Solved: An ac voltage source V = Vosi11^ + 90) is

How would you arrange two fla t circular coils so that their mutual inductance was (a) greatest, (b) least (without separating them by a great distance)?

Suppose the second coil of N2 turns in Fig. 30-2 were moved so it was near the end of the solenoid. How would this affect the mutual inductance?

Would two coils with mutual inductance also have selfinductance? Explain.

Is the energy density inside a solenoid greatest near the ends of the solenoid or near its center?

I f you are given a fixed length of wire, how would you shape it to obtain the greatest self-inductance? The least?

Does the emf of the battery in Fig. 30-6a affect the time needed for the LR circuit to reach (a) a given fraction of its maximum possible current, (b ) a given value of current?

A circuit with large inductive time constant carries a steady current. I f a switch is opened, there can be a very large (and sometimes dangerous) spark or arcing over. Explain.

A t the instant the battery is connected into the LR circuit of Fig. 30-6a, the emf in the inductor has its maximum value even though the current is zero. Explain.

What keeps an LC circuit oscillating even after the capacito r has discharged completely?

Is the ac current in the inductor always the same as the current in the resistor of the LRC circuit of Fig. 30-13?

When an ac generator is connected to an LRC circuit, where does the energy come from ultimately? Where does it go? How do the values o f L , C, and R affect the energy supplied by the generator?

In an ac LRC circuit, if X L > X c , the circuit is said to be predominantly inductive. And if X c > X L , the circuit is said to be predominantly capacitive. Discuss the reasons for these terms. In particular, do they say anything about the relative values of L and C at a given frequency?

Do the results of Section 30-8 approach the proper expected results when co approaches zero? What are the expected results?

Under what conditions is the impedance in an LR C circuit a minimum?

Is it possible for the instantaneous power output of an ac generator

In an ac LRC circuit, does the power factor, cos <fi, depend on frequency? Does the power dissipated depend on frequency?

Describe briefly how the frequency of the source emf affects the impedance of (a) a pure resistance, (b ) a pure capacitance, (c) a pure inductance, (d) an LR C circuit near resonance (R small), (e) an LRC circuit far from resonance (R small).

Discuss the response of an LRC circuit as R > 0 when the frequency is (a) at resonance, (b) near resonance, (c) far from resonance. Is there energy dissipation in each case? Discuss the transformations of energy that occur in each case.

An LR C resonant circuit is often called an oscillator circuit. What is it that oscillates?

Compare the oscillations of an LRC circuit to the vibration of a mass m on a spring. What do L and C correspond to in the mechanical system?

(II) A long straight wire of radius R carries current I uniformly distributed across its cross-sectional area. Find the magnetic energy stored per unit length in the interior of this wire.

(II) A fter how many time constants does the current in Fig. 30-6 reach within (a) 5.0%, (b ) 1.0%, and (c) 0.10% of its maximum value?

(II) How many time constants does it take for the potential difference across the resistor in an LR circuit like that in Fig. 30-7 to drop to 3.0% of its original value?

(II) It takes 2.56 ms for the current in an LR circuit to increase from zero to 0.75 its maximum value. Determine (a) the time constant of the circuit, (b) the resistance of the circuit if L = 31.0 mH.

(II) (a) Determine the energy stored in the inductor L as a function of time for the LR circuit of Fig. 30-6a. (b) After how many time constants does the stored energy reach 99.9% of its maximum value?

(II) In the circuit of Fig. 30-27, determine the current in each resistor , I2, / 3) at the moment (a) the switch is closed, (b) a long time after the switch isclosed. A fter the switch has been closed for a long time, and then reopened, what is each current (c) just after it is opened, (d) after a long time?

(II) (a) In Fig. 30-28a, assume that the switch S has been in position A for sufficient time so that a steady current I0 = V0/R flows through the resistor R. A t time t = 0, the switch is quickly switched to position B and the current through R decays according to I = /0 e~^T. Show that the maximum emf %nax induced in the inductor during this time period equals the battery voltage Vo- (b ) In Fig. 30-28b, assume that the switch has been in position A for sufficient time so that a steady current 70 = V0/R flows through the resistor R. A t time t = 0, the switch is quickly switched to position B and the current decays through resistor R' (which is much greater than R) according to I = Ioe~t/T'. Show that the maximum emf %nax induced in the inductor during this time period is (R '/R )V 0. I f R ' = 55R and V0 = 120 V, determine ^nax. [When a mechanical switch is opened, a high-resistance air gap is created, which is modeled as R r here. This Problem illustrates why high-voltage sparking can occur if a currentcarrying inductor is suddenly cut o ff from its power source.]

(II) You want to turn on the current through a coil of selfinductance L in a controlled manner, so you place it in series with a resistor R = 2200 Cl, a switch, and a dc voltage source Vq = 240 V. A fte r closing the switch, you find that the current through the coil builds up to its steady-state value with a time constant r. You are pleased with the currents steady-state value, but want r to be half as long. What new values should you use for R and V0?

(II) A 12-V battery has been connected to an LR circuit for sufficient time so that a steady current flows through the resistor R = 2.2 k ll and inductor L = 18 mH. A t t = 0, the battery is removed from the circuit and the current decays exponentially through R. Determine the emf % across the inductor as time t increases. A t what time is % greatest and what is this maximum value (V)?

( I ll) Two tightly wound solenoids have the same length and circular cross-sectional area. But solenoid 1 uses wire that is 1.5 times as thick as solenoid 2. (a) What is the ratio of their inductances? (b) What is the ratio of their inductive time constants (assuming no other resistance in the circuits)?

(I) The variable capacitor in the tuner of an AM radio has a capacitance of 1350 pF when the radio is tuned to a station at 550 kHz. (a) What must be the capacitance for a station at 1600 kHz? (b) What is the inductance (assumed constant)? Ignore resistance.

(I) (a) I f the in itia l conditions of an LC circuit were I = 70 and <2 = 0 at t = 0, write Q as a function of time, (ib) Practically, how could you set up these initial conditions?

(II) In some experiments, short distances are measured by using capacitance. Consider forming an LC circuit using a parallel-plate capacitor with plate area A, and a known inductance L. (a) I f charge is found to oscillate in this circuit at frequency / = o i/lir when the capacitor plates are separated by distance x, show that x = 4ir2A e 0f 2L. (b) When the plate separation is changed by Ax, the circuits oscillation frequency w ill change by A f. Show that A x /x ~ 2(A/ / / ) . (c) I f / is on the order of 1 MHz and can be measured to a precision of A f = 1 Hz, with what percent accuracy can x be determined? Assume fringing effects at the capacitors edges can be neglected.

(II) A 425-pF capacitor is charged to 135 V and then quickly connected to a 175-mH inductor. Determine (a) the frequency of oscillation, (b) the peak value of the current, and (c) the maximum energy stored in the magnetic field of the inductor

(II) A t t = 0, let Q = Qo, and 1 = 0 in an L C circuit. (a) A t the first moment when the energy is shared equally by the inductor and the capacitor, what is the charge on the capacitor? (b) How much time has elapsed (in terms of the period T)1 30-6 LC Oscillations

(II) A damped LC circuit loses 3.5% of its electromagnetic energy per cycle to thermal energy. I f L = 65 mH and C = 1.00 /aF, what is the value of RI

II) In an oscillating LRC circuit, how much time does it take for the energy stored in the fields of the capacitor and inductor to fa ll to 75% of the in itia l value? (See Fig. 30-13; assume R V 4 L /C .)

Ill) How much resistance must be added to a pure LC circuit (L = 350 mH, C = 1800 pF) to change the oscillators frequency by 0.25%? W ill it be increased or decreased?

(I) A t what frequency w ill a 32.0-mH inductor have a reactance of 660 fl?

(I) What is the reactance of a 9.2-juF capacitor at a frequency of (a) 60.0 Hz, (b) 1.00 MHz?

I) Plot a graph of the reactance of a 1.0-/xF capacitor as a function of frequency from 10 Hz to 1000 Hz.

(I) Calculate the reactance of, and rms current in, a 36.0-mH radio coil connected to a 250-V (rms) 33.3-kHz ac line. Ignore resistance.

(II) A resistor R is in parallel with a capacitor C, and this parallel combination is in series with a resistor R ' . If connected to an ac voltage source of frequency co, what is the equivalent impedance of this circuit at the two extremes in frequency (a) co = 0, and (b) a> = oo?

(II) What is the inductance L of the primary of a transformer whose input is 110 V at 60 Hz and the current drawn is 3.1 A? Assume no current in the secondary.

(II) (a) What is the reactance of a 0.086-/xF capacitor connected to a 22-kV (rms), 660-Hz line? (b) Determine the frequency and the peak value of the current.

(II) A capacitor is placed in parallel with some device, B, as in Fig. 30-18b, to filte r out stray high-frequency signals, but to allow ordinary 60-Hz ac to pass through with little loss. Suppose that circuit B in Fig. 3 0 -18b is a resistance R = 490 H connected to ground, and that C = 0.35 /xF. What percent of the incoming current w ill pass through C rather than R if it is (a) 60 Hz; (b) 60,000 Hz?

(II) A current I = 1.80 cos 3111 (I in amps, t in seconds, and the angle is in radians) flows in a series LR circuit in which L = 3.85 mH and R = 1.35 k ft. What is the average power dissipation?

(I) A 10.0-kft resistor is in series with a 26.0-mH inductor and an ac source. Calculate the impedance of the circuit if the source frequency is (a) 55.0 Hz; (b ) 55,000 Hz.

(I) A 75-0 resistor and a 6.8-juF capacitor are connected in series to an ac source. Calculate the impedance of the circuit if the source frequency is (a) 60 Hz; (b) 6.0 MHz.

(I) For a 120-V, 60-Hz voltage, a current of 70mA passing through the body for 1.0 s could be lethal. What must be the impedance of the body for this to occur?

(II) A 2.5-kft resistor in series with a 420-mH inductor is driven by an ac power supply. A t what frequency is the impedance double that of the impedance at 60 Hz?

(II) (a) What is the rms current in a series RC circuit if R = 3.8 k ft, C = 0.80 ^iF, and the rms applied voltage is 120 V at 60.0 Hz? (b) What is the phase angle between voltage and current? (c) What is the power dissipated by the circuit? (id) What are the voltmeter readings across R and C?

II) An ac voltage source is connected in series with a 1.0-/zF capacitor and a 750-11 resistor. Using a digital ac voltmeter, the amplitude of the voltage source is measured to be 4.0 V rms, while the voltages across the resistor and across the capacitor are found to be 3.0 V rms and 2.7 V rms, respectively. Determine the frequency of the ac voltage source. Why is the voltage measured across the voltage source not equal to the sum of the voltages measured across the resistor and across the capacitor?

(II) Determine the total impedance, phase angle, and rms current in an LRC circuit connected to a 10.0-kHz, 725-V (rms) source if L = 32.0 mH, R = 8.70 kO, and C = 6250 pF.

(II) (a) What is the rms current in a series LR circuit when a 60.0-Hz, 120-V rms ac voltage is applied, where R = 965 O and L = 225 mH? (b) What is the phase angle between voltage and current? (c) How much power is dissipated? (d ) What are the rms voltage readings across R and L?

(II) A 35-mH inductor with 2.0-0 resistance is connected in series to a 26-/zF capacitor and a 60-Hz, 45-V (rms) source. Calculate (a) the rms current, (b) the phase angle, and (c) the power dissipated in this circuit.

(II) A 25-mH coil whose resistance is 0.80 0 is connected to a capacitor C and a 360-Hz source voltage. If the current and voltage are to be in phase, what value must C have?

(II) A 75-W lightbulb is designed to operate with an applied ac voltage of 120 V rms. The bulb is placed in series with an inductor L, and this series combination is then connected to a 60-Hz 240-V rms voltage source. For the bulb to operate properly, determine the required value for L. Assume the bulb has resistance R and negligible inductance.

(II) In the LRC circuit of Fig. 30-19, suppose I = / 0 sin cot and V = Vosin(&jf + 4>)- Determine the instantaneous power dissipated in the circuit from P = IV using these equations and show that on the average, P = jV 0/0 cos </>, which confirms Eq. 30-30.

(II) An LRC series circuit with R = 150 O, L = 25 mH, and C = 2.0/zF is powered by an ac voltage source of peak voltage V0 = 340 V and frequency / = 660 Hz. (a) Determine the peak current that flows in this circuit. (b) Determine the phase angle of the source voltage relative to the current, (c) Determine the peak voltage across R and its phase angle relative to the source voltage, (d) Determine the peak voltage across L and its phase angle relative to the source voltage. (e) Determine the peak voltage across C and its phase angle relative to the source voltage.

(II) An LR circuit can be used as a phase shifter. Assume that an input source voltage V = V0 sin(2'irft + <J>) is connected across a series combination of an inductor L = 55 mH and resistor R. The output of this circuit is taken across the resistor. I f V0 = 24 V and / = 175 Hz, determine the value of R so that the output voltage VR lags the input voltage V by 25. Compare (as a ratio) the peak output voltage with Vq .

(I) A 3800-pF capacitor is connected in series to a 26.0-/aH coil of resistance 2.00 O. What is the resonant frequency of this circuit?

(I) What is the resonant frequency of the LRC circuit of Example 30-11? A t what rate is energy taken from the generator, on the average, at this frequency?

(II) An LRC circuit has L = 4.15 mH and R = 3.80 kO. (a) What value must C have to produce resonance at 33.0 kHz? (b) What w ill be the maximum current at resonance if the peak external voltage is 136 V?

(II) The frequency of the ac voltage source (peak voltage Vq) in an LRC circuit is tuned to the circuits resonant frequency /o = 1 /( 2 'ttV lC ) . (a) Show that the peak voltage across the capacitor is VCo = V0 Tq/2 tt t) , where T0(= l / / 0) is the period of the resonant frequency and r = RC is the time constant for charging the capacitor C through a resistor R. (b) Define (3 = T0/( 2 ttt) s o that VCQ = (3Vq. Then /3 is the amplification of the source voltage across the capacitor. I f a particular LRC circuit contains a 2.0-nF capacitor and has a resonant frequency of 5.0 kHz, what value of R w ill yield p = 125?

(II) Capacitors made from piezoelectric materials are commonly used as sound transducers ( speakers ). They often require a large operating voltage. One method for providing the required voltage is to include the speaker as part of an LRC circuit as shown in Fig. 30-29, where the speaker is modeled electrically as the capacitance C = 1.0 nF. Take R = 35 0 and L = 55 mH. (a) What is the resonant frequency / 0 for this circuit? (b) I f the voltage source has peak amplitude 66. V 0 = 2.0 V at frequency f = fo, find the peak voltage VCo across the speaker (i.e., the capacitor C). (c) Determine the ratio VCo/V q.

(II) (a) Determine a formula for the average power P dissipated in an LRC circuit in terms of L, R, C, co, and V0. (b) A t what frequency is the power a maximum? (c) Find an approximate formula for the width of the resonance peak in average power, A co, which is the difference in the two (angular) frequencies where P has half its maximum value. Assume a sharp peak.

(II) (a) Show that oscillation of charge Q on the capacitor of an LR C circuit has amplitude (b) A t what angular frequency, co', w ill Q0 be a maximum? (c) Compare to a forced damped harmonic oscillator (Chapter 14), and discuss. (See also Question 20 in this Chapter.)

(II) A resonant circuit using a 220-nF capacitor is to resonate at 18.0 kHz. The air-core inductor is to be a solenoid with closely packed coils made from 12.0 m of insulated wire 1.1 mm in diameter. How many loops w ill the inductor contain?

(II) The output of an electrocardiogram amplifier has an impedance of 45 kO. It is to be connected to an 8.0-0 loudspeaker through a transformer. What should be the turns ratio of the transformer?

A 2200-pF capacitor is charged to 120 V and then quickly connected to an inductor. The frequency of oscillation is observed to be 17 kHz. Determine (a) the inductance, (b) the peak value of the current, and (c) the maximum energy stored in the magnetic field of the inductor.

A t t = 0, the current through a 60.0-mH inductor is 50.0 mA and is increasing at the rate of 78.0 mA/s. What is the initial energy stored in the inductor, and how long does it take for the energy to increase by a factor of 5.0 from the initial value?

A t time t = 0, the switch in the circuit shown in Fig. 30-30 is closed. A fter a sufficiently long time, steady currents I i , I2, and / 3 flow through resistors R l t R2, and R3, respectively. Determine these three currents.

(a) Show that the self-inductance L of a toroid (Fig. 30-31) of radius r0 containing N loops each of diameter d is fiQ N 2d 2 L ------ 8/q if r0 >5> d. Assume the field is uniform inside the toroid; is this actually true? Is this result consistent with L for a solenoid? Should it be? (b) Calculate the inductance L of a large toroid if the diameter of the coils is 2.0 cm and the diameter of the whole ring is 66 cm. Assume the field inside the toroid is uniform. There are a total of 550 loops of wire.

A pair of straight parallel thin wires, such as a lamp cord, each of radius r, are a distance apart and carry current to a circuit some distance away. Ignoring the field within each wire, show that the inductance per unit length is ( f x j7 r)ln [(f - r ) /r \.

Assuming the Earths magnetic field averages about 0.50 X 10_4T near the surface of the Earth, estimate the total energy stored in this field in the first 5.0 km above the Earths surface.

(a) For an underdamped LRC circuit, determine a formula for the energy U = UE + UB stored in the electric and magnetic fields as a function of time. Give answer in terms of the in itia l charge Q0 on the capacitor, (b) Show how dU /d t is related to the rate energy is transformed in the resistor, I 2R.

An electronic device needs to be protected against sudden surges in current. In particular, after the power is turned on the current should rise to no more than 7.5 mA in the first 75 /xs. The device has resistance 15012 and is designed to operate at 33 mA. How would you protect this device?

The circuit shown in Fig. 30-32a can integrate (in the calculus sense) the input voltage Vin, if the time constant L /R is large compared with the time during which V\n varies. Explain how this integrator works and sketch its output for the square wave signal input shown in Fig. 30-32b. [Hint: Write Kirchhoffs loop rule for the circuit. M ultiply each term in this differential equation (in I) by a factor eRtlL to make it easier to integrate.]

Suppose circuit B in Fig. 30-18a consists of a resistance R = 550 H. The filte r capacitor has capacitance C = 1.2 /xF. W ill this capacitor act to eliminate 6.0-Hz ac but pass a high-frequency signal o f frequency 6.0 kHz? To check this, determine the voltage drop across R for a 130-mV signal of frequency (a) 60 Hz; (b ) 6.0 kHz.

An ac voltage source V = Vosi11^ + 90) is connected across an inductor L and current I = 70 sin (cot) flows in this circuit. Note that the current and source voltage are 90 out of phase, (a) Directly calculate the average power delivered by the source_over one period T of its sinusoidal cycle via the integral P = J ^V I d t/T . (b) Apply the relation P = ^rms Kmscos 4> t this circuit and show that the answer you obtain is consistent with that found in part (a). Comment on your results.

A circuit contains two elements, but it is not known if they are L , R, or C. The current in this circuit when connected to a 120-V 60-Hz source is 5.6 A and lags the voltage by 65. What are the two elements and what are their values?

A 3.5-kIl resistor in series with a 440-mH inductor is driven by an ac power supply. A t what frequency is the impedance double that of the impedance at 60 Hz?

(a) What is the rms current in an RC circuit if R = 5.70 k ft, C = 1.80 /jlF, and the rms applied voltage is 120 V at 60.0 Hz? (b ) What is the phase angle between voltage and current? (c) What is the power dissipated by the circuit? (d) What are the voltmeter readings across R and C?

An inductance coil draws 2.5 A dc when connected to a 45-V battery. When connected to a 60-Hz 120-V (rms) source, the current drawn is 3.8 A (rms). Determine the inductance and resistance of the coil.

The Q-value of a resonance circuit can be defined as the ratio of the voltage across the capacitor (or inductor) to the voltage across the resistor, at resonance. The larger the Q factor, the sharper the resonance curve w ill be and the sharper the tuning, (a) Show that the Q factor is given by the equation Q = (1 /R )'S /L /C . (b ) A t a resonant frequency / 0 = 1.0 MHz, what must be the value of L and R to produce a Q factor of 350? Assume that C = 0.010 /jlF.

Show that the fraction of electromagnetic energy lost (to thermal energy) per cycle in a lightly damped (R2 4L/C) LRC circuit is approximately A U _ 2ttR _ 1tt_ U ~ La) ~ <2 The quantity Q can be defined as Q = Lco/R, and is called the Q-value, or quality factor, of the circuit and is a measure of the damping present. A high Q-value means smaller damping and less energy input required to maintain oscillations.

In a series LRC circuit, the inductance is 33 mH, the capacitance is 55 nF, and the resistance is 1.50 k ft. A t what frequencies is the power factor equal to 0.17?

In our analysis of a series LRC circuit, Fig. 30-19, suppose we chose V = Vo sin cot. (a) Construct a phasor diagram, like that of Fig. 30-21, for this case, (b) Write a formula for the current /, defining all terms.

A voltage V = 0.95 sin 754? is applied to an LRC circuit (I is in amperes, t is in seconds, V is in volts, and the angle is in radians) which has L = 22.0 mH, R = 23.2 k ft, and C = 0.42 fiF. (a) What is the impedance and phase angle? (b) How much power is dissipated in the circuit? (c) What is the rms current and voltage across each element?

Filter circuit. Figure 30-33 shows a simple filte r circuit designed to pass dc voltages with minimal attenuation and to remove, as much as possible, any ac components (such as 60-Hz line voltage that could cause hum in a stereo receiver, for example). Assume V[n = V1 + V2 where V1 is dc and V2 = V20 sin cot, and that any resistance is very small, (a) Determine the current through the capacitor: give amplitude and phase (assume R = 0 and X L > X c). (b) Show that the ac component of the output voltage, V2out5 equals (Q/C) V i, where Q is the charge on the capacitor at any instant, and determine the amplitude and phase of V2 out (c) Show that the attenuation of the ac voltage is greatest when X c X L, and calculate the ratio of the output to input ac voltage in this case, (id) Compare the dc output voltage to input voltage.

Show that if the inductor L in the filte r circuit of Fig. 30-33 (Problem 91) is replaced by a large resistor R, there w ill still be significant attenuation of the ac voltage and little attenuation of the dc voltage if the input dc voltage is high and the current (and power) are low.

A resistor R, capacitor C, and inductor L are connected in parallel across an ac generator as shown in Fig. 30-34. The source emf is V = Vq sin cot. Determine the current as a function of time (including amplitude and phase): (a) in the resistor, (b ) in the inductor, (c) in the capacitor. (d ) What is the total current leaving the source? (Give amplitude Iq and phase.) (e) Determine the impedance Z defined as z = Vq/ I q. ( /) What is the power factor?

Suppose a series LR C circuit has two resistors, R i and R2, two capacitors, C\ and C2, and two inductors, L i and L 2 , all in series. Calculate the total impedance of the circuit.

Determine the inductance L of the primary of a transformer whose input is 220 V at 60 Hz when the current drawn is 4.3 A. Assume no current in the secondary.

In a plasma globe, a hollow glass sphere is filled with low-pressure gas and a small spherical metal electrode is located at its center. Assume an ac voltage source of peak voltage V0 and frequency / is applied between the metal sphere and the ground, and that a person is touching the outer surface of the globe with a fingertip, whose approximate area is 1.0 cm2. The equivalent circuit for this situation is shown in Fig. 30-35, where R G and R P are the resistances of the gas and the person, respectively, and C is the capacitance formed by the gas, glass, and finger. (a) Determine C assuming it is a parallel-plate capacitor. The conductive gas and the persons fingertip form the opposing plates of area A = 1.0 cm2. The plates are separated by glass (dielectric constant K = 5.0) of thickness d = 2.0 mm. (b ) In a typical plasma globe, / = 12 kHz. Determine the reactance X c of C at this frequency in M ft. (c) The voltage may be Vq = 2500 V. With this high voltage, the dielectric strength of the gas is exceeded and the gas becomes ionized. In this plasma state, the gas emits light ( sparks ) and is highly conductive so that R q X c . Assuming also that R P X c , estimate the peak current that flows in the given circuit. Is this level of current dangerous? (d) If the plasma globe operated at / = 1.0 MHz, estimate the peak current that would flow in the given circuit. Is this level of Lowcurrent dangerous?

You have a small electromagnet that consumes 350 W from a residential circuit operating at 120 V at 60 Hz. Using your ac multimeter, you determine that the unit draws 4.0 A rms. What are the values of the inductance and the internal resistance?

An inductor L in series with a resistor R, driven by a sinusoidal voltage source, responds as described by the following differential equation: Vq sin cot = d l L + dt RI. Show that a current of the form I = I0 sin(<*tf - <f>) flows through the circuit by direct substitution into the differential equation. Determine the amplitude of the current (/0) and the phase difference (f> between the current and the voltage source.

In a certain LRC series circuit, when the ac voltage source has a particular frequency / , the peak voltage across the inductor is 6.0 times greater than the peak voltage across the capacitor. Determine / in terms of the resonant frequency fQ of this circuit.

For the circuit shown in Fig. 30-36, V = VQ sin cot. Calculate the current in each element of the circuit, as well as the total impedance. [Hint: Try a trial solution of the form / = I0 sin(cot + <f>) for the current leaving the source.]

To detect vehicles at traffic lights, wire loops with dimensions on the order of 2 m are often buried horizontally under roadways. Assume the self-inductance of such a loop is L = 5.0 mH and that it is part of an LRC circuit as shown in Fig. 30-37 with C = 0.10 /xF and R = 45 O. The ac voltage has frequency / and rms voltage Vrms. (a) The frequency / is chosen to match the resonant frequency / 0 of the circuit. Find / 0 and determine what the rms voltage (VR)Tms across the resistor w ill be when / = / 0. (b) Assume that f, C, and R never change, but that, when a car is located above the buried loop, the loops self-inductance decreases by 10% (due to induced eddy currents in the cars metal parts). Determine by what factor the voltage (VR)Tms decreases in this situation in comparison to no car above the loop. [Monitoring (VR)rms detects the presence of a car.]

For the circuit shown in Fig. 30-38, show that if the condition R \R 2 = L/C is satisfied then the potential difference between points a and b is zero for all frequencies.

(II) The RC circuit shown in Fig. 30-39 is called a low-pass filter because it passes low-frequency ac signals with less attenuation than high-frequency ac signals, (a) Show that the voltage gain is A = Vout/Kjn = 1/(47T2f 2R2C2 + l ) i (b) Discuss the behavior of the gain A for / > 0 and / oo. (c) Choose R = 8500 and C = 1.0 X 10_6F, and graph log A versus log / with suitable scales to show the behavior of the circuit at low and high frequencies.

(II) The RC circuit shown in Fig. 30-40 is called a high-pass filter because it passes high-frequency ac signals with less attenuation than low-frequency ac signals, (a) Show that the voltage gain is A = Vouj V m = 2TrfRC/(4'jr2f 2R 2C2 + l) l. (b) Discuss the behavior of the gain A for / 0 and / oo. (c) Choose R = 850 0 and C = 1.0 X 10_6F , and then graph log A versus log / with suitable scales to show the behavior of the circuit at high and low frequencies.

( I ll) Write a computer program or use a spreadsheet program to plot / rms for an ac LRC circuit with a sinusoidal voltage source (Fig. 30-19) with = 0.100 V. For L = 50 /jlH and C = 50 /jlF , plot the / rms graph for (a) R = 0.10 O, and (b) R = 1.0 O from co = 0.1w0 to co = 3.0w0 on the same graph.