Solved: In an ac circuit, why is the average power for an

Household electric power in most of western Europe is supplied at 240 V, rather than the 120 V that is standard in the United States and Canada. What are the advantages and disadvantages of each system?

The current in an ac power line changes direction 120 times per second, and its average value is zero. Explain how it is possible for power to be transmitted in such a system.

In an ac circuit, why is the average power for an inductor and a capacitor zero, but not for a resistor?

Equation (31.14) was derived by using the relationship i = dq>dt between the current and the charge on the capacitor. In Fig. 31.9a the positive counterclockwise current increases the charge on the capacitor. When the charge on the left plate is positive but decreasing in time, is i = dq>dt still correct or should it be i = -dq>dt? Is i = dq>dt still correct when the right-hand plate has positive charge that is increasing or decreasing in magnitude? Explain.

Fluorescent lights often use an inductor, called a ballast, to limit the current through the tubes. Why is it better to use an inductor rather than a resistor for this purpose?

Equation (31.9) says that vab = L di>dt (see Fig. 31.8a). Using Faradays law, explain why point a is at higher potential than point b when i is in the direction shown in Fig. 31.8a and is increasing in magnitude. When i is counterclockwise and decreasing in magnitude, is vab = L di>dt still correct, or should it be vab = -L di>dt? Is vab = L di>dt still correct when i is clockwise and increasing or decreasing in magnitude? Explain

Is it possible for the power factor of an L-R-C series ac circuit to be zero? Justify your answer on physical grounds.

In an L-R-C series circuit, can the instantaneous voltage across the capacitor exceed the source voltage at that same instant? Can this be true for the instantaneous voltage across the inductor? Across the resistor? Explain

In an L-R-C series circuit, what are the phase angle f and power factor cosf when the resistance is much smaller than the inductive or capacitive reactance and the circuit is operated far from resonance? Explain

When an L-R-C series circuit is connected across a 120-V ac line, the voltage rating of the capacitor may be exceeded even if it is rated at 200 or 400 V. How can this be?

In Example 31.6 (Section 31.4), a hair dryer is treated as a pure resistor. But because there are coils in the heating element and in the motor that drives the blower fan, a hair dryer also has inductance. Qualitatively, does including an inductance increase or decrease the values of R, Irms, and P?

A light bulb and a parallel-plate capacitor with air between the plates are connected in series to an ac source. What happens to the brightness of the bulb when a dielectric is inserted between the plates of the capacitor? Explain

A coil of wire wrapped on a hollow tube and a light bulb are connected in series to an ac source. What happens to the brightness of the bulb when an iron rod is inserted in the tube?

A circuit consists of a light bulb, a capacitor, and an inductor connected in series to an ac source. What happens to the brightness of the bulb when the inductor is omitted? When the inductor is left in the circuit but the capacitor is omitted? Explain.

A circuit consists of a light bulb, a capacitor, and an inductor connected in series to an ac source. Is it possible for both the capacitor and the inductor to be removed and the brightness of the bulb to remain the same? Explain.

Can a transformer be used with dc? Explain. What happens if a transformer designed for 120-V ac is connected to a 120-V dc line?

An ideal transformer has N1 windings in the primary and N2 windings in its secondary. If you double only the number of secondary windings, by what factor does (a) the voltage amplitude in the secondary change, and (b) the effective resistance of the secondary circuit change?

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?

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?

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 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 vt has an rms value Irms = 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: Irms or Irav? Explain, using graphs of i 2 and of the rectified current

The voltage across the terminals of an ac power supply varies with time according to Eq. (31.1). The voltage amplitude is V = 45.0 V. What are (a) the root-mean-square potential difference Vrms and (b) the average potential difference Vav between the two terminals of the power supply?

A capacitor is connected across an ac source that has voltage amplitude 60.0 V and frequency 80.0 Hz. (a) What is the phase angle f for the source voltage relative to the current? Does the source voltage lag or lead the current? (b) What is the capacitance C of the capacitor if the current amplitude is 5.30 A?

An inductor with L = 9.50 mH is connected across an ac source that has voltage amplitude 45.0 V. (a) What is the phase angle f for the source voltage relative to the current? Does the source voltage lag or lead the current? (b) What value for the frequency of the source results in a current amplitude of 3.90 A?

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 mH and C = 3.50 mF, 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 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-H inductor at frequencies of 60.0 Hz and 600 Hz. (b) Compute the reactance of a 2.50@mF capacitor at the same frequencies. (c) At what frequency is the reactance of a 0.450-H inductor equal to that of a 2.50@mF capacitor?

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

A Radio Inductor. You want the current amplitude through a 0.450-mH inductor (part of the circuitry for a radio receiver) to be 1.80 mA when a sinusoidal voltage with amplitude 12.0 V is applied across the inductor. What frequency is required?

A 0.180-H inductor is connected in series with a 90.0@ resistor and an ac source. The voltage across the inductor is vL = -112.0 V2sin31480 rad>s2t4. (a) Derive an expression for the voltage vR across the resistor. (b) What is vR at t = 2.00 ms?

A 250@ resistor is connected in series with a 4.80@mF capacitor and an ac source. The voltage across the capacitor is vC = 17.60 V2sin31120 rad>s2t4. (a) Determine the capacitive reactance of the capacitor. (b) Derive an expression for the voltage vR across the resistor.

A 150@ resistor is connected in series with a 0.250-H inductor and an ac source. The voltage across the resistor is vR = 13.80 V2cos31720 rad>s2t4. (a) Derive an expression for the circuit current. (b) Determine the inductive reactance of the inductor. (c) Derive an expression for the voltage vL across the inductor

You have a 200@ resistor, a 0.400-H inductor, and a 6.00@mF capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 V and an angular frequency of 250 rad>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 f of the source voltage with respect to the current? Does the source voltage lag or lead the current? (e) Construct the phasor diagram.

The resistor, inductor, capacitor, and voltage source described in Exercise 31.14 are connected to form an L-R-C series circuit. (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What is the phase angle of the source voltage with respect to the current? Does the source voltage lag or lead the current? (d) What are the voltage amplitudes across the resistor, inductor, and capacitor? (e) Explain how it is possible for the voltage amplitude across the capacitor to be greater than the voltage amplitude across the source

A 200@ resistor, 0.900-H inductor, and 6.00@mF capacitor are connected in series across a voltage source that has voltage amplitude 30.0 V and an angular frequency of 250 rad>s. (a) What are v, vR, vL, and vC at t = 20.0 ms? Compare vR + vL + vC to v at this instant. (b) What are VR, VL, and VC? Compare V to VR + VL + VC. Explain why these two quantities are not equal. 31.17 . In an L-R-C series circuit, the rms voltage across the resistor is 30.0 V, across the capacitor it is 90.0 V, and across the inductor it is 50.0 V. What is the rms voltage of the source?

A resistor with R = 300 and an inductor are connected in series across an ac source that has voltage amplitude 500 V. The rate at which electrical energy is dissipated in the resistor is 286 W. What is (a) the impedance Z of the circuit; (b) the amplitude of the voltage across the inductor; (c) the power factor?

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

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

(a) Show that for an L@R@C series circuit the power factor is equal to R>Z. (b) An L-R-C series circuit has phase angle -31.5o . The voltage amplitude of the source is 90.0 V. What is the voltage amplitude across the resistor?

(a) Use the results of part (a) of Exercise 31.21 to show that the average power delivered by the source in an L@R@C series circuit is given by Pav = I 2 rms R. (b) An L@R@C series circuit has R = 96.0 , and the amplitude of the voltage across the resistor is 36.0 V. What is the average power delivered by the source?

An L-R-C series circuit with L = 0.120 H, R = 240 , and C = 7.30 mF carries an rms current of 0.450 A with a frequency of 400 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-Hz ac source that has Vrms = 80.0 V. The circuit has a resistance of 75.0 and an impedance at this frequency of 105 . What average power is delivered to the circuit by the source?

A series ac circuit contains a 250@ resistor, a 15-mH inductor, a 3.5@mF capacitor, and an ac power source of voltage amplitude 45 V operating at an angular frequency of 360 rad>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?

A series ac circuit contains a 250@ resistor, a 15-mH inductor, a 3.5@mF capacitor, and an ac power source of voltage amplitude 45 V operating at an angular frequency of 360 rad>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 XC of the capacitor is 200 and the voltage amplitude across the capacitor is 600 V. The circuit has R = 300 . What is the voltage amplitude of the source?

Analyzing an L-R-C Circuit. You have a 200@ resistor, a 0.400-H inductor, a 5.00@mF capacitor, and a variablefrequency ac source with an amplitude of 3.00 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 rad>s? At this frequency, will the source voltage lead or lag the current?

An L-R-C series circuit is constructed using a 175@ resistor, a 12.5@mF capacitor, and an 8.00-mH inductor, all connected across an ac source having a variable frequency and a voltage amplitude of 25.0 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 (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 , L = 0.400 H, and C = 6.00 * 10-8 F. When the ac source operates at the resonance frequency of the circuit, the current amplitude is 0.500 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 V and angular frequency 50.0 rad>s, a resistor with R = 400 , an inductor with L = 3.00 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 , L = 0.750 H, and C = 0.0180 mF. The source has voltage amplitude V = 150 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 mF 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 , L = 0.350 H, and C = 0.0120 mF. (a) What is the resonance angular frequency of the circuit? (b) The capacitor can withstand a peak voltage of 670 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?

In an L-R-C series circuit, L = 0.280 H and C = 4.00 mF. The voltage amplitude of the source is 120 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?

Off to Europe! You plan to take your hair dryer to Europe, where the electrical outlets put out 240 V instead of the 120 V seen in the United States. The dryer puts out 1600 W at 120 V. (a) What could you do to operate your dryer via the 240-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 V?

A Step-Down Transformer. A transformer connected to a 120-V (rms) ac line is to supply 12.0 V (rms) to a portable electronic device. The load resistance in the secondary is 5.00 . (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-V line would draw the same power as the transformer? Show that this is equal to 5.00 times the square of the ratio of primary to secondary turns

A Step-Up Transformer. A transformer connected to a 120-V (rms) ac line is to supply 13,000 V (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 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 mA? (c) What current rating should the fuse in the primary circuit have?

A coil has a resistance of 48.0 . At a frequency of 80.0 Hz the voltage across the coil leads the current in it by 52.3. Determine the inductance of the coil.

When a solenoid is connected to a 48.0-V dc battery that has negligible internal resistance, the current in the solenoid is 5.50 A. When this solenoid is connected to an ac source that has voltage amplitude 48.0 V and angular frequency 20.0 rad>s, the current in the solenoid is 3.60 A. What is the inductance of this solenoid?

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

Five infinite-impedance voltmeters, calibrated to read rms values, are connected as shown in Fig. P31.40. Let R = 200 , L = 0.400 H, C = 6.00 mF, and V = 30.0 V. What is the reading of each voltmeter if (a) v = 200 rad>s and (b) v = 1000 rad>s?

A parallel-plate capacitor having square plates 4.50 cm on each side and 8.00 mm apart is placed in series with the following: an ac source of angular frequency 650 rad>s and voltage amplitude 22.5 V; a 75.0@ resistor; and an ideal solenoid that is 9.00 cm long, has a circular cross section 0.500 cm in diameter, and carries 125 coils per centimeter. What is the resonance angular frequency of this circuit? (See Exercise 30.15.)

A toroidal solenoid has 2900 closely wound turns, cross-sectional area 0.450 cm2 , mean radius 9.00 cm, and resistance R = 2.80 . Ignore the variation of the magnetic field across the cross section of the solenoid. What is the amplitude of the current in the solenoid if it is connected to an ac source that has voltage amplitude 24.0 V and frequency 495 Hz?

A series circuit has an impedance of 60.0 and a power factor of 0.720 at 50.0 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?

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 450 W?

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

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?

A High-Pass Filter. One application of L-R-C series circuits is to high-pass or low-pass filters, which filter out either the low- or high-frequency components of a signal. A high-pass filter is shown in Fig. P31.47, where the output voltage is taken across the L-R combination. (The L-R combination represents an inductive coil that also has resistance due to the large length of wire in the coil.) Derive an expression for \(V_{out}/V_s\), the ratio of the output and source voltage amplitudes, as a function of the angular frequency \(\omega\) of the source. Show that when \(\omega\) is small, this ratio is proportional to \(\omega\) and thus is small, and show that the ratio approaches unity in the limit of large frequency.

A Low-Pass Filter. Figure P31.48 shows a low-pass filter (see Problem 31.47); the output voltage is taken across the capacitor in an L-R-C series circuit. Derive an expression for Vout>Vs, the ratio of the output and source voltage amplitudes, as a function of the angular frequency v of the source. Show that when v is large, this ratio is proportional to v-2 and thus is very small, and show that the ratio approaches unity in the limit of small frequency.

An L-R-C series circuit is connected to an ac source of constant voltage amplitude V and variable angular frequency v. (a) Show that the current amplitude, as a function of v, is I = V 2R2 + 1vL - 1>vC22 (b) Show that the average power dissipated in the resistor is P = V2 R>2 R2 + 1vL - 1>vC22 (c) Show that I and P are both maximum when v = 1>2LC, the resonance frequency of the circuit. (d) Graph P as a function of v for V = 100 V, R = 200 , L = 2.0 H, and C = 0.50 mF. Compare to the light purple curve in Fig. 31.19. Discuss the behavior of I and P in the limits v = 0 and v S q.

An L-R-C series circuit is connected to an ac source of constant voltage amplitude V and variable angular frequency v. Using the results of Problem 31.49, find an expression for (a) the amplitude VL of the voltage across the inductor as a function of v and (b) the amplitude VC of the voltage across the capacitor as a function of v. (c) Graph VL and VC as functions of v for V = 100 V, R = 200 , L = 2.0 H, and C = 0.50 mF. (d) Discuss the behavior of VL and VC in the limits v = 0 and v S q. For what value of v is VL = VC? What is the significance of this value of v?

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

In an L-R-C series circuit, the phase angle is 40.0, with the source voltage leading the current. The reactance of the capacitor is 400 , and the resistance of the resistor is 200 . The average power delivered by the source is 150 W. Find (a) the reactance of the inductor, (b) the rms current, (c) the rms voltage of the source.

An L-R-C series circuit has R = 500 , L = 2.00 H, C = 0.500 mF, and V = 100 V. (a) For v = 800 rad>s, calculate VR, VL, VC, and f. Using a single set of axes, graph v, vR, vL, and vC as functions of time. Include two cycles of v on your graph. (b) Repeat part (a) for v = 1000 rad>s. (c) Repeat part (a) for v = 1250 rad>s.

The L-R-C Parallel Circuit. A resistor, an inductor, and a capacitor are connected in parallel to an ac source with voltage amplitude V and angular frequency v. Let the source voltage be given by v = Vcosvt. (a) Show that each of the instantaneous voltages vR, vL, and vC at any instant is equal to v and that i = iR + iL + iC, where i is the current through the source and iR, iL, and iC are the currents through the resistor, inductor, and capacitor, respectively. (b) What are the phases of iR, iL, and iC with respect to v? Use current phasors to represent i, iR, iL, and iC. 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 I = 1IR 2 + 1IC - IL22. (d) Show that the result of part (c) can be written as I = V>Z, with 1>Z = 111>R22 + 3vC - 11>vL24 2.

The impedance of an L-R-C parallel circuit was derived in Problem 31.54. (a) Show that at the resonance angular frequency \(\omega_0 = 1/\sqrt{LC}\), the impedance Z is a maximum and therefore the current through the ac source is a minimum. (b) A \(100-\Omega\) resistor, a \(0.100-\mu F\) capacitor, and a 0.300-H inductor are connected in parallel to a voltage source with amplitude 240 V. What is the resonance angular frequency? For this circuit, what is (c) the maximum current through the source at the resonance frequency; (d) the maximum current in the resistor at resonance; (e) the maximum current in the inductor at resonance; (f) the maximum current in the branch containing the capacitor at resonance?

A 400@ resistor and a 6.00@mF capacitor are connected in parallel to an ac generator that supplies an rms voltage of 180 V at an angular frequency of 360 rad>s. Use the results of Problem 31.54. Note that since there is no inductor in this circuit, the 1>vL term is not present in the expression for 1>Z. Find (a) the current amplitude in the resistor; (b) the current amplitude in the capacitor; (c) the phase angle of the source current with respect to the source voltage; (d) the amplitude of the current through the generator. (e) Does the source current lag or lead the source voltage?

An L-R-C series circuit consists of a 2.50@mF capacitor, a 5.00@mH inductor, and a 75.0@ resistor connected across an ac source of voltage amplitude 15.0 V having variable frequency. (a) Under what circumstances is the average power delivered to the circuit equal to 1 2VrmsIrms? (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?

An L-R-C series circuit has \(R=60.0\ \Omega,\ L=0.800\mathrm{\ H},\text{ and }C=3.00\times 10^{-4}\mathrm{\ F}\text{. }\) The ac source has voltage amplitude 90.0 V and angular frequency 120 rad/s. (a) What is the maximum energy stored in the inductor? (b) When the energy stored in the inductor is a maximum, how much energy is stored in the capacitor? (c) What is the maximum energy stored in the capacitor?

In an L-R-C series circuit, the source has a voltage amplitude of 120 V, R = 80.0 , and the reactance of the capacitor is 480 . The voltage amplitude across the capacitor is 360 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.

In an L@R@C series ac circuit, the source has a voltage amplitude of 240 V, R = 90.0 , and the reactance of the inductor is 320 . The voltage amplitude across the resistor is 135 V. (a) What is the current amplitude in the circuit? (b) What is the voltage amplitude across the inductor? (c) What two values can the reactance of the capacitor have? (d) For which of the two values found in part (c) is the angular frequency less than the resonance angular frequency? Explain

A resistance R, capacitance C, and inductance L are connected in series to a voltage source with amplitude V and variable angular frequency v. If v = v0, the resonance angular frequency, find (a) the maximum current in the resistor; (b) the maximum voltage across the capacitor; (c) the maximum voltage across the inductor; (d) the maximum energy stored in the capacitor; (e) the maximum energy stored in the inductor. Give your answers in terms of R, C, L, and V.

The Resonance Width. Consider an L-R-C series circuit with a 1.80-H inductor, a \(0.900-\mu \mathrm{F}\) capacitor, and a \(300-\Omega\) resistor. The source has terminal rms voltage \(V_{\mathrm{rms}}=60.0 \mathrm{~V}\) 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, \(I_{\text {rms- }-0}\) ? (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 \(I_{\mathrm{rms}-0}\) 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 L-R-C series circuit draws 220 W from a 120-V (rms), 50.0-Hz ac line. The power factor is 0.560, and the source voltage leads the current. (a) What is the net resistance R of the circuit? (b) Find the capacitance of the series capacitor that will result in a power factor of unity when it is added to the original circuit. (c) What power will then be drawn from the supply line?

A coworker of yours was making measurements of a large solenoid that is connected to an ac voltage source. Unfortunately, she left for vacation before she completed the analysis, and your boss has asked you to finish it. You are given a graph of 1>I 2 versus v2 (Fig. P31.64), where I is the current in the circuit and v is the angular frequency of the source. A note attached to the graph says that the voltage amplitude of the source was kept constant at 12.0 V. Calculate the resistance and inductance of the solenoid

You are analyzing an ac circuit that contains a solenoid and a capacitor in series with an ac source that has voltage amplitude 90.0 V and angular frequency v. For different capacitors in the circuit, each with known capacitance, you measure the value of the frequency vres for which the current in the circuit is a maximum. You plot your measured values on a graph of v 2 res versus 1>C (Fig. P31.65). The maximum current for each value of C is the same, you note, and equal to 4.50 A. Calculate the resistance and inductance of the solenoid.

You are given this table of data recorded for a circuit that has a resistor, an inductor with negligible resistance, and a capacitor, all in series with an ac voltage source: f 1Hz2 80 160 Z 1 2 15 13 F 12 -71 67 Here f is the frequency of the voltage source, Z is the impedance of the circuit, and f is the phase angle. (a) Use the data at both frequencies to calculate the resistance of the resistor. (Hint: Use the results of Exercise 31.21.) Calculate the average of these two values of the resistance, and use the result as the value of R in the rest of the analysis. (b) Use the data at 80 Hz and 160 Hz to calculate the inductance L and capacitance C of the circuit. (c) What is the resonance frequency for the circuit, and what are the impedance and phase angle at the resonance frequency?

In an L-R-C series circuit the current is given by i = I cosvt. The voltage amplitudes for the resistor, inductor, and capacitor are VR, VL, and VC. (a) Show that the instantaneous power into the resistor is pR = VRI cos2vt = 1 2 VRI11 + cos 2vt2. What does this expression give for the average power into the resistor? (b) Show that the instantaneous power into the inductor is pL = -VLIsinvt cosvt = - 1 2 VLIsin 2vt. What does this expression give for the average power into the inductor? (c) Show that the instantaneous power into the capacitor is pC = VCIsinvt cosvt = 1 2VCIsin 2vt. What does this expression give for the average power into the capacitor? (d) The instantaneous power delivered by the source is shown in Section 31.4 to be p = VI cosvt1cosfcosvt - sinfsinvt2. Show that pR + pL + pC equals p at each instant of time

(a) At what angular frequency is the voltage amplitude across the resistor in an L-R-C series circuit at maximum value? (b) At what angular frequency is the voltage amplitude across the inductor at maximum value? (c) At what angular frequency is the voltage amplitude across the capacitor at maximum value? (You may want to refer to the results of Problem 31.49.)

What is the dc impedance of the electrode, assuming that it behaves as an ideal capacitor? (a) 0; (b) infinite; (c) 12 * 104 ; (d) 12 * 106 .

If the electrode oscillates between two points 20 mm apart at a frequency of 15000>p2Hz, what is the electrodes impedance? (a) 0; (b) infinite; (c) 12 * 104 ; (d) 12 * 106

The signal from the oscillating electrode is fed into an amplifier, which reports the measured voltage as an rms value, 1.5 nV. What is the potential difference between the two extremes? (a) 1.5 nV; (b) 3.0 nV; (c) 2.1 nV; (d) 4.2 nV.

If the frequency at which the electrode is oscillated is increased to a very large value, the electrodes impedance (a) approaches infinity; (b) approaches zero; (c) approaches a constant but nonzero value; (d) does not change.