766 CHAPTER 23 Circ uits

SUMMARY The goal of Chapter 23 has been to understand the fundamental physical principles that govern electric circuits.

GENERAL PRINCIPLES

Kirchhoff's loop law

For a closed loop:

Ass ign a directi on to the current.

Add potential differences arou nd the loop:

Kirchhoff's junction law

For ajunction:

IMPORTANT CONCEPTS

Series elements

A series connection has no j unction. The current in each element is the same.

Resistors in series can be reduced to an equivalent resistance:

~ Rcq= RI+ R2+ RJ+· R, R,

Capacitors in series can be reduced to an equivaJem capacitance:

-jHHf-c, c, c,

APPLICATIONS

RCcircuits

The discharge of a capacitor through a resi stor is an exponent ial decay:

.6. Vc = (.6. Vdoe- lfT

The time constant for the decay is

Th"";""--J .... A

I ++ ++ do~e~ atl = O . 0

\ c \ -- -- : dVc I ;

(AVc)(j Afterl ~ O, the current d ischarges the capaci tor.

/

R

o+-~--==~ 1= 0

Analyzing Circuits PREPARE Draw a circu it diagram.

SOLVE Break 'he circlIi, do w,,:

Reduce the ci.rcuit to the smallest poss ible number of equivalent resistors.

Find the current and potential difference.

Rebuild the circll;t:

• Find current and potential difference for each resistor.

ASSESS Verify that

The sum of the potential differences across series resistors matches that for the equivalent resistor.

The sum of the currents through parallel resistors matches that for the equivalent resistor.

Parallel elements

Elements connected in parallel are co,nnected by wires at both ends. The pOiential difference across each element is the same.

Resistors in parallel can be reduced to an eq ui valent resi stance:

Capacitors in parallel can be reduced to an equivalent capacitance:

Ceq = C1 + C2 + C3 +.

Electricity in the nervous system dVmcmtnn< (mV)

Cell s in the nervous system maintain a negative potential inside the cell membrane. When tri ggered, the membrane depolarizes and generates an action pOlemial.

An action potential travels as a wave along the axon of a neuron. More rapid saltatory conduction can be achieved by insulating the axon with myelin , causing the action potential to jump from node to node.

+40

I----,c++~-__c-__c- I (ms) o 2 3

I---- Depo!arization

- 70 f-_L -- -_-_--_ _

>-__ 7 ""=X

tMP)TM For homewo~k assig~ed on MasteringPhysics, go to ;,:,.:,/ www.mastenngphyslcs.com

Problem difficulty is labeled as I (straightforward) to 11 111 (challenging).

QUESTIONS

Conceptual Questions

1. The tip o f a fl ashlight bulb is touching the top o r a 3 V bauely as shown in Figure Q23. J . Does the bulb light? Why or why not?

+ 3V

FIGURE a23.1

---

- /~

FIGURE Q23 .2

2. A tl ashlig ht bu lb is connected to a batte ry and is glowing; the

c ircuit is shown in Figure Q 23 .2 . Is curren t 12 greater than , less than, or equal to c urre nt I I? Explain.

3. Current lin flows into three resistors connected together one afler the other as shown in Figure Q23 .3. The accompanying graph shows the value of the pote ntial as a fu ncti o n of pos ition.

a . Is l oul greater than , less than , or equal to I,n? Ex plain. b. Rank in order, from largest to smallest , the three resistances

R I , R2, and R3• Explain .

R, Rl RJ

I .. ~/OIJ I

v

b FIGURE 023.3 Posi tion

4. The c irc uit in F igure Q23.4 has two res istors, w ith R[ > R2•

W hich res istor diss ipates the larger amount of power? Ex plain.

R,

R,

FIGURE 023.4 FIGURE 023.5

5. The circuit in Figure Q23.5 has a battery and two resistors, w ith RI > R2. W hich resistor d iss ipates the larger amount o f power? Explain .

6 . In the circuit shown in Figure Q23.6, bulbs A and B are glowing. Then the switch is closed. W hat happens to each bulb? Does it get bri ghter, stay the same, get d immer, or go out? Ex plain.

Questions 767

Problems labeled INT integrate significant material from earlier

chapters; BID are of biological or medical interest.

A

B

FIGURE 023 .6 FIGURE 023 .7

7 . Figure Q23.7 shows two c ircuits. The two batteries are identical and the fo ur resistors all have exactly the same res islance.

a. Is d Vab larger than, smaller than, or equal to d Vcd? Ex plain.

b. Rank in order, fro m largest to smallest, the curre nts I I' 12, and 13• Ex plain .

8. Figure Q23.8 shows two c ircui ts. The two batteries are identicaJ and the four resis tors all have exactly the same res istance.

a. Compare d Vah • d Vcd , and d Yd' Are they a ll the same? I r not, ran k thcm in order from largcst to smallest. Explain .

b. Ran k in order, fro m largest to smallest, the ri ve currents I I to 15• Ex pl ain .

,

d

FIGURE 023 .8

9. a. In Fig ure Q23.9, what fractio n of current I goes through the

3 n res istor? b. If the 9 n res istor is replaced w ith a larger resistor, w ill the

frac ti o n o f curre nt go ing thro ugh the 3 n res istor increase, decrease, or stay the same?

9<1

FIGURE 023 .9

R 50n R

~

FIGURE 023.10

10. Two of the three resistors in Figure Q23 10 are unknown b ut

equal. Is the tota l res istance between po ints a and b less than, greater than, or

equai lo 50 n? Ex pla in. 11 . Two of the three res istors in Figure Q23. 11

are u nk nown b ut equal. Is the to tal res istance between points a and b less tha n, greate r than, or equal to 200 n? Explain.

R

200n

b R

FIGURE 023 .11

768 CHAPTER 23 Circuits

12. Rank in order, from largest to smallest, the currents I I: 12 , and I} in the circuit diagram in Figure Q23. 12.

RI =3R

c FIGURE 023 .12 FIGURE 023.13

13. The three bulbs in Figure Q23.13 are identical. Rank the bulbs from brightest to dimmest. Explain.

14. The four bu lbs in Figure Q23.14 are identical. Rank the bulbs from brightest to dimmest. Explain.

D

E

FIGURE 023 .14 FIGURE 023.15

15. Figure Q23.15 shows five identical bu lbs connected to a bat­tery. AU the bulbs are glowing. Rank the bulbs from brightest to dimmest. Explain.

16. a. The three bu lbs in Figure Q23. 16 are identical. Rank the bulbs from brightest to dimmest. Explain.

b. Suppose a wire is connected between points I and 2. What happens to each bulb? Docs it get brighter, stay the same, get dimmer, or go out? Explain.

A R R

I

8 C A B 2 2

FIGURE 023 .16 FIGURE 023.17

17. Ini tially, bulbs A and B in Figure Q23. 17 are both glowing. Bu lb B is then removed from its socket. Docs removing bulb B cause the potential difference D. VI! between points I and 2 to

increase, decrease, stay the same, or become zero? Explain. 18. a. Consider the points a and b in Figure Q23. 18. Is the potential

difference .6. V3b between points a and b zero? If so, why? If not, which point is more positive?

b. If a wire is con nected between points a and b, does it carry a current? If so, in which direction- to the right or to the left? Explain.

R 2R R

R

FIGURE 023.18

2 '-------'-----'

FIGURE 023 .19

19. Initially the lightbulb in Figure Q23 .1 9 is glowing. It is then removed from it s socket. a. What happens to the current I when the bulb is removed?

Docs it increase, stay the same, or decrease? Explain. b. What happens to the potential difference .6. VI2 between

points I and 2? Docs it increase, stay the same, decrease, or become zero? Explain.

20. A voltmeter is ( incorrectly) inserted into a circuit as shown in Figure Q23.20. a. What is the current in the c ircuit? b. What does the voltmeter read? c. How wou ld you change the c ircu it to correctly connect the

voltmete r to measure the potential difference across the resi stor?

10 fl

1.0kfl 9.0 V --==-

9.0 V

v S.On

FIGURE 023.20 FIGURE 023.21

2 1. An ammeter is (incorrectly) inserted into a circu it as shown in Figure Q23.2 1. a. What is the current through the 5.0 n resistor? b. How wou ld you change the c ircu it to correctly connect the

ammeter to measure the current through the 5.0 n res istor? 22. Rank in order, from largest to smallest, the equ ivalent capaci­

tances (Ceq)1 to (Ceq)4 of the four groups of capacitors shown in Figure Q23.22.

c c cc

c c c

-1HHf-2 3

FIGURE 023.22

23. Figure Q23.23 shows a circuit con­sisting of a battery, a switch, two identical lightbulbs, and a capacitor that is initially uncharged. a. Imm ediately after the switch is

closed, are e ither or both bulbs glowing? Explain.

4

b. If both bulbs are glowing, which FIGURE 023.23

is brighter? Or are they equaUy bright? Explain.

H

c

c. For any bulb (A or B or both) that lights up immediately after the switch is closed, does it s brightness increase with time, decrease with time, or remain unchanged? Explain.

24. Figure Q23.24 shows the volt- 6.Vc age as a fu nct ion of lime across a capacitor as it is discharged (separately) through three dif­ferent resistors. Rank in order, from largest to smalles t, the values of the resistances RI toR). FIGURE 023.24

25. A charged capacitor could be connected to two identical resis~ tors in either of the two ways shown in Figure Q23.25. Which configuration wilt discharge of the capacitor in the shortest time once the swi tch is closed? Explain.

FtGURE 023 .25

26. A flashing light is controlled by the charging and discharging of an RC circuit. If the light is flashing too rapidly, describe two changes that you could make to the circuit to reduce the flash rate.

27. A device 10 make an electrical measurement of skin moisture BID has electrodes that form two plates of a capacitor; the skin is the

dielectric between the plates. Adding moisture to the ski n means adding water, which has a large dielectric constant. If a circuit repeatedly charges and discharges the capacitor to deter~ mine the capacitance, how will an increase in skin moisture affect the charging and discharging time? Explain.

28. Consider the model of nerve conduct ion in myelinated axons BID presented in the chapter. Suppose the distance between the

nodes of Ranvier was halved for a particular axon. a. How would thi s affect the resistance and the capacitance of

one segment of the axon? b. How would this affect the time constant for the charging of

one segment? c. How would thi s affect the signal propagation speed for the

axon? 29. Adding a myelin sheath to an axon results in faster signal prop­BID agat ion. It also means that less energy is required for a signal to

propagate down the axon. Explain why thi s is so.

Multiple-Choice Questions

30. I What is the CU1Ten t in the circuit of Figure Q23.30? A. 1.0A B. 1.7 A C. 2.SA D.4.2A

4.00

[0 V --==-3 1. 1 Which resistor in Figure Q23.30 diss i ~ 6.00

pates the most power? A. The 4.0 f1 resistor. B. The 6.0 f1 resistor. FIGURE 023 .30

C. Both dissipate the same power.

PROBLEMS

Section 23.1 Circuit Elements and Diagrams

I. II Draw a circuit diagram for the c ircuit of Figure P23. 1.

lOon

75n

FIGURE P23 .1 FIGURE P23 .2

Problems 769

32. tt Normally, household lightbulbs are connected in parallel to a power supply. Suppose a 40 Wand a 60 W lightbulb are, instead, connected in series, as shown in Figure Q23.32. Which bulb is brighter? A. The 60W bulb. DOW

t20 v --==-(jOW

B. The 40 W bulb. FIGURE 023.32

C. The bulbs are equally bright. 33. tt l A metal wire of resistance R is cut into two pieces of equal

length. The two pieces are connected together side by side. What is the resistance of the two connected wires?

34.

A. RI4 B. RI2 C. R D.2R E.4R I What is the value of resis­tor R in Figure Q23.34? A.4.00 B.120 C. 360 D. no E. 96 n

6.V=8.0V Ion 15 n R

35 . I Two capacitors are con-nected in seri es. They are FIGURE 023 .34

then reconnected to be in parallel. The capacitance of the parallel combination A. Is less than that of the series combination. B. Is more than that of the series combination. C. Is the same as that of the series combination. D. Could be more or less than that of the series combination

depending on the values of the capacitances. 36. I If a cell's membrane thickness doubles but the ceU stays the

same size, how do the res istance and the capac itance of the ce ll BID membrane change?

A. The resistance and the capacitance would increase. B. The resistance would increase, the capacitance would decrease. C. The resistance wou ld decrease, the capacitance would increase. D. The resistance and the capacitance would decrease.

37 . III If a ce ll' s diameter is reduced by 50% without chang ing the membrane thickness, how do the resistance and capacitance of

BID the ceU membrane change? A. The resistance and the capacitance would increase. B. The resistance would increase, thecapacilance would decrease. C. The resistance would decrease, the capacitance would

increase. D. The resistance and the capacitance would decrease.

2. Draw a circuit diagram for the ci rcuit of Figure P23.2. 3. Draw a circu it diagram for the circuit of Figure P23.3.

FIGURE P23 .3

770 CHAPTER 23 Circuits

Section 23.2 Kirchhoff's Laws

4. II In Figure P23.4, what is the current in the wire above the junction? Does charge flow toward or away from the j unct ion?

~6V+ I

20 + 50 lOV

--==-- 3.0 V

2.0 n 2

4 3

FIGURE P23 .4 FIGURE P23 .5

5. II The lightbulb in the ci rcuit diagram of Figure P23.5 has a resi stance of 1.0 fl. Consider the potential difference between pairs of points in the figure . a. What are the values of d V12 ' d V23 , and d VJ..I? b. What are the values if the bulb is removed?

6. I a. What are the magnitude and direction of the currenl in the 30 n resi stor in Figure P23.6?

b. Draw a graph of the potential as a function of the distance traveled through the c ircu it , traveling clockwise from V = 0 V at the lower left corner. See Figure P23.9 for an example of such a graph.

~ ~ --==-- 9.0 V 6.0 V --==-- --==-- 3.0 V 6.0 V --==--T T T T

FIGURE P23 .6 FIGURE P23.7

7. II a. What are the magnitude and direction of the curren t in the 18 n resistor in Figure P23.7?

b. Draw a graph of the potential as a function of the distance traveled through the c ircuit, traveling clockwise from V = 0 V at the lower left corner. See Figure P23.9 for an example of such a graph.

8. I a. What is the potential difference across each resistor in Figure P23.8?

b. Draw a brraph of the potential as a function of the di stance trave led through the circu it , traveling clockwise from V = 0 V at the lower left corner. See Figure P23.9 for an example of such a graph.

IOn 15 V 20 0

FIGURE P23.8

V(V)

6

4

2

0 "-------''-"-----Posilion

FIGURE P23.9

9. I The current in a c ircu it with only one battery is 2 .0 A. Figure P23.9 shows how the potential changes when go ing around the c ircuit in the clockwise direction, starting from the lower left corner. Draw the circuit diagram.

Section 23.3 Series and Parallel Circuits

10. I What is the equi valent resistance of each group of res istors shown in Figure P23. 10?

(b)

(a)

~ 2.0 0 3.0 0 6.00

~ ~3 0n

3.0n

FIGURE P23.10

11 . I What is the eq ui valent res istance of each group of res istors shown in Figure P23. 11 ?

FIGURE P23 .11

12. 1111 An 80-cm-long wire is made by welding a I.O-mm-diameter, INT 20-cm-long copper wire to a I.O-mm-diameter. 60-cm-long iron

wire. What is the resistance of the composite wire? 13. I You have a collection of 1.0 kf! resistors. How can you connect

four of them to produce an equivalent resistance of 0.25 kfl? 14. I You have a co llection of six 1.0 kf! resistors. What is the

smallest resistance you can make by combining them? IS . I You have three 6.0 f1 resistors and one 3.0 f! resistor. How can

YOll connect them Lo produce an equivalent resistance of 5.0 n? 16. II You have six 1.0 k!1 resistors. How can you connect them to

produce a lotal equi valent resistance of 1.5 kf!?

Section 23.4 Measuring Voltage and Current

Section 23.5 More Complex Circuits

17. II What is the equivalent res istance between points a and b in Figure P23. 17?

42 n

b

FIGURE P23.17

40 n

Ion

30 n

Ion

b

FIGURE P23.18

60 n

18. I What is the equiva lent res istance between points a and b in Figure P23.18?

19. II The currents in two res istors in a c ircuit are shown in Figure P23 19. What is the vallie of res istor R?

2.5 A 3.5 V

3.0 V --==--R l

4.5 V --==--1500

200 n 11.5 A

2.0 V

FIGURE P23 .19 FIGURE P23.20

20. II Two balteries suppl y current to the c irc ui t in Figure P23.20. The figure shows the potent ial difference across two of the resistors and the value of the third res istor. What curren t is sup­plied by the batteries?

2 1. I Part of a c ircuit is shown in Figure P23.2 1. a. What is the current through the 3.0 n res istor? b. What is the value of the current I?

6 V = 5.0V + .'---;-;;-;C-"-';7;""-'----;;-~' -Ion 150 R

2.0 n 13.0 A 3.0 n ~ 1 - IOOmA

FIGURE P23.21 FIGURE P23.22

22. I What is the value of resistor R in Figure P2'3.22? 23. II What are the res istance R and the emf of the battery in

Figure P23.23?

24. 25.

3.0n 4.5 n

R R

E -=-3.0A 2.0A Ion

FIGURE P23 .23 FIGURE P23.24

II The ammeter in Figure P23.24 reads 3.0 A. Find / 1' /2' and [. III Find the current through and the poten tial d iffe rence across each resistor in Figure P23.25.

FIGURE P23.25

26. III Find the current th roug h and the potent ial d ifference across each resistor in Figure P23.26.

27.

FIGURE P23.26

II For the c i.rcuil shown in Figure P23.27, find the current through and the potent ial d ifference across each res istor. Place your results in a table for ease of read ing.

,-~-, 6.0n

6.0 n ISH

24 v -=-4.00

FIGURE P23 .27

Problems 771

28. 1111 In the c ircu it of Figure P23.28, what arc the va lues of 6. V14•

Ll V24 , and Ll V34?

2 3

5.0n 5.0n 5.0n

IOV Io n Ion 5.0n

FIGURE P23.28 4

29. II For the c ircuit shown in Figure P23.29, find the current th rough and the potent ial d ifference across each resistor. Place your results in a table for ease of reading.

30.

3 1.

3.o n 16 n

9.0 V

FIGURE P23 .29 FIGURE P23.30

II A photoresistor, whose res istance decreases with li ght inten­sity, is connected in the c irc uit of Figure P23.30. On a sunny day, the photoresistor has a resistance of 0.56 fl . On a cloudy day, the resistance ri ses to 4.0 kfl . At night , the res istance is 20k!}. a. What does the voltmeter read for each of these conditions? b. Docs the voltmete r reading increase or decrease as the li ght

in tensity increases? A photoresistor, whose res istance

decreases with light intens ity, is con­nected in the c ircu it of Figure P23.3 1. a. Draw a c ircuit diagram to illus­

trate how you would use a vo lt­meter and an ammeter to determine the res istance of the photoresistor in thi s c ircuit.

b. What do the two mete rs read

9.0V

FIGURE P23 .31

when the resistance of the photoresistor is 2.5 kn ?

Section 23.6 Capacitors in Parallel and Series

32. I A 6.0.uF capacitor, a IO,uF capac itor, and a 16,uF capac itor are connec ted in pill·aIle!. What is their equi valen t capacitance?

33. I A 6.0.uF capacitor, a 10 ,uF capac itor, and a 16,uF capac itor are con nected in series. What is their equi valen t capac itance?

34. I You need a capacitance of 50 ,uF, but you don't happen to have a 50.uF capacitor. You do have a 30,uF capacitor. What add itional capacitor do you need to produce a total capac itance of 50 ,uF? Should you join the two capac itors in para ll el or in series?

35. I You need a capacitance of 50 ,uFo but you don ' t happen to have a 50.uF capac itor. You do have a 75,uF capac ito r. What add itional capacitor do you need 10 produce a total capac itance of 50 ,uF? Should you join the two capac itors in parallel or in series?

772 CHAPTER 23 Circ uits

36. II What is the eq ui va len t capacitance of the three capac itors in Figure P23.36?

FIGURE P23 .36 FIGURE P23 .37

37. I What is the equiva lent capac itance of the three capacitors in Figure P23 .37?

38. III For the circuit of Figure P23 .38, a. What is the equivalent capac itance? b. How much charge flows through the battery as the capac i­

tors are be ing charged?

12 V

L ~ 4.0~F 1'_V _- '0 F :r: -. 11

TL __ --'T 1.0 ~F FIGURE P23 .38 FIGURE P23 .39

39. III For the circuit of Figure P23.39, a. What is the equi valent capac itance? b. What is the charge of each of the capac itors?

Section 23.7 RC Circuits

40. II What is the time constant for the di scharge of the capac itor in Figure P23.40?

r 1.0 l1F

t.OkO

I.OkO

FIGURE P23 .40 FIGURE P23 .41

4 1. II What is the time constant for the discharge of the capac itor in Figure P23.4 1?

42. III! Capaci tors won' t hold a charge indefini tely; as time goes on, charge gradually migrates from the pos itive to the negat ive plate. We can model thi s as a discharge of the capac itor through an inte rnal "leakage resistance." A 0.47 F capac itor charged to 2.5 V wi ll ini tially discharge with a leakage current of 0.25 rn A. a . What is the leakage res istance? b. How long will it take for the capac itor voltage to drop to

1.0 V? 43. III! A 10 J.LF capac itor ini tiall y charged to 20 J.LC is di scharged

through a 1.0 kf! resistor. How long does it take to reduce the capac itor's charge to 10 J-LC?

44. II The switch in Figure P23 .44 has been in posit ion a for a long time. It is changed to pos ition b at I = 0 s. What are the charge Q on the capac-

n"1 9.0V~50fl

itor and the curre nt I through FIGURE P23 .44

the resistor (a) immediately after the switch is closed? (b) At t = 50 J.Ls? (c) At t = 200 J.Ls?

Section 23.8 Electricity in the Nervous System

45. I A 9.0-nm-thick cell membrane undergoes an action poten ti al BID that fo llows the curve in the table on page 76 1. What is the

strength of the electric field ins ide the membrane just before the acti on potential and at the peak of the depo larizat ion?

46. III A ce l.l membrane has a res istance and a capacitance and thus BID a characte ri st ic time cons tant. What is the time constant of a

9.0-nm-thick membrane surround ing a 0.040-mm-diameter spherical celt?

47. I Changing the thickness of the myel in sheath surrounding an BID axon changes its capac itance and thus the conduct ion speed. A

myel inated nerve fi ber has a conduction speed of 55 m/s. If the spac ing between nodes is 1.0 mm and the res istance of seg­ments between nodes is 25 MO , what is the capacitance of each segment?

48. III A particular mye li nated axon has nodes spaced 0.80 mm BID apart. The resistance between nodes is 20 MH ; the capacitance

of each insulated segment is 1.2 pF. What is the conduction speed of a nerve impu.lse along thi s axon?

49. I To measure signal propagation in a nerve in the arm, the BID nerve is triggered near the armpi t. The peak of the action poten­

ti al is measured at the elbow and then, 4.0 ms later, 24 em away from the e lbow at the wri st. a. What is the speed of propagati on along thi s nerve? b. A determination of the speed made by measuring the time

between the application o f a stimulus at the armpit and the peak of an act ion potent ial at the elbow or the wrist would be inaccurate. Explain the problem with thi s approach, and why the noted techniq ue is preferable.

50. II A myel inated axon conduc ts nerve impul ses at a speed of BID 40 mls. What is the signal speed if the thickness of the mye li n

sheath is halved but no other changes are made to the axon?

General Problems

5 1. II How much power is di ss ipated by INT each res istor in Figure P23.5 1?

FIGURE P23 .51

RI = 12 n

-=- 9.0V

R! = 15 n

52. 1111 Two 75 W (120 V) lightbu lbs are wired in series, then the INT combination is connected to a 120 V supply. How much power

is diss ipated by each bul b? 53. 11 11 The corroded contacts in a lightbulb socket have 5.0.n total INT res istance. How much actual power is di ss ipated by a 100 W

(l20V) lightbu lb screwed into thi s socket? 54. 1111 A real battery is not just an emf. We can model a real 1.5 V INT batte ry as a 1.5 V emf in series with a resistor known as the

" internal resistance," as shown in Figure P23.54 . A typical bat-

c te ry has 1.0 n internal res istance due to

imperfections that limit curren t th rough the battery. W hen there's no current through the battery, and thus no vol tage drop across the inlernaJ res istance, the potenti al diffe rence between its termi na ls is 1.5 V, the val ue of

l.on

1.5 v

the emf. Suppose the te rmin a ls of thi s bal- FIGURE P23 .54 tery are connec ted to a 2 .0 fl resistor. a. What is the potenti al difference between the terminals of the

baHery? b. What fraction of the battery's power is diss ipated by the

internal res istance? 55. 1111 For the real battery shown in Figure P23.54, calcu late the

power diss ipated by a res istor R con nected to the battery when (a) R = 0.25 n, (b) R = 0.50 n, (e) R = 1.0 n, (d) R = 2.0 H , and (e) R = 4.0 n. (Your [csults should suggesllhat max imum power d iss ipat ion is achieved when the ex ternal resistance R equals the internal resistance. This is true in general.)

56. 1111 Batteri es are rec harged by connecti ng them to a power sup­INT ply (i.e., another battery) of greater e mf in such a way that the

c urre nt flows ill to the pos itive terminal o f the batLery being recharged, as was shown in Exam ple 23. 1. This reverse current thro ugh the batte ry re ple ni shes it s c he mica ls. The current is kept fa irl y low so as not to overheat the battery being recharged by d iss ipating ene rgy in its internal res istance. a. Suppose the real battery o f Figure P23.54 is rechargeable.

What emf power supply should be used for a 0 .75 A recharg­ing CtliTen t?

b. If thi s power supply c harges the battery for 10 mi nutes, how much energy goes inlO the battery? How much is d issipated

as thermal energy in the internal res istance? 57. 111 When two resistors are connected in parallel across a battery o f

unknown voltage, one resistor calTies a CUlTen t of 3.2 A whi le the second carries a CUlTen t of 1.8 A. What current will be supplied by the same battery if these two resistors are connec ted to it in series?

58. II The 10 n resistor in Figure P23.58 is d iss ipati ng 40 W of INT power. How much power are the other two resistors d iss ipating?

I j 5.0n

R 20~F c;; Ion 20n

20V

FIGURE P23.58 FIGURE P23.59

59. jjji At Ihi s instant, the current in the circuit of Figure P23.59 is tNT 20 rn A in the d irecti on shown and the capacitor charge is

200 /-Lc. What is the resistance R? 60. I What is the equ ivalent res is- 100 0 [00 0 100 n

tance between po ints a and b in Figure P23.60?

FIGURE P23.60

loon loon

loon

b

6 1. You have three 12 n res isto rs. Draw di agrams showing how you could arrange all three so that their equi valen t resistance is (a) 4 .0 H, (b) 8.0 n, (e) 18 H, and (d) 36 n.

Problems 773

62. 1111 A 9 .0 V battery is connec ted to a wire made of th ree seg­INT ments of diffe rent metals connected one after another: 10 cm of

copper wire, the n 12 c m of iron wire, then 18 cm of tu ngsten wire. All o f the wires are 0 .26 mm in di ameter. Find the pote n­ti al d ifference across each piece of wire.

63. 111 You have a dev ice that needs a vol tage reference of 3.0 V, but you have on ly a 9.0 V battery. Fortunately, you also have several 10 k!1 resistors. Show how you can use the res istors and the bat­tery to make a c ircuit that provides a potential difference of3.0 V.

M. I There is a current o f 0 .25 A in the c ircuit of Figure P23.M. tNT a. What is the di rect ion of the cUlTent? Explain .

b. What is the val ue of the res istance R? c. What is the power di ss ipated by R? d. Make a graph of pote nti al versus position, start ing from

V = 0 V in the lower le ft corner and proceeding clockwise. See Figure P23.9 for an example.

6.0f! 12f!

~ '-'+---'-Your measuring circuit

-=- 6.0V [2V -=- R 500 ~A ammelel

Lwvv-----J 50.0f!

FIGURE P23.64 FIGURE P23.65

65. JI A circ uit you're build ing needs an ammeter that goes fro m o mA to a fu ll-scale read ing of 50.0 mA. Un fortunately, the on ly ammeter in the storeroom goes from 0 p.,A to a full ­scale reading of only 500 p.,A. Fortunately, you can make thi s ammeter work by pUlling it in a meas uring circuit , as shown in Figure P23.65 This lets a certain frac ti on of the current pass through the meter; knowing tbi s value, you can deduce the total curre nt. Assume that the amme ter is ideal. a. What value of R must you use so that the meter wi II go to

fu ll scale when the CUlTe nt I is 50.0 mA? Hint: When 1= 50.0 mA, the ammeter should be read ing its max imum val ue . b. What is the equi valent res istance of your measuring c irc uit?

66. II A c ircuit you're bui ld ing needs a vo ltmeter that goes from o V to a full -sca le read ing of 5.0 V. Un fo rtunately, the only meter in the storeroom is an ammeter that goes from 0 /-LA to a ful l-scale read ing of 500 p.,A. It is poss ible to use thi s mete r to measure vo ltages by pu tt ing in a measuring circ ui t as shown in Figure P23.66. What value of R must you use so that the meter wilJ go to full scale when the potenti al d iffe re nce .6. V is 5 .0 V? Assume that the ammeter is ideal.

R 6V

Your measuring circuit

500 ~A ammeter

FIGURE P23.66

4.00 6.0n

24 V 240

8.0n 240

FIGURE P23.67

67. II For the c ircuit show n in Figure P23.67, fin d the curre nt through and the potential d ifference across each res istor. Place your resul ts in a table for ease of read ing.

68. II You have three capac itors. Draw di agrams showing how you could arrange all three so that the ir equi valent capac itance is (a) 4 .0 p.F, (b) 8.0 p.F, (e) 18 p.F, and (d) 36p.F.

774 CHAPTER 23 Circ uits

69.

70. IN!

7 1.

72.

II Initi ally, the switch in Figure P23.69 is in pos ition a and capac i­tors C2 and C] are uncharged. Then the switch is nipped to pos ition b. Afte rward, what are the charge on and the potenti al d iffe r-

FIGURE P23 .69

e nce across each capac itor? II The capacitor in an RC c irc ui t with a time constant of I S ms is charged to 10 V. The capacitor beg ins to d ischarge at t = 0 s. a. At what time will the charge on the capac itor be reduced to

hal f its initi al value? b. At what time will the energy stored in the capac itor be

reduced to half its initial val ue? What val ue res istor will di scharge a LO tLF capac itor to 10%

of its initi al charge in 2.0 ms? 1111 The charg ing circuit for the flas h system of a camera uses a 100 tLF capac itor that is c harged from a 250 V power su\?p ly. What is the most res istance that can be in series with the capac i­tor if the capacitor is to charge to at least 87% of its fin al volt ­age in no more than 8.0 s?

73. II A capac itor is d ischarged through a 100 n res istor. The d is­charge current decreases to 25% of its ini tial value in 2 .5 ms. What is the val ue of the capac itor?

74. III A 50 tLF capac itor thaI had been charged to 30 Y is d ischarged through a resistor. Figure P23.74 shows the capac itor vo ltage as a function of time. What is the val ue of the resistance?

6. Vc (V) Opens at I = Os

30 \

20 60n

Ion

10 lOOV

40n

0 I (ms) 2.0 JlF

0 2 3

FIGURE P23 .74 FIGURE P23 .75

75. III! The switch in Figure P23.75 has been closed for a very long time. a. What is the charge on the capac itor? b. The switch is opened at t = a s. At what time has the charge

on the capac itor decreased to 10% of its ini tial value?

76. Illll ln termittent Switch

windshie ld w ipers

( use a vari able resis-tor in an RC c ircu it loon VOl riable

resistor to set the delay between success ive 12V " b

passes of the wipers. 47kfi l00J,.(F

A ty pica l circui t is shown in Figure P23.76. When the switch closes, the

FIGURE P23 .76

capac itor (initially uncharged) beg ins to charge and the poten tial at point b begins to increase. A sensor measures the potential di f­ference between po ints a and b, tr iggeri ng a pass o f the wipers when Vb = Va' (Another pill1 of the circuit, not shown, discharges the capacitor at this time so that the cycle can start again .) a . W hat value of the variable res istor will g ive 12 seconds

from the start of a cycle to a pass of the wipers?

b. To decrease the time, should the variable res istance be increased or decreased?

77. III In Example 23. 14 we estimated the capacitance of the cell BID membrane LO be 89 pF, and in Example 23 .1 5 we found that

approx imately 10,000 Na ! ions fl ow through an ion channel when it opens. Based on thi s in formati on and what YOLi learned in thi s chapter about the acti on potenti al, esti mate the total num ber o f sodium ion channels in the membrane of a nerve cell .

78. 1111 The g iant axon of a sq uid is 0 .5 mm in d iameter, 10 em long. BID and not mye li nated. Un myel inated ce ll me mbranes behave as INT capac itors with 1 tLF of capac itance per square centimeter o f

me mbrane area. When the axon is charged to the - 70 mY rest­ing poten tial, what is the energy stored in thi s capac itance?

79. II A ce ll has a 7.0-nm- thic k mcm brane with a total mc mbrane BID area of 6 .0 X 10- 9 m2

.

a. We can mode l the ce ll as a capacitor, as we have secn. What is the magn itude of the charge on each "pla te" when the membrane is at its resti ng potenti al of -70 mY?

b. How many sod ium ions does thi s charge correspond to?

Passage Problems

The Delibrillator BID

A defibriU ator is des igned to pass a large current through a patient's torso in order to SLOp dangerous heart rh ythms. Its key part is a capacitor that is charged to a high voltage. The pat ient 's torso plays the role o f a res istor in an RC c irc ui t. When a switch is closed, the capac itor d ischarges through the patient's torso. Ajo lt from a de fi b­rill ator is intended to be intense and rapid; the max imum curren t is very large. so the capac itor d ischarges quic kl y. T hi s rapid pul se dcpolarizcs the heart, stopping all electrical acti vity. Th is allows the heart's in ternal nerve c irc ui try to reestablish a health y rh ythm.

A typical defibrill ator has a 32 tLF capac itor charged to 5000 V. The electrodes connected to the patient are coated with a conduct ing gel that reduces the res istance of the skin to where the e ffect ive res is­tance of the patien t' s torso is 100 n. 80. I Which pair of graphs in Figure P23.80 best represe nts the

capaci tor vo ltage and the current through the torso as a function of time after the switch is closed?

I

A.

o~ o

B.

c.

D.

FIGURE P23 .80

81. I For the va lues noted in the passage above , what is the time constant for the discharge of the capac itor? A. 3.2 JLS B. 160 JLs C. 3.2 TnS D. 160 TnS

82. 1 If a patient receives a series of jolts, the resistance of the torso may increase. How does such a change affect the initial cunent and the time constant of subsequent jol ts? A. The initial current and the time constant both increase. B. The initial current decreases, the time constant increases. C. The initiaJ current increases, the time constant decreases. D. The initial current and the time constant both decrease.

83. 1 In some cases, the defibrillator may be charged to a lower vol tage. How will this affect the time constant of the discharge? A. The time constant will increase. B. The time constant will not change. e. The time constant will decrease.

Electric Fish BID IN!

The voltage produced by a single nerve or muscle ce lJ is quite small , but there are many species of fi sh that use multiple action poten­tials in series to produce sig­nifi cant voltages. The electric organs in these fi sh are composed of speciali zed disk-shaped cells called eiectroc)'te!>". The cell at rest has the usual potential dif­ference between the inside and the outside. but the net potential difference across

the cell is zero. An electro­cyte is connected to nerve fibers that initially trigger a depolarization in one side of the ce ll but not the other. For the very short time of this depolarization, there is a net potent ial difference across the cell, as shown in Figure P23.84. Stacks of

v

Electrolyte during depolarization of one side of cell

Sodium channels o(lt!n oilihis ~ide only.

\

+ - -

+---.--1--\------+-+--x

6. V",II , - - - -\ - - - - - - - - - --

There is a net volt;lge across the cell.

FIGURE P23.84

Stop to Think 23.1: A, B, and D. These three are the same circu it because the logic of the connections is the same. In each case, there is a j unct ion that connects one side of each circuit element and a second junction that connects the other side. In C, the functioning of the circuit is changed by the extra wire connecting the two sides of the capacitor.

Stop to Think 23.2: C == D > A == B. The two bulbs in se ries are of equal brightness, as are the two bulbs in paraUel. But the two bulbs in ser ies have a larger resistance than a single bulb, so there will be less current through the bulbs in series than the bulbs in parallel.

Stop to Think 23.3: C. The voltmeter must be connected in parallel with the resistor, and the ammeter in series.

Problems 775

these cells connected in series can produce a large total vol tage. Each stack can produce a smaJJ CUlTent; for more total current, more stacks are needed, connected in parallel. 84. I In an elec tri c ee l, each e lec trocyte can develop a voltage of

150 m V for a short time. For a total voltage of 450 V, how many elec trocytes must be connected in series? A. 300 B. 450 C. 1500 D. 3000

85. I An electric ee l produces a pulse of curren t of 0.80 A at a volt­age of 500 V. For the short time of the pulse. what is the instan­taneous power? A. 400W B. 500W C. 625W D. 800W

86. I Electric ee ls live in fresh water. The torpedo ray is an elec­tric fish that li ves in sa lt water. The electrocytes in the ray are grouped differently than in the ee l; each stack of electrocytes has fewer cells, bUlthere are more stacks in parallel. Which of the following best explains the ray's electrocyte arrange­ment? A. The lower resistivity of salt water requires more current but

lower voltage. B. The lower resistivity of salt water requires more voltage but

lower current. e. The higher resistivity of sa lt water requires more current but

lower voltage. D. The higher resistivity of salt water requires more vol tage but

lower CUlTent. 87. I The eleClric catfish is another electric fish lhat produces a

vo ltage pulse by means of stacks of e lectrocytes. As the fish grows in length, the magnitude of the voltage pulse the fish pro­duces grows as well. The besl explanation for this change is that, as the fish grows, A. The voltage produced by each electrocyte increases. B. More electrocytes are added to each stack. e. More stacks of eleclrocytes are added in parallel 10 the ex ist­

ing stacks. D. The thickness of the electrocytes increases.

Stop to Think 23.4: A > B > C == D. All (he cunent from the bat­tery goes through A, so it is brightest. The current divides at the junc­ti on, but not equally. Because B is in parallel with C + D, but has half the resistance of the two bulbs together, twice as much current travels through B as through C + D. So B is dimmer than A but brighter than C and D. C and D arc equally bright because of conservation of curre nt.

Stop to Think 23 .5: (Ccq)B > (Ccq),\ > (Ccq)c, Two capacitors in parallel have a larger capac itance than either alone; two capacitors in series have a smaller capacitance than either alone.

Stop to Think 23.6: B. The two 2 n resistors are in series and equiv­alent to a 4 n resistor. Thus 'T == Re = 4 s.