of the gas when it is in amagnetic field of 1.2 T.

••44 Figure 32-38 gives the magne-tization curve for a paramagnetic ma-terial. The vertical axis scale is set by a ! 0.15, and the horizontal axis scaleis set by b ! 0.2 T/K. Let msam be themeasured net magnetic moment of asample of the material and mmax bethe maximum possible net magneticmoment of that sample. According toCurie’s law, what would be the ratio msam/mmax were the sampleplaced in a uniform magnetic field of magnitude 0.800 T, at a tem-perature of 2.00 K?

•••45 Consider a solid containing N atoms per unit volume,each atom having a magnetic dipole moment . Suppose the direc-tion of can be only parallel or antiparallel to an externally ap-%:

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B:

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B:

B:

%:

Fig. 32-36 Problem 32.

E2

E1 E1

(a) (b)

Fig. 32-37Problems 37 and 71.

S

Axis L

N x

Fig. 32-38 Problem 44.

M/M

max

a

0Bext/T (T/K)

b

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886 CHAPTE R 32 MAXWE LL’S EQUATION S; MAG N ETI S M OF MATTE R

18.0 cm is connected to a source of emf ! ! !m sin vt, where !m !220 V and v ! 130 rad/s. The maximum value of the displacementcurrent is id ! 7.60 mA. Neglect fringing of the electric field at theedges of the plates. (a) What is the maximum value of the current iin the circuit? (b) What is the maximum value of d"E /dt, where "E

is the electric flux through the region between the plates? (c) Whatis the separation d between the plates? (d) Find the maximumvalue of the magnitude of between the plates at a distance r !11.0 cm from the center.

sec. 32-6 Magnets•30 Assume the average value of the vertical component of Earth’smagnetic field is 43 mT (downward) for all of Arizona, which has anarea of 2.95 # 105 km2. What then are the (a) magnitude and (b) di-rection (inward or outward) of the net magnetic flux through the restof Earth’s surface (the entire surface excluding Arizona)?

•31 In New Hampshire the average horizontal component ofEarth’s magnetic field in 1912 was 16 mT, and the average inclina-tion or “dip” was 73°. What was the corresponding magnitude ofEarth’s magnetic field?

sec. 32-7 Magnetism and Electrons•32 Figure 32-36a is a one-axisgraph along which two of the al-lowed energy values (levels) of anatom are plotted. When the atom isplaced in a magnetic field of 0.500 T,the graph changes to that of Fig. 32-36b because of the energy associ-ated with . (We neglect .)Level E1 is unchanged, but level E2

splits into a (closely spaced) tripletof levels. What are the allowed val-ues of associated with (a) energylevel E1 and (b) energy level E2? (c) In joules, what amount of en-ergy is represented by the spacing between the triplet levels?

•33 If an electron in an atom has an orbital angularmomentum with , what are the components (a) Lorb,z and(b) morb,z? If the atom is in an external magnetic field that hasmagnitude 35 mT and is directed along the z axis, what are (c) theenergy Uorb associated with and (d) the energy Uspin associatedwith ? If, instead, the electron has , what are (e) Lorb,z, (f)morb,z, (g) Uorb, and (h) Uspin?

•34 What is the energy difference between parallel andantiparallel alignment of the z component of an electron’s spinmagnetic dipole moment with an external magnetic field of magni-tude 0.25 T, directed parallel to the z axis?

•35 What is the measured component of the orbital magnetic di-pole moment of an electron with (a) and (b) ?

•36 An electron is placed in a magnetic field that is directedalong a z axis. The energy difference between parallel and antipar-allel alignments of the z component of the electron’s spin magneticmoment with is 6.00 # 10$25 J.What is the magnitude of ?

sec. 32-9 Diamagnetism•37 Figure 32-37 shows a loopmodel (loop L) for a diamagneticmaterial. (a) Sketch the magneticfield lines within and about the mate-rial due to the bar magnet. What is

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the direction of (b) the loop’s net magnetic dipole moment ,(c) the conventional current i in the loop (clockwise or counter-clockwise in the figure), and (d) the magnetic force on the loop?

•••38 Assume that an electron of mass m and charge magnitude emoves in a circular orbit of radius r about a nucleus. A uniformmagnetic field is then established perpendicular to the plane ofthe orbit. Assuming also that the radius of the orbit does notchange and that the change in the speed of the electron due to field

is small, find an expression for the change in the orbital magneticdipole moment of the electron due to the field.

sec. 32-10 Paramagnetism•39 A sample of the paramagnetic salt to which the mag-netization curve of Fig. 32-14 applies is to be tested to see whether itobeys Curie’s law.The sample is placed in a uniform 0.50 T magneticfield that remains constant throughout the experiment.The magneti-zation M is then measured at temperatures ranging from 10 to 300K. Will it be found that Curie’s law is valid under these conditions?

•40 A sample of the paramagnetic salt to which the magnetiza-tion curve of Fig. 32-14 applies is held at room temperature (300K). At what applied magnetic field will the degree of magnetic sat-uration of the sample be (a) 50% and (b) 90%? (c) Are these fieldsattainable in the laboratory?

•41 A magnet in the form of a cylindrical rod has alength of 5.00 cm and a diameter of 1.00 cm. It has a uniform mag-netization of 5.30 # 103 A/m.What is its magnetic dipole moment?

•42 A 0.50 T magnetic field is applied to a paramagnetic gaswhose atoms have an intrinsic magnetic dipole moment of 1.0 #10$23 J/T. At what temperature will the mean kinetic energy oftranslation of the atoms equal the energy required to reverse sucha dipole end for end in this magnetic field?

••43 An electron with kinetic energy Ke travels in a circular paththat is perpendicular to a uniform magnetic field, which is in thepositive direction of a z axis. The electron’s motion is subject onlyto the force due to the field. (a) Show that the magnetic dipole mo-ment of the electron due to its orbital motion has magnitude m !Ke /B and that it is in the direction opposite that of . What are the(b) magnitude and (c) direction of the magnetic dipole moment ofa positive ion with kinetic energy Ki under the same circum-stances? (d) An ionized gas consists of 5.3 # 1021 electrons/m3 andthe same number density of ions. Take the average electron kineticenergy to be 6.2 # 10$20 J and the average ion kinetic energy to be7.6 # 10$21 J. Calculate the magnetization of the gas when it is in amagnetic field of 1.2 T.

••44 Figure 32-38 gives the magne-tization curve for a paramagnetic ma-terial. The vertical axis scale is set by a ! 0.15, and the horizontal axis scaleis set by b ! 0.2 T/K. Let msam be themeasured net magnetic moment of asample of the material and mmax bethe maximum possible net magneticmoment of that sample. According toCurie’s law, what would be the ratio msam/mmax were the sampleplaced in a uniform magnetic field of magnitude 0.800 T, at a tem-perature of 2.00 K?

•••45 Consider a solid containing N atoms per unit volume,each atom having a magnetic dipole moment . Suppose the direc-tion of can be only parallel or antiparallel to an externally ap-%:

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