Physics 4C Spring 2016 Final Exam

Name:

June 20, 2016

Please show your work! Answers are not complete without clear reasoning. When askedfor an expression, you must give your answer in terms of the variables given in the questionand/or fundamental constants.

Answer as many questions as you can, in any order. Calculators are allowed. Books andnotes are not allowed. Use any blank space to answer questions, but please make sure it isclear which question your answer refers to.

g = 9.8 ms−2

c = 3.00× 108 m/s

atmospheric pressure P0 = 1.013× 105 Pa

density of water ρw = 1000 kg/m3

ρair = 1.20 kg m−3 (sea level, 20◦C)

Fahrenheit to Celsius:

([◦F] − 32) ÷ 1.8 =[◦C]

Celsius to Fahrenheit:

([◦C] × 1.8) + 32 =[◦F]

kB = 1.38× 10−23 J/K

NA = 6.022× 1023

R = 8.314 J mol−1 K−1

mp = 1.67× 10−27 kg

σ = 5.6696× 10−8 W m−2 K−4

1 cal = 4.186 J

I0 = 1.00× 10−12 W m−2

Trigonometric Identities

sin2 θ + cos2 θ = 1

sin(α± β) = sinα cos β ± cosα sin β

cos(α± β) = cosα cos β ∓ sinα sin β

sinα + sin β = 2 cos(α−β

2

)sin(α+β

2

)cosα + cos β = 2 cos

(α−β

2

)cos(α+β

2

)sin(θ + π

2

)= cos θ

cos(θ + π

2

)= − sin θ

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1. A small spherical balloon is submerged in water. Assume the water temperature doesnot change with depth.

(a) Derive an expression for the buoyant force on the balloon, as a function of thedepth h below the surface, the volume Vi of the balloon at the surface, the pressureP0 at the surface, and the density ρw of the water. [5 pts]

(b) At what depth is the buoyant force one quarter the surface value (the value whenthe balloon is just fully submerged)? [4 pts]

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2. A hypodermic syringe contains water. The barrel of the syringe has a cross-sectionalarea A, and the needle has a cross-sectional area a. In the absence of a force on theplunger, the pressure everywhere is P0. A force F acts on the plunger, making medicinesquirt horizontally from the needle. Determine the speed of the water as it leaves theneedle’s tip. [6 pts]

Problems 443

4.00 m 3 1.50 m. Assume the density of the air to be constant at 1.20 kg/m3. The air inside the building is at atmospheric pressure. What is the total force exerted by air on the windowpane? (b) What If? If a second skyscraper is built nearby, the airspeed can be espe-cially high where wind passes through the narrow sepa-ration between the buildings. Solve part (a) again with a wind speed of 22.4 m/s, twice as high.

55. A hypodermic syringe contains a medicine with the density of water (Fig. P14.55). The barrel of the syringe has a cross-sectional area A 5 2.50 3 1025 m2, and the needle has a cross-sectional area a 5 1.00 3 1028 m2. In the absence of a force on the plunger, the pressure everywhere is 1.00 atm. A force F

S of magnitude 2.00 N

acts on the plunger, making medicine squirt hori-zontally from the needle. Determine the speed of the medicine as it leaves the needle’s tip.

A

a

FS vS

Figure P14.55

Additional Problems

56. Decades ago, it was thought that huge herbivorous dinosaurs such as Apatosaurus and Brachiosaurus habit-ually walked on the bottom of lakes, extending their long necks up to the surface to breathe. Brachiosaurus had its nostrils on the top of its head. In 1977, Knut Schmidt-Nielsen pointed out that breathing would be too much work for such a creature. For a simple model, consider a sample consisting of 10.0 L of air at absolute pressure 2.00 atm, with density 2.40 kg/m3, located at the surface of a freshwater lake. Find the work required to transport it to a depth of 10.3 m, with its tempera-ture, volume, and pressure remaining constant. This energy investment is greater than the energy that can be obtained by metabolism of food with the oxygen in that quantity of air.

57. (a) Calculate the absolute pressure at an ocean depth of 1 000 m. Assume the density of seawater is 1 030 kg/m3 and the air above exerts a pressure of 101.3 kPa. (b) At this depth, what is the buoyant force on a spherical submarine having a diameter of 5.00 m?

58. In about 1657, Otto von Guericke, inventor of the air pump, evacuated a sphere made of two brass hemi-spheres (Fig. P14.58). Two teams of eight horses each could pull the hemispheres apart only on some trials and then “with greatest difficulty,” with the resulting

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one of the droplets to determine the speed at which the water leaves the ground. (b) What If? Model the rising stream as an ideal fluid in streamline flow. Use Bernoulli’s equation to determine the speed of the water as it leaves ground level. (c) How does the answer from part (a) compare with the answer from part (b)? (d) What is the pressure (above atmospheric) in the heated underground chamber if its depth is 175 m? Assume the chamber is large compared with the geyser’s vent.

Section 14.7 Other Applications of Fluid Dynamics

51. An airplane is cruising at altitude 10 km. The pressure outside the craft is 0.287 atm; within the passenger compartment, the pressure is 1.00 atm and the temper-ature is 208C. A small leak occurs in one of the window seals in the passenger compartment. Model the air as an ideal fluid to estimate the speed of the airstream flowing through the leak.

52. An airplane has a mass of 1.60 3 104 kg, and each wing has an area of 40.0 m2. During level flight, the pressure on the lower wing surface is 7.00 3 104 Pa. (a) Suppose the lift on the airplane were due to a pressure differ-ence alone. Determine the pressure on the upper wing surface. (b) More realistically, a significant part of the lift is due to deflection of air downward by the wing. Does the inclusion of this force mean that the pressure in part (a) is higher or lower? Explain.

53. A siphon is used to drain water from a tank as illus-trated in Figure P14.53. Assume steady flow without friction. (a) If h 5 1.00 m, find the speed of outflow at the end of the siphon. (b) What If? What is the limita-tion on the height of the top of the siphon above the end of the siphon? Note: For the flow of the liquid to be continuous, its pressure must not drop below its vapor pressure. Assume the water is at 20.08C, at which the vapor pressure is 2.3 kPa.

h

y

!

vS

Figure P14.53

54. The Bernoulli effect can have important consequences for the design of buildings. For example, wind can blow around a skyscraper at remarkably high speed, creating low pressure. The higher atmospheric pres-sure in the still air inside the buildings can cause win-dows to pop out. As originally constructed, the John Hancock Building in Boston popped windowpanes that fell many stories to the sidewalk below. (a) Sup-pose a horizontal wind blows with a speed of 11.2 m/s outside a large pane of plate glass with dimensions

Q/C

P0P

RFS

FS

Figure P14.58

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water ice aluminum

Specific heat capacity 4186 J/(kg ◦C) 2050 J/(kg ◦C) 900 J/(kg ◦C)

Latent heat of fusion 3.33× 105 J/kg same as water 3.97× 105 J/kg

Latent heat of vaporization 2.26× 106 J/kg same as water 1.14× 107 J/kg

3. To answer this question you may need some of the information in the table above. A50.0-g ice cube at −5◦C is placed in 800 g of water at 25.0◦C in an aluminum container,of mass 30 g also at 25.0◦C. The container and contents are thermally isolated fromtheir surroundings. What is the equilibrium temperature of the mixture? [8 pts]

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4. An ideal monatomic gas initially at Pi, Vi, and Ti at point A is taken through a cycleas shown.

(a) What is the change in internal energy of the gas over the complete cycle? [1 pt]

(b) Find an expression for the net work done on the gas per cycle. [4 pts]

(c) This cycle is used as a heat engine. Find Qh, the heat transferred into the gas inthe cycle. [5 pts]

(d) Find Qc the heat transferred out of the gas in the cycle. [5 pts]

(e) Find the efficiency of the heat engine in terms of Pi, Vi , Ti, and constants. [3 pts]

(f) If this cycle were run in reverse as a refrigerator, what would its coefficient ofperformance be? [2 pts]

620 Chapter 20 The First Law of Thermodynamics

40. In Figure P20.40, the change in internal energy of a gas that is taken from A to C along the blue path is 1800 J. The work done on the gas along the red path ABC is 2500 J. (a)  How much energy must be added to the system by heat as it goes from A through B to C ? (b) If the pressure at point A is five times that of point C, what is the work done on the system in going from C to D? (c) What is the energy exchanged with the surroundings by heat as the gas goes from C to A along the green path? (d) If the change in internal energy in going from point D to point A is 1500 J, how much energy must be added to the system by heat as it goes from point C to point D?

41. An ideal gas initially at Pi , Vi , and Ti is taken through a cycle as shown in Figure P20.41. (a) Find the net work done on the gas per cycle for 1.00 mol of gas initially at 0°C. (b)  What is the net energy added by heat to the gas per cycle?

42. An ideal gas initially at Pi , Vi , and Ti is taken through a cycle as shown in Figure P20.41. (a) Find the net work done on the gas per cycle. (b) What is the net energy added by heat to the system per cycle?

Section 20.7 Energy Transfer Mechanisms in Thermal Processes 43. A glass windowpane in a home is 0.620 cm thick and

has dimensions of 1.00 m 3 2.00 m. On a certain day, the temperature of the interior surface of the glass is 25.0°C and the exterior surface temperature is 0°C. (a) What is the rate at which energy is transferred by heat through the glass? (b) How much energy is trans-ferred through the window in one day, assuming the temperatures on the surfaces remain constant?

44. A concrete slab is 12.0 cm thick and has an area of 5.00 m2. Electric heating coils are installed under the slab to melt the ice on the surface in the winter months. What minimum power must be supplied to the coils to maintain a temperature difference of 20.0°C between the bottom of the slab and its surface? Assume all the energy transferred is through the slab.

45. A student is trying to decide what to wear. His bed-room is at 20.0°C. His skin temperature is 35.0°C. The area of his exposed skin is 1.50 m2. People all over the world have skin that is dark in the infrared, with emis-sivity about 0.900. Find the net energy transfer from his body by radiation in 10.0 min.

46. The surface of the Sun has a temperature of about 5 800 K. The radius of the Sun is 6.96 3 108 m. Calcu-late the total energy radiated by the Sun each second. Assume the emissivity of the Sun is 0.986.

B C

DA

P

Pi

3Pi

Vi 3ViV

Figure P20.41 Problems 41 and 42.

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31. Consider the cyclic process depicted in Figure P20.30. If Q is negative for the process BC and DE int is nega-tive for the process CA, what are the signs of Q , W, and DE int that are associated with each of the three processes?

32. Why is the following situation impossible? An ideal gas undergoes a process with the following parameters: Q 5 10.0 J, W 5 12.0 J, and DT 5 22.00°C.

33. A thermodynamic system undergoes a process in which its internal energy decreases by 500 J. Over the same time interval, 220 J of work is done on the system. Find the energy transferred from it by heat.

34. A sample of an ideal gas goes through the process shown in Figure P20.34. From A to B, the process is adiabatic; from B to C, it is isobaric with 345 kJ of energy entering the system by heat; from C to D, the process is isothermal; and from D to A, it is isobaric with 371 kJ of energy leaving the system by heat. Deter-mine the difference in internal energy E int ,B 2 E int ,A.

1

3

P (atm)

0.09 0.2 0.4 1.2

A

CB

D

V (m3)

Figure P20.34

Section 20.6 Some Applications of the First Law of Thermodynamics 35. A 2.00-mol sample of helium gas initially at 300 K, and

0.400 atm is compressed isothermally to 1.20 atm. Not-ing that the helium behaves as an ideal gas, find (a) the final volume of the gas, (b) the work done on the gas, and (c) the energy transferred by heat.

36. (a) How much work is done on the steam when 1.00 mol of water at 100°C boils and becomes 1.00 mol of steam at 100°C at 1.00 atm pressure? Assume the steam to behave as an ideal gas. (b) Determine the change in internal energy of the system of the water and steam as the water vaporizes.

37. An ideal gas initially at 300 K undergoes an isobaric expansion at 2.50 kPa. If the volume increases from 1.00 m3 to 3.00 m3 and 12.5 kJ is transferred to the gas by heat, what are (a) the change in its internal energy and (b) its final temperature?

38. One mole of an ideal gas does 3 000 J of work on its surroundings as it expands isothermally to a final pressure of 1.00 atm and volume of 25.0 L. Determine (a) the initial volume and (b) the temperature of the gas.

39. A 1.00-kg block of aluminum is warmed at atmospheric pressure so that its temperature increases from 22.0°C to 40.0°C. Find (a) the work done on the aluminum, (b) the energy added to it by heat, and (c) the change in its internal energy.

W

M

M

W

P

VCD

BA

Figure P20.40

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5. A block with a speaker bolted to it is connected to a spring having spring constant kand oscillates as shown on a frictionless table. The equilibrium point of the spring isa position x = 0. The total mass of the block and speaker is m, and the amplitude ofthe block-speaker’s motion is A. The speaker emits sound waves of frequency f . Youmay use v as the speed of sound in air.

528 Chapter 17 Sound Waves

amplitude of this unit’s motion is 0.500 m. The speaker emits sound waves of frequency 440 Hz. Deter-mine (a) the highest and (b) the lowest frequencies heard by the person to the right of the speaker. (c)If the maximum sound level heard by the person is 60.0 dB when the speaker is at its closest distance d 5 1.00  m from him, what is the minimum sound level heard by the observer?

m

d

k

Figure P17.41 Problems 41 and 42.

42. Review. A block with a speaker bolted to it is connected to a spring having spring constant k and oscillates as shown in Figure P17.41. The total mass of the block and speaker is m, and the amplitude of this unit’s motion is A. The speaker emits sound waves of frequency f. Determine (a) the highest and (b) the lowest frequen-cies heard by the person to the right of the speaker. (c) If the maximum sound level heard by the person is b when the speaker is at its closest distance d from him, what is the minimum sound level heard by the observer?

43. Expectant parents are thrilled to hear their unborn baby’s heartbeat, revealed by an ultrasonic detector that produces beeps of audible sound in synchroniza-tion with the fetal heartbeat. Suppose the fetus’s ven-tricular wall moves in simple harmonic motion with an amplitude of 1.80 mm and a frequency of 115 beats per minute. (a) Find the maximum linear speed of the heart wall. Suppose a source mounted on the detector in contact with the mother’s abdomen produces sound at 2 000 000.0 Hz, which travels through tissue at 1.50 km/s. (b) Find the maximum change in frequency between the sound that arrives at the wall of the baby’s heart and the sound emitted by the source. (c) Find the maximum change in frequency between the reflected sound received by the detector and that emitted by the source.

44. Why is the following situation impossible? At the Summer Olympics, an athlete runs at a constant speed down a straight track while a spectator near the edge of the track blows a note on a horn with a fixed frequency. When the athlete passes the horn, she hears the fre-quency of the horn fall by the musical interval called a minor third. That is, the frequency she hears drops to five-sixths its original value.

45. Standing at a crosswalk, you hear a frequency of 560 Hz from the siren of an approaching ambulance. After the ambulance passes, the observed frequency of

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with an identical mower the same distance away. This situation annoys you greatly because the total sound now has twice the loudness it had when only one neigh-bor was mowing.

Section 17.4 The Doppler Effect 37. An ambulance moving at 42 m/s sounds its siren whose

frequency is 450 Hz. A car is moving in the same direc-tion as the ambulance at 25 m/s. What frequency does a person in the car hear (a) as the ambulance approaches the car? (b) After the ambulance passes the car?

38. When high-energy charged particles move through a transparent medium with a speed greater than the speed of light in that medium, a shock wave, or bow wave, of light is produced. This phenomenon is called the Cerenkov effect. When a nuclear reactor is shielded by a large pool of water, Cerenkov radiation can be seen as a blue glow in the vicinity of the reactor core due to high-speed electrons moving through the water (Fig. 17.38). In a particular case, the Cerenkov radiation pro-duces a wave front with an apex half-angle of 53.08. Calculate the speed of the electrons in the water. The speed of light in water is 2.25 3 108 m/s.

39. A driver travels northbound on a highway at a speed of 25.0  m/s. A police car, traveling southbound at a speed of 40.0 m/s, approaches with its siren producing sound at a frequency of 2 500 Hz. (a) What frequency does the driver observe as the police car approaches? (b) What frequency does the driver detect after the police car passes him? (c) Repeat parts (a) and (b) for the case when the police car is behind the driver and travels northbound.

40. Submarine A travels horizontally at 11.0 m/s through ocean water. It emits a sonar signal of frequency f 5 5.27 3 103 Hz in the forward direction. Submarine B is in front of submarine A and traveling at 3.00 m/s rela-tive to the water in the same direction as submarine A. A crewman in submarine B uses his equipment to detect the sound waves (“pings”) from submarine A. We wish to determine what is heard by the crewman in submarine B. (a) An observer on which submarine detects a frequency f 9 as described by Equation 17.19? (b) In Equation 17.19, should the sign of vS be positive or negative? (c) In Equation 17.19, should the sign of vO be positive or negative? (d) In Equation 17.19, what speed of sound should be used? (e) Find the frequency of the sound detected by the crewman on submarine B.

41. Review. A block with a speaker bolted to it is con-nected to a spring having spring constant k 5 20.0 N/m and oscillates as shown in Figure P17.41. The total mass of the block and speaker is 5.00 kg, and the

Figure P17.38

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ent o

f Ene

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Phot

o Re

sear

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s, In

c.

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AMT

(a) Find an expression for the highest frequency heard by the person to the right ofthe speaker. [3 pts]

(b) Find an expression for the lowest frequency heard by the person to the right ofthe speaker. [3 pts]

(c) At what position of the block is the highest frequency heard by the observer? Atwhat position is the lowest frequency heard? Is the block moving in each case?Which way? [3pts]

(d) If the maximum sound level heard by the person is β when the speaker is at itsclosest distance d from him, find an expression for the minimum sound level heardby the observer. [4 pts]

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6. Two transverse sinusoidal waves combining in a medium are described by the wavefunctions

y1 = 4.00 sin 2π(x+ 0.600t) y2 = 4.00 sin 2π(x− 0.600t)

where x, y1, and y2 are in centimeters and t is in seconds. Determine the maximumtransverse position of an element of the medium at

(a) x = 0.125 cm, [3 pts]

(b) x = 0.750 cm. [3 pts]

(c) Find the three smallest values of x corresponding to antinodes. [3 pts]

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7. An object 10.0 cm tall is placed at the zero mark of a meterstick. A lens located atsome point on the meterstick creates an image of the object that is upright, 4.00 cmtall, and located at the 18.0-cm mark of the meterstick.

(a) Is the lens converging or diverging? [1 pt]

(b) Is the image real or virtual? [1 pt]

(c) At what point on the meterstick is the lens? [5 pts]

(d) What is the lens’s focal length? [3 pts]

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8. Suppose light strikes a single slit of width a at an angle β from the perpendiculardirection as shown in the diagram.

View of slit for (a) and (b): View of slit and screen for part (b):

1184 Chapter 38 Diffraction Patterns and Polarization

in Figure P38.10. Show that Equation 38.1, the condition for destructive interference, must be modified to read

sin udark 5 m l

a 2 sin b

m 5 61, 62, 63, . . .

11. A diffraction pattern is formed on a screen 120 cm away from a 0.400-mm-wide slit. Monochromatic 546.1-nm light is used. Calculate the fractional intensity I/Imax at a point on the screen 4.10 mm from the center of the principal maximum.

12. Coherent light of wavelength 501.5 nm is sent through two parallel slits in an opaque material. Each slit is 0.700 mm wide. Their centers are 2.80 mm apart. The light then falls on a semicylindrical screen, with its axis at the midline between the slits. We would like to describe the appearance of the pattern of light visible on the screen. (a) Find the direction for each two-slit interference maximum on the screen as an angle away from the bisector of the line joining the slits. (b) How many angles are there that represent two-slit interfer-ence maxima? (c) Find the direction for each single-slit interference minimum on the screen as an angle away from the bisector of the line joining the slits. (d) How many angles are there that represent single-slit interference minima? (e) How many of the angles in part (d) are identical to those in part (a)? (f) How many bright fringes are visible on the screen? (g) If the intensity of the central fringe is Imax, what is the inten-sity of the last fringe visible on the screen?

13. A beam of monochromatic light is incident on a single slit of width 0.600 mm. A diffraction pattern forms on a wall 1.30 m beyond the slit. The distance between the positions of zero intensity on both sides of the central maximum is 2.00 mm. Calculate the wavelength of the light.

Section 38.3 Resolution of Single-Slit and Circular Apertures

Note: In Problems 14, 19, 22, 23, and 67, you may use the Rayleigh criterion for the limiting angle of resolu-tion of an eye. The standard may be overly optimistic for human vision.

14. The pupil of a cat’s eye narrows to a vertical slit of width 0.500 mm in daylight. Assume the average wavelength of the light is 500 nm. What is the angular resolution for horizontally separated mice?

15. The angular resolution of a radio telescope is to be 0.100° when the incident waves have a wavelength of 3.00 mm. What minimum diameter is required for the telescope’s receiving dish?

16. A pinhole camera has a small circular aperture of diam-eter D. Light from distant objects passes through the aperture into an otherwise dark box, falling on a screen at the other end of the box. The aperture in a pinhole camera has diameter D 5 0.600 mm. Two

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Section 38.2 Diffraction Patterns from Narrow Slits 1. Light of wavelength 587.5 nm illuminates a slit of width

0.75 mm. (a) At what distance from the slit should a screen be placed if the first minimum in the diffraction pattern is to be 0.85 mm from the central maximum? (b) Calculate the width of the central maximum.

2. Helium–neon laser light (l 5 632.8 nm) is sent through a 0.300-mm-wide single slit. What is the width of the central maximum on a screen 1.00 m from the slit?

3. Sound with a frequency 650 Hz from a distant source passes through a doorway 1.10 m wide in a sound-absorbing wall. Find (a) the number and (b) the angu-lar directions of the diffraction minima at listening positions along a line parallel to the wall.

4. A horizontal laser beam of wavelength 632.8 nm has a circular cross section 2.00 mm in diameter. A rect-angular aperture is to be placed in the center of the beam so that when the light falls perpendicularly on a wall 4.50 m away, the central maximum fills a rectangle 110 mm wide and 6.00 mm high. The dimensions are measured between the minima bracketing the central maximum. Find the required (a) width and (b) height of the aperture. (c) Is the longer dimension of the cen-tral bright patch in the diffraction pattern horizontal or vertical? (d) Is the longer dimension of the aperture horizontal or vertical? (e)  Explain the relationship between these two rectangles, using a diagram.

5. Coherent microwaves of wavelength 5.00 cm enter a tall, narrow window in a building otherwise essentially opaque to the microwaves. If the window is 36.0 cm wide, what is the distance from the central maximum to the first-order minimum along a wall 6.50 m from the window?

6. Light of wavelength 540 nm passes through a slit of width 0.200 mm. (a) The width of the central maxi-mum on a screen is 8.10 mm. How far is the screen from the slit? (b)  Determine the width of the first bright fringe to the side of the central maximum.

7. A screen is placed 50.0 cm from a single slit, which is illuminated with light of wavelength 690 nm. If the dis-tance between the first and third minima in the dif-fraction pattern is 3.00 mm, what is the width of the slit?

8. A screen is placed a distance L from a single slit of width a, which is illuminated with light of wavelength l. Assume L .. a. If the distance between the minima for m 5 m1 and m 5 m2 in the diffraction pattern is Dy, what is the width of the slit?

9. Assume light of wavelength 650 nm passes through two slits 3.00 mm wide, with their centers 9.00 mm apart. Make a sketch of the combined diffraction and inter-ference pattern in the form of a graph of intensity versus f 5 (pa sin u)/l. You may use Figure 38.7 as a starting point.

10. What If? Suppose light strikes a single slit of width a at an angle b from the perpendicular direction as shown

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aub

Figure P38.10

1162 Chapter 38 Diffraction Patterns and Polarization

a Fraunhofer diffraction pattern. A bright fringe is observed along the axis at u 5 0, with alternating dark and bright fringes on each side of the central bright fringe. Until now, we have assumed slits are point sources of light. In this section, we abandon that assumption and see how the finite width of slits is the basis for under-standing Fraunhofer diffraction. We can explain some important features of this phenomenon by examining waves coming from various portions of the slit as shown in Figure 38.5. According to Huygens’s principle, each portion of the slit acts as a source of light waves. Hence, light from one portion of the slit can interfere with light from another portion, and the resultant light intensity on a viewing screen depends on the direction u. Based on this analysis, we recognize that a diffraction pattern is actually an interference pattern in which the different sources of light are different portions of the single slit! Therefore, the diffraction patterns we discuss in this chapter are applications of the waves in interference analysis model. To analyze the diffraction pattern, let’s divide the slit into two halves as shown in Figure 38.5. Keeping in mind that all the waves are in phase as they leave the slit, consider rays 1 and 3. As these two rays travel toward a viewing screen far to the right of the figure, ray 1 travels farther than ray 3 by an amount equal to the path difference (a/2) sin u, where a is the width of the slit. Similarly, the path difference between rays 2 and 4 is also (a/2) sin u, as is that between rays 3 and 5. If this path difference is exactly half a wavelength (corresponding to a phase difference of 180°), the pairs of waves cancel each other and destructive interference results. This cancel-lation occurs for any two rays that originate at points separated by half the slit width because the phase difference between two such points is 180°. Therefore, waves from the upper half of the slit interfere destructively with waves from the lower half when

a2

sin u 5l

2

or, if we consider waves at angle u both above the dashed line in Figure 38.5 and below,

sin u 5 6l

a

Dividing the slit into four equal parts and using similar reasoning, we find that the viewing screen is also dark when

sin u 5 62 l

a

Likewise, dividing the slit into six equal parts shows that darkness occurs on the screen when

sin u 5 63 l

a

Pitfall Prevention 38.1Diffraction Versus Diffraction Pattern Diffraction refers to the general behavior of waves spread-ing out as they pass through a slit. We used diffraction in explaining the existence of an interference pattern in Chapter 37. A diffraction pattern is actually a misnomer, but is deeply entrenched in the lan-guage of physics. The diffraction pattern seen on a screen when a single slit is illuminated is actually another interference pattern. The interference is between parts of the incident light illuminating dif-ferent regions of the slit.

Figure 38.4 (a) Geometry for analyzing the Fraunhofer diffrac-tion pattern of a single slit. (Draw-ing not to scale.) (b) Simulation of a single-slit Fraunhofer diffrac-tion pattern.

Slit

min

min

min

min

max

max

max

Incomingwave Viewing screen

u

The pattern consists of a central bright fringe flanked by much weaker maxima alternating with dark fringes.

a b

L

Each portion of the slit acts as a point source of light waves.

a

a/2

a/2

2

3

2

5

4

1

u

The path difference between rays 1 and 3, rays 2 and 4, or rays 3 and 5 is (a/ 2) sin u.

sin ua

Figure 38.5 Paths of light rays that encounter a narrow slit of width a and diffract toward a screen in the direction described by angle u (not to scale).

y=0

y

(a) Find the conditions on the angle θ for destructive interference. (If you define avariable that can take discrete values, specify what those values are.) [5 pts]

(b) Suppose a screen is placed parallel to the slit at a perpendicular distance L awayfrom the slit. Let y = 0 be the point on the screen immediately across from thecenter of the slit. Find an expression for y, the position of the central brightmaximum of the diffraction pattern. (Hint: you may like to consider what valuesof θ give the first dark fringes on either side of the central maximum.) [5 pts]

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9. A flat piece of flint glass is held stationary and horizontal above the flat top end of avertical acrylic rod of length L that has its lower end rigidly fixed. The thin film ofair (refractive index 1) between the rod and glass is observed to be bright by reflectedlight when it is illuminated by light of wavelength λ. As the temperature is slowlyincreased by ∆T , the film changes from bright to dark and back to bright k times.

(a) The refractive index of the flint glass piece is ng = 1.60 and the refractive index ofacrylic is na = 1.45. For a bright reflection, what condition relates the thickness,t, of the air gap to the wavelength, λ? [2 pts]

(b) Find an expression for the coefficient of linear expansion of the acrylic. [5 pts]

(c) If the rod and glass where submerged in benzene liquid, refractive index nb = 1.50,what would be the condition relating the gap thickness, t, to the wavelength fora bright reflection? [3 pts]

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–Extra Workspace–

11

–Extra Workspace–

12

P = FA

P = P0 + ρgh

B = ρgV

A1v1 = A2v2

P + 12ρv2 + ρgy =const.

∆L = αLi ∆T

∆V = βVi ∆T

PV = nRT

PV = NkBT (nR = NkB)

n = mM

Q = C ∆T

Q = cm∆T

Q = mL

∆Eint = W +Q

Won gas = −∫ VfViP dV

W = nRT ln(ViVf

)P = kA

∣∣dTdx

∣∣P = kA

(Th−TcL

)P = σAeT 4

P = 23NVK̄

K̄ = 12m0v2 = 3

2kBT

Ktot,trans = 32NkBT = 3

2nRT

e = 1− 1(V1/V2)(γ−1)

Q = nCV ∆T

Q = nCP ∆T

∆Eint = nCV ∆T

γ = CP

CV

PV γ = const.

TV γ−1 = const.

nV (E) ∝ e−E/kBT

Nv ∝ v2e−m0v2/2kBT

vrms =√

3kBTm0

∆S =∫ fi

dQr

T

∆S = nCv ln(TfTi

)+ nR ln

(VfVi

)S = kB lnW

e = Weng

|Qh|= |Qh|−|Qc|

|Qh|= 1− |Qc|

|Qh|

COP (cooling) = |Qc|W

COP (heating) = |Qh|W

e = Th−TcTh

= 1− TcTh

∫ v+dv

vNv dv =

∫ v+dv

v4πN

(m0

2πkBT

)3/2

v2e−m0v2/2kBT dv

13

f = 1T

k = 2πλ

ω = 2πf

T = 2π√

mk

T = 2π√

Lg

∂2y∂x2

= 1v2∂2y∂t2

y(x, t) = f(x∓ vt)

v = fλ = ωk

v =√

Tµ

y(x, t) = A sin (kx− ωt+ φ)

vy = ∂y∂t

ay = ∂2y∂t2

P = 12µω2A2v

y(x, t) = [2A sin(kx)] cos(ωt)

v = (331 m/s)√

1 + TCel

273

B = − ∆P∆V/Vi

y(t) =∑∞

n=1

(An sin(2πnft) +Bn cos(2πnft)

)

v =√

Bρ

s(x, t) = smax cos(kx− ωt)

∆P (x, t) = (∆Pmax) sin(kx− ωt)

∆Pmax = B smaxk = ρvωsmax

I = Pavg

A

Pavg = 12ρvω2As2

max

λn = 2Ln

fn = nv2L

λ2n+1 = 4L(2n+1)

f2n+1 = (2n+1)v4L

|L2 − L1| = nλ

|L2 − L1| = (2n+1)λ2

fbeat = |f1 − f2|

β = 10 log10

(II0

)f ′ =

(v±v0v∓vs

)f

sin θ = vvs

Mach number = vsv

E = hf

n = cv

n1 sin θ1 = n2 sin θ2

sin θc = n2

n1

M = h′

h

f = R2

1p

+ 1q

= 1f

n1

p+ n2

q= n2−n1

R

1f

= (n− 1)(

1R1− 1

R2

)I ∝ E2

d sin θ = mλ

d sin θ = (m+ 12)λ

I = Imax cos2(πd sin θλ

)2nt = (m+ 1

2)λ

2nt = mλ

r ≈√

mλRn

sin θ = mλa

I = Imax

(sin(πa sin θ/λ)πa sin θ/λ

)2

θmin = 1.22 λD

I = Imax cos2 θ

tan θp = n2

n1

14