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h7b_s06_mt2_soln - El University of California at Berkeley...

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Unformatted text preview: El University of California at Berkeley Physics H7B Professor Boggs Spring 2006 Midterm Examination #2 9:30-11:00 am, 21 March 2006, 2 LeConte Print Name SO i U’i' I on 5 Discussion Section# or Time Signature Student ID# This exam is closed book, but you are allowed one 8.5" x 11" (double-sided) page of handwritten notes. You may use a calculator. Remember to circle all of your final answers. Read through the entire exam to start. Work to maximize your credit ~~ try to obtain at least partial credit on every part of every problem. Do your work clearly so we can easily follow. Show all work, using the front and back sides of this exam paper. If you do not show relevant work for any part of a problem, you will not be awarded any credit, even if the answer is correct. If y0u recognize that an answer does not make physical sense and you do not have time to find your error, write that you know that the answer cannot be correct and explain how you know this to be true. (We will award some credit for recognizing there is an error.) Do not get bogged down in algebra -- if you have enough equations to solve for your unknowns and are running out of time, box the equations, state how you would finish, and move on (you can go back and complete the algebra later if you have time). And if you have questions about the interpretation of a problem, please ask! [it I. (20 Points) A small bead of mass m and test charge +q is constrained to move along a thin frictionless rod of length 2L. Two positive charges +0 are placed at each end of the rod. (a) Calculate the force on +q as a function of x (relative to the center) along the rod. (b) Set up the differential equation of motion for this bead. (Do not solve.) (c) Show that for x << L. the equation of motion describes simple harmonic motion. (d) Calculate the period of oscillation for the head in this limit. C4) C01. X LLL/ M J‘QJ‘U‘Q/ a Swat” Fara + I 774m on!» com petunia dim. 61'. 01c M0 Io» g5_ cllx __ #1402 6 jg: '— mr,‘ (Ii-:62 E 3 I4 “w T a»; Mclin ---—' ngchS 6‘26 4' 0(6)- “3/ w j (be? P Dram (ll/M OHM +st m lad”) 0"?) "‘9‘ ‘2“ if; a: ”9.916: #37. x d+1 mL'l' MKS ‘ fiH‘érw-dh'tbi 30“ (CUM LE”; NUNCJ 'dia‘f' '“F ["94 CJJICUFQ‘ILCJ #24 Poi-adv»? ems/‘33, [1(3) 9.“ Hal: ejsfémy x43 ”My; 4., fl“; mmmm. I film/{cm ma, Gan. (7/le UL?) aroma] (:0: [1(1): Uo-l- Kid—g- .. -90 / :. ’e-E £51m” 4+5?" *5.“ "a": Mo” K QXLm' fo) 4w,”- pn+ (a)J we, km ,Df- 5 LII/Owl '1 1 L 0x ,4 ”We £1-16 1113! (LI*X1)A:O (L H) «=6 / = 129.} »0 “” W'i‘ihaew L, L3 El 2. (20 points) A thin disk is centered with its axis along the y-axis as shown below. The disk carries a charge Q, uniformly distributed on its surface. (a) Calculate the electric potential along the y-axis (relative to infinity). (b) Calculate the electric field along the y-axis (check signs for y>0, y<0). (0) Verify this electric field is correct in the limit y>>R. (Do this only for y>0.) (d) Verify this electric field is correct in the limit y<<R. (Do this only for y>0.) [Hintz fig-diff =- N‘xz + a! ] 0/) “Fa; y '7") R J JQ‘CHJ-ez :: E;- (R1 50 fl” 6)! ” _‘ Y flue/n Z'HO’({ EL) : 220 __ Y _ ’ZG l g '— - —-1 “' and 20‘ n -~ V 6 H W)“ 71[-—;:~:g+emj A: \%L ‘ (by Jrolflflj a” gut +10 [Ming 40mm 1‘" / +14, own/#610“). 14 Looks ALL aflfilxz—vza _ ._....._—/ I f: ) 2V0 (To (€619,125 010th) - I+ [mics [steam chnfie w - n+6)“ " 5M“ 192/ (H5) + T + 1pm, ecci 4-“ WM} 7% Wm WW 05 4m ‘ .- ’ PHCO) '1‘ Ht Wt) - «((0) + P (0) e + 33’ e L 5—4;) 6» Jr. 3. (20 points) A very large, thin plane has a uniform surface charge density 0. Touching it on the right is a slab of thickness d with a uniform charge density p. “PPM—‘7 M (“an 0' d (a) Calculate the electric field to the left of the thin plane. (b) Calculate the electric field to the right of the thick slab. (0) Calculate the electric field everywhere inside the slab. (Take x=0 at the slab center.) .’ Fm» M [9in G Fmat's 0M+Mwéj$ Lc/ ‘i’v'f/LQ/ 270- F01 fLL slab) U‘fl/ Gauss" W" A @6,5Ac%9%¢ :7 25A: meclja €7- 0.de (owwk) [it"r'jlple v-L (nods‘y #IS Gifaa‘l'l‘j: ._ 2 j — fir ._—.q1rO~e,.c-:a ZQA‘WPAtzx e= Limo x (mm LE; F...—-——a Lil 4. (20 points) Three concentric conducting spherical shells have radii a, b, c such that a < b < c. Initially, the inner shell is uncharged, the middle shell has a positive charge +Q, and the outer shell has a negative charge —Q. (a) Find the electric potential of the three conducting shells, ¢(a), (Mb), ¢(c). (b) The inner and outer shells are now connected by a thin conducting wire that is insulated as it passes through the middle shell. What is the final charge on each shell? Fm. a single, 5W} LPG"): 5%— out-smlfl, Mal LPG): % {a “$546,449 1'15 ram (wkm=lQh "1 #14 1/10er5 01: .Hu, ¢,th_ O r7C. a\ P0» Tints mcrjum‘i'ronj 6-: 9;; C7r7b O L_,‘7(‘ g0) 411473 (p10 ,l— MGM-y) W, Com ,fl'l'efirai'g ‘Pf‘l’a‘ -f:b E 'ols +0 7cm”! [PM (me'd (F‘- ‘ 9i C/emr f?“ 03mg 646°" “0% +Lb+ M, gig! hut. LP: ~95. 1, Q; (2} (mm; LPq 1,9 Inward?) {6 JR 7+ '\ Rem, +0 f’°”.‘”'F'“‘+‘/; 1,“; +L4+ n :flhlw fliuw W 4”” “W5 “WW0 m W"; - 9(— ac ______________,___._____—a - " a (1-13 (-—-’ ‘ +1, Hun} um, N‘fntd HK {’DKO‘OT—O‘ZK‘ > /(N0 - -9 ’— in) C2 ‘ Q \/ Minus 9b,, ' Ly am PM, FM“; OJ+I< Look 51+ hmfl'nfl )eraJ-url Fm— 0-90) Clo“- 0 mm} (la: 61, {MAN/In Wit“, W95” H" ”Lug, 90+ 41¢ Fomaéwv Q rcO ewe +0 0) yw m4 Iw my dar‘fi‘g' Mm. Fm, 0.905} Gaza—Q9; 0%.: ’O(l-%) WLu-l-vb Adm-(.115 ) a eplbxiw’e (Zapacd‘uL ”444‘ LP “6 +11. MIN/v 6W 9/2,?“ 147w? +0 0 may}? fl‘fl.’ €dfl+ton *0 pincer” ’13-'52, up 1m“ 14d“ a9 ab 0"“ I" /: LP. 9- 6. kph 051':-4kpa+ GLPb 9'"?— L175 ”-é rid-{i : ”gt—Q 314+, KP‘GO and 0‘5: 0: go 0,“: ’L' 5/“ I: 60) 741° 50322 by)“, C? 99 (ft! {42, 0M7‘M sm {4 a'f' ,m-rqm-I'y) i‘? #9 6W fig. 6‘ 2’5W 9de;uf QFZ’LanJ (is /# éAA‘IL-llf! w .l S. (20 points) Consider the circuit below. (a) In steady state, what is the voltage across the capacitor? (Hint: what is dQ/dt on the capacitor in steady state?) (b) At t=0, the battery is disconnected. Give the current flowing from the capacitor as a function of time. (c) How much energy is dissipated as this capacitor discharges? u "+ I” 0x) in dual? 6+4?) Ibo wrW+ 10/90“ +1.”th +114 W’) n” 50 on “a”; I I r5 #157. J 1" is gait—x— KHWL 5v m.LI-1 E ‘ '1' 9'7 €013 % 60E 5 3(1/ LN U 302. M Q pawl/E} pas/6+?» Gina-o WU" 1/0””? 0U.” 90 1':- [Amp LFWQ’J—s W794. 1"” 441‘ 504””, qoil = go I]- CL) g-I; :11}; .. ’- "" ‘2- A/ "" :3 (Er-“00¢ if‘I‘L' IA”IJ) ‘LP- 5' F) .Lzfi— "é—AWP Vflaaovi:%ro: {GEL-Mop: f1? again, (,0; _---‘ C‘loflF l'ol 1.0 U Ct: lO/F ) (Zr. 33.3)),2. j m Vaz'LOV (22190” H’v'f' m UaH-abg, across "HQ, Wu‘f‘w jn gar/”1 M C/kom’f' - AL P i. is My ua "C ,2: Re a So 17%)— ,Q g -..i?-_— 5 IQ): 0-6 9, 333.3}; Amps ...
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