Hw3_solution - Sd/Q'H'on ECE604 Homework 3 Out Tuesday Due Tuesday February 1 2005 Note There is now a class website at

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Unformatted text preview: Sd/Q'H'on ECE604 Homework 3 Out: Tuesday, January 25, 2005 Due: Tuesday, February 1, 2005 Note: There is now a class website at http://shay.ecn.purdue.edu/~ee604/. Problem sets, solutions, and other handouts will be posted there. These problems are from Chapter 1 of the text by Ramo, Whinnery and Van Duzer (3rd edition, 1994) 1) 2) 3) 4) 5) 6) 7) 7.9a 7.11a parts (i) and (ii) only. Assume both functions have period L. 7.1 lb 7.12a 7.12b 7 .12e Note: Make sure you can identify all the important boundary conditions before you jump into the math. A conductor on a circuit board has width W and carries a current I. The conductivity of the conductor is 0'. There is a small hole of radius a at the center of the conductor. The thickness t of the conductor is small compared to a. (Therefore this problem may be treated as a two-dimensional problem in cylindrical coordinates.) Assume the current is V DC. You may also assume that the width of the circuit board W is large compared to the hole radius (W>>a). Let the coordinate origin be at the center of the hole. (i) Sketch qualitatively the sheet current density J S (units are A/m) in the vicinity of the hole. (The sheet current density is the uniform current density J integrated over the thickness of the conductor, that is J S = J Xt ) (ii) Derive an expression for J 8. Express your answer in terms of I, W, r and 4). (iii) What is the maximum magnitude of JS? Express your answer in terms of I and W. (The answer does not include the hole radius a.) Note that the maximum should be located along the x = 0 line (where the current is maximally constricted). (iv) Give an expression for the sheet surface charge density (units Coulomb/m) along the edge of the hole. . I . 3. = 'L‘la In calindrical coordinates WtUn :9,- O, ‘ L . ‘ - Letting 31‘: 20') Rat); ’ves R3? ‘ ' u: J¢z I a" c. We Have two AWec-ential eQuafions . ., 31, F; + n‘F‘4, = o (harmonic «2%. Wrist» solqtion )._ |=¢=.c3¢os'nd> + C4-sfinnd) ' 4H] _ _ - ' ' dz -'J-PA [c.rnr"-'+c,,r(-n)e _ . Also ésww- - a a? ., . tEhzc.r"+ ht. 50 given ‘For'ms u"; Wow-1:5: ‘ I Illa (.i) we“) #M)=Va(I—3fi-‘_—), 0<x<-‘% % =V93é—I), géxcL . __.;_ o % L 3%; ¥wiscven30 b,.=o {’Oralln 1 4(1)=o.o+,.§ancos3='—EEZ‘- L ‘ an = t Mum = affix)“ = 2.;3fvm- ELL) dx = N5 g an =. €- ‘fOL-kx)cos-2-%l‘ Ax = %J?(I—%)cos2“:*dx b = 5.32 {E 2%.: sin — % fi)*[cos2i—£"+“L“swt°‘]j} #6) = M;— +§u :‘V;§(l—cosnrc)ws 22? no _— Too:vo3.:_, __5:x_41_. ‘ _ fi ~ __ 'eunction is neither Odd nor even ~11“): 0.. + imam”? + mam”? )dx (3.. = t fan)“ = t3 vaIL—Ax — 3% M = Zargmwwsxax =%<:%)‘J:2n—Lumz%lae—%x—) = +£auaw§xfi= 2X1m=<ww —w ..I All 0m=0 except filo-48) isodd except {70.— a level surPt o1: 5%) = 3.: fi¥cx)s.im2".'_"‘dx = Jimjt—zfigah 171" 43451) = 41—?)me = _ __ i‘”: [it EJ521153) T t | b {:00 I 5m re tmtio l o a " sine reflsentabion a) 4m Rx): a. + émwaTIf—x \ I / Iao=kjiflka1dx= “f? -a,\\ , a\ - ,m x an: %jo§“%mnalxax ' ' = if[%“—’T‘§-T~[~°%%Fl+?ifl z 0., '- o (set-ks has onlg one term: 400 =$Dn%) = «(u-nufi‘ “SC'H’UTCJ ‘ =b Mad repeated aeros in a direction so use {10m Q, = (A cosh kx—‘r bsimh kxxcmskfl -+ as». k5) %§=Qat'g=o,b. '.Cao.ond.h=1‘—E— Even sstmetrI-a in x so 550, ‘ Then i095): gen At 1:0. 3 v.= gumsmflfifi-swggg E‘Pahd b-C- at ‘K=a. in Fourier sine Series as in 5%,1QIICI4) wzun c» V0 and o.->b~, 9(5):" Eguabing tine M7“ term; 4,95: an CO‘hD%a’- - o MEL-FE" ‘ Kg _.'. @C“,b)=n§4%¥c‘ “so, alga. smnb ’ .(zb ,._ i > “ kl: b," SOIUL‘Z HA SUP’MVDSVJ’IM a +w° 90(047‘843‘ WY‘4 w—flfi ‘5 WNW" as ‘ - "€°(’Ous_l ,.§,,(o,.3) _;.._ ,. . frag) . 'ngw" _ :(YM): ._ I . 54(4)" I I t (YA) 339????“ '96: *9?“ ‘ 1': _' -. ‘ 1’7 Cw) fi‘ilérg) SQ—g‘ffl‘éjffiddnxw .. I I ' @rur fl ‘.' ' : ~ ~. I = g (’7? '- 5/0,?) was; ) w my): gm? ) . a ' Ir l' I?" 7Lg~u(a,'7) CFC: . 'HZb it 3 .av, '5‘” , § § Obtain “total sdufion as §= §.+§; when: £13 I‘=() ‘=O so‘ubion 09 Rob. 'I-lza— and E is soltglfion 09 POHern shown “" a: °' in 4.6%. Far 7.5;, need repeated. zeros in x direction . I §m = (Aeoskx +Bsuhlgx)(c.coshk5 +Dsu'nh kg) _B_.__C__. Potarb'al 53mmetfic in 7‘ so 830. §=o at x =10. 50 K: (n—i)% In b—direction Q has-odd flmmh—b about 3=% , '. sit (15.0 and L. = u mun-Jiv—‘a’éIanhEM-agcamgfl and g = :2. 0'... do: [Ln-éfiglszhh [op-g3 §C5~§fl Need Series in cosines got '90:) {lot- b.c. 03!: 53:0 #00 = z anoos 01- fine?- AdaPfinS . 141(3) an = 35%}. #00 cos Ch-szé' 4’? m, = %L33wscu-é)%dx =' The mtehivg In“h term in ‘bwo c» shah [(n‘é)§7(‘%)] : (n- 4:; )1: ‘ -I)"" » v. (-0" slim [Ch-t)§(|3"%)[ 1n: 1 -J— §;= g (n_4£)n s- '1[(n_‘§)132:1 90901 231 Note: period is 4a, > $5330 142.3 0' To 5.9), need PeriodSc. 'm x: §=o is“ it" =' (A (:05ka + Bsa'm kx)CCwos-h L5 +‘ 195th us) Since §(0.5)=o'and §Cx,o)=o,‘~A=.-:c.=.o ’ O '5 i=0 5 cas‘mg'xsflfl I m:' N - ‘%L»?k,¢"_°°‘k“s‘f"hk5=9. Thus lax-3:; (VI 066) :.§ =:..§°”$3”§'§*5‘”“€% . . To expand b,c~. ox ab; (52 _ .1-ncm with 0.929. and. c—wlo'f kx>.=4:¥5‘.§,%:%_ ' I ‘ 3. Casinh’gg’rb = . 0'- c'”= pafig Wing’s ' ' ~~rL~ wow a; M m M\@ ?¥.-€wid- bDI-fivxdcmgl 0; ~44"; WW W7“?- _~P6'\ XLq bol‘ 0* :o (<74, ,rw'm‘zom A L J :5 “wow +6 @4103on +m~vw1=n§ £91449; 0,} mum, , I . 65mm. 3 >G'E w/uqh51 “090.523; S‘mgcwu. yaw 67(03 ~figxx~vwm3 4A”9°I_ 7; M37” Sit-317:3 flog/em; NV... v-4 ‘ I I FM_g/v*~. 13:??? IL. .1..wL_ $§.\‘"‘. . _ H . —_§ «9.. p2 l . ._ .V . _ 39_1_V.\5.F.45_.5t§{$~ _ ., .. .. .. _. _. .3 .V M. ,_ '. gfliaéééf I . . Qwél f . f, ' ‘ Mfg->40“ I 3:: , . ' . (4.) . {9/31 a J, ., ._ .. ...4U.e:[.(._m44w" 4 Gas 42w“— cuv .. if”? . O. f???) 4.. 7‘? hi.“ .47??? 7 77”“4: V. _. [9.9 :9 . 9&5’7‘12’44’44 9‘"? WWW” 74> . MW??? «C W¢~ : bib—'7) . ,. L ., . ,_._.... _..-_ “ ..._‘_. ._ .m, A. .~....... __ _.. .. . ....~-...., m. M ..,.,_,._.~..._..___ _.-_<.a,_§«_w€ “Ehw‘a. <ég:_.;sw.‘;:. .. “3:61.”..WM -0-..” ~~V.-,.....u an-..“ .. ,._ (flui- $4? gzéna";“”;ge.;g;.g;ie; QQJQQCQ > ‘ V i _ ,. . _ . ta _ ( a _ 4. “71‘ . _ m. ,. ,. . ’ " " (J5, MM NM {f . __ .1 A. _ _ _ _ ‘_ W = ‘32 CM ...
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This note was uploaded on 12/12/2010 for the course ECE 604 taught by Professor Staff during the Spring '08 term at Purdue University-West Lafayette.

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Hw3_solution - Sd/Q'H'on ECE604 Homework 3 Out Tuesday Due Tuesday February 1 2005 Note There is now a class website at

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