898 CHAPTER 2 8 Direct Current Circuits Section 28.4 RCCircuits WEB 29. Consider a series RC circuit (see Fig. 28.16) for which R = 1.00 ~In, C = 5.00 J.LF, and e = 30.0 V. Find (a) the time constant ofthe circuit and (b) the maxi- mum charge on the capacitor after the switch is closed. (c) If the s"itch is closed at t = 0, find the current in the resistor 10.0 slater. liYe Dead 30. A 2.00-nF capacitor ith an initial charge of 5.10 J.LC is battery battery discharged through a 1.30-kfl resistor. (a) Calculate the current through the resistor 9.00 J.LS after the resistor is Figure P28.25 connected across the terminals of the capacitor. (b) What charge remains on the capacitor after 8.00 J.LS? For the network shown in Figure P28.26, show that the (c) What is the maximum current in the resistor? resistance R ab = 17 u. · 27 n 31. A fully charged capacitor stores energy U0 . How much energy remains when its charge has decreased to half its original Yalue? 32. In the circuit of Figure P28.32, switch S has been open for a long time. It is then suddenly closed. Determine the time constant (a) before the switch is closed and (b) after the switch is closed. (c) If the switch is closed at t = 0, determine the current through it as a function of time. Figure P28.26 50.0kQ ----~~.-----.-------------, ~For the circuit shnwn in Figure P28.27, calculate (a) the current in the 2.00-!1 resistor and (b) the potential dif- _ [ 10.0 Jls _jllO.O,uF ference between points a and b. · 1--------_ __________.l._l------ll~ lOOkQ Figure P28.32 b [[ The circuit shown in Figure P28.33 has been connected a for a long time. (a) vVhat is the voltage across the capac- itor? (b) If the battery is disconnected, how long does it take the capacitor to discharge to one-tenth its initial 8.00 v 6.00 Q voltage? Figure P28. 27 ~ywon -~ f~"" Calculate the power delivered to each of the resistors shown in Figure P28.28. 1o.ov- I /~ 2.0Q 1 4.00 ~/ 2.00 Q Figure P28. 33 50V T 4.on 4.on _ 20v ~~~~_j . 34. A 4.00-Mfl resistor and a 3.00-J.LF capacitor are con- nected in series with a 12.0-V power supply. (a) What is lL___________ _ L_ _ _ _ _ _ the time constant for the circuit? (b) Express the cur- rent in the circuit and the charge on the capacitor as Figure P28.28 functions of time.
Problems 1037 A current pulse is fed to the partial circuit shown in Fig- A ure P32.27. The current begins at zero, then becomes 10.0 A between t = 0 and t = 200 f.LS, and then is zero s once again. Determine the current in the inductor as a function of time. E L l(t) 10.0~ 200 flS R _____l, 10 0 --I(_t:_OO_Q---...lf ~I Figure P32.29 single ideal inductor having 1/ Leq = 1/ L1 + 1/L2. (c) Now consider two inductors L 1 and L 2 that have Figure P12.27 nonzero internal resistances R 1 and R2, respectively. As- sume that they are still far apart so that their magnetic fields do not influence each other. If these inductors 28. One application of an RL circuit is the generation of are connected in series, show that they are equivalent to time-varying high voltage from a low-voltage source, as a single inductor having Leq = L1 + L2 and Req = shown in Figure P32.28. (a) What is the current in the R 1 + R 2 . (d) If these same inductors are now con- circuit a long time after the Sitch has been in position nected in parallel, is it necessarily true that they are A? (b) Now the switch is thrown quickly from A to B. equivalent to a single ideal inductor having 1/ Leq = Compute the initial voltage across each resistor and the l/L 1 + l/L2and1/Req = 1/Rl + 1/R2?Explain inductor. (c) How much time elapses before the voltage your answer. across the inductor drops to 12.0 V? Section :52. ::5 Energy in a Magnetic Field A s 31. Calculate the energy associated with the magnetic field of a 200-turn solenoid in which a current of 1. 75 A pro- duces a flux of 3. 70 X 1o- 4 T · m 2 in each turn. 5 0 The magnetic field inside a superconducting solenoid is 4.50 T. The solenoid has an inner diameter of 6.20 em 12.0V 2.00H and a length of 26.0 em. Determine (a) the magnetic energy density in the field and (b) the energy stored in the magnetic field within the solenoid. f33J An air-core solenoid with 68 turns is 8.00 em long and has a diameter of 1.20 em. How much energy is stored 12.0 Q in its magnetic field when it carries a current of0.770 A? At t = 0, an emf of 500 Vis applied to a coil that has an Figure P32.28 inductance of 0.800 H and a resistance of 30.0 !1. (a) Find the energy stored in the magnetic field when ·wEB [I] A 140-mH inductor and a 4.90-!1 resistor are connected the current reaches half its maximum value. (b) Mter with a switch to a 6.00-V battery, as shown in Figure the emf is connected, how long does it take the current P32.29. (a) If the switch is thrown to the left (connect- to reach this value? ing the battery), how much time elapses before the cur- WEB~ On a clear day there is a 100-V/m vertical electric field rent reaches 220 rnA? (b) What is the current in the in- near the Earths surface. At the same place, the Earths ductor 10.0 s after the switch is closed? (c) Now the magnetic field has a magnitude of 0.500 X 10- 4 T. switch is quickly thrown from A to B. How much time Compute the energy densities of the two fields. elapses before the current falls to 160 rnA? 36. An RL circuit in which L = 4.00 Hand R = 5.00 !1 is ~ Consider two ideal inductors, L 1 and L 2 , that have zero connected to a 22.0-V battery at t = 0. (a) What energy internal resistance and are far apart, so that their mag- is stored in the inductor when the current is 0.500 A? netic fields do not influence each other. (a) If these in- (b) At what rate is energy being stored in the inductor ductors are connected in series, show that they are when I= 1.00 A? (c) Whatpower is being delivered t£i equivalent to a single ideal inductor having the circuit by the battery when I= 0.500 A? ~ Leq = L1 + L 2 . (b) If these same two inductors are 37. A 10.0-V battery, a 5.00-!1 resistor, and a 10.0-H inductor connected in parallel, show that they are equivalent to a are connected in series. Mter the current in the circuit
Problems 1039 The switch in Figure P32.54 is connected to point a for The capacitor initially has no charge. Determine (a) the a long time. After the switch is thrown to point b, what voltage across the inductor as a function of time, are (a) the frequency of oscillation of the LC circuit, (b) the voltage across the capacitor as a function of (b) the maximum charge that appears on the capacitor, time, and (c) the time when the energy stored in the (c) the maximum current in the inductor, and (d) the _ capacitor first exceeds that in the inductor. total energy the circuit possesses at t = 3.00 s? 62. All inductor having inductance Land a capacitor hav- ing capacitance C are connected in series. The current in the circuit increases linearly in time as described by I= Kt. The capacitor is initially uncharged. Determine (a) the voltage across the inductor as a function of time, (b) the voltage across the capacitor as a function of time, and (c) the time when the energy stored in theca- pacitor first exceeds that in the inductor. Figure P32. 54 .. ~:"A capacitor in a series LC circuit has an initial charge Q _ ~. and is being discharged. Find, in terms of Land C, thewes D An LC circuit like that illustrated in Figure 32.14 con- flux through each of the N turns in the coil, when the sists of a 3.30-H inductor and an 840-pF capacitor, ini- charge on the capacitor is Q/2. tially carrying a 105-(LC charge. At t = 0 the switch is 64. The toroid in Figure P32.64 consists of N turns and has thrown closed. Compute the following quantities at a rectangular cross-section. Its inner and outer radii are t = 2.00 ms: (a) the energy stored in the capacitor; a and b, respectively. (a) Show that (b) the energy stored in the inductor; (c) the total en- fLoN2h b ergy in the circuit. L= ln- 217 a (Optional) (b) Using this result, compute the self-inductance of a Section :52.6 The RLCCircuit 500-turn toroid for which a = 10.0 em, b = 12.0 em, 56. In Figure 32.19, let R = 7.60 !1, L = 2.20 mH, and and h = 1.00 em. (c) In Problem 14, an approximate C = 1.80 (LF. (a) Calculate the frequency of the damped formula for the inductance of a toroid with R >> r was oscillation of the circuit. (b) What is the critical resis- derived. To get a feel for the accuracy of that result, use tance? the expression in Problem 14 to compute the approxi- ~ Consider an LC circuit in which L = 500 mH and mate inductance of the toroid described in part (b). C = 0.100 (LF. (a) What is the resonant frequency w 0 ? Compare the result with the answer to part (b). (b) If a resistance of 1.00 k!l is introduced into this cir- cuit, what is the frequency of the (damped) oscillations? (c) What is the percent difference between the two fre- quencies? 58. Show that Equation 32.29 in the text is Kirchhoffs loop rule as applied to Figure 32.19. Electrical oscillations are initiated in a series circuit con- taining a ca:r,acitance C, inductance L, and reslstance R. (a) If R<< ~4L/C (weak damping), how much time elapses before the amplitude of the current oscillation Figure P32.64 falls off to 50.0% ofits initial value? (b) How long does it take the energy to decrease to 50.0% of its initial 65. (a) A flat circular coil does not really produce a uniform value? magnetic field in the area it encloses, but estimate the · self-inductance of a flat circular coil, with radius Rand N turns, by supposing that the field at its center is uniform ADDITIONAL PROBLEMS over its area. (b) A circuit on a laboratory table consists Initially, the capacitor in a series LC circuit is charged. A of a 1.5-V battery, a 270-!1 resistor, a switch, and three 30- switch is closed, allowing the capacitor to discharge, and cm-long cords connecting them. Suppose that the circuit after time t the energy stored in the capacitor is one- is arranged to be circular. Think of it as a flat coil with fourth its initial value. Determine L if Cis known. one turn. Compute the order of magnitude of its self- inductance and (c) of the time constant describing how A 1.00-mH inductor and a 1.00-(LF capacitor are con- fast the current increases when you close the switch. nected in series. The current in the circuit is described 66. A soft iron rod (0 m = 800 fLo) is used as the core of a by I= 20.0t, where tis in seconds and I is in amperes. solenoid. The rod has a diameter of 24.0 mm and is
Problems 1069 4. In the simple ac circuit shotvn in Figure 33.1, R = 70.0 0 WEB 11. For the circuit shmm in Figure 33.4, ~ vmax = 80.0 V, and ~v = ~ Vmax sin wt. (a) If t:..vR = 0.250 ~ Vmax for the w = 65.011 rad/s, and L = 70.0 mH. Calculate the cur- first time at t = 0.010 0 s, what is the angular frequency rent in the inductor at t = 15.5 ms. of the generator? (b) What is the next value oft for 12. A 20.0-mH inductor is connected to a standard outlet which ~VR = 0.250~ Vmax? (t:.. Vnns = 120 V, J= 60.0 Hz). Determine the energy ~The current in the circuit shown in Figure 33.1 equals stored in the inductor at t = (1/180) s, assuming that 60.0% of the peak current at t = 7.00 ms. Vvbat is the this energy is zero at t = 0. ·smallest frequency of the generator that gives this cur- Review Problem. Determine the maximum magnetic rent? flux through an inductor connected to a standard out- 6. Figure P33.6 shows three lamps connected to a 120-Vac let (~Vrms = 120V, J= 60.0 Hz). (rms) household supply voltage. Lamps 1 and 2 have 150-W bulbs; lamp 3 has a 100-W bulb. Find the rms Section 33.4 Capacitors in an ac Circuit current and the resistance of each bulb. 14. (a) For what frequencies does a 22.0-p,F capacitor have a reactance below 175 0? (b) Over this same frequency Lamp Lamp Lamp range, what is the reactance of a 44.0-p,F capacitor? 1 2 3 15. What maximl1m current is delivered by a 2.20-p,F capac- itor when it is connected across (a) a North American outlet having ~Vrms = 120 V and f= 60.0 Hz? (b) a Eu- ropean outlet having ~ Vrms = 240 V and f = 50.0 Hz? 16. A capacitor Cis connected to a power supply that oper- ates at a frequency Jand produces an rms voltage t:.. V. What is the maximum charge that appears on either of the capacitor plates? l!1.J What maximum current is delivered by an ac generator Figure P11.6 with ~ Vmax = 48.0 V and f = 90.0 Hz when it is con- nected across a 3.70-p,F capacitor? 7. An audio amplifier, represented by the ac source and 18. A 1.00-mF capacitor is connected to a standard outlet resistor in Figure P33.7, delivers to the speaker alternat- (~ Vrms = 120 V, J= 60.0 Hz). Determine the current ing voltage at audio frequencies. If the source voltage in the capacitor at t == (1/180) s, assuming that at t = 0 has an amplitude of 15.0 V, R = 8.20 0, and the the energy stored in the capacitor is zero. speaker is equivalent to a resistance of 10.4 0, what time-averaged power is transferred to it? Section 33.5 The RLC Series Circuit [ill An inductor (L = 400 mH), a capacitor ( C = 4.43 p,F), and a resistor (R = 500 0) are connected in series. R A 50.0-Hz ac generator produces a peak current of ~ 250 rnA in the circuit. (a) Calculate the required peak voltage~ Vmax· (b) Determine the phase angle by which the current leads or lags the applied voltage. 20. At what frequency does the inductive reactance of a 57.0-p,H inductor equal the capacitive reactance of a 57.0-p,F capacitor? Figure P11. 7 21. A series ac circuit contains the following components: R = 150 0, L = 250 mH, C = 2.00 p,F, and a generatorSection 33.3 Inductors in an ac Circuit with~ Vmax = 210 V operating at 50.0 Hz. Calculate the 8. An inductor is connected to a 20.0-Hz powe~ply (a) inductive reactance, (b) capacitive reactance, that produces a 50.0-V rms voltage. What inductance is (c) impedance, (d) maximum current, and (e) phase needed to keep the instantaneous current in the circuit angle between current and generator voltage. below 80.0 rnA? 22. Asinusoidalvoltage~v(t) = (40.0V) sin(100t) is illJIn a purely inductive ac circuit, such as that shown in applied to a series RLC circuit with L = 160 mH, Figure 33.4, ~ Vmax = 100 V. (a) If the maximum cur- C = 99.0 p,F, and R = 68.0 0. (a) What is the imped- rent is 7.50 A at 50.0 Hz, what is the inductance L? ance of the circuit? (b) What is the maximum current? (b) At what angular frequency w is the maximum cur- (c) Determine the numerical values for Imax• w, and¢ rent 2.50 A? in the equation i(t) =/max sin(wt- ¢). 10. An inductor has a 54.0-0 reactance at 60.0 Hz. What is WEB~ An RLCcircuit consists of a 150-0 resistor, a 21.0-p,F ca- the maximum current when this inductor is connected pacitor, and a 460-mH inductor, connected in series to a 50.0-Hz source that produces a 100-Vrms voltage? with a 120-V, 60.0-Hz power supply. (a) What is the
Problems 1071 R, the average power delivered to the diode circuit shown in Figure P33.35. 43.1 A transformer has N 1 = 350 turns and N 2 = 2 000 turns. If the input voltage is ~ v( t) = (170 V) cos w t, what rms voltage is developed across the secondary coil? Diode / 44. A step-up transformer is designed to have an output voltage of 2 200 V (rms) when the primary is connected 2R across a 110-V (rms) source. (a) If there are 80 turns on the primary winding, how many turns are required on the secondary? (b) If a load resistor across the sec- ondary draws a current of 1.50 A, what is the current in / Diode the primary under ideal conditions? (c) If the trans- former actually has an efficiency of 95.0%, what is the L _ - - - - - - - 1 ......._, 1--------_j current in the primary when the secondary current is ~v 1.20 A? Figure P33.35 In the transformer shown in Figure P33.45, the load re- sistor is 50.0 !1. The turns ratio N 1 : N 2 is 5: 2, and the source voltage is 80.0 V (rms). If a voltmeter across theSection :5:5. 7 Resonance in a Series RLC Circuit load measures 25.0 V (rms), what is the source resis- 36. The tuning circuit of an AM radio contains an LC com- tance Rs? bination. The inductance is 0.200 mH, and the capaci- tor is variable, so the circuit can resonate at any fre- quency between 550 kHz and 1 650 kHz. Find the range of values requiredfor C. ~ An RLC circuit is used in a radio to tune in to an FM station broadcasting at 99.7 MHz. The resistance in the circuit is 12.0 !1, and the inductance is 1.40 pH. What capacitance should be used? I L __________ I _j 38. A series RLC circuit has the following values: L = 20.0 mH, C = 100 nF, R = 20.0 !1, and~ Vmax = 100 V, Figure P33.45 with ~v = ~ Vrnax sin wt. Find (a) the resonant frequency, (b) the amplitude of the current at the resonant fre- 45. The secondary voltage of an ignition transformer in a quency, (c) the Qofthe circuit, and (d) the amplitude of furnace is 10.0 kV. When the primary operates at an rms the voltage across the inductor at resonance. voltage of 120 V, the primary impedance is 24.0 !1 and 3!}. A 10.0-!1 resistor, a 10.0-mH inductor, and a 100-p,F ca- the transformer is 90.0% efficient. (a) What turns ratio pacitor are connected in series to a 50.0-V (rms) source is required? What are (b) the current in the secondary and (c) the impedance in the secondary? having variable frequency. What is the energy delivered to the circuit during one period if the operating fre- 4 7. A transmission line that has a resistance per unit length quency is twice the resonance frequency? of 4.50 X 10- 4 !11m is to be used to transmit 5.00 MW 4(L A resistor R, an inductor L, and a capacitor Care con- over 400 mi (6.44 X 105 m). The output voltage of the nected in series to an ac source of rms voltage ~ V and generator is 4.50 kV. (a) What is the power loss if a variable frequency. What is the energy delivered to the transformer is used to step up the voltage to 500 kV? circuit during one period if the operating frequency is (b) What fraction of the input power is lost to the line twice the resonance frequency? under these circumstances? (c) What difficulties would · be encountered on attempting to transmit the 5.00 MW Compute the quality factor for the circuits described in at the generator voltage of 4.50 kV? Problems 22 and 23. Which circuit has ~sharper reso- nance? (Optional) Section :5:5.9 Rectifiers and FiltersSection :5:5.8 The Transformer and Power Transmission 48. The RC low-pass filter shown in Figure 33.23 has a resis- 42. A step-down transformer is used for recharging the bat- tance R = 90.0 !1 and a capacitance C = 8.00 nF. Calcu- teries of portable devices such as tape players. The turns late the gain (~ vout I~ Yin) for input frequencies of ratio inside the transformer is 13:1, and it is used with (a) 600Hz and (b) 600kHz. 120-V (rms) household service. If a particular ideal wes ~ The RC high-pass filter shown in Figure 33.22 has are- transformer draws 0.350 A from the house outlet, what sistance R = 0.500 !1. (a) What capacitance gives an (a) voltage and (b) current are supplied to a tape output signal that has one-half the amplitude of a 300- player from the transformer? (c) How much power is Hz input signal? (b) What is the gain ( ~ Vout I~ Yin) for delivered? a 600-Hz signal?