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# FWLec2 - A F Peterson Notes on Electromagnetic Fields Waves...

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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 r = + x y 2 2 (2.1) f = Ê Ë Á ˆ ¯ ˜ arctan y x (2.2) z = z (2.3) and x = r f cos (2.4) y = r f 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 < f < p ! Thus, it may be necessary to add or subtract p radians or 180 degrees to place f 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 r f and z as depicted in Figure 3. At any point, the vector ˆ r points in the direction of increasing r , while ˆ f points in the direction of increasing f . 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 A A A A z z = + + ( ˆ ) ˆ ( ˆ ) ˆ ( ˆ)ˆ r r f f (2.7) We observe that the components along ˆ , ˆ , ˆ r f and z are dot products of the vector A with those principal unit vectors. As an example, the unit vector ˆ x can be expressed as ˆ ( ˆ ˆ ) ˆ ( ˆ ˆ ) ˆ ( ˆ ˆ)ˆ (cos ) ˆ ( sin ) ˆ ( )ˆ cos ˆ sin ˆ x x x x z z z = + + = + - + = - r r f f f r f f f r f f 0 (2.8) while the unit vector ˆ y can be expressed as ˆ ( ˆ ˆ ) ˆ ( ˆ ˆ ) ˆ ( ˆ ˆ)ˆ (sin ) ˆ (cos ) ˆ ( )ˆ sin

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

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