Prelim 1 Solutions - PRELIM 1 ECE 303 28 September 2004...

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Unformatted text preview: PRELIM 1 ECE 303 28 September 2004 NAME 50/»:7‘1 ‘W, SECTION 1. (30 points) Consider a capacitor consisting of two thin conducting spherical shells of radii a and b (with b > a). The space between the shells is filled with two different dielectrics, as shown, with each occupying half the space. If the inner conductor is charged to +V volts and the outer conductor is grounded (at zero volts), what is the total charge on the two conductors and what is the capacitance of the capacitor? ’TW we (2 W J0 7105: 5449027 arbh‘pmj, ‘ " -—‘ ' r : , f ) )XEflmstQ 13,!) Fzr ‘33 Sgrqm‘tVLr? awn”. 5, )EL W M as a 60m 19+; 3 W (J‘p‘ +1115, 40’? g 0'? 1 ) ‘X— az H'rrr'L-FD, 3 Qonc/ : “74F” >>3’,//—/as:;=; W 33,")3/03L41/r7- ‘ D —- 'SGLF $dfi/r):fi!f):2 :,J.——>’?2 e) 5, 6 6! I. “y b b a — .52“ ;-J V“ éwa : Jae at ’ ta 4 ’ i a? 4 6 5V 1 7” 6 , ZTM H- 3/6’>_/; @494“ : 21-11(05‘ +2Tm2/gz, rum/as! (/+ z/éI> ‘ K V E +0 gammy- moss) t“ [is] 217' fél+el> F C: @4194 ; , 2: /“ (Hi1 x d MrfltoapZ bus: '24 parallel mp” CU CL 95((11 [14W +36 r Lam +kp CQ/locr/‘vattfl 4%6 RH ff (aphids?) Z x“. b a ’“ ,- ,$- :16 : p/pr: 42 [12}: p" 75%" ’L? "Im'éri ' V age?“ figs-ff? J; a b 5 ; LNré , 5 “$4 W 9(vw a Cgfllyv Zy/qfiybj o : #77“ K fl :3. Chm [’é‘flffij éfél-tél) F (4.9%) v V 2 PRELlM l ECE 303 28 September 2004 2. (40 points) Suppose there is a perfectly conducting, infinite (in the x and y directions), grounded (at zero potential) plane at 2:0, and that there is a charge +Q at the point (x21, y=0, z=l) and a charge +2Q at the point (x23, y=0, z=3). Derive (a) an expression for the electric potential V(x, y, z) everywhere in space, and (b) find the surface charge density on the plane. Assume the medium above the conductor is free space. ‘ v:7 , Use 'w WP’WMO 4. '2‘? '\.Q 2"” a) ‘Zfi’ I (O'E'rrm : - ,K a) r ' '1¢ a , W .91 1/77 63 L+£‘J’_£ R R _. ___/-- ~G) ‘ 3 til? fo’7 &) ’ 477—6, g?“ W3 2 M1 , + L v/ 3 r-q)’ 8X—I)L+§1+E") 3'1 [(¥*3)‘+5L*(Z'3) L ‘HTéo 7 9/7 22 0 I I __ ____.__;___——-—--"""“L I! “LOC’JJ’WL; +(2—+D%la fiK—fif'th +C?‘*3) 3 L J b) (469???) S J?" (350) : — 6° a2‘ 1 i220 ad) ("©sz") - we)!” L 1. 7, 3 1 L 5/), MT [(X’ULWHE’JPJVL + £003) +H1+ (-3) 3A 2pm] +7 +03 3 _.. Z [”‘/1)(L)(3) 9 [(y’})z.+'7n.+(3)1] l1 ‘ ' ' "—'——-'— ——-——-"'~ L I é + 1 'L 3fL _ y) +7 p3 : (X_,)1+L)2+lj3/L +[CY-}y.+|71+7]3/2_ [ohm +|1 4U [(X 3)+ j PRELIM 1 ECE 303 28 September 2004 NAME SECTION 3. (30 points) Shown at left is a sphere, centered at the origin, with a radius a, a permittivity 8 that is different from so, and with a constant charge density pv everywhere in the shaded region, i.e., everywhere except in the spherical hole, which has a radius of a/4 and is centered at x=a/4. Find the electric field everywhere on the x-axis (only). The hole and the region outside the sphere have the permittivity of free space. fl”. 0< )5 qr (Pf—y—cptgf, “an X '> O 3 r3 I, r . 31" ,D‘m o‘V‘Pc-f) égmxlp” t’mlxlm H9? 31 " trait“ @ 5'14”? 312 = E'[x—%)(—,gr) 4% x—% 54/, — _i (am: ‘Y"‘"’ {7%) way? WWW/WM _ 3W1 ler/L,’ _ ‘4? NW aJJ’ W ,2 com-hrLu‘h-AQ; +0 Dx 901W? Effc 1% Hum ' “9375‘” 5 5 (m7ln5 ¥¢rw1 9l+5 f bps W ' 3 Y““/‘* carnct. . , ,I. Q E1- 6° .1— é kg.» @ ~q3x<el E : K 'The chm}; of 6/9160 [J— X ‘ 4" (X'W‘t) j 10”” = [150 6"" 0’17!qu an flue absrmg 3 P'l‘flf. E: L D ...
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