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1 CSIR-UGC NET/JRF JUNE - 6 PHYSICAL SCIENCES OOKLET - [A] PART. The raius of onvergene of the Taylor series epansion of the funtion (). The value of the ontour integral the anti-lokwise iretion, is 4z e () osh ( ) aroun, is z i aroun the unit irle C traverse in C osh ( z) sinh ( z) 8 tanh. The Gauss hypergeometri funtion F( a, b, ; z ), efine by the Taylor series epansion aroun z as F( a, b, ; z) satisfies the equation relation n F( a, b, ; z) F( a, b, ; z) z ab F( a, b, ; z) F( a, b, ; z) z ab ab () F( a, b, ; z) F( a, b, ; z) z ab F( a, b, ; z) F( a, b, ; z) z a ( a )... ( a n ) b ( b )... ( b n ) z ( )... ( n ) n! 4. Let X an Y be two inepenent ranom variables, eah of whih follow a normal istribution with the same stanar eviation, but with means an, respetively. Then the sum X Y follows a istribution with two peaks at an mean an stanar eviation normal istribution with mean an stanar eviation () istribution with two peaks at an mean an stanar eviation normal istribution with mean an stanar eviation 5. Using imensional analysis, Plank efine a harateristi temperature T P from powers of the gravitional onstant G, Plank s onstant h, oltzmann onstant k an the spee of light in vauum. The epression for T P is proportional to n 5 h k G h k G G () 4 h k hk G

2 6. Let (, t) an (, t ) be the oorinate systems use by the observers O an O, respetively. Observer O moves with a veloity v along their ommon positive -ais. If t an t are the linear ombinations of the oorinates, the Lorentz transformation relating O an O takes the form an, an () an, an 7. A ball of mass m, initially at rest, is roppe from a height of 5 meters. If the oeffiient of restitution is.9, the spee of the ball just before it hits the floor the seon time is approimately (take g = 9.8 m/s ) 9.8 m/s 9. m/s () 8.9 m/s 7. m/s 8. Four equal harges of Q eah are kept at the verties of a square of sie R. A partile of mass m an harge Q is plae in the plane of the square at a short istane a ( R) from the enter. If the motion of the partile is onfine to the plane, it will unergo small osillations with an angular frequeny Q R m Q R m () Q R m Q 4 R m 9. The Hamiltonian of a system with generalize oorinate an momentum ( q, p) is H p q. A solution of the Hamiltonian equation of motion is (in the following A an are onstants) A At At A p e, q e At At p Ae, q e At A At A () p Ae, q e A t A t p Ae, q e. Two parallel plate apaitors, separate by istanes an. respetively, have a ieletri material of ieletri onstant. inserte between the plates, an are onnete to a battery of voltage V. The ifferene in harge on the seon apaitor ompare to the first is 66 % % ().% %. The half spae regions an are fille with ieletri meia of ieletri onstants an respetively. There is a uniform eletri fiel in eah part. In the right half, the eletri fiel makes an angle to the interfae. The orresponing angle in the left half satisfies < > E sin sin () tan tan E tan tan sin sin

3 . The - an z-omponents of a stati magneti fiel in a region are ( ) an y z, respetively. Whih of the following solutions for its y-omponent is onsistent with the Mawell equations? y y y y () y y ( ) y y. A magneti fiel is z ˆ in the region an zero elsewhere. A retangular loop, in the yplane, of sies l (along the -iretion) an h (along the y-iretion) is inserte into the region from the region at a onstant veloity v vˆ. Whih of the following values of l an h will generate the largest EMF? l 8, h l 4, h 6 () l 6, h 4 l, h 4. The state of a partile of mass m in a one-imensional rigi bo in the interval to L is given by the normalise wavefuntion 4 4 ( ) sin sin. L 5 L 5 L If its energy is measure, the possible outomes an the average value of energy are, respetively. () h 7, h an h ml ml 5 ml h 9, h an h ml ml ml h 9, h an h 8mL ml 4 ml h 7, h an h 8mL ml ml 5. If L ˆ, L ˆ an L ˆ are the omponents of the angular momentum operator in three imensions, the y z ommutator Lˆ, ˆ ˆ ˆ LLyL z may be simplifie to i L ˆ ˆ Lz Ly i Lˆ Lˆ Lˆ z y () i L ˆ ˆ Lz Ly 6. Suppose that the Coulomb potential of the hyrogen atom is hange by aing an inverse-square ze term suh that the total potential is V ( r ) r g, where g is a onstant. The energy eigenvalues r E nlm in the moifie potential epen on n an l, but not on m epen on n but not on l an m () epen on n an m, but not on l epen epliitly on all three quantum numbers n, l an m 7. The eigenstates orresponing to eigenvalues E an E of a time-inepenent Hamiltonian are an respetively. If at t, the system is in a state ( t ) sin os the value of ( t) ( t) at time t will be ( E sin E os ) E E iet / iet / () e sin e os ie t e sin e os / ie t /

4 4 8. The speifi heat per moleule of a gas of iatomi moleules at high tmeperatures is 8k.5k () 4.5 k k 9. When an ieal monatomi gas is epane aiabatially from an initial volume V to V, its temperature hanges from T to T. Then the ratio T / T is / () 4. A bo of volume V ontaining N moleules of an ieal gas, is ivie by a wall with a hole into two ompartments. If the volume of the smaller ompartment is V/, the variane of the number of partiles in it, is N N 9 / () N 4. A gas of non-relativisti lassial partiles in one-imension is subjete to a potential V ( ) N, where is a onstant). The partition funtion is kt 4m h m h 8m h () m h 4. The epenene of urrent I on the voltage V of a ertain evie is given by I I V V where I an V are onstants. In an eperiment the urrent I is measure as the voltage V applie aross the evie is inrease. The parameters V an I an be graphially etermine as the slope an the y-interept of the I -V graph the negative of the ratio of the y-interept an the slope, an the y-interept of the () the slope an the y-interept of the I -V graph I -V graph the negative of the ratio of the y-interept an the slope, an the y-interept of the I -V graph 4. In the shemati figure given below, assume that the propagation elay of eah logi gate is t gate. A +5V The propagation elay of the iruit will be maimum when the logi inputs A an make the transition (,) (,) (,) (,) () (, ) (,) (, ) (,)

5 5 44. Given the input voltage V i, whih of the following waveforms orretly represents the output voltage V in the iruit shown below? V i.5.5 t V i.5 V 5 K 5 K K K V o.. V o t V o t.. () V o t 45. The intensity istribution of a re LED on an absorbing layer of material is a Gaussian entere at V o the wavelength 66nm an with nm. If the absorption oeffiients varies with wavelength as K ( ), where an K are positive onstants, the light emerging from the absorber will be blue shifte retaining the Gaussian intensity istribution blue shifte with an asymmetri intensity istribution () re shifte retaining the Gaussian intensity istribution re shifte with an asymmetri intensity istribution t

6 6 PART - C ik 46. What is the Fourier transform e f ( ) of elta-funtion? ik ik 47. The integral equation (, t) t k e ( ) k m i ik ( ) i ( tt) (, t) is equivalent to the ifferential equation t () m i (, t) (, t) 6 n f ( ) ( ) ( ), where ( ) is the Dira n k i n k i () t t m i (, t) (, t) m i (, t) (, t) t m i (, t) (, t) 48. A part of the group multipliation table for a si element group G { e, a, b,,, f } is shown below. (In the following e is the ientity element of G). The entries, y an z shoul be e a b f e e a b f a a b e b b e f y z f a, y an z, y a an z (), y an z a a, y an z f 49. In fining the roots of the polynomial f ( ) 4 5 using the iterative Newton-Raphson metho, the initial guess is taken to be. In the net iteration its value is nearest to ()

7 7 5. For a partile of energy E an momentum p (in a frame F), the rapiity y is efine as y E p E p ln. In a frame F moving with veloity v (,, ) with respet to F, the rapi- ity y will be y y ln ( ) y y ln () y y ln y y ln 5. A anonial transformation ( q, p) ( Q, P) is mae through the generating funtion F( q, P) q P on the Hamiltonian are onstants. The equations of motion for ( Q, P ) are () H ( q, p) p q 4 4 q, where an P Q an P Q P P Q an P Q Q 4P Q an 5. The Lagrangian of a system moving in three imensions is Q P P Q an P Q L m m ( ) k k( ) The inepenent onstant(s) of motion is/are energy alone only energy, one omponent of the linear momentum an one omponent of the angular momentum () only energy an one omponent of the linear momentum only energy an one omponent of the angular momentum 5. Consier a sphere S of raius R whih arries a uniform harge of ensity. A smaller sphere S R of raius a is ut out an remove from it. The enters of the two spheres are separate by the R vetor b nˆ, as shown in the figure. S S b r P The eletri fiel at a point P insie S is R n ˆ R ( r na ˆ ) a () R n ˆ 6 a r R

8 8 54. The values of the eletri an magneti fiels in a partiular referene frame (in Gaussian units) are E ˆ 4 yˆ an zˆ, respetively. An inertial observer moving with respet to this frame measures the magnitue of the eletri fiel to be E 4. The magnitue of the magneti fiel measure by him is 5 9 () 55. A loop of raius a, arrying a urrent I, is plae in a uniform magneti fiel. If the normal to the loop is enote by ˆn, the fore F an the torque T on the loop are F an T a I nˆ F an 4 I T () F an ˆ 4 I T I n F an T I 56. A wavelength has a square ross-setion of sie a. For the TM moes of wavevetor k, the transverse eletromagneti moes are obtaine in terms of a funtion (, y) whih obeys the equation y k (, y) with the bounary onition ( a, y) (, a). The frequeny of the lowest moe is given by () k k 4 a a k k a 4a 57. Consier a partile of mass m in a potential V ( ) m g os k. The hange in the groun state energy, ompare to the simple harmoni potential m, to first orer in g is k g ep m k g ep m () k g ep m 58. The energy levels for a partile of mass m in the potential V ( ) approimation b k g ep 4m, etermine in the WK m E V ( ) n, where a, b are the turning points an n,,,...), are a En n 4 m / En n 4 m / () En n 4 m / En 4 m n /

9 9 59. A partile of mass m moves in one-imension uner the influene of the potential V ( ) ( ), where is a positive onstant. The unertainty in the prout ( ) ( p) in its groun state is () 4 6. The groun state energy of a partile of mass m in the potential V ( ), estimate using the 6m / normalize trial wavefuntion ( ) e is /4 4 [Use e an e ]. 4 m / 8 m / () 6. Consier a gas of Cs atoms at a number ensity of atoms/. When the typial inter-partile istane is equal to the thermal e-roglie wavelength of the partiles, the temperature of the gas is nearest to (Take the mass of a Cs atom to be.7 6 kg). m / 8m / 9 5 K 7 K () 8 K K 6. The internal energy E( T ) of a system at a fie volume is foun to epen on the temperature T as E( T) at bt 4. Then the entropy ( ) S T, as a funtion of temperature, is at bt at 4bT () at bt at bt 4 6. A raioative element X eays to Y, whih in turn eays to a stable element Z. The eay onstant from X to Y is, an that from Y to Z is. If, to begin with, there are only N atoms of X, at short times ( t / as well as / ) the number of atoms of Z will be Nt ( ) N t () ( ) N t ( )Nt 64. Two ompletely overlapping semi-irular parallel plates omprise a apaitive transuer. One of the plates is rotate by an angle of º relative to their ommon enter. Ignoring ege effets, the ratio, In : I o of sensitivity of the transuer in the new onfiguration with respet to the original one, is 8 : 9 : () 7 : 8 5 : The state iagram that etets three or more onseutive s in a serial bit stream is Reset S / S / Reset S / S / S / S / S / S /

10 Reset S / S / Reset S / S / () S / S / S / S / 66. The eay onstants f p of the heavy pseuosalar mesons, in the heavy quark limit, are relate to their masses values m p m p by the relation f p a m p, where a is an empirial parameter to be etermine. The p 64 6 MeV an f 8 5 MeV orrespon to unorrelate measurement of a meson. The error on the estimate of a is 75 (MeV) / 9 (MeV) / () (MeV) / 4 (MeV) / 67. Consier eletrons in graphene, whih is a planar monatomi layer of arbon atoms. If the ispersion relation of the eletrons is taken to be ( k) k (where is onstant) over the entire k-spae, then the Fermi energy F epens on the number ensity of eletrons as F / F () 68. Suppose the frequeny of phonons in a one-imensional hain of atoms is proportional to the wavevetor. If n is the number ensity of atoms an is the spee of the phonons, then the Debye frequeny is n n n () n 69. The ban energy of an eletron in a rystal for a partiular k-iretion has the form ( k) A os ka, where A an are positive onstants an ka. The eletron has a holelike behaviour over the following range of k : F ka ka () ka / F / ka 4 7. The groun state eletroni onfiguration of Ti is [Ar] 4s. Whih state, in the stanar spetrosopi notations, is not possible in this onfiguration? F S () D P 7. In a normal Zeeman effet eperiment using a magneti fiel of strength. T, the splitting between the omponents of a 66 nm spetral line is pm pm () 8 pm 6 pm

11 7. The separation between the energy levels of a two-level atom is ev. Suppose that 4 atoms are in the groun state an 7 atoms are pumpe into the eite state just before lasing starts. How muh energy will be release in a single laser pulse? 4.6 J.4 J () 98 J 48 J 7. In the large haron ollier (LHC), two equal energy proton beams traverse in opposite iretions along a irular path of length 7 km. If the total enter of mass energy of a proton-proton pair is 4 TeV, whih of the following is the best approimation for the proper time taken by a proton to traverse the entire path? ns. µs (). ns. µs 74. Let E S enote the ontribution of the surfae energy per nuleon in the liqui rop moel. The ratio S 7 64 Al : S Zn E E is : 4: () 5: : 75. Aoring to the shell moel, the nulear magneti moment of the 7 Al nuleus is (Given that for a proton g, g 5.586, an for a neutron g, g.86 ). l s.9 N 4.44 N () 4.79 N l s