Electrical Field Theory: Vector Analysis in Different Coordinate Systems
Explore the principles of Vector Analysis in Cartesian, Cylindrical, and Spherical coordinate systems as applied to Electrical Field Theory. Learn how to calculate differential lengths, areas, and volumes, and solve practical examples under the guidance of Prof. Dr. Ahmed Mohamed El-Sawy.
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1 Electrical Field Theory Directed By: Prof. Dr. Ahmed Mohamed El-Sawy
Lecture One Lecture One Vector Analysis Differential length, area, and volume: A. Cartesian Coordinates B. Cylindrical Coordinates C. Spherical Coordinates Prof. Dr. Ahmed Mohamed Al-Sawy Electromagnetic Field Theory 2
Vector Analysis Vector Analysis Differential length, area, and volume: Differential elements in length, area, and volume are useful in vector calculus. They are defined in the Cartesian, cylindrical, and spherical coordinate systems. Prof. Dr. Ahmed Mohamed Al-Sawy Electromagnetic Field Theory 3
Vector Analysis Vector Analysis A. Cartesian Coordinates (1) Differential displacement is given by (2) Differential normal area is given by (3) Differential volume is given by Prof. Dr. Ahmed Mohamed Al-Sawy Electromagnetic Field Theory 4
Vector Analysis Vector Analysis B. Cylindrical Coordinates (1) Differential displacement is given by (2) Differential normal area is given by (3) Differential volume is given by Prof. Dr. Ahmed Mohamed Al-Sawy Electromagnetic Field Theory 5
Vector Analysis Vector Analysis C. Spherical Coordinates (1) Differential displacement is given by (2) Differential normal area is given by (3) Differential volume is given by Prof. Dr. Ahmed Mohamed Al-Sawy Electromagnetic Field Theory 6
Vector Analysis Vector Analysis Example Example 1 1 Consider the object shown in the following figure. Calculate: (a) The distance BC (b) The distance CD (c) The surface area ABCD (d) The surface area ABO (e) The surface area AOFD (f ) The volume ABDCFO Prof. Dr. Ahmed Mohamed Al-Sawy Electrical Field Theory 7
Vector Analysis Vector Analysis Example Example 1 1 Solution. The points are transformed from Cartesian to cylindrical coordinates as follows: ?2+ ?2 ? = ? = tan 1? ? Prof. Dr. Ahmed Mohamed Al-Sawy Electrical Field Theory 8
Vector Analysis Vector Analysis Example Example 1 1 (a) Along BC, dl=dz ; hence, (b) Along CD, dl = d and = 5 ,so (c) For ABCD, dS = d dz, = 5 .Hence Prof. Dr. Ahmed Mohamed Al-Sawy Electrical Field Theory 9
Vector Analysis Vector Analysis Example Example 1 1 (d) For ABO, dS = d d and z=0 ,so (e) For AOFD, dS=d dz and = 0 ,so (f) For volume ABDCFO, dv= d dz d . Hence, Prof. Dr. Ahmed Mohamed Al-Sawy Electrical Field Theory 10
