PHYSICS 222 FALL 2006 FINAL EXAM SOLUTIONS DECEMBER 11, 2006 Problems 61-80 are worth 4 points each 61. An electron of mass 9.11 × 10 −31 kg and charge −1.60 × 10 −19 C is moving with speed 3.20 × 10 6 m/s in a circle of radius 0.182 mm in vacuum due to a uniform magnetic field directed perpendicular to the plane of the orbit. The magnitude of the magnetic field is ____ T. A. 0.1 B. 0.2 C. 0.5 D. 3 E. 9 B = mv |q|R = (9.11 × 10 −31 kg)(3.20 × 10 6 m/s) (1.60 × 10 −19 C)(1.82 × 10 −4 m) = 0.100 T . 62. A long thin wire carries a current of 9.55 A and is bent into two long straight sections and a circular arc of radius 1.00 cm as shown. All three sections lie in the x-y plane. Point P is at the center of the arc and the angle subtended by the arc at point P is (π/3) rad. The magnetic field at point P is ____ . -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 y (cm) x (cm) current P A. (100 µT) kˆ B. −(100 µT) kˆ C. (5 µT) kˆ D. −(5 µT) kˆ E. (50 µT) iˆ From the Biot-Savart law, the straight wire sections do not produce a magnetic field at point P. Using the circular right-hand rule, the direction of the magnetic field at P due to the circular current arc is in the −kˆ direction, which is pointed into the page. The magnitude of the field at P due to the circular arc is B = µ o I φ 4πR = (4π × 10 −7 T·m/A)(9.55 A)[(π/3)rad] 4π(0.0100 m) = 1.00 × 10 −4 T = 100 µT . 63. The ac current through a 2.00 µF capacitor has an angular frequency of 5000 rad/s and an amplitude of 3.00 A. The amplitude of the ac voltage across the capacitor is ____ V. A. 50 B. 100 C. 200 D. 300 E. 400 X C = 1 ω C = 1 (5000 rad/s)(2.00 × 10 −6 F) = 100 Ω . V C = I X C = (3.00 A)(100 Ω) = 300 V . 64. A train is traveling along a straight track in the +x direction at 50 m/s. A car is traveling at 30 m/s in the +x direction towards the train on the road next to the track (the train is in front of the car). The train emits a whistle at 540 Hz. The frequency heard by the driver of the car is ____ Hz. (The speed of sound in air is 340 m/s) A. 512 B. 526 C. 540 D. 554 E. 568 The positive direction for the speeds of the source S and listener L is the direction from the listener to the source, which is the +x direction. Thus v L = +30 m/s and v S = +50 m/s. Then f L = f S v + v L v+ v S = (540 Hz) 340 m/s + 30 m/s 340 m/s + 50 m/s = 512 Hz . 65. A monochromatic light beam is incident from glass into air. The wavelength of the light beam in the glass is 400 nm. The wavelength of the light beam after passing into the air is ____ nm. (The index of refraction of the glass is 1.50 and the index of refraction of air is 1.00) A. 270 B. 400 C. 440 D. 530 E. 600 λ 0 = nλ = (1.50)(400 nm) = 600 nm . 66. In a two-slit interference experiment, monochromatic light of wavelength 600 nm is normally incident on the two slits. A point P is situated at the second order intensity maximum on the screen. If the distance from point P to slit 1 is r 1 and the distance from point P to slit 2 is r 2 , then |r 1 − r 2 | = ___ nm. A. 300 B. 600 C. 1200 D. 1500 E. 1800 For constructive interference, |∆r| = |r 1 − r 2 | = |m|λ = 2(600 nm) = 1200 nm . 67. A laser emits an average light power of 1.60 mW at a wavelength of 620 nm. The number of photons per second emitted by the laser is ____ . A. 1 × 10 19 B. 2 × 10 18 C. 3 × 10 17 D. 4 × 10 16 E. 5 × 10 15 E = 1240 eV·nm λ(nm) = 1240 eV·nm 620 nm = 2.00 eV = 3.20 × 10 −19 J . N/t = P ave E = 1.60 × 10 −3 W 3.20 × 10 −19 J = 5.00 × 10 15 /s . 68. The electron in a hydrogen atom undergoes a transition from a state with principal quantum number 4 to a state with principal quantum number 2. The energy of the photon emitted is ____ eV. A. 0.45 B. 1.35 C. 2.55 D. 3.75 E. 4.85 E = E i − E f = − 13.6 eV n i 2 + 13.6 eV n f 2 = − 13.6 eV 4 2 + 13.6 eV 2 2 = 2.55 eV . 69. In an electron diffraction experiment, an electron beam is normally incident on a crystal surface where the rows of atoms are separated by 0.300 nm. If the second order electron diffraction peak is at a diffraction angle of 78.0 degrees, the de Broglie wavelength of the electrons in the beam is ____ nm. A. 0.03 B. 0.06 C. 0.09 D. 0.12 E. 0.15 λ = d sinθ m = (0.300 nm) sin(78.0˚) 2 = 0.147 nm . 70. An electron moves along the x-axis. In the region between x = 0.00 m and x = 5.00 m, the wave function of the electron can be approximated as ψ(x) = (0.250 m −3/2 )x. The probability that the electron is between x = 0.00 m and x = 2.00 m is ____ . A. 0.05 B. 0.09 C. 0.13 D. 0.17 E. 0.21 P(0.00 m, 2.00 m) = ∫ 0.00 m 2.00 m |ψ(x)| 2 dx = (0.250 m −3/2 ) 2 ∫ 0.00 m 2.00 m x 2 dx = 0.167 . 71. A proton of mass 1.67 × 10 −27 kg is confined inside a one-dimensional box of width 1.50 × 10 −15 m. The kinetic energy of the proton in its ground state is ____ J. A. 1.5 × 10 −11 B. 2.5 × 10 −13 C. 3.5 × 10 −15 D. 4.5 × 10 −17 E. 5.5 × 10 −19 E 1 = h 2 8mL 2 = (6.626 × 10 −34 J·s) 2 8(1.67 × 10 −27 kg)(1.50 × 10 −15 m) 2 = 1.46 × 10 −11 J ( = 91.3 MeV) . 72. The electron in a hydrogen atom is in a 3d state. The magnitude of the orbital angular momentum of the electron is ____ . (planckover2pi is Planck's constant divided by 2π) A. 0.5 planckover2pi B. 1.5 planckover2pi C. 2.5 planckover2pi D. 3.5 planckover2pi E. 4.5 planckover2pi L = √ lscript(lscript + 1) planckover2pi = √ 2(2 + 1) planckover2pi = 2.45 planckover2pi . 73. The 4f electron subshell of an atom can accommodate ____ electrons. A. 4 B. 8 C. 12 D. 14 E. 26 N = 2(2lscript + 1) = 2[2(3) + 1] = 14 . 74. At 0˚C, the diameter of a brass ball is 3.0120 cm, and the diameter of a hole in a brass plate is 3.0000 cm. In order for the ball to fit through the hole in the plate, first the plate has to be heated to a temperature of at least ____ ˚C. (The linear thermal expansion coefficient of brass is 2.0 × 10 −5 / ˚C) A. 200 B. 300 C. 400 D. 500 E. 600 ∆T = ∆L α L 0 = 0.0120 cm (2.0 × 10 −5 / ˚C)(3.0000 cm) = 200 ˚C . T = T 0 + ∆T = 0 ˚C + 200 ˚C = 200 ˚C . 75. The amount of heat absorbed by 10.0 g of liquid water as it is heated from 10 ˚C to 20 ˚C is ____ J. (The specific heat of ice is 2100 J/kg·˚C. The heat of fusion of water is 334 kJ/kg. The specific heat of liquid water is 4190 J/kg·˚C. ) A. 210 B. 420 C. 630 D. 840 E. 1050 Q = mc water ∆T = (0.0100 kg)(4190 J/kg·˚C)(20 ˚C − 10˚C) = 419 J . 76. The rms speed of the molecules in a certain gas at a temperature of 390 K is 450 m/s. The molecular weight (molar mass) of the gas is ____ g/mol. A. 16 B. 24 C. 32 D. 40 E. 48 M = 3RT v rms 2 = 3(8.314 J/mol·K)(390 K) (450 m/s) 2 = 0.0480 kg/mol = 48.0 g/mol . 77. One mole of an ideal monatomic gas is taken through a closed cycle on a p-V diagram. During this cycle, the net work done by the gas is 50 J. The net amount of heat absorbed by the gas during the cycle is ____ J. A. 0 B. 20 C. 50 D. 100 E. 300 Closed cycle: ∆U = 0 = Q − W, so Q = W = 50 J . 78. An ideal monatomic gas initially has a volume of 60.0 L. The gas is compressed at constant pressure to a new volume of 30.0 L, during which 60.0 L·atm of work is done on the gas. The pressure of the gas is ____ atm. A. 1 B. 2 C. 3 D. 4 E. 5 p = W by gas ∆V = −60.0 L·atm 30.0 L − 60.0 L = 2.00 atm . 79. One mole of an ideal diatomic gas with C V = 5R/2 initially has a temperature of 277 K. Then the gas is adiabatically compressed to a state where the final temperaure is 366 K. The work done by the gas during this compression process is ____ L·atm. (C V is the molar specific heat at constant volume and R is the molar gas constant) A. −11.4 B. 15.8 C. −15.8 D. −18.3 E. 18.3 W = −∆U = −nC V ∆T = −n(5R/2)∆T = −(1.00 mol)[5(0.08208 L·atm/mol·K)/2](366 K − 277 K) = −18.3 L·atm . 80. The coefficient of performance of a refrigerator in a kitchen is 2.50. During one cycle of operation of the refrigerator, the refrigerator removes 10.0 J of heat from the inside of the refrigerator. The amount of work delivered to the refrigerator during the cycle is ____ J. A. 4 B. 7 C. 10 D. 16 E. 22 |W| = |Q C | K = 10.0 J 2.50 = 4.00 J. Problems 81-100 are worth 2 points each 81. Maxwell's equations for electric and magnetic fields and their relations to charges and currents in otherwise empty space are: (1) ∫ O E → · dA → = Q encl /ε o (3) ∫ O B → · dlscriptlscriptlscriptlscript → = µ o i C + ε o dΦ E dt encl (2) ∫ O B → · dA → = 0 (4) ∫ O E → • dlscriptlscriptlscriptlscript → = − dΦ B dt Which two of these equations together predict the existence of electromagnetic waves? A. (1), (2) B. (2), (3) C. (2), (4) D. (3), (4) E. (1), (4) 82. Which one of the following five statements about an ideal inductor is true? A. The inductance of an inductor increases as the current through it increases. B. The inductance of an inductor decreases as the number of turns of wire in it increases. C. The ac current through an inductor is in-phase with the ac voltage across the inductor. D. The energy stored in the magnetic field of an inductor is proportional to the current through the inductor. E. The reactance of an inductor increases as the frequency of the ac current through it increases. 83. A transverse wave pulse on a stretched string has a wave function given by y(x, t) = x 0 1 + (x + vt) 2 /x 1 2 , where x 0 , x 1 , and v are positive constants. The pulse is moving in the ____ direction. A. iˆ B. − iˆ C. jˆ D. −jˆ E. kˆ 84. At a particular point in space at a particular instant of time, the electric field in an electromagnetic plane wave in vacuum is in the iˆ direction. If the direction of propagation of the wave is in the kˆ direction, then the direction of the magnetic field in the wave at that position and time is in the ____ direction. A. iˆ B. − iˆ C. jˆ D. −jˆ E. −kˆ The direction of propagation of the wave is in the direction of E → × B → at each point in space and at each instant of time. 85. An object is positioned on the principal axis of a thin converging lens. The distance of the object from the lens is twice the focal length of the lens. The image of the object is ____ A. real, inverted, and the same size as the object. B. real, inverted, and larger than the object. C. real, erect, and the same size as the object. D. virtual, inverted, and smaller than the object. E. virtual, erect, and larger than the object. 1 s' = 1 f − 1 s = 1 f − 1 2f = 1 2f , so s' = 2f = s . m = − s' s = − 2f 2f = −1. s' is positive, which means that the image is real. The lateral magnification m is negative, so the image is inverted. The magnitude of m is 1, which means that the lateral size of the image is the same as for the object. 86. In an x-ray diffraction experiment, x-rays are incident on a crystal in which the atomic planes are separated by a distance d. In order to observe any diffraction intensity peaks at diffraction order m ≠ 0, the x-ray wavelength has to be less than or equal to ____ . A. d/2 B. d C. 3d/2 D. 2d E. 5d/2 sinθ = mλ 2d , where m = 1, 2, ... Thus to get a solution for θ requires that λ ≤ 2d . 87. The absolute temperature (in K) of a black body is T o , and the total electromagnetic wave power radiated by the body is P o . If the temperature of the body increases to 2T o , the radiated power is now ____ P o . A. 1 B. 2 C. 4 D. 8 E. 16 The power emitted is proportional to T 4 , so if T increases by 2, the power increases by 2 4 = 16 . 88. The kinetic energy of each electron in an electron beam is K 0 and the de Broglie wavelength of the electrons is λ 0 . If the kinetic energy of each electron in the electron beam doubles, the de Broglie wavelength of the electrons is now ____ . A. 2λ 0 B. 4λ 0 C. λ 0 / √ 2 D. λ 0 /2 E. λ 0 /4 electrons: λ = 1.23 nm √ eV √ K(eV) , so if K doubles, λ becomes λ 0 / √ 2. 89. What principle or law prevents all the electrons in an atom from all occupying the same lowest energy state of the atom? A. Heisenberg uncertainty principle B. Pauli exclusion principle C. Law of conservation of energy D. Law of conservation of linear momentum E. Law of conservation of angular momentum 90. A copper cube of size L × L × L has a hole drilled in it. When the temperature of the block increases, each edge length L increases and the diameter of the hole ____ A. decreases. B. remains the same. C. increases. Every linear dimension of the block, including the diameter of the hole, increases when the temperature of the block increases. 91. An "ideal gas" is ____ A. a gas in which the potential energy of interaction between the molecules can be ignored. B. a monatomic gas. C. a gas in which the molecules do not rotate or vibrate. D. a gas consisting of 21% O 2 and 79% N 2 . E. an extremely high purity gas. 92. When 8.17 grams of liquid water at 0.00 ˚C is converted to solid ice at 0.00 ˚C, the final entropy of the H 2 O minus the initial entropy of the H 2 O is ____ J/K. The specific heat of (liquid) water is 4.190 kJ/kg·˚C. The specific heat of ice is 2.100 kJ/kg·˚C. The heat of fusion of water is 334 kJ/kg. A. −2 B. 6 C. −6 D. −10 E. 10 ∆S = Q T = −m L f T = −(0.00817 kg)(334 kJ/kg) 273.15 K = −0.010 kJ/K = −10 J/K . Laboratory Final Exam Solutions The Laboratory Final Exam problems follow these solutions 93. The test coil is placed in a constant magnetic field. From Faraday's law of induction, no EMF is induced in the test coil because the magnetic field is not changing with time. Thus the answer is "A. 0.000 V." 94. The electrons boiled off from the filament are accelerated by the electric field between the filament and the anode that is caused by the potential difference between the filament and the anode. Thus the primary cause of the kinetic energy of the electrons emerging from the slit in the anode is "D. The potential difference between the filament and the cylindrical anode." 95. Because the three bulbs are in series, the current I through each bulb is the same. The filament of each bulb has the same resistance R, so the power dissipated in each bulb P = I 2 R is the same. Since the brightness of a bulb is determined by the power dissipated in the filament, each bulb is equally bright. If the 3 bulbs are replaced by 1 bulb, then the resistance of the circuit is smaller and the current through the 1 bulb is larger than for 3 bulbs, so the power dissipated in the 1 bulb is larger than in each of the 3 bulbs. Thus the single bulb is brighter than either of the equally bright 3 bulbs. Thus the answer is "B. All the bulbs are equally bright, but of less brightness than that of a single bulb when connected directly across the battery." 96. The fundamental frequency of vibration of the wire is inversely proportional to its length. For a length of 80 cm, it is given that the fundamental frequency is 96 Hz. When the length is decreased to 60 cm while keeping the linear mass density and the tension in the wire the same, the new fundamental frequency is (96 Hz)[(80 cm)/60 cm)] = 128 Hz, which is answer A. 97. The most common use and application of an oscilloscope is "C. An instrument for measuring potential difference (PD) and displaying a graph of PD vs. time". The graph is on the oscilloscope screen. 98. An incandescent light bulb produces a continuous spectrum of light as in graph D rather than a discrete line spectrum as in A. Thus the answer is D. 99. In order to increase the resonant frequency of an LC or LRC circuit, either the inductance of the coil must be reduced or the capacitance of the capacitor must be reduced. Thus the magnetic core must be moved out of the coil or the movable vanes of the capacitor must be moved counter clockwise, which is answer D. 100. The current through the heater is determined by measuring the voltage across the precision "shunt" resistor that is in series with the heater. Thus the current through the heater is I = (0.0385 V)/(0.0100 Ω) = 3.85 A. Then the power dissipated in the heater is P = IV = (3.85 A)(8.86 V) = 34.1 W, which is answer B. PHYSICS 222 - Fall, 2006 Laboratory Final This laboratory final consists of 8 questions, #93 to #100, each worth 2 points. ________________________________________________________________________________ 93. A small test coil (200 turns, 2 cm 2 cross-sectional area) is placed at the center of a solenoid, much as you did in lab. Assume that the solenoid circuit is somewhat different from that which you used, having a DC power supply adjusted to produce a magnetic field within the solenoid of 2 Tesla. Estimate the EMF induced in the test coil when it is located at rest at the center of the solenoid. Rd Bk solenoid (15 cm long) V Digital Scope ch #1 test coil - small coil wound on rod DC power supply A) 0.000 V B) 0.080 V C) 800 V D) 2.0 V E) Cannot know the EMF without knowing the current in the solenoid circuit. ________________________________________________________________________________ 94. In the electron beam experiment, the electron tube contained two electrodes: anode narrow slit electron beam filament a. a straight, glowing filament (perpendicular to the page in the figure on the right); b. a cylindrical anode that surrounded the filament. On the side of the anode was a slit through which a beam of electrons emerged. What was the primary cause of the kinetic energy of the emerging electrons? A) The magnetic field generated by the currents in the two large Helmholtz coils; B) The high temperature of the glowing filament; C) The potential difference between the two ends of the glowing filament; D) The potential difference between the filament and the cylindrical anode; E) None of the above. 95. In the circuit shown at the right, three identical light bulbs are connected to a battery. Compare the brightness of these bulbs to that of a single bulb when connected alone directly across the battery. A) All the bulbs are equally bright, and of the same brightness as that of a single bulb connected directly across the battery. B) All the bulbs are equally bright, but of less brightness than that of a single bulb when connected directly across the battery. 6 V 3 12 C) All three bulbs are less bright than a single bulb connected directly to the battery, and among the three, bulb #1 is brightest, #2 slightly less bright, and #3 the least bright. D) All three bulbs are less bright than a single bulb connected directly to the battery, and among the three, bulb #3 is brightest, #2 slightly less bright, and #1 the least bright. E) Bulb #1 has the same brightness as that of a single bulb connected directly across the battery, but bulbs #2 and #3 are both dimmer, but of the same brightness. 96. In the vibrating wire experiment, the wire passed over a pulley and supported a mass of 3 kg. The horizontal portion of the wire was held by two supports, one at the 0 cm mark and the other at the 80 cm mark of a wooden rule as indicated in the figure. When the wire was plucked gently near the 40 cm mark, the fundamental vibration mode of the wire was about 96 Hz. 080 Assume now that the wire rests firmly upon a wooden fret that is inserted at the 20 cm mark. If the wire is now plucked gently at the 50 cm mark, what is the likely dominant frequency of vibration for the wire? A) 128 Hz B) 24 Hz C) 72 Hz D) 96 Hz E) One cannot predict without knowing the mass per unit length of the wire. ___________________________________________________________________________ 97. Which of the following best describes an oscilloscope and its most common application? A) An instrument for producing voltages and making graphs. B) An instrument for displaying sine waves. C) An instrument for measuring potential difference (PD) and displaying a graph of PD vs. time. D) An instrument for producing sine functions and measuring their frequency. E) None of the above describes a common application of an oscilloscope. _______________________________________________________________________________ 98. If a small detector sensitive to visible light is placed in the plane of the cross-hairs of an optical grating spectrometer, such as used in 222 lab, the intensity, I, of the light diffracted by the grating at various angles, θ, can be measured. If such measurements were made for radiation produced by a ordinary incandescent light bulb with a glowing filament, such as the straight filament lamp which you used in lab, which of the following graphs best represents the features of the data that might be collected. ________________________________________________________________________________ I Θ I Θ I Θ I Θ I Θ AB C DE 99. Consider the simple radio receiver that you worked with in lab. Note that you could move the (magnetic) ferrite core of the inductor into or out of its coil of wire, and rotate the movable set of plates of the capacitor CW (clockwise) or CCW (counter- clockwise). If the radio is “tuned” to WOI (640 kHz), how should the magnetic core be moved (with respect to the coil) or the movable plates be rotated from the position shown, to change the resonant frequency of the radio toward that necessary to receive KASI (1430 kHz)? fixed set of plates rotatable plates magnetic core coil A. into, CW B. into, CCW C. out of, CW D. out of, CCW E. There is insufficient information to answer this question. 100. In the Thermal Measurements lab, the circuit shown at the right was used. If the values shown below were obtained, what is the best estimate for the heater power? (in Joules/second) V V DC power supply heater about 2 Ω shunt 0.0100 Ω Heater P.D. Shunt P.D. 8.86 V 0.0385 V A) 39. B) 34.1 C) 0.0886 D) 8.86 E) Cannot be determined since the circuit current is not given. _____________________________________________________________________________________ David Johnston Final_Exam_F_06_Solutions