The generator voltage in Figure 29-40 is given by (a) For

A coil in an ac generator rotates at How much time elapses between successive peak emf values of the coil

If the rms voltage in an ac circuit is doubled, the peak voltage is (a) doubled, (b) halved, (c) increased by a factor of (d) not changed

If the frequency in the circuit shown in Figure 29-27 is doubled, the inductance of the inductor will (a) double, (b) not change, (c) halve, (d) quadruple

If the frequency in the circuit shown in Figure 29-27 is doubled, the inductive reactance of the inductor will (a) double, (b) not change, (c) halve, (d) quadruple

If the frequency in the circuit in Figure 29-28 is doubled, the capacitive reactance of the circuit will (a) double, (b) not change, (c) halve, (d) quadruple

(a) In a circuit consisting solely of an ac generator and an ideal inductor, are there any time intervals when the inductor receives energy from the generator? If so, when? Explain your answer. (b) Are there any time intervals when the inductor supplies energy back to the generator? If so when? Explain your answer

(a) In a circuit consisting of a generator and a capacitor, are there any time intervals when the capacitor receives energy from the generator? If so, when? Explain your answer. (b) Are there any time intervals when the capacitor supplies power to the generator? If so, when? Explain your answer

(a) Show that the SI unit of inductance multiplied by the SI unit of capacitance is equivalent to seconds squared. (b) Show that the SI unit of inductance divided by the SI unit of resistance is equivalent to seconds

Suppose you increase the rotation rate of the coil in the generator shown in the simple ac circuit in Figure 29-29. Then the rms current (a) increases, (b) does not change, (c) may increase or decrease depending on the magnitude of the original frequency, (d) may increase or decrease depending on the magnitude of the resistance, (e) decreases.

If the inductance value is tripled in a circuit consisting solely of a variable inductor and a variable capacitor, how would you have to change the capacitance so that the natural frequency of the circuit is unchanged? (a) Triple the capacitance. (b) Decrease the capacitance to one-third of its original value. (c) You should not change the capacitance. (d) You cannot determine how to change the capacitance from the data given.

Consider a circuit consisting solely of an ideal inductor and an ideal capacitor. How does the maximum energy stored in the capacitor compare to the maximum value stored in the inductor? (a) They are the same and each is equal to the total energy stored in the circuit. (b) They are the same and each is equal to half of the total energy stored in the circuit. (c) The maximum energy stored in the capacitor is larger than the maximum energy stored in the inductor. (d) The maximum energy stored in the inductor is larger than the maximum energy stored in the capacitor. (e) You cannot compare the maximum energies based on the data given because the ratio of the maximum energies depends on the actual capacitance and inductance values.

True or false: (a) A driven series circuit that has a high factor has a narrow resonance curve. (b) A circuit consists solely of a resistor, an inductor, and a capacitor, all connected in series. If the resistance of the resistor is doubled, the natural frequency of the circuit remains the same. (c) At resonance, the impedance of a driven series combination equals the resistance (d) At resonance, the current in a driven series circuit is in phase with the voltage applied to the combination.

True or false: (a) Near resonance, the power factor of a driven series circuit is close to zero. (b) The power factor of a driven series circuit does not depend on the value of the resistance. (c) The resonance frequency of a driven series circuit does not depend on the value of the resistance. (d) At resonance, the peak current of a driven series circuit does not depend on the capacitance or the inductance. (e) For frequencies below the resonant frequency, the capacitive reactance of a driven series circuit is larger than the inductive reactance. (f) For frequencies below the resonant frequency of a driven series circuit, the phase of the current leads (is ahead of) the phase of the applied voltage.

You may have noticed that sometimes two radio stations can be heard when your receiver is tuned to a specific frequency. This situation often occurs when you are driving and are between two cities. Explain how this situation can occur.

True or false: (a) At frequencies much higher than or much lower than the resonant frequency of a driven series circuit, the power factor is close to zero. (b) The larger the resonance width of a driven series circuit is, the larger the factor for the circuit is. (c) The larger the resistance of a driven series circuit is, the larger the resonance width for the circuit is.

An ideal transformer has turns on its primary and turns on its secondary. The average power delivered to a load resistance connected across the secondary is when the primary rms voltage is The rms current in the primary windings can then be expressed as (a) (b) (c) (d)

True or false: (a) A transformer is used to change frequency. (b) A transformer is used to change voltage. (c) If a transformer steps up the current, it must step down the voltage. (d) A step-up transformer steps down the current. (e) The standard household wall-outlet voltage in Europe is about twice that used in the United States. If a European traveler wants her hair dryer to work properly in the United States, she should use a transformer that has more windings in its secondary coil than in its primary coil. (f ) The standard household wall-outlet voltage in Europe is about twice that used in the United States. If an American traveler wants his electric razor to work properly in Europe, he should use a transformer that steps up the current.

ENGINEERING APPLICATION The impedances of motors, transformers, and electromagnets include both resistance and inductive reactance. Suppose that the phase of the current to a large industrial plant lags the phase of the applied voltage by when the plant is under full operation and using of power. The power is supplied to the plant from a substation from the plant; the line voltage at the plant is The resistance of the transmission line from the substation to the plant is The cost per kilowatt-hour to the company that owns the plant is $0.14, and the plant pays only for the actual energy used. (a) Estimate the resistance and inductive reactance of the plants total load. (b) Estimate the rms current in the power lines and the rms voltage at the substation. (c) How much power is lost in transmission? (d) Suppose that the phase at which the current lags the phase of the applied voltage is reduced to 18 by adding a RLC bank of capacitors in series with the load. How much money would be saved by the electric utility during one month of operation, assuming the plant operates at full capacity for each day? (e) What must be the capacitance of this bank of capacitors to achieve this change in phase angle?

A 100-W lightbulb is screwed into a standard 120-V-rms. socket. Find (a) the rms current, (b) the peak current, and (c) the peak power.

A circuit breaker is rated for a current of at a voltage of (a) What is the largest value of peak current that the breaker can carry? (b) What is the maximum average power that can be supplied by this circuit?

What is the reactance of a inductor at (a) (b) and (c)

An inductor has a reactance of at (a) What is its inductance? (b) What is its reactance at

At what frequency would the reactance of a capacitor equal the reactance of a inductor?

What is the reactance of a 1.00-nF capacitor at (a) (b) and (c)

A20-Hz ac generator that produces a peak emf of is connected to a capacitor. Find (a) the peak current and (b) the rms current.

At what frequency is the reactance of a capacitor (a) (b) and (c)

Acircuit consists of two ideal ac generators and a resistor, all connected in series. The potential difference across the terminals of one of the generators is given by and the potential difference across the terminals of the other generator is given by where (a) Use Kirchhoffs loop rule and a trigonometric identity to find the peak current in the circuit. (b) Use a phasor diagram to find the peak current in the circuit. (c) Find the current in the resistor if and the amplitude of is increased from to

(a) Show that has units of inverse seconds by substituting SI units for inductance and capacitance into the expression. (b) Show that (the expression for the factor) is dimensionless by substituting SI units for angular frequency, inductance, and resistance into the expression.

(a) What is the period of oscillation of an circuit consisting of a 2.0-mH coil and a capacitor? (b) A circuit that oscillates consists solely of an capacitor and a variable ideal inductor. What inductance is needed in order to tune this circuit to oscillate at

An circuit has capacitance and inductance A second circuit has capacitance and inductance and a third circuit has capacitance and inductance (a) Show that each circuit oscillates with the same frequency. (b) In which circuit would the peak current be greatest if the peak voltage across the capacitor in each circuit were the same?

A capacitor is charged to and is then R L connected across an ideal 10-mH inductor. (a) How much energy is stored in the system? (b) What is the frequency of oscillation of the circuit? (c) What is the peak current in the circuit?

A coil with internal resistance can be modeled as a resistor and an ideal inductor in series. Assume that the coil has an internal resistance of and an inductance of A capacitor is charged to and is then connected across coil. (a) What is the initial voltage across the coil? (b) How much energy is dissipated in the circuit before the oscillations die out? (c) What is the frequency of oscillation of the circuit? (Assume the internal resistance is sufficiently small that it has no impact on the frequency of the circuit.) (d) What is the quality factor of the circuit?

An inductor and a capacitor are connected, as shown in Figure 29-30. Initially, the switch is open, and the left plate of the capacitor has charge The switch is then closed. (a) Plot both versus and versus on the same graph, and explain how it can be seen from these two plots that the current leads the charge by (b) The expressions for the charge and for the current are given by Equations 29-38 and 29-39, respectively. Use trigonometry and algebra to show that the current leads the charge by

Acircuit consists of a resistor, an ideal 1.4-H inductor, and an ideal 60-Hz generator, all connected in series. The rms voltage across the resistor is and the rms voltage across the inductor is (a) What is the resistance of the resistor? (b) What is the peak emf of the generator?

Acoil that has a resistance of has an impedance of when driven at a frequency of What is the inductance of the coil?

ENGINEERING APPLICATION A two-conductor transmission line simultaneously carries a superposition of two voltage signals, so the potential difference between the two conductors is given by where and where and A1.00-H inductor and a shunt resistor are inserted into the transmission line as shown in Figure 29-31. (Assume that the output is connected to a load that draws only an insignificant amount of current.) (a) What is the voltage at the output of the transmission line? (b) What is the ratio of the low-frequency amplitude to the high-frequency amplitude at the output?

A coil is connected to a 120-V-rms, 60-Hz line. The average power supplied to the coil is and the rms current is Find (a) the power factor, (b) the resistance of the coil, and (c) the inductance of the coil. (d) Does the current lag or lead the voltage? Explain your answer. (e) Support your answer to Part (d) by determining the phase angle.

A 36-mH inductor that has a resistance of is connected to an ideal ac voltage source whose output is given by where is in seconds. Determine (a) the peak current in the circuit, (b) the peak and rms voltages across the inductor, (c) the average power dissipation, and (d) the peak and average magnetic energy stored in the inductor.

A coil that has a resistance and an inductance has a power factor equal to when driven at a frequency of What is the coils power factor if it is driven at

Aresistor and an inductor are connected in parallel across an ideal ac voltage source whose output is given by as shown in Figure 29-32. Show that (a) the current in the resistor is given by (b) the current in the inductor is given by and (c) the current in the voltage source is given by I _ I where Ipeak _ Epeak>Z.

Figure 29-33 shows a load resistor that has a resistance of connected to a high-pass filter consisting of an inductor that has an inductance inductor and a resistor that has resistance The output of the ideal ac generator is given by Find the rms currents in all three branches of the circuit if the driving frequency is (a) and (b) Find the fraction of the total average power supplied by the ac generator that is delivered to the load resistor if the frequency is (c) and (d) 2000 Hz. SSM

An ideal ac voltage source whose emf is given by and an ideal battery whose emf is are connected to a combination of two resistors and an inductor (Figure 29-34), where and Find the average power delivered to each resistor if the driving frequency is (a) 100 Hz, (b) 200 Hz, and (c) 800 Hz.

An ac circuit contains a resistor and an ideal inductor connected in series. The rms voltage drop across the series combination is and the rms voltage drop across the inductor alone is What is the rms voltage drop across the resistor?

ENGINEERING APPLICATION The circuit shown in Figure 29-35 is called an high-pass filter because it transmits input voltage signals that have high frequencies with greater amplitude than it transmits input voltage signals that have low frequencies. If the input voltage is given by show that the output voltageis where VH _ Vin peakV >21 _ (vRC)_2 . (Assume that the output is connected to a load that draws only an insignificant amount of current.) Show that this result justifies calling this circuit a high-pass filter.

(a) Find an expression for the phase constant in Problem 44 in terms of and (b) What is the value of in the limit that (c) What is the value of in the limit that (d) Explain your answers to Parts (b) and (c).

SPREADSHEET Assume that in Problem 44, and (a) At what frequency is That particular frequency is known as the 3-dB frequency, or for the circuit. (b) Using a spreadsheet program, make a graph of versus where is the frequency. Make sure that the scale extends from at least 10% of the 3-dB frequency to ten times the 3-dB frequency. (c) Make a graph of versus for the same range of frequencies as in Part (b). What is the value of the phase constant when the frequency is equal to the 3-dB frequency?

A slowly varying voltage signal is applied to the input of the high-pass filter of Problem 44. Slowly varying means that during one time constant (equal to there is no significant change in the voltage signal. Show that under those conditions the output voltage is proportional to the time derivative of This situation is known as a differentiation circuit.

We can describe the output from the high-pass filter from Problem 44 using a decibel scale: where is the output in decibels. Show that for The frequency at which is known as (the 3-dB frequency). Show that for the output drops by if the frequency is halved.

Show that the average power dissipated in the resistor of the high-pass filter of Problem 44 is given by

One application of the high-pass filter of Problem 44 is as a noise filter for electronic circuits (a filter that blocks out lowfrequency noise). Using a resistance value of find a value for the capacitance for the high-pass filter that attenuates a 60-Hz input voltage signal by a factor of 10, that is, so

ENGINEERING APPLICATION The circuit shown in Figure 29-36 is an example of a low-pass filter. (Assume that the output is connected to a load that draws only an insignificant amount of current.) (a) If the input voltage is given by show that the output voltage is where (b) Discuss the trend of the output voltage in the limiting cases and

(a) Find an expression for the phase angle for the low-pass filter of Problem 51 in terms of and (b) Find the value of in the limit that and in the limit that Explain your answer.

SPREADSHEET Using a spreadsheet program, make a graph of versus input frequency and a graph of phase angle versus input frequency for the low-pass filter of Problems 51 and 52. Use a resistance value of and a capacitance value of

A rapidly varying voltage signal is applied to the input of the low-pass filter of Problem 51. Rapidly varying means that during one time constant (equal to there are significant changes in the voltage signal. Show that under those conditions the output voltage is proportional to the integral of with respect to time. This situation is known as an integration circuit.

E N G I N E E R I N G APPLICATION The circuit shown in Figure 29-37 is a trap filter. (Assume that the output is connected to a load that draws only an insignificant amount of current.) (a) Show that the trap filter acts to reject signals in a band of frequencies centered at (b) How does the width of the frequency band rejected depend on the resistance

E N G I N E E R I N G APPLICATION A half-wave rectifier for transforming an ac voltage into a dc voltage is shown in Figure 29-38. The diode in the figure can be thought of as a one-way valve for current. It allows current to pass in the forward direction (the direction of the arrowhead) only when is at a higher electric potential than by (i.e., whenever The resistance of the diode is effectively infinite when is less than Plot two cycles of both input and output voltages as a function of time, on the same graph, assuming the input voltage is given by

ENGINEERING APPLICATION The output of the rectifier of Problem 56 can be further filtered by putting its output through a low-pass filter as shown in Figure 29-39a. The resulting output is a dc voltage with a small ac component (ripple) shown in Figure 29-39b. If the input frequency is and the load resistance is find the value for the capacitance so that the output voltage varies by less than 50 percent of the mean value over one cycle.

The generator voltage in Figure 29-40 is given by (a) For each branch, what is the peak current and what is the phase of the current relative to the phase of the generator voltage? (b) At the resonance frequency there is no current in the generator. What is the angular frequency at resonance? (c) At the resonance frequency, find the current in the inductor and the current in the capacitor. Express your results as functions of time. (d) Draw a phasor diagram showing the phasors for the applied voltage, the generator current, the capacitor current, and the inductor current for the case where the frequency is higher than the resonance frequency.

Acircuit consists of an ideal ac generator, a capacitor, and an ideal inductor, all connected in series. The charge on the capacitor is given by where (a) Find the current in the circuit as a function of time. (b) Find the capacitance if the inductance is (c)Write expressions for the electrical energy the magnetic energy and the total energy as functions of time.

ENGINEERING APPLICATION One method for determining the compressibility of a dielectric material uses a driven circuit that has a parallel-plate capacitor. The dielectric is inserted between the plates and the change in resonance frequency is determined as the capacitor plates are subjected to a compressive stress. In one such arrangement, the resonance frequency is when a dielectric of thickness and dielectric constant is placed between the plates. Under a compressive stress of the resonance frequency decreases to Find the Youngs modulus of the dielectric material. (Assume that the dielectric constant does not change with pressure.)

Figure 29-41 shows an inductor in series with a parallel-plate capacitor. The capacitor has a width of and a gap of A dielectric that has a dielectric constant of 4.8 can be slid in and out of the gap. The inductor has an inductance of When half the dielectric is between the capacitor plates (when the resonant frequency of this combination is (a) What is the capacitance of the capacitor without the dielectric? (b) Find the resonance frequency as a function of for

A circuit consists of an ideal ac generator, a capacitor, and an resistor, all connected in series. The output of the generator has a peak emf of and the armature of the generator rotates at Find (a) the power factor, (b) the rms current, and (c) the average power supplied by the generator.

Show that the expression gives the correct result for a circuit containing only an ideal ac generator and (a) a resistor, (b) a capacitor, and (c) an inductor. In the expression P is the average power supplied by the generator, is the root-mean-square value of the emf of the generator, is the resistance, is the capacitance, and is the inductance. [In Part (a), in Part (b), and in Part (c), ]

A series circuit that has an inductance of a capacitance of and a resistance of is driven by an ideal ac voltage source that has a peak emf of Find (a) the resonant frequency and (b) the root-mean-square current at resonance. When the frequency is find (c) the capacitive and inductive reactances, (d) the impedance, (e) the root-mean-square current, and (f) the phase angle.

Find (a) the factor and (b) the resonance width (in hertz) for the circuit in Problem 64. (c) What is the power factor when

ENGINEERING APPLICATION FM radio stations typically operate at frequencies separated by Thus, when your radio is tuned to a station operating at a frequency of the resonance width of the receiver circuit should be much smaller than so that you do not receive signal from stations operating at adjacent frequencies. Assume your receiving circuit has a resonance width of When tuned in to that particular station, what is the factor of your circuit?

A coil is connected to a 60-Hz ac generator with a peak emf equal to At this frequency, the coil has an impedance of and a reactance of (a) What is the peak current in the coil? (b) What is the phase angle between the current and the applied voltage? (c) A capacitor is put in series with the coil and the generator. What capacitance is required so that the current is in phase with the generator emf? (d) What is the peak voltage measured across the capacitor?

An ideal 0.25-H inductor and a capacitor are connected in series with an ideal 60-Hz generator. A digital voltmeter is used to measure the rms voltages across the inductor and capacitor independently. The voltmeter reading across the capacitor is and that across the inductor is (a) Find the capacitance and the rms current in the circuit. (b) What is the rms voltage across the series combination of the capacitor and the inductor?

In the circuit shown in Figure 29-42 the ideal generator produces an rms voltage of when operated at What is the rms voltage between points (a) and (b) and (c) and (d) and and (e) and

When an series circuit is connected to a 120-V-rms, 60-Hz line, the rms current in the circuit is and the current leads the line voltage by (a) Find the average power supplied to the circuit. (b) What is the resistance in the circuit? (c) If the inductance in the circuit is find the capacitance in the circuit. (d) Without changing the inductance, by how much should you change the capacitance to make the power factor equal to 1? (e) Without changing the capacitance, by how much should you change the inductance to make the power factor equal to 1?

SPREADSHEET Plot the circuit impedance versus the angular frequency for each of the following circuits: (a) a driven series circuit, (b) a driven series circuit, and (c) a driven series circuit.

In a driven series circuit, the ideal generator has a peak emf equal to the resistance is and the capacitance is The inductance can be varied from to 40.0 mH by the insertion of an iron core in the solenoid. The angular frequency of the generator is If the capacitor voltage is not to exceed find (a) the peak current and (b) the range of inductances that is safe to use.

A certain electrical device draws an rms current of at an average power of when connected to a 120-V-rms, 60-Hz power line. (a) What is the impedance of the device? (b) What series combination of resistance and reactance would have the same impedance as this device? (c) If the current leads the emf, is the reactance inductive or capacitive?

A method for measuring inductance is to connect the inductor in series with a known capacitance, a known resistance, an ac ammeter, and a variable-frequency signal generator. The frequency of the signal generator is varied and the emf is kept constant until the current is maximum. (a) If the capacitance is the peak emf is the resistance is and the rms current in the circuit is maximum when the driving frequency is what is the value of the inductance? (b) What is the maximum rms current?

Aresistor and a capacitor are connected in parallel across an ac generator (Figure 29-43) that has an emf given by (a) Show that the current in the resistor is given by (b) Show that the current in the capacitor branch is given by (c) Show that the current in the generator is given by where and

Figure 29-44 shows a plot of average power versus generator frequency for a series RLC circuit driven by an ac generator. The average power is given by Equation 29-56. The full width at half-maximum, is the width of the resonance curve between the two points, where is one-half its maximum value. Show that for a sharply peaked resonance, and that (Equation 29-58). Hint: The half-power points occur when the denominator of Equation 29-56 is equal to twice the value it has at resonance; that is, when Let and be the solutions of this equation. Then, show that v _ v2 _ v1 _ R>L.

Show by direct substitution that (Equation 29-43b) is satisfied by where and is the charge on the capacitor at t _ 0. Qt _ 2L>R, v_ _ 21>(

ENGINEERING APPLICATION One method for measuring the magnetic susceptibility of a sample uses an circuit consisting of an air-core solenoid and a capacitor. The resonant frequency of the circuit without the sample is determined and then measured again with the sample inserted in the solenoid. Suppose you have a solenoid that is long, is in diameter, and has of fine wire. You have a sample that is inserted in the solenoid and completely fills the air space. Neglect end effects. (a) Calculate the inductance of the empty solenoid. (b) What value for the capacitance of the capacitor should you choose so that the resonance frequency of the circuit without a sample is exactly (c) When a sample is inserted in the solenoid, you determine that the resonance frequency drops to Use your data to determine the samples susceptibility.

A rms voltage of is required for a device whose impedance is (a) What should the turns ratio of a transformer be, so that the device can be operated from a line? (b) Suppose the transformer is accidentally connected in reverse with the secondary winding across the line and the load across the primary. How much rms current will then be in the primary winding?

Atransformer has in the primary and in the secondary. (a) Is this a step-up or a step-down transformer? (b) If the primary is connected to a 120-V-rms voltage source, what is the open-circuit rms voltage across the secondary? (c) If the primary rms current is what is the secondary rms current, assuming negligible magnetization current and no power loss?

The primary of a step-down transformer has and is connected to a 120-V-rms line. The secondary is to supply at Find (a) the rms current in the primary and (b) the number of turns in the secondary, assuming 100 percent efficiency.

An audio oscillator (ac source) that has an internal resistance of and an open-circuit rms output voltage of is to be used to drive a loudspeaker coil that has a resistance of (a) What should be the ratio of primary to secondary turns of a transformer so that maximum average power is transferred to the speaker? (b) Suppose a second identical speaker is connected in parallel with the first speaker. How much average power is then supplied to the two speakers combined?

The distribution circuit of a residential power line is operated at This voltage must be reduced to for use within residences. If the secondary side of the transformer has how many turns are in the primary?

A resistor that has a resistance carries a current given by where (a) What is the rms current in the resistor? (b) If what is the average power delivered to the resistor? (c) What is the rms voltage across the resistor?

Figure 29-45 shows the voltage versus time for a squarewave voltage source. If (a) what is the rms voltage of this source? (b) If this alternating waveform is rectified by eliminating the negative voltages, so that only the positive voltages remain, what is the new rms voltage? SSM

What are the average values and rms values of current for the two current waveforms shown in Figure 29-46?

In the circuit shown in Figure 29-47, where and Find the maximum, minimum, average, and rms values of the current in the resistor.

Repeat Problem 87 if the resistor is replaced by a capacitor.

A circuit consists of an ac generator, a capacitor and an ideal inductorall connected in series. The emf of the generator is given by (a) Show that the charge on the capacitor obeys the equation (b) Show by direct substitution that this equation is satisfied by where (c) Show that the current can be written as where for and for where is the resonance frequency. SSM