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# practice final solutions - ECE 134" Final Exam Fa112006...

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Unformatted text preview: ECE 134 " Final Exam Fa112006 FINAL EMM K Name; S You have 3 HOURS to complete this exam. A one—page note sheet and calculator is allowed, otherwise there should be no books, electronic devices, or other materials in your purview. When you are ﬁnished, check that your name appears prominently and legibly on the front page. The exam is worth 100 points total. IMPORTANT POINTS FOR GRADING: SHOW ALL WORK. ANSWERS GIVEN WITHOUT CLEAR SUPPORTING EVIDENCE OF INDEPENDENT WORK WILL BE IGNORED. CIRCLE ANSWERS. THOSE NOT CLEARLY MARKED AS YOUR FINAL ANSWER WILL BE IGNORED. BE NEAT. ILLEGIBLE OR SLOPPY WORK WILL BE IGNORED. I am greatly astonished when I consider the weakness of my mind and its proneness to error. Rene Descartes, 1 7m century French philosopher and scientist Page 1 of9 ECE 134 Final Exam Fall 2006 Problem 1 (10 points) A vacuum—tube diode is shown at right. The cathode is hot and emits electrons into the region between two parallel plates, where they travel towards the anode. In operation at a ﬁxed anode voltage V0, the system reaches an steady—state described by a charge cathode distribution ,0(x). The plates are separated by a distance a, and the electrostatic potential between the plates is found to be x 4/3 (DOC) : V0 £_] a a) Find the electric ﬁeld vector E between the plates. P, 73 \ E 4 (25’ (an ~: ‘— b) Find the charge density p(x) between the plates. 0) Assume that the electrons near the cathode are initially at rest and have a mass In. What is the ﬁnal electron velocity at the anode after accelerating through the potential V0? (Hint: assume all the otential ener is converted to kinetic ener . v p gy _ gy) are; m 142. c Rf. , V l \(e. l baud-V3 \ I cd‘ (FLA ed 6 d) The moving electrons constitute a current ﬂow in the device. Find the current density at the anode. " ' Page 2 of9 ECE 134 Final Exam Fall 2006 Problem 2 (15 points) A pair of coaxial conducting cylinders is shown below, with radii of a and b for the inner and outer conductors, respectively. The surface charge density on the inner conductor is p, [C/mz]. The region between the conductors is ﬁlled with a semi-insulating material with a permittivity 8 and conductivity 0. a) Using Gauss' law, ﬁnd an expression for the electric ﬁeld between the conductors 9J4 @ (a < r < b); for this part of the problem, assume that L —> oo . 6&3 93mme‘lma C3 (1 {Facing} 4&6) cl 5 Qemal >> 6 Z 65 % b) Find an expression relating the voltage between the conductors to the surface charge p5 (take the outer conductor as ground potential) b b A ‘ v )Erav: 85‘“ 91;: a 6 Y c) Use your result from (b) to ﬁnd an expression for the capacitance per unit length for the structure, C’ _ A (2TB L .t .' n 6” " 2 s l K 173% Chm/82L 6/“ \V‘“ ' 6 _ ,_ (:2, 95??” 1- C: T, 6 (1) Use your result from (c) to ﬁnd an expression for the resistance per unit length, R' b ('2’ \$ 6 _ 6 m f? l Ixx /6\ ' F> lQ ” t 2 2 a TT% L Page 3 of9 ECE 134 Final Exam Fall 2006 Problem 3 (15 points) A resistor is formed from a thin resistive material of thickness t and conductivity 0. The shape is an angular section of an annulus with inner radius a and outer radius b, and subtends an angle 6. A battery is connected to high conductivity Ohmic contacts at the ends as shown. ( a) Assuming the total current in the resistor is I, ﬁnd the current density :Jr (r) in the J . . . '- . material for a < r < b . You do not need to solve a differential equatlon for this problem. / v: c s b) From your answer in part (a), ﬁnd the electric ﬁeld {Er (r); in the material, and express your answer in terms of the sheet resistance :r . ‘ ' \ ‘ .1” . T __ K- A :/_> / : S : —————- t \ ﬁts «a Page 4 of9 VA (3;. X? ECE 134 Final Exam Fall 2006 Problem 4 (15 points) Holiday lighting systems use strings of small incandescent light bulbs like the one shown here. The I R ‘ bulbs are designed to operate at 2.4V AC (rms), such , . «J 3 that a series connection of ~50 bulbs can be powered a I . from a 120V AC power source (120Vrms = 170V peak). a) The filament has a resistance R during operation, and a surface area of A, for radiative emission. Write the appropriate steady-state equation for the filament temperature T , assuming the bulb is driven by an AC voltage source with an rms amplitude of V0. (Assume radiative cooling only, and an ambient temperature of 7;, = 300 K ) .2. v, a u.» : egg-As(rri§\ {Z b) The ﬁlament is modeled as a cylindrical wire of length 3 , diameter d, and resistivity ,0, express the power balance equation from (a) in terms of these parameters, and solve for the length 3 that is required to achieve a certain filament temperature. (1:):1/5'7’2ﬂ rrrll MM <T“-“1'a“\ 43> t: l c) Suppose the ﬁlament is made of tungsten with a wire diameter of d = 46pm. Find the required filament length for operation at V0 2 2.4Vrms and T =2000K. At this temperature the resistivity and emissivity of tungsten are ,0 = 57 {.19 ~ cm and e = 0.26, respectively. The W Stephan-Boltzmann constant is 058 = 5.6703 ><10_8 /m2 ~K4] l WEI/[A “(l/£5? Nimbus] @2 2.ZX{6 m Fﬁu‘ﬁ b00w\c\ %4 (At C1 l‘eVUj/R d) What is the total time—averaged power dissipation for a string of 50 of these bulbs, assuming they are connected in series? 7 I C.“ law‘s ’ \Q? 7.§§IL :> \ CD Finn/L: .Sbi‘Pb-Mi‘c‘ E? W Page 5 of9 0.7 t. w t ‘- ‘ ECE 134 Final Exam Fall 2006 Problem 5 (15 points) The toroid shown below uses an annular core of high-permeability material with a square cross section. The average radius of the toroid is a, and N turns of wire carrying a current I wrap around the core. A small section of the core is removed to form an air—gap of height g as shown. a) Find the magnetic ﬂux in the gap. A if j N! 1 MM M _\ %_ \_—"—— k€ 3 g? 72m) F ﬂo I w in “rm , 733a? m (P (P : b) Find the self-inductance of this structure W / mi 2 L: T:\\Ll\l 0 8S c) Find an expression for the magnetostatic force between the opposite faces of the core in the gap region. Is this force attractive or repulsive? Page 6 of9 ECE 134 Final Exam Fall 2006 Problem 6 (10 points) This problem revisits the “rail gun” that we examined in homework #7. A metal bar of mass m=0.2 kg slides freely over two conducting rails separated by a distance W=l meter. A uniform magnetic ﬁeld of strength B=0.l Tesla is applied between the rails as shown below. At time t=0, a switch is thrown to connect a battery of voltage V=10 Volts across the rails through a R=IOOQ resistor. The magnetostatic force on the bar causes it to move, but as it moves a ﬂux-cutting emf is generated that opposes this motion. bar of mass m. a) From your knowledge of Netwon’s laws (F 2 m5 ), magnetic forces, and Faraday’s law, derive the differential equation governing the velocity V of the bar. ,— A \T A Vx /-\ g MCLSMC+JC £0ng (M Ear-g 11 X -, l/ 3 M {36 a. W mm/ m .Acuaeé emf vméeﬁxglw : \éﬁw "b- 3amea§loi W (96792065 (kacés Tim “irredij e I... f “ 6y. . _ l "L — L—v “*3 avx Val WSW) 'GVELLO“3rZ—/@m m ‘ K b) If the rails are long enough, the bar eventually reaches a constant terminal velocity v/ . Find an expression for this steady-state terminal velocity, and evaluate it numerically for the given parameters. {:1 " SW/ (CG—— F O I II 7 7 1". Page 7 of9 ECE 134 Final Exam Fall 2006 Problem 7 (10 points) _/</x 2/40 A plane wave propagates in a certain non—magnetic material, and the magnetic ﬁeld vector is given by I? = 0.059 cos(108t —0.52) A/m a) Find the wave velocity ::|D% (RA/S G c) Find the magnitude and direction of the electric ﬁeld (show units) r 0 g 0¢O§§ a J 4/ ~ Cmwi Trips {\Dl/wowr QONR: l H ,‘ Hill/i 21:75:} ijﬂ ll”: . Mfg/I 22>”: Page 8 of9 ECE 134 Final Exam Fall 2006 Problem 8 (10 points) Consider the transmission-line circuit below, where Z0 2509 and 8r :1 . The load is a capacitor with C=2 pF, and the length is E = 1/8 at a frequency of 03:10IO rad/s. [ -—> Ze :l-C I" .o’r :r Zin a) Find the impedance Zm % r a AZ : vjfo IL V/ VAC / \Do Z-X‘D B \ l ’rk "W L) A @Q 9 2’: <3 i A \l %H3%%“W enw:i 33m > {40 _ Ca %M*© ( ébﬁl H_\# b) Find the input impedance again if the frequency is doubled to w=2>< l 0IO rad/s So W @113 T92 (“girder—WM"? , g) ' Llaé (Bl/(M) / \lN’Zt/wg'I/YZ’VAGX 1 . 2 I V l ’2 ‘—’/ A ‘ Z S __Q_ p:)LOC ' A) Page 9 of9 ...
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practice final solutions - ECE 134" Final Exam Fa112006...

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