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vè;k; 1 oS|qr vkos'k rFkk {ks=k 1.1 Hkwfedk ge lHkh dks] fo'ks"kdj 'kq"d ekSle esa] LosVj vFkok laf'y"V oL=kksa dks 'kjhj ls mrkjrs le; pV&pV dh èofu lquus vFkok fpuxkfj;k¡ ns[kus dk vuqHko gksxkA efgykvkssa osQ oL=kksa tSls ikWfy,LVj lkM+h osQ lkFk rks ,slh ?kVuk gksuk izk;% vfuok;Z gksrk gSA D;k...

vè;k; 1 oS|qr vkos'k rFkk {ks=k 1.1 Hkwfedk ge lHkh dks] fo'ks"kdj 'kq"d ekSle esa] LosVj vFkok laf'y"V oL=kksa dks 'kjhj ls mrkjrs le; pV&pV dh èofu lquus vFkok fpuxkfj;k¡ ns[kus dk vuqHko gksxkA efgykvkssa osQ oL=kksa tSls ikWfy,LVj lkM+h osQ lkFk rks ,slh ?kVuk gksuk izk;% vfuok;Z gksrk gSA D;k vkius dHkh bl ifj?kVuk dk Li"Vhdj.k [kkstus dk iz;kl fd;k gS\ fo|qr foltZu dk ,d vU; lkekU; mnkgj.k vkdk'k esa xtZu osQ le; rfM+r fn[kkbZ nsuk gSA fo|qr >VosQ osQ laosnu dk vuqHko gesa ml le; Hkh gksrk gS tc ge fdlh dkj dk njokTk+ k [kksyrs gSa vFkok tc ge viuh cl dh lhV ij f[kldus osQ i'pkr mlesa yxh yksgs dh NM+ dks idM+rs gSaA bu vuqHkoksa osQ gksus osQ dkj.k gekjs 'kjhj esa ls gksdj mu oS|qr vkos'kksa dk fol£tr gksuk gS tks fo|qrjks/h i`"Bksa ij jxM+ osQ dkj.k ,d=k gks tkrs gSaA vkius ;g Hkh lquk gksxk fd ;g oS|qr vkos'k (fLFkjoS|qr) osQ mRiUu gksus osQ dkj.k gSA bl vè;k; rFkk vxys vè;k; esa Hkh ge blh fo"k; ij ppkZ djsaxsA fLFkj ls rkRi;Z gS og lc oqQN tks le; osQ lkFk ifjo£rr vFkok xfre; ugha gksrkA fLFkjoS|fq rdh osQ varxZr ge fLFkj vkos'kksa }kjk mRiUu cyksa] {ks=kksa rFkk foHkoksa osQ fo"k; esa vè;;u djrs gSaA 1.2 oS|qr vkos'k bfrgkl osQ vuqlkj yxHkx 600 bZ- iwoZ gqbZ bl rF; dh [kkst dk Js;] fd Åu vFkok js'keh&oL=k ls jxM+k x;k ,sEcj gydh oLrqvksa dks vkd£"kr djrk gS] xzhl ns'k osQ feysVl osQ fuoklh FksYl dks tkrk gSA ^bysfDVªflVh* 'kCn Hkh xzhl dh Hkk"kk osQ 'kCn bysDVªkWu ls O;qRiUu gqvk gS ftldk vFkZ ,sEcj gSA ml le; inkFkks± osQ ,sls cgqr ls ;qxy Kkr Fks tks ijLij jxM+s  HkkSfrdh tkus ij Hkwls osQ frudksa] ljoaQMs dh xksfy;ksa] dkxTk+ osQ NksVs VqdM+kas vkfn gydh oLrqvksa dks vkd£"kr dj ysrs FksA vki bl izdkj osQ izHkko dk vuqHko vius ?kj ij fuEufyf[kr fØ;kdyki }kjk dj ldrs gSaA li+Qs n dkxT+k dh yach iryh ifêð;k¡ dkVdj mu ij /hjs ls bLrjh dhft,A bu ifêð;ksa dks Vsyhfo”ku osQ insZ vFkok oaQI;wVj osQ ekWfuVj osQ fudV ykb,A vki ns[ksaxs fd ifêð;k¡ insZ dh vksj vkd£"kr gks tkrh gSaA okLro esa os oqQN {k.k rd insZ ls fpidh jgrh gSaA ;g Hkh izsf{kr fd;k x;k fd ;fn Åu vFkok js'ke osQ diM+s ls jxM+h gqbZ nks dk¡p dh NM+ksa dks ,d&nwljs osQ fudV yk,¡ rks os ,d&nwljs dks izfrd£"kr djrh gSa ¹fp=k 1.1(a)ºA Åu dh os yfM+;k¡ vFkok fp=k 1.1 NM+sa rFkk ljoaQMs dh xksfy;k¡% ltkrh; vkos'k ,d&nwljs js'ke osQ diM+s osQ os VqdM+s ftuls bu NM+ksa dks jxM+k dks izfrd£"kr rFkk fotkrh; vkos'k ,d&nwljs dks vkd£"kr djrs gSaA x;k Fkk] os Hkh ijLij ,d&nwljs dks izfrd£"kr djrs gSa ijarq dk¡p dh NM+ rFkk Åu ,d&nwljs dks vkd£"kr djrs gSaA blh izdkj] fcYyh dh lewj ls jxM+h gqbZ nks IykfLVd dh NM+sa ,d&nwljs dks izfrd£"kr djrh gSa ¹fp=k 1.1(b)º ijarq lewj dks vkd£"kr djrh gSaA blosQ foijhr] IykfLVd http://www.physics.ucla.edu/travoltage/HTML/staticElectricity.htm dh NM+sa dk¡p dh NM+ksa dks vkd£"kr djrh gSa ¹fp=k 1.1(c)º rFkk flYd vFkok Åu ftlls dk¡p dh NM+ksa dks jxM+k x;k Fkk] dks izfrd£"kr djrh gSaA dk¡p dh NM+ lewj dks izfrd£"kr djrh gSA ljy fLFkj oS|qrdh; iz;ksxksa ls lacaf/r izHkkoh ltho fp=k.k ;fn lewj ls jxM+h gqbZ fdlh IykfLVd dh NM+ dks js'ke vFkok uk;ykWu osQ /kxksa ls yVdh gqbZ nks NksVh ljoaQMs dh xksfy;ksa (vktdy ge ikWfy,LVjhu dh xksfy;k¡ Hkh mi;ksx dj ldrs gSa) ls Li'kZ djk nsa] rks ;s xksfy;k¡ ,d&nwljs dks izfrd£"kr djrh gSa ¹fp=k 1.1(d)º rFkk Lo;a NM+ ls Hkh izfrd£"kr gksrh gSaA ;gh izHkko ml le; Hkh fn[kkbZ nsrk gS tc ljoaQMs dh xksfy;ksa dks js'ke ls jxM+h dk¡p dh NM+ ls Li'kZ djkrs gSa ¹fp=k 1.1(e)ºA ;g ,d ukVdh; izs{k.k gS fd dk¡p dh NM+ ls Li'kZ dh gqbZ ljoaQMs dh xksyh nwljh IykfLVd NM+ ls Li'kZ dh xbZ ljoaQMs dh xksyh dks vkd£"kr djrh gS ¹fp=k 1.1(f)ºA o"kks± osQ iz;kl rFkk lko/kuhiwodZ fd, x, iz;ksxksa ,oa muosQ fo'ys"k.kksa }kjk ljy izrhr gksus okys ;s rF; LFkkfir gks ik, gSaA fofHkUu oSKkfudksa }kjk fd, x, cgqr ls lko/kuhiw.kZ vè;;uksa osQ i'pkr ;g fu"d"kZ fudkyk x;k gS fd ,d jkf'k gksrh gS] ftls oS|qr vkos'k dgrs gSa vkSj ;g osQoy nks izdkj osQ gh gks ldrs gSaA oS|qr vkos'k dgykus okyh jkf'k osQ osQoy nks izdkj gh gksrs gSaA ge dgrs gSa fd IykfLVd ,oa dk¡p dh NM+] js'ke] lewj] ljoaQMs dh xksfy;k¡ vkfn ¯iM fo|qUe; gks x, gSaA jxM+us ij ;s oS|qr vkos'k v£tr dj ysrs gSaA ljoaQMs dh xksfy;ksa ij fd, x, ;s iz;ksx ;g lq>krs gSa fd vkos'k nks izdkj osQ gksrs gSa rFkk ge ;g ikrs gSa fd (i) ltkrh; vkos'k ,d&nwljs dks izfrd£"kr rFkk (ii) fotkrh; vkos'k ,d&nwljs dks vkd£"kr djrs gSAa ;s iz;ksx ;g Hkh izn£'kr djrs gSa fd Li'kZ djus ij] vkos'k NM+ ls ljoaQMs dh xksyh esa LFkkukarfjr gks tkrs gSaA ;g dgk tkrk gS fd ljoaQMs dh xksfy;k¡ Li'kZ }kjk fo|qUe; vFkok vkosf'kr (vkfo"V) gksrh gSaA og xq.k tks nks izdkj osQ vkos'kksa esa Hksn djrk gS] vkos'k dh /qzork dgykrk gSA HkkSfrdh tc dk¡p dh NM+ dks js'ke ls jxM+rs gSa rks NM+ ,d izdkj dk vkos'k v£tr djrh gS rFkk js'ke nwljs izdkj dk vkos'k v£tr djrk gSA ;g mu lHkh oLrqvksa osQ ;qxy osQ fy, lR; gS tks fo|qUe; gksus osQ fy, ijLij jxM+s tkrs gSaA vc ;fn fo|qUe; dk¡p dh NM+ dks ml js'ke osQ laioZQ esa ykrs gSa ftlls mls jxM+k x;k Fkk] rks os vc ,d&nwljs dks vkd£"kr ugha djrsA ;s vc vU; gydh oLrqvksa dks Hkh vkd£"kr vFkok izfrd£"kr ugha djrs tSlk fd ;s fo|qUe; gksus ij 2 dj jgs FksA  oS|qr vkos'k rFkk {ks=k bl izdkj] jxM+us osQ i'pkr oLrqvksa }kjk v£tr vkos'k vkosf'kr oLrqvksa dks ,d&nwljs osQ laioZQ esa ykus ij yqIr gks tkrk gSA bu iz{s k.kksa ls vki D;k fu"d"kZ fudky ldrs gS\a ;g rks osQoy bruk crkrk gS fd oLrqvksa }kjk v£tr fotkrh; vkos'k ,d&nwljs osQ izHkko dks fu"iQy dj nsrs gSaA blhfy, vesfjdh oSKkfud csatkfeu izsaQdfyu us vkos'kksa dks /ukRed rFkk ½.kkRed dgkA ge tkurs gSa fd tc ge fdlh /ukRed la[;k dks mlh ifjek.k dh ½.kkRed la[;k ls tksM+rs gSa rks ;ksxiQy 'kwU; gksrk gSA vkos'kksa dks /ukRed rFkk ½.kkRed uke nsus osQ ihNs Hkh ;gh roZQ jgk gksxkA ifjikVh osQ vuqlkj dk¡p dh NM+ vFkok fcYyh osQ lewj ij vkos'k /ukRed dgykrk gS rFkk IykfLVd&NM+ vFkok js'ke ij vkos'k ½.kkRed dgykrk gSA tc fdlh oLrq ij dksbZ vkos'k gksrk gS rks og oLrq fo|qUe; vFkok vkosf'kr (vkfo"V) dgh tkrh gSA tc ml ij dksbZ vkos'k ugha gksrk rc mls vukosf'kr dgrs gSaA fo|qr rFkk pqc a dRo dk ,dhdj.k izkphu dky esa fo|qr rFkk pqacdRo i`FkoQ fo"k; le>s tkrs FksA fo|qr osQ varxZr dk¡p dh NM+ksa] fcYyh osQ lewj] cSVjh] rfM+r vkfn osQ vkos'kksa ij ppkZ gksrh Fkh tcfd pqacdRo osQ varxZr pqacdksa] ykSg&Nhyu] pqacdh; lqbZ vkfn esa vU;ksU; fØ;k dk o.kZu fd;k tkrk FkkA lu 1820 bZ- esa MsuekoZQ osQ oSKkfud vkWLVsZM us ;g ik;k fd pqacdh; lqbZ osQ fudV j[ks rkj ls fo|qr /kjk izokfgr djus ij lqbZ fo{ksfir gks tkrh gSA ,sfEi;j rFkk iSQjkMs us bl izs{k.k dk ;g dgdj iks"k.k fd;k fd xfr'khy vkos'k pqacdh; {ks=k mRiUu djrs gSa rFkk xfr'khy pqacd fo|qr /kjk mRiUu djrs gSaA fo|qr rFkk pqacdRo esa ,dhdj.k rc LFkkfir gqvk tc LdkWVySaM osQ HkkSfrdfon eSDlosy rFkk gkWySaM osQ HkkSfrdfon yksjsat us ,d fl¼kar izfrikfnr fd;k ftlesa mUgksaus ;g n'kkZ;k fd ;s nksuksa fo"k; ,d&nwljs ij fuHkZj gSaA bl {ks=k dks oS|qrpqacdRo dgrs gSaA gekjs pkjksa vksj dh vf/dka'k ifj?kVukvksa dh O;k[;k oS|qrpqacdRo osQ varxZr dh tk ldrh gSA oLrqr% os lHkh cy ftuosQ fo"k; esa ge fopkj djrs gSa] tSls ?k"kZ.k inkFkZ dks la;ksftr j[kus osQ fy, muosQ ijek.kqvksa osQ chp yxus okyk jklk;fud cy ;gk¡ rd fd lthoksa dh dksf'kdkvksa esa gksus okyh izfØ;kvksa dh O;k[;k djus okys cyksa dk mn~Hko Hkh oS|qrpqacdh; cyksa ls gqvk gSA oS|qrpqacdh; cy izo`Qfr osQ ewy cyksa esa ls ,d gSA eSDlosy us pkj lehdj.k izLrqr fd, ftudh Dykfldy oS|qrpqacdRo esa ogh Hkwfedk gS] tks ;kaf=kdh esa U;wVu dh xfr oQh lehdj.kksa rFkk xq#Rokd"kZ.k fu;e dh gSA mUgksaus ;g Hkh izekf.kr fd;k fd izdk'k dh izo`Qfr oS|qrpqacdh; gS rFkk bldh pky osQoy fo|qr rFkk pqacdh; ekiksa }kjk izkIr dh tk ldrh gSA mUgksaus ;g Hkh nkok fd;k fd izdkf'kdh osQ foKku rFkk fo|qr ,oa pqacdRo esa izxkus dk iz;kl djsa fd fuEufyf[kr iz;ksx dks djus ij D;k gks ldrk gSA (i) fo|qrjks/h LVSaMksa ij j[ks nks /krq osQ xksyksa A ,oa B dks fp=k 1.4 (a) esa n'kkZ, vuqlkj ,d nwljs osQ laioZQ esa ykb,A (ii) ,d /ukosf'kr NM+ bu xksyksa esa fdlh ,d (ekuk A) osQ fudV ykb, rFkk ;g lko/kuh cjfr, fd NM+ xksys ls Li'kZ u djsA xksys osQ eqDr bysDVªkWu NM+ dh vksj vkd£"kr gksrs gSaA xksys B osQ fiNys i`"B ij /ukos'k dk vkf/D; gks tkrk gSA nksuksa izdkj osQ vkos'k /krq osQ xksyksa esa vkc¼ jgrs gSa] iyk;u ugha dj ikrsA vr% os i`"Bksa ij gh fp=k 1.4(b) esa n'kkZ, vuqlkj jgrs gSaA xksys A osQ ck,¡ i`"B ij ½.kkos'k dk vkf/D; rFkk xksys osQ nk,¡ i`"B ij /ukRed vkos'k dk vkf/D; gksrk gSA rFkkfi xksys osQ lkjs bysDVªkWu xksys A osQ ck,¡ i`"B ij lafpr ugha gksrsA tSls gh xksys A osQ ck,¡ i`"B ij bysDVªkWu ,d=k gksus 'kq: gksrs gSa vU; bysDVªkWu buosQ }kjk izfrd£"kr gks tkrs gSaA oqQN gh {k.k esa NM+ osQ vkd"kZ.k osQ dkj.k fØ;k rFkk lafpr vkos'kksa osQ dkj.k izfrd"kZ.k osQ chp lkE; LFkkfir gks tkrk gSA fp=k 1.4(b) lkE;koLFkk dks n'kkZrk gSA bl izfØ;k dks vkos'k dk izsj.k dgrs gSa rFkk ;g yxHkx rkRdkfyd gSA tc rd dk¡p dh NM+ xksys osQ fudV jgrh gS rc rd fp=k esa n'kkZ, vuqlkj lafpr vkos'k i`"B ij jgrk gSA ;fn NM+ dks gVk ysrs gSa rks vkos'kksa ij dksbZ cká cy dk;Z ugha djrk rFkk os viuh mnklhu voLFkk esa okil ykSV tkrs gSaA (iii) fp=k 1.4(c) esa n'kkZ, vuqlkj dk¡p dh NM+ dks ck,¡ xksys osQ fudV j[krs gq, nksuksa xksyksa dks ,d&nwljs ls oqQN i`FkoQ dhft,A ,slk djus ij nksuksa xksys fotkrh; vkos'kksa }kjk vkosf'kr gksdj ,d&nwljs dks vkd£"kr djrs gSaA (iv) NM+ dks gVk yhft,A xksyksa ij vkos'k fp=k 1.4(d) esa n'kkZ, vuqlkj Lo;a dks iquO;ZofLFkr dj ysrs gSaA vc nksuksa xksyksa osQ chp i`Fkdu vf/d dhft,A ,slk djus ij fp=k 1.4(e) esa n'kkZ, vuqlkj vkos'k xksyksa ij ,dleku :i ls forfjr gks tkrk gSA fp=k 1.4 nks xksyksa osQ lkFk bl izfØ;k esa /krq osQ xksys vkosf'kr gks tkrs gSaA /ukosf'kr dk¡p dh NM+ osQ vkos'k dh izsj.k }kjk vkos'kuA dksbZ {kfr ugha gksrh] tks laioZQ }kjk vkosf'kr djus dh izfØ;k osQ foijhr gSA tc fdlh fo|qUe; NM+ dks gydh oLrqvksa osQ fudV ykrs gSa rks ;gh izHkko gksrk gSA NM+ 6 oLrqvksa osQ ikl okys i`"B ij fotkrh; vkos'k izsfjr djrh gS rFkk ltkrh; vkos'k oLrqvksa osQ  oS|qr vkos'k rFkk {ks=k nwj okys Hkkx ij igq¡p tkrk gS ¹,slk rc Hkh gksrk gS tc gydh oLrq pkyd ugha gksrhA ;g izfØ;k fdl izdkj gksrh gS bldk Li"Vhdj.k ckn esa vuqPNsn 1.10 rFkk 2.10 esa fd;k x;k gS ]A nksuksa izdkj osQ vkos'kksa osQ osaQnzksa esa oqQN i`Fkdu gksrk gSA ge tkurs gSa fd ltkrh; vkos'kksa esa izfrd"kZ.k rFkk fotkrh; vkos'kksa esa vkd"kZ.k gksrk gSA rFkkfi] cy dk ifjek.k vkos'kksa osQ chp nwjh ij fuHkZj djus osQ dkj.k vkd"kZ.k cy izfrd"kZ.k cy dh rqyuk esa vf/d gksrk gSA iQyLo:i dkxT+k osQ VqdM+s] ljoaQMs dh xksyh vkfn gyosQ gksus osQ dkj.k NM+ dh vksj ¯[kp vkrs gSaA mnkgj.k 1.1 vki fdlh /krq osQ xksys dks Li'kZ fd, fcuk oSQls /ukosf'kr dj ldrs gSa\ HkkSfrdh gy fp=k 1.5(a) esa fdlh fo|qrjks/h /krq osQ LVSaM ij dksbZ vukosf'kr /krq dk xksyk j[kk http://www.physicsclassroom.com/mmedia/estatics/estatic/ifsr.cfm izsj.k }kjk nks xksyksa osQ fudk; osQ vkos'ku laca/h izHkkoh ltho fp=k.k n'kkZ;k x;k gSA fp=k 1.5(b) esa n'kkZ, vuqlkj bl xksys osQ fudV dksbZ ½.kkosf'kr NM+ ykb,A xksys osQ eqDr bysDVªkWu izfrd"kZ.k osQ dkj.k nwj tkdj nwjLFk fljs ij ,d=k gks tkrs gSaA ikl dk fljk bysDVªkWuksa dh deh osQ dkj.k /ukosf'kr gks tkrk gSA tc xksys dh /krq osQ eqDr bysDVªkWuksa ij yxus okyk usV cy 'kwU; gks tkrk gS rks vkos'kksa osQ forj.k dh ;g izfØ;k can gks tkrh gSA fdlh pkyd rkj }kjk xksys dks Hkwlai£dr dhft,A bysDVªkWu Hkwfe esa izokfgr gks tk,¡xs tcfd ikl osQ fljs dk /ukos'k] NM+ osQ ½.kkos'k osQ vkd"kZ.k cy osQ dkj.k fp=k 1.5(c) esa n'kkZ, vuqlkj c¼ jgsxkA xksys dk HkwlaioZQ rksM+ nhft,A ikl osQ fljs ij /ukos'k dh c¼rk cuh jgrh gS ¹fp=k 1.5(d)ºA fo|qUe; NM+ dks gVk yhft,A fp=k 1.5(e) esa n'kkZ, vuqlkj /ukos'k xksys osQ i`"B ij ,dleku :i ls iSQy tkrk gSA fp=k 1.5 bl iz;ksx esa /krq dk xksyk izsj.k dh izfØ;k }kjk vkosf'kr gks tkrk gS rFkk NM+ viuk dksbZ vkos'k ugha [kksrhA izsj.k }kjk /krq osQ xksys dks ½.kkosf'kr djus dh izfØ;k osQ Hkh ;s gh pj.k gksrs mnkgj.k 1.1 gSa] blesa /ukosf'kr NM+ xksys osQ lehi ykbZ tkrh gSA bl izdj.k esa bysDVªkWu Hkwfe ls xksys esa ml le; LFkkukarfjr (izokfgr) gksrs gSa tc rkj }kjk xksys dks Hkwlai£dr fd;k tkrk gSA D;k vki bldk dkj.k Li"V dj ldrs gSa\ 7  HkkSfrdh 1.5 oS|qr vkos'k osQ ewy xq.k geus ;g ns[kk gS fd nks izdkj osQ vkos'k gksrs gSaµ /ukos'k rFkk ½.kkos'k rFkk buesa ,d&nwljs osQ izHkko dks fujLr djus dh izo`fÙk gksrh gSA vc] ge ;gk¡ oS|qr vkos'k osQ vU; xq.kksa dk o.kZu djsxa As ;fn vkosf'kr oLrqvksa dk lkb”k muosQ chp dh nwjh dh rqyuk esa cgqr de gksrk gS rks ge mUgsa ¯cnq vkos'k ekurs gSaA ;g eku fy;k tkrk gS fd oLrq dk laiw.kZ vkos'k vkdk'k esa ,d ¯cnq ij laosaQfnzr gSA 1.5.1 vkos'kksa dh ;ksT;rk vc rd geus vkos'k dh ifjek.kkRed ifjHkk"kk ugha nh gS_ bls ge vxys vuqHkkx esa le>saxsA varfje :i esa ge ;g ekusxa s fd ,slk fd;k tk ldrk gS vkSj fiQj vkxs cko loZizFke vaxzs”k iz;ksxdrkZ iSQjkMs }kjk [kksts x, fo|qr vi?kVu osQ izk;ksfxd fu;eksa ls izkIr gqvk FkkA lu~ 1912 esa fefydu us bls okLro esa izk;ksfxd :i ls fun£'kr fd;k FkkA ek=kdksa dh varjkZ"Vªh; iz.kkyh (SI) esa vkos'k dk ek=kd owQykWe gS] ftldk izrhd C gSA ,d owQykWe dks fo|qr /kjk osQ ek=kd osQ inksa esa ifjHkkf"kr fd;k tkrk gS ftlosQ fo"k; esa vki vxys vè;k; esa lh[ksaxsA bl ifjHkk"kk osQ vuqlkj] ,d owQykWe og vkos'k gS tks fdlh rkj esa 1 A (,ssfEi;j) /kjk 1 lsoaQM rd izokfgr djrk gS ¹HkkSfrdh dh ikB~;iqLrd d{kk 11] Hkkx 1 dk vè;k; 2 nsf[k,ºA bl iz.kkyh esa] vkos'k dh ewy bdkbZ e = 1.602192 × 10–19 C bl izdkj] –1C vkos'k esa yxHkx 6 × 1018 bysDVªkWu gksrs gSaA fLFkjoS|qfrdh esa brus fo'kky ifjek.k osQ vkos'kksa ls ;nk&dnk gh lkeuk gksrk gS vkSj blhfy, ge blosQ NksVs ek=kdksa 1 µC (ekbØksowQykWe) = 10–6 C vFkok 1 mC (feyhowQykWe) = 10–3 C dk mi;ksx djrs gSaA ;fn osQoy bysDVªkWu rFkk izksVkWu gh fo'o esa vkos'k osQ ewy ek=kd gSa rks lHkh izsf{kr vkos'kksa dks e dk iw.kk±d xq.kt gksuk pkfg,A bl izdkj ;fn fdlh oLrq esa n1 bysDVªkWu rFkk n2 izksVkWu gSa rks ml oLrq ij oqQy vkos'k n2 × e + n1 × (–e) = (n2 – n1) e gSA pw¡fd n1 rFkk n2 iw.kk±d gSa] budk varj Hkh ,d iw.kk±d gSA vr% fdlh oLrq ij vkos'k lnSo e dk iw.kk±d xq.kt gksrk gS ftls e osQ pj.kksa esa gh ?kVk;k vFkok cus osQ fy, vkb, ge {ks=kiQy vkSj ?ku dks.k (tks dks.k dk rhu foekvksa esa O;kidhdj.k gS) osQ chp laca/ LFkkfir djus dk iz;kl djrs gSaA ;kn dhft, nks foekvksa esa fdlh (lery) dks.k dh ifjHkk"kk fdl izdkj dh tkrh gSA eku yhft, dksbZ NksVk vuqizLFk js[kk vo;o ∆l ¯cnq O ls r nwjh ij j[kk tkrk gSA rc O ij ∆l }kjk varfjr dks.k dk lfUudVu ∆θ = ∆l/r osQ :i esa fd;k tk ldrk gSA blh izdkj] rhu foekvksa esa fdlh NksVs yacor {ks=k ∆S }kjk nwjh r ij varfjr ?ku dks.k* dks ∆Ω = ∆S/r2 O;Dr fd;k tk ldrk gSA ge tkurs gSa fd fdlh fn, x, ?ku dks.k esa vjh; {ks=k js[kkvksa dh la[;k leku gksrh gSA fp=k 1.16 esa vkos'k ls r1 rFkk r2 nwfj;ksa ij fLFkr nks ¯cnqvksa P1 rFkk P2 osQ fy, ?ku dks.k ∆Ω }kjk P1 ij varfjr {ks=k vo;o r12 ∆Ω rFkk P2 ij varfjr {ks=k vo;o r22 ∆Ω gSA bu {ks=k vo;oksa dks dkVus okyh js[kkvksa dh la[;k (eku yhft, n) leku gSA vr% ,dkad {ks=k vo;o dks dkVus okyh {ks=k js[kkvksa dh la[;k P1 ij n/( r12 ∆Ω) rFkk P2 ij n/( r22 ∆Ω) gSA bl izdkj Li"V gS fd {ks=k js[kkvksa dh la[;k vkSj blhfy, {ks=k&izcyrk Li"V :i ls 1/r 2 ij fuHkZj gSA {ks=k js[kkvksa osQ fp=k.k dh [kkst iSQjkMs us vkosf'kr foU;klksa osQ pkjksa vksj fo|qr {ks=k dk ekul izR;{khdj.k djus osQ ,d varnZ'khZ vxf.krh; mik; dks fodflr djus osQ fy, dh FkhA iSQjkMs us bUgsa cy js[kk,¡ dgk FkkA ;g in fo'ks"kdj pqca dh; {ks=kksa osQ izdj.k osQ fy, oqQN Hkzked gSA buosQ fy, vf/d mfpr in {ks=k js[kk,¡ (oS|qr vFkok pqacdh;) gS ftls geus bl iqLrd esa viuk;k gSA bl izdkj fo|qr {ks=k js[kk,¡ vkos'kksa osQ vfHkfoU;kl osQ pkjksa vksj fo|qr {ks=k osQ fp=kkRed fu:i.k dk ,d mik; gSA O;kid :i esa] fo|qr {ks=k js[kk ,d ,slk oØ gksrh gS ftlosQ fdlh Hkh ¯cnq ij [khapk x;k Li'khZ ml ¯cnq ij yxs usV cy dh fn'kk dks fu:fir djrk gSA bl oØ osQ fdlh ¯cnq ij] Li"V :i ls] Li'khZ }kjk fo|qr {ks=k dh nks laHkkfor fn'kkvksa esa ls dksbZ ,d * ?ku dks.k 'kaoqQ dh ,d eki gSA R f=kT;k osQ xksys okys fn, x, 'kaoqQ osQ ifjPNsn ij fopkj dhft,A 'kaoqQ osQ ?ku dks.k ∆Ω dh ifjHkk"kk bls ∆S/R2 osQ cjkcj ekudj djrs gSa] ;gk¡ ∆S 'kaoqQ }kjk xksys ij 24 dkVk x;k {ks=kiQy gSA  oS|qr vkos'k rFkk {ks=k fn'kk n'kkZus osQ fy, oØ ij rhj dk fpÉ vafdr djuk vko';d gSA {ks=k js[kk ,d fno~QLFkku oØ vFkkZr rhu fn'kkvksa esa oØ gksrh gSA fp=k 1.17 esa oqQN ljy vkos'k foU;klksa osQ pkjksa vksj {ks=k js[kk,¡ n'kkZ;h xbZ gSaA tSlk fd igys o.kZu fd;k tk pqdk gS] ;s {ks=k js[kk,¡ rhu foeh; fno~QLFkku esa gSa ;|fi fp=k esa bUgsa osQoy ,d ry esa n'kkZ;k x;k gSA ,dy /ukos'k osQ dkj.k {ks=k js[kk,¡ f=kT;r% (vjh;) cfgeqZ[kh gksrh gSa tcfd ,dy ½.kkos'k osQ dkj.k {ks=k js[kk,¡ f=kT;r% vareqZ[kh gksrh gSaA nks /ukos'kksa (q, q) osQ fudk; osQ pkjksa vksj dh {ks=k js[kk,¡ ikjLifjd izfrd"kZ.k dk ,d ltho fp=k.k izLrqr djrh gSa tcfd ifjek.k esa leku nks fotkrh; vkos'kksa (q, –q) osQ fudk;] vFkkZr fdlh f}/qzo osQ pkjksa vksj {ks=k js[kk,¡ vkos'kksa osQ chp Li"V ikjLifjd vkd"kZ.k n'kkZrh gSaA {ks=k js[kk,¡ oqQN egRoiw.kZ lkekU; xq.kksa dk ikyu djrh gSaµ (i) {ks=k js[kk,¡ /ukos'k ls vkjaHk gksdj ½.kkos'k ij lekIr gksrh gSAa ;fn vkos'k ,dy gS rks ;s vuar ls vkjaHk vFkok vuar ij lekIr gks ldrh gSAa (ii) fdlh vkos'k eqDr {ks=k esa] {ks=k js[kkvksa dks ,sls larr oØ ekuk tk ldrk gS tks dgha ugha VwVrsA (iii) nks {ks=k js[kk,¡ ,d&nwljs dks dnkfi ugha dkVrhaA (;fn os ,slk djsa rks izfrPNsnu ¯cnq ij {ks=k dh osQoy ,d fn'kk ugha gksxh] tks fujFkZd gSA (iv) fLFkjoS|qr {ks=k js[kk,¡ can ywi ugha cukrhaA ;g fo|qr {ks=k dh laj{k.kkRed izo`Qfr ls vuq'kkflr gS (vè;k; 2 nsf[k,)A 1.10 oS|qr ÝyDl fdlh dS {ks=kiQy osQ NksVs i`"B ls mlosQ vfHkyacor v osx ls izokfgr gksus okys fdlh nzo osQ izokg ij fopkj dhft,A nzo osQ izokg dh nj bl {ks=k ls izfr ,dkad le; esa xqtjus okys vk;ru v dS }kjk izkIr gksrh gS rFkk ;g ml ry ls xqtjus okys nzo osQ ÝyDl dks fu:fir djrh gSA ;fn bl ry (i`"B) ij vfHkyac nzo osQ izokg dh fn'kk vFkkZr v osQ lekarj ugha gS] vkSj buosQ chp θ dks.k curk gS rks v osQ yacor ry esa iz{ksfir {ks=kiQy v dS cos θ gksxkA vr% i`"B dS ls ckgj tkus okyk ÝyDl v. n̂ dS gksrk gSA fo|qr {ks=k osQ izdj.k osQ fy,] ge ,d lerqY; jkf'k dks ifjHkkf"kr djrs gSa vkSj bls oS|qr ÝyDl dgrs gSaA rFkkfi gesa ;g è;ku esa j[kuk pkfg, fd nzo&izokg osQ izdj.k osQ foijhr ;gk¡ izko`Qfrd fu;eksa osQ vuqlkj iz{s k.k ;ksX; jkf'k dk dksbZ izokg ugha gSA mijksDr o£.kr fo|qr {ks=k js[kkvksa osQ fp=k.k esa geus ns[kk fd fdlh ¯cnq ij {ks=k osQ vfHkyacor j[ks ,dkad {ks=kiQy ls xqtjus okyh {ks=k js[kkvksa dh la[;k ml ¯cnq ij fo|qr {ks=k izcyrk dh eki gksrh gSA bldk rkRi;Z ;g gS fd ;fn ge fdlh ¯cnq ij E osQ vfHkyacor dksbZ ∆S {ks=kiQy dk NksVk leryh; vo;o j[ksa rks blls xqtjus okyh {ks=k js[kkvksa dh la[;k E ∆S osQ vuqØekuqikrh* gSA vc eku yhft, ge {ks=kiQy vo;o dks fdlh * ;g dguk mfpr ugha gS fd {ks=k js[kkvksa dh la[;k E∆S osQ cjkcj gSA okLro esa fp=k 1.17 fofHkUu vkos'k forj.kksa osQ dkj.k {ks=k js[kkvksa dh la[;k ,slk fo"k; gS tks ge fdruh {ks=k js[kk,¡ [khapus dk p;u djrs gSa] ij fuHkZj gSA vr% fofHkUu ¯cnqvksa ij fn, x, {ks=kiQy ls xqtjus okyh {ks=k js[kkvksa {ks=k js[kk,¡A dh vkisf{kd la[;k osQ Kkr gksus esa gh budh HkkSfrd lkFkZdrk gSA 25  HkkSfrdh dks.k θ ij >qdk nsrs gSaA Li"V gS vc bl {ks=kiQy vo;o ls xqtjus okyh {ks=k js[kkvksa dh la[;k ?kV tk,xhA pw¡fd E osQ vfHkyacor {ks=kiQy vo;o ∆S dk iz{ksi ∆S cosθ gS] vr% ∆S ls xqtjus okyh {ks=k js[kkvksa dh la[;k E ∆S cosθ osQ vuqØekuqikrh gSA tc θ = 90° gksrk gS rks {ks=k js[kk,¡ ∆S osQ lekarj gks tkrh gSa vkSj blls dksbZ Hkh {ks=k js[kk ugha xqtjrh (fp=k 1.18 nsf[k,)A cgqr ls lanHkks± esa {ks=kiQy vo;o osQ ifjek.k osQ lkFk&lkFk mldk fno~QfoU;kl Hkh egRoiw.kZ gksrk gSA mnkgj.k osQ fy,] fdlh ty&izokg esa fdlh ¯jx ls xqtjus okys ty dk ifjek.k LokHkkfod :i ls bl ckr ij fuHkZj djrk gS fd vki ty /kjk esa bls fdl izdkj idM+s gq, gSaA ;fn vki bls ty&izokg osQ vfHkyacor j[krs gSa rks vU; lHkh fno~QfoU;klksa dh rqyuk esa bl foU;kl esa ¯jx ls vf/dre ty xqtjsxkA fp=k 1.18 E rFkk n̂ osQ chp >qdko θ ij blls ;g fu"d"kZ fudyrk gS fd {ks=kiQy&vo;o dks lfn'k ÝyDl dh fuHkZjrkA osQ leku ekuuk pkfg,A blesa ifjek.k osQ lkFk fn'kk Hkh gksrh gSA leryh; {ks=k dh fn'kk oSQls fu£n"V dh tk,\ Li"V :i ls ry ij vfHkyac ry dk fno~QfoU;kl fu£n"V djrk gSA bl izdkj leryh; {ks=k lfn'k dh fn'kk blosQ vfHkyac osQ vuqfn'k gksrh gSA fdlh ofØr i`"B osQ {ks=kiQy dks fdlh lfn'k ls oSQls lac¼ fd;k tkrk gS\ ge ;g dYiuk djrs gSa fd ofØr i`"B cgqr ls NksVs&NksVs {ks=kiQy vo;oksa esa foHkkftr gSA buesa ls izR;sd NksVk {ks=kiQy vo;o leryh; ekuk tk ldrk gS vkSj igys Li"Vhdj.k osQ vuqlkj blls lfn'k lac¼ fd;k tk ldrk gSA ;gk¡ ,d lafnX/rk ij è;ku nhft,A fdlh {ks=kiQy vo;o dh fn'kk mlosQ vfHkyac osQ vuqfn'k gksrh gSA ijarq vfHkyac nks fn'kk,¡ laosQr dj ldrk gSA fdlh {ks=kiQy vo;o ls lac¼ lfn'k dh fn'kk dk p;u fdl izdkj fd;k tkrk gS\ bl leL;k dk lek/ku bl lanHkZ esa mfpr oqQN ifjikfV;ksa osQ fu/kZj.k }kjk fd;k x;k gSA can i`"Bksa osQ izdj.kksa osQ fy, ;g ifjikVh vfr ljy gSA fdlh can i`"B osQ izR;sd {ks=kiQy vo;o ls lac¼ lfn'k dh fn'kk cfgeqZ[kh vfHkyac dh fn'kk ekuh tkrh gSA blh ifjikVh dk mi;ksx fp=k 1-19 esa fd;k x;k gSA bl izdkj] fdlh can i`"B osQ fdlh ¯cnq ij {ks=kiQy vo;o lfn'k ∆S dk eku ∆S n̂ gksrk gS] ;gk¡ ∆S {ks=kiQy lfn'k dk ifjek.k rFkk n̂ bl ¯cnq ij cfgeqZ[kh vfHkyac dh fn'kk esa ,dkad lfn'k gSA vc ge oS|qr ÝyDl dh ifjHkk"kk ij vkrs gSaA fdlh {ks=kiQy vo;o ∆S ls xqtjus okys oS|qr ÝyDl ∆φ dh ifjHkk"kk bl izdkj djrs gSa % ∆φ = E.∆S = E ∆S cosθ (1.11) tks igys dh Hkk¡fr bl {ks=kiQy vo;o dks dkVus okyh {ks=k js[kkvksa dh la[;k osQ vuqØekuqikrh fp=k 1.19 vfHkyac gSA ;gk¡ θ {ks=k vo;o ∆S rFkk E osQ chp dk dks.k gSA iwoksZDr ifjikVh osQ vuqlkj can i`"B osQ n̂ rFkk ∆S dks ifjHkkf"kr fy, θ {ks=k&vo;o ij cfgeqZ[kh vfHkyac rFkk E osQ chp dk dks.k gSA è;ku nhft,] ge O;atd djus dh ifjikVhA E ∆S cosθ ij nks > 2a) gS] rks q ,oa –q osQ dkj.k {ks=k yxHkx fujLr gks tkrs gSaA vr% vf/d nwfj;ksa ij fdlh oS|qr f}/qzo osQ dkj.k fo|qr {ks=k 1/r2 (,dy vkos'k q osQ dkj.k fo|qr {ks=k dh r ij fuHkZjrk) ls Hkh vf/d xfr ls ean gksrk tkrk gSA ;g xq.kkRed /kj.kk uhps fn, x, lqLi"V ifjdyu ls mRiUu gqbZ gS% 1.11.1 oS|qr f}/qzo osQ dkj.k {ks=k vkos'kksa osQ ;qxy (–q rFkk q) osQ dkj.k fno~QLFkku esa fdlh ¯cnq ij fo|qr {ks=k owQykWe fu;e rFkk vè;kjksi.k fl¼kar ls Kkr fd;k tk ldrk gSA fuEufyf[kr nks izdj.kksa osQ ifj.kke ljy gSa% (i) tc ¯cnq f}/qzo osQ v{k ij gS] (ii) tc ¯cnq f}/zqo osQ fo"kqorh; ry] vFkkZr f}/qzo v{k osQ osaQnz ls xqtjus okys f}/qzo v{k osQ yacor ry esa gSA fdlh O;kid ¯cnq P ij fo|qr {ks=k] vkos'k –q osQ dkj.k P ij fo|qr {ks=k E –q rFkk vkos'k +q osQ dkj.k P ij fo|qr {ks=k E+q dks lfn'kksa osQ lekarj prqHkqZt fu;e }kjk la;ksftr djosQ izkIr fd;k tkrk gSSA (i) v{k ij fLFkr ¯cnqvksa osQ fy, eku yhft, ¯cnq P f}/qzo osQ osaQnz ls q dh vksj fp=k (1.20a) esa n'kkZ, vuqlkj r nwjh ij gS] rc q E −q = − ˆ p [1.13(a)] 4 πε 0 (r + a )2 ;gk¡ p̂ f}/qzo v{k (–q ls q dh vksj) osQ vuqfn'k ,dkad lfn'k gSA lkFk gh q E +q = ˆ p [1.13(b)] 4 π ε 0 (r − a )2 P ij oqQy fo|qr {ks=k gS 27  HkkSfrdh q 1 1 E = E +q + E −q = − ˆ p 4 π ε 0 (r − a )2 (r + a )2 q 4a r = ˆ p (1.14) 4 π ε o ( r 2 − a 2 )2 r >> a osQ fy, 4qa E= ˆ p (r >> a) (1.15) 4πε 0r 3 (ii) fo"kqorh; ry ij fLFkr ¯cnqvksa osQ fy, nks vkos'kksa +q rFkk –q osQ dkj.k fo|qr {ks=kksa osQ ifjek.k q 1 E +q = [1.16(a)] 4 πε 0 r 2 + a 2 q 1 E –q = [1.16(b)] 4 πε 0 r + a 2 2 leku gSaA fp=k 1.20 (a) v{k ij fLFkr fdlh ¯cnq] (b) f}/zqo E+q rFkk E–q dhfn'kk,¡ fp=k 1.20(b) esa n'kkZ;h xbZ gSAa Li"V gS fd osQ fo"kqorh; ry ij fLFkr fdlh ¯cnq ij f}/qzo dk f}/qoz v{k osQ vfHkyacor vo;o ,d&nwljs dks fujLr dj nsrs gSAa f}èkzoq fo|qr {ks=kA f}/qzo vk?kw.kZ p lfn'k gS ftldk ifjek.k v{k osQ vuqfn'k vo;o la;ksftr gks tkrs gSaA oqQy fo|qr {ks=k p̂ osQ p = q ×2a gS rFkk fn'kk foijhr gksrk gSA vr% –q ls q dh vksj gSA E = – (E +q + E –q ) cosθ p̂ 2qa =− ˆ p (1.17) 4 π ε o (r 2 + a 2 )3 / 2 vf/d nwfj;ksa (r >> a) ij 2qa E=− pˆ (r >> a ) (1.18) 4 π εo r 3 lehdj.kksa (1.15) rFkk (1.18) ls Li"V gS fd vf/d nwfj;ksa ij f}/qzo {ks=k esa q rFkk a i`Fkd :i ls lfEefyr ugha gksrs_ ;g buosQ la;qDr xq.kuiQy qa ij fuHkZj djrk gSA blls f}/qzo vk?kw.kZ dh ifjHkk"kk dk laosQr feyrk gSA fdlh oS|qr f}/qzo osQ f}/qzo vk?kw.kZ lfn'k p dh ifjHkk"kk bl izdkj dh tk ldrh gS% p = q ×2a p̂ (1.19) vFkkZr ;g ,d lfn'k gS ftldk ifjek.k vkos'k q rFkk i`Fkdu 2a (vkos'kksa q, –q osQ ;qxy osQ chp dh nwjh) rFkk fn'kk –q ls q dh vksj gksrh gSA p osQ inksa esa] fdlh f}/qzo dk fo|qr {ks=k vfèkd nwfj;ksa ij ,d ljy :i ys ysrk gSA f}/qzo v{k osQ fdlh ¯cnq ij 2p E= (r >> a) (1.20) 4πε or 3 fo"kqorh; ry osQ fdlh ¯cnq ij p E=− (r >> a) (1.21) 28 4πε or 3  oS|qr vkos'k rFkk {ks=k bl egRoiw.kZ rF; ij è;ku nhft, fd f}/qzo {ks=k vf/d nwfj;ksa ij 1/r 2 osQ :i esa ugha oju~1/r 3 osQ :i esa ean gksrk gSA blosQ vfrfjDr f}/qzo {ks=k dk ifjek.k rFkk fn'kk osQoy nwjh r ij gh fuHkZj ugh g a S oju~ ;s lfn'k r rFkk f}/qoz vk?kw.kZ p osQ chp osQ dks.k ij Hkh fuHkZj djrs gSA ge mlosQ ckjs esa lksp ldrs gSa& tc f}/zqo vkeki 2a 'kwU; dh vksj vxzlj gksrk tkrk gS] rc vkos'k q vuar dh vksj vxzlj bl izdkj gksrk tkrk gS fd xq.kuiQy p = q ×2a ,d fu;r ifjfer la[;k gksrh gSA bl izdkj osQ f}/qzo dks ¯cnq f}/qzo dgrs gSaA fdlh ¯cnq f}/qzo osQ fy, lehdj.k (1.20) rFkk (1.21) r osQ lHkh ekuksa osQ fy, lR; rFkk ;FkkFkZ gSaA 1.11.2 f}/qzoksa dh HkkSfrd lkFkZdrk vf/dka'k v.kqvksa esa /ukos'kksa rFkk ½.kkos'kks*a osQ oasQnz ,d gh LFkku ij gksrs gSaA blhfy, buosQ f}/zqo vk?kw.kZ 'kwU; gksrs gSaA CO2 rFkk CH4 v.kq blh izdkj osQ gSaA fo|qr {ks=k vkjksfir fd, tkus ij ;s f}/qzo vk?kw.kZ fodflr dj ysrs gSa ijarq oqQN v.kqvksa eaas /ukos'kksa rFkk ½.kkos'kksa osQ osaQnz laikrh ugha gksrsA vr% fo|qr {ks=k dh vuqifLFkfr esa Hkh budk viuk LFkk;h f}/qzo vk?kw.kZ gksrk gSA bl izdkj osQ v.kqvksa dks /qzfor v.kq dgrs gSaA ty dk v.kq] H2O, bl izdkj osQ v.kqvksa dk ,d mnkgj.k gSA fofo/ inkFkZ fo|qr {ks=k dh mifLFkfr vFkok vuqifLFkfr esa jkspd xq.k rFkk egRoiw.kZ vuqiz;ksx izLrqr djrs gSaA mnkgj.k 1.10 ±10 µC osQ nks vkos'k ,d&nwljs ls 5.0 mm nwjh ij fLFkr gSaA (a) bl f}èkqzo osQ v{k ij f}/qzo osQ osaQnz O ls fp=k 1.21(a) esa n'kkZ, vuqlkj] /ukos'k dh vksj 15 cm nwjh ij fLFkr fdlh ¯cnq P ij rFkk (b) f}/qoz osQ v{k osQ vfHkyacor O ls] fp=k 1.21(b) esa n'kkZ, vuqlkj xqtjus okyh js[kk ls 15 cm nwjh ij fLFkr fdlh ¯cnq Q ij fo|qr {ks=k Kkr dhft,A mnkgj.k 1.10 fp=k 1.21 gy % (a) ¯cnq P ij vkos'k +10 µC osQ dkj.k {ks=k * /ukRed ¯cnq vkos'kksa osQ laxzg dks osaQnz dh ifjHkk"kk lagfr osaQnz dh gh Hkk¡fr dh tkrh gS ftlosQ vuqlkj qi ri i. rcm = i qi 29  HkkSfrdh 10 −5 C 1 = × 4 π (8.854 × 10 −12 C 2 N −1 m −2 ) (15 − 0.25)2 × 10 −4 m 2 = 4.13 × 106 N C–1 BP osQ vuqfn'k ¯cnq P ij vkos'k –10 µC osQ dkj.k {ks=k 10 –5 C 1 = × −12 2 −1 4 π (8.854 × 10 C N m ) −2 (15 + 0.25)2 × 10 −4 m 2 = 3.86 × 106 N C–1 PA osQ vuqfn'k A rFkk B ij fLFkr nks vkos'kksa osQ dkj.k P ij ifj.kkeh fo|qr {ks=k = 2.7 × 105 N C–1 BP osQ vuqfn'k gSA bl mnkgj.k esa vuqikr OP/OB dkiQh vf/d (=60) gSA vr%] fdlh f}/qzo osQ v{k ij vR;fèkd nwjh ij fLFkr fdlh ¯cnq ij fo|qr {ks=k Kkr djus osQ fy, lh/s gh lw=k osQ mi;ksx }kjk ge blh osQ lfUudV ifj.kke dh vk'kk dj ldrs gSaA 2a i`Fkdu osQ ± q vkos'kksa ls cus f}èkqzo osQ fy, f}/qzo osQ v{k osQ oasQnz ls r nwjh ij fo|qr {ks=k dk ifjek.k 2p E = (r/a >> 1) 4πε 0r 3 ;gk¡ p = 2a q f}/zqo vk?kw.kZ dk ifjek.k gSA f}/zqzo v{k ij fo|qr {ks=k dh fn'kk lnSo f}/qzo vk?kw.kZ lfn'k osQ vuqfn'k (vFkkZr –q ls q dh vksj) gksrh gSA ;gk¡] p =10–5 C × 5 × 10–3 m = 5 × 10–8 C m vr% 2 × 5 × 10 −8 C m 1 E = −12 2 −1 −2 × (15) × 10 −6 m 3 3 = 2.6 × 105 N C–1 4 π (8.854 × 10 C N m ) f}/qzo vk?kw.kZ dh fn'kk AB osQ vuqfn'k gS] rFkk ;g ifj.kke iwoZ ifj.kke osQ dkiQh fudV gSA (b) ¯cnq B ij fLFkr + 10 µC vkos'k osQ dkj.k ¯cnq Q ij fo|qr {ks=k 10 −5 C 1 = 4 π (8.854 × 10 −12 C2 N −1 m −2 ) × [152 + (0.25)2 ] × 10 −4 m 2 = 3.99 × 106 N C–1 BQ osQ vuqfn'k ¯cnq A ij fLFkr –10 µC vkos'k osQ dkj.k Q ij fo|qr {ks=k 10 −5 C 1 = × 4 π (8.854 × 10 −12 C2 N −1 m −2 ) [152 + (0.25)2 ] × 10 −4 m 2 = 3.99 × 106 N C–1 QA osQ vuqfn'k Li"V gS fd bu nks leku ifjek.k osQ cyksa osQ OQ fn'kk osQ vuqfn'k ?kVd ,d&nwljs dks fujLr djrs gSa ijarq BA osQ lekarj fn'kk osQ vuqfn'k ?kVd la;ksftr gks tkrs gSaA vr% A rFkk B ij fLFkr nks vkos'kksa osQ dkj.k Q ij ifj.kkeh fo|qr {ks=k 0.25 =2× × 3.99 × 106 N C –1 BA osQ vuqfn'k 2 15 + (0.25)2 mnkgj.k 1.10 = 1.33 × 105 N C–1 BA osQ vuqfn'k (a) dh gh Hkk¡fr f}/qo z osQ v{k osQ vfHkyacor fdlh ¯cnq ij f}/qoz fo|qr {ks=k osQ fy, lh/s gh lw=k osQ mi;ksx }kjk ge blh ifj.kke dh vis{kk dj ldrs gSaµ 30  oS|qr vkos'k rFkk {ks=k p E = (r/a >> 1) 4 π ε0 r3 5 × 10−8 C m 1 = × −12 2 4 π (8.854 ×10 C N m )–1 –2 (15) × 10 −6 m 3 3 mnkgj.k 1.10 = 1.33 × 105 N C–1. bl izdj.k esa fo|qr {ks=k dh fn'kk vk?kw.kZ lfn'k dh fn'kk osQ foijhr gSA rFkkfi izkIr ifj.kke igys izkIr gq, ifj.kke osQ vuq:i gSaA 1.12 ,dleku cká {ks=k esa f}/zqo fp=k 1.22 esa n'kkZ, vuqlkj ,dleku fo|qr {ks=k E esa j[ks f}/qzo vk?kw.kZ p osQ LFkk;h f}èkqzo (LFkk;h f}èkqzo ls gekjk rkRi;Z ;g gS fd p dk E ls Lora=k vfLrRo gS_ ;g E }kjk izsfjr ugha gqvk gSA) ij fopkj dhft,A ;gk¡ vkos'k q ij qE rFkk –q ij –qE cy yx jgs gSaA pw¡fd E ,dleku gS vr% f}/zqo ij usV cy 'kwU; gSA ijarq vkos'kksa esa i`Fkdu gS] vr% cy fHkUu ¯cnq ij yxs gSa] ftlosQ ifj.kkeLo:i f}/qzo ij cy vk?kw.kZ dk;Z djrk gSA tc usV cy 'kwU; gS rks cy vk?kw.kZ (cy ;qXe) ewy ¯cnq ij fuHkZj ugha gksrkA bldk ifjek.k izR;sd cy osQ ifjek.k fp=k 1.22 ,dleku fo|qr {ks=k rFkk cy;qXe dh Hkqtk (nks izfrlekarj cyksa osQ chp yacor nwjh) osQ xq.kuiQy osQ cjkcj esa f}/qzoA gksrk gSA cy vk?kw.kZ dk ifjek.k = q E × 2 a sinθ = 2 q a E sinθ bldh fn'kk dkxT+k osQ ry osQ vfHkyacor blls ckgj dh vksj gSA p×E dk ifjek.k Hkh p E sinθ gS rFkk bldh fn'kk dkxT+k osQ i`"B osQ vfHkyacor ckgj dh vksj gSA vr% τ = p×E (1.22) ;g cy vk?kw.kZ f}/qzo dks {ks=k E osQ lkFk lajsf[kr djus dh izo`fÙk j[ksxkA tc p {ks=k E osQ lkFk lajsf[kr gks tkrk gS rks cy vk?kw.kZ 'kwU; gksrk gSA tc {ks=k ,dleku ugha gksrk rc D;k gksrk gS\ Li"V gS] ml izdj.k esa usV cy 'kwU;srj gks ldrk gSA blosQ vfrfjDr] O;kid :i ls fudk; ij igys dh gh Hkk¡fr ,d cy vk?kw.kZ dk;Z djsxkA ;gk¡ O;kid izdj.k varxzLZ r gS] vr% vkb, ,slh ljy fLFkfr ij fopkj djrs gSa ftlesa p {ks=k E osQ lekarj vFkok izfrlekarj gSA nksuksa gh izdj.kksa esa usV cy vk?kw.kZ rks 'kwU; gksrk gS ijarq ;fn E ,dleku ugha gS rks f}/qzo ij ,d usV cy yxrk gSA fp=k 1.23 Lor% Li"Vhdj.k djrk gSA bls vklkuh ls ns[kk tk ldrk gS fd tc p {ks=k E osQ lekarj gS rks f}/zqo ij c 0 gS rks A dh fn'kk ,dkad lfn'k â osQ leku gksxh rFkk ;fn A < 0 gS rks A dh fn'kk â dh fn'kk osQ foijhr gksxhA tc ge ½.ksrj ekuksa rd lhfer j[kuk pkgrs gSa rks ge izrhd A dk mi;ksx djrs gSa rFkk bls A dk ekM~;wYl (ekikad) dgrs gSaA bl izdkj A ≥ 0 gksrk gSA ;g Hkh è;ku nhft, fd mijksDr ppkZ esa ;|fi i`"B (λl ) }kjk ifjc¼ vkos'k dks gh osQoy lfEefyr fd;k x;k Fkk] ijarq fo|qr {ks=k E leLr rkj ij vkos'k osQ dkj.k gSA lkFk gh ;g dYiuk dj ysuk fd rkj dh yackbZ vuar gS bl dYiuk osQ fcuk ge fo|qr {ks=k E dks csyukdkj xkmlh; i`"B osQ vfHkyacor ugha ys ldrsA rFkkfi] yacs rkj osQ csyukdkj Hkkxksa osQ pkjksa vksj tgk¡ vaR; izHkko (end effects) dh mis{kk dh tk ldrh gS] fo|qr {ks=k osQ fy, lehdj.k (1.32) lfUudVr% lgh gSA 1.15.2 ,dleku vkosf'kr vuar lery pknj osQ dkj.k fo|qr {ks=k eku yhft, fdlh vuar lery pknj (fp=k 1.30) dk ,dleku i`"Bh; vkos'k ?kuRo σ gSA ge x-v{k dks fn, x, ry osQ vfHkyacor ekurs gSaA lefefr osQ vuqlkj fo|qr {ks=k y rFkk z funsZ'kkadksa ij fuHkZj ugha djsxk rFkk bldh izR;sd ¯cnq ij fn'kk x-fn'kk osQ lekarj gksuh pkfg,A ge xkmlh; i`"B dks fp=k esa n'kkZ, vuqlkj A vuqizLFkdkV {ks=kiQy osQ vk;rkdkj lekarj "kV~iQyd tSlk ys ldrs gSa (oSls rks csyukdkj i`"B ls Hkh ;g dk;Z gks ldrk gS)A tSlk fp=k ls n`f"Vxkspj gksrk gS] osQoy nks iQyd 1 rFkk 2 gh ÝyDl esa ;ksxnku nsaxs_ fo|qr {ks=k js[kk,¡ vU; iQydksa osQ lekarj gSa vkSj os blhfy, oqQy ÝyDl esa ;ksxnku ugha nsrhaA i`"B 1 osQ vfHkyacor ,dkad lfn'k –x fn'kk esa gS tcfd i`"B 2 osQ vfHkyacor ,dkad lfn'k +x fn'kk esa gSA vr%] nksuksa i`"Bksa ls xqtjus okys ÝyDl E.∆S cjkcj gSa vkSj la;ksftr gks tkrs gSaA fp=k 1.30 ,dleku vkosf'kr vuar vkos'k pknj osQ blhfy, xkmlh; i`"B ls xqtjus okyk usV ÝyDl 2 EA gSA i`"B }kjk fy, xkmlh; i`"BA ifjc¼ vkos'k σA gSA blhfy, xkml fu;e }kjk gesa ;g lac/ a izkIr 38 gksrk gS  oS|qr vkos'k rFkk {ks=k 2 EA = σA/ε0 vFkok E = σ/2ε0 lfn'k :i esa σ E= ˆ n (1.33) 2ε 0 ;gk¡ n̂ ry osQ vfHkyacor blls nwj tkrk gqvk ,dkad lfn'k gSA ;fn σ /ukRed gS rks E ry ls cfgeZq[kh rFkk ;fn σ ½.kkRed gS rks E ry ls vareqZ[kh gksrk gSA è;ku nhft, xkml fu;e osQ vuqiz;ksx ls gesa ,d vfrfjDr rF; ;g izkIr gksrk gS fd E, x ij Hkh fuHkZj ugha gSA fdlh ifjfer cM+h leryh; pknj osQ fy, lehdj.k (1.33), fljksa ls nwj leryh; pknj osQ eè;orhZ {ks=kksa esa lfUudVr% lR; gSA 1.15.3 ,dleku vkosf'kr irys xksyh; [kksy ds dkj.k fo|qr {ks=k eku yhft, R f=kT;k ds irys xksyh; [kksy dk ,dleku i`"Bh; vkos'k ?kuRo σ gS (fp=k 1.31)A Li"V :i ls bl fLFkfr esa xksyh; lefefr gSA fdlh fcanq P ij pkgs og Hkhrj gS vFkok ckgj] fo|qr {ks=k osQoy r ij fuHkZj dj ldrk gS (;gk¡ r [kksy osQ osaQnz ls ml fcanq rd dh f=kT; nwjh gSA)rFkk bl {ks=k dks vjh; (vFkkZr èkzqokarj osQ vuqfn'k) gksuk pkfg,A (i) [kksy osQ ckgj fo|qr {ks=kµ[kksy osQ ckgj /zqokarj r osQ fdlh fcanq P ij fopkj dhft,A fcanq P ij E dk ifjdyu djus osQ fy, ge osaQnz O rFkk f=kT;k r osQ fcanq P ls xqtjus okys xksys dks xkmlh; i`"B ekurs gSaA fn, x, vkos'k foU;kl osQ lkis{k bl xksys ij fLFkr izR;sd fcanq lerqY; gSA (xksyh; lefefr ls gekjk ;gh vfHkizk; gSA) blhfy, xkmlh; i`"B osQ izR;sd fcanq ij fo|qr {ks=k dk leku ifjek.k E gS rFkk izR;sd fcanq ij /zqokarj osQ vuqfn'k gSA bl izdkj izR;sd fcanq ij E rFkk ∆S lekarj gSa rFkk izR;sd vo;o ls xq”kjus okyk ÝyDl E ∆S gSA lHkh ∆S dk ladyu djus ij xkmlh; i`"B ls xq”kjus okyk ÝyDl E ×4 π r 2 gSA ifjc¼ vkos'k σ ×4 π R 2 gSA xkml fu;e ls σ 2 E ×4 π r2 = ε 4 π R 0 σ R2 q vFkok E= 2 = ε0 r 4 π ε0 r 2 ;gk¡ q = 4 π R2 σ xksyh; [kksy ij oqQy vkos'k gSA q lfn'k :i esa] E = 4 πε r 2 rˆ (1.34) fp=k 1.31 fdlh ¯cnq osQ fy, tks 0 (a) r > R, (b) r < R ij gS] xkmlh; ;fn q > 0 gS rks fo|qr {ks=k cfgeqZ[kh gksrk gS rFkk ;fn q < 0 gS rks fo|qr {ks=k i`"BA vareqZ[kh gksrk gSA rFkkfi] ;g [kksy osQ osaQnz O ij fLFkr vkos'k q }kjk mRiUu fo|qr {ks=k gSA vr% [kksy osQ ckgj fLFkr fcanqvksa ij ,dleku vkosf'kr xksyh; [kksy osQ dkj.k fo|qr {ks=k bl izdkj dk gksrk gS] tSls fd [kksy dk leLr vkos'k mlosQ osQa nz ij fLFkr gSA (ii) [kksy osQ Hkhrj fo|qr {ks=kµfp=k 1.31(b) esa fcanq P [kksy osQ Hkhrj gSA bl izdj.k esa Hkh xkmlh; i`"B P ls xq”kjus okyk og xksyk gS ftldk osaQnz O gSA igys fd, x, 39  HkkSfrdh ifjdyuksa dh gh Hkk¡fr xkmlh; i`"B ls xqtjus okyk ÝyDl E ×4 π r 2 gSA rFkkfi bl izdj.k esa] xkmlh; i`"B esa dksbZ vkos'k ifjc¼ ugha gSA rc xkml fu;e osQ vuqlkj E ×4 π r 2 = 0 vFkkZr E = 0 (r < R ) (1.35) vFkkZr ,dleku vkosf'kr irys xksyh; [kksy* osQ dkj.k mlosQ Hkhrj fLFkr lHkh fcanqvksa ij fo|qr {ks=k 'kwU; gSA ;g egRoiw.kZ ifj.kke xkml fu;e dk izR;{k fu"d"kZ gS tks dwykWe fu;e ls izkIr gqvk gSA bl fu;e dk izk;ksfxd lR;kiu dwykWe fu;e esa 1/r2 dh fuHkZjrk dh iqf"V djrk gSA mnkgj.k 1.13 ijek.kq osQ izkjafHkd izfr:i esa ;g ekuk x;k Fkk fd vkos'k Ze dk fcanq vkeki dk /ukRed ukfHkd gksrk gS tks f=kT;k R rd ,dleku ?kuRo osQ ½.kkos'k ls f?kjk gqvk gSA ijek.kq iw.kZ :i esa fo|qr mnklhu gSA bl izfr:i ds fy, ukfHkd ls r nwjh ij fo|qr {ks=k fdruk gS\ fp=k 1-32 ijek.kq dk izkjafHkd izfr:iA gy% fp=k 1-32 esa bl izfr:i dk vkos'k forj.k n'kkZ;k x;k gSA pw¡fd ijek.kq mnklhu (ukfHkd esa vkos'k Z e + ½.kkos'k) gS] vr% R f=kT;k osQ ,dleku xksyh; vkos'k forj.k esa oqQy ½.kkRed vkos'k –Ze gksuk pkfg,A blls gesa rqjar gh ½.kkRed vkos'k ?kuRo ρ izkIr gks tkrk gSA pw¡fd gekjs ikl dqy 'kwU; vkos'k gksuk pkfg,] vr% 4 πR3 ρ = 0 – Ze 3 3 Ze vFkok ρ = − 4 π R 3 ukfHkd ls r nwjh ij fLFkr fcanq P ij fo|qr {ks=k E(r) Kkr djus osQ fy, ge xkml fu;e dk mi;ksx djrs gSAa vkos'k forj.k dh xksyh; lefefr osQ dkj.k fo|qr {ks=k E(r) dk ifj.kke osQoy f=kT; nwjh ij fuHkZj djrk gS] ;gk¡ r dh fn'kk dk dksbZ vFkZ ugha gskrkA bldh fn'kk ewy fcanq ls P dh vksj /zqokarj r ds vuqfn'k (vFkok foijhr) gSA Li"V :i ls bl izdj.k esa xkmlh; i`"B ,d xksyh; i`"B gS ftldk osQa nz ukfHkd gSA ;gk¡ ge nks fLFkfr;ksa r < R rFkk r > R ij fopkj djrs gSaA (i) r < R : xksyh; i`"B }kjk ifjc¼ oS|qr ÝyDl mnkgj.k 1.13 φ = E (r ) × 4 π r 2 ;gk¡ E (r ), r ij fo|qr {ks=k dk ifjek.k gSA bldk dkj.k ;g gS xksyh; xkmlh; i`"B ds fdlh Hkh fcanq ij fo|qr {ks=k dh fn'kk i`"B osQ ml fcanq ij vfHkyacor gksrh gS rFkk bldk ifjek.k i`"B osQ lHkh fcanqvksa ij leku gksrk gSA * bldh rqyuk HkkSfrdh ikB~;iqLrd d{kk 11 osQ vuqHkkx 8-5 esa of.kZr ,dleku nzO;eku [kksy ls 40 dhft,A  oS|qr vkos'k rFkk {ks=k ;gk¡ xkmlh; i`"B }kjk ifjc¼ vkos'k q /ukRed ukfHkdh; vkos'k rFkk r f=kT;k osQ xksys esa fo|eku ½.kkRed vkos'k gS 4 πr3 vFkkZr q = Ze + ρ 3 igys izkIr vkos'k ?kuRo ρ dk eku izfrLFkkfir djus ij gesa izkIr gksrk gSA r3 q = Ze−Ze R3 rc xkml fu;e ls gesa izkIr gksrk gSA Z e 1 r E (r ) = − ; r < R 4 π ε0 r 2 R3 ;gk¡ fo|qr {ks=k f=kT;r% cfgeqZ[kh funf'kZr gSA (ii) r > R : bl izdj.k esa] pw¡fd ijek.kq mnklhu gS] xksyh; xkmlh; i`"B }kjk ifjc¼ vkos'k 'kwU; gSA bl izdkj] xkml fu;e ls mnkgj.k 1.13 E (r) × 4 π r2 = 0 vFkok (r) = 0; r > R r = R, ij nksuksa izdj.kksa ls leku ifj.kke] E = 0 izkIr gksrk gSA lefefr lafØ;kvksa ds fo"k; esa HkkSfrdh esa gekjk izk;% fofHkUu lefefr ds fudk;ksa ls lkeuk gksrk gSA bu lefefr;ksa ij fopkj djuk lh/s&lhèkss ifjdyuksa dh rqyuk esa dgha vf/d 'kh?kzrk ls ifj.kke rd igq¡pus esa lgk;d gksrk gSA mnkgj.k ds fy,] y-z ry osQ vuqfn'k vuar yackbZ dh vkos'k dh ,dleku pknj (i`"Bh; vkos'k ?kuRo σ) ij fopkj dhft,A ;g fudk; vifjofrZr jgrk gS ;fn (a) y-z ry osQ lekarj fdlh Hkh fn'kk esa LFkkukarfjr fd;k tkrk gS] (b) x-v{k osQ ifjr% fdlh Hkh dks.k ij ?kw.kZu djk;k tkrk gSA ftl izdkj ;g fudk; bl izdkj dh lefefr lafØ;kvksa ds v/hu vifjofrZr jgrk gS] blh izdkj blosQ xq.k/eZ Hkh vifjofrZr jgus pkfg,A fo'ks"k :i ls] bl mnkgj.k esa] fo|qr {ks=k E vifjofrZrr gksuk pkfg,A y-v{k ds vuqfn'k LFkkukarjh; lefefr ;g n'kkZrh gS fd fdlh fcanq (0, y1, 0) ij fo|qr {ks=k (0, y2, 0) ij fo|qr {ks=k osQ leku gksuk pkfg,A blh izdkj z-v{k osQ vuqfn'k LFkkukarjh; lefefr ;g n'kkZrh gS fd nks fcanqvksa (0, 0, z1) rFkk (0, 0, z2) ij fo|qr {ks=k leku gksuk pkfg,A x- v{k osQ pkjksa vksj ?kw.kZu lefefr dk mi;ksx djosQ ge ;g fu"d"kZ fudky ldrs gSa fd E y-z ry osQ yacor gksuk pkfg,] vFkkZr ;g x- fn'kk osQ lekarj gksuk pkfg,A vc fdlh ,slh lefefr dks lkspus dk iz;kl dhft, tks vkidks ;g crk, fd fo|qr {ks=k dk ifjek.k ,d fu;rkad gS] tks x- funs'Z kkad ij fuHkZj ugha djrkA bl izdkj ;g dgk tk ldrk gS fd fdlh vuar yackbZ dh ,dleku pkyd pknj osQ dkj.k mRiUu fo|qr {ks=k dk ifjek.k fno~QLFkku esa gj ¯cnq ij leku gksrk gSA rFkkfi pknj osQ nksuksa vksj fo|qr {ks=k dh fn'kk ,d&nwljs osQ foijhr gksrh gSA bldh rqyuk mu ifj.kkeksa ls dhft, tks dwykWe fu;e ds mi;ksx }kjk lh/s ifjdyu dj ifj.kke ij igq¡pus osQ fy, vko';d gksaA 41  HkkSfrdh lkjka'k 1. fo|qr rFkk pqacdh; cy ijek.kqvksa] v.kqvksa rFkk LFkwy nzO; osQ xq.kèkeks± dk fuèkkZj.k djrs gSaA 2. ?k"kZ.k fo|qr osQ ljy iz;ksxksa osQ vkèkkj ij ;g fu"d"kZ fudkyk tk ldrk gS fd izÑfr esa nks izdkj osQ vkos'k gksrs gSaA ltkrh; vkos'kksa esa izfrd"kZ.k rFkk fotkrh; vkos'kksa esa vkd"kZ.k gksrk gSA ifjikVh osQ vuqlkj] js'ke ls jxM+us ij dk¡p dh NM+ ij èkukos'k gksrk gS_ rFkk iQj ls jxM+us ij IykfLVd dh NM+ ij ½.kkos'k gksrk gSA 3. pkyd vius esa ls gksdj ljyrk ls oS|qr vkos'k dh xfr gksus nsrs gSa tcfd fo|qrjksèkh ,slk ugha djrsA èkkrqvksa esa xfr'khy vkos'k bysDVªkWu gksrs gSa_ oS|qr vi?kV~;ksa esa èkuk;u rFkk ½.kk;u nksuksa gh xfr djrs gSaA 4. oS|qr vkos'kksa osQ rhu xq.kèkeZ gksrs gSa % DokaVehdj.k] ;ksT;rk rFkk laj{k.kA oS|qr vkos'k osQ DokaVehdj.k ls rkRi;Z gS fd fdlh oLrq dk oqQy vkos'k (q) lnSo gh vkos'k osQ ,d ewy DokaVe (e) osQ iw.kk±dh xq.kt vFkkZr q = n e gksrk gS] tgk¡ n = 0, ±1, ±2, ±3,.... gSA izksVkWu rFkk bysDVªkWu ij Øe'k% +e rFkk –e vkos'k gksrs gSaA LFkwy vkos'kksa ftuosQ fy, n ,d vR;fèkd cM+h la[;k gksrh gS] esa vkos'k osQ DokaVehdj.k dh mis{kk dh tk ldrh gSA oS|qr vkos'kksa dh ;ksT;rk ls gekjk rkRi;Z ;g gS fd fdlh fudk; dk oqQy vkos'k ml fudk; osQ lHkh ,dkdh vkos'kksa dk chtxf.krh; ;ksx (vFkkZr ;ksx djrs le; muosQ fpÉksa dks è;ku esa j[kdj) gksrk gSA oS|qr vkos'kksa osQ laj{k.k ls gekjk rkRi;Z ;g gS fd fdlh fo;qDr fudk; (isolated system) dk oqQy vkos'k le; osQ lkFk vifjofrZr jgrk gSA bldk vFkZ ;g gS fd tc ?k"kZ.k }kjk oLrq,¡ vkosf'kr dh tkrh gSa rks vkos'kksa dk ,d oLrq ls nwljh oLrq esa LFkkukarj.k gksrk gS] ijarq bl izfØ;k esa u rks dksbZ vkos'k mRiUu gksrk gS vkSj u gh u"V gksrk gSA 5. owQykWe fu;e % nks ¯cnq vkos'kksa q1 rFkk q2 osQ chp ikjLifjd fLFkj oS|qr cy] vkos'kksa osQ xq.kuiQy q1q2 osQ vuqØekuqikrh rFkk muosQ chp dh nwjh r21 osQ oxZ osQ O;qRØekuqikrh gksrk gSA xf.krh; :i esa k (q1q2 ) F21 = q2 ij q1 osQ dkj.k yxus okyk cy = 2 r̂21 r21 1 ;gk¡ r̂21 vkos'k q1 ls q2 dh fn'kk esa ,dakd lfn'k gS rFkk k = vkuqikfrdrk 4 πε 0 fLFkjkad gSA SI ek=kdksa es]a vkos'k dk ek=kd owQykWe gSA fu;rkad ε 0 dk izk;ksfxd eku gS ε 0 = 8.854 × 10–12 C2 N–1 m–2 k dk lfUudV eku k = 9 × 109 N m2 C–2 gksrk gSA 6. fdlh izksVkWu rFkk fdlh bysDVªkWu osQ chp oS|qr cy rFkk xq#Rokd"kZ.k cy dk vuqikr gS] k e2 ≅ 2.4 × 1039 G m em p 7. vè;kjksi.k fl¼kar % ;g fl¼kar bl xq.kèkeZ ij vkèkkfjr gS fd nks vkos'kksa osQ chp yxus okys vkd"khZ vFkok izfrd"khZ cy fdlh rhljs (vFkok vfèkd) vfrfjDr vkos'k dh mifLFkfr ls izHkkfor ugha gksrAs vkos'kksa q1, q2, q3... osQ fdlh lewg osQ fy, fdlh vkos'k (tSls q1) ij cy] q1 ij q2 osQ dkj.k cy] q1 ij q3 osQ dkj.k cy vkfn&vkfn osQ 42  oS|qr vkos'k rFkk {ks=k lfn'k ;ksx osQ cjkcj gksrk gSA izR;sd ;qxy osQ fy, ;g cy igys of.kZr nks vkos'kksa osQ fy, owQykWe osQ fu;e }kjk gh O;Dr fd;k tkrk gSA 8. fdlh vkos'k foU;kl osQ dkj.k fdlh ¯cnq ij fo|qr {ks=k E fdlh NksVs èkukRed ijh{k.k vkos'k (test charge) q dks ml ¯cnq ij j[kus ij mlosQ }kjk vuqHko fd, tkus okys cy dks ml vkos'k osQ ifjek.k }kjk foHkkftr djus ij izkIr gksrk gSA fdlh ¯cnq vkos'k q osQ dkj.k fo|qr {ks=k dk dksbZ ifjek.k q / 4 π ε 0 r 2 gksrk gS_ ;fn q èkukRed gS rks ;g {ks=k vjh; (f=kT;h;) cfgeqZ[kh gksrk gS rFkk ;fn q ½.kkRed gS rks vjh; (f=kT;h;) vareqZ[kh gksrk gSA owQykWe cy dh Hkk¡fr fo|qr {ks=k Hkh vè;kjksi.k fl¼kar dks larq"V djrk gSA 9. fo|qr {ks=k js[kk ,slk oØ gS ftlosQ fdlh Hkh ¯cnq ij [khapk x;k Li'khZ] oØ osQ ml ¯cnq ij fo|qr {ks=k dh fn'kk crkrk gSA {ks=k js[kkvksa dh lkisf{kd ladqyrk fofHkUu fcanqvksa ij fo|qr {ks=k dh lkisf{kd rhozrk dks bafxr djrh gSA izcy fo|qr {ks=k esa js[kk,¡ ikl&ikl rFkk nqcZy {ks=k esa ;s ,d-nwljs ls dkiQh nwj gksrh gSaA ,dleku (vFkok fu;r) fo|qr {ks=k esa {ks=k js[kk,¡ ,d-nwljs ls ,dleku nwjh ij lekarj ljy js[kk,¡ gksrh gSAa 10. {ks=k js[kkvksa dh oqQN egRoiw.kZ fo'ks"krk,¡ bl izdkj gSaµ (a) vkos'keqDr fno~QLFkku esa {ks=k js[kk,¡ larr oØ gksrh gSa tks dgha ugha VwVrhaA (b) nks {ks=k js[kk,¡ ,d&nwljs dks dnkfi ugha dkV ldrhaA (c) fLFkjoS|qr {ks=k js[kk,¡ èkukos'k ls vkjaHk gksdj ½.kkos'k ij lekIr gks tkrh gSaµ ;s lao`r ik'k (can ywi) ugha cuk ldrhaA 11. oS|qr f}èkzqo ifjek.k esa leku fotkrh; nks vkos'kksa q rFkk –q, ftuosQ chp i`Fkdu 2a gks] dk ;qXe gksrk gSA blosQ f}èkzqo vk?kw.kZ lfn'k p dk ifjek.k 2qa gksrk gS rFkk ;g f}èkzqo v{k –q ls q dh fn'kk esa gksrk gSA 12. fdlh oS|rq f}èkzo q dk blosQ fuj{kh;@fo"kqorh; lery (vFkkZr blosQ v{k osQ yacor rFkk blosQ osQanz ls xqtjus okys lery) ij blosQ osQanz ls r nwjh ij fo|qr {ks=kµ −p 1 E= 4 π ε o ( a 2 + r 2 )3 / 2 −p ≅ , (r >> a osQ fy,) 4 π ε or 3 f}èkzqo v{k ij osQanz ls r nwjh ij f}èkzqo fo|qr {ks=k 2 pr E= 4π ε 0 (r 2 − a 2 )2 2p ≅ (r >> a osQ fy,) 4π ε 0r 3 bl rF; ij è;ku nsuk pkfg, fd fdlh f}/qzo dk fo|qr {ks=k 1/r 3 ij fuHkZj gksrk gS] tcfd fdlh ¯cnq vkos'k osQ dkj.k fo|qr {ks=k 1/r 2 ij fuHkZj gksrk gSA 13. fdlh ,dleku fo|qr {ks=k E esa dksbZ oS|rq f}èkzo q ,d cy vk?kw.kZ τ dk vuqHko djrk gSA τ = p ×E ijarq fdlh usV cy dk vuqHko ugha djrkA 14. fo|qr {ks=k E dk fdlh y?kq {ks=kiQy&vo;o ∆S ls xqtjus okyk ÝyDl ∆φ = E. ∆S lfn'k {ks=kiQy&vo;o ∆S = ∆S n̂ ;gk¡ ∆S {ks=kiQy&vo;o dk ifjek.k rFkk n̂ {ks=kiQy&vo;o osQ yacor f=kT; ,dakd lfn'k gS ftls dkiQh NksVs {ks=k osQ fy, leryh; ekuk tk ldrk gSA fdlh can i`"B osQ 43  HkkSfrdh {ks=kiQy&vo;o osQ fy,] n̂ dks ifjikVh osQ vuqlkj cfgeZ[q kh vfHkyac dh fn'kk ekuk tk ldrk gSA 15. xkml fu;e % fdlh can i`"B S ls gksdj xq”kjus okys fdlh fo|qr {ks=k dk ÝyDl ml i`"B S }kjk ifjc¼ oqQy vkos'k dk 1/ε 0 xquk gksrk gSA ;g fu;e fo|qr {ks=k osQ fuèkkZj.k esa fo'ks"k :i ls rc mi;ksxh gksrk gS tcfd vkos'k forj.k esa ljy lefefr gksA (i) ,dleku jSf[kd vkos'k ?kuRo λ dk iryk vuar yackbZ dk lhèkk rkj λ E= ˆ n 2 πε0 r ;gk¡ r ¯cnq dh rkj ls yacor nwjh gS rFkk n$ ml ¯cnq ls xq”kjus okys rkj osQ vfHkyacor ry esa f=kT; ,dakd lfn'k gSA (ii) ,dleku i`"Bh; vkos'k ?kuRo σ dh iryh vuar lery pknj σ E= ˆ n 2ε0 ;gk¡ n̂ lery osQ vfHkyacor ik'oZ osQ nksuksa vksj cfgeZq[kh ,dakd lfn'k gSA (iii) ,dleku i`"Bh; vkos'k ?kuRo σ osQ irys xksyh; [kksy (;k dks'k) q E= rˆ (r ≥ R ) 4π ε 0 r 2 E=0 (r < R) ;gk¡ r xksyh; [kksy osQ osQna z ls ¯cnq dh nwjh rFkk R [kksy dh f=kT;k gSA [kksy dk oqQy vkos'k q gSA tgk¡ q = 4πR 2σ gSA [kksy osQ ckgj fdlh ¯cnq ij vkosf'kr [kksy osQ dkj.k fo|qr {ks=k bl izdkj gksrk gS tSls fd leLr vkos'k [kksy osQ osQanz ij gh osaQfnzr gSA ;gh ifj.kke fdlh ,dleku vk;ru vkos'k ?kuRo osQ Bksl xksys osQ fy, Hkh lR; gksrk gSA [kksy osQ vanj lHkh ¯cnqvksa ij fo|qr {ks=k 'kwU; gksrk gSA HkkSfrd jkf'k izrhd foek,¡ ek=kd fVIi.kh lfn'k {ks=kiQy-vo;o ∆S [L2] m2 ∆S = ∆S n̂ fo|qr {ks=k E [MLT A ] –3 –1 Vm –1 oS|qr ÝyDl φ [ML T A ] 3 –3 –1 Vm ∆φ = E. ∆S f}/zqo vk?kw.kZ p [LT A] Cm ½.kkos'k ls /ukos'k dh vksj fufnZ"V lfn'k vkos'k ?kuRo jSf[kd λ [L–1 T A] C m–1 vkos'k@yackbZ i`"Bh; σ –2 [L T A] Cm –2 vkos'k@{ks=kiQy vk;ru ρ –3 [L T A] C m–3 vkos'k@vk;ru 44  oS|qr vkos'k rFkk {ks=k fopkj.kh; fo"k; 1- vkidks ;g vk'p;Z gks ldrk gS fd ukfHkd esa izksVkWu ftu ij /ukos'k gS] lHkh ,d lkFk dls jgdj dSls lek, gq, gSaA os ,d&nwljs ls nwj D;ksa ugha mM+ tkrs\ vki ;g lh[ksaxs fd ,d rhljs izdkj dk ewy cy Hkh gS ftls izcy cy dgrs gSa vkSj ;gh cy izksVkWuksa dks ,d lkFk ck¡/s j[krk gSA ijarq nwjh dk og ifjlj ftlesa ;g cy izHkkoh gksrk gS] cgqr NksVk ~ 10-14 m gSA ;Fkk;Z :i esa ;gh ukfHkd dk lkb”k gSA lkFk gh DokaVe ;kaf=kdh ds fu;eksa ds vuqlkj bysDVªkWuksa dks izksVkWuksa ds 'kh"kZ ij vFkkZr ukfHkd ds Hkhrj cSBus dh vuqefr Hkh ugha gSA blh ls ijek.kqvksa dks mudh jpuk izkIr gS vkSj mudk izÑfr esa vfLrRo gSA 2. owQykWe cy rFkk xq#Rokd"kZ.k cy leku O;qRØe oxZ fu;e dk ikyu djrs gSaA ijarq xq#Rokd"kZ.k cy dk osQoy ,d gh fpÉ (lnSo vkd"khZ) gksrk gS] tcfd owQykWe cy osQ nksuksa fpÉ (vkd"khZ rFkk çfrd"khZ) gks ldrs gSa] ftlls fo|qr cyksa dk fujlu gks ldrk gSA ;gh dkj.k gS fd vR;ar nqcZy cy gksus ij Hkh xq#Ro cy izÑfr esa ,d izcy rFkk vf/d O;kid cy gks ldrk gSA 3. ;fn vkos'k osQ ek=kd dks owQykWe osQ fu;e }kjk ifjHkkf"kr djuk gS rks owQykWe osQ fu;e esa vkuqikfrdrk fLFkjkad k ,d p;u dk fo"k; gSA ijarq SI ek=kdksa esa fo|qr èkkjk osQ ek=kd ,sfEi;j (A) dks mlh osQ pqacdh; çHkko (,sfEi;j fu;e) }kjk ifjHkkf"kr fd;k tkrk gS rFkk vkos'k osQ ek=kd (owQykWe) dks osQoy (1C = 1A s) }kjk ifjHkkf"kr fd;k tkrk gSA bl çdj.k esa k dk eku LoSfPNd ugha gS_ ;g yxHkx 9×109 N m2 C−2 gSA 4. fLFkjkad k dk vR;fèkd cM+k eku] vFkkZr fo|qr çHkko dh n`f"V ls vkos'k osQ ek=kd (1C) dk cM+k vkdkj (eku) bl dkj.k ls gS D;ksafd (tSlk fd ¯cnq 3 esa mYys[k fd;k tk pqdk gS) vkos'k osQ ek=kd dks pqacdh; cyksa (fo|qrokgh rkjksa ij yxs cyksa) osQ inksa esa ifjHkkf"kr fd;k x;k gS tks fd O;kid :i ls oS|qr cyksa dh rqyuk esa cgqr nqcZy gksrs gSaA ;gh dkj.k gS fd] 1 ,sfEi;j pqacdh; çHkkoksa osQ fy, ;qfDrlaxr ek=kd gS] 1 C = 1 A s oS|qr çHkkoksa osQ fy, ,d vR;fèkd cM+k ek=kd gSA 5. vkos'k dk ;ksT;rk xq.kèkeZ dksbZ lqLi"V xq.kèkeZ ugha gSA ;g bl rF; ls lacafèkr gS fd oS|qr vkos'k ls dksbZ fn'kk lac¼ ugha gksrh] vkos'k ,d vfn'k jkf'k gSA 6. vkos'k osQoy ?kw.kZu osQ varxZr gh vfn'k (vFkok vpj@vifjorZuh;) ugha gS_ ;g vkisf{kd xfr esa Hkh funs'Z k isQz eksa osQ fy, vpj gSA ;g dFku izR;sd vfn'k ds fy, lnSo lR; ugha gSA mnkgj.kkFkZ] xfrt mQtkZ ?kw.kZu osQ varxZr vfn'k gS] ijarq ;g vkisf{kd xfr;ksa esa funsZ'k izsQeksa osQ fy, vpj ugha gS] vFkok bl n`f"V ls rks nzO;eku Hkh vpj ugha gSA 7. fdlh fo;qDr fudk; osQ oqQy vkos'k dk laj{k.k ,d ,slk xq.kèkeZ gS tks ¯cnq 6 osQ varxZr vkos'k dh vfn'k çÑfr ij fuHkZj ugha djrkA laj{k.k fdlh fn, x, funs'Z k izsQe esa le; dh vifjorZuh;rk dh vksj laosQr djrk gSA dksbZ jkf'k vfn'k gksrs gq, Hkh lajf{kr ugha gks ldrh (fdlh vçR;kLFk la?k^ esa xfrt ÅtkZ dh Hkk¡fr)A blosQ foijhr] lfn'k jkf'k dk laj{k.k Hkh gks ldrk gS (mnkgj.k osQ fy, fdlh fo;qDr fudk; dk dks.kh; laosx laj{k.k)A 8. oS|qr vkos'k dk DokaVehdj.k çÑfr dk ewy fu;e gS ftldh vHkh rd O;k[;k ugha dh tk ldh gSA jkspd rF; ;g gS fd lagfr osQ DokaVehdj.k dk dksbZ ln`'k (vuq:i) fu;e ugha gSA 9. vè;kjksi.k fl¼kar dks lqLi"V ugha ekuuk pkfg, vFkok bls lfn'kksa osQ ;ksx osQ fu;e osQ leku ugha ekuuk pkfg,A ;g fl¼kar nks ckrsa crkrk gS% fdlh vkos'k ij nwljs vkos'k osQ dkj.k cy fdUgha vU; vkos'kksa dh mifLFkfr osQ dkj.k çHkkfor ugha gksrk rFkk ;gk¡ 45  HkkSfrdh dksbZ vfrfjDr f=k&fiaM] prq%&fiaM vkfn cy ugha gksrs tks osQoy rHkh mRiUu gksrs gSa tc nks ls vfèkd vkos'k gksaA 10. fdlh fofoDr vkos'k foU;kl osQ dkj.k fofoDr vkos'kksa dh vofLFkfr ij fo|qr {ks=k ifjHkkf"kr ugha gSA larr vk;ru vkos'k forj.k osQ fy, ;g forj.k esa fdlh Hkh ¯cnq ij ifjHkkf"kr gksrk gSA fdlh i`"Bh; vkos'k forj.k osQ fy, fo|qr {ks=k i`"B osQ vkj&ikj fofPNUu gksrk gSA 11. fdlh vkos'k foU;kl ftlesa oqQy vkos'k 'kwU; gS] osQ dkj.k lHkh fcanqvksa ij fo|qr {ks=k 'kwU; ugha gksrk_ mu nwfj;ksa osQ fy, tks vkos'k foU;kl osQ vkdkj dh rqyuk esa cM+h gSa] blosQ {ks=k esa deh 1/r 2 ls Hkh vf/d rhoz xfr ls gksrh gS] tks fd fdlh ,dy vkos'k osQ fo|qr {ks=k dh fo'ks"krk gSA ,d oS|qr f}èkqzo bl rF; dk ,d ljyre mnkgj.k gSA vH;kl 1.1 ok;q esa ,d&nwljs ls 30 cm nwjh ij j[ks nks NksVs vkosf'kr xksyksa ij Øe'k% 2 × 10−7C rFkk 3 × 10−7 C vkos'k gSaA muosQ chp fdruk cy gS\ 1.2 0.4 µC vkos'k osQ fdlh NksVs xksys ij fdlh vU; NksVs vkosf'kr xksys osQ dkj.k ok;q esa 0.2 N cy yxrk gSA ;fn nwljs xksys ij 0.8 µC vkos'k gks rks (a) nksuksa xksyksa osQ chp fdruh nwjh gS\ (b) nwljs xksys ij igys xksys osQ dkj.k fdruk cy yxrk gS\ 1.3 tk¡p }kjk lqfuf'pr dhft, fd k e 2/G memp foekghu gSA HkkSfrd fu;rkadksa dh lkj.kh ns[kdj bl vuqikr dk eku Kkr dhft,A ;g vuqikr D;k crkrk gS\ 1.4 (a) ¶fdlh oLrq dk oS|qr vkos'k DokaVhÑr gS]¸ bl çdFku ls D;k rkRi;Z gS\ (b) LFkwy vFkok cM+s iSekus ij oS|qr vkos'kksa ls O;ogkj djrs le; ge oS|qr vkos'k osQ DokaVehdj.k dh mis{kk oSQls dj ldrs gSa\ 1.5 tc dk¡p dh NM+ dks js'ke osQ VqdM+s ls jxM+rs gSa rks nksuksa ij vkos'k vk tkrk gSA blh çdkj dh ifj?kVuk dk oLrqvksa osQ vU; ;qXeksa esa Hkh çs{k.k fd;k tkrk gSA Li"V dhft, fd ;g çs{k.k vkos'k laj{k.k fu;e ls fdl izdkj lkeatL; j[krk gSA 1.6 pkj ¯cnq vkos'k qA = 2 µC, qB = – 5 µC, qC = 2 µC rFkk qD = – 5 µC, 10 cm Hkqtk osQ fdlh oxZ ABCD osQ 'kh"kks± ij vofLFkr gSAa oxZ osQ osQæa ij j[ks 1 µC vkos'k ij yxus okyk cy fdruk gS\ 1.7 (a) fLFkjoS|qr {ks=k js[kk ,d larr oØ gksrh gS vFkkZr dksbZ {ks=k js[kk ,dk,d ugha VwV ldrhA D;ksa\ (b) Li"V dhft, fd nks {ks=k js[kk,¡ dHkh Hkh ,d&nwljs dk çfrPNsnu D;ksa ugha djrha\ 1.8 nks ¯cnq vkos'k qA = 3 µC rFkk qB = – 3 µC fuokZr esa ,d&nwljs ls 20 cm nwjh ij fLFkr gSAa (a) nksuksa vkos'kksa dks feykus okyh js[kk AB osQ eè; ¯cnq O ij fo|qr {ks=k fdruk gS\ (b) ;fn 1.5 × 10−9 C ifjek.k dk dksbZ ½.kkRed ijh{k.k vkos'k bl ¯cnq ij j[kk tk, rks ;g ijh{k.k vkos'k fdrus cy dk vuqHko djsxk\ 1.9 fdlh fudk; esa nks vkos'k qA = 2.5 × 10−7 C rFkk qB = –2.5 × 10−7 C Øe'k% nks ¯cnqvksa A : (0, 0, –15 cm) rFkk B : (0, 0, +15 cm) ij vofLFkr gSaA fudk; dk oqQy vkos'k rFkk oS|qr f}èkzqo vk?kw.kZ D;k gS\ 1.10 4 × 10−9 C m f}èkqzo vk?kw.kZ dk dksbZ oS|qr f}èkqzo 5 × 104 N C−1 ifjek.k osQ fdlh ,dleku fo|qr {ks=k dh fn'kk ls 30° ij lajsf[kr gSA f}èkqzo ij dk;Zjr cy vk?kw.kZ dk ifjek.k ifjdfyr dhft,A 1.11 Åu ls jxM+s tkus ij dksbZ ikWyhFkhu dk VqdM+k 3×10−7 C osQ ½.kkos'k ls vkosf'kr ik;k x;kA (a) LFkkukarfjr (fdl inkFkZ ls fdl inkFkZ esa) bysDVªkWuksa dh la[;k vkdfyr dhft,A 46 (b) D;k Åu ls ikWyhFkhu esa lagfr dk LFkkukarj.k Hkh gksrk gS\  oS|qr vkos'k rFkk {ks=k 1.12 (a) nks fo|qrjksèkh vkosf'kr rk¡cs osQ xksyksa A rFkk B osQ osaQæksa osQ chp dh nwjh 50 cm gSA ;fn nksuksa xksyksa ij i`Fkd&i`Fkd vkos'k 6.5 × 10−7 C gSa] rks buesa ikjLifjd fLFkjoS|qr çfrd"kZ.k cy fdruk gS\ xksyksa osQ chp dh nwjh dh rqyuk esa xksyksa A rFkk B dh f=kT;k,¡ ux.; gSaA (b) ;fn çR;sd xksys ij vkos'k dh ek=kk nks xquh rFkk xksyksa osQ chp dh nwjh vkèkh dj nh tk, rks çR;sd xksys ij fdruk cy yxsxk\ 1.13 eku yhft, vH;kl 1.12 esa xksys A rFkk B lkb”k esa loZle gSa rFkk blh lkb”k dk dksbZ rhljk vukosf'kr xksyk igys rks igys xksys osQ laioZQ] rRi'pkr nwljs xksys osQ laioZQ esa ykdj] var esa nksuksa ls gh gVk fy;k tkrk gSA vc A rFkk B osQ chp u;k çfrd"kZ.k cy fdruk gS\ 1.14 fp=k 1.33 esa fdlh ,dleku fLFkjoS|rq {ks=k eas rhu vkosf'kr d.kksa osQ iFkfpÉ (tracks) n'kkZ, x, gSaA rhuksa vkos'kksa osQ fpÉ fyf[k,A buesa ls fdl d.k dk vkos'k&lagfr vuqikr (q/m) vfèkdre gS\ fp=k 1.33 1.15 ,dleku fo|qr {ks=k E = 3 × 103 î N/C ij fopkj dhft,A (a) bl {ks=k dk 10 cm Hkqtk osQ oxZ osQ ml ik'oZ ls ftldk ry yz ry osQ lekarj gS] xqtjus okyk ÝyDl D;k gS\ (b) blh oxZ ls xqtjus okyk ÝyDl fdruk gS ;fn blosQ ry dk vfHkyac x-v{k ls 60° dk dks.k cukrk gS\ 1.16 vH;kl 1.15 osQ ,dleku fo|qr {ks=k dk 20 cm Hkqtk osQ fdlh ?ku ls (tks bl çdkj vfHkfoU;kflr gS fd mlosQ iQyd funsZ'kkad ryksa osQ lekarj gSa) fdruk usV ÝyDl xqtjsxk\ 1.17 fdlh dkys ckWDl osQ i`"B ij fo|qr {ks=k dh lkoèkkuhiwoZd yh xbZ eki ;g laosQr nsrh gS fd ckWDl osQ i`"B ls xqtjus okyk usV ÝyDl 8.0 × 103 N m2/C gSA (a) ckWDl osQ Hkhrj usV vkos'k fdruk gS\ (b) ;fn ckWDl osQ i`"B ls usV cfgeZq[kh ÝyDl 'kwU; gS rks D;k vki ;g fu"d"kZ fudkysaxs fd ckWDl osQ Hkhrj dksbZ vkos'k ugha gS\ D;ksa] vFkok D;ksa ugha\ 1.18 fp=k 1.34 esa n'kkZ, vuqlkj 10 cm Hkqtk osQ fdlh oxZ osQ osQaæ ls Bhd 5 cm Å¡pkbZ ij dksbZ +10 µC vkos'k j[kk gSA bl oxZ ls xqtjus okys oS|qr ÝyDl dk ifjek.k D;k gS\ (laosQr % oxZ dks 10 cm fdukjs osQ fdlh ?ku dk ,d iQyd ekfu,A) fp=k 1.34 47  HkkSfrdh 1.19 2.0 µC dk dksbZ ¯cnq vkos'k fdlh fdukjs ij 9.0 cm fdukjs okys fdlh ?kuh; xkmlh; i`"B osQ osQaæ ij fLFkr gSA i`"B ls xqtjus okyk usV ÝyDl D;k gS\ 1.20 fdlh ¯cnq vkos'k osQ dkj.k ml ¯cnq dks osQæa ekudj [khaps x, 10 cm f=kT;k osQ xksyh; xkmlh; i`"B ij oS|qr ÝyDl –1.0 × 103 N m2/CA (a) ;fn xkmlh; i`"B dh f=kT;k nks xquh dj nh tk, rks i`"B ls fdruk ÝyDl xqtjsxk\ (b) ¯cnq vkos'k dk eku D;k gS\ 1.21 10 cm f=kT;k osQ pkyd xksys ij vKkr ifj.kke dk vkos'k gSA ;fn xksys osQ osaQæ ls 20 cm nwjh ij fo|qr {ks=k 1.5 × 103 N/C f=kT;r% vareZq[kh (radially inward) gS rks xksys ij usV vkos'k fdruk gS\ 1.22 2.4 m O;kl osQ fdlh ,dleku vkosf'kr pkyd xksys dk i`"Bh; vkos'k ?kuRo 80.0 µC/m2 gSA (a) xksys ij vkos'k Kkr dhft,A (b) xksys osQ i`"B ls fuxZr oqQy oS|qr ÝyDl D;k gS\ 1.23 dksbZ vuar jSf[kd vkos'k 2 cm nwjh ij 9 × 104 N C−1 fo|qr {ks=k mRiUu djrk gSA jSf[kd vkos'k ?kuRo Kkr dhft,A 1.24 nks cM+h] iryh èkkrq dh IysVas ,d&nwljs osQ lekukarj ,oa fudV gSAa buosQ Hkhrjh iQydksa ij] IysVksa osQ i`"Bh; vkos'k ?kuRoksa osQ fpÉ foijhr gSa rFkk budk ifjek.k 17.0 ×10–22 C/ m2 gSA (a) igyh IysV osQ cká {ks=k esa] (b) nwljh IysV osQ cká {ks=k esa] rFkk (c) IysVksa osQ chp esa fo| rq {ks=k E dk ifjek.k ifjdfyr dhft,A vfrfjDr iz'u 1.25 fefydu rsy cw¡n ç;ksx esa 2.55 × 104 N C−1 osQ fu;r fo|qr {ks=k osQ çHkko esa 12 bysDVªkWu vkfèkD; dh dksbZ rsy cw¡n fLFkj j[kh tkrh gSA rsy dk ?kuRo 1.26 g cm−3 gSA cw¡n dh f=kT;k dk vkdyu dhft, (g = 9.81 m s–2; e = 1.60 × 10 –19 C)A 1.26 fp=k 1.35 esa n'kkZ, x, oØksa esa ls dkSu laHkkfor fLFkjoS|qr {ks=k js[kk,¡ fu:fir ugha djrs\ + ++ ++ ++ + + + + + + + + pkyd + + + + + ++ + + + + + + +++ + (a) 48  oS|qr vkos'k rFkk {ks=k fp=k 1.35 1.27 fno~QLFkku osQ fdlh {ks=k esa] fo|qr {ks=k lHkh txg z-fn'kk osQ vuqfn'k gSA ijarq fo|qr {ks=k dk ifjek.k fu;r ugha gS] blesa ,dleku :i ls z-fn'kk osQ vuqfn'k 105 N C–1 çfr ehVj dh nj ls o`f¼ gksrh gSA og fudk; ftldk ½.kkRed z-fn'kk esa oqQy f}èkqzo vk?kw.kZ 10−7 C m osQ cjkcj gS] fdruk cy rFkk cy vk?kw.kZ vuqHko djrk gS\ 1.28 (a) fdlh pkyd A ftlesa fp=k 1.36 (a) esa n'kkZ, vuqlkj dksbZ dksVj@xqgk (Cavity) gS] dks Q vkos'k fn;k x;k gSA ;g n'kkZb, fd leLr vkos'k pkyd osQ cká i`"B ij izrhr gksuk pkfg,A (b) dksbZ vU; pkyd B ftl ij vkos'k q gS] dks dksVj@xqgk (Cavity) esa bl çdkj èk¡lk fn;k tkrk gS fd pkyd B pkyd A ls fo|qrjksèkh jgsA ;g n'kkZb, fd pkyd A osQ cká i`"B ij oqQy vkos'k Q + q gS ¹fp=k 1.36 (b)ºA (c) fdlh lqxkz gh midj.k dks mlosQ i;kZoj.k osQ çcy fLFkjoS|r q {ks=kksa ls ifjjf{kr fd;k tkuk gSA laHkkfor mik; fyf[k,A fp=k 1.36 1.29 fdlh [kks[kys vkosf'kr pkyd esa mlosQ i`"B ij dksbZ fNæ cuk;k x;k gSA ;g n'kkZb, fd fNæ esa fo|qr {ks=k (σ/2ε 0) n̂ gS] tgk¡ n̂ vfHkyacor fn'kk esa cfgeZq[kh ,dakd lfn'k gS rFkk σ fNæ osQ fudV i`"Bh; vkos'k ?kuRo gSA 1.30 xkml fu;e dk mi;ksx fd, fcuk fdlh ,dleku jSf[kd vkos'k ?kuRo λ osQ yacs irys rkj osQ dkj.k fo|qr {ks=k osQ fy, lw=k çkIr dhft, ¹laoQs r % lhèks gh owQykWe fu;e dk mi;ksx djds vko';d lekdyu dk eku fudkfy,ºA 1.31 vc ,slk fo'okl fd;k tkrk gS fd Lo;a çksVkWu ,oa U;wVªkWu (tks lkekU; æO; osQ ukfHkdksa dk fuekZ.k djrs gSa) vkSj vfèkd ewy bdkb;ksa ftUgsa DokoZQ dgrs gSa] osQ cus gSaA çR;sd çksVkWu rFkk U;wVªkWu rhu Dokdks± ls feydj curk gSA nks çdkj osQ DokoZQ gksrs gSa % ^vi* DokoZQ (u }kjk fufnZ"V) ftu ij +(2/3) e vkos'k rFkk ^Mkmu* DokoZQ (d }kjk fufnZ"V) ftu ij (–1/3) e vkos'k gksrk gS] bysDVªkWu ls feydj lkekU; æO; cukrs gSaA (oqQN vU; çdkj osQ DokoZQ Hkh ik, x, gSa tks fHkUu vlkekU; çdkj dk æO; cukrs gSaA) çksVkWu rFkk U;wVªkWu osQ laHkkfor DokoZQ la?kVu lq>kb,A 49  HkkSfrdh 1.32 (a) fdlh ;kn`fPNd fLFkjoS|qr {ks=k foU;kl ij fopkj dhft,A bl foU;kl dh fdlh 'kwU;&fo{ksi fLFkfr (null-point, vFkkZr~ tgk¡ E = 0) ij dksbZ NksVk ijh{k.k vkos'k j[kk x;k gSA ;g n'kkZb, fd ijh{k.k vkos'k dk larqyu vko';d :i ls vLFkk;h gSA (b) bl ifj.kke dk leku ifjek.k rFkk fpÉksa osQ nks vkos'kksa (tks ,d&nwljs ls fdlh nwjh ij j[ks gSa) osQ ljy foU;kl osQ fy, lR;kiu dhft,A 1.33 izkjaHk esa x-v{k osQ vuqfn'k vx pky ls xfr djrh gqbZ nks vkosf'kr IysVksa osQ eè; {ks=k esa m nzO;eku rFkk –q vkos'k dk ,d d.k izos'k djrk gS (fp=k 1-33 esa d.k 1 osQ leku)A IysVksa dh yackbZ L gSA bu nksuksa IysVksa osQ chp ,dleku fo|qr {ks=k E cuk, j[kk tkrk gSA n'kkZb, fd IysV osQ vafre fdukjs ij d.k dk ÅèokZ/j fo{ksi qEL2/(2m vx2) gSA (d{kk 11 dh ikB~;iqLrd osQ vuqHkkx 4.10 esa o£.kr xq#Roh; {ks=k esa iz{ksI; dh xfr osQ lkFk bl d.k dh xfr dh rqyuk dhft,)A 1.34 vH;kl 1.33 esa o£.kr d.k dh bysDVªkWu osQ :i esa dYiuk dhft, ftldks vx = 2.0 × 106 m s–1 osQ lkFk iz{ksfir fd;k x;k gSA ;fn 0.5 cm dh nwjh ij j[kh IysVksa osQ chp fo|qr {ks=k E dk eku 9.1 × 102 N/C gks rks Åijh Iys V ij bys D Vª k W u dgk¡ Vdjk,xk\ (|e|=1.6 × 10–19 C, me = 9.1 × 10–31 kg.) 50