College Physics

Hugh D. Young Philip W. Adams

Chapter 22

Alternating Current - all with Video Answers

Educators

Chapter Questions

Problem 1

You have a special light bulb with a very delicate wire filament. The wire will break if the current in it ever exceeds $1.50 \mathrm{~A},$ even for an instant. What is the largest root-mean-square current you can run through this bulb?

Ze-Han Lee

Ze-Han Lee

Numerade Educator

Problem 2

An electric motor is being powered with a voltage amplitude of $310 \mathrm{~V}$ at $60.0 \mathrm{~Hz}$. The motor draws a current amplitude of $10.0 \mathrm{~A}$. Find (a) the root-mean-square voltage, (b) the root-mean-square current, and (c) the average power consumed by the motor.

Vishal Gupta

Numerade Educator

Problem 3

A capacitance $C$ and an inductance $L$ are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If $L=5.00 \mathrm{mH}$ and $C=3.50 \mu \mathrm{F}$, what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?

Averell Hause

Carnegie Mellon University

Problem 4

(a) Compute the reactance of a $0.450 \mathrm{H}$ inductor at frequencies of $60.0 \mathrm{~Hz}$ and $600 \mathrm{~Hz}$. (b) Compute the reactance of a $2.50 \mu \mathrm{F}$ capacitor at the same frequencies. (c) At what frequency is the reactance of a $0.450 \mathrm{H}$ inductor equal to that of a $2.50 \mu \mathrm{F}$ capacitor?

Mohit Khurana

Texas A&M University

Problem 5

You are designing an amplifier circuit that will operate in the frequency range from $20 \mathrm{~Hz}$ to $20,000 \mathrm{~Hz}$. For the design to work, the reactance of a particular inductor in the circuit cannot exceed $100 \Omega$. What is the largest inductance that can be used?

Vishal Gupta

Numerade Educator

Problem 6

A $2.20 \mu \mathrm{F}$ capacitor is connected across an ac source whose voltage amplitude is kept constant at $60.0 \mathrm{~V}$, but whose frequency can be varied. Find the current amplitude when the angular frequency is (a) $100 \mathrm{rad} / \mathrm{s} ;$ (b) $1000 \mathrm{rad} / \mathrm{s} ;$ (c) $10,000 \mathrm{rad} / \mathrm{s}$.

Ryan Hood

Numerade Educator

Problem 7

The voltage amplitude of an ac source is $25.0 \mathrm{~V},$ and its angular frequency is $1000 \mathrm{rad} / \mathrm{s}$. Find the current amplitude if the capacitance of a capacitor connected across the source is (a) $0.0100 \mu \mathrm{F}$ (b) $1.00 \mu \mathrm{F}$,
(c) $100 \mu \mathrm{F}$.

Averell Hause

Carnegie Mellon University

Problem 8

Find the current amplitude if the self-inductance of a resistanceless inductor that is connected across the source of the previous problem is (a) $0.0100 \mathrm{H},$ (b) $1.00 \mathrm{H},$ (c) $100 \mathrm{H}$.

Ryan Hood

Numerade Educator

Problem 9

A sinusoidal ac voltage source in a circuit produces a maximum voltage of $12.0 \mathrm{~V}$ and an $\mathrm{rms}$ current of $7.50 \mathrm{~mA}$. Find (a) the voltage and current amplitudes and (b) the rms voltage of this source.

Averell Hause

Carnegie Mellon University

Problem 10

A $65 \Omega$ resistor, an $8.0 \mu F$ capacitor, and a $35 \mathrm{mH}$ inductor are connected in series in an ac circuit. Calculate the impedance for a source frequency of (a) $300 \mathrm{~Hz}$ and (b) $30.0 \mathrm{kHz}$

Vishal Gupta

Numerade Educator

Problem 11

In an $R-L-C$ series circuit, the rms voltage across the resistor is $30.0 \mathrm{~V},$ across the capacitor it is $90.0 \mathrm{~V},$ and across the inductor it is $50.0 \mathrm{~V}$. What is the rms voltage of the source?

Averell Hause

Carnegie Mellon University

Problem 12

A $1500 \Omega$ resistor is connected in series with a $350 \mathrm{mH}$ inductor and an ac power supply. At what frequency will this combination have twice the impedance that it has at $120 \mathrm{~Hz}$ ?

Ryan Hood

Numerade Educator

Problem 13

(a) Compute the impedance of a series $R-L-C$ circuit at angular frequencies of $1000,750,$ and $500 \mathrm{rad} / \mathrm{s} .$ Take $R=200 \Omega$, $L=0.900 \mathrm{H},$ and $C=2.00 \mu \mathrm{F} .$ (b) Describe how the current amplitude varies as the angular frequency of the source is slowly reduced from $1000 \mathrm{rad} / \mathrm{s}$ to $500 \mathrm{rad} / \mathrm{s}$. (c) What is the phase angle of the source voltage with respect to the current when $\omega=1000 \mathrm{rad} / \mathrm{s}$ ? (d) Construct a phasor diagram when $\omega=1000 \mathrm{rad} / \mathrm{s}$

Vishal Gupta

Numerade Educator

Problem 14

A $200 \Omega$ resistor is in series with a $0.100 \mathrm{H}$ inductor and a $0.500 \mu \mathrm{F}$ capacitor. Compute the impedance of the circuit and draw the phasor diagram (a) at a frequency of $500 \mathrm{~Hz}$, (b) at a frequency of $1000 \mathrm{~Hz}$. In each case, compute the phase angle of the source voltage with respect to the current and state whether the source voltage lags or leads the current.

Ryan Hood

Numerade Educator

Problem 15

A hair dryer is designed to produce $1200 \mathrm{~W}$ of heating power. It draws a maximum instantaneous current of 14 A. Find (a) the maximum instantaneous power, (b) the rms current, (c) the rms voltage, and (d) the resistance of the hair dryer.

Vishal Gupta

Numerade Educator

Problem 16

A series $R-L-C$ circuit is connected to a $120 \mathrm{~Hz}$ ac source that has $V_{\mathrm{rms}}=80.0 \mathrm{~V}$. The circuit has a resistance of $75.0 \Omega$ and an impedance of $105 \Omega$ at this frequency. What average power is delivered to the circuit by the source?

Ryan Hood

Numerade Educator

Problem 17

The circuit in Problem 13 carries an rms current of 0.250 A with a frequency of $100 \mathrm{~Hz}$. (a) What is the average rate at which electrical energy is converted to heat in the resistor? (b) What average power is delivered by the source? (c) What is the average rate at which electrical energy is dissipated (converted to other forms) in the capacitor? in the inductor?

Averell Hause

Carnegie Mellon University

Problem 18

A series ac circuit contains a $250 \Omega$ resistor, a $15 \mathrm{mH}$ inductor, a $3.5 \mu \mathrm{F}$ capacitor, and an ac power source of voltage amplitude $45 \mathrm{~V}$ operating at an angular frequency of $360 \mathrm{rad} / \mathrm{s}$. (a) What is the power factor of this circuit? (b) Find the average power delivered to the entire circuit. (c) What is the average power delivered to the resistor, to the capacitor, and to the inductor?

Ze-Han Lee

Numerade Educator

Problem 19

An ac series $R-L-C$ circuit contains a $120 \Omega$ resistor, a $2.0 \mu \mathrm{F}$ capacitor, and a $5.0 \mathrm{mH}$ inductor. Find (a) the resonance angular frequency and (b) the length of time that each cycle lasts at the resonance angular frequency.

Vishal Gupta

Numerade Educator

Problem 20

(a) At what angular frequency will a $5.00 \mu \mathrm{F}$ capacitor have the same reactance as a $10.0 \mathrm{mH}$ inductor? (b) If the capacitor and inductor in part (a) are connected in an $L-C$ circuit, what will be the resonance angular frequency of that circuit?

Ryan Hood

Numerade Educator

Problem 21

In an $R-L-C$ series circuit, $R=150 \Omega, L=0.750 \mathrm{H},$ and $C=0.0180 \mu \mathrm{F}$. (a) What is the resonant frequency of the circuit in rad/s? (b) Suppose you replace the inductor with one that has an inductance of $L=0.25 \mathrm{H}$. What value of capacitance would be needed in order for the resonant frequency to remain unchanged?

Vishal Gupta

Numerade Educator

Problem 22

You need to make a series ac circuit having a resonance angular frequency of $1525 \mathrm{rad} / \mathrm{s}$ using a $138 \Omega$ resistor, a $10.5 \mu \mathrm{F}$ capacitor, and an inductor. (a) What should be the inductance of the inductor, and (b) what is the impedance of this circuit when you use it with an ac voltage source having an angular frequency of $1525 \mathrm{rad} / \mathrm{s} ?$

Vishal Gupta

Numerade Educator

Problem 23

A series circuit consists of an ac source of variable frequency, a $115 \Omega$ resistor, a $1.25 \mu \mathrm{F}$ capacitor, and a $4.50 \mathrm{mH}$ inductor. Find the impedance of this circuit when the angular frequency of the ac source is adjusted to (a) the resonance angular frequency, (b) twice the resonance angular frequency, and (c) half the resonance angular frequency.

Vishal Gupta

Numerade Educator

Problem 24

In a series $R-L-C$ circuit, $R=400 \Omega, L=0.350 \mathrm{H},$ and $C=0.0120 \mu \mathrm{F}$. (a) What is the resonance angular frequency of the circuit? (b) The capacitor can withstand a peak voltage of $550 \mathrm{~V}$. If the voltage source operates at the resonance frequency, what maximum voltage amplitude can it have if the maximum capacitor voltage is not exceeded?

Ryan Hood

Numerade Educator

Problem 25

In a series $R-L-C$ circuit, $L=0.200 \mathrm{H}, C=80.0 \mu \mathrm{F},$ and the voltage amplitude of the source is $240 \mathrm{~V}$. (a) What is the resonance angular frequency of the circuit? (b) When the source operates at the resonance angular frequency, the current amplitude in the circuit is 0.600 A. What is the resistance $R$ of the resistor? (c) At the resonance frequency, what are the peak voltages across the inductor, the capacitor, and the resistor?

Averell Hause

Carnegie Mellon University

Problem 26

In an $R-L-C$ series circuit, $R=300 \Omega, L=0.400 \mathrm{H},$ and $C=6.00 \times 10^{-8} \mathrm{~F}$. When the ac source operates at the resonance frequency of the circuit, the current amplitude is 0.500 A. (a) What is the voltage amplitude of the source? (b) What is the amplitude of the voltage across the resistor, across the inductor, and across the capacitor? (c) What is the average power supplied by the source?

Ryan Hood

Numerade Educator

Problem 27

A solenoid has a resistance of $48.0 \Omega$ and an inductance of $0.150 \mathrm{H}$. If a $100 \mathrm{~Hz}$ voltage source is connected across the solenoid, determine the phase angle between the voltage and the current. Does the voltage lead the current or lag the current?

Vishal Gupta

Numerade Educator

Problem 28

A large electromagnetic coil is connected to a $120 \mathrm{~Hz}$ ac source. The coil has resistance $400 \Omega,$ and at this source frequency the coil has inductive reactance $250 \Omega$. (a) What is the inductance of the coil? (b) What must the rms voltage of the source be if the coil is to consume an average electric power of $800 \mathrm{~W} ?$

Vishal Gupta

Numerade Educator

Problem 29

A parallel-plate capacitor having square plates $4.50 \mathrm{~cm}$ on each side and $8.00 \mathrm{~mm}$ apart is placed in series with an ac source of angular frequency $650 \mathrm{rad} / \mathrm{s}$ and voltage amplitude $22.5 \mathrm{~V}$, a $75.0 \Omega$ resistor, and an ideal solenoid that is $9.00 \mathrm{~cm}$ long, has a circular cross section $0.500 \mathrm{~cm}$ in diameter, and carries 125 coils per centimeter. What is the resonance angular frequency of this circuit? (See Problem 34 in Chapter $21 .)$

Vishal Gupta

Numerade Educator

Problem 30

At a frequency $\omega_{1},$ the reactance of a certain capacitor equals that of a certain inductor. (a) If the frequency is changed to $\omega_{2}=2 \omega_{1},$ what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger?
(b) If the frequency is changed to $\omega_{3}=\omega_{1} / 3,$ what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger?

Ryan Hood

Numerade Educator

Problem 31

An electrical engineer is designing an $R-L-C$ circuit for use in a ham radio receiver. He is unsure of the value of the inductance in the circuit, so he measures the resonant frequency of his circuit using a few different values of capacitance. The data he obtains are shown in the table.
$$
\begin{array}{cc}
\hline \text { Capacitance (nF) } & \text { Frequency (kHz) } \\
\hline 0.2 & 560 \\
0.4 & 395 \\
0.7 & 300 \\
1.0 & 250 \\
\hline
\end{array}
$$
Make a linearized graph of the data by plotting the square of the resonance frequency as a function of the inverse of the capacitance. Using a linear "best fit" to the data, determine the inductance of his circuit.

Vishal Gupta

Numerade Educator

Problem 32

Consider the circuit sketched in Figure $22.22 .$ The source has a voltage amplitude of $240 \mathrm{~V}$, $R=150 \Omega,$ and the reactance of the capacitor is $600 \Omega .$ The voltage amplitude across the capacitor is $720 \mathrm{~V}$. (a) What is the current amplitude in the circuit? (b) What is the impedance? (c) What two values can the reactance of the inductor have?

Ryan Hood

Numerade Educator

Problem 33

In a series $R-L-C$ circuit, the components have the following values: $L=20.0 \mathrm{mH}, C=140 \mathrm{nF},$ and $R=350 \Omega .$ The generator has an rms voltage of $120 \mathrm{~V}$ and a frequency of $1.25 \mathrm{kHz}$. Determine (a) the power supplied by the generator; and (b) the power dissipated in the resistor.

Averell Hause

Carnegie Mellon University

Problem 34

(a) Show that for an $R-L-C$ series circuit the power factor is equal to $R / Z$. (Hint: Use the phasor diagram; see Figure $22.13 \mathrm{~b}$.) (b) Show that for any ac circuit, not just one containing pure resistance only, the average power delivered by the voltage source is given by $P_{\mathrm{av}}=I_{\mathrm{rms}}^{2} R$

Sailesh Mohanty

Sailesh Mohanty

Numerade Educator

Problem 35

In an $R-L-C$ series circuit, the magnitude of the phase angle is $54.0^{\circ},$ with the source voltage lagging the current. The reactance of the capacitor is $350 \Omega,$ and the resistor resistance is $180 \Omega .$ The average power delivered by the source is $140 \mathrm{~W}$. Find (a) the reactance of the inductor; (b) the rms current; (c) the rms voltage of the source.

Averell Hause

Carnegie Mellon University

Problem 36

In a series $R-L-C$ circuit, $R=300 \Omega, X_{C}=300 \Omega,$ and $X_{L}=500 \Omega .$ The average power consumed in the resistor is $60.0 \mathrm{~W}$ (a) What is the power factor of the circuit? (b) What is the rms voltage of the source?

Ryan Hood

Numerade Educator

Problem 37

In a series $R-L-C$ circuit, the phase angle is $40.0^{\circ},$ with the source voltage leading the current. The reactance of the capacitor is $400 \Omega$, and the resistance of the resistor is $200 \Omega .$ The average power delivered by the source is $150 \mathrm{~W}$. Find (a) the reactance of the inductor, (b) the rms current, (c) the rms voltage of the source.

Averell Hause

Carnegie Mellon University

Problem 38

A $100.0 \Omega$ resistor, a $0.100 \mu \mathrm{F}$ capacitor, and a $300.0 \mathrm{mH}$ inductor are connected in series to a voltage source with amplitude $240 \mathrm{~V}$.
(a) What is the resonance angular frequency? (b) What is the maximum current in the resistor at resonance?
(c) What is the maximum voltage across the capacitor at resonance?
(d) What is the maximum voltage across the inductor at resonance?
(e) What is the maximum energy stored in the capacitor at resonance? In the inductor?

Ryan Hood

Numerade Educator

Problem 39

Consider the same circuit as in the preceding problem, with the source operated at an angular frequency that is twice the resonant frequency. Calculate the following ratios: (a) the circuit's impedance at $2 \omega_{0}$ to its impedance at resonance; (b) the maximum current at $2 \omega_{0}$ to the maximum current at resonance; (c) the maximum voltage across the resistor at $2 \omega_{0}$ to the maximum voltage across the resistor at resonance.

Linda Winkler

Numerade Educator

Problem 40

What is the de impedance of the electrode, assuming that it behaves as an ideal capacitor?
A. 0
B. Infinite
C. $\sqrt{2} \times 10^{4} \Omega$
D. $\sqrt{2} \times 10^{6} \Omega$

Saman Zulfiqar

Numerade Educator

Problem 41

If the electrode oscillates between two points $20 \mu \mathrm{m}$ apart at a frequency of $(5000 / \pi) \mathrm{Hz}$, what is the electrode's impedance?
A. 0
B. Infinite
C. $\sqrt{2} \times 10^{4} \Omega$
D. $\sqrt{2} \times 10^{6} \Omega$

Vishal Gupta

Numerade Educator

Problem 42

The signal from the oscillating electrode is fed into an amplifier, which reports the measured voltage as an rms value, $1.5 \mathrm{nV}$. What is the potential difference between the two extremes?
A. $1.5 \mathrm{nV}$
B. $3.0 \mathrm{nV}$
C. $2.1 \mathrm{nV}$
D. $4.2 \mathrm{nV}$

Ze-Han Lee

Numerade Educator

Problem 43

If the frequency at which the electrode is oscillated is increased to a very large value, what happens to the electrode's impedance? The impedance
A. approaches infinity.
B. approaches zero.
C. approaches a constant but nonzero value.
D. does not change.

Mohit Khurana

Texas A&M University