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Chapter 36 Homework
Due: 8:00am on Wednesday, April 28, 2010
Note: To understand how points are awarded, read your instructor's Grading Policy.
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Alternating Voltages and Currents
The voltage supplied by a wall socket varies with time, reversing its polarity with a constant frequency, as shown in the graph.
Part A
What is the rms value
Hint A.1
of the voltage plotted in the graph?
RMS value of a quantity with sinusoidal time dependence
A quantity that varies with time as
(or
) has a maximum value equal to
and an rms value given by
.
Hint A.2
Find the maximum value of the voltage
Hint not displayed
Express your answer in volts.
ANSWER:
= 120
Correct
This is the standard rms voltage supplied to a typical household in North America.
Part B
When a lamp is connected to a wall plug, the resulting circuit can be represented by a simplified AC circuit, as shown in the figure. Here the lamp has been
replaced by a resistor with an equivalent resistance = 120 . What is the rms value
of the current flowing through the circuit?
Hint B.1
Ohm's law in AC circuits
Ohm's law can still be applied to an AC circuit, provided the values used to describe all physical quantities are consistent. For example, Ohm's law can be written using maximum values of voltage and current, or
alternatively using rms quantities.
Express your answer in amperes.
ANSWER:
= 1.00
Correct
Part C
What is the average power
Hint C.1
dissipated in the resistor?
Average power in an AC circuit
Hint not displayed
Express your answer in watts.
ANSWER:
= 120
Correct
The instantaneous power dissipated in the resistor can be substantially higher than the average power. However, since the voltage supplied to the resistor varies in time, so does the instantaneous power.
Therefore, a better estimate of the energy dissipated in an AC circuit is given by the average power. For example, the power rating on light bulbs is in fact the average power dissipated in the bulb.
Problem 36.7
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A 0.65
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capacitor is connected across an AC generator that produces a peak voltage of 23
.
Part A
What is the peak current to and from the capacitor if the emf frequency is 100
?
Express your answer using two significant figures.
ANSWER:
−3
= 9.2×10
Correct
Part B
What is the peak current to and from the capacitor if the emf frequency is 100
?
Express your answer using two significant figures.
ANSWER:
= 9.2
Correct
Problem 36.14
A high-pass
filter with a crossover frequency of 1000 Hz uses a
resistor.
Part A
What is the value of the capacitor?
ANSWER:
1.59
Correct
Determining Inductance from Voltage and Current
An inductor is hooked up to an AC voltage source. The voltage source has EMF
and frequency . The current amplitude in the inductor is
.
Part A
What is the reactance
Hint A.1
of the inductor?
Definition of reactance
Hint not displayed
Express your answer in terms of
and
.
ANSWER:
=
Correct
Part B
What is the inductance
Hint B.1
of the inductor?
Reactance in terms of
and
Hint not displayed
Express your answer in terms of
, , and
.
ANSWER:
=
Correct
Part C
What would happen to the amplitude of the current in the inductor if the inductance
Hint C.1
were doubled?
How to approach the problem
Hint not displayed
ANSWER:
There would be no change in the amplitude of the current.
The amplitude of the current would be doubled.
The amplitude of the current would be halved.
The amplitude of the current would be quadrupled.
Correct
Problem 36.20
An inductor is connected to a 15 kHz oscillator. The peak current is 65 mA when the rms voltage is 6.0 V.
Part A
What is the value of the inductance
?
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ANSWER:
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1.39
mH
Correct
Problem 36.52
Part A
In the figure, what is the current supplied by the emf when the frequency is very small?
ANSWER:
0.100
A
Correct
Part B
What is the current supplied by the emf when the frequency is very large?
ANSWER:
0.200
A
Correct
L-R-C AC Circuits
Learning Goal: To understand basic calculations involving L-R-C ac circuits.
Because the currents and voltages vary, ac circuits are more complex than dc circuits. Consider a circuit consisting of a resistor of resistance
, an inductor of inductance
, and a capacitor of capacitance
connected
in series to an ac power source. As with all circuit components connected in series, the same current flows through each of these elements. This is an ac
circuit, so the current is changing with time. The current at a time can be found using the relationship:
, where is the maximum current and
(
is the frequency of the current source).
The relationship between the current and voltage in an ac circuit works according to Ohm's law. Consider just the resistor in the circuit. Because the current
changes in time, the voltage across the resistor
also changes. However, both and
will be at a maximum at the same time. The maximum voltage
across the resistor is given by
.
Recall that an inductor is designed to oppose any change in current in the circuit. Although an inductor has no resistance, there is a potential difference
and
do not reach maximum values at the same time. The voltage reaches a maximum before the current.
across the ends of the inductor. Unlike the case for the resistor,
The maximum potential difference across the inductor is
.
By defining the quantity
as the inductive reactance
,
can be rewritten as
. This equation is similar to Ohm's law.
Part A
An L-R-C circuit, operating at 60
Hint A.1
, has an inductor with an inductance of
, a capacitance of
, and a resistance of 0.329
. What is the inductive reactance of this circuit?
Find
Hint not displayed
Enter your answer numerically in ohms.
ANSWER:
Because
= 0.577
Correct
is proportional to the frequency, low-frequency ac currents pass through an inductor more easily than high-frequency ac currents. Hence circuits containing inductors are often used as filters that
allow low frequencies but not high frequencies to pass. These filters are called low-pass filters.
A capacitor is designed to store energy by allowing charge to build up on its plates. Although a capacitor has no resistance in an ac circuit, there is a potential difference
maximum values of and
do not occur at the same time. The voltage reaches a maximum after the current.
across the plates of the capacitor. The
The maximum potential difference across the inductor is
.
By defining the quantity
as the capacitive reactance
,
can be rewritten as
. As with the case for the inductor, this equation is similar to Ohm's law.
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Part B
What is the capacitive reactance of the circuit in Part A?
Enter your answer numerically in ohms.
ANSWER:
Because
= 0.159
Correct
is inversely proportional to the frequency, high-frequency ac currents pass through a capacitor more easily than low-frequency ac currents. Hence circuits containing capacitors are often used as
filters that allow high frequencies but not low frequencies to pass. These filters are called high-pass filters.
The capacitive reactance, the inductive reactance, and the resistance of the circuit can be combined to give the impedance
and is similar to the equivalent resistance that can be found from various resistors in a dc circuit. Because
and
and
of the circuit. The impedance is a measure of the total reactance and resistance of the circuit
do not reach maximum values at the same time as , the impedance is not found by adding
,
,
. Instead, the impedance is found using
.
The analogy to Ohm's law is then
.
Part C
What is the total impedance of the circuit in Parts A and B?
Enter your answer numerically in ohms.
ANSWER:
In ac circuits,
and
= 0.532
Correct
are not measured directly. Instead, ac ammeters and voltmeters are designed to measure the root-mean-square values of
and
:
,
which can be used in the equation
to yield
.
Part D
If this circuit were connected to a standard 120
ac outlet, what would the rms current in the circuit be?
Enter your answer numerically in amperes.
ANSWER:
= 226
Correct
The current is high because the total impedance is relatively low. Actually, plugging such a circuit into a 120-
outlet would most likely burn out the circuit elements.
A frequent application of L-R-C ac circuits is the tuning mechanism in a radio. The L-R-C ac circuit will have a resonant frequency that depends on both the inductance and capacitance of the circuit according to the
formula
.
This is the frequency at which the impedance is the smallest, which causes the largest current to appear in the circuit for a given
.
The radio picks up this resonant frequency and suppresses signals at other frequencies. A variable capacitor in this circuit causes the resonant frequency of the circuit to change. When you tune the radio you are
adjusting the value of the capacitance in the circuit and hence the resonant frequency.
Part E
To see whether the L-R-C ac circuit from Part A would be suitable for a tuner in a radio, find the resonant frequency of this circuit.
Enter your answer numerically in hertz.
ANSWER:
= 31.5
Correct
This frequency does not correspond to either the standard AM or FM band.
Average Power in an L-R-C Circuit
A circuit consists of a resistor (resistance
), inductor (inductance
), and capacitor (capacitance
are ideal, so that the only resistance in the circuit is due to the resistor. Also assume that
) connected in series with an AC source supplying sinusoidal voltage
. Assume that all circuit elements
is the resonant frequency of the circuit.
Part A
What is the average power
Hint A.1
supplied by the voltage source?
Find the instantaneous power
Hint not displayed
Hint A.2
Average value of a periodic function
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Hint not displayed
Express your answer in terms of any or all of the following quantities:
,
,
, and
.
ANSWER:
=
Correct
Problem 36.29
The heating element of a hair drier dissipates 1560 W when connected to a 120 V/60 Hz power line.
Part A
What is its resistance?
ANSWER:
9.23
Correct
Problem 36.49
A series RLC circuit consists of a 105
resistor, a 0.100
inductor, and a 21.0
capacitor. It is attached to a 120 V/60 Hz power line.
Part A
What is the peak current ?
ANSWER:
1.24
A
Correct
Part B
What is the phase angle ?
ANSWER:
-40.2
Correct
Part C
What is the average power loss?
ANSWER:
80.1
W
Answer Requested
Resonating RLC Series Circuit Ranking Task
Six series RLC circuits are described below, where
is the resistance,
the inductance, and
the capacitance of the circuits.
Part A
Rank these circuits on the basis of their resonance frequencies.
Hint A.1
Resonance and the phase between voltage and current
Hint not displayed
Hint A.2
Resonance and impedance
Hint not displayed
Hint A.3
Equating inductive and capacitive reactance at resonance
Hint not displayed
Rank from largest to smallest. To rank items as equivalent, overlap them.
ANSWER:
View
Correct
Part B
Each circuit is driven at its resonance frequency by a 100
Hint B.1
AC power supply. Rank these circuits on the basis of their maximum current.
Current at resonance
Hint not displayed
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Hint B.2
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Maximizing maximum current
Hint not displayed
Rank from largest to smallest. To rank items as equivalent, overlap them.
ANSWER:
View
Correct
Problem 36.63
A motor attached to a 120 V/60 Hz power line draws an 8.10 A current. Its average energy dissipation is 760 W.
Part A
What is the power factor?
ANSWER:
0.782
Correct
Part B
What is the rms resistor voltage?
ANSWER:
93.8
V
Correct
Part C
What is the motor's resistance?
ANSWER:
11.6
Correct
Part D
How much series capacitance needs to be added to increase the power factor to 1.0?
ANSWER:
287
Correct
Score Summary:
Your score on this assignment is 109%.
You received 71.79 out of a possible total of 75 points, plus 9.95 points of extra credit.
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