F.t l. (a) When a batteryis connectedto the platesof a capacitorC, it storesa chargeof Q. What is the voltage of the battery? (b) If the samecapacitoris connectedto anotherbatteryand Q'of chargeis storedon the capacitor,what is the voltage of the battery?
(^) C=[L + r/=4n v z '-
fu.'&",p*^*1 V,
c.
r,L 2. An air-filled parallel-platecapacitorhasplatesof aread separated by distanced. The capacitoris connectedto a batteryof voltageV. (a) Find the valueof its capacitance. (b) What is the chargeon the capacitor?(c) (c) What is the magnitudeofthe uniform elechic field betweenthe plates?
(2a ) , L'/ = eA
T
:_
Cbt (= _g V
= ) Q =C v
=[td.)v i
d. I
\
ER) from its center.(a) What is its surfaoechargedensity?(b) What is its capacitance?
I \
-4
G*,,,*"i ""{ q f
\
-*-"
.rt
Qv*IA l
-
(at .
eEu- t'^
s {o^,,t .
Gr";
= >8 . . f f i Y ' E
& E,t a =4nRtr .
tE
{' ltrtv +ii R,
=Q
l Z
fr \ r
= 5"LFJ-
tu, t o . w f ' E = A
(b)
-l-. d.? L e *Tt&T
SVq"l'Ge?E
q )., rR-J-G./i v+ V= ) * f f i 6 r ' - J - \ , - L ]"F =
,
"C =&. : ffi€,"R 4 V
l+r*" rJ g4ert.GefVf*,* *qta q K-
El-ofs C =Uv
?6 5. Two capacitors,C1 andCz, are connectedin parallel,andthe resultingcombinationis connectedto a batteryof voltageV. (a) Find the equivalentcapacitance of the combination. (b) Findthe potentialdifferenceacrosseachcapacitor. (c) Find the chargestoredon each
{*
Lr* rza'ot) tb y,f u/t,a$ l? \
tu I
(a).
a
Lg{f
a -C\-t-'r-z'
cofc(';tr'Cn, ?ol^y,^lJ;14.,-^* ,.cr'.s-po'rl^ ;
Cc),
tb saz^t- -
v\ /
I
:
(=q-re=CV -/
V
''' d"^"X* skd"
c),^y
I tA"
C , =C , V -,..9.
c-v :--
r,1 6. Two capacitorsC1 andCzarenow connectedin seriesandto a batteryof voltageV. (a) Find the equivalentcapacitance ofthe combination.(b) Find the potentialdifference acrosseachcapacitor.(c) Find the chargeon eachcapacitor.
v
c^)
=*L+t- +
_l ref(
I |
Cr+e
C,
hf"{V,
- !
-
L€ff t
-:
e +c. \1
CI. CC.
-----
C,t C-
C
cs.) Uwy th*"L )" C, : ch*y s+o"e&,h C-: dnry sb"qA Cq+ "n cv (c= Cr{e'V +:,& ) C,C,V ---Cft C.
cr)&'
C = -I
V +
r' u / c= g C,V Cft cz-
C,-= * ?"+a,t;^,tarqLos( pule^kiil- aaY6s C,
=
A : C,
C,V CrtC-.
r'8 7. Findthe following.
S,m*+-6i
-n
I ll-r I farr
Cr$sf
LrJ
fatut
"vMv (a) the equivalentcapacitanceof the capacitorsin the figure above (b) the chargeon eachcapacitor (c) the potentialdiffeienceacrosseachcapacitor
cq)
cl
CJ
)
tlfl Ir v?& -LT - ,"
Czrc3 cl
[,1,1 V
.-L=!t + Cqf C, C'rCr " f C C . t C r+) C t < C c ' + G \
C,cc,*G ) t= C, + zCC.f Cs )
+
C.CCrf Cs )
CerF= C,Cc.+cr) C,tzCCz+Ct)
0l , Cl"-,y
c^ CI c CLr,,'n"rvr C-* Cs ; Clr^,y 6>t Cr{f
: C , L C ,ct t J Ct* z[CL+cr]
v
CC=$ +e-cv)
f,q "Tld, J"a- ;q ,rl"? Ahre* C. ""^^ot C' Ne *]('t '{'u y"^*c")
fru't, k)
I
)"f&,r"'*'*'
t. nA
Yof€'*Aolt acrose
n
L I
( (',-( * -d.
d,,
=
V
c-l
C, \C"*GJ
ev"*)
u/r,
C,*z fcr+Cil = f C . +C r 1 Ct+ZLCL+qJ ?r+*,1,"!. At( * +vtcl a'( Yts s C . ?t*1."1" "U(k ,rt 4 c.ctrtrs Cs
€
-€
('
,i pA " acrcss *l't 6[!t"< v 1
p,A. arras s C t
= C,Vr zf.cr+crJ v * T c.t c'7 v. ' Ec-tCJv c, t zCc,t Cr7 t ', C
, , t tV/
C,tzLQtJ
*&r'T'b
&,
T*t-voLr.r.o\dY
Cb) (",,*L*rl)
\ \JWrt"
r'A -'-slo"sL;nC'--C'V'. (-(r',tu#SPg* ta' + r/ __ M Lrr_z_ 4-
--l
Cr+zV,1CrJ :
,-L*r" shve&inCj
GzU"rcl
\ I
P,ro 8. Considerthe systemof capacitorsshownin the figure below.
(a) Find the equivalentcapacitance of the system. (b) Find the chargeon eachcapacitor. ,
(c) Find the potentialdifferenceacrosseachcapacitor.
f
s{
(d) Find the total energystoredby the group.
C,
l
'-rHrr
A^).
\
c
*
(
; -
:>
CttCt
rq.-+{ \---i,l:--)
_
tf = (;c!
r-{ F1
Cr1.Cs
= ft-'d Ctf(t
It't,-
I
V
V.
'" C"{r t t L r C{t
* eG
@
c,crLc.tcvT +G) * crcoLc, fC,+CI fC,rCvJ / l - r A
I b ). (.Uro"y M C, d
E
:
Cs ryf cr,4 U6"f*6^4Fe* C,Cs t t v CttCs
Cl"-f m Cz: *^yo "^ Cr i '
cof,or[ ;1",,^n" f C,& C:
('r'* -"& =c{)
\
- dro,'p,*" 4k "hw co'7oalo'"t . f C - & c r CCv ,t :
t:fu-:-
tr)" f"l**'l
a;ft^"^-u- 4cnszsCr F
P.{l *
_@,_L Cr
C$(-t T
,
;*.V \
t l
\
!
r &. acvalos C. ,tllrn, ", ft^h"a C-
: C,q-V"L C.tCv
N*A*-h]f*,***
ar46,sCt c
C.
--=-
Cr G r*CY Cr
CrL\0,"ru
a,ca6sg C(t3 y = C.C,r.V I Cy 6tCv
sJ"ftd, ir.n^!1.
=*c*V'
F*
Czf(,,
tvt
-€J-
GCiv.*L P*^"J" tWcL
C+
r
ca,pa(qJ-r.s
- J,etcrTczr1(1+&eqfeftC-s\ V' e r
. c , V C,+Cs - t s '
c- .v
(.tCr F4
P.la 9. Threecapacitorsareconnectedto a batteryas shownin the figure below. Their capacitances areC1:3C, C2: C, andCz: 5C.
rT-l (a) What is the equivalentcapacitance of this setof capacitors(in termsof C)? (b) Statethe rankingof the capacitorsaccordingto the chargethey store,from largestto smallest.(c) Rank the capacitorsaccordingto the potentialdifferencesacrossthem,from largestto smallest.(d) tf @\s increased,what happensto the chargestoredby eachof the capacitors?
3C
(^)
I
l.-l -L 6 c
t__t .
)
=-l-+ SC
C4{ r l
6C
-
=*t 6Crr'
: : .
i s syl;t l-r*\-"-
W;^Cr S^'*
-L
lC"o=3c
C b A '"'Cl
'
s l * o ' a s / 4 4 ,**i
C ''" at' Ct . "
,-1"6"r=d)^
C3 YC. t ,', G t+6!c"Y4cv('1'"n^ , ' n& ,
l4u^ C'-
,\
CO. C, wrrlCt h"r* IN"t-Swnp yrte^*,^,L J;VWq *u
C,
lk
u"rn
Crorrd'C,
-
clrra'rp ota co-rl;nesl "\rr%*C,< Gt)-./ *f^*l:: 1fC.o,,a,a. C=6c),h C--%rV-€.
,' - fotw+rr*[ a-cossCt ; luXt^ (gq ,;o.r C ;s J- so"I!^ W W;' samt-A ) V
i Ul . k
C,
@zpa*e+l''"'-"n- o{
c*nL'nn{
ino',oo n-ol,
;
Y'13
Ct *itl. i 'IoLQe'"l '
C, **t
) s*tt te C, a A(C, ,C, cnnL1^r-( , '.
lr^*'. -')'*'A" '1'*f
CrV,
, ;t C.+ G i,2,.,-oao-c*, V, h* l" lnuraoaCr)areconnectedin series,anda batteryof voltageV (b) Find the is connectedacrossthe two capacitors.(a) Find the equivalentcapacitance. (c) Find the energystoredin eachindividual energystoredin this equivalentcapacitance. capacitor.(d) Showthat the sumof thesetwo energiesis the sameasthe energyfound in part (b). (e) Will this equalityalwaysbe true, or doesit dependon the numberof capacitorsand (f) If the samecapacitorswereconnectedin parallel,what potential their capacitances? differencewould be requiredacrossthem so that the combinationstoresthe sameenergyas in part (a)?(g) Which capacitorstoresmore energyin this situation,Cr or Cz?
cq.)
_l_ _ L +
crrr
C,
t,+
_J-
r Ls*C2-
=
&
r r
L-r LZ
f r
4
4t.
I\./) = L rlvvr l L Z
(-c) Q*tf
* tv",lc itl- tY
r l . . U -
=
\-tL-&
-etr
2 f
-
.
/
l,?C #
z-
L | 9'c'1V . z L ctce-J
sl^'c/J"nC'
in C, ) Jt--#
Sh^Qi&
* c]t^,,|*
Lz &
Cr
i nt
C'rr
= ! f f ri 1 =
(
z p*c"1"
t
t I lr
@ f,,ru J--,,-+
n -1*2-
!
e.
2
Lc n c l c -
> CJ f c,c"v f'.L
= l-
Cntc.VT
z rct;r
./oll
4
Sl"no
