This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: SO/u‘l‘t'on ECE604 Homework 3
Out: Tuesday, January 27, 2004 (Session 5)
Due: Tuesday, February 3, 2004 (Session 7)
[Students with tape delays: due according to session number] These problems are from the text by Ramo, Whinnery and Van Duzer (3rd edition, 1994).
1) 7.12a 2) 7.126 3) 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 twodimensional problem in cylindrical coordinates.) Assume the current is
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 3 (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 ><t ) (ii) Derive an expression for J5. Express your answer in terms of I, W, I and (1). (iii) What is the maximum magnitude of J S? Express your answer in terms of I and W.
(The answer does not include the hole radiLis 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. 4) a) Shown below are two infinitely long, very thin insulated conducting plates which are held at potentials V0 and O as indicated. (The plates are infinite in the zdirection and
semi—infinite in the other dimension.) seq“
YT \ q ¢
06‘. ‘.
__.______.._—l—_. __—>
v=0 x Determine the potential distribution for O < (l) < 0c and for oc < (1) < 211:. b) Shown below are two infinitely long, very thin conducting plates separated by a dielectric of permittivity 8, which are held at potentials V0 and O as indicated. (The plates
are infinite in the z—direction and finite in the x—y plane.) (i) Neglecting the effects of fringing, determine the capacitance of the structure per unit
length. (ii) Neglecting the effects of fringing and without using the relationship WE = % C V2, determine the energy stored in the structure per unit length. Show that your result is
in fact equal to %C V2. 5) To measure the conductivity 6 of a semiconducting wafer of thickness t, a hemispherical
electrode of radius a is alloyed into one side of the wafer as shown below. The other
surface is grounded and serves as the second terminal. Assuming that the contacts are ohmic and perfect conductors, derive an expression for the conductivity, 0, as a function
of the measured resistance (R). Since the alloyed electrode is very small and the wafer is very thick, you can assume that
the current is uniformly distributed over the hemispherical alloyed contact. 6) 2.3b
7) 2.30 8) 2.4e 7.12:1. ‘ §=° ' =5 Nccd repeated items in 5 direction so use {lam E, = (A (lush kx+ bsbnln kxIGmskg + Dan kg)
8.0? i=0. at 13:04:, ascend h= mb Evan gamma(:13 in X so 850,
Then @595): Egan ooshmglsannga
At x=o.: v.= Ecnwshﬂﬁg—sawmgg Expand he. at x=o. in qurzer sine Series as m
. ‘ 4v 
a, 1ll(l4)WIU1 0“ V0 and a—pb; ¥(5)'=n Eguating ‘Une nun term; 4%"; = (1n gosh Lg“
: 4v. cpsh&x sanmtgg . J. =nw n7: a b
19%.
352:0 ’ . .. .
'Hzc 0' To b_.._c., need Won panache, m x:
§=o I in = (A cosh!“ + Bsim hxXCwosh L5 + psam k5)
5V r I
1 .Since ¢(0.5)=o and. §£*,o)=o,vAacao
E "‘ cns'unkaJmh k5 '
O E’ b V .. . I h r":
'0 a3xL=t.k¢"°°‘k“sV”hk5=9. Thus ka‘aa (ma.16) .. I ' “It  m
3 g =’n§d'u W1“ xwh m5 T6 3W b.c. ad: Bab, USe THCHJ with 9.5291. and.
‘ ' I V __§_ . nngs
CbVoZ ¥c»<>,=“‘%édn9”m, ' '
4..'_\L.———.——'—g'
:. c" = OI" en a "‘1; SUV!“ ha: V AVo ‘. hn_
1§3n§9_hm 51m"‘—EE'h u. “"34 *' m Mm wan, as m, m m \ . , > V ‘ ‘
M W {31 75*4‘32/[4 nbr¢4_ goggnotm“ b; 44.44 J
at?“ raga XL“ 0 2° ‘6!“ ((74. —T‘M,M‘loﬁr‘em(’ ; MWW Jrc “At0:4.“ +(V‘L‘V aﬁ— $0unta~1 ,4 I I
CS’mw 3 =trE 002mm SM‘OIAW 41500.93 M1» 3%”. :(‘aban )9 ~AA‘VWI’5 4AM?“ _ 7: Aa'i‘ 9E6?) [Mg/mi Wm. I v Q) prme" ,.nv»rr~.°5“.4‘¥= [99M‘QCW7 “M‘HV‘WI‘TSL' ("k—«2k mo, Tkw‘mﬁQ. , . C}n"g’l;’\°¢‘<> aéewdcw .4"? awﬁ?‘ +A5¥M¥yéy R chm? (M'er — w: a“ mum My 3 .
, 3 WW? w ,
, .VW.. 534?? 1’93 (18)“).
7. weskmwﬂ rf’WFﬁS‘iS, was”? 59733???“ wow; ‘9“. . 6M1. 6'84? “CM‘WHUIE V;_J'#9“‘,Q‘1 f5'%‘ ’22; ‘
QWW—m) ,7 L2.) ﬁwk— '33:; £0.15 J“s 1
TB X0 “Ml. $u\d"\ VL§:O L $913;ch A”. “buﬂé
VQméI‘X‘éw ' “ES. Jim/1J7; Calnimijclyao‘ 2L low/1 5v} IfJ (55 Jug/ac; ,c/LmSL igﬁm’fé‘zr ‘ [VG/é: " .Z’UL 625/“%/<M“ 74/
sew (4.5% 09.7 (x) :4 ave/7w ék/ﬁw‘”  ghﬂé'kﬁ'ﬁw G’af (ich owl/"ﬁt 1‘? 2:61;»? ‘77”; XS chﬂuﬂ ﬂ .4949 ark." .17.”. 1‘99 _
5,50; 6.x. 74/749254 zuﬁ'xg 4ggmolm7 {Ax/Q4 "ram: o 217)! 75% W07, (at, 7/71,; 3 1'44;nywa
W #4179 Aug. 6,33 &M#.wq. TA 0J7 7%,“ 7%“ §aluc (725530 wl 67/”; “a meﬂm‘io 44.2.
Mfazﬁﬁhvg 743 {363% 7"; _ Mwﬁmbm Vf‘lu‘“ 5A0;’5/ 059‘“, béG‘T‘y‘c‘ﬂ K550 w (719; _ ¢ : i 2:) egg; .f ' ,, _. (.381 NW N e: ¢>i7~2 ~~+\ussmz<a)
mg; w; my .osﬁmxu as we 'QMQM ML ! _. Uw onél"r*4"‘.. (0“ ,4“ W35” £w~gwx dim» Caz/“‘75 9+. (_ _ f" ’ Jog @571 h ' ;;‘,ég“ Q4442. VD he) I . {7.1 D ._ s h... ‘voﬁvLJJR win away I W q! gamma,4;
Lot‘th 5 cm» S—c‘u+w ﬁvﬁC/El" w: 07Lr€J~<ngan¢9 a ax 049w: @W>= VOW?)
a) Fa #0541?“ ﬁg”) :iv“ ' E” deJ? wax—(‘55 LET“ Fa'oxscﬁmz‘. E¢;"}3—¢  r‘ 2; Fa dégééﬂr‘ BED/2 “ lr‘ 2,;er N044. “I bxqu Pde4Ml7$$~€~g®MLa
xg M“ \Khivdlwl cg #k&.&&p.Le. 5) 5mm 2572a‘ #6 5%???” CMM @Pmﬂmgf gay ﬁlm/van Saw/C6
MA, 51‘: WNW 2mg; ( ray/«cm; fie 71mm?)
4? " {#6 WWW Fanf' P émée mic/en a;
I x 46 416% ~ 47:6/W‘
We @13wa k *2 il/ Mm: 1/ is fie Maggie 5mm *3’ “d
7‘56 fW/Cp/m, WMC Zifkﬂidxg 7% WM W‘gMIWV
m ﬁg gmM/CP/We (w my 2%: WW6) 24%) N i > 7
1/” Emaw'E/Wﬂ): $4516 mm) NWOL (a; 74) “ — 'e For 1/ W" W144 Ma. Jro’E and mtg/“fa aver Ma
ﬁmw‘ﬂl pfaﬁc 75k 7“an éWM. Z 742/4 ng 5a 7443 W/W WP ixlyféﬁh (WWW;
f5 ﬁééw'8/gza 1" "ﬂithﬂfféi .
1:4XWW.%7:7513V2 : “’gkgxnfﬁ o0
Zﬂﬂxyw~é~é72me(7”x?)ﬂ% 4f)[ (22ij / éﬁy _ ’o/t w 1., mi 1  w zo’ﬁ’f '7L_ (fir
’4 7%ﬂ (tawdx 2%:zm’v’é/D ) ’W‘EL‘E"? .— 7Z€ V9“ . 2i,__£,, ,.
, . g ﬁmég ﬂ 47fé£ (r? 155% (A)? C‘M‘a‘xbo ﬂa “fla.— I‘M‘jM’Q J» y} 4RWJV «S 4;!“qu r‘oqr“ a 3% <1? > 3.310.; one side. (Sea €69.) Z 3" Wen rem.“ 09' sohkb“ 90" _ 4 
T \‘5 3‘ b3 . ‘ ‘ '9 E7 Frum AﬁVOX‘A
Ex. 8355. Axial gpmponent ts Haw: , ,_ 56:.
I ‘\ V 1 L ‘ \r
k“%)ones\de. 33“” \“S‘a‘ktf‘ ‘ a ‘ "‘ b" —>
where, mg: %_ and Y': £03432 ,/ ' ‘3
. _ x _ /
Br Qwr ﬂies; “2‘5 2°” “& M‘s’so up
bag, 4‘51. Hg__,_q____.= $11.03
“£03137 {axea? {ohi=2 ‘W KQZ+12)JZQ3‘+%Z’
61.30.: qum‘f per uniat WARM: n1. Wm ’ h _
Fm,“ FOP ten? 1 “'12 '. 0 0 . g ‘ 0 0' Liz
Jr to“ , "“ ”““oa";’°:k“~.'“ swag" (n11, aim?“ ’ 4,0 , a (033,?) 2)“)I2
H: gn‘Eo?’ '47 3‘ 7 " 'L'
2 . 0  1.. 7 849. MW'V ‘m ugMdvKCALML "\Oﬁ be. uKsRcLer 'Q '
smmzwn 0‘1)" a Nisan? w. hrs'k' Ms‘xder wkndm.
9% W o. mam M90m3 wwwlm Mada '
“4AA 2mm —&‘ om simmer9&ma _ ' SMA 3mm 3. “(9:1‘31 ~= osrsc din“
dﬁ‘é‘a  ,rx
P a
or ‘m GAMMA. M'ma'ﬂlx, “4+: ~“q3W: Jig: .13—
“h: “Quack m. — M wee} m mu own we. r‘ as M am oh am) _
' n I
M2 " ‘ “5%) “#99 or “¢.=%'$‘“9°_=‘39;§+ , ogr'go
 _,_ 1 ‘3 l ' ' ' '
‘ "‘3‘: “As/a" ">5 “*3 933$? atm—«>‘*“}3 ’ "b Ha," H'P'W‘?‘=—94—H 3*“ :41” 2‘ , OSr'éb bz‘;:':a3=' 2a,;ea'3‘hfi ‘ “>5  V 50 3“ hokﬁ Hg= Hg++Hg.=4% . Hx=Hx++Hx=o~. £35k: is oonstamt (Vcrtiml)
‘m hole L Losr'é b) '
In conductor, 04x4a—b and Va+b<x<¢ , {5:0 . H*=o , H%=—4§‘;— wig» =4;[1—ﬁh§aﬂ ~external ‘bo conductor, H‘>0, 5:0
1' 3. .— 7. a.
Hx=0. H'o=%.4 (333) = "—3: % 1:51:31 ...
View
Full Document
 Fall '08
 Staff
 Electromagnet, DC., Ramo, Egan ooshmglsannga, 4v. cpsh&x sanmtgg

Click to edit the document details