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

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. It was shown that if phase delay is included in the derivation for the radiation of the electric dipole with oscillating charge magnitudes, then classical solution used in most of the textbooks is invalid. Under the standard assumptions that charge separation distance is much smaller than wavelength and non-zero phase delay between charge oscillations: a) classical solution incorrectly describes the radiation from the electric dipole; b) intensity of transverse electromagnetic waves are on par with intensity of longitudinal electric waves; 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.