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Unformatted text preview: ECE 329 Introduction to Electromagnetic Fields Spring 09 University of Illinois Goddard, Peck, Waldrop, Kudeki Exam 1 Thursday, Feb 19, 2009 — 7:00—8:15 PM Name: 
l L wﬁw _l
Faction: H 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. Froblem 1 (25 points)
Problem 2 (25 points)
Problem 3 (25 points)
Problem 4 (25 points) H.— TOTAL 100 points) 1. The objective of this problem is to determine the magnetic ﬂux density B generated by a dc current
distribution ﬂowing in an infinite slab of width W extending from y = —VV/2 to y = W /2 that is
parallel to the xz—plane. Volumetric current distribution is given by J = ~J0§ A /'nr2 inside the slab
and zero outside. Your expressions for B should be given in terms of Jo, W , and p0, and any relevant
coordinate variables. . henV
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W
a) (10 pts) Using the speciﬁc Amperian loop shown, that extends from x = 3:1 to :0 = 1132 in the a:
direction, and from y = —VV to y = +W in the y direction, use the static—case Ampere’s Law to solve for the expression for the magnetic ﬁeld (magnitude and direction) on the planes located
at y = —W and y = +VV . Show your work. ~ /\
l ; )< 53.“) y} =w
T m2 3
Q 7W Ma w
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I“ a. 7:3 / ‘3 b) (5 pts) Now shrink the Amperian loop in the y—dimension to extend from y = —W/4 to y =
+W/4. Determine the expression for the magnetic ﬁeld (magnitude and direction) on the planes
located at y = —W/4 and y = +l'V/4. Show your work. Replaw: W—> WV; :4 We earl/mim f,“ g
’— p ~
B: y/M—LOZW LynnETI b"% A
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c) (5 pts) Determine the expression for the magnetie‘iﬁeld on the front surface (y = +VV / 2) of the slab.
[“30“ 93 3C»! 2'22“. / Mam» made/(a) d) (5 pts) Determine the expression for the magnetic field at y = 0. i=0 wr =0 Mm 1G,) Mm
WWWL.%
//____ a) (10 pts) Assuming that all ﬁelds are static, show that H = $52 +3337 — z 2 A/ In is realizable as a
magnetic ﬁeld intensity in vacuum (where B = HoH) and ﬁnd the current density J associated
with this ﬁeld. ﬁ‘ID’ 9*, HC ’£:¢(¢‘S VFW—1’1“ A A /\ /\ A /\
,. A x ‘1 1*— =xCo)—~)(v)+e.(l~o)—z LL% :3 1—: V‘H 1‘. 9/3‘ 2/91.I 96% W
. )k )L ’% b) (5 pts) A positive charge, Q, is located above the zyplane. What is the downward displacement
ﬂux fey—plane D  dS through the entire Icyplane? Explain why in 1 sentence. Hint: Think about the total ﬂux 395 D  d8 = Q emanating from the charge. ”54;: Q hccwk. W7 ﬂMMI/twa— 5M £2994 Q
3L M Few W Q < L; fn'l'v
.. 10‘; W M 5x. Yr
MY ’l‘Na Mkaa‘} cro€f¢o ’l'W" fir’4 HM [ate«~44 X/L 4&ch «Man «,4; ind—w m Wag.
.«éi’ 'W—ebw. / c) (10 pts) Two unknown charges, Q1 and Q2 are located at (:c.y,z) =(1, 0, 0) and (—1, O, 0),
respectively, as shown below. The displacement ﬂux through the yz—plane in the +33 direction is
2 C. The flux through the plane 3/ = 1 in the +11) direction is 3 C. Using your argument from
part (b), and superposition, ﬁnd the unknown charges. (Partial credit will be given for correctly
writing out the two equations needed for the solution). chgj“QI 523:4"
Z 7. =7
Ql:“5Cr
”5;: L+QJ
L 2 5,4,4 in WW. 3. A static magnetic flux density in a region of space is given by B = 2 58m Wb/Inz. A rectangular loop
C is initially located in the xy—plane with vertices at (56,31, z) =(0, 0, 0), (1, O, 0), (1, 2, 0), and (0: 2,
0) and moves with a velocity V = 22?: m/s. a) (10 pts) Find the expression for magnetic ﬂux \IJ(t) linking the loop C’ described above. {H “gef 1x43 31‘0(Z+L— ”)3“: ){f‘b Wk (/ b) (10 pts) Use \Il(t) from part (a) to determine the induced emf of the loop in the counterclockwise
direction when viewed from above. 8: — {if : —2o (e—‘)e2+ \/ <1 mew—m2: «‘y 2.1 c ﬁreWW
MT / $0.4M» aAvm/c “7&9.”th
' We ‘f‘ovk AS: inwh? c) (5 pts) If the velocity of the loop were V = 2:0 1n/s instead, what would the induced emf be?
What if V = 2 2 ni/s? Explain your answer. Era 501% (MM 2% loécomw «far V:2§ M]; 8: 23 m/J'
gﬁmx ‘l’ is W Iwm‘m'l 4. A plane electromagnetic wave with a sinusoidal electric wave ﬁeld E(y,t) = 2 10 sin(27r x 10% — 5y) %, {it : lo 9,;‘(wt—(L '3) W“?
is propagating in vacuum where 9’3 = c and c = 3 x 108 m/s. a) (2 pts) Determine the oscillation period T of the wave at y = 0 in units of s. wsznxloq’fj :7 'T: 747?; EALa slog—1.3%
.S w zrrxuo’l‘ b) (2 pts) Determine wavenumber [3 in rad / m units. } c) (2 pts) Determine wavelength A in units of m. — 2n  M
(W'— 922%:37309’ / 7‘ d) (5 pts) Sketch E2 (3/ t) vs 25 for y — O and y— — % over two oscillation periods T. In each case
label the axes appropriately in relexant units “ 0 in wt E%(o,t)= lo LJ«(~'€)/ Ci— {oi—m «(ed/'9):
gait) ( 9A ( (5‘0) 5 23:9,,4(2? I): :locy...((w(t:r I) f) (4 pts) Determine \Ax EA /\ /\
 a a a = x 21a 2 x Hi0 «sewed %»
VxE: 0 3/99 ° 9
0 E11 / g) (3 pts) Determine _%_7 where B is the magnetic v» ave ﬁeld that accompanies E(y, t) ng=' 9;. :7 ..9$ ; 9[($10m(w1’{59)j,v. 51,... FM (4) /"‘ :1
3C / h) (4 pts) Determine H(y, t) that accompanies E(y,.t) 0Q Vailah 17”—
o‘t‘ H obwnaLm. bk? IVL: {quL/ M .3 s“ ‘
31:; WZWK J’o $219.1, ErH rﬂA'l’S‘ a~ Wﬁfwakclmew ‘3 7‘ M :3 ug’kw 1%
cute” 4L
”5% .5 / Tr? ...
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 Spring '08
 FRANKE
 Electromagnet

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