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1 Physics 14 Midterm Exam Fall 016 Last Name: First Name: Net D: Discussion Section: Discussion T: cademic ntegrity: The following actions or activities during a University Examination are grounds for disciplinary action up to and including expulsion: Giving assistance to or receiving assistance from another student Using unauthorized materials, including unauthorized electronic devices (phones, tablets, smart watches, laptops, etc) Turn off and put away all internetcapable electronic devices Calculators may not be shared Exam Duration: You have 15 hours (90 minutes) to complete this examination Examination Type: This is a Midterm Exam This is a closedbook examination Put away all notes, textbooks, and study aids nstructions: Use only a # (HB) pencil to fill in the answer sheet 1 Name: a Print your Last Name in the designated spaces on the answer sheet Fill in the corresponding circles below each character in your Last Name c Print your First nitial in the designated spaces on the answer sheet Fill in the corresponding circle below you First nitial NetD: a Print your NetD in the designated spaces on the answer sheet Fill in the corresponding circles below each character in your NetD Your UN: a Print you UN in the STUDENT NUMBER spaces on your answer sheet Fill in the corresponding circle below each number in your UN Sign your full name on THE STUDENT SGNTURE line on the answer sheet

2 Physics 1 Midterm Exam Fall Print your Discussion Section on the SECTON line of your answer sheet Do not fill in the SECTON box Your Exam Test Form Mark the circle for in the TEST FORM box on your answer sheet Your exam should have 1 and formula sheets Count the number of pages in your exam booklet now Grading Policy: This exam is worth points There are three types of questions: MC5: multiplechoicefiveanswer questions, each worth 6 points Credit will be granted as follows: (a) f you mark only one answer and it is the correct answer, you earn 6 points (b) f you mark two answers, one of which is the correct answer, you earn 3 points (c) f you mark three answers, one of which is the correct answer, you earn points (d) f you mark no answers, or more than three, you earn 0 points MC3: multiplechoicethreeanswer questions, each worth 3 points No partial credit Mark only one answer (a) f your marked answer is the correct answer, you earn 3 points (b) f your marked answer is wrong answer or if you mark no answers, you earn 0 points TF: truefalse questions, each worth points No partial credit Mark only one answer (a) f your marked answer is the correct answer, you earn points (b) f your marked answer is wrong or you marked neither answer, you earn 0 points

3 Page 1 of 16 PHYS 14 Exam 14 Fall 016 Midterm () The next two questions pertain to the situation described below Suppose there are two waves; one with functional form y 1 (x, t) 1 cos(kx + ωt) and another with functional form y (x, t) cos(kx O_0 + ωt + ϕ) The variables are: 1 4, 3, and ϕ will be varied in the following questions 1) What is the amplitude of the combined wave y 1 + y if ϕ 0? a 5 b 5 c O 7 d 1 e 49, L tot, t z 4157 ) What is the total intensity of the combined wave y 1 + y if ϕ π/? a 1 b 7 c 5 d 5 e 49 0 to / z 3, 4 ^ T 5

4 Page of 16 The next two questions pertain to the situation described below 53mm ( at ) n :, R fu 's ) REE Li n ) T 41T bit 81T lot fu 56 ) D ,11 Edith v n la L Consider the Michelson interferometer drawn above Light from a laser source, with wavelength λ 53 nm and intensity source, enters the interferometer from the left The source light is divided by a 50:50 beam splitter, traverses the two arms (top arm and right arm), and finally is recombined and directed to the intensity detector at bottom The two arms have identical lengths from beam splitter to mirror, L 1 m nitially, with both arms completely empty (ie, vacuum), all of the input light exits out of the lower port, and an intensity source is detected 5 Professor Pug decides to slowly (and uniformly) fill the right arm of the interferometer with an unknown gas s the concentration of this gas is increased from 0 to c max, Prof Pug observes that the detected intensity ( Detector ) goes from a maximum value to a minimum and back to a maximum 5 times (as shown in the inset plot) 3) What is the difference in the value of the index refraction of this gas at maximum concentration, n max, from that of vacuum? n other words, what is the value of n max 1? Oa b c d e C m ) Z ( n )

5 Page 3 of 16 4) magine that the unknown gas, in addition to changing the index of refraction in the right arm, also leads to a proportional loss of intensity in that arm with increasing concentration (with 100% intensity loss in the right arm at c max and 50% intensity loss at c max /) n this scenario, which of the following plots (labeled ac) would best describe the dependence of the detected lower port intensity on the gas concentration? M b # z, 1 h o Plot () Plot (b) Plot () '

6 07 Page 4 of 16 The next three questions pertain to the situation described below ' Vol all J L 0 m N 5 L a 0 x 10 b m d a U O x O b n 415 nm screen is set up 0 m away from a diffraction grating, with 5 slits 0 µm wide and with a distance 100 µm between the centers of the slits t is irradiated with light of wavelength 415 nm t produces a pattern o similar to that pictured O 5) What is the distance between the maximum and the first zero of the diffraction envelope? a 5 m Ob 007 m c 0415 m d 0008 m e m a sin On, ma a Omm sin CE ) Ymir L tan Qin ' O in

7 Page 5 of 16 6) How wide are the interference peaks? That is, what's the distance from the central peak to the first zero in the total signal? ' ' n a m b 0 m c 001 m d O m e m it dsin% sin ) :O sin y L tan O 7) What is the minimal change in wavelength that we could detect when looking at the first order maximum of the pattern? m1 ) a m Ob m c 0007 m d m e m DR Fm N' m mm

Page 6 of 16 The next two questions pertain to the situation described below Consider the figure below laser beam with diameter D beam 0 cm and wavelength of λ 650 nm is directed towards, and

8 Page 6 of 16 The next two questions pertain to the situation described below Consider the figure below laser beam with diameter D beam 0 cm and wavelength of λ 650 nm is directed towards, and "overfills", a smaller spherical lens (circular aperture diameter D lens 4 cm) having a focal length of 10 cm D on, " Dz 4cm # e onize, DY by oq_d Day 9L 10cm an Orin 3f # ( ham ' ' 397mm 8) What is the smallest size (diameter) of the light beam as it is focused by the spherical lens? a 0793 μm b 397 μm c 13 μm d 163 μm e 813 μm 9) Which of the following actions would decrease the minimum size of the focused laser beam? NOTE: answer choices ae are found below (action #1) decrease the laser wavelength (action #) make the laser beam smaller, so that its diameter matches that of the lens X (action #3) increase the laser wavelength X D 0mm a action #1 and action # b action #3 only 61 c action # only d action #1 only O e action # and action #3 L tanks m ' )

9 Page 7 of 16 week 4 problem The mass of the muon is 07 times greater than the mass of the electron Muons have the same charge as electrons n electron and a muon are each in the respective ground states of two identical 1D potential wells Compare the debroglie wavelengths of the particles, and the wavelength of a photon needed to excite the particle to the first excited state: E hi ' H n is L Emu Beeler 10) Epnot Oa λ electron debroglie O λ muon debroglie, λ photon to excite electron < Oλ photon to excite muon b λ electron debroglie < λ muon debroglie, λ photon to excite electron > λ photon to excite muon c λ electron debroglie < λ muon debroglie, λ photon to excite electron λ photon to excite muon d λ electron debroglie O λ muon debroglie, λ photon to excite electron $ λ photon to excite muon e λ electron debroglie > λ muon debroglie, λ photon to excite electron λ photon to excite muon Y sin z, # ) L The next three questions pertain to the situation described below a 0 3m 50in mf 3540 n s wave is incident on two small slits as shown in the diagram You can assume that the source is very far away and so the wave is planar The diagram is not to scale There is a detector located a position y measured from the horizontal line crossing the midpoint of the center of the two slits, as shown in the diagram The slits are spaced 3 m apart and the screen is 50 m away from the slits The wave is propagating with wavelength m and velocity 340 m/s

10 Page 8 of 16 11) What is the frequency of the wave? f v Oa 170 Hz b 680 Hz c 338 Hz d 34 Hz e Hz f 340/51705 ' 170 He 1) What is the minimal value of y (greater than zero) for which the intensity is at a maximum? a 177 m b 0667 m c 034 m d 073 m O e 447 m sin O m X a sin ' ( %) y L tan O L tan sin m ' C Nd ) 13) What is the minimal value of y (greater than zero) for which the intensity is at a minimum? " a 447 m b 073 m c 034 m d 177 m e 0667 m dsino 4 d 't since a T Y L tan sin ' ( Ea )

11 mu The next two questions pertain to the situation described below and E et 7 Page 9 of 16 u 0004C Monochromatic light with wavelength λ is incident upon a gold plate (work function ϕ 51 ev) t is observed that electrons are ejected from the surface of the gold plate, traveling at speeds approaching 0004 c (where c is the speed of light) 14) What is the value of the wavelength λ? a 43 nm b 100 nm c 405 nm d 135 nm e 303 nm O 15) What is the de Broglie wavelength of electrons ejected at speeds of 0004 c? a he d 140 EV To t femur nm 5 lev t ( 0511M 004$ Oa 061 nm b 303 nm c 540 nm p X HO0004C 511M he 0511M e V nm 0511 New O 004

10µm D 600mm Xl Page 10 of 16 photon of a wavelength 600 nm passes through a slit of width 10 µm nitially it is going horizontally but after passing through a slit the photon propagation direction

12 10µm D 600mm Xl Page 10 of 16 photon of a wavelength 600 nm passes through a slit of width 10 µm nitially it is going horizontally but after passing through a slit the photon propagation direction acquires some angular distribution Use Heisenberg uncertainty to estimate the angular spread of the photon propagation direction 16) The angular spread (in radians) of the momentum of the photon after passing through the slit is: Py y p O a 0001 b 1 c 01 O d 001 e tano 5 Py DX nets Dpx Dpx # Spx ~ Dp' z o x Recall that the surface area of a sphere with radius R is 4 π R To warm up on a cold winter day in Urbana, Prof Pug holds his paws (having a combined front surface area of 00 m ) in front of a small 40 Watt infrared lamp (with 100% efficiency), at a distance of meters away The lamp illuminates uniformly in all directions, as shown in the figure below, and its emission can be approximated as having a common wavelength of μm µm 40W Epi,o+ H M now Ppag Psy P 17) How many photons from the lamp hit the front of Prof Pug's paws every second? sphere crook ::i ) a 006 b c d Oe Npnut Piping F phot Pee P sphere h p C

13 Page 11 of 16 Week 4 finite potential well extends from x0 to xl as shown in the figure The well is divided into 3 regions Region is to the left of the well, region is in the well and region is to the right of the well We consider a bound state of this well #X 1 18) Compare the probabilities, corresponding to different energy states, to find the particle outside the well, namely at x11l The probability for the ground state is denoted P1 and the probability for the first excited state is denoted P a P10 and P>0 b P10 and P0 c P1 < 0 and P > 0 Od P > P1 e P1 > P

14 Page 1 of 16 ind d distinguishable 4 µ L coherent beam of metastable helium atoms passes through a pattern of three slits with common spacing d Far away from these slits (distance L >> d, L >> λ, the wavelength of the atoms) the interference pattern shown in the figure above is detected on a phosphor screen When considered alone, each of these equalsized slits contributes an "intensity" of atoms 0 on the detection screen We then place a detector near the central slit This detector can discern when the atoms pass through this central opening, but cannot distinguish which of the outer slits the atoms pass through for those events 19) Which of the following plots best reflects the interference pattern that will be measured in the presence of this detector? d since ma you one math now +, # O Plot () Plot () Plot ()

15 Page 13 of 16 0) This detector works by "scattering'' photons from a laser focused to the central slit, and the laser directly opposes the incident velocity of the atoms For this detector to unambiguously detect an atom passing through the central slit, each atom needs to "scatter'' 10 photons of wavelength 1083 nm Consider that for each "scattering'' event, the atom absorbs the momentum of one photon Which condition best describes the change in atomic momentum for detection events? a Δp kg m/s b Δp kg m/s c Δp kg m/s Pphrt he 0 a Ptot 10 pphux ' 45ns in kghls

16 Page 14 of 16 Week 4 The next three questions pertain to the situation described below one dimensional quantum particle having the same mass as the electron is placed in an infinite potential well 09 nm wide nitially it is in its ground state (n1), but a laser is used to excite it to its third excited state (n4) 1) Which of the following wavelengths is the closest to the wavelength of the laser that does this? DE DE a 581 nm b 698 nm (gh )C4, Oc 177 nm H d 505 nm BE e 446 nm 8 ( kg ( ( J s ) ) Compare the magnitude of Ψ n (center), relating to the probability of finding the particle in the area just near the center of the well, for the n4 state versus the n state a P middle (n4) P middle (n) b P middle (n4) P middle (n) c P middle (n4) P middle : } sinln ) > < y (n) rt 3) nstead of a particle having the same mass as an electron, we now put a particle with the mass of a muon (which is 07 times heavier than the electron) in the infinite potential well Compare the energy difference between the n state and the n1 state, ΔE 1, for these two cases W M Me 0,9 um amp (3 108) a ΔE 1 (muon) < ΔE 1 (electron) b ΔE 1 (muon) ΔE 1 (electron) c ΔE 1 (muon) > ΔE 1 (electron)

17 Page 15 of 16 week 4 The next two questions pertain to the situation described below Consider an electron in a small carbon nanotube with length 1 μm We shine light onto the nanotube, and the electron is excited from its initial energy state to a higher one 4) Which of the following could describe the electron s wavefunction after the photon absorption? ssume you can approximate the nanotube as a finite potential well x X F x! at x X a or N b ll except, N and K c H or M d o L or J e G or K xx U

18 DX ' Page 16 of 16 5) Let s say we happen to actually find the electron in the first 10nm part of the tube What can you say about the subsequent likely maximum velocity of the electron (the component along the axis of the nanotube)? The answer, to an order of magnitude is, a ~1000 m/s b ~0 c 0 ~10,000 m/s Dxbp DX ~ 10mm ~ ts / mbv nets Dv MX 1) v ~ ( kg ) ( loxiu 9N ) ~ 11,570ms

19 Physics 14 Common Formulae S Prefixes Power Prefix Symbol 10 9 giga G 10 6 mega M 10 3 kilo k milli m 10 6 micro 10 9 nano n 10 1 pico p Physical Data and Conversion Constants speed of light c m/s Planck constant h J s ev s Planck constant / J s ev s electron charge e C energy conversion 1 ev J conversion constant hc 140 ev nm Jm useful combination h /m e 1505 ev nm Bohr radius a (4 ) / me 0059 nm o o e Rydberg energy 4 hcr m e /(4 ) ev Coulomb constant 1/ (4 o ) N m / C vagadro constant N / mole electron mass m e kg 0511 MeV/c proton mass m p kg 9383 MeV/c neutron mass m n kg 9396 MeV/c hydrogen atom mass m H kg Electron magnetic moment J/T ev/t Proton magnetic moment p J/T ev/t Trigonometric identities sin cos 1 cos cos cos cos sin sin sin cos cos( ) coscossinsin sin( ) sincoscos sin sin( t ) sin( t ) sin( t ) B Bcos C ( here is the external angle) e o Waves, Superposition 1 k f T v f f k General relation for and :, Two sources: max 1 +, min 1 Two sources, same 1 : 41 cos ( /) where / nterference: Slits, holes, etc Farfield pathlength difference: r1r dsin Phase difference: dsin d d y if small L Principal maxima: dsinmax m m0,1,, N slit: Single slit: Single slit: sin( N / ) N 1 where d sin / sin( / ) a a sin a sin m with m 1,, 3 min a asin a a y L sin( / ) / 1 0 with a sin / slit: 0 / a or hole: 0 1 / D c pprox grating resolution: 1 Nm Quantum laws, facts UNVERSL: p k h/ Ehf Light: Ehf hc/ pc Slow particle: KE mv / p / m h /m Photoelectric effect: KEmax evstop h f UNVERSL: xp Et x *( x) ( x) ( x) b P ( x) dx, a xb ab a (Slow) particle in fixed potential U: ( xt, ) ( xt, ) U( x) ( x) i m x t 1 Fall 01

Physics 14 Common Formulae Quantum stationary states (energy eigenstates): (,) xt () xe it where E ( x) U( x) ( x) ( x) E( x) m x n 1D box: n L where n 1, n n ( x) sin x for 0 xl L L n h En n En m L

definite p: ikx ( t) (,) xt e with k /m T Ge KL Tunneling m m K U 0 E U 0 E h ngular momentum and magnetism Orbital: Lz m where m 0, 1,, L ( 1) where 0,1,, Spin: Sz where ms where m 1 s Magnetic

20 Physics 14 Common Formulae Quantum stationary states (energy eigenstates): (,) xt () xe it where E ( x) U( x) ( x) ( x) E( x) m x n 1D box: n L where n 1, n n ( x) sin x for 0 xl L L n h En n En m L 8mL Box, 3D: 1 (*last part*) (,, ) sin n xyz x sin n y sin n z abc a b c h n1 n n3 En ( 1, n, n3) ( ) 8m a b c Simple Harmonic Oscillator (SHO): 1 En n where n 0,1, k m Free slow particle with definite p: ikx ( t) (,) xt e with k /m T Ge KL Tunneling m m K U 0 E U 0 E h ngular momentum and magnetism Orbital: Lz m where m 0, 1,, L ( 1) where 0,1,, Spin: Sz where ms where m 1 s Magnetic energy: U B dbz e Force: Fz z where z Sz dz m tomic orbital filling E E G 16 1 U0 U0 e Hlike atom Ze potential Ur () r 1 ( Ze) 1 1 me Z En 4 a n (4 ) n 4 o o o Z ev n Ground state: 1 r/ ao 1s (, r, ) a e Radial density for sstate: Pr () dr 4 r () r dr Form of n, l, m eigenstate: (, r, ) R () r Y (, ) Y 00 nm n m 3 o 1 3, Y10 cos, 4 4 Y 11 3 sin 8 i e Fall 01

Physics 214 Midterm Exam Spring Last Name: First Name NetID Discussion Section: Discussion TA Name:

Physics 214 Midterm Exam Spring 215 Last Name: First Name NetID Discussion Section: Discussion TA Name: Instructions Turn off your cell phone and put it away. Keep your calculator on your own desk. Calculators