% Coulomb's Law of Electrostatics? CIn Vector aay lonsides 4uo charges +9 ond -4, separated by distance “w', mee foc dnd onrted mgr ae Fi, = fore exerted om qa by de ti fe Fa = kate (Rm) hI we = Kote |(%-G) | + AR t maselEeR) (AE) meg ne) “athe Hi xeason fenaat | Cr eeN) posto a Simla, Fovee applied by Fu, Fav =Rbits (R27) twi® = kude i 7”) | vit Here, we can clearly observe tet > fare Fa re: 3rd law of newton & valid dectrostatics also. a Electiic Field due to dipole on axis: We have two opposite charges separated by a distance of 242 which takes it a dipole . A is a yandom point on axis at a distance ‘y' -from centre of dipole . Now, Feld dueio “4! ab A —> Ey=-ky Similarly FPeld due to +9 ot A — ym onhere -ve stan denotes the dizection lich fs axti-prol Hence, [E=KE] ys hctod of dpe tite oh Jos! we & Toxque on dipale in external ffeld> wre showt un electrtc dipsle with charges ap 4-1, 2 a feta. of 34 pti. uniter eectyic Held. CE’ Dipole enakes an avgle © oath electate field. Ey Fr =-46 — force on charge 2, a R- ge oo on charge which means dre ke aes or dipole. fs equal & i magnitude and opposite % Avec o e,f0.erf. Iaene a velo Weg B coupes [PTT AS couple 15 acting on dipole , soit pmducee ongue tude Ree Meee Gib creer alerts: aki) = F x(t) = 4€ xQtsing) T= Pesta] (“ Pegen) Px vi cae Gse 1: when O=0"— 2. str0=0 ; which meas [T=0 this condition % called Stable eiltbiam because when the. dipole is Aiplced fram tic afentation, Te always comes back te same. configuration Gases when O=Ig0" >. staleo"=0 5 which mean: | see BE gonltion fs cated roll eg ee erage ont, dilaced ole never comes batl Rentation trated tt altgrs: Tuelt pes to the field. a OeB: hen O-88 9. sinfote; whh ntans Ts wmsiown, 5 ane 1 oe tt (ee tto oe Saree4 Gniss law Verification using Coulomb's Law’ We know, the net electate feld an a closed surface (ap) $5 1 Hmes He pet ‘chavge enclosed by the svxfkce- A Fact” aay geal Veriffeatfon s " Acording to electric flux, fe= §, E& - Edscose e kro, trtensily of speiie pat id Fl at same distance from ds charge q will ymain Cons abe for spherfcal sviface O=0° so dectefc flux gy = EGjds cose” G = Ef ds (ns fds wears area = Una?) ges Eur? — O Now, dueording to qulowb’s fad, €- 1 & Unk o* Gam TH@, we get Hye hy fear oy | 4 La(enclesed chooge) eo" est% Electric freld due to a seaight fong charged conductor Electric Held che toa shaight urifoumly changed Tn eta, density. (oncid erat ‘A on this ent crs oe a part of Aerath ‘fon ts Untform Se Anusstam surface will be eylindyical th i thts case. Lat dA be the smal) ates on tg snfte As conductor fs, positfvely ch Sec tea esdtaity cea trea oe eae Now, bi-fedawso [0-90] i 4é O.-fedheso [0-90] ep Gy-Sedn ese — Co-o!] Coma gd Hence, Net fox + Hees Br g.tp, = Jedn cosdo + fedn as do's fede coor ° * 0 + feda d= €A * [debra fi7otal cowed avea of surface -2nxl] We, Acc. 4e Gauss Lavo + from, Here, we. can clearly see ba 1 7 So, Graphically = Re% Electric Feld due to tnfinfte plane ite of density. 'o-! i eae love cha at tenth sn density -X-) ire Sheet Y | Draw a Gaussfam cylindey < axed of yadfus + Take 3 sample small sucfaces da’ at ©,O£ Total flux, B= Git dards = Jedm caso! + fed coso! + feda cosa =feda + [edd +0 = EA+ER ae) Kee fo gauss Lavy goon at from @4Q S 2€A- oA2% Potential ut a point due to pofrt charge? Let there be a point P at a dictance'r from +§ char: Electric potential ‘means workdone 40 bring a unft tye 4o the point oh Wepse) = J Fen dv e i harge from Infinfte W { KAW) de cos@ an i) kQ fdr [P0- m0" + estes’ 1) aes eee acon “KQ fd at nF Wip-ray= KO- on ga ili int te " 5 Polentfal due to dipole @ At a point on axfal (tne? Consider o dipole wrth charges +44-2 expavated by a distance of '20', Nestrt Pittttae ave word nasal '4 atamd: nage te algae So, Potential at P due to ty, V, = a7 dve to-gy Ver “KE + SiMe potential ot Py Wai =i) (V9 (eal irae= Ka Gra) - kG a) ¥ear + Gag Hen, Naia> KET = 1 short dipole (y>>). 4 ipa ee we] art the ti At a point on Sqvatovtal tne eae Lek feve be a point ‘Plat adistence.‘y' on equatvid fw," | i Se papa “Eee ener fotential at Pdvetory, \,= Ke iors dye to -y, > bly) fa*+a* Ss, Net potential at Py Y= Wy) = Kh (ok oa oa Na Oa ae DO Hence, elechfc potential due to dipole at any polit on ey. Line wit be O. (© At ony axbitrosy polit * Jet A be any avbitaasy point at a distances’ from Centye of dipole ma ing, an angle © with dipole ants. Pest bs the ff My = Tf ove. Lah aeindig fold oy fetter emprere_L 8 -& frie as. chown. S04 Big Ther, point A Ber on axial Itne of dipole with dipde moment *pcose’ fo, poterttal at a due to tric Component = Kipcos@)| = and potnt A es on equatorial (fae of dipole uifth dipole moment ‘psind’ Wugl disused dese Als ot ef He os ptt ke fot rpm vill be cexo.Hene, Vy = pss 40 y = Koso q A ¥ ral Relation between cectric field and Potential ® ae Consider wo equfpotential te A ard & sepasated a done often fe petal sae fa=V stele uta A be Vas Vedv Now, Work done fe displace unit positive charge from Bto A: d= Fae costa” d= -Fdx is é= &| eer 8, dw -Edx—@ al ky duds (a-Ve ip, fot Ea daa w= dv -® fom 0 £@ 4 tua ties EE] fel x 3 Potential Energy of System of +00 point ap (in plstpce of E-f-) Toitialy there were no charge. at A ard 8 _ & te Fist we'll baing 4, from co $07 werk clone 949° place aa a A phan th Pelco rie Sept Now, we'll tring q. from 2 to & (and tn this case 9, fs alveody aA) %, potential at 8 dvete 4, at Ay Ve- kK Or a —-® vy Werk done 4o plae q2 FBI waste 01 place a “29 (Ha) [Fg fond] We = kageArd as we knots um of work done 7s equal to the potential energy of system 4 Pentel Grergyhe 0 + Kd Y He ei wil 5 Rotenftal Energy of System of fu0 charges to an extemal electric fteld> let potential ot A and B be Vp ard Vo vespectvely. u Now, Workdone to place q,at A ES. ally was mot here) |,’ Ua Wa= Iva —O (6 Work done to place 9, at & We= bYet k2d2 -® v va Net woxk = camer TEIN Aa Ned kG = find as we know, this work done fs eqal to potentfal energy of system. U= qivat$sVor k4192 frst ott H Capacitance of a parallel plate capaciter [without dielectyfe] lonsides. 9 pavalel plate capacitor of plate aven A, and RecaeeeeeRe etty s Let to- be the surface hogs density. Electyic field outstde capac®y plates “i's zezo, Now, the electric ffeld Frside the tapactor plate, Ele (a appltcaton of gauss Lav of charge. plate) & FotenHal dfftererve between the plates, V= Ed awed 3 Ve £4 -@We brow thet, C= 3 Sa (froma) » | C= Ate aad Constdex 0 lel plate capacitor of pte ave Moecioemtat decree ete ee let to~ he the surface charge density... The gap behueen the plate fies! “ttn aidlectat substance having dfelectete. constart K The clectrte fidd between plates woftl be: E=o— e-& —® (se=8) A&K Se Potential difference befween the. phe Ve Ed Ve A ing ©) 3 at © [vstay @) Wow, Gpactance, c= © v . i ox, [C= ck] ot [c'= KC C! = capacttace. with dielectric C= capacttance wofihout dielectste K = dielectric constant of the medfum nt After fnserng dfelectate medium fh between, the plates of a capactior, THs Eopaditance fncreases by'k' times Eel capaditeme ahere,% Capadtance fn Poaalel & Consider two Capacitors conmected fq pevallel combination as sown ty te fps [ Tn parallel, combinaffon potential difference acess all re! capacttors yearns (sare but distibvtion of charge | +0. -& across’ each capacitor will be df ftevent. ve = Q,+Q. > ice C3 e=ev) al oxex' (e4G) > (are wi oe The effective capacttance of a combinaiton of ‘n/ capacitors fin pavalie) combfnation ts algebyfc som of capacitance of each Capatitoss. % Copacttors in Sewfest eo 4 Consider two capacitors ave connected fn sestee 3] [fl combination fn ga Ureult with capacitance | E E Gand cy vecpectively as shown i Figuee. a SE EISSE Slee allel Jn series combination the potential difference acyos 6 each capacitor fs different. but 1s balon of SSE thexge. ‘vermuins Same- + BeBe + Wye eae aid t pL Which means the effective capacitance of a combination of ‘n’ cepacitors In sevfes tsi eS ee Gf er) ca 165 on % Enengy Stored fn Capactox AVD expression for Cnevgy density’ rareeeeierou amovirt of charge tunsferved by the Sovace from ne ts Then work done ly te source ts dw=vdg [*? ve deo] 4 donkd, [ego]++ Total work done by the source ts trmsferring amount of chirge wefdw ane [Edy a Wel [ody decaeee. 9 Wet 1 ow aaa tise we 1 Gor) [29 rev] a Wed Cty z . Nowa, te work done fs Tn the form of- potenHal energy , ise: + U> Lev" z ov, UL ay 2 > |U2 iw be! ENERGY DENSITY > Te. petental enevgy pe unit volume of a capacitor fs knoion as Eregy density. 9 = -@ te Enevay densi: =U nna "4 ji My volume = tev ke at eye = 2 And + ag = & =| AK = 1 EWE Kd 2 Ard BA Wt : i++ Total work done by the source ts trmsferring amount of chirge wefdw ane [Edy a Wel [ody decaeee. 9 Wet 1 ow aaa tise we 1 Gor) [29 rev] a Wed Cty z . Nowa, te work done fs Tn the form of- potenHal energy , ise: + U> Lev" z ov, UL ay 2 > |U2 iw be! ENERGY DENSITY > Te. petental enevgy pe unit volume of a capacitor fs knoion as Eregy density. 9 = -@ te Enevay densi: =U nna "4 ji My volume = tev ke at eye = 2 And + ag = & =| AK = 1 EWE Kd 2 Ard BA Wt : i4 &e* Tf any medium fs there between plates of a capacitor, MeL GE*& Obtain an expyession for Deift Velocity of electrons Diift velodty Bs the velocfty with which electwns fr a conductor ave datfted towards tne posittve terminals of the poterHal source. We know that fa condictos there axe N number of electrons. AntHally y without any electric Held the electsms fh the conductor move randomly with Some velocity (Vi) Mee | 4 Fey paee @ Now, when Gn electoie Held ts applied across the conductor; the force applied on a electron by the electric ftald fs = Fe-eE Cz Frac] amo = -e6 Ci F-ma) aL ol fe —@ where, as acceleration of €% twards te tesminal. ro- mass of tHe election. Tt we take ‘C’ +0 be the average aelaxatfon Hime (the Hime fnterval between any two svccesstye colitston) then by frst equation of molfon, ils Cis OC) oq Ve 0 + (-2£) (1) a a [We ~eEL cahere a> Wag = dif t velocity oe & Relation between Curvent ond drift velocity © ——— aoe] lanstder 0 tonducton of Length L ond axea of cxoss-secKon ais iE ‘Al and 'n’ be the nee went per unit volume. = Neon Total chovge, @=nAle “+ cusatnt Yn the Conductor, r= meats) Cv-L)2% Sevies combination of Resistance? Two resistors of veststance Ri and Re ave Connectel| fn sevfes. As ne know, fa. sexes. combination custent: ts Hise sane byt’ voltage fs different” acyoss tne components Ve V+ Va vtny ohm law, IR= IR, +LRe R= Les &) hee LR oe ae Liaw pai AS PEE EEL v r Ss for ‘a! no of vesstane Tn sevfes, Rokr ReR, 1 --- ~~ Ry x Povalle! combination of Recfsternce? Two vetisors Ry and Rp awe (0) witty a bakery of YoHage 'V As we know, in ao & nected fr parallel ae 1 . EB pet ie rae te ert a oe Components of tre cPrcut i; a jiehth Bish en R RR WAAL 4) Kine | obo so for ‘n' veststors fn parallel, 1 =i zt been Relation between Internal vesistance, devmival potential dif}. avd Emp Consfdex a cell of emf C wth faternal vestelamce ‘s" tonmected fo the external cesistance(@). The cure in the ctseutt fs? IE —® |r Terminal potential dt Fference ve 1k —® Ha ee fetal ecictonceNow, @ can be watten as > leas E ( cots-mo iy) inde sage fete ) fos ve [v= E+Iy] which fs the velotton bln €,v end. % Cells fn sevtes : Consfder two cells with ey Sars ond Rady + Gnternal veriatance and ¥, vespecttvely. Connected Tn sevfes. ee (for vse) We we, f a turwent fs same but potential across component 1s ae Vs (ET s) + (6-15) = (€+€2)- (14,+1%) * Greeters a) Noo, we know Vey = xg [¥q —® Comparing © 4@, % Cells fm Pavallel loneidex tie cals of emf E.ard Es with fnteand resishance 4 andr, sespectyely conmected th parallel we a fo parallel combinatton polenta) . te-samel bik lcanert wth be sberae e ig components - . Es *)) Ve €-2 eee eae fs | —viheL a v(Bpty< Stern oy We eea Viel Caer Ohta x) at 1 Comparing 4nis ust Ve bey - Ley, = Get bh eter ord | Vey = Ba pert atte x Wheatstone Qridge » Wheatstone ba? pie fs an anangement. of for vesistance. used “fo determine © resistance one tesistors in terms of ofhex tmee reststs. For « balanced betdge, Ve= Vo (0s H figve) Now, appliig. Kivehoffs we an loop ADBAS L-R-1,P=0 > 1?-2e —® Now, ara Kérchoffts wle on loop Bede Q-1,5=0 4 at =~ @) dividirgey ® ABs we get > e ¥ ae at Ri s fou! get : aaa %& fhe condttion for bala ae a* Frading unknown ‘setistarce. sing Slide wie bridges Principle of teterbvidge and nding unknown reilstance Pafnefple wheat stone bovfdge As shotan fn fegurey Ic Ke unknown vesistance S> known resistance Move, the Jockey G) on whe AC of Dngth 2 4o obtain the null point (i-e- ze Teng Pant aRtnoreter)" Le pt: 8 We ml ptt an ae Nes As the bridge is balanced therefore, by Wheatsone bsfdge principle + LES & Is = cc - Rap gar 5 (00-2) pul fame te% Magnetic ffeld a} the centye of q chevlay loop cary ing comer + Consides lax curxent ca loop co: oe Sethe See a atu centvo of this loop. L tone ap Realy dle loc ond te Applying Biod Savas law, we get dB= -e (tLe) > Tas mvc ue Integeorg, both de, oe ge {de =f be Lae bn Med fae 5 B= ri ‘ x27 ("Eat means tot crcumfomny ot G= MeL | p oe yo % Magnetfe Field |on the axts of a cieculax covsent Soop Confdex q ctreular loop of vadius ‘x’ the axts of tre ctreulae Loop ot which Illes eal avert eee oe the ciresliy Soop ond the distance between the Loop and the point Neording fy Biot Savarts law, AB= Mo Tdd sind 4n woSo, the magnete ffeld at P due 4o cuavert element: dt : ole = He Idk sin 90° in Jo" a abe tne gate dt" ar Tata) Magnet field ot P dve to (yrsent element 1d (3 wLdt) de! ae aL sene adel = ae aut Gn (aan) we can see —> Hexe, AB =-dB! Resolvin d& %n two Components we find Hnat cos® component for tuo Fane sca opposite. Termeni ts zich he, ae , thot mg ne te field Sntensity at P wil be only die to stn® component Frese fone tobal magnetic. fleld due othe. cole «sl B = Sde stro Be Lt sh fig, gn B= Metso (Yo “Glan fg ier Gn ate) [ae i B= _belo OR (a2) (atexe)* ae J = re ss B= Been | yt Ht «>a , thin a es neglegi ibe Be Mot 2G) oe " Ele 2%x Ampere's Ctreuttal Law’ Te states that the fine fh al of magnelie field Intensity o closed | Fa eA eee ream me OE Gre feet =Uet Petr tonsider a stealeht comluctor cmnsying as shown tn the. sat ge gh eae hs ound the conductor . ty Band dt axe tn some direction so. angle bebiten them i0. a jeu = fedl cso’ = feat = Bfad * MeL ant — (Jud means dremferena 27) fail] te a WS Solent MF: due to ce long stradght ceseent casrying. covduetor Application of Ampere's CErecital Law fF Tovotd % Magette Feld due to an fntinttely Jeong steatght cvavent casnying. Conducta: Wee given a sholght woive of « Cross-seckonal yadus ‘a’ coxtying. steade See er ae cee eae et er teases Now, we have 49 calculate magneHe Held at a distance + from centre. But heve we'll have 3casess- G Wa, Re point lies outside wine GW) 1= 05 ie point Lies on the wie R Gib v jeu = hl » $Odl cos0"= Mot 9B ddd = a B(2nw hel baa * (for ¥70) Err x=a__ at pot Pa 1d he He Pe epee ee sh Stintlenly Mike Ost weil get — (ase tts To Hind the hi fo dee alee of Me ei wing ney a Grevlar Soop : cs Fadh= vee a Bg date sue 4 B(2np)= we_Ly? “ef 4 B2n “ule =~ [B= Ute noe werB Magnetic Freld due 4o Solenoid ren, see Ne. of urns = Magnetic. {veld Tnstde. tre solenoid Ps uniform ard sha mF. outside the golenotd ts weak (almost zed) Consider a close loop ABCD. ffs - [eu [ea [ee + (edt & Here, — [Bedl-0 [B ovkside 0] cb [Pate feai-o [81 at] Let a solenotd consists of ‘n’ no. of tus. unit Lrath and carry Cnvent I. of ht tt ee Hence, fEdl - $50 -0 0-0 Aeording to Ampere’ law + peat - uot Herel pled es ave. present 4 * O(L) = WeNI —fom © Be Vent iiB-wnl where, = no. of urns per unit Length te, Graph Using Ampere’s deco Case) Inside ol Law, obtained tre magnetic fell thee 4 Tosoid lots eh fiom Ampere's uw > x) PBdt - uel, Ge 6) nee, [Teo ¢ Bato > B-0 (ase D) Behween fre forvs trom Ampere’ lnwi- $B AL =e In fat hr) $Bdt sO = be ly B ft - MeN I Bre) = MoT = Hen a Sey = Hew] ev [Beuund] [acy 2 se 0) outside + (at fa) $@dt = us Te BO % Foxce acting on a corvent. covnying, conductor placed to MP - (ontider a condoctor of Lenath & and area of section A taasying, ete T pl * sy in a mogneHe field ot an gs shown. If number densfty of elections th the <—>/* ductor Ps n, then total no. of eectvore in the conducty fs 2 AnAs the force acting one. cecton tt $=CW® sinO whee Vu fs the datft velocity of electrons So the fotal fore acting on the conductor = Alnf = Afn(evs sine) = Ane Va)£b sin Lal = Ib stn0] wt GRretion ca be Helene by Senn eft and le 2 Fone between oo parallel strafght conductors Grvwying wrtent se ( Confer wo tottatte long. sfatyht condoctrsncovsying consents I, ard Za. tn the Same divecton « “hey oe held parallel 40 each other at a distance 'x', Stnge magnate Hell te produce de coment fiaigh eth onducler eset ‘each condd tor exberfences. a foace and, the force will be IAB sirO New, magnedie Feld ot P due to vent I, 8\- Mol, —@® “ane r TTS As the coment casryfr ing conductor Y. fs fh the magnetic Hdd 21, therefore the funit length of Y wil expettens a forte given by— . a 9 = GB: stngo" [12 Ld nit Joye) G= Ble«l (+ shee) Fe = Wht e Magnetic Field due to curate Ip at point @ B= Nol: —® ant Streflaly conductor X_ sil alco experience afore F dete I, cunsent, roy | FiBL, Stoo [ted (omit lep)] Fy= Ba Lisinto” F= MIL antWe can obsewe thal F, act perpendfeular to X and diverted towards Y. Hence, nich tract each ne. X and Y att he gery J Bs By bit en cuevtnt sill be Sn opposite divections, the conductors will tepel each offer and ma gritvds will be same as derived — abo e. 5) Bs. By Same cumtnt divection —> attraction Th Ee Spaite uoent dix —+ vepuléfon % ToRdvé acting on a cuveent coveying Soopbetl fn unfferm mF. (xectang lax) + yng Soop bo fo m4 Hence When a rectangular cuvaert Careying (ofl & placed fra uni el Hall Soen te ee tagalog oe dns nat ekptatene ane ae Fy Magnelic Force on a currerrt carsying. conductor E= Lib stra ——on avms ADL CD only bot field enerts no forte on the fwo aims Adand BL of oop hecavse Bis onttpavaliel to I | SPP The, mag etic Held is peperialey fo He em AS mete AB of "the Loop and exerts a force F on ity Which fs Arrected fato fhe plane of the loop: a Gffert.) 0 Fiz E0Bsingo" 9 LAB Steflavly, the magnetic feld exerts a force & on vm CD,which is directed oot Ree ene ats a force fy which i a te fas I0B- 6 Ts, the net fore on the foop ts 2e00 (os sald caslien)Bot ad we can see there will a toaqve on the Loop due the pale of forces F, ard fe ork fee Now, Consides the case when the plare f the Leop, i net akong the magnetic ffeld and makes on omgle wlth tt: Let the angle between the field and the normal to the Goth ke ahple 8. £-818, Theforee on epee ae ard MIA M= KIA (Hee k-\) 9 [o=TA) for mach borne, OT NIR ¥ Conversion of Galvanometes fate Ammeter® Galvanometes. can he. converted fnto ammeter by Connecting @ smal] Resistance S(shunt) fn parallel “wth the galvarometer Be s (shunt) = rox tuwert shrough gahanometer T= ammetes vonge & = falvaromeler ” Resstonce As $ and G ave connected th. pmallel, s(I-1y) - LR 3% Conversion of Galvanomeses fnto Voltmeter = Galvanometer con be converted nto voltmeter by Connecting high cesfstance fh Series. = urtvent tena gahenoretor R= hh vesistontee Y= €tenal potential Rye Galvanometes vesistante total vesistone = Re Re Nw, ace to chris lato, Vo 1y (yt) Yam R=L- 5 && Moffonal emf ot Induced Eme : Consider a vectangulars conducting Loop Pars =i a fn re ple of Fe paper tn fonth we |[® * * ge ath =» conductor PQ fs free to more EE * pt a ay let the vod F % moved towards aly ht Gis eld ett with & Constont yeloute'y’ assume —— ta there is no loss of enesly due 4 fricHon & am Q let PQ is moved ‘x distance towards waht , the avea erelosed by Loop PQRS fncreases. > Axed (A)= Lx [ Therefore, the Omount of ete flue Linked vith tre Roop hereases. the Toa Ain-emf % tndoced fn flux Hersough aelals ep ase = BAX (oso" ee 4 Induced Eme fn the (il Ps > e- dB at ] t- betx [fom ©, = BA (de) ==BLV| yg |e Hof chang. dtplacemtnt Bes [ae os Forte on the usive (external) f= BLL sinao" L Fens — fe gy z ale % Induced emf due +e rotation of Rod fn Magnelfe Rede et Consider a metallic. vod of ferath't! fs placecl ito ‘magnetic fats sl tn ‘te fy af Aven Covered by re vod or xotating by atm. aryl I c by sted var by dn age : 1os for 1 unt rotation (axea)= 4 zi £ | ete pad spt) Ss for 0 argle rotaton = 42 ax Axea will be, AlkO -@© Now, flux tnsough oven A, Pe BA es ate) Tnduced €MP frthe red, &= ‘i 3 Gelf- Induction of dolenwid + Consider o solenofd hav we anes witn dengtn Lard cross-section grea A, I isthe cursent ae Tt. So, these ree rragnetic field at a girtn paint tn sGreptaent it by Nove, The, magneHe Hox pex tun “a be equell to ae 4 Band grea ef each ten. = HN EA + Total sack wil be given by product of fiw eel in cia and $= WNI,N mf d- mer —O £ And, wt ako know, d= Lr —O t fom@ 1M Lf ~ Montz” Yl This Ps ae -Prdvetane of a solenaid.2% Mutual frductance of two Solenoidss ome tereder oo on org golenaide 5 ard So each of Lent ALN), ond re a fc me tums Pndhe solensi $) and a aspect! fs wound closely over Sy, so both tne solenoids a cons ideved hee ae oreta of Cross Section T fe the covrent flowing rough S). ee, ibe reagnetic. field & prduced at omy point frside solensid S due to cunt 1 Is B= Mut —@ Xn tums — And, the. magnetic fluc Gnked with each tum of S ie equal to B,A Tota! fragme Hc fx Ainked with saleneid So having Nl, tums ts d= BAN » 9. (Vout) anh a +6 CRE) bt p= mt, —® Le te ett of mel lc 2 fom © 49 > M1,- “(umn)a * [m= MeN.NAA v Pa ae And, HE the Coxe {6 filed witha vragnttte malaial of permeability M-3% AC Voltage applfed toa Resistance: Griider a Vesistor of-yesistor, of xasistance R fe connected fn sérfes with a drtult containing Altemating EMF — fasinwt —O) “current, Hrrough the dheutt,, TG 5 1 Gsout q Te 1, sin ot TH aed te ye conpariny D2) we can say that , there & no phase difference Peheen wget ond Emp Wave foam singly fsinvt | ------ -- ——— ee Melgaen 2% AC weltoye applied 40 an Inductos? n Goncider an fdyctor of Pnductance ‘L’ comnected fn sexes with a drcuit contatoing. Altemating EMF — Easinwt —O € by streat An eme ay frduce fn the frductor due to the cunent I, ee re fewording 42 Lenz lav, the tndued emf will oppose the alternating emer. we ah “aoe =-e a ee az) E= Lat a Lag a dI= Edt 9 dI= fo sinwt dt £fos fotal coment Rrtegrating both side , fare [ig tnae dt 4 te Ee (as) a If ee coswt Sin(-0)=tose Tage Sl (stn(g 4) sin Ge) = -sin8 OL +i Ge sin(wt-3) —® when ain (WEB) eH be Ly Hoe Lule pak valve eB © [i- Lo stn (wot-g ieee Pees D) ae on Comparing @L@ , we see that Land & hwe diferent phase fe phase difference behueen Lard € d= ot-A +a 9B B wblage Leod> wrrent. Have diageam for Tard € ek ie. 6 I aval €© AC Vollage applied fo 0 Capacitor > eialain Consider a capacttos of tapacttane ‘C' ts owed nes fn series Cortalrg AC of me af Epsineat ©. && shat -@ sees The maximum voliage of he capacttos wit be equal to €mF of the AC. fig eon Capacitor, Vv es ee f esinwt fs vee) Instantaneous current tthe cireit I= ot at I> d (cE sinat) dt ai- Ce drat) a T= C&W cosh ate cla (ctor + et) —») feo, L fll be mox(peak) when (Sr ewe) wii) become 4 nh I,=¢&w 0+ [amin] ys 2 Ue Compass d that there Ys a phase difference betwen Londé Le eee ee erate eter ie kau ree 4 Ss Phase dtfference between Tad e, B= wh rg wh oe& Here, voltage drop across aes al ed ard elle fr uctor Pe- Ver IR uw Ver 1X tO sca WA1X wd E Phaser Fa gram for LCR chreutt > Consider, 86 1s the total voltage supplied $n the, cheuth, Nt ARS In the above phasor dfagsam , et ¥, > \e oo M=(y-u) —® Now, wHaze acvoss all the Component, V= [1M FVa Ve (R= De + GY Ve IPT RT Vr L [LOK Ry vet baste tz= EE ! zh ee ot ort7% Resonating frequency fn sevfes Lek efycuit + Resonnnce occuxs when inductive veactance becomes eyval 4o apacttfve. ‘reac tance. t= Xe ol = 1 we we Le we we 2ny= te we Er we 2nJEC \\te ae , Xp will become equal to Xe ond wesonarnce wil) coca, ayid the Pence fs Known as yesonating reg) ty. % Average fower fn LCR Circutt + We know that a voltage €-&stniot applied to a sextes RLC dreutt drives « Covrent in the chruit given by. f= K8n(wt-f) 5 whee =u g ch Trnstontaneout power by the soune is B= ton (reas p= El > Snot xf sin(wt-O (3 fob [asd cas (aut A] =o. a ie ae ae is given by the overage of the two tens tn But we can See that only the second term fs dine dependent. TH ayerage fil be zexo("s positive half Gof the Cosine camels the negative seuond half) A P= frie wos t =(&) (E) oA [<4-d-a] Ps vis Time COS. abex fneagy Stored th an Inductor 3 an Constdes an frductor of inductance L comnected to a voltage source E ak shown fr Fie. peed fs we krowy Pe aa [° ea) it de, ae aw a fete 42) dw= [Idi —@ z Integrating both cfées, fdw= [urd ( = trax cent Inthe creat) We L fis Pye “fe me LL? q Tis work % stored th the chrouit as magreHic poker fad) ‘ 7 ao eat: é [U=tun pe:% Relation between CitHfeal angle and refractive fadex of @ medium’ lonsider ‘ sos a vay travelling from denser medium (1) 5& 3H" Mending fo snellS loao> sinc = (L) sin 90" M Sine = 1 ai ti sinc Ree + Refeaction at a spherical Suaface? wn ae hows xefraction by convex. ee Ke AC let o£, and be the omle made by © $ TE Me CiBoad F e te e <— way with, the prindple Onis The novmal drawn from the Conver vefracting Sustace passes through the. Contre of dein co: AN dines te Matha rom pole. Mind Se rection of tncident veay Bs taken te Now, In Aomc, 3° In Acmi, 8,=44p °. Now, By snellis lao’ My $108) = Ha Sin O., 4% 0,8 axe - small, = Sin®,2 8, and sin@, =O, Mi, = Pla) = ie ia MESO! Here, eee 8 gre eu small jm B= Des b fon be a 209 Mh bk) = Me-(B-h)*& Lens Makes formula + Constdes. 0 convex Lens (thick) , Leh arn objet fs placed on the patnciple axis at ‘0 Te rage formed’ by the convex ihtek “hey, Ps at 1) Be eatiee trough fPrst surface (AOC)= Tf sysface ADC fs mot present then Image will te formed at I, os aon fn the Tie a According 49 xefraction formula: MW-N - me _ mM —O PR vo 2) Further, sefracton trough setond svefare (Apc) tne STA ie is not Pattee eT, will behave Like object He the trate Ep tend iat vill be formed at wn on fF Now, According to ao formula, mom 2 _ We _@ Re a he Adding © 4© + Mone. Mane. n, TR “Rae > Yom, Mom - mn ee LL a fesm — (men). (1-2) Re a vi u+ os (z,-&)-4¢ ty! = FEB) lal % Refraction Aevough Perce’ Consider, 0 tefangulor point, a a vay of Hirt charkes on tHe face At ert oe a ism and then refracted by HS ’ Mace AG jowards the fast of the, al prism fe cot ae in OR ts refracted ¢ ty the fase AC dway the novi 3 i ae a 4, and per yefracton be AC wes pect = om oF st of ee Tn AQNR, Ly + £4, +ZONR = ign" —@ In 4uadvilateral AONR , eee +90" = 360 LA +LONR = 3¢0"-180" A +LONR = 120" ——© fom O£@ 9 lx+ Qa + Lake =A + Lhe, Lorin = LA Az ¥j+ %|-@) a or Allo, f= Si a $= G-a)+ (e-8) a §= (Pre) -— Gas)paEtte ek Gomi) w [re=tra] —@ when , §= $e ster fe Btn J G@ bevomes, rer=A Tauak ~ A} -® And, 0, @ bewmes 9 its bra = eh [ole t dash ® flow, Aucnding fo. srels Law, vuhere. kn 8s -wefanaie Padex ot ) ae uw (ied present fn tne prisin a (SecA) tal tt (fn O10) A)26 fosFHon ond width of the Fringe fr Tnterfesence® The distance feween any fut consecutive, bright 7 40 doak fri Theat ee eae desk Fringe Ts equal tothe width of a bagel fenge. Consider Ihe faom two slit Siard S apni ‘at point P ‘ a ot on the scaten bgt ard dk Finger | ad ne be the distance between two shits Sj mds © D be the distance between cit ome screen. 2 oo aa Now, at point P the path difference of fwo waves is + Ax= 3 P-SP —@ wie MSPs a sitene? In. D528", a iPS, in DSBP, 1 = D+(y-dP_O P= SB BP ae “Eraye Now, ef) - 4 SPS P= D% (Yd De +(y-d)* GP-S) (Ses SP) = by ed (60-5) (SP45P) = 2yd Assorring Pvay clue 49.0, such drat SP &S2P=D 2 Dev)bx = 2h Aoon-Byd axe yd brenda ya nA nad (ose L) fox Maxton, Wea ; shen neo ,y=0 (Central bight fringe) n=l) grab (ist BF) pan > ye DAR Oh OF)Coed) fix miniena: act 4 ye (2n-l ir eb hen, n=) grab Cist De) med 5 yr aap. [2r4 oF] fami 1 47 Gnade (o™ 9) se Alternate Dark & Bright fringes appear. ee fox _fainge width’ Te fifo beeen ee Datgnt fringes gives the ringe width Pate = ds = (ne) Ap nA oe Snilaaly, for bathe. -> MeCHAPTER HLL? Dual Nature of Radickion and Mattes a de- Broglie tquattons i ee f Frepeenty) 4 wavelength A) propagates tr vaccum, = hy] —® Awording to Gnstein mass-enerey equivalence , E= me?] —O Congariry OLD , wa ie Now, momentum of each photon fs 5 eo v Le Mc of a8 p 2 (2 P Pe (t 4-5] 7 — tits 18 de-brogite.es”, Let us fake on Samples e accelerated Fh a polentfal dif}. ‘v' , then its KE can be written as: Keev' The Lineat momentum ¢ Ke of © moving. with velocity * ove \p= mv | 4 k= Lm omolfplyi ing ‘en! both sides: mk= ia 2mk= re Square x00 both side? (2m = [mer my = [2mk Now, frm de-bvegle ers Ae + Ae b, invSubditobing he 6:68x107""Js m= 41X 107% = 16X10 5* aa sce pane e rN jen atoms, devive ine ¢ expression for total Se nce ne merece mel a on au of, mass m and chavge @ - EL yevelving. vgith velocity Y around 4 bles 4 "ern omic number °z. Then the Centatpetal \ xtoviced oy the. election fs provid Bt @ en He -fo er atorsiae laters nud leur r and elechon oral ng to equation: N Au to Bohee fostulates mvt = yh -© =) ontreS tht, oO mie ne” “kzet my = welt x 4m eet 2 = rhs po eta wok —® Gadtus of n™ orbit Now, Yeloutty of e” fn stalfonary oxbits mvt = yh 21 mvhf = oh mn ze~ an ye zee gy tants | wre GYdloity oF 2 tan energy letNow, Energy of € in Stationary oxbits & ke. = Lave ) Pe= bY de ¥v a ihm (get HF = kke) (ze) 2 2nh fo: 7 Ge te 24 a ae Sth & Pe = earn ee G) TE= Ke + PE Te = —mzet gone