FWLec2 - A. F. Peterson: Notes on Electromagnetic Fields...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 9/04 Fields & Waves Note #2 Cylindrical & Spherical Coordinates Objectives: Review the cylindrical and spherical coordinate system. Discuss the conversion of coordinates and the projection of vectors between these systems and the Cartesian system. In a number of situations, boundaries of a problem under consideration coincide with the cylindrical or spherical coordinate system. It is usually simpler to analyze that problem using a coordinate system that conforms to the geometry than it is to force the analysis through using the Cartesian system. Cylindrical Coordinates A point described by the Cartesian coordinates ( x , y z ) can also be expressed using the cylindrical coordinates ( r , f z ) as illustrated in Figure 1. The transformations between these are easily obtained using trigonometry and are =+ xy 22 (2.1) = Ê Ë Á ˆ ¯ ˜ arctan y x (2.2) z = z (2.3) and x = rf cos (2.4) y = sin (2.5) z = z (2.6) These transformations are straightforward, but numerical calculations for the inverse tangent in (2.2) require care to ensure that the resulting angle is correct. This is a consequence of the multi-valued nature of the inverse tangent function (Figure 2). The typical calculator restricts the result to the principal branch of the tangent function, – p/2 < q p/2 , when the actual cylindrical angles encompass the range – p < < p ! Thus, it may be necessary to add or subtract p radians or 180 degrees to place on the proper branch. The process of converting a point from one coordinate system to another is distinct from the process of projecting vectors from one system to another, as explained in the following section.
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A. F. Peterson: Notes on Electromagnetic Fields & Waves 9/04 Projection of vectors between the Cartesian and cylindrical systems In order to define vectors directly in the cylindrical system, we introduce the principal unit vectors ˆ , ˆ , ˆ rf and z as depicted in Figure 3. At any point, the vector ˆ r points in the direction of increasing , while ˆ f points in the direction of increasing . These three unit vectors are mutually perpendicular at any point. A vector can be expressed directly in terms of the principal unit vectors using the relation AA A A z z = + + ( ˆ ) ˆ ( ˆ ) ˆ ( ˆ)ˆ rr ff (2.7) We observe that the components along ˆ , ˆ , ˆ and z are dot products of the vector A with those principal unit vectors.
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This note was uploaded on 01/27/2011 for the course ECE 3025 taught by Professor Citrin during the Spring '08 term at Georgia Institute of Technology.

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FWLec2 - A. F. Peterson: Notes on Electromagnetic Fields...

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