Overview

In Chapter 2, equations (2.3) and (2.4) give the electromagnetic fields due to a source on the z axis at the origin of coordinates. In Appendix A, we find what happens for various re-orientations of the source. In this Appendix, we find how to deal with a source moved away from the origin. Figure B.1 shows a field point at and a source point at . The distance between them is the line R. The propagation part of the field expression is:

Figure B.1: Sketch showing general field point at and source point at .

The parallel-ray approximation is that the field point is so far away from both the origin and the source point that the lines for r and R are essentially parallel. This leads to:

where d is the length of the projection of on . This is the vector dot product of with the unit vector for .

Now replacing R with r d in (B.1),

The result has the propagation function based at the origin, an amplitude-shift factor, and a phase-shift factor. Because d is tiny compared to r, usually the d/ r term is dropped. However, ?d may be a significant angle and so the phase term is kept in cases where this is true or where it makes the small difference between large subtracted terms.

Appendix B Problems

• B.1 Suppose there are two...