A.,u;;=
Physics48 (Fatl201l =::-Chapter27: Circuits
" It is good to have an end to journey toward; but it is thejourney that matters, in the end. ,'
UrsulaK. Le Guin
"Nobod1,made a greater mistake than he who did nothing becausehe could only do a little."
Edmund Burke
Reading: pages705-724
Outline:
+ emf andterminal voltage
= combinationsof resistors
series
parallel
= Kirchoffs rules
junctionrule
loop rule
:+ calculatingpotential difference
= ammetersand voltmeters(PowerPoint)
= RC circuits
chmginganddischarginga capacitor
time constant
ProblemSolvingTechniques
You shouldunderstandthe definition of emf in termsof the work doneon a charseasit moves
throughan emf device.You shouldalsoknow that real emf deviceshaveboth erifs and intemal
resistances
and,becausetheydo, the terminal potentialdifferenceis not the sameasthe emf if
chargeis passingthroughthe device.Be awarein eachinstanceif the currentis in the direction of
the emf or in the oppositedirection.The terminalpotentialdifferencedependson the relative
directionof the crirrent.
You shouldbe able to write down Kirchhoffs loop equationfor a singleloop and solveitTor an
uknown (the current,a resistarce,or an emf). If the currentis known,you ihould be ableto
computethe potential differencebetweenany two points on the circuit. You shouldaisobe ableto
computethe rate with which an emf suppliesenergyand the rate with which a resistordissipates
energy'Rememberthat for anyelectricaldevicethe rate with which energyis hansferredto or from
the deviceis the productofthe currentthroughit andthe potentialdifferenceacrossit.
You shouldknow how to write and solveKirchhoffs junction andloop equationsfor multiloop
circuits. Draw a currentarrow in eachbranchof the circuit andiabel eacharrow with a different
symbol.Write thejunction equationsfirst. The numberof equationswill be onelessthan the number
ofjunctions. Thenwrite the loop equations.The numberwill be suchthat the total numberof
equationsis equalto the numberof branchesin the circuit. Solvethejunction and loop equationsfor
the unknowns.
Someproblemsaskyou to calculatethe equivalentresistancewhentwo or more resistorsare
connectedin parallel or series.You shouldalsoknow how to useequivalentresistancesto solve
circuit problems.Rememberthat for a seriescormection,the currentis the samein all the resistors
andis the sameasthe currentin the equivalentresistor.The potentialdifferenceacrossthe
equivalentresistoris the sumofthe potentialdifferencesacrossthe resistorsof the combination.For
a parallelconnection,the potentialdifferenceacrossall the resistorsis the sameandis the sameas
the potentialdifferenceacrossthe equivalentresistor.The currentin the equivalentresistoris the
sumof the currentsin the resistorsof the combination.
Someproblemsdealwith a resistorand capacitorin a seriescircuit, with the capacitoreither
chargingor discharging.You shouldknow how to solveproblemsusingthe expressionfor the
chargeon the capacitorasa functionoftime; q(t) : C6(1 - e-') for a chargingcapacitotand
q(t) = qpe-/t for a dischargingcapacitor.Heret = RC is the capacitivetime constant.Also know how
to computethe cunent in the circuit andthe potentialdifferenceacrossthe capacitor.Someproblems
askfor the time or the capacitivetime constatt.If q and qsaxegiven, write q/qo= e-o"andtake the
naturallogarithmof both sides.The result is ln(q/qs): - t/t. You cannow solve for t or t.
,J
Questionsand ExampleProblemsfrom Chapter 27
Question2
maze.In the figure, al1the resistorshave
Res-monster
a resistanceof 4.0 O andall tle (ideai) batterieshave
an emf of 4.0 V. What is the currenttlrough resistor
R? (Ifyou canfind the properloop throughthis maze,
you cananswertlie questionwith a few secondsof
menlalcalculation.) l-l-;;-/
d'ot1
I L=
|
4"oV+ Q.ov+,i"or/ - 4.orl |"oV*i(+.ost).o
f
D.n
" /)
>l'
= 1.oA
Problem1
-f
In tfre figure below, , = Dy. andE z.= g v. (a) what is the direction
of the crmentin the resistor?
(b) which batreryis doing positive work? (c) which point,
z or B, is at the higherpotentiar?
{
I
t,
i,,C_c,_.r,.l..ti_.,l
f r--)4..,r, Lq'
--------
( L,\
) --i-L/nq .F* .''tr",""*^.1 *j*.-.*
ei i, _!i*fi2q
/-ko...,-,-",.,o
(c)
?-"} B
Problem 2
;y'<p
c*ur*l
( rctt'v,,pnJ ;sr:,. -L'-
t
11 .,:, _!_,,e,ri* .t*,u"rt*j_
JrJ-r.*_ $r-.,., :,7"
)
A
7, 9-r>'u*.,. i;i;-":.--."t-? j
6 -^,-, ! 4!*,
Thl figurebelowshowsa portionof a circiit
throughrihich therei. u
I = 6.ooa. irc
resistancesareRq: p, - 2.00R3= 2.00& * +.oo ol ffiat is
",*"rit
the equivalentresistance
of the four
resistors.
A,= At= {"o+-r.At-
f,r:
; "c'sr'^
= e-6ol)-
-p
= a-od-L
* 41 o",i4.1,.-,
-t r,.,L :
J*
4"-, = .4=n2,, = l , - o o l r t A " o b , L _
A r," P,A 3,.pM Walf&\
i
- -'
JA'a
n,
--L
F,2
!_ooa,
t
q.obiL
lJ
^.\
:..
4,a = A'Do-rL
6,^
Br, -.* >^ yar,aLl-( 1
4,a{
I
Aotq
-'-F
:
t)
/ 11),34
J-
8,^
+ -J-
f,r,
*l
,.oDJL
zl"oo1.,r i ti:9
l"33_-L,
Problem3
A circuit cqntainingfive resistorsconnectedto a batterywith a 12.0V emf is shownin the figure
below. (a) What is the equivalentresistanceof the five resistors?(b) What is the voltageacrossand
\-'l
the currentthrougheachresistor?
I
6.0e2
n
(\
'/1
'|
,+H"
I
4
h
E-axzJLlPf
I
=-'Lo-L=
B,^
e.
3.0e2
n"
n
R, R^
'
-Ll
6.o-ru' l:rr-
5.0Q
A,^= 4,==:. o l
*,-- -S
Ar^n R, tn,tpt^ta !
R,^z=
Ar^* R,
B,^, -' 4'oLt+ 4'o-n-= s+
Au+Buu,a:i)cz: 4*s = Bq"Bs
._J--*--l
Arr= 3.ol-r s.o.1 . 7+
flesf-l
lL----4l
ll'l
, r--'------------
/,03
+
/
4,,,N'
\ b-/
Brs *
|
Fa,\r{-QIsl :
=
4't.*,
= J,J_
t"o[
v
V,,./p
.
t t\?-,
-- rav = t.5A
g.OlL
Vz- i, Bu = ( t,5,{)(r"oa)
Lz= l"5A
V,t= i,'8,' = (l.s'n)({"orr)= G.oV
L t 2 , =L z --
|
= /' . 5F A A t
(pz
V" = 6:9i
V,=Vo=V,a= G . o t f
1',=l"oA
i,=V,/R.=9'?u^---.1r',=
l.oA/
6"o-IL
"
i^=Vy'n*=;*E=r.;t
irr- Vry't
"/Ar,
= lev '/s.or.-/'5A
Lq-- L9 = Lq5
-3lV,- ?"5V/
V, =,r4, = (r.s*)13.orr-)
=-l&
V,=isA, = (t.5N)(s.orr-)
= ?.1
,1,*
-i
Br,
4,.r", ' Q,osl^
t,osL
r=vs=il"ov
- rea -
t
= /" 54
:rs
Problem4
(a) what arethe sizeand direction of cunenl i1 in
the figure below?(can you answerthis making
only mentalcalculations?)@) At what mte i, ti" tutt#ropplying
energy?
\-/
t\t
l---1
I
a/1.
r---t
t,
r4^
4.0 Q
l).v
12V
A^
|
t)
/'\ |
4.0a
II
tt
?
"
tl
p
lxv =-z
?8'^
, I l;.3
ti
tt
A,oB^ a^4w) po.,o-t&-{
A ,r,' ?.o-tr
4o + B u aJ@)r\ -4r;4rr
i
A,^" . P,,^.A.-- !,.o-ru o/.2rL6.0sr_
V,^.A - Jlv
=
t,^r?3
"
/ H,^z
: e.oA
6_oJL
-\
7
*Y:
.l-
)v\ .4uri4.4 9^"re ,Vb _4_.{iryv;-uaud
= (X-on) ( a"a,sr)=
V^ =/''o4,*
oe A""t ao
a-D4
;
zt-oV
: q"oY
(b) J}to ^rili ^t ,tU' \ pp $"tr?
v
f)
Y=
A4 Aa Irl"pr+inga,pffi
),Dh\j'w t ,u W.ct'-.l^*"J fr"b'I\
/
\
\
att \/,^rr\
",-"fr,/(
t). v
)
*_-)
pv !r.fr0^y
;" Wr*, .UZ.
(|,n
=e"o*)
Problem 5
In the figure below, the ideal batterieshaveemfs E r : 10.0V and E: = 0.5008 r, andthe
resistancesare each4.00 O. What is the currentin (a) resistor2 and(b) resistor3?
__-J rz.
tr[1
tl
f ' /i.r
I L,
/{i
4,
Ilr
ls,
C)
(r)
+
+
C)-
h"')'1) -toa :
t,-/,8,*i"A"--o
+ i. (9,oorr)= a
loV- c',(4.oorr-)
'gwn ual* -l-rp :
- i"Bo - /nA' - t^ --o
t 3 (u""ust-)*/..(t,o.n-) + S-oV --o
,\Do
(lJ
\ro.
)6^Dn lI) --a
L1+[a=La
1r)
(a)
lo - 1t,+fl t^ =o
";
a,*r;t
L7-1,
tr;
=o
lo-'ltr+q(i"-ir)
Et4iat{i.=o
iB - ir- t,
-f-+
Lg=
(t)
lo+qi*-Di,=6
5+ ri i^' 4 U^-C,) = o
(:)
5 r 6c'a- 1t, =o
\
)
9-'-,-
,.,-
'-::::-
prd (E) --Ur)
fo-t{"'^-(lo+lLa*)=o
-4
4t^-16c"=O
+\D\ (q)
16* 4 [t-8;,
--O
---)
t,
(S)
l
19*,,o/"
lD -8.', =6
t
L4=o
o,,|
=)g!
8.oJL
= 1.a54
-t.)54
L ^ n^"1
L
W@,Y
L
| . 15 A t i "fi;,fr )^^;&-, ^auur*"a )
Problem6
Y: E, :.1.9ov, n' = 5.00o,,Rz
=2.00dt,tu = 4.00o, andborh
3-1-ll^ry:,1"]:'ll3 :.: 3..,00
enerry
is
dissipated
in(dnr,.@j
n:.mdGftszwhatis
:r1Tl::T,111T"H,1,^*X,:::1":l
the
power of (d) battery 1 gnd (e) bauery2?
&^, *af nep :
/^
C,-Lzhu1Lt.
1 1t = o
3"ooV_ i= (+.oa_rr_)
+ c", {9-oo-tl) = d
1
-/, B,- c""
A. - e, =o
-
z, ( 5.uc-n) - c'*(aaorr-)- i " o o V = c i
''
3atzfr^. *1
J
ct
*4...c:,1= d
(3)
t:.
--l
L
"'/
.'
F^f
,LtrvrO t aJ
3-4(ir.i,)n 5,",=o
3- 4;^+ 9 c",=g ----a i,=
I
(.r)-', (:)
E ItlJ]*t;"+
*t*^,=,
l=o
Ao/, ia. ?;. = l/q - |
\\-l-
C,=-0,^(a3A
|l
= 0.l5? - (-o.ar.:A)-: ir''t',
).
aiArlr^od
dt l^-a4142i, )a
"4Y/w\p
is = o,vrtA
f\
-t nA
\, )
O. l5F\^l
Problem7
What arethp sizesanddirectionsof cunent(a) i1 and (b) h in the figure below?At what rateis
energ/ beingtransfenedat (c) the 16V batteryand (d) the 8.0 V battery,and for each,is enerry
being suppliedor absorbed?The batteriesareideal.
2.0
3
iul
A,^=
1.0e)<
A
= -&;o'
-J.,=*)-
+
'5
8.0c):
-
---J
iA .,3 + Ll.rrqArl\,rr.a
'1I
8.0()
8.0v
t,-i^
8,2-'4"o1Y
Brrz, = B,r' B,, ff,,
Avzq = 4.olr* )..olt-+ a.oll * g.oru
.l
-----'
1?.3'1+5h
WM
4nzqs
^^
1l
12.0a
x2 >8.0 Q
- *p^",/.ed
l+)'-^^
(J
)_
A,ry
*J
= -_l__ + _lv'asL 7'Drr
As
= 4'oJ\
+6 ;
)?.395
-= L1'oJt+4'ott- = 3'os)A.o^;-'--> Artzqsre 4rr:cs +Av
Wu*P
^^*'LM
8,ov
l.0sr
:
'w
lt,V-l;(r,o-L)--a
* 8,oV + l b V -i*(:'ao) - f.(lon1
--J
z'" (z"ost+1"d_D)
b4
t,= tp*tu= 4"o* --->
( b')
A,^uuu+ R* n^^';er
V"rr"
--->
/rrur" = L",
-
d' eU l\
= tausfirz,t5= (A.oA)("r,drr) =
7.oV
-->
-- V,..rr, = 8.oV
Abz,1+Bu h WAAlfuL
Vru, V.
I ozt= v,^,'/ nor,: t'oYz,on= lra\ :,.
*1
A,n+ 4o * B, .;u,,.avv-u
gn
V12'=114611
(.) ?=i? -
=
t,^= iu = t, = a113q l.oA
= (1.o4)(4"oL) -* 4.ort >
4.ov=
__:-.-i^--vy'B^=
7,Dlt
cl
(d-) ? =cre--- (a.on)(r.ov)
V
= {u =
=
I
Problen 8
h the figurg below, if the potentialat poht P is 100V, what is the potontial at point Q?
--
v'i
rzt-+Al
5.() Q
| "]'Tl?tror'/ n io\IY
rurt'ff
1\-/f
tl't't'(1
/sov-;(a.or) -sov-i(z.oIz)=6
rooV=I(t.os,-*3.o-n)
,";P!
r.lJif-J,,
2.0 Q
i
I
t\
(a.or.')= Vr
V" * l5oV- (xoA)
f:&.:"
rc.:lact*tt't/-
j&p '.
(
I
i
+ lsov-4ov: V, ---+ V; = V, - lrov
I
Vq= (toov)-traV:l-t"il
,.rodhr*;
h-p :
r/
/n-a\/?.-\. * ra,r
5oV -- Vp
v. '(0'on)(3'orr')
Vq + 6or,/t5ov -- V, --:)
!
\'l
t
I
V. = 4 _1toV:
Problem 9
_
,"y_
In anRC seriescircut, 6: 12.0V , R =1.40 Mf), andC = 1.80pF. (a) Calculatethetime constant.
(b) Find the naximum chargethat will appearon the capacitorduring charging.(c) How long doesil
takefor the chargeto build up to 16.0ltc?
6 = 1r'or/
A = l'9o'lotrr-
_
\--
I
I
|
(o) 't---AC = ( f.roxro'*) (r.so*1o*opy
/ = ,..51.
Arys
l2=
c=r.'o'ro-";
(t)
|
I
%=CL(t-Ct/^")=.b"(1-;t/a.S g.=Ct
,g"=(r.fo).ro-bF)(rr.ov)
--. 9. = A;;l
W; Al. Q4
(.) q =%"(t-€.'t*1: 6o(t-€*/"1,n q/%"-l-;'/'
er/o = l-Vq" -
-c/e =9""(t-Vg,)u
I
I
I
I
t= - ? U(r-UqS
I
= 3.1os
'^.(| *H) _-_
f
t __(a.Srs)
lt=,1"t
I
Problem10
What multiple of the time constantr givesthe time takenby an initially unchargedcapacitorin an
RC seriescircuit to be charged lo 99.0%;o
of its final charge?
q c'a+D"r:f'oy
)
%=
Ct
q"=cCv
( t - €*/"-)
z=Ac
q=?"it-;"/")
I\
\ ;r*-,rr-n** %=O"q9%"
o.e9x" = %"( t- {t1") --.--,0"?q = l-e-'/'
dt/'=
l-D.11 - o"ol
't/z
= J'" (o"or)
-A7'1( ;*/')
-!)
t
I
t = tl"u!z
Problem11
A 3.00Mf) resistorand a 1.00.pFcapacitorareconnectedin serieswith an ideal batteryof emf
E= 4.00 V. At 1.00s afterihe connectionis made,what arethe ratesat which (a) the ciarge of the
capacitoris increasing,(b) energyis being storedin the capacitor,(c) thermalenergr is appearingin v
the resistor,and (d) energris beingdeliveredby the battery?
A= 3,oo"lDo-rr-
^ rn1,"c,,/,ot: %
C=l,oo'ro-6F
e = Q,ooV
.L =Ac = 3,60,
(u)
.-
t=AX/lt=7Bp1t/xe
(aJ o* t=l.oo:, [ = (\iill'n)
!.4
\ - *t
"-
: t (/.)
P= ;V
= Ct (t-s*/a'1
-
9.55^1d'7
n*t=ro,,.&=cr (l-uqa)
/^ -r ,)11-rr l^1.33^lD9-l
Q= (t'oo"ro*r)(q'oov)lf- ;,o*"1
P= l.og^lo"W
(c)
(d)
ro'-u)=F.r,.rc-"q
P = ;'B = (q.s5^1d1c/e)'(u.oo'
P=;L
= (t.55^lo1c1s)/a.oov)=
rnab-:
t==z)
3.tra*lO-bVi/