Exam 1 2004 Solutions

Exam 1 2004 Solutions - PHYSICS 13 HOUR EXAM 1 FALL 2004...

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Unformatted text preview: PHYSICS 13 HOUR EXAM 1 FALL 2004 NAME _________ __§_Q_L.ZI_J_IIQ&§____ calculators. e al ur NY 'I r‘ r i UsetuLEoLmNaLandflala Relativistic Energy = mo c2/\/(1-uZ/c2), where mo is the rest mass; mo c2=Rest Energy Relativistic momentum = m, u/V(1-uZ/c2) Lorentz Transformation frame 5’ moves in the +x direction with V as seen in S: x=y(x’+Vt’), y=y’, z=z’, t=y(t’+Vx’/c2) where y=1/\/(1-V2/c2) Lorentz contraction: L=Lo/y Time dilation: t=~yt0 Addition of velocities for an object moving at u’)( and u’y in S’: ux=(u’x +V)/,(1+Vu’x/c2), u,=u’y/(1+Vu’x/c2) Doppler Effect: f’ = f ’[\/(1+v/c) / V(1-v/c)] f = frequency (or v , Greek nu) Compton scattering: 7U - 7i. = (h/mec)(1-cose); (h/mec)=2.43x10"2m Quantum: E=hf Photoelectric. effect: hf = mvmaxz/Z + <I>(work function) de Broglie: k=hlp General waves: 7tf=vphase Light: kf=c Bohr atom (single electron): En=—ZZEo/n2, with Eo=13.61 eV=k2e4m/2h 2 transitions: hfnm= En — Em Heisenberg’s Uncertainty: AxApz h/Z Constants: Velocity of light: c = 3.00x 1O8 meter/sec Planck’s constant: h = 6.63 x 10‘34 J sec= 4.14 x 10'15 eV sec h = h/Zn =1 .05x10-34 J sec=6.58x10*‘6 eV sec hc=1.97x10'7 eV m hc = 1.24 x 10‘6 eV m Rest energy of electron = mec2 = 0.511 MeV Units; Energy: 1 J(oule) = 1 Kg mZ/sec2 ; 1 eV(electron Volt) = 1.6x10"9 J; 1 MeV = 106 eV = 1.78x10'3o Kgc2 & 1 MeV/c2 = 1.78x10'30 Kg Grades: points /possible points I /50 ll / 40 III /50 IV / 60 Total ________________ _____ —_ NAME________________________ I. Multiple choice -- circle the onebest answer. 1) The Heisenberg Uncertainty Principle 3. verifies the wave nature of EM radiation. @uts a lower limit on momentum uncertainty for given position uncertainty. c. puts an upper limit on momentum uncertainty for given position uncertainty. d. verifies the particle nature of electrons. e. puts an upper limit on position uncertainty for given momentum uncertainty. 2) The “twin paradox” is not a paradox. It is resolved because a. the travelling twin does not actually age slower. b. the stationary twin is not always in an inertial frame of reference. @the travelling twin is not always in an inertial frame of reference. d. the slowing down of moving clocks is an illusion. e. the stationary twin is actually younger when the twins are reunited. ‘ 3) Which of the following isnoJ: true of Blackbody Radiation? ' a. The radiated power decreases as the wavelengths become very short for fixed T. b. The maximum power occurs at shorter wavelengths for rising T. c. The power spectrum demonstrates the existence of quanta of energy with E=hf. ‘ The power spectrum demonstrates the ultraviolet catastrophe. e. The Classical (Stefan) law that radiated power cc T4 applies. 4) A satellite is in a distant stable orbit around the earth. In accord with the Equivalence Principle, a set of simple, small, brief experiments conducted by astronauts insderejalelljlemould be consistent with the conclusion that a. the inertial mass of objects increases in time. b. light travels along hyperbolic paths. c. light speed increases when traveling perpendicular to the motion. his is an inertial frame of reference, far from external gravitational forces. e. measuring sticks in the cabin appear shorter in the direction of motion compared to the direction perpendicular to the motion. 5) A neutron has rest energy 940 MeV and average lifetime at rest of 15 minutes. Suppose a beam of neutrons is accelerated to an energy of 9,400 MeV. Which statement is true of a neutron in the beam? a) Its speed is a small fraction of c. b) lts momentum is m(proton) c. c) lts rest mass increases. d) lts lifetime in the lab is 1.5 minutes. .@Its lifetime in the lab is 150 minutes. ’ II. Suppose there are two space stations at rest relative to each other and separated by a distance of _LKm. Call their “rest frame” 5'. Their clocks are synchronized and at time t'=0 both clock faces light up. A rocket is traveling at Qfioalong the line connecting the two space stations. At time t=O (in the rocket’s frame of reference S) the rocket reaches the first station just as its clock lightsup. 3. According to the rocket, how far away is the other space station? -._'__~Z>/ .— / ~Ll /- A X~F®l~1¥ [.4175 gL— 055m |l.b. According to the rocket (i.e. in the rocket’s frame of reference), at what time does the second clock face light up? Does this second event occur before, at the same time, or after the first event? 3 __ ’U‘/ -——,_ 151/__§§ MM f~i<<f~~9€> ~6X ~ lfxsxgxmgmé ': —- QSNOEec before (52‘ aren‘t (in Kocketfmme> c. Suppose a second rocket is going at 0.6c in the oppositefltectian, and coincides with the first rocket a-t‘the second rocket’s time 0 (Le. all the three reference frames’ clocks read zero when their origins coincide). At whatiimefioesihesemndjmnt occur according to this rocket’s time measurement? Is it before, simultaneous with, or after the first event? I__ / v I —6 - . “t, - 75“? +33): +2.57% see after lit event / \ (m 2% new {we v,/:Hv{:~0|bc> d. According to the first rocket, what is. the velocity of the second rocket? /I Z ‘ a. ‘ r—L—dC 30,83, 7); HIE” i+(.z,)-V l.3b ——9’ CO... Ill. Consider the Bohr model of the singly ionized Helium atom (Helium has 2:2). a. What are the three lowest allowed energies of the electron? Call these E1, E2 , E3. Use eV units. Z2; tag—215.0, rw exam E; «4366/ E,:“S$‘fe\/ b. What are the djfletenmnergieiof photons that can be emitted when the electron undergoes transitions starting from the third level E3 to lower energy levels? 3% at: gfiq: l3,b~—L,O :7'5 3/ as l 3 g—e.:§‘hLl-Iill23 W)in 3 s7! fiEf‘EES‘H‘L’O :leev c. Which of the precedingjtansitions re ults in theétsigbesLfitemmncy photon? What is that itequency? g?! r eases Mi"? 5 flour» “‘5 b “g r “E” : 5&5)?le 61.3% ‘5 M7 ND]. 56' d. When this highest frequency light is incident on a certain metal, the photoelectric effect occurs. The wotkiunctiomof the metal is 5.0 eV. What is the stopping_potential for this case? h 1C : KEW + Velaf‘ rem: Mi : (Lei—3.0M lll.e. Assume that same frequency light is emitted from a distant star, traveling away from earth at 1/2 the speed of light. What Wwould be seen on earth? I r . "I: \p ‘j [/1 .‘_ 7L _ ’5’ "I 3C i+% % If? g (Mng 566 IV. An electron, not localized in space, is described by a wavelike solution to the Schrodinger wave equation. Suppose a free electron, with (non-relativistic; kinetic energy of 1000 eV, has a wavefunction given by ‘ ‘P(x,t) = A sin(kx-(nt) . where k=2ulA is the wavenumber, m=21tf is the angular frequency and A is an unspecified constant. a. What is the relation between the wavenumber k and the kinetic energy? :JWKZ [Becadse RIM ;M b. Find thenumecicaliamefiik for this 1000 eV electron wave. 1 l 1 x - IWOeV :: L if : (19.) 1.1: .‘17xlD7eV'M la OW MC“ 40 a 051mb 9v) 7- (3,7‘l7)‘/D eV/m“) k7” k : (03 a 8V jar. Lax [OH NH Emir/0 “ex/m ,___/ c. Find the numerical value of co for this electron wave. _ — 3 ,, _ (036V hg ‘ K“) K ’0 6v ‘7 £0- (05 flo'lbeVoszc : WWW” 5%” f) lV.d. What is the value of the velocity of this electron wave? ‘U’ '2 ‘: ': i0. : [lily/Olga", ,DI'M_] : 33(3)” EM 594 ;L e. If the kinetic energy is that of a classical particle, what value of velocity would f. At time t=0, what is the probability of finding this electron within an interval of i dx/2 from the position x= 10'”m? Express yOur result in terms of A and dx. I‘M/05W“ '3 (“Hibwflonw : A“M(wo"l/o> all! '3/\ M54795) M :Afiw ...
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This note was uploaded on 03/10/2008 for the course PHY 13 taught by Professor Garyr.goldstein during the Fall '04 term at Tufts.

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Exam 1 2004 Solutions - PHYSICS 13 HOUR EXAM 1 FALL 2004...

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