Chapter 31

You have a special light bulb with a very delicate wire filament. The wire will break if the current in it ever exceeds 1.50 \(\mathrm{A}\) , even for an instant. What is the largest root-mean-square current you can run through this bulb?

A sinusoidal current \(i=I \cos \omega t\) has an rms value \(I_{\mathrm{rms}}=\)2.10 A. (a) What is the current amplitude? (b) The current is passed through a full-wave rectifier circuit. What is the rectified average current? (c) Which is larger: \(I_{\text { ms or }} I_{\text { rav }} ?\) Explain, using graphs of \(i^{2}\) and of the rectified current.

A capacitance \(C\) and an inductance \(L\) are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If \(L=5.00 \mathrm{mH}\) and \(C=3.50 \mu \mathrm{F}\) , what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?

Kitchen Capacitance. The wiring for a refrigerator contains a starter capacitor. A voltage of amplitude 170 \(\mathrm{V}\) and frequency 60.0 Hz applied across the capacitor is to produce a current amplitude of 0.850 A through the capacitor. What capacitance \(C\) is required?

(a) Compute the reactance of a \(0.450-\mathrm{H}\) inductor at frequencies of 60.0 \(\mathrm{Hz}\) and 600 \(\mathrm{Hz}\) . (b) Compute the reactance of a \(2.50-\mu \mathrm{F}\) capacitor at the same frequencies. (c) At what frequency is the reactance of a \(0.450-\mathrm{H}\) inductor equal to that of a \(2.50-\mu \mathrm{F}\) capacitor?

(a) What is the reactance of a \(3.00-\mathrm{H}\) inductor at a frequency of 80.0 \(\mathrm{Hz}\) ? (b) What is the inductance of an inductor whose reactance is 120\(\Omega\) at 80.0 \(\mathrm{Hz}\) ? (c) What is the reactance of a \(4.00-\mu \mathrm{F}\) capacitor at a frequency of 80.0 \(\mathrm{Hz}\) ? (d) What is the capacitance of a capacitor whose reactance is 120\(\Omega\) at 80.0 \(\mathrm{Hz} ?\)

A Radio Inductor. You want the current amplitude through a \(0.450-\mathrm{mH}\) inductor (part of the circuitry for a radio receiver) to be 2.60 \(\mathrm{mA}\) when a sinusoidal voltage with amplitude 12.0 \(\mathrm{V}\) is applied across the inductor. What frequency is required?

A 0.180 -H inductor is connected in series with a \(90.0-\Omega\) resistor and an ac source. The voltage across the inductor is \(v_{L}=-(12.0 \mathrm{V}) \sin [(480 \mathrm{rad} / \mathrm{s}) t]\) . (a) Derive an expression for the voltage \(v_{R}\) across the resistor. (b) What is \(v_{R}\) at \(t=2.00 \mathrm{ms} ?\)

\(\mathrm{A} 250-\Omega\) resistor is connected in series with a \(4.80-\mu \mathrm{F}\) capacitor and an ac source. The voltage across the capacitor is \(v_{C}=(7.60 \mathrm{V}) \sin [(120 \mathrm{rad} / \mathrm{s}) t] .\) (a) Determine the capacitive reactance of the capacitor. (b) Derive an expression for the voltage \(v_{R}\) across the resistor.

A \(150-\Omega\) resistor is connected in series with a \(0.250-\mathrm{H}\) inductor and an ac source. The voltage across the resistor is \(v_{R}=(3.80 \mathrm{V}) \cos [(720 \mathrm{rad} / \mathrm{s}) t]\) . ( a) Derive an expression for the circuit current. (b) Determine the inductive reactance of the inductor. (c) Derive an expression for the voltage \(v_{L}\) across the inductor.

You have a \(200-\Omega\) resistor, a \(0.400-\mathrm{H}\) inductor, and a \(6.00-\mu \mathrm{F}\) capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 \(\mathrm{V}\) and an angular frequency of 250 \(\mathrm{rad} / \mathrm{s}\) . (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What are the voltage amplitudes across the resistor and across the inductor? (d) What is the phase angle \(\phi\) of the source voltage with respect to the current? Does the source voltage lag or lead the current? (e) Construct the phasor diagram.

\(\mathrm{A} 200-\Omega\) resistor, a \(0.900-\mathrm{H}\) inductor, and a \(6.00-\mu \mathrm{F}\) capacitor are connected in series across a voltage source that has voltage amplitude 30.0 \(\mathrm{V}\) and an angular frequency of 250 \(\mathrm{rad} / \mathrm{s}\) . (a) What are \(v, v_{R}, v_{L},\) and \(v_{C}\) at \(t=20.0 \mathrm{ms}\) ? Compare \(v_{R}+\) \(v_{L}+v_{C}\) to \(v\) at this instant. (b) What are \(V_{R}, V_{L},\) and \(V_{C} ?\) Compare \(V\) to \(V_{R}+V_{L}+V_{C} .\) Explain why these two quantities are not equal.

In an \(L-R-C\) series circuit, the rms voltage across the resistor is \(30.0 \mathrm{V},\) across the capacitor it is \(90.0 \mathrm{V},\) and across the inductor it is 50.0 \(\mathrm{V} .\) What is the rms voltage of the source?

A resistor with \(R=300 \Omega\) and an inductor are connected in series across an ac source that has voltage amplitude 500 \(\mathrm{V}\) . The rate at which electrical energy is dissipated in the resistor is 216 \(\mathrm{W}\) (a) What is the impedance \(Z\) of the circuit? (b) What is the amplitude of the voltage across the inductor? (c) What is the power factor?

The power of a certain \(\mathrm{CD}\) player operating at 120 \(\mathrm{V} \mathrm{rms}\) is 20.0 \(\mathrm{W}\) . Assuming that the CD player behaves like a pure resistor, find (a) the maximum instantaneous power; (b) the rms current; \((\mathrm{c})\) the resistance of this player.

In an \(L-R-C\) series circuit, the components have the following values: \(L=20.0 \mathrm{mH}, C=140 \mathrm{nF},\) and \(R=350 \Omega .\) The generator has an rms voltage of 120 \(\mathrm{V}\) and a frequency of 1.25 \(\mathrm{kHz}\) . Determine (a) the power supplied by the generator and (b) the power dissipated in the resistor.

An \(L-R-C\) series circuit with \(L=0.120 \mathrm{H}, R=240 \Omega\) and \(C=7.30 \mu\) F carries an rms current of 0.450 \(\mathrm{A}\) with a frequency of 400 \(\mathrm{Hz}\) (a) What are the phase angle and power factor for this circuit? (b) What is the impedance of the circuit? (c) What is the rms voltage of the source? (d) What average power is delivered by the source? (e) What is the average rate at which electrical energy is converted to thermal energy in the resistor? (f) What is the average rate at which electrical energy is dissipated (converted to other forms) in the capacitor? (g) In the inductor?

An \(L_{-} R-C\) series circuit is connected to a \(120-\mathrm{Hz}\) ac source that has \(V_{\mathrm{rms}}=80.0 \mathrm{V} .\) The circuit has a resistance of 75.0\(\Omega\) and an impedance at this frequency of 105\(\Omega .\) What average power is delivered to the circuit by the source?

A series ac circuit contains a \(250-\Omega\) resistor, a 15 -m \(\mathrm{H}\) inductor, a \(3.5-\mu \mathrm{F}\) capacitor, and an ac power source of voltage amplitude 45 \(\mathrm{V}\) operating at an angular frequency of 360 \(\mathrm{rad} / \mathrm{s}\) , (a) What is the power factor of this circuit? (b) Find the average power delivered to the entire circuit. (c) What is the average power delivered to the resistor, to the capacitor, and to the inductor?

In an \(L_{-} R-C\) series circuit the source is operated at its resonant angular frequency. At this frequency, the reactance \(X_{C}\) of the capacitor is 200\(\Omega\) and the voltage amplitude across the capacitor is 600 \(\mathrm{V}\) . The circuit has \(R=300 \Omega .\) What is the voltage amplitude of the source?

Analyzing an \(L \cdot R-C\) Circuit. You have a \(200-\Omega\) resistor, a \(0.400-\mathrm{H}\) inductor, a \(5.00-\mu \mathrm{F}\) capacitor, and a variable-frequency ac source with an amplitude of 3.00 \(\mathrm{V} .\) You connect all four elements together to form a series circuit. (a) At what frequency will the current in the circuit be greatest? What will be the current amplitude at this frequency? (b) What will be the current amplitude at an angular frequency of 400 \(\mathrm{rad} / \mathrm{s} ?\) At this frequency, will the source voltage lead or lag the current?

An \(L-R-C\) series circuit is constructed using a \(175-\Omega\) resistor, a \(12.5-\mu \mathrm{F}\) capacitor, and an \(8.00-\mathrm{mH}\) inductor, all connected across an ac source having a variable frequency and a voltage amplitude of 25.0 \(\mathrm{V}\) (a) At what angular frequency will the impedance be smallest, and what is the impedance at this frequency? (b) At the angular frequency in part (a), what is the maximum current through the inductor? (c) At the angular frequency in part (a), find the potential difference across the ac source, the resistor, the capacitor, and the inductor at the instant that the current is equal to one-half its greatest positive value. (d) In part \((\mathrm{c}),\) how are the potential differences across the resistor, inductor, and capacitor related to the potential difference across the ac source?

In an \(L-R-C\) series circuit, \(R=300 \Omega, L=0.400 \mathrm{H},\) and \(C=6.00 \times 10^{-8} \mathrm{F} .\) When the ac source operates at the resonance frequency of the circuit, the current amplitude is 0.500 \(\mathrm{A}\) . (a) What is the voltage amplitude of the source? (b) What is the amplitude of the voltage across the resistor, across the inductor, and across the capacitor? (c) What is the average power supplied by the source?

An \(L-R-C\) series circuit consists of a source with voltage amplitude 120 \(\mathrm{V}\) and angular frequency \(50.0 \mathrm{rad} / \mathrm{s},\) a resistor with \(R=400 \Omega,\) an inductor with \(L=9.00 \mathrm{H},\) and a capacitor with capacitance \(C .\) (a) For what value of \(C\) will the current amplitude in the circuit be a maximum? (b) When \(C\) has the value calculated in part (a), what is the amplitude of the voltage across the inductor?

In an \(L-R-C\) series circuit, \(R=150 \Omega, L=0.750 \mathrm{H},\) and \(C=0.0180 \mu \mathrm{F} .\) The source has voltage amplitude \(V=150 \mathrm{V}\) and a frequency equal to the resonance frequency of the circuit. (a) What is the power factor? (b) What is the average power delivered by the source? (c) The capacitor is replaced by one with \(C=\) 0.0360\(\mu \mathrm{F}\) and the source frequency is adjusted to the new resonance value. Then what is the average power delivered by the source?

In an \(L-R-C\) series circuit, \(R=400 \Omega, L=0.350 \mathrm{H},\) and \(C=0.0120 \mu \mathrm{F}\) (a) What is the resonance angular frequency of the circuit? (b) The capacitor can withstand a peak voltage of 550 \(\mathrm{V}\) . If the voltage source operates at the resonance frequency, what maximum voltage amplitude can it have if the maximum capacitor voltage is not exceeded?

A series circuit consists of an ac source of variable frequency, a \(115-\Omega\) resistor, a \(1.25-\mu F\) capacitor, and a \(4.50-\mathrm{mH}\) inductor. Find the impedance of this circuit when the angular frequency of the ac source is adjusted to (a) the resonance angular frequency; (b) twice the resonance angular frequency; (c) half the resonance angular frequency.

In an \(L-R-C\) series circuit, \(L=0.280 \mathrm{H}\) and \(C=\) 4.00\(\mu \mathrm{F}\) . The voltage amplitude of the source is 120 \(\mathrm{V}\) . (a) What is the resonance angular frequency of the circuit? (b) When the source operates at the resonance angular frequency, the current amplitude in the circuit is 1.70 A. What is the resistance \(R\) of the resistor? (c) At the resonance angular frequency, what are the peak voltages across the inductor, the capacitor, and the resistor?

A Step-Down Transformer. A transformer connected to a \(120-\mathrm{V}(\mathrm{rms})\) ac line is to supply 12.0 \(\mathrm{V}(\mathrm{rms})\) to a portable electronic device. The load resistance in the secondary is 5.00\(\Omega .\) (a) What should the ratio of primary to secondary turns of the transformer be? (b) What rms current must the secondary supply? (c) What average power is delivered to the load? (d) What resistance connected directly across the \(120-\mathrm{V}\) line would draw the same power as the transformer? Show that this is equal to 5.00\(\Omega\) times the square of the ratio of primary to secondary turns.

A Step-Up Transformer. A transformer connected to a \(120-\mathrm{V}(\mathrm{rms})\) ac line is to supply \(13,000 \mathrm{V}(\mathrm{rms})\) for a neon sign. To reduce shock hazard, a fuse is to be inserted in the primary circuit; the fuse is to blow when the rms current in the secondary circuit; exceeds 8.50 \(\mathrm{mA}\) . (a) What is the ratio of secondary to primary turns of the transformer? (b) What power must be supplied to the transformer when the rms secondary current is 8.50 \(\mathrm{mA}\) ? (c) What current rating should the fuse in the primary circuit have?

off to Europe! You plan to take your hair dryer to Europe, where the electrical outlets put out 240 \(\mathrm{V}\) instead of the 120 \(\mathrm{V}\) seen in the United States. The dryer puts out 1600 \(\mathrm{W}\) at 120 \(\mathrm{V}\) . (a) What could you do to operate your dryer via the \(240 \mathrm{V}\) line in Europe? (b) What current will your dryer draw from a European outlet? (c) What resistance will your dryer appear to have when operated at 240 \(\mathrm{V} ?\)

A coil has a resistance of 48.0\(\Omega .\) At a frequency of 80.0 \(\mathrm{Hz}\) the voltage across the coil leads the current in it by \(52.3^{\circ} .\) Determine the inductance of the coil.

A toroidal solenoid has 2900 closely wound turns, cross-sectional area \(0.450 \mathrm{cm}^{2},\) mean radius \(9.00 \mathrm{cm},\) and resistance \(R=2.80 \Omega .\) The variation of the magnetic field across the cross section of the solenoid can be neglected. What is the amplitude of the current in the solenoid if it is connected to an ac source that has voltage amplitude 24.0 \(\mathrm{V}\) and frequency 365 \(\mathrm{Hz}\) ?

An \(L-R-C\) series circuit has \(C=4.80 \mu \mathrm{F}, L=0.520 \mathrm{H}\) and source voltage amplitude \(V=56.0 \mathrm{V}\) . The source is operated at the resonance frequency of the circuit. If the voltage across the capacitor has amplitude \(80.0 \mathrm{V},\) what is the value of \(R\) for the resistor in the circuit?

A large electromagnetic coil is connected to a 120 -Hz ac source. The coil has resistance \(400 \Omega,\) and at this source frequency the coil has inductive reactance 250\(\Omega\) (a) What is the inductance of the coil? (b) What must the rms voltage of the source be if the coil is to consume an average electrical power of 800 \(\mathrm{w} ?\)

A series circuit has an impedance of 60.0\(\Omega\) and a power factor of 0.720 at 50.0 \(\mathrm{Hz}\) . The source voltage lags the current. (a) What circuit element, an inductor or a capacitor, should be placed in series with the circuit to raise its power factor? (b) What size element will raise the power factor to unity?

An \(L \cdot R-C\) series circuit has \(R=300 \Omega .\) At the frequency of the source, the inductor has reactance \(X_{L}=900 \Omega\) and the capacitor has reactance \(X_{C}=500 \Omega .\) The amplitude of the voltage across the inductor is 450 V. (a) What is the amplitude of the voltage across the resistor? (b) What is the amplitude of the voltage across the capacitor? (c) What is the voltage amplitude of the source? (d) What is the rate at which the source is delivering electrical energy to the circuit?

In an \(L-R-C\) series circuit, \(R=300 \Omega, X_{C}=300 \Omega\) and \(X_{L}=500 \Omega .\) The average power consumed in the resistor is 60.0 \(\mathrm{W}\) . (a) What is the power factor of the circuit? (b) What is the rms voltage of the source?

An \(L-R-C\) series circuit consists of a \(50.0-\Omega\) resistor, a \(10.0-\mu \mathrm{F}\) capacitor, a \(3.50-\mathrm{mH}\) inductor, and an ac voltage source of voltage amplitude 60.0 \(\mathrm{V}\) operating at 1250 \(\mathrm{Hz}\) . (a) Find the current amplitude and the voltage amplitudes across the inductor, the resistor, and the capacitor. Why can the voltage amplitudes add up to more than 60.0 \(\mathrm{V}\) (b) If the frequency is now doubled, but nothing else is changed, which of the quantities in part (a) will change? Find the new values for those that do change.

At a frequency \(\omega_{1}\) the reactance of a certain capacitor equals that of a certain inductor. (a) If the frequency is changed to \(\omega_{2}=2 \omega_{1},\) what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (b) If the frequency is changed to \(\omega_{3}=\omega_{1} / 3,\) what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (c) If the capacitor and inductor are placed in series with a resistor of resistance \(R\) to form an \(L-R-C\) series circuit, what will be the resonance angular frequency of the circuit?

An \(L-R-C\) series circuit is connected to an ac source of constant voltage amplitude \(V\) and variable angular frequency \(\omega\) (a) Show that the current amplitude, as a function of \(\omega,\) is $$I=\frac{V}{\sqrt{R^{2}+(\omega L-1 / \omega C)^{2}}}$$ (b) Show that the average power dissipated in the resistor is $$P=\frac{V^{2} R / 2}{R^{2}+(\omega L-1 / \omega C)^{2}}$$ (c) Show that \(I\) and \(P\) are both maximum when \(\omega=1 / \sqrt{L C},\) the resonance frequency of the circuit. (d) Graph \(P\) as a function of \(\omega\) for \(V=100 \mathrm{V}, R=200 \Omega, L=2.0 \mathrm{H},\) and \(C=0.50 \mu \mathrm{F}\) . Compare to the light purple curve in Fig. \(31.19 .\) Discuss the behavior of \(I\) and \(P\) in the limits \(\omega=0\) and \(\omega \rightarrow \infty\) .

In an \(L-R-C\) series circuit the magnitude of the phase angle is \(54.0^{\circ},\) with the source voltage lagging the current. The reactance of the capacitor is \(350 \Omega,\) and the resistor resistance is 180\(\Omega .\) The average power delivered by the source is 140 \(\mathrm{W} .\) Find (a) the reactance of the inductor; (b) the rms current; (c) the rms voltage of the source.

The \(L \cdot R-C\) Parallel Circuit. A resistor, inductor, and capacitor are connected in parallel to an ac source with voltage amplitude \(V\) and angular frequency \(\omega\) . Let the source voltage be given by \(v=V \cos \omega t .\) (a) Show that the instantaneous voltages \(v_{R}, v_{L},\) and \(v_{C}\) at any instant are each equal to \(v\) and that \(i=i_{R}+i_{L}+i_{C},\) where \(i\) is the current through the source and \(i_{R},\) \(i_{L},\) and \(i_{C}\) are the currents through the resistor, the inductor, and the capacitor, respectively. (b) What are the phases of \(i_{R}, i_{L},\) and \(i_{C}\) with respect to \(v ?\) Use current phasors to represent \(i, i_{R}, i_{L},\) and \(i_{C}\) In a phasor diagram, show the phases of these four currents with respect to \(v\) . (c) Use the phasor diagram of part (b) to show that the current amplitude \(I\) for the current \(i\) through the source is given by \(I=\sqrt{I_{R}^{2}+\left(I_{C}-I_{L}\right)^{2}}\) (d) Show that the result of part (c) can be written as \(I=V / Z,\) with \(1 / Z=\sqrt{1 / R^{2}+(\omega C-1 / \omega L)^{2}}\)

A \(100-\Omega\) resistor, a \(0.100-\mu \mathrm{F}\) capacitor, and a \(0.300-\mathrm{H}\) inductor are connected in parallel to a voltage source with amplitude 240 \(\mathrm{V}\) (a) What is the resonance angular frequency? (b) What is the maximum current through the source at the resonance frequency? (c) Find the maximum current in the resistor at resonance. (d) What is the maximum current in the inductor at resonance? (e) What is the maximum current in the branch containing the capacitor at resonance? (f) Find the maximum energy stored in the inductor and in the capacitor at resonance.

You want to double the resonance angular frequency of an L-R-C series circuit by changing only the pertinent circuit elements all by the same factor. (a) Which ones should you change? (b) By what factor should you change them?

An \(L \cdot R-C\) series circuit consists of a \(2.50-\mu \mathrm{F}\) capacitor, a 5.00 -mH inductor, and a \(75.0-\Omega\) resistor connected across an ac source of voltage amplitude 15.0 \(\mathrm{V}\) having variable frequency. (a) Under what circumstances is the average power delivered to the circuit equal to \(\frac{1}{2} V_{\mathrm{Vms}} I_{\mathrm{rms}} ?\) (b) Under the conditions of part (a), what is the average power delivered to each circuit element and what is the maximum current through the capacitor?

In an \(L-R-C\) series circuit, the source has a voltage amplitude of \(120 \mathrm{V}, R=80.0 \Omega,\) and the reactance of the capacitor is 480\(\Omega .\) The voltage amplitude across the capacitor is 360 \(\mathrm{V}\) . (a) What is the current amplitude in the circuit? (b) What is the impedance? (c) What two values can the reactance of the inductor have? (d) For which of the two values found in part (c) is the angular frequency less than the resonance angular frequency? Explain.

An \(L \cdot R-C\) series circuit has \(R=500 \Omega, L=2.00 \mathrm{H}\) \(C=0.500 \mu \mathrm{F},\) and \(V=100 \mathrm{V} .\) (a) For \(\omega=800 \mathrm{rad} / \mathrm{s},\) calculate \(V_{R}, V_{L}, V_{C},\) and \(\phi .\) Using a single set of axes, graph \(v, v_{R}, v_{L},\) and \(v_{C}\) as functions of time. Include two cycles of \(v\) on your graph. (b) Repeat part (a) for \(\omega=1000 \mathrm{rad} / \mathrm{s} .\) (c) Repeat part (a) for \(\omega=1250 \mathrm{rad} / \mathrm{s}\)

Consider an \(L-R-C\) series circuit with a \(1.80-\mathrm{H}\) inductor, a \(0.900-\mu F\) capacitor, and a \(300-\Omega\) resistor. The source has terminal rms voltage \(V_{\text { rms }}=60.0 \mathrm{V}\) and variable angular frequency \(\omega\) . (a) What is the resonance angular frequency \(\omega_{0}\) of the circuit? (b) What is the rms current through the circuit at resonance, \(I_{\mathrm{rms}-0} ?(\mathrm{c})\) For what\ two values of the angular frequency, \(\omega_{1}\) and \(\omega_{2},\) is the rms current half the resonance value? (d) The quantity \(\left|\omega_{1}-\omega_{2}\right|\) defines the resonance width. Calculate \(I_{\text { rms- } 0 \text { and the resonance width for }}\) \(R=300 \Omega, 30.0 \Omega,\) and 3.00\(\Omega .\) Describe how your results compare to the discussion in Section \(31.5 .\)

An inductor, a capacitor, and a resistor are all connected in series across an ac source. If the resistance, inductance, and capacitance are all doubled, by what factor does each of the following quantities change? Indicate whether they increase or decrease: (a) the resonance angular frequency; (b) the inductive reactance; (c) the capacitive reactance. (d) Does the impedance double?