634 CHAPTER 20 ELECTRIC CIRCUITS

Section 20.9 Internal Resistance

13. A battery has an emf of V and an internal resistance of r. What

resistance R, when connected across the terminals of this battery, will

cause the terminal voltage of the battery

to be (a) (b)

(c) (d) (e)

Section 20.10 Kirchhoff’s Rules

15. When applying Kirchhoff’s rules,

one of the essential steps is to mark each

resistor with plus and minus signs to label

how the potential changes from one end

of the resistor to the other. The circuit in

the drawing contains four resistors, each

marked with the associated plus and

R ! 1

4rR ! rR ! 4r

R ! 2rR ! 1

2r1

2V?

minus signs. However, one resistor is marked incorrectly. Which one

is it? (a) (b) (c) (d)

Section 20.12 Capacitors in Series and in Parallel

18. Three capacitors are identical, each having a capacitance C.

Two of them are connected in series. Then, this series combination

is connected in parallel with the third capacitor. What is the equiva-

lent capacitance of the entire connection? (a) (b) (c) 3C(d) (e)

Section 20.13 RC Circuits

20. The time constant of an RC circuit is 2.6 s. How much time t is

required for the capacitor (uncharged initially) to gain one-half of its

full equilibrium charge?

3

2C2

3C

1

3C1

2C

R4R3R2R1

Note to Instructors: Most of the homework problems in this chapter are available for assignment via an online homework management program suchas WileyPLUS or WebAssign, and those marked with the icon are presented in WileyPLUS using a guided tutorial format that provides enhanced interactivity. See Preface for additional details.

Note: For problems that involve ac conditions, the current and voltage are rms values and the power is an average value, unless indicated otherwise.

ssm Solution is in the Student Solutions Manual.This icon represents a biomedical application.

www Solution is available online at www.wiley.com/college/cutnell

PROBLEMS

Section 20.1 Electromotive Force and Current,

Section 20.2 Ohm’s Law

1. A defibrillator is used during a heart attack to restore the heart

to its normal beating pattern (see Section 19.5). A defibrilla-

tor passes 18 A of current through the torso of a person in 2.0 ms.

(a) How much charge moves during this time? (b) How many elec-

trons pass through the wires connected to the patient?

2. An especially violent lightning bolt has an average current of

1.26 " 103

A lasting 0.138 s. How much charge is delivered to the

ground by the lightning bolt?

3. ssm A fax machine uses 0.110 A of current in its normal mode of

operation, but only 0.067 A in the standby mode. The machine uses

a potential difference of 120 V. In one minute (a) how much more

charge passes through the machine in the normal mode than in the

standby mode, and (b) how much more energy is used?

4. Suppose that the resistance between the walls of a biological

cell is 5.0 " 109 #. (a) What is the current when the poten-

tial difference between the walls is 75 mV? (b) If the current is com-

posed of Na$

ions (q ! $e), how many such ions flow in 0.50 s?

5. The heating element of a clothes dryer has a resistance of 11 #and is connected across a 240-V electrical outlet. What is the current

in the heating element?

* 6. The resistance of a bagel toaster is 14 #. To prepare a bagel,

the toaster is operated for one minute from a 120-V outlet. How

much energy is delivered to the toaster?

* 7. A resistor is connected across the terminals of a 9.0-V battery,

which delivers 1.1 " 105

J of energy to the resistor in six hours. What

is the resistance of the resistor?

*8. A car battery has a rating of 220 ampere %hours (A %h). This rating

is one indication of the total charge that the battery can provide to a

circuit before failing. (a) What is the total charge (in coulombs)

that this battery can provide? (b) Determine the maximum current

that the battery can provide for 38 minutes.

**9. ssm A beam of protons is moving toward a target in a particle ac-

celerator. This beam constitutes a current whose value is 0.50 &A.

(a) How many protons strike the target in 15 s? (b) Each proton has

a kinetic energy of 4.9 " 10'12

J. Suppose the target is a 15-gram

block of aluminum, and all the kinetic energy of the protons goes into

heating it up. What is the change in temperature of the block that re-

sults from the 15-s bombardment of protons?

Section 20.3 Resistance and Resistivity

10. The resistance and the magnitude of the current depend on

the path that the current takes. The drawing shows three situations in

which the current takes different paths through a piece of material.

Each of the rectangular pieces is made from a material whose resis-

tivity is ( ! 1.50 " 10'2 #%m, and the unit of length in the drawing

is L0 ! 5.00 cm. Each piece of material is connected to a 3.00-V bat-

tery. Find (a) the resistance and (b) the current in each case.

11. Two wires are identical, except that one is aluminum and one is

copper. The aluminum wire has a resistance of 0.20 #. What is the

resistance of the copper wire?

12. A cylindrical copper cable carries a current of 1200 A. There is

a potential difference of 1.6 " 10'2

V between two points on the ca-

ble that are 0.24 m apart. What is the radius of the cable?

13. In Section 12.3 it was mentioned that temperatures are often

measured with electrical resistance thermometers made of platinum

wire. Suppose that the resistance of a platinum resistance thermometer

is 125 # when its temperature is 20.0 )C. The wire is then immersed in

boiling chlorine, and the resistance drops to 99.6 #. The temperature

V

+–

+ – –V+

V

2L0 L04L0

2L0 L04L0

2L0 L0

4L0

(a) (c)(b)

+ –

+ –

+ –

+ – + –

+–

+

–

+ –

5.0 V

7.0 V

3.0 V

2.0 V

R4

I4

R1

I1

I3

I5 I6

R2

R3

I2

Question 15

2762T_ch20_599-641.qxd 9/12/08 3:02 PM Page 634

PROBLEMS 635

coefficient of resistivity of platinum is ! " 3.72 # (C°) . What isthe temperature of the boiling chlorine?

14. High-voltage power lines are a familiar sight throughout thecountry. The aluminum wire used for some of these lines has a cross-sectional area of 4.9 # 10$4 m2. What is the resistance of ten kilo-meters of this wire?

15. ssm www Two wires have the same length and the same resis-tance. One is made from aluminum and the other from copper. Obtainthe ratio of the cross-sectional area of the aluminum wire to that ofthe copper wire.

16. A large spool in an electrician’s workshop has 75 m of insulation-coated wire coiled around it. When the electrician connects a batteryto the ends of the spooled wire, the resulting current is 2.4 A. Someweeks later, after cutting off various lengths of wire for use in repairs,the electrician finds that the spooled wire carries a 3.1-A currentwhen the same battery is connected to it. What is the length of wireremaining on the spool?

*17. The temperature coefficient of resistivity for the metal gold is0.0034 (C%)$1, and for tungsten it is 0.0045 (C%)$1. The resistance ofa gold wire increases by 7.0% due to an increase in temperature. Forthe same increase in temperature, what is the percentage increase inthe resistance of a tungsten wire?

*18. A toaster uses a Nichrome heating wire. When the toaster isturned on at 20 %C, the initial current is 1.50 A. A few seconds later,the toaster warms up and the current now has a value of 1.30 A. Theaverage temperature coefficient of resistivity for Nichrome wire is4.5 # (C°) . What is the temperature of the heating wire?

*19. ssm www A wire has a resistance of 21.0 &. It is melteddown, and from the same volume of metal a new wire is made thatis three times longer than the original wire. What is the resistanceof the new wire?

*20. The filament in an incandescent light bulb is made from tung-sten. The light bulb is plugged into a 120-V outlet and draws a cur-rent of 1.24 A. If the radius of the tungsten wire is 0.0030 mm, howlong must the wire be?

** 21. An aluminum wire is hung between two towers and has a lengthof 175 m. A current of 125 A exists in the wire, and the potential dif-ference between the ends of the wire is 0.300 V. The density of alu-minum is 2700 kg/m3. Find the mass of the wire.

Section 20.4 Electric Power

22. The heating element in an iron has a resistance of 24 &. The ironis plugged into a 120-V outlet. What is the power delivered to theiron?

23. A cigarette lighter in a car is a resistor that, when activated, isconnected across the 12-V battery. Suppose that a lighter uses 33 Wof power. Find (a) the resistance of the lighter and (b) the currentthat the battery delivers to the lighter.

24. A blow-dryer and a vacuum cleaner each operate with a voltageof 120 V. The current rating of the blow-dryer is 11 A, and that of thevacuum cleaner is 4.0 A. Determine the power consumed by (a) theblow-dryer and (b) the vacuum cleaner. (c) Determine the ratio ofthe energy used by the blow-dryer in 15 minutes to the energy usedby the vacuum cleaner in one-half hour.

25. There are approximately 110 million households that use TVs inthe United States. Each TV uses, on average, 75 W of power and isturned on for 6.0 hours a day. If electrical energy costs $0.12 perkWh, how much money is spent every day in keeping 110 millionTVs turned on?

$110$4

$110$3 26. Each of the four circuits in the drawing consists of a singleresistor whose resistance is either R or 2R, and a single batterywhose voltage is either V or 2V. The unit of voltage in each circuitis V " 12.0 V and the unit of resistance is R " 6.00 &. Determine(a) the power supplied to each resistor and (b) the current deliv-ered to each resistor.

27. ssm In doing a load of clothes, a clothes dryer uses 16 A of cur-rent at 240 V for 45 min. A personal computer, in contrast, uses 2.7 Aof current at 120 V. With the energy used by the clothes dryer, howlong (in hours) could you use this computer to “surf” the Internet?

*28. A piece of Nichrome wire has a radius of 6.5 # m. It is usedin a laboratory to make a heater that uses 4.00 # 102 W of powerwhen connected to a voltage source of 120 V. Ignoring the effect oftemperature on resistance, estimate the necessary length of wire.

*29. An electric heater used to boil small amounts of water consistsof a 15-& coil that is immersed directly in the water. It operates froma 120-V socket. How much time is required for this heater to raise thetemperature of 0.50 kg of water from 13 %C to the normal boilingpoint?

*30. The rear window of a van is coated with a layer of ice at 0 %C.The density of ice is 917 kg/m3. The driver of the van turns on therear-window defroster, which operates at 12 V and 23 A. The de-froster directly heats an area of 0.52 m2 of the rear window. What isthe maximum thickness of ice coating this area that the defroster canmelt in 3.0 minutes?

*31. ssm Tungsten has a temperature coefficient of resistivity of0.0045 (C%)$1. A tungsten wire is connected to a source of constantvoltage via a switch. At the instant the switch is closed, the tempera-ture of the wire is 28 %C, and the initial power delivered to the wire isP0. At what wire temperature will the power that is delivered to thewire be decreased to P0?

Section 20.5 Alternating Current

32. According to Equation 20.7, an ac voltage V is given as a func-tion of time t by V " V0 sin 2'f t, where V0 is the peak voltage and fis the frequency (in hertz). For a frequency of 60.0 Hz, what is thesmallest value of the time at which the voltage equals one-half of thepeak value?

33. The rms current in a copy machine is 6.50 A, and the resistance ofthe machine is 18.6 &. What are (a) the average power and (b) thepeak power delivered to the machine?

34. The current in a circuit is ac and has a peak value of 2.50 A.Determine the rms current.

35. ssm The average power used by a stereo speaker is 55 W.Assuming that the speaker can be treated as a 4.0-& resistance, findthe peak value of the ac voltage applied to the speaker.

36. A 550-W space heater is designed for operation in Germany,where household electrical outlets supply 230 V (rms) service. Whatis the power output of the heater when plugged into a 120-V (rms)electrical outlet in a house in the United States? Ignore the effects oftemperature on the heater’s resistance.

12

10$4

+ –

V(a)

R

+ –

V(b)

2R 2R

+ –

2V 2V(c)

R

+ –

(d)

2762T_ch20_599-641.qxd 9/12/08 3:02 PM Page 635

37. Review Conceptual Example 7 as an aid in solving this prob-lem. A portable electric heater uses 18 A of current. The manufac-turer recommends that an extension cord attached to the heaterreceive no more than 2.0 W of power per meter of length. What isthe smallest radius of copper wire that can be used in the extensioncord? (Note: An extension cord contains two wires.)

* 38. The recovery time of a hot water heater is the time requiredto heat all the water in the unit to the desired temperature. Supposethat a 52-gal (1.00 gal ! 3.79 " m3) unit starts with cold waterat 11 °C and delivers hot water at 53 °C. The unit is electric and uti-lizes a resistance heater (120 V ac, 3.0 #) to heat the water. Assumingthat no heat is lost to the environment, determine the recovery time(in hours) of the unit.

*39. ssm A light bulb is connected to a 120.0-V wall socket. Thecurrent in the bulb depends on the time t according to the relation I ! (0.707 A) sin [(314 Hz)t]. (a) What is the frequency of the alter-nating current? (b) Determine the resistance of the bulb’s filament.(c) What is the average power delivered to the light bulb?

**40. To save on heating costs, the owner of a greenhouse keeps 660 kgof water around in barrels. During a winter day, the water is heatedby the sun to 10.0 $C. During the night the water freezes into ice at0.0 $C in nine hours. What is the minimum ampere rating of an elec-tric heating system (240 V) that would provide the same heating effectas the water does?

Section 20.6 Series Wiring

41. ssm The current in a series circuit is 15.0 A. When an additional8.00-# resistor is inserted in series, the current drops to 12.0 A. Whatis the resistance in the original circuit?

42. A 60.0-W lamp is placed in series with a resistor and a 120.0-Vsource. If the voltage across the lamp is 25 V, what is the resistanceR of the resistor?

43. ssm Three resistors, 25, 45, and 75 #, are connected in series, anda 0.51-A current passes through them. What are (a) the equivalent resistance and (b) the potential difference across the three resistors?

44. An 86-# resistor and a 67-# resistor are connected in seriesacross a battery. The voltage across the 86-# resistor is 27 V. What isthe voltage across the 67-# resistor?

45. The current in a 47-# resistor is 0.12 A. This resistor is in serieswith a 28-# resistor, and the series combination is connected acrossa battery. What is the battery voltage?

*46. Multiple-Concept Example 9 reviews the concepts that are im-portant to this problem. A light bulb is wired in series with a 144-#resistor, and they are connected across a 120.0-V source. The powerdelivered to the light bulb is 23.4 W. What is the resistance of thelight bulb? Note that there are two possible answers.

*47. ssm An extension cord is used with an electric weed trimmer thathas a resistance of 15.0 #. The extension cord is made of copper wirethat has a cross-sectional area of 1.3 " m2. The combined lengthof the two wires in the extension cord is 92 m. (a) Determine the re-sistance of the extension cord. (b) The extension cord is plugged intoa 120-V socket. What voltage is applied to the trimmer itself?

*48. One heater uses 340 W of power when connected by itself to abattery. Another heater uses 240 W of power when connected by it-self to the same battery. How much total power do the heaters usewhen they are both connected in series across the battery?

10%6

10%3

**49. Two resistances, R1 and R2, are connected in series across a 12-Vbattery. The current increases by 0.20 A when R2 is removed, leavingR1 connected across the battery. However, the current increases by just0.10 A when R1 is removed, leaving R2 connected across the battery.Find (a) R1 and (b) R2.

Section 20.7 Parallel Wiring

50. What resistance must be placed in parallel with a 155-# resistorto make the equivalent resistance 115 #?

51. ssm A 16-# loudspeaker and an 8.0-# loudspeaker are con-nected in parallel across the terminals of an amplifier. Assuming thatthe speakers behave as resistors, determine the equivalent resistanceof the two speakers.

52. The drawing shows three different resistors in two different

636 CHAPTER 20 ELECTRIC CIRCUITS

circuits. The battery has a voltage of V ! 24.0 V, and the resistors havevalues of R1 ! 50.0 #, R2 ! 25.0 #, and R3 ! 10.0 #.(a) For the circuit on the left, determine the current through and the volt-age across each resistor. (b) Repeat part (a) for the circuit on the right.

53. The drawing shows a circuit that contains a battery, two resis-tors, and a switch. What is the equivalent resistance of the circuitwhen the switch is (a) open and (b) closed? What is the totalpower delivered to the resistors when the switch is (c) open and(d) closed?

54. For the three-way bulb (50 W, 100 W, 150 W) discussed inConceptual Example 11, find the resistance of each of the two fila-ments. Assume that the wattage ratings are not limited by significantfigures, and ignore any heating effects on the resistances.

55. ssm Two resistors, 42.0 and 64.0 #, are connected in parallel.The current through the 64.0-# resistor is 3.00 A. (a) Determinethe current in the other resistor. (b) What is the total power suppliedto the two resistors?

56. Two identical resistors are connected in parallel across a 25-Vbattery, which supplies them with a total power of 9.6 W. While thebattery is still connected, one of the resistors is heated so that its re-sistance doubles. The resistance of the other resistor remains un-changed. Find (a) the initial resistance of each resistor and (b) thetotal power delivered to the resistors after one resistor has beenheated.

57. Help with problems of this kind is available in InteractiveSolution 20.57 at www.wiley.com/college/cutnell. A coffee cupheater and a lamp are connected in parallel to the same 120-V outlet.Together, they use a total of 111 W of power. The resistance of theheater is 4.0 " 102 #. Find the resistance of the lamp.

9.00 V

Switch

R1 = 65.0 ! R2 = 96.0 !

R1

R3

R2+–

+– VV R1 R2 R3

(a) (b)

2762T_ch20_599-641.qxd 9/12/08 3:02 PM Page 636

PROBLEMS 637

*58. The drawing shows two circuits, and the same battery is

used in each. The two resistances RA in circuit A are the same, and

the two resistances RB in circuit B are the same. Knowing that the

same total power is delivered in each circuit, find the ratio RB/RA

for the circuits in the drawing.

64. Find the equivalent resistance between points A and B in the

drawing.

65. ssm Determine the equivalent resistance between the points A

and B for the group of resistors in the drawing.

66. Find the equivalent resistance between the points A and B in the

drawing.

67. A 60.0-! resistor is connected in parallel with a 120.0-! re-

sistor. This parallel group is connected in series with a 20.0-!resistor. The total combination is connected across a 15.0-V bat-

tery. Find (a) the current and (b) the power delivered to the

120.0-! resistor.

*68. Each resistor in the three circuits in the drawing has the same

resistance R, and the batteries have the same voltage V. The values for

R and V are 9.0 ! and 6.0 V, respectively. Determine the total power

delivered by the battery in each of the three circuits.

R R

R R

V

+ –

Circuit A

R

R R

V

+ –

Circuit B

R R

R

R

V

+ –

Circuit C

R

R R

R1 = 16 ! R2 = 8 !

R3 = 48 !

A BR4 = 26 !

A

B

20.0 ! 9.0 ! 8.0 !

4.0 !

6.0 !

3.0 !

A

B

4.00 ! 3.00 ! 2.00 !

1.00 !

3.00 !

6.00 !2.00 !

RA RA

V

+ –

Circuit A

RB

RB

V

+ –

Circuit B

*59. ssm The total current delivered to a number of devices connected

in parallel is the sum of the individual currents in each device. Circuit

breakers are resettable automatic

switches that protect against a dan-

gerously large total current by

“opening” to stop the current at a

specified safe value. A 1650-W

toaster, a 1090-W iron, and a

1250-W microwave oven are

turned on in a kitchen. As the draw-

ing shows, they are all connected

through a 20-A circuit breaker

(which has negligible resistance) to

an ac voltage of 120 V. (a) Find

the equivalent resistance of the

three devices. (b) Obtain the total

current delivered by the source and

determine whether the breaker will “open” to prevent an accident.

*60. A cylindrical aluminum pipe of length 1.50 m has an inner radius

of 2.00 " m and an outer radius of 3.00 " m. The interior

of the pipe is completely filled with copper. What is the resistance of

this unit? (Hint: Imagine that the pipe is connected between the ter-

minals of a battery and decide whether the aluminum and copper

parts of the pipe are in series or in parallel.)

**61. The rear window defogger of a car consists of thirteen thin wires

(resistivity # 88.0 " !$m) embedded in the glass. The wires

are connected in parallel to the 12.0-V battery, and each has a length

of 1.30 m. The defogger can melt 2.10 " kg of ice at 0 %C into

water at 0 %C in two minutes. Assume that all the power delivered to

the wires is used immediately to melt the ice. Find the cross-sectional

area of each wire.

Section 20.8 Circuits Wired Partially in Series and Partially in Parallel

62. The circuit in the drawing contains three identical resistors.

10&2

10&8

10&3

10&3

120 V

20 A

Circuitbreaker

Each resistor has a value of 10.0 !. Determine the equivalent resis-

tance between the points a and b, b and c, and a and c.

63. ssm A 14-! coffee maker and a 16-! frying pan are connected

in series across a 120-V source of voltage. A 23-! bread maker is

also connected across the 120-V source and is in parallel with the series

combination. Find the total current supplied by the source of voltage.

Rc

RR

a b

2762T_ch20_599-641.qxd 9/12/08 3:02 PM Page 637

*69. Eight different values of resistance can be obtained by connect-

ing together three resistors (1.00, 2.00, and 3.00 !) in all possible

ways. What are the values?

*70. Concept Simulation 20.5 at www.wiley.com/college/cutnellprovides some background pertinent to this problem. Determine the

power supplied to each of the resistors in the drawing.

*71. ssm www The current in the 8.00-! resistor in the drawing is

0.500 A. Find the current in (a) the 20.0-! resistor and in (b) the

9.00-! resistor.

**72. The circuit shown in the draw-

ing is constructed with six identical

resistors and an ideal battery. When

the resistor R4 is removed from the

circuit, the current in the battery de-

creases by 1.9 A. Determine the re-

sistance of each resistor.

Section 20.9 Internal Resistance

73. ssm A battery has an internal resistance of 0.50 !. A number of

identical light bulbs, each with a resistance of 15 !, are connected in

parallel across the battery terminals. The terminal voltage of the bat-

tery is observed to be one-half the emf of the battery. How many

bulbs are connected?

74. A new “D” battery has an emf of 1.5 V. When a wire of negli-

gible resistance is connected between the terminals of the battery,

a current of 28 A is produced. Find the internal resistance of the

battery.

75. A battery has an internal resistance of 0.012 ! and an emf of

9.00 V. What is the maximum current that can be drawn from the bat-

tery without the terminal voltage dropping below 8.90 V?

76. When a light bulb is connected across the terminals of a battery,

the battery delivers 24 W of power to the bulb. A voltage of 11.8 V

20.0 !

8.00 !

16.0 !

9.00 !

18.0 !

+ –

120 V

R2 = 576 !

R3 = 576 !

R1 = 576 !

+ –

A

exists between the terminals of the battery, which has an internal re-

sistance of 0.10 !. What is the emf of the battery?

*77. A battery delivering a current of 55.0 A to a circuit has a

terminal voltage of 23.4 V. The electric power being dissipated by

the internal resistance of the battery is 34.0 W. Find the emf of the

battery.

*78. When a “dry-cell” flashlight battery with an internal resistance of

0.33 ! is connected to a 1.50-! light bulb, the bulb shines dimly.

However, when a lead-acid “wet-cell” battery with an internal resis-

tance of 0.050 ! is connected, the bulb is noticeably brighter. Both

batteries have the same emf. Find the ratio Pwet/Pdry of the power

delivered to the bulb by the wet-cell battery to the power delivered

by the dry-cell battery.

Section 20.10 Kirchhoff’s Rules

79. ssm Consider the circuit in the

drawing. Determine (a) the magni-

tude of the current in the circuit and

(b) the magnitude of the voltage be-

tween the points labeled A and B.

(c) State which point, A or B, is at the

higher potential.

80. Using Kirchhoff’s loop rule, find the value of the current

638 CHAPTER 20 ELECTRIC CIRCUITS

5.0 !

30.0 V

A 27 !

12 !

8.0 !

10.0 V

B

I in part c of the drawing, where R " 5.0 !. (Note: Parts a and b of

the drawing are used in the online tutorial help that is provided for

this problem in the WileyPLUS homework management program.)

81. For the circuit shown in the drawing, find the current I through the

2.00-! resistor and the voltage V of the battery to the left of this resistor.

82. The drawing shows a

portion of a larger circuit.

Current flows left to right in

each resistor. What is the cur-

rent in the resistor R?

!4.00

3.00 A

!8.00

!2.00

!6.00 24.0 V

V

+ –

+ –

I

R

RA A BB

RV V+–

I

I I

+ –

+ –3.0 ! 4.0 !36 V 12 V

– +

(a) (b)

(c)

I1 = 3.00 A

R1 = 2.70 !

I2

R2 = 4.40 !

R

R1 R2

R4

R5 R6

R3+– 35 V

2762T_ch20_599-641.qxd 9/12/08 3:02 PM Page 638

PROBLEMS 639

83. Determine the current (magnitude and direction) in the 8.0- and

2.0-! resistors in the drawing.

*84. Concept Simulation 20.3 at www.wiley.com/college/cutnell allows you to verify your answer for this problem. Find the current in

the 4.00-! resistor in the drawing. Specify the direction of the current.

*85. ssm Determine the voltage across the 5.0-! resistor in the draw-

ing. Which end of the resistor is at the higher potential?

**86. None of the resistors in the circuit shown in the drawing is con-

nected in series or in parallel with one another. Find (a) the current

I5 and the resistances (b) R2 and (c) R3.

Section 20.11 The Measurement of Current and Voltage

87. ssm A galvanometer has a full-scale current of 0.100 mA and a

coil resistance of 50.0 !. This instrument is used with a shunt resis-

tor to form a nondigital ammeter that will register full scale for a cur-

rent of 60.0 mA. Determine the resistance of the shunt resistor.

88. The coil of wire in a galvanometer has a resistance of RC " 60.0 !.

The galvanometer exhibits a full-scale deflection when the current

through it is 0.400 mA. A resistor is connected in series with this com-

bination so as to produce a nondigital voltmeter. The voltmeter is to

have a full-scale deflection when it measures a potential difference of

10.0 V. What is the resistance of this resistor?

6.0 A

R2

9.0 A75.0 V

12.0 A

R1 = 4.0 !

I5

+–

R4 = 2.0 ! R5 = 2.2 !

R3

5.0 !

10.0 !

2.0 V

10.0 !

+– 10.0 V 15.0 V

+–

+–

+

–6.00 V

9.00 V3.00 V+–

+–

2.00 ! 8.00 !

4.00 !

R1 = 8.0 !

V2 = 12 V

+–

+ –

R2 = 2.0 !

V1 = 4.0 V

89. Nondigital voltmeter A has an equivalent resistance of 2.40 # 105 !

and a full-scale voltage of 50.0 V. Nondigital voltmeter B, using the

same galvanometer as voltmeter A, has an equivalent resistance of

1.44 # 105 !. What is its full-scale voltage?

90. A galvanometer with a coil resistance of 12.0 ! and a full-scale

current of 0.150 mA is used with a shunt resistor to make a nondigital

ammeter. The ammeter registers a maximum current of 4.00 mA.

Find the equivalent resistance of the ammeter.

*91. ssm Two scales on a nondigital voltmeter measure voltages up

to 20.0 and 30.0 V, respectively. The resistance connected in series

with the galvanometer is 1680 ! for the 20.0-V scale and 2930 ! for

the 30.0-V scale. Determine the coil resistance and the full-scale cur-

rent of the galvanometer that is used in the voltmeter.

**92. In measuring a voltage, a voltmeter uses some current from the

circuit. Consequently, the voltage measured is only an approxima-

tion to the voltage present when the voltmeter is not connected.

Consider a circuit consisting of two 1550-! resistors connected in

series across a 60.0-V battery. (a) Find the voltage across one of

the resistors. (b) A nondigital voltmeter has a full-scale voltage of

60.0 V and uses a galvanometer with a full-scale deflection of 5.00 mA.

Determine the voltage that this voltmeter registers when it is con-

nected across the resistor used in part (a).

Section 20.12 Capacitors in Series and in Parallel

93. Two capacitors are connected in parallel across the terminals of a

battery. One has a capacitance of 2.0 $F and the other a capacitance

of 4.0 $F. These two capacitors together store 5.4 # 10%5

C of charge.

What is the voltage of the battery?

94. Three capacitors are connected in series. The equivalent capaci-

tance of this combination is 3.00 $F. Two of the individual capaci-

tances are 6.00 $F and 9.00 $F. What is the third capacitance (in $F)?

95. ssm Three capacitors have identical geometries. One is filled

with a material whose dielectric constant is 2.50. Another is filled

with a material whose dielectric constant is 4.00. The third capacitor

is filled with a material whose dielectric constant & is such that this

single capacitor has the same capacitance as the series combination

of the other two. Determine &.

96. Two capacitors are connected to a battery. The battery volt-

age is V " 60.0 V, and the capacitances are C1 " 2.00 $F and

C2 " 4.00 $F. Determine the total energy stored by the two capaci-

tors when they are wired (a) in parallel and (b) in series.

97. Determine the equivalent capacitance between A and B for the

group of capacitors in the drawing.

98. You have three capacitors: C1 " 67 $F, C2 " 45 $F, and

C3 " 33 $F. Determine the maximum equivalent capacitance you can

obtain by connecting two of the capacitors in parallel and then connect-

ing the parallel combination in series with the remaining capacitor.

99. A 2.00-$F and a 4.00-$F capacitor are connected to a 60.0-V

battery. What is the total charge supplied to the capacitors when they

are wired (a) in parallel and (b) in series with each other?

*100. A 3.00-$F and a 5.00-$F capacitor are connected in series

across a 30.0-V battery. A 7.00-$F capacitor is then connected in par-

allel across the 3.00-$F capacitor. Determine the voltage across the

7.00-$F capacitor.

A

B

5.0 Fµ 24 Fµ

4.0 Fµ 12 Fµ

6.0 Fµ 8.0 Fµ

2762T_ch20_599-641.qxd 10/30/08 5:38 PM Page 639

2nd

*101. ssm A sheet of gold foil (negligible thickness) is placed be-tween the plates of a capacitor and has the same area as each of theplates. The foil is parallel to the plates, at a position one-third of the way from one to the other. Before the foil is inserted, the capaci-tance is C0. What is the capacitance after the foil is in place? Expressyour answer in terms of C0.

**102. The drawing shows twocapacitors that are fully charged (C1 ! 2.00 "F, q1 ! 6.00 "C; C2 ! 8.00 "F, q2 ! 12.0 "C).The switch is closed, and charge flows until equilibrium is reestablished (i.e., until both capacitors have the same voltageacross their plates). Find the resulting voltage across either capacitor.

Section 20.13 RC Circuits

103. ssm In a heart pacemaker, a pulse is delivered to theheart 81 times per minute. The capacitor that controls this

pulsing rate discharges through a resistance of 1.8 # 106 $. Onepulse is delivered every timethe fully charged capacitorloses 63.2% of its originalcharge. What is the capaci-tance of the capacitor?

104. The circuit in the draw-ing contains two resistors andtwo capacitors that are con-nected to a battery via aswitch. When the switch is

closed, the capacitors begin to charge up. What is the time constantfor the charging process?

105. An electronic flash attachment for a camera produces a flash byusing the energy stored in a 150-"F capacitor. Between flashes, thecapacitor recharges through a resistor whose resistance is chosen sothe capacitor recharges with a time constant of 3.0 s. Determine thevalue of the resistance.

*106. Concept Simulation 20.4 at www.wiley.com/college/cutnellprovides background for this problem and gives you the opportunityto verify your answer graphically. How many time constants mustelapse before a capacitor in a series RC circuit is charged to 80.0% ofits equilibrium charge?

*107. For one approach to problems like this one, consult InteractiveSolution 20.107 at www.wiley.com/college/cutnell. Four identicalcapacitors are connected with a resistor in two different ways. Whenthey are connected as in part a of the drawing, the time constant tocharge up this circuit is 0.72 s. What is the time constant when theyare connected with the same resistor, as in part b?

640 CHAPTER 20 ELECTRIC CIRCUITS

C2++ +

Switch

+–– ––

+ +– – C1

R

C C

C C

+ –

R

C

C

C

+ –

C

(b)(a)

+ –

!4.0 k

!2.0 k

Switch

3.0 Fµ

6.0 Fµ

Problem 104

ADDITIONAL PROBLEMS

108. A portable CD player operates with a voltage of 4.5 V, and itspower usage is 0.11 W. Find the current in the player.

109. ssm Suppose that two capacitors (C1 and C2) are connected inseries. Show that the sum of the energies stored in these capacitors isequal to the energy stored in the equivalent capacitor. [Hint: The en-

ergy stored in a capacitor can be expressed as q2/(2C).]

110. A battery charger is connected to a dead battery and delivers a cur-rent of 6.0 A for 5.0 hours, keeping the voltage across the battery termi-nals at 12 V in the process. How much energy is delivered to the battery?

111. ssm In the Arctic, electric socks are useful. A pair of socksuses a 9.0-V battery pack for each sock. A current of 0.11 A is drawnfrom each battery pack by wire woven into the socks. Find the resis-tance of the wire in one sock.

112. Interactive LearningWare 20.1 at www.wiley.com/college/cutnell provides background for this problem. Find the magnitudeand direction of the current in the 2.0-$ resistor in the drawing.

1.0 !

3.0 ! 4.0 V

2.0 ! 1.0 V+ –

– +

113. A coil of wire has a resistance of 38.0 $ at 25 %C and 43.7 $ at55 %C. What is the temperature coefficient of resistivity?

114. The circuit in thedrawing shows two re-sistors, a capacitor, anda battery. When the ca-pacitor is fully charged,what is the magnitude qof the charge on one of its plates?

115. The coil of a galvanometer has a resistance of 20.0 $, and itsmeter deflects full scale when a current of 6.20 mA passes throughit. To make the galvanometer into a nondigital ammeter, a 24.8-m$shunt resistor is added to it. What is the maximum current that thisammeter can read?

116. Multiple-Concept Example 9 discusses the physics principlesused in this problem. Three resistors, 2.0, 4.0, and 6.0 $, are con-nected in series across a 24-V battery. Find the power delivered toeach resistor.

*117. ssm The circuit inthe drawing contains fiveidentical resistors. The 45-V battery delivers 58 W of power to the circuit. What is the resis-tance R of each resistor?

+–V = 12 V

!R2 = 2.0

!R1 = 4.0 C = 9.0 Fµ

+–

45 V

R1

R2

R3

R4

R5

2762T_ch20_599-641.qxd 9/28/08 11:17 AM Page 640

2nd

ADDITIONAL PROBLEMS 641

*118. A resistor (resistance ! R) is connected first in parallel and thenin series with a 2.00-" resistor. A battery delivers five times as muchcurrent to the parallel combination as it does to the series combina-tion. Determine the two possible values for R.

*119. ssm Interactive Solution 20.119 at www.wiley.com/college/cutnell provides one approach to problems like this one. Three resis-tors are connected in series across a battery. The value of each resis-tance and its maximum power rating are as follows: 2.0 " and 4.0 W,12.0 " and 10.0 W, and 3.0 " and 5.0 W. (a) What is the greatestvoltage that the battery can have without one of the resistors burningup? (b) How much power does the battery deliver to the circuit in (a)?

*120. A 75.0-" and a 45.0-" resistor are connected in parallel. Whenthis combination is connected across a battery, the current deliveredby the battery is 0.294 A. When the 45.0-" resistor is disconnected,the current from the battery drops to 0.116 A. Determine (a) theemf and (b) the internal resistance of the battery.

*121. ssm www Two wires have the same cross-sectional area andare joined end to end to form a single wire. One is tungsten, whichhas a temperature coefficient of resistivity of # ! 0.0045 (C°) . The$1

other is carbon, for which # ! $0.0005 (C°) . The total resistanceof the composite wire is the sum of the resistances of the pieces. Thetotal resistance of the composite does not change with temperature.What is the ratio of the lengths of the tungsten and carbon sections?Ignore any changes in length due to thermal expansion.

*122. Two cylindrical rods, one copper and the other iron, are identi-cal in lengths and cross-sectional areas. They are joined end to end toform one long rod. A 12-V battery is connected across the free endsof the copper–iron rod. What is the voltage between the ends of thecopper rod?

*123. ssm A 7.0-%F and a 3.0-%F capacitor are connected in seriesacross a 24-V battery. What voltage is required to charge a parallelcombination of the two capacitors to the same total energy?

**124. A digital thermometer employs a thermistor as the temper-ature-sensing element. A thermistor is a kind of semi-

conductor and has a large negative temperature coefficient ofresistivity #. Suppose that # ! $0.060 (C°) for the thermistor in adigital thermometer used to measure the temperature of a patient. Theresistance of the thermistor decreases to 85% of its value at the normalbody temperature of 37.0 &C. What is the patient’s temperature?

$1

$1

2762T_ch20_599-641.qxd 9/12/08 3:02 PM Page 641