11) A thin, uniform rod of mass M is supported by two vertical strings, as shown below.

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1 Fall 2007 Qualifie Pat II 12 minute questions 11) A thin, unifom od of mass M is suppoted by two vetical stings, as shown below. Find the tension in the emaining sting immediately afte one of the stings is seveed. 12) A small object moving on a cicula obit aound a sta suddenly explodes into two identical pieces. Right afte the explosion, as obseved fom the sta, the pieces move off with equal speeds, and the angle between thei diections of motion is θ. Ae the tajectoies of the pieces ellipses, paabolas, o hypebolas a) when θ = 60 o ; b) when θ = 90 o? Give a quantitative explanation fo you answes. 13) A classical gas in d dimensions has fee enegy F = cvt d +1 at high tempeatues. Hee, c is some positive numeical constant of ode 1. a) Show that this implies a elation between enegy density e = E/V and the entopy density s = S/V of the fom e = c s α, whee c is also a numeical constant, and detemine α in tems of d. b) The specific heat is defined byc V = T ds dt V. Given the equation of state fom pat a), what is the ange of values of α fo which the specific heat is negative? c) Is thee a dimension d fo which the specific heat of a classical gas at high tempeatues becomes negative? ( ) = αe 3 and Ω B ( E ) = βe 4. 14) Conside two systems A and B with density of states Ω A E These two systems ae put into contact with each othe so that they can exchange heat, and they come to themal equilibium. The total enegy of the combined system is E 0. a) How is the enegy distibuted between these systems in equilibium (i.e. what is the enegy of the system A, and what is the enegy of system B, in tems of the total enegy E 0? b) What is the equilibium tempeatue (in tems of the total enegy E 0 )?

2 15) A piece of p-doped silicon has a caie density n = cm 3 and the dimensions Δx = 10 mm, Δy = 2 mm, Δz = 1 mm. The magnetic field B z = 1T is applied in the z-diection, the cuent I x = 1A flows in the x-diection, and the voltage V y is measued. a) Wite down the equilibium foce condition that detemines V y. b) Expess the cuent density j x in tems of the caie density n and the caie velocity v x. c) Find V y in volts. 16) Suppose a gounded conducting sphee of adius a is immesed in a unifom electic field E = E 0ˆ z. The esulting electic potential is V (,θ ) = E 1 a 3 0 cosθ in spheical coodinates. Find the suface chage distibution on the sphee as a function of θ. 17) The adial momentum in spheical coodinates can be epesented by the Hemitian opeato p R = h 1 i. a) Find a solution ψ () of the eigenvalue poblem p R ψ ( ) = hkψ ( ). b) Give a physical intepetation of this wavefunction. 18) The Hamiltonian fo a thee level quantum system is a 0 b H ˆ = 0 c 0 b 0 a whee a, b, and c ae eal numbes. You may assume that thee ae no degeneate states. a. Find the eigenvalues and eigenvectos of Ĥ. b. The system is pepaed in the state 1 ψ ( 0) = 0 0 Find the column vecto ψ ( t). 19) A pion of mass m π decays into an electon and an antineutino with masses m e, and m v.find the velocity of the antineutino in the est fame of the electon in tems of the masses of the thee paticles. Intepet you answe in the case whee m ν =0.

3 20) Conside an electomagnetic plane wave at nomal incidence to an inteface between two media. The polaizations of the incident (i), eflected (), and tansmitted (t) waves ae shown below. The electic pemittivity in the two egions is ε 1 and ε 2, and the magnetic pemeability is µ 1 and µ 2. k B i E E i B 1 k i 2 B t E t k t a. Wite down an equation elating the amplitudes of the electic fields, E 0i, E 0 and E 0t at the bounday between the two media. b. Wite down an equation elating the amplitudes of the magnetic fields, B 0i, B 0 and B 0t at the bounday between the two media. c. Wite down equations elating the following pais of quantities: i. B 0i and E 0i ii. B 0 and E 0 iii. B 0t and E 0t. The elations in a, b, and c can be used to deive the amplitude eflection coefficient fo a wave taveling fom medium 1 into medium 2: n 1 E 0 µ1 n 2 µ2 = E 0i n 1 µ1 + n 2 µ2 d. Feites can be designed to have thei elative electic pemittivity equal to thei elative magnetic pemeability: ε/ε 0 =µ/µ 0. Detemine the value of the amplitude eflection coefficient fo a wave taveling fom vacuum into one of these mateials.

4 21) The band gap E g in semiconducto nanopaticles inceases as the paticle dimension d becomes smalle. The band gap can be measued using optical absoption spectoscopy with light of sufficient enegy. The figue shows a seies of optical absoption specta fo a seies of CdSe nanopaticle samples with diffeent paticle diametes. a) Fo the absoption spectum indicated by the aow, estimate the wavelength, λ t, at the absoption theshold and fom this the band gap in ev. b) A simple model fo this effect elates E g to d, the bulk band gap, E g o, and the effective masses ( m * e and m * h ) of the electons and holes: E g (ev) = E 0 g (ev) + 2h2 π 2 1 ed 2 * m + 1 *, whee S.I. units ae used fo all values e m h except the band gap. Fo bulk CdSe, E o g =1.74 ev, m * e = 0.13 m e and m h * = 0.3 m e, whee the fee electon mass. Calculate the pedicted size of the nanopaticle sample. c) Assuming a 5% eo on you estimate of λ t, and no eos on the othe paametes, what is the eo on the calculated paticle size? Is the model in ageement with expeiment?

5 22) A non-magnetic medium has a dielectic constant that depends upon fequency ω and is given by ε = ε 0 1 A ω 2 ( ), whee A is a constant and ε 0 is the pemittivity of fee space. a) Wite down the dispesion elation fo an electomagnetic wave, which expesses ω as a function of wave numbe k. b) Find the citical fequency below which a wave launched into the medium will not popagate though the medium, but athe will be evanescent o decaying. c) Calculate the popagation distance ove which the amplitude of the wave decays by a facto of e. 23) The total numbe of conduction electons pe unit length in a quantum wie is /m. Assume that the electons move feely in one dimension in a wie of length L. What is the Femi enegy of the wie in electon volts (ev)? (Hint: Recall that the wavelength must satisfy nλ = L, whee λ is the wavelength and n is an intege.) 24) a) Give the definition of a semiconducto. How does a semiconducto diffe fom a metal? b) What is the mechanism fo the electical conductivity of an intinsic (undoped) semiconducto at low tempeatues, k B T << Δ, whee Δ is the enegy gap between the valence band and the conduction band? c) Give a physical agument to explain the appoximate tempeatue dependence of the conductivity when k B T << Δ. 25) The Sun convets potons into He though a seies of eactions amounting to 4 p 4 He + 2e + + 2v e. The sola constant descibing the powe of the sola adiation at Eath is P = 1400 W/m 2. The enegy eleased pe eaction coesponds to the binding enegy of He (28.3 MeV) a) What is the neutino flux at Eath? In othe wods, how many neutinos aive at the Eath s suface (m -2 sec -1 )? b) Give a ough estimate of how the powe of the neutino flux compaes to the sola optical adiation powe. Assume that the aveage enegy of a neutino is 0.3 MeV.