CAPACITANCE C V W M L T. THE CAPACITANCE OF A SPHERICAL CONDUCTOR When a charge Q is given to a isolated spherical conductor then its potential rises.

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1 J-Physics PITN ONPT OF PITN apaciance of a conucor is a measure of abiliy of he conucor o sore charge on i. When a conucor is charge hen is poenial rises. The increase in poenial is irecly proporional o he charge given o he conucor. The consan is known as he capaciy of he conucor. apaciance is a scalar quaniy wih imension Uni :- fara, coulomb/vol T M L T 4 W M L T The capaciy of a conucor is inepenen of he charge given or is poenial raise. I is also inepenen of naure of maerial an hickness of he conucor. Theoreically infinie amoun of charge can be given o a conucor. u pracically he elecric fiel becomes so large ha i causes ionisaion of meium surrouning i. The charge on conucor leaks reucing is poenial. TH PITN OF SPHIL ONDUTO When a charge is given o a isolae spherical conucor hen is poenial rises. 4 4 If conucor is place in a meium hen meium 4 4 r apaciance epens upon : Size an Shape of onucor Surrouning meium Presence of oher conucors nearby O NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 ONDNS/PITO The pair of conucor of opposie charges on which sufficien quaniy of charge may be accommoae is efine as conenser. Principle of a onenser I is base on he fac ha capaciance can be increase by reucing poenial keeping he charge consan. onsier a conucing plae M which is given a charge such ha is poenial rises o hen Le us place anoher ienical conucing plae N parallel o i such ha charge is inuce on plae N (as shown in figure). If is he poenial a M ue o inuce negaive charge on N an is he poenial a M ue o inuce posiive charge on N, hen ' ' Since ' < (as he inuce negaive charge lies closer o he plae M in comparison o inuce posiive charge). ' > Furher, if N is earhe from he ouer sie (see figure) hen " ( he enire posiive charge flows o he earh) " " " >> M M N M N If an ienical earhe conucor is place in he viciniy of a charge conucor hen he capaciance of he charge conucor increases appreciable. This is he principle of a parallel plae capacior.

2 J-Physics NGY STOD IN HGD ONDUTO/ PITO Le is capaciance of a conucor. On being connece o a baery. I charges o a poenial from zero poenial. If q is charge on he conucor a ha ime hen q. Le baery supplies small amoun of charge q o he conucor a consan poenial. Then small amoun of work one by he baery agains he force exere by exsiing charge is q W q q W q q q W where is he final charge acquire by he conucor. This work one is sore as poenial energy, so U ( ) U GOLDN KY POINTS s he poenial of he arh is assume o be zero, capaciy of earh or a conucor q q conncee o earh will be infinie cual capaciy of he arh F Work one by baery W b (charge given by baery) (emf) bu nergy sore in conucor so 5% energy supplie by he baery is los in form of hea. DISTIUTION OF HGS ND LOSS OF NGY When wo charge conucors are connece by a conucing wire hen charge flows from a conucor a higher poenial o ha a lower poenial. This flow of charge sops when he poenial of wo conucors became equal. Le he amouns of charges afer he conucors are connece are ' an ' respecively an poenial is hen (efore connecion) ommon poenial ccoring o law of onservaion of charge ommon poenial afer connecion ' (fer connecion) ' before connecion afer connecion NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

3 J-Physics harges afer connecion ' ( : Toal charge on sysem) ' aio of he charges afer reisribuion ' ' (in case of spherical conucors) Loss of energy in reisribuion When charge flows hrough he conucing wire hen energy is los mainly on accoun of Joule effec, elecrical energy is convere ino hea energy, so change in energy of his sysem, U U f U i U ( ) Here negaive sign inicaes ha energy of he sysem ecreases in he process.sold XMPLS x am ple x am ple conucor ges a charge of 5 when i is connece o a baery of e.m.f. 5. alculae capaciy of he conucor. apaciy of he conucor 6 5 F 5 The capaciy of a spherical capacior in air is 5 F an on immersing i ino oil i becomes F. alculae he ielecric consan of oil. Dielecric consan of oil meium r. 5 air NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 x am ple raio acive source in he form of a meal sphere of iameer 3 m emis paricles a a consan rae of 6.5 paricles per secon. If he source is elecrically insulae, how long will i ake for is poenial o rise by. vol, assuming ha 8% of emie paricles escape from he surface. apaciance of sphere ae o escape of charge from surface F /s herefore q (8 9 ) an q s

4 J-Physics x am ple The plaes of a capacior are charge o a poenial ifference of an hen connece across a resiser. The poenial ifference across he capacior ecays exponenially wih respec o ime. fer one secon he poenial ifference beween he plaes of he capacior is 8. Wha is he fracion of he sore energy which has been issipae? nergy losses U Fracional energy loss U U () (8) () 8 () 9 5 x am ple Two uniformly charge spherical rops a poenial coalesce o form a larger rop. If capaciy of each smaller rop is hen fin capaciy an poenial of larger rop. When rops coalesce o form a larger rop hen oal charge an volume remains conserve. If r is raius an q is charge on smaller rop hen 4 r an q quaing volume we ge apaciance of larger rop harge on larger rop r3 /3 r ' 4 /3 q Poenial of larger rop ' ' / 3 /3 P LLL PL T PITO (i) ap aciance I consiss of wo meallic plaes M an N each of area a separaion. Plae M is posiively charge an plae N is earhe. If r is he ielecric consan of he maerial meium an is he fiel a a poin P ha exiss beween he wo plaes, hen M P N area M ge effec N I sep : Fining elecric fiel [ ] r r II sep : Fining poenial ifference III sep : Fining capaciance q r 4 r r q q ( an ) If meium beween he plaes is air or vacuum, hen r so r K (where r K ielecric consan) NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

5 J-Physics ( i i) Force beween he plaes The wo plaes of capacior arac each oher because hey are opposiely charge. lecric fiel ue o posiive plae Force on negaive charge is F Magniue of force F Force per uni area or energy ensiy or elecrosaic pressure SPHIL PITO ( i ) Ouer sphere is earhe F u p When a charge is given o inner sphere i is uniformly isribue on is surface charge is inuce on inner surface of ouer sphere. The charge inuce on ouer surface of ouer sphere flows o earh as i is groune. for r < an for r > Poenial of inner sphere 4 s ouer surface is earhe so poenial 4 4 Poenial ifference beween plaes 4 ( ) So 4 (in air or vacuum) In presence of meium beween plae 4 r NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 (ii) Inner sphere is earhe Here he sysem is equivalen o a spherical capacior of inner an ouer raii an respecively an a spherical conucor of raius in parallel. This is because charge given o ouer sphere isribues in such a way ha for he ouer sphere. harge on he inner sie is ' an harge on he ouer sie is So oal capaciy of he sysem. 4 ( ) q ou q in ' ( ') on ouer Surface (') ( ') ' ve an Insie ve charges are equal '

6 J-Physics YLINDIL PITO When a charge is given o inner cyliner i is uniformly isribue on is surface. charge is inuce on inner surface of ouer cyliner. The charge inuce on ouer surface of ouer cyliner flows o earh as i is groune lecrical fiel beween cyliners Poenial ifference beween plaes r / r r r n apaciance log e ( / ) In presence of meium r log e ( / ) x am ple The sraosphere acs as a conucing layer for he earh. If he sraosphere exens beyon 5 km from he surface of earh, hen calculae he capaciance of he spherical capacior forme beween sraosphere an earh's surface. Take raius of earh as 64 km. The capaciance of a spherical capacior is 4 ab b a b raius of he op of sraosphere layer 64 km 5 km 645 km m a raius of earh 64 km m x am ple F cylinrical capacior has wo co-axial cyliners of lengh 5 cm an raii.5 cm an.4 cm. The ouer cyliner is earhe an he inner cyliner is given a charge of 3.5 µ. Deermine he capaciance of he sysem an he poenial of he inner cyliner. 5 cm 5 m; a.4cm.4 m ; b.5 cm.5 m; q 3.5 µ apaciance b.33 log a log F Since he ouer cyliner is earhe, he poenial of he inner cyliner will be equal o he poenial ifference beween hem. Poenial of inner cyliner, is q NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

7 J-Physics GOLDN KY POINTS If one of he plaes of parallel plae capacior slies relaively han ecrease (s overlapping area ecreases). If boh he plaes of parallel plae capacior are ouche each oher resulan charge an poenial became zero. lecric fiel beween he plaes of a capacior is shown in figure. Non-uniformiy of elecric fiel a he bounaries of he plaes is negligible if he isance beween he plaes is very small as compare o he lengh of he plaes. M ge effec N uniform in he cenre non-uniform a he eges OMINTION OF PITO apacior in series: In his arrangemen of capaciors he charge has no alernaive pah(s) 3 o flow. (i) The charges on each capacior are equal 3 i.e. 3 3 (ii) The oal poenial ifference across is share by he capaciors in he inverse raio of he capaciances 3 If S is he ne capaciance of he series combinaion, hen S 3 S 3 apaciors in parallel In such in arrangemen of capaciors he charge has an alernaive pah(s) o flow. NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 (i) (ii) The poenial ifference across each capacior is same an equal he oal poenial applie. i.e The oal charge is share by each capacior in he irec raio of he capaciances. 3 If P is he ne capaciance for he parallel combinaion of capaciors : P 3 P

8 J-Physics GOLDN KY POINTS For a given volage o sore maximum energy capaciors shoul be connece in parallel. If N ienical capaciors each having breakown volage are joine in (i) series hen he break own volage of he combinaion is equal o N (ii) parallel hen he breakown volage of he combinaion is equal o. Two capaciors are connece in series wih a baery. Now baery is remove an loose wires connece ogeher hen final charge on each capacior is zero. If N ienical capaciors are connece hen series, N parallel N In D capacior's offers infinie resisance in seay sae, so here will be no curren flows hrough capacior branch. x am ple x am ple apacior,, 4,... are connece in parallel, hen wha will be heir effecive capaciance? Le he resulan capaciance be resulan 4... [ 4... ] n infinie number of capaciors of capaciance, 4, 6... are connece in series hen wha will be heir resulan capaciance? Le he equivalen capaciance of he combinaion eq eq (his is G. P.series) s) S a r firs erm a, common raio r 4 eq 4 eq 3 4 FFT OF DILTI The insulaors in which microscopic local isplacemen of charges akes place in presence of elecric fiel are known as ielecrics. Dielecrics are non conucors upo cerain value of fiel epening on is naure. If he fiel excees his limiing value calle ielecric srengh hey lose heir insulaing propery an begin o conuc. Dielecric srengh is efine as he maximum value of elecric fiel ha a ielecric can olerae wihou breakown. Uni is vol/mere. Dimensions M L T 3 Polar ielecri cs In absence of exernal fiel he cenres of posiive an negaive charge o no coincie-ue o asymmeric shape of molecules. ach molecule has permanen ipole momen. The ipole are ranomly oriene so average ipole momen per uni volume of polar ielecric in absence of exernal fiel is nearly zero. In presence of exernal fiel ipoles ens o align in irecion of fiel. x. Waer, lcohol, O, H, NH 3 8 NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

9 Non polar i elecri cs J-Physics In absence of exernal fiel he cenre of posiive an negaive charge coincies in hese aoms or molecules because hey are symmeric. The ipole momen is zero in normal sae. In presence of exernal fiel hey acquire inuce ipole momen. x. Nirogen, Oxygen, enzene, Mehane Polarisai on : The alignmen of ipole momens of permanen or inuce ipoles in he irecion applie elecric fiel is calle polarisaion. Polarisaion vecor P This is a vecor quaniy which escribes he exen o which molecules of ielecric become polarize by an elecric fiel or oriene in irecion of fiel. P he ipole momen per uni volume of ielecric n p where n is number of aoms per uni volume of ielecric an p is ipole momen of an aom or molecule. P n p qi qi i inuce surface charge ensiy. Uni of P is /m Dimension is L T i i i - Le,, be elecric fiel, poenial ifference an capaciance in absence of ielecric. Le,, are elecric fiel, poenial ifference an capaciance in presence of ielecric respecively. lecric fiel in absence of ielecric lecric fiel in presence of ielecric i i i NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 apaciance in absence of ielecric apaciance in presence of ielecric 9 i The ielecric consan or relaive permiiviy K or r From K i i ( ) an K K i ( i K ) i i

10 J-Physics PIT Y OF DIFFNT ONFIGU TION In case of parallel plae capacior If capacior is parially fille wih ielecric When he ielecric is file parially beween plaes, he hickness of ielaric slab is ( < ). If no slab is inrouce beween he plaes of he capacior, hen a fiel given by, exiss in a space. On insering he slab of hickness, a fiel exiss insie he slab of hickness an r a fiel exiss in remaining space ( ). If is oal poenial hen ( ) Dielecric consan r r r q r r q If meium is fully presen beween he space. r...(i) r me Now from equaion (i) meium r r If capacior is parialy fille by a conucing slab of hickness (< ). r for conucor r DISTN ND DIISION Y DILTI Disance Division (i) Disance is ivie an area remains same. (ii) apaciors are in series. r (iii) Iniviual capaciances are, r These wo in series r r Special case : If r r r r r r r r r r r r r r NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

11 J-Physics rea Division (i) rea is ivie an isance remains same. (ii) apaciors are in parallel. (iii) Iniviual capaciances are r r r r r These wo in parallel so ( r r ) r Special case : If Then r r ariable Dielecric onsa n : If he ielecric consan is variable, hen equivalen capaciance can be obaine by selecing an elemen as per he given coniion an hen inegraing. (i) If ifferen elemens are in parallel, hen, where capaciance of selece ifferenial elemen. (ii) If ifferen elemen are in series, hen is solve o ge equivalen capaciance. FO ON DILTI IN PITO onsier a ifferenial isplacemen x of he ielecric as shown in figure always keeping he ne force on i zero so ha he ielecric moves slowly wihou acceleraion. Then, W lecrosaic W F, where W F enoes he work one by exernal agen in isplacemen x F x x Force exere by an exernal agen W F W lecrosaic W F U F.x W F.x F x NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 x am ple This is also rue for he force beween he plaes of he capacior. If he capacior has baery connece o i, hen as he p.. across he plaes is mainaine consan. F. x parallel plae capacior is half fille wih a ielecric (K) of mass M. apacior is aache wih a cell of emf. Plaes are hel fixe on smooh insulaing horizonal surface. bulle of mass M his he ielecric elasically an is foun ha ielecric jus leaves ou he capacior. Fin spee of bulle. v M b a

12 J-Physics Since collision is elasic elociy of ielecric afer collision is v. Dielecric will move an when i is coming ou of capacior a force is applie on i by he capacior F x U b(k ) Which ecreases is spee o zero, ill i comes ou i ravels a isance a. b(k )a ab(k ) Mv v M / GOLDN KY POINTS x am ple capacior has wo circular plaes whose raius are 8cm an isance beween hem is mm. When mica (ielecric consan 6) is place beween he plaes, calculae he capaciance of his capacior an he energy sore when i is given poenial of 5 vol. rea of plae r (8 ). m an mm 3 m apaciy of capacior Poenial ifference nergy sore r F 5 vol 9 U (.68 ) (5). 5 J NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

13 J-Physics x am ple parallel-plae capacior is forme by wo plaes, each of area cm, separae by a isance of mm. ielecric of ielecric consan 5. an ielecric srengh.9 7 /m is fille beween he plaes. Fin he maximum charge ha can be sore on he capacior wihou causing any ielecric breakown. If he charge on he capacior he surface charge ensiy an he elecric fiel K. This elecric fiel shoul no excee he ielecric srengh.9 7 /m. if he maximum charge which can be given is hen K.9 7 /m cm m (5.) ( ) (8.85 ) (.9 7 ) x am ple The isance beween he plaes of a parallel-plae capacior is.5 m. fiel of 3 4 /m is esablishe beween he plaes. I is isconnece from he baery an an uncharge meal plae of hickness. m is insere ino he (i) before he inroucion of he meal plae an (ii) afer is inroucion. Wha woul be he poenial ifference if a plae of ielecric consan K is inrouce in place of meal plae? (i) In case of a capacior as (/), he poenial ifference beween he plaes before he inroucion of meal plae k (ii) Now as afer charging baery is remove, capacior is isolae so q consan. If ' an ' are he capaciy an poenial afer he inroucion of plae q '' i.e., ' ' NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 n as an ' ( ) ( / K ), ' ( ) ( / K ).5. So in case of meal plae as K, M.5. k.5 (.5.) ( / ) n if insea of meal plae, ielecric wih K is inrouce D.5.35 k.5 3

14 J-Physics HGING & DISHGING OF PITO harging D i s c h a r g i ng When a capacior, resisance, baery, an When a charge capacior, resisance an keys is key is conece in series an key is close, hen conece in series an key is close. Then energy sore in capacior is use o circulae curren in he circui. S S harge a any insan harge a any insan I e e ime consan e / ime consan [ e ].63 e.368 So, in charging, charge increases o So, in ischarging, charge ecreases o 36.8% of he 63.% of charge in he ime equal o. iniial charge in he ime equal o. urren a any insan urren a any insan / i / i e / {i /} i / i e {i /} Poenial a any insan Poenial a any insan ( e ) e x am ple Fin he ime consan for given circui if 4,, 3F an 6F. Given circui can be reuce o : F 3 6, Time consan (3) ( 6 ) 6s NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

15 J-Physics x am ple capacior of.5 F is charge hrough a series resisor of 4M. In wha ime he poenial rop across he he capacior will become 3 imes ha of he resisor. (Given : n.693) 4M.5 F ( e / ) ( e / ) 3 4 e / 4 e / 4e / n4 n4 n s NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 5

16 J-Physics SOM WOKD OUT XMPLS x a m p l e # The swich in circui shifs from o when > /3 an goes back o from when < /3. The volmeer reas volage as ploe. Wha is he perio T of he wave form in erms of an? olage /3 /3 T ime () n3 () n () n3 (D) 3 n3 During ime ' ' capacior is ischarging wih he help of resisor '' q q e / / [ e ] ns. () s 3 ; 3 ; n During ime ' ' capacior is charging wih he help of baery. q q (-e / ) or (-e / ) as 3 ; 3 ; n T n x a m p l e # Seven capaciors, each of capaciance F are o be connece o obain a capaciance of / F. Which of he following combinaions is possible? x a m p l e # 3 () 5 in parallel in series () 3 in parallel 4 in series 5(F) in series wih 6 () 4 in parallel 3 in series (D) in parallel 5 in series F, F in series wih F, F eq In he circui shown, if poenial of is, hen poenial of is - () 5/3 () /3 () 5/3 (D) 5 F ns. () F F NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

17 J-Physics Given circui can be reuce as harge on capaciors 3 F ns. () F Now x a m p l e # 4 The plaes of very small size of a parallel plae capacior are charge as shown.the force on he charge paricle of charge ' q ' a a isance ' ' from he capacior is : ( ssume ha he isance beween he plaes is << ) q q () Zero () 3 q () 3 q (D) 3 4 ns. () ssume capacior as ipole an use F q, kp 3,p r x a m p l e # 5 Twelve inenical capaciors each of capaciance are connece as shown in figure. () The effecive capaciance beween P an T is 6 5 () The effecive capaciance beween P an U is 4 3 () The effecive capaciance beween P an is 7 (D) ll of he above saemens are incorrec P T S U W ns. (,,) T NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 For () : P 6 For () : Given circui can be rawn as qualven capaciance beween P an 5, eff W P T S 7 U 4 U P U

18 J-Physics For () : P W T S U x a m p l e # 6 If a baery be connece across he erminals P an, from symmery W an S u quivalen capaciance 5 () 5 7 varying volage is applie beween he erminals, so ha he volage across he capacior varies as shown in he figure Then. 3 5 D () The volage beween he erminals an D is consan beween an 3 () The curren in he resisor is beween an 3 () The curren in he resisor beween an is wice he curren beween 3 an 5 (D) None of hese When he capacior volage is consan is charge is consan. No curren in he resisor. ns. (D) lso q is ouble beween an compare o 3 an 5 x a m p l e # 7, an are hree large, parallel conucing plaes, place horizonally. an are rigily fixe an earhe. is given some charge. Uner elecrosaic an graviaional forces, may be () in equilibrium miway beween an. () in equilibrium if i is closer o han o. () in equilibrium if i is closer o han o. (D) can never be in sable equilibrium. Solui on ns. (, D) s an are earhe, hey are connece o each oher. Hence, ' ' an ' ' are wo capaciors wih he same poenial ifference. If is closer o han o hen he capaciance is >. The upper surface of will have greaer charge han he lower surface. s he force of aracion beween he plaes of a capacior is proporional o, here will be a ne upwars force on. This can balance is weigh. 8 NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

19 J-Physics x a m p l e # 8 Suy he following circui iagram in figure an mark he correc opion(s) () The poenial of poin a wih respec o poin b when swich S is open is 6. () The poins a an b, are a he same poenial, when S is opene. () The charge flows hrough swich S when i is close is 54 (D) The final poenial of b wih respec o groun when swich S is close is When S is opene : c a c b 6 b a harges flown afer S is close : q , q harges flown hrough S afer i is close : Final poenial of b is 6 a 6 3 c 8 S b 6F 6F a a 6 8. S c 8 b 6. F 3. F ns. () S b q q q 3 q 7F 8F xample# 9 o In he circui shown in figure, he baery is an ieal one wih emf. The capacior is iniially uncharge. The swich S is close a ime. S / 5/ / 9. The charge on he capacior a ime is- () e () e 3 () e 5 (D) e 9 NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65. The curren in a ime is- () e 3 (). Wha is is limiing value a? () () e 3 9 () () e 6 3 (D) (D) e 6 3 3

20 J-Physics 9. ns. () In seay sae capacior volage / alculaion of ime consan ( ) effecive resisance across 3 / 5/ c q q e, q q e 3 /. ns. (D) 5 i q / 5/ P where i 3 q v e,i e e e i /. ns. (), i xample# o 4 Following figure shows he iniial charges on he capacior. fer he swich S is close, fin - S F F. harge on capacior () () 5 () (D) None of hese 3. harge on capacior () () 5 () (D) None of hese 4. Work one by baery () 5 J () J () 5 J (D) None of hese NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

21 J-Physics,3 ns. (), () q q 4. ns. (D) w b q b () charge flown hrough he baery is zero xample# 5 o 7 In he circui shown, capacior has capaciance F when fille wih ielecric slab (k ). apacior an are air capaciors an have capaciances 3F an 3 6F respecively. S S alculae he energy supplie by baery uring process of charging when swich S is close alone. ().34 J ().648 J ().5 J (D).3 J 6. Swich S is opene an S is close. The charge on capacior is () 4 () 8 () 8 (D) 7. Now swich S is opene, slab of is remove. noher i elecric slab k which can jus fill he space in, is insere ino i an hen swich S is close. The charge on capacior is () 9 () 4 () 8 (D) 7 5. ns. () q nergy supplie by baery q.648 J. 6. ns () NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 quivalen of & F ommon poenial F q on ns. (D) vol F q36 36 ommon poenial aaine afer S is close is 9 vol. 4F q 9 q F F 6 F 6 F

22 J-Physics x a m p l e # 8 ll capaciors given in column-i have capaciance of F. olumn-i (ircui ) olumn- II (apaciance ) () (P) 4 F 3 () () 3 F () () 5 F 8 (D) (S) 5 F 3 (T) None of hese Soluion ns. ()(S), ()(), ()(), (D)( ) x a mp l e #9 There are six plaes of equal area an separaion beween he plaes is (<<) are arrange as shown in figure. The equivalen capaciance beween poins an 5, is. Then fin he value of. ns. eraw he circui 5 3,6 4 / eq NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

23 x a m p l e # If charge on 3F capacior is 3. Fin he charge on capacior of capaciance in. J-Physics 6F 3F ns. 9 Poenial ifference across 3F P.D. across 6 F harge on 6F 6 Toal charge on combinaion of 6F an 3F 9 Therefore charge on 9 x a m p l e # In he given circui fin energy sore in capaciors in mj. F F 6F 3F.4 NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 S o l u i o n ns. 6 eq 6 F eq mj nergy sore 3 3 xample# Two parallel plae capaciors wih area are connece hrough a conucing spring of naural lengh in series as shown. Plaes P an S have fixe posiions a separaion. Now he plaes are connece by a baery of emf as shown. If he exension in he spring in equilibrium is equal o he separaion beween he plaes, fin he spring consan k. P 3 S

24 J-Physics Le charge on capaciors be q an separaion beween plaes P an an an S be x a any ime isance beween plaes P an an an S is same because force acing on hem is same. apaciance of capacior P, x P ' x ' S x apaciance of capacior S, x From KL q q q x his momen exension in spring, y x. Force on plae owars P, F q 8x 8x Spring force on plae ue o exension in spring, F ky equilibrium, separaion beween plaes exension in spring Thus x y x x...(i) an F F 3...(ii) From eq. (i) an (ii), 8x ky kx x 8K / 3...(iii) From eq. (i) an (iii), 3 8K 7 3 k 8( ) xample#3 block of mass m kep on a rough horizonal surface is connece o a ielecric slab of mass m/6 an ielecric consan K by means of a ligh an inexensible sring passing over a fixe pulley as shown in fig. The ielecric can compleely fill he space beween he parallel plae capacior of plae are an separaion beween he plaes kep in verical posiion. Iniially swich S is open an lengh of he ielecric insie he capacior is b. 4 s The coefficien of fricion beween he block an he surface is 4. Ignore any oher fricion. (a) Fin he minimum value of he emf of he baery so ha afer closing he swich he block will move (b) If min fin he spee of he block when he ielecric compleely fills he space beween he plaes of he capacior. b NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65

25 J-Physics (a) The forces acing on he ielecric are elecrosaic aracive force of fiel of capacior an is weigh. The M M block will slip when F mg > Mg F g g 4 6 Mg (K ) min Mg Mg (K - ) 6 (K ) (b) Now min. In his case he block will accelerae Dielecric : F mg T ma...(i) an lock : T Mg Ma...(ii) eq. (i) an (ii) give F (m M)g a m M s F (K ) (K ) 4 6 Mg (K ) Mg Thus M Mg g 3g 6 38 a g 7M From equaion of moion, v as v 38g 76 ( b) 7 v g( b) 7 NOD6 ()\Daa\4\Koa\J-vance\SMP\Phy\Uni-7\apaciance\nglish\heory.p65 5