. Given tht Numer of eletron HPTER PITOR Net hrge Q.6 9.6 7 The net potentil ifferene L..6 pitne v 7.6 8 F.. r 5 m. m 8.854 5.4 6.95 5 F... Let the rius of the is R re R D mm m 8.85 r r 8.85 4. 5 m.5 m mm. m V 6V Q? 8.854.5. 8.854.5 Q V 6.8. W Q V.8 6 8 J. 5. Plte re 5 m.5 m Seprtion mm m Potentil v v () We now v.6 8.85.. () Then erese to mm mm m 8.85.5 v.5 9 5998.5 m 6 Km 7.784..6 F 8.85.5.65. The extr hrge given to plte (.65.).. 6. F, 4 F, 6 F V V 4 6 F 6 F 4, 4 48, 6 7 5 m. m mm V
pitor 7. F F 4 F V V The euivlent pity. 4 4 4 () 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. 8. 8 F, 4 F, 4 F e ( ) 8 8 4 F 6 Sine & re prllel & re in series with So, 8 6 48 4 6 4 4 6 4 9. () 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
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..
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 4 8 4 8 6 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 8 6 5 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
pitor ( ) ( ) ( )( ) ( ) ( ) ( )( ) 6 8. () ( )( ) In(R L /R ) e.4 8.85 In [In.69] 8.7 8 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 4 4 4 5 6. V 6 S / V Initilly when s is not onnete, 5 eff 5 4.66 4 fter the swith is me on, Then eff 5 Q 5 5 5 4 Now, the initil hrge will remin store in the store in the short pitor Hene net hrge flowing 5 4.66 4. 4..5
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.. 98 8.85 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
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 9..67 7..67 x.898 4 5.449 4 x x 4.898.5449.896 F F 9 4 5.449 4 5 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 6 48 8 8 5. () 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
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 /8 4 4 4 7 7 7. V F 4 F F /8 4/8 /8 /8 F F 4/8 /8 / / y str Delt onversion eff 8 F F 5 9 5 8 4 4 F 6 F F.8
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 8 6 8 8 4 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 8 8 6 6 6 8 f.9
pitor 7. f f f f 4 5 6 7 8 8. 5 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 4 4 8 4 4 4 8 4 8 4 4 8 6 4 f The vlue of is 4 f 6 6.
.. 8. 8. 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 ( ) 7 6.5 6 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 96 4 4 9 F. s three pitor re rrnge is series 4 So, e 9 8 9 The totl hrge to pitor 8 9 8 8 The hrge of single Plte 8 8 6 8.6 6.6. 4. () 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 5 5 7 8 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..5.5.5.5.5.5.5.5.
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.48 6 8.568.48 F V 8.568 4.84 V V 8.568 5.7 V Energy store in eh pitor E (/) V (/) (4.84) 8.5 84 PJ E (/) V (/) 5 (.7) 7.5 7.5 PJ 7. 4 F, 6 F, V V E. pitor e 4 6.4 4 6 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 48 96 J 8. F 6 F For & 4 4 5 F E 4 4 5 For & 4 4 e 5 F 8 J 8 mj 4 f 6 f V V V 4 4 5 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..
4. Initil hrge store V 6 4 6 Let the hrges on & 4 pitors e & respetively There, V 4. or 4 6 8 6 8 6 6 6 E (/) V 8 (/) E (/) V 8 (/) 4 4 4. 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 ( ) 8.85. 4 564.97 7.7 The neessry eletro stti energy store in uil volume of ege m infront of the plne 65 6 5.65 4 J 44. re m m seprtion mm m i i 4 So, flown out. ie, i f. () So, 8.85 6..6 () Energy sore y ttery uring the proess v.6.7 J () efore the proess E i (/) i v (/) 8.85 44.7 J fter the fore E i (/) f v (/) 8.85 44 6.5 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
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. 47. 5 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
pitor 48. () V V Let the effetive potentil V V V V 7 4.6 5 6 The new hrge V 5 4.6.8 n V 6 4.6 6. () U (/) V U (/) V U f (/) V ( ) (/) (4.6) (5 6) 4.5 6 J.45 mj ut U i.44.4.87 The loss in KE.87.45.7687.77 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 5 6 44 44 5 J.44 mj [hve 5 f V E V] 49. m m 4 m m m 4 t t 8.85 t 4 4 4.6 9 F.4 nf 5. Dieletri onst. 4 F.4 nf, V 6 V hrge supplie V.4 9 6 8.5 9 hrge Inue ( /) 8.5 9 (.5) 6.9 9 6.4 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 8.585 88 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
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 () () 5 5 5. 5 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 8.85 8.85 5 () 5 6 f. V 6 V Q V 5 f f 8. 6 f V 6 V Q V 8. 6 6 5 F hrge flown Q Q F 4 8.85 9 6 54. Let the pitnes e & net pitne Now 4.45 44.5 p. 55. 4 m 4 m m m V 6 V t.5 5 4 m 5 4 8.85 6 4 5 5 5 6 4 4 6 4 8.85.8 8. F. 4 m pitor 5 6 mm 4 mm.6
pitor t t 56. () re Seprtion 8.85 5 4 4 5 5 4 5.4 4.5 / () 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
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
6. pitne F 4 F P. V () V 4 5 5 5 m Dieletri onstnt.5 () New.5.5 4 F New p. 5.5 4 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.. 5 5 6. Here we shoul onsier pitor n in series 4 ( ) 4 ( ) ( ) ( ) 4 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
pitor 64. Q V V 4 ( ) ( ) v 6 v m V (v / ) 6 Q v 6 ( ) ( ) 4 4 4 4 m 8 f ( ) ( ) 4 8 8 f 8 4 8.854 65. m m 66. m m V 4 V The pitne 4 4.45 m 8.85 8.85 The energy store (/) V (/) (4) 548.8 The fore ttrtion etween the pltes 548.8.54 7 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 ( ).
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.
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 ( ).