EUF. Joint Entrance Examination for Postgraduate Courses in Physics

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1 EUF Joint Entrance Examination for Postgraduate Courses in Physics For the second semester of 06 April 06, 06 Part Instructions Do not write your name on the test. It should be identified only by your candidate number EUFxxx). This test contains questions on: classical mechanics, quantum mechanics, and statistical mechanics. All questions have the same weight. The duration of this test is 4 hours. Candidates must remain in the exam room for a minimum of 90 minutes. The use of calculators or other electronic instruments is not permitted during the exam. Answer each question on the corresponding sheet of the answer booklet. The sheets with answers will be reorganized for correction. If you need more answer space, use the extra sheets in the answer booklet. Remember to write the number of the question Qx) and your candidate number EUFxxx) on each extra sheet. Extra sheets without this information will be discarded. Use separate extra sheets for each question. Do not detach the extra sheets. If you need spare paper for rough notes or calculations, use the sheets marked scratch at the end of the answer booklet. Do not detach them. The scratch sheets will be discarded and solutions written on them will be ignored. Do not write anything on the list of constants and formulae. Return both this test and the list of constants and formulae at the end of the test. Have a good exam!

2 Q6. An insulating sphere of radius a has a uniform charge density and total charge Q. Ahollow conducting uncharged sphere, whose inner and outer radii are b and c, respectively,isconcentric with the insulating sphere, as shown in the figure below. a) Find the magnitude of the electric field in the regions: i) r<a;ii)a<r<b;iii)b<r<cand iv) r>c. b) Find the induced charge per unit area on the inner and outer surfaces of the conductor. c) Sketch the graph of the magnitude of the electric field E as a function of r. Identify in your graph each one of the regions listed in item a). Q7. Consider Maxwell s equations in di erential form and solve each item below. a) Derive, showing every step, the wave equations in vector form for both the electric and the magnetic fields in vacuum. Remember: r r V) =rr V) r V and r r V) =0 b) Write the wave equation for a generic scalar function f~r, t )and,bycomparingwiththe expressions obtained in a), obtain the speed of propagation for both fields. c) A possible solution of the equations obtained in a) is a linearly polarized plane wave. Consider an electromagnetic field solution of this kind, which propagates in the ẑ direction. If! is the angular frequency, k is the wave number, E 0 and B 0 are the amplitudes of the electric and magnetic fields, respectively, write explicitly the magnitude and direction of ~ E and ~ B as functions of position and time. d) Consider now Maxwell s equations in the presence of charges and currents and derive the equation that relates the electric charge and electric current densities continuity equation). What conservation law is expressed mathematically by this equation? Q8. Consider a spin-/ particle under the influence of a uniform magnetic field ~ B = Bẑ. The Hamiltonian for this problem is given by Ĥ = BŜz, where is a constant. Let the states +i and i be such that Ŝz ±i = ±~/) ±i. a) What are the eigenvalues of the Hamiltonian? b) At time t =0thestateoftheparticleisgivenby 0)i =[ +i i] / p. Calculate the state of the particle at any time t>0. c) Calculate the average value of the operators Ŝx, Ŝ y,andŝz for any given time t 0. Remember that Ŝx ±i =~/) i e Ŝy ±i = ±i~/) i. d) What is smallest value of t>0forwhichtheparticlereturnstoitsinitialstate?

3 Q9. When two events in space-time are separated in space by the vector x ˆx + y ŷ + z ẑ and in time by t, theinvariantintervalbetweenthem,whosevalueisindependentoftheinertial reference frame, is defined as s x + y + z c t. a) Events ) and ) occur in di erent positions x,y,z )andx,y,z ), respectively, in an inertial frame S) and are such that the invariant interval is positive. Is there an inertial frame in which these events occur at the same point in space? Justify. b) Under the same conditions as in a), could event ) have been caused by event )? Justify your answer analyzing the propagation of a signal from ) to ) with velocity V ~ = V xˆx + V y ŷ + V z ẑ. c) A clock is at rest in an inertial frame S 0 ), which moves with velocity V ~ with respect to S). i) What is the sign of the invariant interval between events defined by two successive positions of the clock hands neglect the spatial dimensions of the clock)? ii) Obtain the relation between the proper time interval t 0 measured in S 0 )andthe time interval t measured in S). d) The spatial separation between a source F and a particle detector D is Lˆx, inthelabframe S). Consider events E F and E D,ofproductionanddetectionofaparticle,respectively. The particle moves from F to D with constant velocity V ~ = V 0ˆx in the lab frame. i) What are the separations in space x and time t between E F and E D in the lab frame? ii) Let L 0 be the distance between F and D in the reference frame of the particle. What are the separations in space x 0 and time t 0 between E F and E D in the reference frame of the particle? iii) Find the relation between L 0 and L. Q0. In a model for a solid crystal it is assumed that the N atoms are equivalent to 3N independent, one-dimensional classical harmonic oscillators of mass m, eachoneoscillatingwiththe same angular frequency! around its equilibrium position. At a distance x from that position, the potential energy is given by U = m! x /. From a few experimental data it is possible to estimate, in terms of the inter-atomic distance at low temperatures d, the root mean square displacement of the atoms when melting occurs. The answers to the items below allow us to obtain this estimate. Assume the crystal is in thermal equilibrium at absolute temperature T. a) The number of states in a cell in phase space x,p) isgivenbydxdp)/h, whereh is Planck s constant. Obtain the partition function of the harmonic oscillator, ZT,!). b) Calculate the average number of oscillators whose position lies between x and x + dx. c) Find an expression for the average potential energy hui per one-dimensional harmonic oscillator. Compare your result with the value expected from the equipartition theorem. d) Let x 0 be the mean square displacement around the equilibrium position when the crystal melts and let f be equal to x 0 /d. Using hui = m! x 0/, obtain an estimate for f for an element whose atomic mass is m = kg, melting temperature is T F =400K, d =0/3) Å=0/3) 0 0 m, and the frequency is such that ~!/k B =300K.

4 EUF Joint Entrance Examination for Postgraduate Courses in Physics For the second semester of 06 April 5-6, 06 LIST OF CONSTANTS AND FORMULAE Do not write anything on this list. Return it at the end of the exam day.

5 Physical constants Speed of light in vacuum Planck s constant Wien s constant c = m/s h = J s = ev s = h/π =, J s = 6, ev s hc 40 ev nm = 40 MeV fm c 00 ev nm = 00 MeV fm W = m K Permeability of free space Magnetic constant) µ 0 = 4π 0 7 N/A = N/A Permittivity of free space Electric constant) ɛ 0 = µ 0 c = F/m = Nm /C 4πɛ 0 Gravitational constant G = N m /kg Elementary charge e = C Electron mass m e = kg = 5 kev/c Compton wavelength λ C =.43 0 m Proton mass m p = kg = 938 MeV/c Neutron mass m n = kg = 940 MeV/c Deuteron mass m d = kg =.876 MeV/c Mass of an α particle m α = kg = 3.77 MeV/c Rydberg constant R H = m, R H hc = 3.6 ev Bohr radius Avogadro s number Boltzmann s constant Gas constant Stefan-Boltzmann constant a 0 = m N A = mol k B = J/K = ev/k R = 8.3 J mol K σ = W m K 4 Radius of the Sun = m Mass of the Sun = kg Radius of the Earth = m Mass of the Earth = kg Distance from the Earth to the Sun =.50 0 m J = 0 7 erg ev =, J Å = 0 0 m fm = 0 5 m Numerical constants π = 3.4 ln = cos30 ) = sin60 ) = 3/ = e =.78 ln 3 =.099 sin30 ) = cos60 ) = / /e = ln 5 =.609 log 0 e = ln 0 =.303

6 Classical Mechanics L = r p dl dt = r F L i = j I ij ω j T R = ij I ijω i ω j I = r dm r = rê r v = ṙê r + r θê θ a = r r θ ) ê r + r θ + ṙ θ ) ê θ r = ρê ρ + zê z v = ρê ρ + ρ ϕê ϕ + żê z a = ρ ρ ϕ ) ê ρ + ρ ϕ + ρ ϕ) ê ϕ + zê z r = rê r v = ṙê r + r θê θ +r ϕ sin θê ϕ a = r r θ ) r ϕ sin θ ê r + r θ + ṙ θ ) r ϕ sin θ cos θ ê θ + r ϕ sin θ + ṙ ϕ sin θ + r θ ) ϕ cos θ ê ϕ E = mṙ + L mr + V r) r V r) = F r )dr V effective = L r 0 mr + V r) R R 0 dr E V r) = L mr m t t 0) θ = L mr d u dθ + u = m L u F /u), u = r ; ) du + u = m [E V /u)] dθ L d dt ) L q k L = 0, L = T V ) d = Q k dt q k Q k = N i= ) d r dt H = F ix x i + F iy y i + F iz z i inertial ) d r = dt f p k q k L; k= rotating + ω q k = H p k ; Q k = V ) dr + ω ω r) + ω r dt rotating ṗ k = H ; H t = L t

7 Electromagnetism E dl + d dt B ds = 0 E + B t = 0 B ds = 0 B = 0 E ds = Q/ɛ 0 = /ɛ 0 ρdv E = ρ/ɛ 0 d E B dl µ 0 ɛ 0 E ds = µ 0 I = µ 0 J ds B µ 0 ɛ 0 dt t = µ 0J F = qe + v B) df = Idl B E = q ê r E = V V = 4πɛ 0 r E dl V = q ê r 4πɛ 0 r F = q q 4πɛ 0 r r ) r r 3 U = 4πɛ 0 q q r r db = µ 0I 4π dl ê r Br) = µ 0 Jr ) r r ) r 4π r r 3 dv B = A Ar) = µ 0 Jr )dv 4π r r J = σe J + ρ t = 0 u = ɛ 0 E E + µ 0 B B S = µ 0 E B P = ρ P P ˆn = σ P M = J M M ˆn = K M D = ɛ 0 E + P = ɛe B = µ 0 H + M) = µh Relativity γ = V /c x = γ x V t) t = γ t V x/c ) v x = v x V V v x /c v y = v y γ V v x /c ) v z = v z γ V v x /c ) E = γm 0 c = m 0 c + K p = γm 0 V E = pc) + m 0 c ) 3

8 Quantum Mechanics i Ψx,t) t = HΨx,t) H = m r r r + ˆL mr + V r) p x = i â = x mω ) ˆp ˆx + i mω [x, p x ] = i â n = n n, â n = n + n + L ± = L x ± il y L ± Y lm θ,ϕ) = ll + ) mm ± ) Y lm± θ,ϕ) L z = x p y y p x L z = i ϕ, [L x,l y ] = i L z E n ) = n δh n E n ) = m n m δh n E n 0) E m 0), φ ) n = m n m δh n E n 0) E m 0) φ 0) m Ŝ = σ σ x = ) 0 0, σ y = ) 0 i i 0, σ z = ) 0 0 ψ p) = π ) 3/ d 3 r e i p r/ ψ r) ψ r) = π ) 3/ d 3 p e i p r/ ψ p) + eâ n=0 Â n n! Modern Physics p = h λ E = hν = hc λ E n = Z hcr H n R T = σt 4 λ max T = W L = mvr = n λ λ = h cos θ) nλ = d sin θ x p / m 0 c E = E np E n) P En), where P E n ) is the distribution function. 4

9 Thermodynamics and Statistical Mechanics du = dq dw du = T ds pdv + µdn df = SdT pdv + µdn dh = T ds + V dp + µdn dg = SdT + V dp + µdn dφ = SdT pdv Ndµ F = U T S G = F + pv H = U + pv Φ = F µn ) V S,N ) p S,N = = ) p S V,N ) V S p,n ) S V T,N ) S p T,N = ) p V,N ) V = p,n ) F p = V T,N ) F S = V,N C V = ) U V,N = T ) S V,N C p = ) H p,n = T ) S p,n Ideal gas: pv = nrt, U = C V T = nc V T, Adiabatic process: pv γ = const., γ = c p /c V = c V + R)/c V S = k B ln W Z = n e βen Z = dγe βeγ) β = /k B T F = k B T ln Z U = β ln Z Ξ = N Z N e βµn Φ = k B T ln Ξ f FD = e βɛ µ) + f BE = e βɛ µ) 5

10 Mathematical results x n e ax dx = n+) n+) n a n π a ) n = 0,,,...) k=0 x k = x x < ) eiθ = cos θ + i sin θ dx a + x ) = ln x + ) x + a / dx a + x ) = x 3/ a x + a ) ln N! N ln N N x dx a + x ) = ln x + ) x x + a 3/ x + a dx x = ) + x ln x dx xx ) = ln /x) a + x dx = a arctan x a x a + x dx = lna + x ) 0 z x e z + dz = x ) Γx) ζx) x > 0) 0 z x e z dz = Γx) ζx) x > ) Γ) = Γ3) = Γ4) = 6 Γ5) = 4 Γn) = n )! ζ) = π 6 π =,645 ζ3) =,0 ζ4) = π4 90 π =,08 ζ5) =,037 π sinmx) sinnx) dx = πδ m,n dx dy dz = ρ dρ dφ dz π cosmx) cosnx) dx = πδ m,n dx dy dz = r dr sin θ dθ dφ Y 0,0 = 4π Y,0 = 3 4π cos θ 3 Y,± = 8π sin θe±iφ Y,0 = 5 3 cos θ ) 5 Y,± = sin θ cos θe±iφ Y 6π 8π 5,± = 3π sin θe ±iφ P 0 x) = P x) = x P x) = 3x )/ General solution to Laplace s equation in spherical coordinates, with azimuthal symmetry: V r,θ) = A l r l + B l r ) P lcos θ) l+ l=0 6

11 V) = 0 f = 0 V) = V) V A ds = A) dv A dl = A) ds Cartesian coordinates A = A x x + A y y + A z z A = Az y A ) y Ax ê x + z z A ) z Ay ê y + x x A ) x ê z y f = f xêx + f y êy + f z êz f = f x + f y + f z Cylindrical coordinates A = ρa ρ ) + A ϕ ρ ρ ρ ϕ + A z z A = [ A z ρ ϕ A ] [ ϕ Aρ ê ρ + z z A ] z ê ϕ + ρ [ ρa ϕ ) ρ ρ ρ ] A ρ ê z ϕ f = f ρ êρ + f + f ρ ϕêϕ z êz f = ρ ρ ρ f ) + f ρ ρ ϕ + f z Spherical coordinates A = r r A r ) r [ A = r sin θ [ + r sin θ + sin θa θ ) + A ϕ ) r sin θ θ r sin θ ϕ sin θa ϕ ) θ A r ϕ r r sin θ ] A θ ê r ϕ ] ra ϕ ) ê θ + r [ ra θ ) r r r ] A r ê ϕ θ f = f r êr + f r θ êθ + f r sin θ ϕêϕ f = r f ) + r r r r sin θ sin θ f ) + θ θ r sin θ f ϕ 7