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homework1 - 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 2007 Homework 1 Due on Aug. 31, 2007 by 5:00 PM Reading Assignments: i) Review the material on cartesian, cylindrical, and spherical co-ordinate systems from your favorite freshman calculus book. Make sure you are comfortable in using these co-ordinate systems. ii) Relevant sections of the online Haus and Melcher book for this week are 2.0-2.6, 3.2, 3.3. 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. Problem 1.1: (warm up) Consider a scalar quantity φ given by the expression, () ( ) 2 2 2 , , z y x a z y x + + = Where a is some constant. a) Find the gradient r of the scalar in the Cartesian co-ordinate system. b) Express the scalar given above in variables of the cylindrical co-ordinate system and then find the gradient r in the cylinderical co-ordinate system. c) Express the scalar given above in variables of the spherical co-ordinate system and then find the gradient r in the spherical co-ordinate system. d) Find the divergence of the gradient for the scalar (i.e. first find the gradient vector and then find the divergence of the gradient vector). Note: The divergence of the gradient, written as ( ) r r , is also more commonly denoted by the Laplacian operator 2 , i.e. ( ) 2 = r r . Problem 1.2: (basic vector calculus review) Consider a vector field F r given by the expression: () z z y y x x z y x F ˆ ˆ ˆ , , 2 + + = r a) Find the total flux associated with the vector F r

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