Chapter 21

What are the peak and rms voltages of a \(120-\mathrm{V}\) ac line and a \(240-\mathrm{V}\) ac line?

How much ac rms current must be in a \(10-\Omega\) resistor to produce an average power of \(15 \mathrm{~W} ?\)

An ac circuit contains a resistor with a resistance of \(5.0 \Omega\). The resistor has an rms current of 0.75 A. (a) Find its rms voltage and peak voltage. (b) Find the average power delivered to the resistor.

\- A hair dryer is rated at \(1200 \mathrm{~W}\) when plugged into a \(120-\mathrm{V}\) outlet. Find \((\mathrm{a})\) its rms current, \((\mathrm{b})\) its peak current, and (c) its resistance.

The voltage across a \(10-\Omega\) resistor varies as \(V=(170 \mathrm{~V}) \sin (100 \pi t) .\) (a) Is the current in the resistor (1) in phase with the voltage, (2) ahead of the voltage by \(90^{\circ},\) or (3) lagging behind the voltage by \(90^{\circ} ?\) (b) Write the expression for the current in the resistor as a function of time and determine the voltage frequency.

An ac voltage is applied to a \(25.0-\Omega\) resistor so that it dissipates \(500 \mathrm{~W}\) of power. Find the resistor's (a) \(\mathrm{rms}\) and peak currents and (b) rms and peak voltages.

An ac voltage source has a peak voltage of 85 and a frequency of \(60 \mathrm{~Hz}\). The voltage at \(t=0\) is zero. (a) If a student measures the voltage at \(t=1 / 240 \mathrm{~s},\) how many possible results are there: (1) one, (2) two, or (3) three? Why? (b) Determine all possible voltages the student might measure.

What are the resistance, peak current, and power level of a computer monitor that draws an rms current of 0.833 A when connected to a 120 -V outlet?

Find the rms and peak currents in a \(40-\mathrm{W}, 120-\mathrm{V}\) lightbulb. What is its resistance?

The current in a resistor is given by \(I=(8.0 \mathrm{~A}) \sin (40 \pi t)\) when a voltage given by \(V=(60 \mathrm{~V}) \sin (40 \pi t)\) is applied to it. (a) What is the resistance value? (b) What are the frequency and period of the voltage source? (c) What is the average power delivered to the resistor?

\- The current and voltage outputs of an operating ac generator have peak values of \(2.5 \mathrm{~A}\) and \(16 \mathrm{~V}\), respectively. (a) What is the average power output of the generator? (b) What is the effective resistance of the circuit it is in?

The current in a \(60-\Omega\) resistor is given by \(I=(2.0 \mathrm{~A}) \sin (380 t) .\) (a) What is the frequency of the current? (b) What is the rms current? (c) How much average power is delivered to the resistor? (d) Write an equation for the voltage across the resistor as a function of time. (e) Write an equation for the power delivered to the resistor as a function of time. (f) Show that the rms power obtained in part (e) is the same as your answer to part (c).

At what frequency does a \(25-\mu \mathrm{F}\) capacitor have a reactance of \(25 \Omega ?\)

A single \(2.0-\mu \mathrm{F}\) capacitor is connected across the terminals of a 60 -Hz voltage source, and a current of \(2.0 \mathrm{~mA}\) is measured on an ac ammeter. What is the capacitive reactance of the capacitor?

What capacitance value would give a reactance of \(100 \Omega\) in a 60 -Hz ac circuit?

How much current is in a circuit containing only a 50- \(\mu \mathrm{F}\) capacitor connected to an ac generator with an output of \(120 \mathrm{~V}\) and \(60 \mathrm{~Hz} ?\)

A single 50 -mH inductor forms a complete circuit when connected to an ac voltage source at \(120 \mathrm{~V}\) and \(60 \mathrm{~Hz}\) (a) What is the inductive reactance of the circuit? (b) How much current is in the circuit? (c) What is the phase angle between the current and the applied voltage? (Assume negligible resistance.)

A variable capacitor in a circuit with a \(120-\mathrm{V}, 60-\mathrm{Hz}\) source initially has a capacitance of \(0.25 \mu \mathrm{F}\). The capacitance is then increased to \(0.40 \mu \mathrm{F}\). (a) What is the percentage change in the capacitive reactance? (b) What is the percentage change in the current in the circuit?

(a) An inductor has a reactance of \(90 \Omega\) in a \(60-\mathrm{Hz}\) ac circuit. What is its inductance? (b) What frequency would be required to double its reactance?

(a) Find the frequency at which a \(250-\mathrm{mH}\) inductor has a reactance of \(400 \Omega\). (b) At what frequency would a \(0.40 \mu \mathrm{F}\) capacitor have the same reactance?

A capacitor is connected to a variable-frequency ac voltage source. (a) If the frequency increases by a factor of \(3,\) the capacitive reactance will be (1) \(3,(2) \frac{1}{3},(3) 9,(4) \frac{1}{9}\) times the original reactance. Why? (b) If the capacitive reactance of a capacitor at \(120 \mathrm{~Hz}\) is \(100 \Omega,\) what is its reactance if the frequency is changed to \(60 \mathrm{~Hz} ?\)

With a single \(150-\mathrm{mH}\) inductor in a circuit with a 60 -Hz voltage source, a current of 1.6 A is measured on an ac ammeter. (a) What is the rms voltage of the source? (b) What is the phase angle between the current and that voltage?

(a) What inductance has the same reactance in a 120 \(\mathrm{V}, 60-\mathrm{Hz}\) circuit as a capacitance of \(10 \mu \mathrm{F} ?\) (b) What would be the ratio of inductive reactance to capacitive reactance if the frequency were changed to \(120 \mathrm{~Hz} ?\)

A circuit with a single capacitor is connected to a \(120-\) \(\mathrm{V}, 60\) -Hz source. (a) What is its capacitance if there is a current of \(0.20 \mathrm{~A}\) in the circuit? (b) What would be the current if the source frequency were halved?

An inductor is connected to a variable-frequency ac voltage source. (a) If the frequency decreases by a factor of \(2,\) the rms current will be (1) \(2,(2) \frac{1}{2},(3) 4,(4) \frac{1}{4}\) times the original rms current. Why? (b) If the rms current in an inductor at \(40 \mathrm{~Hz}\) is \(9.0 \mathrm{~A},\) what is its rms current if the frequency is changed to \(120 \mathrm{~Hz} ?\)

A coil in a 60 -Hz circuit has a resistance of \(100 \Omega\) and an inductance of \(0.45 \mathrm{H}\). Calculate (a) the coil's reactance and (b) the circuit's impedance.

A series \(\mathrm{RC}\) circuit has a resistance of \(200 \Omega\) and a capacitance of \(25 \mu \mathrm{F}\) and is driven by a \(120-\mathrm{V}, 60\) -Hz source. (a) Find the capacitive reactance and impedance of the circuit. (b) How much current is drawn from the source?

A series RL circuit has a resistance of \(100 \Omega\) and an inductance of \(100 \mathrm{mH}\) and is driven by a \(120-\mathrm{V}, 60-\mathrm{Hz}\) source. (a) Find the inductive reactance and the impedance of the circuit. (b) How much current is drawn from the source?

A series \(\mathrm{RC}\) circuit has a resistance of \(250 \Omega\) and a capacitance of \(6.0 \mu \mathrm{F}\). If the circuit is driven by a \(60-\mathrm{Hz}\) source, find (a) the capacitive reactance and (b) the impedance of the circuit.

A series \(R C\) circuit has a resistance of \(100 \Omega\) and a capacitive reactance of \(50 \Omega\). (a) Will the phase angle be (1) positive, (2) zero, or (3) negative? Why? (b) What is the phase angle of this circuit?

A series RLC circuit has a resistance of \(25 \Omega,\) an inductance of \(0.30 \mathrm{H},\) and a capacitance of \(8.0 \mu \mathrm{F}\). (a) At what frequency should the circuit be driven for the maximum power to be transferred from the driving source? (b) What is the impedance at that frequency?

In a series RLC circuit, \(R=X_{C}=X_{L}=40 \Omega\) for a particular driving frequency. (a) This circuit is (1) inductive, (2) capacitive, (3) in resonance. Explain your reasoning. (b) If the driving frequency is doubled, what will be the impedance of the circuit?

(a) An RLC series circuit is in resonance. Which one of the following can you change without upsetting the resonance: (1) resistance, (2) capacitance, (3) inductance, or (4) frequency? (b) A resistor, an inductor, and a capacitor have values of \(500 \Omega, 500 \mathrm{mH},\) and \(3.5 \mu \mathrm{F},\) respectively. They are connected in series to a power supply of \(240 \mathrm{~V}\) with a frequency of \(60 \mathrm{~Hz} .\) What values of resistance and inductance would be required for this circuit to be in resonance (without changing the capacitor)?

(a) What is the resonance frequency of an RLC circuit with a resistance of \(100 \Omega,\) an inductance of \(100 \mathrm{mH},\) and a capacitance of \(5.00 \mu \mathrm{F} ?\) (b) What is the resonance frequency if all the values in part (a) are doubled?

A tuning circuit in a radio receiver has a fixed inductance of \(0.50 \mathrm{mH}\) and a variable capacitor. (a) If the circuit is tuned to a radio station broadcasting at \(980 \mathrm{kHz}\) on the AM dial, what is the capacitance of the capacitor? (b) What value of capacitance is required to tune into a station broadcasting at \(1280 \mathrm{kHz}\) ?

A coil with a resistance of \(30 \Omega\) and an inductance of \(0.15 \mathrm{H}\) is connected to \(\mathrm{a} 120-\mathrm{V}, 60\) -Hz source. \((\mathrm{a})\) Is the phase angle of this circuit (1) positive, (2) zero, or (3) negative? Why? (b) What is the phase angle of the circuit? (c) How much rms current is in the circuit? (d) What is the average power delivered to the circuit?

A small welding machine uses a voltage source of \(120 \mathrm{~V}\) at \(60 \mathrm{~Hz}\). When the source is operating, it requires \(1200 \mathrm{~W}\) of power, and the power factor is \(0.75 .\) (a) What is the machine's impedance? (b) Find the rms current in the machine while operating.

A series circuit is connected to \(a 220-V, 60\) -Hz power supply. The circuit has the following components: a \(10-\Omega\) resistor, a coil with an inductive reactance of \(120 \Omega\), and a capacitor with a reactance of \(120 \Omega\). Compute the rms voltage across (a) the resistor, (b) the inductor, and (c) the capacitor.

A series RLC circuit has a resistance of \(25 \Omega\), a capacitance of \(0.80 \mu \mathrm{F},\) and an inductance of \(250 \mathrm{mH}\). The circuit is connected to a variable-frequency source with a fixed rms voltage output of \(12 \mathrm{~V}\). If the frequency that is supplied is set at the circuit's resonance frequency, what is the rms voltage across each of the circuit elements?

A series \(\mathrm{RLC}\) circuit with a resistance of \(400 \Omega\) has capacitive and inductive reactances of \(300 \Omega\) and \(500 \Omega\) respectively. (a) What is the power factor of the circuit? (b) If the circuit operates at \(60 \mathrm{~Hz},\) what additional capacitance should be connected to the original capacitance to give a power factor of unity, and how should the capacitors be connected?

A series RLC circuit has components with \(R=50 \Omega\), \(L=0.15 \mathrm{H},\) and \(C=20 \mu \mathrm{F}\). The circuit is driven by a \(120-\mathrm{V}, 60-\mathrm{Hz}\) source. (a) What is the current in the circuit, expressed as a percentage of the maximum possible current? (b) What is the power delivered to the circuit, expressed as a percentage of the power delivered when the circuit is in resonance?

A series RLC radio receiver circuit with an inductance of \(1.50 \mu \mathrm{H}\) is tuned to an FM station at \(98.9 \mathrm{MHz}\) by adjusting a variable capacitor. When the circuit is tuned to this station, (a) what is its inductive reactance? (b) What is its capacitive reactance? (c) What is its capacitance?

A circuit connected to a \(110-\mathrm{V}, 60\) -Hz source contains a \(50-\Omega\) resistor and a coil with an inductance of \(100 \mathrm{mH}\). Find (a) the reactance of the coil, (b) the impedance of the circuit, (c) the current in the circuit, and (d) the power dissipated by the coil, and (e) calculate the phase angle between the current and the applied voltage.

A \(1.0-\mu \mathrm{F}\) capacitor is connected to \(\mathrm{a} 120-\mathrm{V}, 60\) - \(\mathrm{Hz}\) source. (a) What is the capacitive reactance of the circuit? (b) How much current is in the circuit? (c) What is the phase angle between the current and the applied voltage? (d) What is the maximum energy stored in the capacitor? (f) What is the power dissipated by this circuit?

An ideal transformer is plugged into a \(12-\mathrm{V}, 60-\mathrm{Hz}\) ac outlet in a motor home, thus enabling the owners to use a \(1500-\mathrm{W}\), 120 - \(\mathrm{V}\) hair dryer. (Ignore any inductance or capacitance associated with the hair dryer.) (a) What type of transformer should be used and what should its turn ratio be? When the hair dryer is in operation, (b) what is its resistance? (c) What are its frequency, \(\mathrm{rms}\) voltage, and rms current? (d) What are its peak current and peak voltage and peak power output? (e) What are the peak power, current, and voltage values on the input side of the transformer?