Transcript DC Circuits (www.bzupages.com)

1
DC Circuits
2
3
26-1 EMF and Terminal Voltage
• To have a current in an electric circuit, we need a
device such as a battery or an electric generator that
transforms one type of energy (chemical, mechanical,
light, etc.) into electric energy.
• Such a device is called a source of electromagnetic
force or emf.
• The potential difference between the terminals of
such a source, when no current flows to an external
circuit, is called the emf
(e ) the source.
4
Terminal Voltage
• r: internal resistance in a battery
• Terminal voltage: Vab = Va – Vb
• When no current is drawn from the battery, the
terminal voltage equals the emf, which is determined
by the chemical reactions in the battery: Vab = E.
• However, when a current I flows from the battery
there is an internal drop in voltage equal to Ir.
• Thus, the terminal voltage (the actual voltage
delivered) is
Vab = e – Ir
5
6
Example 26-1
Battery with internal resistance.
A 65.0-W resistor is connected to the terminals of a
battery whose emf is 12.0 V and whose internal
resistance is 0.5 W. Calculate (a) the current in the
circuit, (b) the terminal voltage of the battery, Vab,
and (c) the power dissipated in the resistor R and the
battery’s internal resistance r.
7
Terminals
Battery
+
_
8
26-2 Resistors in Series and in
Parallel
9
Definition of Resistors in Series
Connected resistances are said to be in series when
a potential difference that is applied across their
combination is the sum of the resulting potential
differences across all resistances.
10
Resistors in Series
The same current runs through each
resistor. Since each resistor may be
different, each can have a different
potential across it.
11
If one bulb burns out
(i.e., the circuit is
broken) all bulbs cease
to function. They get
no current.
12
Resistors in Series
Req = R1 + R2 + R3
13
Definition of Resistors in Parallel
Connected resistors are said to be in parallel when
a potential difference that is applied across their
combination results in that same potential
difference across each resistance.
14
Resistors in Parallel
The current splits at a
junction. Therefore, I
= I 1 + I2 + I3 .
Each resistor is
connected directly to
the terminals of the
battery. Therefore,
each has the same
potential difference.
15
If one bulb burns out
(i.e. the circuit is
broken) the other
devices keep operating.
They still get current.
16
Resistors in Parallel
1
1
1
1
=
+
+
Req R1
R2 R3
17
Conceptual Example 26-2
Series or parallel?
(a) The light bulbs in the figure are identical and have
a resistance R. Which configuration produces more
light? (b) Which way do you think the headlights in a
car are wired?
18
19
Example 26-3
Series and parallel resistors.
Two 100-W resistors are connected (a) in series, and
(b) in parallel, to a 24.0-V battery. What is the
current through each resistor and what is the
equivalent resistance of each circuit?
20
21
Example 26-4
Circuit with series and parallel.
How much current flows from the battery shown?
22
23
Example 26-5
Current in one branch.
How much current is flowing through the 500-W
resistor shown?
24
25
Conceptual Example 26-6
Bulb brightness in a circuit.
The circuit shown has three identical light bulbs, each
of resistance R. How will the brightness of bulbs A
and B compare with that of bulb C?
26
27
IC = IA + IB
IA
IC
IB
28
Example 26-7
Resistor “ladder.”
Estimate the equivalent resistance of the “ladder” of
100-W resistors shown in figure (a). In other words,
what resistance would an ohmmeter read if it were
connected between points A and B? (b) What is the
current through each of the three resistors on the left
if a 50.0 V battery is connected between points A and
B?
29
Series
(a)
30
Parallel
31
Series
32
Parallel
33
26-3 Kirchhoff’s Rules
• Kirchhoff’s first or junction rule:
At any junction point, the sum of all currents
entering the junction must equal the sum of all
currents leaving the junction.
• Kirchhoff’s second or loop rule:
The sum of the changes in potential around any
closed path of a circuit must be zero.
34
The potential (voltage) is said to drop going across
a resistor in the direction of current. The charges
lose energy in the form of heat (P = I2R).
+
Vba = - IR
_
+
Vba
(fi)
35
I = .017 A
The algebraic
sum of the
changes in
potential must be
zero.
36
Example 26-8
Using Kirchhoff’s rules.
Calculate the currents I1, I2, and I3 in each of the
branches of the circuit shown.
37
Junction rule: I3 = I1 + I2
38
Start
Loop
1
Loop 1: a – h – d – c – b – a
Loop Rule
39
Loop 2: a – h – d – e – f – g - a
Start
Loop 2
40
Solving Problems with Kirchhoff’s
Rules
Kirchhoff’s Rules
1. Label + and – for each battery. The long side of a
battery symbol is +.
2. Label the current in each branch of the circuit with a
symbol and an arrow. The direction of the arrow
can be chosen arbitrarily. If the current is actually
in the opposite direction, it will come out minus in
the solution.
3. Apply Kirchhoff’s junction rule at each junction,
and the loop rule for one or more loops.
41
Problem Solving
4. In applying the loop rule, follow each loop in one
direction only. Pay careful attention to signs.
(a) For a resistor, the sign of the potential difference
is negative if your chosen loop direction is the same
as the chosen current direction through that resistor
and vice versa.
(b) For a battery, the sign of the potential difference
is positive if your loop moves from the negative
terminal toward the positive and vice versa.
5. Solve the equations algebraically for the unknowns.
42
Example 26-9
Wheatstone bridge.
A Wheatstone bridge is a type of “bridge circuit”
used to make measurements of resistance. The
unknown resistance to be measured, Rx, is placed in a
circuit with accurately known resistances R1, R2, and
R3. One of these, R3, is a variable resistor which is
adjusted so that when the switch is closed
momentarily, the ammeter A shows zero current
flow. (a) Determine Rx in terms of R1, R2, and R3. (b)
If a Wheatstone bridge is “balanced” when R1 = 630
W, R2 = 972 W, and R3 = 42.6 W, what is the value of
the unknown resistance?
43
R3 is adjusted so that
ammeter reads 0 amps
when switch is closed.
Therefore, the potential at
B = potential at D.
Therefore,VAB = VAD
44
EMF’s in Series and in Parallel; Charging
a Battery
Vca = 1.5 V + 1.5 V = 3.0V
45
EMF’s in Series and in Parallel; Charging
a Battery
Vca = 20 V – 12 V = 8.0 V
46
EMF’s in Series and in Parallel; Charging
a Battery
•Provides more energy when large
currents are needed.
•Each of the batteries in parallel
produce one half the total current, so
the loss due to internal resistance is
less and the batteries will last longer.
V = e - Ir
47
Example 26-10
Jump starting a car.
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 an internal resistance of 0.020 W.
Suppose the weak battery has an emf of 10.1 V and
an internal resistance of 0.10 W. Each copper jumper
cable is 3.0 m long and 0.50 cm in a diameter, and
can be attached as shown. Assume the starter motor
can be represented as a resistor RS = 0.15 W.
Determine the current through the motor (a) if only
the weak battery is connected to it and (b) if the good
battery is also connected.
48
(a)
I1 = I3 = I
Rs = 0.15 W
49
(b)
Junction
50
Example 26-11
Jumper cables reversed.
What would happen if the jumper cables of Example
26-10 were mistakenly connected in reverse, the
positive terminal of each battery connected to the
negative terminal of the other battery? Why could
this be dangerous?
51
RJ = 0.0026 W
52
26-4 Circuits Containing a Resistor and a
Capacitor (RC Circuits)
• Until now the circuits have had a steady current. That
is, one that does not change in time.
• Add a capacitor.
Qmax
• When circuit is switched on, current flows to
capacitor until it is fully charged. That is, Q = Q0
• When fully charged Vc across capacitor = e of
battery.
• Open switch. Capacitor discharges. That is, Q = 0.
53
Charging a
capacitor
Switch is closed at t = 0.
e
54
Charging an RC Circuit
Vc = V0 (1-e-t/RC)
V0 = e
Qc = Q0 (1-e-t/RC)
Q0 = Ce
I = I0
e-t/RC
I0
e
=
R
55
t = RC (time constant)
56
Discharging a
capacitor
Switch closes at t =
0, taking battering
out of circuit.
e
57
Discharging an RC Circuit
Vc = V0 e-t/RC
V0 = e
Qc = Q0 e-t/RC
Q0 = Ce
I = I0
e-t/RC
I0
e
=
R
58
59
Example 26-12
RC circuit, with emf. (charging a capacitor)
The capacitance in the previous circuit is C = 0.30
mF, the total resistance is 20kW, and the battery emf is
12 V. Determine (a) the time constant, (b) the
maximum charge the capacitor could acquire, (c) the
time it takes for the charge to reach 99 percent of this
value, (d) the current I when the charge Q is half its
maximum value, (e) the maximum current, and (f )
the charge Q when, the current I is 0.20 its maximum
value.
60
61
62
Example 26-13
Discharging and RC circuit.
In the RC circuit shown, the battery has fully charged
the capacitor, so Q0 = CE. Then at t = 0 the switch is
thrown from position a to b. The battery emf is 20.0
V, and the capacitance C = 1.02 mF. The current I is
observed to decrease to 0.50 of its initial value in 40
ms. (a) What is the value of R? (b) What is the value
of Q, the charge on the capacitor, at t = 0? (c) What
is Q at t = 60 ms?
63
64
65
66
Homework Problem 1
Calculate the terminal voltage for a battery with an
internal resistance of 0.900 W and an emf of 8.50 V
when the battery is connected in series with (a) a
68.0-W resistor, and (b) a 680-W resistor.
67
Homework Problem 4
What is the internal resistance of a 12-V car battery
whose terminal voltage drops to 9.8 V when the
starter draws 60 A? What is the resistance of the
starter?
68
Homework Problem 8
Suppose that you have a 500-W, a 900-W, and a 1400
W resistor. What is (a) the maximum, and (b)
minimum resistance you can obtain by combining
these?
69
Homework Problem 10
Three 1.20-kW resistors can be connected together in
four different combinations of series and/or parallel
circuits. What are the four ways and what is the net
resistance in each case?
70
Homework Problem 19
Consider the network of resistors shown. Answer
qualitatively: (a) What happens to the voltage across
each resistor when the switch S is closed? (b) What
happens to the current through each when the switch
is closed? (c) What happens to the output of the
battery when the switch is closed? (d) Let R1 = R2 =
R3 = R4 = 100 W and V = 45.0 V. Determine the
current through each resistor before and after closing
the switch. Are your qualitative predictions
confirmed?
71
72
73
Homework Problem 25
Determine the magnitudes and directions of the
currents through R1 and R2 in the figure.
74
75
Homework Problem 29
Determine the current through each of the resistors in
the figure.
76
77
Homework Problem 42
The RC switch in the figure has R = 6.7 kW and C =
6.0 mF. The capacitor has a voltage V0 at t = 0, when
the switch is closed. How long does it take the
capacitor to discharge to 1.0 percent of its initial
voltage?
78
79
Homework Problem 43
How long does it take for the energy stored in a
capacitor in a series RC circuit, shown in the figure,
to reach half its maximum value? Express your
answer in terms of the time constant t = RC.
80
-
81
*26-5 DC Ammeters and Voltmeters
• An ammeter is used to measure current.
• A voltmeter is used to measure voltage.
82
83
84
85
86
87
*26-6 Transducers and the Thermocouple
• A transducer is a device that converts one type of
energy into another.
• A high-fidelity loudspeaker is one kind of
transducer—it transforms electric energy into sound
energy.
88
89
90
91