College Physics

Hugh D. Young

Chapter 22

Alternating Current - all with Video Answers

Educators

Chapter Questions

Problem 1

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

Averell Hause

Averell Hause

Carnegie Mellon University

Problem 2

$\cdot$ The plate on the back of a certain computer scanner says that
the unit draws 0.34 A of current from a $120 \mathrm{V}, 60$ Hz line.
Find (a) the root-mean-square current, (b) the current ampli-
tude, (c) the average current, and (d) the average square of the
current.

Ryan Hood

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

$\bullet$ (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?

Ryan Hood

Numerade Educator

Problem 5

$\bullet$ A radio inductor. You want the current amplitude through
a $0.450-\mathrm{mH}$ inductor (part of the circuitry for a radio receiver)
to be 2.60 $\mathrm{m}$ A when a sinusoidal voltage with amplitude 12.0 $\mathrm{V}$
is applied across the inductor. What frequency is required?

Averell Hause

Carnegie Mellon University

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

$\bullet$ 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) $0.0100 \mu \mathrm{F},(\mathrm{b}) 1.00 \mu \mathrm{F},(\mathrm{c}) 100 \mu \mathrm{F}$

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) $1.00 \mathrm{H},(\mathrm{c}) 100 \mathrm{H}$ .

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 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

$\cdot \mathrm{A} 65 \Omega$ resistor, an 8.0$\mu \mathrm{F}$ 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 $\mathrm{kHz}$ .

Ryan Hood

Numerade Educator

Problem 11

$\bullet$ 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

$\bullet$ (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 rad/s to 500 rad/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 $\mathrm{A} 200 \Omega$ 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

$\bullet$ The power of a certain $\mathrm{CD}$ player operating at 120 $\mathrm{V} \mathrm{rms}$ is 20.0 $\mathrm{W} .$ Assuming that the $\mathrm{CD}$ player behaves like a pure
resistance, find (a) the maximum instantaneous power, (b) the
rms current, and (c) the resistance of this player.

Averell Hause

Carnegie Mellon University

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

$\bullet$ The circuit in Problem 13 carries an rms current of 0.250 $\mathrm{A}$
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$\mu$ 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?

Ryan Hood

Numerade Educator

Problem 19

$\cdot$ 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 reso-
nance angular frequency and (b) the length of time that each
cycle lasts at the resonance angular frequency.

Averell Hause

Carnegie Mellon University

Problem 20

$\bullet$ (a) At what angular frequency will a 5.00$\mu$ 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

$\bullet$ In an $R-L-C$ series circuit, $R=150 \Omega, L=0.750 \mathrm{H},$ and
$C=0.0180 \mu \mathrm{F}$ . The source has voltage amplitude $V=150 \mathrm{V}$
and a frequency equal to the resonance frequency of the circuit. (a) What is the power factor? (b) What is the average power delivered by the source? (c) The capacitor is replaced
by one with $C=0.0360 \mu \mathrm{F}$ and the source frequency is adjusted to the new resonance value. Then what is the average
power delivered by the source?

Averell Hause

Carnegie Mellon University

Problem 22

$\cdot$ You need to make a series ac circuit having a resonance angular frequency of 1525 rad/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 $\mathrm{Hz}$ ?

Ryan Hood

Numerade Educator

Problem 23

A series circuit consists of an ac source of variable frequency, a 115$\Omega$ resistor, a 1.25$\mu$ 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.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 24

$\bullet$ 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

$\bullet$ 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

$\bullet$ 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 $\mathrm{A}$ . (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

$\bullet$ A 125$\Omega$ resistor, an 8.50$\mu \mathrm{F}$ capacitor, and an 11.2 $\mathrm{mH}$
inductor are all connected in parallel across an ac voltage source of variable frequency. (a) At what angular frequency will the impedance have its maximum value, and (b) what is that value?

Averell Hause

Carnegie Mellon University

Problem 28

For the circuit in Figure $22.23, R=300 \Omega, L=0.500 \mathrm{H}$ and $C=0.600 \mu \mathrm{F}$ . The voltage amplitude of the source is 120 $\mathrm{V} .$ (a) What is the resonance frequency of the circuit? (b) Sketch the phasor diagram at the resonance frequency. (c) At the resonance frequency, what is the current amplitude through the source? (d) At the resonance frequency, what is the current amplitude through the resistor? Through the inductor? Through the branch containing the capacitor?

Ryan Hood

Numerade Educator

Problem 29

$\bullet$ For the circuit in Figure $22.23, R=200 \Omega, L=0.800 \mathrm{H}$ and $C=5.00 \mu \mathrm{F} .$ When the source is operated at the resonance frequency, the current amplitude in the inductor is 0.400 A. Determine the current amplitude (a) in the branch containing the capacitor and (b) through the resistor.

Averell Hause

Carnegie Mellon University

Problem 30

$\bullet$ (a) Use the phasor diagram for a parallel $R-L-C$ circuit (see Figure 22.21 ) to show that the current amplitude $I$ for the current i through the source is given by $I=\sqrt{I_{R}^{2}+\left(I_{C}-I_{L}\right)^{2}}$ (b) Show that the result of part (a) can be written as $I=V / Z,$ with $1 / Z=\sqrt{1 / R^{2}+(\omega C-1 / \omega L)^{2}}$

Ryan Hood

Numerade Educator

Problem 31

$\bullet$ A coil has a resistance of 48.0$\Omega .$ At a frequency of 80.0 $\mathrm{Hz}$ , the voltage across the coil leads the current in it by $52.3^{\circ} .$ Determine the inductance of the coil.

Averell Hause

Carnegie Mellon University

Problem 32

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 electrical power of 800 $\mathrm{W} ?$

Ryan Hood

Numerade Educator

Problem 33

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 36 in Chapter $21 . )$

Averell Hause

Carnegie Mellon University

Problem 34

\bullet 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 35

$\bullet$ Five voltmeters, calibrated to read rms values, are connected as shown in Figure $22.22 .$ Let $R=200 \Omega, L=0.400 \mathrm{H}$ and $C=6.00 \mu \mathrm{F}$ . The source voltage amplitude is $V=30.0 \mathrm{V}$ What is the reading of each voltmeter if $($ a) $\omega=200 \mathrm{rad} / \mathrm{s}$ (b) $\omega=1000 \mathrm{rad} / \mathrm{s} ?$

Averell Hause

Carnegie Mellon University

Problem 36

$\bullet$ 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 37

$\bullet$ 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 kHz. Determine (a) the power supplied by the generator; and (b) the power dissipated in the resistor.

Averell Hause

Carnegie Mellon University

Problem 38

$\bullet($ 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 $P_{\mathrm{av}}=I_{\mathrm{mm}}^{2} R .$

Ryan Hood

Numerade Educator

Problem 39

$\bullet$ 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 40

$\bullet$ 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 41

$\bullet$ 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 42

A 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? (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 43

$\bullet$ Consider the same circuit as in the previous problem, with the source operated at an angular frequency of 400 $\mathrm{rad} / \mathrm{s}$ . (a) What is the maximum current in the resistor? (b) What is the maximum voltage across the capacitor? (c) What is the maximum voltage across the inductor? (d) What is the maximum energy stored in the capacitor? in the inductor?

Averell Hause

Carnegie Mellon University

Problem 44

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

Ryan Hood

Numerade Educator

Problem 45

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

Averell Hause

Carnegie Mellon University

Problem 46

The signal from the oscillating electrode is fed into an amplifier, which reports the measured voltage as an rms value, $V_{\text { rms. }}$ However, the number of interest for analyzing currents driven by the cell is the peak-to-peak voltage difference $\left(V_{\mathrm{pp}}\right),$ that is, the voltage difference between the two extremes of the electrodes excursion. What is the value of $V_{\mathrm{pp}}$ in terms of $V_{\mathrm{rms}} ?$
A. $V_{\mathrm{rms}} / \sqrt{2} \quad \mathrm{B} . V_{\mathrm{rms}} / 2 \sqrt{2}$ C. $\sqrt{2} V_{\mathrm{rms}} \quad$ D. 2$\sqrt{2} V_{\mathrm{rms}}$

Ryan Hood

Numerade Educator