Physics for
Scientists and
Engineers
A Strategic Approach
with Modern Physics
Fourth Edition
Global Edition
Chapter 28 Lecture
RANDALL D. KNIGHT
Chapter 28 Fundamentals of Circuits
IN THIS CHAPTER, you will learn the fundamental
physical principles that govern electric circuits.
© 2017 Pearson Education, Ltd.
Slide 28-2
Chapter 28 Preview (1)
© 2017 Pearson Education, Ltd.
Slide 28-3
Chapter 28 Preview (2)
© 2017 Pearson Education, Ltd.
Slide 28-4
Chapter 28 Preview (3)
© 2017 Pearson Education, Ltd.
Slide 28-5
Chapter 28 Preview (4)
© 2017 Pearson Education, Ltd.
Slide 28-6
Chapter 28 Preview (5)
© 2017 Pearson Education, Ltd.
Slide 28-7
Chapter 28 Preview (6)
© 2017 Pearson Education, Ltd.
Slide 28-8
Chapter 28.1 Circuit Elements and Diagrams
© 2017 Pearson Education, Ltd.
Slide 28-9
Circuit Diagrams (1)
§ The top figure shows a
literal picture of a
resistor and a capacitor
connected by wires to a
battery.
§ The bottom figure is a
circuit diagram of the
same circuit.
§ A circuit diagram is a
logical picture of what is
connected to what.
© 2017 Pearson Education, Ltd.
Slide 28-10
Circuit Elements
© 2017 Pearson Education, Ltd.
Slide 28-11
Circuit Diagrams (2)
§ A circuit diagram replaces pictures of the circuit
elements with symbols.
§ The longer line at one end of the battery symbol
represents the positive terminal of the battery.
§ The battery’s emf is shown beside the battery.
§ + and – symbols, even though somewhat redundant,
are shown beside the terminals.
© 2017 Pearson Education, Ltd.
Slide 28-12
QuickCheck 28.1
Does the bulb light?
A. Yes
B. No
C. I’m not sure.
© 2017 Pearson Education, Ltd.
Slide 28-13
QuickCheck 28.1
Does the bulb light?
A. Yes
B. No Not a complete circuit
C. I’m not sure.
© 2017 Pearson Education, Ltd.
Slide 28-14
Chapter 28.2 Kirchhoff’s Law and the Basic Circuit
© 2017 Pearson Education, Ltd.
Slide 28-15
Kirchhoff’s Junction Law
§ For a junction, the law of
conservation of current
requires that:
where the Σ symbol
means summation.
§ This basic conservation
statement is called
Kirchhoff’s junction
law.
© 2017 Pearson Education, Ltd.
Slide 28-16
Kirchhoff’s Loop Law
§ For any path that starts
and ends at the same
point,
§ The sum of all the
potential differences
encountered while
moving around a loop
or closed path is zero.
§ This statement is
known as Kirchhoff’s
loop law.
© 2017 Pearson Education, Ltd.
Slide 28-17
Tactics: Using Kirchhoff’s Loop Law (1)
© 2017 Pearson Education, Ltd.
Slide 28-18
Tactics: Using Kirchhoff’s Loop Law (2)
© 2017 Pearson Education, Ltd.
Slide 28-19
The Basic Circuit
© 2017 Pearson Education, Ltd.
§ The most basic electric
circuit is a single resistor
connected to the two
terminals of a battery.
§ Figure (a) shows a literal
picture of the circuit
elements and the
connecting wires.
§ Figure (b) is the circuit
diagram.
§ This is a complete circuit,
forming a continuous path
between the battery
terminals.
Slide 28-20
Analyzing the Basic Circuit
ℰ − 𝐼𝑅 = 0
ℰ
𝐼=
𝑅
∆𝑉! = 𝐼𝑅 = ℰ
© 2017 Pearson Education, Ltd.
Slide 28-21
Example 28.1 Two Resistors and Two Batteries
© 2017 Pearson Education, Ltd.
Slide 28-22
∆𝑉"#$ % = +ℰ% ; ∆𝑉"#$ & = −ℰ&
∆𝑉'() % = −𝐼𝑅% ; ∆𝑉'() & = −𝐼𝑅&
© 2017 Pearson Education, Ltd.
Slide 28-23
© 2017 Pearson Education, Ltd.
Slide 28-24
Chapter 28.3 Energy and Power
© 2017 Pearson Education, Ltd.
Slide 28-25
Lightbulb Puzzle #1
§ The figure shows two
identical lightbulbs in a circuit.
§ The current through both
bulbs is exactly the same!
§ It’s not the current that the
bulbs use up, it’s energy.
§ The battery creates a potential difference, which
supplies potential energy to the charges.
§ As the charges move through the lightbulbs, they
lose some of their potential energy, transferring the
energy to the bulbs.
© 2017 Pearson Education, Ltd.
Slide 28-26
Energy and Power
§ The power supplied by a battery is
§ The units of power are J/s or W.
§ The power dissipated by a resistor is
§ Or, in terms of the potential drop across the resistor,
© 2017 Pearson Education, Ltd.
Slide 28-27
Power Dissipation in a Resistor
§ A current-carrying resistor dissipates power because the
electric force does work on the charges.
© 2017 Pearson Education, Ltd.
Slide 28-28
Kilowatt Hours (千瓦小時; 度)
§ The product of watts and seconds is
joules, the SI unit of energy.
§ However, most electric companies
prefers to use the kilowatt hour, to
measure the energy you use each
month.
§ Examples:
§ A 4000 W electric water heater uses
40 kWh of energy in 10 hours.
§ A 1500 W hair dryer uses 0.25 kWh of energy
in 10 minutes.
© 2017 Pearson Education, Ltd.
Slide 28-29
Chapter 28.4 Series Resistors
© 2017 Pearson Education, Ltd.
Slide 28-30
Series Resistors (1)
§ The figure below shows two resisters connected in
series between points a and b.
§ The total potential difference between points a and b
is the sum of the individual potential differences
across R1 and R2:
© 2017 Pearson Education, Ltd.
Slide 28-31
Series Resistors (2)
§ Suppose we replace R1 and R2 with a single resistor
with the same current I and the same potential
difference ΔVab.
§ Ohm’s law gives resistance between points a and b:
© 2017 Pearson Education, Ltd.
Slide 28-32
Series Resistors (3)
§ Resistors that are aligned end to end, with no
junctions between them, are called series resistors
or, sometimes, resistors “in series.”
§ The current I is the same through all resistors placed
in series.
§ If we have N resistors in series, their equivalent
resistance is
© 2017 Pearson Education, Ltd.
Slide 28-33
Ammeters
§ Figure (a) shows a simple
one-resistor circuit.
§ We can measure the
current by breaking the
connection and inserting
an ammeter in series.
§ The resistance of the
ammeter is negligible.
§ The potential difference
across the resistor must be
ΔVR = IR = 3.0 V.
§ So the battery’s emf must
be 3.0 V.
© 2017 Pearson Education, Ltd.
Slide 28-34
Real Batteries (1)
§ Real batteries have what is called an internal
resistance, which is symbolized by r.
© 2017 Pearson Education, Ltd.
Slide 28-35
Real Batteries (2)
§ A single resistor connected to a real battery is in
series with the battery’s internal resistance, giving
Req = R + r.
ℰ
ℰ
𝐼=
=
𝑅(* 𝑅 + 𝑟
𝑅
∆𝑉! = 𝐼𝑅 =
ℰ
𝑅+𝑟
∆𝑉"#$
𝑟
𝑅
= ℰ − 𝐼𝑟 = ℰ −
ℰ=
ℰ
𝑅+𝑟
𝑅+𝑟
© 2017 Pearson Education, Ltd.
Slide 28-36
A Short Circuit
§ The figure shows an ideal
wire shorting out a battery.
§ If the battery were ideal,
shorting it with an ideal wire
(R = 0 Ω) would cause the
current to be infinite!
§ In reality, the battery’s internal
resistance r becomes the only
resistance in the circuit.
§ The short-circuit current is
© 2017 Pearson Education, Ltd.
Slide 28-37
Example 28.6 A Short-Circuited Battery
© 2017 Pearson Education, Ltd.
Slide 28-38
Chapter 28.6 Parallel Resistors
© 2017 Pearson Education, Ltd.
Slide 28-39
Parallel Resistors (1)
§ The figure below shows two resisters connected in
parallel between points c and d.
§ By Kirchhoff’s junction law, the input current is the sum
of the current through each resistor: I = I1 + I2
© 2017 Pearson Education, Ltd.
Slide 28-40
Parallel Resistors (2)
§ Suppose we replace R1 and R2 with a single resistor
with the same current I and the same potential
difference ΔVcd.
§ Ohm’s law gives resistance between points c and d:
© 2017 Pearson Education, Ltd.
Slide 28-41
Parallel Resistors (3)
§ Resistors connected at both ends are called parallel
resistors or, sometimes, resistors “in parallel.”
§ The potential differences ΔV are the same across all
resistors placed in parallel.
§ If we have N resistors in parallel, their equivalent
resistance is
© 2017 Pearson Education, Ltd.
Slide 28-42
Voltmeters
§ Figure (a) shows a simple
circuit with a resistor and a real
battery.
§ We can measure the potential
difference across the resistor by
connecting a voltmeter in
parallel across the resistor.
§ The resistance of the voltmeter
must be very high.
§ The internal resistance is
© 2017 Pearson Education, Ltd.
Slide 28-43
Chapter 28.7 Resistor Circuits
© 2017 Pearson Education, Ltd.
Slide 28-44
Example 28.10 Analyzing a Two-Loop Circuit
© 2017 Pearson Education, Ltd.
Slide 28-45
© 2017 Pearson Education, Ltd.
Slide 28-46
© 2017 Pearson Education, Ltd.
Slide 28-47
© 2017 Pearson Education, Ltd.
Slide 28-48
Example 28-X1: Using Kirchhoff’s rules.
Calculate the currents I1, I2, and I3 in the three
branches of the circuit in the figure.
© 2017 Pearson Education, Ltd.
Slide 28-49
(Important !)
© 2017 Pearson Education, Ltd.
Slide 28-50
© 2017 Pearson Education, Ltd.
Slide 28-51
© 2017 Pearson Education, Ltd.
Slide 28-52
© 2017 Pearson Education, Ltd.
Slide 28-53
© 2017 Pearson Education, Ltd.
Slide 28-54
Example 28-x2: Jump starting a car. (Practice by yourself!)
A good car battery is being used to
jump start a car with a weak battery.
The good battery has an emf of 12.5
V and internal resistance 0.020 Ω.
Suppose the weak battery has an
emf of 10.1 V and internal resistance
0.10 Ω. Each copper jumper cable is
3.0 m long and 0.50 cm in diameter,
and can be attached as shown.
Assume the starter motor can be
represented as a resistor Rs = 0.15 Ω.
Determine the current through the
starter motor (a) if only the weak
battery is connected to it, and (b) if
the good battery is also connected.
© 2017 Pearson Education, Ltd.
Slide 28-55
© 2017 Pearson Education, Ltd.
Slide 28-56
© 2017 Pearson Education, Ltd.
Slide 28-57
Chapter 28.8 Getting Grounded
© 2017 Pearson Education, Ltd.
Slide 28-58
Getting Grounded
§ The earth itself is a conductor.
§ If we connect one point of a
circuit to the earth by an ideal
wire, we can agree to call
potential of this point to be
that of the earth: Vearth = 0 V
§ The wire connecting the circuit to the earth is not part
of a complete circuit, so there is no current in this wire!
§ A circuit connected to the earth in this way is said to be
grounded, and the wire is called the ground wire.
§ The circular prong of a three-prong plug is a connection to
ground.
© 2017 Pearson Education, Ltd.
Slide 28-59
A Circuit That Is Grounded
§ The figure shows a circuit
with a 10 V battery and two
resistors in series.
§ The symbol beneath the
circuit is the ground symbol.
§ The potential at the ground
is V = 0.
§ Grounding the circuit allows
us to have specific values
for potential at each point in
the circuit, rather than just
potential differences.
© 2017 Pearson Education, Ltd.
Slide 28-60
Example 28.11 A Grounded Circuit
© 2017 Pearson Education, Ltd.
Slide 28-61
Chapter 28.8 RC Circuits
© 2017 Pearson Education, Ltd.
Slide 28-62
RC Circuits (1)
§ The figure shows a
charged capacitor, a
switch, and a resistor.
§ At t = 0, the switch closes
and the capacitor begins
to discharge through the
resistor.
§ A circuit such as this, with resistors and capacitors,
is called an RC circuit.
§ We wish to determine how the current through the
resistor will vary as a function of time after the switch
is closed.
© 2017 Pearson Education, Ltd.
Slide 28-63
RC Circuits (2)
§ The figure shows an RC
circuit, some time after
the switch was closed.
§ Kirchhoff’s loop law
applied to this circuit
clockwise is
§ Q and I in this equation are the instantaneous values
of the capacitor charge and the resistor current.
§ The resistor current is the rate at which charge is
removed from the capacitor:
© 2017 Pearson Education, Ltd.
Slide 28-64
RC Circuits (3)
§ Knowing that I = –dQ/dt, the loop law for a simple
closed RC circuit is
𝑑𝑄
1
=−
𝑑𝑡
𝑄
𝑅𝐶
§ Rearranging and integrating:
where the time constant τ is
© 2017 Pearson Education, Ltd.
Slide 28-65
RC Circuits (4)
§ The charge on the capacitor
of an RC circuit is
where Q0 is the charge at t = 0,
and τ = RC is the time constant.
§ The capacitor voltage is directly
proportional to the charge, so
where ΔV0 is the voltage at t = 0.
§ The current also can be found to
decay exponentially:
© 2017 Pearson Education, Ltd.
Slide 28-66
QuickCheck 28.18
Which capacitor discharges
more quickly after the switch
is closed?
A. Capacitor A
B. Capacitor B
C. They discharge at
the same rate.
D. Can’t say without
knowing the initial
amount of charge.
© 2017 Pearson Education, Ltd.
Slide 28-67
QuickCheck 28.18
Which capacitor discharges
more quickly after the switch
is closed?
Smaller time constant τ = RC
A. Capacitor A
B. Capacitor B
C. They discharge at
the same rate.
D. Can’t say without
knowing the initial
amount of charge.
© 2017 Pearson Education, Ltd.
Slide 28-68
Charging a Capacitor (1)
When the switch
is closed, the
capacitor will begin
to charge. As it does,
the voltage across it
increases, and the
current through the
resistor decreases.
© 2017 Pearson Education, Ltd.
Slide 28-69
© 2017 Pearson Education, Ltd.
Slide 28-70
© 2017 Pearson Education, Ltd.
Slide 28-71
Charging a Capacitor (2)
§ Figure (a) shows a circuit
that charges a capacitor.
§ The capacitor charge and
the circuit current at time t
are
where I0 = /R and τ = RC.
§ This “upside-down decay”
is shown in figure (b).
© 2017 Pearson Education, Ltd.
Slide 28-72
The voltage across the capacitor is VC = Q/C:
The quantity RC that appears in the exponent
is called the time constant of the circuit:
© 2017 Pearson Education, Ltd.
Slide 28-73
The current at any time t can be found by
differentiating the charge:
© 2017 Pearson Education, Ltd.
Slide 28-74
QuickCheck 28.20
The red curve shows how
the capacitor charges after
the switch is closed at t = 0.
Which curve shows the
capacitor charging if the
value of the resistor is
reduced?
© 2017 Pearson Education, Ltd.
Slide 28-75
QuickCheck 28.20
The red curve shows how
the capacitor charges after
the switch is closed at t = 0.
Which curve shows the
capacitor charging if the
value of the resistor is
reduced?
Smaller time constant.
Same ultimate amount
of charge.
© 2017 Pearson Education, Ltd.
Slide 28-76
Note: Electric Hazards (1)
Most people can “feel” a current of 1 mA; a few mA
of current begins to be painful. Currents above 10 mA
may cause uncontrollable muscle contractions(收縮),
making rescue difficult. Currents around 100 mA
passing through the torso (軀幹) can cause death by
ventricular fibrillation (心室纖維化).
Higher currents may not cause fibrillation, but can
cause severe burns (燃燒).
Household voltage can be lethal (致命) if you are wet
and in good contact with the ground. Be careful!
© 2017 Pearson Education, Ltd.
Slide 28-77
Note: Electric Hazards (2)
A person receiving a
shock has become part
of a complete circuit.
© 2017 Pearson Education, Ltd.
Slide 28-78
Note: Electric Hazards (3)
The safest plugs are those with three prongs (叉);
they have a separate ground line.
Here is an example of household wiring – colors
can vary, though! Be sure you know which is the
hot wire before you do anything.
© 2017 Pearson Education, Ltd.
Slide 28-79
Homework
§ 28.) 5, 11, 24, 28, 45, 51, 56, 63, 64, 66,
73, 83.
© 2017 Pearson Education, Ltd.
Slide 28-80