Chapter 23

The reactance of a capacitor is \(68 \Omega\) when the ac frequency is \(460 \mathrm{~Hz}\). What is the reactance when the frequency is \(870 \mathrm{~Hz} ?\)

A capacitor is connected across the terminals of an ac generator that has a frequency of \(440 \mathrm{~Hz}\) and supplies a voltage of \(24 \mathrm{~V}\). When a second capacitor is connected in parallel with the first one, the current from the generator increases by 0.18 A. Find the capacitance of the second capacitor.

Two identical capacitors are connected in parallel to an ac generator that has a frequency of \(610 \mathrm{~Hz}\) and produces a voltage of \(24 \mathrm{~V}\). The current in the circuit is \(0.16 \mathrm{~A}\). What is the capacitance of each capacitor?

A circuit consists of a \(3.00-\mu F\) and a \(6.00-\mu F\) capacitor connected in series across the terminals of a 510 -Hz generator. The voltage of the generator is \(120 \mathrm{~V}\). (a) Determine the equivalent capacitance of the two capacitors. (b) Find the current in the circuit.

A capacitor is connected across an ac generator whose frequency is \(750 \mathrm{~Hz}\) and whose peak output voltage is \(140 \mathrm{~V}\). The rms current in the circuit is \(3.0 \mathrm{~A}\). (a) What is the capacitance of the capacitor? (b) What is the magnitude of the maximum charge on one plate of the capacitor?

A capacitor (capacitance \(C_{1}\) ) is connected across the terminals of an ac generator. Without changing the voltage or frequency of the generator, a second capacitor (capacitance \(C_{2}\) ) is added in series with the first one. As a result, the current delivered by the generator decreases by a factor of three. Suppose the second capacitor had been added in parallel with the first one, instead of in series. By what factor would the current delivered by the generator have increased?

A 0.047 -H inductor is wired across the terminals of a generator that has a voltage of 2.1 V and supplies a current of 0.023 A. Find the frequency of the generator.

At what frequency (in \(\mathrm{Hz}\) ) are the reactances of a \(52-\mathrm{mH}\) inductor and a \(76-\mu \mathrm{F}\) capacitor equal?

Two ac generators supply the same voltage. However, the first generator has a frequency of \(1.5 \mathrm{kHz},\) and the second has a frequency of \(6.0 \mathrm{kHz}\). When an inductor is connected across the terminals of the first generator, the current delivered is 0.30 A. How much current is delivered when this inductor is connected across the terminals of the second generator?

An 8.2-mH inductor is connected to an ac generator \((10.0 \mathrm{~V} \mathrm{rms}, 620 \mathrm{~Hz})\). Determine the peak value of the current supplied by the generator.

An \(8.2-\mathrm{mH}\) inductor is connected to an ac generator \((10.0 \mathrm{~V} \mathrm{rms}, 620 \mathrm{~Hz})\) Determine the peak value of the current supplied by the generator.

An ac generator has a frequency of \(7.5 \mathrm{kHz}\) and a voltage of \(39 \mathrm{~V}\). When an inductor is connected between the terminals of this generator, the current in the inductor is \(42 \mathrm{~mA}\). What is the inductance of the inductor?

A 30.0 -mH inductor has a reactance of \(2.10 k \Omega\). (a) What is the frequency of the ac current that passes through the inductor? (b) What is the capacitance of a capacitor that has the same reactance at this frequency? The frequency is tripled, so that the reactances of the inductor and capacitor are no longer equal. What are the new reactances of (c) the inductor and (d) the capacitor?

Two inductors are connected in parallel across the terminals of a generator. One has an inductance of \(L_{1}=0.030 \mathrm{H}\), and the other has an inductance of \(L_{2}=0.060 \mathrm{H}\). A single inductor, with an inductance \(L,\) is connected across the terminals of a second generator that has the same frequency and voltage as the first one. The current delivered by the second generator is equal to the total current delivered by the first generator. Find \(L\).

A series RCL circuit includes a resistance of \(275 \Omega,\) an inductive reactance of \(648 \Omega\), and a capacitive reactance of \(415 \Omega\). The current in the circuit is 0.233 A. What is the voltage of the generator?

A \(2700-\Omega\) resistor and a \(1.1-\mu F\) capacitor are connected in series across a generator \((60.0 \mathrm{~Hz}, 120 \mathrm{~V}) .\) Determine the power delivered to the circuit.

A light bulb has a resistance of \(240 \Omega\). It is connected to a standard wall socket (120 V, \(60.0 \mathrm{~Hz}\) ). (a) Determine the current in the bulb. (b) Determine the current in the bulb after a \(10.0-\mu \mathrm{F}\) capacitor is added in series in the circuit. (c) It is possible to return the current in the bulb to the value calculated in part (a) by adding an inductor in series with the bulb and the capacitor. What is the value of the inductance of this inductor?

Multiple-Concept Example 3 reviews some of the basic ideas that are pertinent to this problem. A circuit consists of a \(215-\Omega\) resistor and a 0.200 -H inductor. These two elements are connected in series across a generator that has a frequency of \(106 \mathrm{~Hz}\) and a voltage of \(234 \mathrm{~V}\). (a) What is the current in the circuit? (b) Determine the phase angle between the current and the voltage of the generator.

A circuit consists of an \(85-\Omega\) resistor in series with a \(4.0-\mu F\) capacitor, the two being connected between the terminals of an ac generator. The voltage of the generator is fixed. At what frequency is the current in the circuit one-half the value that exists when the frequency is very large?

In one measurement of the body's bioelectric impedance, values of \(Z=4.50 \times 10^{2} \Omega\) and \(\phi=-9.80^{\circ}\) are obtained for the total impedance and the phase angle, respectively. These values assume that the body's resistance \(R\) is in series with its capacitance \(C\) and that there is no inductance \(L .\) Determine the body's resistance and capacitive reactance.

Refer to Interactive Solution \(\underline{23.23}\) at for help with problems like this one. A series RCL circuit contains only a capacitor \((C=6.60 \mu \mathrm{F}),\) an inductor \((L=7.20 \mathrm{mH}),\) and a generator (peak voltage \(=32.0 \mathrm{~V},\) frequency \(=1.50 \times 10^{3} \mathrm{~Hz}\) ). When \(t=0 \mathrm{~s},\) the instantaneous value of the voltage is zero, and it rises to a maximum one-quarter of a period later. (a) Find the instantaneous value of the voltage across the capacitor/inductor combination when \(t=1.20 \times 10^{-4} \mathrm{~s}\). (b) What is the instantaneous value of the current when \(t=1.20 \times 10^{-4} \mathrm{~s}\) ? (Hint: The instantaneous values of the voltage and current are, respectively, the vertical components of the voltage and current phasors.)

When a resistor is connected by itself to an ac generator, the average power delivered to the resistor is \(1.000 \mathrm{~W}\). When a capacitor is added in series with the resistor, the power delivered is \(0.500 \mathrm{~W}\). When an inductor is added in series with the resistor (without the capacitor), the power delivered is \(0.250 \mathrm{~W}\). Determine the power delivered when both the capacitor and the inductor are added in series with the resistor. Section 23.4 Resonance in Electric Circuits

A series RCL circuit has a resonant frequency of \(690 \mathrm{kHz}\). If the value of the capacitance is \(2.0 \times 10^{-9} \mathrm{~F}\), what is the value of the inductance?

The resonant frequency of a series \(\mathrm{RCL}\) circuit is \(9.3 \mathrm{kHz}\). The inductance and capacitance of the circuit are each tripled. What is the new resonant frequency?

A series RCL circuit is at resonance and contains a variable resistor that is set to \(175 \Omega\). The power delivered to the circuit is \(2.6 \mathrm{~W}\). Assuming that the voltage remains constant, how much power is delivered when the variable resistor is set to \(5620 ?\)

A \(10.0-\Omega\) resistor, a \(12.0-\mu \mathrm{F}\) capacitor, and a \(17.0\) -mH inductor are connected in series with a 155-V generator. (a) At what frequency is the current a maximum? (b) What is the maximum value of the rms current?

A \(10.0-\Omega\) resistor, a \(12.0-\mu F\) capacitor, and a 17.0 -mH inductor are connected in series with a \(155-\mathrm{V}\) generator. (a) At what frequency is the current a maximum? (b) What is the maximum value of the rms current?

A series RCL circuit has a resonant frequency of \(1500 \mathrm{~Hz}\). When operating at a frequency other than \(1500 \mathrm{~Hz}\), the circuit has a capacitive reactance of \(5.0 \Omega\) and an inductive reactance of \(30.0 \Omega .\) What are the values of (a) \(L\) and (b) \(C ?\)

The resonant frequency of an RCL circuit is \(1.3 \mathrm{kHz},\) and the value of the inductance is \(7.0 \mathrm{mH}\). What is the resonant frequency (in \(\mathrm{kHz}\) ) when the value of the inductance is 1.5 \(\mathrm{mH} ?\)

Suppose you have a number of capacitors. Each is identical to the capacitor that is already in a series RCL circuit. How many of these additional capacitors must be inserted in series in the circuit, so the resonant frequency triples?

In a series RCL circuit the dissipated power drops by a factor of two when the frequency of the generator is changed from the resonant frequency to a nonresonant frequency. The peak voltage is held constant while this change is made. Determine the power factor of the circuit at the nonresonant frequency.

When a resistor is connected across the terminals of an ac generator \((112 \mathrm{~V})\) that has a fixed frequency, there is a current of \(0.500 \mathrm{~A}\) in the resistor. When an inductor is connected across the terminals of this same generator, there is a current of \(0.400 \mathrm{~A}\) in the inductor. When both the resistor and the inductor are connected in series between the terminals of this generator, what is (a) the impedance of the series combination and (b) the phase angle between the current and the voltage of the generator?

A capacitor is attached to a 5.00 -Hz generator. The instantaneous current is observed to reach a maximum value at a certain time. What is the least amount of time that passes before the instantaneous voltage across the capacitor reaches its maximum value?

In a series circuit, a generator \((1350 \mathrm{~Hz}, 15.0 \mathrm{~V})\) is connected to a \(16.0-\Omega\) resistor, a \(4.10-\mu \mathrm{F}\) capacitor, and a \(5.30-\mathrm{mH}\) inductor. Find the voltage across each circuit element.

Multiple-Concept Example 3 reviews some of the concepts needed for this problem. An ac generator has a frequency of \(4.80 \mathrm{kHz}\) and produces a current of \(0.0400 \mathrm{~A}\) in a series circuit that contains only a \(232-\Omega\) resistor and a \(0.250-\mu F\) capacitor. Obtain (a) the voltage of the generator and (b) the phase angle between the current and the voltage across the resistor/capacitor combination.

The current in an inductor is \(0.20 \mathrm{~A},\) and the frequency is \(750 \mathrm{~Hz}\). If the inductance is \(0.080 \mathrm{H},\) what is the voltage across the inductor?

A \(40.0-\mu F\) capacitor is connected across a 60.0 -Hz generator. An inductor is then connected in parallel with the capacitor. What is the value of the inductance if the rms currents in the inductor and capacitor are equal?

A circuit consists of a resistor in series with an inductor and an ac generator that supplies a voltage of \(115 \mathrm{~V}\). The inductive reactance is \(52.0 \Omega,\) and the current in the circuit is 1.75 A. Find the average power delivered to the circuit.

When the frequency is twice the resonant frequency, the impedance of a series \(\mathrm{RCL}\) circuit is twice the value of the impedance at resonance. Obtain the ratios of the inductive and capacitive reactances to the resistance; that is, obtain (a) \(X_{\mathrm{L}} / R\) and (b) \(X_{\mathrm{C}} / R\) when the frequency is twice the resonant frequency.

(a) Does the capacitance of a parallel plate capacitor increase or decrease when a dielectric material is inserted between the plates? (b) Is the equivalent capacitance of two capacitors in parallel greater or smaller than the capacitance of either capacitor alone? (c) A capacitor is connected between the terminals of an ac generator. When a second capacitor is connected in parallel with the first, does the current supplied by the generator increase or decrease? Explain your answers. Problem Two parallel plate capacitors are identical, except that one of them is empty and the other contains a material with a dielectric constant of 4.2 in the space between the plates. The empty capacitor is connected between the terminals of an ac generator that has a fixed frequency and rms voltage. The generator delivers a current of 0.22 A. What current does the generator deliver after the other capacitor is connected in parallel with the first one? Verify that your answer is consistent with your answers to the Concept Questions.

(a) An inductance \(L_{1}\) is connected across the terminals of an ac generator, which delivers a current to it. Then a second inductance \(L_{2}\) is connected in parallel with \(L_{1}\). Does the presence of \(L_{2}\) alter the current in \(L_{1}\) ? (b) The generator delivers a current to \(L_{2}\) as well as \(L_{1}\). Would the removal of \(L_{1}\) alter the current in \(L_{2} ?(\mathrm{c})\) Is the current delivered to the parallel combination greater or smaller than the current to either inductance alone? Give your reasoning. Problem An ac generator has a frequency of \(2.2 \mathrm{kHz}\) and a voltage of \(240 \mathrm{~V}\). An inductance \(L_{1}=6.0 \mathrm{mH}\) is connected across its terminals. Then a second inductance \(L_{2}=\) \(9.0 \mathrm{mH}\) is connected in parallel with \(L_{1}\). Find the current that the generator delivers to \(L_{1}\) and to the parallel combination. Check to see that your answers are consistent with your answers to the Concept Questions.

Concept Questions (a) An inductance \(L_{1}\) is connected across the terminals of an ac generator, which delivers a current to it. Then a second inductance \(L_{2}\) is connected in parallel with \(L_{1}\). Does the presence of \(L_{2}\) alter the current in \(L_{1} ?\) (b) The generator delivers a current to \(L_{2}\) as well as \(L_{1} .\) Would the removal of \(L_{1}\) alter the current in \(L_{2} ?\) (c) Is the current delivered to the parallel combination greater or smaller than the current to either inductance alone? Give your reasoning. Problem An ac generator has a frequency of \(2.2 \mathrm{kHz}\) and a voltage of \(240 \mathrm{~V}\). An inductance \(L_{1}=6.0 \mathrm{mH}\) is connected across its terminals. Then a second inductance \(L_{2}=\) \(9.0 \mathrm{mH}\) is connected in parallel with \(L_{1} .\) Find the current that the generator delivers to \(L_{1}\) and to the parallel combination. Check to see that your answers are consistent with your answers to the Concept Questions.

(a) An ac circuit contains only a resistor and a capacitor in series. Is the phase angle \(\phi\) between the current and the voltage of the generator positive or negative, and how is the impedance \(Z\) of the circuit related to the resistance \(R\) and the capacitive reactance \(X_{\mathrm{C}}\) ? (b) An ac circuit contains only a resistor and an inductor in series. Is the phase angle \(\phi\) positive or negative, and how is the impedance \(Z\) of the circuit related to the resistance \(R\) and the inductive reactance \(X_{\mathrm{L}}\) ? Account for your answers. Problem A series circuit has an impedance of \(192 \Omega\), and the phase angle is \(\phi=-75.0^{\circ}\). The circuit contains a resistor and either a capacitor or an inductor. Find the resistance \(R\) and the capacitive reactance \(X_{\mathrm{C}}\) or the inductive reactance \(X_{\mathrm{L}}\), whichever is appropriate.

Concept Questions (a) An ac circuit contains only a resistor and a capacitor in series. Is the phase angle \(\phi\) between the current and the voltage of the generator positive or negative, and how is the impedance \(Z\) of the circuit related to the resistance \(R\) and the capacitive reactance \(X_{\mathrm{C}} ?\) (b) An ac circuit contains only a resistor and an inductor in series. Is the phase angle \(\phi\) positive or negative, and how is the impedance \(Z\) of the circuit related to the resistance \(R\) and the inductive reactance \(X_{\mathrm{L}}\) ? Account for your answers. Problem A series circuit has an impedance of \(192 \Omega\), and the phase angle is \(\phi=-75.0^{\circ} .\) The circuit contains a resistor and either a capacitor or an inductor. Find the resistance \(R\) and the capacitive reactance \(X_{\mathrm{C}}\) or the inductive reactance \(X_{\mathrm{L}},\) whichever is appropriate.

Concept Questions (a) A series RCL circuit contains three elements - a resistor, a capacitor, and an inductor. Which of the elements, on average, consume(s) power? (b) How is the current in the circuit related to the generator voltage and the impedance of the circuit? (c) What is the impedance of the circuit at resonance? (d) At resonance, how is the average power consumed related to the generator voltage and the resistance? Problem The resistor in a series RCL circuit has a resistance of \(92 \Omega\), while the voltage of the generator is \(3.0 \mathrm{~V}\). At resonance, what is the average power delivered to the circuit?

The capacitance in a series RCL circuit is \(C_{1}\). The generator in the circuit has a fixed frequency that is less than the resonant frequency of the circuit. By adding a second capacitance \(C_{2}\) to the circuit, it is desired to reduce the resonant frequency so that it matches the generator frequency. (a) To reduce the resonant frequency, should the circuit capacitance be increased or decreased? (b) Should \(C_{2}\) be added in parallel or in series with \(C_{1}\) ? Give your reasoning. Problem The initial circuit capacitance is \(C_{1}=2.60 \mu \mathrm{F}\), and the corresponding resonant frequency is \(f_{01}=7.30 \mathrm{kHz}\). The generator frequency is \(5.60 \mathrm{kHz}\). What is the value of the capacitance \(C_{2}\) that should be added so that the circuit will have a resonant frequency that matches the generator frequency?

Concept Questions The capacitance in a series RCL circuit is \(C_{1}\). The generator in the circuit has a fixed frequency that is less than the resonant frequency of the circuit. By adding a second capacitance \(C_{2}\) to the circuit, it is desired to reduce the resonant frequency so that it matches the generator frequency. (a) To reduce the resonant frequency, should the circuit capacitance be increased or decreased? (b) Should \(C_{2}\) be added in parallel or in series with \(C_{1}\) ? Give your reasoning. Problem The initial circuit capacitance is \(C_{1}=2.60 \mu \mathrm{F},\) and the corresponding resonant frequency is \(f_{01}=7.30 \mathrm{kHz}\). The generator frequency is \(5.60 \mathrm{kHz}\). What is the value of the capacitance \(C_{2}\) that should be added so that the circuit will have a resonant frequency that matches the generator frequency?

Part \(a\) of the drawing shows a resistor and a charged capacitor wired in series. When the switch is closed, the capacitor discharges as charge moves from one plate to the other. Part \(b\) shows a plot of the amount of charge remaining on each plate of the capacitor as a function of time. (a) What does the time constant \(\tau\) of this resistor-capacitor circuit physically represent? (b) How is the time constant related to the resistance \(R\) and the capacitance \(C ?(\mathrm{c})\) In part \(c\) of the drawing, the switch has been removed and an ac generator has been inserted into the circuit. What is the impedance \(Z\) of this circuit? Express your answer in terms of the resistance \(R\), the time constant \(\tau\), and the frequency \(f\) of the generator. Problem The circuit elements in the drawing have the following values: \(R=18 \Omega\) \(V_{\mathrm{rms}}=24 \mathrm{~V}\) for the generator, and \(f=380 \mathrm{~Hz}\). The time constant for the circuit is \(\tau=3.0 \times 10^{-4} \mathrm{~s}\). What is the rms current in the circuit?

Concept Questions Part \(a\) of the drawing shows a resistor and a charged capacitor wired in series. When the switch is closed, the capacitor discharges as charge moves from one plate to the other. Part \(b\) shows a plot of the amount of charge remaining on each plate of the capacitor as a function of time. (a) What does the time constant \(\tau\) of this resistor-capacitor circuit physically represent? (b) How is the time constant related to the resistance \(R\) and the capacitance \(C ?\) (c) In part \(c\) of the drawing, the switch has been removed and an ac generator has been inserted into the circuit. What is the impedance \(Z\) of this circuit? Express your answer in terms of the resistance \(R,\) the time constant \(\tau,\) and the frequency \(f\) of the generator. Problem The circuit elements in the drawing have the following values: \(R=18 \Omega\) \(V_{\mathrm{rms}}=24 \mathrm{~V}\) for the generator, and \(f=380 \mathrm{~Hz}\). The time constant for the circuit is \(\tau=3.0 \times 10^{-4} \mathrm{~s} .\) What is the rms current in the circuit?

A charged capacitor and an inductor are connected as shown in the drawing (this circuit is the same as that in Figure \(23-16 a\) ). There is no resistance in the circuit. As Section \(23.4\) discusses, the electrical energy initially present in the charged capacitor then oscillates back and forth between the inductor and the capacitor. (a) What is the amount of electrical energy initially stored in the capacitor? Express your answer in terms of its capacitance \(C\) and the magnitude \(q\) of the charge on each plate. (b) A little later, this energy is transferred completely to the inductor (see Figure \(23-16 b\) ). Write down an expression for the energy stored in the inductor. Give your answer in terms of its inductance \(L\) and the magnitude \(I_{\max }\) of the maximum current in the inductor. (c) If values for \(q, C\), and \(L\) are known, how could one obtain a value for the maximum current in the inductor? Remember that energy is conserved. Problem The initial charge on the capacitor has a magnitude of \(q=2.90 \mu \mathrm{C}\). The capacitance is \(C=3.60 \mu \mathrm{F}\), and the inductance is \(L=75.0 \mathrm{mH}\). (a) What is the electrical energy stored initially in the charged capacitor? (b) Find the maximum current in the inductor.