AP C E&M Practice Exam

AP C E&M Practice Exam - Advanced Placement Program...

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Unformatted text preview: Advanced Placement Program AP® Physics C: Electricity and Magnetism Practice Exam The questions contained in this AP® Physics C: Electricity and Magnetism Practice Exam are written to the content specifications of AP Exams for this subject. Taking this practice exam should provide students with an idea of their general areas of strengths and weaknesses in preparing for the actual AP Exam. Because this AP Physics C: Electricity and Magnetism Practice Exam has never been administered as an operational AP Exam, statistical data are not available for calculating potential raw scores or conversions into AP grades. This AP Physics C: Electricity and Magnetism Practice Exam is provided by the College Board for AP Exam preparation. Teachers are permitted to download the materials and make copies to use with their students in a classroom setting only. To maintain the security of this exam, teachers should collect all materials after their administration and keep them in a secure location. Teachers may not redistribute the files electronically for any reason. © 2008 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. Visit the College Board on the Web: www.collegeboard.com. Contents Directions for Administration ............................................................................................ ii Section I: Multiple-Choice Questions ................................................................................ 1 Section II: Free-Response Questions .............................................................................. 13 Student Answer Sheet for Multiple-Choice Section ...................................................... 20 Multiple-Choice Answer Key........................................................................................... 21 Free-Response Scoring Guidelines.................................................................................. 22 The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 5,000 schools, colleges, universities, and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT®, and the Advanced Placement Program® (AP®). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns. Visit the College Board on the Web: www.collegeboard.com. AP Central is the official online home for the AP Program: apcentral.collegeboard.com. -i- AP® Physics C: Electricity and Magnetism Directions for Administration The AP Physics C: Electricity and Magnetism Exam is one and one-half hours in length and consists of a multiple-choice section and a free-response section. • The 45-minute multiple-choice section contains 35 questions and accounts for 50 percent of the final grade. • The 45-minute free-response section contains 3 questions and accounts for 50 percent of the final grade. Students should be given a 10-minute warning prior to the end of each section of the exam. A 10-minute break should be provided before Section I is begun if students are taking Physics C: Mechanics immediately before Electricity and Magnetism. The actual AP Physics C Exams are administered in one session, Mechanics first followed by Electricity and Magnetism. Students taking only one of the exams will have the most realistic experience if both sections are completed in one session. Similarly, students taking both Physics C exams will have the most realistic experience if both exams are completed in one session and a complete morning or afternoon is available to administer them. If a schedule does not permit one time period for administration, it would be acceptable to administer Mechanics on one day and Electricity and Magnetism on a subsequent day, or to further break things up and administer Section I and Section II of each exam on subsequent days. Many students wonder whether or not to guess the answers to the multiple-choice questions about which they are not certain. It is improbable that mere guessing will improve a score. However, if a student has some knowledge of the question and is able to eliminate one or more answer choices as wrong, it may be to the student’s advantage to answer such a question. • The use of calculators is permitted only on Section II. Straightedges or rulers are allowed on both parts of the exam. • It is suggested that the practice exam be completed using a pencil to simulate an actual administration. • Teachers will need to provide paper for the students to write their free-response answers. Teachers should provide directions to the students indicating how they wish the responses to be labeled so the teacher will be able to associate the student’s response with the question the student intended to answer. • The 2008–2009 AP Physics C table of information is included as a part of Section I. The table and the AP Physics C equation lists are included with Section II. The equation lists are not allowed for Section I. If you use these exams in subsequent years you should download the newer versions of the table and lists from AP Central. • Remember that students are not allowed to remove any materials, including scratch work, from the testing site. -ii- Section I Multiple-Choice Questions -1- TABLE OF INFORMATION FOR 2008 and 2009 CONSTANTS AND CONVERSION FACTORS Proton mass, m p = 1.67 ¥ 10 -27 kg Electron charge magnitude, Neutron mass, mn = 1.67 ¥ 10 -27 kg 1 electron volt, 1 eV = 1.60 ¥ 10 -19 J Electron mass, me = 9.11 ¥ 10 -31 kg Speed of light, Universal gravitational constant, Acceleration due to gravity at Earth’s surface, Avogadro’s number, N 0 = 6.02 ¥ 10 23 mol -1 R = 8.31 J (mol iK) Universal gas constant, e = 1.60 ¥ 10 -19 C c = 3.00 ¥ 108 m s G = 6.67 ¥ 10 -11 m 3 kgis2 g = 9.8 m s2 Boltzmann’s constant, k B = 1.38 ¥ 10 -23 J K 1 u = 1.66 ¥ 10 -27 kg = 931 MeV c 2 1 unified atomic mass unit, h = 6.63 ¥ 10 -34 J is = 4.14 ¥ 10 -15 eV is Planck’s constant, hc = 1.99 ¥ 10 -25 J im = 1.24 ¥ 103 eV i nm ⑀0 = 8.85 ¥ 10 -12 C2 N im 2 Vacuum permittivity, Coulomb’s law constant, k = 1 4 p⑀0 = 9.0 ¥ 109 N im 2 C2 m0 = 4 p ¥ 10 -7 (T im) A Vacuum permeability, Magnetic constant, k ¢ = m0 4 p = 10 -7 (T im) A 1 atm = 1.0 ¥ 105 N m 2 = 1.0 ¥ 105 Pa 1 atmosphere pressure, UNIT SYMBOLS Factor meter, kilogram, second, ampere, kelvin, PREFIXES Prefix Symbol m kg s A K mole, hertz, newton, pascal, joule, mol Hz N Pa J watt, coulomb, volt, ohm, henry, W C V W H farad, tesla, degree Celsius, electron-volt, F T ∞C eV VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES q 30 0 37 45 53 60 90 10 9 giga G sin q 0 12 35 2 2 45 3 2 1 106 mega M cosq 1 3 2 45 2 2 35 12 0 103 kilo k tan q 0 3 3 34 1 43 3 • 10 -2 centi c 10 -3 milli m 10 -6 micro m 10 -9 nano n 10 -12 pico p The following conventions are used in this exam. I. Unless otherwise stated, the frame of reference of any problem is assumed to be inertial. II. The direction of any electric current is the direction of flow of positive charge (conventional current). III. For any isolated electric charge, the electric potential is defined as zero at an infinite distance from the charge. -2- PHYSICS C: ELECTRICITY AND MAGNETISM SECTION I Time—45 minutes 35 Questions Directions: Each of the questions or incomplete statements below is followed by five suggested answers or completions. Select the one that is best in each case and place the letter of your choice in the corresponding box on the student answer sheet. 1. Two negative point charges are a distance x apart and have potential energy U. If the distance between the point charges increases to 3x, what is their new potential energy? (A) (B) (C) (D) (E) Questions 3-4 9U 3U U U 3 U 9 _______________________________________ Two concentric spherical surfaces are drawn around an isolated positive charge +q located at their center, as shown above. The inner surface has a radius that is 1 2 that of the outer surface. 3. If the total electric flux passing through the inner surface is f , what is the total electric flux passing through the outer surface? 2. An electric field is produced by the very long, uniformly charged rod drawn above. If the strength of the electric field is E1 at a distance r1 from the axis of the rod, at what distance from the axis is E the field strength 1 ? 4 (A) (A) (B) (C) (D) (E) r1 4 f 4 f 2 f 2f 4f 4. If the electric field strength at the inner surface is E, what is the electric field strength at the outer surface? r (B) 1 2 (C) 2r1 (A) (B) (C) (D) (E) (D) 4r1 (E) 16r1 E 4 E 2 E 2E 4E GO ON TO THE NEXT PAGE. -3- 6. A charge +q is placed at the center of a tetrahedron whose faces are all equilateral triangles, as shown above. What is the flux of the electric field through one face of the tetrahedron? (A) 0 5. Sphere X of mass M and charge +q hangs from a string as shown above. Sphere Y has an equal charge +q and is fixed in place a distance d directly below sphere X. If sphere X is in equilibrium, the tension in the string is most nearly (B) (C) q ⑀0 q 4⑀ 0 (D) 4⑀0 q (A) Mg (E) The flux through one face cannot be determined from the information given. _______________________________________ kq d kq (C) Mg d kq 2 (D) Mg + 2 d kq 2 (E) Mg - 2 d (B) Mg + 7. Two concentric metal spheres X and Y are shown above. X carries a positive charge, and Y is connected to ground. True statements include which of the following? I. The electric field inside X is zero. II. The electric field outside Y is zero. III. The charge density on both spheres is the same. (A) (B) (C) (D) (E) I only III only I and II only II and III only I, II, and III GO ON TO THE NEXT PAGE. -4- 9. The capacitor C in the circuit shown above is initially uncharged. The switch S is then closed. Which of the following best represents the voltage VR across the resistor R as a function of time t ? 8. A small positively charged sphere is lowered by a nonconducting thread into a grounded metal cup without touching the inside surface of the cup, as shown above. The grounding wire attached to the outside surface is disconnected and the charged sphere is then removed from the cup. Which of the following best describes the subsequent distribution of excess charge on the surface of the cup? (A) (B) (A) Negative charge resides on the inside surface, and no charge resides on the outside surface. (B) Negative charge resides on the outside surface, and no charge resides on the inside surface. (C) Positive charge resides on the inside surface, and no charge resides on the outside surface. (D) Positive charge resides on the outside surface, and no charge resides on the inside surface. (E) Negative charge resides on the inside surface, and positive charge resides on the outside surface. (C) (D) (E) GO ON TO THE NEXT PAGE. -5- Questions 10-11 13. The three lightbulbs in the circuit above are identical, and the battery has zero internal resistance. When switch S is closed to cause bulb 1 to light, which of the other two bulbs increase(s) in brightness? An electric circuit consists of a 12 V battery, an ideal 10 A fuse, and three 2 W resistors connected as shown above. (A) (B) (C) (D) (E) 10. What would be the reading on a voltmeter connected across points A and C ? (A) 12 V (B) 6 V (C) 3 V (D) 2 V (E) 0 V, since the fuse would break the circuit 14. A length of wire of resistance R is connected across a battery with zero internal resistance. The wire is then cut in half and the two halves are connected in parallel. When the combination is reconnected across the battery, what happens to the resultant power dissipated and the current drawn from the battery? 11. What would be the reading on an ammeter inserted at point B ? (A) 9 A (B) 6 A (C) 3 A (D) 2 A (E) 0 A, since the fuse would break the circuit _________________________________________ Power Current (A) No change No change (B) Doubles Doubles (C) Quadruples Doubles (D) Doubles Quadruples (E) Quadruples Quadruples ________________________________________ 12. A fixed voltage is applied across the length of a tungsten wire. An increase in the power dissipated by the wire would result if which of the following could be increased? (A) (B) (C) (D) (E) Neither bulb Bulb 2 only Bulb 3 only Both bulbs It cannot be determined without knowing the emf of the battery. The resistivity of the tungsten The cross-sectional area of the wire The length of the wire The temperature of the wire The temperature of the wire’s surroundings 15. In the circuit shown above, the 10 mF capacitor is initially uncharged. After the switch S has been closed for a long time, how much energy is stored in the capacitor? (A) 0 mJ (B) 100 mJ (C) 250 mJ (D) 500 mJ (E) 1000 mJ GO ON TO THE NEXT PAGE. -6- Questions 18-19 16. An electron e - moving in the plane of the page is injected into a uniform magnetic field B that is perpendicular to the page, as shown above. Upon entering the field, the electron takes a path that is (A) straight through the field (B) clockwise, circular, and in the plane of the page (C) counterclockwise, circular, and in the plane of the page (D) circular and curved out of the page (E) circular and curved into the page _________________________________________ A fixed charge distribution produces the equipotential lines shown in the figure above. 18. Which of the following expressions best represents the magnitude of the electric field at point P ? (A) (B) 10 V 0.04 m (C) 25 V 0.14 m (D) 25 V 0.04 m (E) 17. In the figure above, two long, straight, insulated wires at right angles in the plane of the page carry currents of I and 2I, as shown. What is the direction of the magnetic field at point P, which is equidistant from the wires and coplanar with them? 10 V 0.14 m 40 V 0.25 m 19. The direction of the electric field at point P is most nearly (A) Into the page (A) (B) (C) (D) (E) (B) Out of the page (C) toward the left toward the right toward the bottom of the page toward the top of the page perpendicular to the plane of the page (D) (E) GO ON TO THE NEXT PAGE. -7- 20. A helium nucleus (charge +2q and mass 4m) and Questions 23-24 a lithium nucleus (charge +3q and mass 7m) are A cloud contains spherical drops of water of radius R and charge Q. Assume the drops are far apart. accelerated through the same electric potential difference, V0. What is the ratio of their resultant K kinetic energies, lithium ? K helium (A) 2 3 (B) 23. The electric field E0 and potential V0 at the surface of each drop is given by which of the following? 6 7 E0 V0 (A) 0 0 (C) 1 (D) 7 6 (B) kQ R kQ R2 (E) 3 2 (C) kQ R2 kQ R (D) 0 kQ R (E) kQ R 0 21. The electric potential in the xy-plane in a certain region of space is given by V ( x, y) = 6 x 2 y - 2 y3 , where x and y are in meters and V is in volts. What is the magnitude of the y-component of the electric field at the point ( -1, 2) ? 24. If two droplets happen to combine into a single larger droplet, the new potential V at the surface of the larger droplet is most nearly equal to (A) 0 V/m (B) 4 V/m (C) 18 V/m (D) 24 V/m (E) 30 V/m _________________________________________ (A) 3V0 (B) 2 V0 (C) 2 1/ 3 2 V0 (D) 21/ 3 V0 22. Two protons and an electron are assembled along a line, as shown above. The distance between the electron and each proton is a. What is the work done by an external force in assembling this configuration of charges? (A) - 2 (E) V0 ke2 a (B) - 3 ke2 2 a (C) 1 ke2 2 a (D) 3 ke2 2 a (E) 3 ke2 a GO ON TO THE NEXT PAGE. -8- Questions 27-28 An emf of 20 V is induced around a metal ring by increasing a uniform magnetic field at a constant rate from zero to a final magnitude of 1.0 ¥ 10 -2 T throughout the region enclosed by the ring. The field direction is perpendicular to the plane of the ring. 25. A point charge −Q is located at the origin, while a second point charge +2Q is located at x = d on the x-axis, as shown above. A point on the x-axis where the net electric field is zero is located in which of the following regions? 27. If the area enclosed by the ring is 4.0 ¥ 10 -3 m 2 , what is the time interval during which the field is increased? (A) - • < x < 0 (B) 0 < x < (C) d 2 (A) 2.0 ms (B) 5.0 ms (C) 10 ms (D) 20 ms (E) 50 ms d < x<d 2 (D) d < x < • (E) No region on the x-axis __________________________________________ 28. If the electrical resistance of the ring is 500 Ω, what is the rate at which energy is dissipated in the ring as the field is increased? (A) 0.040 W (B) 0.80 W (C) 1.25 W (D) 25 W (E) 1.0 ¥ 10 4 W 26. An inductor of inductance L is connected in series with a resistor of resistance R, a battery of emf e , and a switch, as shown above. When the switch is closed, the current I in the circuit increases with time, approaching the value I max . What is Imax ? (A) R L e (B) RL e (C) e RL (D) e R (E) L e GO ON TO THE NEXT PAGE. -9- 29. Two long, straight wires are parallel to and equidistant from the y-axis, as shown above left. Each carries current I in opposite directions, resulting in a magnetic field of magnitude B0 at the origin. If the wires are each bent into right angles and placed as shown above right, what is the magnitude of the magnetic field at the origin? (A) Zero (B) Between zero and (C) B0 2 B0 2 (D) Between B0 and B0 2 (E) B0 __________________________________________________________________________________ 30. A length of wire carrying a steady clockwise current I is bent to form the triple circular loop X above. An identical length of the same wire is bent less tightly to form the single loop Y of larger radius, which carries the same current I. The ratio of the magnetic field strength at the center of loop Y to the magnetic field strength at the center of loop X is (A) 1 9 (B) 1 3 (C) 1 (D) 3 (E) 9 GO ON TO THE NEXT PAGE. -10- 32. A long wire of radius R carries a current I, as shown above, with a current density J = ar that increases linearly with the distance r from the center of the wire. Which of the following graphs best represents the magnitude of the magnetic field B as a function r ? 31. The diagram above shows the cross section of a long cable that has an inner wire of radius R surrounded by a conducting sheath of outer radius 2R. The wire and the sheath carry currents in opposite directions but with the same uniform current density J. What is the magnitude of the magnetic field at the surface of the outer conductor? (A) (A) Zero 1 m RJ (B) 4 0 1 (C) m RJ 2 0 3 (D) m RJ 4 0 (E) m0 RJ (B) (C) (D) (E) GO ON TO THE NEXT PAGE. -11- 34. Which of the following statements contradicts one of Maxwell’s equations? (A) A changing magnetic field produces an electric field. (B) A changing electric field produces a magnetic field. (C) The net magnetic flux through a closed surface depends on the current inside. (D) The net electric flux through a closed surface depends on the charge inside. (E) The electric field due to an isolated stationary point charge is inversely proportional to the square of the distance from the charge. 33. In the circuit drawn above, the switch S is initially open, and the capacitor C is charged with the polarity indicated. The switch is then closed, and the capacitor begins discharging through the resistor. Which of the following is true of the current that is subsequently induced in the circular wire loop near the long, straight wire AB ? 35. A student building a circuit wishes to increase the frequency of an oscillator consisting of a capacitor of capacitance C and an inductor of inductance L. Which of the following would accomplish this objective? (A) It is counterclockwise and constant. (B) It is counterclockwise and increases with time. (C) It is counterclockwise and decreases with time. (D) It is clockwise and increases with time. (E) It is clockwise and decreases with time. I. II. III. IV. (A) (B) (C) (D) (E) Increase L Increase C Decrease L Decrease C I only I or II I or IV II or III III or IV STOP END OF ELECTRICITY AND MAGNETISM SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON ELECTRICITY AND MAGNETISM SECTION I ONLY. DO NOT TURN TO ANY OTHER TEST MATERIALS. -12- Section II Free-Response Questions -13- TABLE OF INFORMATION FOR 2008 and 2009 CONSTANTS AND CONVERSION FACTORS Proton mass, m p = 1.67 ¥ 10 -27 kg Electron charge magnitude, Neutron mass, mn = 1.67 ¥ 10 -27 kg 1 electron volt, 1 eV = 1.60 ¥ 10 -19 J Electron mass, me = 9.11 ¥ 10 -31 kg Speed of light, Universal gravitational constant, Acceleration due to gravity at Earth’s surface, Avogadro’s number, N 0 = 6.02 ¥ 10 23 mol -1 R = 8.31 J (mol iK) Universal gas constant, e = 1.60 ¥ 10 -19 C c = 3.00 ¥ 108 m s G = 6.67 ¥ 10 -11 m 3 kgis2 g = 9.8 m s2 Boltzmann’s constant, k B = 1.38 ¥ 10 -23 J K 1 u = 1.66 ¥ 10 -27 kg = 931 MeV c 2 1 unified atomic mass unit, h = 6.63 ¥ 10 -34 J is = 4.14 ¥ 10 -15 eV is Planck’s constant, hc = 1.99 ¥ 10 -25 J im = 1.24 ¥ 103 eV i nm ⑀0 = 8.85 ¥ 10 -12 C2 N im 2 Vacuum permittivity, Coulomb’s law constant, k = 1 4 p⑀0 = 9.0 ¥ 109 N im 2 C2 m0 = 4 p ¥ 10 -7 (T im) A Vacuum permeability, Magnetic constant, k ¢ = m0 4 p = 10 -7 (T im) A 1 atm = 1.0 ¥ 105 N m 2 = 1.0 ¥ 105 Pa 1 atmosphere pressure, UNIT SYMBOLS meter, kilogram, second, ampere, kelvin, PREFIXES m kg s A K mole, hertz, newton, pascal, joule, mol Hz N Pa J watt, coulomb, volt, ohm, henry, W C V W H farad, tesla, degree Celsius, electron-volt, F T ∞C eV VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES Prefix Symbol q 0 30 37 45 53 60 giga G sin q 0 12 35 2 2 45 3 2 1 106 mega M cosq 1 3 2 45 2 2 35 12 0 103 kilo k tan q 0 3 3 34 1 43 3 • 10 -2 centi c -3 milli m 10 -6 micro m 10 -9 nano n -12 pico p Factor 10 10 10 9 90 The following conventions are used in this exam. I. Unless otherwise stated, the frame of reference of any problem is assumed to be inertial. II. The direction of any electric current is the direction of flow of positive charge (conventional current). III. For any isolated electric charge, the electric potential is defined as zero at an infinite distance from the charge. -14- ADVANCED PLACEMENT PHYSICS C EQUATIONS FOR 2008 and 2009 MECHANICS u = u0 + at 1 2 at 2 x = x0 + u0 t + u 2 = u0 2 + 2 a ( x - x0 )  F = Fnet = ma F= dp dt J = Ú F dt = Dp p = mv Ffric £ m N W = ÚF K = 1 2 mu 2 P = dW dt ∑ dr P = Fؒv DUg = mgh ac = = = = = = = = = = L = m= N = P = p = r = r = T = t = U= u = W= x = m= q = t = w= a= a F f h I J K k ELECTRICITY AND MAGNETISM acceleration force frequency height rotational inertia impulse kinetic energy spring constant length angular momentum mass normal force power momentum radius or distance position vector period time potential energy velocity or speed work done on a system position coefficient of friction angle torque angular speed angular acceleration u2 = w2r r Fs = - kx  t = t net = I a E= F q ÚE UE Us = 1 2 kx 2 rcm =  mr  m Ts = 2 p L = r ¥ p = Iw Tp = 2 p FG = - Gm1m2 UG = - Gm1m2 r w = w0 + at 1 2 at 2 1 q1q2 = qV = 4 p⑀0 r C = Q V C = k ⑀0 A d  Ci Cp = i 1 1 = Cs i Ci r2 area magnetic field capacitance distance electric field emf force current current density inductance length number of loops of wire per unit length number of charge carriers per unit volume power charge point charge resistance distance time potential or stored energy electric potential velocity or speed resistivity N = P = Q= q = R = r = t = U= V= u = r= fm = magnetic flux k = dielectric constant dQ dt 1 1 QV = CV 2 2 2 ÚB ∑ d ᐉ = m0 I m0 I d ᐉ ¥ r 4p r 3 dB = F= I = Neud A Bs = m0 nI V = IR g 1 2 Iw 2 i = = = = = e= F = I = J = L = = n = E = rJ m k u = rw q  rii 1 4 p⑀0 A B C d E r R= A 2p 1 = f w T = ⑀0 dV dr E =V = Q dA = ∑ Uc = I = Ú r 2 dm =  mr 2 q = q0 + w0 t + 1 q1q2 4 p⑀0 r 2 I = t=r¥F K = F = fm = Ú B ∑ dA Rs = ˆ r  Ri 1 = Rp =- e = -L 1 ÂR i d fm dt e i i P = IV FM = qv ¥ B -15- Ú I dᐉ ¥ B UL = dI dt 1 2 LI 2 ADVANCED PLACEMENT PHYSICS C EQUATIONS FOR 2008 and 2009 GEOMETRY AND TRIGONOMETRY Rectangle A = bh Triangle A= 1 bh 2 Circle A = pr 2 C = 2pr Parallelepiped V = wh Cylinder A= C= V= S = b = h = = w= r = CALCULUS area circumference volume surface area base height length width radius df d f du = dx du dx d n ( x ) = nx n -1 dx d x (e ) = e x dx d (1n x ) = 1 dx x d (sin x ) = cos x dx d (cos x ) = - sin x dx V = pr 2 Úx S = 2pr + 2pr 2 Úe Sphere V = 4 3 pr 3 Ú Right Triangle a c b c tan q = x dx = e x dx = ln x x Ú sin x dx = - cos x a 2 + b2 = c2 cos q = dx = Ú cos x dx = sin x S = 4pr 2 sin q = 1 x n + 1 , n π -1 n +1 n c a 90° q a b b -16- PHYSICS C: ELECTRICITY AND MAGNETISM SECTION II Time—45 minutes 3 Questions Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions, which are worth 15 points each. The parts within a question may not have equal weight. All final numerical answers should include appropriate units. Credit depends on the quality of your solutions and explanations, so you should show your work. Credit also depends on demonstrating that you know which physical principles would be appropriate to apply in a particular situation. Therefore, you should clearly indicate which part of a question your work is for. E&M. 1. A parallel-plate capacitor X, with plate area A and plate separation d, is filled with air and charged by connecting the capacitor through a switch to a power supply of emf e , as shown above. (a) Determine the magnitude of the charge q0 on each plate of the capacitor, in terms of the given quantities and fundamental constants. The switch is now flipped to the right, so that capacitor X is disconnected from the power supply and is instead connected to an uncharged parallel-plate capacitor Y. Capacitor Y has the same plate separation but twice the plate area and is filled with a material having dielectric constant 3. (b) Calculate the equilibrium charges qX and qY on capacitors X and Y, expressing your answers in terms of the initial charge q0 . Once the charges on the two capacitors have reached equilibrium, they are disconnected from one another. The dielectric is then removed from capacitor Y. (c) Indicate whether the energy stored in capacitor Y as the dielectric is removed increases, decreases or remains the same. ___ Increases ___ Decreases ___ Remains the same Justify your answer. (d) Indicate whether your answer to (c) requires that the electric force on the dielectric as it is removed pulls the dielectric into the capacitor, pushes it out, or is zero. ___ Pulls it in ___ Pushes it out ___ Is zero Justify your answer. (e) A resistor of resistance R is now connected to the two sides of capacitor Y. The capacitor is allowed to discharge through the resistor. Derive the equation describing the charge on capacitor Y as a function of time, expressed in terms of R, d, A, the initial charge qY on capacitor Y, and fundamental constants. GO ON TO THE NEXT PAGE. -17- E&M. 2. Electrons created at the filament at the left end of the tube represented above are accelerated through a voltage V0 and exit the tube. The electrons then move with constant speed to the right, as shown, before entering a region in which there is a uniform electric field between two parallel plates separated by a distance D. The electrons enter the field at point P, which is a distance y0 from the bottom plate, and are deflected toward that plate. Express your answers to the following in terms of V0 , D, y0 , and fundamental constants. (a) Calculate the speed of the electrons as they exit the tube. (b) i. Calculate the magnitude of the electric field required to cause the electrons to land the distance y0 from the edge of the plate. ii. Indicate the direction of the electric field. ____ To the left ____ Toward the top of the page ____ Into the page ____ To the right ____ Toward the bottom of the page ____ Out of the page Justify your answer. (c) Calculate the potential difference between the two plates required to produce the electric field determined in part (b). Suppose the electric field between the plates is replaced by a magnetic field and the electrons are to strike the lower plate at the same distance y0 from the edge of the plate. (d) i. Calculate the magnitude of the magnetic field. ii. Indicate the direction of the required magnetic field. ____ To the left ____ Toward the top of the page ____ Into the page ____ To the right ____ Toward the bottom of the page ____ Out of the page Justify your answer. GO ON TO THE NEXT PAGE. -18- E&M. 3. An astronaut on another planet erects a tall, thin, vertical, nonconducting, frictionless cylinder of radius r0 at one of the planet’s magnetic poles. Assume the magnetic field has an upward vertical component with a magnitude that varies with altitude z as Bz = B0 (1 - kz ) , where k is a constant. The horizontal component of the field has the constant value BH at the surface of the cylinder and points radially outward from the axis of the cylinder. A gold ring of resistance R that just fits around the cylinder is released from rest at some height above the ground. Assume air friction is negligible. Express all algebraic answers in terms of B0 , BH , k, r0 , R, and fundamental constants. (a) Determine the magnetic flux through the ring as a function of z . (b) Indicate whether the direction of the induced current in the ring, as seen from above the cylinder, is clockwise or counterclockwise. ___ Clockwise ___ Counterclockwise Justify your answer. (c) Calculate the power dissipated in the ring at the instant it is falling with speed u . (d) Calculate the magnitude of the net upward force on the ring due to the horizontal component of the field at the instant the ring is falling with speed u . (e) Is there a net force on the ring due to the vertical component of the field? Explain your reasoning. STOP END OF EXAM -19- Name: ____________________________________ AP® Physics C: Electricity and Magnetism Student Answer Sheet for Multiple-Choice Section No. 1 Answer No. 31 2 32 3 33 4 34 5 35 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 -20- Answer AP® Physics C: Electricity and Magnetism Multiple-Choice Answer Key No. 1 Correct Answer D No. 31 Correct Answer C 2 D 32 C 3 C 33 E 4 A 34 C 5 E 35 E 6 C 7 C 8 B 9 D 10 A 11 C 12 B 13 C 14 E 15 D 16 B 17 B 18 B 19 A 20 E 21 C 22 B 23 C 24 C 25 A 26 D 27 A 28 B 29 C 30 A -21- AP® Physics C: Electricity and Magnetism Free-Response Scoring Guidelines General Notes about AP Physics Practice Exam Scoring Guidelines 1. The solutions contain a common method of solving the free-response questions and the allocation of points for the solutions. Some also contain a common alternate solution. They are typical of draft guidelines developed before student solutions are available. Teachers should feel free to make modifications based on their students’ responses. 2. The scoring guidelines typically show numerical results using the value g = 9.8 m s2 , but use of 10 m s2 is of course also acceptable. Solutions usually show numerical answers using both values when they are significantly different. The following rules apply to the official scoring of AP Physics Exams. 3. All correct methods of solution receive appropriate credit for correct work. 4. Generally, double penalty for errors is avoided. For example, if an incorrect answer to part (a) is correctly substituted into an otherwise correct solution to part (b), full credit will usually be awarded. One exception to this may be cases when the numerical answer to a later part should be easily recognized as wrong, e.g., a speed faster than the speed of light in vacuum. 5. Implicit statements of concepts normally receive credit. For example, if use of the equation expressing a particular concept is worth 1 point, and a student’s solution contains the application of that equation to the problem but the student does not write the basic equation, the point is still awarded. However, when students are asked to derive an expression, it is normally expected that they will begin by writing one or more fundamental equations, such as those given on the AP Physics Exam equation sheet. For a description of the use of such terms as “derive” and “calculate” on the exams, and what is expected for each, see “The Free-Response Sections⎯Student Presentation” in the AP Physics Course Description. 6. Strict rules regarding significant digits are usually not applied to numerical answers. However, in some cases answers containing too many digits may be penalized. In general, two to four significant digits are acceptable. Numerical answers that differ from the published answer due to differences in rounding throughout the question typically receive full credit. Exceptions to these guidelines usually occur when rounding makes a difference in obtaining a reasonable answer. For example, suppose a solution requires subtracting two numbers that should have five significant figures and that differ starting with the fourth digit (e.g., 20.295 and 20.278). Rounding to three digits will lose the accuracy required to determine the difference in the numbers, and some credit may be lost. -22- AP® Physics C: Electricity and Magnetism Free-Response Scoring Guidelines Question 1 15 points total (a) Distribution of points 2 points For using the expressions relating charge to emf and capacitance to capacitor dimensions q0 = C e C = ⑀0 A d For the correct answer ⑀ A q0 = 0 e d (b) 1 point 1 point 5 points For correct use of conservation of charge q X + qY = q0 For indicating that at equilibrium the voltage is the same across each capacitor q q VX = VY = X = Y CX CY 1 point For a correct relationship between C X and CY 3⑀ (2 A) 6⑀0 A CY = 0 = = 6C X d d Combining these relationships to solve for q X q - qX qX = 0 CX 6C X 1 point 6q X = (q0 - q X ) For correct answers for both capacitors 1 q X = q0 7 6 qY = q0 7 Note: 1 point partial credit will be given if the correct charge for only one of the capacitors is given. -23- 1 point 2 points AP® Physics C: Electricity and Magnetism Free-Response Scoring Guidelines Question 1 (continued) Distribution of points (c) 2 points For correctly indicating that the stored energy increases as the dielectric is removed For a valid justification For example: Removing the dielectric will decrease the capacitance of capacitor Y by a factor of 3. 1 point 1 point The charge on the capacitor stays the same. So the stored energy U = q 2 2C will increase by a factor of 3. (d) 3 points For correctly indicating that the dielectric will be pulled in For a valid justification For example: Removing the dielectric increases the stored energy. This means that positive work was done on the system to remove it. This requires an external force to pull the dielectric material out, acting opposite to the electric force. So the electric force pulls the dielectric in. Note: Full credit can be given if correct reasoning is used, even if an incorrect conclusion is reached based on an incorrect answer in part (c). (e) 1 point 2 points 3 points For a correct expression of the voltage loop equation for the circuit q = IR CY q dq =- R CY dt A general form for the solution of this equation can be determined by integration or deduced by inspection. For giving the proper exponential form of the solution q(t ) = qi e - t RC Using the initial condition at t = 0 q(0) = qi = qY For the correct answer q(t ) = qY e -td 2⑀0 RA -24- 1 point 1 point 1 point AP® Physics C: Electricity and Magnetism Free-Response Scoring Guidelines Question 2 15 points total (a) Distribution of points 2 points For correct application of conservation of energy 1 m u 2 = eV0 2 e For the correct answer 2eV0 u = me 1 point 1 point (b) (i) 4 points For a correct application of the kinematic equation and Newton’s second law in the vertical direction 1 eE Dy = at 2 where Dy = y0 , a = 2 me Combining these equations and solving for t 1 eE 2 y0 = t 2 me 1 point t = 2 y0 me eE For a correct application of the kinematic equation in the horizontal direction Dx = u0 x t where Dx = y0 1 point For indicating that u0 x is the answer from part (a) 1 point 2eV0 2 y0 me me eE For the correct answer 4V E = 0 y0 1 point y0 = (ii) 2 points For correctly indicating that the electric field is directed toward the top of the page For a correct justification For example: Negative charges are accelerated in the direction opposite the electric field. Since the acceleration here is downward, the field must be upward. -25- 1 point 1 point AP® Physics C: Electricity and Magnetism Free-Response Scoring Guidelines Question 2 (continued) Distribution of points (c) 1 point V = Ed For the correct answer 4V0 D V = y0 1 point (d) (i) 4 points For equating the magnetic force to the centripetal force m u2 eu B = e y0 me u ey0 For using the result for u from part (a) m 2eV0 B= e ey0 me For the correct answer 2 meV0 B= 2 ey0 2 points B= (ii) 1 point 1 point 2 points For correctly indicating that the magnetic field must be directed into the page For a correct justification For example: Using the right-hand rule, a field out of the page would produce an initial downward force on positively charged particles. Since these are electrons, the field must be in the opposite direction. -26- 1 point 1 point AP® Physics C: Electricity and Magnetism Free-Response Scoring Guidelines Question 3 15 points total (a) Distribution of points 2 points fm = Bz A For using the correct area 1 point A= For the correct answer fm = B0 (1 - kz ) p r02 p r02 (b) 1 point 3 points For correctly indicating that the current is clockwise For a correct justification For example: The vertical component of the magnetic field and thus the flux is increasing with time because the magnetic field increases with decreasing z. The induced current will act to oppose the change in flux and create a downward field. 1 point 2 points Note: 1 point partial credit can be earned for stating just one of the two ideas above. (c) 5 points For indicating the need to take the derivative of the flux to calculate the induced emf For correctly taking the derivative df e = - dtm = p r02 B0 k dz dt dz For correctly equating with the velocity u dt df e = - dtm = p r02 B0 ku For applying a valid power equation P = e 1 point 1 point 1 point 1 point 2 R For the correct answer 2 p 2 r04 B0 k 2 u 2 P = R 1 point -27- AP® Physics C: Electricity and Magnetism Free-Response Scoring Guidelines Question 3 (continued) Distribution of points (d) 3 points F = I Bradial For the correct determination of the path length = 2 p r0 For the correct substitution of the current I e = p r 2 B0 ku I = 0 R R B ku F = p r02 0 (2 p r0 ) BH R For the correct answer 3 kB B u F = 2 p 2 r0 0 H R (e) 1 point 1 point 1 point 2 points For correctly indicating that there is no net force due to the vertical component of the magnetic field For a correct justification For example: For any pair of points on the ring located opposite each other, there are equal but opposite forces acting on the ring that cancel. -28- 1 point 1 point ...
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