Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 33

Electromagnetic Oscillations and Alternating Current - all with Video Answers

Educators

Chapter Questions

Problem 1

Current Is Zero Suppose that the inductive time constant for the circuit of Fig. $33-19$ is $37.0 \mathrm{~ms}$ and the current in the circuit is zero at time $t=0 \mathrm{~s}$. At what time does the rate at which energy is dissipated in the resistor equal the rate at which energy is being stored in the inductor?

Katie Mcalpine

Katie Mcalpine

Numerade Educator

Problem 2

Consider the Circuit Consider the circuit of Fig. 33-19. In terms of the inductive time constant, at what instant after the battery is connected will the energy stored in the magnetic field of the inductor be half its steady-state value?

Km Neeraj

Numerade Educator

Problem 3

Coil Connected in Series A coil is connected in series with a $10.0 \mathrm{k} \Omega$ resistor. A $50.0 \mathrm{~V}$ battery is applied across the two devices, and the current reaches a value of $2.00 \mathrm{~mA}$ after $5.00 \mathrm{~ms}$. (a) Find the inductance of the coil. (b) How much energy is stored in the coil at this same moment?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 4

Rates A coil with an inductance of $2.0 \mathrm{H}$ and a resistance of $10 \Omega$ is suddenly connected to a resistanceless battery with $\mathscr{E}=$ $100 \mathrm{~V}$. At $0.10 \mathrm{~s}$ after the connection is made, what are the rates at which (a) energy is being stored in the magnetic field, (b) thermal energy is appearing in the resistance, and (c) energy is being delivered by the battery?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 5

Prove That Prove that, after switch $\mathrm{S}$ has been thrown from $a$ to $b$, all the energy stored in the inductor will ulti-mately appear as thermal energy in the resistor.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 6

Energy Delivered For the circuit of assume that $\mathscr{E}=$ $10.0 \mathrm{~V}, R=6.70 \Omega$, and $L=5.50 \mathrm{H} .$ The battery is connected at time $t=0$ s. (a) How much energy is delivered by the battery during the first $2.00 \mathrm{~s}$ ? (b) How much of this energy is stored in the magnetic field of the inductor? (c) How much of this energy is dissipated in the resistor?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 7

Energy Density A solenoid that is $85.0 \mathrm{~cm}$ long has a cross-sectional area of $17.0 \mathrm{~cm}^{2}$. There are 950 turns of wire carrying a current of $6.60$ A. (a) Calculate the energy density of the magnetic field inside the solenoid. (b) Find the total energy stored in the magnetic field there (neglect end effects).

Vishal Gupta

Numerade Educator

Problem 8

Toroidal Inductor A toroidal inductor with an inductance of $90.0 \mathrm{mH}$ encloses a volume of $0.0200 \mathrm{~m}^{3}$. If the average energy density in the toroid is $70.0 \mathrm{~J} / \mathrm{m}^{3}$, what is the current through the inductor?

Vishal Gupta

Numerade Educator

Problem 9

Magnitude of $E$ -Field What must be the magnitude of a uniform electric field if it is to have the same energy density as that possessed by a $0.50$ T magnetic field?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 10

Interstellar Space The magnetic field in the interstellar space of our galaxy has a magnitude of about $10^{-10} \mathrm{~T}$. How much energy is stored in this field in a cube 10 light-years on edge? (For scale, note that the nearest star is $4.3$ light-years distant and the radius of our galaxy is about $8 \times 10^{4}$ light-years.)

Katie Mcalpine

Numerade Educator

Problem 11

Length of Copper Wire A length of copper wire carries a current of 10 A, uniformly distributed through its cross section. Calculate the energy density of (a) the magnetic field and (b) the electric field at the surface of the wire. The wire diameter is $2.5 \mathrm{~mm}$, and its resistance per unit length is $3.3 \Omega / \mathrm{km}$.

Katie Mcalpine

Numerade Educator

Problem 12

Circular Loop A circular loop of wire $50 \mathrm{~mm}$ in radius carries a current of $100 \mathrm{~A} .$ (a) Find the magnetic field strength at the center of the loop. (b) Calculate the energy density at the center of the loop.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 13

What Is the Capacitance 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 14

Maximum Charge 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 15

Total Energy 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 $2.90 \mu \mathrm{C},(\mathrm{a})$ what is the total energy in the circuit and
(b) what is the maximum current?

jm

Jonathan Millis

Numerade Educator

Problem 16

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

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 17

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

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 18

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

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 19

Energy 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 $175 \mu \mathrm{C}$. Find (a) the mass, (b) the spring constant, (c) the maximum displacement, and (d) the maximum speed for a mechanical system with the same period.

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 20

Loudspeakers $L C$ 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 $6.7 \mu \mathrm{F}$ capacitor to produce a frequency of $10 \mathrm{kHz}$, which is near the middle of the audible range of frequencies?

Ben Nicholson

Numerade Educator

Problem 21

Initially a Maximum 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 22

Single Loop 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. $33-21 a$. Show that regardless of the sequence of these circuit elements in the loop, the bahavior of this circuit is identical to that of the simple $L C$ circuit shown in Fig. 33-21b. (Hint: Consider the loop rule and see Problem 7 in Chapter 32.)

Ben Nicholson

Numerade Educator

Problem 23

Maximum Voltage 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 $3.0 \mathrm{~V}$.
(a) What is the maximum charge on the capacitor? (b) What is the maximum current through the circuit? (c) What is the maximum energy stored in the magnetic field of the coil?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 24

Maximum Potential Difference In an oscillating $L C$ 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 $50.0 \mathrm{~mA}$. (a) What is the inductance $L$ ?
(b) What is 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 25

Switch Is Thrown In the circuit shown in Fig. 33-22 the switch is kept in position $a$ for a long time. It is then thrown to position $b$. (a) Calculate the frequency of the resulting oscillating current. (b) What is the amplitude of the current oscillations?

Keshav Singh

Numerade Educator

Problem 26

One Inductor, Two Capacitors You are given a $10 \mathrm{mH}$ inductor and two capacitors, of $5.0 \mu \mathrm{F}$ and $2.0 \mu \mathrm{F}$ capacitance. List the oscillation fre- quencies that can be generated by connecting these elements in various combinations.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 27

Variable Capacitor 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 ratio of maximum to minimum frequencies may be obtained with such a capacitor? (b) 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 may be adjusted. What should be the capacitance of this added capacitor, and what inductance should be used to obtain the desired range of frequencies?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 28

Energy Stored in Magnetic Field In an oscillating $L C$ circuit, $75.0 \%$ of the total energy is stored in the magnetic field of the inductor at a certain instant. (a) In terms of the maximum charge on the capacitor, what is the charge there at that instant? (b) In terms of the maximum current in the inductor, what is the current there at that instant?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 29

Capacitor Is Charging In an oscillating $L C$ circuit, $L=25.0 \mathrm{mH}$ and $C=7.80 \mu \mathrm{F}$. At time $t=0 \mathrm{~s}$ the current is $9.20 \mathrm{~m} \mathrm{~A}$, the charge on the capacitor is $3.80 \mu \mathrm{C}$, and the capacitor is charging. (a) What is the total energy in the circuit? (b) What is the maximum charge on the capacitor? (c) What is 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 $t=0 \mathrm{~s}$. What then is $\phi$ ?

MG

Miguel Angel Garcia Chavez

Numerade Educator

Problem 30

Varied by a Knob 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 31

Oscillating $L \boldsymbol{C}$ Circuit In an oscillating $L C$ circuit, $L=3.00 \mathrm{mH}$ and $C=2.70 \mu \mathrm{F}$. At $t=0 \mathrm{~s}$ the charge on the capacitor is zero and the current is $2.00$ A. (a) What is the maximum charge that will appear on the capacitor? (b) In terms of the period $T$ of oscillation, how much time will elapse after $t=0$ until the energy stored in the capacitor will be increasing at its greatest rate? (c) What is this greatest rate at which energy is transferred to the capacitor?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 32

Angular Frequency 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 Section $28-$ 4 and Problem 7 in Chapter 32.)

Ben Nicholson

Numerade Educator

Problem 33

Current as Function of Time In an oscillating $L C$ circuit with $C=64.0 \mu \mathrm{F}$, the current as a function of time is given by $i=(1.60 \mathrm{~A})$ $\sin \left[\begin{array}{llll}(2500 & \mathrm{rad} / \mathrm{s}) & t+0.680 & \mathrm{rad}]\end{array}\right.$
where $t$ is in seconds. (a) How soon after $t=0 \mathrm{~s}$ will the current reach its maximum value? What are
(b) the inductance $L$ and $(\mathrm{c})$ the total energy?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 34

$\begin{array}{lll}\text {Three } & \text { Identical } & \text { Inductors }\end{array}$ Three identical inductors $L$ and two identical capacitors $C$ are connected in a two-loop circuit as shown in Fig. 33-23. (a) Suppose the currents are as shown . What is the current in the middle inductor? Write the loop equations and show that they are satisfied if the current oscillates with angular frequency $\omega=1 / \sqrt{L C}$. (b) Now suppose the currents are as shown in Fig. $33-23 b$. What is the current in the middle inductor? Write the loop equations and show that they are satisfied if the current oscillates with angular frequency $\omega=$ $1 / \sqrt{3 L C}$. Because the circuit can oscillate at two different frequencies, we cannot find an equivalent single-loop $L C$ circuit to replace it.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 35

Capacitor One, Capacitor Two , capacitor 1 with $C_{1}=$ $900 \mu \mathrm{F}$ is initially charged to $100 \mathrm{~V}$ and capacitor 2 with $C_{2}=100 \mu \mathrm{F}$ is uncharged. The inductor has an inductance of $10.0 \mathrm{H}$. Describe in detail how one might charge capacitor 2 to $300 \mathrm{~V}$ by manipulating switches $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 36

Damped $L C$ Consider a damped $L C$ circuit. (a) Show that the damping term $e^{-R t / 2 L}$ (which involves $L$ but $\operatorname{not} C$ ) can be rewritten in a more symmetric manner (involving $L$ and $C$ ) as $e^{-\pi R(\sqrt{C / L}) t T}$. Here $T$ is the period of oscillation (neglecting resistance). (b) Using
(a), show that the SI unit of $\sqrt{L / C}$ is the ohm. (c) Using (a), show that the condition that the fractional energy loss per cycle be small is $R \ll \sqrt{L / C}$.

Lainey Roebuck

Numerade Educator

Problem 37

What Resistance 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 38

Single-Loop Circuit 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 $N=5,10$, and 100 .

Ben Nicholson

Numerade Educator

Problem 39

Oscillating Series $R L C$ 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 40

No Charge on Capacitor At time $t=0 \mathrm{~s}$ there is no charge on the capacitor of a series $R L C$ circuit but there is current $I$ through the inductor. (a) Find the phase constant $\phi$ in Eq. 33-31 for the circuit. (b) Write an expression for the charge $q$ on the capacitor as a function of time $t$ and in terms of the current amplitude and angular frequency $\omega^{\prime}$ of the oscillations.

Raj Bala

Numerade Educator

Problem 41

Fraction of Energy Lost In an oscillating series $R L C$ circuit, show that the fraction of the energy lost per cycle of oscillation, $\Delta U / U$, 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 42

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

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 43

AC Generator A $50.0 \mathrm{mH}$ inductor is connected as to an ac generator with $\left|\mathscr{E}^{\max }\right|=30.0 \mathrm{~V}$. What is the amplitude of the resulting alternating current if the frequency of the emf is (a) $1.00 \mathrm{kHz}$ and $(\mathrm{b}) 8.00 \mathrm{kHz}$ ?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 44

Frequency of emf Is A $50 \Omega$ resistor is connected as to an ac generator with $\left|\mathscr{E}^{\max }\right|=30.0 \mathrm{~V}$. What is the amplitude of the resulting alternating current if the frequency of the $\mathrm{emf}$ is (a) $1.00 \mathrm{kHz}$ and (b) $8.00 \mathrm{kHz}$ ?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 45

At What Frequency (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 $C$.

Sachin Rao

Numerade Educator

Problem 46

When the Current Is Maximum An ac generator has emf $\mathscr{E}=$ $\mathscr{E}^{\max } \sin \omega^{\mathrm{dr}} t$, with $\mathscr{E}^{\max }=25.0 \mathrm{~V}$ and $\omega^{\mathrm{dr}}=377 \mathrm{rad} / \mathrm{s} .$ It is con-
nected 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 47

At What Time An ac generator has emf $\mathscr{E}=\mathscr{E} \max \sin \left(\omega^{\mathrm{dr}} t-\right.$
$\pi / 4)$, where $\mathscr{E}^{\max }=30.0 \mathrm{~V}$ and $\omega^{\mathrm{dr}}=350 \mathrm{rad} / \mathrm{s}$. The current pro-
duced in a connected circuit is $i(t)=I \sin \left(\omega^{\mathrm{dr}} t-3 \pi / 4\right)$, where $I=$ $620 \mathrm{~m}$ A. (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 48

Generator from Above The ac generator of Problem 46 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?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 49

Find $\boldsymbol{Z}, \boldsymbol{\phi}$, and $I$ (a) Find $Z, \phi$, and $I$ for the situation of Touchstone Example $33-4$ with the capacitor removed from the circuit, all other parameters remaining unchanged. (b) Draw to scale a phasor diagram like that of Fig. $33-15 d$ for this new situation.

Averell Hause

Carnegie Mellon University

Problem 50

Find $\boldsymbol{Z}, \boldsymbol{\phi}$, and $\boldsymbol{I}$ Two (a) Find $Z, \phi$, and $I$ for the situation of Touchstone Example $33-4$ with the inductor removed from the circuit, all other parameters remaining unchanged. (b) Draw to scale a phasor diagram like that of Fig. $33-15 d$ for this new situation.

Averell Hause

Carnegie Mellon University

Problem 51

Find $Z, \phi$, and $I$ Three (a) Find $Z, \phi$, and $I$ for the situation of Touchstone Example $33-4$ with $C=70.0 \mu \mathrm{F}$, the other parame-ters remaining unchanged. (b) Draw a phasor diagram like that of Fig. 33 $15 d$ for this new situation and compare the two diagrams closely.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 52

Adjustable Frequency , a generator with an adjustable frequency of oscillation is connected to a variable resistance $R$, a capacitor of $C=5.50 \mu \mathrm{F}$, and an inductor of inductance $L .$ The amplitude of the current produced in the circuit by the generator is at half-maximum level when the generator's frequency is $1.30$ or $1.50 \mathrm{kHz}$. (a) What is $L ?(\mathrm{~b})$ If $R$ is increased, what happens to the frequencies at which the current amplitude is at half-maximum level?

Sachin Rao

Numerade Educator

Problem 53

At Resonance In an $R L C$ circuit, can the amplitude of the voltage across an inductor be greater than the amplitude of the generator emf? Consider an $R L C$ circuit with $\mathscr{E}^{\max }=10 \mathrm{~V}, R=10 \Omega, L=$ $1.0 \mathrm{H}$, and $C=1.0 \mu \mathrm{F}$. Find the amplitude of the voltage across the inductor at resonance.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 54

Emf Is Maximum When the generator emf in Touchstone Example $33-4$ is a maximum, what is the voltage across (a) the generator, (b) the resistance, (c) the capacitance, and (d) the inductance? (e) By summing these with appropriate signs, verify that the loop rule is satisfied.

Keshav Singh

Numerade Educator

Problem 55

Unknown Resistance 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 56

Capacitive Reactance An ac generator with $\mathscr{E}^{\max }=220 \mathrm{~V}$ and operating at $400 \mathrm{~Hz}$ 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 $X_{C},(\mathrm{~b})$ the impedance $Z$, and $(\mathrm{c})$ 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 57

Half-Width An $R L C$ circuit such as that of Fig. $33-11$ has $R=5.00 \Omega, C=20.0 \mu \mathrm{F}, L=1.00 \mathrm{H}$, and $\mathscr{E}^{\max }=30.0 \mathrm{~V}$. (a) At
what angular frequency $\omega^{\mathrm{dr}}$ will the current amplitude have its maximum value, as in the resonance curves of Fig. 33-17? (b) What is this maximum value? (c) At what two angular frequencies $\omega_{1}^{\mathrm{dr}}$ and $\omega_{2}^{\mathrm{dr}}$ will the current amplitude be half this maximum value?
(d) What is the fractional half-width $\left[=\left(\omega_{1}^{\mathrm{dr}}-\omega_{2}^{\mathrm{dr}}\right) / \omega\right]$ of the resonance curve for this circuit?

Keshav Singh

Numerade Educator

Problem 58

Generator in Series An ac generator is to be connected in series with an inductor of $L=2.00 \mathrm{mH}$ and a capacitance $C .$ You are to produce $C$ by using capacitors of capacitances $C_{1}=4.00 \mu \mathrm{F}$ and $C_{2}=6.00 \mu \mathrm{F}$, either singly or together. What resonant frequencies can the circuit have, depending on how you use $C_{1}$ and $C_{2} ?$

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 59

Fractional Half-Width Show that the fractional half-width (see Problem 57 ) of a resonance curve is given by
$$
\frac{\Delta \omega^{\mathrm{dr}}}{\omega}=\sqrt{\frac{3 C}{L}} R
$$
in which $\omega$ is the angular frequency at resonance and $\Delta \omega^{\mathrm{dr}}$ is the width of the resonance curve at half-amplitude. Note that $\Delta \omega^{\mathrm{dr}} / \omega$ increases with $R$, as Fig. $33-17$ shows. Use this formula to check the answer to Problem $57 \mathrm{~d}$.

Keshav Singh

Numerade Educator

Problem 60

Adjustable Frequency Two In Fig. 33-26, 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 7 in Chapter 32.) What happens to the resonant frequency if (b) the value of $R$ is increased, (c) the value of $L_{1}$ is increased, and (d) capacitance $C_{3}$ is removed from the circuit?

Sachin Rao

Numerade Educator

Problem 61

Thermal Energy 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 62

AC Voltmeter 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 $100 \mathrm{~V}(\mathrm{rms}) ;$ it gives the same reading in volts in each case. What is this reading?

Ben Nicholson

Numerade Educator

Problem 63

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

Ben Nicholson

Numerade Educator

Problem 64

Give or Take (a) For the conditions in Problem $46 \mathrm{c}$, is the generator supplying energy to or taking energy from the rest of the circuit? (b) Repeat for the conditions of Problem $48 \mathrm{c}$.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 65

Average Rate of Dissipation Calculate the average rate of energy dissipation in the circuits of Problems $43,44,49$, and 50 .

Sheh Lit Chang

University of Washington

Problem 66

Energy Is Supplied Show that the average rate at which energy is supplied to the circuit of Fig. $33-11$ can also be written as $\langle P\rangle=$ $(\mathscr{\mathrm { r }} \mathrm{rs})^{2} R / Z^{2}$. Show that this expression for average power gives reasonable results for a purely resistive circuit, for an $R L C$ circuit at resonance, for a purely capacitive circuit, and for a purely inductive circuit.

Sailesh Mohanty

Sailesh Mohanty

Numerade Educator

Problem 67

Air Conditioner 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. (a) Calculate the impedance of the air conditioner. (b) Find the average rate at which energy is supplied to the appliance.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 68

Oscillating $R L C$ In a series oscillating $R L C$ circuit, $R=16.0 \Omega$, $C=31.2 \mu \mathrm{F}, L=9.20 \mathrm{mH}$, and $\mathscr{E}=\left|\mathscr{E}^{\max }\right| \sin \omega^{\mathrm{dr}} t$ with $\left|\mathscr{E}^{\max }\right|=$
$45.0 \mathrm{~V}$ and $\omega^{\mathrm{dr}}=3000 \mathrm{rad} / \mathrm{s}$. For time $t=0.442 \mathrm{~ms}$ find (a) the rate
at which energy is being supplied by the generator, (b) the rate at which the energy in the capacitor is changing, (c) the rate at which the energy in the inductor is changing, and (d) the rate at which energy is being dissipated in the resistor. (e) What is the meaning of a negative result for any of (a), (b), and (c)? (f) Show that the results of $(\mathrm{b}),(\mathrm{c})$, and $(\mathrm{d})$ sum to the result of $(\mathrm{a})$

Ben Nicholson

Numerade Educator

Problem 69

Black Box 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 elements and connections we do not know. Measurements outside the box reveal that
$$
\mathscr{E}(t)=(75.0 \mathrm{~V}) \sin \omega^{\mathrm{dr}}
$$
and
$$
i(t)=(1.20 \mathrm{~A}) \sin \left(\omega^{\mathrm{dr}} t+42.0^{\circ}\right) .
$$
(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? An inductor? A resistor ? (f) At what average rate is energy delivered to the box by the generator? (g) Why don't you need to know the angular fre-
quency $\omega^{\mathrm{dr}}$ to answer all these questions?

BS

Bonny Sahouin

Numerade Educator

Problem 70

Average Rate 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 have tacitly assumed that $r=0$.)

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 71

Energy Is Dissipated In an $R L C$ circuit such as that of assume that $R=5.00 \Omega, L=60.0 \mathrm{mH} f^{\mathrm{dr}}=60.0 \mathrm{~Hz}$, and
$\left|\mathscr{E}^{\max }\right|=30.0 \mathrm{~V}$. For what values of the capacitor would the average rate at which energy is dissipated in the resistance be (a) a maximum and (b) a minimum? (c) What are these maximum and minimum energy dissipation rates? What are (d) the corresponding phase angles and (e) the corresponding power factors?

Sachin Rao

Numerade Educator

Problem 72

Light Dimmer 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 $\left.L^{\max }\right)$ connected in series with the lightbulb $\mathrm{B}$ as shown in Fig. 33-29. The electrical supply is $120 \mathrm{~V}$ (rms) at $60.0 \mathrm{~Hz}$; the lightbulb is rated as "120 V, 1000 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 $R^{\max }$ ) instead of an inductor? If so, what $R^{\max }$ is required? Why isn't this done?

Ben Nicholson

Numerade Educator

Problem 73

Sinusoidal Voltage , $R=15.0 \Omega, C=4.70 \quad \mu \mathrm{F}$, and $L=$
$25.0 \mathrm{mH}$. The generator provides a sinusoidal voltage of $75.0 \mathrm{~V}(\mathrm{rms})$ and frequency $f=550 \mathrm{~Hz}$.
(a) Calculate the rms current.
(b) Find the rms voltages $\Delta V_{a b}, \Delta V_{b c}$, $\Delta V_{c d}, \Delta V_{b d}, \Delta V_{a d} \cdot$ (c) At what average rate is energy dissipated by each of the three circuit elements?

Keshav Singh

Numerade Educator