Ias Mains Physics 2011
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_....;..
Sl. No.
Jl,,~
-
-
-
0 PHYSICS
Paper-I
ITime Allowed
: Three Hours
I
IMaximum Marks : 300 I
INSTRUCTI ONS
Each question is printed both in Hindi and in English. Answers must be written in the medium specified in the Admission Certificate issued to you, which must be stated clearly on the cover of the answer-book in the space .provided for the purpose. No ·marks will be given for the answers written in a medium other than that specified in the Admission Certificate. Candidates should attempt Question Nos. 1 and 5 which are compulsory, and any three of the remaining questions selecting at least one question from each Section. The number of marks carried by each question is indicated, at the end of the question. Assume suitable data if considered necessary and indicate· the same Clearly. usual their carry Symbols/ notations meanings, unless otherwise indicated. tm'f ~ :
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Sectio n-A 1.
Answe r all the six. parts below
1Qx6=60
(a)
With an approp riate diagra m, show that in the Ruther ford scatter ing, the orbit of the particl e is a hyperb ola. Obtain an expres sion for impac t param eter.
(b)
Imagin e that a rigid body is rotatin g about a fixed point with angula r velocit y Assum ing that the coordi nate axes coincid e with the princip al axes, if T stands for kinetic energy and G for .extern al torque acting on the body, show that
w.
a.r . . . .
-=G ·w dt
(c)
A spaces hip measu res 50 m in length on ground and it measu red a length of 49·7 m in space as observ ed from the ground . Find out the speed of the spaces hip.
(d)
Write down the one-di mensio nal harmo nic oscilla tor differe ntial equati on under dampi ng and its solutio n for the lightly dampe d .condit ion, with the meani ngs of symbo ls. Determ ine the depen dent energy in the lightly dampe d condit ion . •
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:..
, •
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(e)
The Fraunhofer- sii).gle-slit diffraction intensity is given by •
2
l=Iostn x x2 1t dy- w1t . h 1' as t h e ,d"1stance "where x = -
AI
from slit to source, d the slit width, y the detector distance and A the wavelength. What is the value of cumulative intensity r~ I(y)dyi (f)
plane wave has the following expression 'for its electric field :
A
->
E = XEox cos(cot- kz+ a)+ yE 0 y cos(cot- kz+ ~) If the phase_ difference is defined as li = ~ -a, under what· conditions do we achieve elliptic polarization? What are the conditions for circular polarization? 2. (a) :Determine the number of' degrees of freedom for a rigid body(i) moving freely in space of three dimensions; (ii) having· iine 'point •fixed; (iii) having· two points 'fixed.
(1:})
30
•
Prove that the time taken by t4e earth to travel over half of its orbit separated by the ininor axis remote from the ·sun is two days more than half a year. Given, the period of the earth is 365 days and eccentricity of the orbit = 1 J60. 10 •
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......-'.
-lt_-=.
-
--
U~
IlL..,.,
~ x = 11}.1dy W
r
->
E
=XEo~. cos(w t- kz +a) +.[JE 0 y cos(oo~- kz + 13) >lft f1l;;j
'fiiill'
'mn t
~=13-a
~· i!
1m!
Wi"' ~ ~ 'f
(q;) 11;'11
w;
it,
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m ~ ~ ~d~dl
~~
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(~)
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(c)
With an app~gpriate .diagram,: deduce the velocity proflle for streamline flow of a liquid through' a capillary of circular cross-section . Deduce· also the fraction of liquid which flows . through the section up to. a distance a/2 from the axis, where a is tlie radius of the 20 •capillary. •
3.
~
!
(a)
Explain the physical significance of group velocity from the concept of phase 15 .velocity with r_elevl¥lt expressions.
(b)
Prove that the group c veloci~y V9 of electrinnagh etic waves in: a dispersive medium with, refractive index. n(A. 0 ) at · wavelength 1., 0 is ,given by
v: -· g -
(c)
c n(A.o)- A. a dn(A.ol dA.o
where cis the free space velocity .of light. the for taken time the Fin:d electromagn etic 'pulse to travel a distance D.
20
When a thin fllm -of a transparent material is put ·behind one o-f the slits in Young's double-slit interference ex_p'eriment, the zero-order· frfuge•moves to the position previously occupied by the ~ourth-order bright fringe. The index. 'of refraction of the mi:n is 'n =1 -2 and the' wavelength 'of llght, A.= sooo A. Determine the thickness of the film.
10
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(d)'
In relation •to a- plane diffraction..-gr ating having 5000 lines .per em and irradiated by light of wavelength 6000 A, answer · the following : (i)
15
What is .thO! highest order spectrum which may be observed?
(ii) If the width of opaque space is
the of that tWice exactly trall_sparent· space, which order of spectra will be absent?
4.
(a)
For calcite, -the refractive indices of ordinary and extraprdinar y .rays are 1·65836 and 1·48641 at A. 0 = 5893 A . . respectively. A left circularly polarized beam ·of ·this wavelength is incident normally on such crystal of thickness 0·005141 mm having its optic axis cut parallel to the surface. What will be the state· of poliuization of the emergent 15 beam?
(b)
Bring out the essential differences between the physical principles of spontaneous and stimulated emission of radiation. Why is it difficult to get efficient lasing action in case of an ideal two-level material system? Can you enhance a. scheme· to propose 15 c;fficiency? Discuss.
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(>r) -f'i;~IL<:i4d<1 fi'I«Jofl ·m~·WN -it,~ 5000
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(i)
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.(c)•
Show :· with :proper •mathematical ru;aJysi_s that .the ratio o~ Einstein's A and B coefficients depends upon the eriergy separatio,;_ .between the two energy levels participating in the optical transitions. What is the physical sigrlificance. of..A coeffJ.c.ient? JustifY the statement, "It is very difficult to develop an X-ray laser". 20+5+5=30
5. Answer all the six parts below : 1
{d) <
10x6=60
Find whether· the discharge of a .:, condenser through the .inductive circuit is oscillatory when C ,; 0 · 1 JlF, L•= 10 mH and R = 200 Q. :If it is oscillatory, .calculate its frequency .
. (b) · The"lfour ru:ms of a Wheatstone' bridge
have the following resistances : AB=100Q, BC=lOQ, CD=5Q, DA=60Q A galvanometer of 15 Q resistance is
conne"ded · across BD: Calculate • 1:1:\.e current through the.galvanometer when a potential difference. . -of. 10 volts is ' ' ·-· maintained across AC. \. ·-· .I
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(TT)
e~a •1fillifl'l li'l~<'l~·~
3ll~t2k ~ ·A'([?.!! .B ~ 'q;r -~· ~l~l
•
fi'l <:'1' H
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'I1ITf
~ 'li"R ~
3il [;, ('
qil ~
mur <'1m qil li'lfuo "Br ~ '1ft.!
"1.% 1l;<m-
.20+5+5=30
10x6=60
('!i) ~-
11
ijmfur q;r
c = o ·1!-lF, ~ ~ Gl<:'lofl t
,L., = 10 - mH .([?.!! R.= -. 200 .Q.
-
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on (<9) 1.%. 101~0~ fir.!
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~ 00 lf'l1 ~' 0'1
lw!· 10 ~
'lil Wd
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----
-- ----
-
-------
(c)
Write down the expreifs ions for Bose-Ei nstein and Fermi-D irac distribution funCtion s, and show how Fehni-D irac distribu tion leads to the explana tion of Pauli's 'exclusi on principl e.
(d)
Find out the total electric potentia l energy of a single spherica l object of uniform charge density p, total charge Q and radius R.
(e)
A long solenoid ·l!a"! 220 turns/ em; its diamete r is 3·2 em. Inside the solenoid at its centre, we place a 130-_turn closepacked coil of diamete r 2· i. cm·"aiong its axjs. The current in the f!Olenoid is increase d from zero to 1·5 ampere s at a steady rate over a p'eriod of 0·16 second. What is the magnitu de of the· induced e.m.f. that appears in the. central coil when the current in the solenoid is being changed ?
(f)
Conside r Maxwel l;s equatio n in differen tial form in media. For j = p = 0, assume .E =E 0 eat and I! ;=1! 0 eat and show that the relevant wave equatio n for ·a. plane wave propaga ting along xodirect ion is iJ~ E iJ 2 D iJD -=!!- +!!C XiJx2 iJt 2 ilt
...
...
where E = Ey and H = HZ. C-DTN- L-QZA/ 25
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t:
= i:oe"t
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6. "(a)
{b)'
An
electrica l 'circuit consists of' a resistan ce R, inducta nce L and f:apacit ance C. in series. J~ a charge is put on the capacito r at some ·instant, determi ne the conditio n that Vc, the voltage across the capacito r, 1s subsequ ently oscillato ry. Derive an express ion for the quality factor Q of the circuit by cpnside rihg the .decay of the oscillati on, using the result that the amplitu de falls by a factor of e in (~) period. 20 A resistor R(~ 6 · 2 Mn) and' a capacito r C(= 2 · 4 !LF) are connect ed' in series and a 12 V battery of negligibl_e internal resistan ce is connect ed across their combinatio11., •
(i)
'
r
•
What is the caP,acithre constan t of this circui,t?
time
(ii) At what time, after the battery is
connect ed, does differen ce across become S'6 v? (c)
the the
potentia l capacito r 5+5=10
Determ ine the t()rql,le .experie nced by an electric dipole of momen t p if placed in '
....
an electric field E in .. nonalig ned state. Also show that the interact ion energy of two dipoles of 'mome11ts. and 2 separat ed by a displace ment is
Pi
U=
1
r
p
1 [... ... ... A)(... A)] 3' Pi · P2 - 3(PJ ·r P2 · r
4 !tEo r
assumin g the express ion for field due to a dipole. 20 C:._DTN-L-QZA/25 14 www.examrace.com
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C-DTN-~ZA/25
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[Pi· p 2 -3(pi · r)(p 2 · r)] ·
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15
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(d)
The electric potential of a grounded conducting sphere 6f radius a m a uniform electric field is given as cfl(r,
e)=
-E 0
+-(;rco·seJ
·Find the surface charge distribution.
7.
10
(a)
Consider a plane 'wave travelling aJong the positive y-direction incident upon a glass of refractive index n =1·.6. Find the transmission, coefficient. 25
(b)
Justify whjcj:l of the four Maxwell's equations imply that there are no magnetic monopoles. How these equations would have been written if they were? 15
(c)
In deiiving the .Rayleigh-Jeans law, we count. the number of modes dn .corresponding to a. wave number k for a photon gas in a cubicai box. Consider a cqbical container of volume V containing such gas in equilibrium. Calculate the differential number of allowable normal modes of frequency w. 20
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a]
~ '[II ~ oir.nrro
10
•
7.
t
~
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(if>) ~ q;il 3'Jq n = 1 · 6
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25
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C-DTN-L-QZA/25
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8.
(a}" Wha t led van der Waal s tO moai fy the
ideal gas equa tion? Usin g the conc epts of critic al temp eratu re Tc, press ure Pc and volum e Vc, show ,that the critic al cons tant for h'!al gas is 8/3. 5+15 =20
a
(b)
(c)
Find out the expre ssion s van der Waal s cons tants a and b.
for 20
Calcu late the value s of van der Waal s cons tants a aiid b for oxyg en with Tc =154 · 2 K, Pc =49 ·' 7 atmo sphe re and R = 80 em 3 atmo sphe re/K. 20
• '
'
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- - ···---
8,, ('!\)
f'R1
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..._,____,_.,_
'liR"T qiO:S(iHRI
'ifi't
I I .....
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~
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Bifl'f>(OI q;r
mil~ "ifiBI ~? SI>IRi'f> (11'1 Tc' 01'! PC wn ~ Vc it ~1f><"1 'IIW q;r 34ti\ ~ ffm q;r Sl>iRI'f>~ 8/3%1
5+15=20
(<9) ql·'miRt it ~ awn bit
iW<
a:ior'f> >mr
<Ji1 f;j !!; I
.('1)
3lT
20
it iW<,
= 49 · 7
Q;i!GIM
liR
f;jldif>l
Q;i!GIM
%,
=
Tc = 154 · 2 K, R
ql-s(qiR! ~
ma
= 80 em 3 a o~ b it 2o
***
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C-DTN-L-QZA
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English version of the Instructions is printed on the front cover of this question paper.
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Sl.N o.
D PHY SICS
Pap er-II
I
Time Allowed : Three Hour s
I I
Maxi mum Mark s : 300
I
INST RUC TION S
Each quest ion is print ed both in Hind i and in English. -Answ ers must be writt en in the medi um speci fied in the .Admis~ion Certificate issue d io you, whic h must be stat,ed clearly on the cover of the answ er-bo ok in the spac e provi ded for the. purpo se. No mark s will be given for the answ ers writt en in a medi um other than that speci fied in the Admi ssion Certificate. Cand idate s shou ld attem pt. Ques tion Nos. 1 .and 5 whic h are compulsory, and any three of the rema ining quest ions selec ting at least pne quest ion fro!Jl each Section. The numb er of mark s carried by each ques tion is indic ated at the end of the question. Symb ols/ Nota tions carry usua l mean ings. Assu me suita ble data if consi dered nece ssary and indic ate the same clearly. Some cons tants are given at the end of quest ions. Vffi ~ :
3l:J;~~fj 'liT J%;
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Sectio n-A 1.
(a)
Calcul ate th<'; wavele ngth of de Broglie waves associ ated with electro ns accele rated throgg h a potent ial differe nce of 200 V. 10
(b)
Norma lize the wave functio n ljl(x) = e-lxl sin ax
(c)
10
a
Let be the vector operat or with . compo nents equal to the Pauli spin . ~ ~ matnc es cr x• cry' and· cr z. lCa and b are the vectors ' in 3-dime nsiona l space, prolle the identit y ~
--).
~
~
.,.
--)-
_.,
-)o
-i-
(cr·a) (cr·b) =a·b+ icr·(a xb)
10
(d)
Discus s the fine structu re of hydrog en atom spectr um. Draw the compo und double t spectr um arising as a result ·of transit ions betwee n 2 P and 2 D levels. 10
(e)
What do you mean by 'term symbo ls'? Obtain term symbo ls for the following sets .. of values of S and L : 10 (i)
S
= ~·
L
=2
(ii) S=1, L =1
(iii) S=:J. 2' L=1 . .(f)
Estima te the size of the' hydrog en atom and the ground -state energy from the uncert ainty princip le. .1 0
C-DTN -L-QZ B/26
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··------
1.
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-2.
---~---------~~~--~~
(a)
Solve the Schro dinge r equat ion for a partic le of mass m in an infini te rectan gular well define d by the poten tial V(x) = 0 , =
00'
osxs L xL •
Obtai n the norma lized eigen functi ons and the corres pondi ng eigenv alues. 25 (b)
Calcu late (ll.x) 2 , where 1\.x = x-(x) .
fc)
The norma lized wave functi on for the electr on "in the groun d state of the hydro gen atom is given by
15
IJI(r) = - 1 , e(-r/ao l (nag) v,
where a 0 is the radius · of the first Bohr orbit. Calcu late (r) and
m.
3.
(a)
Show that
2
20
S 1 , 2 P 1 and 2 P3 levels of 2 2 2
sodiu m spect rum are split in the ratio of 3 : 1 : 2 due to anom alous Zeem an effect. 30 (b)
On the basis of three princi pal. mome nts of inerti a I A, I 8 and Ic each about X, Y and Z axes respec tively , how can you classi fy molec ules? 30
.
4. (a)
(i)
Treati ng a diatom ic riwlec ule as a simpl e harmo nic oscill ator, obtain its vibrat ional, energ y ,levels. 20
C-DT N-L-Q ZB/26
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2.
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=oo,
0' 5, x 5, L
xL
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5
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(ii) The obser ved vibra tiona l frequ ency... :
of the CO mole cule 1s 13 6. 42 x 1'0 Hz. Wha t is the effect ive force cons tant of this mole cule? (Mas s of carbo n atom = 12 u and mass of oxyg en atom = 1'6 u, wher e U'is atom ic mass unit) 10 (b)
(i)
Disc uss pure rotat ional spec tra of linea r molec L!les, 25
(ii) Wha t is Lamb shift?
5
Sect ion- B 5,
(a)
Show that syste m.
nucle us
IS
a
quan tum 10
(b)
Deriv e Brag g diffra ction . cond ition in vecto rial form usmg incid ent and diffra cted wave vecto rs and recip rocal lattic e vecto r. 10
(c)
Find the total kinet ic energ y o[ elect ron and antie lectr on neutr ino emitt ed in beta deca y of free 'neut ron. (The neutr on-p roton mass differ ence 1s l·30 MeV and mass of elect ron 1s 0·51 MeV) 10
(d)
An elect ron beam of
4
keV is diffra cted throu gh a Brag g angle of 16° for the first maxi ma. If. the· energ y ds incre ased ·to 16 keV, .find .the corre spon ding Brag g angle for diffra ction . 10
C-DT N-Jr. QZB /26
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fiiJ co
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6 · 42 x 1013 Hz
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C-DTN-l.r-QZB/26
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(e)
(f)
Exp lain cons erva tion of bary on num ber . .ComiT)ent on stab ility pf prot on.
10
Sim plify the logic al expr essi on AB+ AB +AB C
usin g a Kar nau gh map .
6.
10
(a}
Wha t is the imp orta nce of stud y of deu tero n? Obt ain the solu tion of Schr iidin ger equa tion for grou nd stat e of deu tero n and show that deu tero n is a loos ely bou nd syst em. 25 (ii) Wha t do you mea n by 'non -cen tral forc es'? 5
(b)
(i)
(i)
(ii)
Wha t are chai n reac tion s? Wha t do you mea n by criti cal size of the core m whi ch chai n reac tion take s plac e? Wha t is criti cal mas s? 20 235
U yiel ds two frag men ts of A = 95
and A = 140. Obt ain the ener gy dist ribu tion of the fissi on prod ucts . Ass ume that the two frag men ts are ejec ted with equa l and oppo site ,mo men tum . 10
7.
(a}
Find an expr essi on for lattic.e spec ific hea t of solid s, and its low and high tem pera ture limi ts. Wha t is Deb ye ·tem pera ture ? 20
C-D TN- L-Q ZB/ 26
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8.
(b)
Explain 'the· ongm of· energy band formation in solids. Show that,in nearly free electron approximation, the energy band gap is 2IVa I, where Va is Fourier transform of periodic potential seen by the valence electrons. 20
(c)
Describe the characteristic properties of a superconductor. Derive London equation for a superconductor and hence explain Meissner effect. 20
(a)
Distinguish 'between intrinsic and extrinsic semiconductors. Show that m a semiconductor, the product of ·concentrations of the two types of charge carriers 1s constant at a given temperature. 30
(b)
Simplify the logical expression (A +B)
and draw the implement it. (c)
El (A+ C) logical
circuit
to 15
Draw the commoncbase amplifier circuit; using an n-p-n transistor and briefly discuss its working. 15
C-DTN~L--QZB/26
10
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C-DTN -L-QZ B/26
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Constants which may be needed Velocity of light in vacuum (c)= 3xto8 ms- 1 Mass of electron ("'e)= 9. 11 X
w- 31
kg
Charge of electron (e) = 1· 602 X 1o- 19
c
Specific charge of electron(,;:,)= 1· 76 x td 1 C kg- 1 1u
=1
a.m. u. = 1· 6605 X
w- 27
kg = 931· 5 MeV
Rest mass energy of electron (11lec2 ) = 0·5110 MeV Permittivity in free space (Eo)= 8· 8542
X
to- 12 c 2 N- 1 m"2
Permeability of free space'(/lo) = 47t xlo- 7 N A-2 Gas constant (R) = 8· 314'J mol-l K- 1 Boltzmann con~tant (ks) = 1· 38lxlo- 23 .J K- 1 Planck constant (h)= 6 · 626 X 10-34 J s (hJ = t· 0546 x 1o-34 J s
Bohr magnetori {11 s'i = 9 · 274 x 1o-24 • J T~ 1 Nuclear magneton {11 N) = 5 · 051 x 1o- 27 J T-l Fine structure constant (a)= 1 f 137 · 03599
1
Mass of proton (mpJ.= 1· 0072766 u = 1· 6726 X w-2 7 kg Mass of neutron (ril,) = 1· 0086652 u =1·6749xto- 27 kg Mass of deuteron ("'rJ) = 2· 013553u Mass of a-particle (171a) = 4· 001506u Mass of 1 ~C = 12· OOOOOOu Mass of 1 ~0 =15·994915u Mass of ~~Sr = 86 · 99999 u Mass of ~He = 4:002603 u
C-DTN-L-QZB/26
12 www.examrace.com
mfuil>r<m~q;riM (c)= 3x!08 ms- 1
*8if?i'l
q;r M4H (m,) = 9 · II x I o-3 ! kg
*8if?i'l
q;r
amm (e) = I· 602 xI o-!9 C
·*<'lif?i'l
q;r
~~
(:J
=
J. 76x 1d 1 C
kg- 1
I u " I a.m.u. = I· 6605 x 10- 27 kg = 931· 5 MeV *<'lif?i'li«T l<"l413 "3>'ill (m,c2 ) = Q·5!10 MeV
:!"'"
anq;m ij Pl~j
:!"'"
anq;m <1:\ qi(>IRHU (Ito)= 47t x10- 7 N A-2
fm~ (R)=8·3 14J mol-l K- 1
•lc<::>ll~'l ffil
oi\o1:
~·~
~ ~·~
Jitl:i;, q;r ""'""' (mp) = 1· 0072766 u = 1· 6726. x 1 o- 27 kg
"@'! q;r
M4H (m, I = I· 0086652 U·
=1·6749 x1o- 27 kg
s%
q;r ""'"'"·("' d)= 2· 013553 u
a-"'hUT q;r l<"l4H ("'a)= 4· 001506u 1
~c q;r
""'""' = 12 · oooooo u
go q;r ""'""' = 15· 994915 u
1
~~Sr
q;r M4H = 86 · 99999 u
~He q;r l<"l4H =4· 002603u
*** C-DTN-~QZB/26
13
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C-DTN-L-QZB
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Note :
English verswn of the Instructions is printed on the front cover of this question paper. www.examrace.com
Sl. No.
Jl,,~
-
-
-
0 PHYSICS
Paper-I
ITime Allowed
: Three Hours
I
IMaximum Marks : 300 I
INSTRUCTI ONS
Each question is printed both in Hindi and in English. Answers must be written in the medium specified in the Admission Certificate issued to you, which must be stated clearly on the cover of the answer-book in the space .provided for the purpose. No ·marks will be given for the answers written in a medium other than that specified in the Admission Certificate. Candidates should attempt Question Nos. 1 and 5 which are compulsory, and any three of the remaining questions selecting at least one question from each Section. The number of marks carried by each question is indicated, at the end of the question. Assume suitable data if considered necessary and indicate· the same Clearly. usual their carry Symbols/ notations meanings, unless otherwise indicated. tm'f ~ :
31:lf~i1 'liT '0'!1
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Sectio n-A 1.
Answe r all the six. parts below
1Qx6=60
(a)
With an approp riate diagra m, show that in the Ruther ford scatter ing, the orbit of the particl e is a hyperb ola. Obtain an expres sion for impac t param eter.
(b)
Imagin e that a rigid body is rotatin g about a fixed point with angula r velocit y Assum ing that the coordi nate axes coincid e with the princip al axes, if T stands for kinetic energy and G for .extern al torque acting on the body, show that
w.
a.r . . . .
-=G ·w dt
(c)
A spaces hip measu res 50 m in length on ground and it measu red a length of 49·7 m in space as observ ed from the ground . Find out the speed of the spaces hip.
(d)
Write down the one-di mensio nal harmo nic oscilla tor differe ntial equati on under dampi ng and its solutio n for the lightly dampe d .condit ion, with the meani ngs of symbo ls. Determ ine the depen dent energy in the lightly dampe d condit ion . •
C-DTN -L-QZ A/25
2 www.examrace.com
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:..
, •
10x6=60 (
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'
-
. .A
.,
1:.
dT --> --. -·=G.· w dt '
.
.
'
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50 m ~ 1!T
(TJ) 1f
.
.
i
('Q) :>!'Ill<:~ 1l ~-~ m<'l 3T1
'&'<
mll
'
~ :>~
.
-~
C-DTN-L-QZA/25
3
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(e)
The Fraunhofer- sii).gle-slit diffraction intensity is given by •
2
l=Iostn x x2 1t dy- w1t . h 1' as t h e ,d"1stance "where x = -
AI
from slit to source, d the slit width, y the detector distance and A the wavelength. What is the value of cumulative intensity r~ I(y)dyi (f)
plane wave has the following expression 'for its electric field :
A
->
E = XEox cos(cot- kz+ a)+ yE 0 y cos(cot- kz+ ~) If the phase_ difference is defined as li = ~ -a, under what· conditions do we achieve elliptic polarization? What are the conditions for circular polarization? 2. (a) :Determine the number of' degrees of freedom for a rigid body(i) moving freely in space of three dimensions; (ii) having· iine 'point •fixed; (iii) having· two points 'fixed.
(1:})
30
•
Prove that the time taken by t4e earth to travel over half of its orbit separated by the ininor axis remote from the ·sun is two days more than half a year. Given, the period of the earth is 365 days and eccentricity of the orbit = 1 J60. 10 •
C:-DTr-/-L-QZA/25
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......-'.
-lt_-=.
-
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U~
IlL..,.,
~ x = 11}.1dy W
r
->
E
=XEo~. cos(w t- kz +a) +.[JE 0 y cos(oo~- kz + 13) >lft f1l;;j
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it,
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C-DT N-L-Q ZA/25
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(c)
With an app~gpriate .diagram,: deduce the velocity proflle for streamline flow of a liquid through' a capillary of circular cross-section . Deduce· also the fraction of liquid which flows . through the section up to. a distance a/2 from the axis, where a is tlie radius of the 20 •capillary. •
3.
~
!
(a)
Explain the physical significance of group velocity from the concept of phase 15 .velocity with r_elevl¥lt expressions.
(b)
Prove that the group c veloci~y V9 of electrinnagh etic waves in: a dispersive medium with, refractive index. n(A. 0 ) at · wavelength 1., 0 is ,given by
v: -· g -
(c)
c n(A.o)- A. a dn(A.ol dA.o
where cis the free space velocity .of light. the for taken time the Fin:d electromagn etic 'pulse to travel a distance D.
20
When a thin fllm -of a transparent material is put ·behind one o-f the slits in Young's double-slit interference ex_p'eriment, the zero-order· frfuge•moves to the position previously occupied by the ~ourth-order bright fringe. The index. 'of refraction of the mi:n is 'n =1 -2 and the' wavelength 'of llght, A.= sooo A. Determine the thickness of the film.
10
C:-DTN-I.ri2 ZA/25
6 www.examrace.com
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-- ~ =-"
(TJ) ~ 6~Rla·'t<:il1R1~
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mm
a.
{
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'
15
(19) ~
mm :
Vg=
c
.
...
'
n(A.o)- A.o dn(A.o)
dA.o ~-c
wn 3lil
~
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"fiG_ i'\ .find~ I
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~?
20
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f'Rft ·'l'd<'fi -4R~~IT- ~
3l4ild'lii!>
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(1'1!1
Y
q;J
>.. = 5ooo A~~~
C-DTN -L-QZ A/25
7
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(d)'
In relation •to a- plane diffraction..-gr ating having 5000 lines .per em and irradiated by light of wavelength 6000 A, answer · the following : (i)
15
What is .thO! highest order spectrum which may be observed?
(ii) If the width of opaque space is
the of that tWice exactly trall_sparent· space, which order of spectra will be absent?
4.
(a)
For calcite, -the refractive indices of ordinary and extraprdinar y .rays are 1·65836 and 1·48641 at A. 0 = 5893 A . . respectively. A left circularly polarized beam ·of ·this wavelength is incident normally on such crystal of thickness 0·005141 mm having its optic axis cut parallel to the surface. What will be the state· of poliuization of the emergent 15 beam?
(b)
Bring out the essential differences between the physical principles of spontaneous and stimulated emission of radiation. Why is it difficult to get efficient lasing action in case of an ideal two-level material system? Can you enhance a. scheme· to propose 15 c;fficiency? Discuss.
C-DTN-'-L-QZA/ 25
8
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.'tWt -mt ililifk( m3TR foHl 'li
.
(i)
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Tf'!T.
6000 Aa{•IM ~
~.- f.i'"'IRi_[{ijd ~ :m 15
::i~d4 ~·
~i (ii) ~ 3141mff Wl'l>l~l
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t1
0·005141 mm
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15
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4il f.! t.{ I 't.{'fi itcit '
C-DTN-L-QZA/25
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.(c)•
Show :· with :proper •mathematical ru;aJysi_s that .the ratio o~ Einstein's A and B coefficients depends upon the eriergy separatio,;_ .between the two energy levels participating in the optical transitions. What is the physical sigrlificance. of..A coeffJ.c.ient? JustifY the statement, "It is very difficult to develop an X-ray laser". 20+5+5=30
5. Answer all the six parts below : 1
{d) <
10x6=60
Find whether· the discharge of a .:, condenser through the .inductive circuit is oscillatory when C ,; 0 · 1 JlF, L•= 10 mH and R = 200 Q. :If it is oscillatory, .calculate its frequency .
. (b) · The"lfour ru:ms of a Wheatstone' bridge
have the following resistances : AB=100Q, BC=lOQ, CD=5Q, DA=60Q A galvanometer of 15 Q resistance is
conne"ded · across BD: Calculate • 1:1:\.e current through the.galvanometer when a potential difference. . -of. 10 volts is ' ' ·-· maintained across AC. \. ·-· .I
C-DTN-L--QZA/25
10
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e~a •1fillifl'l li'l~<'l~·~
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-
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C-DTN-l.r-QZA/25.
1'1
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----
-- ----
-
-------
(c)
Write down the expreifs ions for Bose-Ei nstein and Fermi-D irac distribution funCtion s, and show how Fehni-D irac distribu tion leads to the explana tion of Pauli's 'exclusi on principl e.
(d)
Find out the total electric potentia l energy of a single spherica l object of uniform charge density p, total charge Q and radius R.
(e)
A long solenoid ·l!a"! 220 turns/ em; its diamete r is 3·2 em. Inside the solenoid at its centre, we place a 130-_turn closepacked coil of diamete r 2· i. cm·"aiong its axjs. The current in the f!Olenoid is increase d from zero to 1·5 ampere s at a steady rate over a p'eriod of 0·16 second. What is the magnitu de of the· induced e.m.f. that appears in the. central coil when the current in the solenoid is being changed ?
(f)
Conside r Maxwel l;s equatio n in differen tial form in media. For j = p = 0, assume .E =E 0 eat and I! ;=1! 0 eat and show that the relevant wave equatio n for ·a. plane wave propaga ting along xodirect ion is iJ~ E iJ 2 D iJD -=!!- +!!C XiJx2 iJt 2 ilt
...
...
where E = Ey and H = HZ. C-DTN- L-QZA/ 25
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= i:oe"t
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=p =0
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ilD il 2 D il 2 E !1U-=!1-+ 2 ()x2
.-+
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E
ot
ot
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C-D TN-I .rQZ A/25
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6. "(a)
{b)'
An
electrica l 'circuit consists of' a resistan ce R, inducta nce L and f:apacit ance C. in series. J~ a charge is put on the capacito r at some ·instant, determi ne the conditio n that Vc, the voltage across the capacito r, 1s subsequ ently oscillato ry. Derive an express ion for the quality factor Q of the circuit by cpnside rihg the .decay of the oscillati on, using the result that the amplitu de falls by a factor of e in (~) period. 20 A resistor R(~ 6 · 2 Mn) and' a capacito r C(= 2 · 4 !LF) are connect ed' in series and a 12 V battery of negligibl_e internal resistan ce is connect ed across their combinatio11., •
(i)
'
r
•
What is the caP,acithre constan t of this circui,t?
time
(ii) At what time, after the battery is
connect ed, does differen ce across become S'6 v? (c)
the the
potentia l capacito r 5+5=10
Determ ine the t()rql,le .experie nced by an electric dipole of momen t p if placed in '
....
an electric field E in .. nonalig ned state. Also show that the interact ion energy of two dipoles of 'mome11ts. and 2 separat ed by a displace ment is
Pi
U=
1
r
p
1 [... ... ... A)(... A)] 3' Pi · P2 - 3(PJ ·r P2 · r
4 !tEo r
assumin g the express ion for field due to a dipole. 20 C:._DTN-L-QZA/25 14 www.examrace.com
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C-DTN-~ZA/25
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15
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(d)
The electric potential of a grounded conducting sphere 6f radius a m a uniform electric field is given as cfl(r,
e)=
-E 0
+-(;rco·seJ
·Find the surface charge distribution.
7.
10
(a)
Consider a plane 'wave travelling aJong the positive y-direction incident upon a glass of refractive index n =1·.6. Find the transmission, coefficient. 25
(b)
Justify whjcj:l of the four Maxwell's equations imply that there are no magnetic monopoles. How these equations would have been written if they were? 15
(c)
In deiiving the .Rayleigh-Jeans law, we count. the number of modes dn .corresponding to a. wave number k for a photon gas in a cubicai box. Consider a cqbical container of volume V containing such gas in equilibrium. Calculate the differential number of allowable normal modes of frequency w. 20
C-DTN-L--QZA/25
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C-DTN-L-QZA/25
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8.
(a}" Wha t led van der Waal s tO moai fy the
ideal gas equa tion? Usin g the conc epts of critic al temp eratu re Tc, press ure Pc and volum e Vc, show ,that the critic al cons tant for h'!al gas is 8/3. 5+15 =20
a
(b)
(c)
Find out the expre ssion s van der Waal s cons tants a and b.
for 20
Calcu late the value s of van der Waal s cons tants a aiid b for oxyg en with Tc =154 · 2 K, Pc =49 ·' 7 atmo sphe re and R = 80 em 3 atmo sphe re/K. 20
• '
'
C-DT N-fM 2ZA/ 25
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5+15=20
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iW<
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20
it iW<,
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ql-s(qiR! ~
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C-DTN-l.r -QZA/25
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C-DTN-L-QZA
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English version of the Instructions is printed on the front cover of this question paper.
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Sl.N o.
D PHY SICS
Pap er-II
I
Time Allowed : Three Hour s
I I
Maxi mum Mark s : 300
I
INST RUC TION S
Each quest ion is print ed both in Hind i and in English. -Answ ers must be writt en in the medi um speci fied in the .Admis~ion Certificate issue d io you, whic h must be stat,ed clearly on the cover of the answ er-bo ok in the spac e provi ded for the. purpo se. No mark s will be given for the answ ers writt en in a medi um other than that speci fied in the Admi ssion Certificate. Cand idate s shou ld attem pt. Ques tion Nos. 1 .and 5 whic h are compulsory, and any three of the rema ining quest ions selec ting at least pne quest ion fro!Jl each Section. The numb er of mark s carried by each ques tion is indic ated at the end of the question. Symb ols/ Nota tions carry usua l mean ings. Assu me suita ble data if consi dered nece ssary and indic ate the same clearly. Some cons tants are given at the end of quest ions. Vffi ~ :
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Sectio n-A 1.
(a)
Calcul ate th<'; wavele ngth of de Broglie waves associ ated with electro ns accele rated throgg h a potent ial differe nce of 200 V. 10
(b)
Norma lize the wave functio n ljl(x) = e-lxl sin ax
(c)
10
a
Let be the vector operat or with . compo nents equal to the Pauli spin . ~ ~ matnc es cr x• cry' and· cr z. lCa and b are the vectors ' in 3-dime nsiona l space, prolle the identit y ~
--).
~
~
.,.
--)-
_.,
-)o
-i-
(cr·a) (cr·b) =a·b+ icr·(a xb)
10
(d)
Discus s the fine structu re of hydrog en atom spectr um. Draw the compo und double t spectr um arising as a result ·of transit ions betwee n 2 P and 2 D levels. 10
(e)
What do you mean by 'term symbo ls'? Obtain term symbo ls for the following sets .. of values of S and L : 10 (i)
S
= ~·
L
=2
(ii) S=1, L =1
(iii) S=:J. 2' L=1 . .(f)
Estima te the size of the' hydrog en atom and the ground -state energy from the uncert ainty princip le. .1 0
C-DTN -L-QZ B/26
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··------
1.
('h) 200 V ~ il; &m i'1lful
t\ ~ cWIT ij;
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-2.
---~---------~~~--~~
(a)
Solve the Schro dinge r equat ion for a partic le of mass m in an infini te rectan gular well define d by the poten tial V(x) = 0 , =
00'
osxs L x
Obtai n the norma lized eigen functi ons and the corres pondi ng eigenv alues. 25 (b)
Calcu late (ll.x) 2 , where 1\.x = x-(x) .
fc)
The norma lized wave functi on for the electr on "in the groun d state of the hydro gen atom is given by
15
IJI(r) = - 1 , e(-r/ao l (nag) v,
where a 0 is the radius · of the first Bohr orbit. Calcu late (r) and
m.
3.
(a)
Show that
2
20
S 1 , 2 P 1 and 2 P3 levels of 2 2 2
sodiu m spect rum are split in the ratio of 3 : 1 : 2 due to anom alous Zeem an effect. 30 (b)
On the basis of three princi pal. mome nts of inerti a I A, I 8 and Ic each about X, Y and Z axes respec tively , how can you classi fy molec ules? 30
.
4. (a)
(i)
Treati ng a diatom ic riwlec ule as a simpl e harmo nic oscill ator, obtain its vibrat ional, energ y ,levels. 20
C-DT N-L-Q ZB/26
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2.
(
~
0 ,
=oo,
0' 5, x 5, L
x
it IDU ~. 3Fio 311~ill'hl( ~ -q S::"'l~H m it ll;'h 'fiUT it full; ~. tllflifi(UI W
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it 30
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4.
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(i)
q>ffifi(UI
>l
i?
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ifi1 ~
C-DTN-L-QZB/26
30
3TIC!
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5
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(ii) The obser ved vibra tiona l frequ ency... :
of the CO mole cule 1s 13 6. 42 x 1'0 Hz. Wha t is the effect ive force cons tant of this mole cule? (Mas s of carbo n atom = 12 u and mass of oxyg en atom = 1'6 u, wher e U'is atom ic mass unit) 10 (b)
(i)
Disc uss pure rotat ional spec tra of linea r molec L!les, 25
(ii) Wha t is Lamb shift?
5
Sect ion- B 5,
(a)
Show that syste m.
nucle us
IS
a
quan tum 10
(b)
Deriv e Brag g diffra ction . cond ition in vecto rial form usmg incid ent and diffra cted wave vecto rs and recip rocal lattic e vecto r. 10
(c)
Find the total kinet ic energ y o[ elect ron and antie lectr on neutr ino emitt ed in beta deca y of free 'neut ron. (The neutr on-p roton mass differ ence 1s l·30 MeV and mass of elect ron 1s 0·51 MeV) 10
(d)
An elect ron beam of
4
keV is diffra cted throu gh a Brag g angle of 16° for the first maxi ma. If. the· energ y ds incre ased ·to 16 keV, .find .the corre spon ding Brag g angle for diffra ction . 10
C-DT N-Jr. QZB /26
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~
fiiJ co
<€1·
6 · 42 x 1013 Hz
f.:t ~ d i'li
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i
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25
10
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(e)
(f)
Exp lain cons erva tion of bary on num ber . .ComiT)ent on stab ility pf prot on.
10
Sim plify the logic al expr essi on AB+ AB +AB C
usin g a Kar nau gh map .
6.
10
(a}
Wha t is the imp orta nce of stud y of deu tero n? Obt ain the solu tion of Schr iidin ger equa tion for grou nd stat e of deu tero n and show that deu tero n is a loos ely bou nd syst em. 25 (ii) Wha t do you mea n by 'non -cen tral forc es'? 5
(b)
(i)
(i)
(ii)
Wha t are chai n reac tion s? Wha t do you mea n by criti cal size of the core m whi ch chai n reac tion take s plac e? Wha t is criti cal mas s? 20 235
U yiel ds two frag men ts of A = 95
and A = 140. Obt ain the ener gy dist ribu tion of the fissi on prod ucts . Ass ume that the two frag men ts are ejec ted with equa l and oppo site ,mo men tum . 10
7.
(a}
Find an expr essi on for lattic.e spec ific hea t of solid s, and its low and high tem pera ture limi ts. Wha t is Deb ye ·tem pera ture ? 20
C-D TN- L-Q ZB/ 26
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8.
(b)
Explain 'the· ongm of· energy band formation in solids. Show that,in nearly free electron approximation, the energy band gap is 2IVa I, where Va is Fourier transform of periodic potential seen by the valence electrons. 20
(c)
Describe the characteristic properties of a superconductor. Derive London equation for a superconductor and hence explain Meissner effect. 20
(a)
Distinguish 'between intrinsic and extrinsic semiconductors. Show that m a semiconductor, the product of ·concentrations of the two types of charge carriers 1s constant at a given temperature. 30
(b)
Simplify the logical expression (A +B)
and draw the implement it. (c)
El (A+ C) logical
circuit
to 15
Draw the commoncbase amplifier circuit; using an n-p-n transistor and briefly discuss its working. 15
C-DTN~L--QZB/26
10
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C-DTN -L-QZ B/26
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Constants which may be needed Velocity of light in vacuum (c)= 3xto8 ms- 1 Mass of electron ("'e)= 9. 11 X
w- 31
kg
Charge of electron (e) = 1· 602 X 1o- 19
c
Specific charge of electron(,;:,)= 1· 76 x td 1 C kg- 1 1u
=1
a.m. u. = 1· 6605 X
w- 27
kg = 931· 5 MeV
Rest mass energy of electron (11lec2 ) = 0·5110 MeV Permittivity in free space (Eo)= 8· 8542
X
to- 12 c 2 N- 1 m"2
Permeability of free space'(/lo) = 47t xlo- 7 N A-2 Gas constant (R) = 8· 314'J mol-l K- 1 Boltzmann con~tant (ks) = 1· 38lxlo- 23 .J K- 1 Planck constant (h)= 6 · 626 X 10-34 J s (hJ = t· 0546 x 1o-34 J s
Bohr magnetori {11 s'i = 9 · 274 x 1o-24 • J T~ 1 Nuclear magneton {11 N) = 5 · 051 x 1o- 27 J T-l Fine structure constant (a)= 1 f 137 · 03599
1
Mass of proton (mpJ.= 1· 0072766 u = 1· 6726 X w-2 7 kg Mass of neutron (ril,) = 1· 0086652 u =1·6749xto- 27 kg Mass of deuteron ("'rJ) = 2· 013553u Mass of a-particle (171a) = 4· 001506u Mass of 1 ~C = 12· OOOOOOu Mass of 1 ~0 =15·994915u Mass of ~~Sr = 86 · 99999 u Mass of ~He = 4:002603 u
C-DTN-L-QZB/26
12 www.examrace.com
mfuil>r<m~q;riM (c)= 3x!08 ms- 1
*8if?i'l
q;r M4H (m,) = 9 · II x I o-3 ! kg
*8if?i'l
q;r
amm (e) = I· 602 xI o-!9 C
·*<'lif?i'l
q;r
~~
(:J
=
J. 76x 1d 1 C
kg- 1
I u " I a.m.u. = I· 6605 x 10- 27 kg = 931· 5 MeV *<'lif?i'l
:!"'"
anq;m ij Pl~j
:!"'"
anq;m <1:\ qi(>IRHU (Ito)= 47t x10- 7 N A-2
fm~ (R)=8·3 14J mol-l K- 1
•lc<::>ll~'l ffil
oi\o1:
~·~
~ ~·~
Jitl:i;, q;r ""'""' (mp) = 1· 0072766 u = 1· 6726. x 1 o- 27 kg
"@'! q;r
M4H (m, I = I· 0086652 U·
=1·6749 x1o- 27 kg
s%
q;r ""'"'"·("' d)= 2· 013553 u
a-"'hUT q;r l<"l4H ("'a)= 4· 001506u 1
~c q;r
""'""' = 12 · oooooo u
go q;r ""'""' = 15· 994915 u
1
~~Sr
q;r M4H = 86 · 99999 u
~He q;r l<"l4H =4· 002603u
*** C-DTN-~QZB/26
13
SJ-900
www.examrace.com
C-DTN-L-QZB
~-14:f-II
1
"!:?rl
1
>rr% Jf.R "W
Jicm-'T:T 'R 3@/&g 1fTVJ'r if; 31Rtfhn 3Pl .fif;W 1fTVJ'r if iff& >n; 3trr w
/'iPrr
;rtf firM I Jf.R ffislrr 1
di<
5 3/f.iwf
~ it Cfi!l-it-q;q ~
Jf.R
g1 '11'(i; JrR1 if it JiriFF; g'lii>( f
3trr c!IM~ 1
>rr% Jf.R if; ~ Wra JNi Jf.R if; aTrr if ~ >n; gI Jrrftq;j
m
if;
arrr if oil ~
>n; gI
Note :
English verswn of the Instructions is printed on the front cover of this question paper. www.examrace.com
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