|| For an $RC$ filter circuit, show that $V_{\text{R}}=V_{\text{C}}={\epsilon }_{\text{}0}\text{/}\sqrt{2}$ at $\omega ={\omega }_{\text{c}}$.

|| Show that Equation 32.27 for the phase angle $\varphi$ of a series $RLC$ circuit gives the correct result for a capacitor-only circuit.

| INT

1. What is the peak current supplied by the emf in FIGURE P32.42?

   FIGURE P32.42

   

   FIGURE P32.42 Full Alternative Text

2. What is the peak voltage across the $3.0\mu \text{F}$ capacitor?

||

1. For an $RC$ circuit, find an expression for the angular frequency at which $V_{\text{R}}=\frac{1}{2}{\epsilon }_{\text{}0}$.

2. What is $V_{\text{C}}$ at this frequency?

||

1. For an $RC$ circuit, find an expression for the angular frequency at which $V_{\text{C}}=\frac{1}{2}{\epsilon }_{\text{}0}$.

2. What is $V_{\text{R}}$ at this frequency?

|| INT A series $RC$ circuit is built with a $12\text{k}\text{Ω}$ resistor and a parallel-plate capacitor with 15-cm-diameter electrodes. A 12 V, 36 kHz source drives a peak current of 0.65 mA through the circuit. What is the spacing between the capacitor plates?

|| INT You have a resistor and a capacitor of unknown values. First, you charge the capacitor and discharge it through the resistor. By monitoring the capacitor voltage on an oscilloscope, you see that the voltage decays to half its initial value in 2.5 ms. You then use the resistor and capacitor to make a low-pass filter. What is the crossover frequency $f_{\text{c}}?$

|| FIGURE P32.47 shows a parallel $RC$ circuit.

1. Use a phasor-diagram analysis to find expressions for the peak currents $I_{\text{R}}$ and $I_{\text{C}}$.

   Hint: What do the resistor and capacitor have in common? Use that as the initial phasor.

2. Complete the phasor analysis by finding an expression for the peak emf current $I$.

FIGURE P32.47

|| The small transformers that power many consumer products produce a 12.0 V rms, 60 Hz emf. Design a circuit using resistors and capacitors that uses the transformer voltage as an input and produces a 6.0 V rms output that leads the input voltage by $45\text{°}$.

|| Use a phasor diagram to analyze the $RL$ circuit of FIGURE P32.49. In particular,

1. Find expressions for $I$, $V_{\text{R}}$, and $V_{\text{L}}$.

2. What is $V_{\text{R}}$ in the limits $\omega \to 0$ and $\omega \to \infty ?$

3. If the output is taken from the resistor, is this a low-pass or a high-pass filter? Explain.

4. Find an expression for the crossover frequency ${\omega }_{\text{c}}$.

FIGURE P32.49

||| INT A series $RL$ circuit is built with a $110\text{Ω}$ resistor and a 5.0-cm-long, 1.0-cm-diameter solenoid with 800 turns of wire. What is the peak magnetic flux through the solenoid if the circuit is driven by a 12 V, 5.0 kHz source?

|| A series $RLC$ circuit consists of a $75\text{Ω}$ resistor, a 0.12 H inductor, and a $30\mu \text{F}$ capacitor. It is attached to a 120 V/60 Hz power line. What are (a) the peak current $I$, (b) the phase angle $\varphi$, and (c) the average power dissipated?

|| A series $RLC$ circuit consists of a $25\text{Ω}$ resistor, a 0.10 H inductor, and a $100\mu \text{F}$ capacitor. It draws a 2.5 A rms current when attached to a 60 Hz source. What are (a) the emf ${\epsilon }_{\text{rms}}$, (b) the phase angle $\varphi$, and (c) the average power dissipated?

|| INT A series $RLC$ circuit consists of a $550\text{Ω}$ resistor, a 2.1 mH inductor, and a 550 nF capacitor. It is connected to a 50 V rms oscillating voltage source with an adjustable frequency. An oscillating magnetic field is observed at a point 2.5 mm from the center of one of the circuit wires, which is long and straight. What is the maximum magnetic field amplitude that can be generated?

| In FIGURE P32.54, what is the current supplied by the emf when (a) the frequency is very small and (b) the frequency is very large?

FIGURE P32.54

|| A series $RLC$ circuit consists of a $50\text{Ω}$ resistor, a 3.3 mH inductor, and a 480 nF capacitor. It is connected to a 5.0 kHz oscillator with a peak voltage of 5.0 V. What is the instantaneous current $i$ when

1. $\epsilon ={\epsilon }_{\text{}0}?$

2. $\epsilon =0\text{ V}$ and is decreasing?

|| The current lags the emf by $30\text{°}$ in a series $RLC$ circuit with ${\epsilon }_{\text{}0}=10\text{ V}$ and $R=50\text{Ω}$. What is the peak current through the circuit?

|| A series $RLC$ circuit consists of a $50\text{Ω}$ resistor, a 3.3 mH inductor, and a 480 nF capacitor. It is connected to a 3.0 kHz oscillator with a peak voltage of 5.0 V. What is the instantaneous emf $\epsilon$ when

1. $i=I?$

2. $i=0\text{A}$ and is decreasing?

3. $i=-I?$

|| Show that the power factor of a series $RLC$ circuit is $\text{cos}\text{}\varphi =R\text{/}Z$.

|| For a series $RLC$ circuit, show that

1. The peak current can be written $I=I_{\text{max}}\cos \varphi$.

2. The average power can be written $P_{\text{}\text{source}}=P_{\text{max}}{\cos }^2\varphi$.

|| The tuning circuit in an FM radio receiver is a series $RLC$ circuit with a $0.200\mu \text{H}$ inductor.

1. The receiver is tuned to a station at 104.3 MHz. What is the value of the capacitor in the tuning circuit?

2. FM radio stations are assigned frequencies every 0.2 MHz, but two nearby stations cannot use adjacent frequencies. What is the maximum resistance the tuning circuit can have if the peak current at a frequency of 103.9 MHz, the closest frequency that can be used by a nearby station, is to be no more than 0.10% of the peak current at 104.3 MHz? The radio is still tuned to 104.3 MHz, and you can assume the two stations have equal strength.

|| A television channel is assigned the frequency range from 54 MHz to 60 MHz. A series $RLC$ tuning circuit in a TV receiver resonates in the middle of this frequency range. The circuit uses a 16 pF capacitor.

1. What is the value of the inductor?

2. In order to function properly, the current throughout the frequency range must be at least 50% of the current at the resonance frequency. What is the minimum possible value of the circuit’s resistance?

| INT Lightbulbs labeled 40 W, 60 W, and 100 W are connected to a 120 V/60 Hz power line as shown in FIGURE P32.62. What is the rate at which energy is dissipated in each bulb?

FIGURE P32.62

| Commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf, three separate wires carry currents for the emfs ${\epsilon }_1={\epsilon }_{\text{}0}\text{ cos }\omega t\text{,}{\epsilon }_2={\epsilon }_{\text{}0}\text{ cos}\left(\omega t+120\text{°}\right)$, and ${\epsilon }_3={\epsilon }_{\text{}0}\text{ cos}\left(\omega t-120\text{°}\right)$ over three parallel wires, each of which supplies one-third of the power. This is why the long-distance transmission lines you see in the countryside have three wires. Suppose the transmission lines into a city supply a total of 450 MW of electric power, a realistic value.

1. What would be the rms current in each wire if the transmission voltage were ${\epsilon }_{\text{}0}=120\text{ V rms}?$

2. In fact, transformers are used to step the transmission-line voltage up to 500 kV rms. What is the current in each wire?

3. Big transformers are expensive. Why does the electric company use step-up transformers?

||| INT A generator consists of a 12-cm by 16-cm rectangular loop with 500 turns of wire spinning at 60 Hz in a 25 mT uniform magnetic field. The generator output is connected to a series $RC$ circuit consisting of a $120\text{Ω}$ resistor and a $35\mu F$ capacitor. What is the average power delivered to the circuit?

||| You’re the operator of a 15,000 V rms, 60 Hz electrical substation. When you get to work one day, you see that the station is delivering 6.0 MW of power with a power factor of 0.90.

1. What is the rms current leaving the station?

2. How much series capacitance should you add to bring the power factor up to 1.0?

3. How much power will the station then be delivering?

|| Commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf $\epsilon ={\epsilon }_{\text{}0}\text{ cos }\omega t$, three separate wires carry currents for the emfs ${\epsilon }_1={\epsilon }_{\text{}0}\text{ cos }\omega t\text{,}{\epsilon }_{\text{}2}={\epsilon }_{\text{}0}\text{ cos}\left(\omega t+120\text{°}\right)$, and ${\epsilon }_3={\epsilon }_{\text{}0}\text{ cos}\left(\omega t-120\text{°}\right)$. This is why the long-distance transmission lines you see in the countryside have three parallel wires, as do many distribution lines within a city.

1. Draw a phasor diagram showing phasors for all three phases of a three-phase emf.

2. Show that the sum of the three phases is zero, producing what is referred to as neutral. In single-phase electricity, provided by the familiar 120 V/60 Hz electric outlets in your home, one side of the outlet is neutral, as established at a nearby electrical substation. The other, called the hot side. is one of the three phases. (The round opening is connected to ground.)

3. Show that the potential difference between any two of the phases has the rms value $\sqrt{3}{\epsilon }_{\text{rms}}$, where ${\epsilon }_{\text{rms}}$ is the familiar single-phase rms voltage. Evaluate this potential difference for ${\epsilon }_{\text{rms}}=120\text{ V}$. Some high-power home appliances, especially electric clothes dryers and hot-water heaters, are designed to operate between two of the phases rather than between one phase and neutral. Heavy-duty industrial motors are designed to operate from all three phases, but full three-phase power is rare in residential or office use.

|| A motor attached to a 120 V/60 Hz power line draws an 8.0 A current. Its average energy dissipation is 800 W.

1. What is the power factor?

2. What is the rms resistor voltage?

3. What is the motor’s resistance?

4. How much series capacitance needs to be added to increase the power factor to 1.0?

||| FIGURE CP32.68 shows voltage and current graphs for a series $RLC$ circuit.

1. What is the resistance $R$?

2. If $L=200\mu \text{H,}$ what is the resonance frequency in Hz?

FIGURE CP32.68

|||

1. Show that the average power in a series $RLC$ circuit is

   $$P_{\text{avg}}=\frac{{\omega }^2({\epsilon }_{\text{rms}}){\text{}}^{2}R}{{\omega }^2{R}^2+L^2{\left({\omega }^2-{\omega }_0{\text{}}^2\right)}^2}$$

2. Prove that the energy dissipation is a maximum at $\omega ={\omega }_0$.

|||

1. Show that the peak inductor voltage in a series $RLC$ circuit is maximum at frequency

   $${\omega }_{\text{L}}={\left({\displaystyle \frac{1}{{{\omega }_0}^2}}-{\displaystyle \frac{1}{2}}{R}^2{C}^2\right)}^{-1\text{/}2}$$

2. A series $RLC$ circuit with ${\epsilon }_{\text{}0}=10.0\text{ V}$ consists of a $1.0\text{Ω}$ resistor, a $1.0\mu \text{H}$ inductor, and a $1.0\mu \text{F}$ capacitor. What is $V_{\text{L}}$ at $\omega ={\omega }_0$ and at $\omega ={\omega }_{\text{L}}?$

||| The telecommunication circuit shown in FIGURE CP32.71 has a parallel inductor and capacitor in series with a resistor.

1. Use a phasor diagram to show that the peak current through the resistor is

   $$I=\frac{{\epsilon }_0}{\sqrt{R^2+{\left(\frac{1}{X_{\text{L}}}-\frac{1}{X_{\text{C}}}\right)}^{-2}}}$$

   Hint: Start with the inductor phasor $v_{\text{L}}$.

2. What is $I$ in the limits $\omega \to 0$ and $\omega \to \infty ?$

3. What is the resonance frequency ${\omega }_0?$ What is $I$ at this frequency?

FIGURE CP32.71

||| Consider the parallel $RLC$ circuit shown in FIGURE CP32.72.

1. Show that the current drawn from the emf is

   $$I={\epsilon }_0\sqrt{{\displaystyle \frac{1}{R^2}}+{\left({\displaystyle \frac{1}{\omega L}}-\omega C\right)}^2}$$

   Hint: Start with a phasor that is common to all three circuit elements.

2. What is $I$ in the limits $\omega \to 0$ and $\omega \to \infty ?$

3. Find the frequency for which $I$ is a minimum.

4. Sketch a graph of $I$ versus $\omega$.

FIGURE CP32.72