JEST PHYSICS h (d) 2 h. 3. If (q, p) is a canonically conjugate pair, which of the following is not a canonically conjugate pair?

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1 JEST PHYSICS 8 PART A : ONE mrk questions:. A bll of mss m strting from rest, flls verticl distnce h before striking verticl spring, which it compresses by length. Wht is the spring constnt of the spring? (Hint: Mesure ll the verticl distnces from the point where the bll first touches the uncompressed spring i.e., set this point s the origin of the verticl xis) mg mgh mg mg () h h h h. An electromgnetic wve of wvelength is incident normlly on dielectric slb of thickness t. If K is the dielectric constnt of the slb, the chnge in phse of the emergent wve compred with the cse of propgtion in the bsence of the dielectric slb is () K t t K. If (q, p) is cnoniclly conjugte pir, which of the following is not cnoniclly conjugte pir? () q, pq / p, pq / pq, q f p, q / f ' p, where f ' p is the derivtive of f p 4. A Germnium diode is operted t temperture of 7 degree C. The diode terminl voltge is.v when the forwrd current is ma. Wht is the forwrd current (in ma) if the terminl voltge is.4v? () When collection of two-level systems is in equilibrium t temperture T, the rtio of the popultion in the lower nd upper levels is :. When the temperture is chnged to T, the rtio is 8 :. Then () T T T T T T T 4T 6. A collection of N intercting mgnetic moments, ech of mgnitude µ, is subjected to mgnetic field H long the z-direction. Ech mgnetic moment hs doubly degenerte level of energy zero, nd two non-degenerte levels of energies µh nd µh respectively. The collection is in therml equilibrium t temperture T. The totl energy E(T, H) of the collection is µhn sinh µh / kbt () cosh µh / k T µhn cosh µh / kbt cosh µh / k T b b µhn cosh µh / kbt sinh µh / kbt µhn cosh µh / k T 7. For which of the following conditions does the integrl m n Pn x re the Legendre polynomils of order m nd n respectively? B P x P x dx vnish for m n () ll m, m n m n is n odd integer m n is non-zero even integer n m, where P m x nd South Delhi : 8-A/, Ji Sri, Ner-IIT Huz Khs, New Delhi-6, Ph : -6858, 6869 North Delhi : -5, Mll Rod, G.T.B. Ngr (Opp. Metro Gte No. ), Delhi-9, Ph: ,

2 8. A block of mss M is moving on frictionless inclined surfce of wedge of mss m under the influence of grvity. The wedge is lying on rigid frictionless horizontl surfce. The configurtion cn be described using the rdius vectors r nd r shown in the figure. How mny constrints re present nd wht re the types? r r () One constrint; holonomic nd scleronomous Two constrints; both re holonomic, one is scleronomous nd rheonomous Two constrints; both re scleronomous; one is holonomic nd the other is non-holonomic Two constrints; both re holonomic nd scleronomous 9. Consider prticle of mss m moving under the effect of n ttrctive centrl potentil given s V kr, where k >. For given ngulr momentum L, r km / L corresponds to the rdius of the possible circulr orbit nd the corresponding energy is E L / 6mr velocity, energy E = E nd ngulr momentum L. () zero mr L mr L. The prticle is relesed from r > r with n inwrd Infinite. A one dimensionl hrmonic oscilltor (mss m nd frequency ) is in stte such tht the only possible outcomes of n energy mesurement re E, E or E, where E n is the energy is the energy of the n-th excited stte. If H is the Hmiltonin of the oscilltor, H / nd probbility tht the energy mesurement yields E is () / /4 /8 H / 4, then the. Wht is the difference between the mximum nd the minimum eigenvlues of system of two electrons whose Hmiltonin is H JS S, where S nd S re the corresponding spin ngulr momentum opertors of the two electrons? () J/4 J/ J/4 J. Two dielectric spheres of rdius R re seprted by distnce such tht R. One of the spheres (sphere ) hs chrge q nd the other is neutrl. If the liner dimensions of the systems re scled up by fctor two, by wht fctor should be chnge on the sphere be chnged so tht the force between the two spheres remin unchnged? () 4 4. An electric chrge distribution produces n electric field r r E e r where nd re constnts. The net chrge within sphere of rdius centred t the origin is () 4 e 4 e 4 4 e e South Delhi : 8-A/, Ji Sri, Ner-IIT Huz Khs, New Delhi-6, Ph : -6858, 6869 North Delhi : -5, Mll Rod, G.T.B. Ngr (Opp. Metro Gte No. ), Delhi-9, Ph: ,

3 4. The chrge density s function of the rdil distnce r is given by R r r R for r < R nd zero otherwise. The electric flux over the surfce of n ellipsoid with xes R, 4R nd 5R centred t the origin is () 4 R 8 9 R If x is n infinitely differentible function, then D x e x e x sin t t cos t / is () x 6. The Lplce trnsform of R, where the opertor d D exp x dx, is e x s () s s s s s 7. exp x sin x dx, where... is Dirc delt distribution, is () cos x 8. The integrl, dx is x () e e e 9. Two of the eigenvlues of the mtrix A e e e e e zero re nd. Wht is the third eigenvlue? () 5 5. Consider two cnoniclly conjugte opertors X nd Y such tht X, Y i I, where I is identify opertor., where ij re complex numbers, nd Q, Q zi, the vlue of If X Q Q, Y Q Q is () i z i / z i z. A quntum prticle of mss m is moving on horizontl circulr pth of rdius. The prticle is prepred in quntum stte described by the wvefunction, 4 cos South Delhi : 8-A/, Ji Sri, Ner-IIT Huz Khs, New Delhi-6, Ph : -6858, 6869 North Delhi : -5, Mll Rod, G.T.B. Ngr (Opp. Metro Gte No. ), Delhi-9, Ph: ,

4 being the zimuthl ngle. If mesurement of the z-component of orbitl ngulr momentum of the prticle is crried out, the possible outcomes nd the corresponding probbilities re () L,, with P, P nd P z Lz with P L, with P nd P z Lz, with P nd P 6. Suppose the spin degree of freedom of two prticles (non-zero rest nd mss nd non-zero spin) is described completely by Hilbert spce of dimension twenty one. Which of the following could be the spin of one of the prticles? () / /. For clssicl system of non-intercting prticles in the presence of sphericlly symmetric potentil wht is the men energy per prticle? is constnt. V r 4 r, () k T B 5 k T B kbt kbt 4. A prticle of mss kg is undergoing smll oscilltion bout the equilibrium point in the potentil x x for x > meters. The time period (in seconds) of the oscilltion is 6 V x (). 5. In thermodynmic process, the volume of one mole of n idel is vried s V T, where is constnt. The dibtic exponent of the gs is. Wht is the mount of het received by the gs if the temperture of the gs increses T in the process? RT () R T PART B : THREE mrks questions: RT RT. In the circuit shown below, the cpcitor is initilly unchrged. Immeditely fter the key K is closed, the reding in the mmeter is 7 ma. Wht will the reding (in ma) be long time lter? R R R A C R South Delhi : 8-A/, Ji Sri, Ner-IIT Huz Khs, New Delhi-6, Ph : -6858, 6869 North Delhi : -5, Mll Rod, G.T.B. Ngr (Opp. Metro Gte No. ), Delhi-9, Ph: ,

5 . Two conductors re embedded in mteril of conductivity 4 ohm-m nd dielectric constnt 8. The resistnce between the two conductors is 6 ohm. Wht is the cpcitnce (in pf) between the conductors? Ignore the deciml prt of the nswer.. An electronic circuit with components performs its intended function successfully with probbility.99 if there re no fulty components in the circuit. The probbility tht there re fulty components is.5. If there re fulty components, the circuit perform successfully with probbility.. The probbility tht the circuit performs successfully is x/. Wht is x? 4. The normlized eigenfunctions nd eigenvlues of the Hmiltonin of prticle confined to move between x in one dimension re n x n x sin n nd En m respectively. Here,,,... Suppose the stte of the prticle is x x x Asin cos where A is the normliztion constnt. If the energy of the prticle is mesured, the probbility to get the result is / s x. Wht is the vlue of x? m 5. A hrmonic oscilltor hs the following Hmiltonin p H m x m 4 It is perturbed with potentil V x. Some of the mtrix elements of the ground stte re given s follows: x C ; x C ; x C ; x 6 C 5 x in terms of its expecttion vlue in where n is the normlized eigenstte of H corresponding to the eigenvlue E n / E nd E denote the energy corretion of Wht is the frction E / E? n. Suppose O to the ground stte nd the first excited stte, respectively. 6. Consider the trnsistor circuit shown in the figure. Assume V.7 V, V 6V nd the lekge current is negligible. Wht is the required vlue of R B in kilo-ohms if the bse current is to be 4 µa? BEQ BB i C R C C + R B B + V CC + V BB i B i E V CE E South Delhi : 8-A/, Ji Sri, Ner-IIT Huz Khs, New Delhi-6, Ph : -6858, 6869 North Delhi : -5, Mll Rod, G.T.B. Ngr (Opp. Metro Gte No. ), Delhi-9, Ph: ,

6 7. A person on Erth observes two rockets A nd B direclty pproching ech other with speeds.8c nd.6c respsectively. At timer when the distnce between the rockets is observed to be 4. 8 m, the clocks of the rockets nd the Erth re synchronized to t = s. The time of collision (in seconds) of the two rockets s mesured in rocket A s frme is x/. Wht is x? 8. If n belin group is constructed with two distinct elements nd b such tht b I, where I is the group identity. Wht is the order of the smllest belin group contining, b nd I? 9. Two prllel rils of rilrod trck re insulted from ech other nd from the ground. The distnce between the rils is meter. A voltmeter is electriclly connected between the rils. Assume the verticl component of the erth s mgnetic field to be. guss. Wht is the voltge developed between the rils when trin trvels t speed of 8 km/h long the trck? Give the nswer in milli-volts.. Consider simple pendulum in three-dimensionl spce. It consists of string length l = cm nd bob mss m = 5 kg ttched to it shown in the figure below. The ccelertion due to grvity is downwrds s shown in the figure with mgnitude g = ms. 6 z y z I = cm m = 5 kg g = ms The pendulum is pulled in the x z plne to position where the string mkes n ngle with the z-xis. It is then relesed with n ngulr velocity rdins per second so tht the ngle the string mkes with z-xis does not chnge with time? PART C : THREE mrks questions:. A theoreticl model for rel (non-idel) gs gives the following expressions for the internl energy (U) nd the pressure (P), / /, U T V V bv T P T, V V bv T 5/ / nd where nd b re constnts. Let V nd T be the initil volume nd initil temperture respectively. If the gs expnds dibticlly, the volume of thegs is proportionl to () T T / T / T. A frictionless, het conducting piston of negligible mss nd the het cpcity divides verticl, insulted cylinder of height H nd cross-sectionl re A into two hlves. Ech hlf contins one mole of n idel gs t temperture T nd pressure P corresponding to STP. The het cpcity rtio C / C is given. A lod of weight W is tied to the piston nd suddenly relesed. After the system comes to equilibrium, the piston is t rest nd the temperture of the gses in the two comprtments re equl. Wht is the finl displcement y of the piston from its initil, ssuming yw TCv? p v South Delhi : 8-A/, Ji Sri, Ner-IIT Huz Khs, New Delhi-6, Ph : -6858, 6869 North Delhi : -5, Mll Rod, G.T.B. Ngr (Opp. Metro Gte No. ), Delhi-9, Ph: ,

7 7 () H H H H. An pprtus is mde from two concentric conducting cylinders of rdii nd b respectively, where < b. The inner cylinder is grounded nd the outer cylinder is t positive potentil V. The spce between the cylinders hs uniform mgnetic field H directed long the xis of the cylinders. Electrons leve the inner cylinderr with zero speed nd trvel towrds the outer cylinder. Wht is the threshold vlue of V below which the electrons cnnot rech the outer cylinder? () eh b 8mc 4. If y x stisfies nd dy y log dx y y for x, then eh b 8mc b y is () eh b 8mc eh b b infinity 5. Consider two coupled hrmonic oscilltors of mss m ech. The Hmiltonin describing the oscilltors is p p H m x x x x m m The eigenvlues of Ĥ re given by (with n nd n being non-negtive integers) () E n n n, n 8mc En, n n n En, n n n E n n n, n. q 6. Consider the Lgrngin, L q of prticle executing oscilltions whose mplitude is A. If p denotes the momentum of the prticle, then 4p is () A q 4 A q A q 4 A q A q 4 A q A q 4 A q 7. A block of mss M rests on plne inclined t n ngle with respect to the horizontl. A horizontl force F Mg is pplied to the block. If µ is the sttic friction between the block nd the plne, the rnge of so tht the block remins sttionry is () µ tn µ µ cot mu µ tn µ mu mu µ cot µ mu mu South Delhi : 8-A/, Ji Sri, Ner-IIT Huz Khs, New Delhi-6, Ph : -6858, 6869 North Delhi : -5, Mll Rod, G.T.B. Ngr (Opp. Metro Gte No. ), Delhi-9, Ph: ,

8 8. A bll comes in from the left with speed (in rbittrry units) nd cuses series of collisions. The other four blls shown the figure re initilly t rest. The initil motion is shown below (the number in the circle indicte the object s reltive mss). This initil velocities of the blls shown in the figure re represented s [,,,, ] A negive sign mens tht the velocity is directed to the left. All collisions re elstic. Which of the following indictes the velocities of the blls fter ll the collisions re completed? () /, /,,,/ /,,,, / /,,,, / 4 /,,,,/ 9. A lrge cylinder of rdius R filled with prticles of ms m. The cylinder spins bout its xis t n ngulr speed rdins per second, providing n ccelertion g for the prticle t the rim. If the temperture T is constnt insider the cylinder, wht is the rtio of ir pressure P t the xis to the pressure P c t the rim? () exp mgr k b T exp mgr mgr k b T k b T. The coordinte q nd the momentum p of prticle stisfy dq dp p, q 4 p dt dt If At is the re of ny region of points moving in the kbt mgr q, p-spce, then the rtio / () exp t exp 4t exp t / 4 A t A is. An idel fluid is subjected to thermodynmic process described by CV nd P n, where is energy density nd P is pressure. For wht vlues of n nd, the process is dibtic if the volume is chnged slowly? (), n, n, n, n. If F x, y x y xy, its Legendre trnsformed function, () u v uv u v uv u v G u v, upto multiplictive constnt, is u v. The elstic wve on stretched rectngulr membrne of size L L in the x-y plne is described by the function x y Asin sin cost L L where A nd re constnts. The speed of the elstic wves is v. The ngulr frequency is 5 v v () L L 4. Consider the wvepcket defined by x dkf k exp ikx Further, f k for k k/ nd f k () 8 k Kx sin x Kx sin K x 5 v L for k K / South Delhi : 8-A/, Ji Sri, Ner-IIT Huz Khs, New Delhi-6, Ph : -6858, 6869 North Delhi : -5, Mll Rod, G.T.B. Ngr (Opp. Metro Gte No. ), Delhi-9, Ph: , v L. Then, the form of normlized x 8 K Kx cos x sin Kx K x is 8

9 5. In n experiment, certin quntity of n idel gs t temperture T, pressure P nd volume V is heted by current flowing through wire for durtion of t seconds. The volume is kept constnt nd the pressure chnges to P. If the experiment is performed t constnt pressure strting with the sme initil conditions, the volume chnges from V to V. The rtio of the specific hets t constnt pressure nd constnt volume is P P V () V V P P P V V V P PV PV PV PV 9 South Delhi : 8-A/, Ji Sri, Ner-IIT Huz Khs, New Delhi-6, Ph : -6858, 6869 North Delhi : -5, Mll Rod, G.T.B. Ngr (Opp. Metro Gte No. ), Delhi-9, Ph: ,

- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students.

- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students. - 5 - TEST 2 This test is on the finl sections of this session's syllbus nd should be ttempted by ll students. Anything written here will not be mrked. - 6 - QUESTION 1 [Mrks 22] A thin non-conducting