Chapter 6 Review
Key Terms
ammeter
instrument that measures current
electromotive force (emf)
energy produced per unit charge, drawn from a source that produces an electrical current
equivalent resistance
resistance of a combination of resistors; it can be thought of as the resistance of a single resistor that can replace a combination of resistors in a series and/or parallel circuit
internal resistance
amount of resistance to the flow of current within the voltage source
junction rule
sum of all currents entering a junction must equal the sum of all currents leaving the junction
Kirchhoff’s rules
set of two rules governing current and changes in potential in an electric circuit
loop rule
algebraic sum of changes in potential around any closed circuit path (loop) must be zero
potential difference
difference in electric potential between two points in an electric circuit, measured in volts
potential drop
loss of electric potential energy as a current travels across a resistor, wire, or other component
RC circuit
circuit that contains both a resistor and a capacitor
shock hazard
hazard in which an electric current passes through a person
terminal voltage
potential difference measured across the terminals of a source when there is no load attached
thermal hazard
hazard in which an excessive electric current causes undesired thermal effects
threewire system
wiring system used at present for safety reasons, with live, neutral, and ground wires
voltmeter
instrument that measures voltage
Key Equations
Terminal voltage of a single voltage source  
Equivalent resistance of a series circuit  
Equivalent resistance of a parallel circuit  
Junction rule  
Loop rule  
Terminal voltage of N voltage sources in series 

Terminal voltage of N voltage sources in parallel 

Charge on a charging capacitor 

Time constant 

Current during charging of a capacitor 

Charge on a discharging capacitor 

Current during discharging of a capacitor 

Summary
6.1 Electromotive Force
 All voltage sources have two fundamental parts: a source of electrical energy that has a characteristic electromotive force (emf), and an internal resistance . The emf is the work done per charge to keep the potential difference of a source constant. The emf is equal to the potential difference across the terminals when no current is flowing. The internal resistance of a voltage source affects the output voltage when a current flows.
 The voltage output of a device is called its terminal voltage and is given by where is the electric current and is positive when flowing away from the positive terminal of the voltage source and is the internal resistance.
6.2 Resistors in Series and Parallel
 The equivalent resistance of an electrical circuit with resistors wired in a series is the sum of the individual resistances: .
 Each resistor in a series circuit has the same amount of current flowing through it.
 The potential drop, or power dissipation, across each individual resistor in a series is different, and their combined total is the power source input.
 The equivalent resistance of an electrical circuit with resistors wired in parallel is less than the lowest resistance of any of the components and can be determined using the formula .
 Each resistor in a parallel circuit has the same full voltage of the source applied to it.
 The current flowing through each resistor in a parallel circuit is different, depending on the resistance.
 If a more complex connection of resistors is a combination of series and parallel, it can be reduced to a single equivalent resistance by identifying its various parts as series or parallel, reducing each to its equivalent, and continuing until a single resistance is eventually reached.
6.3 Kirchhoff’s Rules
 Kirchhoff’s rules can be used to analyze any circuit, simple or complex. The simpler series and parallel connection rules are special cases of Kirchhoff’s rules.
 Kirchhoff’s first rule, also known as the junction rule, applies to the charge to a junction. Current is the flow of charge; thus, whatever charge flows into the junction must flow out.
 Kirchhoff’s second rule, also known as the loop rule, states that the voltage drop around a loop is zero.
 When calculating potential and current using Kirchhoff’s rules, a set of conventions must be followed for determining the correct signs of various terms.
 When multiple voltage sources are in series, their internal resistances add together and their emfs add together to get the total values.
 When multiple voltage sources are in parallel, their internal resistances combine to an equivalent resistance that is less than the individual resistance and provides a higher current than a single cell.
 Solar cells can be wired in series or parallel to provide increased voltage or current, respectively.
6.4 Electrical Measuring Instruments
 Voltmeters measure voltage, and ammeters measure current. Analog meters are based on the combination of a resistor and a galvanometer, a device that gives an analog reading of current or voltage. Digital meters are based on analogtodigital converters and provide a discrete or digital measurement of the current or voltage.
 A voltmeter is placed in parallel with the voltage source to receive full voltage and must have a large resistance to limit its effect on the circuit.
 An ammeter is placed in series to get the full current flowing through a branch and must have a small resistance to limit its effect on the circuit.
 Standard voltmeters and ammeters alter the circuit they are connected to and are thus limited in accuracy.
 Ohmmeters are used to measure resistance. The component in which the resistance is to be measured should be isolated (removed) from the circuit.
6.5 RC Circuits
 An circuit is one that has both a resistor and a capacitor.
 The time constant for an circuit is
 When an initially uncharged ( at ) capacitor in series with a resistor is charged by a dc voltage source, the capacitor asymptotically approaches the maximum charge.
 As the charge on the capacitor increases, the current exponentially decreases from the initial current:
 If a capacitor with an initial charge is discharged through a resistor starting at then its charge decreases exponentially. The current flows in the opposite direction, compared to when it charges, and the magnitude of the charge decreases with time.
6.6 Household Wiring and Electrical Safety
 The two types of electric hazards are thermal (excessive power) and shock (current through a person). Electrical safety systems and devices are employed to prevent thermal and shock hazards.
 Shock severity is determined by current, path, duration, and ac frequency.
 Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards.
 The threewire system guards against thermal and shock hazards, utilizing live/hot, neutral, and ground wires, and grounding the neutral wire and case of the appliance.
 A ground fault circuit interrupter (GFCI) prevents shock by detecting the loss of current to unintentional paths.
Answers to Check Your Understanding
6.1 If a wire is connected across the terminals, the load resistance is close to zero, or at least considerably less than the internal resistance of the battery. Since the internal resistance is small, the current through the circuit will be large,
The large current causes a high power to be dissipated by the internal resistance (
). The power is dissipated as heat.
6.2 The equivalent resistance of nine bulbs connected in series is
The current is
If one bulb burns out, the equivalent resistance is
and the voltage does not change, but the current increases (
). As more bulbs burn out, the current becomes even higher. Eventually, the current becomes too high, burning out the shunt.
6.3 The equivalent of the series circuit would be
which is higher than the equivalent resistance of the parallel circuit
The equivalent resistor of any number of resistors is always higher than the equivalent resistance of the same resistors connected in parallel. The current through for the series circuit would be
which is lower than the sum of the currents through each resistor in the parallel circuit,
This is not surprising since the equivalent resistance of the series circuit is higher. The current through a series connection of any number of resistors will always be lower than the current into a parallel connection of the same resistors, since the equivalent resistance of the series circuit will be higher than the parallel circuit. The power dissipated by the resistors in series would be
which is lower than the power dissipated in the parallel circuit
6.4 A river, flowing horizontally at a constant rate, splits in two and flows over two waterfalls. The water molecules are analogous to the electrons in the parallel circuits. The number of water molecules that flow in the river and falls must be equal to the number of molecules that flow over each waterfall, just like sum of the current through each resistor must be equal to the current flowing into the parallel circuit. The water molecules in the river have energy due to their motion and height. The potential energy of the water molecules in the river is constant due to their equal heights. This is analogous to the constant change in voltage across a parallel circuit. Voltage is the potential energy across each resistor.
The analogy quickly breaks down when considering the energy. In the waterfall, the potential energy is converted into kinetic energy of the water molecules. In the case of electrons flowing through a resistor, the potential drop is converted into heat and light, not into the kinetic energy of the electrons.
6.5 1. All the overhead lighting circuits are in parallel and connected to the main supply line, so when one bulb burns out, all the overhead lighting does not go dark. Each overhead light will have at least one switch in series with the light, so you can turn it on and off. 2. A refrigerator has a compressor and a light that goes on when the door opens. There is usually only one cord for the refrigerator to plug into the wall. The circuit containing the compressor and the circuit containing the lighting circuit are in parallel, but there is a switch in series with the light. A thermostat controls a switch that is in series with the compressor to control the temperature of the refrigerator.
6.6 The circuit can be analyzed using Kirchhoff’s loop rule. The first voltage source supplies power:
The second voltage source consumes power:
6.7 The current calculated would be equal to
instead of
The sum of the power dissipated and the power consumed would still equal the power supplied.
6.8 Since digital meters require less current than analog meters, they alter the circuit less than analog meters. Their resistance as a voltmeter can be far greater than an analog meter, and their resistance as an ammeter can be far less than an analog meter. Consult Figure 6.4.3 and Figure 6.4.2 and their discussion in the text.
Conceptual Questions
6.1 Electromotive Force
1. What effect will the internal resistance of a rechargeable battery have on the energy being used to recharge the battery?
2. A battery with an internal resistance of
and an emf of
is connected to a load resistor
As the battery ages, the internal resistance triples. How much is the current through the load resistor reduced?
3. Show that the power dissipated by the load resistor is maximum when the resistance of the load resistor is equal to the internal resistance of the battery.
6.2 Resistors in Series and Parallel
4. A voltage occurs across an open switch. What is the power dissipated by the open switch?
5. The severity of a shock depends on the magnitude of the current through your body. Would you prefer to be in series or in parallel with a resistance, such as the heating element of a toaster, if you were shocked by it? Explain.
6. Suppose you are doing a physics lab that asks you to put a resistor into a circuit, but all the resistors supplied have a larger resistance than the requested value. How would you connect the available resistances to attempt to get the smaller value asked for?
7. Some light bulbs have three power settings (not including zero), obtained from multiple filaments that are individually switched and wired in parallel. What is the minimum number of filaments needed for three power settings?
6.3 Kirchhoff’s Rules
8. Can all of the currents going into the junction shown below be positive? Explain.
9. Consider the circuit shown below. Does the analysis of the circuit require Kirchhoff’s method, or can it be redrawn to simplify the circuit? If it is a circuit of series and parallel connections, what is the equivalent resistance?
10. Do batteries in a circuit always supply power to a circuit, or can they absorb power in a circuit? Give an example.
11. What are the advantages and disadvantages of connecting batteries in series? In parallel?
12. Semitractor trucks use four large
batteries. The starter system requires
while normal operation of the truck’s other electrical components utilizes
How could the four batteries be connected to produce
? To produce
? Why is
6.5 RC Circuits
16. A battery, switch, capacitor, and lamp are connected in series. Describe what happens to the lamp when the switch is closed.
17. When making an ECG measurement, it is important to measure voltage variations over small time intervals. The time is limited by the
constant of the circuit—it is not possible to measure time variations shorter than
How would you manipulate
and
in the circuit to allow the necessary measurements?
6.6 Household Wiring and Electrical Safety
18. Why isn’t a short circuit necessarily a shock hazard?
19. We are often advised to not flick electric switches with wet hands, dry your hand first. We are also advised to never throw water on an electric fire. Why?
Problems
6.1 Electromotive Force
20. A car battery with a
emf and an internal resistance of
is being charged with a current of
Note that in this process, the battery is being charged. (a) What is the potential difference across its terminals? (b) At what rate is thermal energy being dissipated in the battery? (c) At what rate is electric energy being converted into chemical energy?
21. The label on a batterypowered radio recommends the use of rechargeable nickelcadmium cells (nicads), although they have a
emf, whereas alkaline cells have a
emf. The radio has a
resistance. (a) Draw a circuit diagram of the radio and its batteries. Now, calculate the power delivered to the radio (b) when using nicad cells, each having an internal resistance of
and (c) when using alkaline cells, each having an internal resistance of
(d) Does this difference seem significant, considering that the radio’s effective resistance is lowered when its volume is turned up?
22. An automobile starter motor has an equivalent resistance of
and is supplied by a
battery with a
internal resistance. (a) What is the current to the motor? (b) What voltage is applied to it? (c) What power is supplied to the motor? (d) Repeat these calculations for when the battery connections are corroded and add
to the circuit. (Significant problems are caused by even small amounts of unwanted resistance in lowvoltage, highcurrent applications.)
23. (a) What is the internal resistance of a voltage source if its terminal potential drops by
when the current supplied increases by
? (b) Can the emf of the voltage source be found with the information supplied?
24. A person with body resistance between his hands of
accidentally grasps the terminals of a
power supply. (Do NOT do this!) (a) Draw a circuit diagram to represent the situation. (b) If the internal resistance of the power supply is
what is the current through his body? (c) What is the power dissipated in his body? (d) If the power supply is to be made safe by increasing its internal resistance, what should the internal resistance be for the maximum current in this situation to be
or less? (e) Will this modification compromise the effectiveness of the power supply for driving lowresistance devices? Explain your reasoning.
25. A
emf automobile battery has a terminal voltage of
when being charged by a current of
(a) What is the battery’s internal resistance? (b) What power is dissipated inside the battery? (c) At what rate (in
) will its temperature increase if its mass is
and it has a specific heat of
assuming no heat escapes?
6.2 Resistors in Series and Parallel
26. (a) What is the resistance of a
a
and a
resistor connected in series? (b) In parallel?
27. What are the largest and smallest resistances you can obtain by connecting a
a
and a
resistor together?
28. An
toaster, a
speaker, and a
lamp are plugged into the same outlet in a
fuse and
circuit. (The three devices are in parallel when plugged into the same socket.) (a) What current is drawn by each device? (b) Will this combination blow the
fuse?
29. Your car’s
headlight and
starter are ordinarily connected in parallel in a
system. What power would one headlight and the starter consume if connected in series to a
battery? (Neglect any other resistance in the circuit and any change in resistance in the two devices.)
30. (a) Given a
battery and
and
resistors, find the current and power for each when connected in series. (b) Repeat when the resistances are in parallel.
31. Referring to the example combining series and parallel circuits and Figure 6.2.6, calculate
in the following two different ways: (a) from the known values of
and
; (b) using Ohm’s law for
In both parts, explicitly show how you follow the steps in the Figure 6.2.7.
32. Referring to Figure 6.2.6, (a) Calculate
and note how it compares with
found in the first two example problems in this module. (b) Find the total power supplied by the source and compare it with the sum of the powers dissipated by the resistors.
33. Refer to Figure 6.2.7 and the discussion of lights dimming when a heavy appliance comes on. (a) Given the voltage source is
the wire resistance is
and the bulb is nominally
what power will the bulb dissipate if a total of
passes through the wires when the motor comes on? Assume negligible change in bulb resistance. (b) What power is consumed by the motor?
34. Show that if two resistors
and
are combined and one is much greater than the other (
), (a) their series resistance is very nearly equal to the greater resistance
and (b) their parallel resistance is very nearly equal to smaller resistance
35. Consider the circuit shown below. The terminal voltage of the battery is
(a) Find the equivalent resistance of the circuit. (b) Find the current through each resistor. (c) Find the potential drop across each resistor. (d) Find the power dissipated by each resistor. (e) Find the power supplied by the battery.
6.3 Kirchhoff’s Rules
36. Consider the circuit shown below. (a) Find the voltage across each resistor. (b)What is the power supplied to the circuit and the power dissipated or consumed by the circuit?
37. Consider the circuits shown below. (a) What is the current through each resistor in part (a)? (b) What is the current through each resistor in part (b)? (c) What is the power dissipated or consumed by each circuit? (d) What is the power supplied to each circuit?
38. Consider the circuit shown below. Find
and
39. Consider the circuit shown below. Find
and
40. Consider the circuit shown below. Find
and
41. Consider the circuit shown below. (a) Find
and
(b) Find the power supplied by the voltage sources. (c) Find the power dissipated by the resistors.
42. Consider the circuit shown below. Write the three loop equations for the loops shown.
43. Consider the circuit shown below. Write equations for the three currents in terms of
and
44. Consider the circuit shown in the preceding problem. Write equations for the power supplied by the voltage sources and the power dissipated by the resistors in terms of
and
45. A child’s electronic toy is supplied by three
alkaline cells having internal resistances of
in series with a
carbonzinc dry cell having a
internal resistance. The load resistance is
(a) Draw a circuit diagram of the toy and its batteries. (b) What current flows? (c) How much power is supplied to the load? (d) What is the internal resistance of the dry cell if it goes bad, resulting in only
being supplied to the load?
46. Apply the junction rule to Junction
shown below. Is any new information gained by applying the junction rule at
?
47. Apply the loop rule to Loop
in the preceding problem.
6.4 Electrical Measuring Instruments
48. Suppose you measure the terminal voltage of a
alkaline cell having an internal resistance of
by placing a
voltmeter across its terminals (see below). (a) What current flows? (b) Find the terminal voltage. (c) To see how close the measured terminal voltage is to the emf, calculate their ratio.
6.5 RC Circuits
49. The timing device in an automobile’s intermittent wiper system is based on an
time constant and utilizes a
capacitor and a variable resistor. Over what range must
be made to vary to achieve time constants from
to
50. A heart pacemaker fires
times a minute, each time a
capacitor is charged (by a battery in series with a resistor) to
of its full voltage. What is the value of the resistance?
51. The duration of a photographic flash is related to an
time constant, which is
for a certain camera. (a) If the resistance of the flash lamp is
during discharge, what is the size of the capacitor supplying its energy? (b) What is the time constant for charging the capacitor, if the charging resistance is
52. A
– and a
capacitor can be connected in series or parallel, as can a
– and a
resistor. Calculate the four
time constants possible from connecting the resulting capacitance and resistance in series.
53. A
resistor, an uncharged
capacitor, and a
emf are connected in series. (a) What is the initial current? (b) What is the
time constant? (c) What is the current after one time constant? (d) What is the voltage on the capacitor after one time constant?
54. A heart defibrillator being used on a patient has an
time constant of
due to the resistance of the patient and the capacitance of the defibrillator. (a) If the defibrillator has a capacitance of
what is the resistance of the path through the patient? (You may neglect the capacitance of the patient and the resistance of the defibrillator.) (b) If the initial voltage is
how long does it take to decline to
55. An ECG monitor must have an
time constant less than
to be able to measure variations in voltage over small time intervals. (a) If the resistance of the circuit (due mostly to that of the patient’s chest) is
what is the maximum capacitance of the circuit? (b) Would it be difficult in practice to limit the capacitance to less than the value found in (a)?
56. Using the exact exponential treatment, determine how much time is required to charge an initially uncharged
capacitor through a
resistor to
of its final voltage.
57. If you wish to take a picture of a bullet traveling at
then a very brief flash of light produced by an
discharge through a flash tube can limit blurring. Assuming
of motion during one
constant is acceptable, and given that the flash is driven by a
capacitor, what is the resistance in the flash tube?
6.6 Household Wiring and Electrical Safety
58. (a) How much power is dissipated in a short circuit of
ac through a resistance of
? (b) What current flows?
59. What voltage is involved in a
short circuit through a
resistance?
60. Find the current through a person and identify the likely effect on her if she touches a
ac source: (a) if she is standing on a rubber mat and offers a total resistance of
; (b) if she is standing barefoot on wet grass and has a resistance of only
61. While taking a bath, a person touches the metal case of a radio. The path through the person to the drainpipe and ground has a resistance of
What is the smallest voltage on the case of the radio that could cause ventricular fibrillation?
62. A man foolishly tries to fish a burning piece of bread from a toaster with a metal butter knife and comes into contact with
ac. He does not even feel it since, luckily, he is wearing rubbersoled shoes. What is the minimum resistance of the path the current follows through the person?
63. (a) During surgery, a current as small as
applied directly to the heart may cause ventricular fibrillation. If the resistance of the exposed heart is
what is the smallest voltage that poses this danger? (b) Does your answer imply that special electrical safety precautions are needed?
64. (a) What is the resistance of a
ac short circuit that generates a peak power of
? (b) What would the average power be if the voltage were
65. A heart defibrillator passes
through a patient’s torso for
in an attempt to restore normal beating. (a) How much charge passed? (b) What voltage was applied if
of energy was dissipated? (c) What was the path’s resistance? (d) Find the temperature increase caused in the
of affected tissue.
66. A short circuit in a
appliance cord has a
resistance. Calculate the temperature rise of the
of surrounding materials, assuming their specific heat capacity is
and that it takes
for a circuit breaker to interrupt the current. Is this likely to be damaging?
Additional Problems
67. A circuit contains a D cell battery, a switch, a
resistor, and four
capacitors connected in series. (a) What is the equivalent capacitance of the circuit? (b) What is the
time constant? (c) How long before the current decreases to
of the initial value once the switch is closed?
68. A circuit contains a Dcell battery, a switch, a
resistor, and three
capacitors. The capacitors are connected in parallel, and the parallel connection of capacitors are connected in series with the switch, the resistor and the battery. (a) What is the equivalent capacitance of the circuit? (b) What is the
time constant? (c) How long before the current decreases to
of the initial value once the switch is closed?
69. Consider the circuit below. The battery has an emf of
and an internal resistance of
(a) Find the equivalent resistance of the circuit and the current out of the battery. (b) Find the current through each resistor. (c) Find the potential drop across each resistor. (d) Find the power dissipated by each resistor. (e) Find the total power supplied by the batteries.
70. A homemade capacitor is constructed of 2 sheets of aluminum foil with an area of
square meters, separated by paper,
thick, of the same area and a dielectric constant of
The homemade capacitor is connected in series with a
resistor, a switch, and a
voltage source. (a) What is the
time constant of the circuit? (b) What is the initial current through the circuit, when the switch is closed? (c) How long does it take the current to reach one third of its initial value?
71. A student makes a homemade resistor from a graphite pencil
long, where the graphite is
in diameter. The resistivity of the graphite is
The homemade resistor is place in series with a switch, a
capacitor and a
power source. (a) What is the
time constant of the circuit? (b) What is the potential drop across the pencil
after the switch is closed?
72. The rather simple circuit shown below is known as a voltage divider. The symbol consisting of three horizontal lines is represents “ground” and can be defined as the point where the potential is zero. The voltage divider is widely used in circuits and a single voltage source can be used to provide reduced voltage to a load resistor as shown in the second part of the figure. (a) What is the output voltage
of circuit (a) in terms of
and
? (b) What is the output voltage
of circuit (b) in terms of
and
?
73. Three
resistors are connect in series with an AAA battery with a rating of
Amphours. (a) How long can the battery supply the resistors with power? (b) If the resistors are connected in parallel, how long can the battery last?
74. Consider a circuit that consists of a real battery with an emf
and an internal resistance of
connected to a variable resistor
(a) In order for the terminal voltage of the battery to be equal to the emf of the battery, what should the resistance of the variable resistor be adjusted to? (b) In order to get the maximum current from the battery, what should the resistance variable resistor be adjusted to? (c) In order for the maximum power output of the battery to be reached, what should the resistance of the variable resistor be set to?
75. Consider the circuit shown below. What is the energy stored in each capacitor after the switch has been closed for a very long time?
76. Consider a circuit consisting of a battery with an emf
and an internal resistance of
connected in series with a resistor
and a capacitor
Show that the total energy supplied by the battery while charging the battery is equal to
77. Consider the circuit shown below. The terminal voltages of the batteries are shown. (a) Find the equivalent resistance of the circuit and the current out of the battery. (b) Find the current through each resistor. (c) Find the potential drop across each resistor. (d) Find the power dissipated by each resistor. (e) Find the total power supplied by the batteries.
78. Consider the circuit shown below. (a) What is the terminal voltage of the battery? (b) What is the potential drop across resistor
?
79. Consider the circuit shown below. (a) Determine the equivalent resistance and the current from the battery with switch
open. (b) Determine the equivalent resistance and the current from the battery with switch
closed.
80. Two resistors, one having a resistance of
are connected in parallel to produce a total resistance of
(a) What is the value of the second resistance? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
81. Two resistors, one having a resistance of
are connected in series to produce a total resistance of
(a) What is the value of the second resistance? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
82. Apply the junction rule at point
shown below.
83. Apply the loop rule to Loop
in the preceding problem.
84. Find the currents flowing in the circuit in the preceding problem. Explicitly show how you follow the steps in the ProblemSolving Strategy in Resistors in Series and Parallel.
85. Consider the circuit shown below. (a) Find the current through each resistor. (b) Check the calculations by analyzing the power in the circuit.
86. A flashing lamp in a Christmas earring is based on an
discharge of a capacitor through its resistance. The effective duration of the flash is