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Physics

John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler

Chapter 23

Alternating Current Circuits - all with Video Answers

Educators


Chapter Questions

03:43

Problem 1

A $63.0-\mu \mathrm{F}$ capacitor is connected to a generator operating at a low frequency. The rms voltage of the generator is $4.00 \mathrm{V}$ and is constant. A fuse in series with the capacitor has negligible resistance and will burn out when the rms current reaches $15.0 \mathrm{A}$. As the generator frequency is increased, at what frequency will the fuse burn out?

Breanna Kloczkowski
Breanna Kloczkowski
Numerade Educator
03:18

Problem 2

Two identical capacitors are connected in parallel to an ac generator that has a frequency of $610 \mathrm{Hz}$ and produces a voltage of $24 \mathrm{V}$. The current in the circuit is 0.16 A. What is the capacitance of each capacitor?

Breanna Kloczkowski
Breanna Kloczkowski
Numerade Educator
04:27

Problem 3

The reactance of a capacitor is $68 \Omega$ when the ac frequency is $460 \mathrm{Hz}$. What is the reactance when the frequency is $870 \mathrm{Hz}$ ?

Breanna Kloczkowski
Breanna Kloczkowski
Numerade Educator
01:56

Problem 4

A capacitor is connected to an ac generator that has a frequency of $3.4 \mathrm{kHz}$ and produces a voltage of $2.0 \mathrm{V}$. The current in the capacitor is $35 \mathrm{mA}$. When the same capacitor is connected to a second ac generator that has a frequency of $5.0 \mathrm{kHz},$ the current in the capacitor is $85 \mathrm{mA}$. What voltage does the second generator produce?

Luis Mendoza
Luis Mendoza
Numerade Educator
04:53

Problem 5

A capacitor is connected across the terminals of an ac generator that has a frequency of $440 \mathrm{Hz}$ and supplies a voltage of $24 \mathrm{V} .$ When a second capacitor is connected in parallel with the first one, the current from the generator increases by 0.18 A. Find the capacitance of the second capacitor.

Breanna Kloczkowski
Breanna Kloczkowski
Numerade Educator
07:37

Problem 6

Two parallel plate capacitors are identical, except that one of them is empty and the other contains a material with a dielectric constant of 4.2 in the space between the plates. The empty capacitor is connected between the terminals of an ac generator that has a fixed frequency and rms voltage. The generator delivers a current of 0.22 A. What current does the generator deliver after the other capacitor is connected in parallel with the first one?

Breanna Kloczkowski
Breanna Kloczkowski
Numerade Educator
03:57

Problem 7

A capacitor is connected across an ac generator whose frequency is $750 \mathrm{Hz}$ and whose peak output voltage is $140 \mathrm{V} .$ The rms current in the circuit is $3.0 \mathrm{A}$. (a) What is the capacitance of the capacitor? (b) What is the magnitude of the maximum charge on one plate of the capacitor?

Breanna Kloczkowski
Breanna Kloczkowski
Numerade Educator
03:52

Problem 8

A capacitor (capacitance $C_{1}$ ) is connected across the terminals of an ac generator. Without changing the voltage or frequency of the generator, a second capacitor (capacitance $C_{2}$ ) is added in series with the first one. As a result, the current delivered by the generator decreases by a factor of three. Suppose that the second capacitor had been added in parallel with the first one, instead of in series. By what factor would the current delivered by the generator have increased?

Supratim Pal
Supratim Pal
Numerade Educator
02:54

Problem 9

An $8.2-\mathrm{mH}$ inductor is connected to an ac generator $(10.0 \mathrm{V} \mathrm{rms},$ $620 \mathrm{Hz}$ ). Determine the peak value of the current supplied by the generator.

Vishal Gupta
Vishal Gupta
Numerade Educator
01:31

Problem 10

An inductor has an inductance of $0.080 \mathrm{H}$. The voltage across this inductor is $55 \mathrm{V}$ and has a frequency of $650 \mathrm{Hz}$. What is the current in the inductor?

Breanna Kloczkowski
Breanna Kloczkowski
Numerade Educator
00:57

Problem 11

An inductor is to be connected to the terminals of a generator (rms voltage $=15.0 \mathrm{V})$ so that the resulting rms current will be $0.610 \mathrm{A} .$ Determine the required inductive reactance.

Breanna Kloczkowski
Breanna Kloczkowski
Numerade Educator
02:39

Problem 12

An ac generator has a frequency of $7.5 \mathrm{kHz}$ and a voltage of $39 \mathrm{V}$. When an inductor is connected between the terminals of this generator, the current in the inductor is $42 \mathrm{mA}$. What is the inductance of the inductor? $?$

Breanna Kloczkowski
Breanna Kloczkowski
Numerade Educator
03:21

Problem 13

A $40.0-\mu \mathrm{F}$ capacitor is connected across a $60.0-\mathrm{Hz}$ generator. An inductor is then connected in parallel with the capacitor. What is the value of the inductance if the rms currents in the inductor and capacitor are equal?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:29

Problem 14

An ac generator has a frequency of $2.2 \mathrm{kHz}$ and a voltage of $240 \mathrm{V} .$ An inductance $L_{1}=6.0 \mathrm{mH}$ is connected across its terminals. Then a second inductance $L_{2}=9.0 \mathrm{mH}$ is connected in parallel with $L_{1} .$ Find the current that the generator delivers to $L_{1}$ and to the parallel combination.

Luis Mendoza
Luis Mendoza
Numerade Educator
01:49

Problem 15

A $30.0-\mathrm{mH}$ inductor has a reactance of $2.10 \mathrm{k} \Omega .$ (a) What is the frequency of the ac current that passes through the inductor? (b) What is the capacitance of a capacitor that has the same reactance at this frequency? The frequency is tripled, so that the reactances of the inductor and capacitor are no longer equal. What are the new reactances of (c) the inductor and (d) the capacitor?

Luis Mendoza
Luis Mendoza
Numerade Educator
01:54

Problem 16

Available in WileyPLUS.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:18

Problem 17

A series $\mathrm{RCL}$ circuit includes a resistance of $275 \Omega,$ an inductive reactance of $648 \Omega,$ and a capacitive reactance of $415 \Omega .$ The current in the circuit is 0.233 A. What is the voltage of the generator?

Luis Mendoza
Luis Mendoza
Numerade Educator
01:44

Problem 18

A series $\mathrm{RCL}$ circuit contains a $47.0-\Omega$ resistor, a $2.00-\mu \mathrm{F}$ capacitor, and a $4.00-\mathrm{mH}$ inductor. When the frequency is $2550 \mathrm{Hz},$ what is the power factor of the circuit?

Luis Mendoza
Luis Mendoza
Numerade Educator
01:26

Problem 19

Multiple-Concept Example 3 reviews some of the basic ideas that are pertinent to this problem. A circuit consists of a $215-\Omega$ resistor and a $0.200-\mathrm{H}$ inductor. These two elements are connected in series across a generator that has a frequency of $106 \mathrm{Hz}$ and a voltage of $234 \mathrm{V}$.
(a) What is the current in the circuit? (b) Determine the phase angle between the current and the voltage of the generator.

Luis Mendoza
Luis Mendoza
Numerade Educator
04:31

Problem 20

An ac series circuit has an impedance of $192 \Omega,$ and the phase angle between the current and the voltage of the generator is $\phi=-75^{\circ} .$ The circuit contains a resistor and either a capacitor or an inductor. Find the resistance $R$ and the capacitive reactance $X_{\mathrm{C}}$ or the inductive reactance $X_{\mathrm{L}}$ whichever is appropriate.

Vishal Gupta
Vishal Gupta
Numerade Educator
01:31

Problem 21

When only a resistor is connected across the terminals of an ac generator (112 V) that has a fixed frequency, there is a current of $0.500 \mathrm{A}$ in the resistor. When only an inductor is connected across the terminals of this same generator, there is a current of $0.400 \mathrm{A}$ in the inductor. When both the resistor and the inductor are connected in series between the terminals of this generator, what are (a) the impedance of the series combination and (b) the phase angle between the current and the voltage of the generator?

Luis Mendoza
Luis Mendoza
Numerade Educator
01:05

Problem 22

A $2700-\Omega$ resistor and a $1.1-\mu$ F capacitor are connected in series across a generator $(60.0 \mathrm{Hz}, 120 \mathrm{V}) .$ Determine the power delivered to the circuit.

Luis Mendoza
Luis Mendoza
Numerade Educator
01:54

Problem 23

Available in WileyPLUS.

Ajay Singhal
Ajay Singhal
Numerade Educator
03:50

Problem 24

Part $a$ of the drawing shows a resistor and a charged capacitor wired in series. When the switch is closed, the capacitor discharges as charge moves from one plate to the other. Part $b$ shows the amount $q$ of charge remaining on each plate of the capacitor as a function of time. In part $c$ of the drawing, the switch has been removed and an ac generator has been inserted into the circuit. The circuit elements in the drawing have the following values: $R=18 \Omega, V_{\mathrm{rms}}=24 \mathrm{V}$ for the generator, and $f=380 \mathrm{Hz}$ for the generator. The time constant for the circuit in part $a$ is $\tau=3.0 \times 10^{-4} \mathrm{s} .$ What is the rms current in the circuit in part $c ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
01:52

Problem 25

A circuit consists of an $85-\Omega$ resistor in series with a $4.0-\mu \mathrm{F}$ capacitor, and the two are connected between the terminals of an ac generator. The voltage of the generator is fixed. At what frequency is the current in the circuit one-half the value that exists when the frequency is very large?

Luis Mendoza
Luis Mendoza
Numerade Educator
04:33

Problem 26

In one measurement of the body's bioelectric impedance, values of $Z=4.50 \times 10^{2} \Omega$ and $\phi=-9.80^{\circ}$ are obtained for the total impedance and the phase angle, respectively. These values assume that the body's resistance $R$ is in series with its capacitance $C$ and that there is no inductance $L .$ Determine the body's resistance and capacitive reactance.

Vishal Gupta
Vishal Gupta
Numerade Educator
01:54

Problem 27

Available in WileyPLUS.

Ajay Singhal
Ajay Singhal
Numerade Educator
00:56

Problem 28

An $84.0-\mathrm{mH}$ inductor and a $5.80-\mu \mathrm{F}$ capacitor are connected in series with a generator whose frequency is 375 Hz. The rms voltage across the capacitor is $2.20 \mathrm{V}$. Determine the rms voltage across the inductor.

Luis Mendoza
Luis Mendoza
Numerade Educator
01:54

Problem 29

Available in WileyPLUS.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:19

Problem 30

A tank circuit in a radio transmitter is a series RCL circuit connected to an antenna. The antenna broadcasts radio signals at the resonant frequency of the tank circuit. Suppose that a certain tank circuit in a shortwave radio transmitter has a fixed capacitance of $1.8 \times 10^{-11} \mathrm{F}$ and a variable inductance. If the antenna is intended to broadcast radio signals ranging in frequency from 4.0 MHz to 9.0 MHz, find the (a) minimum and (b) maximum inductance of the tank circuit.

Luis Mendoza
Luis Mendoza
Numerade Educator
00:48

Problem 31

A series RCL circuit has a resonant frequency of $690 \mathrm{kHz}$. If the value of the capacitance is $2.0 \times 10^{-9} \mathrm{F},$ what is the value of the inductance?

Luis Mendoza
Luis Mendoza
Numerade Educator
01:12

Problem 32

The power dissipated in a series RCL circuit is $65.0 \mathrm{W}$, and the current is 0.530 A. The circuit is at resonance. Determine the voltage of the generator.

Luis Mendoza
Luis Mendoza
Numerade Educator
01:18

Problem 33

A $10.0-\Omega$ resistor, a $12.0-\mu \mathrm{F}$ capacitor, and a $17.0-\mathrm{mH}$ inductor are connected in series with a $155-\mathrm{V}$ generator. (a) At what frequency is the current a maximum?
(b) What is the maximum value of the rms current?

Luis Mendoza
Luis Mendoza
Numerade Educator
02:02

Problem 34

The capacitance in a series $\mathrm{RCL}$ circuit is $C_{1}=2.60 \mu \mathrm{F},$ and the corresponding resonant frequency is $f_{01}=7.30 \mathrm{kHz} .$ The generator frequency is $5.60 \mathrm{kHz}$. What is the value of the capacitance $C_{2}$ that should be added to the circuit so that the circuit will have a resonant frequency that matches the generator frequency? Note that you must decide whether $C_{2}$ is added in series or in parallel with $C_{1}$.

Luis Mendoza
Luis Mendoza
Numerade Educator
01:13

Problem 35

A series $\mathrm{RCL}$ circuit is at resonance and contains a variable resistor that is set to $175 \Omega$. The power delivered to the circuit is $2.6 \mathrm{W}$. Assuming that the voltage remains constant, how much power is delivered when the variable resistor is set to $562 \Omega ?$

Luis Mendoza
Luis Mendoza
Numerade Educator
01:04

Problem 36

The resonant frequency of an $\mathrm{RCL}$ circuit is $1.3 \mathrm{kHz},$ and the value of the inductance is $7.0 \mathrm{mH}$. What is the resonant frequency (in $\mathrm{kHz}$ ) when the value of the inductance is $1.5 \mathrm{mH} ?$

Luis Mendoza
Luis Mendoza
Numerade Educator
01:42

Problem 37

A series $\mathrm{RCL}$ circuit has a resonant frequency of $1500 \mathrm{Hz}$. When operating at a frequency other than $1500 \mathrm{Hz},$ the circuit has a capacitive reactance of $5.0 \Omega$ and an inductive reactance of $30.0 \Omega .$ What are the values of $(\mathrm{a}) L$ and $(\mathrm{b}) C ?$

Luis Mendoza
Luis Mendoza
Numerade Educator
01:24

Problem 38

In a series $\mathrm{RCL}$ circuit the generator is set to a frequency that is not the resonant frequency. This nonresonant frequency is such that the ratio of the inductive reactance to the capacitive reactance of the circuit is observed to be 5.36. The resonant frequency is 225 Hz. What is the frequency of the generator?

Luis Mendoza
Luis Mendoza
Numerade Educator
01:54

Problem 39

Available in WileyPLUS.

Ajay Singhal
Ajay Singhal
Numerade Educator
04:41

Problem 40

A charged capacitor and an inductor are connected as shown in the drawing (this circuit is the same as that in Animated Figure 23.16a). There is no resistance in the circuit. As Section 23.4 discusses, the electrical energy initially present in the charged capacitor then oscillates back and forth between the inductor and the capacitor. The initial charge on the capacitor has a magnitude of $q=$ $2.90 \mu \mathrm{C} .$ The capacitance is $C=3.60 \mu \mathrm{F},$ and the inductance is $L=75.0 \mathrm{mH}$
(a) What is the electrical energy stored initially by the charged capacitor?
(b) Find the maximum current in the inductor.

Vishal Gupta
Vishal Gupta
Numerade Educator
05:57

Problem 41

In the absence of a nearby metal object, the two inductances $\left(L_{\mathrm{A}}\right.$ and $\left.L_{\mathrm{B}}\right)$ in a heterodyne metal detector are the same, and the resonant frequencies of the two oscillator circuits have the same value of $630.0 \mathrm{kHz}$ When the search coil (inductor $\mathrm{B}$ ) is brought near a buried metal object, a beat frequency of $7.30 \mathrm{kHz}$ is heard. By what percentage does the buried object increase the inductance of the search coil?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:54

Problem 42

Available in WileyPLUS.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:54

Problem 43

Available in WileyPLUS.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:20

Problem 44

A circuit consists of a resistor in series with an inductor and an ac generator that supplies a voltage of 115 V. The inductive reactance is $52.0 \Omega$, and the current in the circuit is 1.75 A. Find the average power delivered to the circuit.

Luis Mendoza
Luis Mendoza
Numerade Educator
01:30

Problem 45

In a series circuit, a generator $(1350 \mathrm{Hz}, 15.0 \mathrm{V})$ is connected to a $16.0-\Omega$ resistor, a $4.10-\mu \mathrm{F}$ capacitor, and a $5.30-\mathrm{mH}$ inductor. Find the voltage across each circuit element.

Luis Mendoza
Luis Mendoza
Numerade Educator
01:04

Problem 46

Review Conceptual Example 5 and Figure $23.13 .$ Find the ratio of the current in circuit I to the current in circuit II in the high-frequency limit for the same generator voltage.

Luis Mendoza
Luis Mendoza
Numerade Educator
01:01

Problem 47

At what frequency (in $\mathrm{Hz}$ ) are the reactances of a $52-\mathrm{mH}$ inductor and a $76-\mu$ F capacitor equal?

Luis Mendoza
Luis Mendoza
Numerade Educator
01:04

Problem 48

The resistor in a series $\mathrm{RCL}$ circuit has a resistance of $92 \Omega$ while the voltage of the generator is $3.0 \mathrm{V} .$ At resonance, what is the average power delivered to the circuit?

Luis Mendoza
Luis Mendoza
Numerade Educator
01:22

Problem 49

Two ac generators supply the same voltage. However, the first generator has a frequency of $1.5\mathrm{kHz}$, and the second has a frequency of $6.0 \mathrm{kHz}$. When an inductor is connected across the terminals of the first generator, the current delivered is 0.30 A. How much current is delivered when this
inductor is connected across the terminals of the second generator?

Luis Mendoza
Luis Mendoza
Numerade Educator
01:15

Problem 50

A series circuit contains only a resistor and an inductor. The voltage $V$ of the generator is fixed. If $R=16 \Omega$ and $L=4.0 \mathrm{mH},$ find the frequency at which the current is one-half its value at zero frequency.

Luis Mendoza
Luis Mendoza
Numerade Educator
01:40

Problem 51

A series RCL circuit contains a $5.10-\mu \mathrm{F}$ capacitor and a generator whose voltage is $11.0 \mathrm{V}$. At a resonant frequency of $1.30 \mathrm{kHz}$ the power delivered to the circuit is $25.0 \mathrm{W}$. Find the values of (a) the inductance and (b) the resistance. (c) Calculate the power factor when the generator frequency is $2.31 \mathrm{kHz}$

Luis Mendoza
Luis Mendoza
Numerade Educator
01:54

Problem 52

Available in WileyPLUS.

Ajay Singhal
Ajay Singhal
Numerade Educator
05:30

Problem 53

Part $a$ of the figure shows a heterodyne metal detector being used. As part $b$ of the figure illustrates, this device utilizes two capacitor/ inductor oscillator circuits, A and B. Each produces its own resonant frequency, $f_{0 \mathrm{A}}=1 /\left[2 \pi\left(L_{\mathrm{A}} C\right)^{1 / 2}\right]$ and $f_{0 \mathrm{B}}=1 /\left[2 \pi\left(L_{\mathrm{B}} C\right)^{1 / 2}\right]$. Any difference between these frequencies is detected through earphones as a beat frequency $\mid f_{0 \mathrm{B}}-$ $f_{0 A} \mid .$ In the absence of any nearby metal object, the inductances $L_{\mathrm{A}}$ and $L_{\mathrm{B}}$ are identical. When inductor $\mathrm{B}$ (the search coil) comes near a piece of metal, the inductance $L_{\mathrm{B}}$ increases, the corresponding oscillator frequency $f_{\mathrm{oB}}$ decreases, and a beat frequency is heard. Suppose that initially each inductor is adjusted so that $L_{\mathrm{B}}=L_{\mathrm{A}},$ and each oscillator has a resonant frequency of $855.5 \mathrm{kHz}$. Assuming that the inductance of search coil $\mathrm{B}$ increases by $1.000 \%$ due to a nearby piece of metal, determine the beat frequency heard through the earphones.

Vishal Gupta
Vishal Gupta
Numerade Educator
08:45

Problem 54

Two parallel plate capacitors are filled with the same dielectric material and have the same plate area. However, the plate separation of capacitor 1 is twice that of capacitor 2. When capacitor 1 is connected across the terminals of an ac generator, the generator delivers an rms current of 0.60 A. Concepts: (i) Which of the two capacitors has the greater capacitance? (ii) Is the equivalent capacitance of the parallel combination $\left(C_{\mathrm{P}}\right)$ greater or smaller than the capacitance of capacitor $1 ?$ (iii) Is the capacitive reactance of $C_{\mathrm{P}}$ greater or smaller than for $C_{1} ?$ (iv) When both capacitors are connected in parallel across the terminals of the generator, is the current from the generator greater or smaller than when capacitor 1 is connected alone? Calculations: What is the current delivered by the generator when both capacitors are connected in parallel across the terminals?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:54

Problem 55

An ac generator has a frequency of $1200 \mathrm{Hz}$ and a constant rms voltage. When a $470-\Omega$ resistor is connected between the terminals of the generator, an average power of $0.25 \mathrm{W}$ is consumed by the resistor. Then, a 0.080-H inductor is connected in series with the resistor, and the combination is connected between the generator terminals. Concepts: (i) In which case does the generator deliver a greater rms current? (ii) In which case is the greater average power consumed by the circuit? Calculations: What is the average power consumed in the inductor-resistor series circuit?

Vishal Gupta
Vishal Gupta
Numerade Educator
06:04

Problem 56

Radio jamming is the intentional disruption or interference of radio communications by overwhelming the intended receivers of the signal with random noise. You and your team have been tasked with jamming a specific radio signal at $720 \mathrm{kHz}$. You have access to a high-powered transmitter, but the part of its circuitry that tunes the broadcast frequency, called the tank circuit, has been damaged. A tank circuit is a series RCL circuit whose resonance frequency determines the frequency broadcasted by the antenna. At your disposal are two $220-\Omega$ resistors, one variable capacitor that ranges from 2.0 to $6.0 \mathrm{nF},$ and four inductors with the following values: $L_{1}=5.0 \times 10^{-6} \mathrm{H}, L_{2}=7.2 \times 10^{-6} \mathrm{H}, L_{3}=6.5 \times 10^{-5} \mathrm{H}$ and $L_{4}=5.4 \times 10^{-6} \mathrm{H} .$ (a) If you set your variable capacitor at the center of its range, what must be the value of the inductance of your RCL circuit so that it resonates at $720 \mathrm{kHz}$ ? (b) How should you configure the available inductors to give you the needed equivalent inductance? (Hint: the rules for adding inductors in series and parallel are the same as for resistors.) (c) With the inductance set as calculated in (a), what resonant frequency range does the variable capacitor provide? (d) The two resistors can be configured to give different equivalent resistance values. How should you configure the resistors in the RCL circuit in order to maximize the current at the resonant frequency? (Refer to Section 23.4.)

Suhas Katkar
Suhas Katkar
Numerade Educator
06:13

Problem 57

A series $\mathrm{RCL}$ circuit is composed of a resistor $(R=220 \Omega),$ two identical capacitors $(C=3.50 \mathrm{nF})$ connected in series, and two identical inductors $\left(L=5.50 \times 10^{-5} \mathrm{H}\right)$ connected in series. You and your team need to determine: (a) the resonant frequency of this configuration.
(b) What are all of the other possible resonant frequencies that can be attained by reconfiguring the capacitors and inductors (while using all of the components and keeping the proper series RCL order)? (c) If you were to design a circuit using only one of the given inductors and one adjustable capacitor, what would the range of the variable capacitor need to be in order to cover all of the resonant frequencies found in (a) and (b)?

Suhas Katkar
Suhas Katkar
Numerade Educator