Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 30

Inductance, Electromagnetic Oscillations, and AC Circuits - all with Video Answers

Educators

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Chapter Questions

Problem 1

(II) A 2.44 -m-long coil containing 225 loops is wound on an iron core (average $\mu = 1850 \mu _ { 0 } )$ along with a second coil of 115 loops. The loops of each coil have a radius of 2.00$\mathrm { cm } .$ If the current in the first coil drops uniformly from 12.0$\mathrm { A }$ to zero in $98.0 \mathrm { ms } ,$ determine: $( a )$ the mutual inductance $M ; ( b )$ the emf induced in the second coil.

Ze-Han Lee

Ze-Han Lee

Numerade Educator

Problem 2

(II) Determine the mutual inductance per unit length between two long solenoids, one inside the other, whose radii are $r _ { 1 }$ and $r _ { 2 } \left( r _ { 2 } < r _ { 1 } \right)$ and whose turns per unit length are $n _ { 1 }$ and $n _ { 2 } .$

Nolan Smyth

Numerade Educator

Problem 3

(II) A small thin coil with $N _ { 2 }$ loops, each of area $A _ { 2 }$ is placed inside a long solenoid, near its center. The solenoid has $N _ { 1 }$ loops in its length $\ell$ and has area $A _ { 1 }$ . Determine the mutual inductance as a function of $\theta ,$ the angle between the plane of the small coil and the axis of the solenoid.

Ze-Han Lee

Numerade Educator

Problem 4

(III) A long straight wire and a small rectangular wire loop lie in the same plane, Fig. $25 .$ Determine the mutual inductance in terms of $\ell _ { 1 } , \ell _ { 2 } ,$ and $w .$ Assume the wire is very long compared to $\ell _ { 1 } , \ell _ { 2 } ,$ and $w ,$ and that the rest of its circuit is veryfar away compared to $\ell _ { 1 } , \ell _ { 2 }$ and $w .$

Anand Jangid

Numerade Educator

Problem 5

(I) If the current in a $280 - \mathrm { mH }$ coil changes steadily from 25.0$\mathrm { A }$ to 10.0$\mathrm { A }$ in $360 \mathrm { ms } ,$ what is the magnitude of the induced emf?

Ze-Han Lee

Numerade Educator

Problem 6

(I) How many turns of wire would be required to make 130 -mH inductance out of a $30.0 -$ cm-long air-filled coil with a diameter of 4.2$\mathrm { cm } ?$

Nolan Smyth

Numerade Educator

Problem 7

(I) What is the inductance of a coil if the coil produces an emf of 2.50$\mathrm { V }$ when the current in it changes from $- 28.0 \mathrm { mA }$ to $+ 25.0 \mathrm { mA }$ in 12.0$\mathrm { ms }$ ?

Ze-Han Lee

Numerade Educator

Problem 8

(II) An air-filled cylindrical inductor has 2800 turns, and it is 2.5$\mathrm { cm }$ in diameter and 21.7$\mathrm { cm }$ long. $( a )$ What is its inductance? (b) How many turns would you need to generate the same inductance if the core were filled with iron of magnetic permeability 1200 times that of free space?

Nolan Smyth

Numerade Educator

Problem 9

(II) A coil has $3.25 - \Omega$ resistance and $440 - \mathrm { mH }$ inductance. If the current is 3.00$\mathrm { A }$ and is increasing at a rate of 3.60$\mathrm { A } / \mathrm { s }$ what is the potential difference across the coil at this moment?

Ze-Han Lee

Numerade Educator

Problem 10

(II) If the outer conductor of a coaxial cable has radius $3.0 \mathrm { mm } ,$ what should be the radius of the inner conductor so that the inductance per unit length does not exceed 55$\mathrm { nH }$ per meter?

Nolan Smyth

Numerade Educator

Problem 11

(II) To demonstrate the large size of the henry unit, a physics professor wants to wind an air-filled solenoid with self-inductance of 1.0$\mathrm { H }$ on the outside of a 12 -cm diameter plastic hollow tube using copper wire with a0.81$\cdot$ mm diameter. The solenoid is to be tightly wound with each turn touching its neighbor (the wire has a thin insulating layer on its surface so the neighboring turns are not in electrical contact). How long will the plastic tube need to be and how many kilometers of copper wire will be required? What will be the resistance of this solenoid?

Ze-Han Lee

Numerade Educator

Problem 12

(II) The wire of a tightly wound solenoid is unwound and used to make another tightly wound solenoid of 2.5 times the diameter. By what factor does the inductance change?

Nolan Smyth

Numerade Educator

Problem 13

(II) A toroid has a rectangular cross section as shown in Fig. $26 .$ Show that the self-inductance is $L = \frac { \mu _ { 0 } N ^ { 2 } h } { 2 \pi } \ln \frac { r _ { 2 } } { r _ { 1 } }$ where $N$ is the total number of turns and $r _ { 1 } , r _ { 2 } ,$ and $h$ are the dimensions shown in Fig. $26 . [$ Hint: Use Ampere's law toget $B$ as a function of $r$ inside the toroid, and integrate. $]$

Ze-Han Lee

Numerade Educator

Problem 14

(II) Ignoring any mutual inductance, what is the equivalent inductance of two inductors connected (a) in series, $( b )$ in parallel?

Nolan Smyth

Numerade Educator

Problem 15

(I) The magnetic field inside an air-filled solenoid 38.0$\mathrm { cm }$ long and 2.10$\mathrm { cm }$ in diameter is 0.600 T. Approximately how much energy is stored in this field?

Ze-Han Lee

Numerade Educator

Problem 16

(I) Typical large values for electric and magnetic fields attained in laboratories are about $1.0 \times 10 ^ { 4 } \mathrm { V } / \mathrm { m }$ and 2.0$\mathrm { T }$ . (a) Determine the energy density for each field and compare. (b) What magnitude electric field would be needed to produce the same energy density as the $2.0 - \mathrm { T }$ magnetic field?

Nolan Smyth

Numerade Educator

Problem 17

(II) What is the energy density at the center of a circular loop of wire carrying a $23.0 -$ A current if the radius of the loop is 28.0$\mathrm { cm } ?$

Ze-Han Lee

Numerade Educator

Problem 18

(II) Calculate the magnetic and electric energy densities at the surface of a 3.0 -mm-diameter copper wire carrying a 15 - A current.

Nolan Smyth

Numerade Educator

Problem 19

(II) For the toroid of Fig. $26 ,$ determine the energy density in the magnetic field as a function of $r \left( r _ { 1 } < r < r _ { 2 } \right)$ and integrate this over the volume to obtain the total energy stored in the toroid, which carries a current $I$ in each of its $N$ loops..

Ze-Han Lee

Numerade Educator

Problem 20

(II) Determine the total energy stored per unit length in the magnetic field between the coaxial cylinders of a coaxial cable (Fig. 5) by using Eq. 7 for the energy density and integrating over the volume. $u =$ energy density $= \frac { 1 } { 2 } \frac { B ^ { 2 } } { \mu _ { 0 } }$

Nolan Smyth

Numerade Educator

Problem 21

(II) A long straight wire of radius $R$ carries current $I$ uniformly distributed across its cross-sectional area. Find the magnetic energy stored per unit length in the interior of this wire.

Ze-Han Lee

Numerade Educator

Problem 22

(II) After how many time constants does the current in Fig. 6 reach within $( a ) 5.0 \% , ( b ) 1.0 \% ,$ and $( c ) 0.10 \%$ of itsmaximum value?

Nolan Smyth

Numerade Educator

Problem 23

(II) How many time constants does it take for the potential difference across the resistor in an $L R$ circuit like that in Fig. 7 to drop to 3.0$\%$ of its original value?

Ze-Han Lee

Numerade Educator

Problem 24

(II) It takes 2.56$\mathrm { ms }$ for the current in an $L R$ circuit to increase from zero to 0.75 its maximum value. Determine $( a )$ the time constant of the circuit, $b$ ) the resistance of the circuit if $L = 31.0 \mathrm { mH } .$

Nolan Smyth

Numerade Educator

Problem 25

(II) $( a )$ Determine the energy stored in the inductor $L$ as a function of time for the $L R$ circuit of Fig. 6a. (b) After how many time constants does the stored energy reach 99.9$\%$ of its maximum value?

Ze-Han Lee

Numerade Educator

Problem 26

(II) In the circuit of Fig. $27 ,$ determine the current in each resistor $\left( I _ { 1 } , I _ { 2 } , I _ { 3 } \right)$ at the moment $( a )$ the switch is closed, (b) a long time after the switch is closed. After the switch has been closed for a long time, and then reopened, what is each current $( c )$ just after it is opened, (d) after a long time?

Lainey Roebuck

Numerade Educator

Problem 27

(II) $( a )$ In Fig. $28 \mathrm { a } ,$ assume that the switch $\mathrm { S }$ has been in position $\mathrm { A }$ for sufficient time so that a steady current
$I _ { 0 } = V _ { 0 } / R$ flows through the resistor $R .$ At time $t = 0 ,$ the
switch is quickly switched to position $\mathrm { B }$ and the current through $R$ decays according to $I = I _ { 0 } e ^ { - t / \tau } .$ Show that the maximum emf $\mathscr { X } _ { \text { max induced in the inductor during this time } }$ period equals the battery voltage $V _ { 0 . }$ (b) In Fig, $28 b ,$ assume that the switch has been in position A for sufficient time so that a steady current $I _ { 0 } = V _ { 0 } / R$ flows through the resistor
$R .$ At time $t = 0 ,$ the switch is quickly switched to position B and the current decays through resistor $R ^ { \prime }$ (which is much greater than $R$ ) according the maximum emf $\varepsilon _ { \text { max } }$
induced in the inductor during this time period is $\left( R ^ { \prime } / R \right) V _ { 0 . }$ If $R ^ { \prime } = 55 R$ and $V _ { 0 } = 120 \mathrm { V } ,$ determine $\mathscr { Y } _ { \max } .$ [When a mechanical switch is opened, a high-resistance air gap is created, which is modeled as $R ^ { \prime }$ here. This Problem illustrates why high-voltage sparking can occur if a current-carrying inductor is suddenly cut off from its power source.

Ze-Han Lee

Numerade Educator

Problem 28

(II) You want to turn on the current through a coil of self- inductance $L$ in a controlled manner, so you place it in series with a resistor $R = 2200 \Omega ,$ a switch, and a dc voltage source $V _ { 0 } = 240 \mathrm { V }$ . After closing the switch, you find that the current through the coil builds up to its steady-state value with a time constant $\tau .$ You are pleased with the current's steady-state value, but want $\tau$ to be half as long. What new values should you use for $R$ and $V _ { 0 } ?$

Nolan Smyth

Numerade Educator

Problem 29

(II) A $12 - \mathrm { V }$ battery has been connected to an $L R$ circuit for
sufficient time so that a steady current flows through the resistor $R = 2.2 \mathrm { k } \Omega$ and inductor $L = 18 \mathrm { mH } .$ At $t = 0$ the battery is removed from the circuit and the current decays exponentially through $R$ . Determine the emf $\mathscr { Y }$ across the inductor as time $t$ increases. At what time is 8 greatest and what is this maximum value (V)?

Ze-Han Lee

Numerade Educator

Problem 30

(III) Two tightly wound solenoids have the same length and circular cross-sectional area. But solenoid 1 uses wire that is 1.5 times as thick as solenoid $2 .$ (a) What is the ratio of theirinductances? (b) What is the ratio of their inductive time constants (assuming no other resistance in the circuits)?

Keshav Singh

Numerade Educator

Problem 31

(I) The variable capacitor in the tuner of an $A M$ radio has a capacitance of 1350 pF when the radio is tuned to a station at 550$\mathrm { kHz }$ (a) What must be the capacitance for a station at 1600$\mathrm { kHz }$ ( b) What is the inductance (assumed constant)? Ignore resistance.

Ze-Han Lee

Numerade Educator

Problem 32

(I) $( a )$ If the initial conditions of an $L C$ circuit were $I = I _ { 0 }$ and $Q = 0$ at $t = 0 ,$ write $Q$ as a function of time. (b) Practically, how could you set up these initial conditions?

Nolan Smyth

Numerade Educator

Problem 33

(II) In some experiments, short distances are measured by using capacitance. Consider forming an $L C$ circuit using a parallel-plate capacitor with plate area $A ,$ and a known inductance $L . ( a )$ If charge is found to oscillate in this circuit at frequency $f = \omega / 2 \pi$ when the capacitor plates are separated by distance $x ,$ show that $x = 4 \pi ^ { 2 } A \epsilon _ { 0 } f ^ { 2 } L$ . (b) When the plate separation is changed by $\Delta x ,$ the circuit's oscillation frequency will change by $\Delta f .$ Show that $\Delta x / x \approx 2 ( \Delta f / f ) . ( c )$ If $f$ is on the order of 1$\mathrm { MHz }$ and can
be measured to a precision of $\Delta f = 1 \mathrm { Hz } ,$ with what percent accuracy can $x$ be determined? Assume fringing effects at the capacitor's edges can be neglected.

Ze-Han Lee

Numerade Educator

Problem 34

(II) A 425 -pF capacitor is charged to 135$\mathrm { V }$ and then quickly connected to a 175 -mH inductor. Determine $( a )$ the frequency of oscillation, $( b )$ the peak value of the current, and $( c )$ the maximum energy stored in the magnetic field of the inductor.

Nolan Smyth

Numerade Educator

Problem 35

(II) At $t = 0 ,$ let $Q = Q _ { 0 } ,$ and $I = 0$ in an $L C$ circuit. (a) At the first moment when the energy is shared equally by the inductor and the capacitor, what is the charge on the capacitor? (b) How much time has elapsed (in terms of the period $T ) ?$

Ze-Han Lee

Numerade Educator

Problem 36

(II) A damped $L C$ circuit loses 3.5$\%$ of its electromagnetic energy per cycle to thermal energy. If $L = 65 \mathrm { mH }$ and $C = 1.00 \mu \mathrm { F } ,$ what is the value of $R ?$

Nolan Smyth

Numerade Educator

Problem 37

(II) In an oscillating $L R C$ circuit, how much time does it take for the energy stored in the fields of the capacitor and inductor to fall to 75$\%$ of the initial value? (See Fig. $13 ;$ assume $R \ll \sqrt { 4 L / C }$ )

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 38

(III) How much resistance must be added to a pure $L C$ circuit $( L = 350 \mathrm { mH } , C = 1800 \mathrm { pF } )$ to change the oscillator's frequency by 0.25$\% ?$ Will it be increased or decreased?

Nolan Smyth

Numerade Educator

Problem 39

(I) At what frequency will a $32.0 - \mathrm { mH }$ inductor have a reactance of 660$\Omega ?$

Ze-Han Lee

Numerade Educator

Problem 40

(I) What is the reactance of a $9.2 - \mu \mathrm { F }$ capacitor at a frequency of $( a ) 60.0 \mathrm { Hz } , ( b ) 1.00 \mathrm { MHz }$ ?

Nolan Smyth

Numerade Educator

Problem 41

(I) Plot a graph of the reactance of a $1.0 - \mu \mathrm { F }$ capacitor as a function of frequency from 10$\mathrm { Hz }$ to 1000$\mathrm { Hz }$ .

Ze-Han Lee

Numerade Educator

Problem 42

(I) Calculate the reactance of, and rms current in, a 36.0 -mH radio coil connected to a $250 - \mathrm { V } ( \mathrm { rms } ) 33.3 - \mathrm { kHz }$ ac line. Ignore resistance.

Nolan Smyth

Numerade Educator

Problem 43

(II) A resistor $R$ is in parallel with a capacitor $C ,$ and this parallel combination is in series with a resistor $R ^ { \prime }$ . If connected to an ac voltage source of frequency $\omega ,$ what is the equivalent impedance of this circuit at the two extremes in frequency $( a ) \omega = 0 ,$ and $( b ) \omega = \infty ?$

Ze-Han Lee

Numerade Educator

Problem 44

(II) What is the inductance $L$ of the primary of a transformer whose input is 110$\mathrm { V }$ at 60$\mathrm { Hz }$ and the current drawn is 3.1$\mathrm { A }$ ? Assume no current in the secondary.

Nolan Smyth

Numerade Educator

Problem 45

(II) $( a )$ What is the reactance of a $0.086 - \mu F$ capacitor connected to a $22 - \mathrm { kV } ( \mathrm { rms } ) , 660 - \mathrm { Hz }$ line? $( b )$ Determine the frequency and the peak value of the current.

Ze-Han Lee

Numerade Educator

Problem 46

(II) A capacitor is placed in parallel with some device, B as in Fig. $18 b ,$ to filter out stray high frequency signals, but to allow ordinary $60 - H z$ ac to pass through with little loss. Suppose that circuit $B$ in Fig. 18$b$ is a resistance $R = 490 \Omega$ connected to ground, and that $C = 0.35 \mu \mathrm { F }$ . What percent of the incoming current will pass through $C$ rather than $R$ if it is $( a )$ it is 60$\mathrm { Hz }$ ; (b) $60,000 \mathrm { Hz }$ ?

Narayan Hari

Numerade Educator

Problem 47

(II) A current $I = 1.80 \cos 377 t \quad ( I$ in amps, $t$ in seconds, and the "angle" is in radians) flows in a series $L R$ circuit in which $L = 3.85 \mathrm { mH }$ and $R = 1.35 \mathrm { k } \Omega .$ What is the average power dissipation?

Ze-Han Lee

Numerade Educator

Problem 48

(I) A $10.0 - \mathrm { k } \Omega$ resistor is in series with a $26.0 - \mathrm { mH }$ inductor and an ac source. Calculate the impedance of the circuit if the source frequency is $( a ) 55.0 \mathrm { Hz } ; ( b ) 55,000 \mathrm { Hz }$ .

Nolan Smyth

Numerade Educator

Problem 49

(I) $\mathrm { A } 75 - \Omega$ resistor and a $6.8 - \mu \mathrm { F }$ capacitor are connected in series to an ac source. Calculate the impedance of the circuit if the source frequency is $( a ) 60 \mathrm { Hz } ; ( b ) 6.0 \mathrm { MHz }$ .

Ze-Han Lee

Numerade Educator

Problem 50

(I) For a $120 - \mathrm { V } , 60 - \mathrm { Hz }$ voltage, a current of 70$\mathrm { mA }$ passing through the body for 1.0 s could be lethal. What must be the impedance of the body for this to occur?

Nolan Smyth

Numerade Educator

Problem 51

(II) A 2.5 -k\Omega resistor in series with a $420 - \mathrm { mH }$ inductor is driven by an ac power supply. At what frequency is the impedance double that of the impedance at 60$\mathrm { Hz }$ ?

Ze-Han Lee

Numerade Educator

Problem 52

(II) $( a )$ What is the rms current in a series $R C$ circuit if $R = 3.8 \mathrm { k } \Omega , \quad C = 0.80 \mu \mathrm { F } ,$ and the rms applied voltage is 120$\mathrm { V }$ at 60.0$\mathrm { Hz }$ ? (b) What is the phase angle between voltage and current? (c) What is the power dissipated by the circuit? (d) What are the voltmeter readings across $R$ and $C$ ?

Nolan Smyth

Numerade Educator

Problem 53

(II) An ac voltage source is connected in series with a $1.0 - \mu \mathrm { F }$ capacitor and a $750 - \Omega$ resistor. Using a digital ac voltmeter, the amplitude of the voltage source is measured to be 4.0$\mathrm { V }$ rms, while the voltages across the resistor and across the capacitor are found to be 3.0$\mathrm { V }$ rms and 2.7$\mathrm { V }$ rms, respectively. Determine the frequency of the ac voltage source. Why is the voltage measured across the voltage source not equal to the sum of the voltages measured across the resistor and across the capacitor?

Ze-Han Lee

Numerade Educator

Problem 54

(II) Determine the total impedance, phase angle, and rms current in an $L R C$ circuit connected to a 10.0 -kHz $725 - \mathrm { V } ( \mathrm { rms } )$ source if $L = 32.0 \mathrm { mH } , \quad R = 8.70 \mathrm { k } \Omega ,$ and $C = 6250 \mathrm { pF }$

Nolan Smyth

Numerade Educator

Problem 55

(II) $( a )$ What is the rms current in a series $L R$ circuit when a 60.0 -Hz, $120 - \mathrm { V }$ rms ac voltage is applied, where $R = 965 \Omega$ and $L = 225 \mathrm { mH } ? \quad$ (b) What is the phase angle between voltage and current? (c) How much power is dissipated? (d) What are the rms voltage readings across $R$ and $L ?$

Ze-Han Lee

Numerade Educator

Problem 56

(II) $\mathrm { A } 35 - \mathrm { mH }$ inductor with $2.0 - \Omega$ resistance is connected in series to a $26 - \mu \mathrm { F }$ capacitor and a $60 - \mathrm { Hz } , 45 - \mathrm { V } ( \mathrm { rms } )$ source. Calculate $( a )$ the rms current, $( b )$ the phase angle, and $( c )$ the power dissipated in this circuit.

Nolan Smyth

Numerade Educator

Problem 57

(II) A $25 - \mathrm { mH }$ coil whose resistance is 0.80$\Omega$ is connected to a capacitor $C$ and a 360 -Hz source voltage. If the current and voltage are to be in phase, what value must $C$ have?

Ze-Han Lee

Numerade Educator

Problem 58

(II) A $75 - \mathrm { W }$ lightbulb is designed to operate with an applied ac voltage of 120$\mathrm { V } \mathrm { rms }$ . The bulb is placed in series with an inductor $L ,$ and this series combination is then connected to a $60 - \mathrm { Hz } 240 \mathrm { -V }$ rms voltage source. For the bulb to operate properly, determine the required value for $L .$ Assume the bulb has resistance $R$ and negligible inductance.

Nolan Smyth

Numerade Educator

Problem 59

(II) In the $L R C$ circuit of Fig. 19 , suppose $I = I _ { 0 } \sin \omega t$ and $V = V _ { 0 } \sin ( \omega t + \phi ) .$ Determine the instantaneous power dissipated in the circuit from $P = I V$ using these equations and show that on the average, $\overline { P } = \frac { 1 } { 2 } V _ { 0 } I _ { 0 } \cos \phi ,$ which confirms Eq. $30 .$

Ze-Han Lee

Numerade Educator

Problem 60

(II) An $L R C$ series circuit with $R = 150 \Omega , L = 25 \mathrm { mH }$ and $C = 2.0 \mu \mathrm { F }$ is powered by an ac voltage source of peak voltage $V _ { 0 } = 340 \mathrm { V }$ and frequency $f = 660 \mathrm { Hz }$ . (a) Determine the peak current that flows in this circuit. (b) Determine the phase angle of the source voltage relative to the current. (c) Determine the peak voltage across $R$ and its phase angle relative to the source voltage. (d) Determine the peak voltage across $L$ and its phase angle relative to the source voltage. (e) Determine the peak voltage across $C$ and its phase angle relative to the source voltage.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 61

(II) An $L R$ circuit can be used as a "phase shifter." Assume that an "input" source voltage $V = V _ { 0 } \sin ( 2 \pi f t + \phi )$ is connected across a series combination of an inductor $L = 55 \mathrm { mH }$ and resistor $R .$ The "output" of this circuit is taken across the resistor. If $V _ { 0 } = 24 \mathrm { V }$ and $f = 175 \mathrm { Hz }$ determine the value of $R$ so that the output voltage $V _ { R }$ lags the input voltage $V$ by $25 ^ { \circ } .$ Compare (as a ratio) the peak output voltage with $V _ { 0 } .$

Ze-Han Lee

Numerade Educator

Problem 62

(I) $\mathrm { A } 3800$ -pF capacitor is connected in series to a $26.0 - \mu \mathrm { H }$ coll of resistance 2.00$\Omega .$ What is the resonant frequency of this circuit?

Nolan Smyth

Numerade Educator

Problem 63

(I) What is the resonant frequency of the $L R C$ circuit of Example 11 of "Inductance, Electromagnetic Oscillations, and AC Circuits"? At what rate is energy taken from the generator, on the average, at this frequency?

Ze-Han Lee

Numerade Educator

Problem 64

(II) An $L R C$ circuit has $L = 4.15 \mathrm { mH }$ and $R = 3.80 \mathrm { k } \Omega$ . (a) What value must $C$ have to produce resonance at 33.0$\mathrm { kHz }$ (b) What will be the maximum current at resonance if the peak external voltage is 136$\mathrm { V } ?$

Nolan Smyth

Numerade Educator

Problem 65

(II) The frequency of the ac voltage source (peak voltage $V _ { 0 }$ ) in an $L R C$ circuit is tuned to the circuit's resonant frequency $f _ { 0 } = 1 / ( 2 \pi \sqrt { L C } ) . ( a )$ Show that the peak voltage across the capacitor is $V _ { C 0 } = V _ { 0 } T _ { 0 } / 2 \pi \tau ) ,$ where $T _ { 0 } \left( = 1 / f _ { 0 } \right)$ is the period of the resonant frequency and $\tau = R C$ is the time constant for charging the capacitor $C$ through a resistor $R . ( b )$ Define $\beta = T _ { 0 } / ( 2 \pi \tau )$ so that $V _ { C 0 } = \beta V _ { 0 } .$ Then $\beta$ is the "amplification" of the source voltage across the capacitor. If a particular $L R C$ circuit contains a 2.0 -nF capacitor and has a resonant frequency of 5.0$\mathrm { kHz }$ , what value of $R$ will yield $\beta = 125 ?$

Ze-Han Lee

Numerade Educator

Problem 66

(II) Capacitors made from piezoelectric materials are commonly used as sound transducers ("speakers"). They often require a large operating voltage. One method for providing the required voltage is to include the speaker as part of an $L R C$ circuit as shown in Fig. $29 ,$ where the speaker is modeled electrically as the capacitance $C = 1.0 \mathrm { nF } .$ Take $R = 35 \Omega$ and $L = 55 \mathrm { mH } .$ (a) What is the resonant frequency $f _ { 0 }$ for this circuit? (b) If the voltage source has peak amplitude $V _ { 0 } = 2.0 \mathrm { V }$ at frequency $f = f _ { 0 } ,$ find the peak voltage $V _ { c 0 }$ across the speaker (i.e., the capacitor $C$ ). (c) Determine the ratio $V _ { C 0 } / V _ { 0 . }$

Nolan Smyth

Numerade Educator

Problem 67

(II) $( a )$ Determine a formula for the average power $\overline { P }$ dissipated in an $L R C$ circuit in terms of $L , R , C , \omega ,$ and $V _ { 0 }$ (b) At what frequency is the power a maximum? (c) Find an approximate formula for the width of the resonance peak in average power, $\Delta \omega ,$ which is the difference in the two (angular) frequencies where $\overline { P }$ has half its maximum value. Assume a sharp peak.

Ze-Han Lee

Numerade Educator

Problem 68

(II) (a) Show that oscillation of charge $Q$ on the capacitor of an $L R C$ circuit has amplitude $Q _ { 0 } = \frac { V _ { 0 } } { \sqrt { ( \omega R ) ^ { 2 } + \left( \omega ^ { 2 } L - \frac { 1 } { C } \right) ^ { 2 } } }$ (b) At what angular frequency, $\omega ^ { \prime } ,$ will $Q _ { 0 }$ be a maximum? $( c )$ Compare to a forced damped harmonic oscillator, and discuss. (See Question 20 in this Chapter.)

Zhaojie Xu

Numerade Educator

Problem 69

(II) A resonant circuit using a 220 -nF capacitor is to resonate at 18.0$\mathrm { kH } z$ . The air-core inductor is to be a solenoid with closely packed coils made from 12.0$\mathrm { m }$ of insulated wire 1.1$\mathrm { mm }$ in diameter. How many loops will the inductor contain?

Ze-Han Lee

Numerade Educator

Problem 70

(II) The output of an electrocardiogram amplifier has an impedance of 45$\mathrm { k } \Omega$ . It is to be connected to an $8.0 - \Omega$ loud speaker through a transformer. What should be the turns ratio of the transformer?

Nolan Smyth

Numerade Educator

Problem 71

A 2200 -pF capacitor is charged to 120$\mathrm { V }$ and then quickly connected to an inductor. The frequency of oscillation is observed to be 17$\mathrm { kHz }$ . Determine $( a )$ the inductance, (b) the peak value of the current, and (c) the maximum energy stored in the magnetic field of the inductor.

Ze-Han Lee

Numerade Educator

Problem 72

At $t = 0 ,$ the current through a 60.0 -mH inductor is 50.0$\mathrm { mA }$ and is increasing at the rate of 78.0$\mathrm { mA } / \mathrm { s } .$ What is the initial energy stored in the inductor, and how long does it take for the energy to increase by a factor of 5.0 from the initial value?

Nolan Smyth

Numerade Educator

Problem 73

At time $t = 0 ,$ the switch in the circuit shown in Fig. 30 is closed. After a sufficiently long time, steady currents $I _ { 1 } , I _ { 2 }$ and $I _ { 3 }$ flow through resistors $R _ { 1 } , R _ { 2 } ,$ and $R _ { 3 } ,$ respectively. Determine these three currents.

Ze-Han Lee

Numerade Educator

Problem 74

(a) Show that the self-inductance $L$ of a toroid (Fig. 31$)$ of radius $r _ { 0 }$ containing $N$ loops each of diameter $d$ is $L \approx \frac { \mu _ { 0 } N ^ { 2 } d ^ { 2 } } { 8 r _ { 0 } }$ if $r _ { 0 } \gg d .$ Assume the field is uniform inside the toroid; is this actually true? Is this result consistent with $L$ for a solenoid? Should it be? (b) Calculate the inductance $L$ of a large toroid if the diameter of the coils is 2.0$\mathrm { cm }$ and the diameter of the whole ring is 66$\mathrm { cm }$ . Assume the field inside the toroid is uniform. There are a total of 550 loops of wire.

Nolan Smyth

Numerade Educator

Problem 75

A pair of straight parallel thin wires, such as a lamp cord, each of radius $r ,$ are a distance $\ell$ apart and carry current to a circuit some distance away. Ignoring the field within each wire, show that the inductance per unit length is $\left( \mu _ { 0 } / \pi \right) \ln [ ( \ell - r ) / r ] .$

Lainey Roebuck

Numerade Educator

Problem 76

Assuming the Earth's magnetic field averages about $0.50 \times 10 ^ { - 4 } \mathrm { T }$ near the surface of the Earth, estimate the total energy stored in this field in the first 5.0$\mathrm { km }$ above the Earth's surface.

Nolan Smyth

Numerade Educator

Problem 77

(a) For an underdamped $L R C$ circuit, determine a formula for the energy $U = U _ { E } + U _ { B }$ stored in the electric and magnetic fields as a function of time. Give answer in terms of the initial charge $Q _ { 0 }$ on the capacitor. (b) Show how $d U / d t$ is related to the rate energy is transformed in the resistor, $I ^ { 2 } R .$

Lainey Roebuck

Numerade Educator

Problem 78

An electronic device needs to be protected against sudden surges in current. In particular, after the power is turned on the current. should rise to no more than 7.5$\mathrm { mA }$ in the first 75 us. The device has resistance 150$\Omega$ and is designed to operate at 33$\mathrm { mA } .$ How would you protect this device?

Nolan Smyth

Numerade Educator

Problem 79

The circuit shown in Fig. 32$\mathrm { a }$ can integrate (in the calculus sense) the input voltage $V _ { \text { in } , \text { if the time constant } L / R \text { is } }$ large compared with the time during which $V _ { \text { in varies. } }$ Explain how this integrator works and sketch its output for the square wave signal input shown in Fig. 32b. IHint. Write Kirchhoff's loop rule for the circuit. Multiply each term in this differential equation (in $I )$ by a factor $e ^ { R t / L }$ to make it easier to integrate.]

Lainey Roebuck

Numerade Educator

Problem 80

Suppose circuit $\mathrm { B }$ in Fig. 18$\mathrm { a }$ consists of a resistance $R = 550 \Omega$ . The filter capacitor has capacitance $C = 1.2 \mu \mathrm { F }$ . Will this capacitor act to eliminate $60 - \mathrm { Hz }$ ac but pass a high-frequency signal of frequency 6.0$\mathrm { kHz }$ ? To check this, determine the voltage drop across $R$ for a $130 - \mathrm { mV }$ signal of frequency $( a ) 60 \mathrm { Hz } ; ( b ) 6.0 \mathrm { kHz }$

Nolan Smyth

Numerade Educator

Problem 81

An ac voltage source $V = V _ { 0 } \sin \left( \omega t + 90 ^ { \circ } \right)$ is connected across an inductor $L$ and current $I = I _ { 0 } \sin ( \omega t )$ flows in this circuit. Note that the current and source voltage are $90 ^ { \circ }$ out of phase. $( a )$ Directly calculate the average power delivered by the source over one period $T$ of its sinusoidal cycle via the integral $P = \int _ { 0 } ^ { T } V I d t / T .$ (b) Apply the relation $P = I _ { \mathrm { rms } } V _ { \mathrm { rms } } \cos \phi$ to this circuit and show that the answer

Ze-Han Lee

Numerade Educator

Problem 82

A circuit contains two elements, but it is not known if they are $L , R ,$ or $C .$ The current in this circuit when connected to a $120 - \mathrm { V } 60$ -Hz source is 5.6$\mathrm { A }$ and lags the voltage by $65 ^ { \circ } .$ What are the two elements and what are their values?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 83

A $3.5 - \mathrm { k } \Omega$ resistor in series with a $440 - \mathrm { mH }$ inductor is driven by an ac power supply. At what frequency is the impedance double that of the impedance at 60$\mathrm { Hz } ?$

Ze-Han Lee

Numerade Educator

Problem 84

(a) What is the rms current in an $R C$ circuit if $R = 5.70 \mathrm { k } \Omega$ , $C = 1.80 \mu \mathrm { F } ,$ and the rms applied voltage is 120$\mathrm { V }$ at 60.0$\mathrm { Hz }$ ? (b) What is the phase angle between voltage and current? (c) What is the power dissipated by the circuit?
(d) What are the voltmeter readings across $R$ and $C ?$

Nolan Smyth

Numerade Educator

Problem 85

An inductance coil draws 2.5$\mathrm { A }$ de when connected to a $45 - \mathrm { V }$ battery. When connected to a $60 - \mathrm { Hz } 120 - \mathrm { V } ( \mathrm { rms } )$ source, the current drawn is 3.8$\mathrm { A }$ (rms). Determine the inductance and resistance of the coil.

Ze-Han Lee

Numerade Educator

Problem 86

The $Q$ -value of a resonance circuit can be defined as the ratio of the voltage across the capacitor (or inductor) to the voltage across the resistor, at resonance. The larger the $Q$ factor, the sharper the resonance curve will be and the sharper the tuning. $( a )$ Show that the $Q$ factor is given by the equation $Q = ( 1 / R ) \sqrt { L / C }$ . $( b )$ At a resonant frequency $f _ { 0 } = 1.0 \mathrm { MHz }$ , what must be the value of $L$ and $R$ to produce a $Q$ factor of 350$?$ Assume that $C = 0.010 \mu \mathrm { F } .$

Nolan Smyth

Numerade Educator

Problem 87

Show that the fraction of electromagnetic energy lost (to thermal energy) per cycle in a lightly damped $\left( R ^ { 2 } < 4 L / C \right) L R C$ circuit is approximately $\frac { \Delta U } { U } = \frac { 2 \pi R } { L \omega } = \frac { 2 \pi } { Q }$ The quantity $Q$ can be defined as $Q = L \omega / R ,$ and is called the $Q$ -value, or quality factor, of the circuit and is a measure of the damping present. A high $Q$ -value means smaller damping and less energy input required to maintain oscillations.

Lainey Roebuck

Numerade Educator

Problem 88

In a series $L R C$ circuit, the inductance is $33 \mathrm { mH } ,$ the capacitance is 55$\mathrm { nF }$ , and the resistance is 1.50$\mathrm { k } \Omega$ . At what frequencies is the power factor equal to 0.17$?$

Nolan Smyth

Numerade Educator

Problem 89

In our analysis of a series $L R C$ circuit, Fig. $19 ,$ suppose we chose $V = V _ { 0 } \sin \omega t . ( a )$ Construct a phasor diagram, like that of Fig. 21 , for this case. (b) Write a formula for the current $I ,$ defining all terms.

Ze-Han Lee

Numerade Educator

Problem 90

A voltage $V = 0.95 \sin 754 t$ is applied to an $L R C$ circuit $( I$ is in amperes, $t$ is in seconds, $V$ is in volts, and the "angle" is in radians) which has $L = 22.0 \mathrm { mH } , R = 23.2 \mathrm { k } \Omega ,$ and $C = 0.42 \mu \mathrm { F } , ( a )$ What is the impedance and phase angle? $( b )$ How much power is dissipated in the circuit? (c) What is the rms current and voltage across each element?

James Kiss

Numerade Educator

Problem 91

Filter circuit. Figure 33 shows a simple filter circuit designed to pass de voltages with minimal attenuation and to remove, as much as possible, any ac components (such as $60 -$ Hz line voltage that could cause hum in a stereo receiver, for example). Assume $V _ { \text { in } } = V _ { 1 } + V _ { 2 }$ where $V _ { 1 }$ is dc and $V _ { 2 } = V _ { 20 } \sin \omega t ,$ and that any resistance is very small. (a) Determine the current through the capacitor: give amplitude and phase (assume $R = 0$ and $X _ { L } > X _ { L }$ (b) Show that the ac component of the output voltage, $\dot { V } _ { 2 \text { out, equals } } ( Q / C ) - V _ { 1 , \text { where } } Q$ is the charge on the capacitor at any instant, and determine the amplitude and phase of $V _ { 2 \text { our. } } ( c )$ Show that the attenuation of the ac voltage is greatest when $X _ { C } \ll X _ { L } ,$ and calculate the ratio of the output to input ac voltage in this case.
(d) Compare the dc output voltage to input voltage.

Lainey Roebuck

Numerade Educator

Problem 92

Show that if the inductor $L$ in the filter circuit of Fig. 33 (Problem 91) is replaced by a large resistor $R ,$ there will still be significant attenuation of the ac voltage and little attenuation of the de voltage if the input de voltage is high and the current (and power) are low.

Lainey Roebuck

Numerade Educator

Problem 93

A resistor $R$ , capacitor $C ,$ and inductor $L$ are connected i parallel across an ac generator as shown in Fig. $34 .$ The source emf is $V = V _ { 0 } \sin \omega t .$ Determine the current as a function of time (including amplitude and phase): $( a )$ in the resistor, $( b )$ in the inductor, $( c )$ in the capacitor. (d) What is the total current leaving the source? (Give amplitude $I _ { 0 }$
and phase.) $( e )$ Determine the impedance $Z$ defined as $Z = V _ { 0 } / I _ { 0 } . ( f )$ What is the power factor?

Lainey Roebuck

Numerade Educator

Problem 94

Suppose a series $L R C$ circuit has two resistors, $R _ { 1 }$ and $R _ { 2 }$ two capacitors, $C _ { 1 }$ and $C _ { 2 } ,$ and two inductors, $L _ { 1 }$ and $L _ { 2 } ,$ all in series. Calculate the total impedance of the circuit.

Nolan Smyth

Numerade Educator

Problem 95

Determine the inductance $L$ of the primary of a transformer whose input is 220$\mathrm { V }$ at 60$\mathrm { Hz }$ when the current drawn is 4.3$\mathrm { A }$ . Assume no current in the secondary.

Ze-Han Lee

Numerade Educator

Problem 96

In a plasma globe, a hollow glass sphere is filled with low-pressure gas and a small spherical metal electrode is located at its center. Assume an ac voltage source of peak voltage $V _ { 0 }$ and frequency $f$ is applied between the metal sphere and the ground, and that a person is touching the outer surface of the globe with a fingertip, whose approximate area is 1.0$\mathrm { cm } ^ { 2 } .$ The equivalent circuit for this situation is shown in Fig. $35 ,$ where $R _ { G }$ and $R _ { P }$ are the resistances of the gas and the person, respectively, and $C$ is the capacitance formed by the gas, glass, and finger.
(a) Determine $C$ assuming it is a parallel-plate capacitor. The conductive gas and the person's fingertip form the opposing plates of area $A = 1.0 \mathrm { cm } ^ { 2 } .$ The plates are separated by glass (dielectric constant $K = 5.0$ ) of thickness $d = 2.0 \mathrm { mm } .$ (b) In a typical plasma globe, $f = 12 \mathrm { kHz }$ . Determine the reactance $X _ { C }$ of $C$ at this frequency in $\mathrm { M\Omega }$ . (c) The voltage may be $V _ { 0 } = 2500 \mathrm { V } .$ With this high voltage, the dielectric strength of the gas is exceeded and the gas becomes ionized. In this "plasma" state, the gas
emits light ( "sparks") and is highly conductive so that $R _ { \mathrm { G } } < X _ { C }$ . Assuming also that $R _ { \mathrm { P } } \ll X _ { C } ,$ estimate the peak current that flows in the given circuit. Is this level of current dangerous? $( d )$ If the plasma globe operated at $f = 1.0 \mathrm { MHz }$ , estimate the peak current that would flow in the given circuit. Is this level of current dangerous?

Nolan Smyth

Numerade Educator

Problem 97

You have a small electromagnet that consumes 350$\mathrm { W }$ from a residential circuit operating at 120$\mathrm { V }$ at 60$\mathrm { Hz }$ . Using your ac multimeter, you determine that the unit draws 4.0$\mathrm { A }$ rms. What are the values of the inductance and the internal resistance?

Ze-Han Lee

Numerade Educator

Problem 98

An inductor $L$ in series with a resistor $R ,$ driven by a sinusoidal voltage source, responds as described by the following differential equation: $V _ { 0 } \sin \omega t = L \frac { d I } { d t } + R I$ Show that a current of the form $I = I _ { 0 } \sin ( \omega t - \phi )$ flows through the circuit by direct substitution into the differential equation. Determine the amplitude of the current $I _ { 0 }$ )
and the phase difference $\phi$ between the current and the voltage source.

Nolan Smyth

Numerade Educator

Problem 99

In a certain $L R C$ series circuit, when the ac voltage source has a particular frequency $f$ , the peak voltage across the inductor is 6.0 times greater than the peak voltage across the capacitor. Determine $f$ in terms of the resonant frequency $f _ { 0 }$ of this circuit.

Ze-Han Lee

Numerade Educator

Problem 100

For the circuit shown in Fig. $36 , V = V _ { 0 } \sin \omega t .$ Calculate the current in each element of the circuit, as well as the total impedance. [Hint: Try a trial solution of the form $I = I _ { 0 } \sin ( \omega t + \phi )$ for the current leaving the source.]

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 101

To detect vehicles at traffic lights, wire loops with dimensions on the order of 2$\mathrm { m }$ are often buried horizontally under roadways. Assume the self-inductance of such a loop is $L = 5.0 \mathrm { mH }$ and that it is part of an $L R C$ circuit as shown in Fig. 37 with $C = 0.10 \mu \mathrm { F }$ and $R = 45 \Omega$ . The ac voltage has frequency $f$ and rms voltage $V _ { \mathrm { rms } }$ . (a) The frequency $f$ is chosen to match the resonant frequency $f _ { 0 }$ of the circuit. Find $f _ { 0 }$ and determine what the rms voltage $\left( V _ { R } \right) _ { \text { Tms across the resistor will be when } }$ $f = f _ { 0 } . ( b )$ Assume that $f , C ,$ and $R$ never change, but that, when a car is located above the buried loop, the loop's self-inductance decreases by 10$\%$ (due to induced eddy currents in the car's metal parts). Determine by what factor the voltage $\left( V _ { R } \right) _ { \text { rms decreases in this situation in } }$ comparison to no car above the loop. [Monitoring $\left( V _ { R } \right) _ { \mathrm { rms } }$ detects the presence of a car.

Lainey Roebuck

Numerade Educator

Problem 102

For the circuit shown in Fig. 38 , show that if the condition $R _ { 1 } R _ { 2 } = L / C$ is satisfied then the potential difference between points a and b is zero for all frequencies.

Nolan Smyth

Numerade Educator

Problem 103

(II) The $R C$ circuit shown in Fig. 39 is called a low-pass filter because it passes low-frequency ac signals with less attenuation than high-frequency ac signals. $( a )$ Show that the voltage gain is $A = V _ { \text { out } } / V _ { \text { in } } = 1 / \left( 4 \pi ^ { 2 } f ^ { 2 } R ^ { 2 } C ^ { 2 } + 1 \right) \frac { 1 } { 2 }$ (b) Discuss the behavior of the gain $A$ for $f \rightarrow 0$ and $f \rightarrow \infty$ . (c) Choose $R = 850 \Omega$ and $C = 1.0 \times 10 ^ { - 6 } \mathrm { F }$ , and graph log $A$ versus log $f$ with suitable scales to show the behavior of the circuit at low and high frequencies.

Ze-Han Lee

Numerade Educator

Problem 104

(II) The $R C$ circuit shown in Fig, 40 is called a high-pass filter because it passes high-frequency ac signals with less attenuation than low-frequency ac signals. (a) Show that the voltage gain is $A = V _ { \text { out } } / V _ { \text { in } } = 2 \pi f R C / \left( 4 \pi ^ { 2 } f ^ { 2 } R ^ { 2 } C ^ { 2 } + 1 \right) \frac { 1 } { 2 }$ (b) Discuss the behavior of the gain $A$ for $f \rightarrow 0$ and $f \rightarrow \infty ( c )$ Choose $R = 850 \Omega$ and $C = 1.0 \times 10 ^ { - 6 } \mathrm { F }$ and then graph log $A$ versus log $f$ with suitable scales to show the behavior of the circuit at high and low frequencies.

Nolan Smyth

Numerade Educator

Problem 105

(III) Write a computer program or use a spreadsheet program to plot $I _ { \mathrm { rms } }$ for an ac $L R C$ circuit with a sinusoidal voltage source (Fig. 19) with $V _ { \mathrm { rms } } = 0.100 \mathrm { V }$ For $L = 50 \mu \mathrm { H }$ and $C = 50 \mu \mathrm { F } ,$ plot the $I _ { \mathrm { rms } }$ graph for (a) $R = 0.10 \Omega ,$ and $( b ) R = 1.0 \Omega$ from $\omega = 0.1 \omega _ { 0 }$ to $\omega = 3.0 \omega _ { 0 }$ on the same graph.

Lainey Roebuck

Numerade Educator