Exam1-sol - Name#1 56 log t o a Notes 0 4 problems 50...

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Unformatted text preview: Name #1 56» log t»: o a February 18, 2009 Notes: 0 4 problems, 50 minutes. 0 A double—sided formula sheet is supplied separately or attached at back. 0 You are also allowed your own one-sided 8 1/2” X 11” formula sheet 0 Please put your name on each page. W WU“ ”G WK}; "5’ y §MVQ1 9)«\1«L5 us ‘i‘Lx Leew'w’i “bu/Mfg 3‘ ‘F‘h WALK) mesh» 9/0 (a 47.5”" Name Problem 1. 30 points Three different vector fields F are sketched below (see next page). The direction of the vector fields is represented by the arrows, and the magnitude of the vector field is represented both by the thickness of the field lines (thicker lines means greater magnitude) and the numbers placed close by to the lines. We also assume that the fields don’t vary in the direction normal to the paper (2). You may assume B = uOH and D = 80E. The charge density (9) and current density g!) are not specified but are allowed to be nonzero. For each vector field F, answer the following: a) Is F a possible solution for the electric field (E) within some bounded region of space within the framework of electrostatics? b) Is F a possible solution for the magnetic field (B) within some bounded region of space within the framework of magnetostatics? In each case you must (briefly) justify your answer to receive credit. Figure and space for answers on following page. E Let/“(Vic “C”’[dl) x ~Fri“ V Ltd {J} .Liz L V0 S 541““ 'QJOQ ’ ”IL M? W V‘CE:O Of‘ (fiqfvccim'i/(a, §E «d2 :0 (vaxa‘whw‘r) ‘ “luf‘i‘aih‘c 'vau); Wt rulw-FL' V“ig:o 6P ~vaj‘y‘r‘m‘4'5/ éE‘CQS 3Q Cevmm’kmt) H 5 (iv—«9x gng b 43'an (7K 8 : £193. . \ umwswlmz‘wwfl ~ 3+ WW1 (u. MMW ,M, 2 . Name F Is F an allowed electric field IsF an allowed magnetic (E)? field (B)? < (.9 thus $‘i/bbmm, €613th §bz€9§£o unafuwfi; vs '7 o ., ~ ., _ @e «m e (mg a». We ’L‘C . ‘7 V00 (3a , x 6»; nee. , , C‘id‘vajfw%’ ' G U,W(¢.Ha, t: nua— E32E49,(K)) E:%H§3({);~H\M\ ._. A A . Vxeagw‘é/M’ #0 —“2>5‘a~ :0. CM l~ OJ lowulx Us} «5 cad—em ‘ L! (2‘5 gkwm , we, W ‘ ' a» “ wV-S-s-er' ‘,1,{,_3 ( ,0 bl: he “dd. 0‘2”“) S‘Méku—v ‘NQ SLLWMI WL 5;“ “gig qéo‘ ® (Nc‘T ALLowE'fi) ‘ vEtyuwaH; 331% 8%) WB“6 «SEK #0. C rte‘l" mllouadx aan emlom ”i“ pl . . . “HUB, [email protected] Wm: ”I Be sure to brlefly Justlfy Your answers! Whom) 0“. WV; QW~ \%WW (5M “(4% Sam-Q Ecfi A» '33. (“with tum: sYIE‘Mhl-Q 3i MQM “Hdzm VKE “'3 L)? fight; ))3 :0 0.32 exalt/diff 444:5 foo amt: fig“ 7:2" , TM Qmfl‘ufiw‘ ‘5 wrMS , Shflw. £5 JAE '0 W JAPVLA-C -‘ if; ”(Ml W {A Ema) (m c Ojvd/U QF‘ISIH . M-f’kuk‘y (5: o lo 4a, U“ Name Problem 2. 25 points In cylindrical coordinates a current distribution is specified as follows: Work out a solution for the magnetic flux density vector (B). You may assume u = no everywhere. A {4- ¢ {5 j g7 mm 434*st H s: i—‘yrlf (,‘f‘\ ¢ “OM, Aw C7 imc’fimwl ‘ 4:55" ":6 :1 XT‘ (SOS c. Zn? (K R (i! :: i3: (smut/3%“? 47M 5 H J C“ t5 .,_ 7:: r“ 2,. W ”1;. 3“? (it .. its“ “1 “n a W €~j§><~\ ‘ A? ~ 42):? {W CS .2 Y“ i; as» u . Ix \ 5N“) “ O . gs} r A <~ a» ~ M: m “2.“, ~ 2:: t HNGQ‘ 3‘ F a i i »« WW 75 “,“25 W” Problem 3. 20 points Let G be a vector field that satisfies both V-G=0 and VXG=0 One possible solution for G is a vector that is spatially uniform, i.e., independent of the spatial coordinates. Write down three additional (i.e., not spatially uniform), linearly independent solutions for G. No detailed derivations are required, but you must at least briefly explain your reasoning. Mm gmw we» and :5 um um? ”~5li { a : 'l’hbm 6m“ VXQL e, (”LCM WW6“ C: ~‘9¢ ,ch sink, \7r6:—‘0, Hum vigma. fisa/fi O/fitéown. flwlegdd Seldhrml vault 4; status Ml gm,” cm region Us (l wwk. @m 516%“ 6‘ i) {Ema ~95: WE X-X‘a ‘1) @= {pmwkx} 53" (‘13) , 4 E; :_\7§——> C; :-K&§&(kv§in(ky_>x “KS‘I‘nLI(kX) 505606)Q 7—113 act—«wag given 6? mel’l’f/"U'S g 50/0 74th , 0M 76" «£907 rc‘JJos/‘A‘i Vat/0k Z '14 (up-,1 1C=l, /<= 7-!‘/ zc am we) «10¢ COLA «[50 099 C7/cmeé1/‘CJ 07 fill/(MIKCJ C-OOfcltnc/h'é pet/whoa}, r [M15 Kl" Ci::“'/Cl'2<_ flaw—{Va 57¢: D=G~3O VXE =0 dUL-ES/‘nselm' 174% M4' MPP7r7j Glam} flea/(‘8 ./ iii) (cf/7 be Name Problem 4. 30 points A perfectly conducting sphere of radius a is placed in a uniform applied field E = E02 . (That is, the field is uniform in the absence of the perfectly conducting sphere.) The sphere is surrounded by free-space. perfect conductor E=E0’Z\ a) Provide a sketch that shows the pattern of electric field lines in the presence of the sphere as well as the distribution of surface charge. Discuss (briefly) how your sketch is consistent with the appropriate boundary conditions. b) Solve for the electrostatic potential <1). $2M» Name Problem 4 Additional Worksheet L”) F03 réq7werl~e~1¢c 5:0. (W (om-[05% View ,3 M WW my) ,4 0056} Out/742]“, @626): *fl’wjé 7L Fm r‘Zq r2 I MW-fl’» ”(fiL 74" 4;? 34/;4 on Cam“ E7QJJ 3 F:[0§)% 1272941“ J/mlcmad. ‘f’AL RM 5&7.me 7%. M 07%” g/VVI’KMJD Loy/“UL?"- és— éflAMr‘cJ Guard/[40:04? Md M/M‘M ’4’ m9 [9% wflw 7%; +0 mm 4mmch CVYLJfijllmgfa) QQMJ—(Fré): 6. fif- P24} K 3 7’97/“0 "'59 Q @56 4 :2 ...
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