898 CHAPTER 2 8 Direct Current Circuits

liYe battery

Dead battery

Figure P28.25

For the network shown in Figure P28.26, show that the · R 27 n resistance ab = 17 u.

Figure P28.26

~For the circuit shnwn in Figure P28.27, calculate (a) the current in the 2.00-!1 resistor and (b) the potential dif­ference between points a and b. ·

b

a

8.00 v 6.00 Q

Figure P28. 27

Calculate the power delivered to each of the resistors shown in Figure P28.28.

2.0Q

50V T 4.on 4.on _ 20v

lL_ __________ _L ______ ~~~~_j . Figure P28.28

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 maxi­mum 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.

30. A 2.00-nF capacitor '\ith an initial charge of 5.10 J.LC is discharged through a 1.30-kfl resistor. (a) Calculate the current through the resistor 9.00 J.LS after the resistor is connected across the terminals of the capacitor. (b) What charge remains on the capacitor after 8.00 J.LS? (c) What is the maximum current in the resistor?

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.

50.0kQ ----~~.-----.-------------,

_[ 10.0\ Jls _jllO.O,uF

1'--------_ __________.l._l------ll~ lOOkQ

Figure P28.32

[[[] The circuit shown in Figure P28.33 has been connected for a long time. (a) vVhat is the voltage across the capac­itor? (b) If the battery is disconnected, how long does it take the capacitor to discharge to one-tenth its initial voltage?

~ywon

-~ f~"'" 1o.ov- 'I /~ 1 4.00 ~/ 2.00 Q

Figure P28. 33

34. A 4.00-Mfl resistor and a 3.00-J.LF capacitor are con­nected in series with a 12.0-V power supply. (a) What is the time constant for the circuit? (b) Express the cur­rent in the circuit and the charge on the capacitor as functions of time.

A current pulse is fed to the partial circuit shown in Fig­ure P32.27. The current begins at zero, then becomes 10.0 A between t = 0 and t = 200 f.LS, and then is zero once again. Determine the current in the inductor as a function of time.

l(t)

10.0~ 200 flS

--I(_t:_OO_Q---...lf ___ __l, 10 0 ~I Figure P12.27

28. One application of an RL circuit is the generation of time-varying high voltage from a low-voltage source, as shown in Figure P32.28. (a) What is the current in the circuit a long time after the S\\itch has been in position A? (b) Now the switch is thrown quickly from A to B. Compute the initial voltage across each resistor and the inductor. (c) How much time elapses before the voltage across the inductor drops to 12.0 V?

A s

12.0V 2.00H

12.0 Q

Figure P32.28

wEB [I] A 140-mH inductor and a 4.90-!1 resistor are connected with a switch to a 6.00-V battery, as shown in Figure P32.29. (a) If the switch is thrown to the left (connect­ing the battery), how much time elapses before the cur­rent reaches 220 rnA? (b) What is the current in the in­ductor 10.0 s after the switch is closed? (c) Now the switch is quickly thrown from A to B. How much time elapses before the current falls to 160 rnA?

~ Consider two ideal inductors, L 1 and L 2 , that have zero

internal resistance and are far apart, so that their mag­netic fields do not influence each other. (a) If these in­ductors are connected in series, show that they are equivalent to a single ideal inductor having Leq = L1 + L2. (b) If these same two inductors are 'connected in parallel, show that they are equivalent to a

Problems 1037

A

s

E L

R

Figure P32.29

single ideal inductor having 1/ Leq = 1/ L1 + 1/ L2.

(c) Now consider two inductors L1 and L 2 that have nonzero internal resistances R 1 and R2, respectively. As­sume that they are still far apart so that their magnetic fields do not influence each other. If these inductors are connected in series, show that they are equivalent to a single inductor having Leq = L1 + L2 and Req =

R 1 + R 2 . (d) If these same inductors are now con­nected in parallel, is it necessarily true that they are equivalent to a single ideal inductor having 1/ Leq =

l/L1 + l/L2and1/Req = 1/Rl + 1/R2?Explain your answer.

Section :52. ::5 Energy in a Magnetic Field

31. Calculate the energy associated with the magnetic field of a 200-turn solenoid in which a current of 1. 75 A pro­duces a flux of 3. 70 X 1 o-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 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 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 inductance of 0.800 H and a resistance of 30.0 !1. (a) Find the energy stored in the magnetic field when · the current reaches half its maximum value. (b) Mter the emf is connected, how long does it take the current to reach this value?

WEB~ On a clear day there is a 100-V /m vertical electric field near the Earth's surface. At the same place, the Earth's magnetic field has a magnitude of 0.500 X 10-4 T. Compute the energy densities of the two fields.

36. An RL circuit in which L = 4.00 Hand R = 5.00 !1 is connected to a 22.0-V battery at t = 0. (a) What energy is stored in the inductor when the current is 0.500 A? (b) At what rate is energy being stored in the inductor when I= 1.00 A? (c) Whatpower is being delivered t£i the circuit by the battery when I= 0.500 A? ~

37. A 10.0-V battery, a 5.00-!1 resistor, and a 10.0-H inductor are connected in series. Mter the current in the circuit

The switch in Figure P32.54 is connected to point a for a long time. After the switch is thrown to point b, what are (a) the frequency of oscillation of the LC circuit, (b) the maximum charge that appears on the capacitor, (c) the maximum current in the inductor, and (d) the total energy the circuit possesses at t = 3.00 s?

Figure P32. 54

wes D An LC circuit like that illustrated in Figure 32.14 con­sists of a 3.30-H inductor and an 840-pF capacitor, ini­tially carrying a 105-(LC charge. At t = 0 the switch is thrown closed. Compute the following quantities at t = 2.00 ms: (a) the energy stored in the capacitor; (b) the energy stored in the inductor; (c) the total en­ergy in the circuit.

(Optional)

Section :52.6 The RLCCircuit

56. In Figure 32.19, let R = 7.60 !1, L = 2.20 mH, and C = 1.80 (LF. (a) Calculate the frequency of the damped oscillation of the circuit. (b) What is the critical resis­tance?

~ Consider an LC circuit in which L = 500 mH and C = 0.100 (LF. (a) What is the resonant frequency w 0 ? (b) If a resistance of 1.00 k!l is introduced into this cir­cuit, what is the frequency of the (damped) oscillations? (c) What is the percent difference between the two fre­quencies?

58. Show that Equation 32.29 in the text is Kirchhoff's loop rule as applied to Figure 32.19. Electrical oscillations are initiated in a series circuit con­taining 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 falls off to 50.0% ofits initial value? (b) How long does it take the energy to decrease to 50.0% of its initial value?

ADDITIONAL PROBLEMS

Initially, the capacitor in a series LC circuit is charged. A switch is closed, allowing the capacitor to discharge, and after time t the energy stored in the capacitor is one­fourth its initial value. Determine L if Cis known.

A 1.00-mH inductor and a 1.00-(LF capacitor are con­nected in series. The current in the circuit is described by I= 20.0t, where tis in seconds and I is in amperes.

Problems 1039

The capacitor initially has no charge. 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 the

_ capacitor first exceeds that in the inductor. 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 theca­pacitor first exceeds that in the inductor.

~':" \_ .. ~.

64.

65.

A capacitor in a series LC circuit has an initial charge Q and is being discharged. Find, in terms of Land C, the flux through each of the N turns in the coil, when the charge on the capacitor is Q/2. The toroid in Figure P32.64 consists of N turns and has a rectangular cross-section. Its inner and outer radii are a and b, respectively. (a) Show that

fLoN2h b L= ln-

217 a

(b) Using this result, compute the self-inductance of a 500-turn toroid for which a = 10.0 em, b = 12.0 em, and h = 1.00 em. (c) In Problem 14, an approximate formula for the inductance of a toroid with R >> r was derived. To get a feel for the accuracy of that result, use the expression in Problem 14 to compute the approxi­mate inductance of the toroid described in part (b). Compare the result with the answer to part (b).

Figure P32.64

(a) A flat circular coil does not really produce a uniform 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 over its area. (b) A circuit on a laboratory table consists of a 1.5-V battery, a 270-!1 resistor, a switch, and three 30-cm-long cords connecting them. Suppose that the circuit is arranged to be circular. Think of it as a flat coil with one turn. Compute the order of magnitude of its self­inductance and (c) of the time constant describing how fast the current increases when you close the switch.

66. A soft iron rod (0 m = 800 fLo) is used as the core of a solenoid. The rod has a diameter of 24.0 mm and is

4. In the simple ac circuit shotvn in Figure 33.1, R = 70.0 0 and ~v = ~ Vmax sin wt. (a) If t:..vR = 0.250 ~ Vmax for the first time at t = 0.010 0 s, what is the angular frequency of the generator? (b) What is the next value oft for which ~VR = 0.250~ Vmax?

~The current in the circuit shown in Figure 33.1 equals 60.0% of the peak current at t = 7.00 ms. Vvbat is the ·smallest frequency of the generator that gives this cur­rent?

6. Figure P33.6 shows three lamps connected to a 120-Vac (rms) household supply voltage. Lamps 1 and 2 have 150-W bulbs; lamp 3 has a 1 00-W bulb. Find the rms current and the resistance of each bulb.

Lamp Lamp Lamp 1 2 3

Figure P11.6

7. An audio amplifier, represented by the ac source and resistor in Figure P33.7, delivers to the speaker alternat­ing voltage at audio frequencies. If the source voltage has an amplitude of 15.0 V, R = 8.20 0, and the speaker is equivalent to a resistance of 10.4 0, what time-averaged power is transferred to it?

R

~

Figure P11. 7

Section 33.3 Inductors in an ac Circuit 8. An inductor is connected to a 20.0-Hz powe~ply

that produces a 50.0-V rms voltage. What inductance is needed to keep the instantaneous current in the circuit below 80.0 rnA?

illJ In a purely inductive ac circuit, such as that shown in Figure 33.4, ~ Vmax = 100 V. (a) If the maximum cur­rent is 7.50 A at 50.0 Hz, what is the inductance L? (b) At what angular frequency w is the maximum cur­rent 2.50 A?

10. An inductor has a 54.0-0 reactance at 60.0 Hz. What is the maximum current when this inductor is connected to a 50.0-Hz source that produces a 100-Vrms voltage?

Problems 1069

WEB 11. For the circuit shmm in Figure 33.4, ~ vmax = 80.0 V, w = 65.011' rad/s, and L = 70.0 mH. Calculate the cur­rent in the inductor at t = 15.5 ms.

12. A 20.0-mH inductor is connected to a standard outlet (t:.. Vnns = 120 V, J= 60.0 Hz). Determine the energy stored in the inductor at t = (1/180) s, assuming that this energy is zero at t = 0. Review Problem. Determine the maximum magnetic flux through an inductor connected to a standard out­let (~Vrms = 120V, J= 60.0 Hz).

Section 33.4 Capacitors in an ac Circuit 14. (a) For what frequencies does a 22.0-p,F capacitor have

a reactance below 175 0? (b) Over this same frequency range, what is the reactance of a 44.0-p,F capacitor?

15. What maximl1m current is delivered by a 2.20-p,F capac­itor when it is connected across (a) a North American outlet having ~Vrms = 120 V and f= 60.0 Hz? (b) a Eu­ropean outlet having ~ Vrms = 240 V and f = 50.0 Hz?

16. A capacitor Cis connected to a power supply that oper­ates 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 with ~ V max = 48.0 V and f = 90.0 Hz when it is con­nected across a 3.70-p,F capacitor?

18. A 1.00-mF capacitor is connected to a standard outlet (~ Vrms = 120 V, J= 60.0 Hz). Determine the current in the capacitor at t == (1/180) s, assuming that at t = 0 the energy stored in the capacitor is zero.

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. 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?

21. A series ac circuit contains the following components: R = 150 0, L = 250 mH, C = 2.00 p,F, and a generator with~ Vmax = 210 V operating at 50.0 Hz. Calculate the (a) inductive reactance, (b) capacitive reactance, (c) impedance, (d) maximum current, and (e) phase angle between current and generator voltage.

22. Asinusoidalvoltage~v(t) = (40.0V) sin(100t) is applied to a series RLC circuit with L = 160 mH, C = 99.0 p,F, and R = 68.0 0. (a) What is the imped­ance of the circuit? (b) What is the maximum current? (c) Determine the numerical values for Imax• w, and¢ in the equation i(t) =/max sin(wt- ¢).

WEB~ An RLCcircuit consists of a 150-0 resistor, a 21.0-p,F ca­pacitor, and a 460-mH inductor, connected in series with a 120-V, 60.0-Hz power supply. (a) What is the

R, the average power delivered to the diode circuit shown in Figure P33.35.

Diode /

/ Diode

2R

L_-------1 ......._, 1--------_j

~v

Figure P33.35

Section :5:5. 7 Resonance in a Series RLC Circuit

36. The tuning circuit of an AM radio contains an LC com­bination. The inductance is 0.200 mH, and the capaci­tor is variable, so the circuit can resonate at any fre­quency 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?

38. A series RLC circuit has the following values: L =

20.0 mH, C = 100 nF, R = 20.0 !1, and~ Vmax = 100 V, with ~v = ~ Vrnax sin wt. Find (a) the resonant frequency, (b) the amplitude of the current at the resonant fre­quency, (c) the Qofthe circuit, and (d) the amplitude of the voltage across the inductor at resonance.

3!}. A 10.0-!1 resistor, a 10.0-mH inductor, and a 100-p,F ca­pacitor are connected in series to a 50.0-V (rms) source having variable frequency. What is the energy delivered to the circuit during one period if the operating fre­quency is twice the resonance frequency?

4(L A resistor R, an inductor L, and a capacitor Care con­nected in series to an ac source of rms voltage ~ V and variable frequency. What is the energy delivered to the circuit during one period if the operating frequency is twice the resonance frequency?

Compute the quality factor for the circuits described in Problems 22 and 23. Which circuit has ~sharper reso­nance?

Section :5:5.8 The Transformer and Power Transmission

42. A step-down transformer is used for recharging the bat­teries of portable devices such as tape players. The turns ratio inside the transformer is 13:1, and it is used with 120-V (rms) household service. If a particular ideal transformer draws 0.350 A from the house outlet, what (a) voltage and (b) current are supplied to a tape player from the transformer? (c) How much power is delivered?

Problems 1071

'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?

44. A step-up transformer is designed to have an output voltage of 2 200 V ( rms) when the primary is connected 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 sec­ondary draws a current of 1.50 A, what is the current in the primary under ideal conditions? (c) If the trans­former actually has an efficiency of 95.0%, what is the current in the primary when the secondary current is 1.20 A? In the transformer shown in Figure P33.45, the load re­sistor 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 the load measures 25.0 V (rms), what is the source resis-tance Rs?

I I L __________ _j

Figure P33.45

45. The secondary voltage of an ignition transformer in a furnace is 10.0 kV. When the primary operates at an rms voltage of 120 V, the primary impedance is 24.0 !1 and the transformer is 90.0% efficient. (a) What turns ratio is required? What are (b) the current in the secondary and (c) the impedance in the secondary?

4 7. A transmission line that has a resistance per unit length of 4.50 X 10-4 !11m is to be used to transmit 5.00 MW over 400 mi (6.44 X 105 m). The output voltage of the generator is 4.50 kV. (a) What is the power loss if a transformer is used to step up the voltage to 500 kV? (b) What fraction of the input power is lost to the line under these circumstances? (c) What difficulties would · be encountered on attempting to transmit the 5.00 MW at the generator voltage of 4.50 kV?

(Optional)

Section :5:5.9 Rectifiers and Filters

48. The RC low-pass filter shown in Figure 33.23 has a resis­tance R = 90.0 !1 and a capacitance C = 8.00 nF. Calcu­late the gain (~ vout I~ Yin) for input frequencies of (a) 600Hz and (b) 600kHz.

wes ~ The RC high-pass filter shown in Figure 33.22 has are­sistance R = 0.500 !1. (a) What capacitance gives an output signal that has one-half the amplitude of a 300-Hz input signal? (b) What is the gain ( ~ Vout I~ Yin) for a 600-Hz signal?