Exam1_Solutions

Exam1_Solutions - Georgia Institute of Technology School of...

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Unformatted text preview: Georgia Institute of Technology School of Electrical and Computer Engineering Midterm Exam #1 Date: September 29, 2009 Course: ECE 3025 Section: RPY, RPK RCC Time: 8:05 AM — 9:25 AM (80 minutes) Name: SOLUT/OMS Guidelines for Midterm Exam #1: 1. Closed book, closed notes, no calculators. 2. The work you show on the exam paper itself is the only work which will be graded. You may use the backs of the quiz pages, if you find you need additional space. 3. Plots, sketches and diagrams must be labeled in detail. 4. Answers must include proper units, whenever applicable, and vector quantities must be clearly indicated using the convention adepted for this course. 5. It is required to show all work in order to receive full credit for your answers, regardless of whether or not the answers are correct. Incorrect answers may be awarded partial credit when apprOpriate. 6. There are 6 pages to this exam (including the cover sheet), plus a vector operator page at the end. Be sure you have not overlooked any problems! _ 1 _ Name: Problem 1: Simple Electrostatics (30 points) a) 10 pts. Censider two point charges in a material with dielectric permittivity e = 10'11 F/m: 41: M] at the point (x=1 m, y=o, 2:0) and -4Tt' nC at the point (x=-1 In, y=o, 2:0]. Derive an expression for the electric field strength as finnction of position along the z-axis. ":3 Qt Q“ "‘3’ .a - -: ———-—-.r—":"“‘ A -5 WE“; .. ,': Ck m E LliTélV-"rd? Oman it“! it'—F:.1a " "‘5'- r" it “F *' CM: 4 %0\z once (if; :. (xx-r» 240.; Tnf i‘ glen? E-moS‘ “5“ 1... {fl + A x _S. 4 m2: Oqfiff 1 ....,_;..;?‘_'_ifla_ a w ,t) a. 4.. ~ as: + 2 0t ‘ We)": .. " “‘2' “i__ LiiTxtS'q - 'D‘W fi 7: EEOC: ax V b) 10 pts. Find an expression for the force due to these two point charges acting on a third point clgrg: of lng at tfimsmon (i=0,y=f:fl,z=0); : _ 2 96 V/JIM Symmesfiry‘ —1‘-‘ A ~r2 — "LOO 4-" ' ~ 2' — i - ——d‘7" a N F ‘— Q L— " l é) x O 2 )4- -—__::_.> E:“L'\XlO—l06fo c) 10 pts. A filamentary line of uniform charge density Q; C / In lies along the z-axis. Find an expression for the differential contributiou to the total electric field at an arbitrary observation point F due to an infinitesimally short section of filamentary charge at point (x=0, y=0, 2:20) with length all. Assume that 20 and all are given in meters. Note: you do not need to find the total electric field. A; a QQMA 3?; _\t 2 E033 Hit—é i F""1o0\?l A :3 A A ’ Zeag 5 XCKKJ" VOW 4' {EFZQ‘E‘G‘E .2» ‘nh E ‘5' we t V9 .v t (E...f..§f>f‘a (x? +72+ (2"25‘f3v2 “A a I as. A” \ .3. ‘ .3. = Kat “III-g E “K H *' (38+ y + (a @9311?” ex 5‘ if} 3.; ‘ j> Qgfily {an __ +— (3 fig}? V- t“ ‘ “7. “ ' H) m "" _.,,...-—-— | ‘ '- W I E We? 99?- —-— (T's—3:1)“ W a“ mam-m: ' ' NaInC: Problem 2: Maxwell’s 1st Equation (30 total points) 2a) 10 pts. An unknown charge distribution yields an electric field profile given by the following -7 2 10 Sin(% J05; V/In. Use Gauss’s Law to find the total charge contained in the it 50 right rectangular prism depicted below, which lies entirely in the first octant and has one vertex at the origin. equation: H?) = mil A 5“? Ky (1,2,3) 6‘ a 093'. D «3) .J r W Z ’ “‘5‘ .3 .3 s; l I”? -’ l) T 6:} l; S I OWL-9,1,? am “gr”; " U; u. as. , J “a E grow” “:1 - — . . . . _ _ _ _ __ gage I: y a; I ll. 3 e A a an n J22,[O7Sl{\(:gx\)0\xe0\figya§fz O D l x: x ’-‘ ’ F? [1:1 ’6 QM 5' » é: : Ti, a: M) C 2b) 5 pts. Use Maxwell’s 11"t equation in point form to determine the charge density as a function of Position. J l? H: :3 ‘ i “In? \i I; .— __ _. ' :57 it"- E m" Fr \7- Q” v1. “Mix? ._.I -_!? ‘-,-- \ "°' z “3 C 056‘? xx} 5/9”“; w 3 _ Name: Problem 2: Maxwell’s 1st Equation (continued) 2c) 5 pts- Using your regult from part 2b), perform a volume integration to obtain the total charge contained in the right rectangular prism depicted above. ' h «33“ m:- If'x— I : : I i I-I III: i I I {VJ . far J. l 1‘ I, .} Lfi(2.~ V\$ f 0 \0 ,3. J N __ 1 : 6mg? ; Megan? a _'3 \ié ' I T7"? . WK _ in“. a 1r 0 7 as 1 cw‘wbs r 2d) 10 pts. Find the electrostatic potential at the point (1,2,3) with respect to the electrostatic potential at the origin. f [9*- 0w {O,o,o\flf L3.— 01,73"). ,2 T‘fl rh A be.» .- \ér\ ?: \ w\ i i K A- J ‘ ? ; g‘;£v \1, “WE r VS 4“1E‘0\y-’L’ ' aflagfl w \ 3 \1 ‘ ‘0 O .4 o I y:b Xs' Xi: 3’6 V32 .32: a . 7i . g “ PE C l Eu; E m‘r no: »’ f H”: " ‘J‘f } A 1 ¥ 5‘ ' f I P .3- ; ‘ ‘- w (Fan ("X f Y \Jhc~ : h i E; ~{ZKK le‘ ‘& ‘qrgggpp ; k 2 (5- El .“*’ z 5/ A: r ..- : "iii- lie.— (osng #1 M“ Vail? The £5: 11’ 6.0. - 4 - Name: Problem 3: Energy and Potential [20 points) 33] 10 pts. If the electric field strength is given by E" (p, (11.2) = pdqg, V/m in cylindrical coordinates, find the work required to move a charge of Q Coulombs through the field along the unit circle from (2 In, 0, 0) to (2 in, 2n, 0) in the clOckwise direction. Is this a conservative field? Justify your answer. we «933$ Aer-wer New; W: '— x} ' 5955) fr? W: are 6‘2 {5? sou For W, _ ‘—__ _--. ---.- —.—-.-..___-’;l :32; j \ |v_,..—-———--——-—- _. ...—.. _—._ . .. -- 3b) 10 pts. Consider instead a single point charge of Q1 Coulombs at (31:0, y=1 In, 2:0). Assuming E = so, write down the expression for the electrostatic potential at every point in space due to this point charge. What is the work required to bring a second charge of “Q2” Coulombs to the point (x=o, y=2 m, 2:0)? V M riffiZ/mfl? Problem 4: Electrical Current (20 points) A conducting wire of length 6 cm lies along the x-axis from x=o to X=6 cm, and has a radius of 1mm in its cross section. The electrostatic potential along the wire is known to be (x / 1r) uV, and the electrica] conductivity of the wire 60 - 1 0‘5 S / In. Find the current density within the conducting wire, and then find the total current. Does the current flow towards or away from the origin? A 40' :- \f A ._ fl 3.. A P... w- fli‘x 1r“ J “ T? m7" ms, t, J i A A“ “.2!- i -l [N l: ’5; 2.5 F‘FEEA :P t E“ greefl :- 1‘5 (cfigfir-{mr‘ \ A r fl ’3‘ 119‘ a ‘3 I max“ KL “7— ” N1;- “- 5 wt“ fl. J" : “12;” fit a 0 :5“ : ’ f {Q j] ___~_. _______ w n _.—.. v _ - .. , - F- S I I '. i -— a f: \,f 5 E w “ a. 1 C -" l f c f “T d? E f \N 8; flag-rm L“I‘d!- I/IE it" S f t; ‘ ...
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Exam1_Solutions - Georgia Institute of Technology School of...

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