hw3solutions

hw3solutions - SO/u‘l‘t'on ECE604 Homework 3 Out:...

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Unformatted text preview: SO/u‘l‘t'on ECE604 Homework 3 Out: Tuesday, January 27, 2004 (Session 5) Due: Tuesday, February 3, 2004 (Session 7) [Students with tape delays: due according to session number] These problems are from the text by Ramo, Whinnery and Van Duzer (3rd edition, 1994). 1) 7.12a 2) 7.126 3) A conductor on a circuit board has width W and carries a current I. The conductivity of the conductor is 0'. There is a small hole of radius a at the center of the conductor. The thickness t of the conductor is small compared to a. (Therefore this problem may be treated as a two-dimensional problem in cylindrical coordinates.) Assume the current is DC. You may also assume that the width of the circuit board W is large compared to the hole radius (W>>a). Let the coordinate origin be at the center of the hole. (i) Sketch qualitatively the sheet current density J 3 (units are A/m) in the Vicinity of the hole. (The sheet current density is the uniform current density J integrated over the thickness of the conductor, that is J S = J ><t ) (ii) Derive an expression for J5. Express your answer in terms of I, W, I and (1). (iii) What is the maximum magnitude of J S? Express your answer in terms of I and W. (The answer does not include the hole radiLis a.) Note that the maximum should be located along the x = 0 line (where the current is maximally constricted). (iv) Give an expression for the sheet surface charge density (units Coulomb/m) along the edge of the hole. 4) a) Shown below are two infinitely long, very thin insulated conducting plates which are held at potentials V0 and O as indicated. (The plates are infinite in the z-direction and semi—infinite in the other dimension.) seq“ YT \ q ¢ 06‘. ‘. __.______.._—l—_. __|—> v=0 x Determine the potential distribution for O < (l) < 0c and for oc < (1) < 211:. b) Shown below are two infinitely long, very thin conducting plates separated by a dielectric of permittivity 8, which are held at potentials V0 and O as indicated. (The plates are infinite in the z—direction and finite in the x—y plane.) (i) Neglecting the effects of fringing, determine the capacitance of the structure per unit length. (ii) Neglecting the effects of fringing and without using the relationship WE = % C V2, determine the energy stored in the structure per unit length. Show that your result is in fact equal to %C V2. 5) To measure the conductivity 6 of a semiconducting wafer of thickness t, a hemispherical electrode of radius a is alloyed into one side of the wafer as shown below. The other surface is grounded and serves as the second terminal. Assuming that the contacts are ohmic and perfect conductors, derive an expression for the conductivity, 0, as a function of the measured resistance (R). Since the alloyed electrode is very small and the wafer is very thick, you can assume that the current is uniformly distributed over the hemispherical alloyed contact. 6) 2.3b 7) 2.30 8) 2.4e 7.12:1. ‘ §=° ' =5 Nccd repeated items in 5 direction so use {lam E, = (A (lush kx+ bsbnln kxIGmskg + Dan kg) 8.0? i=0. at 13:04:, ascend h= mb Evan gamma-(:13 in X so 850, Then @595): Egan ooshmglsannga- At x=o.: v.= E-cnwshflfig—sawmgg Expand he. at x=o. in qurzer sine Series as m . ‘ 4v - a, 1-ll(l4-)WIU1 0“ V0 and a—pb; ¥(5)'=n Eguating ‘Une nun term; 4%"; = (1n gosh Lg“- : 4v. cpsh&x sanmtgg . J. =nw n7: a b 19%. 352:0 ’ . .. . 'Hzc 0' To b_.._c., need Won panache, m x: §=o I in = (A cosh!“ + Bsim hxXCwosh L5 + psam k5) 5V r I 1 .Since ¢(0.5)=o and. §£*,o)=o,vAacao E "‘ cns'unkaJmh k5 ' O E’ b V .. . I h r": '0 a3xL=t.k¢"°°‘k“sV”hk5=9. 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hw3solutions - SO/u‘l‘t'on ECE604 Homework 3 Out:...

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