Que stions
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;,:,.:,/
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Problems labeled
chapters;
INT
767
integrate significant material from earlier
BID are of biological
or medical interest.
Problem difficulty is labeled as I (straig htforward) t o 11111 (challenging).
QUESTIONS
Conceptual Questions
1. T he tip o f a fl ashlight bulb is touc hing the top o r a 3 V bauely as
shown in Fi gure Q23. J . Does the bulb light? Why or why not?
A
B
--+
FIGURE 023 .6
3V
- /~
FIGURE Q23 .2
FIGURE a23 .1
2. A tl ashli g ht bu lb is connected to a batte ry a nd is glow in g; the
c irc uit is shown in Fi gure Q 23 .2 . Is c urren t 12 greater than , less
than, or eq ual to c urre nt I I? Explain.
3. Curre nt lin flows into three resistors co nnected togeth er one
afler the othe r as s hown in Figure Q23 .3. The accompany ing
graph shows the value of the pote ntial as a fu ncti o n of pos iti on.
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 , a nd R3 • Explain .
R,
Rl
FIGURE 023 .7
7 . Figure Q23.7 shows two c irc uits. The two batteries are identical
and the fo ur resistors all have exactly the same res isla nce.
a. Is d Vab larger tha n, sm aller than, or equal to d Vcd? Ex plain.
b. Rank in order, fro m largest to s mallest, the c urre nts I I' 12,
a nd 13 • Ex pl ain .
8. Figure Q23.8 shows two c irc ui ts. The two batteries are identicaJ
and the four resis tors all have exactly the same res ista nce.
a. Compare d Vah • d Vcd , a nd d Yd' Are th ey a ll the same ? I r
not, ran k thc m in order from largcst to smallest. Expl ain .
b. Ran k in order, fro m largest to smallest, the ri ve c urrents I I to
15 • Ex pl ain .
,
RJ
I ..~/OIJI
v
FIGURE 023 .3
b
d
FIGURE 023 .8
Posi tion
4. T he c irc uit in F igure Q23.4 has two res istors, w ith R[ > R2 •
W hic h res istor diss ipates the larger amount of power? Ex plain.
9. a. In Fi g ure Q23. 9, wh at fractio n of c urrent I goes through the
3 n res istor?
b. If the 9 n res istor is re pl aced w ith a larger resistor, w ill the
frac ti o n o f c urre nt go ing thro ugh the 3 n res istor inc rease,
decrease, or stay the same?
R
R,
R
9<1
FIGURE 023 .9
FIGURE 023 .4
50n
~
R,
FIGURE 023 .5
5. The circ uit in Fi gure Q23.5 has a battery and two resistors, w ith
RI > R2. W hi c h resistor d iss ipa tes the larger a mount o f power?
Expl ain .
6 . In the circuit shown in Fi gure Q23.6, bulbs A a nd B are glowing.
The n the sw itc h is closed. W hat ha ppens to each bulb ? Does it
get bri ghte r, stay the same, get d immer, or go out? Ex pl ain.
FIGURE 023 . 10
10. Two of the three resistors in Fi gure Q23 10 are unknown b ut
equal. Is the tota l res istance between
R
po ints a a nd b less tha n, greate r th a n, or
eq uai lo 50
Ex pla in.
200n
11 . Two of the three res istors in Figure Q23. 11
are u nk nown b ut eq ual. Is the to tal
b
R
res ista nce between points a and b less
tha n, greate r th a n, or eq ual to 200
FIGURE 023 .11
Exp lain.
n?
n?
768
CHAPTER 23
Circuits
12. Rank in order, from largest to smallest, the curre nts 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.
19. Initi ally the lightbulb in Figure Q23 .1 9 is glowing. It is then
removed from it s soc ket.
a. What happe ns to the current I whe n the bulb is removed?
Docs it increase, stay the same, or decrease? Exp lain.
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 volt meter is (incorrectly) in serted 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 co nnect the
voltmete r to measure the potential difference across the
resi stor?
10
fl
1.0kfl
9.0 V --==-
D
9.0 V
FIGURE 023 .14
15. Figure Q23.15 shows fiv e identical bu lbs con nected to a battery. AU the bu lbs 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 con nected between points I and 2. What
happens to each bu lb? Docs it get brighter, stay the same, get
dimmer, or go out? Explain.
R
c
c
cc
H
c c c
8
B
A
C
2
-1HHf2
2
R
2R
R
R
'-------'-----'
FIGURE 023 .19
3
4
c
FIGURE 023 .22
FIGURE 023 .17
17. Ini tially, bu lbs A and B in Figu re Q23. 17 are both glowing.
Bu lb B is then removed from its soc ket. 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. V 3b between point s 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?
Exp lain.
FIGURE 023 .18
2 1. An am meter is (in correctly) in serted 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 o rder, from largest to smalle st, the equ ivalent capacitance s (Ceq)1 to (Ceq)4 of the four groups of capac itors shown in
Figu re Q23.22.
R
I
FIGURE 023 .16
FIGURE 023.21
FIGURE 023 .20
FIGURE 023 .15
A
S.On
v
E
2
23. Figure Q23.23 shows a circuit consisting of a battery, a sw itch, two
identical li ghtbulbs, and a capacitor
that is initially uncharged.
a. Imm ediately after the sw itch is
closed, are e ither or both bulbs
glowing? Exp lain.
b. If both bulbs are glowing, which FIGURE 023.23
is brighter? Or are they equaUy bright? Explain.
c. For any bu lb (A or B or bot h) that lights up immediately
after the sw itc h is closed, does it s bright ness increase with
time, decrease with time, or remain unchanged? Exp lain.
24. Figure Q23.24 shows the volt- 6.Vc
age as a fu nct ion of lime across
a capacitor as it is disc harged
(separately) through three different resistors. Rank in orde r,
from largest to sma lles t, the
values of th e resistances RI
FIGURE 023 .24
toR).
Problems
25. A charged capacitor could be connected to two identical resis~
tors in either of the two ways shown in Figure Q2 3.25. Which
configuration wilt discharge of the capacitor in the shortest time
once the swi tch is closed? Explain.
FtGURE
023 .25
26. A flashing li ght 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 meas ureme nt of skin moisture
BID has electrodes that form two plates of a capac itor; the skin is the
di electric between the plates. Adding moi sture to the ski n
means adding water, which has a large dielectric constant. If a
circ uit repeatedly charges and discharges the capacitor to deter~
mine the capac itance, how will an increase in skin moisture
affect the c harging and discharging time? Explain.
28. Consider the model of nerve cond uct io n 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 capac itance of
one seg ment of the axon?
b. How would this affect the time constant for the chargin g 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 requ ired 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
3 1. 1 Which resistor in Figure Q23.30
pates the most power?
A. The 4.0 f1 resistor.
B. The 6.0 f1 resistor.
C. Both dissipate the same power.
4.00
[0 V
--==-
diss i ~
6.00
FIGURE
023 .30
769
32. tt Normally, household lightbulbs are
OW
connected in parallel to a power supply.
Suppose a 40 Wand a 60 W lightbulb are, t20 v
instead, con nected in series, as show n in
(jOW
Figure Q23.32. Which bulb is brighter?
A. The 60W bulb.
FIGURE 023 .32
B. The 40 W bulb.
C. The bulbs are eq ually bright.
33. ttl A metal wire of resistance R is c ut into two pieces of equa l
len gth. The two pieces are connected together side by side.
What is the resistance of the two connected wires?
A. RI4
B. RI2
C. R
D.2R
E.4R
34. I What is the value of resistor R in Figure Q23.34?
D
--==-
A.4.00
B.120
C. 360
D.
6.V=8.0V
Ion
15
no
I. II Draw a circuit diagram for the c irc uit of Figure P23. 1.
2.
3.
Draw a circu it diagram for the ci rcuit of Figure P23.2.
Draw a circu it diagram for the circuit of Figure P23.3.
lOon
75n
FIGURE
P23 .1
FIGURE
P23 .2
R
E. 96 n
35 . I Two capacitors are co nnected in seri es. They are FIGURE 023 .34
then reconnected to be in
parallel. The capac itance of the parallel combination
A. Is less than that of the series combination.
B. Is more than that of the series comb ination.
C. Is the same as that of the series comb ination.
D. Could be more or less than that of the series combination
depending on the values of the capac itances.
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 c hang in g the
membrane thickness, how do the resistance and capac itan ce of
BID the ceU membrane chan ge?
A. The resistance and the capac itance would increase.
B. The resistance would increase, thecapacilance would decrease.
C. The resistance would decrease, the capacitan ce would
increase.
D. The resistance and the capac itance would decrease.
PROBLEMS
Section 23.1 Circuit Elements and Diagrams
n
FIGURE
P23 .3
770
CHAPTER
23
Circuits
Section 23.2 Kirchhoff's Laws
(b)
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 ?
~
2.0 0
~
6V +
I
20
50
2.0 n
+
3.0n
11 . I What is the eq ui valent res istance of each group of res istors
shown in Figure P23. 11 ?
3
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 val ues 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 Fi gure P23.6?
b. Draw a graph of the potential as a fun ctio n of the distance
tra veled through the c ircu it , tra veling clockwise from
V = 0 V at the lower left corner. See Figure P2 3.9 for an
example of such a graph.
~
--==--
~
--==--
--==-- 9.0 V
T
6.00
FIGURE P23 .10
4
FIGURE P23 .4
3.0 0
2
--==-- 3.0 V
lOV
~
~ 3 0n
(a)
6.0 V
--==--
T
T
FIGURE P23 .6
3.0 V
6.0 V
T
FIGURE P23 .11
12. 1111 An 80-cm-long wire is made by we lding a I.O-mm-di ameter,
INT 20-cm-long copper wire to a I.O-mm-d iameter. 60-cm-long iro n
wire. What is the resistance of the compos ite wire?
13. I You have a collectio n of 1.0 kf! resistors. How can you con nect
four of them to produce an equ ivale nt resistance of 0.25 kfl?
14. I You have a co ll ection 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 eq ui valent resistance of 1.5 kf!?
FIGURE P23 .7
7. II a. What are the magn itude and direction of the cu rren t in the Section 23.4 Measuring Voltage and Current
18 n resistor in Figure P23.7?
Section 23.5 More Complex Circuits
b. Draw a graph of the potential as a function of the distance
traveled through the c ircuit, tra velin g clockwise from 17. II What is the equivalent res istance between points a and b in
Figure P23. 17?
V = 0 V at the lower left corner. See Fi gure P2 3.9 for an
example of such a graph.
8. I a. What is the potential difference across eac h resistor in
Ion
Figure P23.8?
60 n
b. Draw a brraph of the potential as a function of the di stance
42 n
trave led through the circu it , traveling clockwise from
30 n
40 n
V = 0 V at the lower left corner. See Figure P23.9 for an
examp le of such a graph.
Ion
V(V)
6
IOn
15 V
b
FIGURE P23 .18
20 0
2
0 "-------''-"-- - - Posilion
FIGURE P23 .8
b
FIGURE P23 .17
4
FIGURE P23 .9
9. I The c urrent in a c ircu it with only on e battery is 2 .0 A.
Figure P2 3.9 shows how the potential chan ges when go ing
around the c irc uit in the clockwise direction, starting from the
lower left corner. Draw the circuit diagram.
18. I What is the eq ui va le nt res istance between points a and b in
Figure P23.18?
19. II The curre nts in two res istors in a c ircuit are shown in
Figure P23 19. What is the vallie of res istor R?
3.5 V
2.5 A
3.0 V
R
200 n
11.5 A
4.5 V
--==--
l--==--
1500
Section 23.3 Series and Parallel Circuits
10. I What is the eq ui valent resistance of eac h gro up of res istors
shown in Figure P23. 10?
2.0 V
FIGURE P23 .19
FIGURE P23 .20
Problem s
20. II Two balteries suppl y c urrent to the c irc ui t in Figure P23.20.
The figure shows th e potent ial difference across two of th e
resistors and the value of the third res istor. What curren t is suppli ed 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
13.0A
~
3.0 n
1 - IOOmA
FIGURE P23 .21
FIGURE P23 .22
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
5.0n
R
E
5.0n
Ion
5.0n
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.
16 n
9.0 V
n
R
-=-
FIGURE P23 .29
3.0A
3
FIGURE P23 .28
3.o n
4.5
5.0n
Io n
IOV
22. I What is the val ue of resistor R in Figure P2'3.22?
23. II What are the res ista nce R and the emf of the battery in
Figure P23.23?
3.0n
771
2.0A
Ion
FIGURE P23 .23
FIGURE P23 .24
24. II The ammeter in Fi gure P23.24 reads 3.0 A. Find / 1' /2' and [.
25. III Find the current throug h a nd the poten tial d iffe re nce across
each resistor in Fi gure P23.25.
FIGURE P23 .25
26. III Find the curre nt th roug h and th e potent ial d ifference across
each resistor in Figure P23.26.
FIGURE P23 .30
30. II A photoresistor, whose res istance decreases with li ght inte nsit y, is connected in the c irc uit of Fig ure P23.30. On a sunny
day, the photoresistor has a resistance of 0.56 fl . On a cl oudy
day, the resistance ri ses to 4.0 kfl . At ni ght , the res istance is
20k!}.
a. What does the voltmeter read for eac h of these conditio ns?
b. Docs the voltmete r reading increase or decrease as the li ght
in ten sity increases?
A photoresistor, whose res istance
3 1.
decreases with light intens ity, is connected in the c ircu it of Figure P23.3 1.
9.0V
a. Draw a c ircuit diagram to illustrate how you would use a vo ltmeter and an amm eter to
determine the res istance of the
FIGU RE P23 .31
photoresistor in thi s c ircuit.
b. What do the two mete rs read
when the resistance of the photoresistor is 2.5 kn ?
Section 23.6 Capacitors in Parallel and Series
FIGURE P23.26
27. II For the c i.rc uil shown in Figure P23.27, find the c urre nt
through and the potent ial d ifference across each res istor. Pl ace
your results in a table for ease of read ing.
,-~-, 6.0n
6.0
24
n
ISH
v -=4.00
FIGURE P23 .27
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 eq ui valen t capacitance?
33. I A 6.0.uF capac itor, 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 capacitan ce of 50 ,uF, but you don't happen to
have a 50.uF capac itor. You do have a 30,uF capacitor. What
add iti onal capacitor do you need to produce a total capac itance
of 50 ,uF? Should you join the two capac ito rs in para ll el o r in
series?
35. I Yo u need a capacitan ce 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 iti onal capacitor do you need 10 produce a total capac itance
of 50 ,uF? Sho uld 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 capac itance of the three capac itors in
Figure P23.36?
FIGURE P23 .36
n"1
44. II The sw itch in Figure
P23 .44 has been in posit ion a
for a long time. It is changed 9.0V ~ 50fl
to pos iti o n b at I = 0 s. Wh at
are the charge Q on the capacitor and the curre nt I thro ugh FIGURE P23 .44
the resisto r (a) immedi ately
after the sw itch is closed? (b) At t = 50 J.Ls? (c) At t = 200 J.Ls?
FIGURE P23 .37
Section 23.8 Electricity in the Nervous System
37. I What is the eq ui va lent capac itance of the three capac itors in
Figure P23 .37?
38. III For the circuit of Figure P23 .38,
a. What is the equ ivalent capac itance?
b. How much charge flows thro ugh the battery as the capac itors are be ing charged?
L
~ 4.0~F
:r: '0
-. 11 F
TL__--'T 1.0 ~F
12 V
1'_V _ -
FIGURE P23 .38
FIGURE P23 .39
39. III For the circuit of Figure P23.39,
a. What is the equi vale nt 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 sc harge of the capac itor in
Fi gure P23.40?
r
t.OkO
1.0 l1 F
45. I A 9.0-n m-thi ck cell me mbrane undergoes an action poten ti al
BID that fo llows the curve in the table o n page 76 1. W hat is the
strength of the electric field ins ide the membrane j ust before the
acti on potential and at the peak of the depo larizat ion?
46. III A ce l.l me mbrane has a res istance and a capac itance and thus
BID a c haracte ri st ic tim e cons tant. Wh at is the time co nstant of a
9.0-n m-thi ck me mbran e surround ing a 0.040- mm -d iameter
spherical celt ?
47. I Changing the thi ckn ess of the myel in sheath surroundin g an
BID axon c hanges its capac itance and thu s the conduct io n speed. A
myel inated nerve fi ber has a con ducti on speed of 55 m/s. If the
spac ing betwee n nodes is 1.0 mm and the res ista nce of segments between nodes is 25 MO , what is the capacitance of each
segme nt?
48. III A partic ular mye li nated axon has nodes spaced 0.80 mm
BID apart. The resistance between nodes is 20 MH ; the capac itance
of each in sul ated segme nt is 1.2 pF. What is the conducti on
speed of a nerve impu.l se along thi s axon?
49. I To meas ure signal propagati o n in a nerve in the arm , the
BID nerve is triggered near the armpi t. The peak of the action potenti al is meas ured 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 o n along thi s nerve?
b. A determ inatio n of the speed made by measuring the time
between the applicati o n o f a stimulus at the armpit and the
pea k of an act ion potent ial at the elbow or the wrist woul d
be inaccurate. Exp lain the prob lem with thi s approach, and
why the noted techniq ue is pre ferabl e.
50. II A myel in ated axo n con duc ts nerve impul ses at a speed of
BID 40 mls. Wh at is the signal speed if the thi ckness of the mye li n
sheath is halved but no other changes are made to the axon?
I.Ok O
General Problems
FIGURE P23 .40
FIGURE P23 .41
5 1. II How muc h power is di ss ipated by
INT each res istor in Figure P23.5 1?
4 1. II What is the time constant for the disc harge of the capac itor in
Fig ure P23.4 1?
42. III! Capaci tors won' t hold a charge indefini tely; as time goes o n,
c harge gradu ally mi grates from the pos itive to th e negat ive
plate. We can model thi s as a discharge of the capac itor through
an inte rn al "leakage resistance." A 0.47 F capac itor charged to
2.5 V wi ll ini tially disc harge 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 volt age to drop to
1.0 V?
43. III! A 10 J.LF capac it or ini tiall y charged to 20 J.LC is di scharged
through a 1.0 kf! resistor. How lo ng does it take to reduce th e
capac itor's charge to 10 J-LC?
RI = 12
-=-
n
9.0V
R! = 15 n
FIGURE P23 .51
52. 1111 Two 75 W ( 120 V) li ghtbu lbs are wired in series, the n the
INT combin ati on is con nected to a 120 V suppl y. How much power
is diss ipated by eac h bul b?
53. 1111 The corroded contacts in a lightbul b socket have 5.0.n total
INT res istance. How much actu al power is di ss ipated by a 100 W
(l20V) li ghtbu lb screwed into thi s soc ket?
54. 1111 A real battery is not just an e mf. 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
" intern al resistance," as show n in Figure P23.54 . A typi cal bat-
Problem s
c
55.
56.
INT
57.
58.
INT
te ry has 1.0 n intern al res ista nce d ue to
imperfecti o ns th at limit curren t th rough th e
batte ry. W hen there's no curre nt throug h the
l.on
battery, and thus no vol tage drop across the
inlernaJ res istance, the po tenti al diffe re nce
1.5 v
between its termi na ls is 1.5 V, the val ue of
the e mf. Su ppose 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 pote nti al difference between the terminals of the
baHery?
b. Wh at fra cti on of the battery's power is diss ipated by th e
internal res istance?
1111 For the real battery shown in Fi g ure P23.54 , calcu late the
power d iss 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 sugges llhat max imum
power d iss ipat ion is achieved wh en the ex ternal resistance R
eq uals the intern al resistance. Thi s is true in ge neral.)
1111 Batteri es are rec harged by co nnecti ng them to a power suppl y (i.e., another battery) of greater e mf in such a way that the
c urre nt fl ows ill to the pos itive termin al o f th e batLery bein g
recharged , as was show n in Exam ple 23. 1. Thi s reverse current
thro ugh the batte ry re ple ni shes it s c he mi ca 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. S uppose 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?
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?
II The 10 n resistor in Fi gure P23.58 is d iss ipati ng 40 W of
power. How much power are the other two resistors d iss ipatin g?
I
j
5.0n
c;;
R
Ion
20n
20~F
773
62. 1111 A 9 .0 V battery is connec ted to a w ire made of th ree segINT ments of diffe rent metals connected one after another: 10 cm of
copper w ire, the n 12 c m of iron w ire, then 18 cm of tu ngsten
w ire. A ll o f the w ires are 0 .26 mm in di ameter. Find the pote nti al d ifference across each piece of w ire.
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? Expl ain .
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 vers us position, start ing fro m
V = 0 V in the lower le ft corner and proceed ing cloc kw ise.
See Fi gure P23.9 for an example.
6.0f!
12f!
~
-=- 6.0V
[2V - = -
'-'+---'- Your measuring
circuit
R
500 ~A ammelel
Lwvv-----J
FIGURE P23 .64
50.0f!
FIGURE P23 .65
65. JI A circ uit you're buil d in g needs an amm eter that goes fro m
o mA to a fu ll-scale read in g of 50.0 m A. Un fortunately, the
on ly ammete r in the storeroo m goes fro m 0 p.,A to a full scale readin g of only 500 p.,A. Fort un ately, you can make thi s
ammeter work by pUllin g it in a meas uring circuit , as show n in
Figure P23. 65 Thi s lets a certain frac ti o n o f the current pass
through the meter; knowing tbi s value, you ca n deduce the total
curre nt. Assume that the amme ter is ideal.
a. What value o f R must you use so that the meter wi II go to
fu ll scale when the CUlTe nt I is 50.0 m A?
Hint: When 1 = 50.0 mA, the ammeter shoul d be read ing its
max imum val ue .
b. What is the eq ui valent res istance of your meas uring c irc uit?
66. II A c ircuit you're bui ld in g needs a vo ltm eter th at goes from
o V to a full -sca le read ing of 5.0 V. Un fo rtunately, the o nly
meter in the storeroom is an am meter th at goes from 0 /-LA to a
ful l-scale read in g of 500 p.,A. It is poss ible to use thi s mete r to
meas ure vo ltages by pu tt in g in a measuring circ ui t as s how n in
Fig ure P23.66. What value of R must you use so that the meter
w ilJ go to full scale when the potenti al d iffe re nce .6. V is 5 .0 V?
Assume that the ammeter is ideal.
20V
FIGURE P23 .59
FIGURE P23 .58
59. jjji At Ihi s in stant, the current in th e circuit of Fi gure P23.59 is
tNT 20 rn A in the d irecti o n shown and the capacitor charge is
200 /-Lc. What is the resistance R?
60. I W hat is the equ ivalent res is100 0
[00 0
100 n
tance betwee n po ints a and b in
loon loon
Figure P23.60?
b
loon
FIGURE P23 .60
6 1.
You have three 12 n res isto rs. Draw di agrams s howin g how
you cou ld arrange all three so th at their eq ui valen t resistance is
(a) 4 .0 H, (b) 8.0 n, (e) 18 H, and (d) 36 n.
4.00
R
6V
Your measuring
circuit
500
FIGURE P23 .66
~A
6.0n
24 V
240
8.0n
240
ammeter
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 eac h res istor. Pl ace
your resul ts in a tab le for ease of read ing.
68. II You have three capac itors. Draw di agrams showing how you
could arrange all three so th at the ir eq ui vale nt 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. II Initi ally, the sw itch
in Fi gure P23. 69 is in
pos ition a and capac itors C 2 and C] are
un charged . Then the
sw itch is nipped to
pos ition b. Afte rward,
what are the charge o n
FIGURE P23 .69
and the p otenti al d iffe re nce across each capac itor?
70. II The capacitor in an RC c irc ui t with a tim e constant of I S ms
IN! is charged to 10 V. The capacitor beg ins to d ischarge at t = 0 s.
a. At what time w ill the charge o n the capac itor be red uced to
hal f its initi al value?
b. At what time w ill the energy stored in the capac itor be
reduced to half its initial val ue?
7 1.
W hat val ue res istor will di scharge a LO tLF capac itor to 10%
of its initi al charge in 2.0 ms?
72. 1111 The charg ing circuit for the flas h system of a camera uses a
100 tLF capac itor th at is c harged from a 250 V power s u\?p ly.
W hat is the most res istance that can be in series with the capac itor if the capac itor 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 isc harged through a 100 n res istor. The d ischarge current decreases to 25% o f 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 isc harged
through a resistor. Figure P23.74 shows the capac itor vo ltage as
a functi on of time. What is the val ue of the resistance?
Opens at I = Os
6. Vc (V)
\
30
60n
20
Ion
lOOV
40n
10
0
2.0 JlF
I (ms)
0
2
3
FIGURE P23 .75
FIGURE P23 .74
75. III! The sw itch in Figure P23.75 has been cl osed for a very lo ng
time.
a. What is the charge on the capac itor?
b. The sw itch is opened at t = s. At what time has the charge
on the capac itor decreased to 10% of its ini tial value?
76. Illll ln termittent
Switch
w indshi e ld w ipers
use a vari able resisVOl riable
loon
tor in an RC c ircu it
resistor
the
to set
delay
b
between success ive 12V
"
passes of the wipers.
l00J,.(F
47kfi
A ty pi ca l circ ui t is
show n in Figure
P23.76. When the
FIGURE P23 .76
sw itch cl oses, the
capac itor (initi ally 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, triggeri 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 seco nds
fro m the start of a cycle to a pass of the w ipers?
77.
BID
78.
BID
INT
79.
BID
b. To decrease the time, should th e variable res istance be
increased or decreased?
III In Example 23. 14 we estimated th e capac itance of the cell
membrane LO be 89 pF, and in Examp le 23 .1 5 we fo und that
approx im ately 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 o n potenti al, esti mate the total
num ber o f sodium ion channels in the membrane o f a nerve cell.
1111 The g iant axon o f a sq ui d is 0 .5 mm in d iameter, 10 em long.
and not mye li nated . Un myel inated ce ll me mbranes behave as
capac itors w ith 1 tLF of capac it ance per sq uare ce ntim eter o f
me mbrane area. When the axon is charged to the - 70 mY restin g poten tial, what is the energy stored in thi s capac itance?
II A ce ll has a 7.0-nm- thi c k mcm brane with a total mc m brane
area of 6 .0 X 10- 9 m 2 .
a. We ca n mode l the ce ll as a capacitor, as we have secn. Wh at
is the magn itude of the charge on eac h "p la 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 thro ugh a patient's
torso in order to SLOp dangerous heart rh ythms. Its key part is a
capac itor that is charged to a hi gh voltage. The pat ient 's torso pl ays
the role o f a res istor in an RC c irc ui t. When a sw itch is closed, the
capac itor d ischarges through the patient's torso. Ajo lt from a de fi brill ator is inten ded to be intense an d rapi d; the max imum curren t is
very large. so the capac itor d ischarges quic kl y. T hi s rapi d 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 typi cal defibrill ator has a 32 tLF capac itor charged to 5000 V.
The electrodes connected to the patient are coated w ith 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 Whi ch 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 sw itch is closed?
a
(
I
A.
o~
o
B.
c.
D.
FIGURE P23 .80
Problems
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 subsequen t j ol ts?
A. The initial curre nt 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 c harged to a lower
vol tage. How will this affect the time consta nt of the discharge?
A. The time constant will increase.
B. The time constant will not chan ge.
e. The time constant will decrease.
Electric Fish BID IN!
The voltage produced by a
sin gle nerve or muscle ce lJ
is quite small , but there are
man y spec ies 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 spec iali zed
disk-shaped cells called
eiectroc)'te!>". The cell at rest
has the usual pote ntial difference between the in side
and the outside. but the net
potential difference across
the cell is zero. An electrocyte is co nnected to nerve
fibers that initiall y 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
Electrolyte during depolarization
of one side of cell
Sodium channels o(lt!n
oilihis ~ide only.
\
+-
-
v
+---.--1--\------+-+--x
,-\
6. V ",II
- - -
- - - - - - - - - --
There is a net volt;lge
across the cell.
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 ca n 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 insta ntaneous 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
grou ped differently than in the ee l; eac h 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 hi gher resistivity of salt water requires more vol tage but
lower CUlTent.
87. I The eleC lri c catfish is another electric fish lhat produces a
vo ltage pulse by mean s 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 chan ge 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 isting stacks.
D. The thickness of the electrocytes increases.
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 fun ctioning 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 sin gle 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.
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 conservatio n 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.