Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

| 46 | Concept Questions (a) Does the capacitance of a parallel plate capacitor increase or decrease when a dielectric material is inserted between the plates? (b) Is the equivalent capacitance of two capacitors in parallel greater or smaller than the capacitance of either capacitor alone? (c) A capacitor is connected between the terminals of an ac generator. When a second capacitor is connected in parallel with the first, does the current supplied by the generator increase or decrease? Explain your answers.
Problem Two parallel plate capacitors are identical, except that one of them is empty and the other contains a material with a dielectric constant of 4.2 in the space between the plates. The empty capacitor is connected between the terminals of an ac generator that has a fixed frequency and rms voltage. The generator delivers a current of 0.22 A. What current does the generator deliver after the other capacitor is connected in parallel with the first one? Verify that your answer is consistent with your answers to the Concept Questions.

| 47 | Concept Questions (a) An inductance L1 is connected across the terminals of an ac generator, which delivers a current to it. Then a second inductance L2 is connected in parallel with L1. Does the presence of L2 alter the current in L1 ? (b) The generator delivers a current to L2 as well as L1. Would the removal of L1 alter the current in L2? (c) Is the current delivered to the parallel combination greater or smaller than the current to either inductance alone? Give your reasoning.
Problem An ac generator has a frequency of 2.2 kHz and a voltage of 240 V. An inductance L1 = 6.0 mH is connected across its terminals. Then a second inductance L2 = 9.0 mH is connected in parallel with L1. Find the current that the generator delivers to L1 and to the parallel combination. Check to see that your answers are consistent with your answers to the Concept Questions.

| 48 | Concept Questions (a) An ac circuit contains only a resistor and a capacitor in series. Is the phase angle between the current and the voltage of the generator positive or negative, and how is the impedance Z of the circuit related to the resistance R and the capacitive reactance XC? (b) An ac circuit contains only a resistor and an inductor in series. Is the phase angle positive or negative, and how is the impedance Z of the circuit related to the resistance R and the inductive reactance XL? Account for your answers.
Problem A series circuit has an impedance of , and the phase angle is . The circuit contains a resistor and either a capacitor or an inductor. Find the resistance R and the capacitive reactance XC or the inductive reactance XL, whichever is appropriate.

| 49 | Concept Questions (a) A series RCL circuit contains three elements—a resistor, a capacitor, and an inductor. Which of the elements, on average, consume(s) power? (b) How is the current in the circuit related to the generator voltage and the impedance of the circuit? (c) What is the impedance of the circuit at resonance? (d) At resonance, how is the average power consumed related to the generator voltage and the resistance?
Problem The resistor in a series RCL circuit has a resistance of , while the voltage of the generator is 3.0 V. At resonance, what is the average power delivered to the circuit?

| 50 | Concept Questions The capacitance in a series RCL circuit is C1. The generator in the circuit has a fixed frequency that is less than the resonant frequency of the circuit. By adding a second capacitance C2 to the circuit, it is desired to reduce the resonant frequency so that it matches the generator frequency. (a) To reduce the resonant frequency, should the circuit capacitance be increased or decreased? (b) Should C2 be added in parallel or in series with C1? Give your reasoning.
Problem The initial circuit capacitance is , and the corresponding resonant frequency is . The generator frequency is 5.60 kHz. What is the value of the capacitance C2 that should be added so that the circuit will have a resonant frequency that matches the generator frequency?

| 51 | Concept Questions Part a of the drawing shows a resistor and a charged capacitor wired in series. When the switch is closed, the capacitor discharges as charge moves from one plate to the other. Part b shows a plot of the amount of charge remaining on each plate of the capacitor as a function of time. (a) What does the time constant τ of this resistor-capacitor circuit physically represent? (b) How is the time constant related to the resistance R and the capacitance C? (c) In part c of the drawing, the switch has been removed and an ac generator has been inserted into the circuit. What is the impedance Z of this circuit? Express your answer in terms of the resistance R, the time constant , and the frequency f of the generator.
Problem The circuit elements in the drawing have the following values: , for the generator, and . The time constant for the circuit is . What is the rms current in the circuit?

| 52 | Concept Questions A charged capacitor and an inductor are connected as shown in the drawing (this circuit is the same as that in Figure 23-16a). There is no resistance in the circuit. As Section 23.4 discusses, the electrical energy initially present in the charged capacitor then oscillates back and forth between the inductor and the capacitor. (a) What is the amount of electrical energy initially stored in the capacitor? Express your answer in terms of its capacitance C and the magnitude q of the charge on each plate. (b) A little later, this energy is transferred completely to the inductor (see Figure 23-16b). Write down an expression for the energy stored in the inductor. Give your answer in terms of its inductance L and the magnitude Imax of the maximum current in the inductor. (c) If values for q, C, and L are known, how could one obtain a value for the maximum current in the inductor? Remember that energy is conserved.
Problem The initial charge on the capacitor has a magnitude of . The capacitance is , and the inductance is . (a) What is the electrical energy stored initially in the charged capacitor? (b) Find the maximum current in the inductor.

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