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Unformatted text preview: Electromagnetics I Midterm Examination 4/25/2005 Duration: 100 minutes A Close Book I ; 5% Dictionary and Calculator Allowed If it appears that necessary information is missing in a problem, make a reasonable assumption if state the assumption. \l: V/ZX 1; I 2 / 7 _ h , Honor code: I neither receive nor give any assistance during the exam. (Write the code on your solution / 3 sheets and sign your name next to it.) ({Kﬁl/YQHQ Y E H \/ Problem 1. (15 %) ' "/ 1&5; /1l< 'Zil’j ( I In spherical coordinates, a vector is expressed as A = 2519 at the position in 17 c )rdinates and one in Cylindrical coordinates. ” Val Problem 2. (15%) W S? V ﬂ Giv n a vector ﬁeld 2 = (l/ r)ci,, + cot 0240 in spherical coordinate%’ ( Find the condition that satisﬁes V-Z = 0. (see the useful formula in backsideiUXE/ El; (b) Can it be realized aka magnetic ﬁeld ? Why ? D _’ a a 0Q /\ /\ (1399 twill GHOWGGG Efﬁe Problem3. (25%) "7/ '2 l ‘ W'j=~§% -~—F:—+ =__7r/ 6,¢ = 7r/ 4. Convert the vector and position to one in Cartesian A spaceship, with charge Q and constant velocity V , is passing through a region of Ed?— L ) uniform electric and magnetic ﬁelds given by E = Eo(2cix + 251), + [72) and / e ea 3 713061; — 2% + 251:) in free space. \A : \(9/5 Show that the electric and magnetic ﬁelds are perpendicular to each other. ¢ (b) Is it possible for the spaceship to pass the ﬁeld region with constant velocity ~ 7 without detouringﬂﬁﬁ‘iiﬁﬂiﬁ)? (Assuming steering power off during the passing) (c) If your answer in (b) is no, state the reasons and go to the next problem. Ifyes, a how do you control the velocity I7 of the spaceship before entering the ﬁeld region. -\ \2 (d) Find the minimum velocity (17),“, in (c). (including the speed and direction) 3 4 (see the backside) D E *1 a H 7% “A K“; ./ léyblem 4. (25%) / (13")A rigid rectangular loop of wire is hung by pivoting one side along the x—ax1s, as V shown below. The loop is free to swing without friction under the inﬂuence of gravity and in the presence of a uniform magnetic ﬁeld B = 805 . The induced electromotive force (emf) around the close loop is behaving a function of time as the loop swings. Find the emf ata =0 in terms of B0, a, b, a) , where a) is the angular while the loop swinging toward the vertical. Does the loop swing faster 0 slower 1am in the absence of the magnetic ﬁeld? Why ? ('0) Repeat (a), but change the magnetic ﬁeld as 1—3 = BOX“ . (c) Repeat (a), but change the magnetic ﬁeld as E = B033 . Figure of Problem 4 P oblem 5. (20%) he rectangular cavity resonator is a box comprising the region of free 0 < x < a , O < y < b , and 0 < z < d , and bounded by metallic walls on all of its six sides. The time-varying electric ﬁeld inside the. resonator is given by — . 7rx . air: E = E0 sm—s1n~——cosa1t&v a . where E0 is a constant and a) is the radian frequency of oscillation. Find the value of a) that satisﬁes both of Maxwell’s curl equations (Ampere’s and Faraday’s Laws). Wl'iat do you ﬁnd the physical. meaning of (a in this problem ? 1 F l 70 ” Useful formula: \l K Y i O / i I .., ii i {141" T k = "sflr‘Aﬂ‘ + —. .—(Au Sinai] + - . . r- m- '_ 1' Sin f) (H) r 51 I: H Hit: \jﬁn sphe ical coordinates. ./ >3 77 {3L (Mir (5% spgéiﬁiin: Q 250 556 ...
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92ä¸‹é›»ç£å­¸æœŸä¸­&egrav

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