Chapter 22

\(\bullet\) You have a special lightbulb 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?

\(\cdot\) The plate on the back of a certain computer scanner says that the unit draws 0.34 A of current from a \(120 \mathrm{V}, 60\) Hz line. Find (a) the root-mean-square current, (b) the current ampli- tude, (c) the average current, and (d) the average square of the 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?

\(\bullet\) (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?

\(\bullet\) 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 \(\mathrm{m}\) A when a sinusoidal voltage with amplitude 12.0 \(\mathrm{V}\) is applied across the inductor. What frequency is required?

A 2.20\(\mu \mathrm{F}\) capacitor is connected across an ac source whose voltage amplitude is kept constant at \(60.0 \mathrm{V},\) but whose frequency can be varied. Find the current amplitude when the angular frequency is (a) \(100 \mathrm{rad} / \mathrm{s} ;\) (b) \(1000 \mathrm{rad} / \mathrm{s} ;\) (c) \(10,000 \mathrm{rad} / \mathrm{s}\) .

\(\bullet\) The voltage amplitude of an ac source is \(25.0 \mathrm{V},\) and its angular frequency is 1000 \(\mathrm{rad} / \mathrm{s} .\) Find the current amplitude if the capacitance of a capacitor connected across the source is (a) \(0.0100 \mu \mathrm{F},(\mathrm{b}) 1.00 \mu \mathrm{F},(\mathrm{c}) 100 \mu \mathrm{F}\)

A sinusoidal ac voltage source in a circuit produces a maximum voltage of 12.0 \(\mathrm{V}\) and an rms current of 7.50 \(\mathrm{mA}\) . Find (a) the voltage and current amplitudes and (b) the rms voltage of this source.

\(\cdot \mathrm{A} 65 \Omega\) resistor, an 8.0\(\mu \mathrm{F}\) capacitor, and a 35 \mathrm{mH}\( inductor are connected in series in an ac circuit. Calculate the impedance for a source frequency of (a) 300 \)\mathrm{Hz}\( and (b) 30.0 \)\mathrm{kHz}$ .

A 1500\(\Omega\) resistor is connected in series with a 350 \(\mathrm{mH}\) inductor and an ac power supply. At what frequency will this combination have twice the impedance that it has at 120 \(\mathrm{Hz} ?\)

\(\bullet\) (a) Compute the impedance of a series \(R-L-C\) circuit at angular frequencies of \(1000,750,\) and 500 \(\mathrm{rad} / \mathrm{s} .\) Take \(R=\) \(200 \Omega, L=0.900 \mathrm{H},\) and \(C=2.00 \mu \mathrm{F}\) . (b) Describe how the current amplitude varies as the angular frequency of the source is slowly reduced from 1000 rad/s to 500 rad/s. (c) What is the phase angle of the source voltage with respect to the current when \(\omega=1000 \mathrm{rad} / \mathrm{s} ?\) (d) Construct a phasor diagram when \(\omega=1000 \mathrm{rad} / \mathrm{s}\)

A \(\mathrm{A} 200 \Omega\) resistor is in series with a 0.100 \(\mathrm{H}\) inductor and a 0.500\(\mu \mathrm{F}\) capacitor. Compute the impedance of the circuit and draw the phasor diagram (a) at a frequency of \(500 \mathrm{Hz},\) (b) at a frequency of 1000 \(\mathrm{Hz}\) . In each case, compute the phase angle of the source voltage with respect to the current and state whether the source voltage lags or leads the current.

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

A series \(R-L-C\) 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 of 105\(\Omega\) at this frequency. What average power is delivered to the circuit by the source?

A series ac circuit contains a 250\(\Omega\) resistor, a 15 \(\mathrm{mH}\) inductor, a 3.5\(\mu\) 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?

\(\bullet\) (a) At what angular frequency will a 5.00\(\mu\) F capacitor have the same reactance as a 10.0 \(\mathrm{mH}\) inductor? (b) If the capacitor and inductor in part (a) are connected in an \(L-C\) circuit, what will be the resonance angular frequency of that circuit?

\(\bullet\) In an \(R-L-C\) 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?

A series circuit consists of an ac source of variable frequency, a 115\(\Omega\) resistor, a 1.25\(\mu\) 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, and (c) half the resonance angular frequency.

\(\bullet\) In a series \(R-L-C\) 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?

\(\bullet\) In a series \(R-L-C\) circuit, \(L=0.200 \mathrm{H}, C=80.0 \mu \mathrm{F},\) and the voltage amplitude of the source is 240 \(\mathrm{V}\) . (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 0.600 A. What is the resistance \(R\) of the resistor? (c) At the resonance frequency, what are the peak voltages across the inductor, the capacitor, and the resistor?

\(\bullet\) In an \(R-L-C\) 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?

\(\bullet\) A 125\(\Omega\) resistor, an 8.50\(\mu \mathrm{F}\) capacitor, and an 11.2 \(\mathrm{mH}\) inductor are all connected in parallel across an ac voltage source of variable frequency. (a) At what angular frequency will the impedance have its maximum value, and (b) what is that value?

\(\bullet\) 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 large electromagnetic coil is connected to a 120 \(\mathrm{Hz}\) 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 \(\mathrm{W} ?\)

\bullet 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?

\(\bullet\) In a series \(R-L-C\) 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 kHz. Determine (a) the power supplied by the generator; and (b) the power dissipated in the resistor.

\(\bullet\) In an \(R-L-C\) 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.

\(\bullet\) In a series \(R-L-C\) 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?

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

A A 100.0\(\Omega\) resistor, a 0.100\(\mu \mathrm{F}\) capacitor, and a 300.0 \(\mathrm{mH}\) inductor are connected in series to a voltage source with amplitude 240 \(\mathrm{V}\) (a) What is the resonance angular frequency? (b) What is the maximum current in the resistor at resonance? (b) What is the maximum current in the resistor at resonance? (c) What is the maximum voltage across the capacitor at resonance? (d) What is the maximum voltage across the inductor at resonance? (e) What is the maximum energy stored in the capacitor at resonance? in the inductor?

What is the dc impedance of the electrode, assuming that it behaves as an ideal capacitor? \(A. 0 \) \(B. Infinite\) \(\quad\) C. \(\sqrt{2} \times 10^{4} \Omega \quad\) D. \(\sqrt{2} \times 10^{6} \Omega\)

The signal from the oscillating electrode is fed into an amplifier, which reports the measured voltage as an rms value, \(V_{\text { rms. }}\) However, the number of interest for analyzing currents driven by the cell is the peak- to-peak voltage difference \(\left(V_{\mathrm{pp}}\right),\) that is, the voltage difference between the two extremes of the electrodes excursion. What is the value of \(V_{\mathrm{pp}}\) in terms of \(V_{\mathrm{rms}} ?\) A. \(V_{\mathrm{rms}} / \sqrt{2} \quad \mathrm{B} . V_{\mathrm{rms}} / 2 \sqrt{2}\) C. \(\sqrt{2} V_{\mathrm{rms}} \quad\) D. 2\(\sqrt{2} V_{\mathrm{rms}}\)