1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 mÙkj vè;k; 1 (a) (a) (d) (b) (c) (a) (a) (c), (d) (b), (d) (b), (d) (c), (d) (a), (c). (a), (b), (c) vkSj (d). 'kwU; –QQ (i) 2, (ii) 24π R14π R2 fo|qr {ks=k ijek.kqvksa dks ck¡/dj mnklhu vfLrRo dj nsrs gSaA vkos'kksa osQ vkfèkD; osQ dkj.k {ks=k mRiUu gksrs gSaA fdlh fo;qDr pkyd osQ vUrjki`"B ij vkos'k&vkf/D; ugha gks ldrkA ugha] fo|qr {ks=k vfHkyEcor gks ldrk gSA rFkkfi] bldk foijhr lR; gSA ik'oZ dk n`'; qq qq1.19 (i) , (ii) , (iii) , (iv) .8ε04ε02ε02ε0 1.20 Al osQ 1 eksyj nzO;eku M esa ijek.kqvksa dh la[;k NA = 6.023 × 1023 m ∴ m nzO;eku osQ Al osQ iSls osQ flDosQ esa ijek.kqvksa dh la[;kN = NA M vc Z = 13, M = 26.9815gAlAl0.75 vr%] N = 6.02 × 1023 ijek.kq/eksy × 26.9815g/ eksy = 1.6733 × 1022 ijek.kq ∴ q = iSls esa èkukos'k = N Ze 22–19= (1.67 × 10)(13) (1.60 × 10C) = 3.48 × 104 C. q = 34.8 kC èkukos'k ;g vkos'k dh ,d fo'kky ek=kk gSA 24q 2 ⎛ 9 Nm ⎞ (3.48 ×10C) 23 1.21 F = = 8.99 ×10 = 1.1 10 × N12 ⎜ 2 ⎟ –4 24πε 0 r1 ⎝ C ⎠ 10 m 2 –2 2F2 r1 (10 m) –8 –8 15 10 ⇒ F = F ×10 =× N== = 1.1 10 2 22 21F1 r2 (10 m) 2 –2 2 3 1 –16 F r (10 m) == =10 2 62F1 r3 (10 m) F = 10–16 F = 1.1 × 107 N. 3 1fu"d"kZ: fcUnq vkos'kksa esa i`Fkd djus ij ;s vkos'k fo'kky cy vkjksfir djrs gSaA oS|qr mnklhurk dks fo{kqCèk djuk vklku ugha gSA 1.22 (i) 'kwU;] lefefr lsA (ii) ,d èkukRed Cs vk;u gVkuk ml vofLFkfr ls ,dy ½.kkRed Cs vk;u tksM+us osQ rqY; gSA rc usV cy 2 e F = 4π∈0 r2 ;gk¡ r = Cl vk;u vkSj fdlh Cs vk;u osQ chp nwjh = 0.346 × 10–9 m vr%] F 9 –19 2 –9 2 (8.99×10 )(1.6 10 ) (0.346×10 ) × = –11 192 10 = × = 1.92 × 10–9 N mÙkj% 1.92 × 10–9 N, A ls Cl– dh vksj funsf'kr 1.23 fcUnq P ij fLFkr vkos'k 2q ij q osQ dkj.k cy ck;ha vksj rFkk –3q osQ dkj.k nk;ha vksj gSA ∴ 2 2 2 2 0 0 2q 6q 4 x 4 (d x ) πε πε = + ∴ (d + x)2 = 3x2 ∴ 2x 2 – 2dx – d2 = 0 P 2q x q –3q = ± d 2 x 3d 2 = + d 2 x = + 3d d (1 2 2 3 ) (½.kkRed fpg~u ysus ij x dk eku q rFkk –3q osQ chp gksxk] vr% ;g ekU; ugha gSA) 1.24 (a) vkos'k A rFkk C èkukRed gSa D;ksafd {ks=k js[kk,¡ buls fudyrh gSaA (b) vkos'k C dk ifjek.k vfèkdre gS D;ksafd blls vfèkdre {ks=k js[kk,¡ lac¼ gSaA (c) (i) A osQ fudVA ½.kkos'k rFkk fdlh fLFkfr osQ chp dksbZ mnklhu fcUnq ugha gSA nks ltkrh; vkos'kksa osQ dksbZ mnklhu fcUnq gks ldrs gSaA fp=k esa ge ;g ns[krs gSa fd vkos'kksa A rFkk C osQ chp ,d mnklhu fcUnq gSA lkFk gh] nks ltkrh; vkos'kksa osQ chp mnklhu fcUnq de ifjek.k osQ vkos'k osQ fudV gh gksrk gSA bl izdkj vkos'k A osQ fudV fo|qr {ks=k 'kwU; gSA 1.25 (a) (i) 'kwU; (ii) (b) mÙkj (a) osQ leku πε 2 0 1 q 4 r osQ vuqfn'k (iii) πε 2 0 1 2q 4 r osQ vuqfn'k 1.26 (a) eku yhft, fo'o dh f=kT;k R gSA ;g ekfu, fd gkbMªkstu ijek.kq ,dleku :i ls forfjr gSaA izR;sd gkbMªkstu ijek.kq ij vkos'k eH = – (1 + y) e + e = – ye = |ye| izR;sd gkbMªkstu ijek.kq dk nzO;eku izksVkWu osQ nzO;eku ~ mp osQ rqY; gSA tc R ij] fdlh gkbMªkstu ijek.kq ij ;fn owQykWe&izfrd"kZ.k xq#Roh; vkd"kZ.k ls vfèkd gks tk, rks foLrkj vkjEHk gks tkrk gSA eku yhft, R ij fo|qr {ks=k E gS] rc 4 π R33N ye 4πR2 E = (xkml fu;e)3εo ye 1N R ˆr 3 εo E (R) = eku yhft, R ij xq#Roh; {ks=k GR gSA rc 43– 4πR2 GR = 4 πG mp πRN 3 4 R3 ρG = – πGm NR GR(R)= –4 πGmρ NR ˆr 3 bl izdkj R ij gkbMªkstu ijek.kq ij owQykWe&cy gS 1 Ny 2e2 ye E(R) = Rrˆ 3 ε o bl ijek.kq ij xq#Rokd"kZ.k cy gS 4π 2 pR R pmG ( )= – GNm R rˆ 3 ijek.kq ij usV cy gS 22⎛ 1 Ny e 4π 2 ⎞ p ⎟F = ⎜ R– GNmR rˆ 3 ε 3⎝ o ⎠ ;g ozQkafrd eku rc gS tc 1 Ny C2 e 24π 2R = GNm R 3 εo3 p 2m ⇒ y 2 = 4πε Gp c o2 e –11 26 –62 7 ×10 ×1.8 ×10 ×81 ×10 9 2 –38 9 ×10 ×1.6 ×10 � 63 × 10 –38 –19 –18 ∴ yC � 8×10 � 10 (b) bl usV cy osQ dkj.k gkbMªkstu ijek.kq fdlh Roj.k dk vuqHko djrk gS tks bl izdkj gksrk gS] fd 2 22dR ⎛ 1Nye 4p ⎞ m = R– GNm 2R r ⎜ p ⎟ dt 2 ⎝ 3 eo 3 ⎠ dR ⎛ 22 ⎞2 2 211Nye 4p 2vFkok 2 =a R tgk¡ a= ⎜ – GNm p ⎟dt m3e3 p ⎝ o ⎠ at –at bldk ,d gy R=Ae +Be pwafd ge dksbZ foLrkj [kkst jgs gSa] B = 0 t∴ R = Aeα . ⇒ R =α Ae αt =α R bl izdkj osx osQUnz ls nwjh osQ vuqozQekuqikrh gSA 1.27 (a) leL;k dh lefefr ls ;g Kkr gksrk gS fd fo|qr {ks=k vjh; gSA r < R okys fcUnqvksa osQ fy, fdlh xksyh; xkml&i`"B ij fopkj dhft,A rc ml i`"B ij Er.dS =ρdv �∫ ε 1 ∫V o 1 r 2 ′3 ′4πr Er = 4πk ∫r dr εo o 14 πk 4 = r εo 4 12∴ Er = kr 4εo 1 2ˆE () r = kr r 4εo r > R, okys fcUnqvksa osQ fy, fdlh r f=kT;k osQ xksyh; i`"B ij fopkj dhft, 1 E .dS = ρdv �∫ r ε ∫ oV R 24πk 34πrE = r dr r ε ∫ o o 4πkR 4 = εo 4 4kR ∴ E = r 4εor 2 E() r = ( /4 k εo )( R 4/r 2)rˆ y 1.28 d -Q q Q (b) nksuksa izksVkWu fdlh O;kl osQ vuqfn'k osQUnz osQ foijhr ik'oksZ ij gksus pkfg,A eku yhft, izksVkWu osQUnz ls r nwjh ij gSa R bl izdkj] 4π kr ′3dr = 2e∫ o 4πk 4∴ R = 2e 4 2e ∴k = 4π R izksVkWu 1 ij cyksa ij fopkj dhft,A vkos'k forj.k osQ dkj.k vkd"kZ.k cy gS e 2e 2 r 2 –e E = – kr 2 rˆ = –4 rˆ r 4εo 4πε oR e 21 rˆizfrd"kZ.k cy gS 4πε o (2r)2 2 22⎛ e 2er ⎞ – rˆusV cy gS ⎜ ⎟4πε 4r 24πε R 4⎝ oo ⎠ ;g usV cy 'kwU; gS] blfy, 2 22 e 2er = 16 πε or 24πε oR 4 4 4R 4 R 4 vFkok] r == 32 8 R⇒ r = (8)1/ 4 bl izdkj] izksVkWu osQUnz ls nwjh r = R ij gksuk pkfg,A48 (a) IysV α osQ dkj.k x ij fo|qr {ks=k gS – Q xˆ S2εo IysV β osQ dkj.k x ij fo|qr {ks=k gS q xˆ S2εo bl izdkj] usV fo|qr {ks=k gS (Q – q)E1 =(–xˆ )2εoS (b) Vdjkus osQ le; IysV β rFkk IysV γ ,d lkFk gSa] vr% leku foHko ij gSaA eku yhft, β ij vkos'k q1 rFkk γ ij vkos'k q2 gSA fdlh fcUnq O ij fopkj dhft,A ;gk¡ fo|qr {ks=k 'kwU; gksuk pkfg,A α osQ dkj.k 0 ij fo|qr {ks=k = – Q xˆ 2εoS qβ osQ dkj.k 0 ij fo|qr {ks=k = –1 xˆ 2εoS y qγ osQ dkj.k 0 ij fo|qr {ks=k = –2 xˆ 2ε So °o – (Q + q2 ) q1 q2∴ += 0 q1 2ε S 2ε S oo ⇒ q1– q2 = Q lkFk gh] q1 + q2 = Q + q ⇒ q1 = Q + q /2 rFkk q2 = q/2 bl izdkj β vkSj γ ij vkos'k ozQe'k% Q + q/2 vkSj q/2 gSaA (c) eku yhft, VDdj osQ i'pkr~ nwjh ij ossx v gSA ;fn IysV γ dk nzO;eku m gS] rc bl isQjs esa vftZr xfrt mQtkZ fo|qr {ks=k }kjk fd, x, dk;Z osQ cjkcj gksuh pkfg,A VDdj osQ i'pkr~ γ ij fo|qr {ks=k gS Q (Q + q /2 ) q/2 E = – xˆ + xˆ = xˆ22ε S 2ε S 2ε Soo o IysV γ osQ eqDr gksus ls VDdj rd fd;k x;k dk;Z F1d gS] ;gk¡ F1 IysV γ ij cy gSA VDdj osQ i'pkr~ blosQ d rd igq¡pus rd fd;k x;k dk;Z F2d, ;gk¡ F2 IysV γ ij cy gSA (Q – q)QF = EQ = 112εoS (q/2 )2 rFkk F2 = Eq /2 = 22εoS ∴ oqQy fd;k x;k dk;Z gS 1 212⎡(Q – q)Q +(q /2 ) ⎤ d =(Q – q /2 )d⎣ ⎦2εoS 2εoS 2 d 2⇒ (1/2) mv =(Q – q /2 )2εoS ⎛ d ⎞1 /2 ∴ v =(Q – q /2 ) ⎜⎟ mε S⎝ o ⎠ Qq[1esu vko's k]2 1.29 (i) F = 2 = 1Mkbu =2 r [1cm] vFkok 1 esu vkos'k = 1 (Mkbu)1/2(cm) vr% [1 esu vkos'k] = [F]1/2 L = [MLT–2]1/2 L = M1/2 L3/2 T–1 [1 esu vkos'k] = M1/2 L3/2 T–1 bl izdkj cgs ek=kdksa esa vkos'k dks M dh 1/2 rFkk L dh 3/2 dh fHkUukRed ?kkrksa esa O;Dr fd;k tkrk gSA (ii) nks vkos'kksa] ftuesa izR;sd dk ifjek.k 1 esu vkos'k rFkk ftuosQ chp i`Fkdu 1 cm osQ chp cy ij fopkj dhft,A rc cy 1 Mkbu = 10–5 N. ;g fLFkfr 10–2m i`Fkdu okys xC ifj.kke osQ nks vkos'kksa osQ rqY; gSA blls izkIr gksrk gS% 1 x 2 F = . 4π∈0 10 –4 tks gksuk pkfg, 1 Mkbu = 10–5 N 2 –9 21 x 1 10 Nm . = 10 –5 ⇒ =bl izdkj –4 224πε 010 4πε 0 x C ftlosQ lkFk x = 1, blls izkIr gksrk gS[3] ×10 9 1 Nm 2 –9 218 2 9 = 10 ×[3] ×10 = [3] ×10 24πε 0C ftlosQ lkFk [3] → 2.99792458, gesa izkIr gksrk gS 1 9 Nm 2 = 8.98755.... ×10 rF;r%4πε 0 C2 1.30 osQUnz O osQ vuqfn'k q ij oqQy cy F F = 2 cos θ= –. 4πε 0r 24πε 0r 2 r x q q 22q 2 x – q d d – q F 2 2 2 3 /2 0 –2 4 ( ) q x d xπε = + 2 3 0 2 – 4 q x kx πε d ≈ = osQ fy, x << d bl izdkj rhljs vkos'k q ij cy foLFkkiu osQ vuqozQekuqikrh gS rFkk og nks vU; vkos'kksa osQ osQUnz dh vksj gSA vr% rhljs vkos'k dh xfr ljy vkorZ xfr gS ftldh vko`fÙk gS 2 3 0 2 4 q k d m m ω πε = = vkSj bl izdkj T 2π ω = md q 1 /2 3 3 0 2 8π ε ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ 1.31 (a) NYys osQ v{k osQ vuqfn'k q dks /hjs ls fn;k x;k èkDdk fp=k (b) esa n'kkZ;h fLFkfr mRiUu djsxkA NYys osQ O;kl osQ nks fljksa ij A rFkk B nks fcUnq gSaA A rFkk B ij js[kk vo;oksa –Q 1 F = 2. .. qA +B 2π R 4πε 0 –Q osQ dkj.k q ij cy2 πR 1 . .cos θ2 r (b) iz'u osQ Hkkx (a) ls d 2z Qqz d 2 z Qq m = – vFkok = – z A RB2 323dt 4πε R dt 4πε 0 0 R NYys dk ry 3 (b)Qq 4πε mR vFkkZr ω 2. vr% 0= 3 T = 2π 4πε 0mR Qq vè;k; 2 2.1 (d) 2.2 (c) 2.3 (c) 2.4 (c) 2.5 (a) 2.6 (c) 2.7 (b), (c), (d) 2.8 (a), (b), (c) 2.9 (b), (c) 2.10 (b), (c) 2.11 (a), (d) 2.12 (a), (b) –Q q 1 z = .. 2 2 2 21/2 πR.4 πε 0 (z + R )( z + R ) q ij NYys osQ dkj.k oqQy cy = (FA+B)(πR) –Qq z = 2 2 3/2 4πε o(z + R ) –Qq � z << R osQ fy,4πε o bl izdkj cy ½.kkRed foLFkkiu osQ vuqozQekuqikrh gSA ,sls cyksa osQ vèkhu xfr ljy vkorZ xfr gksrh gSA Z NYys dk v{k oqQy vkos'k –Q (a) NYys dk v{k q 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 U 2.22z z P r Q (c) vkSj (d) vfèkd mPp foHko gk¡] ;fn vkeki fHkUu gSaA ugha pw¡fd fo|qr {ks=k laj{kh gS] nksuksa izdj.kksa esa fd;k x;k dk;Z 'kwU; gksxkA eku yhft, ;g lR; ugha gSA rc i`"B osQ rqjUr Hkhrj i`"B dh rqyuk esa foHko fHkUu gksuk pkfg, ftlosQ iQyLo:i dksbZ foHko izo.krk gksuh pkfg,A bldk ;g vFkZ gqvk fd i`"B osQ vUreZq[kh vFkok cfgeqZ[kh {ks=k js[kk,¡ gksuh pkfg,A pw¡fd i`"B] lefoHko i`"B gS] nwljs fljs ij ;s js[kk,¡ nqckjk i`"B ij ugha gks ldrhaA bl izdkj ;g osQoy rHkh laHko gS tc {ks=k js[kkvksa osQ nwljs fljs Hkhrj vkos'kksa ij gksa] tks vk/kj rF; osQ ijLij fojks/h gSaA vr% Hkhrj leLr vk;ru leku foHko ij gksuk pkfg,A C de gks tk,xhA 1 2lafpr mQtkZ= CV vkSj blfy, vf/d gks tk,xhA2 fo|qr {ks=k vf/d gks tk,xkA lafpr vkos'k leku jgsxkA V de gks tk,xkA fo|qr {ks=k osQ vuqfn'k vkosf'kr pkyd ls vukosf'kr pkyd dh vksj osQ fdlh Hkh iFk ij fopkj dhft,A bl iFk ij foHko fujUrj de gksxkA vukosf'kr pkyd ls vuUr dh vksj osQ vU; iFk ij foHko vkSj ?kVsxkA ;g visf{kr rF; dks fl¼ djrk gSA −qQU= 4πε 0R 1 + z 2 R2 z osQ lkFk fLFkfrt mQtkZ esa ifjorZu dks fp=k esa n'kkZ;k x;k gSA foLFkkfir vkos'k – q nksyu djsxkA ek=k xzkiQ dks ns[kdj ge dksbZ fu"d"kZ ugha fudky ldrsA = 4πε 0 R2 + z 2 2.24 js[kk ls nwjh r ij foHko Kkr djus osQ fy, fo|qr {ks=k ij fopkj dhft,A lefefr }kjk ge ;g ikrs gSa fd {ks=k js[kk,¡ cfgeZq[kh vjh; gksuh pkfg,A f=kT;k r rFkk yEckbZ l dk dksbZ xkmlh; i`"B [khafp,A rc 1 Q2.23 V 1 E.dS =λl�∫ε0 vFkok Er2πrl = 1 λl ε0 λ⇒ Er = 2πε 0r vr%] ;fn f=kT;k r 0 gS] rc r 0 ∫ λ r Vr ( ) – Vr () =− E.d l = ln 0 2πε r r00 fdlh fn, x, V osQ fy,] r 2πε 0 0ln =− [Vr ( ) – Vr ( )] r λ0 − 2πε 0Vr ( 0)/ λ+2πε 0 ()/ λ⇒ r = r0e .e Vr lefoHko i`"B csyukdkj gSa ftudh f=kT;k gS 2πε [V ( )– V (r )] λ r = r e− 0 r 0 02.25 eku yhft, ry ewy fcUnq ls nwjh x gSA rc fcUnq P ij foHko gSA 1 q 1 q−1/2 1/2 4πε 22 4πε 220 ⎡⎣(x + d/2 )+ h ⎤⎦ 0 ⎡⎣(x − d /2 )+h ⎤⎦ ;fn ;g foHko 'kwU; gS] rks 1 1 = 1/2 1/2 ⎡ 22 ⎤⎡ 22 ⎤⎣(x + d /2 )+h ⎦⎣(x − d /2 )+ h ⎦ vFkok (x-d/2)2 + h2 = (x+d/2)2 + h2 2 222⇒ x − dx + d /4 = x + dx + d /4 Or, 2dx = 0 ⇒ x = 0 ;g ry x = 0 dk lehdj.k gSA 2.26 eku yhft, ;g U dh vafre oksYVrk gSA ;fn laèkkfj=k dh ijkoS|qr osQ fcuk èkkfjrk C rc laèkkfj=k ij vkos'k gS Q1 = CU ijkoS|qr gksus ij laèkkfj=k dh èkkfjrk εC gksrh gSA blfy, laèkkfj=k ij vkos'k gS Q2 =εCU =αCU 2 tks laèkkfj=k vkosf'kr Fkk ml ij vkjfEHkd vkos'k gS Q0 = CU0 vkos'kksa osQ laj{k.k ls Q0 = Q1 + Q2 vFkok CU0 = CU + ε CU2 = oksYV 4 −1± = 4 pwafd U èkukRed gS 625 −1 24 U = == 6V 44 2.27 tc pfozQdk ryh dks Li'kZ dj jgh gS rc leLr ifêðdk lefoHko ifêðdk gSA dksbZ vkos'k q′ pfozQdk dks LFkkukUrfjr gks tkrk gSA pfozQdk ij fo|qr {ks=k V = d V ∴ q′= −ε0 πr 2 d pfozQdk ij dk;Zjr cy gS 2VV −× q′=ε πr 2 02dd ;fn pfozQdk dks mQij mBkuk gS] rc 2 ε V πr 2 = mg 02d mgd 2 ⇒ V = 2πε r0 1 ⎧ qq qq qq ⎫dd u d ud2.28 U= – –⎨ ⎬4πε rrr ⎭0 ⎩ 9×10 9 –19 2 =8 –15 (1.6×10 ){(1 3) 2–(2 3)(13)–(2 3)(1 3) }10⎧14 ⎫ –14 = 2.304 × 10–13 ⎨ – ⎬ = –7.68×10 J ⎩99 ⎭= 4.8 × 105 eV = 0.48 MeV = 5.11 × 10–4 (mnc2) 2.29 lEioZQ ls iwoZ % Q1 =σ.4 π R2 Q2=σ .4 π (2 R2)= 4(σ .4 π R2 )= 4Q lEioZQ osQ i'pkr~ Q ′+ Q ′= Q + Q = 5Q ,1212 1 =5 (σ .4 π R2) leku foHko ij gksaxs ′′ QQ12 = R 2R ∴Q2 =2Q ′ . ′ ∴ 3Q1 ′=5(σ.4 π R2 ) ∴ Q ′= 5(σ.4 R 2 ) Q′= 10 σ.4 R 2 )13 π vkSj 23 (π 5 ∴σ=53 σ vkSj ∴σ 2 = 6 σ1 C1 KK1 2 2.30 vkjEHk esa : vkSjV ∝ 1 ,V 1 + V2 = E E = 9VC ⇒V1=3V vkSj V2=6V ∴ Q = CV = 6C×3 =18µ C1 11 Q2=9 µ C vkSj Q3= 0 ckn esa% Q =Q′+ Q223 Q2lkFk gh% C2V + C3V = Q2 ⇒V = =(3 2)V C +C 23 z ′ ′Q2 = 92µC vkSj Q3 = 92µC Q2.31 σ= 2π R 2.32 z q1 d O q2 –d U 2.33 O 2q 2 U = 04πε 0d 1 σ .2 πr dr dU = 0 r + z4πε 22 πσ R2rdr ∴ U = 4πε O ∫ 22 0 r + z 2πσ R2πσ ⎡ 22 ⎤⎡ 22 ⎤= r + z = R +z –z⎢ ⎥⎢ ⎥4πε 0 ⎣⎦O4πε 0 ⎣⎦ 2Q ⎡ 22 ⎤= R +z –z2 ⎢ ⎥⎣ ⎦4πε 0R qq12 = 0+ 22 22 x + y +(z – d )2 x +y +(z + d)2 q –q12∴= 22 22 x + y +(z – d )2 x + y +(z + d )2 bl izdkj] oqQy foHko 'kwU; gksus osQ fy, q1 rFkk q2 osQ fpg~u foijhr gksus pkfg,A oxZ vkSj ljy djus ij gesa izkIr gksrk gS ⎡ q )2 +1⎤(q222 1 x + y + z + ⎢ 22 ⎥ (2 zd )+ d2 =0 ⎢(q q2 ) –1⎦⎥⎣ 1 ⎛⎡ q12 + q12 ⎤⎞ ;g ml xksys dk lehdj.k gS ftldk osQUnz ⎜0,0,–2 d ⎢ 22 ⎥⎟ ij gSA⎜⎟q – q è;ku nhft, ;fn q1 = –q2 ⎝⎣ 11 ⎦⎠ ⇒ rc z = 0, eè; fcUnq ls xqtjus okyk ry gSA 1 ⎪⎧ –q 2–q 2 ⎪⎫ U =+⎨⎬4πε ⎪(d – x ) d – x ⎪0 ⎩⎭ –q 22d U = 4πε 0 (d 2– x 2 ) dU –q 2.2 d 2x = . dx 4π∈ 220 d( – x )2 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 dU x = 0 ij = 0dx x = 0 dksbZ larqyu fcUnq gSA 22 ⎡ 28x 2 ⎤d U ⎛ –2 dq ⎞ – 2 =⎜ ⎟⎢ 2 2 223 ⎥ dx ⎝ 4π∈0 ⎢(d – x 2 ) (d – x ) ⎥⎠⎣ ⎦ ⎛ –2 dq 2 ⎞ 12⎡ 22 2 ⎤=⎜⎟ 2(d – x ) –8 x2 23 ⎝ 4π∈0 ⎠ (d – x ) ⎣⎦ x = 0 ij 2dU ⎛ –2 dq 2 ⎞⎛ 1 ⎞ 2 = ⎜ (2 d )2 ⎟⎜ 6 ⎟ , tks < 0.dx ⎝ 4π∈0 ⎠⎝ d ⎠ vr% ;g vLFkk;h larqyu gSA (b) (a) (c) (b) (a) (a) (b), (d) (a), (d) (a), (b) (b), (c) (a), (c) vè;k; 3 tc dksbZ bysDVªkWu fdlh lafèk dh vksj xeu djrk gS rks ,dleku E osQ vfrfjDr og lekU;r% lafèk osQ rkjksa osQ i`"B ij lafpr vkos'kksa (tks viokg osx vd dks fu;r j[krs gSaA) dk lkeuk Hkh djrk gSA ;s fo|qr {ks=k mRiUu djrs gSaA ;s {ks=k laosx dh fn'kk ifjofrZr dj nsrs gSaA foJkfUr dky bysDVªkWuksa ,oa vk;uksa osQ osxksa ij fuHkZj gksus osQ fy, ckè; gSaA vuqizLFk fo|qr cy bysDVªkWu osQ osx dks 1mm/s dksfV dh pkyksa }kjk izHkkfor djrs gS] tks dksbZ lkFkZd izHkko ugha gSA blosQ foijhr] T esa ifjofrZr osxksa esa 102 m/s dksfV osQ izHkko mRiUu djrk gSA ;g τ esa lkFkZd izHkko yk ldrk gSA [ρ = ρ(E,T ) gS ftlesa E ij fuHkZjrk mis{k.kh; gS] lkekU; vuqiz;qDr oksYVrkvksa osQ fy,]A 3.14 OghVLVksu lsrq esa 'kwU; fo{ksi fof/ dk ;g ykHk gS fd xSYosuksehVj dk izfrjksèk larqyu fcUnq dks izHkkfor ugha djrk rFkk izfrjksèkksa ,oa xSYosuksehVj esa izokfgr èkkjk rFkk xSYosuksehVj osQ vkUrfjd izfrjksèk dks Kkr djus dh dksbZ vko';drk ugha gksrh vkSj fdj[kksiQ fu;e dk ifjiFk ij vuqiz;ksx djosQ vKkr izfrjksèk] RvKkr] ifjdfyr fd;k tk ldrk gSA vU; fofèk;ksa esa ges izfrjksèkksa rFkk xSYosuksehVj esa izokfgr lHkh èkkjkvksa rFkk xSYosuksehVj osQ vkUrfjd izfrjksèk dh ifj'kq¼ ekiksa dh vko';drk gksxhA 3.15 èkkrq dh eksVh ifêð;ksa dk fuEu izfrjksèk gksrk gS ftls 'kqU;&fo{ksi fcUnq ij foHkoekih rkj dh yEckbZ esa lfEefyr djus dh vko';drk ugha gksrhA gesa osQoy lhèks [k.Mksa (izR;sd 1 yEck) osQ vuqfn'k rkjksa dh yEckbZ ekiuh gksrh gS ftls ehVj LosQy }kjk vklkuh ls ekik tk ldrk gSA vkSj ;g eki ifj'kq¼ gksrh gSA 3.16 nks ckrks ij fopkj djus dh vko';drk gksrh gS% (i) èkkrq dk ewY;] rFkk (ii) èkkrq dh vPNh pkydrkA vfèkd ewY; gksus osQ dkj.k ge pk¡nh dk mi;ksx ugha djrsA blosQ i'pkr vPNs pkydksa esa rk¡ck o ,syqfefu;e mi;ksx gksrs gSaA 3.17 feJkrqvksa osQ izfrjksèk dk rki xq.kkad fuEu (fuEu rki lqxzkg~;rk) rFkk izfrjksèkdrk mPp gksrh gSA 3.18 'kfDr {k; PC = I2RC ;gka] RC la;kstd rkjksa dk izfrjksèk gS 2P P = RC C2V 'kfDr {k; PC de djus osQ fy, 'kfDr lapj.k mPp oksYVrk ij fd;k tkuk pkfg,A 3.19 ;fn R esa o`f¼ dj nsa] rks rkj ls izokfgr èkkjk de gks tk,xh vkSj bl izdkj foHko izo.krk Hkh de gks tk,xh] ftlosQ dkj.k larqyu yEckbZ vfèkd gks tk,xhA vr% 'kwU; fo{ksi fcUnq J fcUnq B dh vksj LFkkukUrfjr gks tk,xkA 3.20 (a) E1 dk èkukRed VfeZuy X ls la;ksftr gS rFkk E1 > E (b) E1 dk Í.kkRed VfeZuy X ls la;ksftr gSA V 3.21 E R I = E ; E =10 I3.22 RR + nR R + n 1+ n 1 + n = 10 = n = n1 n +11 + n n = 10. 11 1 RRR RU; wU;Ure wre U;Ure Ure wU;Uw3.23 =+ ....... + , =++ ....... +> 1 RR RRRR Rp 1 nP 12 n vkSj RS = R1 + ...... + R n ≥ R.vfèkdrefp=k (b) esa Rmin fp=k (a)esa èkkjk dks iznku fd, tSlk gh rqY; ekxZ iznku djrk gSA ijUrq blosQ lkFk&lkFk 'ks"k (n – 1) izfrjksèkdksa osQ }kjk (n – 1) ekxZ iznku fd, x, gSaA fp=k (b) esa fo|qr èkkjk > fp=k (a) esa fo|qr èkkjkA fp=k (b) esa izHkkoh izfrjksèk < RminA Li"V :i ls nwljk ifjiFk vfèkd izfrjksèk ogu djus ;ksX; gSA vki fp=k (c)rFkk fp=k (d)dk mi;ksx djosQ R s > R max fl¼ dj ldrs gSaA RRRmin maxmax Rmin V V V V (a) (b) (c) (d) A B6–4 3.24 I == 0.2A 2 + 8 E1 E2E1 osQ fljksa ij foHkokUrj = 6 – 0.2 × 2 = 5.6 V E 2 osQ fljksa ij foHkokUrj = VAB = 4 + .2 × 8 = 5.6 V fcUnq B fcUnq A ls mPp foHko ij gSA E + E I =3.25 R + r + r12 2EV = E – Ir = E – r = 0111 r + r + R12 3.26 3.27 R Veff Reff 3.28 2Er vFkok E = 1 r + r + R12 2r11 = r + r + R12 r + r + R = 2r121 R = r – r12 ρl =RA –3 2π(10 × 0.5) ρl RB = –32 –32 π[(10 ) – (0.5 ×10 )] –3 2 –3 2RA = (10 ) –(0.5 ×10 ) = 3 : 1RB (.5 ×10 –3 )2 fp=k esa n'kkZ, vuqlkj ge fdlh Hkh 'kk[kk R osQ leLr usVooZQ dks ,d ljy ifjiFk esa ifj.kr djus dh lksp ldrs gSaA ViHz kkoh rc R ls izokfgr èkkjk I = Riz+ RHkkoh foeh; :i esa V= V(V1, V, ...... V ) dh foek oskYVrk dh gS rFkk R= RizHkkoh izHkkoh 2nizHkkoh izHkkoh (R1, R2, ....... Rm) dh foek izfrjksèk dh gSA vr% ;fn lc esa n-xquh o`f¼ gks tkrh gS] rc u;k u;k V = nV ,R = nR iziHkkoh ziHkkoh Hkkoh ziHkkoh zvkSj R izHkkoh = nR. bl izdkj èkkjk leku jgrh gSA fdj[kksiQ osQ lafèk fu;e dk vuqiz;ksx djus ij I = I + I12 fdj[kksiQ osQ ik'k fu;e ls izkIr gksrk gS 10 = IR + 10I1....(i) 2 = 5I2 – RI = 5 (I1 – I) – RI 4 = 10I1 – 10I – 2RI..... (ii) ⎛ 10 ⎞(i) – (ii) ⇒ 6 = 3RI + 10I vFkok 2 = I ⎜R+ ⎟⎝ 3 ⎠ R R 2V I1 II2 I2 10� Reff 10V Veff 2 = (R+ R )I dh V= (R + R)I izHkkoh izHkkoh izHkkohvkSj V = 2VizHkkoh10 R = ΩizHkkoh3 3.29 mi;qDr 'kfDr = 2ek=kd/?kaVk = 2kW = 2000J/s P 2000 I == ; 9A V 220 rkj esa 'kfDr {k; = RI2 J/s l 2 –8 10 = ρ I = 1.7 ×10 ×× 81 J/sA π×10 –6 � 4 J/s = 0.2% ρ Al rkj esa 'kfDr {k; = 4 Al =1.6 × 4 = 6.4J/s=0.32% ρ Cu 3.30 eku yhft, foHkoekih osQ rkj dk izfrjksèk R′ gS] rc 10 × R′ < 8 ⇒ 10R ′ < 400 + 8R′ 50 +R ′ 2R ′ < 400 vFkok R′ < 200Ω 10 × R′ > 8 ⇒ 2R′> 80 ⇒ R′> 40 10 + R′ 310 × R′ 4 < 8 ⇒ 7.5R′ < 80 + 8R′ 10 + R ′ R′ > 160 ⇒ 160 < R′ < 200 bldh 400 cm ij foHkoikr > 8V bldh 300 cm ij foHkoikr < 8V φ × 400 > 8V (φ→ foHkokUrj) φ × 300 < 8V φ > 2V/m 2 < 2 V/m3 6 3.31 (a) I = = 1 A = nevd A61 1–4 vd = 29 –19 –6 =×10 m/s10 ×1.6 ×10 ×10 1.6 12xfrt mQtkZ = m vd × nAl 2 e1 –31 1 –8 29 –6 –1 –17 =× 9.1 ×10 ××10 ×10 ×10 ×10 ; 2 ×10 J 2 2.56 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 (b) vkseh {k; = RI2 = 6 × 12 = 6 J/s 2×10 –17 s � 10 –17 s esa bysDVªkWu dh leLr xfrt mQtkZ u"V gks tk,xhA6(d) (a) (a) (d) (a) (d) (a), (b) (b), (d) (b), (c) (b), (c), (d) (a), (b), (d) vè;k; 4 2 mv pqEcdh; {ks=k osQ yEcor xeu djus okys vkosf'kr d.k osQ fy,% = qvB R qB v ∴ ==ω mR ⎡qB ⎤⎡ v ⎤ –1 ∴ [] = == Tω [] ⎢ ⎥⎢⎥⎣ m ⎦⎣ R⎦ dW= F.d l = 0 . dt = 0⇒ Fv ⇒ Fv . = 0 F, osx fuHkZj gksuk pkfg, ftldk vFkZ ;g gS fd F rFkk v osQ chp dks.k 90° dk gSA ;fn v ifjofrZr gksrk gS (fn'kk esa) rks F Hkh (fn'kk esa) bl izdkj ifjo£rr gksxk ftlls mijksDr 'krZ iwjh gks tk,A pqEcdh; cy funsZ'k izsQe ij fuHkZj gS rFkkfi blls mRiUu usV Roj.k tM+Roh; funsZ'k izsQeksa osQ fy, funsZ'k izsQe ij fuHkZj ugha djrk (vukisf{kdh; HkkSfrdh)A 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 d.k ,dkUrjr% Rofjr ,oa eafnr gksxkA vr% nksuksa Mh esa iFk dh f=kT;k vifjofrZr jgsxhA O2ij I1 osQ dkj.k pqEcdh; {ks=k y-v{k osQ vuqfn'k gSA nwljk rkj y-v{k osQ vuqfn'k gS] vr% cy 'kwU; gSA 1ˆˆ0B = ( + +ˆ) µ I ijk 42R ⎡ eB ⎤ωdksbZ foekghu jkf'k ughaA []T –1 = [] = ⎢ ⎥⎣ m ⎦ ˆˆE = E i,E > 0, B = B k0 0 0 d l1 ij d l2 osQ dkj.k cy 'kwU; gSA d l ij d l osQ dkj.k cy 'kwU;srj gSA21iG (G + R1) = 2 (0 – 2V) ifjlj osQ fy, i (G + R + R ) = 20, (0 – 2V) ifjlj osQ fy,G12rFkk i (G + R+ R+ R) = 200, 200V ifjlj osQ fy,G12 3izkIr gksrk gS R1 = 1990Ω R2 = 18 kΩ rFkk R3 = 180 kΩ F = BIl sin θ = BIl µoI B = 2πh QP µoIl 2 F = mg = 2πh 2 –7 µo Il 4π×10 ×250 ×25 ×1 h == 2πmg 2π×2.5 ×10 –3 ×9.8 = 51 × 10–4 m h = 0.51 cm tc pqEcdh; {ks=k dk;Zjr ugha gS] rc ∑τ= 0 Mgl = WloqQ.Myh 500 g l = WloqQ.Myh W = 500 × 9.8 NoqQ.Myhtc pqEcdh; {ks=k yxk fn;k tkrk gS] rc Mgl + mgl = Wl + IBL sin 90°loqQ.Myh mgl = BILl BIL 0.2 × 4.9 ×1×10 –2 –3 m == = 10 kg g 9.8 =1g V d VldB F =ilB = lB τ= F1 = 0 4.24 11 0 , 1 R 22 22R V d VldB F = i lB = 0 lB τ2 = F2 = 0 2 22R 22 42 R τ =τ –τ12 τ= B nˆB F1nˆF2 F2 V0 vxz n`f"V i'p n`f"V 4.25 pwafd B x-v{k osQ vuqfn'k gS] o`Ùkh; d{kk osQ fy, nks d.kksa osQ laosx y -z ry esa gSaA eku yhft, bysDVªkWu rFkk izksVkWu osQ laosx ozQe'k% p1 rFkk p2 gSaA ;s nksuks R f=kT;k osQ o`Ùk dks fu:fir djrs gSaA ;s nksuks foijhr fn'kk osQ o`Ùkksa dk fu:i.k djrs gSaA eku yhft, p1 y v{k ls θ dks.k cukrk rks p2 dks Hkh bruk gh dks.k cukuk pkfg,A buosQ vius futh osQUnzksaZ dks laosxksa osQ yEcor rFkk R nwjh ij gksuk pkfg,A eku yhft, bysDVªkWu dk osQUnz Ce ry ikWthVªkWu dk osQUnz Cp ij gSA Ce osQ funsZ'kkad gSa Ce ≡ (0,– sin , R θR θ cos ) Cp osQ funsZ'kkad gSa y Cp ≡ (0,– R sin θ, 3 R–R cos θ ) 2 ;fn nksuksa osQ osQUnzksa osQ chp dh nwjh 2R ls vfèkd gS] rks bu nksuksa osQ o`Ùk ijLij O;kiu ugha djsaxsA eku yhft, Cp rFkk Ce osQ chp dh nwjh d gS] rc x 2 22 ⎛ 3 ⎞d = (2 Rsin θ ) + ⎜ R –2R cos θ ⎟⎝ 2 ⎠ 92 22 222 = 4R sin θ+ R –6R cos θ+ 4R cos θ 4 92 22 = 4R + R –6 R cos θ 4 pw¡fd d dks 2R ls vfèkd gksuk pkfg, d2 > 4R2 9222 2⇒ 4R + R –6 R cos θ> 4R 4 ⇒ > 6cos θ9 4 3vFkok cos θ< 8 4.26 {ks=kiQy A= 23 4 a , A = a2, A = 23 3 4 a fo|qr èkkjk I lcosQ fy, leku gS pqEcdh; vk?kw.kZ m = n I A 2= 3m Ia ∴ 3a2I 23 3a I (è;ku nhft,% m xq.kksÙkj Js.kh esa gSA) n = 4 n = 3 n = 2 4.27 (a) B (z) z -v{k ij leku fn'kk esa laosQr djrk gS] blhfy, J (L), L dk ,d :ih o`f¼ iQyu gSA (b) J(L) + ifjjs[kk C ij cM+h nwfj;ksa ls ;ksxnku ∴ tSls&tSls L → ∞ = µ0I cM+h nwfj;ksa ls ;ksxnku → 0 D;ksafd 3(B � 1/r ) 0( )J ∞ − µ I (c) z B = 2 0 2 2 3 /2 2( ) IR z R µ + 2 0 2 2 3 /2 – – 2( )z IR B dz dz z R µ∞ ∞ ∞ ∞ = + ∫ ∫ ;fn z = R tanθ dz = R sec2 θ d θ /2 0 0 – – /2 cos 2z I B dz d I π π µ θ θ µ ∞ ∞ ∴ = =∫ ∫ (d) B(z)oxZ < B (z)o`Ùkh; oqQ.Myh ( ) ( )L Lℑ ℑ∴ < oxZ oÙ` kh; oQq.Myh ijUrq (b) esa fn, x, rdks± dk mi;ksx djus ij ( ) ( ) ℑ ℑ∞ = ∞oxZ o`Ùkh; oqQ.Myh 4.28 iG .G = (i1 – iG) (S1+ S2+ S3) tci1 = 10mA iG (G + S1) = (i2 –iG) (S2+ S3) tc i2 = 100mA rFkk iG (G + S1+S2) = (i3 – iG) (S3) tc i3 = 1A ls izkIr gksrk gS S1 = 1Ω, S2 = 0.1Ω rFkk S3 = 0.01Ω –L –L 4.29 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 (a) 'kwU; µ i (b) 0 AO osQ yEcor~ ckb± fn'kk esa2πR (c) µ0 i AO osQ yEcor~ ckb± fn'kk esaπR (c) (a) (c) (b) (b) (a), (d) (a), (d) (a), (d) (a), (c), (d) (b), (c), (d) eh µp ≈ µ≈ ,h =vkSj e2mp 2me 2π µe >> µp D;ksafd m p >> m e Bl =µ Ml =µ (I + I ) vkSj H = 0 = I0 0M Ml = IM = 106 × 0.1 = 105 A ρN 28g/22.4Lt 3.5 –3 χα==×10 ?kuRo ρ A vc = 1.6 × 10–4 ρCu 8g/c c 22.4 χN –4 = 5×10 χCu vr% ;gk¡ izeq[k vUrj ?kuRo osQ dkj.k gSA izfr pqEcdRo bysDVªkWuksa dh d{kh; xfr osQ dkj.k gksrk gS tks vuqiz;qDr {ks=k osQ foijhr pqEcdh; vk?kw.kZ mRiUu djrk gSA blfy, ;g rki ls vfèkd izHkkfor ugha gksrkA NS vè;k; 5 eh h 5.15 5.16 5.17 5.18 5.19 5.20 5.21 vuqpqEcdRo vkSj yksg pqEcdRo ijek.oh; pqEcdh; vk?kw.kksZa osQ vuqiz;qDr {ks=k dh fn'kk esa lajs[k.k osQ dkj.k gksrk gSA rki o`f¼ gksus ij ;g lajs[k.k fo{kksfHkr gks tkrk gS ftlosQ iQyLo:i nksuksa dh pqEcd'khyrk rki o`f¼ osQ lkFk ?kV tkrh gSA (i) pqEcd ls nwj z (ii) pqEcdh; vk?kw.kZ ck,a ls nk,a µ03mr .ˆ ˆB = 3,m = mk 4π r y ds = rr ˆ. 2sin θdθ d 0 ≤θ ≤π,0 ≤φ ≤ π µ0m 3cos θ 2Ñ∫ B.ds = ∫ 3 r sin θ dθ x4π r = 0 (θ lekdyu osQ dkj.k) N usV m = 0. ek=k laHkor% fp=k (b) esa n'kkZ;h xbZ gSA S S m E (r) = c B (r), p = . f}èkzqoksa osQ nzO;eku vkSj tM+Ro vk?kw.kZ leku gSaAc I 11 m 1T = 2π I ′= × I rFkk m′= . T ′= TmB 24 N22 N NM+ ls xqtjus okyhB dh fdlh js[kk ij fopkj dhft,A ;g cUn gksuh pkfg,A eku yhft, C ,sfEi;jh&ik'k gSA S 0 . . 0 P P Q Q d d µ = >∫ ∫ B H l l C . 0 PQP d =∫ H lÑ P Q . 0 Q p d <∫H l P → Q NM+ osQ Hkhrj gSA vr% H vkSj dl osQ chp dk dks.k vfèkd dks.k gSA S N (i) z-v{k osQ vuqfn'k B 0 4 µ π = 3 2 r m 0 0 3 2 11 . 2 – 4 2 2 R R a a dz m d m z R µ µ π π ⎛⎛ ⎞ = = ⎜ ⎟ ⎜⎝ ⎠ ⎝∫ ∫B l (ii) f=kT;k R osQ pkSFkkbZ o`Ùk osQ vuqfn'k – 2 1 a ⎞⎟⎠ 0 0 4 B µ π/ = 3 ˆ– . R m θ ( )0 3 – –sin 4 m R µ θ π = z 0 2. 4 m d R µ π =B l sin dθ θ 2 0 2 0 B. 4 m dl R π µ π =∫ ur uur R a1 O a R x x (iii) x-v{k osQ vuqfn'k 0 3 – 4 m x µ π ⎛ ⎞ = ⎜ ⎟⎝ ⎠ B . 0d =∫B l (iv) f=kT;k a osQ pkSFkkbZ o`Ùk osQ vuqfn'k .d =B l 0 2 – sin 4 m d a µ θ θ π , =∫ .B ld π µ θ θ π ∫ 2 0 2 0 – – sin 4 m d a µ π = 0 2 – 4 m a lHkh dks tksM+us ij] . 0 C d =∫ B lÑ 5.22 χ foekghu gSA χ ml pqEcdh; vk?kw.kZ ij fuHkZj djrk gS tks H ijek.oh; bysDVªkWuksa ls buosQ vkos'kksa e }kjk la;ksftr gksrk gSA m ij bldk izHkko èkkjk I ls gksdj gksrk gS ftlesa ‘e’ dk nwljk dkjd lfEefyr gksrk gSA la;kstu 2 0 " "eµ “vkos'k” Q dh foek ij fuHkZj ugha djrkA χ 2 0e m v Rα β γ = µ 2 0cµ 2 1 c = 2 0 e ε 2 1 ~ c 2 0 . ~ e R Rε 2 c mQtkZ foLrkj [ χ ] = M0L0T0Q0 = 3 –2 0 2 –2 ML T L L QT M L T β α γ⎛ ⎞ ⎜ ⎟⎝ ⎠ –1,α β= 0,γ= –1= χ = 2 0 e mR µ –6 –38 –30 –10 10 10 ~ 10 10 × × –4~10 2 2 1/2 5.23 (i) =µ0 m (4cos θ+ sin θ )B 4π R3 2B = 3cos 2 θ+1, θ =π ij U;wureA22⎛ µ0 ⎞ 2 m⎜ 3 ⎟⎝ 4πR ⎠ pqEcdh; fuj{k ij U;wure gSAB BV = 2cot θ(ii) tan (ufr dks.k) = BH π θ= ij ufr dks.k 'kwU; gks tkrk gSA iqu% fcanqiFk] pqEcdh; fuj{k gSA2 BV(iii) tc = 1 rc ufr dks.k ± 45° gSABH 2 cot θ = 1 θ = tan–12 fcUnqiFk gSA 5.24 layXu fp=k ij è;ku nhft,A 1. fcUnq P ry S esa gS (lqbZ mÙkj dh vksj laosQr djsxh) fnDikr dks.k = 0; P Hkh ,d pqEcdh; fuj{k gSA ∴ ufr dks.k = 0 2.Q pqEcdh; fuj{k ij gS ∴ ufr dks.k = 0 ijUrq fnDikr dks.k = 11.3° LL 5.25 n = n = 122π R 4a m = n IA1 m = n IA11222 L LL2 = I π R = Ia = Ia 2π R 4a 4 2MR I1 = (O;kl ls xqtjus okys fdlh v{k osQ ifjr% tM+Ùo vk?kw.kZ)2 2Ma I = 2 12 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 2 1ω = 1m B 2 2ω = 2m B 1I 2I 1m = 2m 1I 2I L LR 2π × I MR 2 = Ia Ma 2 4 ⇒ a = 3 4 π R 2 12 vè;k; 6 (c) (b) (a) (d) (a) (b) (a), (b), (d) (a), (b), (c) (a), (d) (b), (c) rkj dk dksbZ Hkh Hkkx xfre; ugha gS vr% xfrd fo|qr okgd cy 'kwU; gSA pqEcd fLFkj gS vr% le; osQ lkFk pqEcdh; {ks=k ifjofrZr ugha gksrkA bldk ;g vFkZ gS fd dksbZ fo|qr okgd cy mRiUu ugha gksrk vr% ifjiFk esa dksbZ èkkjk izokfgr ugha gksxhA èkkjk c<+ tk,xhA tSls gh rkjksa dks ,d nwljs ls nwj [khapk tkrk gS fjDr LFkkuksa ls ÝyDl dk {kj.k gksrk gSA ysat+ osQ fu;e osQ vuqlkj izsfjr fo|qr okgd cy bl deh dk fojksèk djrk gS ftls fo|qr èkkjk esa o`f¼ }kjk iwjk fd;k tkrk gSA èkkjk ?kV tk,xhA ifjukfydk esa yksg ozQksM j[kus ij pqEcdh; {ks=k esa o`f¼ gksrh gS vkSj ÝyDl c<+ tkrk gSA ysat+ osQ fu;e osQ vuqlkj izsfjr fo|qr okgd cy dks bl o`f¼ dk fojksèk djuk pkfg, ftls èkkjk esa deh }kjk izkIr fd;k tkrk gSA vkjEHk esa èkkrq osQ oy; ls dksbZ ÝyDl ugha xqtj jgk FkkA èkkjk izokfgr gksrs gh oy; ls ÝyDl xqtjus yxrk gSA ysat+ osQ fu;e osQ vuqlkj izsfjr fo|qr okgd cy bl o`f¼ dk fojksèk djsxk vkSj ;g rc gks ldrk gS tc oy; ifjukfydk ls nwj tk,A bldk foLr`r fo'ys"k.k fd;k tk ldrk gS (fp=k 6.5)A ;fn ifjukfydk esa èkkjk n'kkZ, vuqlkj gS rks ÝyDl (vèkkseq[kh) esa o`f¼ gksrh gS vkSj blls okekorZ (oy; osQ 'kh"kZ ls ns[kus ij) xfr mRiUu gksxhA tSls gh èkkjk 6.15 6.16 6.17 6.18 6.19 dk izokg ifjukfydk esa izokfgr èkkjk osQ foijhr gksrk gS] ;s ,d nwljs dks izfrdf"kZr djsaxs rFkk oy; mQij dh vksj xfr djsxkA tc ifjukfydk esa izokfgr fo|qr èkkjk esa deh gksrh gS] rks oy; esa èkkjk dh fn'kk] ifjukfydk osQ leku gh gksrh gSA bl izdkj ;gk¡ ,d vèkkseq[kh cy yxsxkA bldk ;g vFkZ gS fd oy; dkMZ cksMZ ij gh jgsxkA dkMZ cksMZ dh oy; ij mifjeq[kh izfrfozQ;k c<+ tk,xhA pqEcd osQ fy,] èkkrq osQ ikbi esa Hkaoj èkkjk,¡ mRiUu gksrh gSaA ;s èkkjk,¡ pqEcd dh xfr dk fojksèk djsaxhA blhfy,] pqEcd dk vèkkseq[kh Roj.k] xq#Roh; Roj.k ls de gksxkA blosQ foijhr] pqEcfdr yksgs dh NM+ esa Hkaoj èkkjk,¡ mRiUu ugha gksaxh vkSj og xq#Roh; Roj.k ls uhps fxjsxkA vr% pqEcd fxjus esa vfèkd le; yxsxkA oy; ls xqtjus okyk ÝyDl φ= Bo (πa 2 )cos ωt ε= B(πa 2)ωsin ωt I = B(πa 2)ωsin ωt/ R fofHkUu le;ksa ij èkkjk dk ifjek.k π B (πa 2 )ω t = ; I = , ˆj osQ vuqfn'k2ω R π t = ; I = 0 ω B πa2 )ω t = 3 π ; I =( , –ˆj osQ vuqfn'k 2 ω R gesa ÝyDl osQ fy, leku mÙkj izkIr gksxkA ÝyDl dks fdlh i`"B (ge fdlh {ks=kiQy Δ A ⊥ ls B rd dN = B Δ A js[kk,¡ [khprs gSa) ls xqtjus okyh pqEcdh; {ks=k js[kkvksa dh la[;k ekuk tk ldrk gSA ftl izdkj B dh js[kk,¡ fnDdky esa u rks vkjEHk gksrh gSa vkSj u gh var gksrh gSa (os cUn ik'k cukrh gSa)A i`"B S1 ls xqtjus okyh js[kkvksa dh la[;k i`"B S2 ls xqtjus okyh js[kkvksa dh la[;k osQ leku gksuh pkfg,A fcUnqfdr js[kk CD osQ vuqfn'k xfrd fo|qr {ks=k (v rFkk B nksuksa osQ yEcor~ rFkk v × B osQ vuqfn'k) = vBPQ osQ vuqfn'k E.M.F. = (yEckbZ PQ)×(PQ osQ vuqfn'k {ks=k) R d × vB cos θ= dvB = cos θ vr% dvB I = vkSj ;gθ ij fuHkZj ugha gSAR y ˆk x ( ,0,0)a BB D P B B C B BQ 6.20 6.21 6.22 y A C v OB x 6.23 y AX B R l Y x C x () tD èkkjk esa vfèkdre ifjorZu dh nj AB esa gSA vr% vfèkdre fojksèkh fo|qr fojksèkh cy izkIr gksus dk le; 5 s < t <10 s osQ chp gSA ⎛ dI ⎞;fn u = –L 1/5 ⎜t = 3s, ij= 1/5 ⎟ = e⎝ dt ⎠ 5 s < t < 10 s ij u1 = –L 3 = –3 L = 3e 55 bl izdkj t = 7 s, ij u2 = –3 e 10s < t < 30s ij 2 L 1 u = L == e3 20 10 2 t > 30s ij u3 = 0 10–2 vU;ksU; izsjdRo == 5mH 2 –3 –3ÝyDl = 5 ×10 × 1= 5× 10 Wb eku yhft,] lekUrj rkj y = 0 rFkk y = d gSaA le; t = 0 ij AB dh fLFkfr x=0 gS vkSj ;g osx vˆi ls xfr djrk gSA le; t ij] rkj dh fLFkfr gS x (t) = vt xfrd e.m.f = (Bo sin ωt)vd ( –ˆj) OBAC osQ vuqfn'k {ks=k esa ifjorZu osQ dkj.k e.m.f o ω () = –B ω cos txt d oqQy e.m.f = –Bd [ωx cos (ωt )+ v sin (ωt )]o Bd OBAC osQ vuqfn'k èkkjk (nf{k.kkorhZ) = o (ωx cos ωt + v sin ωt )R i osQ vuqfn'k vko';d cy Bd ˆ= o (ωx cos ωt + v sin ωt )× d × Bo sin ωt R 22Bd o= (ωx cos ωt + v sin ωt ) sin ωt R (i) eku yhft, le; t ij rkj dh fLFkfr x = x (t) gSA ÝyDl = B (t) lx (t) dφ dB () t tlv . () E = – = – lx () t – B () t dt dt (nwljk in xfrd fo|qr okgd cy ls gS) 1 I = E R lB () t ⎡ dB ⎤ ˆcy = – lx () – B () lv () it tt⎢ ⎥R ⎣ dt ⎦ 22 22dx lBdB lB dx m 2 = – x () t – dt Rdt Rdt 2 22dB dx lB dx (ii) = 0, += 0 dt dt 2 mR dt 22dv lB + v = 0 dt mR 22⎛ –lBt ⎞ v = A exp ⎜⎟⎝ mR ⎠ t = 0 ij, v = u v (t) = u exp (–l2B2t/mR) 22 2 22 2 222Blv () t Bl (iii) IR = 2 × R = u exp(–2 l Bt /mR ) RR t 22 22 22Bl mR 'kfDr {k; = I Rdt = u ⎡ –( l B t/mR ) ⎤∫ 22 ⎣1– e ⎦ 0 R 2lB mm = u 2– v 2 () t 22 = xfrt mQtkZ esa deh 6.24 le; t = 0 vkSj t =π osQ chp NM+ OP Hkqtk BD ls lEioZQ cuk,xhA eku yhft, NM+4ω ⎛ π ⎞dh lEioZQ dh yEckbZ OQ fdlh le; t ⎜0 < t < ⎟ ij x gSA {ks=kiQy ODQ ls⎝ 4ω ⎠ xqtjus okyk ÝyDl gS φ= B 1 QD×OD= B 1 l tan θ×l 22 A2l B12=B l ;gk¡ θ =ωt 2 P l dφ 1 ε Qbl izdkj mRiUu emf dk ifjek.k gS ε= = Bl 2ω sec 2ωt izokfgr èkkjk gS I= dt 2 R ;gk¡ R NM+ dh lEioZQ okyh yEckbZ dk izfrjksèk gS CO l D λl R =λ x = cos ωt C O R P B D ∴ 2 21 B B sec cos 2 2 cos l l I t t l t ω ω ω ω λ λ ω = = vUrjky 3 4 t π π < < ω ω esa NM+ Hkqtk AB osQ lEioZQ esa gSA eku yhft, NM+ osQ lEioZQ okys Hkkx dh yEckbZ (OQ) x gSA rc OQBD ls xqtjus okyh ÝyDl gS& 2 2 1 2 tan l Bl ⎛ ⎞ φ = +⎜ ⎟ θ⎝ ⎠ ;gk¡ θ = ωt bl izdkj mRiUu emf dk ifjek.k gS 2 2 2 1 sec 2 tan d t Bl dt t φ ω ε = = ω ω izokfgr èkkjk gS sin = = = t I R x l ε ε ε ω λ λ = 1 2 sin Bl t ω λ ω vUrjky 3 t π π < < ω ω ij NM+ Hkqtk OC dks Li'kZ djsxhA rc OQABD ls xqtjus okyk ÝyDl gS 2 2 2 – 2tan l Bl t ⎛ ⎞φ = ⎜ ⎟ ω⎝ ⎠ CO bl izdkj emf dk ifjek.k gS 22dφ Bωl sec ωt ε= = dt 2tans ωt εε 1Blω I== = R λx 2 λ sin ωt 6.25 rkj ls nwjh r ijB µ I B ( ) ={ks=k ro (dkxt+ osQ cfgeqZ[kh)2πr dr x rkj ls nwjh r ij pkSM+kbZ dr dh fdlh ifêðdk ij fopkj dhft, D r C ywi ls oqQy ÝyDl gS%x o I () tµoI xdr µoI xÝyDl = l = ln ∫2π r 2π x ox o 1 dI ε µol λ x == I = ln Rdt R 2π Rx0 6.26 ;fn ik'k esa izokfgr èkkjk I (t) gS] rc 1 dφ It () = R dt BA L1 ;fn le; t esa izokfgr vkos'k Q gS] rc L+ x2 dQ dQ 1 dφ () ==It or dt dt Rdt D C x 1 I (t)lekdyu djus ij Q (t1) – Q (t2 )= ⎣⎡φ(t1) – φ(t2 )⎦⎤ R L + x µ 2 dx ′ φ(t1)= L1 oI (t1 )∫2π x′ x µ L L + x o 1 2= I (t1) ln 2π x vkos'k dk ifjek.k µ LL + x o 12Q = ln [I –0 ]2π xo µ LI L + xo11 ⎛ 2 ⎞ = ln ⎜⎟2π ⎝ x ⎠ 2B.πa6.27 2 bE = .. ;gk¡ E ik'k osQ pkjksa vksj mRiUu fo|qr {ks=k gSAπ EM F = Δt ⎡ Bπa 2 ⎤cy vk?kw.kZ = b × cy = Q Eb = Qb⎢⎥⎣2πbΔt ⎦ 2Ba = Q 2Δt ;fn dks.kh; laosx esa ifjorZu ΔL gS] rc 2Ba ΔL = cy vk?kw.kZ × Δt = Q 2 vafre dks.kh; laosx = 0 vafre dks.kh; laosx 2 2 2 QBa = mb ω = 2 22 QBa mb ω = 6.28 2 2 d x m dt ( )cos sin – cos B d dx mg Bd R dt θ ⎛ ⎞ = θ × θ⎜ ⎟⎝ ⎠ dv dt g= sin –θ B d v mR 2 2 2(cos )θ dv dt B d v mR 2 2 2(cos )+ θ g= sin θ v 2 2 2 2 2 2 sin exp – (cos ) cos g B d A t B d mR mR θ ⎛ ⎞ = + θ⎜ ⎟⎛ θ ⎞ ⎝ ⎠ ⎜ ⎟⎝ ⎠ [tgk¡ A ,d fLFkjkad gS ftldk eku vkjafHkd voLFkkvksa ls fu/kZfjr gksrk gSA] = 2 2 2 2 2 2 sin 1 – exp – (cos ) cos mgR B d t B d mR ⎛ ⎞θ ⎛ ⎞ θ⎜ ⎟⎜ ⎟θ ⎝ ⎠⎝ ⎠ 6.29 ;fn laèkkfj=k ij vkos'k Q (t) gS (è;ku nhft,] èkkjk izokg A ls B dh vksj gS)] rc I vBd R = – Q RC C X Y S B B B A v d Q dQ vBd RC dt R ⇒ + = ∴ [ ] – / – /1– t RC t RC Q vBdC Ae Q vBdC e = + ⇒ = (le; t = 0 ij Q = 0 = A = –vBdc) –t / RC vBd I e R = 6.30 – dI L vBd IR dt + = dI L IR vBd dt + = – /2 I = vBd + Ae Rt XA R S BB v d t = 0 ij I = 0 ⇒ A = – vBd L BR B I ( )– /1– Rt L vBd e R = B B B 6.31 dφ dt = ÝyDl esa ifjorZu dh nj = (πl2) B o l dz dt = IR 2 ol B I v R π λ = izfr lsd.M mQtkZ {k; = I2 R = 2 22 2 ol B v R (π λ) ;g fLFkfrt mQtkZ esa ifjorZu dh nj ls izkIr gksuk pkfg, = m g dz dt = mgv (v= fu;r gksus osQ dkj.k xfrt mQtkZ fu;r gS) bl izdkj mgv = 2 0 2 2l B v R (π λ ) vFkok v = 2 )2 o mgR ( l Bπ λ 6.32 fdlh ifjukfydk osQ dkj.k pqEcdh; {ks=k 0B = µ nI NksVh oqQ.Myh esa pqEcdh; ÝyDl φ = NBA ;gk¡ A 2= πb vr% =e φ–d dt = – ( )d NBA dt e = d B N b dt 2 ( )– π = d N b nI dt 2 0 – ( )π µ t 2 dI = – N πb µ0n dt 22 2 = –Nn πµ 0bd (mt + C) = –µ0Nn πb 2mt dt e = –µ0Nn πb22mt ½.kkRed fpg~u izsfjr emf dk ifjek.k fp=k esa n'kkZ, vuqlkj le; osQ lkFk ifjofrZr gksrk gSA vè;k; 7 7.1 (d) 7.2 (c) 7.3 (c) 7.4 (b) 7.5 (c) 7.6 (c) 7.7 (a) 7.8 (a), (d) 7.9 (c), (d) 7.10 (a), (b), (d) 7.11 (a), (b), (c) 7.12 (c), (d) 7.13 (a), (d) 7.14 pqEcdh; mQtkZ xfrt mQtkZ osQ ln`'k rFkk oS|qr mQtkZ fLFkfrt mQtkZ osQ ln`'kA 7.15 mPp vko`fÙk ij] la/kfj=k ≈ y?kq iFk (fuEu izfr?kkr) rFkk izsjd [kqyk ifjiFk (mPp izfr?kkr) Z ≈ R1 + R3 tSlk rqY; ifjiFk esa n'kkZ;k x;k gSA 7.16 (a) gk¡] ;fn nksuksa ifjiFkksa esa rms oksYVrk leku gS rks vuqukn fLFkfr esa LCR esa rms èkkjk mruh gh gksxh ftruh R ifjiFk esaA (b) ugha] D;ksafd R ≤ Z, vr% Ia ≥ Ib 7.17 gk¡] ugha 7.18 cSaM pkSM+kbZ mu vko`fÙk;ksa osQ laxr gS ftu ij 1 Im = Imax ≈ 0.7 I 2 maxIm (A) ;g fp=k esa n'kkZ;k x;k gS Δω = 1.2 – 0.8 = 0.4 rad/s 7.19 Irms = 1.6 A fp=k esa fcUnqfdr js[kk }kjk fu:firA 1 2 3 A) –3 0 –1 –2 I ( T 2T t 7.20 ½.kkRed ls 'kwU; fiQj èkukRed] vuqukn vko`fÙk ij 'kwU;A 7.21 (a) A (b) 'kwU; (c) L vFkok C vFkok LC 7.22 a.c. èkkjk dh fn'kk lzksr dh vko`fÙk osQ lkFk cnyrh gS rFkk vkd"kZ.k cy dk vkSlr eku 'kwU; gks tkrk gS] vr% a.c. osQ lanHkZ esa ,fEi;j dks fdlh ,sls xq.k osQ inksa esa ifjHkkf"kr fd;k tkuk pkfg, tks èkkjk dh fn'kk ij fuHkZj u djrk gksA twy dk mQ"eu izHkko ,d ,slk gh xq.k gS vr% bldk mi;ksx a.c. osQ rms eku dks ifjHkkf"kr djus osQ fy, fd;k tk ldrk gSA 7.23 XL = ωL = 2pfL = 3.14Ω ; 3.3Ω ωL tan φ= = 3.14 R φ= tan –1 (3.14) ; 72° 72 ×π ; rad. 180 �(rad/s) le;i'prk Δt =φ ω 72 ×π 1 == s 180 × 2π× 50 250 7.24 PL = 60W, IL = 0.54A 60 VL == 110V 0.54 VªkaliQkWeZj vipk;h gS rFkk fuxZr oksYVrk fuos'k oksYVrk dh vkèkh gS] vr%] 1i = × I2= 0.27A p2 7.25 laèkkfj=k dh IysVksa osQ chp osQ vUrjky dk izfrjksèk vuUr gksus osQ dkj.k blls gksdj fn"Vèkkjk izokfgr ugha gks ldrhA laèkkfj=k dh IysVksa osQ chp tc izR;korhZ èkkjk yxkbZ tkrh gS rks bldh IysVsa ckjh&ckjh ls vkosf'kr vkSj vukosf'kr gksrh gSaA laèkkfj=k ls gksdj izokfgr gksus okyh èkkjk blh ifjorhZ oksYVrk (;k vkos'k) dk ifj.kke gSA vr% ;fn oksYVrk vfèkd nzqr xfr ls ifjofrZr gksrh gS rks laèkkfj=k ls vfèkd èkkjk izokfgr gksxhA bldk fufgrkFkZ ;g gS fd laèkkfj=k }kjk izLrqr izfr?kkr vko`fÙk c<+kus ls de gksrk gS% bldk eku gksrk gS 1/ωC 7.26 izsjd vius fljksa osQ chp ysUt osQ fu;e osQ vuqlkj fojksèkh fo|qr okgd cy fodflr djosQ vius esa ls izokfgr gksus okyh èkkjk dk fojksèk djrk gSA izsfjr oksYVrk dh èkzqork bl izdkj gksrh gS fd fo|eku èkkjk dk Lrj cuk jg losQA ;fn èkkjk de gksrh gS rks izsfjr emf dh èkzqork bl izdkj gksxh fd èkkjk c<+ losQ vkSj ;fn èkkjk c<+rh gS rks izsfjr emf dh èkzqork blosQ foijhr gksxhA D;ksafd izsfjr oksYVrk èkkjk ifjorZu dh nj osQ lekuqikrh gksrh gSA èkkjk ifjorZu dh nj vfèkd gksus ij vFkkZr vko`fÙk vfèkd gksus ij èkkjk izokg osQ izfr izsjd dk izfr?kkr vfèkd gks tk,xkA vr% izsjd dk izfr?kkr vko`fÙk osQ lekuqikrh gksrk gS vkSj bldk eku ωL. }kjk O;Dr fd;k tkrk gSA V2 50,000 7.27 'kfDr P = ⇒ = 25 = Z Z 2000 Z2 = R2 + (XC – XL)2 = 625 XC– X L3 tan φ= = – R 4 2 ⎛ 3 ⎞2 25 625 = R + – R = R⎜⎟⎝ 4 ⎠ 16 R2 = 400 ⇒ R = 20Ω XC – XL = –15Ω V 223 I == � 9A Z 25 IM = 2 × 9=12.6 A ;fn R, XC, XL lHkh dks nksxquk dj fn;k tk, rks tan φ esa dksbZ ifjorZu ugha gksrkA Z dks nksxquk djsa rks èkkjk vkèkh gks tkrh gSA 7.28 (i) Cu osQ rkjksa dk izfrjksèk] R l 1.7×10 –8 ×20000 =ρ = 2 = 4Ω A ⎛ 1⎞ –4 π × ×10 ⎜⎟⎝ 2⎠ 10 6 4220 V ij I : VI = 106 W ; I == 0.45 × 10 A 220 RI2 = {kfr {k;= 4 × (0.45)2 × 108 W> 106 W ;g fofèk lapj.k osQ fy, mi;ksx esa ugha ykbZ tk ldrhA (ii) V′I′ = 106 W = 11000 I′ 12I ′= ×10 1.1 12 44RI′= × 4×10 =3.3×10 W 1.21 3.3 ×10 4 izfr'kfDr {k; = =3.3% 10 6 v sin ωt m7.29 Ri = v sin ωt i = 1 m 1 R q dq 2 2 + L 22 = v sin ωt Cdt m Let q2 = qm sin (ωt + φ) ⎛ q ⎞ q m 2 sin( ωt +φ) = v sin ωt m ⎜ – Lω ⎟ m⎝ C ⎠ qm = vm , φ= 0; 1 – ω2L > 01 2 C– Lω C v 1 vR = m ,φ=π Lω2– > 0 Lw 2–1 C C dq i2 = 2 =ωqm cos( ωt +φ)dt i1 ,oa i2 leku dyk esa ugha gSaA ekuk fd 1– ω2L > 0 C vm sin ωt vm 12 1i + i =+ cos ωt R Lω – cω tgk¡ A sin ωt + B cos ωt = C sin (ωt + φ ) C cos φ = A, C sin φ = B; C = A2 + B2 1 ⎡v 2 22v ⎤ mm ωi + i = sin( t +φ )vr%] 12 ⎢ 2 + 2 ⎥ ⎣ R [ωl –1/ ωC] ⎦ φ= tan–1 R XL – XC 1/ 2 1 ⎧ 11 ⎫ =+⎨ 22 ⎬ Z ⎩R (L ω – 1/ ωC) ⎭ di 2 qi di d ⎛ 12 ⎞7.30 Li + Ri += vi ; Li = = izsjd esa laxzghr mQtkZ ifjorZu dh nj⎜ Li ⎟ dt c dt dt ⎝ 2 ⎠ Ri2 = twy mQ"eu {k; qd ⎛ q 2 ⎞ i = laèkkfj=k esa laxzghr mQtkZ ifjorZu dh nj⎜⎟C dt ⎝ 2C ⎠vi = izpkyd cy }kjk mQtkZ laHkj.k dh njA ;g mQtkZ iz;qDr gksrh gS (i) vkseh; {k; (ii) laxzghr mQtkZ o`f¼ esaA T TTd ⎛ 1 q 2 ⎞ 2dt 2 + Ri dt = vidt ∫⎜ i + ⎟∫ ∫dt ⎝ 2 C ⎠0 00 T 0 + (+ve ) = vidt ∫ 0 T vidt > 0 ;g rHkh vkSj osQoy rHkh laHko gS tc dyk&vUrj] tks vpj gksrk gS]∫ 0 U;wudks.k gksA 2dq dq q7.31 (i) L 2 + R += vm sin ωt dt dt C ekuk q = qm sin (ω t + φ) = – qm cos (ωt + φ ) i = i sin (ω t + φ ) = qω sin (wt + φ )mm vv X – Xmm –1 ⎛⎞i == ;φ= tan CL⎜⎟m 22ZR +(XC – XL ) ⎝ R ⎠ 2 11 ⎡ vm ⎤(ii) UL = Li 2 = L ⎢ 22 ⎥ 20sin ( ωt +φ)22 ⎢ R + X – X ) ⎥⎣ CL 0 ⎦ 2 1 q 21 ⎡ vm ⎤ 1 C ⎢ 22 ⎥ 2 20U == cos ( ωt +φ )2 C 2C ⎢ R + (X – X ) ⎥ ω⎣ CL ⎦ (iii) Lora=k NksM nsus ij ;g ,d LC nksfy=k gSA laèkkfj=k vukosf'kr gksrk tk,xk vkSj lEiw.kZ mQtkZ L esa pyh tk,xhA ;g ozQe myVk gksxk vkSj ckj&ckj ;g izfozQ;k nksgjkbZ tkrh jgsxhA vè;k; 8 8.1 (c) 8.2 (b) 8.3 (b) 8.4 (d) 8.5 (d) 8.6 (c) 8.7 (c) 8.8 (a), (d) 8.9 (a), (b), (c) 8.10 (b), (d) 8.11 (a), (c), (d) 8.12 (b), (d) 8.13 (a), (c), (d) 8.14 D;ksafd oS|qrpqacdh; rjaxsa lery /zqfor gksrh gSa] blfy, vfHkxzkgh ,sUVsuk rjax osQ oS|qr@pqacdh; Hkkx osQ lekarj gksuk pkfg,A 8.15 ekbozQksoso dh vko`fÙk ty osQ v.kqvksa dh vuqukn vko`fÙk ls esy [kkrh gSA dq 8.16 i = i == –2 πq ν sin2 πν t dt CD 0 8.17 vko`fÙk ?kVkus ij izfr?kkr Xc = 1 c<+sxk tks pkyu /kjk dks ?kVk,xkA bl fLFkfr esa ωC iD = iC; vr% foLFkkiu /kjk de gks tk,xhA 2 8 –82 8.18 1 B 13× 10 × (12 ×10 )Iav = c 0 =× –6 = 1.71 W /m 2 2 µ0 2 1.26 × 10 E y 8.20 fo|qrpqacdh; rjaxsa fofdj.k nkc yxkrh gSaA /weosQrq dh iw¡N lkSj fofdj.k osQ dkj.k gSA µ 2I µ 1 µ dφ0 D 00 EB = ==ε 4πr 4πr 2πr 0 dt µε d00 2 = (Eπr )2πr dt µε r dE = 00 2 dt 8.22 (a) λ 1 → ekbozQksoso(lw{e rjaxsa) λ2 → ijkcSaxuh rjaxsa λ3 → X-fdj.ksa λ4 → vojDr rjaxsaa (b) λ< λ<λ< λ3 2 4 1 (c) lw{e rjaxsa (ekbozQksoso)-jMkj ijkcSaxuh rjaxsa & ykfld us=k 'kY;rk X-fdj.ksa -vfLFkHkax ozQeoh{k.k vojDr rjaxsa -izdk'kh; lapkj 2 1 T 2 28.23 S = c ε E × B cos ( kx – ω ) D;ksafd S = c ε0(E × B)t dt av 0 00 T 0 ∫ 2 1 = c ε EB × 000 2 2 ⎛ E0 ⎞ 1 ⎛ E0 ⎞ 00 ⎜⎟⎜ ⎟= c ε E × Q c = ⎝ c ⎠ 2 ⎝ B0 ⎠ 12 =ε Ec 200 E2 ⎛ 1 ⎞ 0 c = = D;ksafd ⎜2µ c0 ⎝ dV 8.24 iD = C dt dV –3 –6 1 10 = 2 10 ×× dt dV 13 =× 10 = 5×10 V /s dt 2 vr% 5 × 102 V/s dk ifjorhZ foHkokUrj yxk dj yf{kr eku dh foLFkkiu èkkjk mRiUu dh tk ldrh gSA 8.25 nkc cy F 1 Δp ΔpP = = = (F == loa xs ifjoruZ dh nj ){k=kiQy sA A Δt Δt 1U (Δpc = U t le; esrjx }kjk inku dh xbmQtk= . =Δ aazZZ)A Δtc I ⎛ U ⎞ = ⎜ rhozrk I = ⎟ c ⎝ AΔt ⎠ 8.26 rhozrk ?kVdj ,d pkSFkkbZ jg tkrh gSA bldk dkj.k gS fd xksyh; {ks=k osQ {ks=kiQy 4π r2 esa lapfjr gksus ij izdk'k iqat dk foLrkj gksrk gS ysfdu ys”kj esa foLrkj ugha gksrk vkSj blfy, rhozrk fLFkj jgrh gSA 8.27 oS|qrpqacdh; rjax dk fo|qr {ks=k nksyk;eku {ks=k gS vkSj fdlh vkosf'kr d.k ij blosQ }kjk mRiUu fo|qr cy Hkh ,slk gh gksrk gSA iw.kZlkaf[;d pozQksa esa vkSlr ysus ij ;g fo|qr cy 'kwU; gS D;ksafd bldh fn'kk izR;sd vk/s pozQ esa ifjofrZr gks tkrh gSA vr% fo|qr {ks=k fofdj.k nkc esa ;ksxnku ugha djrkA λ eˆ s ˆE = j2πε oa 8.28 µ i o ˆB= i2π a µλ v o ˆ= i2πa 11 ⎛ λ ˆjs µo λv ⎞ S= (E×B) = ⎜ ˆj× iˆ⎟ µo µo ⎝ 2πε oa 2πa ⎠ 2–λ v ˆk= 224πε 0a 8.29 eku yhft, IysVksa osQ chp esa nwjh d gSA rc fo|qr {ks=k gksxk E= Vo sin(2 πν t)A pkyu èkkjkd 1 ?kuRo vkse osQ fu;e }kjk izkIr gksxh& Jc = sE = ρ E 1 Vo V0 ⇒ J c = sin (2πν t ) = sin(2 πν t)ρ d ρd c= J sin 2πν t o Vtgk¡ ij J 0 c = 0 ρd ∂ E ∂ foLFkkiu /kjk ?kuRo izkIr gksxk Jd =ε =ε {Vo sin(2 πν t)}dt dt d ε 2πν Vo = cos(2 πvt ) d d ∂ E ∂ V oJ =ε=ε { sin(2 πν t)}dt dt d ε 2πν V = o cos (2 πν t)d d d2πνε V 0 = Jo cos(2πν t ), tgk¡ J0 = d 2πνε Vo ρd d V oo = . =2πνερ = 2π×80 εν ×0.25 = 4πε ν ×10 o 10 ν 4 = = 910 9 9× 8.30 (i) foLFkkiu /kjk ?kuRo fuEu laca/ ls Kkr fd;k tk ldrk gS dEJ =εD 0 dt ∂ ⎛ s ⎞ ˆ= εµ Icos (2πν t). ln k0 00 ⎜⎟∂t ⎝ a ⎠ 1 ⎛ s ⎞ = 2 I02πν 2 (– sin (2πν t )) ln ⎜⎟ kˆ c ⎝ a ⎠ ⎛ν ⎞2 ⎛ a ⎞ ˆ= ⎜⎟ 2π I0 sin (2πν t) ln ⎜⎟k ⎝ c ⎠⎝ s ⎠ 2π ⎛ a ⎞ ˆ= 2 I0ln ⎜⎟ sin (2πν t ) k λ ⎝ s ⎠ (ii) Id = J sdsdθ∫ D 2π a ⎛ a ⎞ = 2 I02π ∫ ln ⎜⎟.sds sin (2πν t ) λ ⎝ s ⎠ s =0 ⎛ 2π ⎞2 a 12 ⎛ a ⎞ = ⎜⎟ I0 ∫ ds l n ⎜⎟.sin (2πν t )⎝ λ ⎠ 2 ⎝ s ⎠ s =0 22 a 22 a ⎛ 2π ⎞⎛ s ⎞⎛ a ⎞ = ⎜⎟ I0 ∫ d ⎜⎟ ln ⎜⎟ .sin (2πν t )4 ⎝ λ ⎠⎝ a ⎠⎝ s ⎠ s =0 2 21 a ⎛ 2π ⎞ = – ⎜⎟ I0 ∫ ln ξ d ξ.sin (2πν t )4 ⎝ λ ⎠ 0 ⎛ a ⎞2 ⎛ 2π ⎞2 =+ ⎜⎟⎜ ⎟ I0 sin2 πν t (lekdy dk eku –1 gS)⎝ 2 ⎠⎝ λ ⎠(iii) foLFkkiu èkkjk 2 d ⎛ a 2π ⎞ dI = ⎜ . ⎟ I0 sin 2 πν t =I 0 sin 2πν t ⎝ 2 λ ⎠ Id 0 ⎛ aπ ⎞2 = ⎜⎟ . I0 ⎝ λ ⎠ x 8.31 (i) E 234 143E=E î x E.dl = E.dl + E.dl + E.dl + E.dl Ñ∫ ∫∫∫∫ 123 4 h dl dl 2 34 1 = E.dl cos 90° + E.dl cos 0 + E.dl cos 90° + E.dl cos 180° 1 2 ∫ ∫∫ ∫ 1 23 4z1 z2 Zdl B=B0 ˆj = E h sin kz [( – t )– (– ωt)] ω sin kz (1)02 1 y x (ii) . dk ewY;koaQu djus osQ fy, izR;sd dk {ks=kiQy ds =∫ B ds h dz dh ifg;ksa ls cus vk;r 1234 ij fopkj djsaA 43 Z 2 B ds 0( ). = Bds cos0 = Bds = B sin kz – ωt hdz dl ∫∫ ∫∫ Z1 1 2 z1 z2 z –Bh dz = o [cos( kz 2–ωt) – cos( kz 1– ωt)] (2) B=By ˆj k y 154 –dφB(iii) Ñ∫ E.dl= dt lehdj.kksa (1) rFkk (2)esa izkIr lac/ksa dk mi;ksx djosQ rFkk ljyhdj.k }kjk gesa izkIr gksxk [( – ωt )– (– ωt )] = Bh ω[( – ωt )– (Ehsinkz sinkz osinkz sinkz – 021 21k ω E = B00 k E0 = cy B0 (iv) �∫ B.dl osQ ewY;koaQu osQ fy, ywi 1234 ij yz ry esa fp=k }kjk n'kkZ, vuqlkj fopkj djsa 2 3 41 B.dl = B.dl + B.dl + B.dl + B.dl Ñ∫ ∫∫∫∫ 1 2 34 234 1 = Bdl cos0 + B dl cos 90° + Bdl cos 180° + Bdl cos 90 °∫∫∫ ∫ 123 4 0[( 1– t )– ( 2– t )] (3)= B h sin kz ω sin kz ω φE = ∫ E.ds dk ewY;koaQu djus osQ fy,] izR;sd {ks=kiQy dh ifg;ksa ls cus vk;r 1234 ij fopkj djsaA Z φE = ∫ E ds = ∫ Eds cos0 = ∫ Eds = ∫ 2 0( 1– ωt hdz . E sin kz ) Z1 –Eh = 0[cos( kz 2– ωt ) – cos( kz 1– ωt )] k dφE Eh 0 ω ∴ [( 1– ωt )–( 2– ωt )] (4)= sin kz sin kz dt k . =µ ⎛ I +ε dφE ⎞⎟ , I = pkyu /kjk∫ B dl⎜00 ⎠Ñ ⎝ dt = 0 fuokZr esa 155 ∴ . =µ00 dφE∫ B dlεÑ dt lehdj.kksa (3) rFkk (4) esa izkIr lac/ksa dk mi;ksx djosQ rFkk ljy djus ij gesa izkIr gksrk gS& ω B = E .µε0 0 00k E0 ω 1 =ysfdu E0/B0 = c, rFkk w = ckB k µε 0 00 1;k cc . = vr%, c = µε 00 +1 0 2 128.32 (i) E -{ks=k dk ;ksxnku gS uE =ε0 E 2 1 B2 B -{ks=k dk ;ksxnku gS u B = 2 µ0 1 21 B2 oqQy mQtkZ ?kuRo u = u + u =ε E + (1)EB 022 µ0 E2 rFkk B2 osQ eku izR;sd fcUnq rFkk izR;sd {k.k ij ifjofrZr gksrs gSaA vr% E2 rFkk B2 osQ izHkkoh eku muosQ dkfyd eku gSaA 2 22 av 0 av (E ) =E [sin (kz –ωt)] (B 2)= (B 2) = B02[sin 2(kz – ωt )]av av avsin2θ rFkk cos2θ osQ xzkiQ vko`Qfr esa le:i gSa ysfdu π/2 ls LFkkukUrfjr gSa] vr% sin2θ rFkk cos2θ osQ vkSlr eku Hkh π osQ fdlh Hkh iw.kZlkaf[;d xq.kt osQ fy, leku gSaA rFkk sin2θ + cos2θ =1 1 vr% lefefr ls sin2θ dk vkSlr = cos2θ dk vkSlr = 2 11222 2∴ (E ) = E and (B ) = Bav 0 av 022 lehdj.k 1 esa izLFkkiu djus ij 1 1 2 2 B0 u =ε E + 40 4 µ 222 2E0 11 B0 E0/cE0 12(ii) gesa Kkr gS = c rFkk c = ∴ = =µε =ε0E0 B µε 4 µ 4µ 4µ 0 040 00 000 1 11 1222 2 u = ε E +ε E =ε E , rFkk I = uc =ε Eav 0000 00 av av 004 42 2 Chapter 9 9.1 (a) 9.2 (d) 9.3 (c) 9.4 (b) 9.5 (c) 9.6 (c) 9.7 (b) 9.8 (b) 9.9 (b) 9.10 (c) 9.11 (a) 9.12 (a), (b), (c) 9.13 (d) 9.14 (a), (d) 9.15 (a), (b) 9.16 (a), (b), (c) 9.17 D;ksafd yky izdk'k osQ fy, viorZukad uhys osQ fy, viorZukad ls de gS] blfy, ysal ij vkifrr lekUrj izdk'k iqat yky izdk'k dh vis{kk uhys izdk'k dh fLFkfr esa v{k dh vksj vf/d eqM+sxkA blfy, yky izdk'k dh vis{kk uhys izdk'k osQ fy, iQksdl nwjh de gksxhA 9.18 lkekU; O;fDr dh fudV n`f"V 25 cm gSA fdlh fcac dks 10 xquk vkof/Zr ns[kus osQ fy, D 25 D ⇒ f == = 2.5 = 0.025mm = m 10 f P = 1 = 40 MkbvkWIVj 0.025 9.19 ughaA ysal dks myVk djus ij izfrfcac dh fLFkfr esa ifjorZu ugha gksxkA (izdk'k dh mRozQe.kh;rk) 9.20 eku yhft, µ2 ls fcac dks ns[kus ij vkHkklh xgjkbZ O1 gSA µ hO = 2 1 µ13 µ3 ls ns[kus ij vkHkklh xgjkbZ O2 gSA µ ⎛ h ⎞ µ ⎛ h µ h ⎞ h ⎛ µµ ⎞3 32 33O =+ O =+ =+2 ⎜ 1 ⎟⎜ ⎟⎜⎟ µ2 ⎝ 3 ⎠ µ2 ⎝ 3 µ13⎠ 3 ⎝ µ2 µ1 ⎠ ckgj ls ns[kus ij vkHkklh mQ¡pkbZ 1 ⎛ h ⎞ 1 ⎡hh ⎛ µ3 µ3 ⎞⎤ = ++O3 = ⎜ +O 2 ⎟ ⎢⎜ ⎟⎥ µ3 ⎝ 3 ⎠µ3 ⎣ 33 ⎝ µ2 µ1 ⎠⎦ Oh ⎛ 111 ⎞ = ++⎜⎟3 ⎝ µ1 µ2 µ3 ⎠ 9.21 U;wure fopyu ij ⎡ (A + D )⎤sin ⎢ m ⎥⎣ 2 ⎦µ= ⎛ A ⎞sin ⎜⎟⎝ 2 ⎠ fn;k gS Dm = A AA2sin cos sin A A22∴µ== = 2cos AA 2sin sin 22 AA 3 ∴ cos = vFkok = 30° ∴ A = 60°222 9.22 eku yhft, fcac osQ nks fljs ozQe'k% fcac nwjh u1= u – L/2 rFkk u2 = u + L/2 ij gSa] ftlls |u–u|= LA eku yhft, nks fljkas osQ izfrfcac vrFkk vij curs gSa] bl izdkj izfrfcac12 12 11 1 fu += ;k v =dh yEckbZ gksxh L′= . D;ksafd , nks fljksa dk izfrfcacv – v12 uv f u – f f (u – L /2 ) f (u + L /2 )gksxk v1 = ij v2 = iju – f – L /2 u – f + L /2 vr% L′=|v –v |= f 2L 12 2(u – f ) × L2/4 D;ksafd fcac NksVk gS rFkk iQksdl ls nwj j[kk x;k gS blfy, ge ik,¡xs L2/4 << (u –f )2 vr% vfUrer% 2f L′= L. (u – f )2 9.23 fp=k osQ lanHkZ esa] nzo Hkjus ls igys vkifrr fdj.k dh fn'kk AM gSA nzo Hkjus osQ i'pkr vkifrr fdj.k dh fn'kk BM gSA nksuksa fLFkfr;ksa esa viofrZr fdj.k AM osQ vuqfn'k ,d gh gSA a + RrFkk sin α= cos(90 – α) = d 2 + (a – R )2 µ(a 2– R2)izfrLFkkfir djus ij gesa izkIr gksxk d = (a + R )2 – µ(a – R)2 ;fn dkVk u tkrk rks fcac eq[; v{k 00′ ls 0.5 cm dh mQ¡pkbZ ij gksrkA 11 1 – = vu f 11 1 1 11∴= += += vuf –50 25 50 ∴ v = 50 cm v 50 vko/Zu m = =– = –1 u 50 vr% izfrfcac izdkf'kd osQUnz ls 50 cm nwj rFkk eq[; v{k ls 0.5 cm uhps cusxkA bl izdkj dVs gq, ysal dh dksj ls xq”kjus okyh X v{k osQ lkis{k izfrfcac osQ funsZ'kkad (50 cm, –1 cm) gSaA 9.25 ysal lw=k 1 11 = – f vu dks ns[kus ij u rFkk v dh mRozQe.kh;rk ls ;g Li"V gSA ,slh nks fLFkfr;k¡ gSa ftuosQ fy, ijns ij izfrfcac cusxkA eku yhft, igyh fLFkfr og gS tc ysal O ij gSA fn;k gS –u + v = D ⇒ u = –(D – v) bls ysal lw=k esa j[kus ij 1 11 += D – vv f v + D –v 1 ⇒ = (D –vv f) ⇒ v2 – Dv + Df = 0 22 ⎛ D– u = –(D – v) = ⎜ ± ⎝ 22 ⎠ DD2–4 Df DD 2–4 Df bl izdkj ;fn fcac nwjh – gS rks izfrfcac + ij gksxkA22 22 DD2–4 Df DD2 –4Df ;fn fcac nwjh + gS rks izfrfcac – ij gksxkA22 22 bu nks fcac nwfj;ksa osQ fy, izdkf'kd osQUnzksa osQ chp dh nwjh gS DD2–4 Df ⎛ 2 ⎞DD – 4D f 2+ – = D –4 Df ⎜ – ⎟22 ⎝ 22 ⎠ ekuk d = D2–4Df Dd Dd ;fn u = + rc izfrfcac gksxk v = – ij22 22 D–d ∴ vko/Zu m1 = D +d D–d D+ d;fn u = rc v = 22 D+ dm 2 ⎛ D+ d ⎞ 2 ∴ vko/Zu m 2 = vr% = ⎜⎟D– dm1 ⎝ D–d ⎠ d 9.26 eku yhft, fMLd dk O;kl d gSA fcanq vn`'; gks tk,xk ;fn fcanq ls i`"B ij vkifrr fdj.ksa2 ozQkafrd dks.k ij gksaA eku yhft, vkiru dks.k i gS 1 rc sin i = µ d /2 vc = tan i h⇒ d = h tan i = h ⎡⎣2 2h ∴d = µ 2 –1 9.27 (i) eku yhft, lkekU; foJkUr us=k osQ fy, nwj fcanq ij {kerk Pf gSA 11 1 rc P = = + = 60 MkbvkWIVjff 0.1 0.02 la'kks/d ysal osQ lkFk nwj fcanq ij fcac nwjh ∞ gSA 11 1 Pf ′==+ = 50D f ′∞ 0.02 p'es osQ lkFk foJkar us=k dh izHkkoh {kerk us=k rFkk p'es osQ ysalksa Pg dk ;ksx gSA ∴ Pf ′ = P + Pfg ∴ P = – 10 D g(ii) lkekU; us=k osQ fy, mldh leatu {kerk 4 MkbvkWIVj gSA eku yhft, fd lkekU; us=k dh fudV n`f"V dh {kerk Pn gS rc 4 = P – Por P = 64 D nf neku yhft, mldk fudV fcanq x n gks] rc 11 1 += 64 vFkok + 50 = 64 x 0.02 x n n 1 = 14 x n ∴ x n = 1 14 ; 0.07m (iii) p'es osQ lkFk n P′ fP′ = 4 54 + = 54 = 1 n x′ 1 1 0.02 n x + = ′ 50 + n x 1 ′ = 4 ∴ nx ′ 1 4 = 0.25m = 9.28 dksbZ fdj.k tks dks.k i ls izos'k djrh gS] AC osQ vuqfn'k funsZf'kr gksxh ;fn iQyd AC ls cuk;k x;k dks.k (φ ) ozQkafrd dks.k ls vf/d gSA ⇒ sin ≥ 1 µ ⇒ cos r ≥ 1 µ vFkok 1 – cos2r ≤ 1 – 2 1 µ B D i.e. sin2r ≤ 1 – 2 1 µ D;ksafd sin i = µ sin r 2 1 µ sin2i ≤ 1 – 2 1 µ ;ksin2i ≤ µ2 – 1 tc i = 2 π rks φ NksVs ls NksVk dks.k gksxkA ;fn ;g ozQkafrd dks.k ls cM+k gS rc lHkh nwljs vkiru dks.k ozQkafrd dks.k ls vf/d gksaxsA vr% 1 ≤ µ2 –1 ;kµ2 ≥ 2 ⇒ µ ≥ 2 9.29 nzo osQ vUnj x rFkk x + dx osQ chp ,d fdj.k osQ fdlh Hkkx ij fopkj dhft,A eku yhft, x ij vkiru dks.k θ gS vkSj eku yhft, ;g irys LraHk esa y mQ¡pkbZ ij izos'k djrh gSA cadu osQ dkj.k ;g dks.k θ + dθ ls y + dy mQ¡pkbZ ij rFkk x + dx fuxZr gksxhA LuSy osQ fu;e ls& µ(y) sin θ = m(y+dy) sin (θ+dθ) ;k µ(y) sinθ ; ( ) d y dy dy µµ ⎛ ⎞ +⎜ ⎟⎝ ⎠ (sinθ cosdθ + cosθ sin dθ ) dµ( )sin θ+µ( )cos y θ θ+ dy ; µ y d sin θ dy y –dµvFkok µ(y) cosθdθ ; dy sin θ dy –1 dµdθ ; dy tan θ µ dy dx ysfdu tanθ = (fp=k ls)dy –1 dµ∴ dθ = dx µ dy –1 dµ d –1 dµ∴θ = dx = d∫µ dy o µ dy dy � + d� (y + dy) dx 9.30 r rFkk r + dr ij nks ryksa ij fopkj djsaA eku yhft, ry r ij izdk'k θ dks.k ls vkifrr gksrk gS rFkk r + dr ls θ +dθ dks.k ls ckgj fudyrk gSA rc Lusy osQ fu;e ls n(r) sinθ = n(r + dr) sin (θ + dθ) ⎛ dn ⎞ ⇒ n(r) sinθ ; ⎜ nr () + dr ⎟ (sinθ cos dθ + cosθ sin dθ )⎝ dr ⎠ ⎛ dn ⎞;⎜ nr () + dr ⎟ (sinθ + cosθ dθ )⎝ dr ⎠ vody xq.ku iQyksa dks NksM+us ij () θ ; () θ+ dn dr sinθ + n(r) cosθdθnrsin nrsin dr dn dθ ⇒ – tan θ= nr () dr dr 2GM ⎛ 2GM ⎞ dθ dθ⇒ tan θ= 1 +≈22 ⎜ 2 ⎟ rc ⎝ rc ⎠ dr dr θ o ∞2GM tan θdr ∴ dθ=∫ 2 ∫ 2 0 c – ∞ r r + dr r R vc r2 = x2 +R2 rFkk tanθ = x 2rdr = 2xdx θ o ∞2GM R xdx dθ=∫ 2 ∫ 3 cx0– ∞ 2 22(x + R ) x = R tan φ jf[k, x = R tan φ dx = R Sec2 φ d φ π /2 22GMR Rsec φ dφ ∴θ = 02 ∫ 33c R sec φ – π /2 π /2 2GM 4GM = cos φ dφ = 2 ∫ 2Rc Rc –π /2 9.31 D;ksafd inkFkZ –1 viorZukad dk gS] θ r ½.kkRed gS rFkk θ′ /ukRed gSA vc = θ = θ′ r θi r r ckgj fudyus okyh fdj.k dk vUnj vkus okyh fdj.k ls oqQy fopyu 4θi gSA fdj.ksa xzkgh IysV rd ugha igq¡psaxh ;fn π 3π ≤ 4θ≤ (dks.k y &v{k ls nf{k.kkorZ ekis x, gSa)2 i 2 π 3π ≤θi ≤88 x vc sin θ= i R π –1 x 3π ≤ sin ≤8 R 8 π x 3π vFkok ≤≤ 8 R 8 Rπ R3πvr% ≤ x ≤ osQ fy, lzksr ls mRlftZr izdk'k xzkgh IysV rd ugha igq¡psxkA88 9.32 (i) S ls P1 rd ikjxeu dk le; gS SP u 2 + b 2 u ⎛ 1 b2 ⎞ t1 = 1 = ; ⎜1+ 2 ⎟ eku yhft, b << u0c cc ⎝ 2 u ⎠ P1 ls O rd ikjxeu dk le; gS PO v 2 + b2 v ⎛ 1 b2 ⎞ t = 1 = ; 12 ⎜ + ⎟ c cc ⎝ 2 v 2 ⎠ ysal ls ikjxeu dk le; gS (n –1) () wb tl = tgk¡ n viorZukad gSA c vr% oqQy le; gS 1 ⎡ 12 ⎛ 11⎞⎤ 1 11 ⎢u +v + b ⎜ + ⎟ + (n –1) () ⎥ . =+ jf[k,t = wb c ⎣ 2 ⎝ uv ⎠⎦ Duv P1 1 ⎛ 1 b2 ⎛ b 2 ⎞⎞ rc t = ⎜u + v ++(n –1) ⎜w0 + ⎟⎟ Sc ⎝ 2 D ⎝ α ⎠⎠ iQjeSV osQ fl¼kUr ls dt b 2( n –1) b = 0 = – db CD cα α= 2( n –1) D vr% ;fn α= 2( n –1) D rks vfHklj.kdkjh ysal cusxkA ;g b ls LorU=k gS vkSj blfy, S ls vkus okyh lHkh mik{kh; fdj.ksa O ij vfHklfjr gksaxh (vFkkZr b << n rFkk b << v fdj.kksa osQ fy,)A D;ksafd 1 = 1 + 1, iQksdl nwjh gSA D uv (ii) bl fLFkfr esa 1⎛ 1 b2 ⎛ k2 ⎞⎞ u +v + +()k ln t = ⎜ n −11 ⎜ ⎟⎟ c ⎝ 2D ⎝ b ⎠⎠ dt b k = 0 = –( n –1) 1 db D b ⇒ b2 = (n – 1) k1D ∴ b = (n –1) kD 1 vr% mQ¡pkbZ ls xqtjus okyh lHkh fdj.ksa izfrfcac cukus esa ;ksxnku nsaxhA fdj.k iFk }kjk cuk;k x;k dks.k b (n –1) kD (n –1) k uv (n –1) ku 11 1β ; = 22 = = vv v (u + v)(u + vv ) vè;k; 10 10.1 (c) 10.2 (a) 10.3 (a) 10.4 (c) 10.5 (d) 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 (a), (b), (d) (b), (d) (a), (b) (a), (b) gk¡ xksyh; xksyh;] i`Foh dh f=kT;k dh rqyuk esa fo'kky f=kT;k ftlls fd ;g yxHkx lery gSA èofu rjaxksa dh vko`fÙk;k¡ 20 Hz ls 20 kHz gksrh gSaA laxr rjaxnS?;Z ozQe'k% 15m rFkk 15mm gSA foorZu izHkko fn[kkbZ nsxk ;fn f>fj;ksa dh pkSM+kbZ a ,slh gks fd a � λ izdk'k rjaxksa osQ fy, rjaxnS?;Z � 10–7mA vr% foorZu izHkko fn[kkbZ nsxk tc – a � 10 7 m tcfd èofu rjaxksas osQ fy, ;s fn[kkbZ nsaxs 15mm < a 15m < nks fcanqvksa osQ chp jSf[kd nwjh l = 2.54 cm 300 ; –2 0.84 10 cm × gSA Z cm nwjh ij ;g dks.k φ : /l z z∴ = l φ = –2 –4 0.84 10 cm 5.8 10 × × : 14.5cm osQoy fo'ks"k fLFkfr;ksa esa tc (III) dh ikfjr v{k (I) ;k (II) osQ lekUrj gS rks dksbZ izdk'k fuxZr ugha gksxkA nwljh lHkh fLFkfr;ksa esa izdk'k fuxZr gksxk D;ksafd (II) dh ikfjr v{k (III) osQ yacor ugha gSA ijkorZu }kjk èzkqo.k rc gksrk gS tc vkiru dks.k czwLVj dks.k osQ cjkcj gks vFkkZr~ n tan θB = 2 tgk¡ n2 < n1n1 tc ,sls ekè;e esa izdk'k xeu djrk gS rks ozQkafrd dks.k gS D;ksafd cM+s dks.kksa osQ fy, |tan θB|>|sin θ c|θB <θC blfy, ijkorZu }kjk fuf'pr :i ls èkzqo.k gksxkA 1.22 λ d = min 2sin β tgk¡ β vfHkn`';d }kjk fcac ij varfjr dks.k gSA 5500 Å osQ izdk'k osQ fy, nsin θc = 2 tgk¡ n2 < n1. n1 1.22 × 5.5 ×10 –7 d = mmin 2sin β 100V ls Rofjr bysDVªkWuksa osQ fy, ns czkXyh rjaxnS?;Z gS h 1.227 λ== = 0.13nm = 0.13 ×10 –9 m p 100 1.22 ×1.3 ×10 –10 ∴ d ' = min 2sin β d 'min 1.3 ×10 –10 –3 ∴= : 0.2 ×10 dmin 5.5 ×10 –7 10.18 T2P = D + x, T1 P = D – x = [D2 + (D – x)2]1/2 SP = [D2 + (D + x)2]1/2 2fufEu"B izkIr gksxk tc λ [D 2 + (D + x)2]1/2 – [D2 + (D – x)2]1/2 = 2 ;fn x = D λ (D2 + 4D2)1/2 = 2 λ (5D2)1/2 = 2 λ .∴ D = 25 10.19 cxSj P osQ A = A ⊥+ A11 120 0A = A + A = A sin( kx –ωt )+A sin( kx – ωt +φ )⊥⊥ ⊥⊥ ⊥ (1) (2) 11 11 11 A = A + A A = A 0 [sin( kx – wt ) + sin( kx – ωt +φ ]11 11 tgk¡ A0, A0 fdlh Hkh fdj.k iaqt osQ ⊥ rFkk 11 èkzqo.kksa esa vk;ke gSaA⊥ 11 ∴ rhozrk 2 22 222A0 A0 }[sin (kx – wt ) (1+cos φ+ 2sin φ )+sin ( kx –ωt ) sin φ]+={ ⊥ 11 vklS r 2 2 ⎛ 1 ⎞ A0 A0 }⎜⎟.2(1+cos φ )+={ ⊥ 11 ⎝ 2 ⎠02 A0 A0= 2 .(1 + cos φ)since vkSlr = vkSlrA ⊥ 11⊥ P osQ lkFk% A12 ⊥ekuk vo#¼ gS 1 22 12rhozrk = = (A + A) + (A )11 11 ⊥ 21 A0 A0= 2 (1 + cos φ)+ .⊥ ⊥ 2 02fn;k gS: I0 = 4 = cxSj iksysjkbtj osQ eq[; mfPp"B rhozrk iksysjkbtj osQ lkFk eq[; mfPp"B ij rhozrk A ⊥ 02 ⎛ 1 ⎞ = A ⎜ 2 + ⎟⊥ ⎝ 2⎠ 5 = 8 I0 iksysjkbtj osQ lkFk eq[; mfPp"B ij rhozrk 02 A ⊥02 = (1–1)+A ⊥ 2 I = 0 8 10.20 iFkkarj = 2d sin θ+ (µ –1) l ∴ eq[; mfPp"B osQ fy, 2d sin θ+ 0.5 l = 0 –l –1 ⎛ d ⎞sin θ0 == ⎜ Q l = ⎟4d 16 ⎝ 4 ⎠ DOP = D tan θ≈ – 16 ∴ 0 izFke fufEu"B osQ fy, λ2d sin θ +0.5 l = ± 2 ±λ/2 – 0.5 l ±λ /2 – λ/8 11 ∴ 1 sin θ1 = ==± – 2d 2λ 4 16 3 θ+èkukRed fn'kk esa: sin = 16 –5½.kkRed fn'kk esa: sin θ= – 16 èkukRed fn'kk esa izFke eq[; mfPp"B dh nwjh + sin θ+ 3 D tan θ= D =D O osQ mQij2 221 – sin θ 16 –3 5 ½.kkRed fn'kk esa nwjh gksxh D tan θ – =2 2 O osQ uhps16 –5 10.21 (i) R1 tks A ls d nwjh ij gS] ij fo{kksHkksa osQ ckjs esa fopkj djsaA eku yhft, A osQ dkj.k R1 ij rjax gSYA = a cos ωtA A ls laosQr dk B ls iFkkUrj λ/2 gS rFkk bl izdkj dykUrj π gSA R2 bl izdkj B osQ dkj.k R1 ij rjax gS yB = a cos( ωt – π ) = –a cos ωt C ls laosQr dk A ls iFkkUrj π gS vkSj bl izdkj dykUrj 2π gSA vr% C osQ dkj.k R1 ij rjax gS yc = a cos ωtA D rFkk A ls �/2 �/2 laosQr osQ chp iFkkUrj gS R1 A B C 2 ⎛ λ ⎞2 �/2d + ⎜⎟ −(d −λ /2 )⎝ 2 ⎠ D 1 /2 λλ⎛⎞ = d ⎜1 + 2 ⎟ − d + ⎝ 4d ⎠ 2 1/ 2 ⎛ λ2 ⎞ λ = d ⎜1 + 2 ⎟ − d + ⎝ 8d ⎠ 2 iFkkUrj :λ vkSj blfy, dykUrj π gSA2 ∴ yD =− a cos ωt R1 ij izkIr gksus okyk laosQr gS y + y+ y+ y= 0AB C D eku yhft, B ls Rij izkIr gksus okyk laosQr gSy= a cos ωtA B rFkk D ij2B 1laosQrksa osQ chp iFkkUrj λ/2 gS ∴ yD = –a1cosωt A rFkk B ij laosQr osQ chp iFkkUrj gS 2 1/2 d 2 ⎛ λ ⎞⎛ λ2 ⎞ : 1 λ2 () +−d = d 1+− d⎜⎟ ⎜⎟ 2⎝ 2⎠⎝ 4d2 ⎠ 8 d 2πλ2 πλ ∴ dykUrj gS . = =φ : 0 8λ d24d vr% yA = a1 cos (ωt-φ) blh izdkj yC = a1 cos (ωt-φ) ∴ R2 }kjk p;fur laosQr gS y+ y+ y+ y= y = 2acos (ωt-φ)A BCD 1 2 22|| = 4a1 cos ( ωt –φ)∴ y ∴ I = 2a12 vr% R1 o`gr laosQr p;u djrk gSA (ii) ;fn B dks cUn dj fn;k tk, R1 p;u djrk gS y = a cos ω t 12I∴ = aR1 2 R2 p;u djrk gS y = a cos ω t 12I∴ = aR 212 bl izdkj R1 rFkk R2 leku laosQr p;u djrs gSaA (iii);fn D dks cUn dj fn;k tk, R1 p;u djrk gS y = a cos ω t 12∴ = aIR1 2 R2 p;u djrk gS y = 3a cos ω t 12∴ = 9aIR2 2 bl izdkj R2, R1 rqyuk esa o`gr laosQr p;u djrk gSA (iv)bl izdkj R1 ij laosQr n'kkZrk gS fd B cUn dj fn;k x;k gS rFkk R2 ij ,d o`gr laosQr n'kkZrk gS D dks cUn dj fn;k x;k gSA 10.22 (i) eku yhft, fd vfHkèkkj.kk lgh gS] rc nks lekUrj fdj.ksa fp=k esa n'kkZ, vuqlkj vxzlj gksrh gSaA eku yhft, ED rjaxkxz dks n'kkZrk gS rks bl ij reke fcanq leku dyk esa gksus pkfg,A leku izdkf'kd iFk yEckbZ osQ lHkh fcanq leku dyk esa gksus pkfg,A vr% – εµ AE = BC – εµ CD ;k BC = εµ (CD − AE ) rr rr rr pwafd BC > 0, CD > AE ;g n'kkZrk gS fd vfHkèkkj.kk ;qfDrlaxr gSA rFkkfi] ;fn izdk'k mlh izdkj vxzflr gksrk gS tSls ;g lkèkkj.k inkFkksZa esa gksrk gS (vFkkZr] pkSFks prqFkkZa'k esa fp=k 2) rc – εµ AE = BC – εµ CD rr rr ;k] BC = εµ (CD − AE )rr D;ksafd AE > CD, BC < O ;g n'kkZrs gq, fd ,slk lEHko ugha gSA vr% vfHkèkkj.kk lgh gSA (ii) fp=k 1 ls BC = AC sin θ rFkk CD-AE = AC sin θ : i r D;ksafd −εµ (AE −CD )= BC rr –n sin θ r = sin θi. 10.23 dks.k i ij vkifrr ,d fdj.k ij fopkj djsaA bl fdj.k dk ,d Hkkx ok;q&fiQYe vUrjki`"B ls ijkofrZr gksrk gS rFkk ,d Hkkx vUnj viofrZr gksrk gSA ;g fiQYe&dk¡p vUrjki`"B ij va'kr% ijkofrZr rFkk va'kr% ikjxr gksrh gSA ijkofrZr fdj.k dk ,d Hkkx fiQYe&ok;q vUrjki`"B ij ijkofrZr gksrk gS rFkk ,d Hkkx r2dh rjg ikjxr r1 osQ lekUrj ikjxr gksrk gSA okLro esa ozQfed ijkorZu rFkk ikjxeu rjax osQ vk;ke dks ?kVkrs jgsaxsA vr%r1 rFkk r2 fdj.ksa O;ogkj ij NkbZ jgsaxhA ;fn vkifrr izdk'k ysal }kjk ikjxfer gks rks r1 rFkk r2 esa fouk'kh O;frdj.k gksuk pkfg,A A rFkk D nksauksa ij ijkorZu fuEu ls mPp viorZukad dh vksj gksaxs vr% ijkorZu ij dksbZ dyk ifjorZu ugha gksxkA r2 rFkk r1osQ chp izdkf'kd iFkkUrj gS fp=k 1 fp=k 2 n (AD + CD) – AB ;fn d fiQYe dh eksVkbZ gS rks d AD = CD = cos r AB = AC sin i AC = d tan r 2 ∴ AC = 2d tan r vr% AB = 2d tanr sini vr% iFkkUrj gS d2n − 2d tan r sin i cos r sin id sin r = 2. − 2d sin i sin r cos r cos r ⎡ 1− sin 2 r ⎤ = 2d sin ⎢⎥sin r cos r⎣⎦ = 2nd cosr bu rjaxksa osQ fouk'kh O;frdj.k osQ fy, ;g λ/2 osQ cjkcj gksuk pkfg,A λ⇒ 2nd cos r = 2 ;k nd cos r = λ/4 oSQejs osQ ysal osQ fy,] lzksr mQèokZèkj ry esa gS vkSj blfy, i � r � 0 λ ∴ nd ; . 4 o o5500 A ⇒ d = ; 1000 A 1.38 × 4 vè;k; 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 (d) (b) (d) (c) (b) (a) (a) (c) (c), (d) (a), (c) (b), (c) (a), (b), (c) (b), (d) 2mα Eαλ /λ= p /p = = 8 :1 pd xp 2mpEp (i) E max = 2hν – φ (ii) ,d gh bysDVªkWu }kjk nks iQksVkWu vo'kksf"kr djus dh izkf;drk vR;Ur fuEu gSA vr% bl izdkj osQ mRltZu ux.; gSaA igyh fLFkfr esa iznÙk (ckgj fudyh) mQtkZ laHkfjr mQtkZ ls de gSA nwljh fLFkfr esa D;ksafd mRlftZr iQksVkWu esa vf/d mQtkZ gksrh gS blfy, inkFkZ dks mQtkZ vkiwfrZ djuh iM+rh gSA LFkk;h inkFkksZa osQ fy, ,slk gksuk laHko ugha gSA ugha] vf/dak'k bysDVªkWu èkkrq esa izdh.kZ gks tkrs gSaA osQoy oqQN gh èkkrq osQ i`"B ls ckgj vkrs gSaA oqQy E fu;r gSA eku yhft, n1 rFkk n2 X-fdj.kksa rFkk n`'; {ks=k osQ iQksVkWu dh la[;k gSA nE= nE11 22 hc hc n = n12 λλ12 n λ11 = n λ22 n11 = n2 500 11.19 laosx èkkrq dks LFkkukarfjr gks tkrk gSA lw{e Lrj ij] ijek.kq iQksVkWu dks vo'kksf"kr djrs gSa rFkk bldk laosx eq[; :i ls ukfHkd rFkk bysDVªkWuksa dks LFkkukarfjr gks tkrk gSA mÙksftr bysDVªkWu mRlftZr gksrk gSA laosx laj{k.k ukfHkd rFkk bysDVªkWuks dks laosx LFkkukarfjr djus osQ fy, laosx laj{k.k osQ ifjdyu dh vko';drk gSA 11.20 vfèkdre mQtkZ = hν – φ ⎛1230 ⎞ 1 ⎛ 1230 ⎞⎜ – φ ⎟ = ⎜ – φ ⎟⎝ 600 ⎠ 2 ⎝ 400 ⎠ 1230 φ= = 1.02eV 1200 11.21 ΔxΔp ;h h 1.05 ×10 –34 Js –25 Δp ;; =1.05 ×10 Δx 10 –9 m 2 –252 2 2 p (1.05 ×10 ) 1.05 –19 1.05 E == –31 =×10 J= eV 2m 2× 9.1 ×10 18.2 18.2 ×1.6 = 3.8 × 10–2eV 11.22 I = nn= nνAA B B nA ν B = 2 = nB ν A iqat B dh vko`fÙk A ls nqxuh gSA hh hh p + p = + ==11.23 pc= if p, p > 0 or p, p< 0A B λλ λλ ABAB AB c c λλ vFkok λ c = AB λA +λB ;fn p> 0, p < 0 vFkok p< 0, p> 0A BA B λ – λ h pc = hB A = λA.λB λc λB.λAλc = λA – λB 11.24 2d sinθ = λ = d =10–10 m h 6.6 ×10 –34 –21 p = = = 6.6 ×10 kgm/s –10 –10 10 10 –242 2 –19 –2 (6.6 ×10 ) 6.6 E =×1.6 ×10 =×1.6 ×10 eV 2×(1.7 ×10 –27 ) 2×1.7 = 20.5 × 10–2eV = 0.21eV 11.25 Na osQ 6 × 1026 ijek.kqvksa dk Hkkj = 23 kg y{; dk vk;ru = (10–4 × 10–3) = 10–7m3 lksfM;e dk ?kuRo = (d) = 0.97 kg/m3 6 × 1026 Na ijek.kqvksa dk vk;ru = 23 m3 = 23.7 m3 0.9723 31 Na ijek.kq dk vk;ru = 26 m = 3.95 × 10–26m3 0.97 × 6×10 10 –7 y{; esa Na ijek.kqvksa dh la[;k = = 2.53 × 1018 3.95 ×10–26 iQksVkWu dh la[;k izfr lsoaQM rFkk 10–4 m2 = n mQtkZ izfr lsoaQM rFkk nhν = 10–4 J × 100 = 10–2 W 1234.5 hν (λ = 660nm osQ fy,) = 600 = 2.05eV = 2.05 × 1.6 × 10–19 = 3.28 × 10–19J 10 –2 n = –19 = 3.05 ×10 16 /s 3.28 ×10 1 17 16 n = ×10 =×3.1 10 3.2 ;fn izfr ijek.kq mRltZu dh izkf;drk P gS] izfr iQksVkWu] iQksVksbysDVªkWu dh izfr lsoaQM mRltZZu dh la[;k 16 18 = P × 3.1 10 × 2.53 ××10 èkkjk = P × 3.1 × 10+16 × 2.53 × 1018 × 1.6 × 10–19 A = P × 1.25 × 10+16 A ;g 100µA osQ cjkcj gksuh pkfg, vFkok 100 ×10 –6 P = 1.25 ×10+16 ∴ P = 8 × 10–21 bl izdkj ,dy ijek.kq ij ,dy iQksVkWu }kjk iQksVks mRltZu dh izkf;drk 1 ls cgqr de gSA (blfy, ,d ijek.kq }kjk nks iQksVkWu dk vo'kks"k.k ux.; gSA) 11 ∞ q 21 q 2 11.26 cká ,tsalh }kjk fd;k x;k dk;Z = + . dx = .∫ 24πε 04 dx 44πε 0d –19 9(1.6 ×10 )× 9 ×10 d = 0.1nm ls] mQtkZ = –10 –19 eV 4(10 ) 1.6 ××10 1.6 × 9 = = 3.6 eVeV 4 11.27 (i) B osQ fy, mPp vko`fÙk ij fujks/h foHko = 0 vr% bldk dk;ZiQyu mPp gS h 2 (ii) W0: gk¡λ 2 (iii) P. π r 2 Δt = W0, Δt = 28.4s d4π d 2πr⎛ dN ⎞(iv) N = ⎜⎟ × 2 ×Δ t = 2 ⎝ dt ⎠ 4πd vè;k; 12 (c) (c) (a) (a) (a) (a) (a) (a), (c) (a), (b) (a), (b) (b), (d) 12.12 (b), (d) 12.13 (c), (d) 12.14 vkbaLVhu osQ nzO;eku&mQtkZ lacaèk ls gesa izkIr gksrk gS% E = mc2. vr% gkbMªkstu osQ ,d ijek.kq dk nzO;eku gS mp +me − B 2, tgk¡B ≈ 13.6 eV caèku&mQtkZ gSA c 12.15 D;ksafd bysDVªkWu osQ nzO;eku dh rqyuk esa nksuksa ukfHkd vR;fèkd Hkkjh gSaA 12.16 D;ksafd bysDVªkWu osQoy oS|qr pqEcdh; :i ls ikjLifjd fozQ;k djrs gSaA 12.17 gk¡] D;ksafd cksgj&lw=k esa osQoy vkos'kksa dk xq.kuiQy fufgr gSA 12.18 ugha] D;ksafd cksgj izfr:i osQ vuqlkj E = –13.6 , vkSj fHkUu&fHkUu mQtkZ osQ bysDVªkWun 2 n fofHkUu n& eku okys Lrjksa ls lEc¼ gksrs gSa% vr% muosQ dks.kh;&laosx fHkUu gksaxs] D;ksafd nh mvr = 2π 4 me 12.19 cksgj osQ lw=k E = – esa 'm' lekuhr nzO;eku gSA H-ijek.kq osQ fy, m ≈ mn 22 e8ε0nh iksftVªksfu;e osQ fy, m ≈ me /2 vr% ,d iksftVªksfu;e osQ fy, E≈ – 6.8eV14me 4 12.20 2e vkos'k okys ukfHkd rFkk –e vkos'k okys bysDVªkWuksa osQ fy,] Lrj gSa& En = – 2 228ε0 nh fuEurj Lrj esa nks bysDVªkWu gksaxs] ftuesa izR;sd dh mQtkZ E rFkk fuEure Lrj dh oqQy mQtkZ –(4×13.6)eV gksxhA 12.21 v = bysDVªkWu dk osx a0 = cksgj f=kT;k ∴,dkad le; esa pozQ.kksa dh la[;k = 2πa0 v 2πa0∴ èkkjk = e v ⎡ 11 ⎤ 12.22 ν= cRZ 22 − 2,U;wure ⎢⎥(n + p ) n⎣⎦ tgk¡ m = n + p, (p = 1, 2, 3, ...) ,oa R fjM~cxZ fu;rkoaQ gSA p << n osQ fy, 2 ⎡ 1 p –2 1 ⎤⎛⎞ν= cRZ U;uw re ⎢ 2 ⎜1+ ⎟ − 2 ⎥⎝⎠⎣n nn ⎦ 2 ⎡ 12p 1 ⎤ν ure = cRZ – −U;w⎢ 232 ⎥⎣nnn ⎦ 22p ⎛ 2cRZ 2 ⎞ ν= cRZ ; pU;uw re 3 ⎜ 3 ⎟ n ⎝ n ⎠ bl izdkj νdk yxHkx ozQe gS& 1, 2, 3...........U;wure 12.23 ckej Js.kh esa Hγ , n = 5 ls n = 2 osQ laozQe.k osQ laxr gSA vr% U;wure Lrj n = 1 osQ bysDVªkWu dks igys n = 5 esa j[kuk pkfg,A vko';d mQtkZ = E1 – E5 = 13.6 – 0.54 = 13.06 eV ;fn dks.kh; laosx lajf{kr jgrk gS] rks iQksVku dk dks.kh; laosx = bysDVªkWu osQ dks.kh; laosx esa ifjorZu = L5– L2 = 5h –2h = 3h= 3×1.06 ×10 –34 = 3.18 × 10–34 kg m2/s me ⎛ me ⎞12.24 H osQ fy, lekuhr nzO;eku =µ = ; m 1– He ⎜⎟ e ⎝ M ⎠1+ m M ⎛ m ⎞⎛ m ⎞⎛ m ⎞e eeD osQ fy, lekuhr nzO;eku =µD ; me ⎜1– ⎟ = me ⎜1– ⎟⎜ 1+ ⎟⎝ 2M ⎠⎝ 2M ⎠⎝ 2M ⎠ hν= (E – E )αµ . vr%] λα 1 ij i j ij µ ;fn gkbMªkstu@M~;wVhfj;e osQ fy, rjaxnS?;Z λ /λ gS λD µH ⎛ me ⎞–1 ⎛ 1 ⎞ HD = ; ⎜1+ ⎟ ; ⎜1– ⎟ λH µD ⎝ 2M ⎠⎝ 2×1840 ⎠ λD =λH × (0.99973) vr% js[kk,¡ gSa 1217.7 Å , 1027.7 Å, 974.04 Å, 951.143 Å 24 µ Ze ⎛ 1 ⎞12.25 ukfHkdh; xfr dks lfEefyr djrs gq,] fLFkj voLFkk esa mQtkZ,¡ gkasxh& En = – ⎜⎟22 28ε0h ⎝ n ⎠ eku fy;k µHgkbMªkstu dk rFkk µHM~;wVhfj;e dk lekuhr nzO;eku gSA rc gkbMªkstu µ e 43 µ e 4 H ⎛ 1 ⎞ Hdh izFke ykbeu js[kk dh vko`fÙk gS hνH = 22 ⎜1– ⎟ = 2 2. vr% laozQe.k8ε0h ⎝ 4 ⎠ 48ε0h 3 µ e 4 dh rjaxnS?;Z gS λ= H 23 A M~;wVhfj;e osQ fy, laozQe.k dh mlh js[kk dhH 48ε0hc 3 µ e 4 rjaxnS?;Z gS λ= H D 2348ε0hc ∴Δ λ =λD – λH vr% vUrj&izfr'kr gS& Δλ λ – λµ – µDH DH100 ×= ×100 =×100 λλ µHH H mM mM eD eH– (m + M )(m + M )eD eH =×100 m eMH /( me + MH ) ⎡⎛ me + MH ⎞ MD ⎤ = –1 ×100 ⎢⎜ ⎟⎥ ⎣⎝ me + MD ⎠ MH ⎦ D;ksafd m e << MH < MD Δλ ⎡ MM ⎛1+ m /M ⎞⎤HD eH×100 =× –1 ×100 ⎢⎜ ⎟⎥ λ MM 1+ m /MH ⎣ DH ⎝ eD ⎠⎦ = ⎡(1 +m /M )(1 + m /M )–1 –1⎤ ×100 ⎣ eH eD ⎦ mm⎡ ee ⎤ ; (1 + – –1 ×100 ⎢⎥MM⎣ HD ⎦ ⎡ 11 ⎤≈ m – × 100 e ⎢⎥MM⎣ HD ⎦ –31 ⎡ 11 ⎤9.110 – ×100 =× ⎢ –27 –27 ⎥⎣1.6725 ×10 3.3374 ×10 ⎦ = 9.1 10–4 = 2.714 × 10–2 % ×[0.5979 – 0.2996 ]×100 12.26 H-ijek.kq esa] ,d fcanq&ukfHkd osQ fy, mv 2 e 21fuEure Lrj:mvr = h , = –. rB rB 24πε 0 ∴ m . =+ ⎜ ⎟2 2 2 mr r 4πε rBB ⎝ 0 ⎠ B h24πε ° ∴ . 20 = rB= 0.51A m e fLFkfrt mQtkZ ⎛ e 2 ⎞ 1 mv 21 h 2 h –KE –⎜⎟. = 27.2 eV ;. == m. = =+13.6eV 22 24πrr 22 mr 2mr R f=kT;k osQ ,d xksyh; ukfHkd osQ fy, ;fn R < r B, ogh ifj.kkeA ;fn R >> rB: bysDVªkWu rB′ (rB′ = u;h cksgj f=kT;k) f=kT;k osQ xksys osQ Hkhrj xfreku gSA ′ ⎛ rB ′3 ⎞ ⎝ 0 ⎠ B BB r oQs Hkhrj vko's k = eB ⎜ 3 ⎟⎝ R ⎠ h 2 ⎛ 4πε 0 ⎞ R 3 ∴rB ′= ⎜ 2 ⎟ 3 m ⎝ e ⎠ rB ′ ° 4 3 ° r′ B = (0.51A). R . R = 10 A ° = 510(A)4 ∴ rB ′ ≈ (510)1/ 4 A °< R. 12 m hh 1xfrt mQtkZ = mv = . = .22 222 m r B ′ 2m rB ′ 22 2⎛ h ⎞⎛ rB ⎞ (0.51) 3.54 = ⎜ ⎟⎜ . ⎟ = (13.6eV) == 0.16eV 2 2 1/2 ⎝ 2mr B ⎠⎝ rB ′ ⎠ (510) 22.6 ⎛ e 2 ⎞⎛ 22 rB ′ –3R ⎞fLFkfrt mQtkZ =+⎜ .⎟⎜ ⎟ ⎝ 4πε 0 ⎠⎝ 2R3 ⎠ 2 22⎛ e 1 ⎞⎛ rB (rB ′ –3 R ⎞ =+ ⎜ . .⎟⎜ 3 ⎟4πε r⎝ 0 B ⎠⎝ R ⎠ ⎡0.51( 510 – 300) ⎤ =+ (27.2eV) ⎢ ⎥⎣ 1000 ⎦ –141 =+ (27.2eV). = –3.83eV 1000 12.27 D;ksafd ukfHkd Hkkjh gS] ijek.kq dk izfrf{kIr laosx ux.; gS rFkk laozQe.k dh oqQy mQtkZ dks vksts&bysDVªkWu dks LFkkukUrfjr eku ldrs gSa D;ksafd Cr esa ,d la;ksth bysDVªkWu gS] mQtkZ voLFkkvksa dks cksgj izfr:i }kjk iznÙk ekuk tk ldrk gSA noha voLFkk dh mQtkZ 21 En = –Z R 2 tgk¡ R fjM~cxZ fu;rkaoQ gS rFkk Z = 24 n 2 ⎛ 1⎞ 3 2 n = 2 ls n = 1 osQ laozQe.k esa eqDr mQtkZ gS ΔE = ZR ⎜1– ⎟ = ZR ⎝⎠ 44 21 n = 4 bysDVªkWu dks mR{ksfir djus osQ fy, vko';d mQtkZ E = ZR 4 16 vr% vksts bysDVªkWu dh xfrt ÅtkZ gS& 2 ⎛ 31 ⎞ 12 . = ZR –KE = ZR ⎜⎟⎝ 4 16 ⎠ 16 11 =× 24 × 24 ×13.6eV 16 = 5385.6 eV 12.28 m p c2 = 10–6 × bysDVªkWu nzO;eku × c2 10–6 ≈× 0.5 MeV –6 –13 ≈ 10 × 0.5 ×1.6 ×10 ≈ 0.8 ×10 –19 J –34 8hhc 10 × 3×10 == –7 2 –19 ≈ 4× 10 m >> cksgj f=kT;k mpc mpc 0.8 ×10 e 2 ⎡ 1 λ ⎤ =+ exp(–λr)F ⎢ 2 ⎥4πε 0 ⎣rr ⎦ h–1 –7 tgk¡ λ= ≈ 4 ×10 m >> rB mc p 1 ∴λ << vFkkZr λr << 1B rB e 2 exp(– λr )Ur () = –. 4πε 0 r h mvr = h ∴ v = mr mv 2 ⎛ e 2 ;g Hkh : =≈ ⎜ ⎞⎡ 1 +λ ⎤⎟⎢ 2 ⎥r ⎝ 4πε 0 ⎠⎣rr ⎦ h2 ⎛ e 2 ⎞⎡ 1 λ ⎤∴= +3 ⎜ ⎟⎢ 2 ⎥mr 4πε ⎣ rr ⎦⎝ 0 ⎠ h 2 ⎛ 22e ⎞ ∴= ⎜⎟ [r +λr ] m 4πε ⎝ 0 ⎠ h 4πε 0;fn λ= 0; r = rB = .2 me 22h e = .r m 4πε 0 B D;ksafd λ–1 >> rB,r = rB +δ j[ksa 22 2∴ rB = rB +δ +λ(rB +δ + 2δ rB ); δ ux.; gSA vFkok 0 =λrB 2 +δ(1 + 2λrB ) δ=λrB 2(1– 2λ rB ) = –λrB 2 D;kfd saλrB << 1 e 2 exp(– λδ – λr )∴ Vr () = –. B 4πε 0 r +B δ e 21 ⎡⎛ δ ⎞⎤ ∴ Vr () = – 1– .(1 – λr )⎢⎜ ⎟ B ⎥4πε 0 rB ⎣⎝ rB ⎠⎦ ≅ (–27.2eV) vifjo£rr jgrk gSA 2 221 21 h hh ⎛ 2δ ⎞xfrt mQtkZ = – mv = m. = 2 = ⎜1– ⎟ 22 mr 2 2( rB +δ )2rB 2 ⎝ rB ⎠ = (13.6eV)1 2[+ λrB ] 22 e hoqQy mQtkZ = – + 2 [1+ 2λrB ]4πε r 2r0 BB = –27.2 + 13.6 1 + 2λ r eV [ B ] mQtkZ esa ifjorZu = 13.6 × 2λrBeV = 27.2 λrBeV 12.29 ekuk ε= 2 +δ qq Rδ R δ 1200 12 –192 9 2+δ 2+δF = . =∧ , tgk¡ qq =∧ , ∧= (1.6 ×10 ) × 9×10 4πε 0 rr 4πε 0 = 23.04 10–29 × 2 mv = r ∧Rδ 20v = 1+δ mr 1/2 m 1/2 +δ /2 nh nh ⎡⎤(i) mvr = nh , r == r⎢ δ ⎥ mv m ⎣ ∧R0 ⎦ 1 ⎡ n2h2 ⎤1– δbls r osQ fy, gy djus ij r n = ⎢ δ ⎥ m ∧ R⎣ 0 ⎦ 1 ⎡ h2 ⎤1– δ n = 1 osQ fy,] rFkk fLFkjkad osQ eku j[kus ij] gesa izkIr gksrk gS r1 = ⎢ δ ⎥ m ∧ R⎣ 0 ⎦ 1 2 –68 ⎡⎤ 2.91.05 ×10 r = 8 × 10–11 = 0.08 nm1 = ⎢ –31 –28 +19 ⎥9.1 10 × 2.3 ×10 ×10 ⎦⎣ × (< 0.1 nm) 1 nh ⎛ m ∧ R δ ⎞1– δ h v = 0 v =(ii) n = nh ⎜⎟ n = 1 osQ fy,, 1 = 1.44 × 106 m/smr 22 mr n ⎝ n h ⎠ 1 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 1 2 –19 (iii) xfrt mQtkZ = mv 1 = 9.43 ×10 J=5.9eV 2 ∧LFkfrt mQtkZ R0 rd = – R0 r δ rdr ∧R 1δ 0 ⎡⎤ Rls r rd fLFkfrt mQtkZ =+∧ R =+ 0 ∫ 2 +δ ⎢ 1+δ ⎥0 Rr –1– δ ⎣r ⎦R00 ∧Rδ ⎡ 11 ⎤0– rd= – ⎢ 1+δ 1+δ ⎥1+δ rR ∧ ⎡ R δ 1 ⎤ ⎣ 0 ⎦ fLFkfrt mQtkZ = – 0–⎢ 1+δ ⎥1 +δ rR⎣ 0 ⎦ ∧ ⎡ R 0 δ 11 +δ ⎤ = –– +⎢ 1 +δ 1 +δ rR⎣ 0 – 1 .9 ∧ ⎡ R 0= – –⎢ –0 .9 –0 .9 r⎣ 2.3 –18 0.9 =×10 [(0.8) –1.9] 0.9 oqQy mQtkZ gS (–17.3 + 5.9) = –11.4 eV vè;k; 13 (c) (b) (b) (a) (a) (b) (b) (a), (b) (b), (d) ⎥ R 0 ⎦ 1 .9 ⎤ ⎥R 0 ⎦ J = – 17.3 eV 13.10 (c), (d) 13.11 ugha] He3 dh caèku ÅtkZ rqyukRed :i ls vfèkd gSA dN dt N 13.12 13.13 B dh vkSlr vk;q de gS D;ksafd B osQ fy, λ dk eku vfèkd gSA 13.14 mÙksftr bysDVªkWu] D;ksafd bysDVªkWfud ÅtkZ Lrjksa dh ÅtkZ dh ijkl eV esa gS] MeV esa ughaA γ -fofdj.k dh ÅtkZ MeV gSA 13.15 nks iQksVku mRiUu gksrs gSa] tks ÅtkZ&laj{k.k gsrq foijhr fn'kkvksa esa xfr djrs gSaA 13.16 izksVku èkukosf'kr gksrs gSa rFkk ,d nwljs dks fo|qrh; :i ls izfrdf"kZr djrsa gSaA 10 ls vfèkd izksVkuksa okys ukfHkd esa ;g izfrd"kZ.k bruk vfèkd gks tkrk gS fd U;wVªkuksa dh vfèkdrk tks osQoy vkd"kZ.k cy mRiUu djrh gS] LFkkf;Ro osQ fy, vko';d gks tkrh gSA t = 0 ijN= Ntcfd N = 0 tSls&tSls le; esa o`f¼ gksrh gS] Ndk pj ?kkrkadh :iAO B A ls iru gksrk gS] B osQ ijek.kqvksa dh la[;k c<+rh gS] vfèkdre gksrh gS vkSj vUr esa ¥ ij 'kwU; gks tkrh gS (pj ?kkrkadh fo?kVu fu;ekuqlkj) 1 R013.18 t = lnλ R 5760 16 5760 4ln = = ln 0.693 120.693 3 5760 4 = ×2.303log = 2391.12 o"kZ 0.693 3 13.19 d nwjh ij i`FkDo`Qr nks oLrqvksa dks vyx djus gsrq vUos"kh flXuy dh rjaxnS?;Z λ, d ls de gksuh pkfg,A vr% U;wfDy;ku osQ Hkhrj vyx&vyx Hkkxksa dk lalwpu djus osQ fy,] bysDVªkWu dh rjaxnS?;Z 10–15 m ls de gksuh pkfg,A λ= h vkSj K ≈ pc ⇒ K ≈ pc = hc p λ 6.63 ×10 34 ×3 ×10 8 = eV –19 –15 1.6 ×10 ×10 = 109 eV. = 1 GeV 13.20 (a) 23 = 11, N = 1211 Na : Z1123 23 ∴ 11 Na dk niZ.k lEHkkjh = 12 Mg (b) D;ksafd Z2 > Z1, Mg dh cUèku ÅtkZ Na ls vfèkd gSA 38 38 38 13.21 S ⎯⎯⎯⎯→ Cl ⎯⎯⎯⎯→ Ar 2.48 h 0.62 h ekuk t le; ij] 38S osQ ikl N 1(t) lfozQ; ukfHkd rFkk 38Cl osQ ikl N2(t ) lfozQ; ukfHkd gSaA dN1 = –λ N = 38 Cl oQs cuus dh nj dt 11 dN2 12 11 = –λ N +λ N dt –λ1tysfdu N = Ne 10 dN2– λ1t 10 22 = –λ Ne –λ N dt λ2t ls xq.kk djosQ iqu% O;ofLFkr djus ije dt λ t λ t (λ –λ )t2 221e dN +λ N e dt =λ N e dt 2 22 10 nksuksa i{kksa dks lekdfyr djus ij λ t N λ (λ –λ )t2 01 21Ne = e + C2λ – λ21 01 21 D;ksafd t = 0, N2 = 0, C=– N λ λ – λ λ 2t N0λ1(λ 2–λ 1)t 2∴ Ne = (e –1) λ – λ21 0 1–λ ,t –λ2tN2 = N λ (e – e )λ – λ21 2vfèkdre fxurh osQ fy, dN = 0 dt ⎛ λ 1 ⎞gy djus ij t = ⎜ln /( λ 1– λ 2)⎟λ⎝ 2 ⎠ 2.48 = ln /(2.48 – 0.62) 0.62 ln4 2.303log4 == 1.86 1.86 = 0.745 s. 13.22 ÅtkZ lja{k.k ls] 22 nE − B = Kn + Kp = p +pp (1)2m 2m laosx lja{k.k ls] E pn + pp = (2) c ;fn E = B, izFke leh- ls izkIr gksrk gS pn = pp = 0 vkSj blfy, f}rh; leh- lUrq"V ugh dh tk ldrh] rFkk izfozQ;k ?kfVr ugha gks ldrhA izfozQ;k osQ ?kfVr gksus osQ fy, ekuk E = B + λ, tgk¡ λ < SpSb , Sn ukfHkd] Sb ukfHkd ls vfèkd LFkk;h gSA (ii) ;g ukfHkd osQ fy, mlh izdkj dh dksf'k; lajpuk dks izn£'kr djrk gS tSlk fd ijek.kq esa gksrk gSA ;g caèku mQtkZ rFkk U;wDyku la[;k osQ chp f[akps oozQ esa mifLFkr f'k[kjksa dh Hkh O;k[;k djrk gSA vè;k; 14 14.1 (d) 14.2 (b) 14.3 (b) 14.4 (d) 14.5 (b) 14.6 (c) 14.7 (b) 14.8 (c) 14.9 (a), (c) 14.10 (a), (c) 14.11 (b), (c), (d) 14.12 (b), (c) 14.13 (a), (b), (d) 14.14 (b), (d) 14.15 (a), (c), (d) 14.16 (a), (d) 14.17 ekfnr fd, tkus okys ijek.kq dk lkbt ,slk gksuk pkfg, fd ;g 'kq¼ v¼Zpkyd osQ fdLVy tkyd dh lajpuk dks rks foo`Qr u djs ijUrq Si ;kGe osQ lkFk lg&la;ksth caèk ljyrkiwoZd fufeZr dj ,d vkos'k okgd dk ;ksxnku dj losQA 14.18 ijek.kq lkbt osQ vuqlkj Sn osQ fy, mQtkZ vUrjky 0 eV, C osQ fy, 5.4 eV, Si osQ fy, 1.1 eV rFkk Ge osQ fy, 0.7eV gSA 14.19 th ugha] D;ksafd lafèk&izfrjksèk dh rqyuk esa oksYVehVj dk izfrjksèk vR;qPp gksuk gh pkfg,] tcfd lafèk izfrjksèk yxHkx vuUr gSA +1V 14.20 0 14.21 (i) 10 × 20 × 30 × 10–3 = 6V (ii) ;fn dc iznk; oksYVrk 5V gS rks vfèkdre fuxZe V cc vr%] V0 = 5V 5V ls vfèkd ugha gks ldrkA 14.22 ugha] vkofèkZr fuxZe osQ fy, okafNr vfrfjDr 'kfDr DC lzksr ls izkIr gksrh gSA 14.23 (i) tsuj lafèk Mk;ksM rFkk lkSj lsy (ii) tsuj Haktd oksYVrk (iii) Q-y?kq iFku èkkjk P-[kqys ifjiFk dh oksYVrk 14.24 vkifrr izdk'k osQ iQksVkWu dh mQtkZ –34 86.6 ×10 ×3 ×10 hv = = 2.06e V–7 –19 6 ×10 ×1.6 ×10 vkifrr fofdj.k] iQksVksMk;ksM }kjk lalwfpr gks losQ blosQ fy, vkifrr fofdj.k iQksVkWuksa dh mQtkZ cSaM&vUrjky ls vfèkd gksuh pkfg,A ;g 'krZ osQoy D2 }kjk iwjh gksrh gSA vr% osQoy D2 gh bu fofdj.kksa dks lalwfpr djsxkA V – V14.25 BB BE =IB R ;fn Rdk eku c<+k;k tk,xk rks Idk eku de gksxkA D;ksafd I = βI, ifj.kke ;g gksxk 1 1B cbfd IC Hkh de gks tk,xk vFkkZr ,ehVj vkSj oksYVehVj osQ ikB~;kad de gks tk,¡xsA OR }kj dk fuxZe uhps nh xbZ lR;eku lkj.kh osQ vuqlkj gksrk gS% A B C 0 0 0 0 1 1 1 0 1 1 1 1 14.27 fuos'k fuxZr A A 0 1 1 0 14.28 rRoh; v¼Zpkydksa osQ mQtkZ&vUrjky ,sls gksrs gSa fd mRltZu IR {ks=k esa gksrk gSA 14.29 lR;eku lkj.kh A B Y 0 0 0 0 1 0 1 0 0 1 1 1 AND }kj P = 0.2A = 200mA 14.30 IZ max = VZ Vs – VZ 2 RS = == 10 Ω Z max I 0.2 14.31 I3 'kwU; gS D;ksafd bl 'kk[kk esa yxk Mk;ksM i'p&ck;flr gSA AB ,oa EF esa ls izR;sd esa izfrjksèk (125 + 25)Ω = 150 Ω gSA 150 D;ksafd AB ,oa EF loZle lekUrj 'kk[kk,¡ gSa] budk izHkkoh izfrjksèk gS] = 75 Ω 2 ∴ ifjiFk esa oqQy izfrjksèk = (75 + 25) Ω = 100 Ω 5 ∴ èkkjk I == 0.05A 1 100 D;ksafd AB vkSj EF osQ izfrjksèk cjkcj gSa rFkk I1 = I2 + I3 + I4, I5 = 0 0.05 ∴ I2 = I4 == 0.025A 2 14.32 D;ksafd Vbe = 0, Rb ij foHkoikr10V gSA 10 ∴ Ib == 25 µ A 400 ×10 3 D;ksafd V = 0, R , ij foHkoikr IR 10V. ceccc ∴ Ic = 10 3 = 3.33 ×10 –3 = 3.33m A .3×10 Ic 3.33 ×10 –3 2∴β= = –6 = 1.33 ×10 = 133. Ib 25 ×10 14.34 fuxZe vfHkyk{kf.kd oozQ osQ fcUnq Q ij] VCE = 8V & IC = 4mA V= IR+ VCCCC CE V – VCC CE R = c IC 16 – 8 Rc = –3 = 2K Ω 4 ×10 pw¡fd V= IR+ VBB BB BE 16 – 0.7 RB = –6 = 510K Ω 30 ×10 I 4 ×10 –3 vc] β= C == 133 IB 30 ×10 –6 RoksYVrk yfCèk =AV = –β C RB 2×10 3 = –133 × 510 ×10 3 = 0.52 'kfDr yfCèk = A = β× p AV 2 RC= – β RB 22 ×10 3 = (133) × = 69 510 ×10 3 14.35 tc fuosf'kr oksYVrk 5V ls vfèkd gksrh gS rks Mk;ksM ls èkkjk izokfgr gksrh gSA tc fuos'k 5V ls de gksrk gS rks Mk;ksM ,d [kqyk ifjiFk gksrk gSA 14.36 (i) ‘n’ {ks=k esa As osQ dkj.k e– dh la[;k ne = ND = 1 × 10–6 × 5 × 1028 ijek.kq/m3 ne = 5 × 1022/m3 vYiak'k okgdksa (gksyksa) dh la[;k 2 162 32 ni (1.5 ×10 ) 2.25 ×10 n == = h 22 22 n 5×10 5 ×10 e nh = 0.45 × 10/m3 blh izdkj tc cksjkWu dk vkjksi.k fd;k tkrk gS] rks ‘p’ izdkj fu£er gksrk gSA ftlesa gksyksa dh la[;k nh = NA = 200 × 10–6 × 5 × 1028 = 1 × 1025/m3 –;g ml n-izdkj dh ijr esa fo|eku e dh la[;k dh rqyuk esa cgqr vfèkd gS ftlesa cksjkWu folfjr fd;k x;k FkkA bl izdkj fu£er ‘p’ {ks=k esa vYika'k okgdksa dh la[;k ni 2 2.25 ×1032 n == e nh 1×10 25 = 2.25 × 107/m3 (ii) vr% i'p ck;flr djus ij n-{ks=k esa fo|eku 0.45 × 1010/m3 gksy] p-{ks=k osQ 2.25 × 107/m3 vYika'k e– dh rqyuk esa i'p lar`fIr èkkjk osQ fy, vfèkd ;ksxnku djsaxsA A 1K� 14.38 CB 14.39 I≈ I∴ I(R+ R) + V =12 VCE C CECERE = 9 – RC = 1.2 KΩ ∴ VE = 1.2 V V = V+ V=1.7 VBEBE V I = B = 0.085 mA 20K 12–1.7 10.3 R = ==108K ΩB IC /β+ 0.085 0.01 +1.085 14.40 I= I+ II= β I(1)EC B C B IR+ V+ IR= V(2)CC CEEE CC RI+ V+ IR= V(3)B BE EE CC lehdj.k (3) ls I ≈ I= β IeC B V –V 11.5 CC BE (R + β R)= V– V, IB == mA ECC BER +β RE 200 lehdj.k (2) ls V–V V–V 2CC CE CC CE RC +RE === (12 – 3)K Ω=1.56K Ω IC β IB 11.5 R = 1.56 –1 = 0.56K C Ω vè;k; 15 15.1 (b) 15.2 (a) 15.3 (b) 15.4 (a) 15.5 (b) 15.6 (c) 15.7 (b) 15.8 (b) 15.9 (c) 15.10 (a), (b), (d) 15.11 (b), (d) 15.12 (b), (c), (d) 15.13 (a), (b), (c) 15.14 (b), (d) 15.15 (i) vuq:i (ii) vuq:i (iii) vadh; (iv) vadh; 15.16 ugha] 30 MHz ls vfèkd vko``fÙk osQ flXuy] vk;ue.My }kjk ijkofrZr ugha gksaxs cfYd varosZèku djsaxsA 15.17 viorZukad] vko`fÙk c<+us osQ lkFk&lkFk c<+rk gS ftldk vFkZ ;g gS fd mPprj vko`fÙk rjaxksa osQ fy, viorZu dks.k de gksrk gS] vFkkZr eqM+uk de gksrk gSA vr% iw.kZ vkUrfjd ijkorZu dh n'kk vfèkd nwjh r; djus ij izkIr gksrh gSA 15.18 Ac + Am = 15, Ac – Am = 3 ∴ 2A = 18, 2A = 12 cmA 2 m m = = ∴ A 3 c 1 = 1MHz 15.19 2π LC 1LC = 62π×10 15.20 vk;ke&ekMqyu esa] okgh rjaxksa dk rRdkfyd foHko ekMqyd rjax foHko }kjk ifjofrZr gksrk gSA lEizs"k.k esa ukW;t flXuy Hkh tqM+ ldrs gSa vkSj xzkgh] ukW;t dks] ekMqysfVax flXuy osQ ,d Hkkx dh Hkk¡fr O;ogkj djrk gSA ijUrq vko`fÙk ekMqyu esa] ekMqyd foHko osQ rRdkfyd foHkokuqlkj okgh rjax vko`fÙk ifjofrZr dh tkrh gSA ;g osQoy feJ.k@ekMqyu Lrj ij fd;k tk ldrk gS] flXuy osQ pSuy es lapj.k osQ nkSjku ughaA vr% ukW;t vko`fÙk ekMqfyr flXuy dks izHkkfor ugha djrhA 15.21 lapj.k iFk esa gqbZ gkfu = – 2 dB km–1 × 5 km = – 10 dB izoèkZd dk oqQy ykHk = 10 dB + 20 dB= 30 dB flxuy dk lEiw.kZ ykHk = 30 dB – 10 dB = 20 dB ⎛ Po ⎞10log⎜ P ⎟ = 12 vFkok Po = Pi × 102 ⎝ i ⎠ = 1.01 mW × 100 = 101 mW 15.22 (i) ijkl = 6 = 16 km2 × 6.4 ×10 ×20 vkPNkfnr {ks=kiQy = 803.84 km2 6 6(ii) ijkl = 2× 6.4 ×10 × 20 + 2× 6.4 ×10 × 25 = (16 + 17.9) km = 33.9 km vkPNkfnr {ks=kiQy = 3608.52 km2 (iii) {ks=kiQy esa izfr'kr o`f¼ (3608.52 – 803.84) = ×100 803.84 = 348.9% 2215.23 d = 2( R + h )mT 8RhT = 2(R+hT)2 (Q dm = 2 4Rh = R2 + h 2 + 2Rh T TT (R – h )2 = 0 TR = hT D;ksafd osQoy vkdk'k rjax vko`fÙk iz;qDr gqbZ gS l << hT vr% osQoy ehukj dh mQ¡pkbZ dk fopkj djsaA f=kfoeh; vkdk'k esa] 6 ,UVsuk&ehukj iz;qDr gksaxsA hT = R 15.24 F1 ijr osQ fy, ⎛ 56 ⎞2 )1/2 or N5 × 106 = 9(N = ⎜ ×10 ⎟ = 3.086 × 1011 m–3 maxmax ⎝ 9 ⎠F2 ijr osQ fy, )1/2 or8 × 106 = 9 (Nmax8⎛ 6 ⎞ 11 –3 ×10 = 7.9 ×10 mN = ⎜⎟ = 7.9 × 1011 m–3 max ⎝ 9 ⎠15.25 ω – ω , ω ,oa ω +ω esa ls osQoy ω +ω ;k ω –ω esa lwpuk fufgr gSA vr% ω + c mc mm c m c m c ω , rFkk ω –ω , nksuksa dks lEizsf"kr djosQ ykxr de dh tk ldrh gSA ω + ω o mc m c m ω –ω c m I 1 ⎛ 1 ⎞ 15.26 (i) = , vr% ln ⎜⎟ = –α x I 4 o ⎝ 4 ⎠ ⎛ ln 4 ⎞vFkok ln4 = ax vFkok x = ⎜⎟⎝ α ⎠ I (ii) 10log10 = –αx tgk¡ 'a' es {kh.ku gS dB/kmI o ;gk¡ I = 1 Io 2 ⎛ 1⎞vFkok 10log ⎜⎟ = –50 α vFkok log2 =5 α⎝⎠2 log 2 0.3010 vFkok α== = 0.0602 B/km d 55 2x 15.27 = oxsle; 2x = 3 × 108 m/s × 4.04 × 10–3s 12.12 × 105 x=m 2 x = 6.06 × 105m = 606 km 22 2 22d = x – hs = (606) –(600) = 7236; d = 85.06 km lzksr rFkk xzkgh osQ eè; nwjh = 2d ≅170 km dm = 22 Rh T 2d = d m 4d2 = 8 RhT 2 7236 d = ≈ 0.565 km 2R = hT 2 × 6400 hT = 565m 15.28 fp=k ls] 100 20 V = = 50V, V = = 10 V max min2 2(i) izfr'kr ekMqyu V – V ⎛ 50 –10 ⎞ 40 max min µ(%) =×100 = ⎜⎟ ×100 = ×100 = 66.67% V + V ⎝ 50 +10 ⎠ 60 max min V + Vmax min (ii) 'kh"kZ okgh foHko V = c2 50 +10 = = 30V 2 (iii) 'kh"kZ lwpuk foHko = V m = µV c 2 =×30 = 20V 3 15.29 (i) vt ( )=A(A m sin ωmt + Am sin ωm2 t + Ac sin ωct)112 ( sin ω t + A sin ω t + A sin 2 + BA ω t)m mm mcc112 2 =( ω t + A sin ω t + A sin ω t)AA sin mmm mcc2211 2 22+ B(( A sin ω t + At ) + A sin ω tmmm cc1 12 +2A (A sin ω t + A sin ω t)cm1 m 1 m 2 c = AA sin ω t + A sin ω t + A sin ω t( )1 mm 2 mc12 c 22 22+ BA sin ω t + A sin ω t + 2AA sin ω t[ sin ω t mm 1 mm2 mm2 mm 212 11 + A2 sin 2 ω t + 2A (A sin ω t sin ω t + A sin ω+ sin ω t]c ccm1 m cm 2 m1 2 c = AA sin ω t + A sin ω t + A sin ω t( 2)m 1 mm 2 m cc1 22 2222 m 1 mm mcc+BA sin ω t + A sin ω t + A[ sin ω t 221 2 AA + m 1 m 2 [cos( ω – ω )t – cos( ω +ω )] t 2 m 2 m 1 m 1 m 2 2 AA cm 2+ [cos( ωc – ωm )t – cos( ωc +ωm )] t 2 11 2 AA cm 1+ [cos( ω – ω )t – cos( ω +ω )] tcm cm2 22 ∴ fo|eku vko`fÙk;k¡ ω ,ω ,ωm1 m 2 c (ω – ω ),( ω +ω )m mmm2112 (ωc – ωm ),( ωc +ωm )11 (ω – ω ),( ω – ω )c m2 cm2 (i) vk;ke dk ω osQ lkFk oozQ fp=k esa n'kkZ;k x;k gSA ω +ω m1 m 2 ω +ω c m1ωm – ωm21 ω –ωc m1 ωωm2 ω – ωωcω +ωm1 cm cm2 2 (ii) tSlk fd ns[kk tk ldrk gS fd vko`fÙk LisDVªe ω c ij lefer ugh gSA ω < ω c ij LisDVªe dk la?kuu fo|eku gSA (iii) vfèkd ekMqyu flXuy osQ tqM+us ij ω <ω c esa vfèkd la?kuu gksrk gS vkSj flXuy osQ fefJr gksus dh lEHkkouk c<+ tkrh gSA (iv) vfèkd flXuyksa dks lfEefyr djus osQ fy, cS.M pkSM+kbZ rFkk ω c c<+kuh pkfg,A ;g izn£'kr djrk gS fd cM+h okgh vko`fÙk vfèkd lwpuk (vfèkd ω m) dk ogu dj ldrh gS vkSj tks ifj.kker% cS.M pkSM+kbZ dks c<+k,xkA 1 –3 15.30 f m = 1.5kHz, = 0.7 ×10 s fm 1 f c = 20MHz, = 0.5 ×10 –7 s fc (i) RC = 103 × 10–8 = 10–5 s 11 vr% << RC < larq"V gksrk gSfc fm vr% ;g foekMqfyr gks ldrh gSA (ii) RC = 104 × 10–8 = 10–4 s 11 ;gk¡ << RC < ff cm vr% ;g Hkh foekMqfyr gks ldrh gSA (iii) RC = 104 × 10–12 = 10–8 s ;gk¡ 1 > RC , vr% ;g foekMqfyr ugha gks ldrhAfc

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