Details

The antenna current is assumed to be a standing wave , where is the propagation constant ( is the angular velocity and is the speed of light). From the charge conservation law, the charge density per unit length is . These complex values represent harmonic quantities. The scalar and vector potentials are determined by

,

.

Using Mathematica's symbolic calculation in vector analysis, it is possible to write down the mathematical equations for the electromagnetic fields and in terms of and directly. The somewhat lengthy expressions in the program are derived by approximating the integrations with the summations over the whole length of antenna parts (the number of divisions is set to eight for the sake of simplicity). The instantaneous fields are calculated by and . Correspondingly, the Poynting vector is calculated by , and the electromagnetic energy density by .

Denoting the wavelength by , snapshots 1, 2, and 3 are cases for , , and , respectively. In the first two cases, the electromagnetic radiation is directed laterally, but in the relatively long antenna for snapshot 3, substantial radiation is produced near the axial directions.

Reference

[1] J. D. Jackson, Classical Electrodynamics, 3rd ed., New York: John Wiley & Sons, 1998.