EEE224_Fall09_midterm2_solutions

EEE224_Fall09_midterm2_solutions - EEE 224 ~— Fall09 —...

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Unformatted text preview: EEE 224 ~— Fall09 — Midterm 2 3 I) (20 pts) (a) Write down the [\fl fundamental postulates of electrostatics in free space in differential form. Describe clearly the quantities with their units in each equation. ( V0 l'l tr V-E= j?!— rE :eleclric field ifl+€/\5l’l:j 6 ~ 3’ ° . [me in r e elm; [Caul/ “DVXEZO tfu.Vou caj 1+3 m) \ Go; permittivity c-F frcc'Jchz (Favan (b) Derive the integral forms of the relations in part (a) by using Divergence and Stokes” theorems. V £3: is :> ‘lV-E‘dsz frvdv T.) 598-0l5201‘0’refl v 6° W 603/52,." 5 6: (atom (MU?- flSE dbl—:5 'M‘l lam 5 H “K O Sigh; jgfifi / “N: H V E5.cl3: ~—~— E‘_ TO VVXE O _.> g x C (C) Is the static electric field is irrotational (i.e., conservative) or not? Support your answer. ‘3 ‘jes , becauje Vx E": 0 (d) Is the static electric field is solenoidal or not? Support your answer. No in frat/J laeccwe 749450 Edit. fvz‘o I’ILM/l :12) E r5 JO/Enode-g (e) By using part (a), decide on which ofthe following vector expressions c0uld be static £77? Justify you answer; i \5 '5 (i) E = 4fir3 63¢ (given in cylindrical coordinates) 19 (ii) E : 1:2 51H+ 3:9 are (given in spherical coordinates) V E: ’4'? gag? : O wnnol [0a yla-lic Ef-FFQH W118 -143; (r599) 3* O 3’ ( a( H V ’E = «(gt 3-K fl‘ 25w 39' gym lei EEE 234 — FallO9 r Midterm 2 4 2) (20 D‘s) . . The figure below shows the planar interface between two iinear lossy media with given parameters. A steady electric current flows in the structure. The current density vectors are assumed to be constant in their respective media, and they are expressed as: J, = a, + 25;}. + 4a: (A/mg) in medium 1 .73 : 5&7‘__+ 131.51“ + ch}: (Ar/ml) in medium 2 Find: {‘5} (a) 02/0" ratio. w) (b) .11 r and .135. if) (c) Find the surface charge density pf at the interface 2 i 0 (in terms of 50 and a] ). (hi-dd) Find the dissipated power density in both media. X fl " '~ Medium 1 Medium 2 :'" h )4: jimomoi+xt 1:1 3* g 6" E4 81 = 280 87 21580 " a «x a u 2 :6 E Taliammofl’kjckoawh‘a—i . - J?“ 2 2— 0'1, inc/ml : W32 dawned: §x+25j _ _ iwmfl = 32% at r. din’a’XE/d? "— gar? 3'25 0‘5 U1 EEE 224 4- Fa1109 i Midterm 2 3) (20 pts) A sgherical capacitor of inner radius 'a’ and outer'radius ‘b’ is filled with a non—uniform dielectric of permittwlty 5 = so [2 The conductors are kept at a potential difference Va. Find: 6(a) E and 5 for (ISRSb. ’k(b) Capacitance C. I k6 (c) Polarization voiume and surface charge densities, p”, and pm. Indicate where they are located. Note: Laplace equaiion is not uscfzd in {his problem because the medial-H is inhon-Iogencous. DSSume +Q M ,Q 0A ouk/ ” USE 60w”! [OHM EEE 224 - Fa1109 — Midterm 2 6 4) (20 pts) The spherical region RS5? is filled with electric charge of unknown density p(R) (COLll/In3). Outside the sphere there is no charge. The electrostatic potential inside the charge cloud is given by '- V(R) 2 V0 ikR when R S a ' p(R) where V0 is a constant in volts, and k is another constant volts/m. 80 89 0- ‘~- ._ J n..-" Assuming permittivity so everywhere, find: (a) E for 153a and RZa. (b) PU?) 5’ (c) The electrostatic energy of the system. g EEE 224 — Fall09 — Midterm 2 7 5) (20 pts) A metal bar of conductivity 0' ‘is bent to form a flat 90° sector of inner radius “a” and outer radius “19’. and thickness as Shown in the figure. (You may call the geometry a quarter—circular washer). Find the resistance between the horizontal surfaces at z I 0 and .z = 1 (between bottom and top flat surfaces), by solving Laplace’s equation. Note that V is a function of: only. Think why, and state your reason! V—O smw‘ \l l5 WORM Eonslaml) oi‘l’ Hue +0}; omol hallo/M 1&1CQS/ 'llm Pokfiliofit «EUflC‘l—JD/l mnm+ be a {LI/U33“:le CHE i" mol Vavtti 3"\Y7’\M%\.;O Laplace 29W Oil—El. :0 ;§ “LV— 2C4 K1) (/5 01 a1 a % Egg; Vtt=03=‘3_{¢) ‘ -:>EE;EEEEEE:lnzi Ul%:+>:vg _ u“ gag _ I \Ito)-2CZ:O ' whit); (ich =\Ig 1) c4: ya if“ "x -- —*"§\l fl ._ "A f 6 E: ‘ O u e; Swfqut {S {more l JC l0 ///_3 Fgosf'l'we . CUIKflWl’ flow I: S :2 g " 610 (3}) . (“6} (circled ll“)m l” lac-HUM. ...
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EEE224_Fall09_midterm2_solutions - EEE 224 ~— Fall09 —...

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