Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 33

Alternating-Current Circuits - all with Video Answers

Educators

Chapter Questions

Problem 1

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

Zulfiqar Ali

Zulfiqar Ali

Numerade Educator

Problem 2

A certain lightbulb is rated at 60.0 W when operating at an rms voltage of 120 V. (a) What is the peak voltage applied across the bulb? (b) What is the resistance of the bulb? (c) Does a 100-W bulb have greater or less resistance than a 60.0-W bulb? Explain.

Zulfiqar Ali

Numerade Educator

Problem 3

An AC power supply produces a maximum voltage $\Delta V_{\max }=$ $100 \mathrm{V} .$ This power supply is connected to a resistor $R=$ 24.0 $\Omega$ , and the current and resistor voltage are measured with an ideal AC ammeter and voltmeter as shown in Figure P33. 3 . An ideal ammeter has. Aero resistance, and an ideal voltmeter has infinite resistance. What is the reading on (a) the ammeter and (b) the voltmeter?

Zulfiqar Ali

Numerade Educator

Problem 4

(a) What is the resistance of a lightbulb that uses an average power of 75.0 $\mathrm{W}$ when connected to a 60.0 $\mathrm{Hz}$ power source having a maximum voltage of 170 $\mathrm{V}$ ? (b) What If? What is the resistance of a $100-\mathrm{W}$ lightbulb?

Zulfiqar Ali

Numerade Educator

Problem 5

The current in the circuit shown in Figure $\mathrm{P} 33.5$ equals 60.0$\%$ of the peak current at $t=7.00 \mathrm{ms}$ . What is the lowest source frequency that gives this current?

Zulfiqar Ali

Numerade Educator

Problem 6

In the AC circuit shown in Figure P33.5, $R=70.0 \Omega$ and the output voltage of the AC source is $\Delta V_{\max }$ sin $\omega t .$ (a) If $\Delta V_{R}=0.250 \Delta V_{\max }$ for the first time at $t=0.0100 \mathrm{s},$ what is the angular frequency of the source? (b) What is the next value of $t$ for which $\Delta V_{R}=0.250 \Delta V_{\max } ?$

Zulfiqar Ali

Numerade Educator

Problem 7

An audio amplifier, represented by the AC source and resistor in Figure P33.7, delivers to the speaker alternating voltage at audio frequencies. If the source voltage has an amplitude of $15.0 \mathrm{V}, R=8.20 \Omega,$ and the speaker is equivalent to a resistance of $10.4 \Omega,$ what is the time-averaged power transferred to it?

Zulfiqar Ali

Numerade Educator

Problem 8

Figure $\mathrm{P} 33.8$ shows three lightbulbs connected to a $120-\mathrm{V}$ $\mathrm{AC}$ (rms) household supply voltage. Bulbs 1 and 2 have a power rating of 150 $\mathrm{W}$ , and bulb 3 has a $100-\mathrm{W}$ rating. Find (a) the rms current in each bulb and (b) the resistance of each bulb. (c) What is the total resistance of the combination of the three lightbulbs?

Zulfiqar Ali

Numerade Educator

Problem 9

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

Zulfiqar Ali

Numerade Educator

Problem 10

In a purely inductive AC circuit as shown in Figure $\mathrm{P} 33.10, \Delta V_{\max }=100 \mathrm{V}$ . (a) The maximum current is 7.50 $\mathrm{A}$ at 50.0 $\mathrm{Hz}$ . Calculate the inductance $L .$ (b) What If? At what angular frequency $\omega$ is the maximum current 2.50 $\mathrm{A}$ ?

Zulfiqar Ali

Numerade Educator

Problem 11

For the circuit shown in Figure $\mathrm{P} 33.10, \Delta V_{\max }=80.0 \mathrm{V}, \omega=$
$65.0 \pi \mathrm{rad} / \mathrm{s},$ and $L=70.0 \mathrm{mH} .$ Calculate the current in the inductor at $t=15.5 \mathrm{ms}$

Zulfiqar Ali

Numerade Educator

Problem 12

An inductor is connected to an AC power supply having a maximum output voltage of 4.00 $\mathrm{V}$ at a frequency of 300 $\mathrm{Hz}$ . What inductance is needed to keep the rms current less than 2.00 $\mathrm{mA}$ ?

Zulfiqar Ali

Numerade Educator

Problem 13

An inductor has a $54.0-\Omega$ reactance when connected to a $60.0-\mathrm{Hz}$ source. The inductor is removed and then connected to a $50.0-\mathrm{Hz}$ source that produces a $100-\mathrm{V}$ rms voltage. What is the maximum current in the inductor?

Zulfiqar Ali

Numerade Educator

Problem 14

A 20.0 -mH inductor is connected to a North American electrical outlet $\left(\Delta V_{\mathrm{rms}}=120 \mathrm{V}, f=60.0 \mathrm{Hz}\right)$ . Assuming the energy stored in the inductor is zero at $t=0,$ determine the energy stored at $t=\frac{1}{180} \mathrm{s}$ .

Zulfiqar Ali

Numerade Educator

Problem 15

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

Zulfiqar Ali

Numerade Educator

Problem 16

The output voltage of an AC source is given by $\Delta v=$ $120 \sin 30.0 \pi t,$ where $\Delta v$ is in volts and $t$ is in seconds. The source is connected across a $0.500-\mathrm{H}$ inductor. Find (a) the frequency of the source, (b) the rms voltage across the inductor, (c) the inductive reactance of the circuit, (d) the rms current in the inductor, and (e) the maximum current in the inductor.

Zulfiqar Ali

Numerade Educator

Problem 17

What is the maximum current in a $2.20-\mu \mathrm{F}$ capacitor when it is connected across (a) a North American electrical outlet having $\Delta V_{\mathrm{rms}}=120 \mathrm{V}$ and $f=60.0 \mathrm{Hz}$ and $(\mathrm{b})$ a European electrical outlet having $\Delta V_{\mathrm{rms}}=240 \mathrm{V}$ and $f=50.0 \mathrm{Hz}^{2}$

Zulfiqar Ali

Numerade Educator

Problem 18

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

Zulfiqar Ali

Numerade Educator

Problem 19

(a) For what frequencies does a $22.0-\mu \mathrm{F}$ capacitor have a reactance below 175$\Omega ?$ (b) What If? What is the reactance of a $44.0-\mu \mathrm{F}$ capacitor over this same frequency range?

Zulfiqar Ali

Numerade Educator

Problem 20

An AC source 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 (a) the capacitive reactance, (b) the rms current, and $(\mathrm{c})$ the maximum current in the circuit. (d) Does the capacitor have its maximum charge when the current has its maximum value? Explain.

Zulfiqar Ali

Numerade Educator

Problem 21

What maximum current is delivered by an AC source with $\Delta V_{\max }=48.0 \mathrm{V}$ and $f=90.0 \mathrm{Hz}$ when connected across a $3.70-\mu \mathrm{F}$ capacitor?

Zulfiqar Ali

Numerade Educator

Problem 22

A capacitor C is connected to a power supply that operates at a frequency $f$ and produces an rms voltage $\Delta V$ . What is the maximum charge that appears on either capacitor plate?

Zulfiqar Ali

Numerade Educator

Problem 23

A $1.00-\mathrm{mF}$ capacitor is connected to a North American electrical outlet $\left(\Delta V_{\mathrm{rms}}=120 \mathrm{V}, f=60.0 \mathrm{Hz}\right) .$ Assuming the energy stored in the capacitor is zero at $t=0,$ determine the magnitude of the current in the wires at $t=\frac{1}{180} \mathrm{s}$ .

Prabhat Tyagi

Numerade Educator

Problem 24

An AC source with $\Delta V_{\max }=150 \mathrm{V}$ and $f=50.0 \mathrm{Hz}$ is connected between points $a$ and $d$ in Figure $\mathrm{P} 33.24$ Calculate the maximum voltages between (a) points $a$ and $b,$ (b) points $b$ and $c,(c)$ points $c$ and $d,$ and (d) points $b$ and $d .$

Zulfiqar Ali

Numerade Educator

Problem 25

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

Zulfiqar Ali

Numerade Educator

Problem 26

At what frequency does the inductive reactance of a $57.0-\mu \mathrm{H}$ inductor equal the capacitive reactance of a $57.0-\mu \mathrm{F}$ capacitor?

Zulfiqar Ali

Numerade Educator

Problem 27

A series AC circuit contains a resistor, an inductor of $150 \mathrm{mH},$ a capacitor of 5.00$\mu \mathrm{F}$ , and a source with $\Delta V_{\max }=$ 240 $\mathrm{V}$ operating at 50.0 $\mathrm{Hz}$ . The maximum current in the circuit is 100 $\mathrm{mA}$ . Calculate (a) the inductive reactance, (b) the capacitive reactance, (c) the impedance, (d) the resistance in the circuit, and (e) the phase angle between the current and the source voltage.

Zulfiqar Ali

Numerade Educator

Problem 28

A sinusoidal voltage $\Delta v=40.0$ sin $100 t,$ where $\Delta v$ is in volts and $t$ is in seconds, is applied to a series $R L C$ circuit with $L=160 \mathrm{mH}, C=99.0 \mu \mathrm{F},$ and $R=68.0 \Omega .$ (a) What is the impedance of the circuit? (b) What is the maximum current? Determine the numerical values for (c) $\omega$ and (d) $\phi$ in the equation $i=I_{\max } \sin (\omega t-\phi)$

Zulfiqar Ali

Numerade Educator

Problem 29

An $R L C$ circuit consists of a $150-\Omega$ resistor, a $21.0-\mu \mathrm{F}$ capacitor, and a $460-\mathrm{mH}$ inductor connected in series with a $120-\mathrm{V}, 60.0-\mathrm{Hz}$ power supply. (a) What is the phase angle between the current and the applied voltage? (b) Which reaches its maximum earlier, the current or the voltage?

Zulfiqar Ali

Numerade Educator

Problem 30

Draw phasors to scale for the following voltages in SI units: (a) 25.0 sin $\omega t$ at $\omega t=90.0^{\circ},$ (b) 30.0 $\sin \omega t$ at $\omega t=60.0^{\circ}$ , and $(\mathrm{c}) 18.0$ sin $\omega t$ at $\omega t=300^{\circ} .$

Zulfiqar Ali

Numerade Educator

Problem 31

In addition to phasor diagrams showing voltages such as in Active Figure 33.15 , we can draw phasor diagrams with resistance and reactances. The resultant of adding the phasors is the impedance. Draw to scale a phasor diagram showing $Z, X_{L}, X_{C},$ and $\phi$ for an AC series circuit for which $R=300 \Omega, C=11.0 \mu \mathrm{F}, L=0.200 \mathrm{H},$ and $f=500 / \pi \mathrm{Hz}$ .

Zulfiqar Ali

Numerade Educator

Problem 32

A $60.0-\Omega$ resistor is connected in series with a $30.0-\mu \mathrm{F}$ capacitor and a source whose maximum voltage is 120 $\mathrm{V}$ , operating at 60.0 $\mathrm{Hz}$ . Find (a) the capacitive reactance of the circuit, (b) the impedance of the circuit, and $\mathrm{c}$ ) the maximum current in the circuit. (d) Does the voltage lead or lag the current? (e) How will adding an inductor in series with the existing resistor and capacitor affect the current? Explain.

Zulfiqar Ali

Numerade Educator

Problem 33

Review. In an $R L C$ series circuit that includes a source of alternating current operating at fixed frequency and voltage, the resistance $R$ is equal to the inductive reactance. If the plate separation of the parallel-plate capacitor is reduced to one-half its original value, the current in the circuit doubles. Find the initial capacitive reactance in terms of $R .$

Zulfiqar Ali

Numerade Educator

Problem 34

An AC voltage of the form $\Delta v=90.0$ sin $350 t,$ where $\Delta v$ is in volts and $t$ is in seconds, 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 (a) the impedance of the circuit, $(\mathrm{b})$ the rms current in the circuit, and $(\mathrm{c})$ the average power delivered to the circuit.

Zulfiqar Ali

Numerade Educator

Problem 35

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

Zulfiqar Ali

Numerade Educator

Problem 36

An AC voltage of the form $\Delta v=100 \sin 1000 t,$ where $\Delta v$ is in volts and $t$ is in seconds, is applied to a series $R L C$ circuit. Assume the resistance is $400 \Omega,$ the capacitance is $5.00 \mu \mathrm{F},$ and the inductance is 0.500 $\mathrm{H.}$ Find the average power delivered to the circuit.

Zulfiqar Ali

Numerade Educator

Problem 37

A series $R L C$ circuit has a resistance of 45.0$\Omega$ and an impedance of 75.0$\Omega$ . What average power is delivered to this circuit when $\Delta V_{\mathrm{rms}}=210 \mathrm{V} ?$

Zulfiqar Ali

Numerade Educator

Problem 38

Why is the following situation impossible? A series circuit consists of an ideal AC source (no inductance or capacitance in the source itself) with an rms voltage of $\Delta V$ at a frequency $f$ and a magnetic buzzer with a resistance $R$ and an inductance $L .$ By carefully adjusting the inductance $L$ of the circuit, a power factor of exactly 1.00 is attained.

Zulfiqar Ali

Numerade Educator

Problem 39

In a certain series $R L C$ circuit, $I_{\mathrm{rms}}=9.00 \mathrm{A}, \Delta V_{\mathrm{rms}}=180 \mathrm{V},$ and the current leads the voltage by $37.0^{\circ} .$ (a) What is the total resistance of the circuit? (b) Calculate the reactance of the circuit $\left(X_{L}-X_{C}\right)$

Zulfiqar Ali

Numerade Educator

Problem 40

Suppose you manage a factory that uses many electric motors. The motors create a large inductive load to the electric power line as well as a resistive load. The electric company builds an extra-heavy distribution line to supply you with two components of current: one that is $90^{\circ}$ out of phase with the voltage and another that is in phase with the voltage. The electric company charges you an extra fee for "reactive volt-amps" in addition to the amount you pay for the energy you use. You can avoid the extra fee by installing a capacitor between the power line and your factory. The following problem models this solution.
In an $R L$ circuit, a $120-\mathrm{V}$ (rms), $60.0-\mathrm{Hz}$ source is in series with a $25.0-\mathrm{mH}$ inductor and a $20.0-\Omega$ resistor. What are $(\mathrm{a})$ the rms current and $(\mathrm{b})$ the power factor? (c) What capacitor must be added in series to make the power factor equal to $1^{2}$ (d) To what value can the supply voltage be reduced if the power supplied is to be the same as before the capacitor was installed?

Zulfiqar Ali

Numerade Educator

Problem 41

A diode is a device that allows current to be carried in only one direction (the direction indicated by the arrow head in its circuit symbol). Find the average power delivered to the diode circuit of Figure $\mathrm{P} 33.41$ in terms of $\Delta V_{\mathrm{rms}}$ and $R .$

Zulfiqar Ali

Numerade Educator

Problem 42

The $L C$ circuit of a radar transmitter oscillates at 9.00 $\mathrm{GHz}$ . (a) What inductance is required for the circuit to resonate at this frequency if its capacitance is 2.00 $\mathrm{pF}$ ? (b) What is the inductive reactance of the circuit at this frequency?

Zulfiqar Ali

Numerade Educator

Problem 43

An $R L C$ circuit is used in a radio to tune into an $\mathrm{FM}$ station broadcasting at $f=99.7 \mathrm{MHz}$ . The resistance in the circuit is $R=12.0 \Omega,$ and the inductance is $L=1.40 \mu \mathrm{H}$ . What capacitance should be used?

Zulfiqar Ali

Numerade Educator

Problem 44

A series $R L C$ circuit has components with the following values: $L=20.0 \mathrm{mH}, C=100 \mathrm{nF}, R=20.0 \Omega,$ and $\Delta V_{\max }=$ $100 \mathrm{V},$ with $\Delta v=\Delta V_{\max } \sin \omega t$ . Find $(\mathrm{a})$ the resonant frequency of the circuit, (b) the amplitude of the current at the resonant frequency, (c) the $Q$ of the circuit, and (d) the amplitude of the voltage across the inductor at resonance.

Zulfiqar Ali

Numerade Educator

Problem 45

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

Zulfiqar Ali

Numerade Educator

Problem 46

A resistor $R,$ inductor $L,$ and capacitor Care connected in series to an AC source of rms voltage $\Delta V$ and variable frequency. If the operating frequency is twice the resonance frequency, find the energy delivered to the circuit during one period.

Zulfiqar Ali

Numerade Educator

Problem 47

Review. A radar transmitter contains an $L C$ circuit oscillating at $1.00 \times 10^{10} \mathrm{Hz}$ . ( a) For a one-turn loop having an inductance of 400 $\mathrm{pH}$ to resonate at this frequency, what capacitance is required in series with the loop? (b) The capacitor has square, parallel plates separated by 1.00 $\mathrm{mm}$ of air. What should the edge length of the plates be? (c) What is the common reactance of the loop and capacitor at resonance?

Zulfiqar Ali

Numerade Educator

Problem 48

A step-down transformer is used for recharging the batteries of portable electronic devices. The turns ratio $N_{2} / N_{1}$ for a particular transformer used in a DVD player is $1 : 13$ . When used with $120-\mathrm{V}$ (rms) household service, the transformer draws an rms current of 20.0 $\mathrm{mA}$ from the house outlet. Find (a) the rms output voltage of the transformer and (b) the power delivered to the DVD player.

Zulfiqar Ali

Numerade Educator

Problem 49

The primary coil of a transformer has $N_{1}=350$ turns, and the secondary coil has $N_{2}=2000$ turns. If the input voltage across the primary coil is $\Delta v=170 \cos \omega t,$ where $\Delta v$ is in volts and $t$ is in seconds, what rms voltage is developed across the secondary coil?

Zulfiqar Ali

Numerade Educator

Problem 50

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 $\mathrm{MW}$ across $400 \mathrm{mi}\left(6.44 \times 10^{5} \mathrm{m}\right) .$ The output voltage of the source is 4.50 $\mathrm{kV}$ . (a) What is the line loss if a transformer is used to step up the voltage to 500 $\mathrm{kV} ?$ (b) What fraction of the input power is lost to the line under these circumstances? (c) What If? What difficulties would be encountered in attempting to transmit the 5.00 $\mathrm{MW}$ at the source voltage of 4.50 $\mathrm{kV}$ ?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

In the transformer shown in Figure $\mathrm{P} 33.51$ , the load resistance $R_{L}$ is 50.0$\Omega$ . The turns ratio $N_{1} / N_{2}$ is 2.50 , and the rms source voltage is $\Delta V_{s}=80.0 \mathrm{V}$ . If a voltmeter across the load resistance measures an rms voltage of $25.0 \mathrm{V},$ what is the source resistance $R_{s} ?$

Zulfiqar Ali

Numerade Educator

Problem 52

A person is working near the secondary of a transformer as shown in Figure $\mathrm{P} 33.52$ . The primary voltage is 120 $\mathrm{V}$ at 60.0 $\mathrm{Hz}$ . The secondary voltage is 5000 $\mathrm{V}$ . The capacitance $C_{s},$ which is the stray capacitance between the hand and the secondary winding, is 20.0 $\mathrm{pF}$ . Assuming the person has a body resistance to ground of $R_{b}=50.0 \mathrm{k} \Omega,$ determine the rms voltage across the body. Suggestion: Model the secondary of the transformer as an AC source.

Zulfiqar Ali

Numerade Educator

Problem 53

The $R C$ high-pass filter shown in Figure $P 33.53$ has a resistance $R=0.500 \Omega$ and a capacitance $C=613 \mu \mathrm{F}$ . What is the ratio of the amplitude of the output voltage to that of the input voltage for this filter for a source frequency of 600 $\mathrm{Hz}$ ?

Ajay Singhal

Numerade Educator

Problem 54

Consider the $R C$ high-pass filter circuit shown in Figure $\mathrm{P} 33.53$ . (a) Find an expression for the ratio of the amplitude of the output voltage to that of the input voltage in terms of $R, C,$ and the AC source frequency $\omega$ . (b) What value does this ratio approach as the frequency decreases toward zero? (c) What value does this ratio approach as the frequency increases without limit?

Zulfiqar Ali

Numerade Educator

Problem 55

One particular plug-in power supply for a radio looks similar to the one shown in Figure 33.20 and is marked with the following information: Input 120 $\mathrm{VAC} 8 \mathrm{W}$ Output 9 $\mathrm{VDC}$ 300 $\mathrm{mA}$ . Assume these values are accurate to two digits. (a) Find the energy efficiency of the device when the radio is operating. (b) At what rate is energy wasted in the device when the radio is operating? (c) Suppose the input power to the transformer is 8.00 $\mathrm{W}$ when the radio is switched off and energy costs $\$ 0.110 / \mathrm{kWh}$ from the electric company. Find the cost of having six such transformers around the house, each plugged in for 31 days.

Zulfiqar Ali

Numerade Educator

Problem 56

Consider the filter circuit shown in Figure P33.56. (a) Show that the ratio of the amplitude of the output voltage to that of the input voltage is
$$\frac{\Delta V_{\mathrm{out}}}{\Delta V_{\mathrm{in}}}=\frac{1 / \omega C}{\sqrt{R^{2}+\left(\frac{1}{\omega C^{2}}\right)}}$$
(b) What value does this ratio approach as the frequency decreases toward zero? (c) What value does this ratio approach as the frequency increases without limit? (d) At what frequency is the ratio equal to one-half?

Zulfiqar Ali

Numerade Educator

Problem 57

A step-up transformer is designed to have an output voltage of 2 200 V (rms) when the primary is connected across a 110-V (rms) source. (a) If the primary winding has exactly 80 turns, how many turns are required on the secondary? (b) If a load resistor across the secondary draws a current of 1.50 A, what is the current in the primary, assuming ideal conditions? (c) What If? If the transformer actually has an efficiency of 95.0%, what is the current in the primary when the secondary current is 1.20 A?

Zulfiqar Ali

Numerade Educator

Problem 58

Why is the following situation impossible? An RLC circuit is used in a radio to tune into a North American AM commercial radio station. The values of the circuit components are $R=15.0 \Omega, L=2.80 \mu \mathrm{H},$ and $C=0.910 \mathrm{pF} .$

Zulfiqar Ali

Numerade Educator

Problem 59

Review. The voltage phasor diagram for a certain series $R L C$ circuit is shown in Figure P33.59. The resistance of the circuit is $75.0 \Omega,$ and the frequency is 60.0 $\mathrm{Hz}$ . Find (a) the maximum voltage $\Delta V_{\text { max }},$ (b) the phase angle (c) the maximum current, (d) the impedance, (e) the capacitance and (f) the inductance of the circuit, and (g) the average power delivered to the circuit.

Zulfiqar Ali

Numerade Educator

Problem 60

Consider a series $R L C$ circuit having the parameters $R=200 \Omega, L=663 \mathrm{mH},$ and $C=26.5 \mu \mathrm{F}$ . The applied voltage has an amplitude of 50.0 $\mathrm{V}$ and a frequency of 60.0 $\mathrm{Hz}$ . Find (a) the current $I_{\max }$ and its phase relative to the applied voltage $\Delta v,(\mathrm{b})$ the maximum voltage $\Delta V_{R}$ across the resistor and its phase relative to the current, (c) the maximum voltage $\Delta V_{C}$ across the capacitor and its phase relative to the current, and (d) the maximum voltage $\Delta V_{L}$ across the inductor and its phase relative to the current.

Aja S

Numerade Educator

Problem 61

Energy is to be transmitted over a pair of copper wires in a transmission line at the rate of 20.0 $\mathrm{kW}$ with only a 1.00$\%$ loss over a distance of 18.0 $\mathrm{km}$ at potential difference $\Delta V_{\mathrm{rms}}=1.50 \times 10^{3} \mathrm{V}$ between the wires. Assuming the current density is uniform in the conductors, what is the diameter required for each of the two wires?

Zulfiqar Ali

Numerade Educator

Problem 62

Energy is to be transmitted over a pair of copper wires in a transmission line at a rate $P$ with only a fractional loss $f$ over a distance $\ell$ at potential difference $\Delta V_{\mathrm{rms}}$ between the wires. Assuming the current density is uniform in the conductors, what is the diameter required for each of the two wires?

Zulfiqar Ali

Numerade Educator

Problem 63

Show that the rms value for the sawtooth voltage shown in Figure $\mathrm{P} 33.63$ is $\Delta V_{\max } / \sqrt{3}$

Zulfiqar Ali

Numerade Educator

Problem 64

A $400-\Omega$ resistor, an inductor, and a capacitor are in series with an AC source. The reactance of the inductor is $700 \Omega,$ and the circuit impedance is $760 \Omega .$ (a) What are the possible values of the reactance of the capacitor? (b) If you find that the power delivered to the circuit decreases as you raise the frequency, what is the capacitive reactance in the original circuit? (C) Repeat part (a) assuming the resistance is 200$\Omega$ instead of 400$\Omega$ and the circuit impedance continues to be 760$\Omega$ .

Zulfiqar Ali

Numerade Educator

Problem 65

A series $R L C$ circuit contains the following components: $R=150 \Omega, L=0.250 \mathrm{H}, C=2.00 \mu \mathrm{F}$ , and a source with $\Delta V_{\max }=210 \mathrm{V}$ operating at 50.0 $\mathrm{Hz}$ . Our goal is to find the phase angle, the power factor, and the power input for this circuit. (a) Find the inductive reactance in the circuit. (b) Find the capacitive reactance in the circuit. (c) Find the impedance in the circuit. (d) Calculate the maximum current in the circuit. (e) Determine the phase angle between the current and source voltage. (f) Find the power factor for the circuit. (g) Find the power input to the circuit.

Linda Winkler

Numerade Educator

Problem 66

A capacitor, a coil, and two resistors of equal resistance are arranged in an AC circuit as shown in Figure P33.66. An AC source provides an emf of $\Delta V_{\text { rms }}=20.0 \mathrm{V}$
at a frequency of 60.0 $\mathrm{Hz}$ . When the double-throw switch $\mathrm{S}$ is open as shown in the figure, the rms current is 183 $\mathrm{mA}$ . When the switch is closed
in position $a,$ the rms current is 298 $\mathrm{mA}$ . When the switch is closed in position $b,$ the rms current is 137 $\mathrm{mA}$ . Determine the values of $(a) R,(b) C,$ and $(c) L .$ (d) Is more than one set of values possible? Explain.

Donald Albin

Numerade Educator

Problem 67

A series $R L C$ circuit consists of an $8.00-\Omega$ resistor, a $5.00-\mu \mathrm{F}$ capacitor, and a $50.0-\mathrm{mH}$ inductor. A variable frequency source applies an emf of 400 $\mathrm{V}$ (rms) across the combination. Assuming the frequency is equal to one-half the resonance frequency, determine the power delivered to the circuit.

Donald Albin

Numerade Educator

Problem 68

A series $RLC$ circuit has resonance angular frequency $2.00 \times 10^{3} \mathrm{rad} / \mathrm{s}$ . When it is operating at some input frequency, $X_{L}=12.0 \Omega$ and $X_{C}=8.00 \mathrm{D} \Omega .$ (a) Is this input frequency higher than, lower than, or the same as the resonance frequency? Explain how you can tell. (b) Explain whether it is possible to determine the values of both $L$ and $C .$ (c) If it is possible, find $L$ and $C .$ If it is not possible, give a compact expression for the condition that $L$ and $C$ must satisfy.

Zulfiqar Ali

Numerade Educator

Problem 69

Review. One insulated conductor from a household extension cord has a mass per length of 19.0 g/m. A section of this conductor is held under tension between two clamps. A subsection is located in a magnetic field of magnitude 15.3 mT directed perpendicular to the length of the cord. When the cord carries an AC current of 9.00 A at a frequency of 60.0 Hz, it vibrates in resonance in its simplest standing-wave vibration mode. (a) Determine the relationship that must be satisfied between the separation d of the clamps and the tension T in the cord. (b) Determine one possible combination of values for these variables.

Zulfiqar Ali

Numerade Educator

Problem 70

(a) Sketch a graph of the phase angle for an $R L C$ series circuit as a function of angular frequency from zero to a frequency much higher than the resonance frequency. (b) Identify the value of $\phi$ at the resonance angular frequency $\omega_{0} .$ (c) Prove that the slope of the graph of $\phi$ versus $\omega$ at the resonance point is 2$Q / \omega_{0}$ .

Donald Albin

Numerade Educator

Problem 71

In Figure $\mathrm{P} 33.71,$ find the rms current delivered by the $45.0-\mathrm{V}$ (rms) power supply when (a) the frequency is very large and (b) the frequency is very small.

Zulfiqar Ali

Numerade Educator

Problem 72

Review. In the circuit shown in Figure P33.72, assume all parameters except C are given. Find (a) the current in the circuit as a function of time and (b) the power delivered to the circuit. (c) Find the current as a function of time after only switch 1 is opened. (d) After switch 1 is and voltage are in phase. Find the capacitance $C$ . Find (e) the impedance of the circuit when both switches are open, (f) the maximum energy stored in the capacitor during oscillations, and (g) the maximum energy stored in the inductor during oscillations. (h) Now the frequency of the voltage source is doubled. Find the phase difference between the current and the voltage. (i) Find the frequency that makes the inductive reactance one-half the capacitive reactance.

Zulfiqar Ali

Numerade Educator

Problem 73

A transformer may be used to provide maximum power transfer between two AC circuits that have different impedances $Z_{1}$ and $Z_{2}$ . This process is called impedance matching. (a) Show that the ratio of turns $N_{1} / N_{2}$ for this transformer is
$$\frac{N_{1}}{N_{2}}=\sqrt{\frac{Z_{1}}{Z_{2}}}$$
(b) Suppose you want to use a transformer as an impedance-matching device between an audio amplifier that has an output impedance of 8.00 $\mathrm{k} \Omega$ and a speaker that has an input impedance of $8.00 \Omega .$ What should your $N_{1} / N_{2}$ ratio be?

Zulfiqar Ali

Numerade Educator

Problem 74

A series $R L C$ circuit is operating at $2.00 \times 10^{3} \mathrm{Hz}$ . At this frequency, $X_{L}=X_{C}=1884 \Omega .$ The resistance of the circuit is $40.0 \Omega .$ (a) Prepare a table showing the values of $X_{L}, X_{C},$ and $Z$ for $f=300,600,800,1.00 \times 10^{3}, 1.50 \times 10^{3}, 2.00 \times$ $10^{3}, 3.00 \times 10^{3}, 4.00 \times 10^{3}, 6.00 \times 10^{3},$ and $1.00 \times 10^{4} \mathrm{Hz}$ (b) Plot on the same set of axes $X_{L}, X_{C},$ and $Z$ as a function of $\ln f .$

Donald Albin

Numerade Educator

Problem 75

Marie Cornu, a physicist at the Polytechnic Institute in Paris, invented phasors in about 1880. This problem helps you see their general utility in representing oscillations. Two mechanical vibrations are represented by the expressions
$$y_{1}=12.0 \sin 4.50 t$$
and
$$y_{2}=12.0 \sin \left(4.50 t+70.0^{\circ}\right)$$
where $y_{1}$ and $y_{2}$ are in centimeters and $t$ is in seconds. Find the amplitude and phase constant of the sum of these functions (a) by using a trigonometric identity (as from Appendix B) and (b) by representing the oscillations as phasors. (c) State the result of comparing the answers to parts (a) and (b). (d) Phasors make it equally easy to add traveling
waves. Find the amplitude and phase constant of the sum of the three waves represented by
$y_{1}=12.0 \sin \left(15.0 x-4.50 t+70.0^{\circ}\right)$
$y_{2}=15.5 \sin \left(15.0 x-4.50 t-80.0^{\circ}\right)$
$y_{3}=17.0 \sin \left(15.0 x-4.50 t+160^{\circ}\right)$
where $x, y_{1}, y_{2},$ and $y_{3}$ are in centimeters and $t$ is in seconds.

Zulfiqar Ali

Numerade Educator

Problem 76

A series $R L C$ circuit in which $R=1.00 \Omega, L=1.00 \mathrm{mH},$ and $C=1.00 \mathrm{nF}$ is connected to an AC source delivering 1.00 $\mathrm{V}$ (rms). (a) Make a precise graph of the power delivered to the circuit as a function of the frequency and (b) verify that the full width of the resonance peak at half-maximum is $R / 2 \pi L .$

Donald Albin

Numerade Educator

Problem 77

The resistor in Figure $\mathrm{P} 33.77$ on page 982 represents the midrange speaker in a three-speaker system. Assume its resistance to be constant at $8.00 \Omega .$ The source represents an audio amplifier producing signals of uniform amplitude $\Delta V_{\text { max }}=10.0 \mathrm{V}$ at all audio frequencies. The inductor and capacitor are to function as a band-pass filter with $\Delta V_{\text { out }} / \Delta V_{\text { in }}=\frac{1}{2}$ at 200 $\mathrm{Hz}$ and at $4.00 \times 10^{3} \mathrm{Hz}$ . Determine the required values of (a) $L$ and (b) $C .$ Find (c) the maximum value of the ratio $\Delta V_{\text { out }} / \Delta V_{\text { in }} ;$ (d) the frequency $f_{0}$ at which the ratio has its maximum value; (e) the phase shift between $\Delta v_{\text { in }}$ and $\Delta v_{\text { out }}$ at 200 $\mathrm{Hz}$ , at $f_{0}$ , and at $4.00 \times 10^{3}$
$\mathrm{Hz} ;$ and $(\mathrm{f})$ the average power transferred to the speaker at
$200 \mathrm{Hz},$ at $f_{0},$ and at $4.00 \times 10^{3} \mathrm{Hz}$ . (g) Treating the filter as a resonant circuit, find its quality factor.

Ajay Singhal

Numerade Educator

Problem 78

An $80.0-\Omega$ resistor and a $200-\mathrm{mH}$ inductor are connected in parallel across a $100-\mathrm{V}$ (rms), $60.0-\mathrm{Hz}$ source. (a) What is the rms current in the resistor? (b) By what angle does the total current lead or lag behind the voltage?

Zulfiqar Ali

Numerade Educator

Problem 79

A voltage $\Delta v=100 \sin \omega t,$ where $\Delta v$ is in volts and $t$ is in seconds, is applied across a series combination of a $2.00-\mathrm{H}$ inductor, a $10.0-\mu \mathrm{F}$ capacitor, and a $10.0-\Omega$ resistor. (a) Determine the angular frequency $\omega_{0}$ at which the power delivered to the resistor is a maximum. (b) Calculate the average power delivered at that frequency. (c) Determine the two angular frequencies $\omega_{1}$ and $\omega_{2}$ at which the power is one-half the maximum value. Note: The $Q$ of the circuit is $\omega_{0} /\left(\omega_{2}-\omega_{1}\right) .$

Donald Albin

Numerade Educator

Problem 80

Figure $\mathrm{P} 33.80$ a shows a parallel $R L C$ circuit. The instantaneous voltages (and rms voltages) across each of the three circuit elements are the same, and each is in phase with
the current in the resistor. The currents in $\mathrm{C}$ and $L$ lead or lag the current in the resistor as shown in the current phasor diagram, Figure $\mathrm{P} 33.80 \mathrm{b}$ . (a) Shown in the rms current delivered by the source is
$$I_{\mathrm{rms}}=\Delta V_{\mathrm{rms}}\left[\frac{1}{R^{2}}+\left(\omega C-\frac{1}{\omega L}\right)^{2}\right]^{1 / 2}$$
(b) Show that the phase angle $\phi$ between $\Delta V_{\text { rms }}$ and $I_{\text { rms is }}$
given by
$$\tan \phi=R\left(\frac{1}{X_{C}}-\frac{1}{X_{L}}\right)$$

Mayukh Banik

Numerade Educator

Problem 81

An AC source with $\Delta V_{\mathrm{rms}}=120 \mathrm{V}$ is connected between points $a$ and $d$ in Figure $\mathrm{P} 33.24$ . At what frequency will it deliver a power of 250 $\mathrm{W}$ ? Explain your answer.

Donald Albin

Numerade Educator