Chapter 31

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?

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?

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 \(1.50 \mu\) 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?

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 \(140 \mu \mathrm{J} ?\)

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.

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 \(5.0 \mathrm{H} ?\)

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?

\(L C\) 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?

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?

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?

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?

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?

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?

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 \(50.0 \mathrm{~mA}\). 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?

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?

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 47 in Chapter \(30 .\)

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 \(\left.\omega^{\prime} \approx \omega .\right)\)

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.

(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 \(C\).

Go 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?

An ac generator has 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 produced in a connected circuit is \(i(t)=I \sin \left(\omega_{d} t-3 \pi / 4\right),\) where \(I=620 \mathrm{~m}\) A. 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?

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?

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?

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.

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)?

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 \((\mathrm{c})\) 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.

(a) In an \(R L C\) circuit, can the amplitude of the voltage across an inductor be greater than the amplitude of the generator emf? (b) Consider an \(R L C\) 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.

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?

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?

An air conditioner connected to a \(120 \mathrm{Vrms}\) 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.

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

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 \(2.60 \mathrm{~A}\) ?

In a series oscillating \(R L C\) circuit, \(R=16.0 \Omega, C=\) \(31.2 \mu \mathrm{F}, L=9.20 \mathrm{mH},\) and \(\mathscr{C}_{m}=\mathscr{E}_{m} \sin \omega_{d} t\) with \(\mathscr{E}_{m}=45.0 \mathrm{~V}\) and \(\omega_{d}=3000 \mathrm{rad} / \mathrm{s} .\) For time \(t=0.442 \mathrm{~ms}\) find \((\mathrm{a})\) 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 \(P_{g} ?\)

Figure \(31-36\) 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 \(\mathscr{E}(t)=(75.0 \mathrm{~V}) \sin \omega_{d} t\) and \(i(t)=(1.20 \mathrm{~A}) \sin \left(\omega_{d} t+42.0^{\circ}\right)\) (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?

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?

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?

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 \(0.30 \Omega /\) cable, and the power of the generator is \(250 \mathrm{~kW}\). 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 \(V_{i}=0.80 \mathrm{kV}\), what are (e) \(\Delta V\) and (f) \(P_{d} ?\)

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?

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?

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?

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?

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 rad. What are the (a) impedance and (b) resistance of the circuit? (c) Is the circuit inductive, capacitive, or in resonance?

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?

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?

An oscillating \(L C\) 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.

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 rad, what are the (a) impedance and (b) resistance of the circuit? (c) Is the circuit predominantly capacitive or inductive?

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?

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?

(a) In an oscillating \(L C\) 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?

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?