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Physics

James S. Walker

Chapter 24

Alternating-Current Circuits

Educators


Problem 1

An ac generator produces a peak voltage of 75 V. What is the rms
voltage of this generator?

João Gabriel A.
Numerade Educator

Problem 2

In many European homes the rms voltage
available from a wall socket is 240 V. What is the maximum voltage in this case?

João Gabriel A.
Numerade Educator

Problem 3

An rms voltage of 120 V produces a maximum current of 4.1 A in
a certain resistor. Find the resistance of this resistor

João Gabriel A.
Numerade Educator

Problem 4

The rms current in an ac circuit with a resistance of 150$\Omega$ is
0.15 A. What are (a) the average and (b) the maximum power con-
sumed by this circuit?

João Gabriel A.
Numerade Educator

Problem 5

A $3.33-\mathrm{k} \Omega$ resistor is connected to a generator with a maximum voltage of 241 $\mathrm{V} .$ Find (a) the average and (b) the maximum power
delivered to this circuit.

João Gabriel A.
Numerade Educator

Problem 6

A "75 -watt" lightbulb uses an average power of 75 $\mathrm{w}$ when connected to an rms voltage of 120 $\mathrm{V}$ . (a) What is the resistance of the
lightbulb? (b) What is the maximum current in the bulb? (c) What
is the maximum power used by the bulb at any given instant of time?

João Gabriel A.
Numerade Educator

Problem 7

An array of solar panels produces 9.45 A of direct current at a potential difference of 192 $\mathrm{V}$ . The current flows into an inverter that produces a $60-\mathrm{Hz}$ alternating current with $V_{\max }=171 \mathrm{V}$ and $I_{\max }=19.5 \mathrm{A}$ (a) What rms power is produced by the inverter? (b) Use the rms values to find the power efficiency $P_{\text { out }} / P_{\text { lin }}$ of the inverter.

João Gabriel A.
Numerade Educator

Problem 8

The relationship $V_{\mathrm{rms}}=V_{\max } / \sqrt{2}$ is valid only for voltages that vary sinusoidally. Find the relationship between $V_{\mathrm{rms}}$ and $V_{\max }$ for the "square-wave" voltage shown in FIGURE 24-30.

João Gabriel A.
Numerade Educator

Problem 9

The reactance of a capacitor is 65$\Omega$ at a frequency of 87 Hz.
What is its capacitance?

João Gabriel A.
Numerade Educator

Problem 10

The capacitive reactance of a capacitor at 60.0 $\mathrm{Hz}$ is 105$\Omega . \mathrm{At}$
what frequency is its capacitive reactance 72.5$\Omega ?$

João Gabriel A.
Numerade Educator

Problem 11

$\cdot \mathrm{A} 105-\mu \mathrm{F}$ capacitor is connected to an ac generator with an rms
voltage of 32.5 $\mathrm{V}$ and a frequency of 125 $\mathrm{Hz}$ . What is the rms cur-
rent in this circuit?

João Gabriel A.
Numerade Educator

Problem 12

The rms voltage across a $0.010-\mu \mathrm{F}$ capacitor is 2.8 $\mathrm{V}$ at a fre-
quency of 75 $\mathrm{Hz}$ . What are (a) the rms and (b) the maximum cur-
rent through the capacitor?

João Gabriel A.
Numerade Educator

Problem 13

The rms current through a $55.5-\mu$ F capacitor is 1.51 A when it is connected to an ac generator. (a) If a second capacitor is connected in series with the first, will the rms current
increase or decrease? (b) Calculate the rms current when a $38.7-\mu \mathrm{F}$
capacitor is connected in series with the $55.5-\mu$ F capacitor.

João Gabriel A.
Numerade Educator

Problem 14

The maximum current in a $22-\mu \mathrm{F}$ capacitor connected to an
ac generator with a frequency of 120 $\mathrm{Hz}$ is 0.15 $\mathrm{A}$ . (a) What is the maximum voltage of the generator? (b) What is the voltage across
the capacitor when the current in the circuit is 0.10 A and increas-
ing? (c) What is the voltage across the capacitor when the current
in the circuit is 0.10 A and decreasing?

João Gabriel A.
Numerade Educator

Problem 15

An rms voltage of 20.5 V with a frequency of 1.00 $\mathrm{kHz}$ is applied to a $0.395-\mu \mathrm{F}$ capacitor. (a) What is the rms current in this circuit? (b) By what factor does the current change if the frequency of the voltage is doubled? (c) Calculate the current for a frequency of 2.00 $\mathrm{kHz}$ .

João Gabriel A.
Numerade Educator

Problem 16

A circuit consists of a $2.00-\mathrm{kHz}$ generator and a capacitor. When the rms voltage of the generator is $0.800 \mathrm{V},$ the rms current in the
circuit is 0.515 $\mathrm{mA}$ (a) What is the reactance of the capacitor at
2.00 $\mathrm{kHz}$ ? ( b) What is the capacitance of the capacitor?
rms voltage is maintained at $0.800 \mathrm{V},$ what is the rms current at 4.00 kHz? At 20.0 kHz?

João Gabriel A.
Numerade Educator

Problem 17

A 0.22 -\muF capacitor is connected to an ac generator with an rms voltage of 12 $\mathrm{V}$ . For what range of frequencies will the rms current
in the circuit be less than 1.0 $\mathrm{mA}$ ?

João Gabriel A.
Numerade Educator

Problem 18

At what frequency will a generator with an rms voltage of 504 $\mathrm{V}$
produce an rms current of 7.50 $\mathrm{mA}$ in a $0.0150-\mu \mathrm{F}$ capacitor?

João Gabriel A.
Numerade Educator

Problem 19

A 22.0 -\muF capacitor is connected to an ac generator with an rms voltage of 118 $\mathrm{V}$ and a frequency of 60.0 $\mathrm{Hz}$ . (a) What is the rms current in the circuit? (b) If you wish to increase
the rms current, should you add a second capacitor in series or
in parallel? (c) Find the value of the capacitance that should be
added to increase the rms current to 1.50 $\mathrm{A} .$

João Gabriel A.
Numerade Educator

Problem 20

Find the impedance of a 60.0 -Hz circuit with a $65.5-\Omega$ resistor
connected in series with a $85.0-\mu \mathrm{F}$ capacitor.

João Gabriel A.
Numerade Educator

Problem 21

An ac generator with a frequency of 125 $\mathrm{Hz}$ and an rms voltage of 42.5 $\mathrm{V}$ is connected in series with a 10.0 $\mathrm{-k} \Omega$ resistor and a
$0.250-\mu \mathrm{F}$ capacitor. What is the rms current in this circuit?

João Gabriel A.
Numerade Educator

Problem 22

The rms current in an $R C$ circuit is 0.72 A. The capacitor in this
circuit has a capacitance of 13$\mu F$ and the ac generator has a fre-
quency of 150 $\mathrm{Hz}$ and an $\mathrm{rms}$ voltage of 95 $\mathrm{V}$ . What is the resis-
tance in this circuit?

João Gabriel A.
Numerade Educator

Problem 23

When an ac generator with a frequency of 180 $\mathrm{Hz}$ and an rms
voltage of 36 $\mathrm{V}$ is connected to an $R C$ circuit, the rms current is
0.28 $\mathrm{A}$ . If the resistance has a value of $54 \Omega,$ what is the value of
the capacitance?

João Gabriel A.
Numerade Educator

Problem 24

A 50.0-Hz generator with an rms voltage of 115 V is connected in series to a 3.12 -k\Omega resistor and a $1.65-\mu$ F capacitor. Find (a) the
rms current in the circuit and (b) the phase angle, $\phi$ , between the
current and the voltage.

João Gabriel A.
Numerade Educator

Problem 25

(a) At what frequency must the circuit in Problem 24 be operated for the current to lead the voltage by $25.0^{\circ}$ (b) Using the frequency found in part (a), find the average power consumed by this
circuit.

João Gabriel A.
Numerade Educator

Problem 26

Find the power factor for an RC circuit connected to a 60.0-Hz generator with an rms voltage of 195 $\mathrm{V}$ . The values of $R$ and $C$ in this circuit are 105$\Omega$ and 82.4$\mu \mathrm{F}$ , respectively.

João Gabriel A.
Numerade Educator

Problem 27

(a) Determine the power factor for an RC cir. cuit with $R=4.0 \mathrm{k} \Omega$ and $C=0.35 \mu \mathrm{F}$ that is connected to an ac
generator with an rms voltage of 24 $\mathrm{V}$ and a frequency of 150 $\mathrm{Hz}$ .
(b) Will the power factor for this circuit increase, decrease, or stay
the same if the frequency of the generator is increased? Explain.

João Gabriel A.
Numerade Educator

Problem 28

The "square-wave" voltage shown in FIGURE $24-31$ is applied to an $R C$ circuit. Sketch the shape of the instantaneous voltage across the capacitor, assuming the time constant
of the circuit is equal to the period of the applied voltage.

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Problem 29

When a long copper wire of finite resistance
is connected to an ac generator, as shown in FIGURE 24-32 (a), a certain amount of current flows through the wire. The wire is now
wound into a coil of many loops and reconnected to the generator,
as indicated in FIGURE 24-32 (b). (a) Is the current supplied to the
coil greater than, less than, or the same as the current supplied
to the uncoiled wire? (b) Choose the best explanation from among the following:
I. More current flows in the circuit because the coiled wire is an
inductor, and inductors tend to keep the current flowing in
an ac circuit.
II. The current supplied to the circuit is the same because the
wire is the same. Simply wrapping the wire in a coil changes
nothing.
III. Less current is supplied to the circuit because the coiled wire acts
as an inductor, which increases the impedance of the circuit.

João Gabriel A.
Numerade Educator

Problem 30

An inductor has a reactance of 56.5$\Omega$ at 85.0 $\mathrm{Hz}$ . What is its reac-
tance at 60.0 $\mathrm{Hz}$ ?

João Gabriel A.
Numerade Educator

Problem 31

What is the rms current in a $97.5-\mathrm{mH}$ inductor when it is con-
nected to a 60.0 -Hz generator with an rms voltage of 135 $\mathrm{V} ?$

João Gabriel A.
Numerade Educator

Problem 32

$\cdot$ What rms voltage is required to produce an rms current of 2.1 $\mathrm{A}$
in a 66 -mH inductor at a frequency of 25 $\mathrm{Hz?}$

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Numerade Educator

Problem 33

$\cdot$ Loudspeaker Impedance A loudspeaker's impedance is listed as 8.0$\Omega$
at 2.0 kHz. Assuming its resistance and capacitance are negligibly
small, what is the inductance of the speaker coil?

João Gabriel A.
Numerade Educator

Problem 34

Fluorescent tamp Ballast An inductor is used to limit the current
through a fluorescent lamp that is operated with an rms voltage of
120 $\mathrm{v}$ at $f=60 \mathrm{Hz}$ . If the maximum rms current should be 0.33 $\mathrm{A}$ , and assuming the lamp has zero resistance, what value of induc-
tance $L$ should be used as the ballast?

João Gabriel A.
Numerade Educator

Problem 35

A $525-\Omega$ resistor and a $295-\mathrm{mH}$ inductor are connected in series with an ac generator with an rms voltage of 20.0 $\mathrm{V}$ and a frequency
of 60.0 $\mathrm{Hz}$ . What is the rms current in this circuit?

João Gabriel A.
Numerade Educator

Problem 36

The rms current in an $R L$ circuit is 0.26 A when it is connected
to an ac generator with a frequency of 60.0 $\mathrm{Hz}$ and an rms voltage of 25 $\mathrm{V}$ (a) Given that the inductor has an inductance of $145 \mathrm{mH},$ what is the resistance of the resistor? (b) Find the rms voltage across the resistor. (c) Find the rms voltage across the
inductor. (d) Use your results from parts (b) and (c) to show that $\sqrt{V_{\mathrm{rms}, R}^{2}+V_{\mathrm{rms},}^{2}}$ is equal to 25 $\mathrm{V}$

João Gabriel A.
Numerade Educator

Problem 37

An ac generator with a frequency of 1.55 kHz and an rms voltage of 20.8 $\mathrm{V}$ is connected in series with a 2.00 -k\Omega resistor and a
$292-\mathrm{mH}$ inductor. (a) What is the power factor for this circuit?
(b) What is the average power consumed by this circuit?

João Gabriel A.
Numerade Educator

Problem 38

An rms voltage of 22.2 V with a frequency
of 1.00 kHz is applied to a 0.290-mH inductor. (a) What is the rms
current in this circuit? (b) By what factor does the current change
if the frequency of the voltage is doubled? (c) Calculate the current for a frequency of 2.00 kHz.

João Gabriel A.
Numerade Educator

Problem 39

A $0.22-\mu \mathrm{H}$ inductor is connected to an ac generator with an rms voltage of 12 $\mathrm{V}$ . For what range of frequencies will the rms current
in the circuit be less than 1.0 $\mathrm{mA}$ ?

João Gabriel A.
Numerade Educator

Problem 40

The phase angle in a certain $R L$ circuit is $68^{\circ}$ at a frequency of 60.0 $\mathrm{Hz}$ . If $R=2.1 \Omega$ for this circuit, what is the value of the
inductance, $L ?$

João Gabriel A.
Numerade Educator

Problem 41

(a) Sketch the phasor diagram for an ac circuit with a $105-\Omega$
resistor in series with a 22.5 -mH inductor. The frequency of the
generator is 60.0 $\mathrm{Hz}$ (b) If the rms voltage of the generator is 120 $\mathrm{V}$ ,
what is the average power consumed by the circuit?

João Gabriel A.
Numerade Educator

Problem 42

A large air conditioner has a resistance of 7.0$\Omega$ and an inductive reactance of 15$\Omega .$ If the air conditioner is powered by a $60.0-\mathrm{Hz}$ generator with an rms voltage of $240 \mathrm{V},$ find (a) the impedance of the air conditioner, (b) its rms current, and (c) the average power consumed by the air conditioner.

João Gabriel A.
Numerade Educator

Problem 43

The "square-wave" voltage shown in Figure $24-31$ is applied to an $R L$ circuit. Sketch the shape of the instantaneous voltage across the inductor, assuming the time constant
of the circuit is much less than the period of the applied voltage.

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Problem 44

An inductor and a capacitor are to be connected to a generator.
Will the generator supply more current at high frequency if the inductor and capacitor are connected in series or in parallel? Explain.

João Gabriel A.
Numerade Educator

Problem 46

An inductor and a capacitor are to be connected to a generator.
Will the generator supply more current at low frequency if the inductor and capacitor are connected in series or in parallel? Explain

João Gabriel A.
Numerade Educator

Problem 46

(a) When the ac generator in FIGURE 24-33 operates at high frequency, is the rms current in the circuit greater than, less than, or the sameas when the generator operates at low frequency? (b) Choose the best explanation from among the following:
I. The current is the same because at high frequency the inductor
is like an open circuit, and at low frequency the capacitor is like
an open circuit. In either case the resistance of the circuit is R.
II. Less current flows at high frequency because in that limit the
inductor acts like an open circuit, allowing no current to flow.
III. More current flows at high frequency because in that limit the
capacitor acts like an ideal wire of zero resistance.

João Gabriel A.
Numerade Educator

Problem 47

(a) When the ac generator in FIGURE 24-34 operates at high frequency, is the rms current in the circuit greater than, less than, or the same as when the generator operates at low
frequency? (b) Choose the best explanation from among the following:
I. The current at high frequency is greater because the higher
the frequency the more charge that flows through a circuit.
II. Less current flows at high frequency because in that limit the
inductor is like an open circuit and current has only one path
to flow through.
III. The inductor has zero resistance, and therefore the resistance
of the circuit is the same at all frequencies. As a result the current is the same at all frequencies.

João Gabriel A.
Numerade Educator

Problem 48

(a) When the ac generator in FilsuRE 24.35
operates at high frequency, is the rms current in the circuit greater
than, less than, or the same as when the generator operates at low
frequency? (b) Choose the best explanation from among the following:
I. The capacitor has no resistance, and therefore the resistance
of the circuit is the same at all frequencies. As a result the current is the same at all frequencies.
II. Less current flows at high frequency because in that limit the
capacitor is like an open circuit and current has only one path
to flow through.
III. More current flows at high frequency because in that limit
the capacitor is like a short circuit and current has two parallel
paths to flow through

João Gabriel A.
Numerade Educator

Problem 49

Find the rms voltage across each element in an $R L C$ circuit with
$R=8.8 \mathrm{k} \Omega, C=0.12 \mu \mathrm{F},$ and $L=28 \mathrm{mH} .$ The generator supplies an rms voltage of 115 $\mathrm{v}$ at a frequency of 60.0 $\mathrm{Hz}$ .

João Gabriel A.
Numerade Educator

Problem 50

What is the impedance of a $1.50-\mathrm{k} \Omega$ resistor, a $135-\mathrm{mH}$ induc-
tor, and a $22.8-\mu \mathrm{F}$ capacitor connected in series with a $60.0-\mathrm{Hz}$ ac
generator?

João Gabriel A.
Numerade Educator

Problem 51

Consider the circuit shown in Figure 24-34. The ac generator in this circuit has an rms voltage of 65 $\mathrm{V}$ . Given that $R=15 \Omega$ and
$L=0.22 \mathrm{mH},$ find the rms current in this circuit in the limit of
(a) high frequency and (b) low frequency.

João Gabriel A.
Numerade Educator

Problem 52

What is the phase angle in an $R L C$ circuit with $R=9.9 \mathrm{k} \Omega$ $C=1.5 \mu \mathrm{F},$ and $L=250 \mathrm{mH} ?$ The generator supplies an rms volt-
age of 115 $\mathrm{V}$ at a frequency of 60.0 $\mathrm{Hz}$ .

João Gabriel A.
Numerade Educator

Problem 53

An ac voltmeter, which displays the rms voltage between the
two points touched by its leads, is used to measure voltages in the circuit shown in FIGURE 24-36. In this circuit, the ac generator has
an rms voltage of 9.00 V and a frequency of 25.0 kHz. The inductance in the circuit is 0.250 $\mathrm{mH}$ , the capacitance is 0.150$\mu \mathrm{F}$ , and
the resistance is 2.50$\Omega .$ What is the reading on a voltmeter when
it is connected to points (a) A and B, (b) B and C, (c) A and C, and
(d) A and D?

João Gabriel A.
Numerade Educator

Problem 54

Consider the ac circuit shown in Figure 2436, where we assume that the values of $R, L,$ and $C$ are the same as in the previous problem, and that the rms voltage of the generator
is still 9.00 $\mathrm{V}$ . The frequency of the generator, however, is doubled
to 50.0 $\mathrm{kHz}$ . Calculate the rms voltage across (a) the resistor, $R$ (b) the inductor, $L,$ and $(\mathrm{c})$ the capacitor, $\mathrm{C}$ ) Do you expect the
sum of the rms voltages in parts (a), and (c) to be greater than,
less than, or equal to 9.00 $\mathrm{V}$ ? Explain.

João Gabriel A.
Numerade Educator

Problem 55

(a) Sketch the phasor diagram for an ac circuit with a $105-\Omega$ resistor in series with a $22.5-\mathrm{mH}$ inductor and a 32.2 -\muF capacitor.
The frequencylof the generator is 60.0 $\mathrm{Hz}$ . (b) If the rms voltage of the generator is $120 \mathrm{V},$ what is the average power consumed by
the circuit?

João Gabriel A.
Numerade Educator

Problem 56

A generator connected to an RLC circuit has an rms voltage of150 $\mathrm{V}$ and an rms current of 39 $\mathrm{mA}$ . If the resistance in the circuit
is 3.3 $\mathrm{k} \Omega$ and the capacitive reactance is $6.6 \mathrm{k} \Omega,$ what is the inductive reactance of the circuit?

João Gabriel A.
Numerade Educator

Problem 57

A manufacturing plant uses 2.22 kW of electric power provided by a $60.0-\mathrm{Hz}$ ac generator with an rms voltage of 485 $\mathrm{V}$ . The plant uses this power to run a number of
high-inductance electric motors. The plant's total resistance is $R=25.0 \Omega$ and its inductive reactance is $X_{L}=45.0 \Omega .$ (a) What is the total impedance of the plant? (b) What is the plant's power factor? (c) What is the rms current used by the plant? (d) What capacitance, connected in series with the power line, will increase the plant's power factor to unity? (e) If the power factor is unity, how much current is needed to provide the 2.22 $\mathrm{kW}$ of power needed by the plant? Compare your answer with the current found in part (c). (Because power-line losses are proportional to the square of the current, a utility company will charge an industrial user with a low power factor a higher rate per kWh than a company with a power factor close to unity.)

João Gabriel A.
Numerade Educator

Problem 58

In an RLC circuit a second capacitor is added in series to the capacitor already present. (a) Does the resonance frequency increase, decrease, or stay the same? (b) Choose the best explanation from among the following:
\begin{equation}\begin{array}{l}{\text { I. The resonance frequency stays the same because it depends }} \\ {\text { only on the resistance in the circuit. }} \\ {\text { II. Adding a capacitor in series increases the equivalent capaci- }} \\ {\text { tance, and this decreases the resonance frequency. }} \\ {\text { II. Adding a capacitor in series decreases the equivalent capaci- }} \\ {\text { tance, and this increases the resonance frequency. }}\end{array}\end{equation}

João Gabriel A.
Numerade Educator

Problem 59

An $R L C$ circuit has a resonance frequency of 1.9 $\mathrm{kHz}$ . If the inductance is 0.13 $\mathrm{mH}$ , what is the capacitance?

João Gabriel A.
Numerade Educator

Problem 60

A magnetron in a kitchen microwave oven resonates at 2.45 GHz in a manner analogous to an $R L C$ circuit. If thecapacitance of its resonator is $4.4 \times 10^{-13} \mathrm{F}$ , what is the value of
its inductance?

João Gabriel A.
Numerade Educator

Problem 61

At resonance, the rms current in an RLC circuit is 5.8 A. If the rms
voltage of the generator is 120 V, what is the resistance, R?

João Gabriel A.
Numerade Educator

Problem 62

The resistance in an RLC circuit is doubled. (a) Does the
resonance frequency increase, decrease, or stay the same? Explain.
(b) Does the maximum current in the circuit increase, decrease, or
stay the same? Explain

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Problem 63

The capacitive reactance in an RLC circuit is determined to be 45$\Omega$ when the inductive reactance is 65$\Omega .$ (a) Is the generator frequency $\omega$ higher or lower than the natu-
ral frequency $\omega_{0} ?$ (b) Is the phase angle $\phi$ positive or negative?
(c) Calculate the phase angle $\phi$ if $R=33 \Omega .$

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Problem 64

The capacitive reactance in an RLC circuit is determined to be 31$\Omega$ when the inductive reactance is 16$\Omega$ . (a) Is the generator frequency $\omega$ higher or lower than the natu-
ral frequency $\omega_{0} ?$ (b) Is the phase angle $\phi$ positive or negative?
(c) Calculate the phase angle $\phi$ if $R=12 \Omega .$

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Problem 65

A $115-\Omega$ resistor, a 67.6 -mH inductor, and a $189-\mu$ F capacitor are connected in series to an ac generator. (a) At what frequency will the current in the circuit be a maximum? (b) At what frequency will the impedance of the circuit be a minimum?

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Problem 66

(a) Find the frequency at which an 18 -\muF capacitor has the same reactance as an $18-\mathrm{mH}$ inductor. (b) What is the reso-
nance frequency of an $L C$ circuit made with this inductor and
capacitor?

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Problem 67

Consider an $R L C$ circuit with $R=105.2, L=518 \mathrm{mH},$ and $C=0.200 \mu F .($ a) At what frequency is this circuit in resonance?
(b) Find the impedance of this circuit if the frequency has the
value found in part (a), but the capacitance is increased to
0.220$\mu \mathrm{F} .$ (c) What is the power factor for the situation described
in part (b)?

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Problem 68

An RLC circuit has a resonance frequency of 155 Hz. (a) If both $L$ and $C$ are doubled, does the resonance frequency increase, decrease, or stay the same? Explain. (b) Find the resonance frequency when $L$ and $C$ are doubled.

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Problem 69

An $R L C$ circuit has a capacitance of 0.29$\mu \mathrm{F}$ . (a) What inductance will produce a resonance frequency of 95 $\mathrm{MHz}$ ? (b) It is
desired that the impedance at resonance be one-fifth the impedance at 11 $\mathrm{kHz}$ . What value of $R$ should be used to obtain this
result?

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Problem 70

Although an incandescent lightbulb
appears to shine with constant intensity, this is an artifact of the
eye's persistence of vision. In fact, the intensity of a bulb's light
rises and falls with time due to the alternating current used in
household circuits. If you could perceive these oscillations, would
you see the light attain maximum brightness 60 or 120 times per
second? Explain.

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Problem 71

An RLC circuit is driven at its resonance frequency. Is its
impedance greater than, less than, or equal to R? Explain.

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Problem 72

Suppose the circuits shown in Figures 24-33
and 24-35 are connected to identical batteries, rather than to ac
generators. (a) Assuming the value of R is the same in the two circuits, is the current in Figure 24-33 greater than, less than, or the same as the current in Figure 24-35? (b) Choose the best explanation from among the following:
I. The circuits have the same current because the capacitor
acts like an open circuit and the inductor acts like a short
circuit.
II. The current in Figure 24-33 is larger because it has more circuit elements, each of which can carry current.
III. The current in Figure 24-35 is larger because it has fewer circuit elements, meaning less resistance to current flow.

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Problem 73

Consider a circuit consisting of a lightbulb
and a capacitor, as shown in circuit 2 of Conceptual Example 24-
10. (a) If the frequency of the generator is increased, does the
intensity of the lightbulb increase, decrease, or stay the same?
(b) Choose the best explanation from among the following:
I. As the frequency increases it becomes harder to force current through the capacitor, and therefore the intensity of the
lightbulb decreases.
II. The intensity of the lightbulb increases because as the frequency becomes higher the capacitor acts more like a short
circuit, allowing more current to flow.
III. The intensity of the lightbulb is independent of frequency
because the circuit contains a capacitor but not an inductor.

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Problem 74

Consider a circuit consisting of a lightbulb and an inductor,
as shown in Conceptual Example 24-14. If the frequency of the
generator is increased, does the intensity of the lightbulb increase,
decrease, or stay the same? Explain

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Problem 75

A $4.40-\mu \mathrm{F}$ and an $8.80-\mu \mathrm{F}$ capacitor are connected in parallel to a $60.0-$ Hz generator operating with an rms voltage of 115 $\mathrm{V}$ . What is
the rms current supplied by the generator?

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Problem 76

A $4.40-\mu \mathrm{F}$ and an $8.80-\mu \mathrm{F}$ capacitor are connected in series to a
60.0 -Hz generator operating with an rms voltage of 115 $\mathrm{V}$ . What is
the rms current supplied by the generator?

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Problem 77

A $12.5-\mu \mathrm{F}$ capacitor and a $47.5-\mu \mathrm{F}$ capacitor are connected to an ac generator with a frequency of 60.0 $\mathrm{Hz}$ . What is the capaci-
tive reactance of this pair of capacitors if they are connected (a) in
parallel or ( b) in series?

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Problem 78

A generator drives an RLC circuit with the voltage V shown
in FIGURE 24-37. The corresponding current I is also shown in the
figure. (a) Is the inductive reactance of this circuit greater than,
less than, or equal to its capacitive reactance? Explain. (b) Is the
frequency of this generator greater than, less than, or equal to the
resonance frequency of the circuit? Explain.

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Problem 79

Trapped on a deserted island, you salvage some copper wire and
a generator to build a makeshift radio transmitter. (a) If you wrap
the wire into an 8-turn coil of radius 3.8 cm and length 25 cm, what
is its inductance? (b) What capacitance do you need to make an
RLC circuit with the coil that oscillates at the emergency beacon
frequency of 406 MHz? (c) If you fashion the parallel-plate capacitor out of two aluminum squares, 5.0 cm on a side,

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Problem 80

When a certain resistor is connected to an ac generator with a maximum voltage of 15 $\mathrm{V}$ , the average power dissipated in the resistor is 22 $\mathrm{W}$ . (a) What is the resistance of the
resistor? (b) What is the rms current in the circuit? (c) We know
that $P_{\mathrm{av}}=I_{\mathrm{rms}}^{2} R,$ and hence it seems that reducing the resistance should reduce the average power. On the other hand, we also
know that $P_{\mathrm{av}}=V_{\mathrm{rms}}^{2} / R,$ which suggests that reducing $R$ increases $P_{\mathrm{av}} .$ Which conclusion is correct? Explain.

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Problem 81

Find the average power consumed by an $R C$ circuit connected to a 60.0 -Hz generator with an rms voltage of 172 $\mathrm{V}$ . The values of $R$ and $C$ in this circuit are 6.30 $\mathrm{k} \Omega$ and 2.05$\mu \mathrm{F}$ , respectively.

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Problem 82

A 1.15 -k\Omega resistor and a $505-\mathrm{mH}$ inductor are connected in series to a 1250 -Hz generator with an rms voltage of 14.2 $\mathrm{V}$ . (a) What
is the rms current in the circuit? (b) What capacitance must be
inserted in series with the resistor and inductor to reduce the rms
current to half the value found in part (a)?

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Problem 83

The phasor diagram for an RLC circuit is shown in FlGURE $24-38$ . (a) If the resistance in this circuit
is $525 \Omega,$ what is the impedance? (b) If the frequency in this circuit is increased, will the impedance increase, or stay the same? Explain.

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Problem 84

Figure 24-39 shows the phasor diagram for an $R L C$ circuit in which the impedance is 337$\Omega .$ (a) What is the resistance, $R,$ in this circuit? (b) Is this circuit driven at a frequency
that is greater than, less than, or equal to the resonance frequency
of the circuit? Explain.

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Problem 85

An $R L C$ circuit has a resistance $R=25 \Omega$ and an inductance $L=160 \mathrm{mH}$ , and is connected to an ac generator with a frequency of 55 $\mathrm{Hz}$ . The phasor diagram for this circuit is shown in Figure $24-39 .$ Find (a) the impedance, $Z,$ and (b) the capacitance, $C,$ for this circuit. (c) If the value of $C$ is decreased, will the impedance of the circuit increase, decrease, or stay the same? Explain.

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Problem 86

You are given a sealed box with two electrical terminals. The box contains a $5.00-\Omega$
resistor in series with either an inductor or a capacitor. When you
attach an ac generator with an rms voltage of 0.750 $\mathrm{V}$ to the terminals of the box, you find that the current increases with increasing frequency. (a) Does the box contain an inductor or a capacitor? Explain. (b) When the frequency of the generator is 25.0 $\mathrm{kHz}$ , the
rms current is 87.2 $\mathrm{mA}$ . What is the capacitance or inductance of
the unknown component in the box?

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Problem 87

An $R L C$ circuit with $R=20.0 \Omega, L=295 \mathrm{mH},$ and $C=49.7 \mu \mathrm{F}$ is connected to an ac generator with an rms voltage of 21.5 $\mathrm{V}$ . Determine the average power delivered to this circuit when the frequency of the generator is (a) equal to the resonance frequency, (b) twice the resonance frequency, and (c) half the resonance frequency.

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Problem 88

The intensity of a lightbulb with a resistance of 120$\Omega$ is controlled by connecting it in series with an inductor whose inductance can be varied from $L=0$ to $L=L_{\max }$ . This "light dimmer" circuit is connected to an ac generator with a frequency of 60.0 $\mathrm{Hz}$ and an rms voltage of 110 $\mathrm{V}$ . (a) What is the average power dissipated in the lightbulb when $L=0$ ? (b) The inductor is now adjusted so that $L=L_{\text { mat }}$ . In this case, the average power dissipated in the lightbulb is one-fourth the value found in part (a). What is the value of $L_{\max } ?$

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Problem 89

An electric motor with a resistance of 15$\Omega$ and an inductance of 53 $\mathrm{mH}$ is connected to a $60.0-\mathrm{Hz}$ ac generator. (a) What is the power factor for this circuit? (b) In order to increase the power factor of this circuit to $0.80,$ a capacitor is connected in series with the
motor and inductor. Find the required value of the capacitance.

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Problem 90

A radio tuning circuit contains an $R L C$ circuit with $R=5.0 \Omega$ and $L=2.8 \mu \mathrm{H}$ (a) What capacitance is needed to produce a resonance frequency of 85 $\mathrm{MHz}$ ?
(b) If the capacitance is increased above the value found in part M(a), will the impedance increase, decrease, or stay the same? Explain. (c) Find the impedance of the circuit at resonance. (d) Find the impedance of the circuit when the capacitance is 1$\%$ higher than the value found in part (a).

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Problem 91

If the maximum voltage in the square wave shown in Figure $24-31$ is $V_{\max },$ what are (a) the average voltage, $V_{\text { av }},$ and (b) the rms voltage, $V_{\mathrm{rms}}$ ?

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Problem 92

An ac generator supplies an rms voltage of 5.00 V to an RL circuit. At a frequency of 20.0 $\mathrm{kHz}$ the rms current in the circuit is $45.0 \mathrm{mA} ;$ at a frequency of 25.0 $\mathrm{kHz}$ the rms current is 40.0 $\mathrm{mA}$ . What are the values of $R$ and $L$ in this circuit?

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Problem 93

An $R C$ circuit consists of a resistor $R=32 \Omega,$ a capacitor $C=25 \mu \mathrm{F},$ and an ac generator with an rms voltage of 120 $\mathrm{V}$ .
(a) At what frequency will the rms current in this circuit be 2.9 $\mathrm{A}$ ?
For this frequency, what are (b) the rms voltage across the resistor, $V_{\mathrm{rms}, R},$ and $(\mathrm{c})$ the rms voltage across the capacitor, $V_{\mathrm{mss}} c^{2}(\mathrm{d})$ Show that $V_{\operatorname{mss}, R}+V_{\text { rms }, C}>120 \mathrm{V}$ but that $\sqrt{V_{\mathrm{rms}, R}^{2}+V_{\mathrm{rms}, c}^{2}}=120 \mathrm{V}$

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Problem 94

At the measured frequency, what is the ratio of the capacitive
reactance of a typical clavus sample to that of verruca?
\begin{equation}\begin{array}{llll}{\text { A. } 0.21} & {\text { B.4.9 }} & {\text { C. } 16} & {\text { D. } 110}\end{array}\end{equation}

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Problem 95

What is the value of the capacitive reactance of the typical clavus sample at 80 $\mathrm{Hz}$ ?
\begin{equation}\begin{array}{lllll}{\text { A. } 96 \Omega} & {\text { B. } 10 \mathrm{k} \Omega} & {\text { C. } 550 \mathrm{k} \Omega} & {\text { D. } 1.7 \mathrm{MS}}\end{array}\end{equation}

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Problem 96

If the sample behaves in a way that can be modeled by a series $R C$ circuit, what is the expected value of the impedance for the typical clavus sample at this frequency?
\begin{equation}\begin{array}{llll}{\text { A. } 96 \Omega} & {\text { B. } 2000 \Omega} & {\text { C.2.0M\Omega }} & {\text { D. } 10 \mathrm{MQ}}\end{array}\end{equation}

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Problem 97

If the sample behaves in a way that can be modeled by a series $R C$ circuit, what is the expected value of the phase angle for the typical clavus sample at this frequency?
\begin{equation}\begin{array}{llll}{\text { A. }-6.0^{\circ}} & {\text { B. }-57^{\circ}} & {\text { C. }-84^{\circ}} & {\text { D. }+33^{\circ}}\end{array}\end{equation}

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Problem 98

Suppose we would like to change the phase angle for this circuit to $\phi=-25.0^{\circ},$ and
we would like to accomplish this by changing the resistor to a
value other than 175$\Omega$ . The inductor is still 90.0 $\mathrm{mH}$ , the capacitor is 15.0$\mu \mathrm{F}$ , the rms voltage is $120.0 \mathrm{V},$ and the ac frequency
is 60.0 $\mathrm{Hz}$ . (a) Should the resistance be increased or decreased?
Explain. (b) Find the resistance that gives the desired phase angle. (c) What is the rms current in the circuit with the resistance found
in part (b)?

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Problem 99

You plan to change the frequency of the generator in this circuit to produce a
phase angle of smaller magnitude. The resistor is still 175$\Omega$ , the
inductor is 90.0 $\mathrm{mH}$ , the capacitor is 15.0$\mu \mathrm{F}$ , and the rms volt-
age is 120.0 $\mathrm{V}$ (a) Should you increase or decrease the frequency?
Explain. (b) Find the frequency that gives a phase angle of $-22.5^{\circ} .$ (c) What is the rms current in the circuit at the frequency found in part (b)?

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