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

The rms output voltage of an AC source is 200 $\mathrm{V}$ and the operating frequency is 100 $\mathrm{Hz}$ . Write the equation giving the output voltage as a function of time.

Donald Albin

Donald Albin

Numerade Educator

Problem 2

(a) What is the resistance of a lightbulb that uses an aver- age 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 3

An AC power supply produces a maximum voltage $\Delta V_{\max }=100 \mathrm{V}$ . This power supply is connected to a $24.0-\Omega$ resistor, and the current and resistor voltage are measured with an ideal AC ammeter and voltmeter as shown in Figure P33.3. What does each meter read? An ideal ammeter has zero resistance and an ideal voltmeter has infinite resistance.

Donald Albin

Numerade Educator

Problem 4

In the simple AC circuit shown in Active Figure $33.2, R=$ 70.0$\Omega$ and $\Delta v=\Delta V_{\max }$ 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 $\Delta v_{R}=0.250 \Delta V_{\max } ?$

Donald Albin

Numerade Educator

Problem 5

The current in the circuit shown in Active Figure 33.2 equals 60.0$\%$ of the peak current at $t=7.00 \mathrm{ms}$ . What is the lowest source frequency that gives this current?

Donald Albin

Numerade Educator

Problem 6

An audio amplifier, represented by the AC source and resistor in Figure P33.6, 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 7

In a purely inductive AC circuit as shown in Active Figure $33.6, \Delta V_{\max }=100 \mathrm{V}$ . (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 $\mathrm{A} ?$

Donald Albin

Numerade Educator

Problem 8

An inductor has a $54.0-\Omega$ reactance at 60.0 $\mathrm{Hz}$ . What is the maximum current if this inductor is connected to a $50.0-\mathrm{Hz}$ source that produces a $100-\mathrm{V}$ rms voltage?

Zulfiqar Ali

Numerade Educator

Problem 9

A For the circuit shown in Active Figure $33.6, \Delta V_{\max }=$ $80.0 \mathrm{V}, \omega=65.0 \pi \mathrm{rad} / \mathrm{s},$ and $L=70.0 \mathrm{mH} .$ Calculate the current in the inductor at $t=15.5 \mathrm{ms}$

Donald Albin

Numerade Educator

Problem 10

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

Donald Albin

Numerade Educator

Problem 11

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

Donald Albin

Numerade Educator

Problem 12

(a) For what frequencies does a $22.0-\mu \mathrm{F}$ capacitor have a reactance below 175$\Omega^{2}$ (b) What If? What is the reactance of a $444.0-\mu \mathrm{F}$ capacitor over this same frequency range?

Donald Albin

Numerade Educator

Problem 13

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 (b) a European electrical outlet having $\Delta V_{\text { rms }}=240 \mathrm{V}$ and $f=$ 50.0 $\mathrm{Hz}^{2}$

Zulfiqar Ali

Numerade Educator

Problem 14

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 15

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 16

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

Narayan Hari

Numerade Educator

Problem 17

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 18

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 19

A series AC circuit contains the following components: a $150-\Omega$ resistor, an inductor of $250 \mathrm{mH},$ a capacitor of $2.00 \mu \mathrm{F},$ and a source with $\Delta V_{\max }=210 \mathrm{V}$ operating at 50.0 $\mathrm{Hz}$ . Calculate the (a) inductive reactance, (b) capacitive reactance, (c) impedance, (d) maximum current, and (e) phase angle between current and source voltage.

Donald Albin

Numerade Educator

Problem 20

A sinusoidal voltage $\Delta v(t)=(40.0 \mathrm{V}) \sin (100 t)$ 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? (c) Determine the numerical values for $I_{\max }, \omega,$ and $\phi$ in the equation $i(t)=$ $I_{\max } \sin (\omega t-\phi)$

Donald Albin

Numerade Educator

Problem 21

An RLC 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?

Donald Albin

Numerade Educator

Problem 22

Four circuit elements-a capacitor, an inductor, a resistor, and an AC source-are connected together in various ways. First the capacitor is connected to the source, and the rms current is found to be 25.1 $\mathrm{mA}$ . The capacitor is disconnected and discharged, and then it is connected in series with the resistor and the source, making the rms current 15.7 $\mathrm{mA}$ . The circuit is disconnected and the capacitor discharged. The capacitor is then connected in series with the inductor and the source, making the rms current 68.2 $\mathrm{mA}$ . After the circuit is disconnected and the capacitor discharged, all four circuit elements are connected together in a series loop. What is the rms current
in the circuit?

Donald Albin

Numerade Educator

Problem 23

A person is working near the secondary of a transformer as shown in Figure P33.23. The primary voltage is 120 $\mathrm{V}$ 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 rms voltage across the body. Suggestion: Model the secondary of the trans- former as an AC source.

Donald Albin

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 $c,(\mathrm{c})$ points $c$ and $d,$ and $(\mathrm{d})$ points $b$
and $d .$

Zulfiqar Ali

Numerade Educator

Problem 25

Draw to scale a phasor diagram and determine $Z, X_{L}, X_{C},$ and $\phi$ for an $\mathrm{AC}$ series circuit for which $R=300 \Omega, C=$ $11.0 \mu \mathrm{F}, L=0.200 \mathrm{H},$ and $f=(500 / \pi) \mathrm{Hz}$

Donald Albin

Numerade Educator

Problem 26

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 $R$ . Explain each step in your solution.

Donald Albin

Numerade Educator

Problem 27

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

Zulfiqar Ali

Numerade Educator

Problem 28

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 29

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 $37.0^{\circ} .(\mathrm{a})$ 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 30

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 a component of current that is 90° out of phase with the voltage as well as with current 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 (a) the rms current and (b) the power factor? (c) What capacitor must be added in series to make the power fact tor 1$?$ (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?

Ajay Singhal

Numerade Educator

Problem 31

Energy is to be transmitted at the rate of 20.0 $\mathrm{kW}$ with only 1.00$\%$ loss over a distance of 18.0 $\mathrm{km}$ at potential difference $\Delta V$ . (a) What is the diameter required for each of the two copper wires in the transmission line? Assume the current density is uniform in the conductors. (b) State how the diameter depends on $\Delta V .$ (c) Evaluate the diameter for $\Delta V=1500 \mathrm{V}$ . (d) If you choose to make the diameter $3.00 \mathrm{mm},$ what potential difference is required?

Dominador Tan

Numerade Educator

Problem 32

A series circuit consists of an AC generator with an rms voltage of 120 $\mathrm{V}$ at a frequency of 60.0 $\mathrm{Hz}$ and a magnetic buzzer with a resistance of 100$\Omega$ and an inductance of 100 $\mathrm{mH}$ . (a) Find the circuit's power factor. (b) Suppose a higher power factor is desired. Can a power factor of 1.00 be achieved by changing the inductance or any other circuit parameters? (c) Show that a power factor of 1.00 can be attained by inserting a capacitor into the original circuit, and find the value of its capacitance.

Donald Albin

Numerade Educator

Problem 33

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 in terms of $\Delta V$ and $R$ the average power delivered to the diode circuit of Figure P33.33.

Donald Albin

Numerade Educator

Problem 34

A radar transmitter contains an $L C$ circuit oscillating at $1.00 \times 10^{10} \mathrm{Hz}$ . (a) What capacitance resonates with a one-turn loop having an inductance of 400 $\mathrm{pH}$ at this frequency? (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?

Donald Albin

Numerade Educator

Problem 35

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

Sachin Rao

Numerade Educator

Problem 36

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, (b) the amplitude of the current at the resonant frequency, $(\mathrm{c})$ the $Q$ of the circuit, and $(\mathrm{d})$ the amplitude of the voltage across the inductor at resonance.

Vishal Gupta

Numerade Educator

Problem 37

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. Find the energy delivered to the circuit during one period if the operating frequency is twice the resonance frequency.

Zulfiqar Ali

Numerade Educator

Problem 38

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

Donald Albin

Numerade Educator

Problem 39

Compute the quality factor for the circuits described in Problems 20 and $21 .$ Which circuit has the sharper resonance?

Donald Albin

Numerade Educator

Problem 40

A step-down transformer is used for recharging the batteries of portable devices such as tape players. The ratio of turns inside the transformer is $13 : 1$ , and the transformer is used with $120-V$ (rms) household service. If a particular ideal transformer draws 0.350 A from the house outlet, what are (a) the voltage and (b) the current supplied to a tape player from the transformer? (c) How much power is delivered?

Donald Albin

Numerade Educator

Problem 41

A transformer has $N_{1}=350$ turns and $N_{2}=2000$ turns. If the input voltage is $\Delta v(t)=(170 \mathrm{V})$ cos $\omega t,$ what rms voltage is developed across the secondary coil?

Zulfiqar Ali

Numerade Educator

Problem 42

A step-up transformer is designed to have an output voltage of 2200 $\mathrm{V}$ (rms) when the primary is connected across a $110-\mathrm{V}$ (rms) source. (a) If the primary winding has 80 turns, how many turns are required on the secondary? (b) If a load resistor across the secondary draws a current of $1.50 \mathrm{A},$ 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 $\mathrm{A}$ ?

Donald Albin

Numerade Educator

Problem 43

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 miles $\left(6.44 \times 10^{5} \mathrm{m}\right) .$ The output voltage of the generator 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 generator voltage of 4.50 $\mathrm{kV} ?$

Donald Albin

Numerade Educator

Problem 44

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 $\mathrm{V}$ AC 8 $\mathrm{W}$ Output 9 $\mathrm{V}$ DC 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 8.0 $\mathrm{W}$ when the radio is switched off and energy costs $\$ 0.135 / \mathrm{kWh}$ from the electric company. Find the cost of having six such transformers around the house, each plugged in for 31 days.

Donald Albin

Numerade Educator

Problem 45

Consider the filter circuit shown in Active Figure 33.22 $\mathrm{a}$ . (a) Show that the ratio of the output voltage to the input voltage is
$$
\frac{\Delta v_{\mathrm{out}}}{\Delta v_{\mathrm{in}}}=\frac{R}{\sqrt{R^{2}+\left(\frac{1}{\omega C}\right)^{2}}}
$$
(b) What value does this ratio approach as the frequency decreases toward zero? What value does this ratio approach as the frequency increases without limit? (c) At what frequency is the ratio equal to one-half?

Donald Albin

Numerade Educator

Problem 46

Consider the filter circuit shown in Active Figure 33.23 $\mathrm{a}$ . (a) Show that the ratio of the output voltage to the input voltage is
$$
\frac{\Delta v_{\mathrm{out}}}{\Delta v_{\mathrm{in}}}=\frac{1 / \omega C}{\sqrt{R^{2}+\left(\frac{1}{\omega C}\right)^{2}}}
$$
(b) What value does this ratio approach as the frequency decreases toward zero? What value does this ratio approach as the frequency increases without limit? (c) At what frequency is the ratio equal to one-half?

Donald Albin

Numerade Educator

Problem 47

A The $R C$ high-pass filter shown in Active Figure 33.22 $\mathrm{a}$ has a resistance $R=0.500 \Omega .$ (a) What capacitance gives an output signal that has one-half the amplitude of a
$300-\mathrm{Hz}$ input signal? (b) What is the ratio $\left(\Delta v_{\text { out }} / \Delta v_{\text { in }}\right)$ for a $600-\mathrm{Hz}$ signal? You may use the result of Problem $45 .$

Donald Albin

Numerade Educator

Problem 48

The $R C$ low-pass filter shown in Active Figure 33.23 $\mathrm{a}$ has a resistance $R=90.0 \Omega$ and a capacitance $C=8.00 \mathrm{nF}$ . Calculate the ratio $\left(\Delta v_{\text { out }} / \Delta v_{\text { in }}\right)$ for an input frequency of (a) 600 $\mathrm{Hz}$ and $(\mathrm{b}) 600 \mathrm{kHz}$ . You may use the result of Problem $46 .$

Donald Albin

Numerade Educator

Problem 49

The resistor in Figure $\mathrm{P} 33.49$ 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 $\Delta V_{\max }=$ 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 4000 $\mathrm{Hz}$ (a) Determine the required values of $L$ and $C$ (b) Find the maximum value of the ratio $\Delta v_{\text { out }} / \Delta v_{\text { in }} .$ (c) Find the frequency $f_{0}$ at which the ratio has its maximum value. (d) Find the phase shift between $\Delta v_{\text { in }}$ and $\Delta v_{\text { out }}$ at 200 $\mathrm{Hz}$ , at $f_{0}$ , and at 4000 $\mathrm{Hz}$ . (e) Find the average power transferred to the speaker at 200 $\mathrm{Hz}$ , at $f_{0},$ and at 4000 $\mathrm{Hz}$ . (f) Treating the filter as a
resonant circuit, find its quality factor.

Donald Albin

Numerade Educator

Problem 50

Show that the rms value for the sawtooth voltage shown in Figure $\mathrm{P} 33.50$ is $\Delta V_{\max } / \sqrt{3} .$

Zulfiqar Ali

Numerade Educator

Problem 51

A $400-\Omega$ resistor, an inductor, and a capacitor are in series with a generator. The reactance of the inductor is $700 \Omega,$ and the circuit impedance is $760 \Omega .$ (a) Explain what you can and cannot determine about the reactance of the capacitor. (b) If you find that the source power decreases as you raise the frequency, what do you know about the capacitive reactance in the original circuit?
(c) Repeat part (a) assuming the resistance is 200$\Omega$ instead of 400$\Omega$ .

Donald Albin

Numerade Educator

Problem 52

A capacitor, a coil, and two resistors of equal resistance are arranged in an AC circuit as shown in Figure $\mathrm{P} 33.52$ . An AC generator provides an emf of 20.0 $\mathrm{V}$ (rms) 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 $1,$ the rms current is 298 $\mathrm{mA}$ . When the switch is closed in position $2,$ the rms current is 137 $\mathrm{mA}$ . Determine the values of $R, C$ and $L .$ Is more than one set of values possible? Explain.

Donald Albin

Numerade Educator

Problem 53

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. Determine the power delivered to the circuit when the frequency is equal to one-half the resonance frequency.

Donald Albin

Numerade Educator

Problem 54

O A series $R L C$ circuit has resonance angular frequency 2000 $\mathrm{rad} / \mathrm{s}$ . When it is operating at some certain frequency, $X_{L}=12.0 \Omega$ and $X_{C}=8.00 \Omega .$ (a) Is this certai
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 this determination is not possible, give a compact expression for the condition that $L$ and $C$ must satisfy.

Donald Albin

Numerade Educator

Problem 55

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

Donald Albin

Numerade Educator

Problem 56

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. Identify the value of $\phi$ at the resonance angular frequency $\omega_{0}$ . 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 57

In Figure $\mathrm{P} 33.57$ , find the rms current delivered by the $45.0-\mathrm{V}$ (rms) power supply when $(\mathrm{a})$ the frequency is very large and $(\mathrm{b})$ the frequency is very small.

Zulfiqar Ali

Numerade Educator

Problem 58

In the circuit shown in Figure $\mathrm{P} 33.58$ (page 950$)$ , assume all parameters except $C$ are given. (a) Find the current as a function of time. (b) Find the power delivered to the circuit. (c) Find the current as a function of time after only switch 1 is opened. (d) After switch 2 is also opened, the current and voltage are in phase. Find the capacitance C. (e) Find the impedance of the circuit when both switches are open. (f) Find the maximum energy stored in the capacitor during oscillations. (g) Find the maxi- mum 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.

Donald Albin

Numerade Educator

Problem 59

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 60

Make an order-of-magnitude estimate of the electric current that the electric company delivers to a town (Fig. P33.60) from a remote generating station. State the data you measure or estimate. If you wish, you may consider a suburban residential community of 20000 people.

Donald Albin

Numerade Educator

Problem 61

Consider a series $R L C$ circuit having the following circuit 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 the following amplitudes. (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 (d) the maximum voltage $\Delta V_{L}$ across the inductor and its phase relative to the current

Donald Albin

Numerade Educator

Problem 62

A voltage $\Delta v=(100 \mathrm{V}) \sin \omega t$ 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 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 63

Impedance matching. Example 28.2 showed that maximum power is transferred when the internal resistance of a DC source is equal to the resistance of the load. A trans- former may be used to provide maximum power transfer between two AC circuits that have different impedances
$Z_{1}$ and $Z_{2} .$ ( a) Show that the ratio of turns $N_{1} / N_{2}$ needed to meet this condition 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?

Dominador Tan

Numerade Educator

Problem 64

A power supply with $\Delta V_{\text { rms }}=120 \mathrm{V}$ 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

Problem 65

Figure $\mathrm{P} 33.65 \mathrm{a}$ shows a parallel $R L C$ circuit, and the corresponding phasor diagram is given in Figure $\mathrm{P} 33.65 \mathrm{b}$ . 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 $C$ and $L$ lead or lag the current in the resistor as shown in
Figure $\mathrm{P} 33.65 \mathrm{b}$ . ( a ) Show that 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 $\Delta V_{\mathrm{rms}}$ and $I_{\mathrm{rms}}$ is
given by
$$
\tan \phi=R\left(\frac{1}{X_{C}}-\frac{1}{X_{L}}\right)
$$

Dominador Tan

Numerade Educator

Problem 66

A certain electric circuit is described by the equations
$$
\begin{array}{c}{\frac{200 \mathrm{V}}{4.00 \mathrm{A}}=\sqrt{(35.0 \Omega)^{2}+\left[\omega(205 \mathrm{mH})-\frac{1}{\omega C}\right]^{2}}} \\ {\omega=2 \pi(100 \mathrm{Hz})}\end{array}
$$
State a problem for which these equations would appear in the solution, giving the data and identifying the unknown. Evaluate the unknown quantity.

Donald Albin

Numerade Educator

Problem 67

A series $R L C$ circuit is operating at 2000 $\mathrm{Hz}$ . 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 $f=300,600,800,1000,1500,2000,3000,$ $4000,6000,$ and 10000 $\mathrm{Hz}$ (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 68

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

Donald Albin

Numerade Educator

Problem 69

O 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
$$
y_{1}=(12.0 \mathrm{cm}) \sin (4.5 t)
$$
and
$$
y_{2}=(12.0 \mathrm{cm}) \sin \left(4.5 t+70^{\circ}\right)
$$
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. State the result of comparing the answers to (a) and (b). (c) Phasors make it equally easy to add traveling waves. Find the amplitude and phase constant of the sum of the three waves represented by
$$
\begin{array}{l}{y_{1}=(12.0 \mathrm{cm}) \sin \left(15 x-4.5 t+70^{\circ}\right)} \\ {y_{2}=(15.5 \mathrm{cm}) \sin \left(15 x-4.5 t-80^{\circ}\right)} \\ {y_{3}=(17.0 \mathrm{cm}) \sin \left(15 x-4.5 t+160^{\circ}\right)}\end{array}
$$

Donald Albin

Numerade Educator