Electrostatics Electric Dipole. Equatorial axis. q 2l

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1 lectosttics lectic Diole () Genel infomtion : System of two equl nd oosite chges seted by smll fixed distnce is clled diole A qutoil xis l q Diole xis (i) Diole xis : Line joining negtive chge to ositive chge of diole is clled its xis It my lso be temed s its longitudinl xis (ii) qutoil xis : eendicul bisecto of the diole is clled its equtoil o tnsvese xis s it is eendicul to length (iii) Diole length : The distnce between two chges is known s diole length (L l) (iv) Diole moment : It is quntity which gives infomtion bout the stength of diole It is vecto quntity nd is diected fom negtive chge to ositive chge long the xis It is denoted s nd is defined s the oduct of the mgnitude of eithe of the chge nd the diole length ie q( l ) Its SI unit is coulomb-mete o Debye ( Debye M L T A C m) nd its dimensions e Note : A egion suounding sttiony electic diole hs electic field only When dielectic is lced in n electic field, its toms o molecules e consideed s tiny dioles Wte (H O), Chloofom (CHCl ), Ammoni (NH ), HCl, CO molecules e some exmle of emnent electic diole H O H () lectic field nd otentil due to n electic diole : It is bette to undestnd electic diole with mgnetic diole 68

2 lectosttics SNo lectic diole Mgnetic diole (i) System of two equl nd oosite chges seted System of two equl nd oosite mgnetic oles by smll fixed distnce ( mgnet) seted by smll fixed distnce (ii) lectic diole moment : q( l ), diected fom (iii) q to q It s SI unit is coulomb mete o Debye Intensity of electic field e qutoil line A q q l e g Mgnetic diole moment : M m( l ), diected fom S to N It s SI unit is mee mete Intensity of mgnetic field m m S N e qutoil line e M g α q α S N l Axil line l Axil line M If, e nd g e thee oints on xil, equtoil nd genel osition t distnce fom the cente of diole on xil oint on equtoil oint (diected fom to q) e (diected fom q to q) on genel oint ( cos ) Angle between nd is ( α) (whee lectic otentil At At g V nd is o, e nd is 8 o, cos V tn α tn ), At e V If, e nd g e thee oints on xil, equtoil nd genel osition t distnce fom the cente of diole on xil oint on equtoil oint µ M π (diected fom S to N) e µ (diected fom N to S) M π µ M on genel oint ( cos ) π Angle between nd M is o, e nd M is 8 o, nd M is ( α) (whee tn α tn ) () Diole (electic/mgnetic) in unifom field (electic/mgnetic) (i) Toque : If diole is lced in n unifom field such tht diole (ie o M ) mkes n ngle with diection of field then two equl nd oosite foce cting on diole constitute coule whose tendency is to otte the diole hence toque is develoed in it nd diole ties to lign it self in the diection of field 69

3 lectosttics Conside n electic diole in lced in n unifom electic field such tht diole (ie ) mkes n ngle with the diection of electic field s shown A mgnetic diole of mgnetic moment M is lced in unifom mgnetic field by mking n ngle s shown q q q A q m S N m () Net foce on electic diole (b) oduced toque τ sin ( τ ) net () Net foce on mgnetic diole net (b) toque τ M sin ( τ M ) (ii) Wok : om the bove discussion it is cle tht in n unifom electic/mgnetic field diole ties to lign itself in the diection of electic field (ie equilibium osition) To chnge it s ngul osition some wok hs to be done Suose n electic/mgnetic diole is ket in n unifom electic/mgnetic field by mking n ngle with the field, if it is gin tun so tht it mkes n ngle with the field, wok done in this ocess is given by the fomul q q M q q W (cos cos ) If o nd ie initilly diole is ket long the field then it tun though so wok done W ( cos) W M(cos cos ) If o nd then W M( cos) (iii) otentil enegy : In cse of diole (in unifom field), otentil enegy of diole is defined s wok done in otting diole fom diection eendicul to the field to the given diection ie if 9 o nd then M M W U (cos 9 cos) U cos W U M(cos 9 cos) U M cos 7

4 lectosttics (iv) quilibium of diole : We know tht, fo ny equilibium net toque nd net foce on ticle (o system) should be zeo We ledy discussed when diole is lced in n unifom electic/mgnetic field net foce on diole is lwys zeo ut net toque will be zeo only when o o 8 o When o ie diole is lced long the electic field it is sid to be in stble equilibium, becuse fte tuning it though smll ngle, diole ties to lign itself gin in the diection of electic field When 8 o ie diole is lced oosite to electic field, it is sid to be in unstble equilibium M M M o 9 O 8 o Stble equilibium Unstble equilibium τ τ mx τ W W W mx U min U U mx o 9 O 8 o Stble equilibium Unstble equilibium τ τ mx M τ W W M W mx M U min M U U mx M (v) Angul SHM : In unifom electic/mgnetic field (intensity /) if diole (electic/mgnetic) is slightly dislced fom it s stble equilibium osition it executes ngul SHM hving eiod of oscilltion If I moment of ineti of diole bout the xis ssing though it s cente nd eendicul to it s length o electic diole : I T π nd o Mgnetic diole : T π I M (vi) Diole-oint chge intection : If oint chge/isolted mgnetic ole is lced in diole field t distnce fom the mid oint of diole then foce exeienced by oint chge/ole vies ccoding to the eltion (vii) Diole-diole intection : When two dioles lced closed to ech othe, they exeiences foce due to ech othe If suose two dioles () nd () e lced s shown in figue then oth the dioles e lced in the field of one nothe hence otentil enegy diole () is U cos then by using d d du, oce on diole () is d 6 Similly foce exeienced by diole () du d 6 so O O q q q 6 7

5 6 Negtive sign indictes tht foce is ttctive nd lectosttics S No Reltive osition of diole oce otentil enegy (i) q q 6 (ttctive) (ii) q q (eulsive) q q (iii) q q (eendicul to ) q Note : Sme esult cn lso be obtined fo mgnetic diole () lectic diole in non-unifom electic field : When n electic diole is lced in nonunifom field, the two chges of diole exeiences unequl foces, theefoe the net foce on the diole is not equl to zeo The mgnitude of the foce is given by the negtive deivtive of the otentil enegy wt du d distnce long the xis of the diole ie d d Due to two unequl foces, toque is oduced which otte the diole so s to lign it in the diection of field When the diole gets ligned with the field, the toque becomes zeo nd then the unblnced foce cts on the diole nd the diole then moves linely long the diection of field fom weke otion of the field to the q stonge otion of the field So in non-unifom electic field (i) Motion of the diole is tnsltoy nd ottoy (ii) Toque on it my be zeo Concets o shot diole, electic field intensity t oint on the xil line is double thn t oint on the equtoil line on electic diole ie xil equtoil It is intesting to note tht diole field deceses much idly s comed to the field of oint chge 7

6 lectosttics q xmles bsed on electic diole xmle: 8 Solution: (d) xmle: 85 If the mgnitude of intensity of electic field t distnce x on xil line nd t distnce y on equtoil line on given diole e equl, then x : y is [AMCT 99] () : (b) : (c) : (d) : Accoding to the question x x / () : y y Thee chges of (q), () nd () e lced t the cones A, nd C of n equiltel tingle of side s shown in the djoining figue Then the diole moment of this combintion is [M MT 99; CMT 99] () q (b) Zeo (c) q q A Solution: (c) (d) q The chge q cn be boken in q, q Now s shown in figue we hve two equl dioles inclined t n ngle of 6 o Theefoe esultnt diole moment will be net cos 6 C 6 O xmle: 86 q An electic diole is lced long the x-xis t the oigin O A oint is t distnce of cm fom this Solution: (b) oigin such tht O mkes n ngle π with the x-xis If the electic field t mkes n ngle with x- xis, the vlue of would be [M MT 997] π π () (b) π tn (c) (d) tn Accoding to question we cn dw following figue π As we hve discussed elie in theoy α So, π tnα tn π tn α tn xmle: 87 An electic diole in unifom electic field exeiences [RT ] () oce nd toque both (b) oce but no toque (c) Toque but no foce (d) No foce nd no toque Solution: (c) In unifom electic field net, τ net xmle: 89 Two oosite nd equl chges 8 coulomb when lced cm wy, fom diole If this diole is lced in n extenl electic field 8 newton/coulomb, the vlue of mximum toque nd the wok done in otting it though 8 o will be [M T 996 Simil to M MT 987] () 6 Nm nd 6 J (c) 6 Nm nd J 7 O y α π/ (b) Nm nd J (d) Nm nd 6 J x

7 Solution: (d) τ mx nd W mx l 8 8 C-m xmle: 9 Solution: (d) So, τ mx 8 8 N-m nd W mx 6 J lectosttics A oint chge lced t ny oint on the xis of n electic diole t some lge distnce exeiences foce The foce cting on the oint chge when it s distnce fom the diole is doubled is () (b) (c) [CMT 99; MNR 986] oce cting on oint chge in diole field vies s whee is the distnce of oint chge fom the cente of diole Hence if mkes double so new foce ' 8 xmle: 9 A oint ticle of mss M is ttched to one end of mssless igid non-conducting od of length L Anothe oint ticle of the sme mss is ttched to othe end of the od The two ticles cy chges q nd esectively This ngement is held in egion of unifom electic field such tht the od mkes smll ngle (sy of bout 5 degees) with the field diection (see figue) Will be minimum time, needed fo the od to become llel to the field fte it is set fee (d) 8 q q ml ml () t π (b) t π q (c) ml t π (d) t π Solution: (b) In the given sitution system oscillte in electic field with mximum ngul dislcement It s time eiod of oscilltion (simil to diole) I T π whee I moment of ineti of the system nd ql ml q Hence the minimum time needed fo the od becomes llel to the field is t T π I Hee L L ML I M M π t ML ql π ML q Ticky exmle: Solution: (b) An electic diole is lced t the oigin O nd is diected long the x-xis At oint, f wy fom the diole, the electic field is llel to y-xis O mkes n ngle with the x-xis then () tn (b) tn (c) 5 o (d) As we know tht in this cse electic field mkes n ngle α with the diection of diole Whee tn α tn Hee α 9 o α 9 Hence tn( 9 ) tn cot tn tn tn Y O tn X 7

8 lectic lux lectosttics () Ae vecto : In mny cses, it is convenient to tet e of sufce s vecto The length of the vecto eesents the mgnitude of the e nd its diection is long the outwd dwn noml to the e d s Ae ds () lectic flux : The electic flux linked with ny sufce in n electic field is bsiclly mesue of totl numbe of lines of foces ssing nomlly though the sufce o lectic flux though n elementy e ds is defined s the scl oduct of e of field ie d ds ds cos Hence flux fom comlete e (S) ds cos S cos If o, ie sufce e is eendicul to the electic field, so flux linked with it will be mx ie mx ds nd if 9 o, min () Unit nd Dimensionl omul N C SI unit (volt m) o m It s Dimensionl fomul (ML T A ) () Tyes : o closed body outwd flux is tken to be ositive, while inwd flux is to be negtive d d s n ody ody ositive flux (A) Negtive-flux () Guss s Lw () Definition : Accoding to this lw, totl electic flux though closed sufce enclosing chge is times the mgnitude of the chge enclosed ie ( enc ) () Gussin Sufce : Guss s lw is vlid fo symmeticl chge distibution Guss s lw is vey helful in clculting electic field in those cses whee electic field is symmeticl ound the souce oducing it lectic field cn be clculted vey esily by the cleve choice of closed sufce tht encloses the souce chges Such sufce is clled Gussin sufce This sufce should ss though the oint whee electic field is to be clculted nd must hve she ccoding to the symmety of souce 75

9 lectosttics eg If suose chge is lced t the cente of hemishee, then to clculte the flux though this body, to encloses the fist chge we will hve to imgine Gussin sufce This imginy Gussin sufce will be hemishee s shown Net flux though this closed body Hence flux coming out fom given hemishee is () Zeo flux : The vlue of flux is zeo in the following cicumstnces (i) If diole is enclosed by sufce (ii) If the mgnitude of ositive nd negtive chges e equl inside closed sufce enc ; q q enc so,, q q 5q (iii) If closed body (not enclosing ny chge) is lced in n electic field (eithe unifom o non-unifom) totl flux linked with it will be zeo Shee in πr out πr T y T d s T d s x d s in out () lux emegence : lux linked with closed body is indeendent of the she nd size of the body nd osition of chge inside it T T T T 76

10 lectosttics (i) If hemisheicl body is lced in unifom electic field then flux linked with the cuved sufce (ii) If hemisheicl body is lced in non-unifom electic field s shown below then flux linked with the cuved sufce R R πr cuved nˆ πr cuved nˆ (v) If chge is ket t the cente of cube (iv) If chge is ket t the cente of fce cone 8 totl ( ) fce edge 6 ist we should enclosed the chge by ssuming Gussin sufce (n identicl imginy cube) totl cube (ie fom 5 fce only) 5 fce Concet In CGS Hence if C chge is enclosed by closed sufce so flux though the sufce will be π π xmle bsed on electic flux nd Guss s xmle: 9 lectic chge is unifomly distibuted long long stight wie of dius mm The chge e cm length of the wie is coulomb Anothe cylindicl sufce of dius 5 cm nd length m symmeticlly encloses the wie s shown in the figue The totl electic flux ssing though the cylindicl sufce is [M T ] () (b) m (c) (d) ( π ) ( π ) 5 cm 77

11 Solution: (b) xmle: 9 lectosttics Given tht chge e cm length of the wie is Since cm length of the wie is enclosed so enc lectic flux emeging though cylindicl sufce A chge is situted t the cone A of cube, the electic flux though the one fce of the cube is [CMT ] Solution: (c) xmle: 9 Solution: () () (b) (c) (d) 6 8 o the chge t the cone, we equie eight cube to symmeticlly enclose it in Gussin sufce The totl flux T Theefoe the flux though one cube will be cube The cube hs six fces nd 8 flux linked with thee fces (though A) is zeo, so flux linked with emining thee fces will Now 8 s the emining thee e identicl so flux linked with ech of the thee fces will be 8 A sque of side cm is enclosed by sufce of shee of 8 cm dius Sque nd shee hve the sme cente ou chges 6 C, 5 6 C, 6 C, 6 6 C e locted t the fou cones of sque, then out going totl flux fom sheicl sufce in N m /C will be () Zeo (b) (6 π) 6 (c) (8π) 6 (d) 6π 6 Since chge enclosed by Gussin sufce is enc ( 5 6 ) so xmle: 9 In egion of sce, the electic field is in the x-diection nd ootionl to x, ie, xˆ i Conside n imginy cubicl volume of edge, with its edges llel to the xes of coodintes The chge inside this cube is () Zeo (b) (c) (d) 6 Solution: (b) The field t the fce ACD x ˆ i lux ove the fce ACD ( x ) The negtive sign ises s the field is diected into the cube The field t the fce GH x )ˆ lux ove the fce GH ( i ( x ) The flux ove the othe fou fces is zeo s the field is llel to the sufces Totl flux ove the cube q whee q is the totl chge inside the cube q Y Z x D A C H G X 78

12 lectosttics Ticky exmle: Solution: (b) In the electic field due to oint chge sheicl closed sufce is dwn s shown by the dotted cicle The electic flux though the sufce dwn is zeo by Guss s lw A conducting shee is inseted intesecting the eviously dwn Gussin sufce The electic flux though the sufce () Still emins zeo (b) Non zeo but ositive (c) Non-zeo but negtive (d) ecomes infinite Due to induction some ositive chge will lie within the Gussin sufce dwn nd hence flux becomes something ositive Aliction of Guss s Lw Guss s lw is oweful tool fo clculting electic field in cse of symmeticl chge distibution by choosing Gussin sufce in such wy tht is eithe llel o eendicul to it s vious fces eg lectic field due to infinitely long line of chge : Let us conside unifomly chged wie of infinite length hving constnt line chge density is chge λ λ Let be oint distnt fom the wie t which the electic field is to be length clculted Dw cylinde (Gussin sufce) of dius nd length l ound the line chge which encloses the chge ( λ l ) Cylindicl Gussin sufce hs thee sufces; two cicul nd one cuved fo sufces () nd () ngle between electic field nd noml to the sufce is 9 o ie, 9 o So flux linked with these sufces will be zeo Hence totl flux will ss though cuved sufce nd it is ds cos (i) Accoding to Guss s lw (ii) quting eqution (i) nd (ii) ds ds xπl l nˆ nˆ π l λ π kλ K 79