# On the Remarkable Features of the Lower Limits of Charge and the Radiated Energy of Antennas as Predicted by Classical Electrodynamics

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## Abstract

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## 1. Introduction

## 2. Expressions for the Current Distribution and the Electromagnetic Radiation Generated by Antennas of Different Lengths Working in the Frequency Domain

## 3. The Total Energy Transmitted by the Antenna as a Function of q and L/λ

**S**of the electromagnetic radiation emitted by the antenna along the radial vector is given by:

**а**is a unit vector defined to be in the direction of increasing r (see Figure 1). The energy radiated over a period of oscillation through a unit area perpendicular to the radial vector at a given distance r, denoted by dU, can be obtained by integrating the Poynting vector over one oscillatory period. The result is given by:

_{r}^{2}and hν where

**ν**is the frequency of oscillation and h is Planck’s constant. Note that hν has the units of energy and, according to the quantum mechanical description, the electromagnetic radiation consists of a large number of photons with energy hν. Once this is done we obtain:

^{−2}C

^{−2}, specifies how the radiation is distributed as a function of the elevation angle θ or its directivity. This quantity is plotted as a function of angle θ for several values of the ratio L/λ in Figure 3. In this calculation, r = 100 km. Observe in these figures how the spatial distribution of the radiation varies as one increases L/λ. For very small values of this parameter, the radiation distribution is bell-shaped and it has its maximum in the direction θ = π/2. As L/λ increases, the spatial distribution of the radiation breaks into many peaks or lobes and the two largest of these move towards the angles θ = 0 and θ = π. However, observe that the radiation is zero at these two angles. As L/λ increases further, the concentration of radiation towards these two angles becomes more and more prominent until, at extreme values of this ratio, the radiation is concentrated completely in the vicinity of θ = 0 and θ = π. Notice, however, that the amplitude of the radiation is zero at these angles.

_{ν}) is given by:

_{ν}increases as the square of the charge oscillating in the antenna. The values of U as a function of L/λ for q equal to electronic charge are plotted in Figure 4. Note, also, that since the energy is proportional to the square of the charge, the energy dissipated by any other charge can be obtained readily from this graph. Observe how the energy increases initially with increasing L/λ and then starts to oscillate around a steady value. The amplitude of the oscillation is small compared to the steady value around which the energy oscillates. Also note the fact that, when the oscillating charge is equal to one electronic charge, the steady value of the energy is about hν. It is important to point out that we have completely neglected the effects of radiation damping in the calculation. As mentioned earlier, the effect of radiation damping is to reduce the current peaks as one moves from the center of the antenna to the end of the antenna. This will lead to a reduction in the emitted energy for a given charge. Thus, in the presence of losses for a given charge, the energy could start decreasing with increasing L/λ instead of remaining constant as depicted in Figure 4. For this reason, the plot is marked in red for values of L/λ larger than 100 to indicate that that region of the graph is valid only in ideal conditions (i.e., a lossless situation) and, in reality, one may find the energy decreases with increasing L/λ instead of remaining constant. Observe, however, that the energy almost reaches its steady state value before reaching the region where these uncertainties become important.

## 4. The Solution to the Analogous Time Domain Problem

## 5. Discussion

_{ν}is the number of photons generated within one period by the antenna under consideration. Quantum mechanics dictates that the smallest amount of energy that can be radiated by an antenna has to be equal to or larger than hν. An interesting question that one may ask in this respect is: what is the minimum oscillating charge, say q

_{c}, that is necessary in a given antenna so that the energy dissipated within a period of oscillation is equal to hν? The answer to this question is given directly by Equation (11) and it can be written as:

^{2}by eΔe (note that in this case q = e). In the case of charges larger than the electronic charge, q > e and Δq ≥ e (note that Δq is the uncertainty in the measurement) the above equation can be written as:

_{e}, becomes comparable to the energy U itself. Thus, for the electronic charge, the left hand side of the relationship given in Equation (21) can be written as:

_{e}, the above equation can be written as:

_{m}, as a function of 1/β, associated with the current pulse required for the energy dissipation to satisfy the time-energy uncertainty principle. Note that the minimum charge that satisfies the uncertainty principle is the electronic charge. Of course, this is obvious because we have shown that Equation (21) gives rise to either Equation (24) or (26).

## 6. Assumptions Made in the Derivations Presented in This Paper

## 7. Conclusions

**ν**is the frequency of oscillation, 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 $U\Delta t\ge \frac{h}{4\pi}\Rightarrow q\ge e$ 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 this result can be reduced to the time energy uncertainty principle in quantum mechanics. In a nutshell, the results based purely on classical electrodynamics show that the smallest charge that can radiate in an antenna working in the frequency domain is controlled by the physical concept of the photon and the smallest charge that can radiate in an antenna working in time domain is controlled by the time-energy uncertainty principle. In both cases the smallest charge ended up being the electronic charge.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

**Figure A2.**(

**a**) The Gaussian current pulse used in the calculation. The amplitude is normalized to unity and the time is normalized with respect to the standard deviation σ; (

**b**) The radiation field at a point located at a distance r on the plane bisecting the path of propagation of the current pulse normally. In the calculation, θ = π/2, β < 1 and distance r satisfies the condition r >> L. Observe that the amplitude of the radiation field is normalized to unity and the time is normalized with respect to the standard deviation σ.

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**Figure 1.**The geometry relevant to the calculation of the electromagnetic fields of an antenna of length L located at the origin. In the diagram

**а**,

_{r}**а**

_{θ}and

**а**

_{φ}are unit vectors in the direction of increasing r, θ and φ.

**Figure 2.**The current distribution along the antenna (z = 0 corresponds to the center of the antenna) for different values of the ratio L/λ.

**Figure 3.**The directional properties, as defined by the parameter D (Equation (9)) of the radiation emitted by antennas for different values of the ratio L/λ. In the calculation, the distance r was fixed at 100 km.

**Figure 4.**The total energy, denoted as a fraction of hν, emitted by the antenna as a function of the ratio L/λ (Equation (10)). The rms value of the oscillating charge is equal to the electronic charge.

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**MDPI and ACS Style**

Cooray, V.; Cooray, G.
On the Remarkable Features of the Lower Limits of Charge and the Radiated Energy of Antennas as Predicted by Classical Electrodynamics. *Atmosphere* **2016**, *7*, 64.
https://doi.org/10.3390/atmos7050064

**AMA Style**

Cooray V, Cooray G.
On the Remarkable Features of the Lower Limits of Charge and the Radiated Energy of Antennas as Predicted by Classical Electrodynamics. *Atmosphere*. 2016; 7(5):64.
https://doi.org/10.3390/atmos7050064

**Chicago/Turabian Style**

Cooray, Vernon, and Gerald Cooray.
2016. "On the Remarkable Features of the Lower Limits of Charge and the Radiated Energy of Antennas as Predicted by Classical Electrodynamics" *Atmosphere* 7, no. 5: 64.
https://doi.org/10.3390/atmos7050064