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) 2
4π 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 qq
1.19 (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 = 26.9815g
AlAl
0.75
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
24
q
2 ⎛ 9 Nm ⎞ (3.48 ×10C) 23
1.21 F = = 8.99 ×10 = 1.1 10 × N
12 ⎜ 2 ⎟ –4 2
4πε 0 r1 ⎝ C ⎠ 10 m
2 –2 2
F2 r1 (10 m) –8 –8 15
10 ⇒ F = F ×10 =× N
== = 1.1 10
2 22 21
F1 r2 (10 m)
2 –2 2
3 1 –16
F r (10 m)
== =10
2 62
F1 r3 (10 m)
F = 10–16 F = 1.1 × 107 N.
3 1
fu"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 ρ
G = – 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 p
mG ( )= – 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 2
R = GNm R
3 εo3 p
2
m
⇒ 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 22
dR ⎛ 1Nye 4p ⎞
m = R– GNm 2R
r ⎜ p ⎟
dt 2 ⎝ 3 eo 3 ⎠
dR ⎛ 22 ⎞
2 2 211Nye 4p 2
vFkok 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
2ˆ
E () 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 3
4πrE = r dr
r ε ∫
o
o
4πkR 4
=
εo 4
4
kR
∴ 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 =
4
π 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 rˆ
izfrd"kZ.k cy gS
4πε o (2r)2
2 22
⎛ e 2er ⎞
– rˆ
usV 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,A
48
(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
q
β osQ dkj.k 0 ij fo|qr {ks=k = –1 xˆ
2εoS
y
q
γ osQ dkj.k 0 ij fo|qr {ks=k = –2 xˆ
2ε S
o °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 = xˆ
22ε S 2ε S 2ε S
oo 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)Q
F = 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 2
1 x 1 10 Nm
. = 10 –5 ⇒ =
bl izdkj
–4 22
4πε 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 2
4πε 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. ..
q
A +B
2π R 4πε 0
–Q
osQ dkj.k q ij cy
2 πR 1
. .cos θ
2
r
(b) iz'u osQ Hkkx (a) ls
d 2z Qqz d 2 z Qq
m = – vFkok = – z A
RB
2 323
dt 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
2.22
z
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 2
lafpr mQtkZ= CV vkSj blfy, vf/d gks tk,xhA
2 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
−qQ
U= 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 Q
2.23 V
1
E.dS =λl
�∫ε
0
vFkok Er2πrl = 1 λl
ε
0
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
r0
0
fdlh fn, x, V osQ fy,]
r 2πε 0
0
ln =− [Vr ( ) – Vr ( )] r λ
0
− 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
0
2.25 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 22
0 ⎡⎣(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
2
VV
−× q′=ε πr 2
02
dd
;fn pfozQdk dks mQij mBkuk gS] rc
2
ε V πr 2 = mg
02
d
mgd 2
⇒ V = 2
πε r
0
1 ⎧ qq qq qq ⎫
dd u d ud
2.28 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
′′
QQ
12
=
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 σ
1
C1
K
K1 2
2.30 vkjEHk esa : vkSjV ∝ 1 ,V 1 + V2 = E E = 9V
C
⇒V1=3V vkSj V2=6V
∴ Q = CV = 6C×3 =18µ C
1 11
Q2=9 µ C vkSj Q3= 0
ckn esa% Q =Q′+ Q
223
Q2
lkFk gh% C2V + C3V = Q2 ⇒V = =(3 2)V
C +C
23
z
′ ′
Q2 = 92
µC vkSj Q3 = 92
µC
Q
2.31 σ= 2
π 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 + z
4πε 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 –z
2 ⎢
⎥⎣ ⎦
4πε 0R
qq
1
2 = 0
+
22 22
x + y +(z – d )2 x +y +(z + d)2
q –q
12
∴
=
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⎤
(q
222 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∈ 22
0 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 = 0
dx 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 x
2 23
⎝ 4π∈0 ⎠ (d – x ) ⎣⎦ x = 0 ij
2
dU ⎛ –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
2
P
P = R
C C
2
V
'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 I
3.22
RR + nR R +
n
1+ n 1 + n
= 10 = n = n
1
n +1
1 +
n
n = 10.
11 1 RRR R
U; wU;Ure wre U;Ure
Ure wU;Uw
3.23 =+ ....... + , =++ ....... +> 1
RR RRRR R
p 1 nP 12 n
vkSj RS = R1 + ...... + R n ≥ R.
vfèkdre
fp=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
RR
R
min
max
max
Rmin
V
V
V
V
(a) (b) (c) (d)
A B
6–4
3.24 I == 0.2A
2 + 8
E1 E2
E1 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 + r
12
2E
V = E – Ir = E – r = 0
111
r + r + R
12
3.26
3.27
R
V
eff
R
eff
3.28
2Er
vFkok E = 1
r + r + R
12
2r1
1 = r + r + R
12
r + r + R = 2r
121
R = r – r
12
ρl
=
RA –3 2
π(10 × 0.5)
ρl
RB =
–32 –32
π[(10 ) – (0.5 ×10 )]
–3 2 –3 2
RA = (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
V
iHz kkoh
rc R ls izokfgr èkkjk I =
Riz+ R
Hkkoh
foeh; :i esa V= V(V1, V, ...... V ) dh foek oskYVrk dh gS rFkk R= R
izHkkoh 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 z
vkSj R izHkkoh
= nR.
bl izdkj èkkjk leku jgrh gSA fdj[kksiQ osQ lafèk fu;e dk vuqiz;ksx djus ij
I = I + I
12
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
I
1
I
I
2
I
2
10�
Reff
10V
V
eff
2 = (R+ R )I dh V= (R + R)I
izHkkoh izHkkoh izHkkoh
vkSj V = 2V
izHkkoh
10
R = Ω
izHkkoh
3
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/s
A 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′
3
10 × 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/m
3
6
3.31 (a) I = = 1 A = nevd A
6
1 1–4 vd = 29 –19 –6 =×10 m/s
10 ×1.6 ×10 ×10 1.6
12
xfrt mQtkZ = m vd × nAl
2 e
1 –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,xhA
6
(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
∴ [] = == T
ω []
⎢ ⎥⎢⎥
⎣ 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
1ˆˆ0
B = ( + +ˆ) µ I
ijk
42R
⎡ eB ⎤
ω
dksbZ foekghu jkf'k ughaA []T –1 = [] = ⎢
⎥
⎣ m ⎦
ˆˆ
E = E i,E > 0, B = B k
0 0 0
d l1 ij d l2 osQ dkj.k cy 'kwU; gSA
d l ij d l osQ dkj.k cy 'kwU;srj gSA
21
iG (G + R1) = 2 (0 – 2V) ifjlj osQ fy, i (G + R + R ) = 20, (0 – 2V) ifjlj osQ fy,
G12
rFkk i (G + R+ R+ R) = 200, 200V ifjlj osQ fy,
G12 3
izkIr 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 = Wl
oqQ.Myh
500 g l = Wl
oqQ.Myh
W = 500 × 9.8 N
oqQ.Myh
tc pqEcdh; {ks=k yxk fn;k tkrk gS] rc
Mgl + mgl = Wl + IBL sin 90°l
oqQ.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 –τ
12
τ=
B
nˆ
B
F1
nˆ
F
2
F2
V
0
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 osQUnzksa
Z
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 θ
R 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
9
2 22
= 4R + R –6 R cos θ
4 pw¡fd d dks 2R ls vfèkd gksuk pkfg, d2 > 4R2
9
222 2
⇒
4R + R –6 R cos θ> 4R
4
⇒
> 6cos θ
9
4
3
vFkok 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 esa
2πR
(c) µ0 i AO osQ yEcor~ ckb± fn'kk esa
πR
(c)
(a)
(c)
(b)
(b)
(a), (d)
(a), (d)
(a), (d)
(a), (c), (d)
(b), (c), (d)
eh
µp ≈ µ≈ ,h =
vkSj e
2mp 2me 2π
µe >> µp D;ksafd m p >> m e Bl =µ Ml =µ (I + I ) vkSj H = 0 = I
0 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
χ
N –4
= 5×10
χ
Cu
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 ˆ
B = 3,m = mk
4π r
y
ds = rr ˆ. 2sin θdθ d 0 ≤θ ≤π,0 ≤φ ≤ π
µ0m 3cos θ 2
Ñ∫ B.ds = ∫ 3 r sin θ dθ x
4π 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 gSaA
c
I 11
m 1T = 2π
I ′= × I rFkk m′= . T ′= T
mB 24
N
22
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
2
B
= 3cos 2 θ+1, θ =π ij U;wureA
22
⎛ µ0 ⎞ 2
m
⎜ 3 ⎟
⎝ 4πR ⎠
pqEcdh; fuj{k ij U;wure gSA
B
B
V = 2cot θ
(ii) tan (ufr dks.k) =
BH
π
θ= ij ufr dks.k 'kwU; gks tkrk gSA iqu% fcanqiFk] pqEcdh; fuj{k gSA
2
BV
(iii) tc = 1 rc ufr dks.k ± 45° gSA
BH
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 =
12
2π R 4a m = n IA1 m = n IA
11222
L LL
2
= I π R = Ia = Ia
2π R 4a 4
2
MR
I1 = (O;kl ls xqtjus okys fdlh v{k osQ ifjr% tM+Ùo vk?kw.kZ)
2
2
Ma
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) = vB
PQ 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 gSA
R
y
ˆ
k
x
( ,0,0)
a
B
B D
P
B
B C
B
B
Q
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 == e
3 20 10 2 t > 30s ij u3 = 0
10–2 vU;ksU; izsjdRo == 5mH
2
–3 –3
ÝyDl = 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
22
Bd
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 ⎤
ˆ
cy = – lx () – B () lv () i
t tt
⎢ ⎥
R ⎣ dt ⎦
22 22
dx lBdB lB dx
m 2 = – x () t –
dt Rdt Rdt
2 22
dB dx lB dx
(ii) = 0, += 0
dt dt 2 mR dt
22
dv 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
2 222
Blv () t Bl
(iii) IR = 2 × R = u exp(–2 l Bt /mR )
RR
t 22
22 22
Bl 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 B
12
=B l ;gk¡ θ =ωt
2
P
l
dφ 1 ε
Q
bl 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
22
dφ Bωl sec ωt
ε= = dt 2tans ωt
εε 1Blω
I== =
R λx 2 λ sin ωt
6.25 rkj ls nwjh r ij
B
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 () t
µoI xdr µoI x
ÝyDl = l = ln
∫
2π r 2π x
o
x
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+ x
2
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 12
Q = ln [I –0 ]
2π xo
µ LI L + x
o11 ⎛ 2 ⎞
= ln
⎜⎟
2π ⎝ x ⎠
2
B.πa
6.27 2 bE = .. ;gk¡ E ik'k osQ pkjksa vksj mRiUu fo|qr {ks=k gSA
π EM F =
Δt ⎡ Bπa 2 ⎤
cy vk?kw.kZ = b × cy = Q Eb = Qb
⎢⎥
⎣2πbΔt ⎦
2
Ba
= Q
2Δt
;fn dks.kh; laosx esa ifjorZu ΔL gS] rc
2
Ba
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
B
B
v d
t = 0 ij I = 0 ⇒ A = – vBd L
B
R
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%]
1
i = × 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 4
220 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′
12
I ′= ×10
1.1
1
2 44
RI′= × 4×10 =3.3×10 W
1.21
3.3 ×10 4
izfr'kfDr {k; = =3.3%
10 6
v sin ωt
m
7.29 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 > 0
1
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 1
i + 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 22
v ⎤
mm ω
i + 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 TT
d ⎛ 1 q 2 ⎞ 2
dt 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
2
dq dq q
7.31 (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 – X
mm –1 ⎛⎞
i ==
;φ= tan CL
⎜⎟
m
22
ZR +(XC – XL ) ⎝ R ⎠
2
11 ⎡ vm ⎤
(ii) UL = Li 2 = L ⎢
22 ⎥ 20
sin ( ωt +φ)22 ⎢ R + X – X ) ⎥
⎣ CL 0 ⎦
2
1 q 21 ⎡ vm ⎤ 1
C ⎢
22 ⎥ 2 20
U ==
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 dφ
0 D 00 E
B = ==ε
4πr 4πr 2πr 0 dt
µε d
00 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< λ
3 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 2
8.23 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 c
0 ⎝
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 Δp
P = = = (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 ˆ
E = j
2πε oa
8.28
µ i
o ˆ
B= i
2π a
µλ v
o ˆ
= i
2πa
11 ⎛ λ ˆjs µo λv ⎞
S= (E×B) = ⎜ ˆj× iˆ⎟
µo µo ⎝ 2πε oa 2πa ⎠
2
–λ v ˆ
k
= 22
4πε 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 èkkjk
d
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
V
tgk¡ 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
o
J =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
dE
J =ε
D 0
dt
∂
⎛ s ⎞
ˆ
= εµ Icos (2πν t). ln k
0 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 ⎞ d
I = ⎜ . ⎟ I0 sin 2 πν t =I 0 sin 2πν t
⎝ 2 λ ⎠
Id 0 ⎛ aπ ⎞2
= ⎜⎟ .
I0 ⎝ λ ⎠
x
8.31 (i)
E 234 1
43
E=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 4
z1 z2 Z
dl
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 ∫∫ ∫∫
Z
1
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 21
k
ω
E = B
00
k
E
0
= cy
B
0
(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 )
Z
1
–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 dlε
Ñ 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 .µε
0 0 00
k E0 ω 1
=
ysfdu E0/B0 = c, rFkk w = ck
B k µε
0 00
1
;k cc . = vr%, c =
µε
00
+1
0 2
12
8.32 (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 0
22 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 av
sin2θ 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
11
222 2
∴ (E ) = E and (B ) = B
av 0 av 0
22
lehdj.k 1 esa izLFkkiu djus ij
1 1 2
2 B0
u =ε E +
40 4 µ
222 2
E0 11 B0 E0/cE0 12
(ii) gesa Kkr gS = c rFkk c =
∴ = =µε =ε0E0
B µε 4 µ 4µ 4µ 0 04
0 00 000
1 11 1
222 2
u = ε E +ε E =ε E , rFkk I = uc =ε E
av 0000 00 av av 00
4 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 = 0.025m
m = 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 h
O = 2
1 µ13
µ3 ls ns[kus ij vkHkklh xgjkbZ O2 gSA
µ ⎛ h ⎞ µ ⎛ h µ h ⎞ h ⎛ µµ ⎞
3 32 33
O =+ 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 ⎠⎦
O
h ⎛ 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 AA
2sin cos
sin A A
22
∴µ== = 2cos
AA
2
sin sin
22
A
A
3
∴ cos = vFkok = 30° ∴ A = 60°
2
22
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 izfrfcac
12 12
11 1 fu
+= ;k v =
dh yEckbZ gksxh L′=
. D;ksafd , nks fljksa dk izfrfcac
v – v
12
uv f u – f
f (u – L /2 ) f (u + L /2 )
gksxk v1 = ij v2 = ij
u – 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%
2
f
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 + R
rFkk 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 ⎠
D
D2–4 Df D
D 2–4 Df
bl izdkj ;fn fcac nwjh –
gS rks izfrfcac +
ij gksxkA
22 22
D
D2–4 Df D
D2 –4Df
;fn fcac nwjh +
gS rks izfrfcac –
ij gksxkA
22 22
bu nks fcac nwfj;ksa osQ fy, izdkf'kd osQUnzksa osQ chp dh nwjh gS
D
D2–4 Df ⎛
2 ⎞
DD – 4D f
2
+
–
= D –4 Df
⎜ –
⎟
22
⎝ 22 ⎠ ekuk d =
D2–4Df Dd Dd
;fn u = + rc izfrfcac gksxk v = – ij
22 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.ksa
2
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 MkbvkWIVj
ff 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 + P
fg
∴ 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 n
eku 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
–dµ
vFkok µ(y) cosθdθ ; dy sin θ dy
–1 dµ
dθ ; 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) cosθdθ
nrsin 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
cx
0– ∞ 2 22
(x + R )
x = R tan φ jf[k,
x = R tan φ
dx = R Sec2 φ d φ
π /2 2
2GMR Rsec φ dφ
∴θ =
02 ∫ 33
c R sec φ
– π /2
π /2
2GM 4GM
= cos φ dφ =
2 ∫ 2
Rc 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 R3π
vr% ≤ x ≤ osQ fy, lzksr ls mRlftZr izdk'k xzkgh IysV rd ugha igq¡psxkA
88
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 << u0
c cc ⎝ 2 u ⎠
P1 ls O rd ikjxeu dk le; gS
PO
v 2 + b2 v ⎛ 1 b2 ⎞
t = 1 = ; 1
2 ⎜ + ⎟
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 + ⎟⎟ S
c ⎝ 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 1
β ; = 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 < n1
n
1
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.
n
1
1.22 × 5.5 ×10 –7
d = m
min 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
2
fufEu"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 0
A = 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 22 222
A0 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 = vkSlr
A ⊥
11
⊥
P osQ lkFk%
A12 ⊥
ekuk vo#¼ gS
1 22 12
rhozrk = = (A + A) + (A )
11 11 ⊥
21
A0 A0
=
2 (1 + cos φ)+ .
⊥ ⊥
2
02
fn;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 ⎠
D
OP = D tan θ≈ –
16
∴ 0
izFke fufEu"B osQ fy,
λ
2d 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
–5
½.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 mQij
2
22
1 – sin θ 16 –3 5
½.kkRed fn'kk esa nwjh gksxh D tan θ – =
2 2 O osQ uhps
16 –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
�/2
d + ⎜⎟ −(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 gSA
2
∴ yD =− a cos ωt
R1 ij izkIr gksus okyk laosQr gS
y + y+ y+ y= 0
AB C D
eku yhft, B ls Rij izkIr gksus okyk laosQr gSy= a cos ωtA B rFkk D ij
2B 1
laosQrksa 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
12
I
∴
= a
R
1
2
R2 p;u djrk gS y = a cos ω t
12
I
∴
= a
R
21
2
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
∴
= a
IR1
2
R2 p;u djrk gS y = 3a cos ω t
12
∴
= 9a
IR2
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
d
2n − 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 o
5500 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 Eα
λ /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= nE
11 22
hc hc
n = n
12
λλ
12
n λ
11
=
n λ
22
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= nν
AA 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< 0
A B
λλ λλ ABAB
AB c c
λλ
vFkok λ c = AB
λA +λB
;fn p> 0, p < 0 vFkok p< 0, p> 0
A 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 3
1 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 = .
∫ 2
4πε 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 eV
eV 4
11.27 (i) B osQ fy, mPp vko`fÙk ij fujks/h foHko = 0 vr% bldk dk;ZiQyu mPp gS
h 2
(ii)