Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 31

Electromagnetic Oscillations and Alternating Current - all with Video Answers

Educators

Chapter Questions

Problem 1

In an oscillating $L C$ circuit with $L=79 \mathrm{mH}$ 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 (a) the first time and (b) the second time?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 2

An ac generator with emf $\mathcal{G}={8}_{\mathrm{m}} \sin \omega_{2} t$, where ${B}_{m}=18.0 \mathrm{~V}$ 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 3

In Fig. 31-19, a generator with an adjustable frequency of oscillation is connected to resistance $R=100 \Omega$, inductances
$L_{1}=9.70 \mathrm{mH}$ and $L_{2}=2.30 \mathrm{mH}$, and capacitances $C_{1}=8.40 \mu \mathrm{F}$, $C_{2}=2.50 \mu \mathrm{F}$, and $C_{3}=3.50 \mu \mathrm{F}$ (a) What is the resonant frequency of the circuit? (Hint See Problem 11 in Chapter $30 .$ ) What happens to the resonant frequency if (b) $R$ is increased, (c) $L_{1}$ is increased, (d) $C_{3}$ is removed from the circuit, and (e) $L_{2}$ is removed?

Sachin Rao

Numerade Educator

Problem 4

A $80.0 \Omega$ resistor is connected as in Fig. 31-8 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) $8.00 \mathrm{kHz}$ ?

Ben Nicholson

Numerade Educator

Problem 5

In Fig. 31-7, set $R=400 \Omega, C=70.0 \mu \mathrm{F}, L=920 \mathrm{mH}, f_{d}=$ $30.0 \mathrm{~Hz}$, and ${\ell}_{\mathrm{m}}=72.0 \mathrm{~V}$. What are (a) $Z$, (b) $\phi$, and (c) $I ?$ (d) Draw a phasor diagram.

Keshav Singh

Numerade Educator

Problem 6

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 $25 \%$ of its initial value. Assume $q=Q$ at $t=0$.

Keshav Singh

Numerade Educator

Problem 7

An $R L C$ circuit such as that of Fig. 31-7 has $R=5.00 \Omega$, $C=20.0 \mu \mathrm{F}, L=2.00 \mathrm{H}$, and ${\ell}_{m}=30.0 \mathrm{~V}$. (a) At what angular frequency $\omega_{d}$ will the current amplitude have its maximum value, as in the resonance curves of Fig. 31-16? (b) What is this maximum value? At what (c) lower angular frequency $\omega_{d 1}$ and (d) higher angular frequency $\omega_{12}$ will the current amplitude be half this maximum value? (e) For the resonance curve for this circuit, what is the fractional half-width $\left(\omega_{\Delta l}-\omega_{12}\right) / \omega$ ?

Keshav Singh

Numerade Educator

Problem 8

An alternating source with a variable frequency, a capacitor with capacitance $C$, and a resistor with resistance $R$ are connected in series. Figure 31-20 gives the impedance $Z$ of the circuit versus the driving angular frequency $\omega_{2}$ the curve reaches an asymptote of $500 \Omega$, and the horizontal scale is set by $\omega_{2}=600 \mathrm{rad} / \mathrm{s}$. The figure also gives the reactance $X_{C}$ for the capacitor versus $\omega_{2}$. What are (a) $R$ and (b) $C$ ?

Ben Nicholson

Numerade Educator

Problem 9

(a) In an $R L C$ circuit, can the amplitude of the voltage across (b) Consider an $R L C$ circuit with emf amplitude ${E}_{m}=10 \mathrm{~V}$, resistance $R=5.0 \Omega$, 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 10

For Fig. 31-21, show that the average rate at which energy is dissipated in resistance $R$ is a maximum when $R$ is equal to the intemal resistance $r$ of the ac generator. (In the text discussion we tacitly assumed that $r=0$.)

Ben Nicholson

Numerade Educator

Problem 11

An air conditioner connected
to a $125 \mathrm{~V}$ rms ac line is equivalent Figure 31-21 Problem 10 . to a $9.20 \Omega$ resistance and a $4.70 \Omega$
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 12

An alternating emf source with a variable frequency $f_{d}$ is connected in series with an $80.0 \Omega$ resistor and a $25.0 \mathrm{mH}$ 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 13

Remove the capacitor from the circuit in Fig. 31-7 and set $R=400 \Omega, L=230 \mathrm{mH}, f_{d}=120 \mathrm{~Hz}$, and $\mathscr{E}_{m}=72.0 \mathrm{~V}$. What are (a) $Z$, (b) $\phi$, and (c) $I ?$ (d) Draw a phasor diagram.

Keshav Singh

Numerade Educator

Problem 14

An alternating source drives a series $R L C$ circuit with an emf amplitude of $10.0 \mathrm{~V}$, 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 15

A coil of inductance $62 \mathrm{mH}$ 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 $82^{\circ}$, what is the resistance of the coil?

Sachin Rao

Numerade Educator

Problem 16

In a series oscillating $R L C$ circuit, $R=12.0 \Omega, C=31.2 \mu \mathrm{F}$, $L=9.20 \mathrm{mH}$, and $\mathscr{E}_{m}=\mathscr{C}_{m} \sin \omega_{d} t$ with $\mathscr{E}_{m}=45.0 \mathrm{~V}$ and $\omega_{d}=3000 \mathrm{rad} / \mathrm{s}$. 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. (c) 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 17

An ac generator has emf $\mathscr{C}={8}_{m} \sin \left(\omega_{\perp} t-\pi / 4\right)$, where $\mathscr{E}_{m}=25.0 \mathrm{~V}$ and $\omega_{d}=270 \mathrm{rad} / \mathrm{s}$. The current produced in a connected circuit is $i(t)=I \sin \left(\omega_{\mu} t-3 \pi / 4\right)$, 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?

Sachin Rao

Numerade Educator

Problem 18

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

Ben Nicholson

Numerade Educator

Problem 19

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 $7.82 \mathrm{~A}$ ?

Sachin Rao

Numerade Educator

Problem 20

An ac generator with emf amplitude $\mathscr{E}_{m}=180 \mathrm{~V}$ 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 21

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 $0.30 \Omega / c a b l e$, and the power of the generator is $300 \mathrm{~kW}$. If $V_{t}=80 \mathrm{kV}$, 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 $V_{f}=0.80 \mathrm{kV}$, what are (c) $\Delta V$ and (f) $P_{d}$ ?

Keshav Singh

Numerade Educator

Problem 22

A single loop consists of inductors $\left(L_{1}, L_{2}, \ldots\right)$, capacitors $\left(C_{1},\right.$, $\left.C_{2}, \ldots\right)$, and resistors $\left(R_{1}, R_{2}, \ldots\right)$ connected in series as shown, for example, in Fig. 31-22a. 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. 31-22b.
(Hint: Consider the loop rule and see Problem 11 in Chapter 30.)
Figure 31-22 Problem 22.

Ben Nicholson

Numerade Educator

Problem 23

Figure 31-23 shows an ac generator connected to a "black box" through a pair of terminals. The box contains an $R L C$ circuit, possibly even a multiloop circuit, whose ele-
ments and connections we do not
know. Measurements outside the box reveal that
$$
\mathscr{E}(t)=(61.4 \mathrm{~V}) \sin \omega_{a} t
$$
and $i(t)=(0.930 \mathrm{~A}) \sin \left(\omega_{u} t+42.0^{\circ}\right) .$
(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 24

What is the capacitance of an oscillating $L C$ circuit if the maximum charge on the capacitor is $2.40 \mu \mathrm{C}$ and the total energy is $140 \mu \mathrm{J} ?$

Ben Nicholson

Numerade Educator

Problem 25

Remove the inductor from the circuit in Fig. 31-7 and set $R=400 \Omega, C=15.0 \mu \mathrm{F}, f_{d}=30.0 \mathrm{~Hz}$, and $\mathcal{E}_{m}=72.0 \mathrm{~V}$. What are (a) $Z$, (b) $\phi$, and (c) $I ?$ (d) Draw a phasor diagram.

Sachin Rao

Numerade Educator

Problem 26

Figure $31-24$ 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 $\operatorname{set}$ at $60.0 \mathrm{~Hz}$. With both switches open, the current leads the emf by $25.0^{\circ}$. With switch $S_{1}$ closed and switch $S_{2}$ still open, the emf leads the current by $20.0^{\circ}$. With both switches closed, the current amplitude is $447 \mathrm{~mA}$. What are (a) $R$, (b) $C$, and (c) $L$ ?
Figure 31-24 Problem $26 .$

Ben Nicholson

Numerade Educator

Problem 27

In an oscillating $L C$ circuit, $L=5.97 \mathrm{mH}$ and $C=4.00 \mu \mathrm{F}$. The maximum charge on the capacitor is $3.00 \mu \mathrm{C}$. Find (a) the maximum current and (b) the oscillation period.

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 28

To construct an oscillating $L C$ system, you can choose from a $10 \mathrm{mH}$ inductor, a $8.0 \mu \mathrm{F}$ capacitor, and a $4.0 \mu \mathrm{F}$ 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 29

In an $R L C$ circuit such as that of Fig. $31-7$ assume that $R=12.0 \Omega$, $L=60.0 \mathrm{mH}, f_{d}=60.0 \mathrm{~Hz}$, and $8_{m}=30.0 \mathrm{~V}$. 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 $(\mathrm{g})$ phase angle and $(\mathrm{h})$ power factor?

Sachin Rao

Numerade Educator

Problem 30

An alternating emf source with a variable frequency $f_{d}$ is connected in series with a $50.0 \Omega$ resistor and a $28.0 \mu \mathrm{F}$ capacitor. The emf amplitude is $12.0 \mathrm{~V}_{-}$(a) Draw a phasor diagram for phasor $V_{R}$ (the potential across the resistor) and phaser $V_{C}$ (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 31

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.

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 32

In an oscillating $L C$ circuit in which $C=6.00 \mu \mathrm{F}$, 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 33

A $85.0 \mathrm{mH}$ inductor is connected as in Fig. 31-12 to an ac generator with $\mathscr{E}_{m}=30.0$ V. What is the amplitude of the resulting alternating current if the frequency of the emf is (a) $1.00 \mathrm{kHz}$ and (b) $5.00 \mathrm{kHz}$ ?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 34

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 $\mathrm{knob}$ turns through $180^{\circ}$. If $L=2.0 \mathrm{mH}$, plot the required capacitance $C$ as a function of the angle of rotation of the knob.

Ben Nicholson

Numerade Educator

Problem 35

A transformer has 400 primary turns and 10 secondary turns. (a) If $V_{p}$ is $120 \mathrm{~V}$ (rms), what is $V_{s}$ with an open circuit? If the secondary now has a resistive load of $27 \Omega$, what is the current in the (b) primary and (c) secondary?

Sachin Rao

Numerade Educator

Problem 36

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 supply lightbulb $\mathrm{B}$, as shown in Fig. 31-25
lightbulb B, as shown in Fig. 31-25.
The electrical supply is $120 \mathrm{~V}$ (rms) at
The cloctrical supply is $120 \mathrm{~V}$ (rms) at $60.0 \mathrm{Hr}$, the lightbulb is rated at $120 \mathrm{~V}, 1200 \mathrm{~W}$. (a) What $L_{\operatorname{mas}}$ is required if the rate of energy dissipation in the lightbulb is to be varied by a factor of 4 from its upper limit of 1200 W? Assume that the resistance of the lightbulb is independent of its temperature. (b) Could one use a variable resistor (adjustable between zero and $R_{\max }$ ) instead of an inductor? (c) If so, what $R_{\max }$ is required? (d) Why isn't this done?

Ben Nicholson

Numerade Educator

Problem 37

In a certain oscillating $L C$ circuit, the total energy is converted from clectrical cnergy in the capacitor to magnetie cnergy in the inductor in $2.50 \mu \mathrm{s}$. 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? (d) In one full cycle, how many times will the electrical energy be maximum?

Keshav Singh

Numerade Educator

Problem 38

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 $125 \mathrm{~V}$ (rms); the meter gives the same reading in volts in each case. What is this reading?

Ben Nicholson

Numerade Educator

Problem 39

In Fig. 31-7, $R=25.0 \Omega, C=4.70 \mu \mathrm{F}$, 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 40

A 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 11 in Chapter 30.)

Ben Nicholson

Numerade Educator

Problem 41

A variable capacitor with a range from 10 to $410 \mathrm{pF}$ 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 42

What is the maximum value of an ac voltage whose rms value is $220 \mathrm{~V}$ ?

Ben Nicholson

Numerade Educator

Problem 43

In an oscillating $L C$ circuit, $L=25.0 \mathrm{mH}$ and $C=2.89 \mu \mathrm{F}$. 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 44

An ac generator has emf $\mathscr{E}=\mathscr{Q}_{m} \sin \omega_{d} t$, with $\mathscr{C}_{m}=30.0 \mathrm{~V}$ 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 $-15.0 \mathrm{~V}$ and increasing in magnitude, what is the current?

Vishal Gupta

Numerade Educator

Problem 45

An electric motor has an effective resistance of $61.0 \Omega$ and an inductive reactance of $52.0 \Omega$ when working under load. The rms voltage across the alternating source is 420 V. Calculate the rms current.

Sachin Rao

Numerade Educator

Problem 46

A $0.25 \mathrm{~kg}$ 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 47

The fractional half-width $\Delta \omega_{\mu}$ of a resonance curve, such as the unes in Fig. 31-16, is the width of the curve at lualf the maxiumum value of $I$. (a) Show that $\Delta \omega_{\mathrm{d}} / \omega=R(3 C / L)^{1 / 2}$, where $\omega$ is the angular frequency at resonance. (b) What happens to the ratio $\Delta \omega_{\mu} / \omega$ with an increase in $R ?$

Keshav Singh

Numerade Educator

Problem 48

$ \quad L C$ 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 $3.4 \mu \mathrm{F}$ capacitor to produce a frequency of $10 \mathrm{kHz}$, which is near the middle of the audible range of frequencies?

Ben Nicholson

Numerade Educator

Problem 49

In Fig. 31-26, $R=14.0 \Omega, C=31.2 \mu \mathrm{F}$, and $L=54.0 \mathrm{mH}$, and the ideal battery has emf $8=34.0 \mathrm{~V}$. 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?
Figure 31-26 Problem 49.

Keshav Singh

Numerade Educator

Problem 50

Figure $31-27$ 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 50 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 $V_{x} / V_{p}$ ? For a step-down transformer, what are the (d) smallest, (e) second smallest, and (f) largest values of $V_{x} / V_{p}$ ?
Figure 31-27
Problem 50 .

Ben Nicholson

Numerade Educator

Problem 51

In an oscillating $L C$ circuit, $L=3.00 \mathrm{mH}$ and $C=3.90 \mu \mathrm{F}$. At $t=0$ the charge on the capacitor is zero and the current is $1.75 \mathrm{~A}$.
(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 52

A $1.50 \mu \mathrm{F}$ capacitor is connected as in Fig. 31-10 to an ac generator with $\mathscr{G}_{m}=24.0 \mathrm{~V}$. 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 53

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 $5.00 \mu \mathrm{C}$, what are (a) the total energy in the circuit, (b) the maximum current, and (c) the period of the oscillations?

Nishant Kumar

Numerade Educator

Problem 54

A single-loop circuit consists of a $7.20 \Omega$ resistor, a $12.0 \mathrm{H}$ inductor, and a $5.60 \mu \mathrm{F}$ 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) $N=5,(\mathrm{~b}) N=10$, and $(\mathrm{c}) N=100 .$

Ben Nicholson

Numerade Educator

Problem 55

In an oscillating $L C$ circuit with $C=64.0 \mu \mathrm{F}$, the current is given by $i=(1.60) \sin (4100 t+0.680)$, 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 56

An oscillating LC circuit has a current amplitude of $7.50 \mathrm{~mA}$, a potential amplitude of $280 \mathrm{mV}$, 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 57

An oscillating $L C$ circuit consisting of a $1.0 \mathrm{nF}$ capacitor and a $9.0 \mathrm{mH}$ 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 58

The current amplitude $I$ versus driving angular frequency $\omega_{\lambda}$ for a driven $R L C$ circuit is given in Fig. $31-28$, where the vertical axis scale is set by $I_{s}=4.00 \mathrm{~A}$. The inductance is $450 \mu \mathrm{H}$, and the emf amplitude is $6.0 \mathrm{~V}$. What are (a) $C$ and (b) $R ?$
Figure 31-28 Problem $58 .$

Ben Nicholson

Numerade Educator

Problem 59

The energy in an oscillating $L C$ circuit containing a $2.50 \mathrm{H}$ 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 60

An alternating source with a variable frequency, an inductor with inductance $L$, and a resistor with resistance $R$ are connected in series. Figure $31-29$ gives the impedance $Z$ of the circuit versus the driving angular frequency $\omega_{6}$, with the horizontal axis scale set by $\omega_{\text {ds }}=3200 \mathrm{rad} / \mathrm{s}$. The figure also gives the reactance $X_{L}$ for the inductor versus $\omega_{d-}$ What are (a) $R$ and (b) $L$ ?
Figure 31-29 Problem 60.

Ben Nicholson

Numerade Educator

Problem 61

What resistance $R$ should be connected in series with an inductance $L=490 \mathrm{mH}$ and capacitance $C=19.0 \mu \mathrm{F}$ for the maximum charge on the capacitor to decay to $85.0 \%$ of its initial value in $50.0$ cycles? (Assume $\omega^{\prime}=\omega$.)

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 62

In an oscillating $L C$ circuit, when $30.0 \%$ 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 63

Using the loop rule, derive the differential equation for an $L C$ circuit (Eq.31-11).

Keshav Singh

Numerade Educator

Problem 64

The frequency of oscillation of a certain $L C$ circuit is $220 \mathrm{kHz}$. 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 65

(a) At what frequency would a $12 \mathrm{mH}$ 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 $C$.

Sachin Rao

Numerade Educator