CHAPTER 31 CAPACITOR

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1 . Given tht Numer of eletron HPTER PITOR Net hrge Q The net potentil ifferene L..6 pitne v F.. r 5 m. m F... Let the rius of the is R re R D mm m 8.85 r r m.5 m mm. m V 6V Q? Q V 6.8. W Q V J. 5. Plte re 5 m.5 m Seprtion mm m Potentil v v () We now v () Then erese to mm mm m v m 6 Km F The extr hrge given to plte (.65.).. 6. F, 4 F, 6 F V V 4 6 F 6 F 4, 4 48, m. m mm V

2 pitor 7. F F 4 F V V The euivlent pity () Let Euivlent hrge t the pitor 4 9. F 6 V V 9. on eh. s this is series omintion, the hrge on eh pitor is sme s the euivlent hrge whih is. () Let the wor one y the ttery W V W W V 6. J F, 4 F, 4 F e ( ) F 6 Sine & re prllel & re in series with So, () V 4 F 8 F 4 F 4 6, re series &, re series s the V is sme t p &. So no urrent pss through p &. p 4 F n 6 F p 5 F () 4 F, 6 F, In se of p &, p 4 F p 6 F & 5 F & 5 F The eution of pitor 5 5 F. R S

3 pitor. V v e [ They re prllel] 5 6 F V. The pitne of the outer sphere. F. F Potentil, V v Let the hrge given to iniviul yliner. V 5 F V 6 F 5 6 V. F The totl hrge given to the inner yliner 44 F. V, Now V R K So, K / R K R 4 R. Similrly 4 R The omintion is neessrily prllel. Hene e 4 R 4 R 4 (R R ) F In this system the pitne re rrnge in series. Then the pitne is prllel to eh other. () The eution of pitne in one row () n three pitne of pity re onnete in prllel The eution of pitne F s the volt pitne on eh row re sme n the iniviul is No. of Totl 6 V pitne 4. Let there re x no of pitors in series ie in row So, x 5 x 4 pitors. Effetive pitne in row 4 Now, let there re y suh rows, So, y 4 y 4 pitor. So, the omintions of four rows eh of 4 pitors..

4 pitor 5. 4 F 8 F F D 6 F 4 F F D 8 F 6 F 4 F 8 F F D 6 F 5 5 () pitor n F 6 8 (i) The hrge on the pitne 8 F 8 4 Q 5 4 The potentil t 4 F 4 4 t 8 F The Potentil ifferene V 6 (ii) Hene the effetive hrge t F 5 F Potentil t F Differene ; Potentil t 6 F 6 5 V 6 5 The potentil t & D is V P R () S It is lne. So from it is lere tht the whet str rige lne. So the potentil t the point & D re sme. So no urrent flow through the point & D. So if we onnet nother pitor t the point & D the hrge on the pitor is zero. 6. e etween & / (The three re prllel) 7. In the figure the three pitors re rrnge in prllel. ll hve sme surfe re First pitne n pitne r pitne e ( ) ( ).4 E / D

5 pitor ( ) ( ) ( )( ) ( ) ( ) ( )( ) 6 8. () ( )( ) In(R L /R ) e In [In.69] PF () Sme s R /R will e sme. 9. Given tht PF F PF F V 4 V 4 4? Let The new hrge PF V The Voltge. Let the new potentil is V fter the flow of hrge, potentil is sme in the two pitor V V 6 S / V Initilly when s is not onnete, 5 eff fter the swith is me on, Then eff 5 Q Now, the initil hrge will remin store in the store in the short pitor Hene net hrge flowing

6 pitor. V.4 F P.4 F Given tht mss of prtile m mg hrge. m Let potentil V.4 The Eution pitne. F The prtile my e in euilirium, so tht the wt. of the prtile ting own wr, must e lne y the eletri fore ting up wr. E Mg Eletri fore E V E mg QV mg.. V 8.85 V. 98 where V Potentil, seprtion of oth the pltes V.4 4 MV.. Let mss of eletron hrge eletron e We now, For hrge prtile to e projete in sie to pltes of prllel plte pitor with eletri fiel E, y E x m where y Vertil istne overe or x Horizontl istne overe Initil veloity From the given t, y V, E R For pitor V s Here hre on pitor., x,? V where Euivlent pitne of the totl rrngement So, V me.6

7 pitor Hene E V ( ) V ( ) Sustituting the t in the nown eution, we get, e V ( )m u u Ve m( ) u Ve m( ) /. The elertion of eletron e eme Me pe The elertion of proton p Mp The istne trvelle y proton X pt () m e e x E E ep x p E The istne trvelle y eletron () From () n () X t x t 4. () x x x M x M p p pe Mp F M x x x F F F F 6 F 6 s the rige in lne there is no urrent through the 5 F pitor So, it reues to similr in the se of () & () s n lso e written s. 6 6 e.5 F () y loop metho pplition in the lose iruit D Q Q Q () F F 4F In the lose iruit D Q Q F 4F Q () From () n () Q Q 48 () n Q 48 n sutrting Q 4Q, n sustitution in eution Q F F 5 F F 6 F F F Q 4 F (Q Q ) 4 F D.7

8 pitor 48 Q Q 48 8 Q Q 48 Q 48, V () Q 4F F 48 4 V V 4 F F V 4 F 4 V 4 V The potentil 4 ( 4) Potentil ifferene V 4 The V V 8 V Left Right () V V 48 8 V 6 F F From the figure it is lere tht the left n right rnh re symmetry n reverse, so the urrent go towrs E from FE sme s the urrent from EDE. The net hrge Q V Q V The potentil t K is zero. () 6 V 4 F E F D V 4 V F F The net potentil V V. V 6. () Net hrge Net pitne / V F 4 F F /8 4/8 /8 /8 F F 4/8 /8 / / y str Delt onversion eff 8 F F F 6 F F.8

9 pitor () F F F 4 F F 4 F F /8 f /8 f 4/8 f f f F / f y str Delt onvensor /8 f / f /8 f 4/8 f f f /8 f /8 f / f /8 f / f 4 f 4 f f /8 f () F 4 F 4/ F 5 F 8/ F 4 F 8 F 4 F 4 F 4 8 ef 4 () 8 F f 4 f 6 f f 4 f 4 f 4 f 6 f 8 f 8 f 4 f 4 f 8 f 8 f 6 f f 4 f f 4 f 8/6 f / f / f 8/6 f ef f.9

10 pitor 7. f f f f n re in series e This is prllel to 6 Whih is series to Whih is prllel to 7 Whih is series to Whih is prllel to 8 This is series to 4 Fig F f Fig 9. Let the euivlent pitne e. Sine it is n infinite series. So, there will e negligile hnge if the rrngement is one n in Fig e ( ) ( ) ( ) (Impossile) So, F 4 f 4 f 4 f 4 f f f f f n 4 f re in series 4 So, 4 Then n f re prllel f f The vlue of is 4 f 6 6.

11 F. 9 F net 8. V.5 V 9.. Given tht pitne F hrge The effetive hrge 8 F V V V. 6. F 7 F 6 net ( ) Potentil V 5 V 7 5 ut potentil n never e ( )ve. So, V 5 V. Here three pitors re forme n eh of x pitor x x 96 f.m. 4 mm 4 m pitne of pitor F. s three pitor re rrnge is series 4 So, e The totl hrge to pitor The hrge of single Plte () When hrge of is introue to the plte, we lso get.5 hrge on the upper surfe of Plte. () Given 5 F 5 9 F 5 8 F Now hrge.5 6 V F V 5. Here given, pitne of eh pitor, 5 f.5 f hrge Q F whih is given to upper plte.5 hrge pper on outer n inner sie of upper plte n.5 of hrge lso see on the mile. () hrge of eh plte.5 pitne.5 f

12 pitor.5 V v V. 5 () The hrge on lower plte lso.5 pitne.5 F.5 V V V. 5 The potentil in V 6. PF F, 5 PF 5 F Effetive 5 5 hrge F V V V V Energy store in eh pitor E (/) V (/) (4.84) PJ E (/) V (/) 5 (.7) PJ 7. 4 F, 6 F, V V E. pitor e The E pitne e.5 F The energy supplie y the ttery to eh plte E (/) V (/).4 48 J The energy supplies y the ttery to pitor J 8. F 6 F For & F E For & 4 4 e 5 F 8 J 8 mj 4 f 6 f V V V V E (/) v (/) 5 () J mj 9. Store energy of pitor 4. J 4. J When then onnete, the hrge shre So tht the energy shoul ivie. The totl energy store in the two pitors eh is J..

13 4. Initil hrge store V Let the hrges on & 4 pitors e & respetively There, V 4. or E (/) V 8 (/) E (/) V 8 (/) hrge Q Rius of sphere R pitne of the sphere 4 R Energy Q 4. Q V 4 R V E E Q 4 R Q 8 R 6 J 8 J [ in spheril shell 4 R] 6 R V 4 R 4. 4 /m m m The energy store in the plne RV [ of igger shell 4 R] 6 m 4 ( ) The neessry eletro stti energy store in uil volume of ege m infront of the plne J 44. re m m seprtion mm m i i 4 So, flown out. ie, i f. () So, () Energy sore y ttery uring the proess v.6.7 J () efore the proess E i (/) i v (/) J fter the fore E i (/) f v (/) J () Worone Fore Distne (e) pitor From () n () we hve lulte, the energy loss y the seprtion of pltes is eul to the wor one y the mn on plte. Hene no het is proue in trnsformer. Q R

14 pitor 45. () efore reonnetion f V 4 V V 4 (efore reonnetion) fter onnetion When f V V V (fter onnetion) (), V V V v () We now V W W v 44 J 4.4 mj The wor one on the ttery. () Initil eletrostti fiel energy Ui (/) V Finl Eletrostti fiel energy U (/) V Derese in Eletrostti Fiel energy (/) V (/) V (/) (V V ) (/) (576 44) 6J Energy 6 j.6 mj (e)fter reonnetion, V v The energy ppere (/) V (/) 44 7 J 7. mj This mount of energy is evelope s het when the hrge flow through the pitor. 46. () Sine the swith ws open for long time, hene the hrge flown must e ue to the oth, when the swith is lose. ef / So E () Worone v () E i E E f (/) E E E E E 4 E E E i E f 4 () The net hrge in the energy is wste s het f V 4 V V 5 4 n 6 f V R V 6 7 Energy store on first pitor E i () Energy store on n pitor E (7) 6 44 J.44 mj 4 J 4. mj E.4

15 pitor 48. () V V Let the effetive potentil V V V V The new hrge V n V () U (/) V U (/) V U f (/) V ( ) (/) (4.6) (5 6) J.45 mj ut U i The loss in KE mj 5 f 4 v 6 f v (i) (ii) When the pitor is onnete to the ttery, hrge Q E ppers on one plte n Q on the other. When the polrity is reverse, hrge Q ppers on the first plte n Q on the seon. hrge Q, therefore psses through the ttery from the negtive to the positive terminl. The ttery oes wor. W Q E QE E In this proess. The energy store in the pitor is the sme in the two ses. Thus the worone y ttery ppers s het in the onneting wires. The het proue is therefore, E J.44 mj [hve 5 f V E V] 49. m m 4 m m m 4 t t 8.85 t F.4 nf 5. Dieletri onst. 4 F.4 nf, V 6 V hrge supplie V hrge Inue ( /) (.5) n Net hrge ppering on one ote surfe 5. Here Plte re m m Seprtion.5 m 5 m Thiness of metl t.4 m 4 m 8.5. n 4.5 m pf t t t (5 4) Here the pitne is inepenent of the position of metl. t ny position the net seprtion is t. s is the seprtion n t is the thiness. m t.4 m m mm m.5

16 5. Initil hrge store 5 Let the ieletri onstnt of the mteril inue e. Now, when the extr hrge flown through ttery is. So, net hrge store in pitor 5 Now or or, V V Deviing () n () we get () () f V 6 V mm m. () the hrge on the ve plte V 5 f 6 V () E V 6V () m t m 5 or m V/M 5 6 When the ieletri ple on it t t () 5 6 f. V 6 V Q V 5 f f 8. 6 f V 6 V Q V F hrge flown Q Q F Let the pitnes e & net pitne Now p m 4 m m m V 6 V t m F. 4 m pitor 5 6 mm 4 mm.6

17 pitor t t 56. () re Seprtion / () similrly / ( ( ) ) ( ) K K ( ) 57. () x ( ) x tn X x tn onsier n elementl pitor of with x our t istne x from one en. It is onstitute of two pitor elements of ieletri onstnts n with plte seprtion xtn n xtn respetively in series R R x x tn ( x tn ) or R x tn x tn (x) (x) ( x )x tn [log e ( ) x tn] tn ( ) [log e ( ) tn log e ] tn ( ) tn n.7

18 pitor R R ( ) ( ) loge loge 58. R ( ) / In s V I. Initilly when swith s is lose Totl Initil Energy (/) V (/) V V () II. When swith is open the pitne in eh of pitors vries, hene the energy lso vries. i.e. in se of, the hrge remins Sme i.e. v eff E In se of eff v v 6 E eff v v v Totl finl energy v 6 Initil Energy Now, Finl Energy 59. efore inserting fter inserting v v v 6 Q Q V v 6 The hrge flown through the power supply Q Q Q V V V ( ) Worone hrge in emf V ( ) ( ) V V ( ) K.8

19 6. pitne F 4 F P. V () V m Dieletri onstnt.5 () New F New p V. [ remins sme fter isonnetion of ttery] () In the sene of the ieletri sl, the hrge tht must hve proue V 4 m () hrge inue t surfe of the ieletri sl ( /) (where ieletri onstnt, hrge of plte) 5 5 m Here we shoul onsier pitor n in series 4 ( ) 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 6. These three metlli hollow spheres form two spheril pitors, whih re onnete in series. Solving them iniviully, for () n () 4 ( for spheril pitor forme y two spheres of rii R > R ) 4RR R R Similrly for () n () eff 4 4 (4 ) ( )( ) ( ) ( ) 4 ( )( ) 4 ( ) 4 6. Here we shoul onsier two spheril pitor of pitne n in series 4 ( ) 4 ( ) pitor.9

20 pitor 64. Q V V 4 ( ) ( ) v 6 v m V (v / ) 6 Q v 6 ( ) ( ) m 8 f ( ) ( ) f m m 66. m m V 4 V The pitne m The energy store (/) V (/) (4) The fore ttrtion etween the pltes N. K M We nows In this prtiulr se the eletrifiel ttrts the ieletri into the pitor with fore Where With of pltes Dieletri onstnt Seprtion etween pltes V E Potentil ifferene. Hene in this se the surfes re fritionless, this fore is ounterte y the weight. So, M E ( ) Mg E ( ) g V ( ).

21 pitor 67. l l K K n n () onsier the left sie The plte re of the prt with the ieletri is y its pitne x n with out ieletri These re onnete in prllel [L x( )] Let the potentil V (L x) U (/) V v L x( ) () Suppose ieletri sl is ttrte y eletri fiel n n externl fore F onsier the prt x whih mes insie further, s the potentil ifferene remins onstnt t V. The hrge supply, () v to the pitor The wor one y the ttery is w v. () v The externl fore F oes wor w e ( f.x) uring smll isplement The totl wor one in the pitor is w w e () v fx This shoul e eul to the inrese v in the store energy. Thus (/) ()v () v fx f v x from eution () v F ( ) V F V ( ) F ( ) For the right sie, V V V F ( ) F ( ) V V F ( ) The rtio of the emf of the left ttery to the right ttery.

22 68. pitne of the portion with ieletris, pitne of the portion without ieletris, ( ) Net pitne ( ) ( ) onsier the motion of ieletri in the pitor. Let it further move istne x, whih uses n inrese of pitne y Q () E The wor one y the ttery w Vg E () E E Let fore ting on it e f Wor one y the fore uring the isplement, x fx Inrese in energy store in the pitor (/) () E () E fx fx (/) () E f ( ) E x ( ) x f E x m f t ( ) E ( ) E ( ) m ( ) ( )m E ( ) (here x ) (l ) t 4m( ) E ( ) E l K pitor l Time perio t 8m( ) E ( ).

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