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1 CSIR-UGC-NET/JRF Jue 0 CSIR-UGC NET/JRF JUNE - 0 PHYSICAL SCIENCES BOOKLET - [A] PART - B. A particle of uit mass moves i a potetial V x b ax, where a ad b are positive costats. x The agular frequecy of small oscillatios about the miimum of the potetial is: 8b 8 a 8 a / b 8 b/a. A sigal of frequecy 0 khz is beig digitized by a A/D coverter. A possible samplig time which ca be used is: 00 s 0 s 60 s 00 s. The electrostatic potetial V(x,y) i free space i a regio where the charge desity is zero is give x by V x, y e f x y. Give that the x-compoet of the electric field, E x, ad V are zero at the origi, f(x) is: x x e 8x x x e 6x x e 8 x e x. Cosider the trasitio of liquid water to steam as water boils at a temperature of 00ºC uder a pressure of atmosphere. Which oe of the followig quatities does ot chage discotiuously at the trasitio? The Gibbs free eergy The iteral eergy The etropy The specific volume 5. The value of the itegral the figure below, is: C z dz z e, where C is a ope cotour i the complex z-plae as show i 5 e e 5 e 5 e e e 6. Which of the followig matrices is a elemet of the group SU()? 5 e e i i i 0 i i 7. For costat uiform electric ad magetic fields E E0 ad B B0, it is possible to choose a gauge such that the scalar potetial ad vector potetial A are give by 0 ad A= B0 r E0. r ad A= B0 r E 0. r ad A 0 0 ad A E0t

2 CSIR-UGC-NET/JRF Jue 0 8. Let a ad b be two distict three-dimesioal vectors. The the compoet of b that is perpedicular to a is give by a b a b a b a b a. b b b b. a a a 9. The wavefuctio of a particle is give by 0 i, where 0 ad are the ormalized eigefctios with eergies E 0 ad E correspodig to the groud state ad first excited state, respectively. The expectatio value of the Hamiltoia i the state is: E 0 E E 0 E 0. A particle is cofied to the regio 0 mea positio of the particle at temperature T is: kbt u 0 B / 0 E E 0 E E 0 x by a potetial which icreases liearly as u x 0 k T u kbt u 0 u0kbt u x. The. Circularly polarized light with itesity I 0 is icidet ormally o a glass prism as show i the figure. The idex of refractio of glass is.5. The itesity I of light emergig from the prism is: I I I I 0.. The acceleratio due to gravity (g) o the surface of Earth is approximately.6 times that o the surface of Mars. Give that the radius of Mars is about oe half the radius of Earth, the ratio of the escape velocity o Earth to that o Mars is approximately: A plae electromagetic wave is propagatig i a lossless dielectric. The electric field is give by,,, 0 ˆ ˆexp 0 E x y z t E x Az ik ct x z, where c is the speed of light i vacuum, E, A 0 ad k 0 are costats ad xˆ ad z ˆ are uit vectors alog the x- ad z-axes. The relative dielectric costat of the medium, r ad the costat A are r ad A r ad A r ad A r ad A

3 CSIR-UGC-NET/JRF Jue 0. Cosider the digital circuit show below i which the iput C is always high (I). The truth table for the circuit ca be writte as A B Z The etries i the Z colum (vertically) are The eergy levels of the o-relativistic electro i a hydroge atom (i.e. i a Coulomb potetial V r / r are give by E m / , where is the pricipal quatum umber, ad the correspodig wave fuctios are give by m, where is the orbital agular mometum quatum umber ad m is the magetic quatum umber. The spi of the electro is ot cosidered. Which of the followig is a correct statemet? There are exactly differet wave fuctios m, for each E m. There are differet wave fuctios m, for each E m. E m does ot deped o ad m for the Coulomb potetial. There is a uique wave fuctio m ad E m. 6. The Hamiltoia of a electro i a costat magetic field B is give by H. B where is a,, deotes the Pauli matrices. Let B / ad I be the positive costat ad uit matrix. The the operator iht/ e simplifies to. cos t i B I si t i. B I cost si t B i. B i. B I sit cost I si t cos t B B 7. The Hamiltoia of a system with degrees of freedom is give by H q,..., q ; p,..., p ; t, B with a explicit depedece o the time t. Which of the followig is correct? Differet phase trajectories caot itersect each other. H always represets the total eergy of the system ad is a costat of the motio. The equatios q i H / pi, p i H / qi are ot valid sice H has explicit time depedece. Ay iitial volume elemet i phase space remais uchaged i magitude uder time evolutio.

4 CSIR-UGC-NET/JRF Jue 0 8. If the perturbatio H ' ax, where a is a costat; is added to the ifiite square well potetial 0 for 0 x V x otherwise The first order correctio to groud state eergy is: a 9. Let p a a x (where = 0,,...) be a polyomial of degree with real coefficiets, defied i the iterval p x p m x dx m, the. If p x ad p x x p0 x p x x 0 a ad p0 x ad p x x 0 ad p x p x x 0. A cavity cotais blackbody radiatio i equilibrium at temperature T. The specific heat per uit volume of the photo gas i the cavity is of the form CV T where is a costat. The cavity is expaded to twice its origial volume ad the allowed to equilibrate at the same temperature T. The ew iteral eergy per uit volume is: T T T PART - C T. Cosider a system of N o-iteractig spis, each of which has classical magetic momet of magitude. The Hamiltoia of this system i a exteral magetic field H N is H. H, where is the magetic momet of the i th spi. The magetizatio per spi at temperature T is: B i H k T H kbt coth kbt H H H sih tah kbt kbt. Which of the followig is a aalytic fuctio of the complex variable z x iy i the domai z? x iy 7 x iy 7 x iy x iy x iy x iy i i

5 CSIR-UGC-NET/JRF Jue 0 5. A particle i oe dimesio moves uder the ifluece of a potetial V(x) = ax 6, where a is a real costat. For large the quatized eergy level E depeds o as: E ~ / E ~ 6/5 E ~. The Lagragia of a particle of charge e ad mass m i applied electric ad magetic fields is give by L m ea. e, where A ad are the vector ad scalar potetials correspodig to the magetic ad electric fields, respectively. Which of the followig statemets is correct? The caoically cojugate mometum of the particle is give by p m p e The Hamiltoia of the particle is give by H A. p e m m L remais uchaged uder a gauge trasformatio of the potetials. Uder a gauge trasformatio of the potetials, L chages by the total time derivative of a fuctio of r ad t. A kr e where A ad k are r positive costats. The electrostatic potetial correspodig to this charge distributio varies with r as 5. A static, spherically symmetric charge distributio is give by r kr r e kr kr e e kr e r r r 6. Cosider two idepedetly diffusig o-iteractig particles i -dimesioal space, both placed at the origi at time t = 0. These particles have differet diffusio costats D ad D. The quatity R t R t where R t ad R t are the positios of the particles at time t, behaves as: 6t D D 6t D D E ~ / 6t D D 6t D D 7. A resistace is measured by passig curret through it ad measurig the resultig voltage drop. If the voltmeter ad the ammeter have ucertaities of % ad %, respectively, the (A) The ucertaity i the value of resistace is: 7.0 %.5 % 5.0 %.0 % (B) The ucertaity i the computed value of the power dissipated i resistace is 7% 5% % 9% 8. I the absece of a applied torque a rigid body with three distict pricipal momets of iertia give by I, I ad I is rotatig freely about a fixed poit iside the body. The Euler equatios for the compoets of its agular velocity,, are I I I I I I,, I I I (A) The equilibrium poits i,, space are,,0,,0, ad 0,,,,0,,0, ad 0,,,0,0, 0,,0 ad 0,0,,,,,, ad 0, 0, 0

6 6 CSIR-UGC-NET/JRF Jue 0 (B) The costats of motio are ad I I I I I w I ad I I I I I I ad ad I I I 9. I a system cosistig of two spi-½ particles labeled ad, let S ad S deote the correspodig spi operators. Here x, y, z ad x, y, z are the three Pauli matrices. (A) I the stadard basis the matrices for the operators S S ad S S respectively, x y y x 0 0, 0 0 i 0 i 0, 0 i 0 i i i 0 0 i i 0, 0 i i 0 0 i i (B) These two operators satisfy the relatio Sx Sy, S y Sx Sz Sz Sx S y, Sy S x i Sz S z 50. Cosider the matrix M i i 0 0 0, i i S S, S S 0 x y y x Sx Sy, S y S x 0 (A) The eigevalues of M are 0,, 0, 0,,,,, (B) The expoetial of M simplifies to (I is the idetity matrix) M e e I M M e I M M M e I M! M e e M 5. The radius of a 6 9 Cu ucleus is measured to be.80 cm. (A) The radius of a 7 Mg ucleus ca be estimated to be.86 0 cm 5. 0 cm.6 0 cm 8.60 cm (B) The root-mea-square (rms) eergy of a ucleo i a ucleus of atomic umber A i its groud state varies as A / A / A / A /

7 CSIR-UGC-NET/JRF Jue The character table of C, the group of symmetries of a equilateral triagle is give below 0 b a c d I the above,, deotes the three classes of C v, cotaiig, ad elemets respectively, ad 0, ad are the characters of the three irreducible represetatios 0, ad of C v. (A) The etries a, b, c ad d i this table are, respectively,,, 0,, 0,,, 0,,,, (B) The reducible represetatio decomposes ito its irreducible represetatios 0,, as with character,0, of Cv Light of wavelegth 660 m ad power of mw is icidet o a semicoductor photodiode with a absorbig layer of thickess of (l ) m. (A) If the absorptio coefficiet at this wavelegth is 0 cm ad if % power is lost o reflectio at the surface, the power absorbed will be 750 W 675 W 50 W 5 W (B)The geerated photo-curret for a quatum efficiecy of uity will be 60 A 00 A A 0 A 5. The magetic field of the TE mode of a rectagular waveguide of dimesios a b as show i the figure is give by H H cos 0. xcos 0. y z 0, where x ad y are i cm. (A) The dimesios of the waveguide are a =. cm, b =.50 cm a = 0.0 cm, b = 0.0 cm a = 0.80 cm, b = 0.60 cm a =.66 cm, b =.5 cm (B) The etire rage of frequecies f for which the TE mode will propagate is: 6.0 GHz < f < 7.5 GHz 7.5 GHz < f < 9.0 GHz 7.5 GHz < f <.0 GHz 7.5 GHz < f

8 8 CSIR-UGC-NET/JRF Jue Cosider the eergy level diagram (as show i the figure below) of a typical three level ruby laser system with Chromium ios per cubic cetimeter. All the atoms excited by the 0. m radiatio decay rapidly to level E which has a lifetime ms (A) Assumig that there is o radiatio of wavelegth 0.7 m preset i the pumpig cycle ad that the pumpig rate is R atoms per cm, the populatio desity i the level N builds up as: t/ N t R e t/ N t R e Rt t/ N t e N t Rt (B) The miimum pump power required (per cubic cetimeter) to brig the system to trasparecy, i.e. zero gai, is.5 kw.6 kw 0.76 kw. kw 56. A flux quatum (fluxoid) is approximately equal to 0 7 gauss-cm. A type II supercoductor is placed i a small magetic field, which is the slowly icreased till the field starts peetratig the 5 supercoductor. The stregth of the field at this poit is 0 gauss (A) The peetratio depth of this supercoductor is 00Å 0Å 000Å Å (B) The applied field is further icreased till supercoductivity is completely destroyed. The stregth 8 5 of the field is ow 0 gauss. The correlatio legth of the supercoductor is: 0Å 00Å 68 Å 000Å 57. A beam of pios is icidet o a proto target, givig rise to the process p (A) Assumig that the decay proceeds through strog iteractios, the total isospi I ad its third compoet I for the decay products, are I, I I, I I, I I, I (B) Usig isospi symmetry, the cross-sectio for the above process ca be related to that of the process p p p p 58. The two dimesioal lattice of graphee is a arragemet of Carbo atoms formig a hoeycomb lattice of lattice spacig a, as show below. The carbo atoms occupy the vertices.

9 CSIR-UGC-NET/JRF Jue 0 9 (A) The Wiger-Seitz cell has a area of a a 6 a (B) The Bravais lattice for this array is a Rectagular lattice with basis vectors d ad d Rectagular lattice with basis vectors c ad c Hexagoal lattice with basis vectors a ad a Hexagoal lattice with basis vectors b ad b a 59. Cosider the decay process v i the rest frame of the. The masses of, ad are M, M (A) The eergy of M M c M (B) The velocity is M M c M M ad zero respectively. is: M r M c M is: M M c M M M M c r M c M M M M c M 60. A arrow beam of X-rays with wavelegth.5å is reflected from a ioic crystal with a fcc lattice structure with a desity of. g cm. The molecular weight is 08 AMU ( AMU=.66 0 g). (A) The lattice costat is: 6.00Å.56Å.00Å.56Å (B) The sie of the agle correspodig to () reflectio is: / / 8 / / 8 6. If a electro is i the groud state of the hydroge atom, the probability that its distace from the proto is more tha oe Bohr radius is approximately A time varyig sigal V i is fed to a op-amp circuit with output sigal V 0 as show i the figure below. The circuit implemets a High pass filter with cutoff frequecy 6 Hz. High pass filter with cutoff frequecy 00 Hz Low pass filter with cutoff frequecy 6 Hz Low pass filter with cutoff frequecy 00 Hz. c

10 0 CSIR-UGC-NET/JRF Jue 0 6. The Hamiltoia of a particle of uit mass movig i the xy-plae is give to be: H xpx ypy x y i suitable uits. The iitial values are give to be (x(0), y(0)) = (, ) ad px 0, py 0,. Durig the motio, the curves traced out by the particles i the xyplae ad the p x p y -plae are Both straight lies A hyperbola a ellipse, respectively A straight lie ad a hyperbola respectively Both hyperbolas 6. Cosider a ideal Bose gas i three dimesios with the eergy-mometum relatio with s > 0. The rage of s for which this system may udergo a Bose-Eistei codesatio at a ozero temperature is: < s < 0 < s < 0 < s < 0 < s < 65. Two gravitatig bodies A ad B with masses m A ad m B, respectively, are movig i circular orbit. Assume that mb ma ad let the radius of the orbit of body A be R A. If the body A is losig mass adiabatically, its orbital radius R A is proportioal to / m A / m A m A m A s p