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® AP Physics C: Electricity and Magnetism 2015 Free-Response Questions © 2015 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. Visit the College Board on the Web: www.collegeboard.org. AP Central is the official online home for the AP Program: apcentral.collegeboard.org. ADVANCED PLACEMENT PHYSICS C TABLE OF INFORMATION CONSTANTS AND CONVERSION FACTORS Proton mass, m p 1.67 10 27 kg Neutron mass, mn 1.67 10 27 kg Electron mass, me 9.11 10 31 kg 6.02 10 23 mol 1 Avogadro’s number, N 0 R Universal gas constant, 8.31 J (mol <K) 1.60 10 19 C 1 electron volt, 1 eV 1.60 10 19 J Speed of light, Universal gravitational constant, Acceleration due to gravity at Earth’s surface, 3.00 108 m s c 6.67 10 11 N<m 2 G kg 2 9.8 m s2 g 1.38 10 23 J K Boltzmann’s constant, k B 1 unified atomic mass unit, 1u 1.66 10 27 kg 931 MeV c 2 Planck’s constant, h 6.63 10 34 J <s 4.14 10 15 eV <s hc 1.99 10 25 J <m 1.24 103 eV < nm e0 8.85 10 12 C2 N < m 2 Vacuum permittivity, 1 4 pe0 Coulomb’s law constant, k m0 Vacuum permeability, m0 4 p Magnetic constant, k 1 atmosphere pressure, UNIT SYMBOLS e Electron charge magnitude, meter, kilogram, second, ampere, kelvin, PREFIXES Factor Prefix Symbol m kg s A K mole, hertz, newton, pascal, joule, 1 atm mol Hz N Pa J 9.0 109 N< m 2 C2 4 p 10 7 (T <m) A 1 10 7 (T< m) A 1.0 105 N m 2 watt, coulomb, volt, ohm, henry, W C V W H 1.0 105 Pa farad, tesla, degree Celsius, electron volt, F T C eV VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES D q 0 30D 37D 45D 53D 60D 90D 109 giga G sin q 0 12 35 2 2 45 3 2 1 106 mega M cos q 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 6 micro m 10 9 nano n 10 12 pico p 10 The following assumptions are used in this exam. I. The frame of reference of any problem is inertial unless otherwise stated. II. The direction of current is the direction in which positive charges would drift. III. The electric potential is zero at an infinite distance from an isolated point charge. IV. All batteries and meters are ideal unless otherwise stated. V. Edge effects for the electric field of a parallel plate capacitor are negligible unless otherwise stated. -2- ADVANCED PLACEMENT PHYSICS C EQUATIONS MECHANICS Ãx Ãx 0 a x t x x0 Ãx 0 t Ãx2 a F J p 1 2 at 2 x Ãx 0 2 a x x x0 2 Fnet m ÇF m dp dt Ô F dt Dp mv F f m FN Ô F dr DE W K 1 2 mà 2 P P dE dt F v DUg ac t a I mg Dh Ã2 r Us rF Çt I x t net I 2 Ô r dm Ç mr à rw L r p 1 2 Iw 2 w0 at q q0 w0 t 1 2 k Dx 2 xmax cos( wt f FE E FE q Ex DV V Q e0 dV dx Ô E dr DV Q C 1 q1q2 4pe0 r C k e0 A d Cp Ç Ci 1 Cs ÇC i 1 i i m k E Tp 2p g I I 1 f R FG Gm1m2 r2 Gm1m2 r Rs 1 Rp P -3- P Q q R r t U = = = = = = = V= v = r = F = k = FM 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 radius or distance time potential or stored energy electric potential velocity or speed resistivity flux dielectric constant qv B Ô B d dQ dt 2p 2p w UC = = = = = = = = = = = = N = q Ç rii i 1 4pe0 qV I F I J L n UE A B C d E ε Ts UG 1 2 at 2 1 q1q2 4pe0 r 2 Ô E dA 2 Iw w acceleration energy force frequency height rotational inertia impulse kinetic energy spring constant length angular momentum mass power momentum radius or distance period time potential energy velocity or speed work done on a system position coefficient of friction angle torque angular speed angular acceleration phase angle k D x 1 QDV 2 r A rJ T Ç mi xi Ç mi xcm K w2r a = E= F = f = h = I = J = K = k = = L = m= P = p = r = T = t = U= v = W= x = m = q = t = w = a = f = Fs ELECTRICITY AND MAGNETISM 1 C DV 2 2 dB F m0 I m0 I d r 4p r 2 Ô I d B Bs m0 nI Nevd A FB Ô B dA DV R e Ô E d e L Ç Ri i 1 ÇR i I DV i UL dI dt 1 2 LI 2 d FB dt ADVANCED PLACEMENT PHYSICS C EQUATIONS GEOMETRY AND TRIGONOMETRY Rectangle A bh 1 bh 2 Circle A pr2 C 2p r s rq Rectangular Solid V wh Cylinder V pr S 2p r 2p r df dx A = area C = circumference V = volume S = surface area b = base h = height = length w = width r = radius s = arc length q = angle Triangle A CALCULUS S d ax e dx aeax q 2 a cos ax d >cos ax @ dx a sin ax Ôe n dx ax dx dx Ôxa 4 3 pr 3 4p r 2 a 2 b2 1 x d >sin ax @ dx Ôx r Right Triangle 1 n 1 x , n 1 n 1 1 ax e a ln x a Ô cos ax dx 1 sin ax a Ô sin ax dx 1 cos ax a c2 sin q a c cos q b c tan q nx n 1 d ln ax dx Sphere V d n x dx s 2 d f du du dx a b c q VECTOR PRODUCTS A B AB cos q AB AB sin q a 90° b -4- 2015 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS 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. Show all your work in this booklet in the spaces provided after each part. E&M.1. A parallel-plate capacitor is constructed of two parallel metal plates, each with area A and separated by a distance D. The plates of the capacitor are each given a charge of magnitude Q, as shown in the figure above. Ignore edge effects. (a) i. On the figure above, draw an arrow to indicate the direction of the electric field between the plates. ii. On the figure above, draw an appropriate Gaussian surface that will be used to derive an expression for the magnitude of the electric field E between the plates. iii. Using Gauss’s law and the Gaussian surface from part (a)-ii, derive an expression for the magnitude of the electric field E between the plates. Express your answer in terms of A, D, Q, and physical constants, as appropriate. The space between the plates is now filled with a dielectric material that is engineered so that its dielectric constant varies with the distance from the bottom plate to the top plate, defined by the x-axis indicated in the Q i , where k is a diagram above. As a result, the electric field between the plates is given by E 0 x D e0 k 0 e A positive constant. Express all algebraic answers to the remaining parts in terms of A, D, Q, k0 , x, and physical constants, as appropriate. (b) Determine an expression for the dielectric constant k as a function of x. © 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -5- 2015 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (c) i. Write, but do NOT solve, an equation that could be used to determine the potential difference V between the plates of the capacitor. ii. Using the equation from part (c)-i, derive an expression for the potential difference VD V0 , where VD is the potential of the top plate and V0 is the potential of the bottom plate. (d) Determine the capacitance of the capacitor. (e) The energy stored in the capacitor that has a varying dielectric is UV . A second capacitor that has a constant dielectric of value k0 is also given a charge Q. The energy stored in the second capacitor is UC . How do the values of UV and UC compare? ____ UV UC ____ UV ! UC ____ UV UC Justify your answer. © 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -6- 2015 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS E&M.2. A student performs an experiment to determine the emf e and internal resistance r of a given battery. The student connects the battery in series to a variable resistance R, with a voltmeter across the variable resistor, as shown in the figure above, and measures the voltmeter reading V as a function of the resistance R. The data are shown in the table below. Trial # 1 2 3 4 5 6 Resistance W 0.50 1.0 2.0 3.0 5.0 10 Voltage V 5.6 7.4 9.4 10.6 10.9 11.4 1 R 1 W 1 V 1 V 2.00 1.00 0.50 0.33 0.20 0.10 0.179 0.135 0.106 0.094 0.092 0.088 (a) i. Derive an expression for the measured voltage V. Express your answer in terms of R, physical constants, as appropriate. e, r, and ii. Rewrite your expression from part (a)-i to express 1 V as a function of 1 R . © 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -7- 2015 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (b) On the grid below, plot data points for the graph of 1 V as a function of 1 R . Clearly scale and label all axes, including units as appropriate. Draw a straight line that best represents the data. (c) Use the straight line from part (b) to obtain values for the following. i. e ii. r (d) Using the results of the experiment, calculate the maximum current that the battery can provide. (e) A voltmeter is to be used to determine the emf of the battery after removing the battery from the circuit. Two voltmeters are available to take this measurement—one with low internal resistance and one with high internal resistance. Indicate which voltmeter will provide the most accurate measurement. ____ The voltmeter with low resistance will provide the most accurate measurement. ____ The voltmeter with high resistance will provide the most accurate measurement. ____ The two voltmeters will provide equal accuracy. Justify your answer. © 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -8- 2015 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS E&M. 3. A circular wire loop with radius 0.10 m and resistance 50 W is held in place horizontally in a magnetic field B directed upward at an angle of 60° with the vertical, as shown in the figure above. The magnetic field in the direction shown is given as a function of time t by B(t ) a 1 bt , where a = 4.0 T and b = 0.20 s 1 . (a) Derive an expression for the magnetic flux through the loop as a function of time t. (b) Calculate the numerical value of the induced emf in the loop. (c) i. Calculate the numerical value of the induced current in the loop. ii. What is the direction of the induced current in the loop as viewed from point P ? ____ Clockwise ____ Counterclockwise Justify your answer. (d) Assuming the loop stays in its current position, calculate the energy dissipated in the loop in 4.0 seconds. (e) Indicate whether the net magnetic force and net magnetic torque on the loop are zero or nonzero while the loop is in the magnetic field. Net magnetic force: ____ Zero ____ Nonzero Net magnetic torque: ____ Zero ____ Nonzero Justify both of your answers. STOP END OF EXAM © 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org. -9-