t\ 5-- <r--
Physics48 (FaII2011)
Chapter 25: Capacitance
"Theanswers
you receivedependuponthequestions
yozlask ,,- ThomasKuhn
"Life is a mirror and will reflect backto the thinker what he thinrrsinto jt ', - ErnestHormes
" Wat we seedependsmainly on what we lookfor.,' _ John
Lubbock
Reading: pages656-674
Outline:
= capacitors
introduction
definitionof capacitance
+ calculatingcapacitance
parallel-plate
capacitor
sphericalcapacitor(readon your own)
cyiindrical capacitor(readon your own)
= combinations
of capacitors
series
parallel
=> energystoredin an electric field
+ dielectrics
capacitors
with dielectrics
atomicview of dielectrics(powerpoint)
Gauss'Law (readon your own)
ProblemSolvingTechniques
To solvemanyproblemsof this chaptel you shouldknow the basicrelafionship
betweenthe
magnitudeq ofthe chargeon eitherplate of a capacitorandthe potential
V
difference
acrossttre
piates:q = cv' You shouldalsoknow how to cilculatethe capacitance
ofa parallel-plate
capacitor,
a cylindrical capacitor,and a sphericalcapacitor,all in termsofttr" g"orrr"t
v"u
f
shbuldalsoknow thatthe capacitance
ofa capacitorthat is filled *iih u ai"tlct"itrr"
i" or"lf""t"r.
a-i-.r""ti"
constantk is c: rc6, wherec6 is the capacitance
ofthe capacitorwithoutthe dietectrrc.
some problems
{eal with capacitorsin seriesor paralel. Ia eachcase,you shouldknow how to
com.pute tle equivalentcapacitance,the chargeon eachcapacitor,andihe potential
differenceacross
eachcapacitor,given the potentialdifference-across
the combination.
v
You shor:ldknow how to computethe enerry storedin a capacitot,given eitherits chargeor
potential difference.You shouldalsoknow how to computethe energydensityat pointsbetweenthe
platesor, in general,at any point within a given electricfield.
Questionsand ExampleProblemsfrom Chapter 25
QuestionI
For eachcircuit in the figure belory,arethe capacitorsconnectedin series,in parallel,or in neither
mode?
-t"
"I
(n)
-A-uv'a4
Question2
The figure below showsthreecircuits, eachconsistingof a switch andtwo capacitors,initially
chargedasindicated.After the switcheshavebeenclosed,in which circuit (if any) will the chargeon
the left-handcapacitor(a) increase,(b) decrease,and (c) remainthe same?
0,,{lou
2n-L
r,{lou
"-l-il
(tl
lc7\
!i.,
-A'amu (
--=
t cV
(^)
( b\
3
t.) O^r,-;} L
r,-.flloo
!(;"1*
_:]
!(;
(3)
.8" apns +^ **4)
V = X/. ^''^,*aJl-'{
Question3
Figurea showsa sircuit with tlree capacitors,and Fig. b showsthe circuit with their equivalent
capacitor,of capacitance
c=, yl.ri9hhgsachargeof 60 pc. without a calculator(and,#possible,
witlout written calculation),find (a) the chargeon and the voltageacror,
ib;
3Ld th"n
(c) the chargeon and (d) the voltageacrosscapacitor1.
"upu"it-
= [o'^rC
$ 'a $a
l0tF
(")
%:_-kes.c
( b) vu-%ul.=
H,
f-=r
c,a:*
f
L:f
Problem I
(d) V,* = %,^k,^=Gouc =
3V=V,=\A
A,o.tF
I
=
E
^::-'
acrlF
30,,rF
(') S '= c'V' = (
ro^^
f-)(3v) - 3gz g
A parallel-plate
capacitorhascircularplatesof 8.20cm radiusand 1.30mm separation.
(a) Calculate
the capacitance.(b) Find the chargefor a potentialdifferenceof 120V.
'=
'"cli..,o"i
')
(?,ri,.ro
"r,,,
a - (t.t5.,' , f
[ , t l r . i D - l , ot\;
r..:
r.
a--
I
c
l"'/3.io-u
Problen 2
Supposethat the two sphericalshellsofa sphericalcapacitorhaveapproximatelyequalradii. Under
theseconditions,the deviceapproximatesa parallel-platecapacitorwith b - a = i. siow that
Eq.25-17(C = 4nx0ab(b - a) doesindeedreduceto Eq. 25-9(C: e6Nd)in this case.
2. jt;*.
l;,-'t
:-!:t ...1.r.\r.,,!-i.,:..1,!il,e4
:!,.,..,1* .._i..i: t
=
I
",
-
-
,f q
;,
-: \--"
-':
i
> o'ta.
4 eD9,q 4J**
l
I
.,Arq -(!- -u-*^)fu*t A-!
fit
arq b= A
b-q - ld
v
-t,/
a
= t"k./ 1
/11
riTqb
Problem 3
In the figure below,the batteryhasa potentialdifferenceof 10.0V andthe five capacitorseachhave
a capacitanceof 10.0FF. What is the chargeon (a) capacitor1 and (b) capacitor2?
V' = toy'
(")
L9
= (lo.oLP)(lo'ov)
Qr= C,V,
e,: loo.ue
:==--
(b)
C, *Co'^^an'ia
+. /"^-/".=G-e"*.
' \-?
Ce>
r---J_^
J_
f
I
-
--->
Ca.+ C,'. rr ga.,"".N{e"Q
:
-)-r
-T'n
a\
La:q I Ls NY\A!^'t Ia -)
Carrs
$..rs
.
\,rlts
=-..-+l
t-
l=
C".r. -- (e-F
|
/ ?.{r
lO.u l-
= C".,'V..r" = (6-t=Xtov) = fuouL ---l
= ffi
v...,=%^',/r^"r
V".={
- 4v'
(aar
=%u=foul
(%r=crvr= '6uc)
=1V
4a= ts
\-r
= tor^g
:--
lov
C.
Ca
C,{
cs
5.r^F t l0r,F
_l* _J-cs
Cr.,
I
Cae.rs
Ill
S,trtr
Cr.., -- C.g *C..
Ca:\ = / Br^F
\- a3'l
I
--
$0uL
6V
i
Problem 4
t-90Oa
is chargedto a potentialdifferenceof 50-V, and the charging
,t
:"t":itor
batreryis
clrscorurected.
Thecapacitoris thencorurected
in parallelwith a secondri"itiiriu"n#geal
capacitor'If the potential differenceacrossthe first capacitor
dropsto 35 v, what is the capacitance
of this secondcapacitor?
/' v - -r
\ - - ' = t^.i.
Yt
Vo-
i
ScV
=) c+)h
lr ),n-,frurL'6,
L t - -|p . , . - :/-* c :i - t - - - . : , .: : , _ . . _ :"r
= C J,
4
)
(Yv-,-',.;r.
..-.'.1't
iutt t te$n-y,,)r,t1,t"
l^
-!
,.tr,r,"r',crl) ">,.'t1-q,'.1 f.:"r;L.",^,:G-.t ts* _.,j,.:,,- :.. ,:,<--_i1,t:.r,J7i
,.rr__,;i _p.,,,r,.u.,
C"o.,"s,1
:i,,"- t,-'i.,:
_
j
> t' J*J.,::
-t
l,'..,,-C, / = .-':,",,-r; ,., *^**$,*:;.,re
?,
.,,"'i-r i .1,-.3.i::.: ,-?! ,4
probtem5
j, j
,r,... ;i.,* ,..r'-.,-Ji,
= 19a:-lrii':'i
C a = %^/,,,. r-, ( \J,-t,//'r'
" =Wlf,
I
t
: 3.0 pF andboth
*.
chargedto a potentialdifferenceofv = 100v bui with opp-ositepolarity
"-uiuino.,
asshown.
switches sl and
52arenow closed.(a) What is now thepotential OifferenceUetweenpoints
a andb? What arenow
the chargeson capacitors(b) I and (c) 2.?
=) asblil"-art^fi1
tt rla+"A,
Xg\,c,a?^i,t1r
a/ll r^1 trDl di-0r{ ..}a #v /vcy}Aq
a7,.+.t' ssq
,^" \9rt -&e.r1e
I
\ v: % =V* = %"/c,.%"'W*
gta
C,v, = (t"our)(roov)
{,'
, Let
= lD o,r.rC
c[a = CaV*
*
%r*
= ( 3 . o , , r p - ) ( 1 6 o v ;- - 3 o o l q
= Aoor,rc -.,",J.l,ri
floo.tL
Cr +Ce. .cr^{vn po.^-"dta!>
,tlrn4zce...-tg
)a c+Wu.*o
Cr^= Cr +Ca"= l.o,,.rF+3.o,uF= 4.o.qtr
V^=%'/",^=
t:# =@
0> S, = C,V, -
(l.orF)q5ov)= eI
=
=
=l_t:9^!
Ca.V"
(z"our)( s,u)
Qa
V,=V*- 5ov
l
Problem6
In the figurebelow,V: 10V, Cr: 10 Y.F,andCz: C: = 20 FF' SwitchS is fust tlrown to the left
sideuntil capacitor1 reachesequilibrium.Thenttre switch is thrown to the right. Whenequilibrium .
is againreached,how much chargeis on capacitor1..
-ugi:
arrd+A
-=---
;-6,^a
{lr
[--f_^
il_:r''
rc.lrn.7.t o-t C,
=
%= C,% (lo,"r)(tov)
= loor,^.L
?-
C a. { C, *
^
4
O'b^d-A"AA
-^
C1s = C^*C,
Ao^^tri AO.r^F
C.3 = {or^F
ry4
ft-g,a^a*-lo raazt^r.A
.\
\'rr---_]*
t-l
L---T
v"= C'Vr *
I
.v
:
%
=
$, t $ra
C*"Vts
V,=V*.=V
C,V= C,V* Cr.V = (c,+c.=)V
( to,*F) ( tov)
/^
/\
\
(Lt r L-lr/
( l o , " l = *f o r F )
V,-' }-"oV
V6= X.oV
= ( to,,F) ( r.ov)
Qer=C'V'
4tt
)O,taC
= C", V^u= (+o-r) (1.ov) = 8 o , ^ , r c
-
Problem 7
Theparallel-.plaes in a capacitor,with a plate areaof 8.50 cnt' andan air-filled separationof
3.00 mm, ae chargedby a 6.00V battery.Theyarethen disconnectedfrom the batteryandpulled
apart(without discharge)to a separationof 8.00mrn. Neglectingfringing, find (a) the potential
differencebetweenthe plates(b) the initial storedenergy,(c) the final storedenergyand (d) the
work requiredto separatethe piates.
\/
C= €.Vl
A- ?,5ocmt- !,gdxl4fqrr
fu e*"ry ^a/-4* fuA,'v,ry
+,nt\.V\tlt fu^A,
= !.@orlt'311
C[ = ?.oon.\
V
= 6.oov
,o"n" ca+-r-b't;a &).caw^h)
ori?"*Ifu --
(3'oo
A.St"lo-'"F
nqQr,,O&a^.oy
^4i'in;^^
V=b.oov
A,a/,\a
)
C= Ct,sS^rt''t7u.')(Y
%=cV
"to-'
''')
= (a,st'1ooF)(r,.oov)= /,5)"\o-tt
-='
(^) v: yc= l' 51,.lo-'t
,ld'?F
'
Q, ? o
V=tGV
Cc) U=/r.C'vi 1.AO,161'"
(U) u -- l( C;V;\ = )/r(l.sr'to-'*F)(a-oov)a
7-'J'------'---1
(d) W= DD=l7.SA^to-"7
= lq.S
Problen
rruurerrl o
8
5J |t
l - 7 c ) lx
l x lo-"
lu
Il ,
IG
'
tt'
)aDacrtof nas a caDacllar
of 50 pF. (a) If eachof its plateshasan areaof '
A parallel-plateair-filled capftibTfficapacildnce
0.35m", what is the sepmation?(b) lf the regionbetweenthe platesis now filled with material
havingr : 5.6, what is the capacitance?
c= sDPF
A = o"35r-,1
K= 5.G
(q) C=e"Vd*
J= €"yc
( 5o* \o-'' i=;
3c',
d = 6.1* lO
(Ul
,\.L,^J* o da.q!-o'Db,^Xx-hrl€&\*As [i"e,ofu, 't&r' rnur-o ca+a]:rj6oo,v.e
,tt ry;vut* !:4
C^,^-=((€.)ftA-K
(n'Ya)
C^,,r*. (s.{e)(sopp) = ATo eF
K C,,A
Problem9
The spacebgtweentwo concentricconductingsphericalshellsof radiiQ: 1.70cm andb = 1.20cm
isfilledwithasubstanceofdielecficconstantK=23.5.ApotentialdifferenceV=73.0Visapplied
acrossthe imrer andouter shells.Determine(a) the capacitanceof the dwice, (b) the free chargeq
on the inner shell, and(c) the chargeq' inducedalongthe surfaceofthe inner shell.
-2
ro^
q=r"/o*lo
"
cq)
/-*\
C"= l rT'r-(;:t
r,
,/.
f abl.
c.)(Co = ?rr118" ( a-uj
)n
6 = l . A o " 1 o 'no;
l<= a3"s
V= ?3.ov
(U)
{ry
looTxio-?F - o.tc?nF
C'
coot',,art -- cV-
c) $-$'=%/Kn
=
( i"o?. 1s-?F) (l;-ev)
%'=%-%/g
= S(l
n'=(t't1"')(t-*") =
| ,/,-\
/r_J
1"%
Problem 10
Two parallelplatesof area100 cm2aregiven chargesof equalmagnitudes8.9 x l0-? C but opposite
signs.The elechic field within the dielectricmaterialfilling the spacebetweenthe platesis
1.4 x 10oV/m. (a) Calculatethe dieleckic constantof the material.(b) Determinethe magritude of
the chargeinducedon eachdielectricsurface.
H
l-l
nil"
ll
l-l
l+l
lt
J+l
Ll
H
f1
fl
fl
fl
tl
ll
fl
d=8'9n{o-?c
e--i"l"lobv/n')
:
^,4m \,
A -- lDo.*"( #.* ) = io'r"
.In
q Au!r'rh"-,A' E,/X =
,ur,wr
ftj
(a)' K==+;r\ --+(' =
?>E
( 8,i 5 iC''
"
( b)
%'%'
= %k
( to
%^') ( r,t*tobufrn)
?{
lK=
r--------
^^A'^a Xt = t"a*oJ'h*"#
%= t,0*n€<mP2&
aur=T.?,.iD-?c
'"u)