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1 Instructor: ALSIN, Michael Course: AP Physics Conductors & Capacitors Dielectrics Circuits RC Circuits Electric Field Magnetic Fields Inductance (LR, LC circuits) Electromagnetism Design Observe, Measure Analyze Data, Errors Gauss’s Law Lab Electrostatics E&M Question Day Date Name (LAST, First):_________________, __________________ Block (circle): 1 2 3 4 5 6 7 8 Date (MM/DD/YY): ___/___/___ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ 99.A.1 99.A.2 99.A.3 00.A.1 00.A.2 00.A.3 01.A.1 01.A.2 01.A.3 02.A.1 02.A.2 02.A.3 03.A.1 03.A.2 03.A.3 04.A.1 04.A.2 04.A.3 05.A.1 05.A.2 05.A.3 06.A.1 06.A.2 06.A.3 07.A.1 07.A.2 07.A.3 08.A.1 08.A.2 08.A.3 09.A.1 09.A.2 09.A.3 10.A.1 10.A.2 10.A.3 11.A.1 11.A.2 11.A.3 12.A.1 12.A.2 12.A.3 Test Analysis Form C 5/11/2012 10:29 AM 1 2 of 2 2 TABLE OF’ INFORMATION DEVELOPED FOR 2012 CONSTANTS AND CON\ ERSION FAQ WRS Proton mass, — 167 10 kg I it) kg 1 electron s di. I eV kg Speed of light. Neutron mass, or Electron mass, In As gadr’ = number. .\ 0 *11 l0 ,(L I th mal -, Bltzmann’ cnsmnt. = l3s . nmi.k lo . 1 u = I Planck’s constant. h = l = Vacuum perIllittis it. constant. k = 1 atmosphere pressure. PREFIXES Factor io S nev ton. A K pascal. joule, G mega NI cosO kilo k tan6 milli 300 10’ I0 C ni - :n S m It) kg = 03 .o3 x l0 J.5 = 4 14 ,. 1 1.09> 10 Jon = C Nin 1(1 mol Hz N Pa J I 124 0(1 x 10 Nnr C = 4r x 10 T.m A u,, 4 = 1 10 1 atm = 1.0 Y watt, coulomb, solt, ohm, henrs, MeV = 10’ eV.mn T.m,, A l0 N/ nf W C V = 1.0 x 101 Pa farad. tesla. degree Celsius. elect! on—s nit. F ‘F C e’ H VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES giga 10 r mole. hertz. Snibo1 centi I ni ke Prefix io- 1 (, — = I 4€ p Magnetic constant. k’ UNIT SYMBOLS I0 1 60 - 5.85 Vacuum permeability. meter. kiloeram. oml, ampere. kelvin. < J K I unitted at’niic mas unit. Coulomb’s law — = I nis era1 era itational en-.tant. Aceciera’ den due to eras it ,t Earta ‘tiflJ.O, - S.’ I i t nrs eral ca’ constant. ‘ 160 Electron charge magnitude, m micro It) nano n 10 pim p 60 90 0 1 4 5 3/2 1 i 2 I 3 5 1 ‘2 4 3 3 L 0 oo The fo1loing con> entions are used in this exam 1. Unless otherwise stated, the frame of reference of any pr&hlein is assumed to be inertial. 11. The direction f an> electric current is the direction of tlk u. of positise charge corn entional current I, 111. For any is lated electric charge, the electric potentid js defined as ze an infinite dismnce from the charge -22 at 3 ADVANCED PLACEMENT PHYSICS C EQUATIONS DES’ ELOPED FOR 2012 MECHANICS a I at c ‘v ,,at 1 a(r 2 r —t 1) ma F JFdr 3 p \p in * fr, <it\ JF.dr = 1 K mu — = cit P r) accelerati n tc ice 1= trequenc y height I— rotational mertia impulse K= kinetic, energy spring .a ntant length angular mc mentum In = mass N = normal lorc p wer momentum I’ = radius or distance position sector 7’ = period time potential energy velocity or speed w = work done on a system 5— pasition coefficient of friction mgh A B C d B qq, r 1 4irE F q F — F fr 1 J B E.JA B— Jr = — — = 1 n ‘ç’q r 1 ql Uj, icA d qq r A = P = Q 1 — q R r = t = V v p 1 — = (1 0= angle torque (0= angular speed angular acceleration phase angle - — ELEC RIC1TY AND MAGNETISM = = = area mgnetic field capautance distance electric field cml trce current current density inductance length number t I I ) ps ci sne per unit length number ol charge arriem per unit solume posser charge point charge resistance distance time potential or stored energy electric potential velocity oi speed resistisity magnetic flux 1 dielectric constant dQ cit = LI = = B*d€=p I —kg = t — t r — a’ F = a — * mr L T rw rxp = I = 2,r I V 77) - — - F pJ Mu A JR IdJxr fId€XB —pnl - JB.dA F 2ir— K 1 ,n Gm t((t r 1 R P R IV Gm,n 00 +144u1 B p 4t RR Jo> 7I (0—0) JB= R=1 A F wJ 11 —mrm r — cos(iut + c = = o 1 1(1 17 t, = 1 U = a’ B £ L’ 11 cit 4 -33 4 ADVANCED PLACEMENT PHYSICS C EQUATIONS DEVELOPEI) FOR 2012 GEOMETRY AND TRIGONOMETR Rectangle A area C = circumference V solume 5 = surface area b =base Ii = height / = length hh 1h 1 — Circle yrr A C = dl di — Triangle A CAL CUL1 S r 2,rr — ci’ do do dr d x ci dx ) fi ax - d (In v) dx radius d Rectangular Solid k = iirh cit (sini) ci Cy finder V S = — r2f 2rc = + 2 2r di 7 Jc Sphere r cit 1’ —-irr I— x J S = sinO = = ccsr = —sinx 1 x nfl , a —l mt sindr Right Triangle x — J cosxdx 4r a +b C = — —cosx = a — C b cosfl — 9O 12 C tanO a — -44 5 1999 PHYSICS C SECTION II, ELECTRICITY AND MAGNETISM 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. Shov all your work in the pink booklet in the spaces provided afler each part. NOT in this green insert. E&M I. An isolated conducting sphere of radius a (a) Determine the charge 0 Q = 0.20 m is at a potential of —2,000 V. on the sphere. The charged sphere is then concentrically surrounded by two uncharged conducting hemispheres of inner radius b 0.40 m and outer radius c 0.50 m, which are joined together as shown above, forming a spherical capacitor. A wire is connected from the outer sphere to ground, and then removed. (b) Determine the magnitude of the electric field in the following regions as a function of the distance r from the center of the inner sphere. i. ii. ijj iv r a < b r < < > a r r c < b < c (c) Determine the magnitude of the potential difference between the sphere and the conducting shell. (d) Determine the capacitance of the spherical capacitor. GO ON TO THE NEXT PAGE 5 6 1999 PHYSICS C—E & M B (out of page) P E&M 2. :::: N, 0 A uniform magnetic field B exists in a region of space defined by a circle of radius a 0.60 m as shown above. The magnetic field is perpendicular to the page and increases out of the page at a constant rate of 0.40T/s. A single circular loop of wire of negligible resistance and radius r 0.90 m is connected to a lightbulb with a resistance R = 5.0 Q, and the assembly is placed concentrically around the region of magnetic field. (a) Determine the emf induced in the loop. (b) Determine the magnitude of the current in the circuit. On the figure above, indicate the direction of the current in the loop at point 0. (c) Determine the total energy dissipated in the lightbulb during a 15 s interval. The experiment is repeated with a loop of radius b 0.40 in placed concentrically in the same magnetic field as before. The same lighthulh is connected to the loop, and the manetic field again increases out of the page at a rate of 0.40 Ts. Neglect any direct effects of the field on the lightbulb itself. (d) State whether the brightness of the bulb will be greater than. less than, or equal to the brightness of the bulb in part (a). Justify your answer. GO ON TO THE NEXT PAGE 6 7 1999 PHYSICS C—E & M V E&M 3. The nonconducting ring of radius R shown above lies in the vz-plane and carries a uniformly distributed positive charge Q. (a) Determine the electric potential at points along the x-axis as a function of x. (b) i. Show that the x-component of the electric field along the x-axis is given by Qx F — 4c ( 0 R ii. What are they- and z- components of the electric field along the v-axis? (C) Determine the following. i. The value of x for which E is a maximum ii. The maximum electric field E GO ON TO THE NEXT PAGE 7 8 1999 PHYSICS C—E & M (d) On the axes below, sketch E versus x for points on the xaxis from x tox = +2R. = —2R I —2R (e) —R An electron is placed at x R = 2R R/2 and released from rest. Qualitatively describe its subsequent motion. STOP END OF SECTION II, ELECTRICITY AND MAGNETISM 8 9 EEEEEEEEEEEEEEEEE 2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS PHYSICS C Section II, ELECTRICITY AND MAGNETISM Time—45 niinutes 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 the pink booklet in the spaces provided after each part, NOT in this green insert. A l02 B&M I Lightbulbs A, B, and C are connected in the circuit shown above, (a) List the bulbs in order of their brightness, from brightest to least bright. If any bulbs have the same brightness, state which ones. Justify your answer. £ =42 V A 10 2 Now a switch S and a 5.0 mH inductor are added to the circuit, as shown above. The switch is closed at time t=O. (b) Determine the currents ‘A’ ‘B’ and ‘c for the following times. i. Immediately after the switch is closed ii. A lone time after the switch is closed Copyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved. AP is a registered trademark of the College Entrance Examination Board. GO ON TO THE NEXT PAGE. -29 10 EEEEEEEEEEEEEEEEE 2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS (C) On the axes below, sketch the magnitude of the potential difference VL across the inductor as a function of time, from immediately after the switch is closed until a long time after the switch is closed. 0 E42V j 1 ° l2Qs. 5.0 l.LF (d) Now consider a similar circuit with an uncharged 5.0 pP capacitor instead of the inductor, as shown above. The switch is again closed at time t = 0. On the axes below, sketch the magnitude of the potential difference Vcap across the capacitor as a function of time, from immediately after the switch is closed until a long time after the switch is closed. ap 1 0 Copyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved. AP is a registered trademark of the College Entrance Examination Board. GO ON TO THE NEXT PAGE. -310 11 EEEEEEEEEEEEEEEEE 2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS E&M2. Three particles, A, B, and C, have equal positive charges Q and are held in place at the vertices of an equilateral triangle with sides of length f, as shown in the figures below, The dotted lines represent the bisectors for each side. The base of the triangle lies on the x -axis, and the altitude of the triangle lies on the y-axis. y V Q x Figure I Figure 2 (a) , the intersection of the three bisectors, locates the geometric center of the triangle and is one 1 i. Point P point where the electric field is zero. On Figure 1 above, draw the electric field vectors EA, EB, and E 1 due to each of the three charges. Be sure your arrows are drawn to reflect the relative magnitude at P of the fields. ii. Another point where the electric field is zero is point P 2 at (0, y ). On Figure 2 above, draw electric 2 field vectors EA, EB, and E at P 2 due to each of the three point charges. Indicate below whether the magnitude of each of these vectors is greater than, less than, or the same as for point P . 1 1 Greater than at P 1 1 Less than at P 1 1 The same as at P EA ‘B Copyright © 2000 College Entrance Examination Board and Educational Testing Service, All rights reserved, AP is a registered trademark of the College Entrance Examination Board, GO ON TO THE NEXT PAGE. -411 12 EEEEEEEEEEEEEEEEE 2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS (h) Explain why the x-component of the total electric field is zero at any point on the v-axis. (c) Write a general expression for the electric potential V at any point on the y-axis inside the triangle in terms of Q, 1, and v. 2 at (d) Describe how the answer to part (c) could be used to determine the v-coordinates of points P 1 and P which the electric field is zero. (You do not need to actually determine these coordinates.) E & M 3. A capacitor consists of two conducting, coaxial, cylindrical shells of radius a and b. respectively, and length L>> b. The space between the cylinders is filled with oil that has a dielectric constant w. Initially both cylinders are uncharged, but then a battery is used to charge the capacitor, leaving a charge +Q on the inner cylinder and —Q on the outer cylinder, as shown above. Let r be the radial distance from the axis of the capacitor. (a) Using Gauss’s law, determine the electric field midway along the length of the cylinder for the following values of r. in terms of the given quantities and fundamental constants. Assume end effects are negligible. i. a<r<b ii. b < r < < L (h) Determine the following in terms of the given quantities and fundamental constants. i. The potential difference across the capacitor ii. The capacitance of this capacitor Copyright © 2000 College Entrance Examination Board and Educational Tcsting Service. All rights reserved. AP is a registered trademark of the College Entrance Examination Board, GO ON TO THE NEXT PAGE. 12 13 EEEEEEEEEEEEEEEEE 2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS (c) Now the capacitor is discharged and the oil is drained from it. As shown above, a battery of emf £ is connected to opposite ends of the inner cylinder and a battery of emf 32 is connected to opposite ends of the outer cylinder. Each cylinder has resistance R. Assume that end effects and the contributions to the magnetic field from the wires are negligible. Using Ampere’s law, determine the magnitude B of the magnetic field midway along the length of the cylinders due to the current in the cylinders for the following values of r. i. a<r<b ii. b<r<<L STOP END OF SECTION II, ELECTRICITY AND MAGNETISM IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON SECTION II, ELECTRICITY AND MAGNETISM, ONLY. DO NOT TURN TO ANY OTHER TEST MATERIALS. Cop) right © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved. AP is a registered trademark of the College Entrance Examination Board. -613 ___ _______ 14 2001 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS PHYSICS C Section II, ELECTRICITY AND MAGNETISM 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 the pink booklet in the spaces provided after each part, NOT in this green insert. 2km Ground Ground • 2 km 3km t+30C Note: Figures not drawn to scale. -‘- —30 C E&M 1. A thundercloud has the charge distribution illustrated above left. Treat this distribution as two point charges, a negative charge of —30 C at a height of 2 km above ground and a positive charge of +30 C at a height of 3 km. The presence of these charges induces charges on the ground. Assuming the ground is a conductor, it can be shown that the induced charges can be treated as a charge of +30 C at a depth of 2 km below ground and a , which is just above the ground 1 charge of—30 Cat a depth of 3 1cm, as shown above right. Consider point P . 1 directly below the thundercloud, and point P , which is 1 km horizontally away from P 2 . 1 (a) Determine the direction and magnitude of the electric field at point P (b) . 2 i. On the diagram above, clearly indicate the direction of the electric field at point P ? 1 ii. How does the magnitude of the field at this point compare with the magnitude at point P Equal Greater Less Justify your answer (c) Letting the zero of potential be at infinity, determine the potential at these points. 1 i. Point P ii. Point “2 . 1 (d) Determine the electric potential at an altitude of 1 km directly above point P (e) Determine the total electric potential energy of this arrangement of charges. Copyright © 2001 by College Entrance Examination Board. All rights reserved, Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. GO ON TO THE NEXT PAGE. 8 14 15 2001 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS 10 > 2 pm 8 6 4 V C (I 20 40 60 $() IOU I 20 140 160 180 20f) Time (mm) E&M2. You have been hired to determine the internal resistance of 8.0 pF capacitors for an electronic component manufacturer. (Ideal capacitors have an infinite internal resistance—that is, the material between their plates is a perfect insulator. In practice, however, the material has a very small, but nonzero, conductivity.) You cannot simply connect the capacitors to an ohmmeter, because their resistance is too large for an ohmmeter to measure. Therefore you charge the capacitor to a potential difference of 10 V with a battery, disconnect it from the battery and measure the potential difference across the capacitor every 20 minutes with an ideal voltmeter, obtaining the graph shown above. (a) Determine the internal resistance of the capacitor. The capacitor can be approximated as a parallel-plate capacitor separated by a 0.10 mm thick dielectric with =5.6. (h) Determine the approximate surface area of one of the capacitor “plates.” (c) Determine the resistivity of the dielectric. (d) Determine the magnitude of the charge leaving the positive plate of the capacitor in the first 100 mm, Cops right © 21XJ1 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. GO ON TO THE NEXT PAGE. 9 15 16 2001 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS ,_-Fixed Horizontal Wires d_____ —Rod (mass in. resistance R) Cable E&M 3. The circuit shown above consists of a battery of ernf 8 in series with a rod of length £, mass in, and resistance R. The rod is suspended by vertical connecting wires of length d, and the horizontal wires that connect to the battery are fixed. All these wires have negligible mass and resistance. The rod is a distance r above a conducting cable. The cable is very long and is located directly below and parallel to the rod. Earth’s gravitational pull is toward the bottom of the page. Express all algebraic answers in terms of the given quantities and fundamental constants. (a) What is the magnitude and direction of the current I in the rod? (b) In which direction must there be a current in the cable to exert an upward force on the rod? Justify your answer. (c) With the proper current in the cable, the rod can be lifted up such that there is no tension in the connecting wires. Determine the minimum current I in the cable that satisfies this situation. (d) Determine the magnitude of the magnetic flux through the circuit due to the minimum current I determined in part (c). END OF SECTION II, ELECTRICITY AND MAGNETISM Copyright © 2001 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 10 16 17 2002 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS PHYSICS C Section II, ELECTRICITY AND MAGNETISM Timc—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 the pink booklet in the spaces provided after each part, NOT in this green insert. ,R 600,,’ ‘R=O.lO m E&M 1. A rod of uniform linear charge density ) +1.5 x 10 C/rn is bent into an arc of radius R is placed with its center at the origin of the axes shown above. 0.10 m. The arc (a) Determine the total charge on the rod. (h) Determine the magnitude and direction of the electric field at the center 0 of the arc. (c) Determine the electric potential at point 0. A proton is now placed at point 0 and held in place. Ignore the effects of gravity in the rest of this problem. (d) Determine the magnitude and direction of the force that must be applied in order to keep the proton at rest. (e) The proton is now released, Describe in words its motion for a long time after its release. Copyright @ 2002 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board, 5 GO ON TO THE NEXT PAGE. 17 ________IC ___ ___Wv___ 18 2002 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS s R—50k2 E&M 2. Your engineering firm has built the RC circuit shown above. The current is measured for the time t after the switch is closed at r = 0 and the best-fit curve is represented by the equation 1(r) milliamperes and t is in seconds. = 5.20 e 110 where I is in (a) Determine the value of the charging voltage V 0 predicted by the equation. (b) Determine the value of the capacitance C predicted by the equation. (c) The charging voltage is measured in the laboratory and found to be greater than predicted in part (a). i. Give one possible explanation for this finding. ii. Explain the implications that your answer to part i has for the predicted value of the capacitance. (d) Your laboratory supervisor tells you that the charging time must be decreased. You may add resistors or capacitors to the original components and reconnect the RC circuit. In parts i and ii below, show how to reconnect the circuit, using either an additional resistor or a capacitor to decrease the charging time. i. Indicate how a resistor may he added to decrease the charging time. Add the necessary resistur and connections to the following diagram. Wv T ii. Instead of a resistor, use a capacitor. Indicate how the capacitor may be added to decrease the charging time, Add the necessary capacitor and connections to the following diagram. T Cops right © 2X)2 h’ Cotlege Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 6 GO ON TO THE NEXT PAGE. 18 19 2002 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS Vertical E&M 3. A circular wire loop with radius 0.10 m and resistance 50 Q is suspended horizontally in a magnetic field of magnitude B directed upward at an angle of 60° with the vertical, as shown above. The magnitude of the field in teslas is given as a function of time t in seconds by the equation B = 4(1 C. 2t). — (a) Determine the magnetic Ilux through the ioop as a function of time. (h) Graph the magnetic flux rn as a function of time Ô,, (T Ofl the axes below. ) 2 m 010 OJ)5 0 - 1 2 4 5 6 7 8 9 10 ‘—i(s) —0,05 —0.10 Copyright © 2002 by College Entrance Examination Board, Alt rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board, 7 GO ON TO THE NEXT PAGE. 19 20 2002 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (c) Determine the magnitude of the induced emf in the loop. (d) i. Determine the magnitude of the induced current in the ioop. ii. Show the direction of the induced current on the following diagram. Vertical (e) Determine the energy dissipated in the ioop from t = 0 to t = 4 s. END OF SECTION II, ELECTRICITY AND MAGNETISM 8 20 21 2003 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS PHYSICS C Section IL ELECTRICITY AND MAGNETISM 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 the pink booklet in the spaces provided after each part. NOT in this green insert. I E&M. 1. A spherical cloud of charge of radius R contains a total charge varies according to the equation p(r) +Q with a nonuniform volume charge density that for r = R and — p = 0 for r > R, where r is the distance from the center of the cloud. Express all algebraic answers in terms of fundamental constants. Q. R. and (a) Determine the following as a function of r for r > R. i. The magnitude E of the electric field ii. The electric potential V (b) A proton is placed at point P shown above and released. Describe its motion for a long time after its release. (c) An electron of charge magnitude e is now placed at point P. which is a distance r from the center of the sphere, and released, Determine the kinetic energy of the electron as a function of r as it strikes the cloud. (d) Derive an expression for Po’ (e) Determine the magnitude E of the electric field as a function of r for r R. Copyright © 2003 h College Entrance Examination Board. All rights reser.ed. Availahle to AP professionals at apcentral.collegeboardcom and to students and parents at swwcollegeboard.cons/apstudents. GO ON TO THE NEXT PAGE. 5 21 22 2003 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS E&M, 2. In the laboratory, you connect a resistor and a capacitor with unknown values in series with a battery of emf E = 12 V You include a switch in the circuit. When the switch is closed at time t = 0, the circuit is completed. and you measure the current through the resistor as a function of time as plotted below. . i(A) 4 8 12 A data-fitting program finds that the current decays according to the equation (s) 16 i(t) = E e -rj4 (a> Using common symbols for the battery, the resistor, the capacitor, and the switch, draw the circuit that you constructed. Show the circuit before the switch is closed and include whatever other devices you need to measure the current through the resistor to obtain the above plot. Label each component in your diagram. (b) Having obtained the curve shown above, determine the value of the resistor that you placed in this circuit. (c) What capacitance did you insert in the circuit to give the result above? Copyright © 2003 by College Entrance Examination Board, All rights reserved. Available to AP professionals at apcentral.coI1egeboardcom and to students and parents at ww.collegeboardcom/apstudents. GO ON TO THE NEXT PAGE. 6 22 23 2003 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS You are now asked to reconnect the circuit with a new switch in such a way as to charge and discharge the capacitor. When the switch in the circuit is in position A, the capacitor is charging; and when the switch is in position B, the capacitor is discharring, as represented by the graph below of voltage V across the capacitor as a function of time. ‘‘c (V) Switch at A Charging Switch at B Discharging 12 6 ,1 t (s) (d) Draw a schematic diagram of the RC circuit that you constructed that would produce the graph above. Clearly indicate switch positions A and B on your circuit diagram and include whatever other devices you need to measure the voltage across the capacitor to obtain the above plot. Label each component in your diagram. Copyright cC 2003 by College Entrance Examinanon Board All rights resersed. Asadable to AP professionals at apcentralcollegeboardeoni and to students and parents at wv ss collegeboardeom!apstudents. GO ON TO THE NEXT PAGE. 7 23 24 2003 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS w S Vertical • V E N Antenna \ Top View E&M. \ Side View 3. An airplane has an aluminum antenna attached to its wing that extends 15 m from wingtip to wingtip. The plane is traveling north at 75 m/s in a region where Earth’s magnetic field has both a vertical component and a northward component, as shown above. The net magnetic field is at an angle of 55 degrees from horizontal and has a magnitude of 6.0 x i- T (a) On the figure below, indicate the direction of the magnetic force on electrons in the antenna. Justify your answer. ni (b) Determine the magnitude of the electric field generated in the antenna. (c) Determine the potential difference between the ends of the antenna. (d) On the figure below, indicate which end of the antenna is at higher potential. -I Copyright © 2003 by College Entrance Examination Board, All rights reserved. Available to AP professionals at apcentral.collegeboardcom and to students and parents at wwwcollegeboardcom/apstudents. GO ON TO THE NEXT PAGE. 8 24 25 2003 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (e) The ends of the antenna are now connected by a conducting wire so that a closed circuit is formed. i. Describe the condition(s) that would be necessary for a current to be induced in the circuit. Give a specific example of how the condition(s) could be created. ii. For the example you gave in i. above, indicate the direction of the current in the antenna on the figure below. END OF SECTION II, ELECTRICITY AND MAGNETISM Copyright © 2003 by College Entrance Examination Board, All rights reserved, Available to AP professionals at apcentralcollegeboardcom and to students and parents at wwwcollegeboard.com/apswdents. 9 25 26 2004 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS PHYSICS C Section II, ELECTRICITY AND MAGNETISM Tinie—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 the booklet in the spaces provided after each part, NOT in this green insert. a Conductor Cross Section E&M. 1. The figure above left shows a hollow, infinite, cylindrical, uncharged conducting shell of inner radius ij and outer radius r-, An infinite line charge of linear charge density +. is parallel to its axis but off center. An enlarged cross section of the cylindrical shell is shown above right. . (a) On the cross section above right, i. sketch the electric field lines, if any, in each of regions I, II, and III and ii. use + and — signs to indicate any charge induced on the conductor. (h) In thes paces below, rank the electric potentials at points a, b. c. d. and e from highest to lowest (1 = highest potential>. If two points are at the same potential. give them the same number. Copyright © 2004 by College Entrance Examination Board. All rights reserved. Visit apcentralcollegeboardcom (for AP professionals) and www,collegeboardcomJapstudcnts (for AP students and parents). GO ON TO THE NEXT PAGE. 5 26 27 2004 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS Cross Section (c) The shell is replaced by another cylindrical shell that has the same dimensions but is nonconducting and carries a uniform volume charge density +p. The infinite line charge, still of charge density +2, is located at the center of the shell as shown above. Using Gauss’s law, calculate the magnitude of the electric field as a function of the distance r from the center of the shell for each of the following regions. Express your answers in terms of the given quantities and fundamental constants. i. ii. iii. I it r<, i ri r > r, Copvriht © 2004 1w College Entrance Examination I3oard. All rights reserved. ip entral o11e.Lhoard oin ((or \P piotes1onals) and ssvs Lollerboard c’mfaptudcnts (for AP stiiduits nid parert GO ON TO THE NEXT PAGE. 6 27 28 2004 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS 14 ‘l2 —.---.-- 20 V -- 1 R lskc2: OPFT 2 0 0 10 15 Time (s) E&M. 2. In the circuit shown above left, the switch S is initially in the open position and the capacitor C is initially uncharged. A voltage probe and a computer (not shown) are used to measure the potential difference across the capacitor as a function of time after the switch is closed. The graph produced by the computer is shown above right. The battery has an emf of 20 V and negligible internal resistance. Resistor R 1 has a resistance of 15 k2 and the capacitor C has a capacitance of 20 jiF. (a) Determine the voltage across resistor R 2 immediately after the switch is closed. (b) Determine the voltage across resistor 1?-, a long time after the switch is closed. . 2 (c) Calculate the value of the resistor K (d) Calculate the energy stored in the capacitor a long time after the switch is closed. Copyright © 2004 by College Entrance Examination Board. All rights nserved. Visit apcentralcollegehoardcorn (for AP professionals) and www.collegeboardcom/apstudents (for AP students and parents), GO ON TO THE NEXT PAGE. 7 28 29 2004 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (e) On the axes below, graph the current in I?, as a function of time from 0 to 15 s. Label the vertical axis with appropriate values. LLLJ LLL JJ ——————-—— i— - L1. L_JJ L LJ_JL L_LL_ —-—-—- Current in R, -•---- ——--—4—4—- ___.4”__ _J L. —-1 ——fr-— -•---- --H----H———— - __4_._I_ —-———— -.-—4—.3-—I.—. JLL 0 10 15 Time (s) Resistor R 2 is removed and replaced with another resistor of lesser resistance. Switch S remains closed for a long time. (f) Indicate below whether the energy stored in the capacitor is greater than, less than, or the same as it was with resistor 1?, in the circuit. Greater than Less than The same as Explain your reasoning. Copyright © 2004 by College Entrance Examination Board. All rights reserved. Visit apcentraLcollegeboarcLcom (for AP professionals) and wwwcollegeboardconifapstudents (for AP students and parents). GO ON TO THE NEXT PAGE. 8 29 30 2004 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS 4e T 3e + I E&M. 3. A rectangular loop of dimensions 3 £ and 4 £ lies in the plane of the page as shown above. A long straight wire also in the plane of the page carries a current I. (a) Calculate the magnetic flux through the rectangular loop in terms of 1, £, and fundamental constants. Starting at time t 1(t) = = 0, the current in the long straight wire is given as a function of time t by e, where 0 I 1 and k are constants. (b) The current induced in the loop is in which direction? Counterclockwise Clockwise Justify your answer. The loop has a resistance R. Calculate each of the following in terms of R, J, k, £, and fundamental constants. (c) The current in the loop as a function of time t (d) The total energy dissipated in the loop from t = 0 to t = END OF SECTION II, ELECTRICITY AND MAGNETISM Copyright © 2004 by College Entrance Examination Board. All rights reserved. Visit apcentraLcollegeboardcom (for AP professionals) and wwwcollegeboar&comfapstudents (for AP students and parents). 9 30 31 2005 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS PHYSICS C Section II, ELECTRICITY AND MAGNETISM 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 the pink booklet in the spaces provided after each part. NOT in this green insert. (in) (3.1 f\J 008 —ê4 (Hi) E&M. 1. Consider the electric field diagram above. (a) Points A, B, and C are all located at y 0.06 rn. i. At which of these three points is the magnitude of the electric field the greatest? Justify your answer. ii. At which of these three points is the electric potential the greatest? Justify your answer. (b) An electron is released from rest at point B. i, Qualitatively describe the electron’s motion in terms of direction, speed. and acceleration. ii. Calculate the electron’s speed after it has moved through a potential difference of 10 V. (c) Points B and C are separated by a potential difference of 20 V. Estimate the magnitude of the electric field midway between them and state any assumptions that you make. (d) On the diagram, draw an equipotential line that passes through point D and intersects at least three electric field lines. Copyright © 2005 by College Entrance Examination Board. All rights reserved. Visit apcentraLcollegeboardcom (for AP professionals) and wwwcollegeboard.com/apstudents (for AP students and parents). GO ON TO THE NEXT PAGE. 5 31 32 2005 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS E&M, 2. In the circuit shown above, resistors 1 and 2 of resistance R 1 and R 2 respectively, and an inductor of inductance L are connected to a battery of emf £ and a switch S. The switch is closed at time t = 0. Express all algebraic answers in terms of the given quantities and fundamental constants. , (a) Determine the current through resistor I immediately after the switch is closed. (b) Determine the magnitude of the initial rate of change of current, d!/dr. in the inductor. (c) Determine the current through the battery a long time after the switch has been closed. (d) On the axes below, sketch a graph of the current through the battery as a function of time. Current Time Some time after steady state has been reached, the switch is opened. (e) Determine the voltage across resistor 2 just after the switch has been opened. Copyright P 2005 1w College Entrance Examination Board, All rights reserved. Visit apcentraLcollegehoard.com (for AP professionals) and www.collegeboard.com/apstudents for AP students and parentsL GO ON TO THE NEXT PAGE. 6 32 33 2005 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS I I.tl IIOhc \lctcr 00 1> (_) c:,; lCml 0 () ) I _4, 30 I 50 6) 7 I \I) I ‘)) 1 H) E&M. 3. A student performs an experiment to measure the magnetic field along the axis of the long, 100-turn solenoid PQ shown above. She connects ends P and Q of the solenoid to a variable power supply and an ammeter as shown. End P of the solenoid is taped at the 0 cm mark of a meterstick. The solenoid can be stretched so that the position of end Q can be varied. The student then positions a Hall probe* in the center of the solenoid to measure the magnetic field along its axis. She measures the field for a fixed current of 3.0 A and various positions of the end Q. The data she obtains are shown below. Trial 1 Position of End (cm) 40 Q Measured Magnetic Field (T) (directed from P to Q) n (turns/rn) 9,70xl0 2 50 7.70xl0 3 60 6.80x10 4 80 4.90xl0 100 4.00xl0’ (a) Complete the last column of the table above by calculating the number of turns per meter. * A Hall Probe is a device used to measure the magnetic field at a point. Copyright C 2005 by College Entrance Examination Board. AU rights reserved. Visit apcentralcollegeboard.com (for AP professionals) and www.collegehoard.com/apstudents (for AP students and parents. GO ON TO THE NEXT PAGE. 7 33 34 2005 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (b) On the axes below, plot the measured magnetic field B versus n. Draw a best-fit straight line for the data points. r 10.0 L.i.. J L_LL .L,L.11 .IjJJ ..Li .JLL.L ; 9.0 I .-.•_4 4. t .-—4-- 1 T 8.0 ._a i ‘“t 70 -— — -t -1— --i———r-—r--r — —.—-4— 4.0 r 2.0 rr’ .,LLi. IJJJ •i•i•• .LLI I 0 LL_ TT1 .IJ 1 5. I -, —‘r—i—i—-1—— I I ‘__—I__ I I 1 I I ; ., ,- - . I I I . . I _a_ _J_J__L_L_ _i,J —-1——I——--—i-— —+—‘t—-I——I—— -—-—+‘—-,‘—-t— I 1 I 50 100 .4... ---—r—i .J .‘ T11 1 A) I rr”” ‘T’1 ‘1- JI —...——— , ; F—f’—-4 —+— —r—- 3.0 — I: 6,u 5.0 , ——— L_L 150 LJ — ------ 200 250 300 , I (c) From the graph, obtain the value of ,u , the magnetic permeability of vacuum. 0 (d) Using the theoretical value of # value of p 0 computed in part (c). = 4ir x lO (T.m)/A, determine the percent error in the experimental END OF SECTION II, ELECTRICITY AND MAGNETISM Copyright 2005 by College Entrance Examination Board, All rights reserved. Viot apcentral.collegeboard.com for 3 Al professionals) and www.collegehoard.com!apstudents for AP students and parents;. 8 34 35 2006 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 the pink booklet in the spaces provided after each part. NOT in this green insert, -Q -Q P E&M 1. The square of side a above contains a positive point charge +Q fixed at the lower left corner and negative point charges —Q fixed at the other three corners of the square. Point P is located at the center of the square. (a) On the diagram, indicate with an arrow the direction of the net electric field at point P. (b) Derive expressions for each of the following in terms of the given quantities and fundamental constants. i.The magnitude of the electric field at point P ii. The electric potential at point P (c) A positive charge is placed at point P. It is then moved from point P to point R, which is at the midpoint of the bottom side of the square. As the charge is moved, is the work done on it by the electric field positive, negative, or zero? Positive Negative Zero Explain your reasoning. (d) i. Describe one way to replace a single charge in this configuration that would make the electric field at the center of the square equal to zero. Justify your answer. ii. Describe one way to replace a single charge in this configuration such that the electric potential at the center of the square is zero but the electric field is not zero. Justify your answer. © 2006 The College Board. All rights reserved. Visit apcentraleoilegehoardcorn (for AP professionals) and wwwcollegeboardcom/apstudents tor students and parents). GO ON TOTHE NEXT PAGE. 5 35 36 2006 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS K. E&M 2. The circuit above contains a capacitor of capacitance C, a power supply of emf E, two resistors of resistances R 1 and R 2 and two switches. S 1 and S . Initially, the capacitor is uncharged and both switches are open. Switch S 2 1 then gets closed at time t = 0. . (a) Write a differential equation that can be solved to obtain the charge on the capacitor as a function of time (b) Solve the differential equation in part (a) to determine the charge on the capacitor as a function of time t. t. Numerical values for the components are given as follows: E = 12 V C=0.060F 1 = R-, = 4700 2 R (c) Determine the time at which the capacitor has a voltage 4.0 V across it. After switch S has been closed for a long time, switch 5, gets closed at a new time r = 0. (d) On the axes below, sketch graphs of the current I in R 1 versus time and of the current 12 in K. versus time, beginning when switch 5, is closed at new time t = 0. Clearly label which graph is I and which is 12 Current rine V 2006 The College Board. All rights reserved. Visit apcentralcollegeboard.com (for Al professionals) and www.eollegehoardcomlapstuaents for students arid parenrs(. GO ON TO THE NEXT PAGE. 6 36 37 2006 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS ix )< X )< X w —H E&M 3. A loop of wire of width w and height/i contains a switch and a battery and is connected to a spring of force constant k, as shown above. The loop carries a current tin a clockwise direction, and its bottom is in a constant, uniform magnetic field directed into the plane of the page. (a) On the diagram of the loop below, indicate the directions of the magnetic forces, if any, that act on each side of the loop. (b) The switch S is opened, and the loop eventually comes to rest at a new equilibrium position that is a distance x from its former position. Derive an expression for the magnitude B 0 of the uniform magnetic field in terms of the given quantities and fundamental constants. © 2006 The College Board. All rights reserved, Visit apcentraLollegeboard.com (for AP professionals) and www.collegeboardcom/apstudents tor students and parents). GO ON TO THE NEXT PAGE. 7 37 ______ ______ ______ 38 2006 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS The spring and loop are replaced with a loop of the same dimensions and resistance R but without the battery and switch. The new loop is pulled upward, out of the magnetic field, at constant speed v 0 Express algebraic answers to the following questions in terms of B . v 0 , R. and the dimensions of the loop. 0 (c) i. On the diagram of the new loop below, indicate the direction of the induced current in the loop as the loop moves upward. f 0 LI x x x x x x x x x x x x x x x x ii. Derive an expression for the magnitude of this current. (d) Derive an expression for the power dissipated in the loop as the loop is pulled at constant speed Out of the field. (e) Suppose the magnitude of the magnetic field is increased. Does the external force required to pull the loop at speed v 0 increase, decrease, or remain the same? Increases Justify your Decreases Remains the same answer, END OF EXAM 200h The College Board. All rights reserved. \ sit apcentraleoileeehoard.eom for AP proteional and wwwcollegehoard.eorrvaestuuents tor tudrnts ani paants 8 38 39 2007 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS PHYSICS C: ELECTRICITY AND MAGNETISM SECTION II Tirne—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 the pink booklet in the spaces provided after each part, NOT in this green insert. Switch - __C=4000,uF E&M I. A student sets up the circuit above in the lab. The values of the resistance and capacitance are as shown, but the constant voltage E delivered by the ideal battery is unknown. At time t = 0, the capacitor is uncharged and the student closes the switch. The current as a function of time is measured using a computer system. and the following graph is obtained. 2.5 2.0 0.5 1 1-I -iI 1 ---- 1.5 1.0 T I —-—— ‘—-----1 --- •t+•••+••+..64,4_ — o 2 4 6 lime (s) 8 , 10 (a) Using the data above, calculate the battery voltage S. (b) Calculate the voltage across the capacitor at time t = 4.0 s. (c) Calculate the charge on the capacitor at t 4,0 s. © .OO7 The College Board. All rights reserved. Visit apcentralcollegeboard.com (for AP professionals) and www.collegehoard.com/apstudents for students and parents). GO ON TO THE NEXT PAGE. -539 ______Greater ______Less ________________________ ______The 40 2007 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (d) On the axes below, sketch a graph of the charge on the capacitor as a function of time. o — 10 Time (s) (e) Calculate the power being dissipated as heat in the resistor at t = 4.0 s. (f) The capacitor is now discharged, its dielectric of constant = 1 is replaced by a dielectric of constant = 3. and the procedure is repeated. Is the amount of charge on one plate of the capacitor at t = 4.0 s now greater than. less than, or the same as before’? Justify your answer. than than same © 2007 The College Board, All rights reserved, Visit apcentral.colIeehoardcom (for AP professionals) and www.collegehoardcom/apstudents for students and parents). GO ON TO THE NEXT PAGE. -640 41 2007 AP® PHYSICS C ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS E&M 2. In the figure above, a nonconducting solid sphere of radius a with charge +Q uniformly distributed throughout its volume is concentric with a nonconducting spherical shell of inner radius 2a and outer radius 3a that has a charge —Q uniformly distributed throughout its volume. Express all answers in terms of the given quantities and fundamental constants. (a) Using Gauss’s law, derive expressions for the magnitude of the electric field as a function of radius r in the following regions. i. Within the solid sphere (r <a) ii. Between the solid sphere and the spherical shell (a <r iii. Within the spherical shell (2a <r iv. Outside the spherical shell (r > < < 2a) 3a) 3a) (b) What is the electric potential at the outer surface of the spherical shell (r (c) Derive an expression for the electric potential difference V — = 3a )? Explain your reasoning. V between points X and Y shown in the figure. 2007 The College Board. All rights reserved. Visit apeentra1collegeboardom for AP professionals and wwco11egeboardcomJapstudents for students and parents GO ON TO THE NEXT PAGE. -741 ______Clockwise ______Counterclockwise 42 2007 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS x x x x x x x x x x x x x x x x x x x x x x x P x v B L x x x x x x x x x x x x Q x x x x x x x x x x x x Top View E&M 3. In the diagram above, a nichrorne wire of resistance per unit length 2 is bent at points P and Q to form horizontal conducting rails that are a distance L apart. The wire is placed within a uniform magnetic field of magnitude B pointing into the page. A conducting rod of negligible resistance, which was aligned with end PQ at time t = 0. slides to the right with constant speed v and negligible friction. Express all algebraic answers in terms of the given quantities and fundamental constants. (a) Indicate the direction of the current induced in the circuit. Justify your answer. (b) Derive an expression for the magnitude of the induced current as a function of time t. (C) Derive an expression for the magnitude of the magnetic force on the rod as a function of time. (d) On the axes below, sketch a graph of the external force f, as a function of time that must be applied to the rod to keep it moving at constant speed while in the field. Label the values of any intercepts. Time o (e) The force pulling the rod is now removed. Indicate whether the speed of the rod increases, decreases, or remains the same. Increases Decreases Remains the same Justify your answ er. END OF EXAM © 2007 The College Board All rights reserved. Visit apcentraLcollegeboardcom (for AP professionals) and www.eollegeboardcorn/apstudents for students and parent©. -842 43 2008 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 the pink booklet in the spaces provided after each part. NOT in this green insert. E&M. 1. A metal sphere of radius a contains a charge +Q and is surrounded by an uncharged, concentric, metallic shell of inner radius b and outer radius c, as shown above. Express all algebraic answers in terms of the given quantities and fundamental constants. (a) Determine the induced charge on each of the following and explain your reasoning in each case. i. The inner surface of the metallic shell ii. The outer surface of the metallic shell (b) Determine expressions for the magnitude of the electric field E as a function of r, the distance from the center of the inner sphere, in each of the following regions. i. r<a ii. a<r<b iii. b<r<c iv. c<r 2008 The College Board. All rights reserved. Visit apcenrralcollegeboard.om tor P protessionals) and svww.collegeboard.corn/apstudents for students and parents. GO ON TO THE NEXT PAGE. -543 44 2008 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (C) On the axes below, sketch a graph of E as a function of r. t I a I) r (d) An electron of mass ,n carrying a charge —e is released from rest at a very large distance from the spheres. Derive an expression for the speed of the particle at a distance lOr from the center of the spheres. V 2008 The College Board, All rights reserved. Visit apcentralcollegehoardcom (for AP professionals) and wwwcollegehoard.com/apstudents tbr students and parents(. GO ON TO THE NEXT PAGE. -644 _____________________ 45 2008 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS R = 1 200Q VV’v 85()()V R=3(X2 R=lOO2 IA Ill E&M, 2. In the circuit shown above, A and B are terminals to which different circuit components can be connected. (a) Calculate the potential difference across R 2 immediately after the switch S is closed in each of the following cases. i. A 50 2 resistor connects A and B. ii. iii. A 40 mH inductor connects A and B. An initially uncharged 0.80 ltF capacitor connects A and B. (b) The switch gets closed at time t = 0. On the axes below, sketch the graphs of the current in the 100 Q resistor R 3 versus time t for the three cases. Label the graphs R for the resistor, L for the inductor, and C for the capacitor. © 200$ The College Board. All rights reserved. Visit apcentral.collegeboard.cum for AP professionals) and www.collegeboardeoniapsrudents for students and parents. GO ON TO THE NEXT PAGE. -745 46 2008 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS R2 Figure 1 E&M. 3. The circular loop of wire in Figure 1 above has a radius of R and carries a current I. Point P is a distance of R/2 above the center of the loop. Express algebraic answers to parts (a) and (b) in terms of R, I, and fundamental constants. (a) i. State the direction of the magnetic field B 1 at point P due to the current in the loop. ii. Calculate the magnitude of the magnetic field B 1 at point P. Axis ? R12 Figure 2 A second identical loop also carrying a Figure 2 above. Figure 3 current I is added at a distance of R above the first loop, as shown in tb) Determine the magnitude of the net magnetic field B, 1 at point P. A small square loop of wire in which each side has a length s is now placed at point P with its plane parallel to the plane of each loop, as shown in Figure 3 above. For parts (c) and (d), assume that the magnetic field between the two circular loops is uniform in the region of the square loop and has magnitude B,, 1 (c) In terms of and s, determine the magnetic flux through the square loop. (d) The square ioop is now rotated about an axis in its plane at an angular speed o. In terms of Bnet, s, and ai, calculate the induced emf in the loop as a function of time t, assuming that the loop is horizontal at t 0. END OF EXAM © 2008 The College Board. All rights reserved. Visit apcentral.collegehoardcom I for AP professionals and www.collegeboardcom/apstudents (for students and parents. -846 47 2009 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, NOT in the green insert. E&M. 1. A spherically symmetric charge distribution has net positive charge Q 0 distributed within a radius of R. Its electric potefltial V as a function of the distance r from the center of the sphere is given by the following. = 1rEJRL 4 V(r)= r 0 4lrE —2 + 3(i-) R for r forr>R Express all algebraic answers in terms of the given quantities and fundamental constants. (a) For the following regions, indicate the direction of the electric field E(r) and derive an expression for its magnitude. i. r < R Radially inward Radially outward ii. r> R Radially inward Radially outward (b) For the following regions, derive an expression for the enclosed charge that generates the electric field in that region, expressed as a function of r, i. r < R ii. r> R (c) Is there any charge on the surface of the sphere (r Yes No If there is, determine the charge. In either case, explain your reasoning. © 2009 The College Board. All rights reserved, Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -547 48 2009 AP’’ PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (d) On the axes below, sketch a graph of the force that would act on a positive test charge in the regions r and r > R. Assume that a force directed radially outward is positive. < R F © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboardcom. GO ON TO THE NEXT PAGE. -648 49 2009 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS jg6 5.0 x 9.OV E&M. 2. A 9.0 V battery is connected to a rectangular bar of length 0.080 m, uniform cross-sectional area 5.0 x l0 rn . 2 and resistivity 4.5 x l0 2.m, as shown above. Electrons are the sole charge carriers in the bar. The wires have negligible resistance. The switch in the circuit is closed at time I = 0. (a) Calculate the power delivered to the circuit by the battery. (b) On the diagram below, indicate the direction of the electric field in the bar. -______ Side View Explain your answer. (c) Calculate the strength of the electric field in the bar. A uniform magnetic field of magnitude 0.25 T perpendicular to the bar is added to the region around the bar, as shown below. 0.080 in m 6 10 x x x ax >< x x x x x x X x x x x )< X X X X X X xx x x x x x x x x Side View (d) Calculate the magnetic force on the bar. (e) The electrons moving through the bar are initially deflected by the external magnetic field. On the diagram below, indicate the direction of the additional electric field that is created in the bar by the deflected electrons. Side View (f The electrons eventually experience no deflection and move through the bar at an average speed of 3.5x l0 rn/s. Calculate the strength of the additional electric field indicated in parue. © 2009 The College Board. All rights reseived, Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -749 _______ 50 2009 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS L jx x J. x X (J) X 1 X X X x X B X X X X E&M, 3. A square conducting loop of side L contains two identical lightbulbs, 1 and 2, as shown above. There is a magnetic field directed into the page in the region inside the loop with magnitude as a function of time t given by B(t) = at + b, where a and b are positive constants. The lightbulbs each have constant resistance R 0 Express . all answers in terms of the given quantities and fundamental constants. (a) Derive an expression for the magnitude of the emf generated in the loop. (b) i. Determine an expression for the current through bulb 2. ii. Indicate on the diagram above the direction of the current through bulb 2. (c) Derive an expression for the power dissipated in bulb 1. Another identical bulb 3 is now connected in parallel with bulb 2, but it is entirely outside the magnetic field, as shown below. L (1) X X x x TX X IX X XX x B 2 X X (d) How does the brightness of bulb 1 compare to what it was in the previous circuit? Brighter — Dimmer The same Justify your answer. © 2009 The College Board, All rights reserved. Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -850 51 2009 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS Now the portion of the circuit containing bulb 3 is removed, and a wire is added to connect the midpoints of the top and bottom of the original ioop, as shown below. L x x x x B x x x x xx x x x x B x 2 ) x (e) How does the brightness of bulb I compare to what it was in the first circuit? Brighter Dimmer The same Justify your answer. END OF EXAM © 2009 The College Board. All rights reserved. Visit the College Board on the Web: wwwcollegeboardcont -951 52 2010 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 the pink booklet in the spaces provided after each part, NOT in this green insert. y A Figure I E&M. 1. A charge +Q is uniformly distributed over a quarter circle of radius R, as shown above. Points A, B, and C are located as shown, with A and C located symmetrically relative to the x-axis. Express all algebraic answers in terms of the given quantities and fundamental constants. (a) Rank the magnitude of the electric potential at points A, B, and C from greatest to least, with number 1 being greatest. If two points have the same potential, give them the same ranking. Justify your rankings. Point P is at the origin, as shown below, and is the center of curvature of the charge distribution. y A +Q1 j / Figure II © 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. GO ON TOTHE NEXT PAGE. -552 53 2010 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (hi Determine an expression for the electric potential at point P due to the charge Q. (c) A positive point charge q with mass in is placed at point P and released from rest. Derive an expression for the speed of the point charge when it is very far from the origin, (d) On the dot representing point P below, indicate the direction of the electric field at point P due to the charge Q. V 4 (e) Derive an expression for the magnitude of the electric field at point P. 30 V E&M. 2. In the circuit illustrated above, switch S is initially open and the battery has been connected for a long time. (a) What is the steady-state current through the ammeter? (b) Calculate the charge on the 10 jiF capacitor. (C) Calculate the energy stored in the 5.0 tiF capacitor. The switch is now closed, and the circuit comes to a new steady state. (d) Calculate the stead-state current through the battery. (e) Calculate the final charge on the 5,0 iF capacitor. (f Calculate the energy dissipated as heat in the 40 12 resistor in one minute once the circuit has reached steady state. 2010 The College Board. Visit the College Board on the Web: www.collegehoard.com. GO ON TO THE NEXT PAGE. -653 54 2010 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS I Lighthulb (resistance R) + I (I E&M. 3. The long straight wire illustrated above carries a current I to the right. The current varies with time t according to the equation I = I Kt, where I and K are positive constants and I remains positive throughout the time period of interest. The bottom of a rectangular loop of wire of width b and height a is located a distance d above the long wire, with the long wire in the plane of the loop as shown. A lightbulb with resistance R is connected in the loop. Express all algebraic answers in terms of the given quantities and fundamental constants. — (a> Indicate the direction of the current in the ioop. Clockwise Counterclockwise Justify your answer. (b) Indicate whether the lightbulb gets brighter, gets dimmer, or stays the same brightness over the time period of interest. Gets brighter .,__Gets dimmer Remains the same Justify your answer. (c) Determine the magnetic field at t long wire. = 0 due to the current in the long wire at distance r from the (d) Derive an expression for the magnetic flux through the loop as a function of time. (e) Derive an expression for the power dissipated by the lightbulb. END OF EXAM © 2010 The College Board, Visit the College Board on the Web: www.collegeboard.com. -754 55 2011 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 the pink booklet in the spaces provided after each part. NOT in this green insert. E&M. 1. A nonconducting, thin, spherical shell has a uniform surface charge density o on its outside surface and no charge anywhere else inside. (a) Use Gauss’s law to prove that the electric field inside the shell is zero everywhere. Describe the Gaussian surface that you use. (b) The charges are now redistributed so that the surface charge density is no longer uniform. Is the electric field still zero everywhere inside the shell? Yes No It cannot be determined from the information given. Justify your answer. Now consider a small conducting sphere with charge below. +Q whose center is at corner A of a cubical surface, as shown B G I I L I) E (c) For which faces of the surface, if any, is the electric flux through that face equal to zero? DEF ABED ABGH EFGH BCFG ADEH Explain your reasoning. (d) At which corner(s) of the surface does the electric field have the least magnitude? (e) Determine the electric field strength at the positions you have indicated in part (d) in terms of fundamental constants, as appropriate. Q, L, and (f) Given that one-eighth of the sphere at point A is inside the surface, calculate the electric tiux through face CDEE. 2011 The College Board, Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -555 56 2011 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE”RESPONSE QUESTIONS 500Q 1 S 25 mF 9,0 V E&M. S 5.0 H 2. The circuit represented above contains a 9.0 V battery, a 25 mF capacitor, a 5.0 H inductor, a 500 Q resistor. and a switch with two positions, S 2 Initially the capacitor is uncharged and the switch is open. 1 and S . (a) in experiment I the switch is closed to position S 1 at time r1 and left there for a long time. i. Calculate the value of the charge on the bottom plate of the capacitor a long time after the switch is closed. ii. On the axes below, sketch a graph of the magnitude of the charge on the bottom plate of the capacitor as a function of time. On the axes, explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate. 11 ‘Time iii. On the axes below, sketch a graph of the current through the resistor as a function of time. On the axes, explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate. Time 201 1 The College Board. \‘isit the College Board on the Web: www.coiiegehoard.org. GO ON TO THE NEXT PAGE. -656 57 2011 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (h) In experiment 2 the capacitor is again uncharged when the switch is closed to position 1 S at time r1 The switch is then moved to position S 2 at time t 2 when the magnitude of the charge on the capacitor plate is 105 inC. allowing electromagnetic oscillations in the LC circuit. i, Calculate the energy stored in the capacitor at time 12 ii. Calculate the maximum current that will be present during the oscillations, iii. Calculate the time rate of change of the current when the charge on the capacitor plate is 50 mC. 2011 The College Board. Visit the College Board on the Web: wwwcollegeboardoro. GO ON TO THE NEXT PAGE. -757 58 2011 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS Cross-sectional View (current into page) E&M. 3. A section of a long conducting cylinder with inner radius a and outer radius b carries a current ! that has a uniform current density, as shown in the figure above. (a) Using Ampere’s law, derive an expression for the magnitude of the magnetic field in the following regions as a function of the distance r from the central axis. i. ii. iii. 1’<(l a <r<b r=2b (b) On the cross-sectional view in the diagram above, indicate the direction of the field at point P, which is at a distance r = 2b from the axis of the cylinder. (c) An electron is at rest at point P. Describe any electromagnetic forces acting on the electron. Justi your answer. \ic th 2i I the College Board. allege Board on the Web: w’. w.ulIehoard.org. GO ON TO THE NEXT PAGE. -858 59 2011 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS Now consider a long, ijd conducting cylinder of radius b carrying a current I. The magnitude of the magnetic field inside this cylinder as a function of r is given by B = poJor/2,rb2 An experiment is conducted using a particular solid cylinder of radius 0.010 m carrying a current of 25 A. The magnetic field inside the cylinder is measured as a function of r, and the data is tabulated below. . Distance r (m) Magnetic Field B (T) 0.002 0,004 1.2 x l0 2.7 x i0 I 0.006 I 0.008 3.6 x l0 4.7 x i0 0.010 6.4 x l0 (d) i. On the graph below, plot the data points for the magnetic field B as a function of the distance r, and label the scale on both axes. Draw a straight line that best represents the data. E z zzzzz — . ..Ezr:zEzEzz . ii. Use the slope of your line to estimate a value of the permeability , 0 t END OF EXAM \‘hit 201 I ‘Ihe College Board. the College Board on the Web: www,coiieeeboard,ore, -959 60 2012 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 IS points each. The parts within a question may not have equal weight. Shoss all your work in this booklet in the spaces provided after each part. V E&M, 1, Two thin, concentric, conducting spherical shells, insulated from each other, have radii of 0.10 m and 0.20 m, as shown above. The inner shell is set at an electric potential of —100 V, and the outer shell is set at an electric potential of +100 V. with each potential defined relative to the conventional reference point. Let Q 1 and Q represent the net charge on the inner and outer shells, respectively, and let r be the radial distance from the center of the shells. Express all algebraic r. and fundamental constants. as appropriate. answers in terms of Q . 1 . (a) Using Gauss’s Law, derive an algebraic expression for the electric field E(r for 0.10 m ht Determine an algebraic expression for the electric field E(r) for r > r < 0.20 m 0.2() m ci Determine an algebraic expression for the electric potential Vr) for r (d < > 0.2() m Using the numerical information given, calculate the value of the total charge Q on the two spherical shells — ÷ (2,,) GO ON TO THE NEXT PAGE. -560 61 2012 AP’ PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (e) On the axes below, sketch the electric field E as a function of r Let the positive direction he radially outward. (J. r(rn) 10 (0 On the axes below, sketch the electric potential V as a function of r. V r(1fl) 0 h U 20 U 0 Ih (oHe BoarcL Boi n Web: GO ON TO THE NEXT PAGE. -661 62 2012 APa PHYSICS C: ELECTRICITY AND MAGNETISM FREERESPONSE QUESTIONS Length in ENl. 2. ii) m .\ plivic %tudent i\he to rneaure the resistlvit ot slightly conductic paper that ha a thickncs of 1 .() The student cut’. a sheet of the conductive paper into strips of idth 00)2 in and varx ing lengths, making live rcsistor labeled RI to R5. Using an ohmmeter, the student measures the resistance of each strip, as shown aho e. The data are recorded below. / RI R5 Length () 0.020 0100 Resistance () b0,000 440,000 Resistor t a) Use the grid below to plot a linear graph of the data points from which the resistivity of the paper can be determined. Include labels and scales for both axes, Draw the straight line that best represents the data. Ib) Using the graph. calculate the resistivity of the paper. ‘.i’’h( ih (i Br 1 ,,. Brd•n :h \\h: . rd’r,. GO ON TO THE NEXT PAGE. -762 63 2012 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS Capas.dtor Battery rue student uses resistors R4 and R5 to build a circuit using wire, a 1.5 V batter, an uncharged 10 iaF capacitor. and an open switch, as shown above, (C) Calculate the time constant of the circuit, (d) At time t = 0, the student closes the switch. On the axes below, sketch the magnitude of the voltage V across the capacitor and the magnitudes of the voltages VR4 and VR5 across each resistor as functions of time r. Clearly label each curve according to the circuit element it represents. On the axes, explicitly label any intercepts, asymptotes, maxima, or minima with values or expressions, as appropriate. 0 Time I C O1 The College Board. icC the College Board on the Web: wwwcollegeboard.org. GO ON TO THE NEXT PAGE. -863 64 2012 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS Crossbar B x x H E&.M. x x L i x. H 3. A closed loop is made of a U-shaped metal wire of negligible resistance and a movable metal crossbar of resistance R, The crossbar has mass in and length L. It is initially located a distance h ) from the other end of the 1 loop. The loop is placed vertically in a uniform horizontal magnetic field of magnitude B( in the direction shown in the figure above. Express all algebraic answers to the questions below in terms of B L, in. h,. 1?. and fundamental constants, as appropriate. . (a) Determine the magnitude of the magnetic flux through the loop when the crossbar is in the position shown. The crossbar is released from rest and slides with negligible friction down the U-shaped wire without losing electrical contact. cb) On the figure below, indicate the direction of the current in the crossbar as it falls. Justify your answer. (c) Calculate the magnitude of the current in the crossbar as it falls as a function of the crossbar’s speed v. (d) Derive, but do NOT solve, the differential equation that could be used to determine the speed crossbar as a function of time t. (e) Determine the terminal speed f UT U of the of the crossbar. If the resistance R of the crossbar is increased, does the terminal speed increase, decrease, or remain the same J)ecreases Increases Remains the same Give a physical pistitication for your answer in terms of the forces on the crossbar. STOP END OF EXAM 2 ‘1 hr College Board. Visit the College Board on the Web: wtsv.eollegeboard.org. -964 65 2013 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 ot 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 L A very long, solid, nonconducting cylinder of radius R has a positive charge of uniform volume density p. A section of the cylinder far from its ends is shown in the diagram above, Let r represent the radial distance from the axis of the cylinder. Express all answers in terms of r. R, p, and fundamental constants, as appropriate. (a) Using Gauss’s law, derive an expression for the magnitude of the electric field at a radius r appropriate Gaussian surface on the diagram. < R. Draw an (b) Using Gauss’s law, derive an expression for the magnitude of the electric field at a radius r> R. (C) On the axes below, sketch the graph of electric field E as a function of radial distance r for r = 0 to r = 2R. Explicitly label any intercepts, asymptotes. maxima, or minima with numerical values or algebraic expressions. as appropriate. 1: 2!? 1? (d) i. Derive an expression for the magnitude of the potential difference between r ii. Is the potential higher at r r = 0 = 0 or r = 0 and r = R. R ? r=R 2013 The Collcee Hoard. \tstt the College Board on the \Veh: www.eollegeboard.org. GO ON TO THE NEXT PAGE. -665 66 2013 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE GUESTIONS e) The nonconducting cylinder is replaced with a conducting cylinder of the same shape and same lin charge density. On the axes below, sketch the electric field F as a function of r for r 0 to r = 2R. Explicitly label an intercepts, asymptotes. maxima, or minima with numerical alues or algebraic expressions, as appropriate. I R ,()i \ 1 he (oIht Boird. the ( lleg Board on the \\ eb: ww e(lleget’ard.org. GO ON TO THE NEXT PAGE. -766 67 2013 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS E&M 2, In a lab, you set up a circuit that contains a capacitor c, a resistor R, a switch S. and a power supply, as shown in the diagram above, The capacitor is initially uncharged. The switch, which is initially open, can be moved to positions A or B. (a) i. Indicate the position to which the switch should be moved to charge the capacitor. A B ii. On the diagram, draw a voltmeter that is properly connected to the circuit in a manner that will allow the voltage to be measured across the capacitor. After a long time you move the switch to discharge the capacitor, and your lab partner starts a stopwatch. You collect the following measurements of the voltage across the capacitor at various times. t(s) V(V) 6 18 30 42 54 252 74 33 10 6 You wish to determine the time constant v of the circuit from the slope of a linear graph. i. Indicate two quantities you would plot to obtain a linear graph. ii. Use the remaining rows in the table above, as needed, to record any quantities that you indicated that are not given. Label each row you use and include units. © 2013 The College Board. Vhit the College Board on the Web: wwwcollegeboardorg. GO ON TO THE NEXT PAGE. -867 68 2013 Al) PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (c) On the axes below, graph the data from the table that will produce a linear relationship. Clearly scale and label all axes including units, if appropriate. Draw a straight line that best fits your data points. L I I I e I r I I ——— 1 I 1 I I I 1 1 I I ——— I I I I I I I I I I I I I I I I (d) From your line in part (c), obtain the alue of the time constant r of the circuit. (e) .iF. Calculate an experimental value for i. In the experiment, the capacitor C had a capacitance of 1.50 1 the resistance R. ii. On the axes in part (c), use a dashed line to sketch a possible graph if the capacitance was greater than 1.50 pP but the resistance R as the same. Justify your answer. GO ON TO THE NEXT PAGE. -‘968 69 2013 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS xxxxxxx X X B xxx xxx x xxx xx x x XXX )( )( xX xxxxxxx E&M 3, The figure above shows a circular loop of area 0.25 m 2 and resistance 12 ] that lies in the plane of the page. A magnetic field of magnitude B directed into the page exists in the area of the loop. The field varies with time t. as shown in the graph below. B (T) B 0 4 8 = 1 ,8e© 12 16 18 (a) i. Derive an expression for the magnitude of the induced emf in the loop as a function of time for the interval t = 0 s to r = 8 s. ii. Calculate the magnitude of the induced current I in the loop at time t = 4 5, © 2013 The College Board. Visit t.he ©oilege Board on the Web: www.coi1egeboardorg. GO ON TO THE NEXT PAGE. -1069 70 2013 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS (b) i. Sketch a graph of the induced current I in the ioop as a function of time i from r axes below. assumine that a counterclockwise (CCW) current is positive. = 0s to r = 18 s on the IA) 0 4 8 12 16 18 Cw ii. For the time interval 12 s to 16 s, justify the direction of the current you have indicated in your graph. (C) Calculate the total energy dissipated in the loop during the first 8 s shown. STOP END OF EXAM 2OiIhe College Board, Visit the College Board on the Web: www.collegeboard.org. -11— 70