University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 30

Electromagnetic Oscillations and Currents - all with Video Answers

Educators

Chapter Questions

Problem 1

A $200-\Omega$ resistor, a $40.0-\mathrm{mH}$ inductor and a $3.0-\mu \mathrm{F}$ capacitor are connected in series with a time-varying source of emf that provides $10.0 \mathrm{~V}$ at a frequency of $1000 \mathrm{~Hz}$. What is the impedance of the circuit?
a) $200 \Omega$
b) $228 \Omega$
c) $342 \Omega$
d) $282 \Omega$

Ajay Singhal

Ajay Singhal

Numerade Educator

Problem 2

For which values of $f$ is $X_{L}>X_{C} ?$
a) $f>2 \pi(L C)^{1 / 2}$
b) $f>(2 \pi L C)^{-1}$
c) $f>\left(2 \pi(L C)^{1 / 2}\right)^{-1}$
d) $f>2 \pi L C$

Ajay Singhal

Numerade Educator

Problem 3

Which statement about the phase relation between the electric and magnetic fields in an LC circuit is correct?
a) When one field is at its maximum, the other is also, and the same for the minimum values.
b) When one field is at maximum strength, the other is at minimum (zero) strength.
c) The phase relation, in general, depends on the values of $L$ and $C$.

Ajay Singhal

Numerade Educator

Problem 4

For the band-pass filter shown in Figure $30.24,$ how can the width of the frequency response be increased?
a) increase $R_{1}$
b) decrease $C_{1}$
c) increase $R_{2}$
d) increase $C_{2}$
e) do any of the above

Linda Winkler

Numerade Educator

Problem 5

The phase constant, $\phi$, between the voltage and the current in an AC circuit depends on the ______.
a) inductive reactance
b) capacitive reactance
c) resistance
d) all of the above

Ajay Singhal

Numerade Educator

Problem 6

The AM radio band covers the frequency range from $520 \mathrm{kHz}$ to $1610 \mathrm{kHz}$. Assuming a fixed inductance in a simple LC circuit, what ratio of capacitance is necessary to cover this frequency range? That is, what is the value of $C_{\mathrm{h}} / C_{\mathrm{l}}$ where $C_{\mathrm{h}}$ is the capacitance for the highest frequency and $C_{1}$ is the capacitance for the lowest frequency?
a) 9.59
b) 0.104
c) 0.568
d) 1.76

Eduard Sanchez

Numerade Educator

Problem 7

In the $\mathrm{RLC}$ circuit in the figure, $R=60 \Omega, L=3 \mathrm{mH}$ $C=4 \mathrm{mF}$, and the source of time-varying emf has a peak voltage of $120 \mathrm{~V}$. What should the angular frequency, $\omega$, be to produce the largest current in the resistor?
a) $4.2 \mathrm{rad} / \mathrm{s}$
b) $8.3 \mathrm{rad} / \mathrm{s}$
c) $204 \mathrm{rad} / \mathrm{s}$
d) $289 \mathrm{rad} / \mathrm{s}$
e) $5000 \mathrm{rad} / \mathrm{s}$
f) $20,000 \mathrm{rad} / \mathrm{s}$

Eduard Sanchez

Numerade Educator

Problem 8

A standard North American wall socket plug is labeled $110 \mathrm{~V}$. This label indicates the __________ value of the voltage.
a) average
b) maximum
c) root-mean-square (rms)
d) instantaneous

Ajay Singhal

Numerade Educator

Problem 9

What is the impedance of a series RLC circuit when the frequency of time-varying emf is set to the resonant frequency of the circuit?

Ajay Singhal

Numerade Educator

Problem 10

Estimate the total energy stored in the $5.00 \mathrm{~km}$ of space above Earth's surface if the average magnitude of the magnetic field at Earth's surface is about $0.500 \cdot 10^{-4} \mathrm{~T}$

Ajay Singhal

Numerade Educator

Problem 11

In a DC circuit containing a capacitor, a current will flow through the circuit for only a very short time, while the capacitor is being charged or discharged. On the other hand, a steady alternating current will flow in a circuit containing the same capacitor but powered by a source of time-varying emf. Does it mean that charges are crossing the gap (dielectric) of the capacitor?

Vishal Gupta

Numerade Educator

Problem 12

In an RL circuit with alternating current, the current lags behind the voltage. What does this mean, and how can it be explained qualitatively, based on the phenomenon of electromagnetic induction?

Eduard Sanchez

Numerade Educator

Problem 13

In Solved Problem 30.1 , the voltage supplied by the source of time-varying emf is $33.0 \mathrm{~V}$, the voltage across the resistor is $V_{R}=I R=13.1 \mathrm{~V}$, and the voltage across the inductor is $V_{L}=I X_{L}=30.3 \mathrm{~V}$. Does this circuit obey Kirchhoff's rules?

Eduard Sanchez

Numerade Educator

Problem 14

Why is RMS power specified for an AC circuit, not average power?

Ajay Singhal

Numerade Educator

Problem 15

Why can't we use a universal charger that plugs into a household electrical outlet to charge all our electrical appliances-cell phone, toy dog, can opener, and so on - rather than using a separate charger with its own transformer for each device?

Eduard Sanchez

Numerade Educator

Problem 16

If you use a parallel plate capacitor with air in the gap between the plates as part of a series RLC circuit in a generator, you can measure current flowing through the generator. Why is it that the air gap in the capacitor does not act like an open switch, blocking all current flow in the circuit?

Ajay Singhal

Numerade Educator

Problem 17

A common configuration of wires has twisted pairs as opposed to straight, parallel wires. What is the technical advantage of using twisted pairs of wires versus straight, parallel pairs?

Eduard Sanchez

Numerade Educator

Problem 18

In a classroom demonstration, an iron core is inserted into a large solenoid connected to an AC power source. The effect of the core is to magnify the magnetic field in the solenoid by the relative magnetic permeability, $\kappa_{\mathrm{m}},$ of the core (where $\kappa_{\mathrm{m}}$ is a dimensionless constant, substantially greater than unity for a ferromagnetic material, introduced in Chapter 28 ) or, equivalently, to replace the magnetic permeability of free space, $\mu_{0}$, with the magnetic permeability of the core, $\mu=\kappa_{\mathrm{m}} \mu_{0}$
a) The measured root-mean-square current drops from approximately 10 A to less than $1 \mathrm{~A}$ and remains at the lower value. Explain why.
b) What would happen if the power source were DC?

Vishal Gupta

Numerade Educator

Problem 19

Along Capitol Drive in Milwaukee, Wisconsin, there are a large number of radio broadcasting towers. Contrary to expectation, radio reception there is terrible; unwanted stations often interfere with the one tuned in. Given that a car radio tuner is a resonant oscillator-its resonant frequency is adjusted to that of the desired station-explain this crosstalk phenomenon.

Keshav Singh

Numerade Educator

Problem 20

A series RLC circuit is in resonance when driven by a sinusoidal voltage at its resonant frequency, $\omega_{0}=(L C)^{-1 / 2}$ But if the same circuit is driven by a square-wave voltage (which is alternately on and off for equal time intervals), it will exhibit resonance at its resonant frequency and at $\frac{1}{3}, \frac{1}{5}$, $\frac{1}{7}, \ldots,$ of this frequency. Explain why.

Keshav Singh

Numerade Educator

Problem 21

Is it possible for the voltage amplitude across the inductor in a series RLC circuit to exceed the voltage amplitude of the voltage supply? Why or why not?

Ajay Singhal

Numerade Educator

Problem 22

Why can't a transformer be used to step up or step down the voltage in a DC circuit?

Ajay Singhal

Numerade Educator

Problem 23

For the $L C$ circuit in the figure. $L=32.0 \mathrm{mH}$ and $C=45.0 \mu \mathrm{F}$. The capacitor is charged to $q_{0}=10.0 \mu \mathrm{C},$ and at $t=0 \mathrm{~s},$ the switch is closed. At what time is the energy stored in the capacitor first equal to the energy stored in the inductor?

Eduard Sanchez

Numerade Educator

Problem 24

30.24 A 2.00 - $\mu$ F capacitor is fully charged by being connected to a 12.0 - $\mathrm{V}$ battery. The fully charged capacitor is then connected to a $0.250-\mathrm{H}$ inductor. Calculate (a) the maximum current in the inductor and (b) the frequency of oscillation of the LC circuit.

Eduard Sanchez

Numerade Educator

Problem 25

An LC circuit consists of a 1.00 -mH inductor and a fully charged capacitor. After $2.10 \mathrm{~ms}$, the energy stored in the capacitor is half of its original value. What is the capacitance?

Ajay Singhal

Numerade Educator

Problem 26

The time-varying current in an LC circuit where $C=10.0 \mu \mathrm{F}$ is given by $i(t)=(1.00 \mathrm{~A}) \sin (1200 . t),$ where $t$ is
in seconds.
a) At what time after $t=0$ does the current reach its maximum value?
b) What is the total energy of the circuit?
c) What is the inductance, $L$ ?

Ajay Singhal

Numerade Educator

Problem 27

A 10.0 - $\mu$ F capacitor is fully charged by a 12.0 - $\mathrm{V}$ battery and is then disconnected from the battery and allowed to discharge through a 0.200 - $\mathrm{H}$ inductor. Find the first three times when the charge on the capacitor is $80.0-\mu \mathrm{C},$ taking $t=0$ as the instant when the capacitor is connected to the inductor.

Eduard Sanchez

Numerade Educator

Problem 28

A 4.00 -mF capacitor is connected in series with a $7.00-\mathrm{mH}$ inductor. The peak current in the wires between the capacitor and the inductor is $3.00 \mathrm{~A}$.
a) What is the total electric energy in this circuit?
b) Write an expression for the charge on the capacitor as a function of time, assuming the capacitor is fully charged at $t=0 \mathrm{~s}$

Ajay Singhal

Numerade Educator

Problem 29

A series RLC circuit has resistance $R$, inductance $L$, and capacitance $C$. At what time does the energy in the circuit reach half of its initial value?

Dominador Tan

Numerade Educator

Problem 30

An RLC oscillator circuit contains a $50.0-\Omega$ resistor and a $1.00-\mathrm{mH}$ inductor. What capacitance is necessary for the time constant of the circuit (the $1 / e$ value) to be equal to the oscillation period? Plot the voltage across the resistor as a function of time.

Eduard Sanchez

Numerade Educator

Problem 31

A 2.00 - $\mu$ F capacitor was fully charged by being connected to a 12.0 - $V$ battery. The fully charged capacitor is then connected in series with a resistor and an inductor: $R=50.0 \Omega$ and $L=0.200 \mathrm{H}$. Calculate the damped frequency of the resulting circuit.

Ajay Singhal

Numerade Educator

Problem 32

An LC circuit consists of a capacitor, $C=2.50 \mu \mathrm{F},$ and an inductor, $L=4.0 \mathrm{mH}$. The capacitor is fully charged using a battery and then connected to the inductor. An oscilloscope is used to measure the frequency of the oscillations in the circuit. Next, the circuit is opened, and a resistor, $R$, is inserted in series with the inductor and the capacitor. The capacitor is again fully charged using the same battery and then connected to the circuit. The angular frequency of the damped oscillations in the RLC circuit is found to be $20 \%$ less than the angular frequency of the oscillations in the LC circuit.
a) Determine the resistance of the resistor.
b) How long after the capacitor is reconnected in the circuit will the amplitude of the damped current through the circuit be $50 \%$ of the initial amplitude?
c) How many complete damped oscillations will have occurred in that time?

Eduard Sanchez

Numerade Educator

Problem 33

At what frequency will a $10.0-\mu \mathrm{F}$ capacitor have reactance $X_{C}=200 . \Omega ?$

Ajay Singhal

Numerade Educator

Problem 34

A capacitor with capacitance $C=5.00 \cdot 10^{-6} \mathrm{~F}$ is connected to an AC power source having a peak value of $10.0 \mathrm{~V}$ and $f=100 . \mathrm{Hz} .$ Find the reactance of the capacitor and the maximum current in the circuit.

Ajay Singhal

Numerade Educator

Problem 35

An inductor with inductance $L=47.0 \mathrm{mH}$ is connected to an AC power source having a peak value of $12.0 \mathrm{~V}$ and $f=1000 . \mathrm{Hz} .$ Find the reactance of the inductor and the maximum current in the circuit.

Eduard Sanchez

Numerade Educator

Problem 36

The figure shows a circuit with a source of constant emf connected in series to a resistor, an inductor, and a capacitor. What is the steady-state current flow through the circuit?

Vishal Gupta

Numerade Educator

Problem 37

A series circuit contains a $100.0-\Omega$ resistor, a $0.500-\mathrm{H}$ inductor, a 0.400 - $\mu$ F capacitor, and a time-varying source of emf providing $40.0 \mathrm{~V}$.
a) What is the resonant angular frequency of the circuit?
b) What current will flow through the circuit at the resonant frequency?

Ajay Singhal

Numerade Educator

Problem 38

A variable capacitor used in an RLC circuit produces a resonant frequency of $5.0 \mathrm{MHz}$ when its capacitance is set to $15 \mathrm{pF}$. What will the resonant frequency be when the capacitance is increased to $380 \mathrm{pF} ?$

Ajay Singhal

Numerade Educator

Problem 39

Determine the phase constant and the impedance of the RLC circuit shown in the figure when the frequency of the time-varying emf is $1.00 \mathrm{kHz}$ $C=100 . \mu \mathrm{F}, L=10.0 \mathrm{mH},$ and $R=100 . \Omega$.

Eduard Sanchez

Numerade Educator

Problem 40

What is the resonant frequency of the series RLC circuit of Problem 30.39 if $C=4.00 \mu \mathrm{F}, L=5.00 \mathrm{mH},$ and $R=1.00 \mathrm{k} \Omega ?$ What is the maximum current in the circuit if $V_{\mathrm{m}}=10.0 \mathrm{~V}$ at the resonant frequency?

Ajay Singhal

Numerade Educator

Problem 41

In a series RLC circuit, $V=(12.0 \mathrm{~V})(\sin \omega t), R=10.0 \Omega$ $L=2.00 \mathrm{H},$ and $C=10.0 \mu \mathrm{F}$. At resonance, determine the voltage amplitude across the inductor. Is the result reasonable, considering that the voltage supplied to the entire circuit has an amplitude of $12.0 \mathrm{~V} ?$

Ajay Singhal

Numerade Educator

Problem 42

An AC power source with $V_{\mathrm{m}}=220 \mathrm{~V}$ and $f=60.0 \mathrm{~Hz}$ is connected in a series RLC circuit. The resistance, $R$, inductance, $L$, and capacitance, $C$, of this circuit are, respectively, $50.0 \Omega, 0.200 \mathrm{H},$ and $0.040 \mathrm{mF}$. Find each of the following quantities:
a) the inductive reactance
b) the capacitive reactance
c) the impedance of the circuit
d) the maximum current through the circuit
e) the maximum potential difference across each circuit element

Eduard Sanchez

Numerade Educator

Problem 43

The series RLC circuit shown in the figure has $R=2.20 \Omega, L=9.30 \mathrm{mH}$
$C=2.27 \mathrm{mF}, V_{\mathrm{m}}=110 \mathrm{~V},$ and
$\omega=377 \mathrm{rad} / \mathrm{s}$
a) What is the maximum current, $I_{\mathrm{m}}$, in this circuit?
b) What is the phase constant, $\phi$, between the voltage and the current?
c) The capacitance, $C$, can be varied. What value of $C$ will allow the largest current amplitude oscillations to occur, and what are the magnitudes of this current, $I_{\mathrm{m}}^{\prime}$, and the phase angle, $\phi^{\prime}$, between the current and the voltage?

Eduard Sanchez

Numerade Educator

Problem 44

Design an RC band-pass filter that passes a signal with frequency $5.00 \mathrm{kHz},$ has a ratio $V_{\text {out }} / V_{\text {in }}=0.500,$ and has an impedance of $1.00 \mathrm{k} \Omega$ at very high frequencies.
a) What components will you use?
b) What is the phase of $V_{\text {out }}$ relative to $V_{\text {in }}$ at the frequency of $5.00 \mathrm{kHz} ?$

Eduard Sanchez

Numerade Educator

Problem 45

Design an RC high-pass filter that rejects $60.0 \mathrm{~Hz}$ line noise from a circuit used in a detector. Your criteria are reduction of the amplitude of the line noise by a factor of 1000 . and total impedance at high frequencies of $2.00 \mathrm{k} \Omega$
a) What components will you use?
b) What is the frequency range of the signals that will be passed with at least $90.0 \%$ of their amplitude?

Eduard Sanchez

Numerade Educator

Problem 46

What is the maximum value of the AC voltage whose root-mean-square value is (a) $110 \mathrm{~V}$ or (b) $220 \mathrm{~V} ?$

Eduard Sanchez

Numerade Educator

Problem 47

The quality factor, $Q$, of a circuit can be defined by $Q=\omega_{0}\left(U_{E}+U_{B}\right) / P .$ Express the quality factor of a series RLC circuit in terms of its resistance $R$, inductance $L$, and capacitance $C .$

Ajay Singhal

Numerade Educator

Problem 48

A label on a hair dryer reads "110V $1250 \mathrm{~W}$." What is the peak current in the hair dryer, assuming that it behaves like a resistor?

Ajay Singhal

Numerade Educator

Problem 49

A radio tuner has a resistance of $1.00 \mu \Omega$, a capacitance of $25.0 \mathrm{nF}$, and an inductance of $3.00 \mathrm{mH}$
a) Find the resonant frequency of this tuner.
b) Calculate the power in the circuit if a signal at the resonant frequency produces an emf across the antenna of $V_{\mathrm{rms}}=1.50 \mathrm{mV}$

Ajay Singhal

Numerade Educator

Problem 50

A circuit contains a $100 .-\Omega$ resistor, a $0.0500-\mathrm{H}$ inductor, a $0.400-\mu \mathrm{F}$ capacitor, and a source of time-varying emf connected in series. The time-varying emf corresponds to $V_{\mathrm{rms}}=50.0 \mathrm{~V}$ at a frequency of $2000 . \mathrm{Hz}$.
a) Determine the current in the circuit.
b) Determine the voltage drop across each component of the circuit.
c) How much power is drawn from the source of emf?

Ajay Singhal

Numerade Educator

Problem 51

The figure shows a simple FM antenna circuit in which $L=8.22 \mu \mathrm{H}$ and $C$ is variable (the capacitor can be tuned to receive a specific station). The radio signal from your favorite FM station produces a sinusoidal time-varying emf with an amplitude of $12.9 \mu \mathrm{V}$ and a frequency of $88.7 \mathrm{MHz}$ in the antenna.
a) To what value, $C_{0}$, should you tune the capacitor in order to best receive this station?
b) Another radio station's signal produces a sinusoidal time-varying emf with the same amplitude, $12.9 \mu \mathrm{V}$, but with a frequency of $88.5 \mathrm{MHz}$ in the antenna. With the circuit tuned to optimize reception at $88.7 \mathrm{MHz}$, what should the value, $R_{0}$, of the resistance be in order to reduce by a factor of 2 (compared to the current if the circuit were optimized for $88.5 \mathrm{MHz}$ ) the current produced by the signal from this station?

Eduard Sanchez

Numerade Educator

Problem 52

The transmission of electric power occurs at the highest possible voltage to reduce losses. By how much could the power loss be reduced by raising the voltage by a factor of $10 ?$

Eduard Sanchez

Numerade Educator

Problem 52

The transmission of electric power occurs at the highest possible voltage to reduce losses. By how much could the power loss be reduced by raising the voltage by a factor of $10 ?$

Eduard Sanchez

Numerade Educator

Problem 53

Treat the solenoid and coil of Solved Problem 29.1 as a transformer
a) Find the root-mean-square voltage in the coil if the solenoid has a root-mean-square voltage of $120 \mathrm{~V}$ and $\mathrm{a}$ frequency of $60 \mathrm{~Hz}$. The length of the solenoid is $12.0 \mathrm{~cm}$.
b) What is the voltage in the coil if the frequency is $0 \mathrm{~Hz}$ (DC current)?

Eduard Sanchez

Numerade Educator

Problem 54

A transformer has 800 turns in the primary coil and 40 turns in the secondary coil.
a) What happens if an AC voltage of $100 . \mathrm{V}$ is across the primary coil?
b) If the initial $A C$ current is $5.00 \mathrm{~A}$, what is the output current?
c) What happens if a DC current at $100 .$ V flows into the primary coil?
d) If the initial DC current is $5.00 \mathrm{~A}$, what is the output current?

Ajay Singhal

Numerade Educator

Problem 55

A transformer contains a primary coil with 200 turns and a secondary coil with 120 turns. The secondary coil drives a current $I$ through a $1.00-\mathrm{k} \Omega$ resistor. If an input voltage $V_{\mathrm{rms}}=75.0 \mathrm{~V}$ is applied across the primary coil, what is the power dissipated in the resistor?

Ajay Singhal

Numerade Educator

Problem 56

Consider the filtered fullwave rectifier shown in the figure. If the frequency of the source of time-varying emf is $60 . \mathrm{Hz},$ what is the frequency of the resulting current?

Eduard Sanchez

Numerade Educator

Problem 57

A voltage $V_{\mathrm{rms}}=110 \mathrm{~V}$ at a frequency of $60 . \mathrm{Hz}$ is applied to the primary coil of a transformer. The transformer has a ratio $N_{\mathrm{p}} / N_{\mathrm{S}}=11$. The secondary coil is used as the source of $V_{\text {emf }}$ for the filtered fullwave rectifier of Problem 30.56
a) What is the maximum voltage in the secondary coil of the transformer?
b) What is the DC voltage provided to the resistor?

Eduard Sanchez

Numerade Educator

Problem 58

A vacuum cleaner motor can be viewed as an inductor with an inductance of $100 . \mathrm{mH} .$ For a $60.0-\mathrm{Hz} \mathrm{AC}$ voltage, $V_{\mathrm{rms}}=115 \mathrm{~V}$, what capacitance must be in series with the motor to maximize the power output of the vacuum cleaner?

Ajay Singhal

Numerade Educator

Problem 59

When you turn the dial on a radio to tune it, you are adjusting a variable capacitor in an LC circuit. Suppose you tune to an AM station broadcasting at a frequency of $1000 . \mathrm{kHz},$ and there is a $10.0-\mathrm{mH}$ inductor in the tuning circuit. When you have tuned in the station, what is the capacitance of the capacitor?

Ajay Singhal

Numerade Educator

Problem 60

A series RLC circuit has a source of time-varying emf providing $12.0 \mathrm{~V}$ at a frequency $f_{0}$, with $L=7.00 \mathrm{mH}$, $R=100 . \Omega,$ and $C=0.0500 \mathrm{mF}$
a) What is the resonant frequency of this circuit?
b) What is the average power dissipated in the resistor at this resonant frequency?

Ajay Singhal

Numerade Educator

Problem 61

What are the maximum values of (a) current and
(b) voltage when an incandescent 60 -W light bulb (at $110 \mathrm{~V})$ is connected to a wall plug labeled $110 \mathrm{~V} ?$

Ajay Singhal

Numerade Educator

Problem 62

A $360-\mathrm{Hz}$ source of emf is connected in a circuit consisting of a capacitor, a $25-\mathrm{mH}$ inductor, and an $0.80-\Omega$ resistor. For the current and voltage to be in phase what should the value of $C$ be?

Ajay Singhal

Numerade Educator

Problem 63

What is the resistance in an RLC circuit with $L=65.0 \mathrm{mH}$ and $C=1.00 \mu \mathrm{F}$ if the circuit loses $3.50 \%$ of its total energy as thermal energy in each cycle?

Eduard Sanchez

Numerade Educator

Problem 64

A transformer with 400 turns in its primary coil and 20 turns in its secondary coil is designed to deliver an average power of $1200 .$ W with a maximum voltage of $60.0 \mathrm{~V}$. What is the maximum current in the primary coil?

Ajay Singhal

Numerade Educator

Problem 65

A $5.00-\mu F$ capacitor in series with a $4.00-\Omega$ resistor is charged with $9.00-\mathrm{V}$ battery for a long time by closing the switch (position $a$ in the figure). The capacitor is then discharged through an inductor $(L=40.0 \mathrm{mH})$ by closing the switch (position $b$ ) at $t=0$
a) Determine the maximum current through the inductor.
b) What is the first time at which the current is at its maximum?

Keshav Singh

Numerade Educator

Problem 66

In the RC high-pass filter shown in the figure, $R=10.0 \mathrm{k} \Omega$ and $C=0.0470 \mu \mathrm{F}$ What is the 3.00 -dB frequency of this circuit (where $\mathrm{dB}$ means basically the same for electric current as it did for sound in Chapter 16 )? That is, at what frequency does the ratio of output voltage to input voltage satisfy $20 \log \left(V_{\text {out }} / V_{\text {in }}\right)=-3.00 ?$

Eduard Sanchez

Numerade Educator

Problem 67

The discussion of $\mathrm{RL}, \mathrm{RC},$ and $\mathrm{RLC}$ circuits in this chapter has assumed a purely resistive resistor, one whose inductance and capacitance are exactly zero. While the capacitance of a resistor can generally be neglected, inductance is an intrinsic part of the resistor. Indeed, one of the most widely used resistors, the wire-wound resistor, is nothing but a solenoid made of highly resistive wire. Suppose a wire-wound resistor of unknown resistance is connected to a DC power supply. At a voltage of $V=10.0 \mathrm{~V}$ across the resistor, the current through the resistor is 1.00 A. Next, the same resistor is connected to an AC power source providing $V_{\mathrm{rms}}=10.0 \mathrm{~V}$ at a variable frequency. When the frequency is $20.0 \mathrm{kHz},$ a current, $I_{\mathrm{rms}}=0.800 \mathrm{~A},$ is measured through the resistor.
a) Calculate the resistance of the resistor.
b) Calculate the inductive reactance of the resistor.
c) Calculate the inductance of the resistor.
d) Calculate the frequency of the AC power source at which the inductive reactance of the resistor exceeds its resistance.

Keshav Singh

Numerade Educator

Problem 68

An RLC circuit has a capacitor, a resistor, and an inductor connected in parallel, as shown in the figure, and connected to a time-varying source of emf providing $V_{\mathrm{rms}}$ at frequency $f$. Find an expression for $I_{\mathrm{rms}}$ in terms of $V_{\mathrm{rms}}, f, L, C,$ and $R$.

Ajay Singhal

Numerade Educator

Problem 69

a) A loop of wire $5.00 \mathrm{~cm}$ in diameter is carrying a current of $2.00 \mathrm{~A}$. What is the energy density of the magnetic field at its center?
b) What current has to flow in a straight wire to produce the same energy density at a point $4.00 \mathrm{~cm}$ from the wire?

Vishal Gupta

Numerade Educator

Problem 70

A 75,000 -W light bulb (yes, there are such things!) operates at $I_{\mathrm{rms}}=200 . \mathrm{A}$ and $V_{\mathrm{rms}}=440 . \mathrm{V}$ in a $60.0-\mathrm{Hz} \mathrm{AC}$ circuit. Find the resistance, $R$, and self-inductance, $L$, of this bulb. Its capacitive reactance is negligible.

Ajay Singhal

Numerade Educator

Problem 71

Show that the power dissipated in a resistor connected to an AC power source with frequency $\omega$ oscillates with frequency $2 \omega$.

Ajay Singhal

Numerade Educator

Problem 72

A $300 .-\Omega$ resistor is connected in series with a $4.00-\mu F$ capacitor and a source of time-varying emf providing $V_{\mathrm{rms}}=40.0 \mathrm{~V}$
a) At what frequency will the potential drop across the capacitor equal that across the resistor?
b) What is the rms current through the circuit when this occurs?

Ajay Singhal

Numerade Educator

Problem 73

An electromagnet consists of 200 loops and has a length of $10.0 \mathrm{~cm}$ and a cross-sectional area of $5.00 \mathrm{~cm}^{2}$. Find the resonant frequency of this electromagnet when it is attached to the Earth (treat the Earth as a spherical capacitor)

Eduard Sanchez

Numerade Educator

Problem 74

Laboratory experiments with series RLC circuits require some care, as these circuits can produce large voltages at resonance. Suppose you have a 1.00 - $\mathrm{H}$ inductor (not difficult to obtain) and a variety of resistors and capacitors. Design a series RLC circuit that will resonate at a frequency (not an angular frequency) of $60.0 \mathrm{~Hz}$ and will produce at resonance a magnification of the voltage across the capacitor or the inductor by a factor of 20.0 times the input voltage or the voltage across the resistor.

Eduard Sanchez

Numerade Educator

Problem 75

A particular RC low-pass filter has a breakpoint frequency of $200 .$ Hz. At what frequency will the output voltage divided by the input voltage be $0.100 ?$

Eduard Sanchez

Numerade Educator

Problem 76

In a certain $\mathrm{RLC}$ circuit, a $20.0-\Omega$ resistor, a 10.0 - $\mathrm{mH}$ inductor, and a $5.00-\mu \mathrm{F}$ capacitor are connected in series with an AC power source for which $V_{\mathrm{rms}}=10.0 \mathrm{~V}$ and $f=100 . \mathrm{Hz} .$ Calculate
a) the amplitude of the current,
b) the phase between the current and the voltage, and
c) the maximum voltage across each component.

Ajay Singhal

Numerade Educator