311_exam3sol_sp10

311_exam3sol_sp10 - ECE 311 — Spring Quarter 2010 Final...

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Unformatted text preview: ECE 311 — Spring Quarter 2010 Final Exam June 9th, 2010 Write your name below and sign the honor pledge “No aid given, received, or observed” if it applies. There are 3 problems on this exam. Exam is open book and notes. Please box or underline your final answers, and remember to include units. Be sure to show all work clearly if you wish to obtain any partial credit. Try to keep your work within the provided space; use the back of a previous sheet if necessary. A partial list of useful formulas is included below for your reference. Name: ' Egg “The Pledge”: Gauss” Law (integral form) 4-[5-51'75' = [Hp VdV differential form: V - D = pV E field path independence 41:?- (TL = 0 differential form: V x E = 0 Potential difference: V(P2)— V(P1) = —f2E- (71 80 = 8.854 x 10—12 F/m P1 Capacitance -Definition C = % Energy inacapacitor éCV2 Dielectric media D = 8E Electrostatic boundary conditions E,1 = Et2 Dnl —Dn2 = pS Ohm’s Law .7 = of? Resistance - Definition R = ; Uniform conductors: RC = 8/0' . . — _ 6 . . _ — _ 6 Charge conservat10n(1ntegral) d — ——a—IIHdeV dIfferentlal form. V -J — —a—t(pV) Power dissipation in a volume P = - .7dV = oHflElde = gruff/lde Definitions of curl and divergence in cylindrical and spherical coordinates: see PS 7 or book fix = —1 (—1 —I—1 ) forn>l = me) n n _ 1 n — 1 — l a x rdesinefndrb = 4n rdefndtb = 21:2 0 0 0 0 Problem 1 (3 parts, 35 pointsl A capacitor is created from two concentric (and hollow) perfectly conducting spheres. The inner sphere has radius 5 cm, while the outer has radius 15 cm. The space between the spheres is __ W30 filled with a dielectric material having a non-constant permittivity of s —a 2 g R spherical coordinate in meters. Assume total charges of +Q and —Q are‘distributed on the surfaces of the inner and outer spheres, respectively. , where R is the Inner radius 5 cm Outer radius 15 cm (a) What components of the electric field should exist in this problem, and what coordinates do those components depend on? Be sure to specify the coordinate system you are using. (3’ points) (b) Find the electric field vector in the region between the spheres in terms of Q - be sure to specify both magnitude and direction. M points) ML (c) Find the capacitance of this device. Give a numerical answer. ($6 points) C= & Sb \x ‘ \/ “M 1* Rb v. “- e:— o an» (amp, V=-g%fi Rea (Axum Ell/tpé‘dg h L“ UN: (wank-M gawk wké‘ :D.\< . \I — - “I " A a Q Q .QML K=M< " :2 %E_(b,\)= Eg- an 3% C: g e, 2 SC. “ flit» b L3 W 5 f \x ‘ m {5 5V 3‘, mbéctfinfl‘D‘ MY] wk ‘ W‘kfifi)“ “kw % \D h) *U‘Q‘L w» mmxouufl (8 9M3) e— ' L, _ m- w vim $9 5‘, w \‘D‘év/L C: CQ°KK 3° VN—zcgfl 7— ‘0 Go 5 Problem 2 (4 parts, 30 points: The x-y plane is an interface between free space (2 > 0) and a PEC material (2 < 0 ). An electric potential V = yz + z2 (Volts) exists in the free space region. (a) Find the electric field associated with this potential in the free space region (8 points). TE; 5N ‘-' ‘3? - gkal— 1%) v[m_ (b) Find the surface charge density on the PEC surface. (8 points) E‘Wm“ S’s/es on: 1:0 M£=Dl E:* a SQ §S=t€°a> (vw‘afilASqu> N K J W\V\O©W3v ‘QKXX 90V}; \owwxo ((Qr 37k) (c) Find the volumetric charge density in the free space region. (8 points) Q‘fD" iv $1 Eu: = %(‘3 7c 41, (wag) Q ‘5‘ 979131 3v 9v: ‘kb > (d) A point charge of 1 nC is moved from the point (x=0,y=0, z=1) to a point that is 10 m away. Specify (x,y,z) coordinates for this point so that no work is required in performing this motion (6 points, note there are many correct answers.) Problem 3 7 arts 5 oints each : Circle the answer which is alwa s true. (1* ch of the following is NOT true about the method of images? t can be used to field the fields produced by charges in the presence of any PEC material - when applicable, it shows that the field outside the PEC material arises from the original charges and a set of “image” charges (c) when considering a point charge above an infinite PEC plane, it shows that field above the plane is that of 3113 electric dipole (d) when applicable, it ensures that there are no tangential electric fields on the PEC surface (e) none of the above (2) Given 2 = J‘cy, V x Z is fix) (a) 0 (b) 1 (c) 5c + j) I\‘(d) —§, (e) none of the above (3) The x-y plane is a boundary between two dielectric materials: region 1 (z > 0 , field E] ) that has a = 280 and region 2 (z < 0 , field E2 ) that has a = 680. Which set of fields below satisfies the appropriate boundary conditions? a E1 = 2, E2 = 32 (b) ‘1 = 32,1?2 = 2 1 = 5:, E2 = 35c (d)1721= 32,3 = 2 (e) none of the above (4) Which of the following statements is NOT true about electrostatic fields? (a) when computing a line integral to evaluate the potential difference between two points in free space, any path connecting the two points@ can be used (b) the curl of the electric field is zero (c) the divergence of the electric field can be non-zero d the electric field inside a perfectly conducting material must be zero @me of the above (5) An air-filled coaxial capacitor has length 10 cm, inner radius 2 mm and outer radius 1 cm. U r: - the no-fringing approximation, the capacitance of this geometry is: w: .46 pF (b) 10.7 pF (0) 824.3 pF (d) 0.35 nF (e) none of the above (6) A medium has conductivity 0' = 0.25 S/m. If an electric field E = 45c + 22 V/m exists in this medium, the resulting current density (in Amps/m2) is (a)5c+22 (b) 165c+82 (d) 42+ (7) The volume V bounded by surface S contains an amount of charge, Q time. Which of the following statements must be true? (e) none of the above NIN) we , that is constant in . 6 current density on surface S must be zero e net current flowing out of surface S must be zero 0) the current density must be only in the normal direction on surface S (d) the current density must be only in tangential directions on surface S (e) none of the above Have a great Summer break! Good luck in ECE 312. ...
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This note was uploaded on 11/11/2011 for the course ECE 311 taught by Professor Johnson,j during the Fall '08 term at Ohio State.

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311_exam3sol_sp10 - ECE 311 — Spring Quarter 2010 Final...

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