This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ECE 529 EMI IN MICROELECTRONICS University at Buffalo Dr. James J. Whalen Lecture 8 2SSep96.~.~ ~ .   ~' .. ..  .•..~ ~ ..., _.. .. ~.7 THE SPHERICAL COORDINATE SYSTEM The spherical coordinate system is also formed by three surfaces. One surface is a cone about the z axis of half angle e. The next surface is a sphere centered on the origin of radius r, and the third surface is a plane perpendicular to the xy plane and rotated about the z axis by angle ¢ in the direction from x to y as shown in Fig. 2.10. Observe that o < t:P < 21T and 0 < e < 1T. A point is defined as the intersection of these three surfaces, P(rl,e1,t:Pl)' Unit vectors an ae, and a¢ point in the direction of increasing coordinate value. Note that all three unit vectors change direction from point to point unlike the three unit vectors of a rectangular coordinate system which are fixed in direction. The three axes are labeled r, e, and ¢ and these are ordered with the right handrule such that a, X ae = a¢. Note that, for example, a, X a¢ =ae. The same symbol r is used in both the cylindrical and the spherical coordinate systems but they have different meaning. Two vectors expressed in this coordinate system at point P can be added at that point since the unit vectors of the corresponding components are parallel at that point. z 8 =8 1 cone ¢ = ¢1 plane r= r 1 sphere \ I", / .. \ I' / . \ I / \ 1/ (1 ~ /...... ~. ..•. / ..•. r~~y / / / / x    C / J 1 Figure 2.10 The spherical coordi nate system illustrating the unit vectors and the location of a point as the intersection of three constantcoordinate surfaces. 2.7 The Spherical Coordinate System .•• 0;) Hence for A = A,», + Aeae + A.pa.p B = B,«; + Beae + B.pa.p the sum is (2.21) The dot and cross products are very similar to those for the rectangular coordinate system: (2.22) (2.23) Observe that the cyclic ordering of the axis labels, r ~ e~ ¢ ~ r ~ e~ ¢ ~ ... can, like the rectangular coordinate system, be used to quickly form the cross product...
View
Full
Document
This note was uploaded on 02/22/2011 for the course EE 529 taught by Professor Dr.j.j.whalen during the Fall '10 term at SUNY Buffalo.
 Fall '10
 Dr.J.J.WHALEN
 Electromagnet, Microelectronics

Click to edit the document details