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Physics 7C Section 1 Fall Semester 2002 Second Midterm
November 4, 2001 (6:157:45 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 handwriting 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 (Emma1 .33?
Speed Of light in vacuum _c I . .300X10“ m/s
Gravitational constant _ G ' 6.67 X 10'11 Nmzilrg2
Avoga'dro’s number NA _ 6.02 X 1023 11101—1
Gas constant . . R _ . ' 8.315 Jlmovi = 1.99 cal/molK
_ = 0.082 aimlit  Boltzmann’s constant k 1.38 X 10”23 ”Ker/11101 K
Charge on electron e 1.60 x 10‘“ C
StefanBoltzmann constant 0' 5.67 X 10‘3W/m2K‘
Permittiwty of tree space so = (lfczpn) 8.85 x 10‘12 (Ez/Nm2
Permeability of free space no 41: x 10—7 T—m/A 
Planck’s constant I: 6.63 x 10‘3‘15  '
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 BlackBody 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 wavematter 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 blackbody 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 lIe+ 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 onedimensional potential barrier of height Vo=2 eV and width L=1.5 nm.
Potential Energy V Page5 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 timedependent 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 timeindependent 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 timeindependent 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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