i'l#:Tff 'fr:il ff iff - Físicadcyt · PDF file1028 CHAPTER 32 • Inductance Section 32.5 Oscillations in an LC Circuit 46. A 1.00-!F capacitor is charged by a 40.0-V power supply

1030 C H A P T E R 3 2 • Inductance

70. In Figure P32.69 take ! " 6.00 V, R 1 " 5.00 #, andR 2 " 1.00 #. The inductor has negligible resistance. Whenthe switch is opened after having been closed for along time, the current in the inductor drops to 0.250 A in0.150 s. What is the inductance of the inductor?

In Figure P32.71, the switch is closed for t $ 0, and steady-state conditions are established. The switch is opened att " 0. (a) Find the initial voltage !0 across L just aftert " 0. Which end of the coil is at the higher potential: a orb? (b) Make freehand graphs of the currents in R 1 andin R2 as a function of time, treating the steady-state direc-tions as positive. Show values before and after t " 0.(c) How long after t " 0 does the current in R2 have thevalue 2.00 mA?

71.

74. An air-core solenoid 0.500 m in length contains 1 000turns and has a cross-sectional area of 1.00 cm2. (a) Ignor-ing end effects, find the self-inductance. (b) A secondarywinding wrapped around the center of the solenoid has100 turns. What is the mutual inductance? (c) The sec-ondary winding carries a constant current of 1.00 A, andthe solenoid is connected to a load of 1.00 k#. Theconstant current is suddenly stopped. How much chargeflows through the load resistor?

75. The lead-in wires from a television antenna are often con-structed in the form of two parallel wires (Fig. P32.75).(a) Why does this configuration of conductors have aninductance? (b) What constitutes the flux loop for thisconfiguration? (c) Ignoring any magnetic flux inside thewires, show that the inductance of a length x of this type oflead-in is

where a is the radius of the wires and w is their center-to-center separation.

L "%0x&

ln ! w ' aa "

S

6.00 k! " 0.400 HL18.0 V

2.00 k!

R1

R2

a

b

Figure P32.71

L

R

C

S 0"

Figure P32.72

7.50 !

450 mH

10.0 V12.0 V

Armature

R

Figure P32.73

TV setI

I

TV antenna

Figure P32.75

72. The open switch in Figure P32.72 is closed at t " 0. Beforethe switch is closed, the capacitor is uncharged, and allcurrents are zero. Determine the currents in L, C, and Rand the potential differences across L, C, and R (a) at theinstant after the switch is closed, and (b) long after it isclosed.

To prevent damage from arcing in an electric motor, adischarge resistor is sometimes placed in parallel with thearmature. If the motor is suddenly unplugged whilerunning, this resistor limits the voltage that appears acrossthe armature coils. Consider a 12.0-V DC motor with anarmature that has a resistance of 7.50 # and an inductanceof 450 mH. Assume the back emf in the armature coils is10.0 V when the motor is running at normal speed. (Theequivalent circuit for the armature is shown in FigureP32.73.) Calculate the maximum resistance R that limitsthe voltage across the armature to 80.0 V when the motor isunplugged.

73.

76. The resistance of a superconductor. In an experiment carriedout by S. C. Collins between 1955 and 1958, a current wasmaintained in a superconducting lead ring for 2.50 yrwith no observed loss. If the inductance of the ring was3.14 ( 10'8 H, and the sensitivity of the experiment was

Review problems. Problems 76 through 79 apply ideasfrom this chapter and earlier chapters to some propertiesof superconductors, which were introduced in Section27.5.


70. In Figure P32.69 take ! " 6.00 V, R 1 " 5.00 #, andR 2 " 1.00 #. The inductor has negligible resistance. Whenthe switch is opened after having been closed for along time, the current in the inductor drops to 0.250 A in0.150 s. What is the inductance of the inductor?

In Figure P32.71, the switch is closed for t $ 0, and steady-state conditions are established. The switch is opened att " 0. (a) Find the initial voltage !0 across L just aftert " 0. Which end of the coil is at the higher potential: a orb? (b) Make freehand graphs of the currents in R 1 andin R2 as a function of time, treating the steady-state direc-tions as positive. Show values before and after t " 0.(c) How long after t " 0 does the current in R2 have thevalue 2.00 mA?

71.

74. An air-core solenoid 0.500 m in length contains 1 000turns and has a cross-sectional area of 1.00 cm2. (a) Ignor-ing end effects, find the self-inductance. (b) A secondarywinding wrapped around the center of the solenoid has100 turns. What is the mutual inductance? (c) The sec-ondary winding carries a constant current of 1.00 A, andthe solenoid is connected to a load of 1.00 k#. Theconstant current is suddenly stopped. How much chargeflows through the load resistor?

75. The lead-in wires from a television antenna are often con-structed in the form of two parallel wires (Fig. P32.75).(a) Why does this configuration of conductors have aninductance? (b) What constitutes the flux loop for thisconfiguration? (c) Ignoring any magnetic flux inside thewires, show that the inductance of a length x of this type oflead-in is

where a is the radius of the wires and w is their center-to-center separation.

L "%0x&

ln ! w ' aa "

S

6.00 k! " 0.400 HL18.0 V

2.00 k!

R1

R2

a

b

Figure P32.71

L

R

C

S 0"

Figure P32.72

7.50 !

450 mH

10.0 V12.0 V

Armature

R

Figure P32.73

TV setI

I

TV antenna

Figure P32.75

72. The open switch in Figure P32.72 is closed at t " 0. Beforethe switch is closed, the capacitor is uncharged, and allcurrents are zero. Determine the currents in L, C, and Rand the potential differences across L, C, and R (a) at theinstant after the switch is closed, and (b) long after it isclosed.

To prevent damage from arcing in an electric motor, adischarge resistor is sometimes placed in parallel with thearmature. If the motor is suddenly unplugged whilerunning, this resistor limits the voltage that appears acrossthe armature coils. Consider a 12.0-V DC motor with anarmature that has a resistance of 7.50 # and an inductanceof 450 mH. Assume the back emf in the armature coils is10.0 V when the motor is running at normal speed. (Theequivalent circuit for the armature is shown in FigureP32.73.) Calculate the maximum resistance R that limitsthe voltage across the armature to 80.0 V when the motor isunplugged.

73.

76. The resistance of a superconductor. In an experiment carriedout by S. C. Collins between 1955 and 1958, a current wasmaintained in a superconducting lead ring for 2.50 yrwith no observed loss. If the inductance of the ring was3.14 ( 10'8 H, and the sensitivity of the experiment was

Review problems. Problems 76 through 79 apply ideasfrom this chapter and earlier chapters to some propertiesof superconductors, which were introduced in Section27.5.


Section 32.5 Oscillations in an LC Circuit

46. A 1.00-!F capacitor is charged by a 40.0-V power supply.The fully charged capacitor is then discharged through a10.0-mH inductor. Find the maximum current in theresulting oscillations.

47. An LC circuit consists of a 20.0-mH inductor and a 0.500-!F capacitor. If the maximum instantaneous currentis 0.100 A, what is the greatest potential difference acrossthe capacitor?

48. In the circuit of Figure P32.48, the battery emf is 50.0 V,the resistance is 250 ", and the capacitance is 0.500 !F.The switch S is closed for a long time and no voltage ismeasured across the capacitor. After the switch is opened,the potential difference across the capacitor reachesa maximum value of 150 V. What is the value of theinductance?

(a) the energy stored in the capacitor; (b) the energy storedin the inductor; (c) the total energy in the circuit.

Section 32.6 The RLC Circuit

54. In Figure 32.21, let R # 7.60 ", L # 2.20 mH, and C #1.80 !F. (a) Calculate the frequency of the damped oscilla-tion of the circuit. (b) What is the critical resistance?

Consider an LC circuit in which L # 500 mH and C # 0.100 !F. (a) What is the resonance frequency $0?(b) If a resistance of 1.00 k" is introduced into this circuit,what is the frequency of the (damped) oscillations? (c) Whatis the percent difference between the two frequencies?

56. Show that Equation 32.28 in the text is Kirchhoff’s looprule as applied to the circuit in Figure 32.21.

57. The energy of an RLC circuit decreases by 1.00% duringeach oscillation when R # 2.00 ". If this resistance isremoved, the resulting LC circuit oscillates at a frequencyof 1.00 kHz. Find the values of the inductance and thecapacitance.

58. Electrical oscillations are initiated in a series circuit con-taining a capacitance C, inductance L, and resistance R.(a) If R %% (weak damping), how much timeelapses before the amplitude of the current oscillation fallsoff to 50.0% of its initial value? (b) How long does it takethe energy to decrease to 50.0% of its initial value?

Additional Problems

59. Review problem. This problem extends the reasoning ofSection 26.4, Problem 26.37, Example 30.6, and Section32.3. (a) Consider a capacitor with vacuum between itslarge, closely spaced, oppositely charged parallel plates.Show that the force on one plate can be accounted for bythinking of the electric field between the plates as exertinga “negative pressure” equal to the energy density of theelectric field. (b) Consider two infinite plane sheets carry-ing electric currents in opposite directions with equal lin-ear current densities Js . Calculate the force per area actingon one sheet due to the magnetic field created by theother sheet. (c) Calculate the net magnetic field betweenthe sheets and the field outside of the volume betweenthem. (d) Calculate the energy density in the magneticfield between the sheets. (e) Show that the force on onesheet can be accounted for by thinking of the magneticfield between the sheets as exerting a positive pressureequal to its energy density. This result for magneticpressure applies to all current configurations, not just tosheets of current.

60. Initially, the capacitor in a series LC circuit is charged. Aswitch is closed at t # 0, allowing the capacitor todischarge, and at time t the energy stored in the capacitoris one fourth of its initial value. Determine L, assuming Cis known.

61. A 1.00-mH inductor and a 1.00-!F capacitor are connectedin series. The current in the circuit is described byI # 20.0t, where t is in seconds and I is in amperes. Thecapacitor initially has no charge. Determine (a) the

!4L/C

55.

R

" L C

S

Figure P32.48

1.00 µF

10.0 #

S

ba

µ

0.100 H

12.0 V

Figure P32.52

A fixed inductance L # 1.05 !H is used in series with avariable capacitor in the tuning section of a radiotele-phone on a ship. What capacitance tunes the circuit to thesignal from a transmitter broadcasting at 6.30 MHz?

50. Calculate the inductance of an LC circuit that oscillates at120 Hz when the capacitance is 8.00 !F.

51. An LC circuit like the one in Figure 32.16 contains an82.0-mH inductor and a 17.0-!F capacitor that initially car-ries a 180-!C charge. The switch is open for t % 0 andthen closed at t # 0. (a) Find the frequency (in hertz) ofthe resulting oscillations. At t # 1.00 ms, find (b) thecharge on the capacitor and (c) the current in the circuit.

52. The switch in Figure P32.52 is connected to point a fora long time. After the switch is thrown to point b, whatare (a) the frequency of oscillation of the LC circuit,(b) the maximum charge that appears on the capacitor,(c) the maximum current in the inductor, and (d) thetotal energy the circuit possesses at t # 3.00 s?

49.

An LC circuit like that in Figure 32.16 consists of a3.30-H inductor and an 840-pF capacitor, initially carrying a105-!C charge. The switch is open for t % 0 and then closedat t # 0. Compute the following quantities at t # 2.00 ms:

53.













Additional Problems




!4L/C

55.

R

" L C

S

Figure P32.48

1.00 µF

10.0 #

S

ba

µ

0.100 H

12.0 V

Figure P32.52





49.


53.













Additional Problems




!4L/C

55.

R

" L C

S

Figure P32.48

1.00 µF

10.0 #

S

ba

µ

0.100 H

12.0 V

Figure P32.52





49.


53.

59. A circular loop of wire of radius r is in a uniform magneticfield, with the plane of the loop perpendicular to thedirection of the field (Fig. P31.59). The magnetic fieldvaries with time according to B(t) ! a " bt, where a and bare constants. (a) Calculate the magnetic flux through theloop at t ! 0. (b) Calculate the emf induced in the loop.(c) If the resistance of the loop is R , what is the inducedcurrent? (d) At what rate is energy being delivered to theresistance of the loop?

60. In Figure P31.60, a uniform magnetic field decreases at aconstant rate dB/dt ! # K, where K is a positive constant.A circular loop of wire of radius a containing a resistanceR and a capacitance C is placed with its plane normal tothe field. (a) Find the charge Q on the capacitor when it isfully charged. (b) Which plate is at the higher potential?(c) Discuss the force that causes the separation of charges.

A rectangular coil of 60 turns, dimensions 0.100 m by0.200 m and total resistance 10.0 $, rotates with angularspeed 30.0 rad/s about the y axis in a region where a 1.00-Tmagnetic field is directed along the x axis. The rotation isinitiated so that the plane of the coil is perpendicular tothe direction of B at t ! 0. Calculate (a) the maximuminduced emf in the coil, (b) the maximum rate of changeof magnetic flux through the coil, (c) the induced emf att ! 0.050 0 s, and (d) the torque exerted by the magneticfield on the coil at the instant when the emf is a maximum.

62. A small circular washer of radius 0.500 cm is held directlybelow a long, straight wire carrying a current of 10.0 A.The washer is located 0.500 m above the top of a table(Fig. P31.62). (a) If the washer is dropped from rest, whatis the magnitude of the average induced emf in the washerfrom the time it is released to the moment it hits the table-top? Assume that the magnetic field is nearly constant overthe area of the washer, and equal to the magnetic field atthe center of the washer. (b) What is the direction of theinduced current in the washer?

61.

63. A conducting rod of length ! moves with velocity v parallelto a long wire carrying a steady current I. The axis of therod is maintained perpendicular to the wire with the nearend a distance r away, as shown in Figure P31.63. Showthat the magnitude of the emf induced in the rod is

&%&!&0Iv2'

ln !1 "!

r "

64. A rectangular loop of dimensions ! and w moves with aconstant velocity v away from a long wire that carries acurrent I in the plane of the loop (Fig. P31.64). The totalresistance of the loop is R. Derive an expression that givesthe current in the loop at the instant the near side is adistance r from the wire.

The magnetic flux through a metal ring varies with time taccording to (B ! 3(at3 # bt2) T ) m2 , with a ! 2.00 s#3

and b ! 6.00 s#2. The resistance of the ring is 3.00 $.Determine the maximum current induced in the ringduring the interval from t ! 0 to t ! 2.00 s.

66. Review problem. The bar of mass m in Figure P31.66 ispulled horizontally across parallel rails by a massless stringthat passes over an ideal pulley and is attached to a

65.

1000 CHAPTE R 31 • Faraday’s Law

B

Figure P31.59

R C

Bin

! ! ! !

! ! ! !

! ! ! !

! ! ! !Figure P31.60

h

I

Figure P31.62

vIR

r w

!

Figure P31.64

r

v

I

!

Figure P31.63

before the top edge of the loop reaches the field, the loopapproaches a terminal speed vT. (a) Show that

(b) Why is vT proportional to R? (c) Why is it inversely propor-tional to B2?

Section 31.7 Maxwell’s Equations44. An electron moves through a uniform electric field

E ! (2.50 i " 5.00 j) V/m and a uniform magnetic fieldB ! (0.400k)T. Determine the acceleration of theelectron when it has a velocity v ! 10.0 i m/s.

A proton moves through a uniform electric field given byE ! 50.0 j V/m and a uniform magnetic field B !(0.200 i " 0.300 j " 0.400k)T. Determine the accelerationof the proton when it has a velocity v ! 200 i m/s.

Additional Problems46. A steel guitar string vibrates (Figure 31.6). The compo-

nent of magnetic field perpendicular to the area of apickup coil nearby is given by

B ! 50.0 mT " (3.20 mT) sin(2# 523 t/s)

The circular pickup coil has 30 turns and radius 2.70 mm.Find the emf induced in the coil as a function of time.

47. Figure P31.47 is a graph of the induced emf versus timefor a coil of N turns rotating with angular speed $ in a uni-form magnetic field directed perpendicular to the axis ofrotation of the coil. What If ? Copy this sketch (on a largerscale), and on the same set of axes show the graph of emfversus t (a) if the number of turns in the coil is doubled;(b) if instead the angular speed is doubled; and (c) if theangular speed is doubled while the number of turns in thecoil is halved.

45.

vT !MgRB2w2

48. A technician wearing a brass bracelet enclosing area0.005 00 m2 places her hand in a solenoid whose magneticfield is 5.00 T directed perpendicular to the plane of thebracelet. The electrical resistance around the circumfer-ence of the bracelet is 0.020 0 %. An unexpected powerfailure causes the field to drop to 1.50 T in a time of20.0 ms. Find (a) the current induced in the bracelet and(b) the power delivered to the bracelet. Note: As this

problem implies, you should not wear any metal objectswhen working in regions of strong magnetic fields.

49. Two infinitely long solenoids (seen in cross section) passthrough a circuit as shown in Figure P31.49. The magni-tude of B inside each is the same and is increasing at therate of 100 T/s. What is the current in each resistor?

50. A conducting rod of length ! ! 35.0 cm is free to slide ontwo parallel conducting bars as shown in Figure P31.50.Two resistors R1 ! 2.00 % and R 2 ! 5.00 % are connectedacross the ends of the bars to form a loop. A constantmagnetic field B ! 2.50 T is directed perpendicularly intothe page. An external agent pulls the rod to the left with aconstant speed of v ! 8.00 m/s. Find (a) the currents inboth resistors, (b) the total power delivered to the resis-tance of the circuit, and (c) the magnitude of the appliedforce that is needed to move the rod with this constantvelocity.

51. Suppose you wrap wire onto the core from a roll of cello-phane tape to make a coil. Describe how you can use a barmagnet to produce an induced voltage in the coil. What isthe order of magnitude of the emf you generate? State thequantities you take as data and their values.

52. A bar of mass m, length d, and resistance R slides withoutfriction in a horizontal plane, moving on parallel rails asshown in Figure P31.52. A battery that maintains aconstant emf & is connected between the rails, and aconstant magnetic field B is directed perpendicularly tothe plane of the page. Assuming the bar starts from rest,show that at time t it moves with a speed

v !&Bd

(1 ' e'B 2d 2t/mR)


(mV)

1 2t(ms)

!

10

5

–5

–10

3

Figure P31.47

0.500 m 0.500 m

0.500 m6.00 "5.00 "

r2 = 0.150 m

BoutBin

r1 = 0.100 m

3.00 "

# ####

##

Figure P31.49

######

######

######

######

######

######

5.00 "

B######

######

2.00 " v

Figure P31.50

d B (out of page) !

Figure P31.52


is closed? (c) What is the direction of the induced currentin R when the current I in Figure P31.28c decreasesrapidly to zero? (d) A copper bar is moved to the rightwhile its axis is maintained in a direction perpendicular toa magnetic field, as shown in Figure P31.28d. If the top ofthe bar becomes positive relative to the bottom, what is thedirection of the magnetic field?

A rectangular coil with resistance R has N turns, each oflength ! and width w as shown in Figure P31.29. The coilmoves into a uniform magnetic field B with constant velocityv. What are the magnitude and direction of the totalmagnetic force on the coil (a) as it enters the magnetic field,(b) as it moves within the field, and (c) as it leaves the field?

29.

31. Two parallel rails with negligible resistance are 10.0 cm apartand are connected by a 5.00-! resistor. The circuit alsocontains two metal rods having resistances of 10.0 ! and15.0 ! sliding along the rails (Fig. P31.31). The rods arepulled away from the resistor at constant speeds of 4.00 m/sand 2.00 m/s, respectively. A uniform magnetic field ofmagnitude 0.010 0 T is applied perpendicular to the planeof the rails. Determine the current in the 5.00 -! resistor.

Section 31.4 Induced emf and Electric Fields32. For the situation shown in Figure P31.32, the magnetic

field changes with time according to the expressionB " (2.00t 3 # 4.00t 2 $ 0.800)T, and r2 " 2R " 5.00 cm.(a) Calculate the magnitude and direction of the forceexerted on an electron located at point P2 when t " 2.00 s.(b) At what time is this force equal to zero?

A magnetic field directed into the page changes with timeaccording to B " (0.030 0t 2 $ 1.40)T, where t is inseconds. The field has a circular cross section of radiusR " 2.50 cm (Fig. P31.32). What are the magnitude anddirection of the electric field at point P1 when t " 3.00 sand r1 " 0.020 0 m ?

34. A long solenoid with 1 000 turns per meter andradius 2.00 cm carries an oscillating current given byI " (5.00 A) sin(100%t). What is the electric field inducedat a radius r " 1.00 cm from the axis of the solenoid?What is the direction of this electric field when the currentis increasing counterclockwise in the coil?

Section 31.5 Generators and Motors

33.

A coil of area 0.100 m2 is rotating at 60.0 rev/s withthe axis of rotation perpendicular to a 0.200-T magneticfield. (a) If the coil has 1 000 turns, what is the maximumemf generated in it? (b) What is the orientation of the coilwith respect to the magnetic field when the maximuminduced voltage occurs?

36. In a 250-turn automobile alternator, the magnetic flux ineach turn is &B " (2.50 ' 10#4 Wb) cos((t), where ( isthe angular speed of the alternator. The alternator isgeared to rotate three times for each engine revolution.When the engine is running at an angular speed of1 000 rev/min, determine (a) the induced emf in thealternator as a function of time and (b) the maximum emfin the alternator.

37. A long solenoid, with its axis along the x axis, consists of200 turns per meter of wire that carries a steady current of15.0 A. A coil is formed by wrapping 30 turns of thin wirearound a circular frame that has a radius of 8.00 cm. Thecoil is placed inside the solenoid and mounted on anaxis that is a diameter of the coil and coincides with they axis. The coil is then rotated with an angular speedof 4.00% rad/s. (The plane of the coil is in the yz plane

35.

w

v

Bin

! ! ! ! ! ! !

! ! ! ! ! ! !

! ! ! ! ! ! !

! ! ! ! ! ! !

! ! ! ! ! ! !!

Figure P31.29

a

R

b

Motion towardthe loop

S

N

Figure P31.30

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!

10.0 " 15.0 "

5.00 " 2.00 m/sB4.00 m/s

Figure P31.31

Bin

! ! ! ! ! ! ! !

! ! ! ! ! ! !

! ! ! ! ! ! !

! ! ! ! ! !

! ! ! ! !

! ! ! ! ! !

! ! ! ! !

r2P2

P1

R

r1

Figure P31.32 Problems 32 and 33.

30. In Figure P31.30, the bar magnet is moved toward theloop. Is Va # Vb positive, negative, or zero? Explain.

Problems 28 and 62 in Chapter 29 can be assigned withthis section.

Problems

essM

Tutoring problem available (at instructor's discretion) in WileyPLUS and WebAssignWorked-out solution available in Student Solutions Manual WWW Worked-out solution is at

o - .ro Number of dots indicates level of problem difficulty ILW lnteractive solution is atAdditional information available in The Flying Circus of Physics and at flyingcircusofphysics.com

sec. 31-2 LC Oscillations, Oualitativelyrl In a certain oscillattng LC circuit, the total energy is con-verted from electrical energy in the capacitor to magneticenergy in the inductor in 1.50 ps. What are (a) the period ofoscillation and (b) the frequency of oscillation? (c) How longafter the magnetic energy is a maximum will it be a maximumagain?ol What is the capacitance of an oscillatrng LC circuit if themaximum charge on the capacitor is I.60 pC and the totalenergy is 140 pJ?of In an oscillating LC circuit, L: 1.10 mH and C - 4.00pF, The maximum charge on the capacitor is 3.00 p,C. Find themaximum current.:d The frequency of oscillation of a certain LC circuit is 200kHz. At time t : 0,, plate A of the capacitor has maximumpositive charge. At what earliest time / > 0 will (a) plate Aagain have maximum positive charge, (b) the other plate ofthe capacitor have maximum positive charge, and (.) theinductor have maximum magnetic field?r$ An oscillating LC circuit consists of a75.0 mH inductorand a 3.60 pF capacitor. If the maximum charge on the capaci-tor is 2.90 p,C,, what are (a) the total energy in the circuit and(b) the maximum current?

sec. 31-3 The Electrical-Mechanical AnalogyrS A 0.50 kg body oscillates in SHM on a spring that, whenextended 2.0 mm from its equilibrium position, has an 8.0 Nrestoring force. What are (a) the angular frequency of oscilla-tion, (b) the period of oscillation, and (c) the capacitance of anLC circuit with the same periodlf L is 5.0 H?..7 The energy in an oscillating LC circuit containing aI.25 H inductor is 5.70 pJ.The maximum charge on the capac-itor is 175 p,C. For a mechanical system with the same period,find the (a) mass, (b) spring constant, (c) maximum displace-ment, and (d) maximum speed. ssm

sec. 31-4 LC Oscillations, Ouantitativelyr$ LC oscillators have been used in circuits connected toloudspeakers to create some of the sounds of electronic music.What inductance must be used with a 6.7 pcF capacitor toproduce a frequency of L0 kHz, which is near the middle ofthe audible range of frequencies?oQ In an oscillating LC circuit with L - 50 mH and C -4.0 p"F,the current is initially a maximum. How long will it takebefore the capacitor is fully charged for the first time? rtw

.10 A single loop consists of inductors (L1, Lr,.. .), capaci-tors (Ct, Cr, . . .), and resistors (Rr, Rr, . . .) connected in seriesas shown, for example, in Fig. 31,-27 a. Show that regardless ofthe sequence of these circuit elements in the loop, the behav-

ior of this circuit is identicalshown in Fig. 3I-27b. (Hint:Problem 47 in Chapter 30.)

to that of the simple LC circuitConsider the loop rule and see

(a)

FlG. 31-27 Problem 10.

oo11 In Fig. 3t-28,R - I4.0 A, C: 6.20 pE, and L - 54.0 ffiH, andthe ideal battery has emf 6 - 34.0V. The switch is kept in position afor a long time and then thrown toposition b. What are the (u) fre-quency and (b) current amplitudeof the resulting oscillations? ttw..12 To construct an oscillating

FlG. 31-28 Problem 11.

LC system, lou can choose from a 10 mH inductor, a 5.0 pFcapacitor, and a 2.0 pF capacitor. What are the (a) smallest,(b) second smallest,(c) second largest, and (d) largest oscilla-tion frequency that can be set up by these elements in variouscombinations?..13 An oscillating LC circuit consisting of a 1.0 nF capaci-tor and a 3.0 mH coil has a maximum voltage of 3.0 V. What

il: i'l#:"Tff 'fr:il ;:X1?: ff iff (61ffi ?: i? Jl'" ilffistored in the magnetic field of the coil? rtw.o14 An inductor is connected across a capacitor whosecapacitance can be varied by turning a knob. We wish to makethe frequency of oscillation of this LC circuit vary linearlywith the angle of rotation of the knob, going from 2 x LOs to4 x 10s Hz as the knob turns through 180'. If L: 1.0 ffiH,plot the required capacitance C as a function of the angle ofrotation of the knob...15 A variable capacitor with a range from 10 to 365 pF isused with a coil to form a variable-frequency LC circuit totune the input to a radio. (a) What is the ratio of maximumfrequency to minimum frequency that can be obtained withsuch a capacitor? If this circuit is to obtain frequencies from0.54 MHz to 1.60 MHz,the ratio computed in (a) is too large.By adding a capacitor in parallel to the variable capacitor, thisrange can be adjusted. To obtain the desired frequency range,(b) what capacitance should be added and (c) what induc-tance should the coil have? ssm www..16 A series circuit containing inductance L1 and capac-itance Cr oscillates at angular frequency ar. A second seriescircuit, containing inductance L2 and capacitance C2, oscillates

Cr L2 Rl C2

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i'l#:Tff 'fr:il ff iff - Físicadcyt · PDF file1028 CHAPTER 32 • Inductance Section 32.5 Oscillations in an LC Circuit 46. A 1.00-!F capacitor is charged by a 40.0-V power supply