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Physics 7C - Fall 2002 - Midterm 2

Physics 7C - Fall 2002 - Midterm 2 - Page 1 of 5 Physics 7C...

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Unformatted text preview: Page 1 of 5 Physics 7C Section 1 Fall Semester 2002 Second Midterm November 4, 2001 (6:15-7:45 pm) Instructions , , ,. 1. This is a closed book exam. You are allowed to bring along only pens, pencils, scientific calculator, and blue books. _ 2. Write your name, Discussion Section # and SIM on the top of all materials you intend to hand in and want to be graded. 3. Read all questions carefully: before attempting them. Questions do not carry egual points. Try the questions you find easier first. Partially credits will be given for equations only if you can indicate how they can be used to solve the problem. 4. While cleanliness and legibility of your hand-writing will not get yOur extra credit, they will help to make Sure that your answers get the credit they deserve. In case you make mistakes be sure to cross them out so they will not be mistaken as your answer. It helps to underline your final answers. Always give yOur answers in the proper units. You are provided with the following constants for yOur reference. There is no guarantee that they will all be required in solving the problems. R.“— ental Constants (Emma-1 .33? Speed Of light in vacuum _c I . .300X10“ m/s Gravitational constant _ G ' 6.67 X 10'11 N-mzilrg2 Avoga'dro’s number NA _ 6.02 X 1023 11101—1 Gas constant . . R _ . ' 8.315 Jlmovi = 1.99 cal/mol-K _ = 0.082 aim-lit - Boltzmann’s constant k 1.38 X 10”23 ”Ker/11101 K Charge on electron e 1.60 x 10‘“ C Stefan-Boltzmann constant 0' 5.67 X 10‘3W/m2-K‘ Permittiwty of tree space so = (lfczpn) 8.85 x 10‘12 (Ez/N-m2 Permeability of free space no 41: x 10—7 T—m/A - Planck’s constant I: 6.63 x 10‘3‘1-5 - ' Electron rest mass me 9.11 X 10—31 kg = 0 000549 11 p t _ = 0.511l‘lvilt.‘.V,/C2 to on rest mass mp 1.6726 X 10‘2"r kg = 1.00728 1.1- N = 938.3 lfle‘i’j’c2 eutron rest mass mll . 1.6749 x 10‘” kg = 1.008665 11 - A1 ' ‘ - = 9139.6REV/"r:2 ornrc mass unit (1 u) 1.6605 X 10—27 kg = 931.5 MeV/c" The binding energy of the electron in the n=1 level in the hydrogen atom=13.6eV. Page 2 of 5 You are given the following eguations for your reference. There is no guarantee that you will need them in solving the problems. 1. Intensity of F raunhofer diffraction of light of wavelength A by a single slit ofwidth a: I=L,[sinou'ot]2 where awnasinflr‘l ( 2. Intensity of Fraunhofer diffraction of light of wavelength A by two slits of same width a separated by distance d: I=Io[sino/a]200525 where CFMSinGIK and B=rtdsin8l7t 3. Intensity of Frau nhofer diffraction of light of wavelength A by a grating of N lines of same width separated by distance d: 1:10[sinl‘siBr'sinB]2 where B=rcdsin9m 4. Resolving power of a diffraction grating of N lines=NAl=Nm where m=order of diffraction. 5. Rayleigh Criterion: two images are just resolvable by a circular lens of diameter D when they are separated by the angle 6:].22MD. 6. Brewster angle 61, between two media of refractive indexes n; and n1: tan 8p=ngfn1. 7. Plan ck’s Radiation Law for a Black-Body Radiator: 2mc21‘5 8mm _] r(t.:r)= 8. Wien’s Displacement Law: 13t1JT=2.90:rIO'3 m.K where T is the temperature and ELF =the wavelength where the intensity of emission is maximum. 9. Planck’s quantum hypothesis: E=hf. 10. De Broglie’s wave-matter duality hypothesis: p=hf7t. 1 1. Rydberg series for emission from the hydrogen atom: %= R[;2—~17] here R=I .0974)¢:107r m"1 and n> m 2 I are both integers. m n 12. Bohr’s model of Hydrogen atom: electron can be stable in stationary orbits with angular momentum quantized into: L=mvr=nhf2n 13. Energy levels of electron in Bohr’s model of hydrogen atom: 11" _ _ me4 i] '9 Seghz- n2 2 l4. Bohr radius of the electron in the n==I energy level :03 = h E; = 0.529x10_m In. more 15. The Kinetic Energy K of the photoelectrons ejected by photons of frequency (f) from a metal of work fimction W is given by: K=hf—W. - - 16. Heisenberg’s Uncertainty Principles: Apr 2 h and AEAI 2 h 17. The time dependent Schrodinger Equation: 18. The time independent Schrodinger Equation: 2 {vi—472 +V(rJ]w(r) = Err/(r) m 19. Probability of tunneling through a barrier of width L and of height V0 T=3XP[-2GL] where G = W—g—j Page 3 of 5 Intensity, Is vs. 9 Figure 1(a) m = l m = 0 ' m = I . - Figure l (b) The difliaction pattern shown in Fig. 1(a) was obtained by shining monochromatic light of wavelength l=600nm normally on an array of narrow slits of identical width and equally spaced from each other along a line perpendicular to the slits. (a) (5 Points) Mare being illuminated by the light? Give the reasons behind younanswer. (h) (5 Points) Estimate the width of each slit? (c) (5 Points) Estimate the separation between the slits? The same light is now incident normally on another array of equally spaced lines forming a difi‘raction grating. The resultant diffraction pattern is shown in Fig. 1(b). rn denotes the order of the diffraction peak. ((1) (5 Points) How many lines of the grating are being illuminated by the light? Give the reasons behind your answer. (e) (5 Points) Suppose the angular separation between the m=0 and m=l diffraction peaks is 20°, what is the separation between the lines in the grating? (I) (5 Points) Suppose the light beam is replaced by an electron beam'and the kinetic energy of the electrons is equal to 1 eV and the same diffraction pattern in Fig. 1 (b) is obtained. What should be the separation between the lines in the grating? Question 2 I 25 Points! Assume that the sun can be approximated as a black-body radiator. (a) (3 Points) Ifthe wavelength at which the intensity of the sun’s radiation is maximum is 500 mn. Estimate the temperature of the surface of the sun. (b) (15 Points) Use the Planck’s radiation law and the Wien’s displacement law to estimate the value of the Planck’s constant h, {Hint the transcendental equation e‘y=[ l-(y/ 5)] has two roots : y=0 and y~5] (c) (7 Points) In a photoelectric experiment shown in the following figure it was found that the photocurrent produced by the metal target P, collected by the electrode C and read by the ammeter A is reduced to zero when the potential applied by the battery is equal to (—Vo)_ The value of this stopping potential depends on the wavelength A of the Page 4 of 5 incident radiation. The value of V0 was found to be 1.4 and 2.254 Volts for l=400 and 550 mm respectively. What are the values of theflwork fim ' n W of the metal tar at and the Planck’s constant 11 determined from this experiment? Light ._ SOIJIOG Question 3 (total: 20 Points! When an electron is removed from a He atom a positively charged l-Ie+ ion is formed._In this ion there is only one electron moving around a doubly charged He nucleus so it is very similar to the hydrogen atom except for the fact that the nuclear charge is —2e (where e is the magnitude of the charge of the electron). (a) (5 Points) Derive the muation which gives the values of [m for the series of emission lines which corresponds to the Balmer series in thehydrowtom. {hint the Coulomb potential between two charges Q and q separated by a distance r is equal to (l f4rteo)(qur)} (b) (5 Points) What is the value of the corresponding Bohr radius in the He+ ion? ((3) (5 Points) What is the potential energy of the electron in the n=2 orbit of the He+ ion? (d) (5 Points) What is the kinetic energy of the electron in the n=2 orbit of the He+ ion? Question 4 (total: 25 Points! The following figure shows a one-dimensional potential barrier of height Vo=2 eV and width L=1.5 nm. Potential Energy V Page-5 of 5 A particle with mass m equal to that of the electron and kinetic energy E equal to 0.5 eV is incident on the barrier from the lefi. (a) (5 Points) If we divide the whole space into three regions I, II and III. Write down (you don’t have to derive it if you know the answer) the physically acceptable solution to the time-dependent Schrodinger equation in all three regions. (Represent the nonnalization constant as C). (b) (7 points) Write down (you don’t have to derive it if you know the answer) the physically acceptable solutions to the time-independent Schrodinger equation in regions I and III. (Represent the normalization constant as A,B,etc). (c) (3 Points) What are the de Broglie wavelengths for this particle in the regions I and (d) (5 Points) Write down (you don’t have to derive it if you know the answer) a physically acceptable solution to the time-independent Schrodinger equation in region II. (Again represent the normalization constant as F). (e) (5 Points) What is the probability of this particle tunneling through the barrier? ----------------- END OF QUESTIONS---—--------------- ...
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