College Physics

Raymond A. Serway, Jerry S. Faughn, Chris Vuille

Chapter 21

Alternating-Current Circuits and Electromagnetic Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

When an AC generator is connected across a $12.0-\Omega$ resistor, the rms current in the resistor is $8.00 \mathrm{~A}$. Find (a) the rms voltage across the resistor, (b) the peak voltage of the generator, (c) the maximum current in the resistor, and
(d) the average power delivered to the resistor.

Sheh Lit Chang

Sheh Lit Chang

University of Washington

Problem 2

A certain lightbulb is rated at $60.0 \mathrm{~W}$ when operating at an rms voltage of $1.20 \times 10^{2} \mathrm{~V}$. (a) What is the peak voltage applied across the bulb? (b) What is the resistance of the bulb? (c) Does a $1.00 \times 10^{2} \mathrm{~W}$ bulb have greater or less resistance than a $60.0$ -W bulb? Explain.

Sheh Lit Chang

University of Washington

Problem 3

An AC power supply that produces a maximum voltage of $\Delta V_{\max }=100 \mathrm{~V}$ is connected to a $24.0-\Omega$ resistor. The current and the resistor voltage are respectively measured with an ideal AC ammeter and an ideal $A C$ voltmeter, as shown in Figure $\mathrm{P} 21.3 .$ What does each meter read? Note that an ideal ammeter has zero resistance and an ideal voltmeter has infinite resistance.

Sheh Lit Chang

University of Washington

Problem 4

Figure $\mathrm{P} 21.4$ shows three lamps connected to a $120-\mathrm{V}$ AC (rms) household supply voltage. Lamps 1 and 2 have $150-\mathrm{W}$ bulbs; lamp 3 has a $100-\mathrm{W}$ bulb. Find the rms current and the resistance of each bulb.

Aja S

Numerade Educator

Problem 5

An audio amplifier, represented by the ACisource and the resistor $R$ in Figure $\mathrm{P} 21.5$, delivers alternating voltages at audio frequencies to the speaker. If the source puts out an alternating voltage of $15.0 \mathrm{~V}$ (rms), the resistance $R$ is $8.20 \Omega$, and the speaker is equivalent to a resistance of $10.4 \Omega$, what is the time-averaged power delivered to the speaker?

Sheh Lit Chang

University of Washington

Problem 6

The output voltage of an AC generator is given by $\Delta v=(170 \mathrm{~V}) \sin (60 \pi t) .$ The generator is connected
across a $20.0-\Omega$ resistor. By inspection, what are the
(a) maximum voltage and
(b) frequency? Find the (c) rms voltage across the resistor, (d) rms current in the resistor,
(e) maximum current in the resistor, and (f) power delivered to the resistor. (g) Should the argument of the sine function be in degrees or radians? Compute the current when $\ell=0.0050 \mathrm{~s}$

Sheh Lit Chang

University of Washington

Problem 7

Show that the SI unit of capacitive reactance $X$, is the ohm.

Sheh Lit Chang

University of Washington

Problem 8

What is the maximum current delivered to a circuit containing a $2.20-\mu \mathrm{F}$ capacitor when it is connected across
(a) a North American outlet having $\Delta V_{\text {rms }}=120 \mathrm{~V}$ and $f=$ $60.0 \mathrm{~Hz}$ and $(\mathrm{b})$ a European outlet having $\Delta V_{\mathrm{rms}}=240 \mathrm{~V}$
and $f=50.0 \mathrm{~Hz}$ ?

Sheh Lit Chang

University of Washington

Problem 9

When a $4.0-\mu \mathrm{F}$ capacitor is connected to a generator whose rms output is $30 \mathrm{~V}$, the current in the circuit is observed to be $0.30 \mathrm{~A}$. What is the frequency of the source?

Salamat Ali

Numerade Educator

Problem 10

An ACgenerator with an output rms voltage of $36.0 \mathrm{~V}$ at a frequency of $60.0 \mathrm{~Hz}$ is connected across a $12.0-\mu \mathrm{F}$ capacitor, Find the (a) capacitive reactance, (b) rms current, and (c) maximum current in the circuit. (d) Does the capacitor have its maximum charge when the current takes its maximum value? Explain.

Sheh Lit Chang

University of Washington

Problem 11

What must be the capacitance of a capacitor inserted in a $60-\mathrm{Hz}$ circuit in series with a generator of $170-\mathrm{V}$ maximum output voltage to produce an rms current output of $0.75 \mathrm{~A} ?$

Sheh Lit Chang

University of Washington

Problem 12

A generator delivers an $\mathrm{AC}$ voltage of the form $\Delta v=$ $(98.0 \mathrm{~V}) \sin (80 \pi t)$ to a capacitor. The maximum current in the circuit is $0.500 \mathrm{~A}$. Find the (a) rms voltage of the generator, (b) frequency of the generator, (c) rms current, (d) reactance, and (e) value of the capacitance.

Averell Hause

Carnegie Mellon University

Problem 13

Show that the inductive reactance $X_{L}$ has SI units of ohms.

Sheh Lit Chang

University of Washington

Problem 14

An AC generator has an output $\mathrm{rms}$ voltage of $78.0 \mathrm{~V}$ at a frequency of $80.0 \mathrm{~Hz}$. If the generator is connected across a $25.0-\mathrm{mH}$ inductor, find the (a) inductive reactance,
(b) rms current, and (c) maximum current in the circuit.

Sheh Lit Chang

University of Washington

Problem 15

An inductor is connected to an AC power supply having a maximum output voltage of $4.00 \mathrm{~V}$ at a frequency of $300.0 \mathrm{~Hz}$. What inductance is needed to keep the rms current less than $2.00 \mathrm{~mA}$ ?

Zulfiqar Ali

Numerade Educator

Problem 16

The output voltage of an AC generator is given by $\Delta v=\left(1.20 \times 10^{2} \mathrm{~V}\right) \sin (30 \pi t) .$ The generator is con-
nected across a $0.500$ - $\mathrm{H}$ inductor. Find the (a) frequency of the generator, (b) rms voltage across the inductor,
(c) inductive reactance, (d) rms current in the inductor,
(c) maximum current in the inductor, and (f) average power delivered to the inductor. (g) Find an expression for the instantaneous current. (h) At what time after $t=$ 0 does the instantancous current first reach $1.00 \mathrm{~A}$ ? (Use the inverse sine function.)

Sheh Lit Chang

University of Washington

Problem 17

Determine the maximum magnetic flux through an inductor connected to a standard outlet $\left(\Delta V_{\text {rms }}=120 \mathrm{~V}\right.$, $f=60.0 \mathrm{~Hz})$.

Salamat Ali

Numerade Educator

Problem 18

A sinusoidal voltage $\Delta v=(80.0 \mathrm{~V}) \sin (150 t)$ is applied to a series $R L C$ circuit with $L=80.0 \mathrm{mH}, C=125.0 \mu \mathrm{F}$ and $R=40.0 \Omega .$ (a) What is the impedance of the circuit?
(b) What is the maximum current in the circuit?

Sheh Lit Chang

University of Washington

Problem 19

A $40.0-\mu \mathrm{F}$ capacitor is connected to a $50.0-\Omega$ resistor and a generator whose rms output is $30.0 \mathrm{~V}$ at $60.0 \mathrm{~Hz}$. Find
(a) the rms current in the circuit, (b) the rms voltage drop across the resistor, (c) the rms voltage drop across the capacitor, and (d) the phase angle for the circuit.

Sheh Lit Chang

University of Washington

Problem 20

An inductor $(L=400 \mathrm{mH})$, a capacitor $(C=4.43 \mu \mathrm{F})$, and a resistor $(R=500 \Omega)$ are connected in series. $\underline{A}$ $50.0-\mathrm{Hz}$ AC generator connected in series to these elements produces a maximum current of $250 \mathrm{~mA}$ in the circuit. (a) Calculate the required maximum voltage $\Delta V_{\max }$
(b) Determine the phase angle by which the current leads or lags the applied voltage.

Sheh Lit Chang

University of Washington

Problem 21

A resistor $\left(R=9.00 \times 10^{2} \Omega\right)$, a capacitor $(C=0.250 \mu \mathrm{F})$, and an inductor $(L=2.50 \mathrm{H})$ are connected in series across a $2.40 \times 10^{2}-\mathrm{Hz}$ AC source for which $\Delta V_{\max }=$
$1.40 \times 10^{2} \mathrm{~V} .$ Calculate (a) the impedance of the circuit,
(b) the maximum current delivered by the source, and
(c) the phase angle between the current and voltage.
(d) Is the current leading or lagging the voltage?

Sheh Lit Chang

University of Washington

Problem 22

A $50.0-\Omega$ resistor, a $0.100-\mathrm{H}$ inductor, and a $10.0-\mu \mathrm{F}$ capacitor are connected in series to a $60.0-\mathrm{Hz}$ source. The rms current in the circuit is $2.75 \mathrm{~A}$. Find the $\mathrm{rms}$ voltages across (a) the resistor, (b) the inductor, (c) the capacitor, and (d) the $R L C$ combination. (e) Sketch the phasor diagram for this circuit.

Sheh Lit Chang

University of Washington

Problem 23

A $60.0-\Omega$ resistor, a $3.00-\mu \mathrm{F}$ capacitor, and a $0.400-\mathrm{H}$ inductor are connected in series to a $90.0-\mathrm{V}(\mathrm{rms}), 60.0-\mathrm{Hz}$ source. Find (a) the voltage drop across the $L C$ combination and (b) the yoltage drop across the $R C$ combination.

Sheh Lit Chang

University of Washington

Problem 24

An AC source operating at $60 \mathrm{~Hz}$ with a maximum voltage of $170 \mathrm{~V}$ is connected in series with a resistor $(R=$ $1.2 \mathrm{k} \Omega)$ and an inductor $(L=2.8 \mathrm{H})$.
(a) What is the maximum value of the current in the circuit? (b) What are the maximum values of the potential difference across the resistor and the inductor? (c) When the current is at a maximum, what are the magnitudes of the potential differences across the resistor, the inductor, and the AC source? (d) When the current is zero, what are the magnitudes of the potential difference across the resistor, the inductor, and the AC source?

Sheh Lit Chang

University of Washington

Problem 25

A person is working near the secondary of a transformer, as shown in Figure $\mathrm{P} 21.25$. The primary voltage is $120 \mathrm{~V}$ (rms) at $60.0 \mathrm{~Hz}$. The capacitance $C_{s}$. which is the stray capacitance between the hand and the secondary winding, is $20.0$ pF. Assuming the person has a body resistance to ground of $R_{b}=50.0 \mathrm{k} \Omega$, determine the $\mathrm{rms}$ voltage across the body. Hint: Redraw the circuit with the secondary of the transformer as a simple $A C$ source.

Sheh Lit Chang

University of Washington

Problem 26

A $60.0-\Omega$ resistor is connected in series with a $30.0-\mu \mathrm{F}$ capacitor and a generator having a maximum voltage of $1.20 \times 10^{2} \mathrm{~V}$ and operating at $60.0 \mathrm{~Hz}$. Find the (a) capacitive reactance of the circuit, (b) impedance of the circuit, and (c) maximum current in the circuit. (d) Does the voltage lead or lag the current? How will putting an inductor in series with the existing capacitor and resistor affect the current? Explain.

Sheh Lit Chang

University of Washington

Problem 27

A series $R L C$ circuit contains the following components: $R=1.50 \times 10^{2} \Omega, L=2.50 \times 10^{2} \mathrm{mH}, C=2.00 \mu \mathrm{F}$,
and a generator with $\Delta V_{\max }=2.10 \times 10^{2} \mathrm{~V}$ operating at $50.0 \mathrm{~Hz}$. Calculate the (a) inductive reactance, (b) capacitive reactance, (c) impedance, (d) maximum current, and (e) phase angle between the current and generator voltage. (f) Calculate the individual maximum voltages across the resistor, inductor, and capacitor.

Sheh Lit Chang

University of Washington

Problem 28

Consider the $R L C$ circuit in Problem $21.27$. When the voltage across the resistor is a maximum, what are the individual voltages across the capacitor and inductor? Explain.

Sheh Lit Chang

University of Washington

Problem 29

An AC source with a maximum voltage of $150 \mathrm{~V}$ and $f=50.0 \mathrm{~Hz}$ is connected between points $a$ and $d$ in Figure P21.29. Calculate the $\mathrm{rms}$ voltages between points
(a) $a$ and $b$, (b) $b$ and $c$, (c) $c$ and $d$, and (d) $b$ and $d$.

Salamat Ali

Numerade Educator

Problem 30

An AC source operating at $60 \mathrm{~Hz}$ with a maximum voltage of $170 \mathrm{~V}$ is connected in series with a resistor $(R=$ $1.2 \mathrm{k} \Omega$ ) and a capacitor $(C=2.5 \mu \mathrm{F})$. (a) What is the maximum value of the current in the circuit? (b) What are the maximum values of the potential difference across the resistor and the capacitor? (c) When the current is zero, what are the magnitudes of the potential difference across the resistor, the capacitor, and the AC source? How much charge is on the capacitor at this instant? (d) When the current is at a maximum, what are the magnitudes of the potential differences across the resistor, the capacitor, and the AC source? How much charge is on the capacitor at this instant?

Sheh Lit Chang

University of Washington

Problem 31

A multimeter in an $R L$ circuit records an rms current of $0.500 \mathrm{~A}$ and a $60.0-\mathrm{Hz} \mathrm{rms}$ generator voltage of $104 \mathrm{~V} . \mathrm{A}$
wattmeter shows that the average power delivered to the resistor is $10.0 \mathrm{~W}$. Determine (a) the impedance in the circuit, (b) the resistance $R$, and (c) the inductance $L$.

Sheh Lit Chang

University of Washington

Problem 32

An $\mathrm{AC}$ voltage of the form $\Delta v=(90.0 \mathrm{~V}) \sin (350 t)$ is applied to a series $R L C$ circuit. If $R=50.0 \Omega, C=25.0 \mu \mathrm{F}$, and $L=0.200 \mathrm{H}$, find the (a) impedance of the circuit,
(b) rms current in the circuit, and (c) average power delivered to the circuit.

Averell Hause

Carnegie Mellon University

Problem 33

Calculate the average power delivered to the circuit described in Problem 21 .

Sheh Lit Chang

University of Washington

Problem 34

A series $R L C$ circuit has a resistance of $22.0 \Omega$ and an impedance of $80.0 \Omega .$ If the $\mathrm{rms}$ voltage applied to the circuit is $160 \mathrm{~V}$, what average power is delivered to the circuit?

Averell Hause

Carnegie Mellon University

Problem 35

An inductor and a resistor are connected in series. When connected to a $60-\mathrm{Hz}, 90-\mathrm{V}$ (rms) source, the voltage drop across the resistor is found to be $50 \mathrm{~V}$ (rms) and the power delivered to the circuit is $14 \mathrm{~W}$. Find (a) the value of the resistance and (b) the value of the inductance.

Salamat Ali

Numerade Educator

Problem 36

Consider a series $R L C$ circuit with $R=25 \Omega, L=$ $6.0 \mathrm{mH}$, and $C=25 \mu \mathrm{F} .$ The circuit is connected to a $10-\mathrm{V}(\mathrm{rms}), 600-\mathrm{Hz}$. AC source. (a) Is the sum of the voltage drops across $R, L$, and $C$ equal to $10 \mathrm{~V}(\mathrm{rms}) ?$ (b) Which is greatest, the power delivered to the resistor, to the capacitor, or to the inductor? (c) Find the average power delivered to the circuit.

Sheh Lit Chang

University of Washington

Problem 37

A resonant circuit in a radio receiver is tuned to a certain station when the inductor has a value of $3.00 \mu \mathrm{H}$ and the capacitor has a value of $2.50 \mathrm{pF}$. The resistance of the circuit is $12 \Omega$. (a) Find the frequency of the radio station. (b) Is there any information given in the problem that is not needed to solve it? Explain.

Sheh Lit Chang

University of Washington

Problem 38

The $L C$ circuit of a radar transmitter oscillates at $9.00 \mathrm{GHz}$. (a) What inductance will resonate with a 2.00-pF capacitor at this frequency? (b) What is the inductive reactance of the circuit at this frequency?

Sheh Lit Chang

University of Washington

Problem 39

The AM band extends from approximately $500 \mathrm{kH} z$ to $1600 \mathrm{kHz}$. If a $2.0-\mu \mathrm{H}$ inductor is used in a tuning circuit for a radio, what are the extremes that a capacitor must reach to cover the complete band of frequencies?

Sheh Lit Chang

University of Washington

Problem 40

Consider a series $R L C$ circuit with $R=15 \Omega, L=200 \mathrm{mH}$, $C=75 \mu \mathrm{F}$, and a maximum voltage of $150 \mathrm{~V}$. (a) What is the impedance of the circuit at resonance? (b) What is the resonance frequency of the circuit? (c) When will the current be greatest: at resonance, at $10 \%$ below the resonant frequency, or at $10 \%$ above the resonant frequency?
(d) What is the rms current in the circuit at a frequency of $60 \mathrm{~Hz}$ ?

Sheh Lit Chang

University of Washington

Problem 41

A $10.0-\Omega$ resistor, a $10.0-\mathrm{mH}$ inductor, and a $100-\mu \mathrm{F}$ capacitor are connected in series to a $50.0-\mathrm{V}(\mathrm{rms})$ source having variable frequency. Find the energy delivered to the circuit during one period if the operating frequency is twice the resonance frequency.

Zulfiqar Ali

Numerade Educator

Problem 42

A series circuit contains a $3.00$ - $\mathrm{H}$ inductor, a $3.00-\mu \mathrm{F}$ capacitor, and a $30.0-\Omega$ resistor connected to a $120-\mathrm{V}$ (rms) source of variable frequency. Find the power delive ered to the circuit when the frequency of the source is
(a) the resonance frequency, (b) one-half the resonance frequency, (c) one-fourth the resonance frequency,
(d) two times the resonance frequency, and (e) four times the resonance frequency. From your calculations, can you draw a conclusion about the frequency at which the maximum power is delivered to the circuit?

Eduard Sanchez

Numerade Educator

Problem 43

The primary coil of a transformer has $N_{1}=250$ turns, and its secondary coil has $N_{2}=1500$ turns. If the input voltage across the primary coil is $\Delta v=(170 \mathrm{~V}) \sin \omega t$, what rms voltage is developed across the secondary coil?

Salamat Ali

Numerade Educator

Problem 44

A step-down transformer is used for recharging the batteries of portable devices. The turns ratio $N_{2} / N_{1}$ for a particular transformer used in a CD player is $1: 13 .$ When used with $120-\mathrm{V}$ (rms) household service, the transformer draws an rms current of $250 \mathrm{~mA}$. Find the (a) rms output voltage of the transformer and (b) power delivered to the CD player.

Averell Hause

Carnegie Mellon University

Problem 45

An AC power generator produces $50 \mathrm{~A}$ (rms) at $3600 \mathrm{~V}$. The voltage is stepped up to $100000 \mathrm{~V}$ by an ideal transformer, and the energy is transmitted through a longdistance power line that has a resistance of $100 \Omega$. What percentage of the power delivered by the generator is dissipated as heat in the power line?

Sheh Lit Chang

University of Washington

Problem 46

A transformer is to be used to provide power for a computer disk drive that needs $6.0 \mathrm{~V}$ (rms) instead of the $120 \mathrm{~V}(\mathrm{rms})$ from the wall outlet. The number of turns in the primary is 400 , and it delivers $500 \mathrm{~mA}$ (the secondary current) at an output voltage of $6.0 \mathrm{~V}(\mathrm{rms}) .$ (a) Should the transformer have more turns in the secondary compared with the primary, or fewer turns? (b) Find the current in the primary. (c) Find the number of turns in the secondary.

Sheh Lit Chang

University of Washington

Problem 47

A transformer on a pole near a factory steps the voltage down from $3600 \mathrm{~V}$ (rms) to $120 \mathrm{~V}$ (rms). The transformer is to deliver $1000 \mathrm{~kW}$ to the factory at $90 \%$ efficiency. Find (a) the power delivered to the primary, (b) the current in the primary, and (c) the current in the secondary.

Sheh Lit Chang

University of Washington

Problem 48

A transmission line that has a resistance per unit length of $4.50 \times 10^{-4} \Omega / \mathrm{m}$ is to be used to transmit $5.00 \mathrm{MW}$ over 400 miles $\left(6.44 \times 10^{5} \mathrm{~m}\right) .$ The output voltage of the generator is $4.50 \mathrm{kV}$ (rms). (a) What is the line loss if a transformer is used to step up the voltage to $500 \mathrm{kV}(\mathrm{rms}) ?$
(b) What fraction of the input power is lost to the line under these circumstances? (c) What difficulties would be encountered on attempting to transmit the $5.00 \mathrm{MW}$ at the generator voltage of $4.50 \mathrm{kV}$ (rms)?

Sheh Lit Chang

University of Washington

Problem 49

The U.S. Navy has long proposed the construction of extremely low frequency (ELF waves) communications systems; such waves could penetrate the oceans to reach distant submarines. Calculate the length of a quarterwavelength antenna for a transmitter generating ELF waves of frequency $75 \mathrm{~Hz}$. How practical is this antenna?

Sheh Lit Chang

University of Washington

Problem 50

(a) The distance to Polaris, the North Star, is approximately $6.44 \times 10^{18} \mathrm{~m}$. If Polaris were to burn out today, how many years would it take to see it disappear? (b) How long does it take sunlight to reach Earth? (c) How long does it take a microwave signal to travel from Earth to the Moon and back? (The distance from Earth to the Moon is $\left.3.84 \times 10^{5} \mathrm{~km} .\right)$

Averell Hause

Carnegie Mellon University

Problem 51

An electromagnetic wave in free space has an electric field of amplitude $330 \mathrm{~V} / \mathrm{m}$. Find the amplitude of the corresponding magnetic field.

Sheh Lit Chang

University of Washington

Problem 52

Experimenters at the National Institute of Standards and Technology have made precise measurements of the speed of light using the fact that, in vacuum, the speed of electromagnetic waves is $c=1 / \sqrt{\mu_{0} \epsilon_{1}}$, where the constants $\mu_{0}=4 \pi \times 10^{-7} \mathrm{~N} \cdot \mathrm{s}^{2} / \mathrm{C}^{2}$ and $\epsilon_{0}=8.854 \times 10^{-12}$
$\mathrm{C}^{2} / \mathrm{N} \cdot \mathrm{m}^{2}$. What value (to four significant figures) does this formula give for the speed of light in vacuum?

Sheh Lit Chang

University of Washington

Problem 53

Oxygenated hemoglobin absorbs weakly in the red (hence its red color) and strongly in the near infrared, whereas deoxygenated hemoglobin has the opposite absorption. This fact is used in a "pulse oximeter" to measure oxygen saturation in arterial blood. The device clips onto the end of a person's finger and has two light-emitting diodes $-\mathrm{a}$ red $(660 \mathrm{~nm})$ and an infrared $(940 \mathrm{~nm})-$ and a photocell that detects the amount of light transmitted through the finger at each wavelength. (a) Determine the frequency of each of these light sources. (b) If $67 \%$ of the energy of the red source is absorbed in the blood, by what factor does the amplitude of the electromagnetic wave change? Hint: The intensity of the wave is equal to the average power per unit area as given by Equation $21.28$.

Sheh Lit Chang

University of Washington

Problem 54

The transmission of light energy as it passes through a solution of light-absorbing molecules is described by the Beer-lambert law $$
I=l_{0} 10^{-c a} \quad \text { or } \quad \log _{10}\left(\frac{I}{I_{0}}\right)=-\epsilon C L
$$
which gives the decrease in intensity $I$ in terms of the distance $L$ the light has traveled through a fluid with a concentration $C$ of the light-absorbing molecule. The quantity $\epsilon$ is called the extinction coefficient, and its value depends on the frequency of the light. (It has units of $\mathrm{m}^{2} / \mathrm{mol} .$ ) Assume the extinction coefficient for $660-\mathrm{nm}$ light passing through a solution of oxygenated hemoglobin is identical to the coefficient for $940-\mathrm{nm}$ light passing through deoxygenated hemoglobin. Also assume $940-\mathrm{nm}$ light has zero absorption $(\epsilon=0)$ in oxygenated hemoglobin and $660-\mathrm{nm}$ light has zero absorption in deoxygenated hemoglobin. If $33 \%$ of the energy of the red source and $76 \%$ of the infrared energy is transmitted through the blood, what is the fraction of hemoglobin that is oxygenated?

Eduard Sanchez

Numerade Educator

Problem 55

The Sun delivers an average power of $1.340 \mathrm{~W} / \mathrm{m}^{2}$ to the top of Earth's atmosphere. Find the magnitudes of $\overrightarrow{\mathbf{E}}_{\text {mat }}$ and $\overrightarrow{\mathbf{B}}_{\text {max }}$ for the electromagnetic waves at the top of the atmosphere.

Sheh Lit Chang

University of Washington

Problem 56

A laser beam is used to levitate a metal disk against the force of Earth's gravity. (a) Derive an equation giving the required intensity of light, $I$, in terms of the mass $m$ of the disk, the gravitational acceleration $g$, the speed of light $c$, and the cross-sectional area of the disk $A$. Assume the disk is perfectly reflecting and the beam is directed perpendicular to the disk. (b) If the disk has mass $5.00 \mathrm{~g}$ and radius $4.00 \mathrm{~cm}$, find the necessary light intensity.
(c) Give two reasons why using light pressure as propulsion near Earth's surface is impractical.

Eduard Sanchez

Numerade Educator

Problem 57

A microwave oven is powered by an electron tube called a magnetron that generates electromagnetic waves of frequency $2.45 \mathrm{GH} z$. The microwaves enter the oven and are reflected by the walls. The standing-wave pattern produced in the oven can cook food unevenly, with hot spots in the food at antinodes and cool spots at nodes, so a turntable is often used to rotate the food and distribute the energy. If a microwave oven is used with a cooking dish in a fixed position, the antinodes can appear as burn marks on foods such as carrot strips or cheese. The separation distance between the burns is measured to be $6.00 \mathrm{~cm}$. Calculate the speed of the microwaves from these data.

Sheh Lit Chang

University of Washington

Problem 58

Assume the solar radiation incident on Earth is $1340 \mathrm{~W} / \mathrm{m}^{2}$ (at the top of Earth's atmosphere). Calculate the total power radiated by the Sun, taking the average separation between Earth and the Sun to be $1.49 \times 10^{11} \mathrm{~m}$.

Sheh Lit Chang

University of Washington

Problem 59

The eye is most sensitive to light of wavelength $5.50 \times 10^{-7} \mathrm{~m}$, which is in the green-yellow region of the yisible electromagnetic spectrum. What is the frequency of this light?

Sheh Lit Chang

University of Washington

Problem 60

A diathermy machine, used in physiotherapy, generates electromagnetic radiation that gives the effect of "deep heat" when absorbed in tissue. One assigned frequency for diathermy is $27.33 \mathrm{MHz}$. What is the wavelength of this radiation?

Averell Hause

Carnegie Mellon University

Problem 61

What are the wavelength ranges in (a) the AM radio band $(540-1600 \mathrm{kHz})$ and (b) the FM radio band $(88-$ $108 \mathrm{MHz}$ )?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 62

An important news announcement is transmitted by radio waves to people who are $100 \mathrm{~km}$ away, sitting next to their radios, and by sound waves to people sitting across the newsroom, $3.0 \mathrm{~m}$ from the newscaster. Who receives the news first? Explain. Take the speed of sound in air to be $343 \mathrm{~m} / \mathrm{s}$

Sheh Lit Chang

University of Washington

Problem 63

Infrared spectra are used by chemists to help identify an unknown substance. Atoms in a molecule that are bound together by a particular bond vibrate at a predictable frequency, and light at that frequency is absorbed strongly by the atom. In the case of the $\mathrm{C}=\mathrm{O}$ double bond, for example, the oxygen atom is bound to the carbon by a bond that has an effective spring constant of $2800 \mathrm{~N} / \mathrm{m}$. If we assume the carbon atom remains stationary (it is attached to other atoms in the molecule), determine the resonant frequency of this bond and the wavelength of light that matches that frequency. Verify that this wavelength lies in the infrared region of the spectrum. (The mass of an oxygen atom is $2.66 \times 10^{-26} \mathrm{~kg}$.

Sheh Lit Chang

University of Washington

Problem 64

A spaceship is approaching a space station at a speed of $1.8 \times 10^{5} \mathrm{~m} / \mathrm{s}$. The space station has a beacon that emits green light with a frequency of $6.0 \times 10^{14} \mathrm{~Hz}$. What is the frequency of the beacon observed on the spaceship? What is the change in frequency? (Carry five digits in these calculations.)

Sheh Lit Chang

University of Washington

Problem 65

While driving at a constant speed of $80 \mathrm{~km} / \mathrm{h}$, you are passed by a car traveling at $120 \mathrm{~km} / \mathrm{h}$. If the frequency of light emitted by the taillights of the car that passes you is $4.3 \times 10^{14} \mathrm{~Hz}$, what frequency will you observe? What is the change in frequency?

Sheh Lit Chang

University of Washington

Problem 66

A speeder tries to explain to the police that the yellow warning lights on the side of the road looked green to her because of the Doppler shift. How fast would she have been traveling if yellow light of wavelength $580 \mathrm{~nm}$ had been shifted to green with a wavelength of $560 \mathrm{~nm}$ ? Note:
For speeds less than $0.03 c$, Equation $21.32$ will lead to a value for the change of frequency accurate to approximately two significant digits.

Sheh Lit Chang

University of Washington

Problem 67

A $50.0-\Omega$ resistor is connected in series with a $15.0-\mu \mathrm{F}$ capacitor and a $60.0-\mathrm{Hz}, 1.20 \times 10^{2}-\mathrm{V}$ (rms) source. Find the (a) impedance of the circuit and (b) rms current in the circuit. (c) What is the value of the inductor that must be inserted in the circuit to reduce the current to onehalf that found in part (b)?

Sheh Lit Chang

University of Washington

Problem 68

The intensity of solar radiation at the top of Earth's atmosphere is $1340 \mathrm{~W} / \mathrm{m}^{3}$. Assuming $60 \%$ of the incoming solar energy reaches Earth's surface and assuming you absorb $50 \%$ of the incident energy, make an order-ofmagnitude estimate of the amount of solar energy you absorb in a 60 -minute sunbath.

Eduard Sanchez

Numerade Educator

Problem 69

A $200-\Omega$ resistor is connected in series with a $5.0-\mu \mathrm{F}$ capacitor and a $60-\mathrm{Hz}, 120-\mathrm{V}$ rms line. If electrical energy costs $\$ 0.080 / \mathrm{kWh}$, how much does it cost to leave this circuit connected for $24 \mathrm{~h} ?$

Salamat Ali

Numerade Educator

Problem 70

A series RLC circuit has a resonance frequency of $2000 / \pi \mathrm{Hz} .$ When it is operating at a frequency of
$\omega>\omega_{0}, X_{L}=12 \Omega$ and $X_{C}=8.0 \Omega .$ Calculate the values of $L$ and $C$ for the circuit.

Sheh Lit Chang

University of Washington

Problem 71

As a way of determining the inductance of a coil used in a research project, a student first connects the coil to a $12.0-\mathrm{V}$ battery and measures a current of $0.630 \mathrm{~A}$. The student then connects the coil to a $24.0-\mathrm{V}(\mathrm{rms}), 60.0-\mathrm{Hz}$ generator and measures an rms current of $0.570 \mathrm{~A}$. What is the inductance?

Salamat Ali

Numerade Educator

Problem 72

(a) What capacitance will resonate with a one-turn loop of inductance $400 \mathrm{pH}$ to give a radar wave of wavelength $3.0 \mathrm{~cm}$ ? (b) If the capacitor has square parallel plates separated by $1.0 \mathrm{~mm}$ of air, what should the edge length of the plates be? (c) What is the common reactance of the loop and capacitor at resonance?

Averell Hause

Carnegie Mellon University

Problem 73

A dish antenna with a diameter of $20.0 \mathrm{~m}$ receives (at normal incidence) a radio signal from a distant source, as shown in Figure $\mathrm{P} 21.73 .$ The radio signal is a continuous sinusoidal wave with amplitude $E_{\max }=0.20 \mu \mathrm{V} / \mathrm{m}$. Assume the antenna absorbs all the radiation that falls on the dish. (a) What is the amplitude of the magnetic field in this wave? (b) What is the intensity of the radiation received by the antenna? (c) What is the power received by the antenna?

Sheh Lit Chang

University of Washington

Problem 74

A particular inductor has appreciable resistance. When the inductor is connected to a $12-\mathrm{V}$ battery, the current in the inductor is $3.0 \mathrm{~A}$. When it is connected to an $\mathrm{AC}$ source with an rms output of $12 \mathrm{~V}$ and a frequency of $60 \mathrm{~Hz}$, the current drops to $2.0 \mathrm{~A}$. What are (a) the impedance at $60 \mathrm{~Hz}$ and (b) the inductance of the inductor?

Averell Hause

Carnegie Mellon University

Problem 75

One possible means of achieving space flight is to place a perfectly reflecting aluminized sheet into Earth's orbit and to use the light from the Sun to push this solar sail. Suppose such a sail, of area $6.00 \times 10^{4} \mathrm{~m}^{2}$ and mass $6000 \mathrm{~kg}$, is placed in orbit facing the Sun. (a) What force is exerted on the sail? (b) What is the sail's acceleration?
(c) How long does it take this sail to reach the Moon, $3.84 \times 10^{8} \mathrm{~m}$ away? Ignore all gravitational effects and assume a solar intensity of $1340 \mathrm{~W} / \mathrm{m}^{2}$. Hint: The radiation pressure by a reflected wave is given by 2 (average power per unit area) $/ c .$

Sheh Lit Chang

University of Washington

Problem 76

The U.S. Food and Drug Administration limits the radiation leakage of microwave ovens to no more than $5.0 \mathrm{~mW} / \mathrm{cm}^{2}$ at a distance of $2.0$ in. A typical cell phone, which also transmits microwaves, has a peak output power of about $2.0 \mathrm{~W}$. (a) Approximating the cell phone as a point source, calculate the radiation intensity of a cell phone at a distance of $2.0 \mathrm{in}$. How does the answer compare with the maximum allowable microwave oven leakage? (b) The distance from your ear to your brain is about 2 in. What would the radiation intensity in your brain be if you used a Bluetooth headset, keeping the phone in your pocket, $1.0 \mathrm{~m}$ away from your brain? Most headsets are so-called Class 2 devices with a maximum output power of $2.5 \mathrm{~mW}$.

Salamat Ali

Numerade Educator