# Ch 31 solutions

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Someproblemsdeal with the analogybetweenan LC,ele,ctricalcircuit and
€.)
4
v'
1lk' and I
system.Rememberthat4 e x' L # m' C
you should know the relationshipbetweenthe maximum chargeon the capacitor and the maxlmum
wdflen in lermsof
cunent:,I- 6oQ.You shouldalsoknow that the totalenergyof t}e circuit canbe
chargeon the
the
maximum
: 8212C = LP/z Someproblemsask you to compute
.irit".-"ftn"r",7
capacitor,given the maximum potential differenceacrossthe plates: Q CV^u*'
of oscillation changes
Someproblemsdeal with dampedcircuits.You shouldknow that the ftequency
the maximum chargeon the capacitor
with the addition ofresistance and that in succeeding.cycles,
the relevantfactorbeing e-"""'
exponentially.
decreases
you shouldknow how to calculatethe reactancesand the impedancefor an RIC seriescircuit:
7-
Xr = aaL,Xc = l/a(rL' and
a'+(xr-
a;ais lhe angularfrequencY.Some
problemsgivethefrequencyfrandyoumustuse
then
emf.If f,,: fi.,,,sin a11t,
you shouldknowhow to computethecurrentgiventhe generator
i=tdlrrtri f),whereI:8./ZurrdtanO:(XL-XdlR.Thintoflasaphasedffirence'Thatis'if
"{'= V,,,sin(aat+ a), theni = I sin(aat+ o - il'
the individual
Giventhe current,you shouldknow how to computethe potentialdifferencesacross
- n/2), andvY= IXL sln(adt+ El2)'
va : lR, vc: D(c sin(roat
circuitelements:
suppliesenergyat the rate PE : i 8 ' theinductorstores
Finally,you shouldknow thatthe generator
generates
tn" tutePL: ivL,thecapacitorstoresenergyat theratePc: lvc' andtheresistor
"irgi';
therial energyat the rate Pn : i2R Theseareall time dependentquantities'
You should
al.egivenor requested'
thermsvaluesof currentandpotentialdifferences
Sometimes
hold for thepotential
- fl, then1,*.:1/ J2 . Similarexpressions
thatif i : I sin(roat
remember
:
differences.You shouldalsorecognizethat equationssuchasv1 IXL sin(a:at
f'L rms - rmszrl.
Questionsand ExampleProblemsfrom Chapter31
QuestionI
leastpositivephase
F-oreachof thecurvesof q(t) in the figurebelowfor anLC circuit,determinethe
constant(Din the equation q = Qcos(at + @) requiredto producethe curve'
wt+\$-:o
qru
r)
c)
----)
(--
- d/*
p>o a',"x*'"XM UY
/<o
n **aA-Poi^{c\}
Question 2
The figure b91owshowsthree oscillating LC circuits with identical inductors and capacitors.Rank the
circuits accordingto the time taken to fully dischargethe capacitorsduring the oscillations,greatest
first.
r,an nar.*"t 1;^1.f:'rAf"Z* r= Y+
w =-L{fr:},'f
..?.
+=*[email protected]
= i\rr
iLC
I
Jb,?,' I
Question3
The figure below showsthe current i and driving emf t for a seriesRLC circuit. (a) Does the current
lead or lag the emf? (b) Is the circuit's load mainly capacitiveor mainly inductive? (c) Is the angular
frequencyro6ofthe emf greaterthan or lessthan the natural angularfrequency co?
\
fr,
t)Jrv\9^v\
)
tr^
JlQ.d.d-a
(t tt
b)
(,
)
W
tt'(Jc E,",.-,,,,)
UJa< ur) X. > X.
,T^dr4.lij
A'ytr^,t,,)n;
Question4
The figure above
ion 3) showsthecurrenti anddrivingemf t for a seriesRLC circuit.Relative
doesthe currentcurveshift leftwardor rightwardanddoesthe amplitudeofthat
to the emf
or decreaseif we slightly increase(a) L, (b) C, and (c) co6?
A./
j (X,,',.,..-.--
, @
At
L)
t-ai r ( tu
o-! ft2Wet L .tor*'',-*,..f...r..'
)
"
( t,/*
wvt&aAer) &4<t.)4y^**)
( X
c)
Problem I
)cg4.4-r-"""^,
-*)
)
In an oscillatingLC circuit,L : 1.10mH andC = 4.00pF. Themaximumchargeon the capacitoris
3.00pC. Find themaximumcurrent.
'A
^ +-l*.n+)'\tEdt p^\y%
I
WhrA
I
^"'^4""fi<t"*^frX/nuJ
_+-
-F- .l-- .
. i'VvnJt her^n
I
,/v.,,?'144
i\
VE
=-
qe/
a/gL
=
\)r= t1.Li\ = y^tt};r^1"**/)
l\
r\
^)r
u= ve*L,rt3= a/1.=
/ra/. = /^ Lr^
7-_L -
A,/--/It.
lLe
-
0X. r'et'(wt+
3 . o o" l o - o c
i;
{ t " ro' 1j ; 1v,or",l"^F)
{rr"
r^n*
\$)
=
Q
*.
ot".'t(*t'6)
= ^"\zr( &^,jr
a\ o.ryt,h
_7-- Tvvaaqrz'&r..*th-94
f = 4"5A*to-'A
* Jr,.t
|
Problem 2)
An oscillatingLC circuit consistingofa 1.0 nF capacitorand a 3.0 mH coil has a maximum voltage
of 3.0 V. what are (a) the maximum charge on the capacitor, (b) the maximum current through the
circuit, and (c) the maximum energystoredin the magneticfield of the coil?
q)
C = t.o, ldi F
f
-pm ^ .-a^.;X"
q = CV
Q = CV = (l.o*to1 F)(a.ov)
= f .o,.g6t'g
3.onC--3.oxlaqc-
V^*a = 3.oY
b) te'e ^*% l) = /a,V- :
Jr: aV. ---Jf = e/C.
r=
(z.ox 16'H) ( t'o " to'rF)
f=
Uu= TrLir----+ \)p,.*= YaLr' :
c)
t/" LI'
(r---Q)
l.'7'lD-'A
'/rG.ortorH)(1.?,rulA)a
U g,"n,,= 4.5 ^ lo-q1Problem3
In anoscillatingLC circuitwith C : 64.0pF, thecurrentasa functionof time is givenby
I in amperes,
, = (1.60)sin(2500t+ 0.680),wheret is in seconds,
(a) How soonafter t : 0 will the currentreachits maximumvalue?What are (b) the inductanceL and
(c) the total energy?
t = - I tt"^,(ut" 6) = 1-t,ooA)-a;r[(rsooi')Z + 0'6sa^'i.l
(o)
(l.*
^rzfrlt L^lA{ -alo(wt"@)=1
uiJ4 1,,^^* /r r#4i^,ur/wr
n\r,1"'\rh^-k
d I.t o A --.5 ;y'
= n/e. --.'
ot"fl
wt*d.Ya
= !-, i = -l.aoA ; fu n,rao.t -hoaq
j;au,t"6)
1 /r/IrI
"t
--/g,q.t\ b^'AL.,'k4 n
\$*;*rq
r/a - o"oroJ
f = TI4:L =
.)
t= 3.5t",16qs
L ^bbo 5 /
( b)
*^
*,
L c ct",,o,^) :
tl
w= /{lz --
Cc)
L_
L---ro}c
=
6F)
(esoor;^(6 '1.o4o
Xoi,! -e^,-nv U=QZa.=/aLf'
| \ l/
/ ->
u - /x/-J
---)
l-= J,"5o*to-3H
Problem4
L :22a wH andcapacitance
in serieswith aninductance
R shouldbe connected
what resistance
of its initidl valuein 50.0
C = 12.0uF for the maximumchargeon the capacitorto decayto 99.0%o
co'o o.)
(Assume
cycles?
-B'L/rt
= Q 2""'^" n-o (-'t"65
4
%
^
2c Q e
= g"r^.4(
ma (vct,d) urt*
= F\ -At/tt
-
4*o
rr
q**" GJe
)^ (s*/a) = -At/^,
- At/tt
'
r
,
*Y.&""
G" ^'^'" 5d-")
-ht/tt-
--------+
o.iq o
g*".,.=
,..,,/tt>
A -
= Q;^'t"
,
J\
'@a(utL-6)
-AY^,
-P^r(o"?q) =
/<= -A7.9" (0.9,
* ,ucp nzeA b;.a 4n 5o 4pA
t = 60_r
= 5o(a-nfZ)
f
= - l(?ao'ro-)t) g\ i:.,1\)
(o.st'os)
(a) At whatfrequencywould a 6.0mH inductoraada 10 pF capacitorhavethe samereactance?
ib) What would the reactancebe? (c) Showthat this ftequencywould be the naturalfrequencyof an
oscillatinecircuitwith the sameL andC.
x"=
d;
zc,u+at,^-h* al,a^*a^r.s
(a) -rrr. ur** f -Ar\ X"= X" ---+
,l\
--b
t LL
, l ( a - o * r c i ')H( r o - i o " F )
(b)
X..
4.
\-../
r,,J
+ L = (4orA
"^A/r)
,/ttl,a
=
,ltndts
),t,^-.1
--,
f= G5ollz
(6,o,.169p) ---r
I
-
= @tL
= QotX-A/s
--1 \$=*/^n
w=).irf
fi
(1o8Aaa1t11
lo ^ td" F/
X " = Av.s-.'ir-
( c ) f\/d*J ^!
a/xn = _,F1
+*Sr-rr,'".J "\$ r"t LC rc.^^r-it : f =
arr [/C
7
\
-- L
( n l l |- |
\Ja--
t
Problem6
- n/4), wheret^= 30.0V andao : 350radls.Theeurrent
hasemf e : t- sin(coat
An acgenerator
- 3nl4),where1= 620mA. At whattime after
producedin a connected
circuit is i(t) : I sin(ro61
=
i 0 does(a) the generatoremf first reacha maximumand(b) the currentfust reacha maximum?
(c) The circuit containsa singleelementotherthanthe generator.Is it a capacitor,an inductor,or a
the case.may be?
t = t,' i;
i
-
(h)
/
).^-rr\ ^r.rfdJ t-"A)
t.
=t
= tp.)Onft
Drt-T/,t = TA
6L
3n/4
wat.
3r/^-,
' ,--d =
1=
t', = 3o-ov
(c.1*1 -='/q)
,a-.v^-.,
= f
(")
U)4 = 35Aa-"1/+
(,n*t-Yo)
3n
--J
lQso^il1,,)
(.?3 r lo-35
-^^'A.4 td+t - 37q = -/r.
(r.F
t =f
ulrt = Sr/u
5-
l_ ,_ srt1..., =
/.r\pa,
(.)
t :
1pso"_ll+)
/ul^/\!^"*glaAa ^\$\ rn, emf
t\t
! ,r*l-*.
|
(r)
-t_ t A t
VL'
/,
l. )1" )o-trs
I
I
E tt-
At"ala7,L"%)
(at^r'*l^a"
llro TcE *"n
Xr'uDoL
L-m
(r...r.z-;
Jwd
L--
( ao.ov)
(r"ro^1614;1 ^4/a)
"sD
--'-)
zl=D")3t'H
Problem 7
Removethe inductor from the cirouit in the figure below and set R = 200 O, C = 15.0 pF,,
: 36.0 V. What are (a) z, (b) { and (c) I? (d}E+a.*?ptra's-draetlaq.
fa:60.0H2, and S,
-ffi,
*t
c-
Arif"C
) ( I 5.0-dlF)
!,n^ (Go.ortz
X" = l77JL
a"L-o
Xr=
t
t = J l'+(X.-X.)"
(q)
t oC
=
-
-\ -r / X.-X" \
)
__ &^-'10-t"?-rL) ---+
p = .t.an \ _7_/
\
AoO-'L /
( D/
{A\(d.^.)^
-tlt.5"
6 =
r---__-..--l
3r"'ov .>lJ=o'135AI
= +=
f= Y--,
7 = ),lo'7s-r-
/ 6a.oot)a(o 1rh)"
z
ab"7rL
Problem8
(a) hr anRLC circuit,canthe amplitudeofthe voltageacrossan inductorbe greaterthanthe
amplitudeof thegeneratorem{?(b) ConsideranRLC circuitwith 6- = 10V, R : 10A, L = 1.0H,
andC: 1.0uF.Find the amplitudeof the voltageacrosstheinductorat resonance.
(")Ml
(/ , t)\
*n A LC ^,,-rr' L r f
&r
5
= :.9'
a-t ,^*"--.-',r.q , X,_ = X.
z
Ao
/-._..-..-._--?
7= tr Ar*(!.-X.)'
Z=a : l,o-st-
(
o
u)d
,u,^uAtp+r,"l*^
d+ =* ==ft
I
.-.-....--.:-l_-:.t ta
\ c-'-
t.
y' (l.otl
V"=f X.=U
X r . o , l o 6 Fl
il+C
Y7-.-
f X. = .D (o"L)
e) (,.2
( t"or+)
Qlk) (rca"",t/,)
=
V, . looat/
(. r.oA)
( V oo,.t/5)qt.otfl
-| ooot/
l
a
Problem9
in series
and.a0.94pF capacitorareconnected
. Acoil.of inductance
88 mII.andunknownresistance
voltage.and
with an altematingemf of fiequoncy 9i30Hz.If thephaseconstantbefweenthe applied
of thecoil?
thecurrentis 75', what is the resistance
L= Tl*lo'tr
C - o"9{Ylo*c-
/" L! --
Jw'^
A=?
l= 93otl*
1e-fi r) -
A=
q= ?5"
\t--
A= Lj.
)ba"r
I
I
(angnc;
J
l\,D'|Yl|I
I
B=
)."rr(t:o#z)( fi"rc9*) - fry1";66;,15o.7
.,1r.,1 ?5'
I A= 9qrL I
rveraserate at which ent
= &,o"I"^" o-d
t.
I
i
t.-, ( €,^,/z)(A/s) "---z
(")
(")
()\
an"fl
ALc,ct^auL Jnga^a'rnar , 7--B (X.=X.)
-*"', o gt"u*t1 ,;,f-.fr^tu /.;tt r-*) A =6
Z-"",Ar/ro
,o.
n\^'/^
v
^\4
)
Jr(\t
\4\4
9aJ
)A
I
)
,la ra alarttzv"rr..'
Problem I I
The figure bglow shows an ac generatorconnectedto a "black box" through.apair ofterminals. The
box containsan RLC circuit, possibly even a multiloop circuit, whose elementsard connectlonswe
do not know. Measurementsoutsidethe box revealthat
8(,)- fl5 0iDsinru6r66 i(l) - (120A)sa(r'ral* +z o')
(a) What is the power factor? (b) Does the current lead or lag the emfl (c) Is the circuit in the box
iargely inductive or largely capacitive?(d) Is the circuit in the box in resonance?(e) Must therebe a
capacitorin the box? (f) An inductor? (g) A resistor?(h) At what averagerate is energydeliveredto
thi box by the generator? (i) Why don't you need to,know the angular frequency w6 to answer all
thesequestions?
i = j|-^,.(ddt-d)
,{r}
*,,\$1
LJ-i
Q
6
Fnd=
)
ptt
^\r.d
&
ooq 7v.r'l/r.Jz? (i8
X. -X.
R
(tcc1
Q
=o'
Arr"'reQ=ln/^
/Ur;""
day,
A*O
/'..Yu,'+-!)< cd4^7'lh,v(,
q n-e;",^tbtat,",) a c*-rtoli)at
\$eT-l-a
l,toA
- - q^'o'
: fo'l;t
tA^AfuAcD+*uh,(J) n* of nu-.o.*n
,
Z=
)
(e) _l,'p'l
C+)l;l
___r
.t
F''-
., .frwg )44e\'[email protected]
t
*
X.
(
(',) ?^*=t^,4., *d = (uyd(r/a)^p
= t/t e
^T,t;a"\$ = |/1(rr.o,r)(t.eo+)(o.r,l3)
= 3?.,1W
Problem 12
A' L: 6Q'0mH'
In an RLC cireuit such as that of the figure below, assumethat R: 5'00
averagerate at which
f^:60.0*z.and tm: 30.0 V. For what valuesof the capacitancewould the
ii#
l;ii,1oj1"'i'i*r'. '"'i'*"e
whatare{c)the
and(b)aminimum?
be(a)amaximum
rate andthe corresponding(d) phasgangleand(e) power factor?^Whatare
;il;;-tr*.1pation
and (h) power factor?
?Oifr" *"rC*.r* dissipationrate andthecorresponding(g) phaseangle
m ln rmo.rr
i."_
-- 6.odJL
e^r.t A
/-= @o"Oxlo-tt
Z^
4
/^
iJ = 60'oHz
9 tYi = 3 o , o r /
\-
(O
, olte
,{" ( X.-X-)'
=
o.rnro t,'aqa X. X.' ) dr u]dL = #r"
t)o
*r=#
-->c=itL =
d+"ru
|
c = (;)6;;G'o**n)
(t)
? = €1-. rq
f-=,r-"**-1
rr l:rdll=1
La --
-
Z \o q,^.naa u."q,n 4^+ (X.-X)
^]*
(x"=*)
o * 4 urar>Ao<4!1A*[email protected]
(€/::ia
(.) ?**= "- ?*-=
:
+
z '\
\
,
\/
\/
\ d,t T,'\ Q = a_!:
2.
=O
(U) p-"ttrry'.A='-q
Da,^oo =n
I
\f/
tll
b,
d = 4:eJ: -= -oD
= coa (-to)
=El
-qo'o\$-
A
Problem13
coil has
supplies100V to thepriinarycoil of a transformerof 50 tums. If the secondary
A generator
voitage?
500tums,whatis the secondary
Vp = loov
Vu=?
Nu = 5o
N"= 5oo
..-
rv/ t - - i / / N " l . f
"P\
/Nl/
V, = (roov\(5"/r")
tv*';t
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