homework4 - School of Electrical and Computer Engineering,...

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1 School of Electrical and Computer Engineering, Cornell University ECE 303: Electromagnetic Fields and Waves Fall 2005 Homework 4 Due on Sep. 23, 2005 by 5:00 PM Reading Assignments: i) Review the lecture notes. ii) Relevant sections of the online Haus and Melcher book for this week are 8.0-8.2, 8.6, 11.0-11.2. Note that the book contains more material than you are responsible for in this course. Determine relevance by what is covered in the lectures and the recitations. The book is meant for those of you who are looking for more depth and details. Solutions of Laplace’s Equation for ECE303 Spherical Coordinate System Cylindrical Coordinate System () A r = φ Constant potential ( ) A r = Constant potential () r A r = Spherically symmetric potential ( ) ( ) r A r ln = Cylindrically symmetric potential () () θ cos r A r = Potential for uniform z- directed E-Field in spherical coordinates ( ) ( ) cos r A r = Potential for uniform x- directed E-Field in cylindrical coordinates () () 2 cos r A r = Potential for point-charge- dipole-like solution with the dipole pointing in the +z-direction () ( ) r A r cos = Potential for a line-charge- dipole-like solution with the dipole pointing in the +x-direction Problem 4.1: (A line-charge-dipole) Consider a line-charge dipole consisting of two infinite line charges, carrying λ + and Coulombs/m of charge, and separated by a distance d as shown. + x y r d
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2 From lecture notes you already know the expression for the potential in cylindrical coordinates. Here you need to show that for distances r >> d , the potential takes the following form (excuse the bad notation where the same symbol is being used for the potential and the angle): () ( ) r d r o φ ε π λ cos 2 = r Problem 4.2: (A perfect metal rod in a uniform electric field)
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homework4 - School of Electrical and Computer Engineering,...

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