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homework4 - 1 School of Electrical and Computer Engineering...

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Unformatted text preview: 1 School of Electrical and Computer Engineering, Cornell University ECE 303: Electromagnetic Fields and Waves Fall 2007 Homework 4 Due on Sep. 21, 2007 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. Table of Solutions of Laplace’s Equation Spherical Coordinate System Cylindrical Coordinate System ( ) A r = r φ Constant potential ( ) A r = r φ Constant potential ( ) r A r = r φ Spherically symmetric potential ( ) ( ) r A r ln = r φ Cylindrically symmetric potential ( ) ( ) θ φ cos r A r = r Potential for uniform z-directed E-Field ( ) ( ) φ φ cos r A r = r Potential for uniform x-directed E-Field ( ) ( ) φ φ sin r A r = r Potential for uniform y-directed E-Field ( ) ( ) 2 cos r A r θ φ = r Potential for point-charge-dipole- like solution oriented along the z-axis ( ) ( ) r A r φ φ cos = r Potential for line-charge-dipole- like solution oriented along the x-axis ( ) ( ) r A r φ φ sin = r Potential for line-charge-dipole- like solution oriented along the y-axis Problem 4.1: (Metal wire over a perfect metal plane) Consider a thin metal wire of radius a over an infinite perfect metal ground plane, as shown in the figure below. The distance of the wire from the ground plane is much larger than the radius of the wire (i.e. a d >> ). The structure is infinite in the z-direction. In this problem you will need to find the inductance per unit length (i.e. the length in the z-direction) between the metal wire and the ground plane. The metal wire is assumed to carry a time-varying current I (“time-varying” in the magnetoquasistatic sense) in the +z-direction. a) Sketch the image metal wire using the method of images. Indicate the orientation and the current that the image wire carries on your sketch. 2 b) Assuming that the entire current I in the metal wire is carried by a line current at the center of the metal wire, find the vector potential ( ) r A r r everywhere in the region outside the perfect metal (use a result in your lecture handouts). c) Using your result in part (b) above, and the vector potential ( ) r A r r , write an expression that relates the current I to the magnetic flux per unit length that passes between the metal wire and the ground plane. d) Find the inductance per unit length L (units: Henry/m) between the metal wire and the ground plane by taking the ratio of the magnetic flux per unit length (calculated in part (c) above) to the current I ....
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homework4 - 1 School of Electrical and Computer Engineering...

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