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329sp09he1sol 06-34-56 - ECE 329 Introduction to...

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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. . hen-V lav - , ”4 iw‘fua’ Wra‘ ( LM *9 é“ 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 "’ 0 o 2" 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 Lynn-ETI b"% A “X (“fl—3’” Mu 7:"254 M 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 zy-plane. What is the downward displacement flux fey—plane D - dS through the entire Icy-plane? 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:“5-Cr ”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 fire-WW MT / $0.4M» aAvm/c “7&9.”th ' We ‘f‘ovk AS: inwh? c) (5 pts) If the velocity of the loop were V = 2:0 1n/s instead, what would the induced emf be? What if V = 2 2 ni/s? Explain your answer. Era 501% (MM 2% loécomw «fa-r V:2§ M]; 8: 23 m/J' gfimx ‘l’ is W Iwm‘m'l 4. A plane electromagnetic wave with a sinusoidal electric wave field E(y,t) = 2 10 sin(27r x 10% — 5y) %, {it : lo 9,;‘(wt—(L '3) W“?- is propagating in vacuum where 9’3 = c and c = 3 x 108 m/s. a) (2 pts) Determine the oscillation period T of the wave at y = 0 in units of s. wsznxloq’fj :7 'T: 747?; EAL-a slog—1.3% .S w zrrxuo’l‘ b) (2 pts) Determine wavenumber [3 in rad / m units. } c) (2 pts) Determine wavelength A in units of m. — 2n- - M (W'— 922%:3730-9’ / 7‘ d) (5 pts) Sketch E2 (3/ t) vs 25 for y — O and y— — % over two oscillation periods T. In each case label the axes appropriately in relexant units “ 0 in wt E%(o,t)= lo LJ«(~'€)/ Ci— {oi—m «(ed/'9): gait) ( 9A ( (5‘0) 5 23:9,,4-(2? I): :locy...((w(t:r I) f) (4 pts) Determine \Ax EA /\ /\ - a a a = x 21a 2 x Hi0 «sewed %» VxE: 0 3/99 ° 9 0 E11 / g) (3 pts) Determine _%_7 where B is the magnetic v» ave field that accompanies E(y, t) ng=' 9;. :7 ..9$ ; 9[($10m(w1’{59)j,v. 5-1,... FM (4) /"‘ :1 3C / h) (4 pts) Determine H(y, t) that accompanies E(y,.t) 0Q Vail-ah 17”— o‘t‘ H obwnaLm. bk? IVL: {quL/ M .3 s“ ‘ 31:; WZWK J’o $219.1, Er-H rflA'l’S‘ a~ Wfifwakclmew ‘3 7‘ M :3 ug’kw 1% cute” 4L ”5% .5 / Tr? ...
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