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Unformatted text preview: ECE 329 Introduction to Electromagnetic Fields Spring 10 University of Illinois Roach, Kim, Goddard, Kim Exam 1 Thursday, Feb 18, 2010 — 7:008:15 PM l 1 Section: 9 AM 12 Noon 1 PM 2 PM I Please clearly PRINT your name in CAPITAL LETTERS and CIRCLE your section in the above boxes. This is a closed book exam and calculators are not allowed. You are allowed to bring notes on a 3x5
index card * both sides of the card may be used. Please show all your work and make sure to include your
reasoning for each answer. All answers should include units wherever appropriate. Problem 1 (25 points) Problem 2 (25 points) Problem 3 (25 points) Problem 4 (25 points) TOTAL (100 points) {wrn (over) 1. girth S (25 pts) Consider a static charge distribution consisting of two inﬁnitesimally thin, parallel sheets of
charge in the z = 0 and z r: 4 planes. The bottom sheet (at z : 0) has an unknown surface charge
density 933 C / m2, while the top sheet has a surface charge density psT of —8 C/m2. The displacement
ﬁeld in the region between the sheets is known to be D : 102 C/m2, which is a superposition of the
ﬁelds generated by the two surface charges. a) (10 pts) Determine the unknown surface charge density 983 of the bottom sheet. L©s§£_@\@ ‘ .7.
@97lZC/m” 5? b) (7 pts) Write an expression for the volumetric charge density 9(33, y, 2) C / m2 in terms of the top
and bottom surface charge densities, psT and 983, and appropriately shifted delta functions. ﬁtter) : 65 Se) + 5; 39,43 C/M; c) (8 pts) Determine D for the region 2 > 4 m. @4271 D 4 N N
Mew)», 1?: =+@=°2>
e5 3 2 O
m L A (\
232% (over\ glu’eﬂ‘ﬂ S' 2. (25 pts) A cylindrical coaxial cable consists of two conducting regions — a central conductor of radius
a and an outer conductor of radius b enclosing the central conductor. A dielectric of permittivity a
ﬁlls in the space between the two coaxial conductors. While the conductor may be considered to be
inﬁnite in length, we would like to consider only a segment of length 2 (>> b). Keeping these in mind
do the following: a) (10 pts) Making the observation that the electric ﬁeld E'should be in the radial direction and
a function of r, apply Gauss’s law in integral form to determine the radial ﬁeld E(r) = Er(r)f._
(Hint: choose a cylindrical surface of radius r and length E as the surface over which the integra
tion of the displacement vector should be performed and assume that the total charge enclosed
in the cylinder is +62.) 5le § 5 EU“)? mt Ertr) 7< ante/«Ad war awkward
{We/OF what; r out WM 9~ . C? Q 45 = Gamma = <9 §
Answer: E,(r) = Q >
) wife/Q r b) (10 pts) By integrating the resulting Er(r determine the voltage difference V between the two
conducting cylinders in terms of a, b, Q, and ﬂ. (Hint: You may want to assume the outer
conductor is grounded and ﬁnd the voltage on the inner nductor, which is biased at voltage v.) a
. ~ Q
V: " ﬁbre” : 21:99. in») Q 0"” (v)
RGYGSL c) (5 pts) Determine the capacitance C of the coaxial cable of length Z. (If you don’t know the
formula, capacitance represents the rate at which the charge stored in the coaxial cable increases
as the voltage V increases.) Q ~ WEQIL
C t T ” Mir/o gzrrrel
,QMU/a.) CF) Answer: C = Lower) gt? (M’ifomg 3. (25 pts) In a certain region of free space7 the Charge density is speciﬁed as 2x
(a2 + $2)2 p(w, y, z) = Q C/m3. a) (4 pts) Considering the problem’s geometry, what is the appropriate form for Poisson’s equation? gMCQ )3 W5 @«vLy am )9 7+3 or (Cl PvaL%,
2 2 r __ dZV : _Jb(><)
v v 4;: .9 3;; ‘ f)
b) (4 pts) Give a brief physical explanation of Why is maximum at a: z 0. X C 0) all ‘HNQ
03159.14 aka/mgr, and WW 05AM 3 Wayk‘ +00 “(0
F a «geld M‘sz '3? dwecﬁwgg I ”O
W , j} aicvgo" . I: u,
Q E K g Orqu/MC‘M (0% M 50 O. (064:4 m_ ‘ ,
o o. __ W SL091 cu} (4:0 Mb “NOre ‘Haed‘ c) (3 pts) In what 22mm does E point at x 2 0?‘ \ ‘~~\\ 515% dag
Obi/vi ha I :1—
” J Wat/W14 \ + d) (4 pts) Give a brief physical explanation of why E = 0 at :r 2 —00.
M ream WSTS‘B 0&3 cc POSF+>Ja wot mega/W Slab wl/xage (law; can al. A (Whetstw 4M2 CW Clint.454,me resembsz a. at: «Km Add ms 40 am , 57W? Wfirf’cﬁe 1665;:
((ﬂﬂnwe‘t‘ W??? Alternative We W QW’W“ (Mid 1'“
theha‘bove 0%servat1 ms to solve o ‘ e) (10 pts) Use r the potenti V(x,y, 2) if the zero of potential 9 S is at :c : ~00. Hint: 2mm 1 _ E {1125:er CMW
/(a2 +$2)2 : ‘a2 +3122 +0 .I—UmMcﬁQ 61A 44w. ‘Mﬁw
dx 1 m (3ng W ame /a2+m2=Earctan(;l—)+c Lew, +0 96) 541948 4. (25 pts) Consider the following spherically symmetric conﬁguration of composite materials in steady
state equilibrium (the ﬁgure belows details a spherical cross section of the geometry):
(i) Region 1: r < a, where r = V162 + 3/2 + 22 is the radial distance from the origin is a perfect
conductor (01 z 00) and holds a net charge of Q = —5 C.
(ii) Region 2: a < r < b, composes of a perfect dielectric shell with e r: 460.
(iii) Region 3: b < r < c, has 6 = 260, (73 = 107 S/m, and holds a net charge of Q = 3 C.
(iv) Region 4: 7° > c, is occupied by free space (6 = 60). Note in all four regions ,u = no. Utilize the integral form of Gauss’s law, 3% DdS = fV pdV, when applicable and what you know about steady—state ﬁelds within conducting materials to deter
mine the electric ﬁeld vector, displacement vector, polarization vector in each region and also the surface charge densities at each interface. a) (7 pts) Determine D, E, and P in region 1. What is the surface charge density, p, in C / m2units,
at r = a? 6 glud; 0’: O”, a ye/feef (cMIMCJW, all charges
UUIHHU MIDMH ’l‘v HM Sodom~ le; ypvsicto :0 Giausgb Lou) gndg: 85w :0 2 650
0 l3: {27: E30 I [W 44443 Can (’WWL MUM? V0 a (cow/umLof //U 9&4ch 3%a2‘e. W SUle‘CQ/ ,4; /5‘ c —Q: v/f/
ﬁrz‘! ’71: 5 WT” m “77"; ML (9 KW clwys W w WA
(over) ﬂ aArAb b) (5 pts) Find D, E, andPin region 2. Us"; Aqausgrs éﬁ’dé 3 Zena : ‘gc CW5? Gaussiaﬁ $144902 05 a 3:510?
(a a? é WIH‘ (aohus r m FM» “you, Sum é 1) I6 {aphid/2 lifecfed 0M fa/al/c/ﬁ) 7% JIP/t/enﬁé? éi/ﬁééf/ii ‘ ’. 7 9 Li {Z:'§C 37)) =15: :11; :c 7 r V r W}, .. __ .. AV ‘ 3‘; _> _ " o A
5%“ f: 22%: " (gr
9: c) (8 pts) Find D, E, and P in region 3. What are the surface charge densities at 69 JM ﬂay“) 3 (bum) , r¢03am w/a My /0?5/M 2)
Val? (5M Melt» ciw  We WSW/u beam; Vail 019 Elm/34 _ a m swig Shh, as w Ma) Eso/ P/ o (‘5? On 7% r: b Surﬁu on mdm‘a Q: +§¢mv and Ho «€Cclwguﬁ r200 'A 1
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This note was uploaded on 04/20/2010 for the course ECE ECE 329 taught by Professor Goddart during the Spring '10 term at University of Illinois at Urbana–Champaign.
 Spring '10
 Goddart

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