College Physics With an Integrated Approach to Forces and Kinematics

Alan Giambattista, Betty McCarthy Richardson , Robert C. Richardson

Chapter 21

Alternating Current - all with Video Answers

Educators

Chapter Questions

Problem 1

A lightbulb is connected to a $120-\mathrm{V}(\mathrm{rms}), 60-\mathrm{Hz}$ source. How many times per second does the current reverse direction?

Yaqub Khan

Yaqub Khan

Numerade Educator

Problem 2

A European outlet supplies $220 \mathrm{~V}$ (rms) at $50 \mathrm{~Hz}$. How many times per second is the magnitude of the voltage equal to $220 \mathrm{~V} ?$

Mayukh Banik

Numerade Educator

Problem 3

A $1500-W$ heater runs on $120 \mathrm{~V} \mathrm{rms}$. What is the peak current through the heater?

Vishal Gupta

Numerade Educator

Problem 4

A circuit breaker trips when the rms current exceeds $20.0 \mathrm{~A}$. How many $100.0$ -W lightbulbs can run on this circuit without tripping the breaker? (The voltage is $120 \mathrm{Vrms}$.)

Mayukh Banik

Numerade Educator

Problem 5

A $1500-\mathrm{W}$ electric hair dryer is designed to work in the United States, where the ac voltage is $120 \mathrm{~V} \mathrm{rms}$. What power is dissipated in the hair dryer when it is plugged into a $240-\mathrm{V} \mathrm{rms}$ socket in Europe? What may happen to the hair dryer in this case?

Vishal Gupta

Numerade Educator

Problem 6

A $4.0-\mathrm{kW}$ heater is designed to be connected to a $120-\mathrm{V}$ rms source. What is the power dissipated by the heater if it is instead connected to a $120-\mathrm{V}$ de source?

Vishal Gupta

Numerade Educator

Problem 7

(a) What rms current is drawn by a $4200-\mathrm{W}$ electric room heater when running on $120 \mathrm{~V} \mathrm{rms}$ ? (b) What is the power dissipation by the heater if the voltage drops to $105 \mathrm{~V} \mathrm{rms}$ during a brownout? Assume the resistance stays the same.

Vishal Gupta

Numerade Educator

Problem 8

A television set draws an rms current of $2.50 \mathrm{~A}$ from a $60-\mathrm{Hz}$ power line. Find (a) the average current, (b) the average of the square of the current, and (c) the amplitude of the current.

Mayukh Banik

Numerade Educator

Problem 9

The instantaneous sinusoidal emf from an ac generator with an rms emf of $4.0 \mathrm{~V}$ oscillates between what values?

Vishal Gupta

Numerade Educator

Problem 10

A hair dryer has a power rating of $1200 \mathrm{~W}$ at $120 \mathrm{~V}$ rms. Assume the hair dryer circuit contains only resistance. (a) What is the resistance of the heating element? (b) What is the rms current drawn by the hair dryer? (c) What is the maximum instantaneous power that the resistance must withstand?

Mayukh Banik

Numerade Educator

Problem 11

Show that over one complete cycle, the average value of a sine function squared is $\frac{1}{2}$.

Yaqub Khan

Numerade Educator

Problem 12

The diagram shows a simplified household circuit. Resistor $R_{1}=240.0 \Omega$
represents a lightbulb; resistor $R_{2}=$ $12.0 \Omega$ represents a hair dryer. The resistors $r=0.50 \Omega$
(each) represent the resistance of the wiring in the walls. Assume that the generator supplies a constant $120.0 \mathrm{~V} \mathrm{rms}$. (a) If the lightbulb is on and the hair dryer is off, find the rms voltage across the lightbulb and the power dissipated by the lightbulb.
(b) If both the lightbulb and the hair dryer are on, find the rms voltage across the lightbulb, the power dissipated by the lightbulb, and the $\mathrm{rms}$ voltage between point $A$ and ground. (c) Explain why lights sometimes dim when an appliance is turned on. (d) Explain why the neutral and ground wires in a junction box are not at the same potential even though they are both grounded.

Mayukh Banik

Numerade Educator

Problem 13

A variable capacitor with negligible resistance is connected to an ac voltage source. How does the current in the circuit change if the capacitance is increased by a factor of $3.0$ and the driving frequency is increased by a factor of $2.0 ?$

Yaqub Khan

Numerade Educator

Problem 14

At what frequency is the reactance of a $6.0-\mu \mathrm{F}$ capacitor equal to $1.0 \mathrm{k} \Omega$ ?

Mayukh Banik

Numerade Educator

Problem 15

A $0.400-\mu \mathrm{F}$ capacitor is connected across the terminals of a variable frequency oscillator. (a) What is the frequency when the reactance is $6.63 \mathrm{k} \Omega ?$ (b) Find the reactance for half of that same frequency.

Yaqub Khan

Numerade Educator

Problem 16

A $0.250-\mu \mathrm{F}$ capacitor is connected to a $220-\mathrm{V} \mathrm{rms}$ ac source at $50.0 \mathrm{~Hz}$. (a) Find the reactance of the capacitor. (b) What is the rms current through the capacitor?

Mayukh Banik

Numerade Educator

Problem 17

A capacitor is connected across the terminals of a 115 $\mathrm{V} \mathrm{rms}, 60.0-\mathrm{Hz}$ generator. For what capacitance is the rms current $2.3 \mathrm{~mA}$ ?

Vishal Gupta

Numerade Educator

Problem 18

Show, from $X_{\mathrm{C}}=1 /(\omega C)$, that the units of capacitive reactance are ohms.

Mayukh Banik

Numerade Educator

Problem 19

A parallel plate capacitor has two plates, each of area $3.0 \times 10^{-4} \mathrm{~m}^{2}$, separated by $3.5 \times 10^{-4} \mathrm{~m}$. The space
between the plates is filled with a dielectric. When the capacitor is connected to a source of $120 \mathrm{~V} \mathrm{rms}$ at $8.0 \mathrm{kHz}$, an rms current of $1.5 \times 10^{-4} \mathrm{~A}$ is measured. (a) What is the capacitive reactance? (b) What is the dielectric constant of the material between the plates of the capacitor?

Vishal Gupta

Numerade Educator

Problem 20

A capacitor (capacitance $=C$ ) is connected to an ac power supply with peak voltage $V$ and angular frequency $\omega$. (a) During a quarter cycle when the capacitor goes from being uncharged to fully charged, what is the average current (in terms of $C, V$, and $\omega) ?[$ Hint $:$ $\left.i_{\mathrm{av}}=\Delta Q / \Delta t .\right]$ (b) What is the rms current? (c) Explain why the average and rms currents are not the same.

Vishal Gupta

Numerade Educator

Problem 21

Three capacitors $(2.0 \mu \mathrm{F}, 3.0 \mu \mathrm{F}, 6.0 \mu \mathrm{F})$ are connected in series to an ac voltage source with amplitude $12.0 \mathrm{~V}$ and frequency $6.3 \mathrm{kHz}$. (a) What are the peak voltages across each capacitor? (b) What is the peak current that flows in the circuit?

Yaqub Khan

Numerade Educator

Problem 22

A capacitor and a resistor are connected in parallel across an ac source. The reactance of the capacitor is equal to the resistance of the resistor. Assuming that $i_{\mathrm{C}}(t)=I \sin \omega t$, sketch graphs of $i_{\mathrm{C}}(t)$ and $i_{\mathrm{R}}(t)$ on the
same axes.

Mayukh Banik

Numerade Educator

Problem 23

A variable inductor with negligible resistance is cor nected to an ac voltage source. How does the current the inductor change if the inductance is increased by factor of $3.0$ and the driving frequency is increased by factor of $2.0 ?$

Yaqub Khan

Numerade Educator

Problem 24

At what frequency is the reactance of a $20.0-\mathrm{mH}$ indu weanal to $188 \Omega ?$

Mayukh Banik

Numerade Educator

Problem 25

What is the reactance of an air core solenoid of length $8.0 \mathrm{~cm}$, radius $1.0 \mathrm{~cm}$, and 240 turns at a frequency of $15.0 \mathrm{kHz} ?$

Yaqub Khan

Numerade Educator

Problem 26

A solenoid with a radius of $8.0 \times 10^{-3} \mathrm{~m}$ and 200 turns $/ \mathrm{cm}$ is used as an inductor in a circuit. When the solenoid is connected to a source of $15 \mathrm{~V} \mathrm{rms}$ at $22 \mathrm{kHz}$, an $\mathrm{rms}$ current of $3.5 \times 10^{-2} \mathrm{~A}$ is measured. Assume the resistance of the solenoid is negligible. (a) What is the inductive reactance? (b) What is the length of the solenoid?

Vishal Gupta

Numerade Educator

Problem 27

A 4.00-mH inductor is connected to an ac voltage source of $151.0 \mathrm{~V} \mathrm{rms}$. If the $\mathrm{rms}$ current in the circuit is $0.820 \mathrm{~A}$, what is the frequency of the source?

Vishal Gupta

Numerade Educator

Problem 28

Two ideal inductors $(0.10 \mathrm{H}, 0.50 \mathrm{H})$ are connected in series to an ac voltage source with amplitude $5.0 \mathrm{~V}$ and frequency $126 \mathrm{~Hz} .$ (a) What are the peak voltages across each inductor? (b) What is the peak current that flows in the circuit?

Mayukh Banik

Numerade Educator

Problem 29

. Suppose that current flows to the left through the inductor in Fig. $21.6 \mathrm{a}$ so that $i$ is negative. (a) If the current is increasing in magnitude, what is the sign of $\Delta i / \Delta t ?$
(b) In what direction is the induced $\mathrm{emf}$ that opposes the increase in current? (c) Show that Eq. $(21-8)$ gives the correct sign for $v_{\mathrm{L}}$. [Hint: $v_{\mathrm{L}}$ is positive if the left side of the inductor is at a higher potential than the right side.]
(d) Repeat these three questions if the current flows to the left through the inductor and is decreasing in magnitude.

Yaqub Khan

Numerade Educator

Problem 30

Suppose that an ideal capacitor and an ideal inductor are connected in series in an ac circuit. (a) What is the phase difference between $v_{\mathrm{C}}(t)$ and $v_{\mathrm{L}}(t) ?[$ Hint: Since they are in series, the same current $i(t)$ flows through both. $]$ (b) If the rms voltages across the capacitor and inductor are $5.0 \mathrm{~V}$ and $1.0 \mathrm{~V}$, respectively, what would an ac voltmeter (which reads rms voltages) connected across the series combination read?

Mayukh Banik

Numerade Educator

Problem 31

The voltage across an inductor and the current through the inductor are related by $v_{1 .}=1 . \Delta i / \Delta t .$ Suppose that $i(t)=I$ sin w. (a) Write an expression for $v_{\mathrm{L}}(t) .$ [Hint:
Use one of the relationships of Eq. $(20-7) .]$ (b) From your expression for $v_{\mathrm{L}}(t)$, show that the reactance of the inductor is $X_{\mathrm{L}}=\omega L .$ (c) Sketch graphs of $i(t)$ and $v_{\mathrm{T}}(t)$ on the same axes. What is the phase difference? Which one leads?

Vishal Gupta

Numerade Educator

Problem 32

Make a figure analogous to Fig. $21.4$ for an ideal inductor in an ac circuit. Start by assuming that the voltage across an ideal inductor is $v_{\mathrm{L}}(t)=V_{\mathrm{L}} \sin \omega t$. Make a graph showing one cycle of $v_{\mathrm{L}}(t)$ and $i(t)$ on the same axes. Then, at each of the times $t=0, \frac{1}{8} T, \frac{2}{8} T, \ldots, T$, indicate the direction of the current (or that it is zero), whether the current is increasing, decreasing, or (instantaneously) not changing, and the direction of the induced emf in the inductor (or that it is zero).

Ajay Singhal

Numerade Educator

Problem 33

A $25.0-\mathrm{mH}$ inductor, with internal resistance of $25.0 \Omega$, is connected to a $110-\mathrm{V}$ rms source. If the average power dissipated in the circuit is $50.0 \mathrm{~W}$, what is the frequency? (Model the inductor as an ideal inductor in series with a resistor.)

Vishal Gupta

Numerade Educator

Problem 34

An inductor has an impedance of $30.0 \Omega$ and a resistance of $20.0 \Omega$ at a frequency of $50.0 \mathrm{~Hz}$. What is the inductance? (Model the inductor as an ideal inductor in series with a resistor.)

Mayukh Banik

Numerade Educator

Problem 35

A $6.20-\mathrm{mH}$ inductor is one of the elements in a simple $R L C$ series circuit. When this circuit is connected to a 1.60-kHz sinusoidal source with an rms voltage of $960.0 \mathrm{~V}$, an $\mathrm{rms}$ current of $2.50 \mathrm{~A}$ lags behind the voltage by $52.0^{\circ}$. (a) What is the impedance of this circuit?
(b) What is the resistance of this circuit? (c) What is the average power dissipated in this circuit?

Vishal Gupta

Numerade Educator

Problem 36

A series combination of a resistor and a capacitor are connected to a $110-\mathrm{V} \mathrm{rms}, 60.0-\mathrm{Hz}$ ac source. If the capacitance is $0.80 \mu \mathrm{F}$ and the $\mathrm{rms}$ current in the circuit is $28.4 \mathrm{~mA}$, what is the resistance?

Vishal Gupta

Numerade Educator

Problem 37

A $300.0-\Omega$ resistor and a $2.5-\mu \mathrm{F}$ capacitor are connected in series across the terminals of a sinusoidal emf with a frequency of $159 \mathrm{~Hz}$. The inductance of the circuit is negligible. What is the impedance of the circuit?

Yaqub Khan

Numerade Educator

Problem 38

A series $R L C$ circuit has a $0.20-\mathrm{mF}$ capacitor, a $13-\mathrm{mH}$ inductor, and a $10.0-\Omega$ resistor, and is connected to an ac source with amplitude $9.0 \mathrm{~V}$ and frequency $60 \mathrm{~Hz}$.
(a) Calculate the voltage amplitudes $V_{\mathrm{L}}, V_{\mathrm{C}}, V_{\mathrm{R}}$, and the phase angle. (b) Draw the phasor diagram for the voltages of this circuit.

Mayukh Banik

Numerade Educator

Problem 39

(a) Find the power factor for the $R L C$ series circuit of Example $21.4$. (b) What is the average power delivered to each element $(R, L, C) ?$

Yaqub Khan

Numerade Educator

Problem 40

A computer draws an rms current of $2.80 \mathrm{~A}$ at an $\mathrm{rms}$ voltage of $120 \mathrm{~V}$. The average power consumption is $240 \mathrm{~W}$. (a) What is the power factor? (b) What is the phase difference between the voltage and current?

Mayukh Banik

Numerade Educator

Problem 41

An $R L C$ series circuit is connected to an ac power supply with a 12-V amplitude and a frequency of $2.5 \mathrm{kHz}$. If $R=220 \Omega, C=8.0 \mu \mathrm{F}$, and $L=0.15 \mathrm{mH}$, what is the average power dissipated?

Yaqub Khan

Numerade Educator

Problem 42

An ac circuit has a single resistor, capacitor, and inductor in series. The circuit uses $100 \mathrm{~W}$ of power and draws a maximum rms current of $2.0 \mathrm{~A}$ when operating at $60 \mathrm{~Hz}$ and $120 \mathrm{~V} \mathrm{rms}$. The capacitive reactance is $0.50$ times the inductive reactance. (a) Find the phase angle.
(b) Find the values of the resistor, the inductor, and the capacitor.

Vishal Gupta

Numerade Educator

Problem 43

Suppose that two sinusoidal voltages at the same frequency are added: $V_{1} \sin \omega t+V_{2} \sin \left(\omega t+\phi_{2}\right)=V \sin (\omega t+\phi)$ A phasor representation is shown in the diagram. (a) Substitute $t=0$ into the equation. Interpret the result by referring to the phasor diagram. (b) Substitute $t=\pi /(2 \omega)$ and simplify using the trigonometric identity $\sin (\theta+\pi / 2)=\cos \theta$. Interpret the result by referring to the phasor diagram.

Yaqub Khan

Numerade Educator

Problem 44

An ac circuit contains a $12.5-\Omega$ resistor, a $5.00-\mu \mathrm{F}$ capacitor, and a $3.60-\mathrm{mH}$ inductor connected in series to an ac generator with an output voltage of $50.0 \mathrm{~V}$ the power factor, and the phase difference between the source voltage and current for this circuit.

Vishal Gupta

Numerade Educator

Problem 45

A $0.48-\mu \mathrm{F}$ capacitor is connected in series to a $5.00-\mathrm{k} \Omega$ resistor and an ac source of voltage amplitude $2.0 \mathrm{~V}$.
(a) At $f=120 \mathrm{~Hz}$, what are the voltage amplitudes across the capacitor and across the resistor? (b) Do the voltage amplitudes add to give the amplitude of the source voltage (i.e., does $V_{\mathrm{R}}+V_{\mathrm{C}}=2.0 \mathrm{~V}$ )? Explain. (c) Draw a phasor diagram to show the addition of the voltages.

Yaqub Khan

Numerade Educator

Problem 46

A series combination of a $22.0-\mathrm{mH}$ inductor and a $145.0-\Omega$ resistor are connected across the output terminals of an ac generator with peak voltage $1.20 \mathrm{kV}$. (a) At $f=1250 \mathrm{~Hz}$, what are the voltage amplitudes across the inductor and across the resistor? (b) Do the voltage amplitudes add to give the source voltage (i.e., does $V_{\mathrm{R}}+V_{\mathrm{L}}=1.20 \mathrm{kV}$ )? Explain. (c) Draw a phasor diagram to show the addition of the voltages.

Mayukh Banik

Numerade Educator

Problem 47

A $3.3-\mathrm{k} \Omega$ resistor is in series with a $2.0-\mu \mathrm{F}$ capacitor in an ac circuit. The rms voltages across the two are the same. (a) What is the frequency? (b) Would each of the rms voltages be half of the $\mathrm{rms}$ voltage of the source? If not, what fraction of the source voltage are they? (In other words, $\left.V_{\mathrm{R}} / 8_{\mathrm{m}}=V_{\mathrm{C}} / 8_{\mathrm{m}}=?\right)[$ Hint: Draw a phasor
diagram.] (c) What is the phase angle between the source voltage and the current? Which leads? (d) What is the impedance of the circuil?

Linda Winkler

Numerade Educator

Problem 48

A $150-\Omega$ resistor is in series with a $0.75-\mathrm{H}$ inductor in an ac circuit. The $\mathrm{rms}$ voltages across the two are the same. (a) What is the frequency? (b) Would each of the rms voltages be half of the rms voltage of the source? If not, what fraction of the source voltage are they? (In other words, $V_{\mathrm{R}} / 8_{\mathrm{m}}=V_{\mathrm{L}} / 8_{\mathrm{m}}=?$ ) (c) What is the phase angle between the source voltage and the current? Which leads? (d) What is the impedance of the circuit?

Mayukh Banik

Numerade Educator

Problem 49

A series circuit with a resistor and a capacitor has a time constant of $0.25 \mathrm{~ms}$. The circuit has an impedance of $350 \Omega$ at a frequency of $1250 \mathrm{~Hz}$. What are the capacitance and the resistance?

Yaqub Khan

Numerade Educator

Problem 50

(a) What is the reactance of a $10.0-\mathrm{mH}$ inductor at the frequency $f=250.0 \mathrm{~Hz}$ ? (b) What is the impedance of a series combination of the $10.0-\mathrm{mH}$ inductor and a $10.0-\Omega$ resistor at $250.0 \mathrm{~Hz}$ ? (c) What is the maximum current through the same circuit when the ac voltage source has ? peak value of $1.00 \mathrm{~V} ?$ (d) By what angle does the current lag the voltage in the circuit?

Mayukh Banik

Numerade Educator

Problem 51

The FM radio band is broadcast between $88 \mathrm{MHz}$ and $108 \mathrm{MHz}$. What range of capacitors must be used to tune in these signals if an inductor of $3.00 \mu \mathrm{H}$ is used?

Yaqub Khan

Numerade Educator

Problem 52

An $R L C$ series circuit is built with a variable capacitor. How does the resonant frequency of the circuit change when the area of the capacitor is increased by a factor of 2 ?

Vishal Gupta

Numerade Educator

Problem 53

A series $R L C$ circuit has $R=500.0 \Omega, L=35.0 \mathrm{mH}$, and $C=87.0 \mathrm{pF}$. What is the impedance of the circuit at resonance? Explain.

Yaqub Khan

Numerade Educator

Problem 54

In an $R L C$ series circuit, these three elements are connected in series: a resistor of $60.0 \Omega$, a $40.0-\mathrm{mH}$ inductor, and a $0.0500-\mathrm{F}$ capacitor. The series elements are connected across the terminals of an ac oscillator with an $\mathrm{rms}$ voltage of $10.0 \mathrm{~V}$. Find the resonant frequency for the circuit.

Vishal Gupta

Numerade Educator

Problem 55

An $R L C$ series circuit is driven by a sinusoidal emf at the circuit's resonant frequency. (a) What is the phase difference between the voltages across the capacitor and inductor? [Hint: Since they are in series, the same current $i(t)$ flows through them.] (b) At resonance, the $\mathrm{rms}$ current in the circuit is $120 \mathrm{~mA}$. The resistance in the circuit is $20 \Omega$. What is the $\mathrm{rms}$ value of the applied $\mathrm{emf}^{?}$ (c) If the frequency of the emf is changed without changing its rms value, what happens to the rms current?

Vishal Gupta

Numerade Educator

Problem 56

. An $R L C$ series circuit has a resistance of $R=325 \Omega$, an inductance $L=0.300 \mathrm{mH}$, and a capacitance $C=33.0 \mathrm{nF}$. (a) What is the resonant frequency? (b) If the capacitor breaks down for peak voltages in excess of $7.0 \times 10^{2} \mathrm{~V}$, what is the maximum source voltage amplitude when the circuit is operated at the resonant frequency?

Mayukh Banik

Numerade Educator

Problem 57

An $R L C$ series circuit has $L=0.300 \mathrm{H}$ and $C=6.00 \mu \mathrm{F}$. The source has a peak voltage of $440 \mathrm{~V}$. (a) What is the angular resonant frequency? (b) When the source is set at the resonant frequency, the peak current in the circuit is $0.560 \mathrm{~A}$. What is the resistance in the circuit?
(c) What are the peak voltages across the resistor, the inductor, and the capacitor at the resonant frequency?

Vishal Gupta

Numerade Educator

Problem 58

Finola has a circuit with a $4.00-\mathrm{k} \Omega$ resistor, a 0.750-H inductor, and a capacitor of unknown value connected in series to a $440.0$ -Hz ac source. With an oscilloscope, she measures the phase angle to be $25.0^{\circ} .$ (a) What is the value of the unknown capacitor? (b) Finola has several capacitors on hand and would like to use one to tune the circuit to maximum power. Should she connect a second capacitor in parallel across the first capacitor or in series in the circuit? Explain. (c) What value capacitor does she need for maximum power?

Mayukh Banik

Numerade Educator

Problem 59

Repeat Problem 38 for an operating frequency of $98.7 \mathrm{~Hz}$. (a) What is the phase angle for this circuit?
(b) Draw the phasor diagram. (c) What is the resonant frequency for this circuit?

Yaqub Khan

Numerade Educator

Problem 60

An $R C$ filter is shown. The filter resistance $R$ is variable between $180 \Omega$ and $2200 \Omega$ and the filter capacitance is $C=0.086 \mu \mathrm{F}$. At what frequency is the output amplitude equal to $1 / \sqrt{2}$ times the input amplitude if $R=($ a) $180 \Omega ?$ (b) $2200 \Omega$ ?
(c) Is this a low-pass or high-pass filter? Explain.

Mayukh Banik

Numerade Educator

Problem 61

In the crossover network of the figure, the crossover frequency is found to be $252 \mathrm{~Hz}$. The capacitance is $C=$ $560 \mu \mathrm{F}$. Assume the inductor to be ideal. (a) What is the impedance of the tweeter branch (the capacitor in series with the $8.0-\Omega$ resistance of the tweeter) at the crossover frequency? (b) What is the impedance of the woofer branch at the crossover frequency? [Hint: The current amplitudes in the two branches are the same.] (c) Find $L_{\text {. }}$ (d) Derive an equation for the crossover frequency $f_{c o}$ in terms of $L$ and $C$.

Yaqub Khan

Numerade Educator

Problem 62

In the crossover network of Problem 61 , the inductance $L$ is $1.20 \mathrm{mH}$. The capacitor is variable; its capacitance can be adjusted to set the crossover point according to the frequency response of the woofer and twecter. What should the capacitance be set to for a crossover point of $180 \mathrm{~Hz}$ ?

Mayukh Banik

Numerade Educator

Problem 63

The circuit shown has a source voltage of $440 \mathrm{~V} \mathrm{rms}$, resistance $R=250 \Omega$, inductance $L=0.800 \mathrm{H}$, and capacitance $C=2.22 \mu \mathrm{F}$. (a) Find the angular frequency $\omega_{0}$ for resonance in this circuit. (b) Draw a phasor diagram for the circuit at resonance. (c) Find these rms voltages measured between various points in the circuit:
$V_{a b}, V_{b c}, V_{c d}, V_{b d}$, and $V_{\text {od }}$. (d) The resistor is replaced with one of $R=125 \Omega$. Now what is the angular frequency for resonance? (e) What is the rms current in the circuit operated at resonance with the new resistor?

Vishal Gupta

Numerade Educator

Problem 64

For a particular $R L C$ series circuit, the capacitive reactance is $12.0 \Omega$, the inductive reactance is $23.0 \Omega$, and the maximum voltage across the $25.0-\Omega$ resistor is $8.00 \mathrm{~V}$. (a) What is the impedance of the circuit?
(b) What is the maximum voltage across this circuit?

Mayukh Banik

Numerade Educator

Problem 65

The phasor diagram for a particular $R L C$ series circuit is shown in the figure. If the circuit has a resistance of $100 \Omega$ and is driven at a frequency of $60 \mathrm{~Hz}$, find (a) the current amplitude, (b) the capacitance, and
(c) the inductance.

Vishal Gupta

Numerade Educator

Problem 66

A portable heater is connected to a $60-\mathrm{Hz}$ ac outlel How many times per second is the instantaneous powe a maximum?

Vishal Gupta

Numerade Educator

Problem 67

What is the rms voltage of the oscilloscope trace of th figure, assuming that the signal is sinusoidal? The cen tral horizontal line represents zero volts. The oscillo scope voltage knob has been clicked into its calibrate

Yaqub Khan

Numerade Educator

Problem 68

A $22-\mathrm{kV}$ power line that is $10.0 \mathrm{~km}$ long supplies the electric energy to a small town at an average rate of 6.0 MW. (a) If a pair of aluminum cables of diameter $9.2 \mathrm{~cm}$ are used, what is the average power dissipated in the transmission line? (b) Why is aluminum used rather than a better conductor such as copper or silver?

Mayukh Banik

Numerade Educator

Problem 69

An $x$ -ray machine uses $240 \mathrm{kV} \mathrm{rms}$ at $60.0 \mathrm{~mA} \mathrm{rms}$ when it is operating. If the power source is a $420-\mathrm{V} \mathrm{rms}$ line, (a) what must be the turns ratio of the transformer?
(b) What is the rms current in the primary? (c) What is the average power used by the $x$ -ray tube?

Vishal Gupta

Numerade Educator

Problem 70

A coil with an internal resistance of $120 \Omega$ and inductance of $12.0 \mathrm{H}$ is connected to a $60.0-\mathrm{H} z, 110-\mathrm{V} \mathrm{rms}$
line. (a) What is the impedance of the coil? (b) Calculate the current in the coil.

Vishal Gupta

Numerade Educator

Problem 71

The field coils used in an ac motor are designed to have a resistance of $0.45 \Omega$ and an impedance of $35.0 \Omega$. What inductance is required if the frequency of the ac source is (a) $60.0 \mathrm{~Hz}$ ? and (b) $0.20 \mathrm{kHz}$ ?

Yaqub Khan

Numerade Educator

Problem 72

A capacitor is rated at $0.025 \mu \mathrm{F}$. How much rms current flows when the capacitor is connected to a $110-V$ rms, $60.0-\mathrm{Hz}$ line?

Mayukh Banik

Numerade Educator

Problem 73

A capacitor to be used in a radio is to have a reactance of $6.20 \Omega$ at a frequency of $520 \mathrm{~Hz}$. What is the capacitance?

Yaqub Khan

Numerade Educator

Problem 74

An alternator supplies a peak current of $4.68 \mathrm{~A}$ to a coil with a negligibly small internal resistance. The voltage of the alternator is $420-V$ peak at $60.0 \mathrm{~Hz}$. When a capacitor of $38.0 \mu \mathrm{F}$ is placed in series with the coil, the power factor is found to be $1.00$. Find
(a) the inductive reactance of the coil and (b) the inductance of the coil.

Vishal Gupta

Numerade Educator

Problem 75

At what frequency does the maximum current flow through a series $R L C$ circuit containing a resistance of $4.50 \Omega$, an inductance of $440 \mathrm{mH}$, and a capacitance of $520 \mathrm{pF}$ ?

Vishal Gupta

Numerade Educator

Problem 76

What is the $\mathrm{rms}$ current flowing in a $4.50-\mathrm{kW}$ motor connected to a $220-\mathrm{V} \mathrm{rms}$ line when (a) the power fac-

Vishal Gupta

Numerade Educator

Problem 77

A variable capacitor is connected in series to an inductor with negligible internal resistance and of inductance $2.4 \times 10^{-4} \mathrm{H}$. The combination is used as a tuner for a radio. If the lowest frequency to be tuned in is $0.52 \mathrm{MHz}$, what is the maximum capacitance required?

Yaqub Khan

Numerade Educator

Problem 78

A large coil used as an electromagnet has a resistance of $R=450 \Omega$ and an inductance of $L=2.47 \mathrm{H}$. The coil is connected to an ac source with a voltage amplitude of $2.0 \mathrm{kV}$ and a frequency of $9.55 \mathrm{~Hz}$. (a) What is the power factor? (b) What is the impedance of the circuit? (c) What is the peak current in the circuit? (d) What is the average power delivered to the electromagnet by the source?

Mayukh Banik

Numerade Educator

Problem 79

An ac series circuit containing a capacitor, inductor, and resistance is found to have a current of amplitude $0.50 \mathrm{~A}$ for a source voltage of amplitude $10.0 \mathrm{~V}$ at an angular frequency of $200.0 \mathrm{rad} / \mathrm{s}$. The total resistance in the circuit is $15.0 \Omega$. (a) What are the power factor and the phase angle for the circuit? (b) Can you determine whether the current leads or lags the source voltage? Explain.

Yaqub Khan

Numerade Educator

Problem 80

A gencrator supplies an average power of $12 \mathrm{MW}$ through a transmission line that has a resistance of $10.0 \Omega$. What is the power loss in the transmission line if the rms line voltage $\mathbb{8}_{\mathrm{ms}}$ is (a) $15 \mathrm{kV}$ and
(b) $110 \mathrm{kV} ?$ What percentage of the total power supplied by the generator is lost in the transmission line in each case?

Vishal Gupta

Numerade Educator

Problem 81

(a) Calculate the rms current drawn by the load in the figure with Problem 80 if $\mathscr{E}_{\mathrm{mss}}=250 \mathrm{kV}$ and the average power supplied by the generator is $12 \mathrm{MW}$. (b) Suppose that the average power supplied by the generator is still $12 \mathrm{MW}$, but the load is not purely resistive; rather, the load has a power factor of $0.86 .$ What is the $\mathrm{rms}$ current drawn? (c) Why would the power company want to charge more in the second case, even though the average power is the same?

Yaqub Khan

Numerade Educator

Problem 82

Transformers are often rated in terms of kilovolt-amps. A pole on a residential street has a transformer rated at $35 \mathrm{kV} \cdot \mathrm{A}$ to serve four homes on the street. (a) If each home has a fuse that limits the incoming current to $60 \mathrm{~A}$ $\mathrm{rms}$ at $220 \mathrm{~V} \mathrm{rms}$, find the maximum load in $\mathrm{kV} \cdot \mathrm{A}$ on the transformer. (b) Is the rating of the transformer adequate? (c) Explain why the transformer rating is given in $\mathrm{kV}$. A rather than in $\mathrm{kW}$.

Linda Winkler

Numerade Educator

Problem 83

A variable inductor can be placed in series with a lightbulb to act as a dimmer. (a) What inductance would reduce the current through a $100-\mathrm{W}$ lightbulb to $75 \%$ of its maximum value? Assume a $120-\mathrm{V} \mathrm{rms}, 60-\mathrm{Hz}$ source. (b) Could a variable resistor be used in place of the variable inductor to reduce the current? Why is the inductor a much better choice for a dimmer?

Linda Winkler

Numerade Educator

Problem 84

A certain circuit has a $25-\Omega$ resistor and one other component in series with a $12-V(\mathrm{rms})$ sinusoidal ac source. The rms current in the circuit is $0.317 \mathrm{~A}$ when the frequency is $150 \mathrm{~Hz}$ and increases by $25.0 \%$ when the frequency increases to $250 \mathrm{~Hz}$. (a) What is the second component in the circuit? (b) What is the current at $250 \mathrm{~Hz}$ ? (c) What is the numerical value of the second component?

Mayukh Banik

Numerade Educator

Problem 85

A $40.0-\mathrm{mH}$ inductor, with internal resistance of $30.0 \Omega$, is connected to an ac source
$$\mathscr{E}(t)=(286 \mathrm{~V}) \sin [(390 \mathrm{rad} / \mathrm{s}) t]$$
(a) What is the impedance of the inductor in the circuit?
(b) What are the peak and $\mathrm{rms}$ voltages across the inductor (including the internal resistance)? (c) What is the peak current in the circuit? (d) What is the average power dissipated in the circuit? (e) Write an expression for the current through the inductor as a function of time.

Vishal Gupta

Numerade Educator

Problem 86

In an $R L C$ circuit, these three elements are connected in series: a resistor of $20.0 \Omega$, a $35.0-\mathrm{mH}$ inductor, and a $50.0-\mu \mathrm{F}$ capacitor. The ac source of the circuit has an rms voltage of $100.0 \mathrm{~V}$ and an angular frequency of $1.0 \times 10^{3} \mathrm{rad} / \mathrm{s}$. Find (a) the reactances of the capacitor and inductor, (b) the impedance, (c) the rms current,
(d) the current amplitude, (c) the phase angle, and
(f) the rms voltages across each of the circuit elements. (g) Does the current lead or lag the voltage?
(h) Draw a phasor diagram.

Linda Winkler

Numerade Educator

Problem 87

(a) What is the reactance of a $5.00-\mu \mathrm{F}$ capacitor at the frequencies $f=12.0 \mathrm{~Hz}$ and $1.50 \mathrm{kHz}$ ? (b) What is the impedance of a series combination of the $5.00-\mu \mathrm{F}$ capacitor and a $2.00-\mathrm{k} \Omega$ resistor at the same two frequencies? (c) What is the maximum current through the circuit of part (b) when the ac source has a peak voltage of $2.00 \mathrm{~V} ?$ (d) For each of the two frequencies, does the current lead or lag the voltage? By what angle?

Mayukh Banik

Numerade Educator

Problem 88

An $R L C$ series circuit is connected to a $240-V$ rms power supply at a frequency of $2.50 \mathrm{kHz}$. The elements in the circuit have the following values: $R=12.0 \Omega, C=$ $0.26 \mu \mathrm{F}$, and $L=15.2 \mathrm{mH}$. (a) What is the impedance of the circuit? (b) What is the rms current? (c) What is the phase angle? (d) Does the current lead or lag the voltage? (e) What are the rms voltages across each circuit element?

Vishal Gupta

Numerade Educator