Long-Range Longitudinal Electric Wave in Vacuum Radiated by Electric Dipole: Part II

TL;DR

This paper extends classical electric dipole radiation theory by analyzing cases where wavelength is much smaller than charge separation, revealing the coexistence of long-range longitudinal electric and transverse electromagnetic waves.

Contribution

It generalizes classical solutions, showing conditions under which longitudinal electric waves dominate and providing new formulas for radiated power in different regimes.

Findings

Classical solution is recovered when charge oscillations have phase delay.

Longitudinal electric waves dominate when wavelength is much smaller than charge separation.

Total radiated power depends on frequency and charge separation distance differently in each regime.

Abstract

In this work we further advance theoretical investigation of radiation by the electric dipole under the assumption that wavelength is much smaller than charge separation distance of an electric dipole, which in turn is much smaller than a distance up to the point of observation. The electric dipole considered in this paper is the one with fixed charge positions, but oscillating charge magnitudes. Specifically, two cases were considered. In the first case phase delay between oscillations of the charge magnitudes was taken into account. In the second case, opposite current wave travel direction was considered. These results generalize the classical solution for electric dipole radiation, and they show that under the above assumptions the electric dipole emits both long-range longitudinal electric and transverse electromagnetic waves. Longitudinal and transverse electric fields emitted by the electric dipole with current wave traveling in both directions are displayed for specific values of the dipole system parameters. It was shown that under the standard assumptions that charge separation distance is much smaller than wavelength and non-zero phase delay between charge oscillations: a) our solution reduces to classical solution; b) transverse electromagnetic waves are dominant while longitudinal electric waves are negligible; c) total radiated power is proportional to the fourth degree of frequency and to the second degree of the charge separation distance. In case wavelength is much smaller than charge separation distance: a) the classical solution is invalid and it overestimates the total radiated power; b) longitudinal electric waves are dominant and transverse electromagnetic waves are negligible; c) total radiated power is proportional to the third degree of frequency and to the charge separation distance. All derived potential and force fields were summarized in the table.