ELETROSTATICS Part II: BASICS

Transcription

1 GROWING WITH ONPTS: Physics LTROSTTIS Prt II: SIS Presence of chrge on ny oject cretes n electrosttic fiel roun it n in turn n electricl potentil is experience roun the oject. This phenomenon hs foun ppliction in storge of chrge on pir of electroes clle pcitors. nlysis of properties of pcitor hs een gretly simplifie with the use of Guss s. s shown elow. onsier n isolte lrge conucting plte crrying chrge +Q, it will estlish n electric fiel. Here, is the uniform electric fiel intensity on oth sies of the chrge plte, while σ is chrge ensity per unit re on the plte. Likewise, electric fiel ue to n isolte lrge plte crrying chrge Q woul e ; or. When these two pltes re rought sufficiently closer to ech other, such tht their seprtion is sufficiently smller thn their re, electric fiel in the interplte spces shll e itive, while eyon he pltes they woul e sutrctive (opposite) in nture. ccoringly, net electric fiel etween equl n oppositely chrge pltes shll e. Such comintion of equl n oppositely chrge prllel pltes is clle Prllel Plte pcitors. +Q -Q +Q -Q +Q -Q lectric iel ue to Isolte +ve n ve hrge Pltes Properties of pcitors: lectric iel When +ve n ve hrge Pltes t istnce Net lectric iel ue to Oppositely hrge Pltes (pcitor) quipotentil Surfce: pcitor hs two chrge crrying surfces of conucting mteril. lectric fiel rives +ve chrge to move in the irection of fiel n ve chrges ginst the irection of the fiel. Moreover, +ve chrge emntes electric from the surfce n ve chrge receive electric fiel into the surfce. ccoringly, chrge insie conuctor moves onto its surfce, which serves s equipotentil surfce, with uniform chrge ensity. Voltge ifference etween Prllel Pltes of pcitor (line integrl of lectric iel long line of seprtion etween prllel pltes of the cpcitor) = Volts pcitnce: Sustituting Vlue of in the Voltge ; here, ; or oulom. Here, is clle pcitnce, in rs. orce etween Pltes of pcitors: Since cpcitor is lwys oppositely chrge n hence ech of pltes will experience force of ttrction ue to the electric fiel crete y the other plte lectric fiel in the inter plte spce is uniform n is the comine effect of chrges on oth the pltes, hence chrge on one plte will fin it self plce in the fiel of the other plte will experience force Newton. nergy Store in pcitor: There re two pproches to etermining energy store in cpcitor. Metho 1: isplcement of plte experiencing force ( ) through istnce will cuse work which will e store s energy insie cpcitor. It is equl to : ( ) Metho 2: It is se on uiling of incrementl chrge n is equl to :

2 l nhncing Properties of pcitors: pcitors hve extensive utility. Iel prllel plte cpcitor s, which require extremely lrge surfce re n very smll seprtion etween pltes. These two conitions pose prcticl limittion in their use to meet its ppliction requirement. ccoringly, there re three pproches to enhnce the cpility of cpcitors viz: ) onfigurtion, ) omintion of pcitors n c) use of ielectric in the inter electroe region. ch of these ) etermintion of pcitnce of lectroe in ifferent onfigurtions: i) Sphericl pcitor. oncentric Hollow Spheres : iel outsie inner conucting sphere, of rius, crrying chrge +Q, t istnce r from its centre is. ll the fiel emnting from inner sphere woul en on Q chrge on the - Q inner surfce of the rius. Therefore, potentil ifference etween Two Sphere is [ ] ccoringly, pcitnce of Sphericl cpcitors is. +Q r V r V It les to Two cses Outer sphere hs rius >> then,. It ehves like n isolte sphere Rius of inner sphere ; then,. It ehves like prllel plte cpcitor. Oppositely hrge Isolte Spheres This configurtion cretes non-uniform electric fiel. n hence nlysis of the prolem shll e one y etermining electric fiel on point P on the line joining centre of the Two sphericl isolte opposite chrges. The net electric fiel Here, n, where is the unit vector long line. Since, the two spheres re of conucting mteril n hence ech, inepenently, woul e equipotentil surfce, n pth for clculting V, cn e chosen se on mthemticl convenience. ccoringly potentil ifference etween Two Spheres shll e: ( ) [ ] [ ]. Here, integrting y prts where, [ ] [ ] n is one using sustitution. It les to n, further, V +Q x P V -Q replcing limits. s n. ccoringly, the two integrls: [ ] [ ]. Re-sustituting isolte spheres ii) ylinricl pcitor [[ ] [ ]] ( ) ( ( ). If n, then ). ccoringly, cpcitnce the Two r.. oncentric Hollow yliners: In this electric fiel t istnce r from the surfce of the inner cyliner is chrge cyliners is:. ccoringly potentil ifference etween the Two concentric, oppositely [ ]. + Q, + Q, V V r r Therefore, cpcitnce of cylinricl cpcitor,.

3 . Isolte Prllel yliners: In this cse cross-sectionl igrm of the two isolte cyliners shll e sme s tht for the Two isolte spheres. The ifference is crete in mthemticl formultion ue to electric fiel t point hving ril istnce from the prllel cyliners. Like the cse of isolte spheres, here electric fiel ( ) n V re clculte se on mthemticl convenience. ccoringly, Here, n, where is the unit vector long line. ccoringly potentil ifference etween Two Spheres shll e: [ ] [ ] [ ] ( ) ( ). Here, is the chrge ensity in oulom per metre n l is length of the cyliners such tht.ccoringly, for the Two prllel cyliners cpcitnce r/m. r/m. If n, then ( ) ) omintion of pcitors: pcitors re use either in prllel, series comintion, n/or oth. ch of the comintion hs its specific utility. i) Series omintion of pcitors: This comintion is shown elow. In this, ech of the pir of pltes of cpcitor crries equl chrge, s chrcteristic to cpcitor When +Q chrge is given to the leftmost pltes of the cpcitor 1, it inuces Q chrge on the opposite plte of the cpcitor 1. This inuce chrge is cuse y the + Q - Q + Q - Q + Q - Q + Q - Q electric fiel prouce y the +Q chrge. This inuction leves +Q chrge on the left plte of cpcitor 2. Likewise, ech of the pir of connecte plte lso crries equl n opposite chrge. This is V 1 V 2 V 3 V 4 ttriute to the fct tht there is no flow or exchnge of chrge V outsie the cpcitor, except in the extreme leftmost n rightmost pltes. These, extreme pltes trnsfer ve chrge through the source. The -ve chrges cn flow esily ue to their inerti eing very smll, this leving +ve chrge ehin. This will ecome more cler s journey of the stuy procees into current electricity. Now ccoring to stnr reltionship,.. Moreover, +. Let is the equivlent cpcitnce of the series comintion of cpcitors shown ove. Then,. This cn e generlize into n expression, ; or,. Series cpcitors re use when voltge to e pplie is lrger thn cpcity of ny iniviul cpcitor ii) Prllel omintion of pcitors: In this pltes connecting +ve hrge n ve hrge re connecte together through conuctor. ccoringly, ll pltes crrying +ve chrge shll e t the sme potentil, likewise ll pltes crrying ve chrge shll e t sme potentil. Nevertheless, chrge on ech pir of pltes of cpcitor shll e regulte y cpcitnce of ech cpcitor. Thus, + Q 1 + Q 2 - Q Q 2. V 2 V Thus, totl +ve chrge, on the on the prllel comintion of cpcitors is: + Q 3 - Q 3. Thus it cn we 3 conclue tht n equivlent cpcitnce of prllel comintion of cpcitors is:... Prllel cpcitors re use when chrge to e hnle is greter thn the cpcity of ny of the cpcitor. iii) Series-Prllel omintion: It cn e ny comintion involving cpcitor connecte in series n prllel in ny sequence n orer. This involves ecomposition of complex comintion of cpcitors into simplest of series n prllel comintions,

4 n then integrting these ecompose elements into the mil solution. To ech of these simple comintions, ove concepts re pplie. It is illustrte through stge-wise solution in n exmple elow where equivlent cpcitnce cross - is to e etermine. Stge 0 (Given Prolem) Stge 1(Solution) Stge 2 (Solution) Stge 3 (Solution) Stge 4 (Soln.) Stge 5 (Soln.) Stge 6 (Soln.) Stge 7 (Soln.) Stge 8 (Soln.) inl (Soln.) ) ielectric n pcitors: hrge crrier conucting surfces re since t ifferent electric potentil they hve to e physiclly seprte n electriclly insulte. This introuction of insulting mteril in the interspce of the electroes of cpcitors rings into ply new phenomenon clle Polriztion with new possiilities to enhnce cpility of cpcitors. These possiilities re ) enhncing voltge withstn cpility, n ) incresing energy ensity of cpcitor. very sustnce is compose of electriclly neutrl toms (where centre of +ve n -ve chrges in n tom re concentric) n molecules (espite oning re so isperse tht the comine effect is electriclly neutrl). ut, in presence of electric fiel, these constituents re electriclly stresse n thus it prouces isplcement of chrges, without isloging them from toms n/or molecules. This kin of isplcement of chrges, within the inter-moleculr spces is clle polriztion. In cse of pure tomic sustnce, polriztion cn e ue to rift of electron orits uner the influence of electric fiel. While, in cse of molecules it cn e ue to reorienttion of moleculr on long the electric fiel. typicl ehviour of ielectric insie cpcitor is shown in the figure elow. 0 pcitor with Vcuum etween Pltes p onstituent lectric iels In Presence of ielectric =0-p pcitor With Reorienttion of Molecules of ielectric in Inter-Plte Region Let, cpcitor with inter-plte spce fille with vcuum hols chrge Q t potentil ifference etween pltes V 0, n the sme cpcitor when fille with ielectric in the inter-plte region require voltge V to hol the sme chrge Q. Since, potentil ifference etween the pltes is ifferent n hence presence of ielectric must e chnging the cpcitnce from originl vlue 0 to some vlue. ccoringly, ; or. In the ove figures 0 is the electric fiel cuse y the potentil ifference etween pltes of cpcitor, while p is the fiel cuse y inuce polriztion in ielectric. Thus net fiel in the region fille y ielectric is. s per Guss s Lw Likewise, ; here, is the net chrge ensity. ccoringly, for seprtion etween pltes, ( ) ( ) permittivity of the ielectric, while, K is the reltive permittivity n is lwys >1. ; or ; here n is the solute permittivity or simply

5 Thus cpcitnce in presence of ielectric of permittivity is ; i.e. increse in cpcitnce for sme imensions y n orer of K. Likewise, in presence of ielectric voltge cross ielectric, for holing sme chrge is is reuce y the orer of its reltive permittivity. The energy ensity (cpcity to store energy per unit volume) is:.. Guss s Lw in ielectric: Revisiting the ielectric fille cpcitor in the perspective of Gussin surfce s shown in the figure elow les to. This cn e written in the nother form of Gussin Surfce integrl in ielectric s :. Here, is clle enclose ree hrge, since it exists on the pltes of cpcitors freely; while is the clle the oun hrge, which is inuce insie the ielectric. This oun chrge is cuse ue to electric fiel prouce within ielectric y ree hrge. It is lso known s Guss s Lw in ielectric. isplcement Vector: Pictoril nlysis of electric fiel in ielectric is shown ove. It is seen tht vectors n re in opposite irections. ccoringly, effective fiel insie ielectric is. ut, y convention polriztion vector is long the +ve chrge crete y polriztion n hence, ; or. ccoringly, expression of Guss s lw using polriztion vector ecomes ( ). Here, is clle isplcement Vector, which tkes into ccount effective chrge n chrge crete y polriztion. References: ( Mking these concepts intuitive requires prcticing with prolem solving for which following references re vise. ) 1. NRT; PHYSIS, Text ook for lss XI n XII (Prt I n II), H.. Verm; oncepts of Physics, (Vil 1 & 2). 3. Resnick, Hlliy, Resnick n Krne; Physics (Vol I n II) Sers & Zemnsky; University Physics with Moern Physics s 0 s p