vH;klksa osQ mÙkj vè;k; 9 9.1 v = –54 cm A izfr¯cc okLrfod] myVk rFkk vko£/r gSA izfr¯cc dk lkb”k 5.0 cm gSA tc u → f, v → ∞; u < f osQ fy, izfr¯cc vkHkklh cusxkA 9.2 v = 6.7 cmA vko/Zu = 5/9, vFkkZr izfr¯cc dk lkb”k 2.5 cm gSA tSls gh u → ∞; v → f (ijarq iQksdl ls vkxs dHkh ugha c<+rk) tcfd m → 0 9.3 1.33; 1.7 cm 9.4 n = 1.51; n = 1.32; n = 1.144; ftlls sin r = 0.6181 vFkkZr r ~ 38° izkIr ga wagwgksrk gSA 9.5 r = 0.8 × tan ic rFkk sin i = 1/1.33 ≅ 0.75 , tgk¡ r lcls cM+s o`Ùk dh f=kT;k ehVj esac gS rFkk ic ikuh&ok;q varjki`"B osQ fy, Økafrd dks.k gSA {ks=kiQy = 2.6 m2 9.6 n ≅ 1.53 rFkk ty esa fizTe osQ fy, Dm ≅ 10o 9.7 R = 22 cm 9.8 ;gk¡ ¯cc vkHkklh rFkk izfr¯cc okLrfod gSA u = +12 cm (¯cc nkfguh vksj gS_ vkHkklh) (a) f = +20 cm A izfr¯cc okLrfod gS rFkk ysal ls 7.5 cm nwj nkfguh vksj gSA (b) f = –16 cm A izfr¯cc okLrfod gS rFkk ysal ls 48 cm nwj nkfguh vksj gSA 9.9 v = 8.4 cm A izfr¯cc lh/k rFkk vkHkklh gSA ;g lkb”k esa NksVk gS] lkb”k = 1.8 cmA tSls u → ∞, v → f (ysfdu f ls vkxs ugha tkrk tcfd m → 0)A è;ku nhft,] tc oLrq vory ysal(f = 21 cm) osQ iQksdl ij j[kh gksrh gS] rc mldk izfr¯cc ysal ls 10.5 cm nwj curk gS (vuar ij ugha curk tSlk fd xyrh ls dksbZ lksp ldrk gS)A 9.10 60 cm iQksdl nwjh dk vilkjh ysalA 9.11 (a) v = –25 cm rFkk f = 6.25 cm ls u = –5 cm; v = (15 – 5) cm =eee0 10 cm izkIr gksrk gS] fO = uO = – 2.5 cm; vko/Zu {kerk = 20 (b) u0 = – 2.59 cm; vko/Zu {kerk = 13.5 9.12 25 cm nwjh ij izfr¯cc cuus osQ fy, usf=kdk dk dks.kh; vko/Zu 25 25 = +1 = 11 ; | ue |= cm = 2 27 cm . ; v = 7.2 cm11 02.5 i`Fkdu nwjh = 9.47 cm; vko/Zu {kerk = 88 9.13 24; 150 cm 9.14 (a) dks.kh; vko/Zu = 1500 (b) izfr¯cc dk O;kl = 13.7 cm 9.15 okafNr ifj.kke Kkr djus osQ fy, niZ.k osQ lehdj.k rFkk niZ.k dh lhek dk iz;ksx dhft,A (a) f < 0 (vory niZ.k); u < 0 (¯cc ckb± vksj) (b) f > 0 osQ fy,_ u < 0 (c) f > 0 (mÙky niZ.k) rFkk u < 0 (d) f < 0 (vory niZ.k)_ f < u < 0 9.16 fiu 5.0 cm Åij mBh gqbZ izrhr gksrh gSA ;g Li"V izdk'k fdj.k vkjs[k }kjk ns[kk tk ldrk gS fd mÙkj dk¡p osQ xqVosQ dh fLFkfr ij fuHkZj ugha djrk (NksVs vkiru dks.kksa osQ fy,)A 9.17 (a) sin i ′ c = 1.44/1.68 ftlls i′ c = 59° izkIr gksrk gSA iw.kZ vkarfjd ijkorZu i > 59° vFkok tc r < r max = 31° ij gksrk gSA vc] , ftlls i max ~ 60° izkIr gksrk gSA bl izdkj dks.k osQ ifjlj 0 < i < 60° dh lHkh vkifrr fdj.kksa dk ikbi esa iw.kZ vkarfjd ijkorZu gksxk (;fn ikbi dh yackbZ ifjfer gS] tks fd O;ogkj esa gksrh gS] rc i ij fuEu lhek ikbi osQ O;kl rFkk mldh yackbZ osQ vuqikr }kjk fu/kZfjr gksxhA) (b) ;fn dksbZ cká vkoj.k ugha gS] tks i′ c = sin–1(1/1.68) = 36.5°A vc] i = 90° osQ fy, r = 36.5° rFkk i ′ = 53.5° gksaxs] tks i′ c ls vf/d gSA bl izdkj ¹ifjlj esa lHkh vkifrr fdj.ksa (53.5° < i < 90°)º iw.kZ vkarfjd ijko£rr gksaxhA 9.18 (a) fdlh lery vFkok mÙky niZ.k osQ ^ihNs* fdlh ¯cnq ij vfHklfjr fdj.ksa niZ.k osQ lkeus ijns ij fdlh ¯cnq ij ijko£rr gks tkrh gSaA nwljs 'kCnksa esa] dksbZ lery niZ.k vFkok mÙky niZ.k vkHkklh ¯cc osQ fy, okLrfod izfr¯cc mRiUu dj ldrk gSA dksbZ mfpr izdk'k fdj.k vkjs[k [khapdj Lo;a dks larq"V dhft,A (b) tc ijko£rr vFkok vio£rr fdj.ksa vilkjh gksrh gSa rks izfr¯cc vkHkklh gksrk gSA vilkjh fdj.kksa dks mfpr vfHklkjh ysal dh lgk;rk ls ijns ij vfHklfjr fd;k tk ldrk gSA us=k dk vkHkklh ysal Bhd ;gh djrk gSA ;gk¡ vkHkklh izfr¯cc ysal osQ fy, ¯cc dh Hkk¡fr dk;Z djrk gS vkSj okLrfod izfr¯cc curk gSA è;ku nhft,] ;gk¡ vkHkklh izfr¯cc dh fLFkfr ij ijns dks vofLFkr ugha fd;k tkrk gSA ;gk¡ dksbZ viokn ugha gSA (c) vf/d yackA (d) yxHkx vfHkyacr% ns[kus dh rqyuk esa frjNs ns[kus osQ fy, vkHkklh xgjkbZ de gks tkrh gSA izs{kd dh fofHkUu fLFkfr;ksa osQ fy, izdk'k fdj.k vkjs[k [khapdj bl rF; dks Lo;a Lohdkj dhft,A (e) ghjs dk viorZukad yxHkx 2.42 gksrk gS tks lkekU; dk¡p osQ viorZukad (yxHkx 1.5) ls dkiQh vf/d gksrk gSA ghjs dk Økafrd dks.k yxHkx24° gS tks dk¡p osQ Økafrd dks.k dh vis{kk dkiQh de gSA dksbZ ghjs dks rjk'kus okyk n{k O;fDr vkiru dks.k (ghjs osQ Hkhrj) osQ cM+s ifjlj 24° ls90° dk ykHk ;g lqfuf'pr djus esa mBk ysrk gS fd ghjs ls ckgj fudyus ls iwoZ izdk'k dbZ iQydksa ls iw.kZ ijko£rr gksµbl izdkj ls fd ghjs dk pednkj izHkko mRiUu gksA 9.19 ijns rFkk oLrq osQ chp fuf'pr nwjh s osQ fy,] ysal lehdj.k ml fLFkfr esa u rFkk v osQ fy, okLrfod gy çnku ugha djrh] tc f dk eku s/4 ls vfèkd gksrk gSA vr% f max = 0.75 m 537 9.20 9.21 9.22 9.23 21.4 cm (a) (i) eku yhft, fd dksbZ lekarj çdk'k&iqat ckb± vksj ls igys mÙky ysal ij vkifrr gksrk gSA rc f1 = 30 cm, u1 = − ∞ ls izkIr gksrk gS v1 = + 30 cmA ;g çfrfcac nwljs ysal osQ fy, vkHkklh ¯cc cu tkrk gSA f2 = – 20 cm, u2 = + (30 – 8) cm = + 22 cm, ftlls v2 = – 220 cm çkIr gksrk gSA lekarj vkifrr fdj.k&iqat nks ysalksa osQ fudk; osQ osaQæ ls 216 cm nwj fdlh ¯cnq ls vilkfjr gksrk çrhr gksrk gSA (ii) eku yhft, fd dksbZ lekarj çdk'k&iqat ckb± vksj ls igys vory ysal ij vkifrr gksrk gSA rc f1 = –20 cm, u1 = – ∞ ls izkIr gksrk gS v1 = – 20 cmA ;g çfrfcac nwljs ysal osQ fy, okLrfod ¯cc cu tkrk gSA f2 = + 30 cm, u2 = – (20 + 8) cm = – 28 cm, ls v2 = – 420 cm izkIr gksrk gSA lekarj çdk'k&iqat nks ysalksa osQ ra=k osQ eè; ¯cnq dh ckb± vksj ls 416 cm nwj fLFkr ¯cnq ls vilfjr gksrk çrhr gksrk gSA Li"V gS fd mÙkj bl ij fuHkZj djrk gS fd ysal ra=k osQ fdl vksj lekarj izdk'k&iqat vkifrr gksrk gSA lkFk gh] gekjs ikl dksbZ ,slh ljy ysal lehdj.k ugha gS tks lHkh u (rFkk v) osQ ekuksa osQ fy,] fudk; osQ fuf'pr fu;rkad osQ inksa esa lR; gksA (fudk; osQ fLFkjkad f1 rFkk f2 rFkk nksuksa ysalksa osQ chp i`Fkdu nwjh }kjk fuèkkZfjr gksrs gSaA) çHkkoh iQksdl nwjh dh èkkj.kk] blfy, bl ra=k osQ fy, vFkZiw.kZ çrhr ugha gksrhA (b) u1 = – 40 cm, f1 = 30 cm ls v1 = 120 cm izkIr gksrk gSA igys (mÙky) ysal osQ dkj.k vkoèkZu dk ifjek.k = 120/40 = 3 u2 = + (120 – 8) cm = + 112 cm (¯cc vkHkklh) 112 × 20 f2 = – 20 cm ls v2 =− cm izkIr gksrk gSA92 vFkkZr nwljs (vory) ysal osQ dkj.k vkoèkZu dk ifjek.k = 20/92 vkoèkZu dk usV ifjek.k = 3 × (20/92) = 0.652 çfrfcac dk lkb”k = 0.652 × 1.5 cm = 0.98 cm ;fn fç”e esa viofrZr fdj.k nwljs iQyd ij Økafrd dks.k ic ij vkifrr gksrh gS rks] igys iQyd ij viorZu dks.k r dk eku (60° – ic ) gksrk gSA vc ic = sin–1 (1/1.524) ~ 41° vr% r = 19° rFkk sin i = 0.4962, rFkk i = sin–1 0.4965 ~ 30°A leku dk¡p osQ cus nks loZle fiz”eksa dks Li'kZ djrs gq, ;fn bl izdkj lek;ksftr fd;k tk, fd muosQ vk/kj ,d nwljs osQ foijhr gksa] rks os ,d dk¡p osQ LySc dh Hkk¡fr dk;Z djsaxs rFkk blls izdk'k iqat u rks fopfyr gksrk gS vkSj u gh fo{ksfir gksrk gS_ ijarq iqat dk ek=k lekarj foLFkkiu gksrk gSA (a) fcuk fo{ksi.k çdk'k&iqat dks fopfyr djus osQ fy,] fdlh inkFkZ tSls Økmu dk¡p dk ,d igyk fç”e yhft, rFkk fdlh mfpr viorZu dks.k dk ¯ÝyV dk¡p dk nwljk fç”e pqfu, ¹nwljs fç”e (¯ÝyV dk¡p) dk viorZu dks.k Økmu dk¡p osQ fç”e ls NksVk yhft, D;ksafd ¯ÝyV dk¡p vis{kkÑr vfèkd fo{ksi.k djrk gSºA bu nksuksa fç”eksa dks ,d&nwljs osQ lkis{k myVk j[kus ij ,d fç”e nwljs fç”e osQ fo{ksi.k dks fujLr dj nsrk gSA (b) fcuk fopyu osQ çdk'k osQ fo{ksi.k osQ fy, ¯ÝyV dk¡p osQ fç”e osQ viorZu dks.k esa o`f¼ dhft, (vfèkd vkSj vfèkd viorZu dks.k osQ ¯ÝyV dk¡p osQ fç”e ysdj ç;kl dhft,) rkfd nksuksa fç”eksa }kjk mRiUu fopyu ,d&nwljs osQ leku rFkk foijhr gksaA (¯ÝyV dk¡p dk viorZu Økmu dk¡p dh vis{kk vfèkd gksus osQ dkj.k vHkh Hkh ¯ÝyV dk¡p osQ fç”e dk viorZu dks.k Økmu dk¡p osQ fç”e dh rqyuk esa NksVk gksrk gS) D;ksafd blesa cgqr ls o.kks± osQ fy, lek;kstu djuk gksrk gS] vr% ;g okafNr mís'; osQ fy, ifj'kq¼ O;oLFkk ugha gksrhA 9.24 oLrqvksa dks vuar ij ns[kus osQ fy, us=k viuh U;wure vfHklfjr {kerk dk mi;ksx djrk gSA ;g {kerk (40 + 20) MkbvkWIVj = 60 MkbvkWIVj gSA blls n`f"ViVy rFkk dkW£u;k us=k ysal osQ chp dh nwjh r dh LFkwy èkkj.kk feyrh gS % (5/3) cmA fdlh ¯cc dks fudV ¯cnq (u = – 25 cm) ij iQksdflr dj n`f"ViVy (v = 5/3 cm) ij çfrfcac çkIr djus osQ fy, iQksdl nwjh −1⎡13⎤25 ⎢ + 25 5 ⎥ cm gksuh pkfg,A= 16 ;g 64 MkbvkWIVj vfHklfjr {kerk osQ rnuq:i gSA rc us=k ysal dh {kerk (64 – 20) MkbvkWIVj = 24 MkbvkWIVj gSA us=k ysal dh leatu dk ifjlj yxHkx 20 ls 24 MkbvkWIVj gksrk gSA 9.25 ughaA fdlh O;fDr osQ us=k ysal dh leatu dh ;ksX;rk ({kerk) lkekU; gksrs gq, Hkh mlesa fudV n`f"V vFkok nh?kZ n`f"V nks"k gks ldrk gSA fudV n`f"V nks"k us=k xksyd osQ lkeus rFkk ihNs ls cgqr NksVk gksus ij mRiUu gksrk gSA O;ogkj esa blosQ lkFk&lkFk us=k ysal Hkh viuh leatu {kerk [kks nsrk gSA tc us=k xksyd dh viuh yackbZ lkekU; gksrh gS ijarq us=k&ysal viuh leatu {kerk dks vkaf'kd :i esa [kks nsrk gS (tSlk vk;q esa o`f¼ gksus ij fdlh Hkh lkekU; us=k esa gks ldrk gS) rc bl n`f"V ^nks"k* dks tjk nwjnf'kZrk dgrs gSa rFkk bldk fujkdj.k nh?kZ n`f"V nks"k dh gh Hkk¡fr fd;k tkrk gSA 9.26 O;fDr dk nwj ¯cnq 100 cm gS] tcfd mldk fudV ¯cnq lkekU; (yxHkx 25 cm) gks ldrk FkkA p'ek yxkus ij vuar ij j[kh oLrq dk vkHkklh çfrfcac 100 cm nwj curk gSA blls ikl dh oLrqvksa] vFkkZr~ tks fd (ftuosQ p'es osQ }kjk çfrfcac) 100 cm vkSj 25 cm osQ chp gSa] rks O;fDr vius us=k ysasl dh leatu {kerk dh ;ksX;rk dk mi;ksx djrk gSA çk;% ;g ;ksX;rk dk vfèkd vk;q gksus ij vkaf'kd ßkl gks tkrk gS (tjk nwjnf'kZrk)A ,sls O;fDr dk fudV ¯cnq 50 cm nwj pyk tkrk gSA oLrqvksa dks 25 cm nwjh ij ns[kus osQ fy, O;fDr dks +2 MkbvkWIVj {kerk osQ p'es dh vko';drk gSA 9.27 v¯cnqdrk uked n`f"V nks"k viorhZ ra=k (dkWfuZ;k $ us=k ysal) gksus ij gksrk gSA ¹us=k izk;% xksyh; gksrk gS] vFkkZr bldh fofHkUu ryksa esa oØrk leku gksrh gS] ijarq v¯cnqdrk dh fLFkfr esa dkWfuZ;k xksyh; ugha gksrhºA orZeku fLFkfr esa] ÅèokZèkj ry dh oØrk i;kZIr gS] vr% ÅèokZèkj /kfj;ksa dk Li"V çfr¯cc jsfVuk ij cu ldrk gSA ijarq {kSfrt ry esa oØrk i;kZIr ugha gS] vr% {kSfrt /kfj;k¡ èkq¡èkyh çrhr gksrh gSaA bl nks"k dh la'kqf¼ ÅèokZèkj osQ vuqfn'k v{k oØrk osQ fl¯yMjh ysal }kjk dh tk ldrh gSA Li"V gS fd ÅèokZèkj ry dh lekarj fdj.ksa dksbZ vfrfjDr viofrZr ugha gksaxh] ijarq tks {kSfrt ry esa gSa] ;fn flfyaMjh i`"B dh oØrk dk p;u mfpr çdkj ls fd;k x;k gks rks flfyaMjh ysal osQ ofØr i`"B ls os okaNuh; vfrfjDr vfHklfjr gks ldrh gSaA 9.28 9.29 9.30 9.31 9.32 1 (a) fudVre nwjh = 4 cm ≈ 42 . cm rFkk nwjre nwjh = 5 cm 6 (b) vfèkdre dks.kh; vkoèkZu = [25/(25/6)] = 6; U;wure dks.kh; vkoèkZu = [25/5] = 5 1 1 1 (a) += ] vFkkZr~ v = – 90 cm v 9 10 vkoèkZu dk ifjek.k = 90/9 = 10 vkHkklh çfrfcac esa çR;sd oxZ dk {ks=kiQy = 10 × 10 × 1 mm2 = 100 mm2 = 1 cm2 (b) vkoèkZu {kerk = 25/9 = 2.8 (c) ugha] fdlh ysal }kjk vkoèkZu rFkk fdlh çdkf'kd ;a=k dh dks.kh; vkoèkZu ¹vFkok vkoèkZu {kerkº nks fHkUu vfHkèkkj.kk,¡ gSaA dks.kh; vko/Zu oLrq osQ dks.kh; lkb”k (tks fd izfr¯cc osQ vkof/Zr gksus ij izfr¯cc osQ dks.kh; lkb”k osQ cjkcj gksrk gSA) rFkk ml fLFkfr esa oLrq osQ dks.kh; lkb”k (tcfd mls fudV ¯cnq 25 cm ij j[kk tkrk gS)] dk vuqikr gksrk gSA bl izdkj] vkoèkZu dk ifjek.k A (v/u)A gksrk gS rFkk vkoèkZu {kerk (25/|u|) gksrh gSA osQoy rc tc çfrfcac fudV ¯cnq ij |v| = 25 cm ij gS rks osQoy rHkh nksuksa jkf'k;k¡ leku gksrh gSaA (a) çfrfcac osQ fudV ¯cnq (25 cm) ij cuus ij vf/dre vko/Zu {kerk izkIr gksrh gSA vr% u = – 7.14 cm (b) vkoèkZu dk ifjek.k = (25/|u|) = 3.5 (c) vkoèkZu {kerk = 3.5 gk¡] vkoèkZu {kerk (tc çfrfcac 25 cm ij curk gS) vkoèkZu osQ ifjek.k osQ leku gksrh gSA vkoèkZu (625 / ) = 2.5.1v = + 2.5 u; vr% 1 11 + −= 2.5uu 10 vFkkZr~ u = – 6 cm |v| = 15 cm vkHkklh çfrfcac lkekU; fudV ¯cnq (25 cm) ls Hkh ikl curk gS rFkk bls us=k Li"V ugha ns[k ldrkA (a) ;fn çfrfcac dk fujis{k lkb”k oLrq osQ lkb”k ls cM+k Hkh gS] rks Hkh çfrfcac dk dks.kh; lkb”k oLrq osQ dks.kh; lkb”k osQ leku gksrk gSA dksbZ vkoèkZd ysal gekjh bl :i esa lgk;rk djrk gS % ;fn vkoèkZd ysal ugha gS rks oLrq 25 cm ls de nwjh ij ugha j[kh tk ldrh_ vkoèkZd ysal gksus ij ge oLrq dks vis{kkÑr cgqr fudV j[k ldrs gSaA oLrq fudV gks rks mldk dks.kh; lkb”k 25 cm nwj j[kus dh rqyuk esa dgha vfèkd gksrk gSA gekjs dks.kh; vkoèkZu ikus ;k miyCèk djus dk ;gh vFkZ gSA (b) gk¡] ;g FkksM+k de gksrk gS] D;ksafd us=k ij varfjr dks.k ysal ij varfjr dks.k ls FkksM+k NksVk gksrk gSA ;fn çfrfcac cgqr nwj gks rks ;g çHkko ux.; gksrk gSA ¹uksV % tc us=k dks ysal ls i`Fkd~ j[krs gSa] rks çFke oLrq }kjk us=k ij varfjr dks.k rFkk blosQ çfrfcac }kjk us=k ij varfjr dks.k leku ugha gksrsA] (c) çFke] vR;ar NksVs iQksdl nwjh osQ ysalksa dh f?klkbZ vklku ugha gSA blls vfèkd egÙoiw.kZ ckr gS fd ;fn vki iQksdl nwjh de djrs gSa rks blls foiFku (xksyh; rFkk o.kZ) c<+ tkrk gSA vr% O;ogkj esa] vki fdlh ljy mÙky ysal ls 3 ;k vfèkd dh vkoèkZu {kerk ugha çkIr dj ldrs gSaA rFkkfi] fdlh foiFku la'kksfèkr ysal ç.kkyh osQ mi;ksx ls bl lhek dks 10 ;k blosQ lfUudV dkjd ls c<+k ldrs gSaA (d) fdlh usf=kdk dk dks.kh; vkoèkZu [(25/f e) + 1] (f e cm esa) gksrk gS ftlosQ eku esa v 1 = f e osQ ?kVus ij o`f¼ gksrh gSA iqu% vfHkn`';d dk vkoèkZu 0 ls |u | (| u |/ f )−10 00 çkIr gksrk gS tks vfèkd gksrk gS ;fn |u0|, f0 ls oqQN vfèkd gksA lw{en'khZ dk mi;ksx vfr fudV dh oLrqvksa dks ns[kus osQ fy, fd;k tkrk gSA vr% |u0| de gksrk gS vkSj rnuqlkj f0 HkhA (e) usf=kdk osQ vfHkn`';d osQ çfrfcac dks ^fuxZe }kjd* dgrs gSaA oLrq ls vkus okyh lHkh fdj.ksa vfHkn`';d ls viorZu osQ i'pkr fuxZe }kjd ls xqtjrh gSaA vr% gekjs us=k ls ns[kus osQ fy, ;g ,d vkn'kZ fLFkfr gSA ;fn ge vius us=k dks usf=kdk osQ cgqr gh fudV j[ksa rks usf=kdk cgqr vfèkd çdk'k dk vfèkxzg.k ugha dj ik,xh rFkk n`f"V&{ks=k Hkh ?kV tk,xkA ;fn ge vius us=k dks fuxZe&}kjd ij j[ksa rFkk gekjs us=k dh iqryh dk {ks=kiQy fuxZe&}kjd osQ {ks=kiQy ls vfèkd ;k leku gks rks gekjs us=k vfHkn`';d ls viofrZr lHkh fdj.kksa dks vfHkx`fgr dj ysaxsA fuxZe&}kjd dk lVhd LFkku lkekU;r% vfHkn`';d ,oa usf=kdk osQ varjky ij fuHkZj djrk gSA tc ge fdlh lw{en'khZ ls] blosQ ,d fljs ij vius us=k dks yxkdj ns[krs gSa rks us=k ,oa usf=kdk osQ eè; vkn'kZ nwjh ;a=k osQ fM”kkbu esa varfuZfgr gksrh gSA 9.33 eku yhft, fd lw{en'khZ lkekU; mi;ksx esa gS vFkkZr çfrfcac 25 cm ij gSA usf=kdk dk dks.kh; vkoèkZu 25 = +1 = 6 5 vfHkn`';d dk vkoèkZu 30 = = 5] vr% 6 11 1 −= 50 u0 125 u . ftlls u0 = – 1.5 cm.; v0 = 7.5 cm ; |ue| = (25/6) cm = 4.17 cm izkIr gksrk gSA vfHkn`';d ,oa usf=kdk osQ chp nwjh (7.5 + 4.17) cm = 11.67 cm gksuh pkfg,A visf{kr vkoèkZu çkIr djus osQ fy, oLrq dks vfHkn`';d ls 1.5 cm nwj j[kuk gksxkA 9.34 (a) m = (f0/f e) = 28 1⎢⎣⎡⎤f f0 f e 0 25 (b) = 33.6+⎥⎦m = 9.35 (a) f0 + f e = 145 cm (b) ehukj }kjk varfjr dks.k = (100/3000) = (1/30) rad; vfHkn`';d }kjk cuk, izfr¯cc ls varfjr dks.k = h/f0 ; f0 = 140 cmA nksuksa dks.kksa osQ ekuksa dh rqyuk djus ij h = 4.7 cm izkIr gksrk gSA(c) usf=kdk dk vko/Zu = 6 vafre izfr¯cc dh Å¡pkbZ = 28 cm 9.36 cM+s niZ.k (vory) }kjk cuk;k x;k izfr¯cc NksVs niZ.k (mÙky) osQ fy, vkHkklh ¯cc dk dk;Z djrk gSA vuar ij j[ks ¯cc ls vkus okyh lekarj fdj.ksa] cM+s niZ.k ls 110 mm nwj iQksdflr gksaxhA NksVs niZ.k osQ fy, vkHkklh ¯cc dh nwjh = (110 –20) = 90 mm gksxhA NksVs niZ.k dh iQksdl nwjh 70 mm gSA niZ.k lw=k dk mi;ksx djus ij ge ns[ksaxs fd izfr¯cc NksVs niZ.k ls 315 mm nwj curk gSA 9.37 ijko£rr fdj.ksa niZ.k osQ ?kw.kZu dks.k ls nksxqus dks.k ij fo{ksfir gksrh gSaA vr% d/1.5 = tan 7°; d = 18.4 cm 9.38 n = 1.33 vè;k; 10 10.1 (a) ijko£rr izdk'k % (rjaxnS?;Z] vko`fÙk] pky vkifrr izdk'k osQ leku gSa) λ = 589 nm, ν = 5.09 × 1014 Hz, c = 3.00 × 108 m s–1 (b) vio£rr izdk'k % (vko`fÙk] vkifrr vko`fÙk osQ leku gS) ν = 5.09 × 1014Hz v = (c/n) = 2.26 × 108 m s–1, λ = (v/ν) = 444 nm 10.2 (a) xksyh; (b) lery (c) lery (cM+s xksys dh lrg dk ,d NksVk {ks=k yxHkx leryh; gksrk gS) 10.3 (a) 2.0 × 108 m s–1 (b) gk¡] D;ksafd viorZukad vkSj blfy, ekè;e esa izdk'k dh pky rjaxnS?;Z ij fuHkZj djrh gS ¹tc dksbZ fof'k"V rjaxnS?;Z ;k izdk'k dk jax u fn;k x;k gks rks ge fn, x, viorZukad dk eku ihys izdk'k osQ fy, ys ldrs gSaºA vc ge tkurs gSa fd cSaxuh izdk'k dk fopyu dk¡p osQ fiz”e esa yky izdk'k ls vf/d gksrk gSA vFkkZr nv > nr blfy,] 'osr izdk'k dk cSaxuh vo;o] yky vo;o ls /heh xfr ls xeu djrk gSA –2 –312 10 ×. ×× 0 28 10 . 10.4 λ= m = 600 nm × .4 14 10.5 K/4 10.6 (a) 1.17 mm (b) 1.56 mm 10.7 0.15° 10.8 tan–1(1.5) ~ 56.3o 542 10.9 5000 Å, 6 × 1014 Hz; 45° 10.10 40 m 10.11 lw=k vλ′ – λ = λ dk mi;ksx djus ls c cvFkkZr v = (λ′ – λ)λ 3×10 8 ×15 = 6563 = 6.86 × 105 m s–1 10.12 U;wVu osQ df.kdk fl¼kar osQ vuqlkj] viorZu esa] fojy ekè;e ls l?ku ekè;e esa izos'k djrs le; vkifrr d.k lrg osQ yacor vkd"kZ.k cy dk vuqHko djrk gSA bldh ifj.kfr osx osQ vfHkyac ?kVd dh o`f¼ esa gksxhA ysfdu i`"B osQ vuqfn'k ?kVd fu;r jgrk gSA bldk rkRi;Z v sin i c sin i = v sin r ;k == n; D;ksafd n > 1, v > c gSA c sin r ;g vo/kj.kk izk;ksfxd ifj.kkeksa osQ fo#¼ gS (v < c)A izdk'k dk rjax fl¼kar iz;ksx laxr gSA 10.13 ¯cnq ¯cc dks osaQnz ysdj niZ.k dks Li'kZ djrs gq, ,d o`Ùk [khafp,A ;g xksyh; rjaxkxz dk ¯cc ls niZ.k ij igq¡pus okyk leryh; Hkkx gSA vc niZ.k dh mifLFkfr ,oa vuqifLFkfr esa t le; osQ ckn mlh rjaxkxz dh bUgha fLFkfr;ksa dks vkjsf[kr dhft,A vki niZ.k osQ nksuksa vksj fLFkr nks ,d tSls pki ik,¡xsA ljy T;kfefr osQ mi;ksx ls] ijko£rr rjaxkxz dk osaQnz (¯cc dk izfr¯cc) niZ.k ls ¯cc dh cjkcj nwjh ij fn[kkbZ nsxkA 10.14 (a) fuokZr esa izdk'k dh pky ,d lkoZf=kd fu;rkad gS tks lwphc¼ dkjdksa esa ls fdlh ij Hkh fuHkZj ugha gSA fo'ks"kr% ;g ,d vk'p;Ztud rF; gS fd ;g lzksr rFkk izs{kd dh lkis{k xfr ij Hkh fuHkZj ugha djrk gSA ;g rF; vkbalVkbu osQ vkisf{kdrk osQ fof'k"V fl¼kar dk ewy vfHkx`ghr gSA (b) izdk'k dh pky dh ekè;e ij fuHkZjrk (i) lzksr dh izo`Qfr ij fuHkZj ugha gS (izdk'k dh pky dk fu/kZj.k ekè;e osQ lapj.k xq.kksa ls gSA ;g rF; vU; rjaxksa osQ fy, Hkh lR; gS] tSls èofu&rjaxksa ,oa ty&rjaxksa vkfn osQ fy,)A (ii) lenSf'kd ekè;e osQ fy, lapj.k fn'kk ij fuHkZj ugha djrk gSA (iii) lzksr rFkk ekè;e dh lkis{k xfr ij fuHkZj ugha djrk ysfdu izs{kd rFkk ekè;e dh lkis{k xfr ij fuHkZj djrk gSA (iv) rjaxnS?;Z ij fuHkZj djrk gSA (v) rhozrk ij fuHkZj ugha djrk (;|fi vf/d rhoz fdj.k&iaqt osQ fy, ;g fLFkfr vfèkd tfVy gS rFkk ;gk¡ gekjs fy, egRoiw.kZ ugha gS)A 10.15 èofu&rjaxksa osQ lapj.k osQ fy, ekè;e vko';d gSA ;|fi (i) rFkk (ii) fLFkfr esa laxr leku lkis{k xfr (lzksr rFkk isz{kd osQ eè;) HkkSfrd :i ls le:ih ugha gS] D;ksafd ekè;e osQ lkis{k izs{kd dh xfr bu nksuksa fLFkfr;ksa esa fHkUu gSA vr%] (i) rFkk (ii) fLFkfr;ksa esa ge èofu osQ fy, MkWIyj osQ lw=kksa dh lekurk dh vis{kk ugha dj ldrsA fuokZr esa izdk'k&rjaxksa osQ fy,] Li"Vr;k (i) rFkk (ii) fLFkfr osQ chp dksbZ Hksn ugha gSA ;gk¡ ek=k lzksr rFkk isz{kd dh lkis{k xfr;k¡ gh vFkZ j[krh gSa rFkk vkisf{kdh; MkWIyj dk lw=k (i) rFkk (ii) fLFkfr osQ fy, leku gSA ekè;e esa izdk'k lapj.k osQ fy, iqu% èofu&rjaxksa osQ leku] nksuksa fLFkfr;k¡ leku ugha gSa rFkk (i) rFkk (ii) fLFkfr;ksa osQ fy, gesa MkWIyj osQ lw=k osQ fHkUu gksus dh vis{kk j[kuh pkfg,A 10.16 3.4 × 10–4 m 10.17 (a) vkdkj ~ λ/d lw=k osQ vuqlkj] vkdkj vk/k jg tkrk gSA rhozrk pkj xquh c<+ tkrh gSA (b) f}&f>jh lek;kstu esa O;frdj.k ¯izQtksa dh rhozrk izR;sd f>jh osQ foorZu iSVuZ }kjk ekMqfyr (modulated) gksrh gSA (c) o`Ùkh; vojks/ osQ fdukjksa ls foo£rr rjaxsa Nk;k osQ osaQnz ij laiks"kh O;frdj.k }kjk iznhIr ¯cnq mRiUu djrh gSaA (d) rjaxksa osQ cM+s dks.k ij foorZu vFkok eqM+us osQ fy, vojks/ksa@}kjdksa dk vkdkj] rjax dh rjaxnS?;Z osQ led{k gksuk pkfg,A ;fn vojks/@}kjd dk vkdkj rjaxnS?;Z dh rqyuk esa cgqr cM+k gS rks foorZu NksVs dks.k ls gksxkA ;gk¡ vkdkj oqQN ehVjksa dh dksfV dk gksrk gSA izdk'k dh rjaxnS?;Z yxHkx 5 × 10–7 m gS] tcfd èofu&rjaxksa_ tSls 1k Hz vko`fÙk okyh èofu dh rjaxnS?;Z yxHkx 0.3 m gSA bl izdkj èofu&rjaxsa foHkktd osQ pkjksa vksj eqM+ ldrh gSa tcfd izdk'k rjaxsa ugha eqM+ ldrhaA (e) U;k;laxrrk dk vk/kj (d) esa mYysf[kr gSA lk/kj.k izdkf'kd ;a=kksa esa iz;qDr }kjdksa dk vkdkj izdk'k dh rjaxnS?;Z ls cgqr cM+k gksrk gSA 10.18 12.5 cm 10.19 0.2 nm 10.20 (a) ,saVhuk }kjk izkIr lh/s laosQr rFkk xq”kjus okys ok;q;ku ls ijko£rr laosQrksa dk O;frdj.kA (b) vè;kjksi.k dk fl¼kar rjaxxfr dks fu;af=kr djus okyh vody (differential) lehdj.k osQ js[kh; pfj=k ls izfrikfnr gSA ;fn y1 vkSj y2 bl lehdj.k osQ gy gSa] rks y1 vkSj y2 dk js[kh; ;ksx Hkh mudk gy gksxkA tc vk;ke cM+s gksa (mnkgj.k osQ fy, mPp rhozrk dk ys”kj fdj.k&iqat) rFkk vjSf[kd izHkko egRoiw.kZ gks rks ;g fLFkfr vkSj Hkh tfVy gks tkrh gS] ftldk le>uk ;gk¡ vko';d ugha gSA 10.21 fdlh ,dy f>jh dks n NksVh f>fj;ksa esa ck¡fV, ftuesa izR;sd dh pkSM+kbZa ′ = a/n gSA dks.k θ = nλ/a = λ/a ′A izR;sd NksVh f>jh ls dks.k θ dh fn'kk esa rhozrk 'kwU; gSA budk la;kstu Hkh 'kwU; rhozrk iznku djrk gSA vè;k; 11 11.1 (a) 7.24 × 1018 Hz (b) 0.041 nm 11.2 (a) 0.34 eV = 0.54 × 10–19J (b) 0.34 V (c) 344 km/s 11.3 1.5 eV = 2.4 × 10–19 J 11.4 (a) 3.14 × 10–19J, 1.05 × 10–27 kg m/s (b) 3 × 1016 iQksVkWu/s (c) 0.63 m/s 11.5 4 × 1021 iQksVkWu/m2 s 11.6 6.59 × 10–34 J s 11.7 (a) 3.38 × 10–19 J = 2.11 eV (b) 3.0 × 1020 iQksVkWu/s 11.8 2.0 V 11.9 ugha] D;ksafd ν <ν o 11.10 4.73 × 1014 Hz 11.11 2.16 eV = 3.46 × 10–19J 11.12 (a) 4.04 × 10–24 kg m s–1 (b) 0.164 nm 11.13 (a) 5.92 × 10–24 kg m s–1 (b) 6.50 × 106 m s–1 (c) 0.112 nm 11.14 (a) 6.95 × 10–25 J = 4.34 µeV (b) 3.78 × 10–28 J = 0.236 neV 11.15 (a) 1.7 × 10–35 m (b) 1.1 × 10–32 m (c) 3.0 × 10–23 m 11.16 (a) 6.63 × 10–25 kg m/s (nksuksa osQ fy,) (b) 1.24 keV (c) 1.51 eV 11.17 (a) 6.686 × 10–21 J = 4.174 × 10–2 eV (b) 0.145 nm 11.18 λ = h/p = h/(hν/c) = c/ν 11.19 0.028 nm 11.20 (a) eV = (m v 2/2) dk mi;ksx dhft, vFkkZr, v = [(2eV/m)]1/2 ; v = 1.33 × 107 m s–1 (b) ;fn ge V = 107 V osQ fy, mlh lw=k dk iz;ksx djsa] rks v = 1.88 × 109 m s–1 vkrk gSA ;g Li"V :i ls xyr gS] D;ksafd dksbZ Hkh nzO; d.k izdk'k osQ osx (c = 3 × 108 m s–1) ls vfèkd osx ls ugha py ldrkA oLrqr% xfrt mQtkZ osQ fy, mijksDr lw=k (mv 2/2) osQoy (v/c) << 1 osQ fy, oS/ gSA cgqr vfèkd pky ij] tc (v/c) osQ yxHkx rqY; (;|fi ges'kk 1 ls de) gksrk gS] rks vkisf{kdh; izHkko&{ks=k osQ dkj.k fuEufyf[kr lw=k oSèk gksrs gSa % vkisf{kdh; laosx p = m v 2oqQy mQtkZ E = m c2 2xfrt mQtkZ K = m c – m octgk¡ vkisf{kdh; nzO;eku m fuEukuqlkj fn;k tkrk gS −12 /⎛2 v  ⎜⎜⎝1−⎟⎟m = m0 2 c⎠m 0 d.k dk fojke nzO;eku dgykrk gSA bu lacaèkksa ls izkIr gksrk gS % 2 22 4)1/2E = (pc + m0cè;ku nhft, fd vkisf{kdh; izHkko&{ks=k esa] tc v/c yxHkx 1 osQ cjkcj gksrk gS] rks 2oqQy mQtkZ E ≥ m 0c (fojke nzO;eku mQtkZ)A bysDVªkWu dh fojke nzO;eku mQtkZ yxHkx 0.51 MeV gksrh gSA blfy, 10 MeV dh xfrt mQtkZ] tks bysDVªkWu dh fojke nzO;eku mQtkZ ls cgqr vfèkd gS] vkisf{kdh; izHkko&{ks=k dks O;Dr djrh gSA vkisf{kdh; lw=kksa osQ iz;ksx ls v (10 MeV xfrt mQtkZ osQ fy,) = 0.999 c 11.21 (a) 22.7 cm (b) ughaA tSlk fd mQij Li"V fd;k x;k gS] 20 MeV dk ,d bysDVªkWu vkisf{kdh; xfr ls pysxkA ifj.kkeLo:i] v&vkisf{kdh; lw=k R = (m0v/eB ) oSèk ugha jgrkA vkisf{kdh; lw=k gS R = p/eB = mv/eB ;k R = mv /(eB 0 11.22 e V = (m v 2/2) rFkk R = (m v/e B) osQ iz;ksx ls (e/m) = (2V/R2 B 2 ); rFkk fn, x, vk¡dM+ksa osQ iz;ksx ls izkIr gksrk gS % (e/m) = 1.73 × 1011 C kg–1 11.23 (a) 27.6 keV (b) 30 kV dh dksfV dkA 11.24 λ = (hc/E) osQ iz;ksx ls] tgk¡ E = 5.1 × 1.602 × 10–10J λ = 2.43 × 10–16 m 11.25 (a) λ = 500 m osQ fy, E = (h c/ λ) = 3.98 × 10–28J izfr lsoaQM mRl£tr iQksVkWuksa dh la[;k –1= 104J s–1/3.98 × 10–28J ∼ 3 × 1031 sge ns[krs gSa fd jsfM;ksiQksVkWu dh mQtkZ cgqr de gS vkSj jsfM;ks iqat esa izfr lsoaQM mRlftZr iQksVkWuksa dh la[;k cgqr vfèkd gSA blfy, ;gk¡ mQtkZ osQ U;wure DokaVe (iQksVkWu) osQ vfLrRo dks misf{kr djus vkSj jsfM;ks rjax dh oqQy mQtkZ dks lrr ekuus ls ux.; =kqfV vkrh gSA (b) ν = 6 × 1014 Hz osQ fy, E ∼ 4 × 10–19J U;wure rhozrk osQ laxr iQksVkWuksa dk vfHkokg (ÝyDl) –2 –1= 10–10 W m–2/4×10–19J = 2.5 × 108 msvk¡[k dh iqryh esa izos'k djus okys iQksVkWuksa dh la[;k izfr lsoaQM = 2.5 × 108 × 0.4 × 10–4 s–1 = 104 s–1A ;|fi ;g iQksVkWuksa dh la[;k (a) dh rjg vR;fèkd ugha gS] fiQj Hkh gekjs fy, ;g dkiQh vfèkd gS] D;ksafd ge dHkh Hkh viuh vk¡[kksa ls iQksVkWuksa dks u rks vyx&vyx ns[k ldrs gSa] u gh fxu ldrs gSaA φ0 11.26 φ0 = h ν – e V0 = 6.7 × 10–19 J = 4.2 eV; ν0 = = 1.0 × 1015 Hz; ν = 4.7 h × 1014 Hz < ν0 osQ laxr λ = 6328Å gSA pkgs yslj osQ izdk'k dh rhozrk fdruh Hkh vfèkd D;ksa u gks] iQksVkslsy bl izdk'k osQ fy, vfØ;k'khy gh jgsxkA 11.27 nksuksa Ïksrksa osQ fy, eV0 = h ν – φ 0 dk mi;ksx dhft,A izFke Ïksr osQ fy, fn, x, vk¡dM+ksa ls] φ 0 = 1.40 eVA vr%] nwljs Ïksr osQ fy, V0 = 1.50 V A 11.28 V0 vkSj v esa vkjs[k [khafp,A vkjs[k dk r ugha gksrh gSA cfYd] mQtkZ vlrr ^DokaVk* osQ :i esa vkrh gS vkSj mQtkZ dk vo'kks"k.k èkhjs&èkhjs ugha gksrkA iQksVkWu ;k rks vo'kksf"kr ugha gksrk gS] ;k yxHkx rkR{kf.kd :i ls bysDVªkWu }kjk vo'kksf"kr gksrk gSA 11.31 λ = 1 Å osQ fy,] bysDVªkWu dh mQtkZ = 150 eV; iQksVkWu dh mQtkZ = 12.4 keV blfy,] leku rjaxnS?;Z osQ fy,] iQksVkWu dh mQtkZ] bysDVªkWu dh mQtkZ ls dkiQh vfèkd gksrh gSA 11.32 (a) λ= h = h p 2 mK blfy, leku K osQ fy,] λ] nzO;eku m osQ lkFk (1/ m ) osQ vuqlkj ?kVrh gSA vc (m n/me) = 1838.6; vr% leku mQtkZ 150 eV osQ fy, (vH;kl 11.31 dh rjg)]  1  U;wVªkWu dh rjaxnS?;Z =   × 10–10 m = 2.33 × 10–12 mA varjkijekf.od 18386 (Interatomic) nwfj;k¡ blls yxHkx lkS xquk cM+h gSaA blfy, 150 eV mQtkZ dk U;wVªkWuiqat foorZu iz;ksxksa osQ fy, mi;qDr ugha gSA (b) λ = (h /3 mkT ) osQ iz;ksx ls λ = 1.45 × 10–10 m, tks fØLVy esa varjkijekf.od nwfj;ksa osQ rqyuh; gSA Li"Vr;k mQij (a) rFkk (b) ls] rkih; U;wVªkWu foorZu iz;ksxksa osQ fy, mi;qDr vUos"kh (d.k) gSaA blfy, mPp mQtkZ osQ U;wVªkWu-iqat dks foorZu osQ fy, iz;qDr djus ls iwoZ rkfir dj ysuk pkfg,A 11.33 λ = 5.5 × 10–12 m λ (ihyk izdk'k) 5.9 × 10–7m foHksnu {kerk] rjaxnS?;Z osQ O;qRØekuqikrh gSA blfy, bysDVªkWu lw{en'khZ dh foHksnu {kerk] izdk'kh; lw{en'khZ dh foHksnu {kerk ls yxHkx 105 xquk gSA O;ogkj esa nwljs (T;kferh;) dkjdksa dk varj bl rqyuk dks FkksM+k lk ifjofrZr dj ldrk gSA 11.34 laosx osQ fy, h 6.63 10 × –34 Js p = = –15 λ 10 m = 6.63 × 10–19 kg m s–1 mQtkZ osQ fy, vkisf{kdh; lw=k osQ iz;ksx ls E 2 222= cp + m0 c 4 = 9 × (6.63)2 × 10–22 + (0.511 × 1.6)2 × 10–26 ∼ 9 × (6.63)2 × 10–22 J2 f}rh; in (fojke nzO;eku mQtkZ) ux.; gks tkrk gSA blfy,] E = 1.989 × 10–10 J = 1.24 BeV vr% Rojd (accelerator) ls fudys bysDVªkWu dh mQtkZ oqQN BeV dh dksfV dh vo'; gksuh pkfg,A h 4 10 3× � 11.35 λ=; mHe = 23 kg osQ iz;ksx ls6 10 ×3 mkT λ = 0.73 × 10–10 m ekè; i`FkDdj.k (nwjh) r = (V/N)1/3 = (kT/p)1/3 T = 300 K, p = 1.01 × 105 Pa osQ fy, r = 3.4 × 10–9 m izkIr gksrk gSA ge ikrs gSa fd r >> λ 11.36 vH;kl 11.35 okyk leku lw=k iz;ksx djus ij λ = 6.2 × 10–9 m tks nh xbZ varjkbysDVªkWfud nwjh ls cgqr vfèkd gSA 11.37 (a) DokoZQ] U;wVªkWu ;k izksVkWu esa ,sls cyksa ls c¡èks ekus tkrs gSa] tks mudks nwj [khapus ij izcy gksrs gSaA blfy, ,slk izrhr gksrk gS fd ;|fi izÑfr esa fHkUukRed vkos'k gks ldrs gSa] rFkkfi izs{k.kh; vkos'k e osQ iw.kZ xq.kt gksrs gSaA (b) fo|qr rFkk pqacdh; {ks=kksa osQ fy, Øe'k% nksuksa ewy laca/ e V = (1/2) m v 2 ;k e E = m a rFkk e B v =m v2/r, izn£'kr djrs gSa fd bysDVªkWu dh xfrdh e ,oa m nksuksa }kjk vyx&vyx fu/kZfjr ugha gksrh] cfYd e/m }kjk fu/kZfjr gksrh gSA (c) fuEu nkcksa ij vk;uksa dh] muosQ laxr bysDVªksMksa ij igq¡pus vkSj /kjk dh jpuk djus dh laHkkouk gksrh gSA lkekU; nkcksa ij] xSl v.kqvksa ls VDdj vkSj iqul±;kstu osQ dkj.k vk;uksa dh ,slh dksbZ laHkkouk ugha gksrhA (d) dk;Z&iQyu] bysDVªkWu dks pkyu cSaM osQ Åijh Lrj ls /krq ls ckgj fudkyus osQ fy, vko';d U;wure ÅtkZ ek=k gSA /krq osQ lHkh bysDVªkWu bl Lrj (ÅtkZ voLFkk) esa ugha gksrsA os Lrjksa dh larr cSaM esa jgrs gSaA ifj.kkeLo:i] ,d gh vkifrr fofdj.k osQ fy,] fofHkUu Lrjksa ls fudys bysDVªkWu] fofHkUu ÅtkZvkssa osQ lkFk fuxZr gksrs gSaA (e) fdlh d.k dh ÅtkZ E (u fd laosx p) dk ije eku ,d ;ksxkRed fLFkjkad osQ vèkhu Lora=k gSA blfy, tgk¡ λ HkkSfrd :i ls egRoiw.kZ gS] ogha ,d bysDVªkWu dh nzO; rjax osQ fy, ν osQ ije eku dk dksbZ lh/k HkkSfrd egRo ugha gksrk gSA blh rjg dyk pky νλ Hkh HkkSfrd d.k ls egRoiw.kZ ugha gSA lewg pky 2dν dE d ⎛ p ⎞ p== ⎜ ⎟= 1d(/ λ ) dp dp 2mm⎝⎠ HkkSfrd :i ls vFkZiw.kZ gSA vè;k; 12 12.1 (a) ls fHkUu ugha (b) VkWelu ekWMy] jnjiQksMZ ekWMy (c) jnjiQksMZ ekWMy (d) VkWelu ekWMy] jnjiQksMZ ekWMy (e) nksuksa ekWMy 12.2 gkbMªkstu ijek.kq dk ukfHkd izksVªkWu gSA bldk æO;eku 1.67 × 10–27kg gS] tcfd vkifrr ,sYI+kQk d.k dk æO;eku 6.64 × 10–27 kg gSA D;ksafd izdh.kZ gksus okys d.k dk æO;eku y{; ukfHkd (izksVkWu) ls vR;f/d gS blfy, izR;{k la?kêð esa Hkh ,sYI+kQk&d.k okil ugha vk,xkA ;g ,slk gh gS tSls fd dksbZ iqQVcky] fojkekoLFkk esa Vsful dh xsan ls Vdjk,A bl izdkj izdh.kZu cM+s dks.kksa ij ugha gksxkA 12.3 820 nm 12.4 5. 6 × 1014 Hz 12.5 13.6 eV; – 27. 2 eV 12.6 9.7 × 10–8 m; 3.1 × 1015 Hz 12.7 (a) 2.18 × 106 m/s; 1.09 × 106 m/s; 7.27 × 105 m/s (b) 1. 52 × 10–16s ; 1. 22 × 10–15 s; 4.11 × 10–15 s 12.8 2.12 × 10–10 m; 4.77 × 10–10m 12.9 ykbeSu Js.kh% 103 nm rFkk 122 nm ckej Js.kh% 665 nm 12.10 2.6 × 1074 12.11 (a) yxHkx leku (b) dkI+kQh de (c) ;g laosQr djrk gS fd izdh.kZu eq[;r% ,d la?k^ osQ dkj.k gS D;ksafd ,d la?kêð dh laHkkouk y{; ijek.kqvksa dh la[;k osQ lkFk jSf[kdr% c<+rh gS vkSj blfy, eksVkbZ osQ lkFk jSf[kdr% c<+rh gSA (d) VkWelu ekWMy esa] ,d la?k^ osQ dkj.k cgqr de fo{ksi gksrk gSA izsf{kr vkSlr izdh.kZu dks.k dh O;k[;k osQoy cgqizdh.kZu dks è;ku esa j[kdj gh dh tk ldrh gSA vr% VkWelu ekWMy esa cgqizdh.kZu dh mis{kk xyr gSA jnjiQksMZ ekWMy esa vfèkdrj izdh.kZu ,d la?k^ osQ dkj.k gksrk gS vkSj cgqizdh.kZu izHkko dh izFke lfUudVu ij mis{kk dh tk ldrh gSA 4πε 0(h /2 π )2 12.12 cksj ekWMy dh izFke d{kk dh f=kT;k a0 ftldk eku gS a0= 2 me e 2;fn ge ijek.kq xq#Roh; cy (Gmpm/r2), }kjk c¡èkk ekurs gSa] rc gesa (e/4 π ε0) osQeLFkku ij GmmizfrLFkkfir djuk pkfg,A vFkkZr cksj ekWMy dh izFke d{kk dh f=kT;kpe G (h /2 π )2 a = ≅ 1.2 × 1029 m gksuh pkfg,A ;g laiw.kZ fo'o osQ vkdfyr vkdkj ls0 Gm m 2 pe dgha vf/d gSA 4 me 4(2n −1)⎡⎤11 1) me ν= − =⎢⎣⎥⎦12.13ε ε(4π)3 2( 3 2 2 3 2 223 1)/2π) − π /2π) −h (h(n (4) 0 4 me (nn n 0 ν≅ 3n osQ vfèkd eku osQ fy,] 32 πε 2 33h /2π)(0 d{kh; vko`fÙk ν = (v/2 π r) gSAcn(h /2π)4 πε0(h /2π)2 2cksj ekWMy esa v = , vkSj r = 2 n gSA mr me me n(h /2π)4 =vr% νc = 23 3 2πmr 32 πε 02(h /2π) n 3 tks n osQ vfèkd eku osQ fy, ν osQ eku osQ leku gSA n ⎛⎜⎜⎞⎟⎟2 e dh foek yackbZ dh foek gSA bldk eku 2.82×10–15(a) jkf'k m gS tks12.14 4πε0 mc 2 ⎝⎠iz:ih ijek.oh; vkeki ls dkI+kQh de gSA 4πε0(h /2π)2 (b) jkf'k 2 dh foek] yackbZ dh foek gSA bldk eku 0.53 × 10–10 m gS me tks ijek.oh; lkb”kksa dh dksfV dk gSA (è;ku nhft, fd foeh; roZQ okLro esa ;g ugha crk ldrs fd gesa lgh lkb”k izkIr djus osQ fy, h osQ LFkku ij 4π vkSj h/2π izfrLFkkfir djuk pkfg,A mv 2 Ze 2 12.15 cksj ekWMy esa] mvr = n ℏ vkSj = 2 r 4π ε 0r 1 Ze 24πε h2 2 02vr% T = mv =;r =2 n 28πε 0r Zem bu lacaèkksa ij fLFkfrt mQtkZ osQ fy, 'kwU; osQ p;u dk dksbZ izHkko ugha gSA vc fLFkfrt mQtkZ osQ 'kwU; Lrj dks vuar ij p;u djus ij V = – (Ze2/4 π ε 0r ) ftlls V = –2T vkSj E = T + V = – T izkIr gksrk gS (a) E dk mn~èk`r eku = – 3.4 eV vuar ij fLFkfrt mQtkZ 'kwU; Lrj osQ izFkkxr p;u ij vkèkkfjr gSA E = – T iz;ksx djus ij] bysDVªkWu dh bl voLFkk esa xfrt mQtkZ + 3.4 eV gSA (b) V = – 2T osQ iz;ksx ls] bysDVªkWu dh fLFkfrt mQtkZ = 6.8 eV izkIr gksrh gSA (c) ;fn fLFkfrt mQtkZ osQ 'kwU; Lrj dk fHkUu rjhosQ ls p;u fd;k tkrk gS rks xfrt mQtkZ vifjofrZr jgrh gSA xfrt mQtkZ dk eku + 3.4 eV] fLFkfrt mQtkZ osQ 'kwU; Lrj osQ p;u ij fuHkZj ugha djrk gSA ;fn fLFkfrt mQtkZ dk 'kwU; Lrj fHkUu 0 rFkk Q2 > 0 ijarq Q2 > 0 dk vFkZ Q1 > 0 vko';d ugha gSA 25 26 13.23 Mg : 9.3%, Mg :11.7%12 12 13.24 ,d ukfHkd AZ X dh U;wVªkWu i`FkDdj.k ÅtkZ Sn osQ fy, lehdj.k gS] A−1 A2S = ⎡m ( X) +m − m ( X)  c n  NZ n NZ  fn, gq, vk¡dM+ksa ,oa c 2 = 931.5 MeV/u dk mi;ksx djus ij ge ikrs gSa 41 27 S ( Ca) =8.36 MeV ,oa S ( Al) 13.06 MeV = n 20 n 13 13.25 209 d 13.26 14 6C osQ mRltZu osQ fy, 223 209 14 2Q = [m ( Ra) − m ( Pb) − m ( C)] cN 88 N 82 N 6 223 209 14 2= [m( Ra) − m( Pb) −m( C)] c = 31.85 MeV88 82 6 4 223 219 42He osQ mRltZu osQ fy,] Q = [m( Ra) − m( Rn) −m( He)] c =5.98 MeV288 86 2 238 140 99 213.27 Q = [m( U) + m − m( Ce) −m( Ru)] c = 231.1 MeV92 n58 44 234 213.28 (a) Q = [m( H) + m( H) − m( He) − m ]c = 17.59 MeV 112 n (b) owQykWe izfrd"kZ.k osQ fujlu osQ fy, vko';d xfrt ÅtkZ = 480.0 KeV480.0 keV = 7.68×10–14 J = 3kT −14 7.68 ×10 −23 −1∴T =−23 (pf¡w dk = 1.381 × 10 JK ) 3 ×1.381 ×10 = 1.85 ×109 K (vko';d rki) – – 13.29 K (β )= 0.284MeV, K (β )= 0.960MeV max 1 max 2 20 2020( )= 2.627 × 10 Hz , νγ )= 0.995 × 10 Hz , νγ = 1.632 × 10 Hz νγ ( ()1 23 13.30 (a) uksV djsa fd lw;Z osQ vH;arj esa pkj 11H ukfHkd feydj (lay;u) ,d 24He ukfHkd cukrs gSa rFkk izfr lay;u yxHkx 26 MeV dh ÅtkZ foeqDr gksrh gSA 1kg gkbMªkstu osQ lay;u esa foeqDr ÅtkZ = 39 ×1026 MeV235 (b) 1kg U osQ fo[kaMu esa foeqDr ÅtkZ = 5.1×1026 MeV921 kg gkbMªkstu osQ lay;u esa foeqDr ÅtkZ] 1 kg ;wjsfu;e osQ fo[kaMu esa foeqDr ÅtkZ dh yxHkx 8 xquh gSA 13.31 3.076 × 104 kg 553 vè;k; 14 14.1 (c) 14.2 (d) 14.3 (c) 14.4 (c) 14.5 (c) 14.6 (b), (c) 14.7 (c) 14.8 v/Zrjax osQ fy, 50 Hz ; iw.kZ rjax osQ fy,100 Hz 14.9 vi = 0.01 V ; IB= 10 µA 14.10 2 V 14.11 ugha (hν dk eku E g ls vf/d gh gS) 14.12 n ≈ 4.95 × 1022; n = 4.75 × 109 ; n-izdkj dk] pw¡fd n >> neh eh laosQr % vkos'k mnklhurk osQ fy, N – N= n – n ; n.n = DAehehbu lehdj.kksa dks gy djus ij] n e = 14.13 1 × 105 14.14 (a) 0.0629 A, (b) 2.97 A, (c) 0.336 Ω (d) nksuksa oksYVrkvksa osQ fy, /kjk I dk eku yxHkx I0 osQ leku gksxk] blls Kkr gksrk gS fd i'pfnf'kd ck;l esa xfrd izfrjks/ dk eku vuar gksxk! 14.16 NOT ; A Y 0 1 1 0 14.17 (a) AND (b) OR 14.18 OR xsV 14.19 (a) NOT, (b) AND vè;k; 15 15.1 (b) 10 kHz dk fofdj.k ugha gksxk (,saVsuk lkb”k)] 1 GHz ,oa 1000 GHz ikj pys tk,¡xsA 15.2 (d) lkj.kh 15.2 nsf[k,A 15.3 (c) n'keyo iz.kkyh larr ekuksa dk leqPp; gSA 15.4 ughaA ftl {ks=k esa lsok,¡ igq¡psaxh mldk {ks=kiQy gS A = p 2 Td = 6 222 ×162 × 6.4 ×10 = 3258 km 7 15.5 µ = 0.75 = m c A A m A 0.75 12 = 9 V = × 15.6 (a) (b) µ =0.5 15.7 pw¡fd AM rjax (A + A sinω t) cos ω t, }kjk O;Dr gksrh gS] bldk vf/drecmm c vk;ke M = A + A gksxk tcfd U;wure vk;ke M = A – A gksxkA vr% ekMqyu1cm 2cm lwpdkad gS] ;fn M2 = 0 rks Li"V :i ls gh m =1 pkgs M1 dk eku oqQN Hkh gksA 15.8 ljyrk dh n`f"V ls ekuk fd vfHkxzkgh flXuy A1 cos (ω c + ω m) t gSA okgd flXuy Ac cos ω ct] vfHkxzkgh LVs'ku ij miyCèk gSA nksuksa flXuyksa dks xq.kk djus ij gesa izkIr gksrk gS] AA cos (ω + ω ) t cos ω t 1ccmc ;fn bl flXuy dks fuEu ikjd fiQYVj ls xqtkjk tk, rks ge ekMqfyr flXuy izkIr dj ysrs gSaA