123 spring 2015 solutions

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Physics 123, Spring 15 Exam 3, page 1

I. Lab Questions

1. [4 pts.] A compound microscope of magnification +200 is to be constructed with two lenses. The objectives focal length is 8 mm. If the tube length (L) is 16 cm, what must be the focal length of the eyepiece?

(a) 25 mm (b) 10 cm (c) 25 cm (d) 10 mm (e) 150 mm

2. [4 pts.] In the ray diagram below, which ray(s) are drawn incorrectly?

CA

B

A

B

C

(a) A (b) B (c) C (d) A and B (e) B and C

3. [4 pts.] An experiment is set up with an optical bench as shown below. Rays are drawn to locate the image of the lamp made by the convex side of the three sided mirror. Which of the labeled dots is closest to the image position?

(a) A (b) B (c) C (d) D (e) E

SourceWhite Light

Slit plate

3sided mirrorLamp locaon

Paper

A

B

C

D

E

fo fo



Grating Screen

CenterlineIncoming

Light

of Slit Screen

Y Magnified View

Light P Q

II. Diffraction & Interference, A. Light passes through a single slit of width a, and strikes a screen, producing a varying intensity. At a certain point (Y) on the screen, light from a point at one side of the slit (P) and light from a point at the other side (Q) are out of phase by exactly one full wavelength so that they add constructively. 4. (4 pts) Which choice best represents the intensity on the screen at point Y for this situation?

(Assume that the maximum central intensity for this case is 1 unit).

A. Zero B. 22

C.

232

D. 22

3

E. None of these / need more information

II. Diffaction & Interference, B. A green laser ( = 540 nm) illuminates a diffraction grating, producing an interference pattern on a distant screen. The first principal maximum is located at an angle of 18 from the centerline between grating and screen. 5. (4 pts) Which choice best represents the

number of slits per mm for this grating? 920 750 570 440 290 A B C D E II. Diffraction & Interference, C. In a multi-slit interference experiment, laser light passes

through 7 equally-spaced, very thin slits and strikes a distant screen. 6. (4 pts) Representing the light passing through the slits with one phasor for each slit, how

many distinct phasor patterns could produce dark spots on the screen?

A. Seven B. Six C. Five D. Four E. None of these



III. Quantum Theory. Laser light of frequency f = 7.501014 Hz beams into a thin cloud of electrons. The reflected light enters a series of detectors that allow experimenters to determine the position of the electrons. 7. (4 pts) Which choice best represents the momentum of a photon of the incoming light?

Zero 8.810-49 kgm/s 2.610-40 kgm/s 1.710-27 kgm/s 5.010-19 kgm/s A B C D E Assume that a series of measurements produces an uncertainty in the positions of these electrons of approximately x = 610-7 m. 8. (4 pts) Which choice best represents the minimum uncertainty in the momentum of one of

these electrons after its position is measured?

910-26 kgm/s 910-29 kgm/s 310-31 kgm/s 310-36 kgm/s Zero A B C D E Suppose that you shine light of frequency f = 1.501014 Hz onto to a certain metal surface in order to eject electrons of maximum kinetic energy 1.50 eV. 9. (4 pts) Which choice best represents the work function of this metal?

A. 2.12 eV. B. 0.88 eV. C. -0.88 eV. D. 0.62 eV. E. None of these, or not possible.



III. Quantum Theory, continued. 10. (4 pts) Which choice best represents the energy released as an electron jumps from the n = 4

level to the n = 2 level in the Bohr model for the Hydrogen atom?

A. 4.25 eV. B. 3.40 eV. C. 2.55 eV. D. 1.89 eV. E. None of these / not possible.

11. (4 pts) How many electrons can fit into the n = 4 level of an atom? Assume that each

electron must be described using only these four quantum numbers: n, , m, and ms.

A. 4 B. 12 C. 18 D. 32 E. None of these



IV. Special Relativity Two identical ships move in opposite directions at constant speeds 0.78c (ship A) and 0.62c (ship B). Each speed is measured by an Earth observer. 12. (4 pts) Which choice best represents the speed of ship B measured by an observer on ship A?

0.94 c 0.79 c 0.31 c 0.16 c 0.11 c A B C D E The length of ship B is 18 m as measured by an observer on Earth. 13. (4 pts) Which choice best represents the proper length of ship B?

11 m 14 m 18 m 23 m 27 m A B C D E The captain of ship B transmits light signals at intervals of 3.0 s by his ships clock. 14. (4 pts) Which choice best represents the time interval between signal transmissions measured

by an observer on Earth?

4.8 s 3.8 s 3.0 s 2.4 s 1.9 s A B C D E

v = 0.78c A Earth

v = 0.62c B



V. Miscellaneous Modern Physics: A. Special Relativity

15. (8 pts) The proton (rest) mass is 271.67 10 kg. If this proton moves through empty space with kinetic energy 104.5 10 J, what is the magnitude of its momentum? Show your work.

For relativistic protons, 2K mc , so 210 27 84.5 10 J (1.67 10 kg) 2.998 10 m/s 2.998 3.00

The speed is found from 2

2

28

2 2 21 1 11 1 2.826 10 m/s

1 vc

v v cc

The proton momentum is then 27 8 182.998(1.67 10 )(2.826 10 ) 1.42 10 kg m/sp mv Note: a decent approximation is 3, v c , but should have 3 sig figs for full credit. B. Phasors for Interference & Diffraction Laser light passes through 4 very narrow, closely-packed, equally spaced, identical slits, then strikes a large, distant screen to form an interference pattern. 16. (6 pts) Sketch a phasor diagram to represent a point on the screen where there is a secondary

maximum. 4 equal phasors, each rotated 135 degrees from the previous phasor, add to form the red vector sum (amplitude). Only the diagram is needed, orientation not important. C. Wave Functions and Probabilities

For a particle in a box, the wavefunctions are: 2( ) sin , 1,2,3...n xL Lx n 17. (5 pts) Sketch the wave function for the value of n given below: Using only the upper half of these curves is probably more correct. 18. (5 pts) Compute the probability of finding the particle in its first excited state within a very

small length x = 0.003L centered on x = L/3. Show your work. For small volumes (length in this case) no need to integrate, just multiply:

2 2 22 2 2( ) sin sin 0.003 0.00453

n x Lx x x LL L L L

Note: n = 2 is the first excited state, and all the Ls cancel. The answer has no units.

n = 1

x L 0

n = 4

x L 0

n = 100

x L 0

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Physics 123A&C, Spring 2015 Exam 3 WO-UWA123X152T-E3(PCQ)Sol.doc

VI. This page is composed of two unrelated parts, A and B. Treat each part as a classical one-dimensional problem (i.e., do not use quantum mechanics). A. [13 pts] A ball is dropped a few meters above the ground. It bounces

elastically from the ground and comes back up to the same height before moving back downward. Ignore air resistance and assume the ball is able to bounce and up and down in this fashion forever, always reaching the same height. Two regions (A and B) and two points (1 and 2) are shown at right. The size of region B is larger than that of region A. After the ball has been bouncing for a long time, its precise position is measured at a random time. i. Is the probability PA of the center of the ball being measured in

region A greater than, less than, or equal to the probability PB of its center being measured in region B? If there is not enough information, state so explicitly. Explain. The speed of the ball is greater at every point in region B than in region A. However, the height of region B is larger than region A, so the relative amounts of time the ball spends in the two regions cannot be compared. Since the amounts of time cannot be compared, the probabilities cannot be compared. Thus, there is not enough information to answer.

ii. Is the probability density 1 at point 1 greater than, less than, or equal to the probability density 1 at point 2? If there is not enough information, state so explicitly. Explain. The amount of time dt that the ball spends in an infinitesimal region of height dy centered at point 1 is greater than that at point 2. Therefore, the probability dP = dt/T that the ball is in the infinitesimal region centered at point 1 is greater than that at point 2. Since the probability dP is greater near point 1, the probability density dP/dy is also greater. Thus, 1 > 2 .

B. [7 pts] A small grain of sand is placed in a random location in a long box. The box is composed of multiple regions of different lengths. Some of these regions are shown at right. Consider regions K and U (not shown): Region K has length LK. It is known that the probability of the

grain of sand being located in region K is PK. Region U has length LU. What is the probability PU that the grain of sand is located in region U? Write your answer only in terms of the given variables (i.e, only in terms of LK, PK, and/or LU). Assume the probability density is constant throughout the box. Explain.

Since the probability density (i.e., the probability per unit length) for the entire box is uniform, it can be calculated by dividing the probability in region K by the length of region K: = PK/LK. The probability in region U can be found by multiplying the probability density and the length of region U: (PK/LK)LU = PKLU/LK.

Region A

Region B

Point 1

Point 2

}

}Ball bounces elastically

and returns to same height

Grain of sand located somewhere in long box

Example region{

123 spring 2015 solutions

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