Fundamentals of Physics, Volume 2

David Halliday & Robert Resnick & Jearl Walker

Chapter 31

Electromagnetic Oscillations and Alternating Current - all with Video Answers

Educators

Chapter Questions

Problem 1

An oscillating $L C$ circuit consists of a $75.0 \mathrm{mH}$ inductor and a $3.60 \mu \mathrm{F}$ capacitor. If the maximum charge on the capacitor is $2.90 \mu \mathrm{C}$, what are (a) the total energy in the circuit and (b) the maximum current?

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Jonathan Millis

Numerade Educator

Problem 2

The frequency of oscillation of a certain $L C$ circuit is $200 \mathrm{kHz}$ At time $t=0$, plate $A$ of the capacitor has maximum positive charge. At what earliest time $t>0$ will (a) plate $A$ again have maximum positive charge, (b) the other plate of the capacitor have maximum positive charge, and (c) the inductor have maximum magnetic field?

Ben Nicholson

Numerade Educator

Problem 3

In a certain oscillating $L C$ circuit, the total energy is converted from electrical energy in the capacitor to magnetic energy in the inductor in $1.50 \mu \mathrm{s}$. What are (a) the period of oscillation and (b) the frequency of oscillation? (c) How long after the magnetic energy is a maximum will it be a maximum again?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 4

What is the capacitance of an oscillating $L C$ circuit if the maximum charge on the capacitor is $1.60 \mu \mathrm{C}$ and the total energy is $140 \mu \mathrm{J}$ ?

Ben Nicholson

Numerade Educator

Problem 5

In an oscillating $L C$ circuit, $L=1.10 \mathrm{mH}$ and $C=4.00 \mu \mathrm{F}$. The maximum charge on the capacitor is $3.00 \mu \mathrm{C}$. Find the maximum current.

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 6

A $0.50 \mathrm{~kg}$ body oscillates in SHM on a spring that, when extended $2.0 \mathrm{~mm}$ from its equilibrium position, has an $8.0 \mathrm{~N}$ restoring force. What are (a) the angular frequency of oscillation, (b) the period of oscillation, and (c) the capacitance of an $L C$ circuit with the same period if $L$ is $5.0 \mathrm{H}$ ?

Ben Nicholson

Numerade Educator

Problem 7

The energy in an oscillating $L C$ circuit containing a $1.25 \mathrm{H}$ inductor is $5.70 \mu \mathrm{J}$. The maximum charge on the capacitor is $175 \mu \mathrm{C}$. For a mechanical system with the same period, find the (a) mass, (b) spring constant, (c) maximum displacement, and (d) maximum speed.

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 8

A single loop consists of inductors $\left(L_1, L_2, \ldots\right)$, capacitors $\left(C_1, C_2, \ldots\right)$, and resistors $\left(R_1, R_2, \ldots\right)$ connected in series as shown, for example, in Fig. 31.9a. Show that regardless of the sequence of these circuit elements in the loop, the behavior of this circuit is identical to that of the simple $L C$ circuit shown in Fig. 31.9b.

Ben Nicholson

Numerade Educator

Problem 9

In an oscillating $L C$ circuit with $L=50 \mathrm{mH}$ and $C=4.0 \mu \mathrm{F}$, the current is initially a maximum. How long will it take before the capacitor is fully charged for the first time?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 10

$LC$ oscillators have been used in circuits connected to loudspeakers to create some of the sounds of electronic music. What inductance must be used with a $6.7 \mu \mathrm{F}$ capacitor to produce a frequency of $10 \mathrm{kHz}$, which is near the middle of the audible range of frequencies?

Keshav Singh

Numerade Educator

Problem 11

A variable capacitor with a range from 10 to $365 \mathrm{pF}$ is used with a coil to form a variable-frequency $L C$ circuit to tune the input to a radio. (a) What is the ratio of maximum frequency to minimum frequency that can be obtained with such a capacitor? If this circuit is to obtain frequencies from $0.54 \mathrm{MHz}$ to $1.60 \mathrm{MHz}$, the ratio computed in (a) is too large. By adding a capacitor in parallel to the variable capacitor, this range can be adjusted. To obtain the desired frequency range, (b) what capacitance should be added and (c) what inductance should the coil have?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 12

In an oscillating $L C$ circuit, when $75.0 \%$ of the total energy is stored in the inductor's magnetic field, (a) what multiple of the maximum charge is on the capacitor and (b) what multiple of the maximum current is in the inductor?

Ben Nicholson

Numerade Educator

Problem 13

In an oscillating $L C$ circuit, $L=3.00 \mathrm{mH}$ and $C=2.70 \mu \mathrm{F}$. At $t=0$ the charge on the capacitor is zero and the current is $2.00 \mathrm{~A}$. (a) What is the maximum charge that will appear on the capacitor? (b) At what earliest time $t>0$ is the rate at which energy is stored in the capacitor greatest, and (c) what is that greatest rate?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 14

To construct an oscillating $L C$ system, you can choose from a $10 \mathrm{mH}$ inductor, a $5.0 \mu \mathrm{F}$ capacitor, and a $2.0 \mu \mathrm{F}$ capacitor. What are the (a) smallest, (b) second smallest, (c) second largest, and (d) largest oscillation frequency that can be set up by these elements in various combinations?

Ben Nicholson

Numerade Educator

Problem 15

An oscillating $L C$ circuit consisting of a $1.0 \mathrm{nF}$ capacitor and a $3.0 \mathrm{mH}$ coil has a maximum voltage of $3.0 \mathrm{~V}$. What are (a) the maximum charge on the capacitor, (b) the maximum current through the circuit, and (c) the maximum energy stored in the magnetic field of the coil?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 16

An inductor is connected across a capacitor whose capacitance can be varied by turning a knob. We wish to make the frequency of oscillation of this $L C$ circuit vary linearly with the angle of rotation of the knob, going from $2 \times 10^5$ to $4 \times 10^5 \mathrm{~Hz}$ as the knob turns through $180^{\circ}$. If $L=1.0 \mathrm{mH}$, plot the required capacitance $C$ as a function of the angle of rotation of the knob.

Ben Nicholson

Numerade Educator

Problem 17

In Fig. 31.10, $R=14.0$ $\Omega, C=6.20 \mu \mathrm{F}$, and $L=54.0 \mathrm{mH}$, and the ideal battery has emf $\mathscr{E}=34.0 \mathrm{~V}$. The switch is kept at $a$ for a long time and then thrown to position $b$. What are the (a) frequency and (b) current amplitude of the resulting oscillations?

Keshav Singh

Numerade Educator

Problem 18

An oscillating $L C$ circuit has a current amplitude of $7.50 \mathrm{~mA}$, a potential amplitude of $250 \mathrm{mV}$, and a capacitance of $220 \mathrm{nF}$. What are (a) the period of oscillation, (b) the maximum energy stored in the capacitor, (c) the maximum energy stored in the inductor, (d) the maximum rate at which the current changes, and (e) the maximum rate at which the inductor gains energy?

Ben Nicholson

Numerade Educator

Problem 19

Using the loop rule, derive the differential equation for an $L C$ circuit (Eq. 31.1.11).

Ralph Maestre

Numerade Educator

Problem 20

$L C$ circuit in which $C=4.00 \mu \mathrm{F}$, the maximum potential difference across the capacitor during the oscillations is $1.50 \mathrm{~V}$ and the maximum current through the inductor is $50.0 \mathrm{~mA}$. What are (a) the inductance $L$ and (b) the frequency of the oscillations? (c) How much time is required for the charge on the capacitor to rise from zero to its maximum value?

Ben Nicholson

Numerade Educator

Problem 21

In an oscillating $L C$ circuit with $C=64.0 \mu \mathrm{F}$, the current is given by $i=(1.60) \sin (2500 t+0.680)$, where $t$ is in seconds, $i$ in amperes, and the phase constant in radians. (a) How soon after $t=0$ will the current reach its maximum value? What are (b) the inductance $L$ and (c) the total energy?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 22

series circuit containing inductance $L_1$ and capacitance $C_1$ oscillates at angular frequency $\omega$. A second series circuit, containing inductance $L_2$ and capacitance $C_2$, oscillates at the same angular frequency. In terms of $\omega$, what is the angular frequency of oscillation of a series circuit containing all four of these elements? Neglect resistance. (Hint: Use the formulas for equivalent capacitance and equivalent inductance; see Module 25.3 and Problem 47 in Chapter 30.)

Ben Nicholson

Numerade Educator

Problem 23

In an oscillating $L C$ circuit, $L=25.0 \mathrm{mH}$ and $C=7.80 \mu \mathrm{F}$. At time $t=0$ the current is $9.20 \mathrm{~mA}$, the charge on the capacitor is $3.80 \mu \mathrm{C}$, and the capacitor is charging. What are (a) the total energy in the circuit, (b) the maximum charge on the capacitor, and (c) the maximum current? (d) If the charge on the capacitor is given by $q=Q \cos (\omega t+\phi)$, what is the phase angle $\phi$ ? (e) Suppose the data are the same, except that the capacitor is discharging at $t=0$. What then is $\phi$ ?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 24

A single-loop circuit consists of a $7.20 \Omega$ resistor, a $12.0 \mathrm{H}$ inductor, and a $3.20 \mu \mathrm{F}$ capacitor. Initially the capacitor has a charge of $6.20 \mu \mathrm{C}$ and the current is zero. Calculate the charge on the capacitor $N$ complete cycles later for (a) $N=5$, (b) $N=10$, and (c) $N=100$.

Ben Nicholson

Numerade Educator

Problem 25

What resistance $R$ should be connected in series with an inductance $L=220 \mathrm{mH}$ and capacitance $C=12.0 \mu \mathrm{F}$ for the maximum charge on the capacitor to decay to $99.0 \%$ of its initial value in 50.0 cycles? (Assume $\omega^{\prime} \approx \omega$.)

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 26

In an oscillating series $R L C$ circuit, find the time required for the maximum energy present in the capacitor during an oscillation to fall to half its initial value. Assume $q=Q$ at $t=0$.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 27

In an oscillating series $R L C$ circuit, show that $\Delta U / U$, the fraction of the energy lost per cycle of oscillation, is given to a close approximation by $2 \pi R / \omega L$. The quantity $\omega L / R$ is often called the $Q$ of the circuit (for quality). A high- $Q$ circuit has low resistance and a low fractional energy loss $(=2 \pi / Q)$ per cycle.
Module 31.3 Forced Oscillations of Three Simple Circuits

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 28

A $1.50 \mu \mathrm{F}$ capacitor is connected as in Fig. 31.3.5 to an ac generator with $\mathscr{E}_m=30.0 \mathrm{~V}$. What is the amplitude of the resulting alternating current if the frequency of the emf is (a) $1.00 \mathrm{kHz}$ and (b) $8.00 \mathrm{kHz}$ ?

Keshav Singh

Numerade Educator

Problem 29

A $50.0 \mathrm{mH}$ inductor is connected as in Fig. 31.3 .7 to an ac generator with $\mathscr{G}_m=30.0 \mathrm{~V}$. What is the amplitude of the resulting alternating current if the frequency of the emf is (a) $1.00 \mathrm{kHz}$ and (b) $8.00 \mathrm{kHz}$ ?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 30

A $50.0 \Omega$ resistor is connected as in Fig. 31.3.3 to an ac generator with $\mathscr{E}_m=30.0 \mathrm{~V}$. What is the amplitude of the resulting alternating current if the frequency of the emf is (a) $1.00 \mathrm{kHz}$ and (b) $8.00 \mathrm{kHz}$ ?

Ben Nicholson

Numerade Educator

Problem 31

(a) At what frequency would a $6.0 \mathrm{mH}$ inductor and a $10 \mu \mathrm{F}$ capacitor have the same reactance? (b) What would the reactance be? (c) Show that this frequency would be the natural frequency of an oscillating circuit with the same $L$ and $C$.

Ryan Hood

Numerade Educator

Problem 32

An ac generator has emf $\mathscr{E}=\mathscr{E}_m \sin \omega_d t$, with $\mathscr{E}_m=25.0 \mathrm{~V}$ and $\omega_d=377 \mathrm{rad} / \mathrm{s}$. It is connected to a $12.7 \mathrm{H}$ inductor. (a) What is the maximum value of the current? (b) When the current is a maximum, what is the emf of the generator? (c) When the emf of the generator is $-12.5 \mathrm{~V}$ and increasing in magnitude, what is the current?

Vishal Gupta

Numerade Educator

Problem 33

An ac generator has emf $\mathscr{E}=\mathscr{E}_m \sin \left(\omega_a t-\pi / 4\right)$, where $\mathscr{E}_m=30.0 \mathrm{~V}$ and $\omega_d=350 \mathrm{rad} / \mathrm{s}$. The current produced in a connected circuit is $i(t)=I \sin \left(\omega_d t-3 \pi / 4\right)$, where $I=620 \mathrm{~mA}$. At what time after $t=0$ does (a) the generator emf first reach a maximum and (b) the current first reach a maximum? (c) The circuit contains a single element other than the generator. Is it a capacitor, an inductor, or a resistor? Justify your answer.
(d) What is the value of the capacitance, inductance, or resistance, as the case may be?

Keshav Singh

Numerade Educator

Problem 34

An ac generator with emf $\mathscr{E}=\mathscr{E}_m \sin \omega_d t$, where $\mathscr{E}_m=25.0 \mathrm{~V}$ and $\omega_d=377 \mathrm{rad} / \mathrm{s}$, is connected to a $4.15 \mu \mathrm{F}$ capacitor. (a) What is the maximum value of the current? (b) When the current is a maximum, what is the emf of the generator? (c) When the emf of the generator is $-12.5 \mathrm{~V}$ and increasing in magnitude, what is the current?

Ben Nicholson

Numerade Educator

Problem 35

A coil of inductance $88 \mathrm{mH}$ and unknown resistance and a $0.94 \mu \mathrm{F}$ capacitor are connected in series with an alternating emf of frequency $930 \mathrm{~Hz}$. If the phase constant between the applied voltage and the current is $75^{\circ}$, what is the resistance of the coil?

Sachin Rao

Numerade Educator

Problem 36

An alternating source with a variable frequency, a capacitor with capacitance $C$, and a resistor with resistance $R$ are connected in series. Figure 31.11 gives the impedance $Z$ of the circuit versus the driving angular frequency $\omega_d$; the curve reaches an asymptote of $500 \Omega$, and the horizontal scale is set by $\omega_{d s}=300 \mathrm{rad} / \mathrm{s}$. The figure also gives the reactance $X_C$ for the capacitor versus $\omega_d$. What are (a) $R$ and (b) $C$ ?

Ben Nicholson

Numerade Educator

Problem 37

An electric motor has an effective resistance of $32.0 \Omega$ and an inductive reactance of $45.0 \Omega$ when working under load. The voltage amplitude across the alternating source is $420 \mathrm{~V}$. Calculate the current amplitude.

Sachin Rao

Numerade Educator

Problem 38

The current amplitude $I$ versus driving angular frequency $\omega_d$ for a driven $R L C$ circuit is given in Fig. 31.12, where the vertical axis scale is set by $I_s=4.00 \mathrm{~A}$. The inductance is $200 \mu \mathrm{H}$, and the emf amplitude is $8.0 \mathrm{~V}$. What are (a) $\mathrm{C}$ and (b) $R$ ?

Ben Nicholson

Numerade Educator

Problem 39

Remove the inductor from the circuit in Fig. 31.3.2 and set $R=200 \Omega, C=15.0 \mu \mathrm{F}, f_d=60.0 \mathrm{~Hz}$, and $\mathscr{E}_m=36.0 \mathrm{~V}$. What are
(a) $Z$, (b) $\phi$, and (c) $I$ ? (d) Draw a phasor diagram.

Sachin Rao

Numerade Educator

Problem 40

An alternating source drives a series $R L C$ circuit with an emf amplitude of $6.00 \mathrm{~V}$, at a phase angle of $+30.0^{\circ}$. When the potential difference across the capacitor reaches its maximum positive value of $+5.00 \mathrm{~V}$, what is the potential difference across the inductor (sign included)?

Ben Nicholson

Numerade Educator

Problem 41

In Fig. 31.3.2, set $R=200 \Omega, C=70.0 \mu \mathrm{F}, L=230 \mathrm{mH}$, $f_d=60.0 \mathrm{~Hz}$, and $\mathscr{E}_m=36.0 \mathrm{~V}$. What are (a) $Z$, (b) $\phi$, and (c) $I$ ?
(d) Draw a phasor diagram.

Keshav Singh

Numerade Educator

Problem 42

An alternating source with a variable frequency, an inductor with inductance $L$, and a resistor with resistance $R$ are connected in series. Figure 31.13 gives the impedance $Z$ of the circuit versus the driving angular frequency $\omega_d$, with the horizontal axis scale set by $\omega_{d s}=1600 \mathrm{rad} / \mathrm{s}$. The figure also gives the reactance $X_L$ for the inductor versus $\omega_d$. What are (a) $R$ and (b) $L$ ?

Keshav Singh

Numerade Educator

Problem 43

Remove the capacitor from the circuit in Fig. 31.3.2 and set $R=200 \Omega, L=230 \mathrm{mH}, f_d=60.0 \mathrm{~Hz}$, and $\mathscr{E}_m=36.0 \mathrm{~V}$. What are (a) $Z$, (b) $\phi$, and (c) $I$ ? (d) Draw a phasor diagram.

Sachin Rao

Numerade Educator

Problem 44

An ac generator with emf amplitude $\mathscr{E}_m=220 \mathrm{~V}$ and operating at frequency $400 \mathrm{~Hz}$ causes oscillations in a series $R L C$ circuit having $R=220 \Omega, L=150 \mathrm{mH}$, and $C=24.0 \mu \mathrm{F}$. Find (a) the capacitive reactance $X_C$, (b) the impedance $Z$, and (c) the current amplitude $I$. A second capacitor of the same capacitance is then connected in series with the other components. Determine whether the values of (d) $X_C$, (e) $Z$, and (f) $I$ increase, decrease, or remain the same.

Keshav Singh

Numerade Educator

Problem 45

In an RLC circuit, can the amplitude of the voltage across an inductor be greater than the amplitude of the generator emf? (b) Consider an RLC circuit with emf amplitude $\mathscr{E}_m=10 \mathrm{~V}$, resistance $R=10 \Omega$, inductance $L=1.0 \mathrm{H}$, and capacitance $C=1.0 \mu \mathrm{F}$. Find the amplitude of the voltage across the inductor at resonance.

Sachin Rao

Numerade Educator

Problem 46

An alternating emf source with a variable frequency $f_d$ is connected in series with a $50.0 \Omega$ resistor and a $20.0 \mu \mathrm{F}$ capacitor. The emf amplitude is $12.0 \mathrm{~V}$. (a) Draw a phasor diagram for phasor $V_R$ (the potential across the resistor) and phasor $V_C$ (the potential across the capacitor). (b) At what driving frequency $f_d$ do the two phasors have the same length? At that driving frequency, what are (c) the phase angle in degrees, (d) the angular speed at which the phasors rotate, and (e) the current amplitude?

Keshav Singh

Numerade Educator

Problem 47

An $R L C$ circuit such as that of Fig. 31.3.2 has $R=5.00 \Omega, C=20.0 \mu \mathrm{F}, L=1.00 \mathrm{H}$, and $\mathscr{E}_m=30.0 \mathrm{~V}$. (a) At what angular frequency $\omega_d$ will the current amplitude have its maximum value, as in the resonance curves of Fig. 31.4.3? (b) What is this maximum value? At what (c) lower angular frequency $\omega_{d 1}$ and (d) higher angular frequency $\omega_{d 2}$ will the current amplitude be half this maximum value? (e) For the resonance curve for this circuit, what is the fractional half-width $\left(\omega_{d 1}-\omega_{d 2}\right) / \omega ?$

Keshav Singh

Numerade Educator

Problem 48

Figure 31.14 shows a driven $R L C$ circuit that contains two identical capacitors and two switches. The emf amplitude is set at $12.0 \mathrm{~V}$, and the driving frequency is set at $60.0 \mathrm{~Hz}$. With both switches open, the current leads the emf by $30.9^{\circ}$. With switch $S_1$ closed and switch $S_2$ still open, the emf leads the current by $15.0^{\circ}$. With both switches closed, the current amplitude is $447 \mathrm{~mA}$. What are (a) $R$, (b) $C$, and (c) $L$ ?

Ben Nicholson

Numerade Educator

Problem 49

In Fig. 31.15, a generator with an adjustable frequency of oscillation is connected to resistance $R=100 \Omega$ inductances $L_1=1.70 \mathrm{mH}$ and $L_2=2.30 \mathrm{mH}$, and capacitances $C_1=4.00 \mu \mathrm{F}, C_2=$ $2.50 \mu \mathrm{F}$, and $C_3=3.50 \mu \mathrm{F}$. (a)
What is the resonant frequency of the circuit? (Hint: See Problem 47 in Chapter 30.) What happens to the resonant frequency if (b) $R$ is increased, (c) $L_1$ is increased, and (d) $C_3$ is removed from the circuit?

Sachin Rao

Numerade Educator

Problem 50

An alternating emf source with a variable frequency $f_d$ is connected in series with an $80.0 \Omega$ resistor and a $40.0 \mathrm{mH}$ inductor. The emf amplitude is $6.00 \mathrm{~V}$. (a) Draw a phasor diagram for phasor $V_R$ (the potential across the resistor) and phasor $V_L$ (the potential across the inductor). (b) At what driving frequency $f_d$ do the two phasors have the same length? At that driving frequency, what are (c) the phase angle in degrees, (d) the angular speed at which the phasors rotate, and (e) the current amplitude?

Ben Nicholson

Numerade Educator

Problem 51

The fractional half-width $\Delta \omega_d$ of a resonance curve, such as the ones in Fig. 31.4.3, is the width of the curve at half the maximum value of $I$. Show that $\Delta \omega_d / \omega=R(3 C / L)^{1 / 2}$, where $\omega$ is the angular frequency at resonance. Note that the ratio $\Delta \omega_d / \omega$ increases with $R$, as Fig. 31.4.3 shows.

Keshav Singh

Numerade Educator

Problem 52

An ac voltmeter with large impedance is connected in turn across the inductor, the capacitor, and the resistor in a series circuit having an alternating emf of $100 \mathrm{~V}$ (rms); the meter gives the same reading in volts in each case. What is this reading?
ase with time.

Ben Nicholson

Numerade Educator

Problem 53

An air conditioner connected to a $120 \mathrm{~V} \mathrm{rms}$ ac line is equivalent to a $12.0 \Omega$ resistance and a $1.30 \Omega$ inductive reactance in series. Calculate (a) the impedance of the air conditioner and (b) the average rate at which energy is supplied to the appliance.

Sachin Rao

Numerade Educator

Problem 54

What is the maximum value of an ac voltage whose rms value is $100 \mathrm{~V}$ ?

Ben Nicholson

Numerade Educator

Problem 55

What direct current will produce the same amount of thermal energy, in a particular resistor, as an alternating current that has a maximum value of $2.60 \mathrm{~A}$ ?

Sachin Rao

Numerade Educator

Problem 56

A typical light dimmer used to dim the stage lights in a theater consists of a variable inductor $L$ (whose inductance is adjustable between zero and $L_{\max }$ ) connected in series with a lightbulb B, as shown in Fig. 31.16. The electrical supply is $120 \mathrm{~V}$ (rms) at $60.0 \mathrm{~Hz}$; the lightbulb is rated at $120 \mathrm{~V}, 1000 \mathrm{~W}$. (a) What $L_{\max }$ is required if the rate of energy dissipation in the lightbulb is to be varied by a factor of 5 from its upper limit of $1000 \mathrm{~W}$ ? Assume that the resistance of the lightbulb is independent of its temperature. (b) Could one use a variable resistor (adjustable between zero and $R_{\text {max }}$ ) instead of an inductor? (c) If so, what $R_{\max }$ is required? (d) Why isn't this done?

Ben Nicholson

Numerade Educator

Problem 57

In an $R L C$ circuit such as that of Fig. 31.3.2 assume that $R=5.00 \Omega, L=60.0 \mathrm{mH}, f_d=60.0 \mathrm{~Hz}$, and $\mathscr{E}_m=30.0 \mathrm{~V}$. For what values of the capacitance would the average rate at which energy is dissipated in the resistance be (a) a maximum and (b) a minimum? What are (c) the maximum dissipation rate and the corresponding (d) phase angle and (e) power factor? What are (f) the minimum dissipation rate and the corresponding (g) phase angle and (h) power factor?

Sachin Rao

Numerade Educator

Problem 58

For Fig. 31.17, show that the average rate at which energy is dissipated in resistance $R$ is a maximum when $R$ is equal to the internal resistance $r$ of the ac generator. (In the text discussion we tacitly assumed that $r=0$.

Ben Nicholson

Numerade Educator

Problem 59

In Fig. $31.3 .2, R=15.0$ $\Omega, C=4.70 \mu \mathrm{F}$, and $L=25.0 \mathrm{mH}$. The generator provides an emf with rms voltage $75.0 \mathrm{~V}$ and frequency $550 \mathrm{~Hz}$ (a) What is the rms current? What is the rms voltage across (b) $R$, (c) $C$, (d) $L$, (e) $C$ and $L$ together, and (f) $R, C$, and $L$ together? At what average rate is energy dissipated by (g) $R$, (h) $C$, and (i) $L$ ?

Keshav Singh

Numerade Educator

Problem 60

In a series oscillating $R L C$ circuit, $R=16.0 \Omega$, $C=31.2 \mu \mathrm{F}, L=9.20 \mathrm{mH}$, and $\mathscr{E}_m=\mathscr{E}_m \sin \omega_d t$ with $\mathscr{E}_m=45.0 \mathrm{~V}$ and $\mathscr{E}_m=3000 \mathrm{rad} / \mathrm{s}$. For time $t=0.442 \mathrm{~ms}$ find (a) the rate $P_g$ at which energy is being supplied by the generator, (b) the rate $P_C$ at which the energy in the capacitor is changing, (c) the rate $P_L$ at which the energy in the inductor is changing, and (d) the rate $P_R$ at which energy is being dissipated in the resistor. (e) Is the sum of $P_C, P_L$, and $P_R$ greater than, less than, or equal to $P_g$ ?

Ben Nicholson

Numerade Educator

Problem 61

Figure 31.18 shows an ac generator connected to a "black box" through a pair of terminals. The box contains an RLC circuit, possibly even a multiloop circuit, whose elements and connections we do not know. Measurements outside the box reveal that
$$
\begin{array}{cc}
& \mathscr{E}(t)=(75.0 \mathrm{~V}) \sin \omega_d t \\
\text { and } \quad i(t)=(1.20 \mathrm{~A}) \sin \left(\omega_d t+42.0^{\circ}\right) \text {. }
\end{array}
$$
(a) What is the power factor? (b) Does the current lead or lag the emf? (c) Is the circuit in the box largely inductive or largely capacitive? (d) Is the circuit in the box in resonance? (e) Must there be a capacitor in the box? (f) An inductor? (g) A resistor? (h) At what average rate is energy delivered to the box by the generator? (i) Why don't you need to know $\omega_d$ to answer all these questions?

Keshav Singh

Numerade Educator

Problem 62

A generator supplies $100 \mathrm{~V}$ to a transformer's primary coil, which has 50 turns. If the secondary coil has 500 turns, what is the secondary voltage?

Ben Nicholson

Numerade Educator

Problem 63

A transformer has 500 primary turns and 10 secondary turns. (a) If $V_p$ is $120 \mathrm{~V}(\mathrm{rms})$, what is $V_s$ with an open circuit? If the secondary now has a resistive load of $15 \Omega$, what is the current in the (b) primary and (c) secondary?

Sachin Rao

Numerade Educator

Problem 64

Figure 31.19 shows an "autotransformer." It consists of a single coil (with an iron core). Three taps $T_i$ are provided.
Between taps $T_1$ and $T_2$ there are 200 turns, and between taps $T_2$ and $T_3$ there are 800 turns. Any two taps can be chosen as the primary terminals, and any two taps can be chosen as the secondary terminals. For choices producing a step-up transformer, what are the (a) smallest, (b) second smallest, and (c) largest values of the ratio $V_s / V_p$ ? For a step-down transformer, what are the (d) smallest, (e) second smallest, and (f) largest values of $V_s / V_p$ ?

Ben Nicholson

Numerade Educator

Problem 65

An ac generator provides emf to a resistive load in a remote factory over a two-cable transmission line. At the factory a step-down transformer reduces the voltage from its (rms) transmission value $V_t$ to a much lower value that is safe and convenient for use in the factory. The transmission line resistance is $0.30 \Omega / \mathrm{cable}$, and the power of the generator is $250 \mathrm{~kW}$. If $V_i=80 \mathrm{kV}$, what are (a) the voltage decrease $\Delta V$ along the transmission line and (b) the rate $P_d$ at which energy is dissipated in the line as thermal energy? If $V_t=8.0 \mathrm{kV}$, what are (c) $\Delta V$ and (d) $P_d$ ? If $V_t=0.80 \mathrm{kV}$, what are (e) $\Delta V$ and (f) $P_d$ ?

Keshav Singh

Numerade Educator

Problem 66

In Fig. 31.17, let the rectangular box on the left represent the (high-impedance) output of an audio amplifier, with $r=1000 \Omega$. Let $R=10 \Omega$ represent the (low-impedance) coil of a loudspeaker. For maximum transfer of energy to the load $R$ we must have $R=r$, and that is not true in this case. However, a transformer can be used to "transform" resistances, making them behave electrically as if they were larger or smaller than they actually are. (a) Sketch the primary and secondary coils of a transformer that can be introduced between the amplifier and the speaker in Fig. 31.17 to match the impedances. (b) What must be the turns ratio?

Ben Nicholson

Numerade Educator

Problem 67

An ac generator produces emf $\mathscr{E}=\mathscr{E}_m \sin \left(\omega_d t-\pi / 4\right)$, where $\mathscr{E}_m=30.0 \mathrm{~V}$ and $\omega_d=350 \mathrm{rad} / \mathrm{s}$. The current in the circuit attached to the generator is $i(t)=I \sin \left(\omega_d t+\pi / 4\right)$, where $I=620 \mathrm{~mA}$. (a) At what time after $t=0$ does the generator emf first reach a maximum? (b) At what time after $t=0$ does the current first reach a maximum? (c) The circuit contains a single element other than the generator. Is it a capacitor, an inductor, or a resistor? Justify your answer. (d) What is the value of the capacitance, inductance, or resistance, as the case may be?

Keshav Singh

Numerade Educator

Problem 68

A series $R L C$ circuit is driven by a generator at a frequency of $2000 \mathrm{~Hz}$ and an emf amplitude of $170 \mathrm{~V}$. The inductance is $60.0 \mathrm{mH}$, the capacitance is $0.400 \mu \mathrm{F}$, and the resistance is $200 \Omega$. (a) What is the phase constant in radians? (b) What is the current amplitude?

Keshav Singh

Numerade Educator

Problem 69

A generator of frequency $3000 \mathrm{~Hz}$ drives a series $R L C$ circuit with an emf amplitude of $120 \mathrm{~V}$. The resistance is $40.0 \Omega$, the capacitance is $1.60 \mu \mathrm{F}$, and the inductance is $850 \mu \mathrm{H}$. What are (a) the phase constant in radians and (b) the current amplitude?
(c) Is the circuit capacitive, inductive, or in resonance?

Sachin Rao

Numerade Educator

Problem 70

A $45.0 \mathrm{mH}$ inductor has a reactance of $1.30 \mathrm{k} \Omega$. (a) What is its operating frequency? (b) What is the capacitance of a capacitor with the same reactance at that frequency? If the frequency is doubled, what is the new reactance of (c) the inductor and (d) the capacitor?

Ben Nicholson

Numerade Educator

Problem 71

An $R L C$ circuit is driven by a generator with an emf amplitude of $80.0 \mathrm{~V}$ and a current amplitude of $1.25 \mathrm{~A}$. The current leads the emf by $0.650 \mathrm{rad}$. What are the (a) impedance and (b) resistance of the circuit? (c) Is the circuit inductive, capacitive, or in resonance?

Sachin Rao

Numerade Educator

Problem 72

A series $R L C$ circuit is driven in such a way that the maximum voltage across the inductor is 1.50 times the maximum voltage across the capacitor and 2.00 times the maximum voltage across the resistor. (a) What is $\phi$ for the circuit? (b) Is the circuit inductive, capacitive, or in resonance? The resistance is $49.9 \Omega$, and the current amplitude is $200 \mathrm{~mA}$. (c) What is the amplitude of the driving emf?

Ben Nicholson

Numerade Educator

Problem 73

A capacitor of capacitance $158 \mu \mathrm{F}$ and an inductor form an $L C$ circuit that oscillates at $8.15 \mathrm{kHz}$, with a current amplitude of $4.21 \mathrm{~mA}$. What are (a) the inductance, (b) the total energy in the circuit, and (c) the maximum charge on the capacitor?

Sachin Rao

Numerade Educator

Problem 74

An oscillating LC circuit has an inductance of $3.00 \mathrm{mH}$ and a capacitance of $10.0 \mu \mathrm{F}$. Calculate the (a) angular frequency and (b) period of the oscillation. (c) At time $t=0$, the capacitor is charged to $200 \mu \mathrm{C}$ and the current is zero. Roughly sketch the charge on the capacitor as a function of time.

Ben Nicholson

Numerade Educator

Problem 75

For a certain driven series $R L C$ circuit, the maximum generator emf is $125 \mathrm{~V}$ and the maximum current is $3.20 \mathrm{~A}$. If the current leads the generator emf by $0.982 \mathrm{rad}$, what are the (a) impedance and (b) resistance of the circuit? (c) Is the circuit predominantly capacitive or inductive?

Sachin Rao

Numerade Educator

Problem 76

A $1.50 \mu \mathrm{F}$ capacitor has a capacitive reactance of $12.0 \Omega$. (a) What must be its operating frequency? (b) What will be the capacitive reactance if the frequency is doubled?

Ben Nicholson

Numerade Educator

Problem 77

In Fig. 31.20, a threephase generator $G$ produces electrical power that is transmitted by means of three wires. The electric potentials (each relative to a common reference level) are $V_1=A \sin \omega_d t$ for wire $1, V_2=A$ $\sin \left(\omega_d t-120^{\circ}\right)$ for wire 2 , and $V_3=A \sin \left(\omega_d t-240^{\circ}\right)$ for wire
3. Some types of industrial equipment (for example, motors) have three terminals and are designed to be connected directly to these three wires. To use a more conventional two-terminal device (for example, a lightbulb), one connects it to any two of the three wires. Show that the potential difference between any two of the wires (a) oscillates sinusoidally with angular frequency $\omega_d$ and (b) has an amplitude of $A \sqrt{3}$.

Sachin Rao

Numerade Educator

Problem 78

An electric motor connected to a $120 \mathrm{~V}, 60.0 \mathrm{~Hz}$ ac outlet does mechanical work at the rate of $0.100 \mathrm{hp}(1 \mathrm{hp}=746 \mathrm{~W})$.
(a) If the motor draws an rms current of $0.650 \mathrm{~A}$, what is its effective resistance, relative to power transfer? (b) Is this the same as the resistance of the motor's coils, as measured with an ohmmeter with the motor disconnected from the outlet?

Ben Nicholson

Numerade Educator

Problem 79

(a) In an oscillating LC circuit, in terms of the maximum charge $Q$ on the capacitor, what is the charge there when the energy in the electric field is $50.0 \%$ of that in the magnetic field? (b) What fraction of a period must elapse following the time the capacitor is fully charged for this condition to occur?

Sachin Rao

Numerade Educator

Problem 80

A series $R L C$ circuit is driven by an alternating source at a frequency of $400 \mathrm{~Hz}$ and an emf amplitude of $90.0 \mathrm{~V}$. The resistance is $20.0 \Omega$, the capacitance is $12.1 \mu \mathrm{F}$, and the inductance is $24.2 \mathrm{mH}$. What is the rms potential difference across
(a) the resistor, (b) the capacitor, and (c) the inductor? (d) What is the average rate at which energy is dissipated?

Ben Nicholson

Numerade Educator

Problem 81

In a certain series $R L C$ circuit being driven at a frequency of $60.0 \mathrm{~Hz}$, the maximum voltage across the inductor is 2.00 times the maximum voltage across the resistor and 2.00 times the maximum voltage across the capacitor. (a) By what angle does the current lag the generator emf? (b) If the maximum generator emf is $30.0 \mathrm{~V}$, what should be the resistance of the circuit to obtain a maximum current of $300 \mathrm{~mA}$ ?

Supratim Pal

Numerade Educator

Problem 82

A $1.50 \mathrm{mH}$ inductor in an oscillating $L C$ circuit stores a maximum energy of $10.0 \mu \mathrm{J}$. What is the maximum current?

Ben Nicholson

Numerade Educator

Problem 83

A generator with an adjustable frequency of oscillation is wired in series to an inductor of $L=2.50 \mathrm{mH}$ and a capacitor of $C=3.00 \mu \mathrm{F}$. At what frequency does the generator produce the largest possible current amplitude in the circuit?
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Sachin Rao

Numerade Educator

Problem 84

A series $R L C$ circuit has a resonant frequency of $6.00 \mathrm{kHz}$. When it is driven at $8.00 \mathrm{kHz}$, it has an impedance of $1.00 \mathrm{k} \Omega$ and a phase constant of $45^{\circ}$. What are (a) $R$, (b) $L$, and (c) $C$ for this circuit?
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Ben Nicholson

Numerade Educator

Problem 85

An $L C$ circuit oscillates at a frequency of $10.4 \mathrm{kHz}$.
(a) If the capacitance is $340 \mu \mathrm{F}$, what is the inductance? (b) If the maximum current is $7.20 \mathrm{~mA}$, what is the total energy in the circuit? (c) What is the maximum charge on the capacitor?

Sachin Rao

Numerade Educator

Problem 86

When under load and operating at an rms voltage of $220 \mathrm{~V}$, a certain electric motor draws an rms current of $3.00 \mathrm{~A}$. It has a resistance of $24.0 \Omega$ and no capacitive reactance. What is its inductive reactance?
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Ben Nicholson

Numerade Educator

Problem 87

The ac generator in Fig. 31.21 supplies $120 \mathrm{~V}$ at $60.0 \mathrm{~Hz}$. With the switch open as in the diagram, the current leads the generator emf by $20.0^{\circ}$. With the switch in position 1, the current lags the generator emf by $10.0^{\circ}$. When the switch is in position 2, the current amplitude is $2.00 \mathrm{~A}$. What are (a) $R$, (b) $L$, and (c) $C$ ?

Keshav Singh

Numerade Educator

Problem 88

In an oscillating $L C$ circuit, $L=8.00 \mathrm{mH}$ and $C=1.40 \mu \mathrm{F}$. At time $t=0$, the current is maximum at $12.0 \mathrm{~mA}$. (a) What is the maximum charge on the capacitor during the oscillations? (b) At what earliest time $t>0$ is the rate of change of energy in the capacitor maximum? (c) What is that maximum rate of change?

Ben Nicholson

Numerade Educator

Problem 89

For a sinusoidally driven series $R L C$ circuit, show that over one complete cycle with period $T$ (a) the energy stored in the capacitor does not change; (b) the energy stored in the inductor does not change; (c) the driving emf device supplies energy $\left(\frac{1}{2} T\right) \mathscr{E}_m I \cos \phi$; and (d) the resistor dissipates energy $\left(\frac{1}{2} T\right) R I^2$.
(e) Show that the quantities found in (c) and (d) are equal.

Keshav Singh

Numerade Educator

Problem 90

What capacitance would you connect across a $1.30 \mathrm{mH}$ inductor to make the resulting oscillator resonate at $3.50 \mathrm{kHz}$ ?
with

Vishal Gupta

Numerade Educator

Problem 91

A series circuit with resistor-inductor-capacitor combination $R_1, L_1, C_1$ has the same resonant frequency as a second circuit with a different combination $R_2, L_2, C_2$. You now connect the two combinations in series. Show that this new circuit has the same resonant frequency as the separate circuits.

Sachin Rao

Numerade Educator

Problem 92

Consider the circuit shown in Fig. 31.22. With switch $S_1$ closed and the other two switches open, the circuit has a time constant $\tau_C$. With switch $\mathrm{S}_2$ closed and the other two switches open, the circuit has a time constant $\tau_L$. With switch $S_3$ closed and the other two switches open, the circuit oscillates with a period $T$. Show that $T=2 \pi \sqrt{\tau C} \tau_L$.

Ben Nicholson

Numerade Educator

Problem 93

Hand-to-hand current. Here are the physiological effects when an ac current is established across a person, say, hand to hand:
$1 \mathrm{~mA}$, perception threshold
$10-20 \mathrm{~mA}$, onset of involuntary muscle contractions
$100-300 \mathrm{~mA}$, heart fibrillation, eventually fatal
$1 \mathrm{~A}$, heart ceases to beat, internal burns produced
Anyone skilled in working with live (energized) ac circuits knows to put one hand behind the back to avoid having both hands in contact with the circuit. Indeed, some people tuck a hand in a back pocket. Figure 31.23 shows a live circuit that a person touches with both hands. The rms voltage is $V_{\mathrm{rms}}=120 \mathrm{~V}$ and the resistance of the conducting pathway through the body is $R_{\text {body }}=300 \Omega$. What is the rms current $I_{\text {rms }}$ through the body if the skin resistance on each hand is (a) $R_{\mathrm{dry}}=100 \mathrm{k} \Omega$ for dry hands and (b) $R_{\text {wet }}=1.0 \mathrm{k} \Omega$ for skin wet with sweat?

Sanjeev Kumar

Numerade Educator

Problem 94

The let-go current. Here is one common danger of electricity in the home and workplace: If a person grabs a live (energized) wire (or some other conducting object), the person may not be able to let go because of involuntary contractions of the hand muscles. Suppose that the pathway of the ac current is then through the bare hand, the body, and the shoes, to a conducting floor. According to experiments, most people can let go of a wire for a rms current of $6 \mathrm{~mA}$ but not for a rms current of $22 \mathrm{~mA}$, dubbed the "let-go" level. Consider the common rms voltage $V_{\text {rms }}=120 \mathrm{~V}$. Assume the hand's skin resistance is $R_{\mathrm{dry}}=100 \mathrm{k} \Omega$ for dry skin and $R_{\text {wet }}=1.0 \mathrm{k} \Omega$ for skin wet with sweat. Take $R_{\text {body }}=300 \Omega$ for the conducting pathway through the body, $R_{\text {boots }}=2000 \Omega$ for common electrician work boots, and $R_{\text {shoes }}=200 \Omega$ for common leather shoes. (a) What is the rms current $I_{\text {rms }}$ through the person for dry skin when wearing the boots, and is it above the let-go level? (b) What are the results if the person has wet skin and is wearing the boots? (c) What are the results if the person has wet skin and is wearing the leather shoes? Even if the current is only somewhat above the let-go level, the involuntary contractions can produce a tighter grip with a greater contact area and sweat production, and the resistances can decrease

Sanjeev Kumar

Numerade Educator