UPSC Main 2012 Optional Subject Physics Paper I Www.upscportal.com

8/13/2019 UPSC Main 2012 Optional Subject Physics Paper I Www.upscportal.com

1/16

GSM XA1\42 ,J

S. F DTN M QIZA

PHYSICS

PaperI

Time Allowed : Three Hours Maximum M arks : 300

INSTRUCTIONS

Each question is printed both in H indi and in En glish.

Answ ers mu st be writ ten in the m edium specified in the

Ad mission C ertificate issued to you, wh ich mu st be stated

clear ly on the cover of the answe r-book in the space

provided for the purpose. No m arks will be given for the

answ ers written in a m edium other than that specified in

the Adm ission Certificate.Can didates should attempt Question Nos. 1 and 5 whichare compulsory, and any three of the remaining questionsselecting at least one question from each Section.

The num ber of marks carried by each question is indicatedat the end o f the question.

Assume suitable data if considered necessary and indicate

the same clearly.Sym bols / nota tions carry thei r usual m eanings , unless

otherwise indicated.

Important Note Whenever a question is being attempted,all its parts/sub-parts must be attempted contiguously. Thism eans that before moving on to the next question to be

attempted, candidates m ust f inish at tempting a l l parts/

sub-parts of the previous question attempted. This is to bestrictly followed.

Pages left blank in the answer-book are to be clearly struckout in ink. Any answers that follow pages left blank may notbe given credit.

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SectionA

1. Answer all the five parts below : 2x5=60

(a) Two identical relativistic particles of restm ass m and kinet ic energy T collidehead -on. W hat is the relative kineticenergy, i.e., the kinetic energy T' of onein the rest fram e of the other?

(b) Define a conservative field. Determine ifthe field given below is conservative innature :

E = c[y 2 i + (2xy + z 2 ) 3+ 2yz1C] V/m

where c is a co nstant.

(c) Calculate the mo m ent of inertia of asolid cone of mass M, height h, verticalhalf-angle a and rad ius of its base R,about an axis passing through its vertexand parallel to its base.

(d) A resistance R and a lossless capacitor Care connected through a sw itch. Thecapacitor is charged to potential ;, andthe sw itch is closed at t = 0. Prove that

the energy stored in the capacitor isequ al to the energy dissipated in theresistor.

(e) A rigid body is spinning with an angularvelocity of 4rs about an axis parallel tothe direction (4 j- 3k) passing through

the point A with OA = 2i +33 - k, where0 is the origin of the coordinate system.Find the magnitude and direction of thelinear velocity of the bo dy at point P

with OP = 4i - 2 j + k.

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2 T h e v o l u m e b e tw e e n tw o c o n c e n t riccondu cting sph erical surfaces of radii a and

b (a < b) i s fi ll ed w i th an inhom ogeneou sdielectric with E =E0 1(1+ cr), where c i s acon s tan t and r i s t he r ad ia l coord ina te .A charge +Q is placed on the inner surface,w h i le t h e o u te r s u r fa c e i s g r o u n d e d .Determine

(a) D in the region a < r < b;

(b) capacitance of the dev ice;

(c) polarization charge density in the regiona < r < b;

(d) surface polarization charg e densities atr = a and r = b 15+15+15+15=60

3 A particle is moving in a central force field onan orbit given by r = ke ae , where k and a areposit ive con stants , r is the radial distanceand 9 is the po lar angle.

(a) Find the force law for the central forcefield. 20

(b) Find 9 (t). 20

(c) Find the total energy. 20

4 (a) W hat is a displacemen t current? Provethat lines of cond uct ion current p lusdisplacemen t current are continuous. 5

F-DTN-M-QIZA/ 1 7




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2. f it a 3 - t b a


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(b) C o n s i d e r a u n i fo r m h a l f -s p h e r e o fradius R and m ass M. The half-sphereis supported by a frictionless horizontal

p l a n e a s s h o w n i n th e f i g u r e . T h e

half-sphere lies in the region z < O .

Y 0 = O rigin

A = Supporting plane

Find the centre of mass of the

half-sphere. 15

(c) Assume E = 0 inside a perfectcondu ctor. Elaborate on any o ther fourelectrostatic properties that arise fromthis property. 15

(d) A p rojectile of mass M explodes, whilein f ligh t , in to th ree f ragm ents . O nefragm ent of mass m l = M / 2 travels inthe original direction o f the projectile.A nother fragm ent of m ass m 2 = M /6t ravels in the op posi te d i rect ion andthe th ird f ragment o f m ass m 3 = M / 3

com es to rest. The energy E, released inthe exp losion, is 5 t im es the kinet icenergy of the p rojecti le at explosion.What are the velocities of thefragments? 15

F-DTN-M-QIZA/ 17




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SectionB

5. Answer all the five parts below : 2x5=60

(a) The m ot ion o f a dam ped m echan ica loscillator is represented b y

Trik + +0x = 0

where at, a and 1 3 are constants. Theosc i l la tor i s c ri ti ca lly dam ped . Thesystem is given an im pulse at x = 0 andt = 0 , resulting in an initial velocity v.

A f te r h o w m u c h t im e th e s y s te mexperiences m aximum displacem ent?

(b) Sh ow tha t a tr ave l li ng w ave on thes tr in g , c l a m p e d o n b o t h t h e e n d s ,undergoes a phase change of n. H ence

obtain the time-independent form of thew ave equation representing a standingw ave on the string.

(c) T he sep a ra t ion b e tw een the s l i ts i s0 .5 mm n oung s ouble-slitexp erime nt. Th e interference pat ternobserved on a screen placed 5 m awayreveals the location of the first

m a x im u m w h ic h is 6 m m f ro m th ecen t re of the pa t te rn . C alcu la te thew ave leng th o f l igh t and sep a ra tionb e t w e e n s e c o n d a n d th i rd b r ig h tfringes.

(d) Sho w that the H elmh ol tz f ree energyo f a s y s te m n e v e r in c r e a s e s in a n yisothermal-isochoric transformation.

F-DTN-M-QIZA/ 1 7




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5. ft t MI Ntr, : 2x5=60

ctl 1-i l fff t e l t -11 11pf

+ a +I3x =0 aim , , cx At

p Pirk t alic , Shincioncr: 31-44144 1 44 *Ix = 0 3 Rt t = o ER 1,ct) 3t*1fa 1 rr sidr t, Nu*

3TR-r n i T U Idr t f a 4 t4H 1

* cuI u q 4 3T14*44 fan 6

(u) < t r( 4if1 4 -{ 47=4 ClUi

it TT cbelicik 61cli t 13ra1a ,-fichkur

444 -1 44?Ri rr4 *rr -A7, t awrriftai r y -4Friur mt

(4) t -M4z 4414 4 4-a k 4t4 TerWCUT

0 .5 m m t l 5 m 'r lf tE R T 7 A p m..0b4r4 ER

it Tr 0)4PIcheff 5191113T14

EIM tleidr t, 14 6 mm ER

t I lAchI41* 3-f1T vflawrt141*Tt O

t s t 7 11 T h 7 1i* I 11i1cht-I-f*r1771

(4) t ift

-TP:13TRrafiT tc0-11.-cw it 4 3.ft -et

FDTNMQIZA/ 17 [ P.T.O.




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(e) Co nsider non-equilibrium situation fora s y s te m i n w h i c h th e p o p u l a ti o ninversion has been ach ieved. Explainthat such a system can be treated as if

i t has negative ab solute temp erature.

6 . (a ) D is tin g u i s h b e tw e e n F r e s n e l a n dFraunhofer classes of diffraction. Show

that the area of each Fresnel half-period

zone is same.

(b) A diffraction grating of w idth 5 cm withslits of width 1 0 -4 cm sepa ra ted by a

distance of 2 x10 -4 cm is illuminated by

l ight of wavelength 550 nm . Wh at willbe the width of the principal maximumin the diffraction pattern? W ould therebe any m issing orders?

(c) Establish the relation ( a vnd p s r

and then d er ive T pp 7, _ 21aT 2

H e n c e s h o w t h a t t h e h e a t c a p a c i tyC p of an idea l gas i s indep ende nt o fpressure P.

7 . (a ) Der ive an exp ress ion for in te rm odaldispersion for a m ultim odal step-indexfibre. 20

20

20

20

F-DTN-M-QIZA/ 17 0




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*ri -A tictolt ft

aial \itichf firt 1N ti

6. (*) fat a * * --4F afR 9)1.3.0( a1T Ala71 r-7T 40-3rra-J-q gil4 ar

44 ken t 51irt 20

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4)-0 *7 Wral *415 cm t -4r*t ft 444it 550 nm WtrA 171 ITT7T giu 9-t1e ft-4I

t1 ft-44 cal 11 iitogilt4 11 1-1- %41Oft? tai al 14 614? 20

el)(al )ID WifcM

ft .T 11* qg ac P ) - T ( 3 2 al) }T. T 2 p 9-6m

.314 T4-6-4v ft 3170 a-x4r zuft-dtc p , Zia Pit +-c4c11effi tI 20

7. *) ell Aura-k 4* 4 -3 3447AI-if ut4441*71-) oq ,sitt, I6N *t -N7 20

F-DTN-M-QIZA/17 [ P.T.O.




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(b) A pulse of X 0 = 600 nm and AX = 10 nmpropag ates through a f ibre which hasa m ater ia l d i spers ion coeff ic ien t o f5 0 p s p e r k m p e r n m a t 6 0 0 n m .

C a l c u la t e th e p u l se b r o a d e n i n g i ntraversing a 10 km length of the fibre. Ifthe pulse w idth at the inpu t of the fibreis 12 ns, wha t will be the pulse width at

the output of the fibre? 10

(c) S h o w th a t b o th F e rm i- D i ra c a n d

B ose -E ins t e in d i s tr ibu t ion s r educeunder certain condition in a form w hich

gives the total num ber of particles as

N = Ank e - p r de

w here A is a constant and 13=1 / k BT.

Fu rther show that th is express ion isjust the sam e as that obtained from the

M axwellian speed distribution. 5+15=30

8 ( a ) U s e m a tr ix m e th o d to o b ta in a n

exp ress ion fo r the foca l l ength of acoaxial com bination of tw o thin lenses

having focal lengths f l and f 2 separatedby distance d . 20

b) Calculate the m inim um thickness of aquar tz p la te w hich would behave as a

quar ter-wave pla te for w avelength oflight, X = 6000 A. The refractive indices

for ordinary and extraordinary rays are

p c, = 1544 and 1- 553. 15

F-DTN-M-QIZA/ 1 7 2




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(c) Th e Einstein theory of specific heat ofsolids gives the expression

3NkBx 2 e xCV -

e x _ 1)2

where x =9

with O E as the Eins te inT

temperature.

M ention Einstein's assum ptions in

deriving i t. Also obtain low- an d

high-tempe rature limiting expres-

sions for it.

(ii) Give schem atic plot of Vkver sus3 B

and com men t on the validity ofsEexpressions in 1) in com parison

w ith experime nts. 15+10=25

F-DTN-M-QIZA/ 17 4




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3NkBx 2e x

c v - (e x _ 1) 2

,716I 374z51 Wftf *fllie E t FITZT x =9ET

t

B)9 -d:m


16/16

F DTN M QIZA

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UPSC Main 2012 Optional Subject Physics Paper I Www.upscportal.com

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