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Unformatted text preview: ECE 329 Introduction to Electromagnetic Fields Spring 08 University of Illinois Goddard, Peck, Waldrop, Kudeki Exam 2 Thursday, March 13, 2008 — 7 zoo8:15 PM Name: .
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l 9 AM 12 Noon 1 PM 2 PM 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 one formula sheet of 8.5 by
11 inch dimensions — both sides of the sheet 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. TOTAL (100 points) 1. Consider plane electromagnetic wave propagation in free space. Magnetic ﬁeld intensity of the wave is given as
H = y2cos(37r~108t — ,Bz) m a) (4 pts) Given that the wave is known to propagate in +2 direction what is the numerical value
of ,8 (stated with proper units)? W‘C a) ﬁ,%: ‘3qu03 =ﬁ'ﬂ/ 7;” me‘ M b) (3 pts) What is the corresponding wavelength? A=gzﬂT=ZM~.
e Tr / c) (7 pts) What is the expression for the accompanying wave electric ﬁeld E (with proper units)? A A Y N‘ “I: ‘7
E = [62.0130 £1740 4,— (42) .E; ) 41.2mm, > x
5.5 ﬁzuou'mBWto‘b—{az} V/Wﬂ. "’E d) (7 pts) What is the instantaneous Poynting vector at t = ﬁg s on 2 = é— m plane? I i , .. 1
P: BxH =€4ﬁoﬁwt (31“0't(5?)L
=€4ioﬂm‘(zwg)=0/ ii. z = 1 m plane? ‘(3 3 4; 4eomaf(2ﬁ  ﬂ) = 3 “y e) (4 pts) In view of energy conservation and the results of part (d), determine if energy is being stored into or released from the electromagnetic ﬁeld at t = 5% s within a cube of % m dimensions located between 2 = % m and 2 = 1 m planes? Brieﬂy state your reasoning. am?” if Mimaai af‘hwq' {ash.1 beacon—c.
 Wei Mule—rt! Fowu 2. (25 pts) A rightcircular polarized electromagnetic wave with a co—sinusoidal time variation and a
frequency of w = 67r  108 rad/s is propagating in +z direction in free space. What is the expression
for the wave electric ﬁeld E(z,t) given in the time—domain if it is known that E(0,0) = 45; V/m?
Your answer should include a brief but detailed explanation of how your expression for E(z, t) agrees
with the wave description given above. {5: w‘zén‘xwlzaﬁ Z 1;?
" A . A ' é . “cZIr v
E : 47 an(£ﬂ 4031;. 2H2) + 4.3 9;».(1? m a) I” WWW.WmWMHy.a.«.weWW.trNW4W.WeWmwemum“.Wimmﬂwwmwwmmwmawummmwwm.w.mmm A ‘ 11H, ..
ZITE'eJ ‘ 2 g “a! 0
‘7 ‘0 ‘" —) H ’?J9i:wm(znlo‘{+2n%_r_r)ﬁ (a m, 69)!qu 4% ‘t In . >X 3. A plane wave with an electric ﬁeld waveform 2’. 9 I 6MB}!
4 D
E : ~1062“ cos(27r  10775 + 27rz):% X (“7 E 3 m is observed to propagate in some material medium with ,u = 10w, Where ,uo = 471'  10—7 H/m. 1
E
i
E
W i
i a) (3 pts) Express the phase velocity of the wave as a vector pointing in the propagation direction. i i i i l i q. % U:21T'l01') S9=2Tr ‘5) U}=‘é;:l° “4/; E % A "i‘  ‘ i .1 F :: azlo m/S 7N... ma Tutu/owns tame/(Am; %
P c... — '5 awed)“ 5
b) (4 pts) Find the propagation constant 7, expressing it in exponential form in terms of magnitude E "
M and phase. , %
l
E
3
i
E
E
E
l
E
3 Tl
"‘ r.) m: 24% a; t , J
7;: x45}..— Za +J an = 217043) arr/h! e,
Ab]: J}; Y“ c) (6 pts) Find the complex impedance n in terms of magnitude 77 and phase ’7‘. Hint: 777 = jw/r. ;a’.[o"'.o.po , o{.4nxla1 , .ﬁ .1
Cf) /’* __. Kiowat) /‘4 'w (4 _
yer; eJ 41“ W '5  t M144
MI
(1) (8 pts) What is the associated magnetic ﬁeld waveform H? A :5 (o e) (4 pts) Explain Whether the material should be considered a perfect dielectric, imperfect dielec
tric, good conductor, perfect conductor, or none of these. W Undudv/ lhCMR o(={i. 4. An inﬁnite dielectric slab with a ﬁnite thickness W in z direction has a uniform DC polarization of
P = 22 C/m2 when the electric ﬁeld outside the slab due to external charges is known to be E0 = $2 V/m. The slab is embedded in free space with permittivity an m mg F/m. M“
’T : M
e l ED3 2050 A a) (7 pts) What is D inside the dielectric slab? Explain your reasoning. 33:43 git  Pram; fro. In. ﬂak b) (8 pts) What is the electric ﬁeld vector E Within the dielectric slab? ’. “— — " ﬂ _' 32' a
D:‘E.F+P —> 5—D P :42 2%, ._% 33% F $9 in 2° c) (5 pts) What is the electric susceptibility Xe of the dielectric if P is known to be linearly related
to E via the relation P = eoer? GELoneg =7 l‘xe, 3) (2% “E? d) (5 pts) What is the permittivity e of the slab in terms of so? 2: 21mg) = aim) = 22%. ...
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 Spring '08
 FRANKE

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