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Sp06-Final-Liphardt-Soln

Sp06-Final-Liphardt-Soln - ,v UNIVERSITY OF CALIFORNIA...

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Unformatted text preview: ,v , UNIVERSITY OF CALIFORNIA, BERKELEY /' i ' ' Physics Department 7b, Liphiardt Spring Term 2006 Final Exam SO [MI—W77“; Name Disc. Section SID #1 1° (20) #2 20 (20) #3 u (20) #4 2° (20) #5 20 (20) Total 10 ° Pmtf The relative weight of each problem is indicated next to the problem number. Most credit will be given for algebraic work. Please do not insert numerical values until you have a final algebraic answer inside a box. If you don’t know a particular constant, use a symbol to get partial credit. It is almost impossible to award partial credit if you insert numbers too early. If you get stuck on one problem, go on to the next and come back to the difficulties later in the exam perlod. Do not leave early until you have completed everything. Do not quit! Never, never, never quit!! As usual, before you start to manipulate equations and numbers you should think for a moment and make sure that you have a general sense of the important features of a problem. Also, once you have computed something, check to see if your answer has the right units, and if it is in the right ballpark. For example, if your answer to any problem is something like it x 10.4 8 Kelvin/meters, you should check your work. !!!!!!!!!!!!GOODLUCK!!!!!!!!!!! (Pr-Mm \ [710 gtsj A WV C: 3/4? a: clay W390 Q = 4x 164C onA i? ConflQdQA 1n 3M6 math (1 mm o€ \vang Q2 20mm Wme hcé Ciro-\‘QV‘ 0/05“ sachcm mdluf r: \ mm, a regisiy‘wlk’ 4,4621me a mag 59 \04 k5) ahA Cx SPQGQ‘L kefi £5023 £31360 A4 Jew, JAM Cgpaddbr ij‘mC *0 CLEC‘WCCS’K firms?“ Hm mQéor, kg: [.38 Moms/K (x. mm 6mg +\\Q M46831 cm 9%: @1de Amp “rm 23V? cth) ACLLU 2 %' ACL : -1 é’ ‘5 £3; a} flac‘l 1% ‘L h“ 5...}. 615 Q _’£ 1’? ‘19“ 2c ‘7 9W5; ‘ /r<c Q o ‘9 , a. confimeé: umL MM g: J20 : 4x10 92m .2” A —“—“ qr x (l mml R: ”1.57a/VIDASZJJ C=3/«F, 62: 41154C,7}F273‘4 CT: QCQM-Eé‘; S: (DJ? 3593‘? 9 b. when ”we ’Uofiaga GU05? Mme Cagoaohlor ff 23 v, wML 35 +913. va‘fcicfe W +019 @065? C- “Ow Muck +V\Qfma\ W33 us AQPwieA \n “We wire 5 W 3.55deng FFOCQ’Y (2 g .4» co) M1 : ELLC Q1) UP 7" 0 WWW 1n Gawac‘for. \"4C 2 1 KAUW AC Q = <4“) > = 2.07m) J WA (OK M 3:? “ML, Je : 7235124“ R9; §:~*/MJ£- 5:.) A UBM \S ’HNL 1;}ch MW SCAM wire? ASSWQ 43m Wm MA ah ‘miJn‘oJ kmfzemflre of; 20° C and 45nd: Wm K ho law mo had +0 xthe QDVIronmn’r Au: MAT: INCH-13%? 2 égg VG MC fl 7”: ”LDC—t ETLQ “0 3 = (20.53 ’0 W03 SOUS/‘QOL T9: (La/53 C 6. WW LCfLQ agha‘ (MICSQBML m3 0? HQ deal/Won? \n fixe wire? Where Cum 3 CEng O‘QMW QFQQLJW decks“: wig 1mm X\\1(CU\O‘ ’2 Airedicmf, TM quflwm fiwrm 3646 910+ each 4.0.10. ConJrfihA-QS‘ CK W o? 3‘:ng Jqo Max/ergg kineficahqaj, So {guy 2 33 1957? 2“” ejedym‘ / -23 _ {Ebb 1‘ -—?z . [.33 HO 37K x ($0,153 4273J§>K : 90%] n0 3 [.919 : 6,08 “5(1‘3 rm Mm? r. MW \S ‘W, chaYggQ 1“ WW 5C+lfl€ umwma ‘m c? “HMS PmCeQS 1. A3: 9;! :2 Mchi (wemfiné areuQKMg T T T‘ wax connedfivxg T: 57;) ,4 %.{3+7,~13.\§§ AS ‘= 10 k3 * 5’00 J/kjlk *9“( 4.0 01¢“? Problem 2. Ionic conductivity [20]. Consider a gas of electrons in a cylindrical vessel with length b, cross sectional area A, and whose caps are metallic disks. Connect these two caps to opposite poles of a battery (voltage V). An E field will be produced between the two plates, and the electrons will drift towards the positive plate: there will be an electrical current. General question: How does this current depend on the voltage difference across the two plates? (21) Let’s get started: assume that the E field is zero, and assume that the electrons interact only via collisions. Derive an expression for 'c, the mean time between collisions. Frontal area of electron, 0, temperature, T, mass of electron, me, density n, Boltzmann constant, kB. Am 952cm LUN‘K Bro/Sfogx cue» 0‘ and \)0v\<0°~“-\'% E \(‘Q\O.k"w ‘Domr- dent-90$ SMQQS W cm egkxcf’v‘rsa ooQqum 90S ’rx‘wm 61F . MU“\‘*:Q\‘6;MQH"Q%MKI‘Y33 mac—t U-O‘W ‘CG‘QQAJMO‘MQ, W w‘i‘ “Wk? Wei/x mug ‘3“: (6?“ ”i Husk 9cm waxyw. { ‘ ‘ ‘ WM':G7 we M We: WM sz‘xocn‘rgdbmnalgcvvw \"i $501363 ‘ J— mvf;%kT g) Utr'; IE. 2. - 7' W‘ (Jr/98 LIT £00m? 0"“? 8°}? ‘C= 9—- : fir§% ‘ ‘6ng \54 reSFmNWa mm (row . , a, v : ‘i~ ”’3‘ Ag 0 3m Mu, we aw, WWI “R “W ““44““? ”Q m" “‘3‘ ”‘03”: 44> 0"2'J,1+L£I:L'r’LCE/Ug’i> TEL=LTTL+U12h§ U? ”32:0 f? "51:20:: ;fi if: Ede-v \ , «\G‘ 6‘0" (b) Now let’s turn the E field on. What is the E field in the vessel, as a function of V? Neglect the contribution to the field by the electrons. balm QM. 93,on WWW “Owl/woken], g (.9 ka (c) What is the drift velocity, vdfifi. of the electrons? Vdfifi. = F * ‘t / m, where F is the force, I is the mean time between collisions, and m is the mass. If you are unsure of your answer to (a), just use 12 as the mean time between collisions. Charge of electron, qe. " ‘ 2:? firm“ “am 3% b 4:) V136: é:- 03:3) 1:17 B mag?“ 43 (y '1 c 9V Smc _. 8L 4L1. (”if 7L5. ” bwf 3%“ M66“ b“ (d) What is the current as a function of the voltage difference across the two plates? \/ 2, \/ _ Q j: U‘dmrbke‘n \ 8W Tong; 4::Atamwem :: Q’VA- b 0" (e) What is the electrical resistance of an electron gas? icfi 7;"? ’: k D" ,g ”Evil-5% MWM—«mwm. «Ex/A #13 i 3.4;3 '1' f.— \06 A...» 3. Austin’s problem [30]. Displacement Current and Poynting Vector in an RC Circuit. An initially uncharged parallel plate capacitor with circular plates of radius rc. and a separation d is connected to a resistor, R, a battery, V, and a switch. At time I: 0, the switch it closed. (a) What is the charge on the capacitor, Q(t), as a fimction of time? Rem“ that a mug! Flake Wm has a wazrtme or“ C= 5% :> C: 663’]: Al: tso, (glam-{hr :‘r ”fill-3574‘! a) 62(0)“- 0 W gf"A3*/§ watery/a :> R are = yae‘tfifowfi} c mv L00, 2qu V- 75—11220 (b) What is the electric field, E(t) between the capacitor plates as a function of time [ignore edge effects] Let the separation of the plates define the z direction. . ‘3‘ 2; 61L“? E-Ecell of am we w/ we we we cr- e = { .1; W Our loop encldfcr q airfare w 1‘0 Pun lém. Fwmere, E (d) What is the magnetic field, B(r, t) everywhere between the capacitor plates as a function of time and the radial distance from the capacitor's axis? 3;an our 5554!.“ F5 (uprruxl’hdk'j) (jl'kJ/.1cm’l7 Sybu‘f’n, (”9 Ca. '1“? § E‘lr‘ t) Th (diff/Maw“ tar/Ml JJLLW’W "5 "i" “Wt 4‘; 0‘ Mm)“ Hut/z M"? )4 § ‘ é 3’5"”) AMP“? La”; M W” ”‘7’ “Fr” (”1 f §°Jf : DJTP Bglffli): Mo (1.01:4? IDrQ’K) Ihc‘:0 f Inf-m : {KW} c) 6 z. V~7zc _~> B(ct’)‘ gag—4R4 ) awr c (e) What is the Poynting Vector, S(r, t) everywhere between the capacitor plates? .a .5 4 l GU 4) 'ro 7"‘é/4c9' S=;,';E"B‘ fibfifizxtamw ‘3 )v _b ’ oV‘ AXA ‘b/c 01:) fi/gKKfifiVU‘e flLéJIQ'T/g—MJCZ 6’)er R (f) Imagine a tube of arbitrary radius r < rC. that sheathes one of the wires leading to a capacitor plate and extends into the region between the plates. The tube is capped at the end that is in the capacitor cavity: you thus have a surface and a boundary loop. How much displacement current pierces this surface, and what is the integral of B dot dl around the boundary loop? a A {‘s «l Hill af‘q (”0. B'E‘eH Grand H4 ow far away CF?»- He flak; J I ‘1 IA I‘flw boa? §= 44-9-9 A“ — "H’ 201‘ 50,444,”, of: 93"” _. “‘L (up +- 111 emu/demo! radw run, Inca“;"g w {r2 (C) SLUUI’J ”244 Find “UV. dn‘sf’flczwnt “AW“; firm?“ 0~"i_dc5“ckof'rfié““ r‘<|";, PQIPMAiwh‘ {aldhd C€4+Qreol 04, “‘2 QXINS am! 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