On the remarkable features of the lower limits of charge and the radiated energy of antennas as predicted by classical electrodynamics

Abstract

Electromagnetic energy radiated by antennas working in both the frequency domain and time domain is studied as a function of the charge associated with the current in the antenna. The frequency domain results, obtained under the assumption of sinusoidal current distribution, show that, for a given charge, the energy radiated within a period of oscillation increases initially with L/λ and then starts to oscillate around a steady value when L/λ > 1. The results show that for the energy radiated by the antenna to be equal to or larger than the energy of one photon, the oscillating charge in the antenna has to be equal to or larger than the electronic charge. That is, U ≥ hν or UT ≥ h ⇒ q ≥ e, where U is the energy dissipated over a period, ν is the frequency of oscillation, T is the period, h is Planck's constant, q is the rms value of the oscillating charge, and e is the electronic charge. In the case of antennas working in the time domain, it is observed that UΔt ≥ h/4π ⇒ q ≥ e, where U is the total energy radiated, Δt is the time over which the energy is radiated, and q is the charge transported by the current. It is shown that one can recover the time-energy uncertainty principle of quantum mechanics from this time domain result. The results presented in this paper show that when quantum mechanical constraints are applied to the electromagnetic energy radiated by a finite antenna as estimated using the equations of classical electrodynamics, the electronic charge emerges as the smallest unit of free charge in nature.